Elliptic Double Affine Hecke Algebras
We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C~ₙ version of t...
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| citation_txt | Elliptic Double Affine Hecke Algebras. Eric M. Rains. SIGMA 16 (2020), 111, 133 pages |
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| description | We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C~ₙ version of the construction to construct a flat noncommutative deformation of the nth symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 111, 133 pages
Elliptic Double Affine Hecke Algebras
Eric M. RAINS
Department of Mathematics, California Institute of Technology, USA
E-mail: rains@caltech.edu
Received December 19, 2019, in final form October 16, 2020; Published online November 05, 2020
https://doi.org/10.3842/SIGMA.2020.111
Abstract. We give a construction of an affine Hecke algebra associated to any Coxeter
group acting on an abelian variety by reflections; in the case of an affine Weyl group, the
result is an elliptic analogue of the usual double affine Hecke algebra. As an application,
we use a variant of the C̃n version of the construction to construct a flat noncommutative
deformation of the nth symmetric power of any rational surface with a smooth anticanonical
curve, and give a further construction which conjecturally is a corresponding deformation
of the Hilbert scheme of points.
Key words: elliptic curves; Hecke algebras; noncommutative deformations
2020 Mathematics Subject Classification: 33D80; 39A70; 14A22
Contents
1 Introduction 1
2 Line bundles on En and their sections 8
3 Coxeter group actions on abelian varieties 26
4 Elliptic analogues of affine Hecke algebras 34
5 Infinite groups 60
6 The (double) affine case 76
7 The C∨Cn case 87
8 The (spherical) C∨Cn Fourier transform 107
9 Deformations of Hilbert schemes 122
References 131
1 Introduction
The origin of this paper was a question of P. Etingof which was conveyed to the author by
A. Okounkov at a 2011 conference,1 to wit whether the author knew of a way to construct
This paper is a contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quan-
tum Field Theory. The full collection is available at https://www.emis.de/journals/SIGMA/elliptic-integrable-
systems.html
1Affine Hecke Algebras, the Langlands Program, Conformal Field Theory and Super Yang–Mills Theory, Centre
International de Rencontres Mathématiques (Luminy), June 2011.
mailto:rains@caltech.edu
https://doi.org/10.3842/SIGMA.2020.111
https://www.emis.de/journals/SIGMA/elliptic-integrable-systems.html
https://www.emis.de/journals/SIGMA/elliptic-integrable-systems.html
2 E.M. Rains
noncommutative deformations of symmetric powers of the complement of a smooth cubic plane
curve. Although the answer was “no” (at the time, see below!), it seemed likely that it should
be possible to extend the approach of [28] (which the author and S. Ruijsenaars had devel-
oped earlier that month) to multivariate difference operators; although this would not answer
the question as posed, it would give analogous deformations associated to the complement of
a smooth biquadratic curve in P1 × P1, represented as algebras of elliptic difference operators
in n variables. The author continued to develop this approach while on a sabbatical that fall
at MIT, eventually coming up with a construction for such deformations for any rational surface
equipped with a smooth anticanonical curve and a rational ruling.
There were, however, a couple of significant issues. One was that the spaces of operators
were cut out by a number of conditions, including in particular certain residue conditions that
only made sense for generic values of the parameters. This would have been merely an annoying
technicality, except that the conditions specifically failed to make sense in the commutative case,
making it rather difficult to consider the family as actually being a deformation. This could be
worked around by considering the family as a whole, or in other words only considering those
operators that could be extended to an open subset of parameter space. However, although this
would indeed give a well-defined family of algebras, it would make the question of flatness even
more difficult, and would in principle even allow the representation in difference operators to
fail to be faithful!
Despite these difficulties, the definition was still well-behaved enough to allow a fair amount
of experimentation. One thing that became clear was that the construction directly led to some
spaces of operators that had been considered in the literature, in particular those associated
to the difference equations for interpolation and biorthogonal functions [23, 24], of particular
interest in the latter case since no satisfying construction was yet known for the full space of
operators. In addition, since these functions degenerate to more familiar functions, to wit the
Macdonald and Koornwinder polynomials, this suggested that the algebras of elliptic difference
operators should degenerate to algebras related to those polynomials. In particular, the latter
algebras can be constructed as spherical algebras of appropriate double affine Hecke algebras,
and P. Etingof suggested to the author that the same might hold at the elliptic level, and give
a possible approach to flatness.
Indeed, the same approach to constructing the algebra of operators (as operators preserving
(locally) holomorphic functions and satisfying appropriate vanishing conditions on the coeffi-
cients) could be fairly easily extended to give a construction of elliptic double affine Hecke alge-
bras. The resulting residue conditions turned out to be essentially those of [11], with again the
caveat that they only make sense when the noncommutative parameter q is non-torsion. In fact,
something slightly stronger is true: the residue conditions are well-behaved as long as one only
considers a sufficiently small interval relative to an appropriate (Bruhat) filtration, with the
constraint on the interval being simply that it act faithfully. This led the author to investigate
that filtration more carefully, leading eventually to the realization that (a) the residue condi-
tions always make sense on rank 1 subalgebras (which are very special cases of the construction
of [11]), and (b) those rank 1 subalgebras always generate a flat algebra, even when q is torsion.
As a result, one could avoid the residue conditions entirely and simply consider the algebra
generated by the rank 1 algebras. It is then relatively straightforward to show that the resulting
family of algebras is flat (and the representation as difference-reflection operators is faithful),
and not too difficult to show that this flatness is inherited by the spherical algebra. Moreover,
much of the theory can be developed for quite general actions of Coxeter groups on abelian
varieties, so that the DAHAs are just the special cases in which the Coxeter group is affine.
In the above discussion, we have neglected a few technical issues. The first is that the
deformations of symmetric powers are not, in general, algebras of operators. The difficulty is
that, with the exception of complements of smooth anticanonical curves in del Pezzo surfaces,
Elliptic Double Affine Hecke Algebras 3
none of the surfaces we wish to deform are actually affine! Since they are only quasiprojective in
general, it is easier to simply deform the symmetric power of the original projective surface (and
then take the appropriate localization if desired). This still requires a choice of ample divisor,
and one then encounters the difficulty that twisting a noncommutative variety by a line bundle
tends to change the noncommutative variety. As a result, the object actually being deformed is
the category of line bundles on the symmetric power.
When it comes to the DAHA, however, the situation is even more complicated. The problem
is that the elliptic DAHA essentially arises by replacing one of the two commutative subalgebras
of the usual DAHA by the structure sheaf of an abelian variety of the form En. In the affine
case considered in [11], the commutative subalgebra is finite over the center of the affine Hecke
algebra, and thus one is naturally led to consider sheaves over the center, or in other words
sheaves on the quotient En/W , where W is the relevant finite Weyl group. Unfortunately, in
the double affine setting, W is replaced by an affine Weyl group, and there is no such quotient
scheme! As a result, the trickiest part of our construction turns out to be simply figuring
out what kind of object we should be constructing. The key idea, coming from earlier work
in noncommutative geometry [2, 37] is that since the elliptic DAHA should have OEn as a
subalgebra, it should have a natural bimodule structure over OEn , and thus correspond to a
quasicoherent sheaf on the product En×En. Subject to some finiteness conditions (satisfied for
any sheaf of meromorphic difference operators), such bimodules form a monoidal category, and
thus we can construct the elliptic DAHA as a monoid object (“sheaf algebra”) in that category.
A final technical issue is that we wish to deform symmetric powers of arbitrary blowups
of P1 × P1 or the Hirzebruch surface F1. Each time we blow up a point, we acquire a new
parameter, and thus our construction needs to admit an unbounded number of parameters.
This is an issue from the standpoint of traditional double affine Hecke algebras, where one
normally has precisely one parameter per root (which must be constant on orbits), plus an overall
parameter q. One partial exception is the C∨Cn case, where one has a total of 5+1 parameters.
This is normally explained by taking a nonreduced root system, so that one has 5 orbits of roots,
but from the standpoint of the actual algebra, this is rather artificial: there is an action of S2×S2
on the parameters that has no effect on the algebra, but moves degrees of freedom between
corresponding short and long roots. It is thus much more natural to view those four parameters
as assigning an unordered pair to each orbit of short roots of the reduced root system. This turns
out to generalize easily to the elliptic setting: we obtain an elliptic DAHA for every assignment
of an effective divisor on E to each orbit of roots. This causes some difficulties in constructing
the spherical algebra, as the usual construction via idempotents fails even generically, but one
can show that the spherical algebra still continues to inherit flatness in this general case.
As we mentioned above, the construction of deformations of Symn(X), where X is a pro-
jective rational surface with a choice of smooth anticanonical curve, depends on a choice of
rational ruling on X. One consequence is that we cannot directly obtain the case X = P2 from
our construction. This can be worked around by blowing up a point but then only considering
those line bundles coming from P2, but this approach leads to a nontrivial question of showing
that the result is independent of the choice of point. Similarly, if X ∼= P1 × P1, then there
are two choices of ruling on X, but deformation theory suggests that the resulting deforma-
tions should be the same (both have the maximum number of parameters). Both questions
turn out to reduce to the existence of a certain generalized Fourier transform in the P1 × P1
case, which is also key to proving the most general form of the flatness result. (The DAHA
only tells us that certain sheaves are flat, so gives only an asymptotic flatness result for their
global sections.) We find that not only is our deformation of the category of symmetric powers
of line bundles on X flat (modulo some possibility of bad parameters in codimension ≥ 2 not
including the original symmetric-power-of-commutative-surface case), but it is invariant under
the action of a Coxeter group of type W (Em+1). (In other words, to first approximation, the
4 E.M. Rains
construction only depends on the underlying surface X and two points q and t of the Jacobian
of the anticanonical curve.) Note that both facts are actually not true for the full original family
of commutative categories, but hold for the subcategory in which we only allow those morphisms
that extend to a neighborhood in the family.
The plan of this paper is as follows. First, in Section 2, we deal with a largely notational issue
that arises due to the fact that we wish to deal with the construction from a purely algebraic
standpoint. The issue is that we need in many cases to deal with twisted versions of our algebras,
in which the coefficients of the operators lie in nontrivial line bundles. To make sense of this,
one must not only describe those line bundles but various maps between tensor products of
pullbacks of those bundles through elements of the Coxeter group, with associated concerns
about compatibility. In the analytic setting, one can avoid most of those issues by constructing
the line bundle via an appropriate automorphy factor on the universal cover, and our objective
in Section 2 is to do something similar in the algebraic setting. The key idea is to replace the
individual curve E by the universal curve over the moduli stack of elliptic curves; it turns out
that one can compute the Picard group of the corresponding stack En in general, with only very
mild rigidification required to make pullbacks and tensor products behave well. In addition,
certain of the line bundles come with natural global sections, which one can use to construct
sections of more general bundles; our most significant result along these lines gives conditions for
a function on the analytic locus described via theta functions to extend to the full moduli stack
(i.e., when it extends in a nice way to elliptic curves over an arbitrary base). We also partially
consider the case of varieties which are isogenous to powers of elliptic curves (i.e., over a moduli
stack of elliptic curves with a cyclic subgroup), and as an application give some results on spaces
of invariant sections of equivariant line bundles on En. The main result along those lines states
that the dimension of the space of invariants is independent of the curve E with only finitely
many possible exceptions (supersingular curves of characteristic dividing the order of the group).
In Section 3, we give some structural results on the main scenario we consider in the sequel,
namely a Coxeter group acting on an abelian variety “by reflections”. In particular, we show
that under reasonable conditions every root of the Coxeter group can be assigned an associated
“coroot” morphism to an elliptic curve, compatibly with the linear relations between roots in
the standard reflection representation (and satisfying suitable notions of positivity!). This is
a key ingredient in our construction, as our parameters will correspond to effective divisors on
those curves. We also show that in the case of a finite Coxeter group, the invariant theory is
better behaved than suggested by the results of Section 2: as long as a certain isogeny (which
is an isomorphism in the most natural cases) has diagonalizable kernel, the invariant theory
continues to behave well even for supersingular curves. This flatness of invariants is a crucial
ingredient in proving flatness of the spherical algebra, and in particular means the C∨Cn case
will be flat over any field.
Section 4 is largely a recapitulation of the construction of [11], in which we associate to
any finite Coxeter group W acting on a family X of abelian varieties a family (the “elliptic
Hecke algebra” for concision, though it should be thought of as affine, with the role of the
commutative subalgebra being played by the structure sheaf OX) of sheaves of algebras on X/W
parametrized by effective divisors on the “coroot” curves. The main difference, apart from
allowing arbitrary numbers of parameters and slightly more general abelian varieties, is that we
replace the residue conditions of [11] by the (equivalent) condition that the operators preserve
the spaces of local sections of the structure sheaf of X on W -invariant open subsets, as the
latter is easier to generalize from a conceptual standpoint. Our main new tool for studying
these algebras is a natural filtration by Bruhat order on W , which allows us to express various
subsheaves as extensions of line bundles in natural ways. In particular, this makes it easy to
show that the algebra is generated by the subalgebras corresponding to simple roots, which will
be a key ingredient of the extension to infinite Coxeter groups, as well as giving a construction
Elliptic Double Affine Hecke Algebras 5
which is easily seen to respect base change. We also prove an important technical lemma
on the space of “invariants” of a module over the elliptic AHA, giving fairly general conditions
on a family of modules guaranteeing that the invariants are well-behaved; this will be the key
lemma in proving flatness of spherical algebras. In addition, we give an analogue of Mackey’s
theorem for the case of parabolic subgroups in which the usual sum over double cosets is replaced
by a filtration.
Section 5 begins with a discussion of sheaf algebras, which we use in place of a sheaf of
algebras on the quotient. Since the corresponding tensor product of bimodules is somewhat
tricky to deal with, we discuss approaches to dealing with this issue (and in particular ways to
describe the maps A⊗ B → C we need in order to discuss algebras or categories). This makes
it relatively straightforward to construct analogues of the affine Hecke algebra in which W is
replaced by an infinite Coxeter group: start with the sheaf algebra of meromorphic reflection
operators and take the sheaf subalgebra generated by the rank 1 Hecke algebras. (The only
tricky aspect comes when we consider twisted forms of the algebra, for which we prove a fairly
general result about “orders” in twisted forms of k(X)[W ] containing OX [W ].) Most of the
results extend immediately from the finite case (with the caveat that any parabolic subgroups
considered should be finite); in particular, the infinite analogue of the Mackey result is precisely
the remaining result we need to show that the spherical algebras are flat.
Section 6 discusses the special case in which W is an affine Weyl group, so that the sheaf
algebras constructed in Section 5 are analogues of double affine Hecke algebras. Apart from
some mild issues about viewing q as a parameter (not the case for the standard construction),
this mainly consists of some observations about the spherical algebra (relative to the associated
finite Weyl group): the fact that the fibers are (sheaf) algebras of difference operators, and are
thus in a natural sense domains, and the fact that the elliptic DAHA is at least generically
Morita equivalent to its spherical algebra (as well as versions in which some of the parameters
have been shifted by q). In addition, we discuss the consequences of the fact that the action
of W̃ fails to be faithful when q is torsion: not only does the sheaf algebra come from a sheaf of
algebras on the quotient by the image of W̃ , but it has a 2n-dimensional center (over which it is
presumably finite). Moreover, under mild conditions on the twisting, we can identify the center
as the spherical algebra of an elliptic DAHA with q = 0 living on an isogenous abelian variety.
Section 7 considers in detail the case that W is the affine Weyl group of type Cn and X is
a particularly nice action of that group, with a view to constructing deformations of symmetric
powers of rational surfaces. In addition to the spherical algebras themselves, one must also
consider certain intertwining bimodules. We prove that these are always flat as sheaf bimodules,
and show that in the case t = 0 the result is indeed a symmetric power of the univariate case
(which was constructed in [26] without reference to DAHAs). This enables us to at least partially
extend the flatness as sheaves to flatness of global sections, by giving a number of cases in which
the sheaves are acyclic. In addition to these general results, our main result in this section
is showing that if we blow up 8 points of P1 × P1, then there is a hypersurface in parameter
space on which the “anticanonical” algebra is an integrable system: it is generated by n + 1
commuting (and self-adjoint) elliptic difference operators in n variables. (One can verify that
this is precisely the integrable system of [6, 7, 16]. In addition, the geometry strongly suggests
the existence of other integrable systems with the same number of parameters, but higher-order
operators.)
Section 8 deals with the question of showing that the construction is mostly independent of
the way in which we represented our rational surface as a blowup of a ruled surface. The key
ingredient is a certain “Fourier transform”. Analytically, this should be represented by the
integral operator with kernel constructed in [27], but there are difficulties in showing this is
well-defined in general (as well as showing that it respects the additional conditions associated
to any points we have blown up). As a result, we construct the transform in several steps. First,
6 E.M. Rains
we give a construction that is manifestly well-defined (and a homomorphism) as long as q is not
torsion, but lives on a certain completion of the algebra of meromorphic difference operators.
Although the construction is in general quite complicated, it is sufficiently well-behaved to allow
us to compute a few special cases. The result in those special cases turns out to be well-defined
even when q is torsion, and this allows us to show that this formal transform extends in general.
Moreover, the special cases we understand are sufficiently close to generating the full algebras
that we can prove that the formal transform restricts to an actual transform. This also gives
us some ability to explore how the algebras behave when we degenerate the elliptic curve, as it
is easy to take limits of the almost-generators. In addition to allowing us to prove a much
stronger flatness result, the Fourier transform also allows us to construct a large collection
of “quasi-integrable” systems, in particular including the aforementioned operators associated
to interpolation and biorthogonal functions.
Section 9 gives some partial results towards “desingularizing” the above construction; i.e.,
giving a deformation of the Hilbert scheme of points of X rather than just the symmetric power.
The basic idea is fairly straightforward: simply include additional bimodules that shift t as well
as the points being blown up. The argument for flatness breaks down in general, even at the level
of sheaves, but we can show both flatness and agreement with Euler characteristics of line bundles
on the Hilbert schemes in a fair number of cases. Moreover, for several of those cases, not only
do we obtain the correct number of global sections, but one can identify those global sections as
global sections on the Hilbert scheme in such a way that they satisfy precisely the same relations.
In particular, this includes the line bundles corresponding to the embedding of Hilbn(X) in
a Grassmannian that takes the ideal sheaf I to the subspace Γ(X; I(D)) ⊂ Γ(X;OX(D)).
(One can also show that the Fourier transform extends, so that again the construction essentially
depends only on X and not the particular way it was obtained from a Hirzebruch surface.)
We close with a summary of some of the various open problems that arose in the course of this
work. (Of course, this is only a small sampling of such problems, as nearly any existing result on
double affine Hecke algebras suggests the existence of a generalization to the elliptic case! We are
also omitting some questions discussed in Sections 2 and 3, as they are peripheral to the main
thrust of the work.) One big collection of questions has to do with the fact that, although we
show that the spherical algebras of the elliptic DAHAs are flat in significant generality, we can
prove almost nothing else about them in general. In particular, we cannot even show that they
are Noetherian (even in the specific C∨Cn-type cases for which we have such strong flatness
results). The approach to such questions in [1] suggests that one should be able to reduce this to
the case in which everything is defined over a finite field, when one expects the spherical algebra
to be finite over its center (which should itself be Noetherian); unfortunately, understanding
the center (when q is torsion) is itself an open problem. (This reduces to understanding the
spherical algebra when q = 0.)
Another natural question is whether the Fourier transform of the usual DAHA extends to the
elliptic level. The construction of Section 8 can be viewed as a partial affirmative answer
to this question in the C∨Cn case, as it constructs a Fourier transform on the spherical algebra.
For n = 1, this at least implicitly leads to a Fourier transform on the elliptic DAHA by using
the appropriate Morita equivalence, but even there it is unclear how to make the transformation
explicit. (There is also a philosophical question: our work suggests that the transform should
really be viewed as living on the spherical algebra, as the analogous transform on the DAHA is
not as well behaved relative to the natural filtration.)
In fact, even in the Cn case, there are still open questions about the Fourier transform,
as the construction of Section 8 only applies to the case in which we have assigned precisely one
parameter to the Dn-type roots. There is evidence suggesting the existence of some other Fourier
transforms related to the versions of the DAHAs of types An and Cn that do not have this “t”
parameter. Indeed, the paper [35] discusses several integral transformations; one appears to
Elliptic Double Affine Hecke Algebras 7
relate two versions of the An spherical algebra, while the other two appear to relate the An
spherical algebra to the Cn spherical algebra. The main obstruction to understanding these
cases is that the lack of a t parameter makes it difficult to control the spaces of global sections,
as we can no longer view the spherical algebras as deformations of a symmetric power. In any
event, this suggests that the existence of isomorphisms between spherical algebras is a much
more subtle question than one might have thought based on the classical theory of double affine
Hecke algebras. Another source of questions about the Fourier transform is the analytic version
constructed in [27]. In addition to the “interpolation kernel”, that paper constructed a few other
functions with some similar properties (the “Littlewood”, “dual Littlewood” and “Kawanaka”
kernels), and it is natural to ask whether those functions interact with the C∨Cn spherical
algebra in any interesting way. That such an interaction should exist is strongly suggested by
the fact that the quadratic transforms proved in that paper were generalizations of results first
proved using the action of affine Hecke algebras on Laurent polynomials.
There are also a number of questions about our deformations of symmetric powers having
to do with taking global sections. One fundamental question has to do with the fact that our
construction, although (mostly) flat and highly symmetrical, does not quite correspond directly
to deformations of symmetric powers of surfaces: we must still make a choice of ample line bundle
in order to obtain an actual projective deformation in the commutative case or a deformation
of the category of sheaves in the noncommutative case. For n = 1, it was shown in [26] that all
such choices give the same result, but it will be difficult to extend those techniques to n > 1.
In addition, one would like to show that the corresponding family of commutative quasiprojective
varieties is at least generically smooth (which experiment suggests is the case). Of course, we also
expect that our conjectural deformation of Hilbert schemes should always be smooth (i.e., in
the noncommutative case, that the corresponding category of coherent sheaves should satisfy
Serre duality). In addition, one would like to have a proof that our family of algebras (or
noncommutative varieties) actually depends on all of the parameters (and is not, say, simply
a base change from a lower-dimensional family); this suggests trying to understand how the
family relates to the infinitesimal deformation theory of the symmetric power. (Note, however,
that the deformation associated to the t parameter is almost certainly not locally trivial. Also,
the fact that replacing t by q − t gives an isomorphic algebra implies that the corresponding
Kodaira–Spencer map will vanish unless we first descend the family to the quotient by this
symmetry, and it is unclear how to do so without breaking the representation via difference
operators.) This would largely be settled if we could show that the q = 0 case of the Hilbert
scheme deformation agreed with the deformation constructed in [26] (as a moduli space of rank 1
sheaves on a noncommutative rational surface).
In any event, a choice of ample divisor allows one to translate each original line bundle into
an actual sheaf on the (noncommutative) variety, and thus produces a saturated version of the
original Hom space (by taking all morphisms between the sheaves). In the univariate case,
the saturated morphisms were still difference operators, and there is a primarily combinatorial
algorithm for computing the resulting dimensions. The question is more subtle in the multi-
variate setting, with two main issues arising. For some line bundles on F1 (including those
coming from line bundles on P2), we can only prove flatness away from a possible bad locus
of codimension ≥ 2, and some new approach to constructing global sections is likely needed to
eliminate this possibility. The other tricky case arises from the elliptic pencil on a deformation
of a non-Jacobian elliptic surface (i.e., in which the elliptic fibration does not have a section).
There, not only do we want to know how many global sections there are (with the conjecture
being that on the hypersurface where the algebra of global sections is nontrivial, it is flat), but
also, by analogy with the Jacobian case, expect that the algebra of global sections will give
a new integrable system, associated to the same parameters as the van Diejen/Komori–Hikami
system, with the addition of a choice of nontrivial torsion point on the elliptic curve.
8 E.M. Rains
Finally, the condition that a projective rational surface have a smooth anticanonical curve
is quite restrictive, and in particular excludes a number of cases in which deformations were
already known. Although the strong version of flatness cannot be expected to extend in general,
one can still expect to have flatness for ample bundles. There are already issues before blowing
up any points, as it appears one must give an analogue of the elliptic DAHA in which the
elliptic curve becomes singular, reducible, or even nonreduced, and this can cause issues with
the Bruhat filtration as well as the generation in rank 1. Beyond that, although it should
be fairly straightforward (especially if the base curve remains integral) to consider blowups
in smooth points of the base curve, this is again a pretty restrictive condition, while blowing
up singular points quickly leads to a combinatorial explosion without some more conceptual
approach.
2 Line bundles on En and their sections
In the sequel, we will quite frequently need to specify line bundles on a power En of an elliptic
curve E or (meromorphic) sections thereof. (Here E will typically be defined over an algebraically
closed field, though much of the discussion works over a general base.) At first glance, the
problem of specifying a line bundle appears nearly trivial. Indeed, given any principally polarized
abelian variety A, we have a short exact sequence
0→ A→ Pic(A)→ NS(A)→ 0,
where the Néron–Severi group NS(A) is naturally isomorphic to the group of endomorphisms
of A which are symmetric under the Rosati involution. If End(E) = Z (which holds generically),
then this becomes
0→ En → Pic(En)→ NS(En)→ 0,
where NS(En) is the group of symmetric n× n integer matrices. Moreover, it turns out (as we
will discuss in more detail below) that this short exact sequence splits, giving us canonical labels
for line bundles up to isomorphism.
This last caveat is quite significant, however; if L1 and L2 represent given classes in Pic(En)
and L3 represents their sum, then there exists an isomorphism L1 ⊗ L2
∼= L3, but this is only
determined up to an overall scalar multiple. Another consequence is that if we have (as we will
below) a group G acting on En, a G-invariant class in Pic(En) need not specify an equivariant
line bundle.
If E is an analytic curve C/〈1, τ〉, there is a standard way to avoid these difficulties, namely
the theory of theta functions. Indeed, given a cocycle z ∈ Z1(π1(En);A(Cn)∗), we can construct
a corresponding line bundle Lz, and these bundles satisfy Lz1 ⊗ Lz2 ∼= Lz1z2 and g∗Lz ∼= Lg∗z.
Moreover, since H1(Z,A(C)∗) = 0, we can arrange for our cocycles to have trivial restriction
along Zn ⊂ 〈1, τ〉n ∼= π1(En). We thus obtain the following description of line bundles on En:
given a symmetric integer matrix Q and constants C1, . . . , Cn ∈ C∗, we could consider the line
bundle on (C/〈1, τ〉)n with local sections given by functions on C satisfying
f(z1, . . . , zi−1, zi + 1, zi+1, . . . , zn) = f(z1, . . . , zn),
f(z1, . . . , zi−1, zi + τ, zi+1, . . . , zn) = Cie
(
−
∑
j
Qijzj
)
f(z1, . . . , zn),
where e(x) := exp
(
2π
√
−1x
)
. These line bundles behave well under tensor product, but in
slightly odd ways under pullback; it turns out that we should not quite trivialize the cocycle
along Zn in general, but instead merely insist that it restrict to an appropriate morphism
Elliptic Double Affine Hecke Algebras 9
Zn → {±1}. This leads us to define the line bundle LQ; ~C on (C/〈1, τ〉)n as the sheaf with local
sections consisting of holomorphic functions satisfying
f(~z + ~xτ + ~y) =
∏
1≤i≤n
(−1)Qii(xi+yi)Cxii
∏
1≤i,j≤n
e(−Qijxi(zj + xjτ/2))f(~z),
for ~x, ~y ∈ Zn. Note that this does not quite solve the problem as stated, since it gives multiple
representatives for each line bundle (multiplying f by the nowhere vanishing entire function e(zi)
multiplies Ci by e(τ)), but still makes it straightforward to control equivariant structures on line
bundles, as well as some of the more gerbe-like structures we need to consider below.
Although this suffices for many purposes, we would like to have an algebraic solution to this
problem. Not only is our construction below essentially algebraic in nature, there are also some
indications that it may prove useful in later work to be able to consider versions defined over
finite fields. (In particular, see the use in [1] of finite field instances of the Sklyanin algebra in
proving the latter is Noetherian, of particular interest given that we do not yet have a proof
that our algebras are Noetherian; see also unpublished work of Bezrukavnikov and Okounkov.)
A first step towards this is to observe that we are not really interested in constructing things
over a particular curve; rather, we wish to have constructions that apply to all curves. In other
words, what we truly want to understand are line bundles on the nth fiber power En of the
universal curve E over the moduli stack M1,1 of elliptic curves.
Remark. For those readers who may be unfamiliar with stacks, it is worth noting that there
is an equivalence between statements about line bundles on En (orM1,1) and statements about
equivariant line bundles on an appropriate scheme-with-group-action. This comes from the fact
that each of these stacks is a quotient stack. Indeed, a choice of a nonzero holomorphic differen-
tial ω on an elliptic curve on which 6 is invertible2 determines a unique expression of the curve as
a Weierstrass curve: the compactification of a curve y2 = x3 + a4x+ a6, with differential dx/2y.
It follows immediately that for any family of elliptic curves E/S with 6 invertible on S, there is
a Gm-torsor T over S such that ET is given by such an equation: simply take T to be the torsor
of nonzero holomorphic differentials! It follows that there is a natural isomorphism between
the (symmetric monoidal) category of line bundles on M1,1[1/6] (or En[1/6]) and the cate-
gory of Gm-equivariant line bundles on Spec(Z[1/6, a4, a6, 1/∆]) (or the corresponding family of
abelian varieties). Indeed, if we are given a Gm-equivariant line bundle on the family of Weier-
strass curves (or, for En, on the appropriate family of abelian varieties), then for any family of
elliptic curves over S (resp. with n additional sections), we obtain an induced Gm-equivariant
bundle on the Gm-torsor T , and this descends naturally to a line bundle on S (satisfying the
requisite compatibility conditions). Conversely, any line bundle on the stack induces a line bun-
dle on the corresponding family of Weierstrass curves, and the compatibility conditions make
this line bundle naturally equivariant. This equivalence would allow one to entirely eliminate
stacks from the discussion below, at the cost of requiring some additional bookkeeping in the
arguments to keep track of equivariance.
The theory of Jacobi forms [8, 13, 14, 17] gives us an approach to this at the analytic level.
Analytically, En is the quotient of Cn ×H by the appropriate action of Z2n o SL2(Z), and thus
again we may specify line bundles via cocycles. This leads to the following definition: Given
a symmetric integer matrix Q (the “level”) and an integer w (the “weight”), we define the line
bundle LQ,w on the complex locus of En to be the sheaf with local sections consisting of functions
f(z1, . . . , zn; τ) such that
f(~z + ~xτ + ~y; τ) = (−1)~x
tQ~x+~ytQ~ye(−~xtQ(~z + ~xτ/2))f(~z; τ),
f(~z/(cτ + d); (aτ + b)/(cτ + d)) = (cτ + d)we(c~z tQ~z/2(cτ + d))f(~z; τ),
2Something similar holds over Z, but with Gm replaced by the group (y, x) 7→
(
a3y + bx + c, a2x + d
)
.
10 E.M. Rains
with ~x, ~y ∈ Zn and
(
a b
c d
)
∈ SL2(Z). (This differs from the usual notion of Jacobi form by virtue
of our not imposing any condition at the cusp; we have also allowed Q to have odd diagonal.)
Lemma 2.1. The line bundle L1,−1(−[0]) on the complex locus of E is trivial, where the divi-
sor [0] is the image of the identity section 0: M1,1 → E.
Proof. We first observe that the function
ϑ(z; τ) :=
(e(z/2)− e(−z/2))
∏
1≤j(1− e(jτ + z))(1− e(jτ − z))∏
1≤j(1− e(jτ))2
=
∑
k∈1/2+Z(−1)k−1/2e(kz + k2τ/2)∑
k∈1/2+Z(−1)k−1/2ke(k2τ/2)
is a global section of L1,−1. This follows immediately from the standard transformation law
for Jacobi theta functions together with the transformation law for the Dedekind eta func-
tion e(τ/24)
∏
1≤j(1 − e(jτ)). (Note that in each case, the overall transformation law involves
complicated arithmetic characters, but these turn out to cancel.) This is holomorphic on C×H,
and for each τ ∈ H is a nonzero function vanishing only on the lattice 〈1, τ〉; thus the corre-
sponding section of L1,−1 has divisor [0], establishing triviality as required. �
Remark. The function ϑ(z; τ) may be expressed in the standard multiplicative notation for
theta functions in elliptic special function theory as ϑ(z; τ) =
−x−1/2θp(x)
(p;p)2
∞
, where x = e(z),
p = e(τ).
It follows in particular that we can extend L1,−1 (or, rather, an algebraic representative of
its isomorphism class!) to the entirety of E : simply take the line bundle OE([0]). Pulling this
back through a homomorphism g : En → E , (x1, . . . , xn) 7→
∑
i gixi, gives an algebraic version
of Lgtg,−1, and thus by taking tensor products an algebraic version of LQ,w in general. Of course,
there is a potential issue here of uniqueness. Up to isomorphism, this is settled by the following.
Proposition 2.2. The Picard group of En is a canonically split extension of Pic(M1,1) ∼= Z/12Z
by the Néron–Severi group of the generic fiber.
Proof. That Pic(M1,1) ∼= Z/12Z is standard (see [10] for an extension to fairly general base
changes), as is the fact that we can represent the restrictions of such line bundles to the complex
locus via sheaves of modular forms of given weight. (The reduction mod 12 then comes from
the fact that the discriminant ∆(τ) is a holomorphic form of weight 12, nowhere vanishing on
the smooth locus.) Moreover, the identity section M1,1 → En gives rise to a splitting of the
natural pullback morphism Pic(M1,1)→ Pic(En). The above construction moreover shows that
the natural map from Pic(En) to the Néron–Severi group of the generic fiber is surjective. (Note
that since M1,1 is a stack, “the generic fiber” does not quite make sense, but it suffices to
base change to some smooth curve covering M1,1, say by imposing a full level 3 or full level 4
structure.)
It thus remains only to show that if L is a line bundle on En which on the generic fiber
is algebraically equivalent to the trivial divisor, then every fiber of L is trivial, and thus L is
the pullback of a line bundle on M1,1. Since an algebraically trivial line bundle on an abelian
variety gives rise to a point of the dual variety, and En is principally polarized via the product
polarization, we find that L induces a section M1,1 → En, and we need merely show that this
is the identity section. The elliptic curve y2 + txy = x3 + t5 over C(t) has trivial Mordell–Weil
group, and thus the pullback of any section M1,1 → En to this elliptic curve is trivial. Since
the corresponding map Spec(C(t))→M1,1 is dominant, it follows that any section M1,1 → En
is generically trivial, and thus everywhere trivial since this is a closed condition. �
Elliptic Double Affine Hecke Algebras 11
Thus each line bundle on En is determined by its weight (the restriction to the zero section
as an element of Pic(M1,1) ∼= Z/12Z) and polarization (the class in the Néron–Severi group
of the generic fiber). It will be convenient going forward to represent the polarization as a sym-
metric integer matrix Q or as the corresponding quadratic polynomial ~zQ~z t/2; the latter will
be particularly convenient when we have assigned names to the coordinates in En.
Let us thus choose for each symmetric integer matrix Q and integer w ∈ Z a line bundle LQ,w
(unique up to isomorphism) which restricts to the bundle of weight w on the identity section and
induces the symmetric endomorphism Q on the generic fiber of En. This very nearly solves our
problem of constructing line bundles with consistent isomorphisms under tensor product and
pullbacks, by virtue of the fact that there are very few global units on En. Indeed, since En is
proper overM1,1, the global units on En are just the global units onM1,1, and these are easily
seen to consist precisely of elements of the form ±∆l for l ∈ Z. But in fact the same thing that
gives us a splitting of the Picard group gives us a natural way to rigidify things completely.
Definition 2.3. For an integer w, a weight w trivialization of a line bundle L on En is an iso-
morphism
0∗L ∼= (0∗ωE/M1,1
)w.
We now enhance our data as follows: LQ,w is not just a line bundle in the appropriate iso-
morphism class; rather, it is such a line bundle together with a choice of weight w trivialization.
We also insist that L0,0 = OEn with the obvious trivialization. (Here we now take w an integer
rather than a class mod 12, as trivializing 0∗ω12
E/M1,1
requires choosing one of ∆ or −∆. This is
not a significant issue, however, and in fact should allow us to extend everything to the cusps
of M1,1.)
Theorem 2.4. There is a family of natural isomorphisms
LQ1,w1 ⊗ LQ2,w2
∼= LQ1+Q2,w1+w2 ,
agreeing with the obvious isomorphisms when (Q1, w1) = (0, 0) or (Q2, w2) = (0, 0), and, for any
linear transformation g : Zn → Zm (with induced homomorphism g : En → Em), a family of natu-
ral isomorphisms
g∗LQ,w ∼= LgtQg,w,
agreeing with the obvious isomorphism when g = 1. Moreover, these are compatible in the sense
that any isomorphism⊗
1≤i≤n
g∗iLQi,wi → L∑i g
t
iQigi,
∑
i wi
constructed from these ingredients is invariant under permutations of the tensor factors and
independent of the order in which the above isomorphisms are applied.
Proof. In either case, not only are the bundles isomorphic, but so are their pullbacks through 0
(e.g., since 0∗g∗ = 0∗), and the respective trivializations actually induce a specific choice of
isomorphism of the pullbacks. Since both isomorphisms are determined up to a global unit
on M1,1, rigidifying the pullback suffices to rigidify the desired isomorphisms. Similarly, the
compatibility conditions certainly hold up to a scalar factor, and pulling back through 0∗ shows
that that scalar must be 1. �
To specify (meromorphic) sections of such line bundles, we would like to have an analogue
of the Jacobi theta function. Here we have the following.
12 E.M. Rains
Lemma 2.5. The line bundle OE([0]) on E has a natural weight −1 trivialization.
Proof. We need an isomorphism
0∗OE([0])⊗ 0∗ωE/M1,1
∼= OM1,1 ,
or equivalently
0∗
(
ωE/M1,1
([0])
) ∼= OM1,1 ,
but this is just adjunction. �
Remark. Note that the isomorphism coming from adjunction simply takes a differential with
simple pole at 0 to its residue. In particular, the natural isomorphism [−1]∗OE([0]) ∼= OE([0])
negates the trivialization.
Definition 2.6. Let L1,−1 be the chosen line bundle with trivialization on E . Then the global
section ϑ ∈ Γ(L1,−1) is the image of 1 under the isomorphism OE([0]) ∼= L1,−1 respecting the
trivialization.
Note that the function ϑ considered above was normalized so that
Resz=0
dz
ϑ(z; τ)
= 1,
and thus the two definitions essentially agree on the analytic locus. (This is not quite correct,
since the analytic definition of L1,−1 only gives something isomorphic to an algebraic bundle,
but the residue condition implies that there is a system of isomorphisms between the algebraic
versions of LQ,w and their analytic versions satisfying appropriate compatibility conditions along
tensor products and pullbacks and taking ϑ to ϑ(; τ).) This gives us the following consequence,
a very powerful way of constructing functions on powers En.
Theorem 2.7. Let cij, 1 ≤ i ≤ l, 1 ≤ j ≤ n and mi, 1 ≤ i ≤ l be integers such
that
∑
1≤i≤lmi = 0 and
∑
1≤i≤lmicijcik = 0 for 1 ≤ j, k ≤ n. Then there is a (unique)
meromorphic function on En, defined on every fiber, which on the complex locus restricts
to
∏
1≤i≤l ϑ
(∑
j cijzj ; τ
)mi.
Proof. We may view each ci as a morphism En → E , and find that
∏
1≤i≤l(c
∗
iϑ)mi is a mero-
morphic section of the line bundle⊗
1≤i≤l
(c∗iL1,−1)mi ∼= L∑
imic
t
ici,−
∑
imi
∼= OEn ,
with both maps canonical. In other words, the given product of sections of line bundles deter-
mines a meromorphic function on En, the divisor of which manifestly has no vertical compo-
nents. �
Remark 2.8. By a very mild abuse of notation, we will write the functions constructed in this
way as
∏
1≤i≤l ϑ(
∑
j cijzj)
mi , and similarly for the analogous meromorphic sections of line bun-
dles LQ,w.
Remark 2.9. Note that since [−1]∗ negates the natural trivialization of OE([0]), we have the
identity ϑ(−z) = −ϑ(z).
Elliptic Double Affine Hecke Algebras 13
Remark 2.10. Of course, if we multiply by a suitable power of ∆, we can construct similar
functions under the weaker assumption that
∑
imi is a multiple of 12. On the other hand,
with the constraints as given, there is a natural limit at the cusp of M1,1, namely the rational
function∏
j
x
−
∑
imicij/2
j
∏
1≤i≤l
(
1−
∏
j
x
cij
j
)mi
on (C∗)n = e(C)n (or more generally Gn
m). Note that
∑
imicij is even since
∑
imic
2
ij = 0, so this
is indeed well-defined. This gives a useful method for sanity-checking calculations, by verifying
that the result is correct in this limit. One can also take a limit as all of the variables approach 0;
this is ill-defined, but blowing up the 0 section gives the function
∏
1≤i≤l
(∑
j
cijyj
)mi
on the exceptional Pn−1 (as long as we avoid the finitely many characteristics in which one of
the factors is identically 0).
It is worth noting that there is an alternate approach to the theorem which, while it does not
help us deal with line bundles, is more powerful in one important respect: it gives a reasonable
algorithm for evaluating such functions at specific points (i.e., on n-tuples of points of specific
elliptic curves, say over a finite field). We choose (to make the appropriate induction work)
a differential ω on E, allowing us to eliminate the constraint
∑
imi = 0. The corresponding
trivialization of the line bundle OE(−d[0]) can then be computed as follows: choose any uni-
formizer u at 0 such that ω/u has residue 1, and then for any function f with multiplicity d
at 0, express f = c(f, ω)ud + O(ud+1) to obtain a “leading coefficient” c(f, ω). It is easy to
see that this leading coefficient is independent of the choice of u, and that c(f, ω) scales in the
appropriate way with ω.
Now, we can characterize the function
∏
1≤i≤l ϑ
(∑
j cijzj
)mi up to a scalar multiple by vie-
wing it as a function of zn with specified divisor. If there are no factors depending only on zn,
then the value of the function along zn = 0 is then a function of the same form, which can be
computed by induction, giving us the requisite scale factor and letting us plug in the specific
value of zn desired. If the only factors depending only on zn are powers of ϑ(zn), then we
use the leading coefficient instead and proceed as before. (The correct leading coefficient is
computed by removing powers of ϑ(zn) and setting zn = 0.) Note that in general we can
compute the leading coefficient as long as every factor ϑ(kzn)m that arises has k invertible
(since ϑ(kz) has leading term ku in characteristic 0). Thus if we can construct functions of the
form ϑ(kz)
ϑ(z)k2 directly, we could eliminate those factors before computing the leading coefficient.
Since ϑ(kz) = −ϑ(−kz), we reduce to the case k > 1. Divisor considerations then tell us
that such a function must be proportional to the k-division polynomial (as defined in [34,
Example 3.7]), and we find in fact that the k-division polynomial has the correct leading term
in characteristic 0, so gives the desired function on general E.
Note that although we are writing ϑ(z) as a function, the fact that it is only a section of a line
bundle means that we must be somewhat cautious when specializing. In particular, consider
the ratio ϑ(u + z)/ϑ(u), a section of the line bundle with polarization z2/2 + uz and weight 0.
The restriction of this line bundle to the hypersurface z = 0 is trivial (and in a natural way
given our global choices of trivializations), and thus ϑ(u + z)/ϑ(u) restricts to a function on
this hypersurface, which we can verify is equal to 1. There is, however, an issue in the analytic
setting, as the algebraic hypersurface z = 0 corresponds to the analytic hypersurfaces z = x+yτ ,
14 E.M. Rains
x, y ∈ Z, and ϑ(u+ x+ yτ ; τ)/ϑ(u; τ) is of course nontrivial for most values of x, y. (For z = 0,
there is still no issue, naturally.)
In contrast, if we restrict the ratio ϑ(u+mz)/ϑ(u) to the hypersurface mz = 0, then there are
difficulties even algebraically. The difficulty here is that although the restriction of the line bun-
dle is trivial on every fiber over E [m], it is not canonically trivial, and thus the restriction could
be a nontrivial line bundle on E [m], and even if trivial, the given section need not restrict to 1.
Indeed, let q ∈ µ2(E[2]) be the character of the natural [−1]-equivariant structure on L1,−1
(i.e., such that the action on the fiber at 0 is trivial) on the fixed subscheme E[2]. Note that
q(0) = 1 and when the characteristic is not 2 is −1 at the remaining 2-torsion points; in general,
it is a µ2-valued quadratic form such that the induced bilinear form is the Weil pairing.
Proposition 2.11. The restriction of the function
ϑ(u− z)ϑ(v − z)ϑ(u+ v + z)
ϑ(u+ z)ϑ(v + z)ϑ(u+ v − z)
to the hypersurface 2z = 0 is given by q(z).
Proof. To compute q, we need merely express L1,−1 as O(D) for a suitable divisor D disjoint
from the 2-torsion, at which point q is the restriction to E[2] of the unique function f with divisor
D − [−1]∗D such that f(0) = 1. Taking D = [u] + [v]− [u+ v] gives the desired result. �
We note the following fact useful for simplifying products of such factors.
Proposition 2.12. For any integer m ≥ 1, the restriction of the function
ϑ(u+mz)ϑ(u+ w)ϑ(v)ϑ(v + w +mz)
ϑ(u)ϑ(u+ w +mz)ϑ(v +mz)ϑ(v + w)
to the hypersurface mz = 0 is 1.
Proof. Indeed, the original function is a rational function on E4, which may be described as the
unique elliptic function of u with the appropriate divisor and taking the value 1 at u = v. On the
hypersurface mz = 0, this elliptic function of u has divisor 0, and is thus the function 1. �
Remark. It follows that the restriction to mz = 0 of
ϑ(u)ϑ(u+ v +mz)
ϑ(u+mz)ϑ(u+ v)
is independent of u and in a suitable sense a homomorphism in v. Of course, again, it is only
a section of a line bundle which is isomorphic to the pullback of a line bundle on E [m], but not
canonically so. In the analytic setting, the corresponding ratio as a function of v indeed gives
a homomorphism from the universal cover; the specific homomorphism depends on the choice
of representative of the given m-torsion point, and for some m-torsion points cannot be made
trivial.
Note that since both lemmas are stated in terms of actual functions on the moduli stack,
we may replace the respective variables by arbitrary homomorphisms En → E .
In addition to line bundles and their sections, we will also need some gerbe-ish structures.
The objects we want to consider are “equivariant gerbes” (more precisely, equivariant structures
on the trivial gerbe; all equivariant gerbes considered in the present work will have trivial
Elliptic Double Affine Hecke Algebras 15
underlying gerbe): a system of line bundles Zg associated to g ∈ G ⊂ GLn(Z) along with
morphisms ζg,h : Zg ⊗
(
g−1
)∗Zh ∼= Zgh making all diagrams
Zg1 ⊗
(
g−1
1
)∗Zg2 ⊗
(
(g1g2)−1
)∗Zg3 −−−−→ Zg1g2 ⊗
(
(g1g2)−1
)∗Zg3y y
Zg1 ⊗
(
g−1
1
)∗Zg2g3 −−−−→ Zg1g2g3
commute. This, of course, is easy to construct: given any 1-cocycle for G valued in pairs (Q,w),
we may take Zg to be LQg ,wg and ζg,h to be the natural isomorphism
ζg,h : LQg ,wg ⊗
(
g−1
)∗LQh,wh ∼= LQg+g−tQhg−1,wg+wh = LQgh,wgh .
The situation becomes more complicated if we want to also construct meromorphic sections of
such gerbes: i.e., a system of meromorphic sections zg ∈ Zg such that ζg,h
(
zg⊗
(
g−1
)∗
zh
)
= zgh.
In this case, we do not have a completely general solution, but there is an important special
case.
Consider first the case G = Z, with generator acting on E2 by (z, q) 7→ (z + q, q). There is
a natural equivariant gerbe with meromorphic section such that Z1 = Lz2/2,−1 and z1 = ϑ(z).
Indeed, we then find more generally that Zk is the natural line bundle of weight −k with
polarization
(z, q)Q(z, q)t/2 = k
z2
2
+
k(k − 1)
2
qz +
k(2k − 1)(k − 1)
6
q2
2
,
and
zk = ϑ(z; q)k :=
∏
0≤i<k
ϑ(iq + z), k ≥ 0,∏
1≤i≤−k
ϑ(−iq + z)−1, k ≤ 0.
We will denote this meromorphic section of an equivariant gerbe by Γq(z), which we refer to as
an “elliptic Gamma function”. Of course, this is even less a function than ϑ, but in the ana-
lytic setting one can indeed replace Γq(z) by a suitable meromorphic solution of the functional
equation Γq(q+z) = ϑ(z)Γq(z). Of course, this only determines Γq up to multiplication by invert-
ible q-periodic functions, so the resulting meromorphic function is far from unique. One such
solution (for q in the upper half-plane) is
Γq(z; τ) :=
(
−
∏
1≤j
(1− e(jτ))2
)−z/q
e(−z(z − q)/4q)
∏
0≤j,k
1− e((j + 1)τ + (k + 1)q − z)
1− e(jτ + kq + z)
;
this depends on a choice of log
(
−
∏
1≤j(1− e(jτ))2
)
, but this will not matter for our purposes.
Here the double product is just the usual elliptic Gamma function [30].
More generally, for given morphisms q, z : En → E , we may pull this formal symbol back
to En. This is tricky to deal with in complete generality, but if we fix q and vary z, we may
consider general products∏
1≤i≤l
Γq(~αi · ~z)mi .
If the corresponding element
∑
imi[~αi] of Z[Hom(En, E)/q] is trivial, then this formal product
may be resolved into a product of ϑ functions using the functional equation. Moreover, since for
16 E.M. Rains
each congruence class the corresponding subproduct may be pulled back from E2, we find that
the resulting product of ϑ functions is independent of any choices that may have been made.
More generally, if G ⊂ GLn(Z) fixes q and the element
∑
imi[~αi] ∈ Z[Hom(En, E)/q], then
the ratio of the formal product and its pullback under g ∈ G will always resolve to a product of ϑ
functions, and thus gives an equivariant-gerbe-with-meromorphic-section which we refer to as
the coboundary of the formal symbol. Note in particular that the hypothesis always holds for the
(translation) subgroup of GLn(Z) that acts trivially on both q and the quotient Hom(En, E)/q;
in the cases of interest (affine Weyl groups), the intersection of G with this subgroup will be
cofinite in both groups, and it will be relatively straightforward to check that individual instances
give rise to sections of gerbes.
We should caution the reader that there is a mild subtlety when it comes to the reflection
principle. Although the product Γq(z)Γq(q−z) corresponds to the trivial equivariant gerbe (it has
polarization 0 and weight −1 (see below), both of which are invariant under z 7→ z + q), the
corresponding meromorphic section is not quite trivial: one finds that Γq(z)Γq(q − z) is negated
by the translation z 7→ z + q.
On the other hand, the multiplication principle Γq(z) =
∏
0≤j<k Γkq(z + jq) for integer k > 0
does work, as both sides truly do correspond to the same gerbe-with-section. This also works for
negative k: Γq(z) =
∏
1≤j≤−k Γkq(z−jq)−1. Using this, one can make sense of products of elliptic
Gamma functions with varying q as long as the different q that appear have a common multiple.
Note in particular the special case Γq(z) = Γ−q(z − q)−1.
We will of course want to know the polarizations and weights of the line bundles associated to a
given such gerbe section, which reduces to knowing the polarization and weight when such a prod-
uct resolves to a product of ϑ functions. There is, in fact, a very simple bookkeeping procedure
for determining this. Define the polarization of Γq(z) to be z(z−q)(2z−q)
12q , and the weight of Γq(z) to
be − z
q , extended to products of pullbacks in the obvious way. By this definition, the polarization
of the formal product Γq(q+z)/Γq(z) is z2/2 and the weight is −1, agreeing with the polarization
and weight of ϑ(z). It follows more generally that the polarization and weight of a product of
powers of elliptic Gamma functions is consistent with the usual notion whenever the product
resolves to a product of ϑ functions. (In particular, if the product resolves and the polarization
and weight are trivial, then it resolves to an honest function on En.) Of course, when considering
the associated gerbe-with-section, only the z-dependent portion of the polarization matters.
One should note here that not every cocycle valued in pairs (Q,w) comes from a product of Γq
symbols; for instance, any product of symbols Γq(az+ bq) with trivial polarization has weight of
the form 12cz/q + d, so that every line bundle in the coboundary has weight a multiple of 12.
(There are also some additional parity issues for n > 1.) Of course, it is conceivable that the
cocycles violating these congruence conditions do not have any consistent family of meromorphic
sections.
Note that if one wishes to convert a product of standard elliptic Gamma functions into
a product of symbols Γq, one must first use the reflection principle to eliminate appearances of p
from the arguments (which, if there is a balancing condition that involves p will require shifting
some of the variables by p first). If the result has
∑
imi
(∑
i ~αi · zi
)2
=
∑
imi
(∑
i ~αi · zi
)
= 0,
then replacing the standard Gamma functions by the explicit meromorphic solution given above
for the functional equation of Γq will have no effect on the resulting meromorphic function.
Since we now have a method for constructing equivariant line bundles on En, it is natural to
ask about the corresponding representations of G on global sections; in particular, we will want
to understand the space of G-invariant global sections. It turns out that if we want to extend
the standard analytic approach to this question to algebraic curves, we will need to extend the
above construction to cover certain abelian varieties isogenous to En.
To be precise, for a positive integer N let X0(N) denote the moduli stack of elliptic curves E1
equipped with a cyclic N -isogeny φ : E1 → EN , with universal curves denoted by E = E1, EN .
Elliptic Double Affine Hecke Algebras 17
(Here “cyclic” is in the sense of [15]; in particular, note that in characteristic p, any isogeny of
degree pk is cyclic.) For each divisor d|N , there is a corresponding factorization φ = φN,d ◦ φd,1,
where φd,1 : E1 → Ed, φN,d : Ed → EN are cyclic isogenies of degrees d and N/d respectively.
For d = p prime, this is constructed as follows: on the locus where the p-part of kerφ is étale, Ep
is the quotient by the p-torsion of kerφ; on the complementary locus, where the p-part of kerφ is
nonreduced, φ1,p is the Frobenius isogeny. (Per [15], this rule gives the correct limit to make Ep
a flat family.) In either case, kerφ1,p ⊂ kerφ, and thus φ factors as required, and we find
that φp,N is again cyclic. We thus have induced factorizations for every d|N , and it is easy to
see that they are all compatible. More precisely, for any pair d1|d2|N , we obtain an isogeny
φd2,d1 : Ed1 → Ed2 , and φd3,d2 ◦ φd2,d1
∼= φd3,d1 . We also, of course, have similarly compatible
isogenies φd1,d2 : Ed2 → Ed1 obtained by dualizing φd2,d1 .
Lemma 2.13. For any d1, d2|N , we have Hom(Ed1 , Ed2) ∼= Z, generated by the composition
φd2,d1 := φd2,gcd(d1,d2) ◦ φgcd(d1,d2),d1
= φd2,lcm(d1,d2) ◦ φlcm(d1,d2),d1
.
Proof. Let d0 = gcd(d1, d2) and d3 = lcm(d1, d2), and let f : Ed1 → Ed2 be any homomorphism.
Then φd0,d2 ◦ f ◦ φd1,d0 is an endomorphism of Ed0 , so must be multiplication by some integer.
By degree considerations, that integer must be a multiple of d2/d0 and d1/d0, and thus (since
these are relatively prime) of d1d2/d
2
0 = d3/d0. Since φd0,d2 ◦φd2,d1 ◦φd1,d0 =
[
d1d2/d
2
0
]
, the first
claim follows. We moreover find that
φd0,d2 ◦ φd2,d3 ◦ φd3,d1 ◦ φd1,d0 = φd0,d3 ◦ φd3,d0 = [d3/d0],
from which the other factorization follows. �
For any sequence of divisors di of N , we may consider the corresponding fiber product of cur-
ves Edi over X0(N); we will generally omit X0(N) from the product notation. As in the N = 1
case, morphisms between such products may be expressed as matrices, with the only difference
being that the ij entry is now a multiple of the natural isogeny φdi,dj .
Proposition 2.14. For any d1, d2|N , there is an isomorphism
Ed1 × Ed2
∼= Egcd(d1,d2) × Elcm(d1,d2).
Proof. Let d0 = gcd(d1, d2), d3 = lcm(d1, d2), and choose a, b so that ad1 − bd2 = d0. We then
easily verify that the morphisms(
aφd0,d1 bφd0,d2
φd3,d1 φd3,d2
)
: Ed1 × Ed2 → Ed0 × Ed3
and (
φd1,d0 −b(d2/d0)φd1,d3
−φd2,d0 a(d1/d0)φd2,d3
)
: Ed0 × Ed3 → Ed1 × Ed2
are inverses, giving the desired isomorphism. �
It follows immediately that any product of curves Ed is isomorphic to one of the form
∏
i Edi ,
where 1|d1| · · · |dn|N .
If we attempt to repeat our N = 1 construction for general N , we encounter two difficulties.
The first is that for each d|N , we may obtain a line bundle on X0(N) by taking the fiber at 1 of the
sheaf of differentials on Ed, but these line bundles are not quite the same. If E1 → Ed is an étale
isogeny, there is no problem: φ∗1,d induces an isomorphism of ωEd and ωE1 , so in particular of
their fibers over 1. However, if E1 → Ed is inseparable, then φ∗1,d actually annihilates ωEd .
18 E.M. Rains
As a result, ωEd |0 and ωE1 |0 differ by a linear combination of components of the fibers of X0(N)
over primes dividing d. Of course, if all we want to do is construct line bundles, this is not
an issue, but this does mean that the construction of functions in this way will be nontrivial.
The more serious difficulty is that it is no longer the case that
∏
i Edi → X0(N) has only the
trivial section. Indeed, we have the following.
Lemma 2.15. Let N be a positive integer, and d|N . Then the group of sections of Ed over X0(N)
consists entirely of 2-torsion. If both d and N/d are odd, the group is trivial; if precisely one is
even, it has rank 1, and if both are even, it has rank 2.
Proof. Since the stack X0(N) has nontrivial stabilizers, any section of Ed must be invariant
under the action of the stabilizer, and is thus preserved by [−1], so is 2-torsion. The structure
of the 2-torsion of Ed then follows by considering the image of the corresponding congruence
subgroup in SL2(Z/2Z). �
Remark. For more general level structures, it follows from [33, Theorem 5.5] that for any
subgroup Γ ⊂ SL2(Z/NZ), the group of sections over the corresponding quotient stack X (N)/Γ
is N -torsion, and isomorphic to the subgroup
(
(Z/NZ)2
)Γ
.
If N is odd, we may thus conclude as before that the pullback of OEd1 ([0]) through φd1,d2
is indeed fiberwise isomorphic to OEd2 ([0])deg(φd1,d2 ). However, if N is even, this is no longer
the case; indeed, the pullback of OE([0]) through a 2-isogeny E′ → E is the tensor product
of OE′(2[0]) by the corresponding 2-torsion line bundle.
It turns out that we can fix this, at the cost of imposing some additional level structure.
Let X0(2N, 2) be the slight reinterpretation of the stack X0(4N) obtained by dividing all of
the subscripts of the isogenous curves by 2; that is X0(2N, 2) classifies cyclic 4N -isogenies
E1/2 → E2N in terms of the curve E1. Now, it follows from the lemma that for each integer d|N ,
the curve Ed has full 2-torsion over X0(2N, 2); in addition to the generators of the kernels
Ed → Ed/2 and Ed → E2d, it also has their sum, which we denote by σd. We thus obtain
a line bundle L̂1,Ed := OEd([σd]) ⊗ 0∗OEd([σd])−1 on Ed with trivial fiber over 0. Note that
again 0∗OEd([σd]) is a nontrivial line bundle, though it is actually far better-behaved than
0∗OEd([0]). Indeed, 0∗OEd([σd]) is trivial away from the locus where σd = 0. This can only
happen in characteristic 2, and only when the 4-isogeny Ed/2 → E2d is multiplication by 2
(corresponding to one of the three components of X0(2, 2) in characteristic 2).
The merit of using this 2-torsion point is that it makes everything compatible.
Lemma 2.16. If d1|d2|N then there are natural isomorphisms
φ∗d2,d1
L̂1,Ed2
∼= L̂d2/d1
1,Ed1
,
φ∗d1,d2
L̂1,Ed1
∼= L̂d2/d1
1,Ed2
.
Proof. The two claims are equivalent via the (Atkin-Lehner) involution X0(2N, 2) ∼= X0(2N, 2)
that replaces E1/2 → E2N by its dual, so it suffices to prove the first. Since the fibers at the
origin of the two bundles are trivial, it suffices to show that the line bundles are isomorphic on
each fiber over X0(2N, 2), at which point we may take the isomorphism respecting the fibers
at the origin. We may also, for convenience, reduce to the case d1 = 1, d2 = N .
Being isomorphic is a closed condition, so we may exclude primes dividing 4N , and in particu-
lar assume that the isogenies are all separable. On a given such curve, the line bundle φ∗N,1L̂1,EN
is represented by the divisor∑
x∈φ−1
N,1(σN )
[x],
Elliptic Double Affine Hecke Algebras 19
and thus we need to show∑
x∈φ−1
N,1(σN )
x = Nσ1,
or equivalently∑
x∈φ−1
N,1(σN−φN,1(σ1))
x = 0.
If N is odd, then φ(σN ) = σ1, and this sum becomes∑
x∈kerφN,1
x = 0
as required. Otherwise, we may factor through E2 and thus reduce to the case N = 2. Then
we find that φ2,1σ1 is the generator of the kernel of φ1,2, and thus σ2 − φ2,1σ1 is the generator
of the kernel of φ2,4. It follows that the preimage of σ2 − φ2,1σ1 consists of the two generators
of the kernel of φ1,4, and these add to 0 as required. �
Remark. It is worth noting here that there are additional curves in the isogeny class of E1,
since after all each of the 2-torsion points σd itself determines a 2-isogeny. It is unclear whether
we can extend the above family of line bundles to the other curves arising in this way.
Corollary 2.17. For any integer a, [a]∗L̂1,Ed
∼= L̂a2
1,Ed.
Proof. This is clearly true for a = 0 and a = −1, so we may assume a > 0. Over X0(2aN, 2),
we have [a] = φd,ad ◦ φad,d, so that the claim follows immediately from the lemma. Since the
isomorphism is natural, it descends to X0(2N, 2) as required. �
Since L̂1,Ed represents the standard principal polarization on Ed, we also have the following.
Lemma 2.18. For any d|N , the bundle
[x1 + x2]∗L̂1,Ed ⊗ [x1]∗L̂−1
1,Ed ⊗ [x2]∗L̂−1
1,Ed
is naturally isomorphic to the Poincaré bundle PEd on Ed × Ed.
Theorem 2.19. For any sequence 1|d1| · · · |dn|N , suppose L1 and L2 are two line bundles
on
∏
i Edi obtained as tensor products of pullbacks of bundles L̂1,Ed through morphisms∏
i Edi → Ed. If L1 and L2 represent the same polarization, then they are naturally isomorphic.
Proof. By the theorem of the cube, it suffices to prove this for n = 2. In this case, it is clear
that the images of the bundles (1×0)∗L̂1,Ed1 , (0×1)∗L̂1,Ed2 and (1×φd1,d2)∗PEd1 span the group
of symmetric endomorphisms of Ed1 × Ed2 , and thus it will suffice to show that any pullback
of L̂1,Ed is isomorphic to the appropriate product of these bundles (again using the triviality at 0
to fix the isomorphism).
Thus consider a morphism ψ := aφd,d1 + bφd,d2 : Ed1 × Ed2 → Ed. We then have
ψ∗L̂1,Ed
∼= (aφd,d1)∗L̂1,Ed ⊗ (bφd,d2)∗L̂1,Ed ⊗ (aφd,d1 × bφd,d2)∗PEd .
Now,
(aφd,d1)∗L̂1,Ed = [a]∗φ∗gcd(d,d1),d1
φ∗d,gcd(d,d1)L̂1,Ed = L̂a
2 gcd(d,d1)2/d1d
1,Ed1
,
and similarly for the second term. We also have
(aφd,d1 × bφd,d2)∗PEd ∼= (1× aφd1,dbφd,d2)∗PEd=(1× abcφd1,d2)∗PEd ∼=
(
(1×φd1,d2)∗PEd
)abc
for a suitable integer c, so that the claim follows. (The first step here is essentially the definition
of the dual isogeny.) �
20 E.M. Rains
Thus for each symmetric Q ∈ End(
∏
i Edi), we obtain a line bundle L̂Q;d1,...,dn on
∏
i Edi , and
these line bundles satisfy the same compatibility relations as for our earlier construction on En.
In general, our two constructions do not agree, but there is one important special case.
Proposition 2.20. Suppose Q ∈ Matn(Z) is a symmetric matrix with even diagonal entries.
Then L̂Q;1,...,1 descends to M1,1, where it is canonically isomorphic to LQ,0.
Proof. Any symmetric integer matrix with even diagonal is not only in the span of pullbacks
of 1, but in fact is in the span of pullbacks of H = ( 0 1
1 0 ), the symmetric endomorphism associated
to the Poincaré bundle P ∼= L̂H;1
∼= LH,0. �
Remark. More generally, if Qii is even whenever di is odd, then L̂
Q;~d
descends to X0(2N);
similarly, if Qii is even whenever N/di is odd, then L̂
Q;~d
descends to X0(N, 2).
It will be convenient to have a somewhat more functorial version of the constructions of En
or the products
∏
i Edi above. For the first, if B is a finitely generated free abelian group,
then we may consider the family of group schemes E ⊗ B. For a specific curve E, we may also
construct E ⊗B, which is simply the corresponding fiber of E ⊗B.
This clearly extends to a functor, which is exact on short exact sequences of free abelian
groups. Note, however, that if we extend it in the obvious way to a functor on the category
of all finitely generated abelian groups, then it is no longer exact. We readily compute the
special cases
E ⊗ Z ∼= E,
Torp(E,Z) = 0, p > 0,
E ⊗ Z/NZ = 0,
Tor1(E,Z/NZ) = E[N ],
Torp(E,Z/NZ) = 0, p > 0
using the obvious projective resolution of Z/NZ. (This, of course, is the expected behavior
for tensoring with a divisible group E.)
Proposition 2.21. Let φ : B → C be a morphism of finitely generated free abelian groups. Then
E ⊗ φ is surjective iff coker(φ) is finite, and injective iff ker(φ) = 0 and coker(φ) is free.
Proof. The four-term sequence
0→ kerφ→ B → C → cokerφ→ 0
is a free resolution, and thus we have an isomorphism coker(E ⊗ φ) ∼= E ⊗ cokerφ and a short
exact sequence
0→ E ⊗ ker(φ)→ ker(E ⊗ φ)→ Tor1(E, coker(φ))→ 0.
The claim follows immediately. �
Proposition 2.22. If E does not have complex (or quaternionic) multiplication (in particular
if E = E), then Hom(B,C)→ Hom(E ⊗B,E ⊗ C) is an isomorphism.
Proof. This reduces to the case B = Zn, C = Zm, and thus to the case B = C = Z, where it
is essentially by definition. �
Corollary 2.23. If E does not have complex multiplication, then the natural map B →
Hom(E,E ⊗B) is an isomorphism, as is the natural map E ⊗Hom(E,E ⊗B)→ E ⊗B.
Elliptic Double Affine Hecke Algebras 21
The dual variety is then easy to compute.
Proposition 2.24. For B free, there is a natural isomorphism (E ⊗B)∨ ∼= E ⊗B∗.
Proof. Indeed, we have E ⊗B ∼= En, so dually (E ⊗B)∨ ∼= En. Thus for E without complex
multiplication, the natural map E⊗Hom
(
E, (E⊗B)∨
)
→ (E⊗B)∨ is an isomorphism. Duality
gives Hom(E, (E⊗B)∨) ∼= Hom(E⊗B,E) ∼= Hom(B,Z), and thus E⊗Hom(B,Z) ∼= (E⊗B)∨
as desired. The isomorphism holds for E = E , and thus for all fibers E. �
This gives the following description of the Néron–Severi group of E⊗B: NS(E⊗B) consists of
symmetric morphisms E⊗B → (E⊗B)∨ ∼= E⊗B∗, and thus of symmetric pairings Q : B⊗B → Z.
We then find as above that any such symmetric pairing (and any weight) induces a line bun-
dle LQ,w on E ⊗B.
Now, suppose B → C is an injective morphism with finite cokernel. Then we have a short
exact sequence
0→ Tor1(E , C/B)→ E ⊗B → E ⊗ C → 0,
where the kernel is a product of groups of the form E[di]. If C/B has exponent N , then we
have Tor1(E , C/B) ∼= E [N ] ⊗Z/NZ C/B, which in turn suggests that we consider the subgroup
κN ⊗Z/NZ C/B, where κN is the kernel of a the cyclic N -isogeny corresponding to a point
of X0(N). This, it turns out, does not quite behave correctly in characteristic dividing N , but
we do have the following.
Proposition 2.25. Let N be a positive integer, and let B be a finitely generated abelian group
of exponent N . Then the group scheme κN ⊗Z/NZ B on X0(N) × Spec(Z[1/N ]) extends in
a natural way to a flat group scheme on X0(N).
Proof. If we choose an isomorphism B ∼=
⊕
Z/diZ, then we certainly have such an extension:
away from N , the group is just
∏
i κdi , where κdi is the kernel of the isogeny E1 → Edi , and
this product makes sense in all characteristics. If B → B′ is a morphism of N -torsion groups,
then the morphism κN ⊗Z/NZ B → κN ⊗Z/NZ B
′ is simply the restriction of the morphism
E[N ]⊗Z/NZ B → E[N ]⊗Z/NZ B
′. The latter morphism is defined in all characteristics, and the
requirement that it restrict to a specific morphism is a closed condition, so is inherited from the
generic case. �
This allows us to define families of abelian varieties over X0(N) as follows: given abelian
groups B, C such that NC ⊂ B ⊂ C, we define EB,C to be the quotient of E ⊗ B by the
subgroup scheme extending κN ⊗Z/NZ C/B. (We will denote this extension by the same tensor
product notation, but caution the reader that it is not the actual tensor product in general.)
Proposition 2.26. If NC1 ⊂ B1 ⊂ C1, NC2 ⊂ B2 ⊂ C2 are pairs of finitely generated free
abelian groups and φ : C1 → C2 is a morphism such that φ(B1) ⊂ B2, then there is an induced
morphism EB1,C1 → EB2,C2, making the construction functorial.
Proof. The condition on φ implies that it induces a morphism C1/B1 → C2/B2, and thus we
have a commutative diagram
0 −−−−→ κN ⊗Z/NZ C1/B1 −−−−→ E ⊗B1 −−−−→ EB1,C1 −−−−→ 0y y
0 −−−−→ κN ⊗Z/NZ C2/B2 −−−−→ E ⊗B2 −−−−→ EB2,C2 −−−−→ 0.
Since the rows are exact and the given vertical arrows are functorial, the claim follows. �
22 E.M. Rains
Of course, given an isomorphism C/B ∼=
∏
i Z/diZ, there is a corresponding isomorphism
EB,C ∼=
∏
i Edi . This makes the following straightforward to verify.
Proposition 2.27. If E1 → EN is a cyclic N -isogeny between curves with no complex multipli-
cation, then the morphisms between corresponding fibers of EB1,C1 and EB2,C2 are precisely those
coming from the previous proposition.
Corollary 2.28. For NC ⊂ B ⊂ C, we have
Hom(E , EB,C) ∼= B,
Hom(EN , EB,C) ∼= C
in such a way that composition with the dual isogeny EN → E induces the inclusion B ⊂ C.
Proposition 2.29. There is a natural isomorphism E∨B,C ∼= EC∗,B∗.
Proof. The previous corollary allows us to canonically identify anything isomorphic to a product
of curves Edi with a variety of the form EB,C . Since EB,C is isomorphic to such a product, its dual
is also of that form, and thus it remains only to compute Hom(E , E∨B,C) and Hom(EN , E∨B,C). �
It follows that the Néron–Severi group of EB,C (or of any fiber without complex multipli-
cation) consists of pairings B ⊗ C → Z which become symmetric when restricted to B ⊗ B;
equivalently, it consists of symmetric pairings Q : B⊗B → Z such that Q(B,NC) ⊂ NZ. From
our construction above, we find that if we base change to X0(2N, 2), then any such symmetric
pairing induces a natural line bundle L̂Q;B,C on EB,C , satisfying the appropriate compatibility
relations.
Suppose now that Q is a positive definite pairing of “level” dividing N (i.e., such that
NB∗ ⊂ QB) Then we have a chain of free abelian groups Q−1NB∗ ⊂ B ⊂ C ⊂ Q−1B∗, giving
rise to an isogeny π : EB,C → EB,Q−1B∗ . The pairing Q still induces an element of the Néron–
Severi group of the quotient, which by degree considerations is now a principal polarization,
represented by the line bundle Θ̂B := L̂Q;B,Q−1B∗ . We moreover find that π∗Θ̂B
∼= L̂Q;B,C .
We also note that the action of E [N ] ⊗
(
Q−1B∗/B
) ∼= Tor1
(
E , Q−1B∗/B
)
on E ⊗ B des-
cends to an action of the quotient
(
E [N ] ⊗
(
Q−1B∗/B
))
/
(
κN ⊗
(
Q−1B∗/B
))
on EB,Q−1B∗ ,
which by the Weil pairing may be identified with an action of the Pontryagin dual Hom
(
κN ⊗(
Q−1B∗/B
)
, µN
)
. If κN is diagonalizable, then this dual is discrete, and may be identified
with Hom
(
Q−1B∗/B,Hom(κN , µN )
)
.
Lemma 2.30. On the locus of X0(2N, 2) where the N -isogeny E1 → EN has diagonalizable
kernel κN , we have natural identifications
Γ
(
EB,C ; L̂Q;B,C
) ∼= ⊕
g∈Hom(Q−1B∗/C,Hom(κN ,µN ))
g∗Γ
(
EB,Q−1B∗ ; Θ̂B
)
,
in which each (1-dimensional!) summand on the right is an eigenspace for the induced action
of ker(EB,C → EB,Q−1B∗).
Proof. We have
Γ
(
EB,C ; L̂Q;B,C
) ∼= Γ
(
EB,C ;π∗Θ̂B
) ∼= Γ
(
EB,Q−1B∗ ;π∗π
∗Θ̂B
)
.
The natural map Θ̂B → π∗π
∗Θ̂B selects a particular eigenspace of the kernel of the isogeny, and
the decomposition as claimed then follows by the structure theory of representations of Heisen-
berg groups, see [32] as well as the exposition in [22]. Here we use the fact that Γ
(
EB,C ; L̂Q;B,C
)
is the unique irreducible representation of the Heisenberg group G
(
L̂Q;B,C
)
on which the cen-
tral Gm acts with weight 1, together with the fact that the commutator pairing on the Heisenberg
group is precisely the Weil pairing, so the different isotypic components for the diagonalizable
kernel are related via the complementary translation subgroup. �
Elliptic Double Affine Hecke Algebras 23
Corollary 2.31. Suppose that the finite group G acts on C, preserving the subgroup B and
the polarization Q : B → B∗ so that G acts on EB,C as automorphisms fixing the identity, with
an induced equivariant structure on L̂Q;B,C . On the locus of X0(2N, 2) where E1 → EN has
diagonalizable kernel, the G-module Γ
(
EB,C ; L̂Q;B,C
)
is isomorphic to the permutation module
arising from the action of G on Hom
(
Q−1B∗/C,Q/Z
)
.
Remark. Here we note that for general free abelian groups B ⊂ C with finite quotient,
Hom(C/B,Q/Z) ∼= Tor1(B∗/C∗,Q/Z) ∼= B∗/C∗,
and thus
Hom
(
Q−1B∗/C,Q/Z
) ∼= C∗/QB ∼= Q−1C∗/B.
Corollary 2.32. Let B be a finitely generated free abelian group and Q : B ⊗ B → Z an even
symmetric pairing of level dividing N , and let E be an elliptic curve which, if supersingular, has
characteristic prime to N . Then Γ(E⊗B;LQ,0) is isomorphic as a G-module to the permutation
module coming from the action of G on Q−1B∗/B.
Proof. The condition on E ensures that we may choose a point of X0(2N, 2) lying over it such
that the cyclic N -isogeny E ∼= E1 → EN has diagonalizable kernel. We may then identify the
G-module Γ(E ⊗B;LQ,0) with Γ
(
EB,B; L̂Q;B,B
)
and thus apply the previous corollary. �
Corollary 2.33. With the same hypotheses, the dimension dim
(
Γ(E ⊗ B;LQ,0)G
)
is equal to
the number of orbits of G in Q−1B∗/B.
Remark. Both conclusions remain valid for supersingular curves of characteristic prime to |G|;
indeed, any 1-parameter family of G-modules over an algebraically closed field containing 1/|G|
is trivial, so the claims follow from the case of ordinary curves.
It turns out that the exclusion of certain supersingular curves above is indeed necessary.
For instance, suppose that B = Z8 and Q is the Gram matrix of the lattice Q8(1) of [5]. This
is a symmetric matrix with even diagonal and elementary divisors 1, 1, 1, 1, 5, 5, 5, 5, and
the corresponding automorphism group GO+
4 (5) acts in the natural way on coker(Q). If E
is the (geometrically unique) supersingular curve of characteristic 5, then one finds that the
induced GO+
4 (5)-module structure on Γ
(
E8;LQ,0
)
is not isomorphic to the given permutation
representation. In this case, the actual invariant subspace does not jump, but we find that
dim
(
Γ
(
E16;LQ⊕Q,0
)GO+
4 (5)) ∼= dim
((
Γ
(
E8;LQ,0
)⊗2)GO+
4 (5))
= 160,
while for all other curves, the invariant space has dimension 156. To compute the action of G
in such supersingular cases, we may use the fact that LQ,0 always descends to the appropriate
quotient and gives an equivariant isomorphism Γ
(
En;LQ,0
) ∼= Γ
(
kerψ∨Q;L′
)
; what fails is that
we no longer have a natural basis (which in the case of the theorem are an eigenbasis for the
action of the diagonalizable group kerψQ) of the latter. There is, in fact, an isomorphism
Γ
(
kerψ∨Q;L′
) ∼= Γ
(
kerψ∨Q;Okerψ∨Q
)
,
since kerψ∨Q is 0-dimensional, but this is not in general equivariant; in general, the action of G
is twisted by some cocycle with values in the unit group of the coordinate ring. When det(Q)
is odd, however, we can use the description of L′ in terms of pullbacks of OEd([σd]) to compute
this cocycle. In the Q8(1) case, this further simplifies, since we only need to know what happens
on α4
5, allowing us to reduce to an evaluation of functions on E4
5 in an appropriate formal
neighborhood of the identity.
24 E.M. Rains
A similar calculation applies to the Gram matrix of
√
3Λ⊥E6
, with its automorphism
group O5(3); in this case, there are also subgroups of O5(3) for which the corollary fails on
the supersingular curve of characteristic 3. We can also obtain a characteristic 2 counterexam-
ple from
√
2Λ⊥E7
; in this case, it is unclear how to compute the cocycle, but we can simply check
that none of the 16 elements of H1
(
Sp6(2);µ2
(
α6
2
))
give rise to the permutation module. (There
is also a subgroup with too many invariants, namely the preimage in W (D6) of the transitive
Alt5 ⊂ S6, which has too many invariants in each of the 16 possible cases.)
We should further note that the requirement that Q have even diagonal is also necessary;
indeed, otherwise the claim already fails for the case Q = 1, G = GL1(Z) for any curve of
characteristic not 2.
Even when E is supersingular of characteristic dividing N , there may still be isogenies
of the form
EB,C → EB,C′
with diagonalizable kernel, which as an abstract group scheme can be (geometrically) identified
with µN ⊗Z/NZC
′/C. Indeed, the only requirement is that |C ′/C| be prime to the characteristic
of E. Making this a canonical identification is somewhat trickier, as the kernel is only naturally
described as the quotient
(κN ⊗Z/NZ C
′/B)/(κN ⊗Z/NZ C/B).
Moreover, the translations moving between the different eigenspaces are only defined up to the
kernel of the descended polarization. We find in general that
Γ(EB,C ;LQ;B,C) ∼=
⊕
g
g∗Γ(EB,C′ ;LQ;B,C′),
where g ranges over the quotient group
Hom
(
Q−1B∗/C,Hom(κN , µN )
)
/Hom
(
Q−1B∗/C ′,Hom(κN , µN )
)
.
We can thus only use this decomposition in understanding group actions when this quotient
group has an equivariant splitting. Luckily, there is an important case when this happens:
if C ′/C is the l-part of Q−1B∗/C for some prime l which is invertible on E, then the quotient
may be identified with the l-part of Hom
(
Q−1B∗/C,Hom(κN , µN )
)
.
Lemma 2.34. With B,C,Q as above and l a prime invertible on E, there is a G-equivariant
isomorphism
Γ(EB,C ;LQ;B,C) ∼=
⊕
H
IndGH ResGH Γ(EB,C′ ;LQ;B,C′),
where H ranges over the point stabilizers in the different orbits of the action of G on the l-part
of Q−1C∗/B and
C ′ =
⋃
k
Q−1B∗ ∩ l−kC.
Here we should note that we have such a reduction for every prime dividing N ; in particular,
if the polarization does not have prime power degree, then we can always choose a prime dividing
the degree of the polarization which is invertible on E, and use the corresponding reduction.
Although we have seen that there can indeed be (finitely many) bad curves for such invariant
theory questions, it turns out that our hypotheses are in fact slightly more restrictive than
Elliptic Double Affine Hecke Algebras 25
they need to be. Suppose we have a finite group G acting on an abelian variety A (fixing the
identity). There are two natural induced abelian subvarieties. The subgroup scheme AG is
still projective, and thus (up to a possible inseparable base change) we may take its reduced
identity component AG0. Equivalently (and without need for base change), we could instead
define AG0 to be the image of the endomorphism
∑
g∈G g ∈ End(A). There is also an almost
complementary subvariety AG giving by the image of the endomorphism |G| −
∑
g∈G g. Both
subvarieties are clearly preserved by G, and since the sum of the endomorphisms is an isogeny,
it follows that we have a natural G-equivariant isogeny AG ×AG0 → A. Any G-equivariant line
bundle on A (with trivial action on the fiber over the identity) pulls back to a G-equivariant
line bundle on AG×AG0, namely L|AG �L|AG0 . Moreover, the action of G on the second factor
is trivial, since it is trivial at the identity.
It turns out that even though this isogeny can fail to have diagonalizable kernel, we can still
use it to reduce questions about G-module structures to AG.
Lemma 2.35. With A, G, L as above, assume that L is ample. Then there is a G-module
isomorphism
Γ(A;L)d ∼= Γ(AG;L|AG)e,
where d =
∣∣AG ∩AG0
∣∣ and e = dim Γ
(
AG0;L|AG0
)
.
Proof. Let K = AG∩AG0 be the kernel of the isogeny AG×AG0 → A. Then we have a natural
isomorphism
Γ(A;L) ∼= Γ
(
AG ×AG0;L|AG � L|AG0
)K ∼= (Γ(AG;L|AG)⊗ Γ
(
AG0;L|AG0
))K
.
Let H be the preimage of K in the Heisenberg group G
(
L−1
AG0
)
. This certainly acts naturally
on Γ
(
AG0;L|AG0
)∗
, but the fact that the line bundle descends to A implies that it also acts
on Γ(AG;L|AG). We thus have a natural isomorphism(
Γ(AG;L|AG)⊗ Γ
(
AG0;L|AG0
))K ∼= HomH
(
Γ
(
AG0;L|AG0
)∗
,Γ(AG;L|AG)
)
,
which by Frobenius reciprocity further becomes
HomH
(
Γ
(
AG0;L|AG0
)∗
,Γ(AG;L|AG)
)
∼= HomG(L|−1
AG0 )
(
Γ
(
AG0;L|AG0
)∗
, Ind
G(L−1
AG0 )
H Γ(AG;L|AG)
)
.
By the structure of Heisenberg representations, we may then conclude that there is a functorial
isomorphism
Γ(A;L)⊗ Γ
(
AG0;L|AG0
) ∼= Ind
G(L−1
AG0 )
H Γ(AG;L|AG)
of G
(
L−1
AG0
)
-modules. Moreover, the splitting K → H is G-invariant, since it could be computed
inside AG0, and thus we may rewrite this as a G× G
(
L−1
AG0
)
-module isomorphism:
Γ(A;L)⊗ Γ
(
AG0;L|AG0
) ∼= Ind
G×G(L−1
AG0 )
G×H Γ(AG;L|AG).
Moreover, the induction functor is exact (since the homogeneous space is affine), as is restriction
toG, and thus if we forget the action of the Heisenberg group, we obtain aG-module isomorphism
Γ(A;L)e ∼= Γ(AG;L|AG)e
2/d,
from which the claim follows. �
Remark 2.36. Note that although d always divides e2, it need not divide e, and vice versa e
can fail to divide d.
Remark 2.37. The reader should note that the notion of an induced module for representations
of group schemes corresponds to what would normally be called a coinduced module.
26 E.M. Rains
3 Coxeter group actions on abelian varieties
One of the major ingredients in the construction of elliptic analogues of double affine Hecke
algebras is a suitable action of an affine Weyl group on a power of an elliptic curve (or more
generally on a variety isogenous to such a power). It will be convenient to work somewhat more
abstractly, and begin with the finite case.
With this in mind, let A be an abelian variety, and suppose the finite Weyl group W acts
faithfully on A (fixing the identity) in such a way that for any reflection r ∈ R(W ), the corre-
sponding fixed subgroup scheme has codimension 1. We will naturally refer to such an action
as an action “by reflections”.
For each reflection r ∈ R(W ), the subgroup 〈r〉 splits A (up to isogeny) as discussed above;
in this case, we have natural subvarieties Ar0 := im(1 + r) and Ar := im(1 − r), and an indu-
ced isogeny Ar0 × Ar → A (with kernel contained in Ar[2]). Since Ar0 by assumption has
codimension 1, we see that Ar is a 1-dimensional abelian variety. In other words, each reflection
in W induces a corresponding elliptic curve Er = Ar contained in A, as the image of the
endomorphism 1− r. We call such a curve the “root curve” associated to r. Applying the same
construction to the dual variety A∨ gives root curves E′r ⊂ A∨, the duals of which we refer to
as “coroot curves”. Note that the coroot curve associated to r can be described directly as the
cokernel of the endomorphism 1 + r. In particular, the endomorphism 1− r factors through E′r,
giving rise to a natural map E′r → Er such that the composition A → E′r → Er → A is 1 − r
and the composition Er → A→ E′r → Er is multiplication by 2.
Fix a system of simple roots S = {α1, . . . , αn} in W , and let E1, . . . , En; E′1, . . . , E
′
n be the
corresponding root and coroot curves, with induced maps ιi : Ei → A, ι′i : A→ E′i, νi : E
′
i → Ei.
The action of si on Ej can be described quite simply:
si ◦ ιj = ιj + (si − 1) ◦ ιj = ιj − ιi ◦ νi ◦ ι′i ◦ ιj .
This suggests that we should define a morphism µij : Ej → Ei as the composition −νi ◦ ι′i ◦ ιj ;
that is, it is the morphism Ej → Ei induced by si − 1.
Lemma 3.1. The curves E1, . . . , En are distinct.
Proof. Suppose otherwise, and reorder the simple roots so that E1 = E2. Then s1s2 6= 1, but
(s1s2 − 1) = (s1 − 1)(s2 − 1) + (s1 − 1) + (s2 − 1),
so that s1s2−1 has image E1 = E2. Since s1s2 fixes E1 = E2, we find that (s1s2)k−1 = k(s1s2−1)
for all k ≥ 1, and thus s1s2 has infinite order, contradicting finiteness of W . �
Lemma 3.2. For i 6= j, the composition µji ◦ µij is multiplication by k ∈ {0, 1, 2, 3}, and if the
composition is 0, then µij = µji = 0.
Proof. Since Ei 6= Ej , the product Ei × Ej is isogenous with its image in A. Define an action
of si, sj on Ei × Ej by
si(xi, xj) = (−xi + µji(xj), xj),
sj(xi, xj) = (xi,−xj + µij(xi)).
The elements si, sj clearly act as involutions on the product, and the actions are compatible
with the actions on A, so that the action of (sisj)
mij −1 induces a homomorphism from Ei×Ej
to the kernel of the map Ei × Ej → A. Since Ei × Ej is proper, reduced, and connected,
and said kernel is finite, we see that this description actually gives an action of the rank 2 Weyl
Elliptic Double Affine Hecke Algebras 27
group 〈si, sj〉. Moreover, since this group is finite, there is a W -invariant polarization on Ei×Ej ,
of the form(
2ri −ψji
−ψ∨ji 2rj
)
,
with 4rirj − deg(ψji) > 0. We then find that ψji = riµji = rjµ
∨
ij , so that
deg(ψji) = ψjiψ
∨
ji = riµjirjµij ,
and thus µjiµij is multiplication by a nonnegative integer less than 4. Moreover, since riµji =
rjµ
∨
ij , we see that if one of µij , µji vanishes, then so does the other. �
Remark. We then readily see that the order of sisj is equal to 2, 3, 4, 6 when µijµji = µjiµij
is equal to 0, 1, 2, 3 respectively.
Proposition 3.3. Let (W,S) be a finite Weyl group, and suppose that E1, . . . , En is a system of
elliptic curves and µij : Ei → Ej, i 6= j, a system of morphisms such that µijµji = 4 cos(π/mij)
2,
with µij = µji = 0 whenever mij = 2. Then there is a faithful action of W on
∏
iEi such that
si(x1, . . . , xn) =
(
x1, . . . , xi−1,−xi +
∑
j 6=i
µji(xj), xi+1, . . . , xn
)
.
Proof. The action of si is clearly an involution, and the braid relations are straightforward
to verify. (This is easy when mij = 2, and for mij > 2 it suffices to show that
(
(sisj)
2+
(2− µijµji)(sisj) + 1
)
(sisj − 1) vanishes, which reduces to a computation in Ei × Ej .) So this
certainly gives an action of W , and it remains only to show that it is faithful.
Since the construction clearly respects products, we may as well assume that W is irreducible.
For any path in the Coxeter diagram of W , we may take the corresponding composition of
morphisms µij ; since µij = 0 iff mij = 2, any such composition will be an isogeny. Moreover,
since the Coxeter diagram is a tree by finiteness of W , we see that any two isogenies Ei → Ej
arising in this way will differ by a positive factor (any time the path backtracks introduces a factor
µijµji > 0). In particular, for any element w ∈ W , the induced map Ei → Ej (apply w then
project onto the jth factor) is an integer linear combination of such isogenies, and in particular
has a corresponding notion of positivity. This allows us to turn any element of W into a real
matrix by taking each such morphism to the appropriately signed square root of its degree.
The consistency of sign ensures that this will give rise to an actual representation of W , and we
can then verify that up to a diagonal change of basis, this is precisely the standard reflection
representation of W . �
Corollary 3.4. Let the finite Weyl group W act faithfully on the abelian variety A by reflec-
tions, with simple root curves E1, . . . , En. Then the induced morphism
∏
iEi → A is made
W -equivariant by the above action, and its kernel is finite and fixed by W .
Proof. The equivariance is obvious by construction, so it remains only to show that the kernelK
is fixed by W . Otherwise, some simple reflection si will act nontrivially on K, and thus (si−1)K
⊂ K is nonzero. But (si−1)K ⊂ Ei, and K∩Ei = 0 since Ei is defined as a subscheme of A. �
The proof of the proposition suggests an extension of this construction to more general
crystallographic Coxeter groups (in particular to affine Weyl groups). Certainly, one could con-
sider an action of the above form associated to any system of morphisms µij , but to relate it
to the standard reflection representation of a Coxeter group, we make the following assump-
tions:
28 E.M. Rains
� The composition µijµji is multiplication by kij ∈ {0, 1, 2, 3, 4}.
� There is a system of positive integers ri such that riµji = rjµ
∨
ij for each i 6= j.
� Any composition µi1i2µi2i3 · · ·µimi1 is multiplication by a nonnegative integer.
We call such a system of curves and morphisms an “elliptic root datum”.
Theorem 3.5. Any elliptic root datum gives rise to a faithful action on
∏
iEi of the Coxeter
group with multiplicities mij given by kij = 4 cos(π/mij)
2, such that si acts as above.
Proof. The conditions on the morphisms ensure that we can faithfully translate the action into
one on a real vector space, taking each morphism to the appropriately signed square root of
its degree. Conjugating by the diagonal matrix with entries
√
ri turns this into the standard
reflection representation of the given Coxeter group. �
Corollary 3.6. For each conjugacy class C of reflections, there is a corresponding elliptic
curve EC equipped with isomorphisms EC ∼= Er, r ∈ C such that the action of W on the set of
compositions EC ∼= Er → A and their negatives can be identified with the action of W on the
corresponding set of roots, with the compositions EC ∼= Er → A corresponding to the positive
roots.
Proof. Indeed, for each simple root αi, we may consider the set of compositions w◦ιi for w ∈W ,
and find that each such composition has the form (βi1, βi2, . . . , βin) in which either every entry
is a nonnegative element of the relevant Hom space (i.e., corresponding to a nonnegative real
number) or every entry is nonpositive. If r = wsiw
−1, then the image of 1 − r is w times the
image of 1−si, and thus w ◦ ιi identifies Ei with Er. Each Er is afforded with precisely two such
identifications, of which we naturally choose the one corresponding to a positive root. We fur-
thermore see that conjugate simple reflections give rise to equivalent systems of identifications
of root curves. �
Corollary 3.7. Relative to the action of W on
∏
iEi arising in this way, any two reflections
have distinct root curves.
Proof. Assuming without loss of generality that W is irreducible, we find that each Ei is
isogenous to E1 in an essentially canonical way (choose the isogeny of smallest degree among
the “positive” isogenies), and then see that two root curves agree iff the corresponding maps
E1 →
∏
iEi correspond to proportional real vectors, making the two reflections agree. �
More generally, if B is an abelian variety with trivial action of W , we could consider the image
B×E1×· · ·×En under a W -equivariant isogeny. We will say that the abelian variety A arising
in this way has an action of W of “root type”. Note that as in the finite case, we may always
arrange for the kernel of the isogeny to be not just preserved by W but fixed elementwise by W ,
as otherwise there will be kernel elements contained in root curves. We will also need the dual
notion: an abelian variety with an action of W is of “coroot type” if its dual is of root type. These
are equivalent for finite groups, or more generally for Coxeter groups with nondegenerate Cartan
matrices, but in the affine case the two notions do not agree. Note that in the coroot type case,
rather than having well-behaved positivity for roots, we have well-behaved positivity for coroots:
for each conjugacy class of reflections, we can choose isomorphisms between the corresponding
coroot curves and a fixed curve E in such a way that the resulting set of maps to E, together
with their negatives, are in equivariant, sign-preserving bijection with the corresponding set of
root vectors.
Elliptic Double Affine Hecke Algebras 29
Consider the case of the affine Weyl group of type Ã2. Since mij = 3, kij = 1, we see
that each µij is an isomorphism, and the positivity assumption forces the isomorphisms to be
consistent. We thus obtain the following faithful action on E3:
s0(x0, x1, x2) = (x1 + x2 − x0, x1, x2),
s1(x0, x1, x2) = (x0, x0 + x2 − x1, x2),
s2(x0, x1, x2) = (x0, x1, x0 + x1 − x2).
This action fixes the diagonal copy of E, but does not fix any morphism to E. It follows that
the corresponding action on the dual variety fixes a morphism to E, but does not fix any curve.
In fact, we see that the dual action takes the form
s0(x0, x1, x2) = (−x0, x0 + x1, x0 + x2),
s1(x0, x1, x2) = (x0 + x1,−x1, x1 + x2),
s2(x0, x1, x2) = (x0 + x2, x1 + x2,−x2),
from which we may see that the corresponding root curves do not even generate E3.
For our purposes, we will in fact prefer actions of coroot type. The main issue with actions
of root type in the affine case is that since there are only finitely many distinct coroots, the kernel
of any given coroot map is fixed by infinitely many reflections. For instance, in the above Ã2
example, both s0 and s1s2s1 fix the hypersurface x1 + x2 = 2x0 pointwise. However, the dual
of the standard model, though of coroot type, is badly behaved for other reasons: the product
of root curves corresponding to the finite Weyl group does not inject, and the image of the
product is fixed by the translation subgroup.
Suppose W̃ = 〈s0, . . . , sn〉 is an affine Weyl group (with associated finite Weyl group W =
〈s1, . . . , sn〉), and that the abelian variety A is equipped with an action of W̃ of coroot type.
The W̃ -invariant subvariety of A∨ has codimension n, and induces by duality a universal equiv-
ariant morphism A → B such that W̃ acts trivially on B and the fibers have dimension n.
(In other words, we may interpret the original action as a family of actions on n-dimensional
varieties.) In contrast, the invariant subvariety of A has codimension n+ 1, and thus its image
in B has codimension 1. Thus if we base change by a suitable isogeny B′ → B, we may arrange
for B to be the product of AW̃0 by an elliptic curve, allowing us to split off that factor and
reduce to the case that B is an elliptic curve E. Now, since W is finite, AW0 has codimension n,
and is thus itself an elliptic curve, which necessarily surjects onto E. Although this curve AW0
is not preserved by W̃ , we may still base change by it, and thus find that the natural action
of W on AW0 ×AW extends to an action of W̃ in such a way that the isogeny AW0 ×AW → A
is equivariant. (Note, however, that the factorization itself is not equivariant; the projection
to AW is not an equivariant map.)
We can describe this action explicitly on generators. Of course, for 1 ≤ i ≤ n, si(z, x) =
(z, si(x)), so only s0 is nontrivial. The root curve associated to s0 is the same as the root
curve associated to the reflection r in the relevant root of W , and the action on 0× AW is the
same as that of r. We thus see that s0(z, x) = (z, r(x) + ζ(z)) for some (nonzero) morphism
ζ : AW0 → Er. Conversely, it is easy to see that any action of this form has coroot type.
We may view this action as a family of actions of W̃ on AW parametrized by z, with the one
caveat being that the action no longer preserves the identity; indeed, the translation subgroup
of W̃ acts (unsurprisingly) as translations of AW . Of course, the action on a given fiber depends
only on the point q := ζ(z), and we easily see that it is faithful precisely when q is non-torsion.
(It follows from the above considerations that this is the typical form of an action of coroot type,
up to base change and twisting by a (AW )W -torsor.)
Note that in this construction, we may as well start with a given action of W and then
adjoin s0. In non-simply-laced cases, one must choose an orbit of roots and then obtain s0 by
30 E.M. Rains
shifting the action of the reflection in the highest root of that orbit by an element q of the
corresponding root curve. This produces an action of the affine Weyl group W n Λ, where Λ
is the free abelian group generated by the root maps in that orbit. It is worth noting that the
choice of orbit is entirely orthogonal to the direction (if any) of the arrows in the finite Dynkin
diagram: e.g., each of the three versions of the finite diagram Bn = Cn gives rise to both an
action of B̃n and an action of C̃n, and similarly in the G2 and F4 cases, there are two natural
extensions to actions of the corresponding affine groups. It is also worth noting that the various
base changes required to put the action in this form can eliminate some of the information
present in the original coroot model; in particular, in the C̃n case (including C̃1 = Ã1), the
special node is connected to the rest of the Dynkin diagram by an arrow, and thus there is
a choice of isogeny in the coroot model. This includes some exotic coroot models in which E0
and En are merely 4-isogenous and the corresponding family of abelian varieties with C̃n action
has no section.
Returning to the finite case, suppose that A/S is a family of abelian varieties (over an
integral base S) equipped with a faithful action of the finite Weyl group W by reflections, and
suppose moreover that we are given a W -invariant ample line bundle L on A. This can be
made equivariant by taking the action on the fiber at 1 to be trivial, and we may then ask
when the map s 7→ dim Γ(As;L)W is constant on S. By the reductions of the previous section,
this reduces to considering the corresponding question for AW , which is very nearly a variety
of the form we considered above. To be precise, the root curves in each irreducible component
of W are isogenous, and since indecomposable finite Weyl groups have at most two conjugacy
classes of reflections, we see that each component is associated to a point of X0, X0(2), or X0(3).
To ensure that we can apply our previous results, we must insist that the induced line bundles
on the root curves be suitable; to wit, we insist that for each reflection, L|Er ∼= L2dr,0;Er for
some positive integer dr, clearly constant on conjugacy classes of involutions. We thus see that
the only possible issues arise when (a) one of the curves Er is supersingular of characteristic
dividing W , or (b) the “root kernel”, i.e., the kernel of
∏
iEi → A, fails to be diagonalizable.
In fact, the first condition turns out not to be necessary.
Lemma 3.8. Let L be a W -invariant ample line bundle on an abelian variety A such that there
are positive integers di such that L|Ei ∼= L2di,0;Ei for each i. Then for any section f ∈ Γ(A;L),
the antisymmetrization
∑
w∈W σ(w)wf vanishes along the divisor
∑
r∈R(W )[ker(r − 1)].
Proof. We may write
∑
w∈W σ(w)w = (1− r)
∑
w∈W0
w, where W0 is the even subgroup of W .
Since the divisors ker(r − 1) are transverse for distinct r, it thus suffices to show that (1− r)f
vanishes along the divisor [ker(r− 1)]. We thus reduce to the case that W has rank 1. In other
words, A is a quotient of a variety B ×E (with r acting trivially on B) obtained by identifying
some subgroup K ⊂ E[2] with a subgroup of B. The action of r lifts to B×E, and we conclude
(by considering how [−1] acts on sections of L2d) that the antisymmetrization of any section of
the pulled back line bundle must vanish on the divisor B × E[2]. This is the preimage of the
divisor (B × E[2])/K, which in turn is precisely the kernel of r − 1 as required. �
This gives us the following possible approach to controlling invariants in such bundles. Let L∆
be the line bundle OA
(∑
r∈R(W )[ker(r−1)]
)
, but equipped with the equivariant structure which
is trivial at the identity. If there is a section g ∈ Γ(AW ;L∆) with nontrivial antisymmetrization,
then the operation f 7→
∑
w∈W σ(w)w(gf)∑
w∈W σ(w)w(g) induces an idempotent on any Γ(A;L) which projects
onto the symmetric subspace. More generally, if we have a family of such varieties such that
such a section g exists locally (or, equivalently, on every fiber), then we obtain such idempo-
tents locally, and thus the spaces Γ(As;L)W are fibers of a vector bundle, implying that their
dimensions are constant.
Elliptic Double Affine Hecke Algebras 31
Theorem 3.9. Suppose A/S is a family of abelian varieties equipped with a faithful action by
reflections of the finite Weyl group W , and let L be a W -invariant ample line bundle on A
such that the restriction to every root curve of every fiber is isomorphic to an even power
of L1. If the root kernel of A is diagonalizable, then the functions s 7→ dim Γ(As;L)W and
s 7→ dim
((∑
w∈W σ(w)w
)
Γ(As;L)
)
are constant on S.
Proof. We first observe that by Lemma 2.35, the claims hold for A iff they hold for AW , a.k.a.
the image of
∏
iEi → A. We may thus without loss of generality assume that the morphism∏
iEi → A is an isogeny such that W acts trivially on the root kernel K. By assumption, K is
diagonalizable, so that Γ
(∏
iEi;L
)
decomposes into K-eigenspaces, and this decomposition
is compatible with the action of W . It then follows by semicontinuity that the claims hold
for A if they hold for
∏
iEi. Since this is a product over the components of W , we may assume
without loss of generality that W is indecomposable.
We may then reduce as discussed to showing that for any elliptic root datum corresponding
to an indecomposable finite Weyl group, the corresponding line bundle L∆ contains a section
with nontrivial antisymmetrization. (There is also the technical, but easy to verify, condition
that L∆|Er ∼= L2dr,0;Er for suitable positive integers.)
Here we may use the classification of finite Weyl groups. The simplest case is W = An,
in which case we may identify
∏
iEi with the subvariety of En+1 on which
∑
i zi = 0. By induc-
tion in n (with trivial base case n = 0), the result holds for n−1, and thus any Sn-anti-invariant
section can be obtained by antisymmetrization over Sn. It thus suffices to show that there
is an Sn-anti-invariant section that when summed over coset representatives of Sn+1/Sn with
appropriate sign gives a nonzero result. Equivalently, by dividing by the appropriate product
of ϑ functions, we need to find an Sn-invariant function with suitable poles that symmetrizes to
a nonzero constant. For auxiliary parameters y1, . . . , yn+2, we may consider the function∏
1≤i≤n+2 ϑ(zn+1 − yi)
∏
1≤i≤n ϑ(Y − zi)∏
1≤i≤n ϑ(zn+1 − zi)
,
where Y =
∑
1≤i≤n+2 yi. Summing this over Sn+1/Sn gives a function with no poles, which must
therefore be constant; on the other hand, of the n + 1 terms that result, all but one vanishes
when zn+1 = Y . We thus find that∑
w∈Sn+1/Sn
w ·
∏
1≤i≤n+2 ϑ(zn+1 − yi)
∏
1≤i≤n ϑ(Y − zi)∏
1≤i≤n ϑ(zn+1 − zi)
=
∏
1≤i≤n+2
ϑ(Y − yi),
which is generically nonzero. (Note that the case n = 1 is a version of the standard addition
law for theta functions.) This identity is a disguised form of a classical theta function identity;
see the discussion around [29, equation (1.22)].
For types B/C/D, we may similarly reduce to lower rank cases, noting that D2 and B1 both
follow from the result for A1. There are four cases to consider: the action of Dn on E ⊗ ΛDn ,
the action of Bn on the same variety, the action of Cn on E ⊗ Zn (following [18], we label the
cases by the dual root system), and the action of BCn on the variety EΛDn ,Zn associated to
a point of X0(2) lying over E. In each case, there is a natural isogeny to E ⊗ Zn, and it turns
out we can choose the function being symmetrized to be the pullback of a function on E ⊗ Zn.
The simplest identity corresponds to the Cn case, valid for all n ≥ 1:∑
w∈Cn/Cn−1
w ·
∏
1≤i≤2n+1 ϑ(zn − yi)
∏
1≤i≤n ϑ(Y + zi)
∏
1≤i<n ϑ(Y − zi)
ϑ(2zn)
∏
1≤i<n ϑ(zn + zi)ϑ(zn − zi)
=
∏
1≤i≤2n+1
ϑ(Y − yi),
32 E.M. Rains
with Y =
∑
1≤i≤2n+1 yi; if we expand this out as a sum of 2n terms, we find that it is simply the
special case (z1, . . . , z2n) 7→ (−zn, . . . ,−z1, z1, . . . , zn) of the S2n/S2n−1 identity. In characteristic
not 2, we may set y2n−2, . . . , y2n+1 to be the four points of E[2] to obtain an identity for Dn,
n > 2: ∑
w∈Dn/Dn−1
w ·
∏
1≤i≤2n−3 ϑ(zn − yi)
∏
1≤i≤n ϑ(Y + zi)
∏
1≤i<n ϑ(Y − zi)∏
1≤i<n ϑ(zn + zi)ϑ(zn − zi)
= ϑ(2Y )
∏
1≤i≤2n−3
ϑ(Y − yi),
where Y =
∑
1≤i≤2n−3 yi. Since this identity is expressed entirely in terms of ϑ, it continues to
hold in characteristic 2. If we only specialize three parameters to the nonzero 2-torsion points,
we instead obtain (for n ≥ 2):
∑
w∈Bn/Bn−1
w ·
∏
1≤i≤2n−2 ϑ(zn − yi)
∏
1≤i≤n ϑ(Y + zi)
∏
1≤i<n ϑ(Y − zi)
ϑ(zn)
∏
1≤i<n ϑ(zn + zi)ϑ(zn − zi)
=
ϑ(2Y )
ϑ(Y )
∏
1≤i≤2n−2
ϑ(Y − yi).
We omit the analogous identity for BCn (obtained from the Cn identity by setting two of the yi
to be the 2-torsion points not in the kernel of φ) due to notational difficulties with using ϑ in the
presence of isogenies, but note that for purposes of the reduction there is no reason we cannot
simply use the Bn identity.
For the seven exceptional cases (two each of G2 and F4 along with the simply laced cases E6,
E7, E8), we observe that the relevant line bundle comes from a polarization of degree a multiple
of 6, and we may thus use Lemma 2.34 to reduce to a smaller group. That is, we obtain
an equivariant isomorphism
Γ(A;L∆) ∼=
⊕
H
IndWH ResWH Γ(A′;L∆),
where H ranges over the point stabilizers in the different orbits of the action of G on an appro-
priate diagonalizable 2- or 3-group. The image of antisymmetrization on the left is thus the
direct sum of terms(∑
h∈H
σ(h)h
)
ResWH Γ(A′;L∆),
and since H is a reflection group in each case, we may apply induction. (In fact, only that term
which is nonzero in characteristic 0 has any hope of contributing).
For instance, for E8, the variety is E8 with polarization given by 30 times the Cartan matrix
of E8. In characteristic not 2, we may use the 2-part of ΛE8/30ΛE8 to split into eigenspaces,
of which only one term survives. We thus reduce to showing that the descended line bundle
on EΛE8
,2ΛE8
∼= E8
2 (with nonproduct polarization) has a nontrivial antisymmetrization under
the stabilizer W (D8). This is 2-isogenous to the standard D8 model, and thus (since the charac-
teristic is not 2) the image of antisymmetrization has the same dimension as in characteristic 0.
A similar reduction using the 3-part reduces to antisymmetrization for W (A8) and proves the
result for any characteristic other than 3. �
Remark 3.10. Results of [18, 31] in characteristic 0 actually compute the structure of the
invariant ring
(
i.e.,
⊕
d Γ
(
EB,C ;LdQ
)
, where Q is the minimal invariant polarization satisfying
Elliptic Double Affine Hecke Algebras 33
the evenness requirement
)
and find that in each case the result is a free polynomial ring in
generators of degrees that can be read off of the coefficients of the highest short (co)root. This
suggests that something similar should hold in arbitrary characteristic. It would be natural in
this context to also consider actions of complex reflection groups on varieties isogenous to En
where j(E) ∈ {0, 1728}, or even quaternionic reflection groups in the case of supersingular curves
of characteristic 2 or 3.
Remark 3.11. The diagonalizability hypothesis is necessary, at least as far as the antisym-
metrization claim is concerned. For example, suppose x ∈ E[3] is a nontrivial 3-torsion point,
and consider the quotient A of the sum 0 subvariety of E3 by the subgroup generated by (x, x, x).
The image in A of the point (0, x,−x) is negated by every reflection, and is thus not contained
in any reflection hypersurface, but still has nontrivial stabilizer Z/3Z ⊂ S3. It follows that
in characteristic 3, the antisymmetrization of any section of an ample line bundle will vanish
at this point, and thus for no ample line bundle is the dimension of the image of antisym-
metrization the same as in characteristic 0 (except, of course, when there are no antisymmetric
elements in characteristic 0). The invariants remain well-behaved, however, as the invariant ring
is the same as that of 〈x〉 on P2; this is a permutation representation, so its Hilbert series is
independent of the characteristic.
In the sequel, we will need some understanding of equivariant gerbes on abelian schemes, and
thus in particular will want to understand H1(W ; Pic(A)). This turns out to be nearly trivial
when W is finite and the map
∏
iEi → A is an isomorphism, in that any nontrivial cohomology
class remains nontrivial when restricted to some simple reflection. (In particular, H1(W ; Pic(A))
is 2-torsion.)
For inductive purposes, we will need to consider a slightly larger class of actions. Given
an elliptic root datum of finite type and a point u ∈
∏
iEi(S), we may define an action of W
on
∏
iEi by
si(x1, . . . , xn) =
(
x1, . . . , xi−1,−xi +
∑
j 6=i
µji(xj) + ui, xi+1, . . . , xn
)
;
this is of course no longer a pointed action of W , but is instead the twist by a class in H1
(
S;AW
)
determined by u. (More precisely, the cocycle corresponding to u is fppf locally a coboundary,
and the nonuniqueness of this representation gives a class in H1
(
S;AW
)
.) (And, of course,
a class in H1
(
S;AW
)
is representable in this form iff the corresponding A-torsor has a section.)
Lemma 3.12. Let W be a finite Weyl group of rank 2, and let X = E1 × E2 be equipped
with an action of the above form such that the root datum has r1 ≥ r2. Then the kernel
of
∑
w∈W (−1)`(w)w on Pic(X) is spanned by π∗1 Pic(E1) and Pic(X)〈s1〉.
Proof. It suffices to prove the corresponding claims for Pic0(X) and NS(X) (modulo the requ-
irement in the latter case that the classes lift to actual bundles still of the required form). Since
Pic0(X) is generated by π∗1 Pic0(E1) and π∗2 Pic0(E2), with the latter s1-invariant, we find that
the antisymmetrizer vanishes on Pic0(X), and the claim is immediate. The Néron–Severi group
may be identified with the space of matrices of the form
Q =
(
a b
b∨ c
)
,
with antisymmetrization(
0 4b
4b∨ 0
)
34 E.M. Rains
when m12 = 2 and(
0 m12
(
bµ−1
21 −
(
bµ−1
21
)∨)
µ21
−m12µ
∨
21
(
bµ−1
21 −
(
bµ−1
21
)∨)
0
)
,
when m12 ∈ {3, 4, 6}. It follows in either case that the vanishing of the antisymmetrization
implies b ∈ Zµ21. (Here for m12 ∈ {3, 4, 6}, we use the fact that deg(µ21) ≤ deg(µ12) and
µ12µ21 ∈ {0, 1, 2, 3}, so that deg(µ21) is squarefree and an isogeny is in Qµ21 iff it is in Zµ21.)
In particular, the kernel of the antisymmetrizer is spanned by the matrices(
1 0
0 0
)
,
(
2 −µ21
−µ∨21 µ∨21µ21
)
,
(
0 0
0 1
)
.
The first matrix is in the pullback of π∗1, while the second and third are s1-invariant (and, in
fact, images of s1-invariant bundles). �
Proposition 3.13. Let X/S =
∏
iEi be equipped with an action of the finite Weyl group W of
the above form, and let z ∈ Z1(W ; Pic(X)) be such that zsi is a coboundary for every i. Then z
is a coboundary.
Proof. Since W is finite, its diagram is a forest, and we may thus order the roots in such a way
that s1 corresponds to a leaf, and thus without loss of generality commutes with s3, . . . , sn.
Moreover, since every component has at most two values of ri, we may arrange to have r1 ≥ ri
for all i. If we view X as a family of abelian varieties over E1, then we see that the action
of W{2,...,n} is still of the above form, and thus the corresponding restriction of z is a coboundary
by induction. We may thus reduce to the case that zs2 = · · · = zsn = 0.
Now, consider the action of 〈s1, s2〉 on X viewed as a family over
∏
3≤i≤nEi, and let v ∈
Pic(X) be such that zs1 = v−s1v. The cocycle condition on zs1 implies that
∑
w∈〈s1,s2〉(−1)`(w)wv
= 0, and thus the lemma implies that we may replace v by the pullback of a class in E1 ×∏
3≤i≤nEi without changing zs1 . Since the cocycle is trivial on W{3,...,n} and s1 commutes with
this subgroup, zs1 must be W{3,...,n}-invariant, and thus a computation in the Néron–Severi group
shows that v is in the subgroup generated by Pic(E1) and Pic
(∏
3≤i≤nEi
)
. Since elements of the
latter have no effect on zs1 , we see that we may take v ∈ π∗1 Pic(E1). This is W{2,...,n}-invariant,
and thus z = ∂v as required. �
Remark. Note that this can fail if we quotient by a subgroup of AW . Indeed, if A = E2 is
given the standard action of W (A2), then the result fails for the induced action on A∨ ∼= E2.
The cocycles in which all bundles have degree 0 (and which are coboundaries in rank 1) are
themselves classified by A∨(S), while a coboundary must be in the image of A(S) under the
invariant polarization, and these are different unless E(S) is 3-divisible. There is no difficulty
with the polarization, however, so although the induction breaks down, it may still be the case
that the claim holds on the Néron–Severi group, which would imply an fppf local version of the
proposition.
4 Elliptic analogues of affine Hecke algebras
Before proceeding to the construction of Hecke algebras associated to general elliptic root data,
it will be helpful to consider the finite case, a generalization of the construction of [11]. Note that
although we work with a finite group, the resulting Hecke algebras are most naturally thought of
as elliptic analogues of affine Hecke algebras, as they include multiplication operators in addition
to reflection operators. (In particular, [11] constructs affine Hecke algebras as degenerations
of a special case of the construction given below.)
Elliptic Double Affine Hecke Algebras 35
Although the approach in [11] via residue conditions can (mostly) be extended to the infinite
case, there are two alternate approaches for which the generalization is more straightforward:
one as a space of operators preserving appropriate holomorphy conditions, the other as the
subalgebra of the algebra of operators generated by the rank 1 subalgebras. Since we will need
to understand the rank 1 case to give the second construction, we begin with the first.
In our application to noncommutative rational varieties below, we will need to be able to
attach an arbitrary finite set of parameters to the endpoint roots of the affine Cn diagram; we will
thus give a version of the general construction in which each conjugacy class of reflections can
be given arbitrarily many parameters. This, of course, includes a case without any parameters
at all, which we consider first.
This “master” Hecke algebra has a third description which does not generalize well to the
infinite case, but is simplest of all to give (and extend to actions of arbitrary finite groups).
Let X be a regular integral scheme, and let G be a finite group acting faithfully on X. Then we
define the master Hecke algebra HG(X) to be the sheaf of algebras on X/G given by HG(X) :=
End(π∗OX), where π : X → X/G is the quotient map.
If π is flat, then HG(X) is the endomorphism ring of a vector bundle, so is in particular
an Azumaya algebra on X/G, and the category of quasicoherent HG(X)-modules is equivalent
to the category of quasicoherent sheaves on X/G. However, this condition holds only rarely;
even in the case when G is a finite Weyl group acting on an abelian variety, this morphism
can easily fail to be flat. For instance, consider the case G = G2 acting on the sum zero
subvariety X of E3 (as permutations and global negation). In characteristic not 2, consider the
point (τ1, τ2, τ1 + τ2) ∈ E3, where τ1, τ2 generate E[2]. This point has stabilizer Z(G), and
is isolated in the subvariety fixed by its stabilizer, and thus we see that its image in X/G is
a singular point (of type A1), and that the quotient morphism fails to be flat in a neighborhood
of that orbit.
There are two prominent cases in which we do have flatness, namely the action of An on the
sum 0 subvariety of En+1 and the action of Cn on En; in each case, the quotient morphism is flat
because it is a quasi-finite morphism between regular schemes (from En to Pn, to be precise).
In general, although HG(X) may not be an Azumaya algebra, we at least know that it
is torsion-free, and thus may be viewed as contained in its generic fiber Endk(X/G)(k(X)).
Since k(X) is Galois over k(X/G), the generic fiber has an alternate description as a twisted
group algebra k(X)[G], giving rise to the following description of HG(X). Denote the natural
action of Aut(X) on k(X) by gf := (g−1)∗f ; we will also use a similar notation for the actions
on line bundles and divisors. It will also be convenient to define (for finite G) Gf = π∗Gf , where
f ∈ k(X/G) and π : X → X/G is the natural quotient.
Proposition 4.1. The master Hecke algebra HG(X) is the subsheaf of the twisted group algebra
k(X)[G] such that for any G-invariant open subset U , Γ(U/G;HG(X)) consists of the operators∑
i cig such that for any G-invariant open V and f ∈ Γ(V ;OX),
∑
i ci
gf ∈ Γ(U ∩ V ;OX).
Proof. Indeed, HG(X) is the subsheaf of Endk(X/G)(k(X)) which on U consists of endomor-
phisms preserving OX |U , or equivalently preserving global sections of OX |V for all invariant
V ⊂ U . The claim then follows by using the twisted group algebra description of the endomor-
phism ring and observing that Γ(V ;OX) ⊂ Γ(U ∩ V ;OX). �
One consequence is that if H ⊂ G, then there is a natural inclusion HH(X) ⊂ HG(X)
(where we conflate HH(X) with its direct image under X/H → X/G). In addition, if α
is an automorphism of X that normalizes G, then there is a corresponding automorphism
of HG(X), either by pulling back through the induced automorphism of X/G, or on opera-
tors as
∑
g cgg 7→
∑
g
αcα−1gαg.
Note that HG(X) clearly contains a copy of the structure sheaf OX as well as the operators g
for each g ∈ G, and thus contains a copy of the twisted group algebra OX [G]. It can, however,
36 E.M. Rains
be bigger than the twisted group algebra. Consider the case of G = µn = 〈s〉 acting on X = A1
in characteristic prime to n by rescaling the variable x. Applying the operator 1 + ζns+ ζ2
ns
2 +
· · ·+ζn−1
n sn−1 to any function which is holomorphic at the origin gives a function which vanishes
to order n−1 at the origin, and thus Hµn
(
A1
)
contains the operator x1−n(1 + ζns+ ζ2
ns
2 + · · ·+
ζn−1
n sn−1
)
not contained in OX [G].
It turns out that this is the typical case in which the coefficients may have poles; more
precisely, the only poles are associated to “complex reflections” in G (relative to the action
on X). It will be useful to consider a more general setting. We begin with a couple of local
results.
Lemma 4.2. Suppose R, S are discrete valuation rings, and suppose ψ1, . . . , ψn : S → R are
distinct finite homomorphisms. Then the module of operators
∑
i ciψi mapping S to R is a free
R-module of rank n.
Proof. Consider the subalgebra R~ψS ⊂ Rn generated by the image of S under (ψ1, . . . , ψn).
The monoid characters ψi : S \ {0} → R∗ are linearly independent over KR and thus over R,
so that R~ψS is free of rank n as an R-module. An operator
∑
i ciψi maps S to R iff the KR-
linear functional ~c maps R~ψS to R, and thus the space of such operators is isomorphic to the
dual Rn. �
Remark. It follows that the quotient by the submodule in which the coefficients are in R has
finite length, equal to the colength of R~ψS as a submodule of Rn.
For n = 1, R~ψS = R, and thus (R~ψS)∗ = R. We can also give an explicit description for
n = 2.
Corollary 4.3. Let ψ1, ψ2 : S → R be finite homomorphisms of discrete valuation rings inducing
the same action on residue fields. Then the operator c1ψ1 + c2ψ2 : k(S) → k(R) maps S to R
iff c1, c2 ∈ det(R~ψS)−1 and c1 + c2 ∈ R.
Proof. The module R~ψS is spanned by elements of the form (ψ1(h), ψ2(h)), or equivalently
by the element (1, 1) and elements of the form (0, ψ2(h) − ψ1(h)), and thus can be expressed
as R(1, 1) + det
(
R~ψS
)
(0, 1). The corresponding condition on the operator is that c1 + c2 ∈ R
and c2 ∈ det
(
R~ψS
)−1
. These conditions imply that c1 ∈ det
(
R~ψS
)−1
as well. �
Remark 4.4. For any f ∈ S∗, since c1ψ1(f) + c2ψ2(f) − ψ1(f)(c1 + c2) = (ψ2(f) − ψ1(f))c2
and ψ2(f) − ψ1(f) ∈ det
(
R~ψ
)
we see that when c2 ∈ det
(
R~ψ
)−1
, the conditions c1 + c2 ∈ R
and c1ψ1(f) + c2ψ2(f) ∈ R are equivalent. This is useful in global situations in which one has
twisted by a line bundle.
Remark 4.5. It follows that (R(ψ1, ψ2)S)∗ = Rψ1 + det
(
R~ψS
)−1
(ψ1 − ψ2).
For n > 2, it is difficult to give an explicit description in general, but in the case of Coxeter
groups, we can generally reduce to the n = 2 case, using the following result. Given a homo-
morphism ψ : S → R of local rings, let ψ̄ : kS → kR denote the corresponding homomorphism
of residue fields. Also, note that for any subset I ⊂ {1, . . . , n}, there is a natural morphism
R~ψS → R(ψi : i ∈ I)S.
Lemma 4.6. Suppose R, S are discrete valuation rings, and suppose ψ1, . . . , ψn : S → R are dis-
tinct finite homomorphisms. Then the algebra R~ψS splits as the direct sum
⊕
σ R(ψi : ψ̄i = σ)S
of local rings, where σ ranges over all morphisms kS → kR.
Elliptic Double Affine Hecke Algebras 37
Proof. The radical of R~ψS is equal to its intersection with mn
R, and the quotient is a product
of copies of kR, with one for each distinct reduction ψ̄i (in particular, the radical is maximal
and R~ψS local iff there is only one such reduction). If R is complete, then we can lift the
idempotents from the reduction to obtain the desired splitting. Moreover, the lifts are unique,
and thus agree with the lifts one would have obtained if working inside the larger algebra Rn, i.e.,
the projections onto the given sets of coordinates. It thus follows that even if R is not complete,
the lifts in the completion of R~ψS agree with the projections, and thus said projections lie
in R~ψS. �
Remark. By duality, the same splitting applies to the module of operators taking S to R.
This leads to the following global result.
Lemma 4.7. Let X, Y be normal integral schemes and let φ1, . . . , φn : X → Y be a collection
of distinct finite morphisms. Let M~φ
be the subsheaf of k(X)n which on an open subset U ⊂ X
consists of those n-tuples (c1, . . . , cn) ∈ k(X)n such that for any open subset V ⊂ Y and any
function f ∈ k(Y ) holomorphic on V , the function
∑
i ciφ
∗
i f is holomorphic on U ∩
⋂
i φ
−1
i V .
Then there is an n-tuple of divisors ∆i ∈ Div(X) such that M~φ
⊂
⊕
iOX(∆i), and each ∆i is
supported on those hypersurfaces on which φj = φi for some j 6= i.
Proof. Let D ⊂ X be a reduced irreducible hypersurface; we need to understand the possible
singularities of the coefficients along D. Note that if U1, U2 are two open subsets meeting D,
then U1 ∩ U2 also meets D, and thus any bound on singularities holding on U1 ∩ U2 also holds
for global sections along U1, U2. We may thus take a limit along those open subsets meeting D.
Similarly, since we are only considering holomorphy along D, we may take a limit over those V
meeting φi(D) for every i. In other words, the condition for (c1, . . . , cn) to be a section of the
base change of M~φ
to the local ring of k(D) is that for any function f which is holomorphic
along φ1(D), . . . , φn(D), the image
∑
i ciφ
∗
i f is holomorphic along D.
Now, given such an n-tuple, let d be the maximum order of pole of a coefficient ci along D.
The condition that
∑
i ciφ
∗
i f be holomorphic only depends on the value of f modulo the inter-
sections of the d-th powers of the maximal ideals at the divisors φi(D), and by the Chinese
remainder theorem, the reductions corresponding to distinct divisors may be chosen indepen-
dently. Taking those reductions to be 0 on all but φj(D) tells us that if the operator
∑
i ciφ
∗
i
preserves holomorphy, then so does
∑
i:φi(D)=φj(D) ciφ
∗
i .
We may as well assume, therefore, that the divisors φi(D) are all equal to the same divisor D′
in Y , and thus reduce to the local case. �
Remark. By mild abuse of notation, if g1, . . . , gn ∈ Aut(X), then M~g will denote the sheaf
corresponding to the n-tuple ~φ = (g−1
1 , . . . , g−1
n ); i.e., the sheaf of operators
∑
i cigi that preserve
holomorphy. This generalizes to the case of operators
∑
i cigiG : k(X/G) → k(X) for a finite
subgroup G ⊂ Aut(X), which act as f 7→
∑
i ci
giGf .
The local result for n = 2 leads to the following result in the global setting. Given two finite
morphisms φ, ψ of normal integral schemes, let [φ = ψ] denote the Cartier divisor obtained
by removing all codimension > 1 components from the subscheme on which φ and ψ agree.
Corollary 4.8. With notation as above, suppose that for each i, the n−1 divisors [φi = φj ] are
mutually transverse. Then M~φ
may be identified with the subsheaf of⊕
i
OX
(∑
j 6=i
[φj = φi]
)
in which the local sections (c1, . . . , cn) satisfy the additional “residue” conditions that ci + cj is
holomorphic along [φi = φj ] for all i 6= j.
38 E.M. Rains
Proof. The condition on the divisors ensures that when we perform the various reductions,
we will end up with local problems involving at most 2 morphisms. The case with one morphism
is trivial (the coefficient is forced to be holomorphic, and this makes the operator preserve
holomorphy), and the case with two is just Corollary 4.3 above. �
Remark. If [φi = φj ] is reduced, then the residue conditions can of course be stated in terms
of the residues of ci and cj with respect to any differential holomorphic and nonvanishing along
[φi = φj ].
Note that we can also right-multiply operators by local sections of OY , and thus obtain
an (OX ,OY )-bimodule structure onM~φ
. Such a structure is equivalent to an OX ⊗OY -module
structure, or equivalently an OX×Y -module structure. We will consider this structure in more
detail when discussing the infinite case, but for the moment we observe the following.
Corollary 4.9. The induced sheaf M~φ
on X × Y is a coherent subsheaf of
⊕
i(1, φi)∗OX(∆i).
Remark. It is worth noting that this sheaf depends only on the set of morphisms {φ1, . . . , φn},
and not on the ordering, and the map from global sections to operators can be reconstructed
from the bimodule structure. As a result, when considering sheaves on X × Y , we allow the
subscript to be a set rather than a sequence.
Corollary 4.10. The algebra HG(X), viewed as an OX×X-module, is coherent, and contained
in the sum
⊕
g∈G(1, g)∗OX(∆), where ∆ is an effective divisor supported on the “reflection
hypersurfaces” of G on X: the irreducible hypersurfaces which are fixed pointwise by some
g ∈ G.
In the simplest elliptic case A1 acting on E, the divisor ∆ is precisely the divisor correspon-
ding to the subscheme E[2]. In characteristic not 2, this subscheme is reduced, and thus the
coefficients have at most simple poles at the 2-torsion points, but in characteristic 2, the coeffi-
cients can have double poles at the 2-torsion points of an ordinary curve, and a quadruple pole
at the origin of a supersingular curve. This, of course, is an artifact of wild ramification; without
that, a “reflection” of order n will admit poles of order at most n − 1 along the corresponding
reflection hypersurfaces.
There is an important variation arising from the interpretation as an OX×X -module: simply
consider the sheaf HG;L1,L2(X) := HG(X)⊗X×X L2 � L−1
1 for invertible sheaves L2, L1 on X.
Since both left- and right-multiplication by sections ofOX/G agree inHG(X), this twisted version
still descends to a sheaf on X/G, and the result moreover has induced compositions
HG;L1,L2(X)⊗X/G HG;L2,L3(X)→ HG;L1,L3(X).
In fact, as a sheaf on X/G, we have HG;L1,L2(X) ∼= HomG/X(π∗L1, π∗L2), with the obvious
induced composition; this is most easily seen by representing L1, L2 by Cartier divisors, and
observing that this turns HG;L1,L2(X) into the subsheaf of the meromorphic twisted group
algebra taking the subsheaf of k(X) corresponding to L1 into the subsheaf corresponding to L2.
When L1 = L2, we omit the second copy of L2, and note that the result is again a sheaf of alge-
bras, with each HG;L1,L2(X) a bimodule that induces a Morita equivalence between HG;L1(X)
and HG;L2(X).
In addition to the isomorphisms of such sheaves arising from isomorphisms of L1, L2, there
are also isomorphisms coming from twisting both sheaves by a suitable invertible sheaf. Let L
be a G-equivariant invertible sheaf on X which “descends in codimension 1”; that is, there is an
open subscheme of X/G containing every codimension 1 point over which L descends to a line
bundle. (Note that this is a local condition and is automatically satisfied on the complement
of the reflection hypersurfaces.) Then for each reflection hypersurface H with inertia group
Elliptic Double Affine Hecke Algebras 39
(i.e., pointwise stabilizer) IH , there is a IH -invariant neighborhood UH of the generic point of H
such that the H-equivariant sheaf LIH is equivariantly isomorphic to OUH . It follows that there
is a natural isomorphism HG;L(X) ∼= HG(X). Indeed, since L is G-equivariant, both algebras
are naturally contained in k(X)[G], and the conditions for any given reflection hypersurface
are the same on both sides. This condition is automatically satisfied by the pullback of an
invertible sheaf on X/G, but this is not necessary. For instance, in the case of W (G2) acting
on the sum 0 subvariety of E3, the quotient is a weighted projective space with generators of
degree 1, 1, 2. The pullback of OP2(1) (from X/W (A2) ∼= P2) does not descend to a line bundle
on X/W (G2) (since OX/W (G2)(1) is not invertible on such a weighted projective space), but does
so if we remove the singular point of X/W (G2) and the three points of X lying over it.
A particularly important instance of twisting by line bundles arises when we consider the
natural involution on operators:
∑
g cgg 7→
∑
g g
−1cg.
Proposition 4.11. This gives a contravariant isomorphism HG(X)op ∼= HG;ωX (X).
Proof. We need to show that
∑
g cgg preserves holomorphic functions iff
∑
g g
−1cg preserves
holomorphic n-forms. By Lemma 4.7, it suffices to prove that the conditions on individual
hypersurfaces are the same, and thus we may fix a hypersurface D and restrict our attention to
the case that every term in the sum gives the same divisor g−1D = D′. By duality, a function f
is holomorphic along D iff ResD fω = 0 for all n-forms ω which are holomorphic along D,
and we may similarly detect holomorphy of n-forms by taking residues against test functions.
In particular, for any function f holomorphic along D, and any n-form ω holomorphic along D′,
we have
ResD′
(∑
g
cg
gf
)
ω =
∑
g
ResD′ cg
gfω =
∑
g
ResD
g−1
cgf
g−1
ω = ResD f ·
(∑
g
g−1
(cgω)
)
.
It follows that
∑
g cg
gf is holomorphic along D′ for all f iff
∑
g
g−1
(cgω) is holomorphic along D
for all ω. �
Let us now turn to the “elliptic” case, in which G is a finite Weyl group W acting by
reflections on an abelian torsor X/S over a normal integral base S. That is, the flat family X/S
is a torsor over an abelian variety A/S, and W acts on X in such a way that the induced action
on A is an action by reflections. Note that since the subvariety XW is the intersection of the
simple reflection hypersurfaces, it has codimension at most n, so is nonempty (the corresponding
intersection is nonempty and transverse in A, and thus the corresponding intersection number
is positive) and thus a torsor over AW ; conversely, any AW -torsor induces a corresponding
family X/S by twisting. The action of W is faithful on every fiber, and this remains true if we
view X as a family over the quotient S′ := X/AW . We will see that the fibers of HW (X) over S
are identified with the master Hecke algebras of the fibers, in a fairly strong way.
Given a reflection r, let [Xr] denote the effective Cartier divisor cut out by the equa-
tion rx = x. Also, let Cr denote the quotient of X by the abelian subvariety (r + 1)A; this,
of course, is a torsor over the corresponding coroot curve E′r. The morphism (r − 1) : X → A
factors through Cr and has image Er, so that there is a morphism Cr → Er compatible with
the isogeny E′r → Er, and thus Cr corresponds to a class in H1(S; ker(E′r → Er)). Note that
in contrast to the coroot curve, we cannot expect to have a natural torsor over the root curve
inside X.
For the rank 1 case, we have the following immediate consequence of Lemma 4.7. Here
a “hyperelliptic curve of genus 1” is a smooth genus 1 curve C with a marked involution such
that the quotient is (geometrically) rational; note that the torsor arising in the rank 1 case is
always a family of such curves-with-involutions.
40 E.M. Rains
Lemma 4.12. Let C/S be a flat family of hyperelliptic curves of genus 1 with G = A1 = 〈s〉
acting by the marked involution. Then for any G-invariant open set U , Γ(U ;HA1(C)) consists
of operators f0 + f1(s− 1) such that f0 ∈ Γ(U ;OC) and f1 ∈ Γ(U ;OC([Cs])).
Remark 4.13. Note that
f0 + f1(s− 1) = (f0 − f1 − sf1) + (1 + s)sf1,
and thus (since f1+sf1 ∈ Γ(U ;OC)) we may also describe Γ(U ;HA1(C)) as the space of operators
f ′0 + (1 + s)f ′1 such that f ′0 ∈ Γ(U ;OC) and f ′1 ∈ Γ(U ;OC([Cs])). This also follows from the
above description of the adjoint once we realize that ωC is the trivial line bundle with equivariant
structure such that s acts as −1.
Remark 4.14. It will be useful in the sequel to know when an A1-equivariant line bundle on C
descends in codimension 1. If L is equivariantly isomorphic toOC(D) for some symmetric Cartier
divisor, then we may write D as a linear combination of divisors D′ and D′′ + sD′′ with D′, D′′
irreducible and D′ = sD′. The latter case is the pullback of the image of D′′ in C/A1, and
thus certainly has no effect on twisting, while if D′ is not a component of [Cs], then its image
in C/A1 is twice a divisor, so that again D′ is a pullback. We are thus left to consider the
linear combinations of reflection hypersurfaces, and thus determine that the condition on D is
precisely that the valuations along reflection hypersurfaces must be even (or no condition at
all in characteristic 2 if the reflection hypersurface is inseparable over S). This, of course, is
for the standard equivariant structure on OC(D); in odd characteristic, we may twist by the
sign character of A1, in which case the condition becomes that the valuations along reflection
hypersurfaces are odd. Note that in any event, a line bundle that descends in codimension 1
will restrict on the generic fiber to a power of the hyperelliptic bundle.
For rank n, we have the following.
Lemma 4.15. The OX/W -algebra HW (X) is generated by the OX/W -subalgebras H〈si〉(X) for
1 ≤ i ≤ n.
Proof. It follows from Lemma 4.7 that HW (X) is generated as a left OX -module by the twisted
group algebra OX [W ] along with the subsheaves arising from operators of the form cww+crwrw
for some reflection r ∈ R(W ). Now, by the above explicit description, each subalgebra H〈si〉(X)
contains OX [si], and thus the algebra they generate contains OX [W ]. But then if we express r
above as w−1
1 siw1 for some simple reflection si, we have
cww + crwrw = cww + cw−1
1 siw1w
w−1
1 siw1w = w−1
1
(
w1cw + w1cw−1
1 siw1w
si
)
w1w,
and find that cww + crwrw is a local section of HW (X) iff
w1cw + w1cww1siw
−1
1
si
is a local section of HW (X), iff it is a local section of H〈si〉(X). The claim follows. �
Remark. Of course, the same argument shows that if G is generated by a collection of cyclic
groups meeting every conjugacy class of (generalized) reflections, thenHG(X) is generated by the
corresponding subalgebras.
We can actually say a great deal more in the case of interest; not only is HW (X) a flat sheaf
in general, but we can in fact express it as an extension of invertible sheaves on X. The key
ingredient is the fact that there is a natural partial order on W (the Bruhat order), the weakest
partial order such that if w′w−1 is a reflection, then w and w′ are comparable and ordered
Elliptic Double Affine Hecke Algebras 41
according to their length. Note that omitting any set of reflections from a reduced word for w
gives an element w′ ≤ w, and classical results on Coxeter groups give the converse: for any
reduced word for w, w′ ≤ w iff some word for w′ (iff some reduced word for w′) can be obtained
by omitting reflections from the chosen reduced word.
Given an order ideal I with respect to Bruhat order (i.e., a subset I ⊂W such that if w ∈ I
and w′ ≤ w, then w′ ∈ I), we may consider the subsheaf HW (X)[I] of HW (X) consisting of
those operators in which the coefficient of w is 0 for w /∈ I. Any chain of order ideals induces
in this way a filtration, and we will show that in the case of a maximal chain, the subquotients
of the filtration are invertible sheaves on X. Let [≤ w] denote the order ideal consisting of
elements ≤ w.
For any element w ∈W , define a divisor Dw :=
∑
r∈R(W ),rw<w[Xr].
Lemma 4.16. Let I be a Bruhat order ideal, and suppose that w is a maximal element of I.
Then there is a short exact sequence
0→ HW (X)[I \ {w}] ⊂ HW (X)[I]→ OX(Dw)→ 0.
Proof. By definition, HW (X)[I \{w}] is the kernel of the “coefficient of w” map on HW (X)[I],
so we first need to show that the coefficient of w is contained in OX(Dw). It follows from
Corollary 4.8 that the coefficient has polar divisor bounded by
∑
w′∈(I\{w})∩R(W )w
[
Xw′w−1]
.
If w′w−1 is a reflection, then w′ is comparable to w, which since w is maximal in I implies that
w′ < w, and thus the bound on the divisor is Dw as required.
It remains only to show that the map is surjective. Choose a reduced word w = s1 · · · sn, and
consider the multiplication map
H〈s1〉(X)⊗ · · · ⊗ H〈sn〉(X)→ HW (X).
Every term in the resulting expansion corresponds to an element in which some (possibly empty)
subset of the simple reflections have been omitted, and thus the image of this multiplication map
is contained in HW (X)[≤ w] ⊂ HW (X)[I]. The image under the leading coefficient map can
then be determined by replacing each factor by its corresponding leading coefficient line bundle.
It thus remains only to verify that
Dw =
∑
1≤i≤n
s1···si−1 [Xsi ] =
∑
1≤i≤n
[Xs1···si−1sisi−1···s1 ].
But this follows from the strong exchange property: the reflections r such that rw < w are
precisely those of the form s1 · · · si−1sisi−1 · · · s1, and these are distinct since such reflections
are naturally bijective with the n = `(w) positive roots that become negative under w. �
Corollary 4.17. For any reduced word w = s1 · · · sn, the multiplication map
H〈s1〉(X)⊗ · · · ⊗ H〈sn〉(X)→ HW (X)[≤ w]
is surjective. Moreover, any product of rank 1 subalgebras is equal to some Bruhat interval.
Proof. It suffices to show that if sw < w, then
H〈s〉 ⊗HW (X)[≤ sw]→ HW (X)[≤ w]
is surjective. The image clearly contains the subsheaf HW (X)[≤ sw], so it suffices to show
surjectivity to the quotient HW (X)[≤ w]/HW (X)[≤ sw]. This, in turn, is an iterated extension
of invertible sheavesOX(Dw′) on X, and thus it suffices to show surjectivity for each subquotient.
That is, if [≤ sw] ⊂ I ⊂ [≤ w] is an order ideal and w′ is a maximal element of I not contained
42 E.M. Rains
in [≤ sw], then we need to show that the intersection of the image with HW (X)[I] surjects
onto LDw′ .
Since sw < w, we may choose a reduced word for w beginning with s, and the subword
description of the Bruhat order then tells us that [≤ w] = [≤ sw] ∪ s[≤ sw]. Since w′ 6≤ sw and
w′ 6= w, it follows that sw′ < sw. We may thus consider the composition
H〈s〉(X)⊗HW (X)[≤ sw′]→ HW (X)[≤ w′]→ HW (X)[I].
The proof of the lemma shows that the composition with the “coefficient of w′” map is surjective
as required.
The second claim follows immediately from the fact that for any word s1 · · · sl, the products
of subwords still form a Bruhat interval (so multiplication maps into the corresponding subsheaf),
and the maximum of that interval can be represented by a reduced subword (so multiplication
surjects). �
Remark. In particular, the closest thing to an analogue of the braid relations in this setting is
the fact that if (sisj)
mij = 1, then the products
H〈si〉(X)H〈sj〉(X) · · · = H〈sj〉(X)H〈si〉(X) · · ·
(with mij terms on each side) agree as subsheaves of HW (X). Indeed, both sides are equal to
the order ideal generated by the longest element of 〈si, sj〉, and thus equal H〈si,sj〉(X).
Corollary 4.18. The construction HW (X) respects base change T → S.
Proof. Let π1 : X ×S T → X be the natural projection; we need to show that π∗1HW (X) ∼=
HW (X ×S T ). In the rank 1 case, this is immediate from the explicit description and the fact
that π∗1
(
OX([Xs])
) ∼= OX([(X ×S T )s]
)
. Since the rank 1 subalgebras generate the full algebra,
this induces a morphism π∗1HW (X) → HW (X ×S T ). (Normally one would need to check
relations, but this is simply the restriction of the corresponding isomorphism for the algebra
of meromorphic operators such that the common polar divisor does not contain any fiber; thus
generators suffice.)
It remains only to show that this morphism is an isomorphism, but this follows from the
existence of compatible filtrations (i.e., coming from a chain of Bruhat order ideals) such that
the induced maps on subquotients are isomorphisms. �
We can also give an alternate description of the adjoint involution. Let w0 be the longest
element of W .
Proposition 4.19. There is a contravariant isomorphism HW (X)op ∼= OX(−Dw0)⊗HW (X)⊗
OX(Dw0).
Proof. It suffices to show that the näıve adjoint
∑
w cww 7→
∑
w w
−1cw on the meromorphic
twisted group algebra restricts to an isomorphism as claimed. Since this is an involution, it redu-
ces to showing the analogous claim for each rank 1 subalgebra. Let U be an open subset on which
the effective Cartier divisor Dw0 is cut out by an equation h = 0. We thus need to show (using
the two descriptions of the rank 1 Hecke algebra and taking the adjoint on the left)
Γ(U ;OX) + (si − 1)Γ
(
U ;OX([Cs])
)
⊂ h
(
Γ(U ;OX) + (si + 1)Γ(U ;L[Cs])
)
h−1.
Given an instance f0 + (si − 1)f1 on the left, conjugating by h gives
f0 − (1 + (h/sih))f1 + (si + 1)(h/sih)f1.
Since siDw0 = Dw0 , we find that h/sih is a unit, and local considerations near [Xsi ] tell us that
1 + (h/sih) vanishes on [Xsi ]. �
Elliptic Double Affine Hecke Algebras 43
Remark 4.20. We could also show that ωX ⊗OX(Dw0) descends in codimension 1. The divi-
sor Dw0 is certainly invariant under every reflection, and thus it remains only to verify the
conditions along reflection hypersurfaces. In characteristic not 2, ωX is equivariantly isomorphic
to the twist of OX by the sign character, and thus the condition is that the divisor must have
odd valuation along the reflection hypersurfaces, while in characteristic 2, there is no need to
twist, and the valuations of separable reflection hypersurfaces must be even. In either case, the
condition is automatically satisfied.
Remark 4.21. It is worth noting that this operation is triangular, in the sense that the image
of the subsheaf corresponding to an order ideal is always the subsheaf corresponding to an order
ideal. This follows immediately from the fact that w 7→ w−1 is an order-preserving automorphism
of the Bruhat poset.
The proof of Theorem 3.9 has the following consequence for our algebras. Here and below,
by the root kernel of X, we mean the root kernel of the corresponding abelian scheme A.
Proposition 4.22. Let X/S be a flat family of abelian torsors equipped with a faithful action
by reflections of the finite Weyl group W , with dim(X/S) = rank(W ). If the root kernel of X is
diagonalizable on S, then S may be covered by open subsets on which HW (X) has a symmetric
idempotent: an idempotent global section which on each fiber has image Γ(Xx;OX)W .
Proof. For any fiber x, choose a global section h of OX(Dw0) with nonzero symmetrization (ref-
lections negate the fiber over 0, so this corresponds to the antisymmetrization of Theorem 3.9),
extend it to a neighborhood of x ∈ S, and observe that the antisymmetrization will remain
nonzero in a possibly smaller, but nonempty, neighborhood. (By the proof of Theorem 3.9, this
is guaranteed to exist for any geometric fiber (over which we can equivariantly trivialize the
torsor), but the existence of an element with nontrivial antisymmetrization in an extension field
implies the existence of such an element over the ground field.) Dividing by the symmetrization
gives a function f with poles at most ∆ such that
∑
w∈W
wf = 1. The proof of Theorem 3.9
shows that the idempotent operator
∑
w∈W wf preserves the space of functions holomorphic on
any given invariant open subset of X, and thus is a section of HW (X) over the given open subset
of S. �
Remark. Of course, we can always ensure the dim = rank condition holds by taking the
parameter space to be X/AW . This condition is needed so that
∑
r∈R(W )[X
r] is ample, allowing
Theorem 3.9 to be applied. For instance, in the case of A1 acting on E2 by swapping the factors,
every global section of HA1(E2) has holomorphic coefficients, and thus in characteristic 2 there is
no symmetric idempotent. The same argument gives a weaker statement without the dimension
condition: for any ample divisor D on X/W , there is (locally on S) a symmetric idempotent
in Γ(X/W ;HW (X)⊗OX/W (D)).
Since HW (X) contains the twisted group algebra of näıvely holomorphic operators, there is in
particular an action of the group on any HW (X)-module, and thus for any such module M which
is (quasi)coherent as an OX/W -module, we could consider the W -invariant subsheaf of M . This
as it stands may not be well-behaved, say for torsion sheaves supported on the reflection hyper-
surfaces. To obtain a better notion, we note that if we view OX as a module over HW (X), then
there is a surjective HW (X)-module morphism
∑
w cww 7→
∑
w cw from HW (X) to OX , with
kernel containing the kernel of the natural morphism OX [W ] → OX . Thus for modules which
are torsion-free as OX -modules, the sheaf HomHW (X)(OX ,M) will agree with the sheaf of W -
invariant sections of M . With this in mind, we define MW as the image of HomHW (X)(OX ,M)
under the “evaluate at 1” morphism.
44 E.M. Rains
Corollary 4.23. If the root kernel of X is diagonalizable, then the functor −W on HW (X)-
modules is exact and commutes with base change.
Proof. If HW (X) has an idempotent of the form
(∑
w w
)
h, then the map HW (X) → OX(
taking
∑
w cww to
∑
cw
)
splits as f 7→ f
(∑
w w
)
h. Since such idempotents exist locally
on S′ = X/AW , it follows that OX is locally projective, and thus the corresponding sheaf Hom
functor is exact. Moreover, it follows that MW =
(∑
w w
)
hM , and this operation clearly
commutes with base change. �
Of course, as it stands, the algebra HW (X) does not bear a terribly strong resemblance to
the more familiar Hecke algebras, due to the lack of any parameters associated to the roots.
Classically, one generally has one parameter for each orbit of roots, but in the classical Cn case
(viewing the affine Hecke algebra as being specified by an action of the finite Hecke algebra on
the space of Laurent polynomials), one effectively has two parameters associated to the endpoint
of the Dynkin diagram. This is traditionally interpreted as arising from the nonreduced root
system BCn, in which the endpoint is associated to two orbits of roots (differing by a factor
of 2). If one looks at the actual action on Laurent polynomials, however, one finds that there is
more symmetry in the parameters than is suggested by this interpretation, making it far more
natural to associate an unordered pair of parameters to the given simple reflection. In fact,
as we mentioned above, for our application, we will need a place to put an unbounded number
of parameters; since there is already an example in which one can assign two parameters to
a root without breaking things, this suggests that we should be able to assign arbitrarily many
parameters to each orbit of roots.
Let us first consider the case of rank 1, so that X is a flat family C/S of hyperelliptic curves
of genus 1. By consideration of the classical A1 and C1 cases, we are led to consider the following
algebra.
Definition 4.24. Let C/S be a flat family of hyperelliptic curves of genus 1 on which A1 = 〈s〉
acts as the marked involution, and let T be an effective Cartier divisor on C not containing any
fiber of C over S. The rank 1 Hecke algebra HA1,T (C) is the subsheaf of HA1(C) such that the
coefficient of s in a local section of HA1,T (C) is a local section of OC([Cs]− T ).
To see that this is an algebra, we note that the local sections of HA1,T (C) are precisely the
operators of the form f0 +f1(s−1) with f0 ∈ Γ(U ;OC), f1 ∈ Γ(U ;OC([Cs]−T ), or equivalently
the operators of the form g0 + (s + 1)sg1 with g0 ∈ Γ(U ;OC), g1 ∈ Γ(U ;OC([Cs] − T )). Thus
the general product of two local sections can be expressed as
(f0 + f1(s− 1))(g0 + (s+ 1)sg1) = f0g0 + f1(sg0 − g0) + f0(g1 + sg1)
+ (f1
sg0 + f0g1)(s− 1),
so that the coefficient of s in the product is again a section of OC([Cs]− T ).
There is an alternate description which makes the algebra property clearer, at the cost of
a mild loss of generality.
Proposition 4.25. The algebra HA1,T (C) is contained in the subalgebra of HA1(C) which pre-
serves the subsheaf OC(−T ) ⊂ OC . If the divisors T and sT have no component in common,
then this subalgebra is equal to HA1,T (C).
Proof. Let U be an invariant open subset on which T is cut out by a single equation h = 0.
Then the space Γ(U ;HA1,T (C)) can be described as the space of operators f0 + (s+ 1)f1
sh such
that f0 ∈ Γ(U ;OC), f1 ∈ Γ(U ;OC([Cs])). Similarly, Γ(U ;OC(−T )) = hΓ(U ;OC), so we need to
Elliptic Double Affine Hecke Algebras 45
show that f0 + (s+ 1)f1
sh preserves hΓ(U ;OC), or equivalently that h−1(f0 + (s+ 1)f1
sh)h ∈
Γ(U ;HA1(C)). Since
h−1(f0 + (s+ 1)f1
sh)h = f0 + h−1(s+ 1)f1(shh) = f0 + sh(s+ 1)f1,
and f0,
sh, (s+ 1)f1 ∈ Γ(U ;HA1(C)), the first claim follows.
Conversely, if f0 + sf1 ∈ Γ(U ;HA1(C)) also preserves OC(−T )|U , then both f0 + sf1 and
h−1(f0 + sf1)h are in Γ(U ;HA1(C)). The first condition implies f1 ∈ Γ(U ;OC([Cs])), while the
second implies f1 ∈ Γ(U ;OC([Cs]−T + sT ). If sT has no component in common with T , so that
OC(T ) ∩ OC(sT ) = OC , then OC([Cs]) ∩ OC([Cs] − T + sT ) = OC([Cs] − T ). In other words,
f1 ∈ Γ(U ;OC([Cs]− T )), so that f0 + sf1 ∈ Γ(U ;HA1,T (C)) as required. �
Remark. It is likely that the condition on T here is slightly stronger than strictly necessary:
the claim most likely continues to hold as long as T ∩ sT is contained in (Xs)red.
Note that we could have used this to prove the algebra property, in the following way. For each
nonnegative integer m, let S′ be the relative symmetric m-th power of C over S, and let C ′ be
the base change of C to S′. There is a corresponding tautological divisor T ′, and our original
data (C/S, T ) (assuming T has degree m over S) is the base change of (C ′/S′, T ′) by the
section S → S′ corresponding to T . The space of operators as described respects base change,
and thus it suffices to prove the algebra property in this larger family. Since T ′ and sT ′ have
no component in common, this follows from the above result. There is one caveat here, though:
although our original description respects base change, the description from the proposition
does not. Indeed, if there is an effective divisor T0 such that T − T0 − sT0 is effective, then
the subalgebra preserving OC(−T ) is the same as that preserving OC(−T + T0 + sT0), but the
corresponding rank 1 Hecke algebras are not the same.
In the above argument, we used the fact that shh is central. This means we could also
have described HA1,T (C) (subject to the given condition on T ) as the subalgebra of HA1(C)
preserving the supersheaf OC(sT ). This symmetry leads to the following.
Proposition 4.26. There is a natural isomorphism
OC(T )⊗HA1,T (C)⊗OC(−T ) ∼= HA1,sT (C).
Proof. Replacing (C/S, T ) by a larger family as necessary, we may assume that T and sT have
no component in common. We then have
HA1,T (C) = HA1(C) ∩ OC(−T )⊗HA1(C)⊗OC(T ),
and thus, conjugating by OC(T ),
OC(T )⊗HA1,T (C)⊗OC(−T ) = HA1(C) ∩ OC(T )⊗HA1(C)⊗OC(−T ).
Replacing T by sT in the alternate description
HA1,T (C) = HA1(C) ∩ OC(sT )⊗HA1(C)⊗OC(−sT )
tells us that
HA1(C) ∩ OC(T )⊗HA1(C)⊗OC(−T ) = HA1,sT (C)
as required. �
46 E.M. Rains
Proposition 4.27. The adjoint isomorphism HA1(C)op ∼= OC(−[Cs]) ⊗ HA1(C) ⊗ OC([Cs])
restricts to a contravariant isomorphism
HA1,T (C)op ∼= OC(−[Cs])⊗HA1,sT (C)⊗OC([Cs]),
inducing a contravariant isomorphism
HA1,T (C)op ∼= OC(T − [Cs])⊗HA1,T (C)⊗OC([Cs]− T ).
With this construction in mind, let ~T be a system of effective Cartier divisors Tα on X
associated to the roots α ∈ Φ(W ), such that Tα never contains a fiber of X and w(Tα) = Twα
for all α ∈ Φ(W ), w ∈ W . Clearly, to specify such a system, it suffices to specify Tα for one
representative of each orbit of roots, subject to the condition that w(Tα) = Tα whenever wα = α.
Although the construction would work in this generality, we will also impose the further condition
that Tα descends to a divisor on the corresponding coroot curve (or, equivalently, is invariant
under translation by any point in (1 + rα)A). This makes the stabilizer condition automatic,
and thus we may specify ~T by specifying effective divisors on the coroot curves associated to
a set of inequivalent simple roots.
We will call such a system ~T of divisors a “system of parameters for W on X”.
Definition 4.28. Let W be a finite Weyl group acting on an abelian torsor X/S by reflections,
and let ~T be a system of parameters for W on X. Then the Hecke algebra HW ;~T (X) is the
subalgebra of HW (X) generated by the rank 1 algebras H〈si〉,Ti(X).
We again have a filtration by Bruhat order, inherited from HW (X), and the subquotients are
again explicit line bundles.
Lemma 4.29. Let I be a Bruhat order ideal, and suppose that w is a maximal element of I.
Then there is a short exact sequence
0→ HW ;~T (X)[I \ {w}] ⊂ HW ;~T (X)[I]→ OX(Dw(~T ))→ 0,
where Dw(~T ) :=
∑
r∈R(W ),rw<w([Xr]− Tαr), with αr the positive root corresponding to r.
Proof. Suppose first that Tα never has a component in common with the discriminant divi-
sor Dw0 . Then an easy induction tells us that the left coefficient of w in any local section
of HW ;~T (X) vanishes on Tαr for every reflection r such that rw < w; this is by a calculation
as in Lemma 4.16 above, except that we must also argue that s1 · · · si−1αi is positive. But this
is again standard Coxeter theory; if it were not positive, then s1 · · · si could not be a reduced
word. The claim then follows as in the no parameter case.
To extend this to bad parameters, we observe (as in the rank 1 case, as we will discuss more
precisely below) that we can always embed our family in a larger family which generically satisfies
the condition on ~T . On the one hand, sinceHW ;~T (X) is generated by a flat family of submodules,
its Hilbert polynomial is lower semicontinuous and is thus bounded above by the sum of the
Hilbert polynomials of the line bundles of the subquotients of the generic Bruhat filtration. Since
we can construct elements of Bruhat intervals with the desired leading coefficients, it follows
that this bound must be tight, and the claim follows in general. �
Remark. We may also write the divisor as Dw(~T ) =
∑
α∈Φ+(W )∩wΦ−(W )([X
rα ]− Tα).
This leads to an alternate description valid under fairly weak conditions on the system of
parameters.
Elliptic Double Affine Hecke Algebras 47
Corollary 4.30. Suppose ~T is such that every Tα is transverse to every reflection hypersurface.
Then HW ;~T (X) may be identified with the subalgebra of HW (X) consisting of local sections∑
w cww such that cw vanishes on
∑
α∈Φ+(W )∩wΦ−(W ) Tα for every w ∈W .
Proof. We showed in the proof of Lemma 4.29 that every section of HW ;~T (X) satisfies the given
vanishing conditions, so it remains only to show that every local section of HW (X) satisfying
the conditions is in fact a local section of HW ;~T (X). Let D =
∑
w cw be such a section (on the
open subset U ⊂ X/W ), and I be the smallest order ideal containing the support of D, with w1
a maximal element of I. Then cw1 is a section of OX(Dw1(~T )), so that by Lemma 4.29, there is
an open covering U = ∪iVi such that on each Vi there is a local section of HW ;~T (X) supported
on I with the same left coefficient of w1. Subtracting this local section gives an element which
by induction is itself a local section of HW ;~T (X). It follows that the restriction of D to each Vi
is a section of HW ;~T (X), and thus D is a local section of HW ;~T (X) as required. �
Similarly, the corollaries carry over immediately.
Corollary 4.31. For any reduced word w = s1 · · · sn, the multiplication map
H〈s1〉, ~T (X)⊗ · · · ⊗ H〈sn〉, ~T (X)→ HW ;~T (X)[≤ w]
is surjective. Moreover, any product of rank 1 subalgebras is equal to some Bruhat interval.
Corollary 4.32. The construction HW ;~T (X) respects base change.
We also have an immediate extension of the adjoint isomorphism.
Proposition 4.33. The adjoint isomorphism HW (X)op ∼= OX(−Dw0) ⊗ HW (X) ⊗ OX(Dw0)
restricts to a contravariant isomorphism
HW ;~T (X)op ∼= OX
(
−Dw0(~T )
)
⊗HW ;~T (X)⊗OX
(
Dw0(~T )
)
.
Proof. Again, it suffices to prove that the adjoint identifies the corresponding rank 1 subalge-
bras, and one finds that twisting by OX
(
Dw0(~T ) −Dsi(
~T )
)
has no effect, so the claim follows
from the rank 1 case. �
One important special case is when Tα = [Xrα ] (which descends to the coroot curve since it
is the preimage of the identity under the composition X → E′r → Er). In that case, we find
that the rank 1 subalgebras are just the twisted group algebras OX [〈s〉], and thus that the full
algebra is itself simply equal to OX [W ].
One disadvantage of the approach via rank 1 subalgebras is that it is not particularly con-
venient when trying to determine whether a given operator is a (local) section of the Hecke
algebra. For this, it will be helpful to have a generalization of Proposition 4.25.
Proposition 4.34. The algebra HW ;~T (X) is contained in the subalgebra of HW (X) preserving
the subsheaf OX
(
−
∑
α∈Φ+(W ) Tα
)
⊂ OX , with equality holding unless there is a root α such
that Tα and T−α have a common component.
Proof. Containment reduces to showing that the rank 1 subalgebras preserve the given sub-
sheaf. Since the simple reflection si permutes the positive roots other than αi, the divisor
Ti−
∑
α∈Φ+(W ) Tα is si-invariant, and has trivial valuation along the components of [Xsi ]. It fol-
lows that on the corresponding rank 1 subalgebra, preserving OX
(
−
∑
α∈Φ+(W ) Tα
)
is equivalent
to preserving OX(−Ti), at which point the claim is just Proposition 4.25.
48 E.M. Rains
Using the Bruhat filtration, we see that equality holds whenever
OX
(
Dw(~T )
)
= OX(Dw) ∩ OX
(
Dw −
∑
α∈Φ+(W )
Tα + w
( ∑
α∈Φ+(W )
Tα
))
.
Since ∑
α∈Φ+(W )
Tα − w
( ∑
α∈Φ+(W )
Tα
)
=
∑
α∈Φ+(W )
Tα −
∑
α∈Φ+(W )
Twα =
∑
α∈Φ+(W )∩wΦ−(W )
(Tα − T−α),
we have equality as long as there is no cancellation, i.e., unless there is a positive root α and
a negative root β such that Tα and Tβ have a common component. If β 6= −α, then the
two divisors are pulled back through different coroot maps, and thus cannot have a common
component, so only the case Tα, T−α is relevant, and the claim follows. �
As in the rank 1 case, the restriction on the divisors is not particularly serious, as we can
always obtain the algebra we want as the base change of a more general family. In particular,
if S′ is an appropriate product of relative symmetric powers of coroot curves, then there is
a corresponding tautological system of parameters ~T ′ on the base change to S′, and the original
system ~T is the pullback along a suitable section S → S′.
Corollary 4.35. There is a natural isomorphism
OX
( ∑
α∈Φ+(W )
Tα
)
⊗HW ;~T (X)⊗OX
(
−
∑
α∈Φ+(W )
Tα
)
∼= HW ;− ~T (X),
where −Tα := T−α.
Another source of isomorphisms is diagram automorphisms.
Corollary 4.36. Let δ be an automorphism of X over S such that composition with δ permutes
the set of positive coroot maps. Then δ normalizes W , and the induced action on HW (X)
preserves HW ;~T (X) for all ~T .
Proof. The assumption on δ (and finiteness of W ) implies that δ preserves the set of simple
coroot maps as well as the divisor
∑
α∈Φ+(W ) Tα. �
Corollary 4.37. Let w0 be the longest element of W . Then the action of w0 on HW (X)
takes HW ;~T to HW ;− ~T .
Proof. Since [−1]w0 acts as a diagram automorphism, it suffices to show the corresponding
fact for [−1], which clearly commutes with W and satisfies
[−1]
( ∑
α∈Φ+(W )
Tα
)
=
∑
α∈Φ+(W )
T−α. �
We now turn to modules over HW ;~T (X). Since we constructed HW ;~T (X) as a space of
operators, this gives rise to a natural left module denoted OX , which as a sheaf on X/W is
the direct image of OX . This works more generally for any W -equivariant line bundle L that
descends in codimension 1, asHW ;0(X) still acts on such bundles. (We also have a corresponding
submodule L
(
−
∑
α∈Φ+(W ) Tα
)
coming from Proposition 4.34, which we will discuss more below.)
An important construction of modules comes from the fact that our algebras are gener-
ated by the rank 1 subalgebras, and thus any parabolic subgroup WI induces a corresponding
Elliptic Double Affine Hecke Algebras 49
parabolic subalgebra HWI ;~T |Φ(WI )
(X) ⊂ HW ;~T (X), which by mild abuse of notation we denote
by HWI ;~T (X). As a result, given a (left) HWI ;~T (X)-module M , we may tensor with HW ;~T (X)
to obtain an induced HW ;~T (X)-module which we denote by IndW ;~T
WI
M , or by IndW ;0
WI
M when
considering the analogous construction for the master Hecke algebra; we also denote the corre-
sponding restriction functor as ResW ;~T
WI
. Note that the restriction of the left module associated
to a line bundle is the left module associated to the same line bundle.
If L is a W -equivariant line bundle on OX that descends in codimension 1, then we have
a locally free module IndW ;~T
1 L, which establishes a Morita autoequivalence of the category
of HW ;~T (X)-modules, which we denote by L⊗−. We of course have a corresponding notion for
right modules, with IndW ;~T
1 L ∼= L⊗HW ;~T (X) ∼= HW ;~T (X)⊗L. Similarly, there is an equivalence
L
(
−
∑
α∈Φ+(W ) Tα
)
⊗− taking HW ;~T (X)-modules to HW ;− ~T (X)-modules.
If we take the restriction of an induced module, we would ordinarily expect the result to split
as a sum over double cosets. This fails even in the case of the regular representation, asHW ;~T (X)
does not naturally split as a direct sum of
(
HWI ;~T (X),HWJ ;~T (X)
)
-bimodules corresponding to
double cosets. The case I = J = ∅ is suggestive however: although the Hecke algebra does not
split as a sum of line bundles indexed by W , our results on the Bruhat filtration come fairly
close. It turns out that there is a natural Bruhat order on (parabolic) double cosets. Indeed,
every double coset WIwWJ has a unique minimal representative, and the restriction of Bruhat
order to the set of such representatives is well-behaved. (See, e.g., [36] and references therein.)
In particular, for any order ideal in the set IW J of minimal representatives, the corresponding
union of double cosets is an order ideal in W . As a result, any order ideal in IW J induces
a corresponding sub-bimodule of HW ;~T (X), and thus a subfunctor of ResW ;~T
WI
IndW ;~T
WJ
.
Given any element w ∈ IW J , the intersections WI ∩ wWJw
−1 and w−1WIw ∩WJ are both
parabolic, giving subsets I(w) ⊂ I, J(w) ⊂ J such that WI(w)
∼= WJ(w), extending in an obvious
way to an isomorphism of the corresponding Hecke algebras.
Lemma 4.38. For any w ∈ IW J , Dw
(
~T
)
is WI(w)-invariant, and has trivial valuation along
the corresponding reflection hyperplanes.
Proof. We recall the expression
Dw
(
~T
)
=
∑
α∈Φ+(W )∩wΦ−(W )
(
[Xrα ]− Tα
)
.
The fact that w is WI -minimal implies that no root of WI appears in this sum, and thus in
particular that no root of WI(w) appears. It thus remains only to show WI(w)-invariance, but
this follows by comparing Dsiw
(
~T
)
and Dwsj
(
~T
)
for reflections si ∈ WI , sj ∈ WJ such that
siw = wsj . �
This ensures that the twisting functor in the following Mackey-type result is well-defined.
Proposition 4.39. Let I, J ⊂ S. Then for any HWJ ;~T (X)-module M and any maximal chain
in the Bruhat order on IW J , the subquotient corresponding to w ∈ IW J in the resulting filtration
of ResW ;~T
WI
IndW ;~T
WJ
M is the HWI ;~T (X)-module
IndWI ;~T
WI(w)
(
OX
(
Dw
(
~T
))
⊗ wResWJ ;~T
WJ(w)
M
)
,
where here w represents the induced isomorphism from the category of HWJ(w), ~T
(X)-modules to
the category of HWI(w), ~T
(X)-modules.
50 E.M. Rains
Proof. Since the description of the subquotient is functorial, it suffices to consider the case
that M = HWJ ;~T (X), or in other words to consider the Bruhat filtration on HW ;~T (X) viewed
as a bimodule. Let O, O ∪ {w} be the elements of the chosen maximal chain that differ by w,
so that we need to understand the quotient of the subsheaf corresponding to WI(O ∪ {w})WJ
by the subsheaf corresponding to WIOWJ . Both of these are bimodules over the respective
Hecke algebras, and the actions commute with projecting onto the vector space of meromorphic
operators supported on WIwWJ . We thus immediately see from Corollary 4.31 that the quotient
is generated by the subsheaf supported on WI(w)wWJ(w) = WI(w)w, and is in fact induced from
the corresponding
(
HWI(w), ~T
(X),HWJ(w), ~T
(X)
)
-bimodule structure. Moreover, one easily verifies
that this bimodule induces the Morita equivalence M 7→ OX
(
Dw
(
~T
))
⊗ wM , from which the
result follows. Note that the fact that w ∈ IW J ensures that Dw
(
~T
)
is WI(w)-invariant and has
trivial valuation along the reflection hypersurfaces corresponding to R(WI(w)), so this twisting
is indeed well-defined. �
Taking I = ∅ gives the following, where we omit ∅ from the notation in ∅W J .
Corollary 4.40. Let I ⊂ S. Then for any HWI ;~T (X)-module M and any maximal chain in
the Bruhat order on W I , the subquotient corresponding to w ∈ W I in the resulting filtration
of IndW ;~T
WI
M is the OX-module OX
(
Dw
(
~T
))
⊗ wM .
Corollary 4.41. The functor IndW ;~T
WI
is exact.
Proof. Indeed, the proof of Proposition 4.39 shows that the HWI ;~T (X)-module HW ;~T (X) has
a filtration by (locally) free modules, so is itself locally projective. �
For finite groups, this exactness arises from the fact that restriction and induction are adjoint
in both directions. This is again not quite true in our setting, but something fairly close is true.
Define
CoindW ;~T
WI
M := HomH
WI ;~T
(X)
(
HW ;~T (X),M
)
,
which is clearly right adjoint to ResW ;~T
WI
. Given a set I of simple roots, let I ′ denote its
image under the diagram automorphism corresponding to w0, and note that the double coset
WI′w0WI = w0WI = WI′w0. Let wI denote the longest element of WI .
Lemma 4.42. For any HWI ;~T (X)-module M , there is a natural isomorphism
IndW ;~T
WI
(M) ∼= CoindW ;~T
WI′
(
OX
(
Dw0wI
(
~T
))
⊗ w0wIM
)
.
Proof. There is a natural morphism
ResW ;~T
WI′
IndW ;~T
WI
(M)→ OX
(
Dw0wI
(
~T
))
⊗ w0wIM,
since the codomain is precisely the top subquotient in the Bruhat filtration of the domain.
By adjunction, this induces a morphism from the induced module to the coinduced module, and
it remains only to show that this is an isomorphism. The image of a local section x ∈ IndW ;~T
WI
(M)
in the coinduced module is the map taking local sections y ∈ ResW ;~T
WI′
HW ;~T (X) to the top
subquotient of yx in the WI′ \ W/WI filtration. The map w 7→ w0wIw
−1 induces an order-
reversing isomorphism from W I to I′W , and thus our putative isomorphism is triangular, and
it suffices to show that it is an isomorphism on the diagonal. This reduces to showing that
Dw0wIw−1
(
~T
)
+ w0wIw
−1
Dw
(
~T
)
= Dw0WI
(
~T
)
Elliptic Double Affine Hecke Algebras 51
for any w ∈W I , which in turn reduces to `(w0wIw
−1) + `(w) = `(w0WI) and thus to `(wwI) =
`(w) + `(WI). �
Remark. Since the transformation being applied to M is invertible, we also have an expression
CoindW ;~T
WI
M ∼= IndW ;~T
WI′
(
OX
(
−Dw0wI
(− ~T ))⊗ w0wIM
)
.
As in the master Hecke algebra case, we again have a module OX coming from the action on
operators, and the restriction to HW ;~T (X) of the natural map HW (X)→ OX is still surjective.
In particular, we may again define MW to be the image of the natural injective morphism
HomH
W ;~T
(X)(OX ,M)→M .
Proposition 4.43. The kernel of the natural morphism HW ;~T (X)→ OX is generated as a left
ideal sheaf by the subsheaves of the form OX([Xsi ]− Ti)(si − 1).
Proof. Let I be the left ideal sheaf so generated. This is clearly contained in the kernel, so
it remains to show that it contains the kernel. Let
∑
w cww be a local section of the kernel,
and suppose w1 is Bruhat-maximal among the elements of W for which cw 6= 0. Since by
definition
∑
w cw = 0, w1 cannot be the identity, and thus has a reduced expression of the form
w1 = s1 · · · sm with m > 0. We thus have a (surjective) multiplication map
H〈s1〉, ~T (X) · · ·H〈sm〉, ~T (X)→ HW ;~T (X)[≤ w1].
Restricting the last tensor factor toOX([Xsm ]−Tm)(sm−1) gives an image in I without changing
the leading coefficient line bundle, and thus there is an element
∑
w≤w1
c′ww of I with c′w = cw.
Subtracting this element makes the order ideal generated by the support of the operator smaller,
and thus the result follows by induction. �
Corollary 4.44. There is an exact sequence of OX/W -modules
0→MW →M →
⊕
1≤i≤n
OX
(
Ti − [Xsi ]
)
⊗M.
Proof. The proposition gives a presentation of OX , and this is just the sheaf Hom from that
presentation to M . �
If M is S-flat, then this tells us that MW is the kernel of a morphism of S-flat sheaves.
Lemma 4.45. Let Y/S be a projective morphism with relatively ample line bundle OY (1), and
suppose φ : M → N is a morphism of S-flat coherent sheaves on Y . If the Hilbert polynomial
of ker(φs) is independent of the point s ∈ S, then the kernel, image, and cokernel of φ are all
S-flat, and the natural map ker(φ)s → ker(φs) is an isomorphism for all s.
Proof. If the Hilbert polynomial of ker(φs) is independent of s, then so is the Hilbert polynomial
of coker(φs) ∼= coker(φ)s. It follows that coker(φ) is S-flat, implying immediately that the image
and kernel are also S-flat (as kernels of surjective morphisms of S-flat sheaves). The final claim
follows using the four-term sequence
0→ Tor2(coker(φ), k(s))→ ker(φ)s → ker(φs)→ Tor1(coker(φ), k(s))→ 0
arising by comparing the two spectral sequences for tensoring the complexM → N with k(s). �
Corollary 4.46. If M is a coherent S-flat HW ;~T (X)-module such that the Hilbert polynomial
of (Ms)
W is independent of s, then MW, M/MW are flat and the natural map
(
MW
)
s
→ (Ms)
W
is an isomorphism for all s.
52 E.M. Rains
If M satisfies the hypothesis, we say that M has “strongly flat invariants”.
Before introducing parameters, we could show that this functor respected base change and
flatness by observing that (subject to diagonalizability of the root kernel) the Hecke algebra had
(locally on X/AW ) idempotents projecting onto MW . Unfortunately, this fails, and quite badly,
in cases with parameters. Indeed, if the divisors Tα are of sufficiently large degree, then the
subquotient corresponding to w in the Bruhat filtration will have negative degree unless w is
the identity, and thus in such a case the fibers of the Hecke algebra cannot have any nonscalar
global sections, let alone symmetric idempotents.
Luckily, S-flatness is local on the source, not the base, and thus the correct condition is not
that there be global symmetric idempotents, but merely that there be local symmetric idempo-
tents.
Lemma 4.47. Let U be a W -invariant open subset. If h ∈ Γ
(
U ;OX
(
Dw0
(
~T
)))
, then(∑
w
w
)
w0h ∈ Γ
(
U ;HW ;~T (X)
)
.
Proof. Suppose first that Tα and T−α have no common component for any root α, so that we
may use Proposition 4.34 to characterize HW ;~T (X). The given operator clearly maps Γ(V ;OX)
to Γ(U ∩ V ;OX)W for any invariant open V , so that
(∑
w w
)
w0h ∈ HW (X). Similarly, if
f ∈ Γ
(
V ;−
∑
α∈Φ+(W ) Tα
)
, then w0hf ∈ Γ
(
V ;Dw0 −
∑
α∈Φ(W ) Tα
)
, so that the symmetrization
vanishes along
∑
α∈Φ(W ) Tα. The claim follows in this case.
For the general case, we base change from the family with universal ~T , and observe that
since Dw0(~T ) is a flat family of divisors, h extends to a local section of OX
(
Dw0
(
~T
))
on a neigh-
borhood of the original base. We thus find that there is a local section of the larger Hecke
algebra that restricts to the desired local section, from which the result follows. �
We say that HW ;~T (X) has a local symmetric idempotent at a point x ∈ X/W if the restriction
of HW ;~T (X) to the local ring at x contains an idempotent of the form
(∑
w w
)
h. By the lemma,
this is equivalent to asking for the restriction of OX
(
Dw0
(
~T
))
to the local ring at the orbit
corresponding to x to contain an element h with
∑
w∈W
wh = 1. Moreover, if there is an
element for which this sum is a unit, then we can divide by the sum to obtain an element
symmetrizing to 1. It follows that if the condition holds on the fiber containing x, then we
still have a local symmetric idempotent at x; that is, the condition of having a local symmetric
idempotent respects base change.
Similarly, we say that HW ;~T (X) is covered by symmetric idempotents if it has a local symmet-
ric idempotent at every point x ∈ X/W . This is too much to hope for even without imposing
parameters, as the A2 example we considered at the end of Section 3 gives an explicit point
where the master Hecke algebra fails to have a local symmetric idempotent. In general, the
most we can say is that there is a (possibly empty) open subset of S such that the base change
is covered by symmetric idempotents. Indeed, the condition to have a symmetric idempotent
at x is open, and X/S is proper, so we can simply take the complement of the image of the
complement of the locus with local symmetric idempotents.
Lemma 4.48. Suppose that the root kernel of X is diagonalizable, and that for any nonnegative
linear dependence
∑
i kiαi = 0 of roots, the intersection ∩iTαi is empty. Then HW ;~T (X) is
covered by symmetric idempotents.
Proof. This is local in S, so we may restrict to an open subset over which HW,0(X) has a glo-
bal symmetric idempotent
∑
w wh; diagonalizability of the root kernel ensures that these open
Elliptic Double Affine Hecke Algebras 53
subsets cover S. For any point x ∈ X, let Dx be the corresponding decomposition group, and
observe that
1 =
∑
w∈W
wh =
∑
g∈Dx
g
( ∑
w∈Dx\W
wh
)
,
and thus there is a section of OX(Dw0) in the local ring at x for which the sum over the
decomposition group is 1.
Now, suppose that x is not contained in Tα for any positive α. Then this local section
of OX(Dw0) at x is in fact a section of OX
(
Dw0 −
∑
α Tα
)
near x, and we can add a section
that vanishes at x in such a way that the resulting section is holomorphic on the orbit Wx and
vanishes at the points of the orbit other than x. It follows that the resulting function symmetrizes
to a unit in the relevant local ring, and thus gives rise to a local symmetric idempotent at Wx.
In general, let Φx be the set of roots such that x ∈ Tα. Since x is contained in the corre-
sponding intersection of divisors Tα, we conclude that the elements of Φx, viewed as real vectors,
cannot satisfy any nonnegative linear dependence. This implies that there is a real linear func-
tional which is negative on Φx, and thus (since all systems of positive roots in a finite Weyl
group are equivalent) that wΦx ⊂ Φ−(X) for some w ∈ W . This implies that wx satisfies the
conditions for the construction of the previous paragraph to apply, and thus that there is a local
symmetric idempotent in a neighborhood of the orbit Wx. �
It is not too hard to see that the empty intersection condition is satisfied on the generic
fiber of the family with universal ~T ; indeed, if we further base change to express each Ti as
a sum of points, then the values of those parameters at a point of such an intersection must
themselves satisfy a nonnegative linear dependence, and there are only finitely many minimal
such dependences to consider. It follows that any family of Hecke algebras is the base change of
a family which is generically covered by symmetric idempotents.
Lemma 4.49. If HW ;~T (X) is covered by symmetric idempotents, then −W is exact and any
coherent HW ;~T (X)-module M has strongly flat invariants.
Proof. Indeed, OX is locally a direct summand of HW ;~T (X), and is therefore locally projective
as before. Exactness follows immediately. Any local section of (Ms)
W on an open subset
supporting a symmetric idempotent extends to a local section of M which can then be projected
to a section of MW restricting to the given section of (Ms)
W . It follows that the natural map(
MW
)
s
→ (Ms)
W is an isomorphism. Since MW is locally a direct summand of M , it is flat,
and thus its fibers have constant Hilbert polynomial as required. �
It turns out that if the generic fiber is covered by symmetric idempotents, then this has
consequences even on those fibers without such a covering.
Lemma 4.50. Suppose that there is a dense open subset of S over which HW ;~T (X) is covered
by symmetric idempotents, and suppose that the module M admits a filtration such that each
subquotient is S-flat with strongly flat invariants. Then M has strongly flat invariants.
Proof. Fix a relatively ample bundle OX/W (1) on X/W . By semicontinuity, for any point
s ∈ S and d� 0, we have
dim
(
Γ
(
X/W, (Mk(S))
W (d)
))
≤ dim
(
Γ
(
X/W, (Ms)
W (d)
))
≤
∑
i
dim
(
Γ
(
X/W, (M i
s)
W (d)
))
=
∑
i
dim
(
Γ
(
X/W,
(
M i
k(S)
)W
(d)
))
,
54 E.M. Rains
where the M i are the subquotients of the given filtration on M . Since the generic fiber
of HW ;~T (X) is covered by symmetric idempotents, −W is exact on the generic fiber and thus
dim
(
Γ
(
X/W,
(
Mk(S)
)W
(d)
))
=
∑
i
dim
(
Γ
(
X/W,
(
M i
k(S)
)W
(d)
))
for d� 0, implying
dim
(
Γ
(
X/W, (Mk(S))
W (d)
))
= dim
(
Γ
(
X/W, (Ms)
W (d)
))
as required. �
To apply this, we will need a family of modules for which we can prove strongly flat invariants
without resorting to idempotents. For I ⊂ S, let wI denote the maximal element of WI .
Proposition 4.51. Let I ⊂ S and let L be a WI-equivariant line bundle that descends in codi-
mension 1. Then we have a natural isomorphism
(
IndWWI
L
)W ∼= (L(Dw0
(− ~T )−DwI
(− ~T )))WI
of OX/W -modules, where the right-hand side denotes the sheaf of WI-invariant sections of the
given line bundle.
Proof. For any HWI ;~T (X)-module M , we have(
IndW ;~T
WI
M
)W ∼= (CoindW ;~T
WI′
(
OX
(
Dw0wI
(
~T
))
⊗ w0wIM
))W
∼=
(
OX
(
Dw0wI
(
~T
))
⊗ w0wIM
)WI′
∼=
(
OX
(
wIw0Dw0wI
(
~T
))
⊗M
)WI
∼=
(
OX
(
Dw0
(− ~T )−DwI
(− ~T ))⊗M)WI .
We may thus reduce to the case W = WI , where the result is immediate. �
Corollary 4.52. Suppose that the root kernel of X is diagonalizable. Then any module of the
form M = IndWWI
L has strongly flat invariants.
Corollary 4.53. Let I, J ⊂ S and let LI , LJ be WI , WJ -equivariant line bundles that descend
in codimension 1. If the root kernel of X is diagonalizable, then
HomH
W ;~T
(X)
(
IndW ;~T
WI
LI , IndW ;~T
WJ
LJ
)
is S-flat and respects base change.
Proof. By extending the family as appropriate (noting that LI and LJ themselves extend),
we may assume that there is a dense open subset U ⊂ S which is covered by symmetric idem-
potents. We observe that
HomH
W ;~T
(X)
(
IndW ;~T
WI
LI , IndW ;~T
WJ
LJ
)
∼= HomH
WI ;~T
(X)
(
LI ,ResW ;~T
WI
IndW ;~T
WJ
LJ
)
∼=
(
L−1
I ⊗ ResW ;~T
WI
IndW ;~T
WJ
LJ
)WI
.
Each subquotient of the Bruhat filtration for
L−1
I ⊗ ResW ;~T
WI
IndW ;~T
WJ
LJ
has strongly flat invariants, and thus the same holds for the HWI ;~T (X)-module itself. It follows
in particular that the module is S-flat and that the construction commutes with base change. �
Elliptic Double Affine Hecke Algebras 55
Proposition 4.54. Let LI , LJ be WI , WJ -equivariant line bundles that descend in codimen-
sion 1. If the root kernel of X is diagonalizable, then there is a natural isomorphism
HomHW (X)
(
IndW ;0
WJ
LJ , IndW ;0
WI
LI
)
∼= Hom
(
(π∗LI)WI , (π∗LJ)WJ
)
which is (contravariantly) compatible with composition.
Proof. Since the root kernel is diagonalizable, both HWI
(X) and HWJ
(X) have symmetric
idempotents eI , eJ on the complement of any W -invariant ample divisor, and these embed as
local sections of End(π∗LI) and End(π∗LJ) respectively. We may thus identify the sections of
the left-hand side as the subspace eJHom(π∗LI , π∗LJ)eI , and this is contravariant with respect
to composition. As a section of End(π∗LI), eI is a projection onto (π∗LI)WI , and similarly
for eJ ; the claim follows immediately. �
Remark 4.55. Note that if LI descends to a line bundle on X/WI , then (π∗LI)WI may be
identified with the direct image of that line bundle.
Remark 4.56. If LI and LJ are trivial and J = ∅, then the right-hand side consists of ope-
rators locally taking WI -invariant holomorphic functions to holomorphic functions, and may
thus be identified with the sheaf M~gWI
, where g1, . . . , gn are a system of coset representatives
for W/WI . More generally, if
∑
wWI∈W/WI
cwWI
wWI is such an operator, requiring that the
image be WJ -invariant is equivalent to requiring that (g − 1)
∑
wWI∈W/WI
cwWI
wWI annihilate
every WI -invariant function, and thus that the operator itself is invariant under left multipli-
cation by elements of WJ . Thus, on any invariant open subset, the given space is determined
by WJ -invariance along with (by Corollary 4.8) the condition that the pole of cwWI
is bounded
by the sum
∑
r:rwWI 6=wWI
[Xr], and the condition that for any reflection, cwWI
+ crwWI
is holo-
morphic along [Xr]. If LI and LJ are nontrivial, but LI descends, then the first two conditions
remain the same, but the residue condition becomes the holomorphy of wfcwWI
+ rwfcrwWI
for
any WI -invariant meromorphic section of LI such that wf is holomorphic on [Xr]. Here, it suf-
fices to check the condition for a single section f as long as both wf and wf−1 are holomorphic
along [Xr].
There is an analogue of the adjoint in this setting.
Corollary 4.57. If the root kernel of X is diagonalizable, then there is an isomorphism
HomHW (X)
(
IndW ;0
WJ
OX , IndW ;0
WI
OX
)
∼= HomHW (X)
(
IndW ;0
WI
OX(Dw0 −DwI ), IndW ;0
WJ
OX(Dw0 −DwJ )
)
,
contravariant with respect to composition.
Proof. Embedding the left-hand side in End(π∗OX) and taking the adjoint there gives an
isomorphism to Hom(e∗Jπ∗OX(Dw0), e∗Iπ∗OX(Dw0)), where e∗I , e
∗
J are the adjoints of the cor-
responding idempotents. If eI =
(∑
w∈WI
w
)
hI , then e∗I = hI
(∑
w∈WI
w
)
, and we then find
that
e∗Iπ∗OX(Dw0) = hI
( ∑
w∈WI
w
)
π∗OX(Dw0) = hI(π∗OX(Dw0 −DwI ))
WI ,
where the second equality follows from Theorem 3.9. We thus have
Hom
(
e∗Jπ∗OX(Dw0), e∗Iπ∗OX(Dw0)
)
∼= Hom
(
hJ(π∗OX(Dw0 −DwJ ))WJ , hI(π∗OX(Dw0 −DwI ))
WI
)
∼= Hom
(
(π∗OX(Dw0 −DwJ ))WJ , (π∗OX(Dw0 −DwI ))
WI
)
,
from which the claim follows. �
56 E.M. Rains
Remark. There is, of course, a version with a pair of line bundles; we omit the details.
Define
HW,WI ,WJ ;~T (X) := HomH
W ;~T
(X)
(
IndW ;~T
WJ
OX , IndW ;~T
WI
OX
)
,
with composition law given by
HW,WJ ,WK ;~T (X)⊗HW,WI ,WJ ;~T (X)→ HW,WI ,WK ;~T (X),
contravariant to the standard composition on Hom sheaves.
Proposition 4.58. If the root kernel of X is diagonalizable, then HW,WI ,WJ ;~T (X) may be
identified with a subsheaf of HW,WI ,WJ
(X), compatibly with composition. Moreover, the cor-
responding operators take WI-invariant sections of the line bundle OX
(∑
α∈Φ−(W )\Φ−(WI) Tα
)
to WJ -invariant sections of the line bundle OX
(∑
α∈Φ−(W )\Φ−(WJ ) Tα
)
. Conversely, if there are
no roots such that Tα and T−α have a common component, then HW,WI ,WJ ;~T (X) is precisely the
subsheaf of HW,WI ,WJ
(X) cut out by this condition.
Proof. Suppose first that HWI ;~T (X) and HWJ ;~T (X) are covered by symmetric idempotents.
This allows us to (locally) embed HW,WI ,WJ ;~T (X) in HW ;~T (X) as in the parameter-free case.
It follows that HW,WI ,WJ ;~T (X) acts on the WI -invariant sections of k(X) in such a way as to
take WI -invariant sections of OX to WJ -invariant sections of OX and WI -invariant sections
of OX
(∑
α∈Φ−(W ) Tα
)
to WJ -invariant sections of OX
(∑
α∈Φ−(W ) Tα
)
.
Since
∑
α∈Φ−(W ) Tα is not WI -invariant, a WI -invariant section of OX
(∑
α∈Φ−(W ) Tα
)
must
lie in the intersection of the images of this bundle under WI , so in particular (taking the inter-
section with the image under wI) in
OX
( ∑
α∈Φ−(W )
Tα
)
∩ OX
( ∑
α∈Φ+(WI)∪Φ−(W )\Φ−(WI)
Tα
)
,
which by hypothesis is OX
(∑
α∈Φ−(W )\Φ−(WI) Tα
)
. The same calculation for J tells us that the
elements of HW,WI ,WJ ;~T (X) act as required.
For a WI -invariant section
∑
w∈W I cwwWI of HW,WI ,WJ ;~T (X) to be contained in
HW,WI ,WJ
(X) is a closed condition, and thus holds in general (extending to the family with
universal parameters as necessary). That it respects the given supersheaves is also a closed
condition, and thus the first claim follows for general parameters.
To show equality under the conditions on Tα, it suffices to compare subquotients in the
respective Bruhat filtrations, and thus to compute the intersection
OX ∩ OX
( ∑
α∈Φ−(W )\Φ−(WJ )
Tα −
∑
α∈Φ−(W )\Φ−(WI)
Twα
)
.
(More precisely, we want to intersect the WJ(w)-invariant subsheaves, but since both sheaves are
equivariant, we may as well take their intersection before passing to invariants.) We may write∑
α∈Φ−(W )\Φ−(WJ )
Tα −
∑
α∈Φ−(W )\Φ−(WI)
Twα =
∑
α∈Φ−(W )
(Tα − Twα)−
∑
α∈Φ−(WJ )
Tα +
∑
α∈Φ−(WI)
Twα.
Here ∑
α∈Φ−(W )
(Tα − Twα) =
∑
α∈Φ+(W )∩wΦ−(W )
(T−α − Tα),
Elliptic Double Affine Hecke Algebras 57
while ∑
α∈Φ−(WI)
Twα −
∑
α∈Φ−(WJ )
Tα =
∑
α∈Φ−(WI)\Φ−(WI∩w−1WJw)
Twα −
∑
α∈Φ−(WJ )\Φ−(WJ∩wWIw−1)
Tα.
The hypotheses ensure that there is no further cancellation, so the intersection is
OX
(
−
∑
α∈Φ+(W )∩wΦ−(W )
Tα −
∑
α∈Φ−(WJ )\Φ−(WJ∩wWIw−1)
Tα
)
,
agreeing with the (T -dependent part of the) line bundle arising in the Bruhat filtration. �
Corollary 4.59. Let ~T be a system of parameters such that every Tα is transverse to every re-
flection hypersurface. If the root kernels of WI and WJ are diagonalizable, then HW,WI ,WJ ;~T ;γ(X)
may be identified with the subsheaf of HW,WI ,WJ ;γ(X) consisting of operators
∑
w cwwWI such
that for every w, cw vanishes on the divisor∑
α∈Φ+(W )∩wΦ−(W )
Tα +
∑
α∈Φ−(WJ )\Φ−(WJ∩wWIw−1)
Tα.
Corollary 4.60. There is an isomorphism
HW,WI ,WJ ;~T (X) ∼= OX
(
Dw0
(− ~T )−DwJ
(− ~T ))⊗HW,WJ ,WI ;~T (X)
⊗OX
(
DwI
(− ~T )−Dw0
(− ~T ))
which is contravariant for the natural composition.
Proof. Extend to universal parameters, write the left-hand side as an intersection, and take
the adjoint of both twists of HW,WI ,WJ ;~T (X). The resulting equality extends as usual to the full
parameter space. �
When I = J , we denote this byHW,WI ;~T (X), and call the resulting sheaf of algebras a spherical
algebra of the Hecke algebra, which is in general a subalgebra of the algebra End
(
(π∗LI)WI
)
corresponding to ~T = 0.
Proposition 4.58 leads in the usual way to a description of the spherical algebra for general ~T
as an intersection of two twists of the master spherical algebra:
HW,WI ,WJ ;~T (X) = HW,WI ,WJ
(X)
∩ OX
( ∑
α∈Φ−(W )\Φ−(WJ )
Tα
)
⊗HW,WI ,WJ
(X)⊗OX
(
−
∑
α∈Φ−(W )\Φ−(WI)
Tα
)
.
In the case of the Hecke algebra, we could twist this description to make the second term
untwisted, and found a relation between the original Hecke algebra and the Hecke algebra
for − ~T . In this case, however, the resulting twisted bimodule is not itself a spherical bimodule,
as the divisors∑
α∈Φ−(W )\Φ−(WI)
Tα and −
∑
α∈Φ+(W )\Φ+(WI)
Tα
do not differ by a W -invariant divisor. One can, however, give a somewhat related description
for the resulting twist, using the other family of HW ;~T (X)-modules associated to line bundles.
We find the following by tracing the various twists.
58 E.M. Rains
Corollary 4.61. We have a natural isomorphism
HomH
W ;~T
(X)
(
IndW ;~T
WJ
OX
(
−
∑
α∈Φ+(WJ )
Tα
)
, IndW ;~T
WI
OX
(
−
∑
α∈Φ+(WI)
Tα
))
∼= O
( ∑
α∈Φ−(W )∪Φ+(WJ )
Tα
)
⊗HW,WI ,WJ ;− ~T (X)⊗O
(
−
∑
α∈Φ−(W )∪Φ+(WI)
Tα
)
,
and in particular the left-hand side respects base change.
This suggests looking at what happens when only one of the two bundles is of the given
form. Unfortunately, this behaves badly for special values of ~T , as the piece associated to any
given Bruhat subquotient involves taking W -invariant sections of a line bundle which is not
itself equivariant, but instead a subbundle of an equivariant bundle cut out by (generically)
non-invariant vanishing conditions. This of course is not a problem when there is a covering by
symmetric idempotents, which suggests looking for an alternate description that gives the same
bimodule on such fibers. This is not too difficult: when HW ;~T (X) has a covering by symmetric
idempotents, the module OX is locally projective (and thus so is every module induced from it),
while the HW ;~T (X)-module OX
(
−
∑
α∈Φ+(W ) Tα
)
is locally projective whenever HW ;− ~T (X) has
a covering by symmetric idempotents. As a result, if L is a twist of either of these by an
equivariant bundle, then the natural map
HomH
W ;~T
(X)
(
L,HW ;~T (X)
)
⊗M → HomH
W ;~T
(X)(L,M)
is an isomorphism. The tensor product still behaves badly in the cases of interest, but in
a different way: for bad values of ~T , it acquires torsion. Thus we may hope that the image of this
natural map will give a strongly flat extension of the Hom sheaf from the symmetric idempotent
locus. This image has a general description which is closely related to the description coming
from symmetric idempotents when they exist.
Proposition 4.62. The image of the natural map
HomH
W ;~T
(X)
(
OX ,HW ;~T (X)
)
⊗H
W ;~T
(X) M →MW
is the same as the image of the map
m 7→
∑
w∈W
wm : OX
(
Dw0
(− ~T ))⊗M →MW .
Proof. The natural map HW ;~T (X)W ⊗M →MW is just the restriction of the map giving the
action of the Hecke algebra on M . The local sections of HW ;~T (X)W have the form
∑
w∈W wh,
where h is a local section of OX
(
Dw0
(− ~T )), and thus the image of the natural map consists
of elements
∑
w∈W whm, immediately giving the desired description. Note here that
∑
w∈W w
is indeed contained in HW ;~T (X)⊗OX
(
−Dw0
(− ~T )), so this is well-defined. �
Taking the analogous result for − ~T and reexpressing everything in terms of the original Hecke
algebra gives the following.
Proposition 4.63. The image of the natural map
HomH
W ;~T
(X)
(
OX
(
−
∑
α∈Φ+(W )
Tα
)
,HW ;~T (X)
)
⊗H
W ;~T
(X) M
→ HomH
W ;~T
(X)
(
OX
(
−
∑
α∈Φ+(W )
Tα
)
,M
)
Elliptic Double Affine Hecke Algebras 59
is naturally isomorphic to the image of the natural map
m 7→
∑
w∈W
wm : OX(Dw0)⊗M →
(
OX
( ∑
α∈Φ+(W )
Tα
)
⊗M
)W
.
Using these, it is quite straightforward to obtain the desired strong flatness results (in those
cases where the Hom sheaf itself is not strongly flat, that is): as images of maps of S-flat sheaves,
the sheaves in question can only get smaller under specialization, so it suffices to show that
the natural limits of the Bruhat subquotients are saturated. Each such subquotient reduces to
the image of
∑
w∈WJ
w on a line bundle of the form L(DwJ ) where L is equivariant and descends
in codimension 1; by the existence of symmetric idempotents when ~T = 0, we conclude that the
image is just LW independently of ~T .
Proposition 4.64. There is a natural strongly flat family of bimodules which for ~T in general
position is given by
HomH
W ;~T
(X)
(
IndW ;~T
WJ
OX
(
−
∑
α∈Φ+(WJ )
Tα
)
, IndW ;~T
WI
OX
)
,
and may be expressed as the intersection
HW,WI ,WJ
(X) ∩ OX
( ∑
α∈Φ−(W )∪Φ+(WJ )
Tα
)
⊗HW,WI ,WJ
(X)⊗OX
(
−
∑
α∈Φ−(W )\Φ−(WI)
Tα
)
,
compatibly with composition.
Proof. We may interpret IndW ;~T
WI
OX as HW,WI ,1;~T (X), and thus view it as a subsheaf of the
parameter-free case HW,WI ,1(X). The operation m 7→
∑
w∈WJ
wm gives a well-defined (surjec-
tive) map
OX(DwJ )HW,WI ,1(X)→ HW,WI ,WJ
(X),
and thus the (strongly flat) subsheaf discussed above of the Hom sheaf may itself be interpreted
as a subsheaf of HW,WI ,WJ
(X).
Twisting gives the analogous subsheaf of
HomH
W ;− ~T (X)
(
IndW ;− ~T
WJ
OX , IndW ;− ~T
WI
OX
(
−
∑
α∈Φ−(WI)
Tα
))
.
Since this is itself a subsheaf of
HomH
W ;− ~T (X)
(
IndW ;− ~T
WJ
OX , IndW ;− ~T
WI
OX
)
= HW,WI ,WJ ;− ~T (X),
the other inclusion follows. Comparing Bruhat quotients gives the desired equality for ~T in
general position. �
Corollary 4.65. There is a natural strongly flat family of bimodules which for ~T in general
position is given by
HomH
W ;~T
(X)
(
IndW ;~T
WJ
OX , IndW ;~T
WI
OX
(
−
∑
α∈Φ+(WI)
Tα
))
,
and may be expressed as the intersection
HW,WI ,WJ
(X) ∩ OX
( ∑
α∈Φ−(W )\Φ−(WJ )
Tα
)
⊗HW,WI ,WJ
(X)⊗OX
(
−
∑
α∈Φ−(W )∪Φ+(WI)
Tα
)
,
compatibly with composition.
60 E.M. Rains
5 Infinite groups
Most of our arguments above regarding the structure of the Hecke algebra boiled down to the
combinatorics of (double) cosets in Coxeter groups and the associated Bruhat order. Indeed,
virtually everything in the above discussion carries over immediately to the case of infinite
Coxeter groups, with one glaring exception: the Hecke algebra was defined as a sheaf of algebras
on the quotient X/W , and there is no such quotient scheme when W is infinite!
Thus the primary (and to first approximation only) issue in generalizing the above construc-
tion is simply to determine what manner of object we will be constructing. Luckily, a suitable
generalization of sheaves of algebras has already appeared in the literature on noncommutative
geometry, namely the notion of a “sheaf algebra”.
We recall the definition from [37, Section 2], generalizing an earlier definition of [2, Section 2].
We first need the notion of a sheaf bimodule: Let X, Y be Noetherian S-schemes of finite
type: An OS-central (OX ,OY )-bimodule is a quasicoherent OX×SY -module M such that the
support of any coherent subsheaf of M is finite over both X and Y (relative to the projections).
We will sometimes shorthand this by saying that M is a sheaf bimodule on X ×S Y . Note
that if X = Spec(RX), Y = Spec(RY ), then a sheaf bimodule on X ×S Y is an (RX , RY )-
bimodule such that Γ(S;OS) is central and such that any finitely generated sub-bimodule is
finitely generated both as a left module and as a right module.
As with ordinary bimodules, there is a notion of tensor product for sheaf bimodules. If M
is a sheaf bimodule on X ×S Y and N is a sheaf bimodule on Y ×S Z, then we can construct
a sheaf bimodule on X ×S Z by pulling back M and N to X ×S Y ×S Z, tensoring, and then
taking the direct image to X×S Z to obtain a sheaf M ⊗Y N . Note that if ∆X/S is the diagonal
in X×SX, then O∆X/S
is a sheaf bimodule on X×SX, and for any sheaf bimodule M on X×SY ,
there is a natural isomorphism O∆X/S
⊗X M ∼= M . Furthermore, this tensor product operation
is naturally associative and agrees with the usual tensor product when the schemes are affine,
see [37].
The tensor product provides the category of (OX ,OX)-bimodules with a natural monoidal
structure, thus allowing one to define a sheaf algebra on X/S to be a monoid object in that
category; that is, a sheaf bimodule A equipped with morphisms O∆X
→ A and A ⊗X A → A
satisfying the obvious axioms. More generally, one may also consider sheaf categories, in which
every object of the category has an associated scheme and the Hom sets are replaced by sheaf
bimodules.
One difficulty in dealing with the above construction is that it is not always easy to work
with local sections of the tensor product of sheaf bimodules. Indeed, since the tensor product is
a direct image, we in general need to choose an affine open covering of Y and look for compatible
systems of elements of the corresponding näıve tensor products. It turns out that for coherent
sheaf bimodules, there is a cleaner approach.
Proposition 5.1. Let M be a sheaf bimodule on X×SY and let N be a sheaf bimodule on Y ×SZ.
Let Spec(R) ∼= V ⊂ Y be an affine open subset and let U ⊂ X, W ⊂ Z be open subsets. If M
is coherent and the preimage of V in the support of M contains the preimage of U , or if N is
coherent and the preimage of V in the support of N contains the preimage of W , then there is
a natural isomorphism
Γ(U ×W ;M ⊗Y N) ∼= Γ(U × V ;M)⊗R Γ(V ×W ;N).
Proof. By symmetry, we may suppose that the constraint on M holds. The sections of a cohe-
rent sheaf bimodule on a product of open subsets depends only on the intersection of those
open subsets on the support of the sheaf bimodule. It follows, therefore, that there is a natural
isomorphism
Γ(U × V ′;M) ∼= Γ(U × (V ∩ V ′);M)
Elliptic Double Affine Hecke Algebras 61
for any open subset V ′. Computing the tensor product via an affine open covering of Y contai-
ning V gives a natural morphism
Γ(U ×W ;M ⊗Y N)→ Γ(U × V ;M)⊗R Γ(V ×W ;N),
and the compatibility conditions ensure that this is an isomorphism as required. �
Remark. For coherent M , there is a maximal U satisfying the hypothesis: take X \ U to be
the image of X of the preimage of Y \ V , and observe that finiteness implies that this image
is closed, so U is open. For quasicoherent M , it is tempting to consider the intersection of
the U ’s corresponding to the coherent subsheaves of M , but of course this will rarely be open.
Of course, if it is open, then taking the limit tells us that the conclusion of the proposition
continues to hold.
Of course, we would like to know that this incorporates the usual notion of a sheaf of algebras
on the quotient.
Proposition 5.2. Let f : X → Y be a finite morphism of Noetherian S-schemes of finite type,
and suppose that A is a quasicoherent sheaf of OY -algebras on Y equipped with an algebra
morphism f∗OX → A. Then A induces a sheaf algebra AX on X/S such that for any open
subsets U, V ⊂ Y , Γ
(
f−1(U)× f−1(V );AX
) ∼= Γ(U ∩ V ;A).
Proof. As usual, we may assume that S is affine. For any affine open subset U ⊂ Y , the sheaf-of-
algebras morphism f∗OX → A induces an algebra morphism Γ(U ; f∗OX)→ Γ(U ;A), and thus
makes Γ(U ;A) a bimodule over Γ(U ; f∗OX) ∼= Γ
(
f−1(U);OX
)
. More generally, if U, V ⊂ Y
are two affine open subsets, then we may use the morphisms Γ(U ; f∗OX) → Γ(U ∩ V ; f∗OX)
and Γ(V ; f∗OX)→ Γ(U ∩ V ; f∗OX) to make Γ(U ∩ V ;A) a
(
Γ
(
f−1(U);OX
)
,Γ
(
f−1(V );OX
))
-
bimodule.
This in particular gives a family of
(
Γ
(
f−1(Ui);OX
)
,Γ
(
f−1(Uj);OX
))
-bimodules associated
to any affine open covering of Y . Since the coefficient rings are commutative, we may reinterpret
this as a Γ
(
f−1(Ui);OX
)
⊗OS Γ
(
f−1(Uj);OX
)
-module structure, and then observe that
Γ
(
f−1(Ui);OX
)
⊗OS Γ
(
f−1(Uj);OX
) ∼= Γ
(
f−1(Ui)×S f−1(Uj);OX×SX
)
.
The open subsets f−1(Ui)×S f−1(Uj) cover X×SX, and their intersections have the same form,
making it easy to see that one has natural and compatible restriction maps. It follows that these
module structures glue together to give a sheaf on X ×S X. Moreover, the sheaf is supported
on the preimage in X ×S X of the diagonal in Y ×S Y , and thus satisfies the requisite finiteness
condition to be a sheaf bimodule.
It remains to see that this is a sheaf algebra. We note that for any open subset of Y , the pair
of open subsets
(
f−1(U), f−1(U)
)
satisfies the hypotheses of Proposition 5.1 for any coherent
subsheaf of AX on either side, from which it easily follows that the morphisms f∗OX → A
and A⊗f∗OX A → A induce a sheaf algebra structure on AX . �
There is a more general version of the approach to tensor products via open subsets that
works for fairly general quasicoherent sheaf algebras and bimodules (including any case where
the schemes are projective); this leads to some simpler alternate arguments and constructions
below, but will not be strictly needed.
The construction of a sheaf algebra from a sheaf of algebras suggests defining an “invariant
open subset” of a sheaf algebra, as an open subset U such that the two preimages of U in the
support of any coherent subsheaf of the sheaf algebra agree. If U is an affine open subset which
is invariant for a given sheaf algebra A, we immediately conclude that the sections on U × U
of A form an algebra equipped with a morphism from Γ(U ;OU ). The difficulty, of course, is
62 E.M. Rains
that a typical sheaf algebra will have no invariant affine opens. Luckily, the usual construction
of a sheaf by gluing depends far more on the subsets being affine than that they be open.
Define an affine localization of X to be a nonempty affine scheme which is the directed limit of
a (possibly infinite) family of open subschemes of X. As with affine opens, an affine localization
is determined by its image in the underlying topological space of X, and if given a family of such
subsets covering X (in a locally finite way: every open subset of X needs to be contained in
a finite union of localizations), the corresponding family of morphisms will be faithfully flat.
It then follows by fpqc descent that we may specify a sheaf (or morphisms of sheaves) using
a covering by affine localizations in place of a covering by affine opens. One similarly finds that
there is a well-behaved notion of “invariant” affine localization, and the restriction of a sheaf
algebra to an invariant affine localization is an algebra.
There is still a difficulty here, in that there is no guarantee that there will always be a lo-
cally finite covering by invariant affine localizations. (For instance, the only invariant affine
localization of the sheaf algebra k
(
P1
)[
PGL2(k)
]
on P1
k
is the field k
(
P1
)
itself.) If there were
an fpqc base change S′ → S such that the pullback to (X ×S S′) ×S′ (X ×S S′) was covered
by invariant localizations, then we could work with those localizations to understand the algebra
structure and then use a further application of fpqc descent to recover the morphisms on X×SX.
Of course, rather than make two separate applications of fpqc descent, we could simply observe
that Ui ⊗S′ Uj → X ×S X give an fpqc covering and do the descent directly. But this tells us
that there was no need for the Ui to cover X ×S S′; all we need is for them to cover X.
Given a scheme S, let Â1
S denote the localization of A1
S obtained as the limit of those open
subschemes that are dense in every geometric fiber (equivalently, that contain the generic point
of every fiber). Although the map Â1
S → A1
S is not an fpqc cover, the composition Â1
S → S is
both fpqc and surjective, and thus an fpqc cover.
This construction is functorial in S and if S → T is a finite morphism with T Noetherian,
then Â1
S
∼= Â1
T ×T S. Note, however, that this construction does not respect open embeddings,
so the obvious way to associate a sheaf of algebras on S to this construction does not always
produce a quasicoherent sheaf.
We observe that if X is projective over a Noetherian ring R, then Â1
X is affine over R. Since
the construction respects closed embeddings, it suffices to consider the case X = PnR. In that
case, we note that the section
∑
i t
ixi of the pullback of OX(1) is invertible, and thus the trivial
line bundle is very ample on Â1
X , making it affine. Since this holds for any embedding of X in
projective space, it follows that any very ample line bundle on X becomes trivial on Â1
X , and
thus (since very ample bundles generate the Picard group) that any line bundle on X becomes
trivial on Â1
X .
Proposition 5.3. Suppose X and Y are projective over the Noetherian affine scheme S, and
let M be a quasicoherent sheaf bimodule on X ×S Y . Let M ′ be the base change of M to Â1
S.
Then the fiber products of Â1
X and Â1
Y with the support of any coherent subsheaf of M ′ are
canonically isomorphic, and the affine localization Â1
X ×Â1
S
Â1
Y of X ×S Y ×S Â1
S is an fpqc
covering of X ×S Y .
Proof. The only thing to observe is that the support of any coherent subsheaf of M ′ is contained
in the base change of the support of a coherent subsheaf of M , and thus the claim reduces to the
fact that the construction Â1 respects finite morphisms. �
Remark. In particular, given bimodules on X ×S Y and Y ×S Z, we can use the corresponding
bimodules on Â1
X ×Â1
S
Â1
Y and Â1
Y ×Â1
S
Â1
Z to control the tensor product.
Thus when X is projective, or more generally when Â1
X is affine, we always have the option to
replace the sheaf algebra with the actual algebra of global sections of M on Â1
X×Â1
S
Â1
X , and des-
Elliptic Double Affine Hecke Algebras 63
cent essentially reduces to checking that the algebra has a description which is independent of
the auxiliary coordinate.
According to Proposition 5.2, the algebras HG(X), HW (X), HW ;~T (X) from the previous sec-
tion may all be interpreted as sheaf algebras on X/S, as can the twisted group algebras k(X)[G],
OX [G]. The latter are quite easy to generalize to the infinite case.
Proposition 5.4 ([37, Lemma 2.8]). Let g be an automorphism of X. Then for any quasico-
herent sheaf M on X,
(
1, g−1
)
∗M is a sheaf bimodule, and for h ∈ Aut(X) and N ∈ coh(X),
we have a natural isomorphism(
1, g−1
)
∗M ⊗X
(
1, h−1
)
∗N →
(
1, (gh)−1
)
∗(M ⊗
gN).
Proof. Since
(
1, g−1
)
∗M is supported on the graph of an automorphism, the same applies
to any coherent subsheaf, and thus it satisfies the requisite finiteness condition to be a sheaf
bimodule.
Now, let U be any affine open subset of X. Then
(
U, g−1(U)
)
satisfy the hypotheses of Pro-
position 5.1, and thus we have
Γ
(
U ×X;
(
1, g−1
)
∗M ⊗X (1, h−1)∗N
)
∼= Γ
(
U × g−1(U);
(
1, g−1
)
∗M
)
⊗Γ(g−1(U);OX) Γ
(
g−1(U)×X;
(
1, h−1
)
∗N
)
∼= Γ(U ;M)⊗Γ(g−1(U);OX) Γ
(
g−1(U);N
)
,
where f ∈ Γ
(
g−1(U);OX
)
acts on Γ(U ;M) as multiplication by gf . With this action, there is
a natural isomorphism
Γ(U ;M)⊗Γ(g−1(U);OX) Γ
(
g−1(U);N
) ∼= Γ(U ;M)⊗Γ(U ;OX) Γ
(
U ; gN
)
given by m⊗ n 7→ m⊗ gn, and thus the result follows. �
Definition 5.5. Let X/S be an Noetherian S-scheme of finite type with integral geometric
fibers, and let G be a group acting on X. Then the “twisted group sheaf algebra” k(X)[G] is
the sheaf⊕
g∈G
(
1, g−1
)
∗k(X)
on X × X (with k(X) denoting the sheaf of meromorphic functions on X which are defined
on the generic point of every geometric fiber of X over S) with sheaf algebra structure induced
by the natural morphisms(
1, g−1
)
∗k(X)⊗X
(
1, h−1
)
∗k(X)→
(
1, (gh)−1
)
∗k(X)
coming from the proposition.
Remark. We could also define this using an affine localization. The affine scheme Â1
X con-
structed in Proposition 5.3 is functorial for AutS(X), and thus has an induced action of G.
We may thus define a twisted group algebra k(X)⊗S OÂ1
X
[G], and this has a natural associated
descent datum. The only nontrivial thing to verify is the fact that cyclic bimodules are finitely
generated on both sides, but this follows easily from the fact that Â1
X is Noetherian: the bimo-
dule OÂ1
X
cggOÂ1
X
is cyclic on both sides, and any other cyclic bimodule is contained in a finite
sum of such bimodules, so is finitely generated on both sides.
64 E.M. Rains
We may similarly define OX [G] to be the sheaf subalgebra
⊕
g∈G
(
1, g−1
)
∗OX , which we
readily verify to contain the image of the identity (a copy of OX) and be preserved by the
multiplication map.
Moreover, we have the following.
Proposition 5.6. Let g1, . . . , gn; h1, . . . , hm be two finite subsets of Aut(X/S), and let
M{g1,...,gn}, M{h1,...,hm}, M{g1h1,...,gnhm} be interpreted as sheaf sub-bimodules of
k(X)[Aut(X/S)]. Then the multiplication on k(X)[Aut(X/S)] restricts to a morphism
M{g1,...,gn} ⊗M{h1,...,hm} →M{g1h1,...,gnhm}.
Proof. Given an open subset V ⊂ X, we may associate an open subset UV =
⋂
i gi(V ), and we
claim that there is an affine open covering Vi of X such that UVi also covers X. Indeed, x ∈ UV
iff g−1
1 (x), . . . , g−1
n (X) ∈ V , and thus if Vx is an affine open neighborhood of this set of points,
we have x ∈ UVx . It thus suffices to specify how the above morphism acts on local sections
on sets of the form UV ×X, for which we note
Γ
(
UV ×X;M{g1,...,gn} ⊗XM{h1,...,hm}
)
∼= Γ
(
UV ×X;M{g1,...,gn}
)
⊗Γ(V ;OV ) Γ
(
V ×X;M{h1,...,hm}
)
.
But the result then follows immediately from the definition ofM{g1,...,gn} as the space of opera-
tors preserving holomorphy. �
Remark. When X/S is projective, or more generally when we have nice affine localizations
as constructed above, then we could define M{g1,...,gn} more simply as the subsheaf of
k(X)[Aut(X/S)] supported on {g1, . . . , gn} and preserving the subring OÂ1
X
.
In particular, if G is any subgroup of the group of automorphisms of X/S, we may define
a sheaf algebra H+
G(X) as the union in k(X)[G] of the sheaves M~g associated to finite subsets
of G. More generally, we will wish to only allow poles on a proper (but G-invariant) subset of
the reflection hypersurfaces, as otherwise in the affine Weyl group case we could acquire poles
along the fibers where the “q” parameter is torsion.
Suppose now that X/S is a family of abelian varieties and that (W,S) is a Coxeter group
(of finite rank, but possibly infinite) equipped with an action on X of coroot type. Then we
define HW (X) to be the sheaf subalgebra of H+
W (X) consisting of operators which are holomor-
phic away from the reflection hypersurfaces corresponding to conjugates of the simple reflections.
Clearly, this agrees with our previous notation, in that when W is finite, HW (X) is the sheaf
algebra on X associated to the sheaf of algebras on X/W we previously denoted by HW (X).
Since W still has a Bruhat order, we may consider the subsheaf HW (X)[I] for any order
ideal I ⊂ W , and if the order ideal is finite, the result will be of the form MI(X) and thus
coherent. In fact, we have the following, by precisely the same argument as Lemma 4.16 and its
corollaries.
Proposition 5.7. If I is a finite order ideal in W and w ∈ I is a maximal element, then there
is a short exact sequence
0→ HW (X)[I \ {w}]→ HW (X)[I]→ (1, w−1)∗OX(Dw)→ 0
of sheaf bimodules, where Dw =
∑
r∈R(W ),rw<w[Xr].
Corollary 5.8. For any reduced word w = s1 · · · sn, the multiplication map
H〈s1〉(X)⊗X · · · ⊗X H〈sn〉(X)→ HW (X)[≤ w]
is surjective.
Elliptic Double Affine Hecke Algebras 65
Corollary 5.9. The construction HW (X) respects base change.
Since any finite subset of W is contained in a finite order ideal, we also obtain the following.
Corollary 5.10. The sheaf algebra HW (X) is the sheaf subalgebra of k(X)[W ] generated by the
sheaf subalgebras H〈s〉(X) for s ∈ S.
Since the action of W on X is of coroot type, we still have a well-defined association of coroot
morphisms to the roots of X, respecting positivity, and thus the notion of a system of parameters
carries over.
Definition 5.11. The (untwisted) Hecke algebra HW ;~T (X) is the sheaf subalgebra of HW (X)
generated by the rank 1 sheaf algebras H〈si〉,Ti(X).
Lemma 4.29 and Corollary 4.31 again carry over immediately, as does the fact that this
construction respects base change. However, the description of the adjoint and the description
of Proposition 4.34 both founder on the fact that they involve a sum over all positive roots.
To deal with this, we will need to generalize the construction further.
We first note that if we are given a G-equivariant gerbe Z (including, of course, the compatible
explicit isomorphisms ζg,h : Zg ⊗ gZh ∼= Zgh), then as in the finite case, there is a corresponding
crossed product algebra: take the sheaf bimodule
⊕
g
(
1, g−1
)
∗Zg with multiplication induced
by ζ. Of course, this also gives a twisted version of k(X)[G] by replacing Zg by its sheaf of
meromorphic sections. Note that if Zg and Z ′g are meromorphically equivalent (i.e., there is
a system of nonzero meromorphic maps between the line bundles which are compatible with
the maps ζ), then this induces an isomorphism between the corresponding meromorphic crossed
product algebras. In particular, the meromorphic crossed product algebra associated to a given
equivariant gerbe is isomorphic to the usual twisted group algebra iff there is a consistent family
of meromorphic sections of Zg, iff the equivariant gerbe has the form Zg = OX(Zg), where Zg
is a cocycle valued in Cartier divisors. (More generally, if one chooses arbitrary meromorphic
sections, one obtains a meromorphic equivariant gerbe in which the line bundles are trivial but
the maps ζg,h are only meromorphic, giving a class in Z2(G; k(X)∗) and making the algebra
a crossed product algebra in the usual sense. We will encounter an example of such a gerbe in
Theorem 6.13 below.) We denote these sheaf algebras as OX [G]Z and k(X)[G]Z respectively.
In this generality, we cannot expect to have a well-defined analogue of HG(X) without
some additional data. For any line bundle L, there is an induced sheaf algebra structure
on HG(X)⊗X×X L�L−1, which is sandwiched between OX [G]∂L and k(X)[G]∂L, with ∂L de-
noting the coboundary gerbe Zg = L⊗ gL−1. This equivariant gerbe is isomorphic to the trivial
equivariant gerbe when L is G-equivariant (more precisely, such an isomorphism specifies a G-
equivariant structure on L), and this will in general give a different sheaf algebra sandwiched be-
tween OX [G] and k(X)[G]. (Consider multiplying the equivariant structure by a character of G.)
We thus start with a more general construction. Given a subsheaf A ⊂ k(X)[G]Z and a subset
S ⊂ G, let A|S denote the subsheaf of operators supported on S (which is a subalgebra if S is
a subgroup). An order in k(X)[G]Z is defined to be a torsion-free sheaf algebra A with generic
fiber k(X)[G]Z such that A|S is coherent for every finite subset S ⊂ G. (Note that if G is finite,
then k(X)[G]Z may be viewed as a central simple algebra over k(X/G), and this is precisely the
usual notion of order.) In particular, OX [G]Z is an order in k(X)[G]Z , and HG(X) is an order
in k(X)[G].
We are interested in a particular subclass of orders, namely those which are left reflexive,
in that each sheaf A|S is reflexive as a left OX -module. The order OX [G]Z is clearly left reflexive,
and since an element of k(X)[G] is a (left) local section of HG(X) iff it is a local section over
every codimension 1 local ring, HG(X) is also left reflexive. (Note that when G is finite but
X → X/G is not flat, HG(X) is left reflexive as an OX module, but not as an OX/G-module.)
This suggests that we should consider left reflexive orders more generally.
66 E.M. Rains
Proposition 5.12. Let A be a left reflexive order in k(X)[G]Z containing OX [G]Z . For any
open subset U ⊂ X, an element D ∈ k(X)[G]Z is in Γ(U × X;A) iff for every codimension 1
point x ∈ U , with inertia group Ix, D ∈ OX,x ⊗OX (A|IxOX [G]Z).
Proof. Since A is left reflexive, we certainly have that D is a local section iff D ∈ OX,x⊗OX A
for all x ∈ U of codimension 1, as this is true for any reflexive sheaf on X, and A inherits it
from its restrictions to finite subsets. We thus need to understand the modules OX,x ⊗OX A|S
for finite S. But then we may proceed as in the proof of Lemmas 4.7 and 4.6 to deduce that
OX,x ⊗OX A|S =
⊕
g∈Ix\G
OX,x ⊗A|Ixg∩S ,
and thus reduce to the case S ⊂ Ixg. Since A|IxZgg ⊂ A|Ixg and A|Ix ⊂ A|IxgZg−1g−1,
we conclude that A|Ixg = A|IxZgg, giving the desired result. �
Proposition 5.13. Let X be a normal integral scheme equipped with an action of the group G
and an equivariant gerbe Z, and for each codimension 1 point x ∈ X, let Ax be an subalgebra
of k(x)[Ix]Z containing OX,x[Ix]Z and such that Ax|S is a free left OX,x-module for every finite
subset S. Suppose moreover that for any g ∈ G, we have Agx = ZggAxZg−1g−1. Then there is
a unique left reflexive order A in k(X)[G]Z such that OX,x ⊗OX A|Ix = Ax for all x.
Proof. For each finite S ⊂ G, we certainly obtain a well-defined left reflexive sheaf bimodule
consisting of operators
∑
g∈S cgg which are in AxOX [G]Z for every x; the point is that just as in
the case of holomorphy-preserving operators, there are only finitely many x such that S meets
some Ixg in more than one element. By the previous proposition, any algebra as described must
contain these sheaves, so it remains only to show that these sheaf bimodules are compatible
under multiplication. That is, if D1 is a section on U × X and D2 is a section on X × V ,
then D1D2 is a section on U × V . We thus need to check that D1D2 is in AxOX [G]Z for every
codimension 1 point x ∈ U . Since this must hold for all U , we may as well take the limit and
thus take D1 to be an element of the left stalk at x. Since this splits as a direct sum, we may
further suppose that D1 is supported on a single coset of the inertia group, so that D1 ∈ AxZgg.
This is equivalent to D1 ∈ ZggAg−1x, and we find
AxZggD2 ⊂ AxOX [G]Z ⊗OV
iff
Ag−1xD2 ⊂ Ag−1xOX [G]Z ⊗OV .
Since D2 is a section of the sheaf, we have
D2 ∈ AyOX [G]Z ⊗OV
for every codimension 1 point y ∈ X, and thus for y = g−1x, and the claim follows from the fact
that Ag−1x is an algebra. �
We thus see that the construction of a reasonable analogue of HG(X) reduces to constructing
an analogue for each inertia group, subject to the compatibility conditions under conjugation.
There are two approaches we might take to this. The first is that if we are given a trivialization
of the gerbe (or, more precisely, its restriction to Ix) along the local ring at x, then this lets
us pull HIx(X) back to an algebra containing OX,x[Ix]Z which we may use as an ingredient
in the above construction. Note that when Ix is finite, we do not actually require Ix to be
a trivialization of the gerbe for this to work; it merely needs to be an approximate trivialization,
Elliptic Double Affine Hecke Algebras 67
since what we are really determining is the quotient OX,x-module OX,x ⊗ HIx(X)/OX,x[Ix]Z .
Equivalently, we may ask for a trivialization over the complete local ring.
A second approach is to simply ask for an order of approximately the same “shape”. This is
particularly feasible in the case of order 2 reflections. Although the result is the same in most
cases, this will be particularly useful for us, as it will be easy to extend to more general base
schemes. With this in mind, let X/S be a normal scheme with an action of an involution s
such that the fixed subscheme Xs is an irreducible hypersurface, and let Zs, ζs be a gerbe, so
that ζs : Zs ⊗ sZs → OX is an isomorphism satisfying ζs = sζs. An obvious “shape” to take for
a larger algebra would be to take operators c1 + css such that c1 ∈ OX([Xs]), cs ∈ Zs([Xs]),
and c1 + hcs ∈ OX,Xs for some rational map h : Zs 99K OX which is holomorphic along Xs.
We may then readily verify that the result is closed under multiplication iff ζs − hsh vanishes
along 2[Xs]. Note that if h− h′ vanishes on Xs, then replacing h by h′ gives the same algebra;
the consistency condition is unchanged since hsh − h′sh′ vanishes along 2[Xs]. (Note that we
may view hsh as the pullback of the norm of h down to X/〈s〉, and this interpretation induces
a natural norm from Hom(Zs,O[Xs]) to Hom(Zs ⊗ sZs,O2[Xs]).) This is very nearly the same
as asking for a trivialization over the complete local ring; indeed, the two notions disagree only
when the residue field is an inseparable extension of the residue field of the s-invariant subring.
In particular, given a line bundle L on X, we have a coboundary gerbe Zs = L ⊗ sL−1, and
there is a natural choice of h, namely h = sff−1|[Xs] for some (any) meromorphic section f
of L which is holomorphic and not identically 0 along Xs. This, of course, simply corresponds
to the usual notion of twisting by a line bundle. This coboundary operation is functorial in
a particularly strong sense: not only is it functorial, but the functor takes any automorphism
to the identity. Thus if instead of a line bundle on X we are given a line bundle LT on the
base change XT to some fppf cover, then all we need for the coboundary to descend to S is
for the two pullbacks of LT to T ×S T to be isomorphic (that is, they need not be compatibly
isomorphic!). Indeed, if LT does not descend, then the obstruction is given by an automorphism
of a pullback to T ×S ×T ×S T , and the coboundary functor turns this into the identity. In par-
ticular, if X/S is projective, any section of the relative Picard scheme gives rise to a well-defined
coboundary.
More generally, given X/S with an action of an involution s such that Xs is nonempty and
everywhere of codimension 1, define a “twisting datum” to be an equivariant gerbe equipped
with a morphism hs : Zs → O[Xs] of norm ζs. Note that if X/S is proper, then ζs is the pullback
of a function on S, so is determined by its restriction to 2[Xs], and thus by hs, which must
merely satisfy the requirement that its norm be invertible and constant. For each fppf T/S, let
Tw0(X/S)(T ) be the group of twisting data with Zs = OX ; when X is proper, this is the set
of global sections of O[Xs]×ST with norm in O∗T , and thus Tw0(X/S) is represented by a group
scheme. There is a natural morphism to this group scheme from the sheaf of groups consisting
of pairs (L, ψ) with ψ : sL ∼= L: take the coboundary twisting datum and use ψ to make the
line bundle trivial. When X/S is projective, this sheaf of groups is itself representable; it is
a Gm-bundle over the s-invariant subscheme of the Picard scheme of X. Moreover, there is
a natural homomorphism from Pic(X/〈s〉) to this scheme given by pulling back and letting ψ
be the natural equivariant structure.
Lemma 5.14. Let C/S be a hyperelliptic curve of genus 1. Then the above morphisms give rise
to a short exact sequence 0→ Pic(C/〈s〉)→ Gm.Pic(C)〈s〉 → Tw0(C)→ 0 of group schemes.
Proof. Since Pic(C/〈s〉) ∼= Z (the free group generated by the isomorphism class O(1)) and
the first map simply doubles degree, we find that it is indeed injective. A pair (L, ψ) induces
the gerbe (OC , ψsψ), so the gerbe is trivial iff ψ makes L equivariant. We then find that h is
given by the restriction of the equivariant structure to Cs, and thus is trivial iff L descends
to the quotient; thus the sequence is exact in the middle.
68 E.M. Rains
It remains to show that the second morphism is surjective. This is essentially a statement
about fppf sheaves and thus we may feel free to base change so that C/S is elliptic. In that case,
both groups map to Gm by taking restrictions to the identity, and it will suffice to show that
the remaining factors are isomorphic. We readily verify that the subgroup of Tw0(E) such that
h(0) = 1 is a finite group scheme of order 8 in every fiber; indeed, it may be identified with the
complete intersection of 3 quadrics in a suitable P3. Similarly, the relevant quotient of Pic(E)〈s〉
may be identified with the disjoint union of two copies of E[2] (for C, this is Pic0(C)[2] and the
torsor of Weierstrass points), so is also a finite flat group scheme of order 8. The image of L1
is (by definition) the quadratic function q ∈ µ2(E[2]) considered above, which in characteristic
not 2 is 1 at 0 and −1 at the nontrivial 2-torsion points. The action of E[2] by translation
induces an action on both groups, and the homomorphism is equivariant. In particular, we may
compute the image of x ∈ E[2](S) in Tw0(E) as the image of x∗L1⊗L−1
1 . Modulo overall scalars,
this is q(z + x)q(z)−1; applying the splitting gives q(z + x)q(z)−1q(x)−1. Since this is precisely
the Weil pairing of z and x, we conclude that the restriction of the coboundary morphism
to E[2] is the Weil pairing, and in particular is injective. Since the image of any element of
this index 2 subgroup scheme is a homomorphism and q is not a homomorphism, we conclude
that the morphism on Pic0(E)[2] ] Pic1(E)〈s〉 is injective, and thus by degree considerations is
surjective. �
Corollary 5.15. Any twisting datum for the action of s on C is isomorphic to a coboundary.
Proof. Since the line bundle Zs has norm OC/s, it must have degree 0, and since multiplication
by 2 is surjective on the group scheme Pic0(C), there is an fppf covering T → S and a line
bundle L of degree 0 on CT such that L ⊗ sL−1 ∼= Zs. If we choose such an isomorphism,
then the coboundary induces a twisting datum with line bundle Zs, and thus we may divide
our original twisting datum by this new twisting datum to obtain a datum with trivial line
bundle. By the lemma, this is the image of (L′, ψ), where L′ is an isomorphism class of line
bundles of degree 0 or 1 and ψ : L′ ∼= sL′. We thus conclude that the base change of the original
twisting datum is isomorphic to the coboundary of L ⊗ L′. The two pullbacks of this bundle
to T ×S T have the same coboundary, so must differ by a pullback from OC/〈s〉, which by degree
considerations must be trivial. Thus L ⊗ L′ corresponds to an S-point of the relative Picard
scheme of C, and the twisting datum is the coboundary of this point. �
We may extend the notion of twisting datum to more general groups by assigning a twisting
datum to each order 2 subgroup that fixes some reflection hypersurface, and insist on the appro-
priate compatibility relations. Again, any point of the relative Picard scheme has a well-defined
coboundary.
Proposition 5.16. Let W be a finite Weyl group and let X/S be an abelian torsor on which W
acts by reflections. If the root kernel of X is trivial, then any twisting datum on X is isomorphic
to a coboundary.
Proof. Let T/S be an fppf cover over which X has a W -invariant section. The induced twisting
datum on each rank 1 parabolic subgroup is the coboundary of a point in the Picard scheme,
but since XT has a section, it is in fact the coboundary of a line bundle on XT . We may
then apply Proposition 3.13 to express the induced class in Z1(W ; Pic(XT )) as a coboundary.
Moreover, we may use the invariant section (which is contained in every [Xs]) to rigidify the
various isomorphisms, and thus express the base changed twisting datum as the product of this
coboundary and a twisting datum with trivial underlying gerbe.
Choose representatives among the simple reflections of the conjugacy classes of reflections,
and observe that for each such si, there is a unique bundle Li on Ei of degree 0 or 1 such
that the restriction of the latter twisting datum on si is the coboundary of π∗iLi. Tensoring
Elliptic Double Affine Hecke Algebras 69
the conjugates of those bundles gives a W -invariant (but not equivariant) line bundle with the
desired twisting datum, and thus expresses the base change of the original twisting datum as
the coboundary of a line bundle LT .
Since the two pullbacks of LT to T ×S T have the same coboundary, they differ by an equi-
variant bundle on XT that descends in codimension 1. Since the polarization of a line bundle
is locally constant, we may arrange for the ratio to have trivial polarization, so be a point
of Pic0(X)(T ×S T ) ∼=
∏
iEi(T ×S T ). The restriction to si of the coboundary of such a point is
essentially just the coboundary of the corresponding point of Ei(T×ST ); since it must be trivial,
we conclude that the two pullbacks of LT are in fact isomorphic, and thus that L descends to
a point of the relative Picard scheme of X/S. �
Remark. This can fail if the root kernel is nontrivial, even when X has a section. However,
there is a flat finite cover T → S over which X can be expressed as the quotient of a torsor with
trivial root kernel, so that we may describe a twisting datum on X as the coboundary of a line
bundle on XT , subject to appropriate descent conditions.
More generally, if W is a Coxeter group and X/S is an abelian torsor with an action of W
of coroot type, then an assignment of twisting data on the rank 1 parabolic subgroups extends
to at most one twisting datum for W . Indeed, we may extend the collection of pairs (Zsi , ζi) to
an equivariant gerbe by choosing a reduced word for each w ∈W and defining
Zw := Zs1 ⊗ s1Zs2 ⊗ s1s2Zs3 ⊗ · · · ⊗ s1···sm−1Zsm .
If all we had was the equivariant gerbe structure, we would also need to specify isomorphisms
corresponding to the different braid relations, which would themselves need to satisfy com-
patibility relations (coming from finite parabolic subgroups of rank ≤ 3).3 Luckily, each braid
relation may be restated as a conjugacy relation between simple reflections, and the isomorphism
corresponding to the braid relation appears linearly in the corresponding consistency condition
on the twisting data. For any rank ≤ 3 finite parabolic subgroup, the different reflection hyper-
surfaces have a nonempty common intersection, and thus the further compatbility conditions
of the gerbe will be automatically satisfied. Thus a collection of rank 1 twisting data extends to
a full twisting datum iff it extends for every finite rank 2 subgroup.
Let γ denote such a twisting datum, and write k(X)[W ]γ for k(X)[W ]Zγ , and let HW ;~T ;γ(X)
denote the sheaf subalgebra generated by the rank 1 algebras Li ⊗H〈si〉,Ti(X)⊗L−1
i , where Li
is any line bundle with coboundary γ|si . Then the usual arguments carry over from the finite
case to give the following.
Proposition 5.17. If I is a finite order ideal in W and w ∈ I is a maximal element, then there
is a short exact sequence
0→ HW ;~T ;γ(X)[I \ {w}]→ HW ;~T ;γ(X)[I]→ (1, w−1)∗
(
Zγ,w ⊗OX
(
Dw
(
~T
)))
→ 0
of sheaf bimodules.
Corollary 5.18. The construction of the sheaf algebra HW ;~T ;γ(X) respects base change.
Corollary 5.19. Let ~T be a system of parameters such that every Tα is transverse to every ref-
lection hypersurface. Then HW ;~T ;γ(X) may be identified with the sheaf subalgebra of HW ;γ(X)
consisting locally of operators
∑
w cww such that for every w, cw vanishes on the divisor∑
α∈Φ+(W )∩wΦ−(W ) Tα.
3In general, one can define a G-equivariant gerbe by giving a line bundle for each generator of G, a morphism
for each relation, subject to a consistency condition for each 3-cell of the classifying space BG. For Coxeter groups,
there is a model of BW with k-cells corresponding to multisets of k simple roots such that the corresponding
parabolic subgroup is finite, and thus the 3-cells come from finite parabolic subgroups of rank ≤ 3.
70 E.M. Rains
Corollary 5.20. Let w ∈W be given by the reduced word w = s1 · · · sl. Then the multiplication
map H〈s1〉, ~T ;γ(X)⊗X · · ·⊗XH〈sl〉, ~T ;γ(X)→ HW ;~T ;γ(X)[≤ w] is surjective. Moreover, any product
of rank 1 subalgebras is equal to some Bruhat interval.
Suppose that D1, . . . , Dn are Cartier divisors such that Zsi = Di − siDi extends to a cocycle
valued in Cartier divisors, and consider the case that γi = ∂OX(Di). Since the action of W
on the group of Cartier divisors is a permutation module, its restriction to any finite subgroup
is induced from a trivial module, so has trivial H1. In particular Z|〈si,sj〉 is a coboundary of
some Dij for any finite rank 2 parabolic subgroup. If Di − Dij has even valuation along any
(separable) component of [Xsi ], and similarly for Dj −Dij , then γi = ∂OX(Dij) and similarly
for γj , and thus we have a compatible extension of twisting data. Note that since Dij is only
determined up to 〈si, sj〉-invariant divisors, we can change its parity along each orbit of 〈si, sj〉-
reflection hypersurfaces independently, and thus if si and sj are not conjugate, this condition
can always be satisfied, and otherwise reduces to a simple parity constraint.
Given any other twisting datum γ, let γ
(
~D
)
denote the twisting datum obtained by tensoring
with the above twisting datum. Since Zsi extends to a cocycle valued in Cartier divisors,
the resulting equivariant gerbe comes with a natural meromorphic equivalence to the original
equivariant gerbe, and thus we have an induced isomorphism k(X)[W ]γ ∼= k(X)[W ]γ( ~D) for
any γ, and may in this way view HW ;~T ;γ( ~D) as a subalgebra of k(X)[W ]γ .
To understand such isomorphisms more generally, we will need to understand cocycles valued
in Cartier divisors. The fact that Hom(W,Z) = 0 implies that any coinduced module for W has
trivial H1. Since Cartier divisors are a sum of induced modules, there can be (and are) cocycles
valued in Cartier divisors which are not coboundaries. However, since the induced modules are
contained in the corresponding coinduced modules, we can always express such a cocycle as
a coboundary in the larger module (of integer-valued functions on the set of irreducible Cartier
divisors). Note that since the typical element of a coinduced module will not have coboundary in
the induced submodule, we need to add the condition that any element of w only changes finitely
many values of the function; naturally, it suffices to verify the condition for the simple reflections.
For instance, if we interpret
∑
α∈Φ+(W ) Tα as giving an integer-valued function on irreducible
Cartier divisors (i.e., the sum over α ∈ Φ+(W ) of the valuation of Tα along the given divisor),
then any element of W only changes finitely many values of the function, and thus we obtain
a well-defined coboundary Zw =
∑
α∈Φ+(W )∩wΦ−(W )(Tα − T−α).
We may also use such formal sums to define (meromorphically trivial) twisting data; if Γ is
an integer-valued function on irreducible Cartier divisors such that Γ − siΓ has finite support,
then we may obtain a divisor Di with the same coboundary on 〈si〉 by restricting Γ to the union
of the support of Γ− siΓ and the components of the reflection hyperplanes. Similarly, if 〈si, sj〉
is finite, then we may obtain a divisor Dij by restricting Γ to the union of the supports of Γ−wΓ
for w ∈ 〈si, sj〉, and find that Dij −Di and Dij −Dj are pullbacks, so that γi = ∂OX(Di) gives
a well-defined twisting datum. We denote the twist of some other γ by this meromorphically
trivial datum by γ(Γ). (More precisely, a twisting datum is determined by Γ along with a choice
of representation of each Γ − siΓ as a coboundary; the above convention can behave badly in
families, but there is always a consistent way to take a limit of the choices of representations as
coboundaries in rank 1.)
We then introduce the notation
OX(Γ)⊗HW ;~T ;γ(X)⊗OX(−Γ)
for HW ;~T ;γ(Γ)(X) viewed as a subalgebra of k(X)[W ]γ . Note that if Γ′ − Γ has finite support,
then
OX(Γ′)⊗HW ;~T ;γ(X)⊗OX(−Γ′)
∼= OX(Γ′ − Γ)⊗ (OX(Γ)⊗HW ;~T ;γ(X)⊗OX(−Γ))⊗OX(Γ− Γ′),
Elliptic Double Affine Hecke Algebras 71
where the outer twist on the right hand side is the usual twist by a line bundle. We may also
define a sheaf OX(Γ′)⊗HW ;~T ;γ(X)⊗OX(−Γ) in this case by OX(Γ′−Γ)⊗(OX(Γ)⊗HW ;~T ;γ(X)⊗
OX(−Γ)).
Proposition 5.21. Let ~T , ~T ′ be two systems of parameters for W on X. Then
HW ;~T+~T ′;γ(X) = OX
(
−
∑
α∈Φ+(W )
T ′α
)
⊗HW ;~T+− ~T ′;γ(X)⊗OX
( ∑
α∈Φ+(W )
T ′α
)
as subalgebras of k(X)[W ]γ.
Proof. This reduces immediately to the corresponding claim in the rank 1 case, where (after
twisting by a line bundle to make γ trivial) it reads
HA1,T+T ′(C) = OC(−T ′)⊗HA1,T+sT ′(C)⊗OC(T ′).
For general parameters (such that no two of T , sT , T ′, sT ′ have a common component), this
is straightforward: it is easy to see that that HA1,T+T ′(C) preserves the subsheaf OC(−T ′),
and HA1,T+sT ′(C) preserves the subsheaf OC(−sT ′), and this gives both inclusions. �
The proof of Proposition 4.34 carries over to give the following.
Proposition 5.22. Suppose that Tα and T−α have no common component for any α ∈ Φ(W ).
Then
HW ;~T ;γ(X) = HW ;γ(X) ∩ OX
(
−
∑
α∈Φ+(W )
Tα
)
⊗HW ;γ(X)⊗OX
( ∑
α∈Φ+(W )
Tα
)
.
We also note the following fact, which allows us to decouple the conditions associated to
different parameters.
Proposition 5.23. Suppose that ~T and ~T ′ are such that Tα and T ′α have no common component
for any α. Then
HW ;~T+~T ′;γ(X) = HW ;~T ;γ(X) ∩HW ;~T ′;γ(X).
Proof. The rank 1 subalgebras on the left are contained in the corresponding subalgebras on the
right, so algebra on the left is certainly contained in the intersection on the right. To see equality,
we use the Bruhat filtration and observe that each subquotient on the left is the intersection of
the corresponding subquotients on the right. �
Remark. This easily gives a version of Proposition 5.22 in which HW ;~T+~T ′;γ(X) is given as an
intersection of two twists of HW ;~T ;γ(X).
The construction of the adjoint in the finite case carries over. Note that the näıve adjoint∑
w cww 7→
∑
w wcw induces a natural isomorphism k(X)[W ]op
γ
∼= k(X)[W ]γ−1 . (In terms of the
sheaf algebra itself, all we are doing is swapping the two factors of X×SX.) To describe how this
acts on the Hecke algebras, it will be helpful to denote the formal sum
∑
α∈Φ+(W )
(
[Xrα ] − ~T
)
by Dw0
(
~T
)
, and similarly for Dw0 . This of course agrees with the usual notation whenever the
longest element w0 ∈W actually exists.
Proposition 5.24. The näıve adjoint on k(X)[W ]γ induces an identity
HW ;~T ;γ(X)op = OX
(
Dw0
(
~T
))
⊗HW ;~T ;γ−1(X)⊗OX
(
−Dw0
(
~T
))
= OX(Dw0)⊗HW ;− ~T ;γ−1(X)⊗OX(−Dw0).
of subalgebras of k(X)[W ]γ−1.
72 E.M. Rains
Proof. Again, this reduces immediately to the rank 1 case. �
Diagram automorphisms of course work as well in the infinite case; the only caveat is that
unlike in the finite case, a diagram automorphism can fail to preserve the parameters and
twisting datum. More generally, if H is a group of automorphisms of X acting as diagram
automorphisms of W and preserving the parameters, and there is an H-equivariant gerbe Zh
such that hγi ∼ γi⊗∂Zh for each i, then the corresponding holomorphic crossed product algebra
normalizes the Hecke algebra, and we can combine them into a larger algebra associated to the
extended Coxeter Group W oH. (In the Cn case we consider in detail below, we will see that
even the requirement that the parameters be invariant can be finessed.)
Suppose A and B are sheaf algebras, on X/S and Y/S respectively. An (A,B)-bimodule is
then simply a sheaf bimodule M on X ×S Y equipped with multiplication maps A⊗XM →M ,
M ⊗Y B → M making the obvious diagrams commute. (Note that the restriction of M to
a compatible pair of localizations is a bimodule over the corresponding restrictions of A and B.)
The tensor product is then defined in the obvious way, so that we may define induced modules.
Restriction is of course also easy to define, though the sheaf form of Frobenius reciprocity is
somewhat tricky, as there are difficulties with defining Hom on sheaf bimodules in general.
(The difficulty is that the category of sheaf bimodules is cocomplete, but not complete, and
Hom from a direct limit is an inverse limit. Thus the Hom of sheaf bimodules will still be
a quasicoherent sheaf on the relevant fiber product scheme, but may fail to satisfy the finiteness
requirement.)
This is not a problem for the analogue of Proposition 4.39; the only change is that M
should be replaced by a suitable bimodule. In the finite case, this is no difficulty: when W is
finite, any HW ;~T (X)-module in the usual sense determines a corresponding
(
HW ;~T (X),OX/W
)
-
bimodule structure.
Proposition 5.25. Suppose I, J ⊂ S are such that the parabolic subgroups WI , WJ
are finite. Then for any
(
HWJ ;~T ;γ(X),OY
)
-bimodule M and any maximal chain in the
Bruhat order on IW J , the subquotient corresponding to w ∈ IW J in the resulting filtration
of ResW ;~T ;γ
WI
IndW ;~T ;γ
WJ
M is the
(
HWI ;~T ;γ(X),OY
)
-bimodule
IndWI ;~T ;γ
WI(w)
Zγ,w
(
Dw
(
~T
))
⊗ wResWJ ;~T ;γ
WJ(w)
M.
We also have a weaker form of Frobenius reciprocity.
Lemma 5.26. For any Y , induction and restriction are adjoint functors between the categories
of
(
HW ;~T ;γ ,OY
)
-bimodules and
(
HWI ;~T ;γ ,OY
)
-bimodules.
Proof. Since both functors are constructed as tensor products, we see that it suffices to con-
struct compatible morphisms
HW ;~T ;γ ⊗HWI ;~T ;γ
HW ;~T ;γ → HW ;~T ;γ
and
HWI ;~T ;γ → ResW ;~T ;γ
WI
HW ;~T ;γ ,
both of which (along with compatibility) follow directly from the fact that HWI ;~T ;γ is a subal-
gebra of HW ;~T ;γ . �
Corollary 5.27. Let M be a coherent
(
HWI ;~T ;γ ,OY
)
-bimodule and N an
(
HW ;~T ;γ ,OZ
)
-bimo-
dule. Then the quasicoherent sheaf HomH
W ;~T ;γ
(
IndW ;~T ;γ
WI
M,N
)
on Y ×S Z is a sheaf bimodule.
Elliptic Double Affine Hecke Algebras 73
Proof. Frobenius reciprocity gives (when Y and Z are affine, which we may certainly reduce to)
HomH
W ;~T ;γ
(
IndW ;~T ;γ
WI
M,N
) ∼= HomH
WI ;~T ;γ
(
M,ResW ;~T ;γ
WI
N
)
,
and the latter is a sheaf bimodule since M is coherent. �
Corollary 5.28. Let MI be a coherent
(
HWI ;~T ;γ , YI
)
-bimodule, MJ a coherent
(
HWJ ;~T ;γ , YJ
)
-
bimodule, and M an
(
HW ;~T ;γ , Y
)
-bimodule. Then there is a natural composition morphism
HomH
W ;~T ;γ
(
IndW ;~T ;γ
WI
MI , IndW ;~T ;γ
WJ
MJ
)
⊗YJ HomH
W ;~T ;γ
(
IndW ;~T ;γ
WJ
MJ ,M
)
→ HomH
W ;~T ;γ
(
IndW ;~T ;γ
WI
MI ,M
)
of (YI , Y )-bimodules, satisfying the obvious associativity relation.
In particular, given an
(
HW ;~T ;γ , Y
)
-bimodule M , we may again define an (X/WI , Y )-bimo-
dule MWI as
HomH
W ;~T ;γ
(
IndW ;~T ;γ
WI
OX ,M
) ∼= HomH
WI ;~T ;γ
(
OX ,ResW ;~T ;γ
WI
M
)
.
This, of course, is essentially just the extension of
(
ResW ;~T ;γ
WI
M
)WI to bimodules in the obvious
way.
Now that we have reasonable definitions, most of the calculations we did in the finite case carry
over. We find that (assuming γ is trivial on WI and WJ) the submodule of ResW ;~T ;γ
WI
IndW ;~T ;γ
WJ
OX
corresponding to any finite Bruhat order ideal has strongly flat invariants for WI , and thus
HW,WJ ,WI ;~T ;γ(X) :=
(
IndW ;~T ;γ
WJ
OX
)WI
is an S-flat sheaf bimodule on X/WI ×S X/WJ , and this construction commutes with base
change. The subquotients in the corresponding Bruhat filtration may all be described in the
following way. For each w ∈ IW J , there is a corresponding line bundle Lw on X/WI(w) (con-
structed from ~T and γ) such that the subquotient is the direct image in X/WI ×S X/WJ
of the (X/WI(w), X/WJ(w))-bimodule (1, w−1)∗Lw. More precisely, the line bundle Lw is the
descent to X/WI(w) of the (WI(w)-equivariant!) line bundle Zγ,w
(
Dw
(
~T
))
⊗ OX
(
DwI
(− ~T ) −
DwI(w)
(− ~T )).
If ΓI , ΓJ are WI , WJ -invariant functions which are even on reflection hypersurfaces and have
finitely supported difference, then for any twisting datum γ which is trivial on WI and WJ ,
OX(ΓJ)⊗HW ;~T ;γ ⊗OX(−ΓI)
becomes a left HWJ ;~T -module and a right HWI ;~T -module, and thus has a corresponding spherical
module which we may denote by
OX(ΓJ)⊗HW,WI ,WJ ;~T ;γ ⊗OX(−ΓI).
The adjoint takes the following form.
Proposition 5.29. If the root kernels of WI and WJ on X are diagonalizable and the twisting
datum γ is trivial on WI and WJ , then there is an isomorphism
HW,WI ,WJ ;γ(X) ∼= OX(Dw0 −DwI )⊗HW,WJ ,WI ;γ−1(X)⊗OX(DwJ −Dw0)
which is contravariant with respect to composition.
74 E.M. Rains
Proof. We may view the element
∑
w w as a section of HWI
(X) ⊗ OX(−DwI ), since it takes
sections of OX(DwI ) to sections of OX . We thus have an embedding
HW,WI ;γ(X)→ HW ;γ(X)⊗OX(−DwI )
acting as∑
w∈W I
cwWI
wwI 7→
∑
w∈W
cwWI
w.
Moreover, if the original operator is a local section of HW,WI ,WJ ;γ(X), then we have
cw′wWI
= w′cwWI
for w′ ∈WJ , so that we may write the image as∑
w′∈WJ
w′
∑
w∈JW
cWJwWI
w.
Taking the adjoint (including the twist by OX(Dw0)) gives∑
w′∈WJ
∑
w∈JW
(−1)`(ww
′)w−1cWJwWI
w′−1 =
∑
w∈WJ
∑
w′∈WJ
(−1)`(ww
′)wcWJw−1WI
w′
in OX(−DwI )⊗HW ;γ−1(X). Right dividing by
∑
w′∈WJ
(−1)`(w
′)w′ gives a section of
OX(−DwI )⊗HW,WJ ,WI ;γ−1(X)⊗OX(DwJ )
as required. Compatibility with composition follows by observing that in a composition, the
factor
∑
w∈WI
w needed in the middle is already present in the other operator, and the factor∑
w∈WJ
(−1)`(w)w that should be removed is needed in the other operator. �
Proposition 5.30. If the root kernels of WI and WJ on X are diagonalizable, then
HW,WI ,WJ ;~T ;γ(X) ⊂ HW,WI ,WJ ;γ(X)
∩ OX
( ∑
α∈Φ−(W )\Φ−(WJ )
Tα
)
⊗HW,WI ,WJ ;γ(X)⊗OX
(
−
∑
α∈Φ−(W )\Φ−(WI)
Tα
)
,
with equality unless there is a root α such that Tα and T−α have a common component.
Proof. Over the locus of S covered by symmetric idempotents, we may use those idempotents
to locally identify HW,WI ,WJ ;~T ;γ(X) with a submodule of HW,WI ,WJ ;γ(X) (using the same idem-
potent to embed both in HW ;γ(X)). This identification is compatible with the identification
of meromorphic fibers, so extends to a global identification on each fiber covered by symmetric
idempotents, and from there to the closure of the symmetric idempotent locus.
Similarly, local idempotents embed HW,WI ,WJ ;~T ;γ(X) in
OX
( ∑
α∈Φ−(W )
Tα
)
⊗HW ;γ(X)⊗OX
(
−
∑
α∈Φ−(W )
Tα
)
,
and the idempotents eliminate the contributions of Tα for α ∈WI , WJ respectively.
To see that the inclusion is tight, we need merely verify that both sides have the same Bruhat
subquotients, which reduces to verifying that the negative part of∑
α∈Φ+(W )∩wΦ−(W )
(T−α − Tα) +
∑
α∈Φ−(WJ )\Φ−(WJ∩wWIw−1)
(Twα − Tα)
has no further cancellation, just as in the finite case. �
Elliptic Double Affine Hecke Algebras 75
Corollary 5.31. Let ~T be a system of parameters such that every Tα is transverse to
every reflection hypersurface. If the root kernels of WI and WJ are diagonalizable, then
HW,WI ,WJ ;~T ;γ(X) may be identified with the sheaf sub-bimodule of HW,WI ,WJ ;γ(X) consisting
of operators
∑
w cwwWI such that for every w, cw vanishes on the divisor∑
α∈Φ+(W )∩wΦ−(W )
Tα +
∑
α∈Φ−(WJ )\Φ−(WJ∩wWIw−1)
Tα.
Corollary 5.32. If the root kernels of WI and WJ on X are diagonalizable, then there is an
isomorphism
HW,WI ,WJ ;~T ;γ(X) ∼= OX
(
Dw0
(− ~T )−DwI
(− ~T ))⊗HW,WJ ,WI ;~T ;γ−1(X)
⊗OX
(
DwJ
(− ~T )−Dw0
(− ~T ))
which is contravariant with respect to composition.
Remark. Of course, we also have analogous results for the other three natural Hom sheaves
discussed above.
We close by considering the analogue in this setting of the residue conditions of [11]. It suffices
to consider the algebras HW,WI ,WJ ;γ(X), since ~T simply imposes generic vanishing conditions
on the coefficients, as already discussed. (And, of course, this includes the Hecke algebras
themselves by taking I = J = ∅.) We may further assume J = ∅, as HW,WI ,WJ ;γ(X) is the
submodule of HW,WI ;γ(X) consisting of WJ -invariant operators.
Since we may embed HW,WI ;γ(X) in HW ;γ(X) using a symmetric idempotent, the fact that
the latter is spanned by submodules
H〈r〉;γ(X)OX [W ]γ
implies something similar for the former: it is spanned by the submodules
H〈r〉;γ(X)
⊕
w∈W I
ZwwWI .
If rwWI = wWI , then we may rewrite the corresponding summand as
ZwwH〈w−1rw〉;γ(X)WI = ZwwWI ,
and thus we may omit any such summand.
If we restrict to a finite subset S ⊂W/WI , then we conclude that
cwWI
∈ Zw
( ∑
r∈R(W ) : rwWI∈S\{wWI}
[Xr]
)
and that there is a residue condition relating cwWI
and crwWI
along [Xr]. Again moving Zww
to the left lets us express this condition in the form
cw + whw−1rwζ
−1
w,w−1rw
crw = 0 ∈ Zw([Xr])|[Xr],
where hw−1rw comes from the root datum on 〈w−1rw〉. Note that we could also determine this
by right dividing by fwwWI , where fw trivializes Zw in a neighborhood of Xr to obtain the
condition in the form
cw + fw
rf−1
w hrζ
−1
r,wcrw = 0 ∈ Zw([Xr])|[Xr].
Of course, these are equivalent (since otherwise γ would violate the compatibility conditions).
Naturally, in the untwisted case, the condition is just that cw+crw is holomorphic (essentially
Corollary 4.3), and the same holds (along [Xr]) if we embed HW,WI ;γ(X) in k(X)[W/WI ] via
an expression of the twisting datum as the coboundary of a formal divisor which is transverse
to the reflection hypersurfaces.
76 E.M. Rains
6 The (double) affine case
The most interesting case for our purposes is when the Coxeter group is an affine Weyl group W̃ .
We actually want to modify the construction (very) slightly in that case, as the abelian variety
being acted on is slightly larger than we would like. That is, rather than have an (n + 1)-
dimensional variety with an invariant map to an elliptic curve, we would prefer to act on the
fibers of that map. If we pull back the sheaf bimodule H
W̃ ;~T ;γ
(X) from X×SX to X×X/A
W̃
X,
then we find (by considering what happens on invariant localizations, say) that the result is still
naturally a sheaf algebra. The group no longer acts faithfully on every fiber, but the various
calculations involving the Bruhat filtration carry over without difficulty, so that we still obtain a
flat family of sheaf algebras generated by the rank 1 subalgebras. One caveat is that Tα and Tβ
need not be transverse for α 6= ±β; if they correspond to the same root of the finite root system,
then their divisors differ only by a translation, which may act trivially on some fibers.
Still, we have the following definition. First, if X is a torsor over the abelian scheme A,
an action of W̃ on X by affine reflections is simply an action such that every simple reflection
fixes a hypersurface and the action on A factors through a faithful action of the corresponding
finite Weyl group. Any such action arises from an action of coroot type by specializing the
parameter q. (Recall that q = ζ(z) gives the image of the origin of a fiber under the special
reflection s0.) In addition, every finite parabolic subgroup still acts faithfully (regardless of q),
and thus in particular we still have good notions of systems of parameters and twisting data.
The one caution is that when expressing twisting data as a coboundary of some formal sum of
divisors, one needs to assume q non-torsion. This is not truly an issue, however, as one can
simply take the limit of the twisting data from the non-torsion case (which simply adds an extra
level of formality to the formal sum). In particular, the cocycles in Cartier divisors associated
to the formal sums
∑
α∈Φ+(W ) Tα and
∑
α∈Φ+(W )[X
rα ] remain cocycles after specializing q.
Definition 6.1. Let W̃ act on X by affine reflections, let ~T be a system of parameters, and
let γ be a twisting datum. The corresponding elliptic double affine Hecke algebra H
W̃ ;~T ;γ
(X) is
the sheaf subalgebra of k(X)
[
W̃
]
γ
generated by the rank 1 subalgebras H〈s〉;~T ;γ(X) for s ∈ S.
Proposition 6.2. The subsheaf of H
W̃ ;~T ;γ
(X) corresponding to any finite Bruhat order ideal is
an S-flat coherent sheaf on X ×S X.
Proposition 6.3. We have
H
W̃ ;~T ;γ
(X) ⊂ H
W̃ ;γ
(X) ∩ OX
(
−
∑
α∈Φ+(W̃ )
Tα
)
⊗H
W̃ ;γ
(X)⊗OX
( ∑
α∈Φ+(W̃ )
Tα
)
,
with equality unless there are α ∈ Φ+
(
W̃
)
, β ∈ Φ−
(
W̃
)
such that Tα and Tβ have a common
component.
Remark. When q is torsion, then in fact each α has infinitely many β (both positive and
negative) such that Tα = Tβ, and thus the hypothesis for equality is never satisfied.
Corollary 6.4. Let ~T be a system of parameters such that every Tα is transverse to every ref-
lection hypersurface. Then H
W̃ ;~T ;γ
(X) may be identified with the sheaf subalgebra of H
W̃ ;γ
(X)
consisting locally of operators
∑
w cww such that for every w, cw vanishes on the divisor∑
α∈Φ+(W )∩wΦ−(W ) Tα.
It will be helpful to understand the possible twisting data in this scenario. If we assume
that W has trivial root kernel, then we may trivialize the twisting datum along W , and it thus
Elliptic Double Affine Hecke Algebras 77
remains to determine the possibilities for the restriction to s0. If W̃ 6= Ãn, then the affine
diagram is a tree with s0 as a leaf, and thus s0 commutes with a rank n− 1 parabolic subgroup
of W . The polarization of Zs0 must be invariant under that subgroup, and since it is also
negated by s0, we find that Zs0 must have trivial polarization. To fully specify the twisting
datum, we need to choose a solution of L ⊗ s0L−1 = Zs0 . If W̃ 6= C̃n, so that s0 is connected
to the finite diagram via an ordinary edge, then we find that the polarization of L is the sum
of a polarization on the connected kernel of the coroot map and a polarization in the image
of 1 + s0; it thus follows that we may replace L itself by a line bundle with trivial polarization
without affecting the root datum.
In other words, for W̃ not of type A or C, any twisting datum is (up to an overall twist by
a line bundle) given by taking the twisting datum along s0 to be the coboundary of a point
in Pic0(X). For type C, there is at most one additional component which may be reached if it
exists by taking the coboundary of the pullback of a degree 1 line bundle on the coroot curve.
For type A, the situation is more complicated, as it turns out that there are in fact twisting
data with nontrivial polarizations. Luckily, these can always be described via cocycles in Cartier
divisors. Type An corresponds to the action of Sn+1 on the sum zero subvariety of En+1, with s0
acting as (z1, zn+1) 7→ (q + zn+1, z1 − q). For any point u ∈ E, we may consider the formal sum∑
j≥0
∑
1≤i≤n+1
[zi = u+ jq].
This is invariant under the finite Weyl group, while its coboundary under s0 is [z1 = u]− [zn+1 =
u + q]. This gives a well-defined twisting datum for generic u, q, and thus extends to all u, q.
As u varies, these cover the corresponding component of the group scheme classifying twisting
data trivial on W , and one readily verifies that this component generates the full group scheme.
In the Ãn case, we could obtain every twisting datum via a cocycle in Cartier divisors. This
is unlikely to hold in general type (albeit without a known counterexample), but something only
slightly weaker is true.
Proposition 6.5. Let X/S be an abelian torsor equipped with an action of W̃ of coroot type such
that W has trivial root kernel. Then any twisting datum on X/S can be fppf locally represented
by a cocycle in Cartier divisors.
Proof. Certainly the twist by a line bundle may be represented as a cocycle in Cartier divisors,
so we reduce to the case that the twisting datum is trivial along W . Let λ0 : X → E′ be
a (nonconstant) homomorphism from X to an elliptic curve, and consider the orbit W̃λ0 of such
maps for general u ∈ E′. There is a point q′ ∈ E′ (determined from q and λ0) such that λ is in
the orbit iff
λ = wλ0 + jq′
for some w ∈W , j ∈ Z. Now, consider the formal sum∑
j≥0
∑
λ∈Wλ0
[wλ0 = u+ jq′]
of divisors on X. This is W -invariant, and its coboundary with respect to s0 has finite support,
so we obtain a well-defined family of cocycles in Cartier divisors.
Apart from type A and C, it will suffice to show that it depends nontrivially on u (since
then we have a nonconstant morphism from E′ to the connected 1-parameter group scheme
parametrizing twisting data). Such dependence is clearly independent of q, so we may assume q
non-torsion, and thus q′ non-torsion. Then shifting u by q′ above subtracts
∑
λ∈Wλ0
[wλ0 = u]
78 E.M. Rains
from the formal sum. This divisor class is W -invariant and ample, so is not s0-invariant, and
thus shifting u has a nontrivial effect on the twisting datum as required.
In type C, the same argument shows that everything in the identity component of the scheme
parametrizing twisting data is fppf locally represented by cocycles in Cartier divisors, and one
can explicitly verify (indeed, see the discussion of the C̃ case below) that when there is another
component, it can still be reached in this way. �
Remark 6.6. The presence of exotic twisting data in type A can be explained by considering the
induced cocycle in polarizations. As discussed in more detail in the C̃n case below, such a cocycle
for arbitrary type may be obtained as the coboundary of a W -invariant rational homogeneous
polynomial of the form p3(~z, q)/q) (with appropriate modifications in the presence of nontrivial
isogenies). The only part that contributes to the polarization of the coboundary is the part of
degree 3 in ~z, and the only indecomposable finite Weyl groups with invariants of degree 3 are
those of type An for n ≥ 2.
Remark 6.7. In the above argument, we used the fact that shifting u by q′ had the effect
of twisting by a W -invariant line bundle, and that this had a nontrivial effect on the twisting
datum. It follows that we do not have a well-defined scheme parametrizing twisting data modulo
twists by line bundles. Indeed, it follows that for q non-torsion, twists by line bundles are dense
in the identity component of the group scheme of twisting data trivial on W .
It is unclear (but likely) if this result holds for actions with nontrivial root kernel. This
fact is useful enough, however, that we will include it as an implicit assumption below; that is,
we will impose as an additional condition that the twisting datum is fppf locally represented by
a cocycle in Cartier divisors, or equivalently (by the above argument) that the restriction of the
twisting datum to W is a coboundary.
The description of H
W̃
(X)
(
or H
W̃ ,W̃I ,W̃J
(X)
)
as holomorphy-preserving operators continues
to hold, as long as W̃ acts faithfully, or more generally for any Bruhat order ideal that injects
in Aut(X). So we may again apply Corollary 4.8 to obtain residue conditions analogous to those
of [11] for the finite case, just as in the non-affine case.
For the twisted case, we note that when q is nontorsion, the algebraH
W̃ ;γ
(X) is still an algebra
of the type constructed in Proposition 5.13, so there is no difficulty in generalizing the proofs
and we still have reasonable residue conditions. (If we represent the twisting datum as a cocycle
in Cartier divisors transverse to the reflection hypersurfaces, then the corresponding embedding
in k(X)
[
W̃
]
is again holomorphy-preserving away from the support of the cocycle, and thus the
residue conditions can still be obtained from Corollary 4.8. Of course, the argument we used in
the non-affine case works equally well!) Any Bruhat interval is flat as q varies, and thus we can
obtain conditions for torsion q by taking limits. This can become quite complicated when the
Bruhat interval is large compared to the order of q, but in the case that the Bruhat interval acts
faithfully, there is no difficulty taking the limit; the poles are at most simple, and one simply has
the usual condition along each reflection hypersurface given by the twisting datum. The same
applies to spherical modules; again, the sheaf on each Bruhat interval can be obtained as the limit
from general q, and the limit is straightforward as long as there is no coefficient such that two
components of its allowed polar divisor coalesce in the limit. In other words, for the WI -invariant
Bruhat interval [≤ wWI ], if the set of reflections {r ∈ R(W ) : rw′WI ∈ [≤ wWI ]\{w′WI}} injects
in Aut(X) for every w′WI ≤ wWI , then the residue conditions are precisely as expected from
the non-affine case.
The most important case for the spherical algebra construction is when W̃I = W is the
corresponding finite Weyl group. In that case, we note that each coset W̃/W contains a unique
translation, and thus we may interpret elements of the spherical algebra as (elliptic) difference
operators, with H
W̃ ,W
(X) for non-torsion q corresponding to difference operators that (locally)
Elliptic Double Affine Hecke Algebras 79
preserve W -invariant holomorphic functions. Note that the Bruhat order on W \W̃/W is simply
the usual dominance order on dominant weights [19].
This has the following consequence. We say that a sheaf algebra is a domain if the product
of a nonzero section on U × V and a nonzero section on V ×W is always a nonzero section
on U ×W .
Proposition 6.8. Suppose that the root kernel for W on X is diagonalizable. Then for any
twisting datum γ which is trivial on W , every fiber of the spherical algebra H
W̃ ,W ;~T ;γ
(X) is
a domain.
Proof. We first note that the inclusion of H
W̃ ,W ;~T ;γ
(X) in k(X)[Λ]γ is injective on fibers. This
follows from the fact that we can compute H
W̃ ,W ;~T ;γ
(X) as the W -invariant submodule of an S-
flat module with strongly flat invariants, and thus the inclusion H
W̃ ,W ;~T ;γ
(X) ⊂ IndW̃ ;~T ;γ
W OX is
injective on fibers; as the induced module injects in the induced module of k(X) and this equals
k(X)[Λ]γ , the desired injectivity follows.
In particular, any local section of a fiber on a product of W -invariant open subsets can
be identified with a W -invariant section of k(X)[Λ]γ . Since this identification is compatible
with the multiplication, it will suffice to show that k(X)[Λ]γ is a domain. Since Λ is a finitely
generated free abelian group, there exist injective homomorphisms Λ→ R, allowing us to define
a total ordering on Λ compatible with the group law. In particular, for any nonzero element∑
λ∈Λ cλ[λ] of k(X)[Λ]γ , there is a corresponding notion of “leading monomial”, defined as
cλ[λ], where λ is the largest element of the support. If f has leading monomial fλ[λ] and g has
leading monomial gµ[µ], then fg has leading monomial ζλµ(fλ ⊗ λgµ), and is therefore nonzero
as required. �
Another important feature of the spherical algebra in the affine case is that it is Morita
equivalent to the DAHA itself, at least for generic parameters. The proof relies on the following
result on two-sided ideals in the DAHA.
Lemma 6.9. Let k be an algebraically closed field, and suppose W̃ acts faithfully on the abelian
torsor Y/k. Let S be a product of symmetric powers of coroot curves of Y , and let ~T be the
corresponding universal system of parameters on X := Y ×S. Then for any ideal sheaf I ⊂ OX ,
there is a dense open subset of S on which I generates H
W̃ ;~T ;γ
(X) as a two-sided ideal.
Proof. Suppose otherwise. We may assume that
I =
(
H
W̃ ;~T ;γ
(X)IH
W̃ ;~T ;γ
(X)
)∣∣
1
,
since both sides generate the same two-sided ideal. If we replace each instance of the DAHA
by the restriction of the DAHA to operators supported entirely on si, it follows that
I ⊃ siI ⊗ OX
(
Ti + siTi
)
.
This in turn induces a weak symmetry condition on the set Z cut out by I: if x ∈ Z, then
either si(x) ∈ Z or x ∈ Ti ∪ siTI , and in those terms our goal is to prove that Z does not meet
the generic fiber.
We claim that this is true for any proper closed subset of X satisfying this condition. Since
the weak symmetry condition is preserved under restriction to the generic fiber of S as well
as under taking Zariski closure, we may assume that every component of Z meets the generic
fiber, and thus by properness that it meets every fiber. Now consider a fiber on which each Ti is
contained (as a set) in [Y si ]. If x is a point of Z in such a fiber, the weak symmetry condition
implies that either si(x) ∈ Z or x ∈ [Y si ] and thus x = si(x). Thus the restriction of Z to such
80 E.M. Rains
a fiber must be si-invariant (as a set!) and thus W̃ -invariant. Since W̃ acts faithfully on Y ,
any W̃ -orbit in Y is Zariski dense, and thus Z must contain every such fiber.
It will thus suffice to prove that Z cannot contain any fiber, and thus obtain a contradiction.
Again, since W̃ acts faithfully, Zk(S) cannot contain any W̃ -orbits, and thus for any point
x ∈ Zk(S), there exists w ∈ W̃ and i ∈ {0, . . . , n} such that wx ∈ Zk(S) but siwx /∈ Zk(S).
It follows that wx ∈ Ti ∪ siTi, and thus Zk(S) is covered by the sets of the form wTi for w ∈ W̃ ,
i ∈ {0, . . . , n}. Since every component of Z meets the generic fiber, Z is covered by the same
sets; since none of these sets contains a fiber, Z cannot contain a fiber, and we are done. �
Proposition 6.10. Suppose W̃ acts faithfully on X such that the root kernel of W on X is
diagonalizable and γ is trivial on W . If ~T is in sufficiently general position, then the categories
of H
W̃ ;~T ;γ
(X)-modules and H
W̃ ,W ;~T ;γ
(X)-modules are equivalent, with the inverse equivalences
given by −W and IndW̃ ;~T ;γ
W OX ⊗H
W̃ ,W ;~T ;γ
(X) −.
Proof. Since we are assuming ~T is in sufficiently general position, we may in particular assume
that the finite Hecke algebra HW ;~T (X) has a covering by symmetric idempotents, so that both
functors are exact. It suffices to check that they are inverses on the regular representation. One
direction is by definition of the spherical algebra, so we reduce to showing that the natural map
φ : IndW̃ ;~T ;γ
W OX ⊗H
W̃ ,W ;~T ;γ
(X) HW̃ ;~T ;γ
(X)W → H
W̃ ;~T ;γ
(X)
is surjective. The image certainly contains the image of
OX ⊗H
W,W ;~T
(X) HW ;~T (X)W → HW ;~T (X).
This map is almost certainly not surjective, but its cokernel is torsion, since the analogous map
for k(X)[W ]γ is surjective. It follows that the image meets OX nontrivially, and thus the same
is true for φ. But then the image of φ is a two-sided ideal in the DAHA meeting OX nontrivially,
and thus the lemma tells us that φ is surjective. �
Remark. More generally, if ~T and ~T ′ are two systems of parameters in sufficiently general
position, the same argument tells us that HW̃ ;− ~T+~T ′;γ(X) is Morita equivalent to
End
(
IndW̃ ;~T+~T ′;γ
W OX
(
−
∑
α∈Φ+(W )
Tα
))
.
(Note that by Proposition 5.21, this is indeed a module, and is cut out by the image of local
symmetric idempotents in HW ;− ~T+~T ′(X).)
In the affine case, the spherical algebra has an additional symmetry. As we mentioned
above, for general Coxeter groups, the usual symmetry replacing ~T by − ~T has an issue in the
spherical algebra case. The proof of that symmetry relied on the fact that
∑
α∈Φ(W ) Tα has
no effect on twisting, so that twisting by
∑
α∈Φ+(W ) Tα and −
∑
α∈Φ−(W ) Tα have the same
effect, letting one move the twist to the other half of the intersection. For the spherical algebra,
this operation instead turns −
∑
α∈Φ−(W )\Φ−(WI) Tα into
∑
α∈Φ+(W )∪Φ−(WI) Tα, and thus does
not give an algebra of the same form
(
but rather involves the other natural representation
OX
(
−
∑
α∈Φ+(WI) Tα
))
. However, in the affine case, it turns out that there actually is a system
of parameters ~T ′ such that∑
α∈Φ+(W̃ )∪Φ−(W )
Tα =
∑
α∈Φ−(W̃ )\Φ−(W )
T ′α.
Elliptic Double Affine Hecke Algebras 81
In both cases, we can break up the sum as a sum over roots of the finite Weyl group W , and
find that on the left-hand side we have a sum over translates of Tα by nonnegative multiples of
some qα, while on the right-hand side we have a sum over translates of T ′α by negative multiples
of qα. We may thus simply take T ′α to be the translate of T−α by qα.
As a result, we find that the algebra
HomH
W̃ ;~T
(X)
(
IndW̃ ;~T ;γ
W OX
(
−
∑
α∈Φ+(W )
Tα
)
, IndW̃ ;~T ;γ
W OX
(
−
∑
α∈Φ+(W )
Tα
))
may be identified with an instance of H
W̃ ,W ;~T
(X) in which the parameters have been translated
by qα. We also obtain a pair of (strongly flat) intertwining bimodules as discussed above in
the finite case. For generic parameters, each of the algebras is Morita equivalent to the original
DAHA by Proposition 6.10 and the remark following, and thus the algebras are Morita equivalent
to each other, with the equivalences induced by the corresponding bimodules. More generally,
if we write ~T = ~T ′ + ~T ′′, then the algebra obtained by translating ~T ′′ but leaving ~T ′ alone
is still generically expressible as the endomorphism ring of an induced representation
(
coming
from OX
(
−
∑
α∈Φ+(W )
~T ′α
))
, so the proof of Proposition 6.10 still gives a Morita equivalence to
the original DAHA for generic parameters, and thus Morita equivalences between the spherical
algebras with shifted parameters. Note that the corresponding DAHAs are themselves Morita
equivalent for generic parameters, since their spherical algebras are Morita equivalent.
Another feature of the double affine case is that the inverse map acts on the poset W W̃W
as a diagram automorphism (which is often trivial). In particular, if we can arrange for the
pullback of the adjoint through the diagram automorphism to have isomorphic twist datum,
then this gives an actual involution of the Hecke algebra, which allows us to consider self-adjoint
operators (and even have a reasonable chance of proving commutativity). For a specific example
of this phenomenon in the C̃n case, see Theorem 7.28 below.
One new phenomenon that arises in the affine case is that the group can fail to act faithfully.
Since the finite Weyl group acts faithfully on A, the kernel is necessarily contained in the
translation subgroup, and we see that there is a kernel precisely when q is torsion. In that
case, the action of W̃ ∼= W n Λ on X factors through a semidirect product of the form W̃q :=
W n (Λ/Λq), where Λq is the sublattice acting trivially (which is a multiple of either Λ or, in
the non-simply-laced case, the lattice generated by the other orbit of roots). One thus finds
that H
W̃ ,~T ;γ
(X) is the sheaf algebra associated to a sheaf of algebras on the quotient X/W̃q.
The centralizer of k(X) in the generic fiber is isomorphic to k(X)[Λq]γ , and thus the center of the
generic fiber is isomorphic to k
(
X/W̃q
)
[Λq]
W
γ . This agrees with the generic fiber of the center,
and thus the center of H
W̃ ,~T ;γ
(X) for q torsion is the coordinate sheaf of an integral X-scheme
of relative dimension n.
When ~T = 0, we can make this quite precise; it turns out that for q torsion, H
W̃ ;γ
(X) is
an algebra à la Proposition 5.13 with G = W̃q acting on an explicitly described (but no longer
projective) scheme X+; its center is thus the structure sheaf of X+/W̃q, and everything is
Noetherian. Clearly, if such a construction exists, X+ must be the relative Spec of the (abelian)
restriction H
W̃ ;γ
(X)|Λq . This is difficult to understand directly (we will in fact give an alternate
purely geometric construction and then verify that its relative coordinate ring is as described),
but luckily it will largely suffice to deal with the Ã1 case.
Thus let C/S be a genus 1 curve equipped with a pair of hyperelliptic involutions s0, s1, and
let γ be a twisting datum for the corresponding action of Ã1 = 〈s0, s1〉 on C. We may describe γ
by giving a pair of line bundles L0 and L1 such that γ|〈si〉 is the coboundary of Li. It is then
straightforward to determine the residue conditions for HÃ1;γ(C) when s1s0 has infinite order.
We find that c(s1s0)k and c(s1s0)k+ls1 have a potential pole along
[
C(s1s0)2k+ls1
]
, subject to the
82 E.M. Rains
condition that
c(s1s0)k
(s1s0)kf + c(s1s0)k+ls1
(s1s0)k+ls1f
is holomorphic on that reflection hypersurface for any local section
f ∈
{
Γ
(
U ;Hom
(⊗
0<i≤l
(s1s0)iL0,
⊗
0≤i≤l
(s1s0)iL1
))
, l ≥ 0,
Γ
(
U ;Hom
(⊗
l<i<0
(s1s0)iL1,
⊗
l<i≤0
(s1s0)iL0
))
, l < 0.
(Of course, the condition is independent of f as long as f is a unit in the local ring at every
component of the reflection hypersurface.)
Now, suppose s1s0 has finite order m, so that the image of 〈s0, s1〉 in Aut(C) is actually
the dihedral group of order 2m. Thus as automorphisms of C, they satisfy the braid rela-
tion (s0s1)m/2 = (s1s0)m/2 (if m is even) or (s0s1)(m−1)/2s0 = (s1s0)(m−1)/2s1 (if m is odd).
In the dihedral group, both sides represent the longest element, and thus the corresponding
Bruhat interval is the entire group. Something similar happens in Ã1: each side generates a
Bruhat interval which is not only faithful, but a setwise section of the map to the dihedral group,
and the two sections agree except on the longest element. Even better, the residue conditions
in the two Bruhat intervals are very nearly the same; indeed, below the top, the conditions are
necessarily the same, while the conditions involving the top element differ only mildly. We thus
obtain the following, where πq : C → C ′ is the quotient by 〈s1s0〉 (which is translation by some
torsion point q), an isogenous curve with induced hyperelliptic involution s (the common action
of s0 and s1). (We write (s0s1)m/2 for (s0s1)(m−1)/2s0 when m is odd, so that we can express
the result in a uniform way for either parity.)
Proposition 6.11. Let U ⊂ C ′ be an s-invariant open subset on which there is an invertible
section g ∈ Γ
(
U ;Nπq
(
L1 ⊗ L−1
0
)∗)
. Then
c(s0s1)m/2(s0s1)m/2 +
∑
w<(s0s1)m/2
cww ∈ Γ
(
π∗qU ;HÃ1;γ(C)
)
iff
π∗q (g/
sg)c(s0s1)m/2(s1s0)m/2 +
∑
w<(s1s0)m/2
cww ∈ Γ
(
π∗qU ;HÃ1;γ(C)
)
.
Proof. This reduces to a straightforward verification that the transformation respects the
residue conditions on every reflection hypersurface. �
Corollary 6.12. With U , g as above, let f0, f1 ∈ Γ
(
U ;OX(π∗q [C
′s])
)
be such that f1−π∗q (g/sg)f0
is holomorphic. Then
f0 + f1(s1s0)m ∈ Γ
(
U ;HÃ1;γ(C)
)
.
Proof. Take a general section of HÃ1;γ(C)[≤(s0s1)m/2], apply the proposition to get a section
supported on [≤(s1s0)m/2], take the difference, and then left-multiply by a suitable operator
h(s1s0)m/2. This gives a general section f0 + f1(s1s0)m such that f1 = π∗q (g/
sg)f0, to which
we may add any f ′0 ∈ Γ(U ;OX). �
Remark. Note that f1 is indeed supposed to be a meromorphic section of the same bundle
of which π∗q (g/
sg) is a local trivialization, namely
⊗
w∈G
w
(
L1 ⊗ L−1
0
)(−1)`(w)
, where G is the
image of Ã1 in Aut(C).
Elliptic Double Affine Hecke Algebras 83
Remark. This points out that in the affine case (in contrast to the situation for more general
Coxeter groups where there is no parameter q), there is something special about Bruhat intervals:
the subsheaves of operators supported on more general finite subsets may fail to be flat.
The restriction of OC [Ã1]γ to the kernel 〈(s1s0)m〉 of the action has a natural geometric
description. Indeed, it may be given in the form
⊕
j∈ZZ
⊗j
(s1s0)m(s1s0)jm, which is easily recog-
nized as the relative structure sheaf of the Gm-bundle over C corresponding to the invertible
sheaf Z(s1s0)m . The corollary tells us that HÃ1;γ(C) contains a larger ring, which is easily rec-
ognized as coming from an affine blowup. To be precise, the restrictions of the various local
sections π∗q (g/
sg) to the union of reflection hypersurfaces induces a section of the Gm-bundle
over that union. Enlarging the algebra has the effect of blowing up that union of sections and
then removing the strict transforms of the fibers over the reflection hypersurfaces.
More generally, let W̃ act on X with kernel Λq. The restriction of the gerbe to Λq in particular
induces a homomorphism Λq → Pic(X/S), or equivalently a class in H1(X; Hom(Λq,Gm)), and
thus a principal Hom(Λq,Gm)-bundle P over X. Each Zλ for λ ∈ Λq comes with an induced
trivialization on the pullback, and this in fact trivializes that portion of the gerbe. Indeed,
this compatibility is clear for any twisting datum specified as a cocycle in Cartier divisors, and
(per our standing assumption) any twisting datum is fppf locally of this form. We thus obtain
a natural W̃q-equivariant gerbe on the torus bundle, on which W̃q acts faithfully, and there is
a natural isomorphism OX
[
W̃
]
Z
∼= π∗OP
[
W̃
]
Z .
The twisting datum itself does not directly lift, for the simple reason that W̃ acts nontrivially
on Λq, and thus in particular the fixed subschemes of the various reflections are codimension 2;
the fixed subscheme of r is a principal Hom(Λq,Gm)〈r〉-bundle over [Xr]. (Note that this is either
a Gn−1
m -bundle or a Gn−1
m × µ2-bundle, with the latter only arising when W n Λq ∼= W (C̃n).)
However, each reflection in W̃q is the image of an infinite collection of reflections in W n Λq,
the set of reflections of a copy of Ã1. Moreover, for any reflection r, the subalgebra of operators
supported on 〈1, r〉 may be computed by conjugating the corresponding algebra for some simple
reflection. We thus find that each such copy of Ã1 actually gives rise to a subalgebra of the
form HÃ1;γ′(X). We thus obtain additional elements as above.
We may think of each such element as a section of OP ([Xr]), and find that the ideal sheaf
generated by all such sections cuts out a Hom
(
Λ
〈r〉
q ,Gm
)
-bundle over [Xr]. (In the untwisted
case, this is the subbundle of the r-fixed scheme containing ([Xr], 1), and in general is uniquely
determined by the requirement that it contain a section determined by the twisting datum.)
Adjoining all such elements to the relative coordinate ring gives a new scheme X+, still affine
overX. Geometrically, X+ is obtained by blowing up the union of all such bundles then removing
the strict transforms of the fibers of P over the reflection hypersurfaces. This operation is W̃q-
equivariant, and thus X+ inherits an action of W̃q. Now, not only does the gerbe lift, but the
twisting datum as well: on X, it is specified by sections of line bundles on the various reflection
hypersurfaces, and each reflection hypersurface of X+ lies over a reflection hypersurface of X,
so we can simply pull back the section. (Again, any potential issues with compatibility may be
reduced to the untwisted case by expressing the twisting datum via Cartier divisors, pulling back
the Cartier divisors to X+ and twisting. In the untwisted case, the possible incompatibilities
coming from the nontriviality of the gerbe maps go away, since the blowup makes those functions
congruent to 1 on the relevant reflection hypersurface.)
We then have the following.
Theorem 6.13. Let X+ be constructed as above, with associated projection π : X+ → X and
induced twisting datum γ+ for the action of W̃q on X+. Then there is a natural isomorphism
H
W̃ ;γ
(X) ∼= π∗HW̃q ;γ+(X+).
84 E.M. Rains
Proof. The map OX
[
W̃
]
γ
→ π∗OX+
[
W̃q
]
γ+ extends to a map on meromorphic operators, and
we find that the restriction to any rank 1 subalgebra of H
W̃ ;γ
is contained in π∗HW̃q ;γ+(X+).
Since these generate the full algebra, we obtain a homomorphism H
W̃ ;γ
(X)→ π∗HW̃q ;γ+(X+),
and it remains only to show that this is surjective.
By construction, the image contains OX+ , and thus for any reflection r, the image contains
any element of the form gπ∗f(1−π∗hr), where g ∈ Γ(π∗U ;OX+), f ∈ Γ
(
U ;OX([Xr])
)
, and h ∈
Γ(U ;Z∗r ) restricts to give the twisting datum along r. For U sufficiently small (but containing
any chosen codimension 1 point), the elements of the form gπ∗f span Γ
(
U ;OX([(X+)r])
)
, and
thus we obtain any element in Γ(U ;OX+) + Γ
(
U ;OX+([Xr])
)
(1 − π∗hr). But this is precisely
the algebra corresponding to 〈1, r〉 in H
W̃q ;γ+(X+), and these subalgebras again generate the
full algebra. �
Corollary 6.14. The center of H
W̃ ;γ
(X) as a sheaf of algebras over X/W̃q is naturally isomor-
phic to the relative coordinate ring of X+/W̃q.
Proof. Since W̃q acts faithfully on X+, the centralizer of OX+ in H
W̃q ,γ+(X+) is precisely OX+ ;
furthermore, for an element of OX+ to commute with sections of Zgg for all g ∈ W̃q, it must
be W̃q-invariant. We thus find that the center is contained in OW̃q
X+ = O
X+/W̃q
. This subalgebra
is clearly central, and thus the claim holds. �
We would also, of course, like to understand the center of the spherical algebra. When
X → X/W is flat, a symmetric idempotent already establishes a Morita equivalence bet-
ween HW (X) and OX/W , and thus forces the center of the spherical algebra to agree with that
of the DAHA itself. Of course, as we noted above, this map is essentially never flat (with the
notable exceptions of the action of W (An) on the sum zero subscheme and the action of W (Cn)
on En by signed permutations), but it turns out to be close enough to let us prove the analogous
result.
Proposition 6.15. Suppose γ is trivial on W and the root kernel of W is diagonalizable. Then
there is a natural isomorphism Z
(
H
W̃ ;γ
(X)
) ∼= Z
(
H
W̃ ,W ;γ
(X)
)
, and both algebras are strongly
flat.
Proof. Let U be the largest W̃q-invariant open subset of X such that every point of U lies
over a regular point of X/W . The morphism U → U/W is a morphism of regular schemes
with 0-dimensional fibers, so is flat, and thus in particular πW∗OU is locally free. It follows
that a covering of U by symmetric idempotents induces a Morita equivalence between HW (U)
and OU/W , and thus between the restrictions of H
W̃ ;γ
(X) and H
W̃ ,W ;γ
(X)
(
viewed as sheaf
algebras on X/W̃q
)
to U/W̃q. In particular, we find that the natural morphism between the
centers is an isomorphism on U/W̃q.
For any other W̃q-invariant open subset V , given any element D ∈ Γ
(
V ;Z(H
W̃ ,W ;γ
(X))
)
,
we may restrict it to V ∩ U and transport it through the isomorphism of centers to obtain an
element D′ ∈ Γ
(
V ∩ U ;Z(H
W̃ ;γ
(X))
)
. By normality of X/W , U contains every codimension 1
point of X, and thus Γ(V ∩ U ;H
W̃ ;γ
(X)) = Γ(V ;H
W̃ ;γ
(X)), so that D′ is actually holomorphic
on V as a section of the DAHA. Since an operator is in the center of the DAHA iff it is in the
center of the meromorphic twisted group algebra, we conclude that D′ ∈ Γ
(
V ;Z(H
W̃ ;γ
(X))
)
,
and thus the natural map from the center of the DAHA to the center of the spherical algebra is
indeed surjective. �
Corollary 6.16. If Λ acts trivially on X, γ is trivial on W , and the root kernel of W is
diagonalizable, then there is a natural isomorphism Z
(
H
W̃ ;γ
(X)
) ∼= HW̃ ,W ;γ
(X).
Elliptic Double Affine Hecke Algebras 85
Remark. More generally, if γ = γ′∂L with γ′ trivial on W , then we can identify Z
(
H
W̃ ;γ
(X)
)
with the twist of H
W̃ ,W ;γ′
(X) by L. In particular, if the root kernel is trivial, we can always do
this fppf locally on S.
Remark. One consequence is that there is a natural Poisson structure on X+/W̃q, since as
a spherical algebra, its relative coordinate ring is the commutative fiber of a family of non-
commutative algebras. Presumably this Poisson structure has a natural description in terms
of the geometry of X+. Note that OX+ is an elliptic analogue of the subalgebra of the usual
DAHA generated by the commutative subalgebras, so it would not be unreasonable to expect
it to have a flat deformation to non-torsion q, corresponding to a pulled back Poisson structure
on X+. This fails, however: the pullback of the Poisson structure on OW̃q
X+ is only meromorphic
on OX+ . Indeed, it agrees with the natural Poisson bracked on OX [Λ]γ , but the bracket of one
of the additional generators with a function with nonzero partial derivative on the reflection
hypersurface will fail to be holomorphic. Note that W̃q-invariance forces this to vanish, so the
restriction of this meromorphic Poisson structure to the W̃q-invariants is holomorphic.
Unfortunately, the trick with Ã1-subalgebras breaks down completely when the system
of parameters is nontrivial. As a result, we cannot give such a simple description of the center.
It turns out, however, that we can reduce to the case q = 0.
Return for the moment to the case ~T = 0. The sublattice Λq is isomorphic as a group with W
action to one of the root lattices of W , and thus the semidirect product W nΛq is itself an affine
Weyl group. (Note that when W̃ is of type B or C, this new affine Weyl group may be of the
other type.) The quotient Λ/Λq acts on both X and X+, and we observe that X+/(Λ/Λq) is
generically a torus bundle over Y = X/(Λ/Λq), and is more precisely an affine blowup of Y in
the corresponding bundles over the reflection hypersurfaces. We thus obtain the following.
Proposition 6.17. There is an induced twisting datum NX/Y (γ) on Y such that the center
of H
W̃ ;γ
(X) is canonically isomorphic to the center of HWnΛq ;NX/Y (γ)(Y ).
Proof. When γ is trivial, this holds with NX/Y (γ) trivial. More generally, if γ is represented
by a cocycle in Cartier divisors, we may obtain NX/Y (γ) by taking the images of the diffe-
rent Cartier divisors, and the resulting twisting datum is independent of the choice of cocycle
representation. �
We can then say the following in general, where NX/Y
~T is again obtained by taking the image
of each Tα under the induced isogeny of coroot curves.
Theorem 6.18. If the root kernel of W is diagonalizable, then the center of H
W̃ ;~T ;γ
(X) is
canonically isomorphic to the center of HWnΛq ;NX/Y ~T ;NX/Y γ
(Y ).
Proof. The claim respects twisting by line bundles, so we may assume that γ is trivial on W ,
and replace the algebra on Y with the spherical algebra HWnΛq ,W ;NX/Y ~T ;NX/Y γ
(Y ). (Both cen-
ters will then be isomorphic to the same spherical algebra.)
Imposing a system of parameters does not change the generic fiber, and thus we have
Z
(
H
W̃ ;~T ;γ
(X)
)
= H
W̃ ;~T ;γ
(X) ∩ Z
(
H
W̃ ;γ
(X)
)
.
When ~T is sufficiently general, we may apply Corollary 6.4 to identify the precise conditions for
an element of Z(H
W̃ ;γ
(X)) to be contained in the smaller Hecke algebra. (Note that determining
whether a negative root of W̃ becomes positive under a translation is quite straightforward.)
In particular, we find as in the discussion above regarding reflection hypersurfaces that the divisor
along which each coefficient must vanish is Λ/Λq-invariant.
(
To be precise, it is a nonnegative
86 E.M. Rains
linear combination of sums of the form
∑
x∈〈qα〉
xTα, where qα generates the group of translations
acting on the corresponding coroot curve.
)
Thus imposing the condition that the coefficient
is Λ/Λq-invariant has no additional effect on the vanishing condition, so that the algebras agree
under the given constraint on ~T .
Since the spherical algebra is strongly flat, any section for special ~T extends to a neighborhood
in parameter space. The corresponding operator in Z
(
H
W̃ ;γ
(X)
)
is thus contained inH
W̃ ;~T ;γ
(X)
for generic parameters, and thus for all parameters. It follows in particular that the image
of the spherical algebra saturates the Bruhat filtration, and is thus surjective as required. �
Remark. In the ~T = 0 case, we were able to use OX+ as an intermediate step to understanding
the center. Is there an analogous geometric description of OX+ ∩H
W̃ ;~T ;γ
(X)? For ~T transverse
to reflection hypersurfaces, we can again describe this intersection via vanishing conditions,
but it does not follow from the above discussion that the intersection is flat, and the obvious
comparison to the spherical module fails, as the latter is not W̃q-invariant. It is likely that
a description in the Ã1 case could be extended to general type.
The same argument gives the following.
Corollary 6.19. If the root kernel of W is diagonalizable and γ is trivial on W , then the center
of H
W̃ ;~T ;γ
(X) is canonically isomorphic to the center of H
W̃ ,W ;~T ;γ
(X).
Note that Z
(
H
W̃ ;~T ;γ
(X)
)
is always a domain (it is contained in the structure sheaf of the
integral scheme X+). We conjecture that it is also Noetherian, and moreover that H
W̃ ;~T ;γ
(X) is
finite over its center, just as in the ~T = 0 case. Assuming the technique of [1] (reducing to finite
fields, where q is always torsion) could be adapted to the case of sheaf algebras, this would be
enough to prove H
W̃ ;~T ;γ
(X) Noetherian even when q was not torsion.
The above description of the DAHA for q torsion can in principle be used to give explicit de-
generations of the residue conditions, though in practice it seems simpler to check the analogous
conditions on H
W̃q ;γ+(X+). For that, the following general reduction (which we implicitly used
above) may be useful. Recall that any reflection r ∈ R(W ) induces a corresponding subgroup
of type Ã1 of W̃ (generated by the reflections in rΛ), and that this gives rise to a corresponding
subalgebra which we denote by HÃ1(r);γ(X).
Proposition 6.20. An element
∑
w cww ∈ k(X)
[
W̃
]
γ
is a local section of H
W̃
(X) iff
its coefficients are holomorphic away from the reflection hypersurfaces and for any reflec-
tion r ∈ R(W ) and any w0 ∈ W , the operator
∑
w∈Ã1(r)w0
cww is a section of the localization
of HÃ1(r);γ(X)Zw0w0 to the corresponding union of reflection hypersurfaces.
Proof. An element is in the DAHA if it is holomorphic away from the reflection hypersur-
faces and is in the DAHA over the localization to the union of reflection hypersurfaces. Since
the reflection hypersurfaces of Ã1(r) for distinct r remain transverse to each other even after
specializing q, the corresponding conditions are independent. The conditions along the reflec-
tion hypersurfaces of Ã1(r) for special q are the limit from the case for general q, where they
agree with the residue conditions on HÃ1(r);γ(X). That algebra is itself a DAHA, and thus the
corresponding conditions remain flat for special q as well. �
One possible application of this reduction is to the construction of degenerations: a limit
of H
W̃ ;γ
(X) should be well-behaved as long as the limits of the various HÃ1(r);γ(X) are well-
behaved and there is no further coalescence of reflection hypersurfaces. (There are, however,
technical issues, in that most natural degenerations will break the normality of X, but one
may be able to finesse this by working over X/W instead, as the quotient will tend to remain
a weighted projective space.)
Elliptic Double Affine Hecke Algebras 87
One such limit of interest is the other natural q → 0 (or q torsion) limit: rather than take the
limit to a twisted group algebra in which the group does not act faithfully, one might instead
take a limit to an algebra of differential-reflection operators; i.e., take the limit in such a way as
to consider how the operators themselves actually act. This would presumably be the correct
way to interpret the algebra of holomorphy-preserving operators if one does not suppress the
poles for q torsion, but is not directly accessible via our techniques. In particular, the resulting
algebra would not be generated by the rank 1 subalgebras, and the corresponding spherical
algebra is almost certainly not a domain when q is a nontrivial torsion element.
7 The C∨Cn case
We now restrict our attention to the case that the affine Weyl group is of type C. This has
a natural action on the family En+1 given as follows:
s0(z1, . . . , zn, q/2) = (q − z1, z2, . . . , zn, q/2),
sn(z1, . . . , zn, q/2) = (z1, . . . , zn−1,−zn, q/2),
while for 1 ≤ i ≤ n − 1, si swaps zi and zi+1. Here we denote the last coordinate in En+1
by q/2, so that q is twice that coordinate. We use this notation since the corresponding family
of actions of C̃n on En (and thus the resulting Hecke algebras) depends only on q, but it will
be convenient (and more symmetric) to be able to divide q by 2. We find for this action that
the simple coroot morphisms are q/2− z1, z1− z2, . . . , zn−1− zn, zn, and thus that this is in fact
an action of coroot type, as required for our theory. We also have an action corresponding to
the diagram automorphism:
ω(z1, . . . , zn) = (q/2− zn, . . . , q/2− z1),
which clearly permutes the simple coroot morphisms as expected.
The root curves are all isomorphic to E , and the simple root morphisms are given by
(−1, 0, . . . , 0), (1,−1, 0, . . . , 0), (0, 1,−1, 0, . . . , 0), . . . , (0, . . . , 0, 1,−1, 0), (0, . . . , 0, 1, 0).
It follows that the root kernel for any finite parabolic subgroup is trivial, so that there will be
no difficulties with invariants and (local) idempotents.
In fact, we have the following.
Proposition 7.1. For any finite parabolic subgroup WI ⊂ C̃n, the quotient En/WI is smooth
over M1,1.
Proof. Any quotient En/WI is a product of symmetric powers of E and a quotient or two of
the form Em/Cm (using the diagram automorphism to identify parabolic subgroups involving s0
with the usual hyperoctahedral group). Symmetric powers of a curve are smooth, so there is no
difficulty. For the quotient by the full hyperoctahedral group, we first observe that the quotient
by the normal subgroup of order 2m is just the product of n copies of the quotient of E by [−1],
a.k.a. P1. We thus find that Em/Cm ∼=
(
P1
)m
/Sm ∼= Pm, giving smoothness as required. �
“Miracle flatness” immediately gives the following, which in particular ensures that the sphe-
rical algebras we consider will be locally free in a suitable sense (i.e., that their direct images
in either copy of En/W are locally free).
Corollary 7.2. For any parabolic subgroup WI ⊂ Cn, the quotient maps En → En/WI and
En/WI → En/Cn are flat.
88 E.M. Rains
Suppose for the moment that ~T0 = ω ~Tn. Then we may consider the extended double affine
Hecke algebra obtained by adjoining OXω to HC̃n;~T (En). This has a natural Z/2Z-grading, and
it will be useful to think of it as corresponding to a (sheaf) category with two objects rather
than a sheaf algebra. That is, we have objects 0 and 1 with endomorphisms given by the sheaf
algebra HC̃n;~T (En) and the remaining morphisms given in either direction by the sheaf bimodule
HC̃n;~T (En)ω. Doing this actually lets us remove the constraint on the parameters: we may
always obtain a sheaf category with two objects by taking
Hom(0, 0) = HC̃n;~T (En), Hom(0, 1) = HC̃n;ω ~T (En)ω,
Hom(1, 0) = HC̃n;~T (En)ω, Hom(1, 1) = HC̃n;ω ~T (En),
whereHC̃n;~T (En)ω denotes the bimodule obtained by twisting the regular bimodule of the DAHA
by the isomorphism induced by ω.
The Bruhat order on the extended affine Weyl group induces a Bruhat filtration on this
category (which agrees with the usual Bruhat filtration coming from viewing each Hom bimodule
as a regular module of some DAHA). Although of course this filtration is useful in itself, we will
also make great use of a filtration obtained from a much coarser order. The category is certainly
generated by the rank 1 subalgebras along with ω, but since ω permutes the rank 1 subalgebras,
we can actually omit the subalgebras corresponding to s0 from the generators. As a result,
we find that the category is generated by the two morphisms corresponding to ω along with
the Hecke algebras in each degree corresponding to the finite Weyl group. We may then define
a filtration on each Hom bimodule by the number of times ω was used; e.g., the degree ≤ d piece
of Hom(0, d mod 2) is the image of
HCn;~T (X)ωHCn;ω ~T (X)ωHCn;~T (X) · · ·
(with d copies of ω and d+ 1 finite Hecke algebras as tensor factors). We can express each finite
Hecke algebra factor as the image of a product of rank 1 algebras (corresponding to the longest
element of Cn), move all of the copies of ω to the end, and then use Corollary 5.20 to express this
in terms of a Bruhat interval in the appropriate Hecke algebra. Moreover, that Bruhat interval is
clearly a union of (W,W ) double cosets, so is determined by the corresponding set of dominant
weights; we find that the condition to be degree ≤ d is simply that the first coefficient of the
dominant weight is ≤ d/2.
This leads to a “graded” (or “compactified”) version of the extended DAHA: take the sheaf
category with objects Z generated by elements ω ∈ Hom(k, k + 1) and algebras HCn;~T (X) ⊂
Hom(2k, 2k), HCn;ω ~T (X) ⊂ Hom(2k+1, 2k+1). (We may think of this as a sort of Rees algebra
corresponding to the filtration, taking into account parity.)
Note that when asking whether two (small) sheaf categories are isomorphic, there are actually
two natural notions. The issue here is that for any automorphism of an object of a category,
there is a corresponding inner automorphism of the category as a whole (and more generally
for any assignment of an automorphism to each object). In our case, any unit on parameter
space gives such an inner automorphism, and since we are primarily interested in the individual
fibers, we should extend that to allow local units. This leads to a notion of twisting objects by
line bundles on the base; note that the resulting sheaf category will still be locally isomorphic
to the original sheaf category. In general, we will often only state that given sheaf categories
are isomorphic locally on the base; this is mainly to save the bookkeeping effort of determining
precisely which line bundles one needs to twist by to make the isomorphism global. In particular,
when specifying line bundles and equivariant gerbes, the z-independent terms of the polarization
and weight will be largely irrelevant, so we can make the simplest consistent choice without
having to worry too much about which choice would make later polarizations simpler.
Elliptic Double Affine Hecke Algebras 89
Of course, we have yet to incorporate a twisting datum. We first note that the underlying
equivariant gerbe induces a cocycle valued in pairs of polarizations and weights. We can embed
the C̃n-module of polarizations in the degree 2 subspace of (1/q)Q[~z, q, ~π] (where ~π corresponds to
additional factors of E which we include to allow some room for further parameters), and similarly
for the C̃n-module of weights. Since q is C̃n-invariant, both rational C̃n-modules are isomorphic
to the corresponding module Q[~z, ~π] obtained by specializing q = 1. We can compute H1 of this
module by restriction to the (finite index) translation subgroup, where the filtration by degree
makes it easy to verify that H1 is trivial. It follows that the given cocycle is the coboundary
of a pair (p3(~z, q, ~π)/q, p1(~z, q, ~π)/q), where p3 and p1 are homogeneous polynomials of degree 3
and 1 respectively. Of course, p3 and p1 are only determined modulo 0-cocycles, but those are
again easily seen to be just the polynomials independent of ~z.
In our case, since our primary interest is in the spherical algebras, we want the twisting
datum to be trivial on Cn. The same must in particular hold for the equivariant gerbe, so
that p3 and p1 must be Cn-invariant polynomials. (More precisely, they must be Cn-invariant
modulo 0-cocycles, but we can then average over Cn without changing the coboundary.) Since
we are allowed to ignore terms independent of ~z, we see that we may as well take p1 = 0 and p3 =
λ(q/2,~π)
q
∑
i
z2
i
2 for some linear polynomial λ with rational coefficients. Imposing the condition
that the coboundary consist of actual polarizations then forces λ to have integer coefficients.
That is, the value of the coboundary at s0 is p3(~z)− s0p3(~z) = λ(q/2, ~π)z1 − λ(q/2, ~π) q2 , which
is integral iff λ is integral since q/2 is a variable.
To extend this to a twisting datum, it suffices to choose a meromorphic section of the restric-
tion to 〈s0〉 and represent the corresponding Cartier divisor as D0 − s0D0 with D0 transverse
to [Xs0 ]. Since D0 is only determined modulo s0-invariant divisors and linear equivalence, it is
equivalent to specify the polarization
k
z1(z1 − q)
2
+
λ(q/2, ~π)z1
2
,
with k ∈ {0, 1}. (Again, the constant term is irrelevant for our purposes.) Note that this imposes
a stronger integrality constraint on λ, which must now have even coefficients.
Of course, we saw above when discussing the elliptic Gamma “function” that not every
suitably integral cocycle has meromorphic sections, even for the translation subgroup. Luckily,
in our case, we can easily write down explicit products of elliptic Gamma functions that do the
trick. To be precise, consider the product∏
1≤i≤n
Γq(a± zi),
where a is linear in q/2 and π. Here and below, we have used the shorthand notation that
multiple arguments to Γq or ϑ represent a product and the appearance of ± in the argument
means that both signs should be used; thus Γq(a± zi) = Γq(a+ zi)Γq(a− zi).
This is Cn-invariant, and has polarization (ignoring the z-independent term)
(2a− q)
∑
i
z2
i
2
.
Moreover, we find
s0
∏
1≤i≤n Γq(a± zi)∏
1≤i≤n Γq(a± zi)
=
ϑ(a− z1)
ϑ(a− q + z1)
,
and thus this corresponds to the twisting datum with D0 = [z1 = a]. This gives the general
degree 1 case, and we may obtain the general degree 0 case by taking a ratio of two such products.
90 E.M. Rains
Such a choice of product induces an embedding of the twisted Hecke algebra in the algebra
of meromorphic difference-reflection operators; equivalently, meromorphic sections of the twisted
Hecke algebra act on formal functions of the form fγ, where γ is the product of Γq symbols.
This lets us extend the holomorphy-preserving property to the twisted case: if f is locally
holomorphic away from the poles of the sections of the cocycle corresponding to γ, then so its
image. This leads to issues where γ has poles, but we have enough choice in how we represent
things that the corresponding invariant localizations give a finite covering.
(
I.e., in the degree 0
case, we may take γ =
∏
1≤i≤n Γq(v + a ± zi)/Γq(v ± zi) with v varying; in degree 1, we take
γ =
∏
1≤i≤n Γq(v ± zi, w ± zi)/Γq(v + w − a± zi) with v, w varying.
)
To include the diagram automorphism, we note that because we extended to a category
above, we may choose a different product of Γq symbols for each object, and need only have
a line bundle Li in each degree such that sections of Liω take k(X)γi to k(X)γi+1. This is easy
enough, since
ω
∏
1≤i≤n Γq(a± zi)∏
1≤i≤n Γq(a− q/2± zi)
=
∏
1≤i≤n
ϑ(a− q/2− zi),
ω
∏
1≤i≤n Γq(a± zi)∏
1≤i≤n Γq(a+ q/2± zi)
=
∏
1≤i≤n
1
ϑ(a− q/2 + zi)
.
In the degree 0 case, this ends up not changing the twisting datum, but in degree 1, the
polarization of Li ends up alternating between positive and negative. This would increase the
number of cases we would need to consider, so it will be convenient to enlarge the category even
further by replacing each object by a sequence of objects, one for each (isomorphism class of)
line bundle on Pn. The Hom bimodule between two objects in the enlarged category is then
just the twist of the original Hom bimodule by the pair of line bundles (inverting the one on the
domain side). The benefit of this is that we can move between the two degree 1 cases by twisting
by OPn(±1), and thus in the enlarged category, there is only one case.
It turns out that even the above category is not quite general enough to include everything
we want to do for the spherical algebras. As a result, we will first focus our attention on the case
in which the endpoints do not have any parameters assigned. We also specialize to the usual
Macdonald-ish case, in which Ti for 1 ≤ i ≤ n−1 is the divisor t+zi−zi+1 = 0. (The definitions
work more generally; in particular, we note that everything works mutatis mutandum for the
case Ti = 0 for 1 ≤ i ≤ n − 1, though oddly enough this “t-free” case turns out to be harder
to understand in a number of respects.) It also turns out that most of the symmetries of the
algebra do not preserve the untwisted case, but do preserve a particular twist; this leads us
to make a somewhat odd-appearing choice in parametrizing twists. The specific basis we use
for the Z2 of objects is inspired by [26] (where it in turn came from the geometry of rational
surfaces); this also informs the choice of parameters (in [26], there was a parameter η which
we have arranged to be 0).
Definition 7.3. The even elliptic DAHA H(n)
η′;q,t (of type C) is the smallest sheaf category
on En+3 with objects Z〈s, f〉 such that
H(n)
η′;q,t(ds+ d′1f, ds+ d′2f) = OPn(d′2)⊗HCn;~Tt
(
En+3
)
⊗OPn(−d′1)
and
H(n)
η′;q,t(ds+ d′1f, (d+ 1)s+ d′2f) ⊃ Lω,
where L is the line bundle with polarization
−(d′1 − d′2 + 1)
∑
i
z2
i − ((n− 1)t+ η′ + (d− d′1 + 1)q)
∑
i
zi + (d− d′1)q2/4.
Elliptic Double Affine Hecke Algebras 91
The odd elliptic DAHA H′(n)
x0;q,t is defined similarly, except that L has polarization
−(2d′1 − 2d′2 + 3)
∑
i
z2
i
2
− ((n− 1)t+ x0 + (3d− 2d′1 + 2)q/2)
∑
i
zi + (3d− 2d′1)q2/8.
Remark. In this case, we were able to choose the z-independent part of the polarization to make
everything globally consistent, where OPn(1) is chosen so as to pull back to the line bundle with
polarization
∑
i z
2
i . We can recover the usual (compactified) elliptic DAHA by restricting the
algebra H(n)
−(n−1)t−q;q,t to the subset Z(s+ f) of objects; if we restrict to even multiples and then
invert the sections 1 ∈ Hom(d(s + f), (d + 2)(s + f)), we recover the uncompactified elliptic
DAHA. We should note that while H(n)
η′;q,t(0, 2s+ 2f) always contains 1, this is only true locally
on the base for H′(n)
x0;q,t(0, 2s + 3f), and it is not possible to fix this without also changing the
particular representative of OPn(1).
We similarly let S(n)
η′;q,t and S ′(n)
x0;q,t denote the corresponding spherical categories (i.e., replacing
each Hom bimodule by the appropriate subquotient). Each Hom bimodule in one of these
categories is a sheaf bimodule on the quotient Pn = En/Cn, every local section of which is
a meromorphic difference operator on En.
Proposition 7.4. The subsheaf corresponding to any Bruhat order ideal in either S(n)
η′;q,t or S ′(n)
x0;q,t
is a coherent sheaf bimodule on Pn ×E3 Pn, and the direct image in either Pn is locally free.
Proof. This reduces to the corresponding statement for the subquotients in the Bruhat filtra-
tion, each of which comes from a line bundle on the quotient by a parabolic subgroup of Cn,
and is therefore flat on the quotient Pn. �
Note that just as for more general Coxeter groups, we can describe the t-independent condi-
tions on sections of these sheaf categories in terms of residue conditions (at least generically).
This does not quite follow from that case, as we are working in an extended affine Weyl group
rather than an affine Weyl group, and of course the twisting makes things trickier.
For H(n)
−q−(n−1)t;q;t(0, ds + d′f), H′(n)
−q−(n−1)t;q,t(0, ds + d′f), the local sections are just local
sections of the untwisted Hecke algebra up to twisting on the left by some OPn(l), and thus
satisfy the usual residue conditions: the coefficients are all meromorphic sections of the same
bundleOPn(1), and for any two elements of the extended affine Weyl group related by a reflection,
the sum of the corresponding coefficients must be holomorphic along the reflection hypersurface.
(Moreover, the polar divisors of the coefficients must be sums of reflection hypersurfaces, with
a given reflection hypersurface appearing only if it appears in a residue condition). For the
spherical algebra, the condition is analogous, giving a residue condition for any pair of distinct
translations which are conjugates by a reflection.
In our case, the twisting is simple enough that we can express the residue conditions explicitly
even in the presence of twisting. For the even case, we find that for sufficiently general u, v,( ∏
1≤i≤n
ϑ(v ± zi)d
′
Γq(−dq/2− u± zi,−dq/2− (n− 1)t− η′ + u± zi)
)−1
×H(n)
η′;q,t(0, ds+ d′f)
∏
1≤i≤n
Γq(−u± zi,−(n− 1)t− η′ + u± zi)
consists of operators with elliptic coefficients and preserving holomorphy (away from divisors
depending on u and v). (This is essentially just performing an elementary transformation as
considered below.) Thus the residue conditions onH(n)
η′;q,t(0, ds+d
′f) may be obtained by gauging
92 E.M. Rains
the untwisted residue conditions. We may then use Propositions 2.11 and 2.12 to simplify
the result. We give the conditions for the spherical category for simplicity; for the DAHA,
the correction is the same except for those reflections preserving the corresponding coset, when
the correction factor is necessarily trivial.
We find that sections of S(n)
η′;q,t(0, ds+ d′f) satisfy the following t-independent residue condi-
tions (in addition to Cn-invariance and the corresponding bounds on poles). Write such a local
section as
∑
~k
c~k(~z)
∏
i T
ki
i , where each ki is congruent to d/2 modulo Z. First, for m 6= k1 + k2,
cm−k2,m−k1,k3,...,kn(~z) +
[
ϑ(u, q+ (n− 1)t+ η′+u+ z1 + z2 +mq)
ϑ(u+ z1 + z2 +mq, q+ (n− 1)t+ η′+u)
]2(m−k1−k2)
ck1,k2,k3,...,kn(~z)
is holomorphic along z1 + z2 + mq = 0, where u is arbitrary subject to u 6= 0 and u 6= −q −
(n− 1)t− η′ and otherwise has no effect on the condition, and similarly when m 6= 2k1,
cm−k1,k2,...,kn(~z) +
[
ϑ(u, q + (n− 1)t+ η′ + u+ 2z1 +mq)
ϑ(u+ 2z1 +mq, q + (n− 1)t+ η′ + u)
]m−2k1
ck1,k2,...,kn(~z)
is holomorphic along 2z1 + mq = 0. (The conditions along other reflection hyperplanes follow
by Cn-invariance.) Moreover, for generic parameters, any operator satisfying these conditions
and the appropriate t-dependent vanishing conditions will be a section of the given Hom sheaf.
(To be precise, we must assume that q is not torsion (or we are in a faithful Bruhat interval) and
that t is not a multiple of q; of course, by flatness, to check that a family of operators is a section
of the corresponding family of categories S, it suffices to verify that this holds generically.)
Similarly, for S ′(n)
x0;q,t(0, ds+ d′f), the correction factors in the residue conditions are[
ϑ(u, q + (n− 1)t+ x0 + u+ z1 + z2 +mq)
ϑ(u+ z1 + z2 +mq, q + (n− 1)t+ x0 + u)
]2(m−k1−k2)
and [
q(z1 +mq/2)
ϑ(u, q + (n− 1)t+ x0 + u+ 2z1 +mq)
ϑ(u+ 2z1 +mq, q + (n− 1)t+ x0 + u)
]m−2k1
,
where we recall that in characteristic not 2, q ∈ µ2(E[2]) is the function taking 0 to 1 and
nontrivial 2-torsion points to −1. (Note that without the appearance of q in the residue con-
ditions, the subcategory S ′ with objects 2Zs + Zf would be a simple reparametrization of the
corresponding subcategory of S.)
By comparison with the univariate case (which we will discuss in more detail shortly), we are
led to define generalizations (“blowups”) of these algebras with even more parameters. Recall
that local sections of the spherical algebras are difference operators. We let Ti denote the
operator that pulls back through zi 7→ zi + q; note that in terms of our convention for how
group elements act, Ti is the same as the action of the translation zi 7→ zi − q. We extend this
to half-integer powers of Ti by using the chosen q/2. Every local section is then a left linear
combination of monomials T
~k :=
∏
i T
ki
i in which all ki are half-integers in the same coset of Z
(determined by the coefficient of s in the degree of the operator in the category).
Definition 7.5. The sheaf category S(n)
η′,x1,...,xm;q,t is the sheaf category on Pn/Em+3 with objects
Z〈s, f, e1, . . . , em〉 defined by taking S(n)
η′,x1,...,xm;q,t(d1s+d
′
1f−r11e1−· · ·−r1mem, d2s+d
′
2f−r21e1−
· · · − r2mem) to be the subsheaf of S(n)
η′;q,t(d1s+ d′1f, d2s+ d′2f) consisting (locally) of operators
D such that the left coefficient of
∏
1≤i≤n T
ki
i vanishes on the divisors zi = xj − (2l− d2 + 1)q/2
for 1 ≤ i ≤ n, 1 ≤ j ≤ m, ki+(d2−d1)/2+r1j ≤ l < r2j and the divisors zi = −xj+(2l−d2+1)q/2
for 1 ≤ i ≤ n, 1 ≤ j ≤ m, −ki + (d2 − d1)/2 + r1j ≤ l < r2j . Similarly, the sheaf category
S ′(n)
x0,x1,...,xm;q,t is the subsheaf category of S ′(n)
x0;q,t satisfying the same vanishing conditions.
Elliptic Double Affine Hecke Algebras 93
Remark. Note that by symmetry, it would have sufficed to impose the first set of vanishing
conditions.
Note that in this definition, we need to impose the vanishing conditions on the family as
a whole; on individual fibers, the condition along individual divisors may be too strong (when the
divisors are reflection hypersurfaces) or too weak (when the divisors are not distinct). As a result,
it is a nontrivial question whether the family is flat, and even when it is flat, it is conceivable
that the individual fibers might fail to inject in the category of meromorphic operators (which
could in turn allow the fibers to acquire zero divisors).
Luckily, since we defined these using the spherical algebra of an elliptic DAHA, there is
an obvious approach to studying these categories: construct them as spherical subquotients of
a suitable subcategory of H(n)
η′;q,t or H′(n)
x0;q,t. There are actually multiple choices one might make
here, as one can view a divisor zi = xj − kq/2 as a pullback from the coroot curve associated
to either endpoint. Although it might seem natural to choose the endpoint that matches the
parity of k, it will be easier for present purposes to consistently use s0. Other choices may lead
to more natural Hecke algebras, however; for instance the classical double affine Hecke algebra
of type C∨Cn corresponds to assigning two parameters to sn and two parameters to s0.
Since we are keeping the parameters away from the finite Weyl group, the vanishing conditions
should be appropriately Cn-equivariant, and thus the vanishing condition for the left coefficient
of w should depend only on wCn and be equivariant on the left. Each coset of wCn has a unique
representative
∏
i T
ki
i , and we readily verify that the vanishing conditions we imposed above
transform well under Cn. We may thus define H(n)
η′,x1,...,xm;q,t by imposing the resulting vanishing
conditions, and similarly for H′(n)
x0,x1,...,xm;q,t.
We note that in addition to the functoriality on fibers implied by the fact that this is defined
over the moduli stack, we also have functoriality with respect to translation by 2-torsion.
Proposition 7.6. If τ is an fppf-local section of E [2], then there are isomorphisms
H(n)
η′,x1,...,xm;q,t
∼= H(n)
η′,x1+τ,...,xm+τ ;q,t,
H(n)
x0,x1,...,xm;q,t
∼= H(n)
x0+τ,x1+τ,...,xm+τ ;q,t.
Proof. Conjugating by the involution (z1, . . . , zn) 7→ (z1+τ, . . . , zn+τ) induces such an isomor-
phism on the generators for m = 0 and acts as described on the additional vanishing conditions
for m > 0. �
Remark 7.7. In fact, we could have defined these categories over the moduli stack of hyperel-
liptic curves of genus 1, at the cost of making the action of Cn slightly more complicated (with sn
acting by zn 7→ s(zn), where s is the hyperelliptic involution), in which case these isomorphisms
follow by functoriality. This would have made a number of later formulas more complicated,
as well as making it more difficult to discuss line bundles and Γq symbols.
Remark 7.8. The above isomorphism involves translation by τ in every degree. If one instead
only translates in the odd degrees, none of the parameters visible in the notation change, but q/2
is replaced by q/2 + τ . In particular, the algebra is indeed independent of the choice of q/2.
We have the following useful “elementary transformation” symmetry. Here and below, we sim-
plify things by observing that our sheaf categories satisfy a natural translation symmetry in
which translation in the group of objects corresponds to translations of the parameters by mul-
tiples of q/2, and thus it suffices to consider Hom bimodules starting at the 0 object.
Proposition 7.9. There are (locally on the base) natural isomorphisms
H′(n)
x0,x1,x2,...,xm;q,t(0, ds+ d′f − r1e1 − · · · − rmem)
94 E.M. Rains
∼=
∏
1≤i≤n
Γq((r1 + (1− d)/2)q − x1 ± zi)
×H(n)
x0−x1,−x1,x2,...,xm;q,t(0, ds+ (d′ − r1)f − (d− r1)e1 − r2e2 − · · · − rmem)
×
∏
1≤i≤n
Γq(q/2− x1 ± zi)−1
and
H(n)
η′,x1,x2,...,xm;q,t(0, ds+ d′f − r1e1 − · · · − rmem)
∼=
∏
1≤i≤n
Γq((r1 + (1− d)/2)q − x1 ± zi)
×H′(n)
η′−x1,−x1,x2,...,xm;q,t(0, ds+ (d+ d′ − r1)f − (d− r1)e1 − r2e2 − · · · − rmem)
×
∏
1≤i≤n
Γq(q/2− x1 ± zi)−1.
Proof. It is easy to see that the corresponding categories of meromorphic operators are iso-
morphic (locally on the base), and the pseudo-conjugation by Γq symbols respects the twisting
data, so the image of the right-hand side in the meromorphic category corresponding to the
left-hand side satisfies the same conditions on the reflection hypersurfaces. The conditions along
the divisors corresponding to xi for 2 ≤ i ≤ m are clearly the same, and it is straightforward to
check that the same holds for i = 1. �
Remark. Since the definition is also clearly invariant under permutations of x1, . . . , xm, we may
apply this symmetry in any xi, and then by composition in any subset of the xi. If the subset
has odd size, then the isomorphism switches between H(n) and H′(n), while even subsets induce
isomorphisms between extended DAHAs of the same parity. In particular, for each of the two
families, there is an action of W (Dm) on the parameter space that extends to an action on the
family of sheaf categories.
If we can show that these sheaf categories have well-behaved Bruhat filtrations (i.e., in which
the subquotients are obtained from the m = 0 case by imposing the vanishing conditions),
then the same will immediately hold for their spherical subquotients. Unfortunately, we cannot
simply copy the arguments we used in the usual elliptic Hecke algebra case; although the upper
bound on the Bruhat subquotients works the same way, the category structure means there is
no longer a canonical way to associate a multiplication map to a reduced word.
There are some special cases which are easy, however. As before, we may restrict our attention
to Hom bimodules starting from the 0 object.
The following is a trivial consequence of the definition.
Proposition 7.10. If rm ≤ 0, then
H(n)
η′,x1,...,xm;q,t(0, ds+ d′f − r1e1 − · · · − rmem)
= H(n)
η′,x1,...,xm−1;q,t(0, ds+ d′f − r1e1 − · · · − rm−1em−1),
H′(n)
x0,x1,...,xm;q,t(0, ds+ d′f − r1e1 − · · · − rmem)
= H′(n)
x0,x1,...,xm−1;q,t(0, ds+ d′f − r1e1 − · · · − rm−1em).
Applying the elementary transformation symmetry in xm means that the case rm ≥ d is also
straightforward to deal with. There is one more case which is nice.
Elliptic Double Affine Hecke Algebras 95
Proposition 7.11. There is a system of parameters ~T and a twisting datum γ such that each
Hom bimodule
H(n)
η′,x1,...,xm;q,t(0, d(2s+ 2f − e1 − · · · − em))
is equal to the corresponding Bruhat interval in HC̃n;~T ;γ
(
En+m+3
)
, and similarly for
H′(n)
x0,x1,...,xm;q,t(0, d(2s+ 3f − e1 − · · · − em)).
Proof. We can rephrase the vanishing conditions for generic parameters as stating that∏
1≤i≤n
1≤j≤m
Γq(q/2− xj ± zi)D
∏
1≤i≤n
1≤j≤m
Γq(q/2− xj ± zi)−1
has holomorphic coefficients. The cocycle in Cartier divisors associated to this product of Γq
symbols is precisely the right form to come from a system of parameters (associated to the orbit
of s0). �
It turns out that it suffices to understand these cases.
Theorem 7.12. For any vector v = ds + d′f − r1e1 − · · · − rmem, the bounds on the Bruhat
subquotients of H(n)
η′,x1,...,xm;q,t(0, v) and H′(n)
x0,x1,...,xm;q,t(0, v) coming from the vanishing conditions
are saturated. In particular, both sheaf categories are locally free, and the map from any Hom
bimodule to the sheaf bimodule of meromorphic operators is injective on fibers.
Proof. We first use the elementary transformation symmetry to replace all the cases with ri ≥ d
with cases with ri ≤ 0 (possibly changing the parity), and thus with ri = 0. In particular,
we observe that the cases d ∈ {0, 1} of the theorem reduce to the subcase r1 = · · · = rm = 0,
and thus to the original elliptic DAHA, where we certainly have saturation. For d > 1, if some
ri = 0, we can simply omit that parameter and thus reduce to a case with smaller m. We thus
find that it suffices to prove saturation when 0 < r1, . . . , rm < d.
We consider the even case H(n), with the odd case H′(n) being entirely analogous. Let
Cm := 2s + 2f − e1 − · · · − em, and suppose by induction that we have saturation for v − Cm.
It will then suffice to show that the image of the multiplication map
H(n)
η′,x1,...,xm;q,t(v − Cm, v)⊗H(n)
η′,x1,...,xm;q,t(0, v − Cm)→ H(n)
η′,x1,...,xm;q,t(0, v)
saturates the bound. Since the first factor is a Bruhat interval in an honest elliptic DAHA, it
in particular has a global section 1, giving an inclusion
H(n)
η′,x1,...,xm;q,t(0, v − Cm) ⊂ H(n)
η′,x1,...,xm;q,t(0, v),
identifying the former with the Bruhat interval of the latter corresponding to the dominant
weight (d/2 − 1, d/2 − 1, . . . , d/2 − 1). The bounds on the subquotients are clearly the same,
and thus we have saturation for any w in this interval.
For the rest of the module, we note that both sides are bimodules over the finite Hecke
algebra, and since the xi parameters have no effect on this algebra, we may reduce to the
case of a Bruhat interval [≤ w] with w a minimal representative of Cn \ C̃n. Let λ(w) be the
corresponding dominant weight. We have already shown that the leading coefficient map is
saturated when λ(w)1 ≤ d/2−1, so without loss of generality may assume λ(w)1 = d/2. It then
follows from the structure of minimal coset representatives that not only is s0w < w, but λ(s0w)
is obtained from λ(w) by reducing some coefficient from d/2 to d/2−1. This changes the bound
on the leading coefficient bundle by precisely the leading coefficient divisor of s0 in the relevant
DAHA, and thus gives saturation as required. �
96 E.M. Rains
Passing to the spherical subquotient gives us the following.
Corollary 7.13. For any vector v = ds + d′f − r1e1 − · · · − rmem, the bounds on the Bruhat
subquotients of S(n)
η′,x1,...,xm;q,t(0, v) and S ′(n)
x0,x1,...,xm;q,t(0, v) coming from the vanishing conditions
are saturated. In particular, both sheaf categories are locally free, and the map from any Hom
bimodule to the sheaf bimodule of meromorphic difference operators is injective on fibers.
As in the general DAHA situation, we would like to understand the centers of the extended
DAHAs and their spherical algebras. There is a technical issue, however, in that the notion
of center is not quite well-defined for general categories. More precisely, one might be inclined
to define the center of a category to be the endomorphism algebra of the identity functor, but
this is too small in our cases. We could fix the “too small” problem by instead taking the center
to be the category with objects the automorphisms of the original category and morphisms given
by natural transformations, but this is too large to be useful. Luckily, there is a happy medium:
we take a suitable action of Zm+2 on the category such that the induced action on objects is
just the usual action by translation.
This action is mostly straightforward: since every Hom space of degree 2s + 2f (or 2s + 3f
in the odd case, at least locally on the base) contains the operator 1, and similarly for any Hom
space of degree ei, any element of the sublattice 〈2s + 2f, e1, . . . , em〉 induces an isomorphism
between corresponding categories. Translation by f is slightly more subtle, but corresponds to
twisting by the equivariant bundle π∗OPn(1), so again induces an isomorphism of categories.
The result is index 2 in Zm+2, which is good enough for many purposes, but since we would
like to be able to identify the center of the extended DAHA with one of our spherical algebras,
we need to fill out the lattice. This is not too difficult in the extended DAHA, which contains
a unique “element” of the form wLwω with w ∈ Cn and wω a translation. This is invertible,
and again induces appropriate isomorphisms of categories. (Note, however, that its square is
not actually compatible with the lattice of automorphisms already described.)
Now, suppose q is torsion, of order l. If we take the lth powers of the above isomorphisms,
then all of the categories that arise can be identified with the original category, and we thus
obtain the desired family of automorphisms (which became consistent after taking lth powers,
as the error was translation by q). Moreover, the automorphism corresponding to ls of the
extended DAHA is W -invariant, and thus extends to an automorphism of the spherical algebra.
We define the “center” of the extended DAHA or its spherical algebra to be the category with
objects lZm+2 arising from the above construction.
The following is straightforward.
Lemma 7.14. If q = 0, then the categories S(n) and S ′(n) are naturally isomorphic to their
centers.
More generally, let πq : E → E′ be the l-isogeny with kernel generated by q.
Theorem 7.15. The centers of H(n)
η′,x1,...,xm;q,t and S(n)
η′,x1,...,xm;q,t are naturally isomorphic to the
pullback of S(n)
πq(η′),πq(x1),...,πq(xm);0,πq(t)
from E′.
Proof. The argument of Theorem 6.18 together with the flatness results for the spherical alge-
bra allows us to reduce to the parameter-free case (defined in the obvious way). We may as
well localize the central section 1 of degree 2s + 2f , as we can then recover the center of the
category from the induced filtration on the center of the localization. Any invariant section
of π∗OPn(l) is clearly central, and we can locally choose an invertible such section to deal with
the automorphisms of degree lf . We obtain a description of the resulting sheaf of categories
with objects Z/2Z as a sheaf algebra over the corresponding OX+ . The even part of the center
is precisely the center of the corresponding DAHA, while the odd part of the center may be
Elliptic Double Affine Hecke Algebras 97
identified with a submodule of the form Z(HC̃n,γ(X))Zwlwl, where wl is given by wlω
l = (wω)l,
where wω is a translation with w ∈ W . Thus the center satisfies strong flatness and saturates
the appropriate Bruhat filtration, so agrees with the spherical algebra. �
To understand the significance of our construction, we will need to understand some special
cases. The case t = 0 is of particular interest, due to the following. Note that the tensor
and symmetric power constructions on modules extend to sheaf bimodules, and thus carry over
to analogous constructions for algebras and categories. (Of course, in the latter cases, one
should take the symmetric subobject, not the quotient object.) The following is an immediate
consequence of the fact that the corresponding An−1 Hecke algebra is just the usual twisted
group algebra.
Proposition 7.16. One has the following isomorphisms (locally on the base):
S(n)
η′,x1,...,xm;q,0
∼= Symn
(
S(1)
η′,x1,...,xm;q,0
)
,
S ′(n)
x0,x1,...,xm;q,0
∼= Symn
(
S ′(1)
x0,x1,...,xm;q,0
)
.
Remark. Note that the tensor product of n univariate difference operators is described as
follows. The ith operator acts on k(En) by pulling back the coefficients from the ith factor and
only translating the ith coordinate. These actions commute, and thus we may compose them to
obtain the tensor product difference operator on k(En). The tensor product of the algebras is
then the image of the tensor product of sheaves under this operation, and the symmetric power
consists of those operators in the tensor product that commute with Sn.
There is a similar description for t = q coming from the following symmetry (essentially
Corollary 5.32, combined with the fact that the conditions associated to x1, . . . , xm are unaf-
fected).
Proposition 7.17. One has the following isomorphisms (locally on the base):
S(n)
η′,x1,...,xm;q,q−t
∼=
∏
1≤i<j≤n
Γp,q(t± zi ± zj)S(n)
η′,x1,...,xm;q,t
∏
1≤i<j≤n
Γp,q(t± zi ± zj)−1,
S(n)
x0,x1,...,xm;q,q−t
∼=
∏
1≤i<j≤n
Γp,q(t± zi ± zj)S(n)
x0,x1,...,xm;q,t
∏
1≤i<j≤n
Γp,q(t± zi ± zj)−1.
We also note the following version of the adjoint symmetry. It will be convenient to express
the adjoint in terms of a formal density; in particular, dT simply represents a formal C̃n-invariant
measure.
Proposition 7.18. The adjoint with respect to the formal density
∏
1≤i<j≤n
Γq(t± zi ± zj)
Γq(±zi ± zj)
∏
1≤i≤n
1
Γq(±2zi)
dT
induces (locally on the base) contravariant isomorphisms
S(n)
η′,x1,...,xm;q,t
∼= S(n)
−η′,−x1,...,−xm;q,t,
S ′(n)
x0,x1,...,xm;q,t
∼= S ′(n)
−x0,−x1,...,−xm;q,t
acting on objects as v 7→ −v.
98 E.M. Rains
Remark. We will refer to this formal adjoint as the “Selberg” adjoint, as the formal density
consists of the interaction terms in the elliptic Selberg integral. More generally, composing the
Selberg adjoint with a sequence of elementary transformations gives an adjunction involving
densities of the form
∏
1≤i≤n
∏
1≤j≤k Γq(uj ± zi)
Γq(±2zi)
∏
1≤i<j≤n
Γq(t± zi ± zj)
Γq(±zi ± zj)
dT,
where the uj depend on the xj and the specific domain and codomain objects. Here we should
think of the operators as mapping between two different inner product spaces, so that the two
formal integrals are against (slightly) different densities.
We were somewhat vague in our descriptions of the subquotients of the Bruhat filtration
above, as the specific divisors that are forced into a given coefficient are somewhat complicated
to describe in general. For the most part, though, the important information about the sub-
quotients is not how they are built up out of divisors, but simply which line bundle one ends
up with in the end. (Recall that the subquotients are obtained from line bundles which are
invariant under some parabolic subgroup WI by descending to X/WI then taking the direct
image to X/Cn.)
Propositions 7.16 and 7.17 make this information relatively straightforward to determine. Any
filtered isomorphism preserves the Bruhat subquotients, and thus the associated polarizations
must be invariant under t 7→ q−t (modulo line bundles on the base, that is). Modulo line bundles
on the base, the t-dependent contribution to the polarization is linear, and thus the t 7→ q − t
symmetry forces it to be trivial. In other words, the subquotients are (up to isomorphism local on
the base) independent of t and thus by the first proposition are determined by the subquotients
for n = 1. More precisely, for t = 0, the line bundle on En associated to a given dominant weight
is the outer tensor product of the univariate subquotients associated to the parts of the weight;
one then descends to the quotient by the stabilizer in Cn of the weight.
Helpfully, it turns out that the univariate case has already been studied. Let ΓS(n), ΓS ′(n)
denote the associated “global section” categories; more precisely, these are sheaves of categories
on Em+3 in which each Hom sheaf is the direct image of the corresponding Hom bimodule.
Since we included twisting by OPn(1) in the definition of the category, one can recover S(n)
and S ′(n) from their global section categories: for each Hom bimodule M in the sheaf category,
the corresponding graded module (relative to the Segre embedding of Pn×Pn) can be extracted
from the global section category.
In [26], two families of categories Sη,η′,x1,...,xm;q,p and S ′η,x0,x1,...,xm;q,p were constructed on an
analytic curve C∗/〈p〉. These have the same group of objects as our categories, and an interpre-
tation of the local sections of the Hom sheaves as difference operators. Moreover, the categories
for general m are cut out from the categories for m = 0 by suitable vanishing conditions, while
the categories for m = 0 are described via explicit generators given in terms of theta functions.
Switching from multiplicative to additive notation and replacing θ by ϑ then extends this to
arbitrary curves. (In fact, [26] gave such an extension by specifying an explicit gauging by
products of Gamma functions that makes everything elliptic, and observing that the elliptic
functions extend. But of course one could do the same gauging in terms of ϑ and Γq symbols, so
the resulting categories are the same.)
Although those operators are not quite C1-symmetric in our sense, they are close: indeed,
each operator formally takes functions invariant under z 7→ (1−d1)q+η−z to functions invariant
under z 7→ (1 − d2)q + η − z. This, of course, is easy enough to fix: if we base change to have
an element η/2 (and recall that we already have an element q/2), then we can compose on both
sides by a suitable translation to make the operator honestly C1-symmetric.
Elliptic Double Affine Hecke Algebras 99
Proposition 7.19. Locally on the base, the global section categories are isomorphic to the
C1-symmetric versions of the categories S, S ′ constructed in [26]. More precisely, if v, w are
arbitrary elements of the object group Z〈s, f, e1, . . . , em〉, then
ΓS(1)
η′,x1,...,xm;q,t(v, w) ∼= S2c,2c+η′,c+x1,...,c+xm;q;E(v, w),
ΓS ′(1)
x0,x1,...,xm;q,t(v, w) ∼= S ′2c,c+x0,c+x1,...,c+xm;q;E(v, w),
where c is a free parameter.
Proof. For m = 0, S2c,2c+η′;q;E and S ′2c,c+x0;q;E are generated in degrees f , s, s+f . The degree f
operators are clearly elements of the corresponding global section category, and the C1-symmetry
along with the fact that the only poles are [Xs1 ] implies the same for the degree s and s + f
operators. Since the subcategory generated in this way saturates the Bruhat filtration, the
categories are actually isomorphic.
For each of the four categories, every Hom sheaf for m > 0 is contained in the appropriate
Hom sheaf for m = 0, and the image of a local section of S or S ′ satisfies the correct vanishing
conditions to be a local section of ΓS(1) or ΓS ′(1). Moreover, the Bruhat filtration and the analo-
gous filtration by order tells us that both Hom sheaves are direct images of vector bundles on P1
with the same Hilbert polynomials, and must therefore be identified by the isomorphism. �
This leads to a particularly nice interpretation of our categories. If the rational surface Xm
is obtained from a Hirzebruch surface X0 by blowing up m points of a smooth anticanonical
curve, then the line bundles on Xm are parametrized by the group Z〈s, f, e1, . . . , em〉. This
then gives rise to a category on this group of objects by taking the full subcategory of coh(Xm)
in which the objects consist of one line bundle of each isomorphism class. We can, of course,
do this over the entire moduli stack of such surfaces, which turns out (at least for m > 0) to be
isomorphic to Em+1. We then obtain a sheaf of categories on this base by taking the appropriate
sheaf version of Hom between line bundles. It was shown in [26] that this sheaf of categories is
precisely the specialization to q = 0 of S or S ′, depending on whether X0 comes from a vector
bundle of even or odd degree. (One caveat here is that the fibers in this category can be slightly
different from the categories associated to individual surfaces; Hom spaces in the latter may
jump in the presence of −2-curves, while the global category is flat.)
The same, therefore, applies to our categories, and thus for all n, S(n)
η′,x1,...,xm;0,0 and S ′(n)
x0,...,xm;0,0
can be interpreted as symmetric powers of rational surfaces. (To be precise, each fiber is equi-
valent to a subcategory of the subcategory of line bundles on such a symmetric power, which
is full whenever the ratio of line bundles is acyclic.) The categories with q = 0 and general t
are thus commutative deformations of such powers (some sort of compactified discrete elliptic
Calogero–Moser spaces), while the categories with general q, t are further noncommutative
deformations.
We can also obtain analogous deformations for P2, though in that case only the global section
category makes sense. If we restrict ΓS ′(n)
x0;q,t to the objects in Z(s+f), then for n = 1, q = t = 0,
we can identify consecutive Hom spaces in such a way as to obtain the polynomial algebra in three
generators. (For n = 1, q 6= 0, we instead get the three-generator Sklyanin algebra of [1, 3],
see [26].) Thus for general n, q = 0, we again obtain a family of commutative deformations
of Symn(P2), and further noncommutative deformations for general parameters.
There are some caveats to the above discussion. One is that since we are including all line
bundles in the construction, there is no canonical way to associate a projective variety for q = 0
or a noncommutative analogue in general: in general, we would need to make an explicit choice
of ample divisor, or make some other choice of what it means for a module over the category
to be torsion (i.e., map to the 0 sheaf). For n = 1, it was shown in [26] that any reasonable
choice of ample divisor (in particular, any divisor which is ample on every Xm) gives rise to the
100 E.M. Rains
same quotient category, and thus there is no difficulty. Unfortunately, the argument there relied
heavily on showing that various product maps are surjective, and the analogous surjectivity fails
for n > 1 even for q = t = 0. We therefore leave this as an open question.
There is also an issue here that, due to some difficulties in applying the Hecke algebra ideas
to the P2 case, we cannot always prove flatness for general ample divisors. For any surface other
than P2, this is not a significant issue, as there will always be a nonempty subcone of the ample
cone for which everything does work as expected. For P2, or in general outside this subcone,
we will at least be able to show that each Hom space is flat outside some finite (and most likely
empty) set of bad pairs (q, t).
We should also note that since the data in each case includes an explicit morphism to a Hir-
zebruch surface, that a priori the category might depend on this map (and, when X0
∼= P1×P1,
on the choice of ruling) and not just on the surface Xm. We will show in the following section
that (just as for n = 1) this is not an issue, but the argument is decidedly nontrivial.
Before proceeding to studying flatness for the global section category, we should note that
there are also interpretations of H(1) and H′(1) in terms of the categories constructed in [26].
The point is that for n = 1, we are taking a spherical algebra relative to a master Hecke algebra.
Since this is the endomorphism algebra of a vector bundle on P1, we immediately find that the
spherical algebra and the DAHA are Morita equivalent. (Note that even for generic parameters,
this does not quite follow from Proposition 6.10, as this new Morita equivalence applies to
the compactified versions of the algebras.) Making this Morita equivalence explicit gives the
following.
Proposition 7.20. Let v, w ∈ Z〈s, f, e1, . . . , em〉. Then
ΓH(1)
η′,x1,...,xm;q(v, w) ∼=
(
ΓS(1)
η′,x1,...,xm;q(v, w) ΓS(1)
η′,x1,...,xm;q(v − 2f, w)
ΓS(1)
η′,x1,...,xm;q(v, w − 2f) ΓS(1)
η′,x1,...,xm;q(v − 2f, w − 2f)
)
,
and similarly for ΓH′(1).
Remark. One can also apply this at the level of sheaf categories on P1. There one finds
(per the analogous statement of [26]) that the spherical sheaf category is the sheaf “Z-algebra”
associated to a noncommutative P1-bundle on P1 [38]. There is, of course, no reason why we
could not apply the Morita equivalence associated to OP1⊕OP1(−2) to any such noncommutative
P1-bundle on P1 and thus obtain an associated DAHA (which will always be a degeneration of
the elliptic DAHA).
In [26], it was shown that the algebras S satisfy a “Fourier transform” symmetry swapping η
and η′ and swapping s and f , which in turn induces a symmetry
ΓS(1)
η′,x1,...,xm;q,0
∼= ΓS(1)
−η′,x1−η′/2,...,xm−η′/2;q,0,
again swapping s and f . (We will show in the next section how to extend this to general n.)
Since the description of ΓH(1) in terms ΓS(1) is not invariant under swapping s and f , this
symmetry does not actually extend to the DAHA itself. It turns out that, at least for m = 0,
this is a consequence of the compactification we performed. Each Hom space of degree 2s+ 2f
contains an element 1 (in a Fourier-invariant way!); if we localize with respect to those elements,
then the objects v and v+ 2s+ 2f of the category become isomorphic, and thus we may replace
v−2f by v+2s in the above description. Using the translation symmetry, we can then subtract 2s
from v and v + 2s at the cost of changing the parameters slightly. But then swapping s and f
recovers the above description of ΓH(1). In other words, the localized DAHAs actually do satisfy
a Fourier transformation symmetry (though it is not clear how to describe it in terms of explicit
Elliptic Double Affine Hecke Algebras 101
operators). It is likely that something similar holds in general (including n > 1), but this will
require a better understanding of the relevant Morita equivalences.
In the univariate setting, one can gain some insight from the results of [21] on the traditio-
nal C∨C1 Hecke algebra. This suggests in general that replacing −2f above by −2f+e1+· · ·+ek
would have the effect of moving the parameters x1, . . . , xk from s0 to sn. If so, then the Fourier
transform would continue to extend to the noncompact elliptic DAHA in the presence of non-t
parameters, but would effectively swap the roles of the two roots vis-à-vis the parameters.
This degeneration also gives strong evidence that the full SL2(Z) action will not extend
to the elliptic DAHA (compact or not). Indeed, if one looks at the action of SL2(Z) on the
corresponding surfaces in the case that the spherical algebra is commutative, one finds that it
relies on the fact that the anticanonical curve at infinity is singular. Each generator of SL2(Z)
blows up a singular point and then blows down a different component of the anticanonical curve,
so that the anticanonical curve has the same shape and its complement has not changed, but
the actual projective surface has. Blowing up a smooth point of the anticanonical curve in
general does change the complement of the anticanonical curve, and thus we cannot expect this
operation to survive to the elliptic level.
As with sheaves in general, when we take global sections in a family of sheaf categories,
the fibers of the global sections can differ considerably from the global sections of the fibers.
That is, there is a natural morphism from each fiber of the global section category to the global
section category of the corresponding fiber, but this morphism can fail to be either injective or
surjective. The failure of injectivity is particularly bad when we consider that the kernel of the
map does not inherit an interpretation in terms of difference operators. In particular, if we have
such a failure of injectivity, then we can no longer be confident that the fiber is a domain.
Each Hom sheaf in the global section category is the direct image of the corresponding Hom
bimodule, and we can factor the direct image through one of the projections Pn × Pn → Pn to
find that the Hom sheaf is the direct image of a vector bundle on Pn. If every fiber of that
vector bundle is acyclic, then Grauert’s theorem tells us that taking the direct image actually
does commute with passing to fibers. It turns out that this holds (modulo some genericity
assumptions in some cases) for a sufficiently large class of degrees to allow us to prove in general
that the map is always injective and that the global section category is flat.
There are two ways to show acyclicity. One is to show that every subquotient of the Bruhat
filtration is acyclic; the other is to use the symmetric power description for t = 0 to deduce
acyclicity for t = 0 and thus in a neighborhood of t = 0 by semicontinuity. In either case,
the bundle is either itself a symmetric power or is built up out of symmetric powers, and thus
we need to understand when such a bundle is acyclic.
Given a sheaf M on a scheme X, we may define a sheaf Symn(M) on the symmetric power
Symn(X) by descending M�n through the quotient by Sn.
Lemma 7.21. Let X be a projective scheme over a field k, and let M be an acyclic sheaf on X.
Then Symn(M) is an acyclic sheaf on Symn(X) for all n ≥ 1.
Proof. First, note that if k has characteristic p > n or 0, then this is immediate, since Symn(M)
is a direct summand of an acyclic sheaf, namely the direct image of the acyclic sheaf M�n.
In general, we proceed by induction on the pair (n, dimX) relative to the product partial
order. Let OX(1) be a very ample divisor on X, and note that Symn(OX(1)) is at least ample
on Symn(X). (It can fail to be very ample!) In particular, there exists l > 0 so that
Symn(M)⊗ Symn(OX(1))l ∼= Symn(M(l))
is acyclic. Choose a nonzero section of OX(l), and use it to embed M as a subsheaf of M(l).
This one-step filtration of M(l) induces a symmetric power filtration F• of Symn(M(l)) such
102 E.M. Rains
that Fm+1/Fm is the direct image on Symn(X) of the sheaf Symm(M(l)/M) � Symn−m(M)
on Symm(X) × Symn−m(X). By induction each subquotient Fm+1/Fm for m > 0 is acyclic,
as each factor is either a symmetric power of lower degree or supported on a lower-dimensional
projective scheme. Since F1 = Symn(M), it follows that Symn(M(l))/ Symn(M) is acyclic, and
thus that Hp(Symn(M)) = 0 for p > 1.
For all m ≥ 0, we have H0(Symn(M(m))) ∼= Symn
(
H0(M(m))
)
, and thus h0(Symn(M(m)))
=
(
h0(M(m))+n−1
n
)
. Applying this to m� 0 lets us compute the Hilbert polynomial of Symn(M),
and then setting m = 0 gives
χ
(
Symn(M)
)
=
(
χ(M) + n− 1
n
)
=
(
h0(M) + n− 1
n
)
= h0
(
Symn(M)
)
.
Since we have already shown that the higher cohomology spaces vanish, this implies that h1 also
vanishes, and the claim follows. �
By considering subquotients for the Bruhat filtration, we obtain the following.
Lemma 7.22. Suppose d′ ≥ d − 1. Then every fiber of S(n)
η′;q,t(0, ds + d′f) is acyclic for the
morphism to parameter space.
Lemma 7.23. Suppose 2d′/3 ≥ d− 1. Then every fiber of S ′(n)
x0;q,t(0, ds+ d′f) is acyclic for the
morphism to parameter space.
Lemma 7.24. Suppose d′ ≥ max(d, d/2+r1) and 0 ≤ r1 ≤ d. Then every fiber of S(n)
η′,x1;q,t(0, ds+
d′f − r1e1) is acyclic for the morphism to parameter space.
Proof. The first two lemmas are straightforward. For the third, note that if r1 ≤ d/2, then
imposing the vanishing conditions subtracts r1 from the degree of the top subquotient, r1 − 1
from the next, etc., until we reach 0, and in each case there is sufficient degree to do this without
becoming negative (or 0, apart from the subquotient supported on P1). For r1 > d/2, we apply
the elementary transformation symmetry to reduce to a r1 ≤ d/2 case with the opposite parity.
Each case gives a convex cone in which we are guaranteed acyclicity, and combining the cones
gives the desired result. �
Proposition 7.25. Suppose v = ds + d′f − r1e1 − · · · − rmem satisfies the inequalities d′ ≥
max(d, d/2+r1), d ≥ r1 +r2 and r1 ≥ r2 ≥ · · · ≥ rm ≥ 0, and either v = 0 or 2d+2d′−r1−r2−
· · · − rm > 0. Then every fiber of S(n)
η′,x1,...,xm;q,t(0, v) is acyclic for the morphism to parameter
space.
Proof. Every subquotient is the direct image under a finite morphism of an outer tensor product
of symmetric powers of subquotients of the n = 1 case. The constraints on v ensure that every
univariate subquotient is acyclic: either (the direct image of) a line bundle of positive degree
on E or a line bundle of nonnegative degree on P1. It follows that every Bruhat subquotient for
general n is acyclic, and thus the same holds for the full Hom space.
To check the univariate assertion, note that for d = 0, the vector bundle is simply OP1(d′),
so there is no problem. For d > 0, the leading subquotient in the filtration comes from a line
bundle on E of degree 2d+ 2d′ − r1 − · · · − rm, so is positive, and the corresponding subsheaf is
the same as the Hom sheaf obtained by subtracting 2s+ 2f − e1−· · ·− em from v (unless d = 1,
when the subsheaf is trivial and there is nothing further to discuss). If m > 1 (the m = 1 case
already having been dealt with), then this subtraction preserves all of the inequalities except
possibly rm ≥ 0 and 2d + 2d′ − r1 − · · · − rm > 0. The first inequality could only be violated
if we had rm = 0, in which case we might as well have omitted that parameter. For the other
inequality, we are adding m − 8 to the left-hand side, so there is no problem if m ≥ 8. But if
m < 8, then the inequality is implied by the other inequalities. �
Elliptic Double Affine Hecke Algebras 103
Note that this proposition is already stronger than it seems, as we can always arrange to
have the inequalities d ≥ r1 + r2, r1 ≥ r2 ≥ · · · ≥ rm ≥ 0 by applying a suitable combination
of elementary transformations and setting negative ri to 0. Indeed, with the exception of the
final inequality, this is just stating that the vector v is in the fundamental chamber for the
corresponding action of W (Dm). In particular, for any vector v, we can use this proposition to
find an explicit d′ such that v+ d′f satisfies acyclicity. (The existence of such a d′ was of course
already guaranteed by Serre vanishing.) If r1 ≤ d/2, then this bound is pretty close to tight
(based on what we know about the n = 1 case), but for r1 ≥ d/2, the following result suggests
that there is considerable room for improvement.
Proposition 7.26. Suppose v = ds + d′f − r1e1 − · · · − rmem satisfies the inequalities d′ ≥
d ≥ r1 + r2; r1 ≥ r2 ≥ · · · ≥ rm ≥ 0; and either v = 0 or 2d + 2d′ − r1 − r2 − · · · − rm > 0.
Then there is a codimension ≥ 2 subscheme of parameter space, not meeting the subschemes
t = 0 or t = q, such that every fiber of S(n)
η′,x1,...,xm;q,t(0, v) on the complement is acyclic for the
morphism to parameter space.
Proof. If n = 1, then these inequalities are enough to guarantee acyclicity, per [26]. It follows
that any fiber with t = 0 satisfies acyclicity, and thus the open subscheme on which acyclicity
holds contains the divisor t = 0. By the t 7→ q − t symmetry, the acyclic locus also contains
the divisor t = q. This pair of divisors is relatively ample over M1,1 for the E2 parametrizing q
and t, and thus their complement contains no closed subscheme of codimension ≤ 1. �
We of course conjecture that the codimension ≥ 2 subscheme is always empty.
Corollary 7.27. Suppose v = ds+d′f−r1e1−· · ·−rmem satisfies the inequalities d′ ≥ d ≥ r1+r2
and r1 ≥ r2 ≥ · · · ≥ rm ≥ 0. Then there is a codimension ≥ 2 subscheme of parameter
space (empty if d′ ≥ d/2 + r1 and never meeting t = 0 or t = q) on the complement of which
ΓS(n)
η′,x1,...,xm;q,t(0, v) is flat and the map to meromorphic difference operators is injective on fibers.
Proof. If v = 0 or 2d + 2d′ − r1 − · · · − rm > 0, then this follows from acyclicity, so suppose
2d + 2d′ − r1 − · · · − rm ≤ 0. This is the degree of the leading subquotient of the univariate
filtration; if it is negative, then this leading subquotient never has a global section, while if it
is 0, the line bundle depends nontrivially on the parameters, and thus generically does not have
a global section. Either way, the direct image of a nontrivial symmetric power of the leading
univariate subquotient will always be 0, and the same holds for an outer tensor product with
such a power.
Consider a Bruhat order ideal (i.e., an order ideal in the poset of dominant weights) contained
in the interval [≤ (d/2, . . . , d/2)]. If this order ideal contains a dominant weight with λ1 = d/2,
then there is such a weight which is a maximal element of the order ideal. Since the subquotient
corresponding to that maximal element has no direct image, removing it has no effect on the
direct image. We thus find that the direct image of the interval [≤ (d/2, . . . , d/2)] is the same
as the direct image of the interval [≤ (d/2 − 1, . . . , d/2 − 1)], and thus we reduce to v − (2s +
2f − e1 − · · · − em) as before. �
Remark. That the bad subscheme has codimension ≥ 2 follows from general considerations;
a failure of injectivity in codimension 1 comes from a local family of operators vanishing on
a hypersurface, and we can always locally divide such a family by a function cutting out the
hypersurface. Thus the true content is that the locus does not meet t = 0 or t = q, and
further has codimension ≥ 2 image in the surface parametrizing q and t. In particular, we have
injectivity over the local ring at any point with t = 0 or t = q, including the point q = t = 0
corresponding to the undeformed case.
104 E.M. Rains
For 0 ≤ m ≤ 7 (or for m = −1, i.e., P2), the corresponding commutative surface is a (pos-
sibly singular) del Pezzo surface with a choice of smooth anticanonical curve and a sequence of
blowdowns to a Hirzebruch surface. The anticanonical embedding of this surface is given by the
graded algebra⊕
d≥0
ΓS(1)
η′,x1,...,xm;q,t(0, d(2s+ 2f − e1 − · · · − em));
we can interpret this as a graded algebra by using the fact that 1 is in
ΓS(1)
η′,x1,...,xm;q,t(d(2s+ 2f − e1 − · · · − em), (d+ 1)(2s+ 2f − e1 − · · · − em))
for any d. This is the Rees algebra of the natural filtration on the spherical algebra⋃
d≥0
ΓS(1)
η′,x1,...,xm;q,t(0, d(2s+ 2f − e1 − · · · − em)),
the coordinate ring of the complement of the chosen smooth anticanonical curve. Taking the
multivariate versions thus gives deformations of the (anticanonically embedded) symmetric pow-
ers of Xm and Xm \ E, and, apart from possible codimension ≥ 2 exceptions for P2, these
deformations are always flat, and every fiber is a domain.
If Xm has −2-curves or m > 7, then the anticanonical divisor is no longer ample, and
thus one can no longer expect to obtain a deformation of Symn(Xm) as a graded algebra,
or of Symn(Xm \ E) as a filtered algebra. This is why we generalized the spherical algebra
construction to the above categories: one needs to work with some non-(pluri)anticanonical
divisor, and it is then easier to include all divisors.
There is an interesting phenomenon that arises for the spherical algebra in the m = 8 case.
The global section algebra in this case is trivial (consisting only of the global section 1), but this
merely reflects the fact that the generic fiber has no nontrivial global sections. The univariate
subquotients in this case are all multiples of x1 + · · · + x8 − 2η′, and thus if this value is
r-torsion, then any subquotient of weight a multiple of r will be trivial. (In terms of surfaces,
this corresponds to the case that X8 is an elliptic surface, in which one fiber consists of r copies
of the chosen anticanonical curve.) As a result, the dimension of global sections of such a fiber
in a given Bruhat interval can in principle be as large as the number of such weights contained
in the interval, or (a priori) as small as 1.
It turns out that at least for r = 1 (i.e., when the elliptic surface has a section), this upper
bound is attained (i.e., the dimension of global sections in a Bruhat interval is equal to the
size of the Bruhat interval), and furthermore those global sections satisfy a surprising property.
Note that it suffices to find n+ 1 global sections of degree 2s+ 2f − e1−· · ·− e8, as we can then
obtain global sections with arbitrary dominant weight by taking products.
Theorem 7.28. On any fiber such that 2η′ = x1 + · · · + x8, the space of global sections
of S(n)
η′,x1,...,x8;q,t(0, 2s + 2f − e1 − · · · − e8) is n + 1-dimensional, and any two global sections
commute.
Proof. Certainly, n + 1 is an upper bound on the number of global sections, since there are
n + 1 subquotients, each of which has a unique global section. The given Hom bimodule is
contained in the Hom bimodule S(n)
η′,x1,...,x7;q,t(0, 2s + 2f − e1 − · · · − e7), and the latter Hom
bimodule satisfies acyclicity. Since the m = 7 bimodule has 2 global sections when n = 1, it has
n + 1 =
(
2+n−1
n
)
global sections when t = 0 and thus (by flatness) in general. We thus need to
show that those global sections are actually global sections of the subsheaf we want.
Let D be such a global section. This is determined by the left coefficients cm of
∏
1≤i≤m T
−1
i
for 0 ≤ m ≤ n + 1, where cm is Sm × Cn−m-invariant. Each cm is a section of a line bun-
dle Lm(Dm), where Lm comes from the equivariant gerbe and Dm comes from the allowed
Elliptic Double Affine Hecke Algebras 105
poles and forced zeros. The allowed poles are somewhat complicated, since we are not assuming
that cm is a leading coefficient, but the forced zeros are the same as they would have been if
it were a leading coefficient. The symmetric power property then tells us that when t = 0,
the forced zeros associated to t must all cancel allowed poles, and any allowed pole associated
to a root of type Dn must be cancelled in this way.
The remaining zeros and poles can be deduced from the univariate case, and we thus find
that cm is a multiple of∏
1≤i<j≤m
ϑ(t− zi − zj , q + t− zi − zj)
ϑ(−zi − zj , q − zi − zj)
∏
1≤i≤m
m<j≤n
ϑ(t− zi ± zj)
ϑ(−zi ± zj)
×
∏
1≤i≤m
∏
1≤j≤7 ϑ(q/2 + xj − zi)
ϑ(−2zi, q − 2zi)
∏
m<j≤n
1
ϑ(−q − 2zj , q − 2zj)
,
in the sense that the ratio is a holomorphic section of the line bundle with polarization∑
1≤i≤m
(z2
i /2− (q/2 + x8)zi) +
∑
m<j≤n
4z2
i ,
modulo line bundles on the base. (The only difference between this and the leading coefficient
of the corresponding Bruhat interval are the factors ϑ(−q − 2zi, q − 2zi) for m < j ≤ n.) This
line bundle has degree 1 in each zi for 1 ≤ i ≤ m, so every holomorphic section has the same
dependence on those variables, which we can read off from the polarization.
We thus conclude that cm/
∏
1≤i≤m ϑ(q/2 + x8 − zi) is independent of z1 through zm, so is
still holomorphic. As a result, we find that every global section of the m = 7 bimodule is also
a global section for m = 8; more precisely, the holomorphy gives it generically, but the condition
is closed, so it holds in general.
It remains to show commutativity. We first note as a sanity check that the n + 1 leading
term operators
∏
1≤i<j≤m
ϑ(t− zi − zj , q + t− zi − zj)
ϑ(−zi − zj , q − zi − zj)
∏
1≤i≤m
m<j≤n
ϑ(t− zi ± zj)
ϑ(−zi ± zj)
∏
1≤i≤m
∏
1≤j≤8 ϑ(q/2 + xj − zi)
ϑ(−2zi, q − 2zi)
T−1
i
commute. It follows that on any fiber with x1 + · · ·+ x8 = 2η′, the global section algebra of the
spherical algebra⋃
d≥0
S(n)
η′,x1,...,x8;q,t(0, d(2s+ 2f − e1 − · · · − e8))
has abelian associated graded algebra; the above leading term operators give one element for each
fundamental weight, so generate the associated graded algebra.
This global section algebra has a particularly nice symmetry: it is preserved by the formal
adjoint with respect to the density
∏
1≤i≤n
∏
1≤j≤8 Γq(q/2 + xj ± zi)
Γq(±2zi)
∏
1≤i<j≤n
Γq(t± zi ± zj)
Γq(±zi ± zj)
dT.
Indeed, this is the composition of the Selberg adjoint and all 8 elementary transformations, with
the total effect on the parameters being η′ 7→ x1 + · · ·+ x8 − η′ = η′. Note that although such
isomorphisms are usually only defined up to a unit, we can eliminate that freedom by insisting
that the adjoint of 1 be 1. This is triangular with respect to Bruhat order, and is trivial
106 E.M. Rains
on the associated graded algebra, since it fixes the generators and the associated graded algebra
is abelian. Since a triangular involution which is 1 on the diagonal is 1, we find that this formal
adjoint acts trivially on the entire global section algebra. Since an algebra consisting entirely
of self-adjoint operators is abelian, we conclude that the generators commute as required. �
Since the n + 1 operators are filtered by Bruhat order, it is natural from an integrable
systems perspective to designate the first nontrivial operator (with leading term ∝ T−1
1 ) as the
Hamiltonian. This has leading coefficient∏
1≤j≤8 ϑ(q/2 + xj − z1)
ϑ(−2z1, q − 2z1)
∏
2≤j≤n ϑ(t− z1 ± zj)∏
2≤j≤n ϑ(−z1 ± zj)
,
which turns out to be a mild reparametrization of the leading coefficient of the Hamiltonian
proposed by van Diejen in [6] (see also [7, equations (3.12)–(3.14)], with the caveat that one
must gauge the operator), and later shown to be integrable in [16]. In fact, one can verify
(we omit the details) that van Diejen’s operator satisfies the appropriate residue and vanishing
conditions to be a global section of S(n)
η′,x1,...,x8;q,t(0, 2s + 2f − e1 − · · · − e8), and thus we have
given a new proof that van Diejen’s Hamiltonian is integrable.
Remark. Since our Hecke algebra methods gave a new proof of the existence of the commut-
ing operators which were constructed in [16], it is natural to wonder whether there might be
applications in the other direction; that is, using their R-matrix based approach to construct
global sections of other Hom sheaves in our spherical DAHA categories. Such a construction
might make it possible to prove flatness in general without having to exclude a codimension ≥ 2
subscheme; if a given Hom bimodule generically has N global sections, then to prove flatness
and injectivity in a neighborhood of a given fiber, it suffices to construct N local sections on a
neighborhood of the fiber such that the restrictions to the fiber are linearly independent.
The connection to elliptic surfaces suggests a possible generalization of this integrable system.
If x1 + · · · + x8 − 2η′, instead of being 0, is a torsion point of order r, then we again find that
there are many trivial Bruhat subquotients, and thus it becomes nontrivial to determine how
many global sections the spherical algebra has. We cannot answer this in general, but we can,
at least, show that the r-torsion condition forces there to be some nontrivial global sections.
Proposition 7.29. Let E be an elliptic curve and η′, x1, . . . , x8, q, t be points of E such that
x1 + · · · + x8 − 2η′ is a torsion point of order r. Then the corresponding fiber of the spherical
algebra
⋃
d S
(n)
η′,x1,...,x8;q,t(0, d(2s + 2f − e1 − · · · − e8)) has a global section of dominant weight
(r, 0, . . . , 0) with nonzero leading term.
Proof. Indeed, every subquotient in the Bruhat filtration for the order ideal [<(r, 0, . . . , 0)]
is acyclic: the bottom subquotient is OPn , while the remaining subquotients are nontrivial
elements of En[r]. �
It is then natural to conjecture that the resulting Hamiltonian is integrable, or more precisely
the following.
Conjecture 7.30. Under the same hypotheses, the fiber of
S(n)
η′,x1,...,x8;q,t(0, r(2s+ 2f − e1 − · · · − e8))
has n+ 1 global sections, all of which commute.
Elliptic Double Affine Hecke Algebras 107
Both parts of the proof for r = 1 fail here: the m = 7 surface has too many global sections, and
the adjoint no longer gives an element of the same spherical algebra. There is some experimental
evidence for this conjecture, however: for n = r = 2, the analogous statement for a suitable
degeneration to a nodal curve holds by a computer calculation. (We will briefly discuss how
to construct such degenerations at the end of the next section.) This statement is, of course,
trivial for n = 1 (given the proposition), but it is worth noting there that the global sections
of S(1)
η′,x1,...,x8;0,0(0, r(2s+2f −e1−· · ·−e8)) are just the pullback of the global sections of OP1(1)
from the base of the elliptic fibration.
8 The (spherical) C∨Cn Fourier transform
Our objective in the present section is to prove the following result.
Theorem 8.1. There is, locally on the base, an isomorphism
ΓS(n)
2c,x1,...,xm;q,t
∼= ΓS(n)
−2c,x1−c,...,xm−c;q,t
acting on objects as ds+ d′f − r1e1 − · · · − rmem 7→ d′s+ df − r1e1 − · · · − rmem and triangular
with respect to the Bruhat filtration. Moreover, this isomorphism commutes (up to local units)
with the Selberg adjoint.
We refer to this isomorphism as the “Fourier transform”: in particular, note that it takes
multiplication operators (of degree d′f) to difference operators (of degree d′s) and (at least on
the parameters) is an involution. (In addition, though we will not be using this fact, the Fourier
transform can be represented in the analytic setting by a formal integral operator [27].)
Before constructing the Fourier transform, we give some consequences. The simplest is that
we can conjugate the symmetry by an elementary transformation.
Corollary 8.2. There is, locally on the base, an isomorphism
ΓS ′(n)
x1+2c,x1,x2,...,xm;q,t
∼= ΓS ′(n)
x1−c,x1+c,x2−c,...,xm−c;q,t
acting on objects as ds+ d′f − r1e1 − r2e2 − · · · − rmem 7→ (d′ − r1)s+ d′f − (d′ − d)e1 − r2e2 −
· · · − rmem.
This also tells us that the deformations of P2 we constructed are independent of x0 (as one
would expect).
Corollary 8.3. The restriction to Z(s+ f) of ΓS ′(n)
x0;q,t is (fppf locally) independent of x0.
Proof. The previous corollary gives (locally) an isomorphism ΓS ′(n)
x0,x0−2c;q,t
∼= ΓS ′(n)
x0−3c,x0−c;q,t.
The action on objects takes d(s+ f) to d(s+ f), so this local isomorphism induces a local iso-
morphism ΓS ′(n)
x0;q,t|Z(s+f)
∼= ΓS ′(n)
x0−3c;q,t|Z(s+f) for any x0 and c. It follows that any two geometric
fibers with the same values of q, t are isomorphic. �
Remark 8.4. More generally, if (E, x0, x1, q, t) is a point of E4 over some scheme S, then we
have an isomorphism ΓS ′(n)
x0;q,t|Z(s+f)
∼= ΓS ′(n)
x1;q,t|Z(s+f) defined Zariski locally on S as long as
x1−x0 ∈ 3E(S). Without this assumption, there may very well be no such isomorphism; indeed
for n = 1, q = 0, these are essentially the homogeneous coordinate rings of the embeddings of C
via [x0] + 2[0] and [x1] + 2[0].
Remark 8.5. When c ∈ E[3], this isomorphism becomes an automorphism, but is quite non-
trivial.
108 E.M. Rains
The most significant consequence is the following. Note here that we are, as usual, taking
global sections before passing to fibers.
Theorem 8.6. For any v ∈ Z〈s, f, e1, . . . , em〉, there is a codimension ≥ 2 subscheme of para-
meter space on the complement of which ΓS(n)
η′,x1,...,xm;q,t(0, v) is flat and the map to meromorphic
difference operators is injective on fibers.
Proof. Applying the Fourier transform has no effect on flatness (since it is an isomorphism), and
the Fourier transform will be constructed via an action on meromorphic difference operators, and
thus injectivity on fibers is also preserved. This allows us to reduce to Corollary 7.27, as in [26].
To be precise, let v = ds + d′f − r1e1 − · · · − rmem. We may apply a permutation and an
even number of elementary transformations to put v into the fundamental chamber for W (Dm).
Moreover, if rm < 0, then we may set it to 0 without changing the sheaf of global sections, and
in this way may arrange to have d ≥ r1 + r2 and r1 ≥ · · · ≥ rm ≥ 0. If d′ ≥ d, then we may
apply Corollary 7.27. If d′ < 0, then we observe that ΓS(n)
η′,x1,...,xm;q,t(0, v) = 0, so the result
again follows. Otherwise, we apply the Fourier transform. Since this strictly decreases d but
keeps it nonnegative, the claim follows by induction. �
Remark. Of course, the codimension ≥ 2 subscheme is the same as that of the appropriate
special case of Corollary 7.27.
Just as elliptic pencils gave rise to integrable systems above, there is something analogous
(if slightly weaker) for rational pencils. Call a small category with object set Z “quasi-abelian”
if there is a commutative graded algebra A such that Hom(j, k) ∼= A[k − j] for all j, k, with
composition given by multiplication.
Corollary 8.7. Suppose v ∈ Z〈s, f, e1, . . . , em〉 is the class of a rational pencil on the rational
surface Xm. Then ΓS(n)
η′,x1,...,xm;q,t|Zv is quasi-abelian, and the corresponding graded algebra is
a free polynomial algebra in n+ 1 generators.
Proof. If v = f , this is easy: the global sections of degree df are just multiplication by Cn-
invariant sections of the bundle with polarization d
∑
i z
2
i , and this is precisely the pullback
of OPn(d). More generally, it follows from the theory of rational surfaces (see [25]) that v
represents a rational pencil iff it is in the orbit of f under the group W (Em+1) generated by the
Fourier transform and W (Dm). �
Just as integrable systems lead to natural eigenvalue equations, such “quasi-integrable” sys-
tems lead to generalized eigenvalue problems. A generalized eigenfunction of a space D of
operators is a function f such that the image Df is 1-dimensional. (We then obtain an asso-
ciated “generalized eigenvalue”, namely the point in P(D) associated to the kernel of the map
D 7→ Df on D.)
Given a quasi-integrable system associated to a rational pencil, we have for each d a map φd
from A[1] to the space of operators, such that φd+1(y)φd(x) = φd+1(x)φd(y). We may then
consider for each d the generalized eigenvalue problem associated to φd(A[1]). For any generalized
eigenfunction fd for φd(A[1]), let fd+1 be a nonzero representative of φd(A[1])fd. Then for
suitable y ∈ A[1], we have
φd+1(x)fd+1 = φd+1(x)φd(y)fd = φd+1(y)φd(x)fd = λd(x)φd+1(y)fd+1
for all x ∈ A[1], and thus fd+1 is a generalized eigenfunction for φd+1(A[1]). More gener-
ally, if V ⊂ A[1] is such that the corresponding generalized eigenvalue problem for φd+1(V ) is
nondegenerate (i.e., for each point of projective space, the corresponding problem has at most
Elliptic Double Affine Hecke Algebras 109
1-dimensional solution space), then any generalized eigenfunction fd for φd(V ) is a generalized
eigenfunction for φd(A[1]), since then φd(y)fd is a generalized eigenfunction for φd+1(V ).
There are two cases of particular interest. In the case v = s+f−e1−e2, η′ = −(n−1)t−q, the
generators of the quasi-integrable system are operators of the form considered in [23], and the
quasi-abelian property turns into the quasi-commutation relation used there. The corresponding
generalized eigenvalue problem is precisely the difference equation [23, Proposition 3.9] satisfied
by the elliptic interpolation functions. (The interpolation kernel of [27] is also a generalized
eigenfunction for essentially the same space of operators, [27, Proposition 3.12].)
The biorthogonal functions of [23, 24] are also generalized eigenfunctions of such a quasi-
integrable system, corresponding to v = 2s+ 2f − e1− e2− e3− e4− 2e5 and η′ = −(n− 1)t− q.
Indeed, the first-order difference operators considered in [24] correspond to products of operators
of degrees s−e5, s+2f−e1−e2−e3−e4−e5 and s+f−ei−ej−e5, 1 ≤ i < j ≤ 4, giving rise to 8
operators of degree 2s+ 2f − e1− e2− e3− e4− 2e5. The biorthogonal functions are generalized
eigenfunctions of the span of these 8 operators, and the generalized eigenvalues are all distinct
points of P7. It thus follows that any biorthogonal function is a generalized eigenfunction for the
full space of operators. (With more effort, one can in fact verify that the generalized eigenvalues
are given by suitable specializations of the leading coefficients; this is a consequence in general
of the fact that the Fourier transform respects leading coefficients.)
We now turn to constructing the Fourier transform. The traditional approach would be to
construct a Fourier transform on the DAHA and then observe that it restricts to a transform on
the spherical algebra. One significant issue that arises here is that although we have a reasonable
facsimile of a presentation, it is at the level of sheaves, not at the level of global sections,
while the Fourier transform does not make sense in terms of sheaves (since it does not preserve
multiplication). Furthermore, most of the rank 1 subalgebras we used to generate the DAHA do
not have any nontrivial global sections (the leading Bruhat subquotient is a generically nontrivial
line bundle of degree 0 in every variable). As a result, it seems unlikely that the Fourier transform
on the DAHA (assuming it exists) would have a construction that was significantly simpler than
the construction we give in the spherical case. Beyond that, there is another issue: as we
discussed above, the description of the rank 1 DAHA via a Morita equivalence to the spherical
algebra strongly suggests that the Fourier transform only exists for the noncompact version of
the DAHA. In other words, the Fourier transform on the DAHA would not respect the filtration
by degree; since this filtration comes from the Bruhat filtration, the latter also could not be
preserved. As a result, even having a Fourier transform for the DAHA would not be enough to
prove the theorem; one also needs to understand why the spherical version is triangular!
We thus wish an approach that works directly with the spherical algebra. Note that since the
action of the Fourier transform on objects preserves Z〈s, f〉, the Fourier transform for m > 0
restricts to a transform of the same sort for m = 0. Moreover, since every Hom sheaf is contained
in one of degree in Z〈s, f〉, it suffices to specify how the transform acts on such sheaves and
show that it preserves the various subsheaves of interest. We thus focus our initial attention on
the case m = 0.
In the univariate setting, the Fourier transform was easy to construct: for generic parameters,
one can give an explicit presentation for the category (with generators of degrees s and f) and
this presentation has an obvious symmetry. Moreover, a slightly larger set of elements generates
the category even without the genericity condition, and one can determine how the transform
must act on those elements by taking a suitable limit.
Although we have analogues of those generators (and will indeed be able to describe their
Fourier transforms explicitly), this approach founders in the multivariate setting for two reasons.
The first is that the operators of degree s and f do not even come close to generating the category
for n > 1 in general: for q = 0, t = 0, the full category is the bihomogeneous coordinate ring
of Symn
(
P1 × P1
)
, while the elements of degrees s and f lie in the subring corresponding to
110 E.M. Rains
the quotient Pn × Pn. If we include elements of degree s + f , the situation is somewhat better
(we will see that these come close enough to generating to be useful), but this only forces us to
confront the fact that we have absolutely no understanding of the relations satisfied by these
elements.
As a result, we will need some way to construct the Fourier transform which is explicitly
a homomorphism. We will do this by constructing a transform on a much larger algebra of
operators, and then show that it preserves the particular subspace we care about. The simplest
way to construct a homomorphism on a category of operators is to apply a gauge transformation:
assign an operator to each object and apply the associated quasi-conjugation.
The first step in constructing such operators is to determine on what spaces they act, and
thus we need to think a bit about where our existing operators act. Define a family of (gerbe)
polarizations
Pd(η
′; q, t) := −((n− 1)t− (d− 1)q + η′)
∑
i
z2
i /q.
If F is the product of a Γq symbol with polarization Pd′1−d1
(η′; q, t) and a rational function on En,
then we can apply any global section of a fiber of S(n)
η′;q,t(d1s+ d′1f, d2s+ d′2f) and the result will
be a rational function times a Γq symbol with polarization Pd′2−d2
(η′; q, t).
Thus the Fourier transform should be given by operators that take functions with polariza-
tion Pd(η
′; q, t) to functions with polarization P−d(−η′; q, t), or equivalently take P0(η′+dq; q, t)
to P0(−η′ − dq; q, t). There are issues in general, but there is one important case in which ope-
rators of this form do indeed exist. Indeed, the simplest way to obtain an operator mapping
Pd(0; q, t) to P−d(0; q, t) would be to take a global section of S(n)
0;q,t(df, ds), assuming such a global
section exists.
For d = 1, this is not too difficult to control, and indeed we can understand global sections
of order 1 in general.
Lemma 8.8. For any point of E3, the corresponding fiber of S(n)
η′;q,t(0, s + d′f) is spanned by
operators of the form
D(n)
q (u0, u1, . . . , u2d′+1; t)
=
∑
σ∈{±1}n
∏
1≤i≤n
∏
0≤r<2d′+2 ϑ(ur + σizi)
ϑ(2σizi)
∏
1≤i<j≤n
ϑ(t+ σizi + σjzj)
ϑ(σizi + σjzj)
∏
1≤i≤n
T
σi/2
i ,
with u0 + · · ·+ u2d′+1 = q + η′.
Proof. The interval [≤ (1/2, . . . , 1/2)] in the Bruhat order consists of a single double coset, and
thus the space of global sections is (up to multiplication by an explicit product of ϑ functions)
the space of Sn-invariant sections of the appropriate line bundle. That space is spanned by
products of the form
∏
1≤i≤n f(zi), where f is a section of the corresponding line bundle on E ,
and any such section can be factored into ϑ functions. �
Proposition 8.9. For any d ≥ 0, the space of global sections of S(n)
0;q,t(df, ds) is 1-dimensional.
Proof. We proceed by induction in d, with the case d = 0 being obvious. Suppose we are given
a nonzero global section D
(n)
q,t (d) ∈ S(n)
0;q,t(df, ds). Then for any u, v, the operator
D(n)
q ((d+ 1)q/2± u; t)D
(n)
q,t (d)
∏
1≤i≤n
ϑ(zi ± v)
−D(n)
q ((d+ 1)q/2± v; t)D
(n)
q,t (d)
∏
1≤i≤n
ϑ(zi ± u)
Elliptic Double Affine Hecke Algebras 111
is a section of S(n)
0;q,t((d − 1)f, (d + 1)s). Moreover, we know the leading coefficient of D
(n)
q,t (d)
up to a scalar multiple, and may therefore verify that both operators have the same leading
coefficient. Since every subquotient below the top of the corresponding univariate vector bundle
has negative degree, none of the multivariate subquotients below the top have polarizations
represented by positive semidefinite matrices. Thus none of those subquotients have any global
sections, let alone symmetric ones. It follows that a section of S(n)
0;q,t((d − 1)f, (d + 1)s) with
vanishing leading coefficient must in fact be 0, and thus
D(n)
q ((d+ 1)q/2± u; t)D
(n)
q,t (d)
∏
1≤i≤n
ϑ(zi ± v) = D(n)
q ((d+ 1)q/2± v; t)D
(n)
q,t (d)
∏
1≤i≤n
ϑ(zi ± u).
Equivalently,
D(n)
q ((d+ 1)q/2± u; t)D
(n)
q,t (d)
∏
1≤i≤n
ϑ(zi ± u)−1
is independent of u. In particular, the apparent u-dependent poles of this product of operators
are not, in fact, singularities, and thus this gives a section of S(n)
0;q,t((d+1)f, (d+1)s) as required.
That this is the only global section up to scalar multiples follows by observing that again all
Bruhat subquotients below the top have indefinite polarizations, while the top subquotient is
trivial. �
Following the above proof, we define D
(n)
q,t (d) by the recurrence
D
(n)
q,t (d+ 1) = D(n)
q ((d+ 1)q/2± u; t)D
(n)
q,t (d)
∏
1≤i≤n
ϑ(zi ± u)−1,
with base case D
(n)
q,t (0) = 1. (Note that D
(n)
q,t (1) = D
(n)
q (; t).) Equivalently, D
(n)
q,t (d) is the unique
global section of S(n)
0;q,t(df, ds) with leading term∏
1≤i<j≤n
Γq(dq + t− zi − zj)
Γq(t− zi − zj)
∏
1≤i≤j≤n
Γq(−zi − zj)
Γq(dq − zi − zj)
∏
1≤i≤n
T
−d/2
i .
Since translation by s + f does not change the parameters, this also gives a global section
of S(n)
0;q,t(d0(s+ f) + df, d0(s+ f) + ds) for any d0.
Corollary 8.10. If q has exact order d, then
D
(n)
q,t (d) =
∑
σ∈{±1}n
∏
1≤i≤n
1
ϑ(2σizi; q)d
∏
1≤i<j≤n
ϑ(t+ σizi + σjzj ; q)d
ϑ(σizi + σizj ; q)d
∏
1≤i≤n
T
−d/2
i .
Proof. By Theorem 7.15, the center of the given spherical algebra is itself a spherical alge-
bra with q = 0. In particular, the center contains a nonzero element mapping f to s, which
becomes an element of S(n)
0;q,t(df, ds) under the isomorphism. By uniqueness, this element is pro-
portional to D
(n)
q,t (d). For this element to be central, every shift that appears must be congruent
to −d/2 modulo d. There is only one Cn-orbit of shifts that survives, so that we may recover
all coefficients from the leading coefficient, and obtain the stated formula. �
Remark. If we gauge by a product of gamma functions before specializing q (à la an elemen-
tary transformation), the interior coefficients will still vanish, giving central sections of degree
d(s+ d′f) for any d′, and establishing that any element of the center with degree of the given
form is obtained in this way. Since the Fourier transform respects leading coefficients, we can
compute how it acts on central elements of degree ds, df , d(s+ f), and thus conclude (following
the argument below) that the Fourier transform respects the isomorphism of Theorem 7.15.
112 E.M. Rains
The following result shows that these operators indeed behave like Fourier transforms.
Proposition 8.11. We have the operator relations
D(n)
q ((d+ 1)q/2± u; t)D
(n)
q,t (d) = D
(n)
q,t (d+ 1)
∏
1≤i≤n
ϑ(zi ± u),
D
(n)
q,t (d)D(n)
q (−dq/2± u; t) =
∏
1≤i≤n
ϑ(zi ± u)D
(n)
q,t (d+ 1),
and, if u0 + u1 + u2 + u3 = (d+ 1)q,
D
(n)
q,t (d)D(n)
q (u0, u1, u2, u3; t) = D(n)
q (u0 + dq/2, u1 + dq/2, u2 + dq/2, u3 + dq/2; t)D
(n)
q,t (d).
Proof. In each case, both sides are sections of the same Hom sheaf of S(n)
0;q,t with the same
leading coefficient, and only the top subquotient has positive semidefinite polarization. �
It turns out that if we adjoined formal inverses of the operators D
(n)
q,t (d) and declared them to
be D
(n)
q,t (−d), then the result would indeed define a Fourier transform on a certain subcategory
of the category with η′ = 0 (in which the Hom sheaves of degree ds+ d′f for d > d′ are replaced
by the images under the Fourier transform of the Hom sheaves of degree d′s+ df). Proving this
directly is somewhat tricky, however, as unlike in the univariate setting, there does not appear
to be a readily accessible test for right divisibility by D
(n)
q,t (d). And, of course, even using the
translation symmetry, this would at best give us a transform for η′ ∈ Zq, which is especially
weak when q is torsion.
The key idea for proceeding further is that the relation
D
(n)
q,t (d)D(n)
q (−dq/2± u; t) =
∏
1≤i≤n
ϑ(zi ± u)D
(n)
q,t (d+ 1)
gives us a system of recurrences that we can use to solve for the coefficients of D
(n)
q,t (d). Indeed,
it follows from this relation that
D
(n)
q,t (d)D(n)
q (−dq/2± u; t)|u=zi = 0
for 1 ≤ i ≤ n. Since the operators are symmetric, let us consider the specialization u = zn.
The coefficient of
∏
i T
ki−(d+1)/2
i in this specialized operator is a linear combination of the left
coefficients of
∏
i T
li−d/2
i in D
(n)
q,t (d) for max(ki − 1, 0) ≤ li ≤ ki. The coefficient in this linear
combination for ~l = ~k is∏
1≤i≤n ϑ(−kiq + zn − zi, kiq + zi + zn)
∏
1≤i<j≤n ϑ(t+ dq − (ki + kj)q − zi − zj)∏
1≤i≤j≤n ϑ(dq − (ki + kj)q − zi − zj)
,
and thus we can solve for the coefficient of
∏
i T
ki−d/2
i in D
(n)
q,t (d), at least generically. In fact,
we find the only difficulty arises when ϑ(−knq) = 0, so if q is not torsion and ~k 6= 0, there will
always be one of the n specializations that allows us to solve for the coefficient of
∏
i T
ki−d/2
i in
terms of coefficients of terms which are smaller in dominance order. In other words, D
(n)
q,t (d) is
determined by the given relation along with the choice of leading coefficient.
The fact that we can control coefficients near the leading coefficients suggests a way to proceed
further: take an appropriate completion! Define a nonarchimedean metric on the Z/2Z-graded
algebra k(X)
[
T1, . . . , Tn,
∏
i T
−1/2
i
]
by∣∣∣∣∑
~k
ck
∏
i
T kii
∣∣∣∣ := max
~k:ck 6=0
exp
(
−
∑
i
ki
)
.
Elliptic Double Affine Hecke Algebras 113
We call an element of the corresponding completion a formal difference operator. We in par-
ticular denote the completion of the subalgebra k(X)[T1, . . . , Tn] by k(X)[[T1, . . . , Tn]]. This
construction of course applies equally well to the case of twisted difference operators, or even
to the corresponding category in which the objects are polarizations Pd(η
′; q, t) with fixed η′.
We will mostly suppress the twisting from the notation.
The major advantage of formal difference operators is that the ring has a large number of
units. Indeed, the usual argument for inverting a commutative formal power series with invertible
constant term applies equally well in the noncommutative setting to give the following.
Proposition 8.12. If D ∈ k(X)[[T1, . . . , Tn]] has nonzero constant term, then D is a unit.
Since
∏
i T
−1/2
i is also clearly invertible, we find that any of the operators D
(n)
q,t (d) are inver-
tible as formal operators. In fact, in the ring of formal operators we can solve for D
(n)
q,t (d) in
terms of D
(n)
q,t (d+1) and in this way define D
(n)
q,t (d) for d < 0. We then find by an easy induction
that D
(n)
q,t (−d) = D
(n)
q,t (d)−1.
In addition to these inner automorphisms, we also have automorphisms coming from gauging
by Γq symbols and translations on E. Let Tω(c) denote the translation of all variables by c, so
that Tω(q/2) =
∏
1≤i≤n T
1/2
i . Then for any Γq symbol Γ of polarization
(η′ − η′′)
∑
i
z2
i /q + 2((n− 1)t+ q + η′)c
∑
i
zi/q,
there is an induced isomorphism
D 7→ ΓTω(c)DTω(−c)Γ−1
from End(P0(η′; q, t))0 (the subspace involving only integer powers of Ti) to End(P0(η′′; q, t))0.
With this in mind, we define a “formal gauging operator” from P0(η′; q, t) to P0(η′′; q, t) to be
an object of the form
ΓTω(c)D,
whereD is a unit in the endomorphism ring. The “leading term” of such an operator is the formal
symbol ΓTω(c)f , where f is the constant term of D. The formal gauging operators form a group,
with a natural subgroup consisting of elements of the form ΘTω(kq/2)D, where Θ is a product
of ϑ symbols (i.e., of invertible formal difference operators). If G1, G2 are formal gauging oper-
ators such that G1G
−1
2 lies in the subgroup of formal difference operators, then for any formal
difference operator D with only integer shifts (and with coefficients having appropriate polariza-
tions), G1DG
−1
2 := (G1G
−1
2 )G2DG
−1
2 will again be a formal difference operator. This extends
to the half-integer case by writing D = Tω(q/2)D′ and G1DG
−1
2 := (G1Tω(q/2)G−1
2 )G2D
′G−1
2 .
In either case, the operation clearly respects multiplication as long as the gauging operators
match up.
Proposition 8.13. There is a unique family of formal gauging operators D(n)
q,t (c) from P0(2c; q, t)
to P0(−2c; q, t) with leading term∏
1≤i≤j≤n
Γq(−zi − zj)
Γq(−2c− zi − zj)
∏
1≤i<j≤n
Γq(t− 2c− zi − zj)
Γq(t− zi − zj)
Tω(c)
such that D(n)
q,t (−dq/2) = D
(n)
q,t (d) for all d ∈ Z. Moreover, if one divides any coefficient of D(n)
q,t (c)
by the leading term, then the only z-independent poles of the resulting meromorphic section
on En+3 are along hypersurfaces for which q is torsion.
114 E.M. Rains
Proof. If such a family of operators exists, then it must satisfy
D(n)
q,t (c)D(n)
q (c± u; t)|u=zi = 0
for 1 ≤ i ≤ n. This gives an algebraic (and triangular) system of equations for the coefficients
of D(n)
q,t (c) which we have already seen has at most one solution (and if it has a solution, the
only z-independent poles are where q is torsion). Since it has a solution on the Zariski dense set
of divisors c ∈ Zq/2, it must have a solution in general. �
Remark. For an analytic approach to constructing such operators, see [27].
Since we understand a Zariski dense subset of these operators, we can immediately deduce
some relations.
Proposition 8.14. The operators D(n)
q,t (c) satisfy the operator identities
D(n)
q (−c± u; t)D(n)
q,t (c+ q/2) = D(n)
q,t (c)
∏
1≤i≤n
ϑ(zi ± u),
D(n)
q,t (c)D(n)
q (c± u; t) =
∏
1≤i≤n
ϑ(zi ± u)D(n)
q,t (c− q/2),
and, if u0 + u1 + u2 + u3 = q + 2c,
D(n)
q,t (c)D(n)
q (u0, u1, u2, u3; t) = D(n)
q (u0 − c, u1 − c, u2 − c, u3 − c; t)D(n)
q,t (c).
We also note the following fact, generalizing the first two identities.
Proposition 8.15. The operators D(n)
q,t (c) satisfy the operator identity∏
1≤i≤n
Γq(t0− d± zi)
Γq(t0 + d± zi)
D(n)
q,t (c+ d)
∏
1≤i≤n
Γq(t0− c± zi)
Γq(t0 + c± zi)
=D(n)
q,t (c)
∏
1≤i≤n
Γq(t0− c− d± zi)
Γq(t0 + c+ d± zi)
D(n)
q,t (d).
In particular, D(n)
q,t (c)−1 = D(n)
q,t (−c).
Proof. Consider the composition∏
1≤i≤n
Γq(t0 + d± zi)
Γq(t0 − d± zi)
D(n)
q,t (c)
∏
1≤i≤n
Γq(t0 − c− d± zi)
Γq(t0 + c+ d± zi)
D(n)
q,t (d)
∏
1≤i≤n
Γq(t0 + c± zi)
Γq(t0 − c± zi)
.
If we substitute
D(n)
q,t (d) = D(n)
q (−d± (t0 − c); t)D(n)
q,t (d+ q/2)
∏
1≤i≤n
ϑ(zi ± (t0 − c))−1,
then apply the easy relation∏
1≤i≤n
Γq(t0 − c− d± zi)
Γq(t0 + c+ d± zi)
D(n)
q (c− d− t0, t0 − c− d; t)
= D(n)
q (c− d− t0, t0 + c+ d; t)
∏
1≤i≤n
Γq(t0 + q/2− c− d± zi)
Γq(t0 + q/2 + c+ d± zi)
,
we can combine the two operators:
D(n)
q,t (c)D(n)
q (c± (t0 + d); t) =
∏
1≤i≤n
ϑ(zi ± (t0 + d))D(n)
q,t (c− q/2),
Elliptic Double Affine Hecke Algebras 115
and find that the result simplifies to the case (c, d, t0) 7→ (c − q/2, d + q/2, t0 + q/2) of the
above composition. In other words, the given operator is invariant under such translations, so
by density is invariant under any translation (c, d, t0) 7→ (c−u, d+u, t0 +u). Taking u = c gives∏
1≤i≤n
Γq(t0 + 2c+ d± zi)
Γq(t0 − d± zi)
D(n)
q,t (0)
∏
1≤i≤n
Γq(t0 − d± zi)
Γq(t0 + 2c+ d± zi)
D(n)
q,t (c+ d) = D(n)
q,t (c+ d),
since D(n)
q,t (0) = 1. �
Remark 8.16. Compare the proof of [23, Theorem 4.1]. The similarity in arguments is not
at all a coincidence: The analytic construction of D(n)
q,t (c) in terms of the interpolation kernel
of [27] implies that one can obtain the elliptic binomial coefficients of [23] as specializations of
the coefficients of D(n)
q,t (c), making [23, Theorem 4.1] a (Zariski dense) special case of the above
relation.
Remark 8.17. The action of the Fourier transform on objects is a reflection in an appropriate
inner product (the intersection form of the surface!), as are the generators of the W (Dm) action.
Each generator has a certain action on operators. If one takes into account the action on
parameters, the generators are involutions, and all relevant braid relations are satisfied, so that
this gives an action of a Coxeter group W (Em+1) on the base of the family. Showing that this
lifts to an action of W (Em+1) on the actual sheaf categories reduces to verifying lifts of each
braid relation, and the only nontrivial such lift reduces to the above identity.
We thus define a Fourier transform on formal difference operators in the following way. If the
formal operator D maps the polarization P0(2c; q, t) to the polarization P0(2c′; q, t), then its
Fourier transform D̂ is the operator
D(n)
q,t (c′)DD(n)
q,t (−c)
mapping P0(−2c; q, t) to P0(−2c′; q, t). There is some choice here (since the polarizations only
depend on 2c, 2c′), but luckily it is not particularly serious.
Lemma 8.18. If τ is a 2-torsion point, then
D(n)
q,t (c+ τ) = Tω(τ)D(n)
q,t (c) = D(n)
q,t (c)Tω(τ).
Proof. Indeed, the recurrence we used to solve for the coefficients of D(n)
q,t (c) is equivariant
under translation by 2-torsion. �
For our purposes, we will always be working in the subcategory with objects P0(−dq+η′; q, t),
and will take c, c′ in the Fourier transform to be the appropriate linear combination of q/2 and
some fixed η′/2. We have, of course, already computed some instances of the Fourier transform:∏
1≤i≤n
ϑ(zi ± u) 7→ D(n)
q (q/2− c± u; t),
D(n)
q (c+ q/2± u; t) 7→
∏
1≤i≤n
ϑ(zi ± u),
and
D(n)
q (u0, u1, u2, q+2c−u0−u1−u2; t) 7→ D(n)
q (u0−c, u1−c, u2−c, q+c−u0−u1−u2; t),
where in each case the input is a (general) section of S(n)
2c;q,t starting from the 0 object, of deg-
ree f , s, and s+ f respectively.
116 E.M. Rains
Theorem 8.19. If c′−c is an integer multiple of q/2, then the Fourier transform is holomorphic;
that is, the Fourier transform of any holomorphic family of operators is a holomorphic family
of operators.
Proof. The only issue is when q is torsion, as otherwise both D(n)
q,t (−c) and D(n)
q,t (c′) are holo-
morphic (in the sense that they have no z-independent poles other than those for q torsion).
Consider a multiplication operator h. If this is Cn-invariant, we can express it as a ratio
of holomorphic Cn-invariant theta functions. The algebra of such theta functions is generated
by functions
∏
1≤i≤n ϑ(u ± zi) = (−1)n
∏
1≤i≤n ϑ(zi ± u), and thus any holomorphic family
of Cn-invariant functions h has holomorphic Fourier transform. (The leading term of the Fourier
transform of an operator is easy to determine, so we find that the Fourier transform of the
denominator is indeed invertible.)
Now, let h be a general multiplication operator. To show that ĥ is holomorphic, we need to
show that every coefficient is holomorphic. The coefficient of
∏
i T
ki
i has denominator dividing∏
1≤j≤max(k1,...,kn) ϑ(jq), and by Hartog’s lemma it suffices to prove that the coefficient is holo-
morphic at the generic point of every component of the corresponding divisor. Each coefficient
is a finite linear combination of shifts of h, and we are evaluating it at a point with generic
(z1, . . . , zn). In particular, none of the points where we are evaluating h are in the same Cn
orbit (though we may be hitting the same point multiple times). It follows that there exists
a Cn-invariant function g such that the corresponding sum for h − g is holomorphic: simply
take g to be a very good approximation near the points where h is being evaluated. Since ĝ is
holomorphic and this coefficient of the Fourier transform of h− g is holomorphic, it follows that
the given coefficient of ĥ is holomorphic as required.
Now, let D be an operator of the form D
(n)
q (c + q/2 ± u; t), which again has a holomorphic
Fourier transform. If q 6= 0, then the space of operators k(X)D
(n)
q (c + q/2 ± u; t)k(X) is
a 2n-dimensional vector space on the left. Indeed, each of the 2n shifts that appear induce
different automorphisms of k(X). It follows that any element of that space has holomorphic
Fourier transform (except possibly where q = 0). Since that space contains elements ∝
∏
i T
±1/2
i
for every combination of signs, we have proved holomorphy of the Fourier transform on a set of
(topological) generators of the ring of twisted formal difference operators. The Fourier transform
is continuous with respect to the nonarchimedean metric, so the result follows in general.
It remains to consider the case q = 0. This splits into two components, depending on
whether q/2 = 0 or q/2 is nontrivial 2-torsion. The latter case reduces to the first, however,
since everything is invariant under translation by 2-torsion. We may thus restrict our attention
to the local ring at the generic point with q/2 = 0. In that case, the special fiber of the ring
of twisted formal difference operators is abelian, since all shifts are trivial. As a result, the
algebra over the local ring picks up an additional (Poisson bracket) operation on operators:
(D1, D2) 7→ (D1D2 − D2D1)/π, where π is a uniformizer. This takes any pair of holomorphic
families of operators to a holomorphic family of operators, and the Fourier transform respects
this operation. We may thus use this operation to construct operators with known holomorphic
Fourier transform. It turns out that the usual proof of independence of automorphisms of fields
can be expressed in terms of this operation, and thus we still obtain the full 2n-dimensional
space of operators. �
Remark. The proof for q = 0 is of course based on the standard fact that an automorphism of
a family of noncommutative algebras preserves the induced Poisson structure on any commuta-
tive fiber.
Of course, the algebra of formal difference operators is far too large, and doesn’t even have
an action of Cn (as it preserves neither the metric nor the topology). So we need to show that
Elliptic Double Affine Hecke Algebras 117
the operators we care about map to operators which not only have finite support, but have Cn
symmetry. Luckily, this is a closed condition, so it suffices to prove it generically.
Lemma 8.20. On the generic fiber, the Z/2Z-graded algebra
⋃
d S
(n)
2c;0,0(0, d(s+ f)) is generated
by S(n)
2c;0,0(0, s+ f).
Proof. In fact, we claim that for d� 0, S(n)
2c;0,0(0, d(s+f)) is spanned by products of d elements
of S(n)
2c;0,0(0, s+ f). Since this contains the spaces for all smaller d of the same parity, the result
will immediately follow.
Since this graded algebra is the homogeneous coordinate ring of Symn(P1 × P1), what we
are in fact claiming is that the ample bundle Symn(OP1×P1(1)) is very ample. In general,
it follows from [4, Section 1.3] that over any field of characteristic 0, Symn(OPm(1)) is very
ample on Symn(Pm), and thus the same holds for the symmetric power of any closed sub-
scheme of Pm. �
Remark. It is likely that this fails in small characteristic. It is certainly the case that the
line bundle Symn(OPm(1)) can fail to be very ample on Symn(Pm); indeed this already happens
for Sym3(P2) in characteristic 3. In addition, even in characteristic 0, the Z-graded algebra is not
generated in degree 1 if n is sufficiently large. Indeed, one has h0(Symn(OP1×P1(1))) =
(
n+3
3
)
,
while h0(Symn(OP1×P1(2))) =
(
n+8
8
)
. So for n � 0, even if we take into account noncommuta-
tivity, there are simply not enough sections of degree 1 for their products to account for every
section of degree 2!
Corollary 8.21. For any d, the Fourier transform induces an isomorphism of stalks
ΓS(n)
2c;q,t(0, d(s+ f))q=t=0
∼= ΓS(n)
−2c;q,t(0, d(s+ f))q=t=0.
Proof. Fix a basis of the global sections of the fiber over the generic point with q = t = 0.
Each such global section can be expressed as a polynomial in sections of degree 1; if we choose
an extension to the stalk for each degree 1 operator that appears, then the result will be a basis
of the stalk of degree d operators in which every element is a polynomial in first-order operators.
It follows that every element of the basis has Fourier transform in
⋃
e≥0 ΓS(n)
−2c;q,t(0, e(s+f))q=t=0,
but the Fourier transform clearly preserves the space of operators
∏
i T
−l/2
i k(X)[[T1, . . . , Tn]] for
each l, and thus the Fourier transform is actually in ΓS(n)
−2c;q,t(0, d(s + f))q=t=0 as required.
The inverse operation is of course just the Fourier transform again. �
Corollary 8.22. For any d, the Fourier transform induces a (local) isomorphism of sheaves of
categories ΓS(n)
2c;q,t|Z(s+f)
∼= ΓS(n)
−2c;q,t|Z(s+f).
Proof. The given Hom sheaves of the global section category are flat and the map to difference
operators is injective on fibers. We may thus identify sections with holomorphic families of
difference operators and apply the Fourier transform to obtain a holomorphic family of formal
difference operators. The generic point of this family is a section of the other global section
category, and thus the family itself is a section. �
Corollary 8.23. For d ≤ d′, the Fourier transform induces a morphism
ΓS(n)
2c;q,t(0, ds+ d′f)→ ΓS(n)
−2c;q,t(0, d
′s+ df).
Proof. If d = 0, this is easy, as the algebra is generated in degree 1, and we know the result
there. More generally, given a section D of ΓS(n)
2c;q,t(0, ds+d′f) and any section g of ΓS(n)
−2c;q,t(d
′s+
118 E.M. Rains
df, d′s + d′f), consider the composition ĝD ∈ ΓS(n)
2c;q,t(0, d
′s + d′f), which makes sense since g
has degree (d′ − d)f . Since the Fourier transform is a covariant involution, we find that ĝD has
Fourier transform gD̂, so that gD̂ is a section of ΓS(n)
−2c;q,t(0, d
′s+d′f) for any g. But this implies
that D̂ is actually a section of ΓS(n)
−2c;q,t(0, d
′s+ df) as required. �
Corollary 8.24. For any d, d′, the sheaf ΓS(n)
2c;q,t(0, ds + d′f) is flat and the map to difference
operators is injective on fibers. Moreover, the Fourier transform induces an isomorphism
ΓS(n)
2c;q,t(0, ds+ d′f) ∼= ΓS(n)
−2c;q,t(0, d
′s+ df)
for all d, d′.
Proof. We already have flatness if d < 0 or d ≤ d′, so suppose d ≥ d′. It suffices to show
injectivity on fibers, as it implies that any Tor1 of the cokernel is 0. Thus, let D ∈ ΓS(n)
2c;q,t(0, ds+
d′f) be a local section such that the corresponding difference operator vanishes on some fiber.
Consider the Fourier transform
ΓS(n)
−2c;q,t(0, d
′s+ df)→ ΓS(n)
2c;q,t(0, ds+ d′f).
The domain is locally free and injective on fibers, and the codomain is at least generically
free of the same rank. Since the Fourier transform is invertible at the level of operators, this
map is injective, and thus generically an isomorphism. It follows that there is a section D′ ∈
ΓS(n)
−2c;q,t(0, d
′s+ df) such that D− D̂′ is generically 0. But this, of course, implies that D′ = D̂.
In particular, the corresponding fiber of D′ vanishes, which means that in a suitable local basis
we have D′ =
∑
i ciDi, in which each ci vanishes on a divisor passing through that fiber. We then
have D =
∑
i ciD̂i with each D̂i a local section of ΓS(n)
2c;q,t(0, ds+d′f). It follows that the section
corresponding to D vanishes at the fiber, so that injectivity holds.
In particular, the Fourier transform induces a morphism in both directions, and thus gives
an isomorphism as required. �
To finish the proof of the theorem, we need to show that the transform respects Bruhat
order, that it respects the vanishing conditions associated to x1, . . . , xm, and that it commutes
with the Selberg adjoint. Each of these have analogous statements for general formal difference
operators, and in the first two cases reduce to the fact that (due to continuity) the Fourier
transform affects leading coefficients in easy to control ways.
For the Bruhat order, we actually obtain a finer (inclusion) partial order in the formal setting.
Proposition 8.25. Let D be a holomorphic family of formal difference operators from P0(2c; q, t)
to P0(2c + lq; q, t). Let S ⊂ Zn ∪ (1/2, . . . , 1/2)Zn be the set of vectors ~v such that for some
~k ∈ Nn, the left coefficient of
∏
i T
vi−ki
i is nonzero, and let Ŝ be the corresponding set for D̂.
Then Ŝ = (l/2, . . . , l/2) + S.
Proof. Conjugating by Tω(c) or a Γq symbol has no effect on the support of an operator,
and multiplication by Tω(lq/2) shifts the support by (l/2, . . . , l/2). The remaining operation
consists of left- and right-multiplication by units in k(X)[[T1, . . . , Tn]], and this clearly preserves
the set S. �
For the vanishing conditions, we have the following. Note that we only consider half of the
vanishing conditions, as in the formal setting it only makes sense to consider conditions on the
leading few terms. Also, for convenience, we only consider the generic case.
Elliptic Double Affine Hecke Algebras 119
Proposition 8.26. Over the generic point (E, x, c, q, t) ∈ E4 and for integers r, l, consider the
space of formal difference operators D mapping P0(2c; q, t) to P0(2c+lq; q, t) such that D
∏
i T
r/2
i
involves only integer shifts. If the left coefficients of both D and∏
1≤i≤n
Γq(x+ rq/2− zi)−1D
∏
1≤i≤n
Γq(x− zi)
are holomorphic along all hypersurfaces of the form zi ∈ x + rq/2 + kq, k ∈ Z, then the left
coefficients of both D̂ and∏
1≤i≤n
Γq(x− c+ (r − l)q/2− zi)−1D̂
∏
1≤i≤n
Γq(x− c− zi)
are holomorphic along all hyperplanes of the form zi ∈ x− c+ (r − l)q/2 + kq, k ∈ Z.
Proof. By definition, we have
D̂ = D(n)
q,t (c+ lq/2)DD(n)
q,t (−c).
There are only countably many hypersurfaces of the form zi = y on which some left coefficient
of D(n)
q,t (−c) and D(n)
q,t (c + lq/2) has a pole (including poles of the meromorphic sections of
equivariant gerbes corresponding to the leading coefficients). Since x is generic, it follows that
all three factors on the right are holomorphic on the given orbits of hypersurfaces, and thus so
is the product.
The claim for∏
1≤i≤n
Γq(x− c+ (r − l)q/2− zi)−1D̂
∏
1≤i≤n
Γq(x− c− zi)
analogously reduces to checking possible poles of∏
1≤i≤n
Γq(x− zi)−1D(n)
q,t (−c)
∏
1≤i≤n
Γq(x− c− zi)
and ∏
1≤i≤n
Γq(x− c+ (r − l)q/2− zi)−1D(n)
q,t (c+ lq/2)
∏
1≤i≤n
Γq(x+ rq/2− zi).
In each case, the gauging only multiplies the coefficients by holomorphic theta functions, so
cannot introduce any new poles. �
To get an analogue for the Selberg adjoint, there is a mild difficulty coming from the fact
that the Selberg adjoint was only defined for Cn-symmetric operators, and the obvious extension
does not make sense for formal operators. Luckily, the formal adjoint with respect to the inner
product∫
f(z1, . . . , zn)g(−z1, . . . ,−zn)
∏
1≤i<j≤n
Γq(t± zi ± zj)
Γq(±zi ± zj)
∏
1≤i≤n
1
Γq(±2zi)
dT
does make sense for formal difference operators and formal gauging operators and agrees with
the Selberg adjoint in the Cn-symmetric case. Using this as the definition of the Selberg adjoint
for formal operators gives the following, which immediately implies consistency of the Fourier
transform with the Selberg adjoint.
120 E.M. Rains
Proposition 8.27. The operators D(n)
q,t (c) are self-adjoint under the Selberg adjoint.
Proof. The Selberg adjoint has the correct leading term, so it suffices to show that
D(n)
q,t (c)adtD(n)
q (c± u; t) =
∏
1≤i≤n
ϑ(zi ± u)D(n)
q,t (c− q/2)adt .
Since
∏
1≤i≤n ϑ(zi ± u) is self-adjoint, this reduces to checking that
D(n)
q (c± u; t)adt = D(n)
q (q/2− c± u; t),
an easy verification. �
Remark. In fact, one has in general
D(n)
q (u0, . . . , u2d′+1; t)adt = D(n)
q (q/2− u0, . . . , q/2− u2d′+1; t),
either by a direct computation or by using the fact that both are sections of the same Hom
sheaf, and with the same leading coefficient.
We mention a couple of further consequences of the proof. First, the fact that the Fourier
transform is determined by its values where we know it explicitly has consequences in the analytic
setting. Indeed, in [27], a kernel function K(n)
c (~x; ~y; q, t) was constructed, with the property that
for D of degree s, f , or s+ f , one had
D~xK(n)
c (~x; ~y; q, t) = D̂adt
~y K
(n)
c (~x; ~y; q, t).
It follows from the above proof and continuity that this holds for all operators which are global
sections of the appropriate Hom spaces. In particular, this applies to operators of degree 2s+2f−
e1−· · ·−e8 (i.e., the van Diejen/Komori–Hikami integrable system considered in Theorem 7.28),
showing that the associated formal integral operator takes eigenvalue equations of this form
to eigenvalue equations of the same form.
Also, we have already mentioned the consequence that the resulting deformations of Symn(P2)
only depend (geometrically) on E, q, and t. It is worth mentioning the specific form that the
given isomorphisms take. The isomorphism ΓS ′(n)
x0;q,t|Z(s+f)
∼= ΓS ′(n)
x1;q,t|Z(s+f) is given (up to
a choice of element (x0 − x1)/3) by gauging by the operator
Gd(x0, x1) :=
∏
1≤i≤n
Γq
(
−(d− 1)q
2
− 2x0 + x1
3
± zi
)
D(n)
q,t ((x0 − x1)/3)
×
∏
1≤i≤n
Γq
(
−(d− 1)q
2
− x0 + 2x1
3
± zi
)−1
in degree d; i.e., the action on morphisms from d1(s+ f) to d2(s+ f) is given by
D 7→ Gd2(x0, x1)DGd1(x0, x1)−1.
In particular, we see that when (x0−x1)/3 is 3-torsion, the resulting automorphism is still quite
nontrivial. In addition, the translation symmetry of the category involves changing x0, and thus
although one can identify it with a graded algebra at the cost of choosing an element q/3, the
resulting graded algebra does not actually have a representation in (finite) difference operators.
If we restrict to the “anticanonical” model, i.e., to Z(3s + 3f), then the Hom space of degree
3s+ 3f contains the 1-dimensional subspace of operators of degree s, spanned by
G0(x0, x0 − 3q/2) =
∏
1≤i≤n
Γq(−x0 ± zi)D(n)
q,t (−q/2)
∏
1≤i≤n
Γq(−q/2− x0 ± zi)−1.
Elliptic Double Affine Hecke Algebras 121
If we adjoin the formal inverse of such an operator, then the result in degree 0 may be identified
with a filtered algebra of formal difference operators. We can include elements of degree not
a multiple of 3 at the cost of choosing q/3 and allowing some Γq factors and shifts by multiples
of q/3. Indeed, Proposition 8.15 tells us (assuming compatible choices when dividing by 3) that
Gd(x1, x2)Gd(x0, x2) = Gd(x0, x2), and thus the various isomorphisms between categories with
parameter x0 + kq/2 are all compatible. It follows that if we compose an operator mapping
d1(s + f) to d2(s + f) with the formal operators giving isomorphisms x0 7→ x0 + d1q/2 and
x0 + d2q/2 7→ x0, then the result will be compatible with compositions and will be the same as
if we only used the spaces with d1 = 0. Of course, even in the univariate setting, the resulting
algebra is not likely to be easy to describe in any direct fashion!
One final thing to mention is that our description of first-order operators in Lemma 8.8 as well
as our description of the operators D(n)
q,t (c) are both quite well suited to considering degenerations
of the tuple (E, c, q, t). In light of the fact that operators of degree s + f are generically very
ample, we can give at least indirect descriptions of the limiting algebras
⋃
d S
(n)
η′;q,t(0, d(s + f))
by specifying their elements of degree 1, and understanding the extension to the whole category
simply requires keeping track of the elements of degree f as well.
Taking the limit can be somewhat tricky in general, as it may be necessary to gauge by
suitable functions before the limit is well-defined. The simplest approach is to choose a suitable
gauge transformation to make the operators elliptic before taking the limit; this introduces
additional parameters which we can then eliminate by a further limit. Indeed, we find that for
any operator D ∈ ΓS(n)
η′;q,t(0, ds+ d′f), the gauge transformation
∏
1≤i≤n
Γq(η
′ − dq/2− d′q + (n− 1)t+ v0 + v1 + v2 ± zi)
Γq(−dq/2 + v0 ± zi,−dq/2 + v1 ± zi,−dq/2 + v2 ± zi)
×D
∏
1≤i≤n
Γq(v0 ± zi, v1 ± zi, v2 ± zi)
Γq(η′ + (n− 1)t+ v0 + v1 + v2 ± zi)
is elliptic (for fixed v0, v1, v2). We can then take the limit as p→ 0 and remove the parameters
by gauging back by an appropriate product of q-Pochhammer symbols (x; q)∞ :=
∏
0≤j(1−qjx).
We obtain a limit ΓS(n)
η′;q,t;∗(0, s+ d′f) (with q, t, η′ and ~z in the multiplicative group) consisting
of operators of the form
∑
σ∈{±1}n
∏
1≤i≤n
zσii f(zσii )
1− z2σi
i
∏
1≤i<j≤n
1− tzσii z
σj
j
1− zσii z
σj
j
∏
1≤i≤n
T
σi/2
i ,
where f(z) is a univariate Laurent polynomial with exponents ranging from 1 − d′ to d′ − 1
satisfying the condition [zd
′−1]f(z) = qη′[z1−d′ ]f(z) on its extreme coefficients. The Fourier
transformation has a corresponding limit, represented by operators D(n)
q,t:∗(c) satisfying
D(n)
q,t:∗(c)
∏
1≤i≤n
((vcd)z±1
i ; q)∞
((v/cd)z±1
i ; q)∞
D(n)
q,t:∗(d)
=
∏
1≤i≤n
((vd)z±1
i ; q)∞
((v/d)z±1
i ; q)∞
D(n)
q,t:∗(cd)
∏
1≤i≤n
((vc)z±1
i ; q)∞
((v/c)z±1
i ; q)∞
(8.1)
with leading term∏
1≤i≤n
θq1/2(−1/zi)
θq1/2(−1/czi)
∏
1≤i≤j≤n
(1/c2zizj ; q)∞
(1/zizj ; q)∞
∏
1≤i<j≤n
(t/zizj ; q)∞
(t/c2zizj ; q)∞
Tω(c)
122 E.M. Rains
and special case
D(n)
q,t:∗(q
−1/2) =
∑
σ∈{±1}n
∏
1≤i≤n
zσii
1− z2σi
i
∏
1≤i<j≤n
1− tzσii z
σj
j
1− zσii z
σj
j
∏
1≤i≤n
T
σi/2
i .
Note that taking v = 0 in (8.1) gives D(n)
q,t:∗(c)D
(n)
q,t:∗(d) = D(n)
q,t:∗(cd), so that we may interpret
D(n)
q,t:∗(c) as a fractional power of the operator for c = q−1/2, which in turn is a lowering operator
appearing in the theory of Koornwinder polynomials. We can further extend this limit to the case
of blowups in sufficiently general position (i.e., with all xi finite) by imposing the appropriate
conditions on the leading coefficients; this is how we tested the n = r = 2 case of Conjecture 7.30.
Another noteworthy limit involves gauging by a translation so as to break the z 7→ 1/z sym-
metry, and taking a limit in the resulting parameter. This gives a Fourier transform represented
by operators satisfying
D(n)
q,t:∗∗(c)
∏
1≤i≤n
((vcd)zi; q)∞
((v/cd)zi; q)∞
D(n)
q,t:∗∗(d) =
∏
1≤i≤n
((vd)zi; q)∞
((v/d)zi; q)∞
D(n)
q,t:∗∗(cd)
∏
1≤i≤n
((vc)zi; q)∞
((v/c)zi; q)∞
with leading term
∏
1≤i≤n
θq1/2(−1/zi)
θq1/2(−1/czi)
Tω(c)
and special case
D(n)
q,t:∗∗(q
−1/2) =
∑
I⊂{1,...,n}
(−1)|I|t|I|(|I|−1)/2
∏
1≤i≤n
z−1
i
∏
i∈I,j /∈I
zj − tzi
zj − zi
∏
i∈I
T
1/2
i
∏
i/∈I
T
−1/2
i ,
a.k.a. the lowering operator for GLn-type Macdonald polynomials. The Hom spaces of deg-
ree s + d′f have similar, if somewhat more complicated forms, obtained by gauging the q−1/2
case of the Fourier transform operator by suitable products of Pochhammer symbols. We omit
the details, except to note that the results again look like operators arising in Macdonald theory.
There are some other symmetry breaking limits (e.g., the image of η′ → 0 under the Fourier
transform); we omit the details. Of course, such symmetry-breaking limits have unfortunate
effects on the Bruhat ordering; for instance, the “leading term” must now incorporate all ∼ 2n
Sn-orbits corresponding to the given Cn-orbit of weights. As a result, in more degenerate cases,
it can be difficult to figure out the correct way to compactify the algebra. This can be fixed in
some cases by realizing that the Sn-symmetric operator is actually a shadow of a Cn-symmetric
operator acting on a power of a reducible curve (a hyperelliptic curve of arithmetic genus 1).
Similarly, there are differential limits living on a power of the nonreduced curve y2 = 0.
9 Deformations of Hilbert schemes
One potential issue with studying the category of sheaves on S, even if one can resolve the un-
certainty in the definition, is that the commutative projective scheme it deforms, the symmetric
power of a surface, is singular. This suggests that one should look for an analogous deformation
in which the symmetric power is replaced by its natural resolution, namely the Hilbert scheme
of points.
The Picard group of the n-point Hilbert scheme of X is still discrete (and isomorphic to
the Néron–Severi group), with Pic(Hilbn(X)) ∼= Pic(X) ⊕ Z for n > 1. The copy of Pic(X)
in Pic(Hilbn(X)) is the pullback of Pic(Symn(X)) ∼= Pic(X), and since the map Hilbn(X) →
Elliptic Double Affine Hecke Algebras 123
Symn(X) is a birational morphism, the global sections of any such bundle will be the same
on either 2n-fold. Thus if we deform the category of line bundles on Hilbn(X), the subcategory
corresponding to Symn(X) should be precisely S. As we saw for the symmetric power, twisting
by line bundles cannot be expected to give an endofunctor, and thus we expect that twisting by
the additional generator of Pic(Hilbn(X)) should also change the parameters.
There is, of course, only one remaining parameter we could reasonably shift, and thus
we should consider what happens as we change t. Consider for the moment the P1 × P1 case.
The sections of S are (for generic t) cut out by the condition that both D and∏
1≤i<j≤n
Γq(t± zi ± zj)D
∏
1≤i<j≤n
Γq(t± zi ± zj)−1
are sections of the corresponding parameter-free category. Incorporating a shift in t here is then
mostly straightforward: we should be considering operators such that both D and∏
1≤i<j≤n
Γq(t+ a2q ± zi ± zj)D
∏
1≤i<j≤n
Γq(t+ a1q ± zi ± zj)−1
are holomorphic for some integers a1 and a2. We should also note the effect on twisting, which
boils down to noting that shifting t acts on the standard polarization as Pd(η
′; q, t + aq) =
Pd−(n−1)a(η
′; q, t).
With this in mind (and incorporating the conditions corresponding to x1, . . . , xm), define
a product of Γq symbols for any element of Pic(Hilbn(X)) = Z〈δ, s, f, e1, . . . , em〉 as follows:
∆x1,...,xm;q,t(aδ + ds+ d′f − r1e1 − · · · − rmem;~z)
=
∏
1≤i<j≤n
Γq(t+ aq ± zi ± zj)
∏
1≤i≤n
1≤j≤m
Γq((rj + (1− d)/2)q − xj ± zi).
Also, let π : Pic(Hilbn(X)) → Z〈s, f〉 be given by π(s) = s, π(f) = f , π(ei) = 0 and π(δ) =
−(n−1)f . Then (for n > 1) S(n)
η′,x1,...,xm;q,t(v, w) is defined to be the subsheaf of S(n)
η′;q(π(v), π(w))
(the spherical algebra of the construction corresponding to the master DAHA) consisting for
generic parameters of the operators D such that ∆x1,...,xm;q,t(w, ~z)D∆x1,...,xm;q,t(v, ~z)
−1 is also
a local section of the correspondingly twisted master spherical algebra. It is easy to see that
when v and w have the same coefficient of δ, this imposes the same vanishing conditions as
the original definition of S, and thus this indeed extends the category to the larger group of
objects. The elementary transformation symmetries automatically extend (using the obvious
definition for S ′(n)), as does the t 7→ q− t symmetry, which now acts nontrivially on the objects
(negating δ); for the Fourier transform, see below.
We should note in passing that this is closely related to a construction valid for general affine
Weyl groups. We have already mentioned that translating part of the system of parameters of
such a DAHA or spherical algebra by q gives a Morita equivalent algebra, and thus in particular
a natural bimodule. More generally, we may translate any point of any Ti by any multiple of q
without affecting generic Morita equivalence, and the associated bimodule can be constructed
as the tensor product of the bimodules for the atomic shifts. As a result, we may construct
a sheaf category in which the objects are the lattice of different ways we may shift the pa-
rameters (i.e., with rank equal to the total number of components of the independent Ti), and
each Hom bimodule is the relevant Morita equivalence. Each of the original bimodules embeds
in H
W̃ ,W ;γ
(X) (or H
W̃ ;γ
(X) in the DAHA version) as operators satisfying appropriate vanishing
conditions and thus the construction has a natural extension to general parameters. (Note that
this construction only allows us to shift those parameters which are associated to roots of W
itself, so to obtain the C∨Cn version in this way we need to associate the ~x parameters to sn
124 E.M. Rains
rather than s0.) This may be thought of as a generalization (to the elliptic level and arbitrary
numbers of parameters) of the construction of [12].
As in the symmetric power case, we would like to show that the above generic conditions
extend to give a strongly flat family of categories (i.e., not only flat but such that each fiber
injects in the algebra of meromorphic difference operators). The first step of the construction
carries over: replacing the t-dependent factor of ∆ by the appropriate nonsymmetric version
gives us a natural corresponding extension of H. Unfortunately, the argument of Theorem 7.12
fails in this case to show flatness of the extended H. The key difference here is that the roots
involved in the conditions corresponding to x1, . . . , xm form a root system of type An1 . If we
make the analogous construction for any other root system, then we will find that there are
some leading terms for which the only vanishing condition involves a non-simple root, and such
that any of the terms covered by that leading term do not have the corresponding vanishing
condition. In particular, since the t-dependent vanishing conditions live on a root system of
type Dn, this argument fails, with the notable exception of n = 2.
Proposition 9.1. The sheaf category S(2)
η′,x1,...,xm;q,t defined above is locally free, and the map
from any Hom bimodule to the sheaf bimodule of meromorphic operators is injective on fibers.
Proof. We argue essentially as in the symmetric power case. We first note that by using
elementary transformations and the analogue for t, that we may arrange for the vanishing
condition on the identity to be trivial. Then for any w in the appropriate Bruhat interval, if
there is any nontrivial vanishing condition associated to t or xi in the interval, then there is
such a condition at w and associated to a simple reflection that makes w shorter. Thus we may
obtain the restriction of the given Hom sheaf to [≤ w] by composing the appropriate rank 1
Hecke algebra (using only those parameter still active) to the restriction to [≤ sw]. �
Remark. For each simple root of W and each associated parameter t or xi, the existence of
a nontrivial vanishing condition of cw along t+α or xi +α is equivalent to the alcove associated
to w being on the appropriate side of a hyperplane orthogonal to α. For n = 2, there is
thus a square (itself a Bruhat order ideal) such that any alcove in the square has no vanishing
condition at t (so we may ignore the parameter) and similarly for each xi. For n > 2, there is
a corresponding shape in which no simple root (or its negative) has a vanishing condition for t;
if one could prove strong flatness for the corresponding Bruhat order ideal, this would imply
strong flatness in general.
There is, however, one more special case in which we can prove strong flatness. This has
to do with the additional symmetry of the spherical algebra in the DAHA case. In particular,
we noted that both the algebra for t and the algebra for t+ q could be obtained from the Hecke
algebra via an analogue of the construction of the spherical algebra, and that moreover there
were analogous constructions of intertwining bimodules. But these intertwiners are precisely the
Hom sheaves we want.
Proposition 9.2. Suppose the coefficient of δ in v is −1, 0, or 1. Then S(n)
η′,x1,...,xm;q,t(0, v) is
locally free and injects on fibers into the sheaf bimodules of meromorphic operators.
Remark. Actually, there are two more cases in which we can prove strong flatness. If a ≥ d,
then the t-dependent vanishing conditions become vacuous, and thus the claim reduces to strong
flatness in the version without a t parameter, where the usual argument works. By the t 7→ q− t
symmetry, this implies strong flatness when a ≤ −d.
Not only is this further evidence that this construction is well-behaved, but this also has seve-
ral useful consequences. The most significant of these is that the construction of the intertwining
Elliptic Double Affine Hecke Algebras 125
bimodules allows us to understand the case t = 0. In particular, let v ∈ Z〈s, f, e1, . . . , em〉, and
consider the Hom sheaf
S(n)
η′,x1,...,xm;q,0(0, δ + v).
By the infinite analogue of Proposition 4.64, these operators are all of the form
∑
w∈Cn wD,
where D is a section of the left twist by OX(DwCn ) of the corresponding spherical module. Since
t = 0, the spherical module is just the tensor product of n copies of the univariate spherical
module M
(1)
η′,x1,...,xm;q(v), and thus the Hom sheaf is spanned by elements of the form( ∑
w∈Cn
w
)
1∏
1≤i≤n ϑ(2zi)
∏
1≤i<j≤n ϑ(zi ± zj)
D1(z1) · · ·Dn(zn)
with each Di in the univariate spherical module. Summing over the normal subgroup of order 2n
turns each Di into a general element of the corresponding spherical algebra, giving∑
π∈Sn
π
1∏
1≤i<j≤n ϑ(zi ± zj)
D′1(z1) · · ·D′n(zn) =
1∏
1≤i<j≤n ϑ(zi ± zj)
det
1≤i,j≤n
D′i(zj)
with each D′i in the univariate spherical algebra. (Note that since these are univariate operators
in distinct variables, the determinant makes sense.) We thus conclude the following.
Proposition 9.3. For any v ∈ 〈s, f, e1, . . . , em〉, there is a natural isomorphism of sheaf bimo-
dules
S(n)
η′,x1,...,xm;q,0(0, δ + v) ∼= ∧nS(1)
η′,x1,...,xm;q,0(0, v).
If q = 0 and v corresponds to an acyclic line bundle on the corresponding rational projective
surface Xm, then the exterior power is also acyclic and we deduce
ΓS(n)
η′,x1,...,xm;0,0(0, δ + v) ∼= ∧nΓ(Xm;OXm(v)).
Any element of either side (which differ only in the product of theta functions corresponding to
the Dn roots) gives a rational function on Xn
m, and the description as an exterior power tells
us that the corresponding map to projective space is in fact a map to a Grassmannian: each
point in Xm determines a linear functional on Γ(Xm;OXm(v)), and the map takes Xn
m to the
n-dimensional span of the corresponding linear functionals.
This is clearly invariant under permutations of the n-tuple, so gives a rational map on the
symmetric power, and thus on the Hilbert scheme. Since we may also view this as the Grass-
mannian of (h0(OXm(v))−n)-dimensional subspaces of Γ(Xm;OXm(v)), we see that the map on
the Hilbert scheme takes a given ideal sheaf I to the subspace Γ(Xm; I(v)), assuming this has
the correct dimension. In particular, for any sufficiently ample v, I(v) will always be acyclic and
globally generated, and thus we obtain an embedding of the Hilbert scheme in the Grassman-
nian. From known facts about acyclicity and Hilbert polynomials of line bundles on the Hilbert
scheme [9], the Plücker coordinates in fact form all global sections of the given line bundle on
the Hilbert scheme. (This line bundle has the form −∆/2 + v, where ∆ is the discriminant, i.e.,
the divisor where the corresponding n-point subscheme is nonreduced.)
We thus conclude that for q = t = 0 and v sufficiently (relatively) ample on Xm, the (left)
vector bundle S(n)
η′,x1,...,xm;0,0(0, δ + v) on Pn may be identified with the direct image of OXm(v)
under the map Hilbn(Xm) → Symn(P1) ∼= Pn. Moreover, since we obtained this identification
by using the actual values of the “operators” in S, these elements satisfy precisely the same
relations as their counterparts on the Hilbert scheme.
126 E.M. Rains
Note that although this indicates that there is a strong relation between our construction and
the Hilbert scheme, it is not quite enough to tell us that S is a true deformation of the Hilbert
scheme: the problem is that there could conceivably be additional global sections involving
larger multiples of δ. For n = 2, we can resolve this issue: it is straightforward (if tedious)
to use the Bruhat filtration to compute the Euler characteristics of Hom sheaves of S, and we
find in particular that when a(−∆/2) + v is acyclic over Pn on the Hilbert scheme, the Euler
characteristic of the Hom sheaf agrees with the Euler characteristic of the corresponding line
bundle on the Hilbert scheme. Since we’ve shown an isomorphism for a large class of very
ample divisors, it follows that there is a cone in which the direct image of the line bundle on
the Hilbert scheme injects in the corresponding Hom sheaf, and must therefore be isomorphic.
We thus conclude that (in a somewhat vague sense) S(2) is indeed the desired deformation of
the Hilbert scheme. (Note that this argument does not apply to the obvious category associated
to Hilb2
(
P2
)
in the absence of additional acyclicity results.)
Unfortunately, the calculation of Euler characteristics of Hom sheaves is not only tedious
but subject to combinatorial explosion: the contribution from each subquotient depends in
a nontrivial and apparently nonuniform way on the corresponding parabolic subgroup, and thus
the general computation requires computing a subsum for each parabolic subgroup, a total of 2n
individual sums. Moreover, the condition under which the divisor on the Hilbert scheme is
acyclic involves an upper bound on a depending on d, d′, etc., and that bound is not preserved
if we subtract 2s+ 2f − e1 − · · · − em; as a result, we cannot expect to simplify the calculation
by using an induction on d and n. A further issue arises in the final step of comparing to the
Hilbert scheme: the formula of [9] is not quite explicit; it depends on a certain universal power
series which is only known in low degree, and thus the Euler characteristics are only known for
small n.
Still, it seems reasonable on the above evidence to conjecture that S indeed provides a flat
deformation of Hilbn(Xm) for every n.
Assuming that this is true, we would like to know that the deformation depends only on the
original rational surface rather than on the way in which we obtained it as a blowup. As in
the symmetric power case, elementary transformations are easy to deal with, and it is only the
analogue of the Fourier transform that we must consider. There is a clear guess as to how
the Fourier transformation should be defined, at least when q is not torsion: use the same
operators Dq,t(c) and simply adjust t as appropriate. This leads to an obvious question, namely
whether this extension of the Fourier transform is still well-defined when q is torsion. As in the
symmetric power case, we can first ask for this to hold for formal difference operators, where it
takes the form
D 7→ D(n)
q,t+aq(c+ bq/2)DD(n)
q,t (−c)
for a, b ∈ Z. Since we know the usual Fourier transform
D̂ = D(n)
q,t (c+ (b+ (n− 1)a)q/2)DD(n)
q,t (−c)
is well-defined, we may factor this as
D(n)
q,t+aq(c+ bq/2)D(n)
q,t (−c− (b+ (n− 1)a)q/2)D̂,
and thus reduce to showing that the formal operator
D(n)
q,t+aq(c+ bq/2)D(n)
q,t (−c− (b− (n− 1)a)q/2)
remains holomorphic when q is torsion. This then easily reduces to the analogous statement for
D(n)
q,t+q(−c− (n− 1)q/2)D(n)
q,t (c).
Elliptic Double Affine Hecke Algebras 127
Indeed, for a > 0, we can factor the operator in question into a operators of the given form,
while for a < 0 we have a similar factorization of the inverse, and the leading term is clearly
nonzero.
Proposition 9.4. The formal operator
D(n)
q,t+q(−c− (n− 1)q/2)D(n)
q,t (c)
is a global section of S(n)
2c;q,t(0, δ + (n− 1)s).
Proof. The Hom sheaf S(n)
2c;q,t(0, δ + (n − 1)s) is a vector bundle, and the description as an
alternating power for t = 0 tells us that it has Euler characteristic 1 and is acyclic away from
a finite set of hypersurfaces, not containing any component on which q is torsion. Similarly,
S(n)
2c;q,t(−s, δ + (n − 1)s) has Euler characteristic n + 1 and is also acyclic on a suitable open
subset. Moreover, a section of either vector bundle with vanishing leading coefficient is 0;
by semicontinuity (and flatness for Bruhat intervals), it suffices to check this for t = 0, where
it again reduces to facts about the univariate case. Let D(c) be any nonzero local section
of S(n)
2c;q,t(0, δ + (n− 1)s). Then as u varies,
D(c)D(n)
q (c± u; t)
spans an n+ 1-dimensional family of sections of S(n)
2c;q,t(−s, δ + (n− 1)s), which must therefore
be everything. Applying the same argument on the left and comparing the dependence of the
leading coefficient on u, we conclude that
D(n)
q ((n− 1)q/2 + c± u; t+ q)−1D(c)D(n)
q (c± u; t)
is a section of S(n)
2c;q,t(−s, δ + (n− 2)s) = S(n)
2c−q;q,t(0, δ + (n− 1)s) and thus
D(n)
q ((n− 1)q/2 + c± u; t+ q)−1D(c)D(n)
q (c± u; t) ∝ D(c− q/2),
with coefficient independent of zi and u.
Now, D(c) has leading term
F (~z; c; q, t)
∏
1≤i≤j≤n
1
ϑ(−zi − zj ; q)n−1
∏
1≤i<j≤n
ϑ(q + t− zi − zj ; q)n−2Tω(−(n− 1)q/2),
with F (~z; c; q, t) holomorphic and Sn-invariant. The relation between D(c) and D(c− q/2) gives
a weak recurrence for the leading coefficient, namely that
F (z1 − q/2, . . . , zn − q/2; c− q/2; q, t)F (z1, . . . , zn; c; q, t)−1
is independent of ~z, and thus F factors as
F (~z; c; q, t) = G(c− ~z; q, t)H(c; q, t),
where we may as well absorb any c-dependence of H into D(c; q, t) and thus make the recurrence
exact.
If we specialize c = −(n−1)q/2, then we still find that S(n)
−(n−1)q;q,t(0, δ+(n−1)s) is generically
1-dimensional by reference to the t = 0 case. Since the operator D
(n)
q,t (n− 1) satisfies all of the
128 E.M. Rains
requisite vanishing conditions, we conclude that D(−(n− 1)q/2) is proportional to D
(n)
q,t (n− 1),
and thus that
G(−(n− 1)q/2− ~z; q, t) ∝
∏
1≤i<j≤n
ϑ(t− zi − zj),
so that (after rescaling) D(c) has leading term
∏
1≤i≤j≤n
1
ϑ(−zi − zj ; q)n−1
∏
1≤i<j≤n
ϑ(q + t− zi − zj ; q)n−2ϑ(2c+ t+ (n− 1)q − zi − zj)
× Tω(−(n− 1)q/2).
We then find that
D(n)
q,t+q(c+ (n− 1)q/2)D(c)
has the same leading term and satisfies the same defining recurrence as D(n)
q,t (c), and thus by the
proof of Proposition 8.13,
D(c) = D(n)
q,t+q(−c− (n− 1)q/2)D(n)
q,t (c),
from which the desired claim immediately follows. �
Remark. When n = 2, this is a first-order operator, and thus one can easily deduce an explicit
formula from the leading coefficient. This turns out to be an operator we have already seen,
albeit with a rather odd change of parameters: D(2)
q,t+q(−c−q/2)D(2)
q,t (c) = D
(2)
q,t+q+2c(1), a special
case of the curious identity D(2)
q,2u1
(u2 − u3)D(2)
q,2u2
(u3 − u1)D(2)
q,2u3
(u1 − u2) = 1. The latter can
be proved by Zariski closure from the case u2 = kq/2 + u3, which in turn follows by induction
in k from the known special case.
The operator considered in the proposition is the Fourier transform of the global section
1 ∈ ΓS(n)
−2c;q,t(0, δ + (n− 1)f).
As mentioned above, the usual t 7→ q − t symmetry extends to a symmetry that negates δ, and
thus we also find that the Fourier transform of the global section∏
1≤i<j≤n
ϑ(t± zi ± zj) ∈ ΓS(n)
−2c;q,t(0,−δ + (n− 1)f)
is a section of ΓS(n)
2c;q,t(0,−δ + (n− 1)s). An easy induction then shows that
D(n)
q,t (−c− a(n− 1)q/2)
∏
1≤i<j≤n
ϑ(t± zi ± zj ; q)aD(n)
q,t+aq(c)
is a section of S(n)
2c;q,t(aδ, a(n − 1)s) for a ≥ 0. For purposes of the following proof, denote this
operator by S
(n)
− (a; c; q, t).
Proposition 9.5. The Fourier transform extends to the Hilbert scheme category.
Elliptic Double Affine Hecke Algebras 129
Proof. As in the symmetric power case, this reduces to showing that the Fourier transform
takes ΓS(n)
2c;q,t(0, aδ + ds + d′f) to ΓS(n)
−2c;q,t(0, aδ + d′s + df). By the t 7→ q − t symmetry, it
suffices to show this for a ≥ 0. Let D be a local section of this category on some open subset of
parameter space. Then∏
1≤i<j≤n
ϑ(t± zi ± zj ; q)aD ∈ ΓS(n)
2c;q,t(0, ds+ (d′ + a(n− 1))f),
S
(n)
− (a; c− (d′ − d)q/2; q, t)D ∈ ΓS(n)
2c;q,t(0, (d+ a(n− 1))s+ d′f),
since each prefactor is itself a section of ΓS(n). Both of these operators have well-behaved Fourier
transforms, and we thus conclude that
S
(n)
− (a;−c+ (d′ − d)q/2; q, t)D̂ ∈ ΓS(n)
2c;q,t(0, (d
′ + a(n− 1))s+ df)∏
1≤i<j≤n
ϑ(t± zi ± zj ; q)aD̂ ∈ ΓS(n)
2c;q,t(0, d
′s+ (d+ a(n− 1))f).
We claim that this implies D̂ ∈ ΓS(n)
2c;q,t(0, aδ + d′s+ df) as required.
It suffices to show this on an open subset, and thus we may assume that c and t are in general
position. The second claim shows that∏
1≤i<j≤n
Γq(aq + t± zi ± zj)D̂
∏
1≤i<j≤n
Γq(t± zi ± zj)−1
=
∏
1≤i<j≤n
Γq(t± zi ± zj)
∏
1≤i<j≤n
ϑ(t± zi ± zj ; q)aD̂
∏
1≤i<j≤n
Γq(t± zi ± zj)−1
is a section of the t-free version of the category, and thus it remains only to show that D̂ itself
is such a section. In other words, we need to show that dividing by ϑ(t ± zi ± zj ; q)a does not
introduce any poles. Now, the operator S
(n)
− (a;−c+ (d′ − d)q/2; q, t) is holomorphic away from
the reflection hypersurfaces, and has leading (left) coefficient∏
1≤i<j≤n ϑ(t− zi− zj ; q)anϑ(t± (zi− zj); q)aϑ(2c+ t+ zi + zj − (d′− d+ a(n− 1))q; q)a∏
1≤i≤j≤n ϑ(− zi− zj ; q)a(n− 1)
.
If we take its inverse as a formal operator, the only poles that can arise are translates of those
originally present and translates of the divisors on which the leading coefficient vanishes. We find
in particular that the inverse remains (since t and c are generic) holomorphic on any divisor of
the form t + zi + zj = kq. Since S
(n)
− (a;−c + (d′ − d)q/2; q, t)D̂ is holomorphic away from the
reflection hypersurfaces, it follows that D̂ itself is holomorphic on the divisors t+ zi + zj = kq,
and thus by symmetry on any divisor t ± zi ± zj = kq. But this is precisely what we needed
to show. �
Remark. As in the symmetric power case, this tells us that the subcategory corresponding
to P2 is indeed (up to fppf local isomorphism) independent of c.
When q = 0, the category is equal to its center (taking the natural extension of the definition
used in the symmetric power case), and thus we expect to obtain a family of commutative
deformations of the Hilbert scheme as t varies. (We similarly expect the center for q torsion
to be the pullback of this family through a suitable base change; this does not quite follow
from the usual arguments, since those depended on strong flatness for the spherical algebra.)
Such a family was constructed in [26] (see also [20] for the case of P2), and thus we naturally
conjecture that they agree under a suitable reparametrization. The significance of this is that the
130 E.M. Rains
construction of [20, 26] is as a moduli space of sheaves on a noncommutative surface (essentially
ProjS(1)), and there are some natural birational transformations between such moduli spaces.
In particular, under certain conditions, the deformed Hilbert scheme is birational to a moduli
space of sheaves with 1-dimensional support, which in turn may be interpreted as a moduli
space of elliptic difference equations. We thus expect that our deformations are similarly closely
related to noncommutative deformations of such moduli spaces.
When q = t = 0 (i.e., the usual Hilbert scheme), this birational map may be described
as follows. Let X be a rational surface with a chosen anticanonical curve Ca, and let D be
a divisor class such that OCa(D) is trivial and the linear system |D| is generically integral.
The curves in this linear system have genus g = D2/2 + 1, and the linear system is itself a Pg.
There is a natural incidence relation between Hilbg(X) and the linear system, i.e., whether the
curve contains the g-point subscheme Z. Each divisor in |D| is incident with a g-dimensional
subscheme of Hilbg(X), and thus Hilbg(X) is birational to the relative Hilbg of the linear system.
For C ∈ |D|, there is a natural map from Hilbg(C) to a compactification of Picg(C) (i.e., torsion-
free sheaves on C), and again this is generically invertible. In other words, the incidence relation
is the graph of a birational map as desired.
There are two things that may go wrong with the map from Hilbg(X) to the relative Picg:
there may be more than one curve in the linear system containing the given subscheme, and
the curve may be reducible (so that the resulting torsion-free sheaf may fail to be stable).
The first issue happens when h0(I(D)) > 1 (where I is the ideal sheaf of Z); since this sheaf
has Euler characteristic 0, we expect this to happen only in codimension ≥ 3. (In fact, given
a particular curve C containing Z, the deformations of C that still contain Z are given by
a subsheaf of the cotangent sheaf corresponding to OC(K −Z), and thus C is moveable iff Z is
moveable in Picg(C), so this is codimension ≥ 2 in the graph of the birational map.) For generic
parameters, the only reducible curves of D will be those meeting Ca and thus having it as
a component. Since |D − Ca| ∼= Pg−1, there are two divisors in Hilbg(X) compatible with such
a reducible curve: either all g points lie on a curve of |D − Ca| or at least one point lies on Ca.
(These correspond to the divisor classes on Hilbg(X) denoted by δ+D−Ca and Ca respectively
relative to the above basis.) For an appropriate choice of stability conditions, the map is well-
defined on the locus where one point lies on Ca, and contracts the other divisor to a subscheme
of codimension ≥ 2.
We thus find that if we remove the unique divisor of class δ + D − Ca from Hilbg(X), then
the result is not only birational to the compactified relative Picg(C), but the birational map
is an isomorphism in codimension 1. The compactified relative Picg(C) is an abelian fibration
over Pg ∼= |D| (with integral fibers), and the derived autoequivalences of the fibers should
extend to derived autoequivalences of Picg(C). One also expects that birational maps which
are isomorphisms in codimension 1 should induce derived equivalences, and thus we expect that
contracting the given divisor on Hilbg(X) should give a projective scheme with a large family
of derived autoequivalences. Moreover, since the noncommutative deformations of a scheme
are functions of the derived category alone, we should expect these derived autoequivalences to
act on the corresponding formal neighborhood in the family corresponding to S(n), and thus
one hopes on the family itself. The result would be an analogue of the derived equivalences
of geometric Langlands, with (symmetric) elliptic difference equations replacing connections.
(For n = 1, such derived equivalences in fact exist, see [26, Section 12].)
One part of the above line of reasoning is the strong suggestion that the divisor δ +D − Ca
should be contractible. Something along these lines holds for S.
Proposition 9.6. For any v ∈ 〈δ, s, f, e1, . . . , em〉, the union⋃
k∈Z
S(n)
η′,x1,...,xm;q,t(0, v + k(δ + (n− 1)f))
Elliptic Double Affine Hecke Algebras 131
is coherent and equal to
S(n)
η′,x1,...,xm;q,t(0, v + (d− a)(δ + (n− 1)f))
= S(n)
η′+(n−1)t,x1,...,xm;q,0(0, v + (d− a)(δ + (n− 1)f))
if v = aδ + ds+ · · · .
Proof. Indeed, this is a nested sequence of sheaf bimodules as k increases, and once k ≥ d− a,
there are no longer any t-dependent vanishing conditions, and thus the sequence stabilizes.
The only remaining dependence on t is in the twisting datum, and thus we may make t = 0 as
long as we adjust η′ accordingly. �
Remark. For the commutative Hilbert scheme, what this is suggesting is that if D is the unique
divisor of class δ+ (n− 1)f , then for any line bundle L, Γ(Hilbn(X) \D;L) is finite-dimensional
(and isomorphic to the space of global sections of some L′). Applying this to powers of an ample
bundle gives a graded algebra the Proj of which has the same homogeneous coordinate ring as
Hilbn(X) \D, and thus the corresponding map contracts D to a subscheme of codimension ≥ 2.
Note that we cannot expect this to be a blow down, but it should be similar: if we take em
rather than δ+ (n− 1)f , then the same construction gives the Hilbert scheme of the blow down
of X.
Remark. The case D − Ca = (n − 1)f (and thus m = 2n + 6) corresponds to a moduli space
of second-order equations with 2n+ 6 singularities.
As this holds even without the constraint that D ·Ca = 0, it seems likely that something ana-
logous should hold for any divisor. To be precise, if w ∈ Z〈s, f, e1, . . . , em〉 is such that OXm(w)
is generically acyclic with n global sections, then we conjecture that the union⋃
k∈Z
S(n)
η′,x1,...,xm;q,t(0, v + k(δ + w))
stabilizes (and is thus coherent) for large k, and moreover that the smallest k for which it stabi-
lizes is a linear function of v, so that the resulting category may be identified with a subcategory
of S. The result would then be invariant under the shift in parameters corresponding to δ +w,
and thus equivalent to a category with t = 0. Something along these lines appears to work
for n = 2 with w = 2s + 2f − e1 − · · · − e7, in that the Euler characteristic of the line bundle
reaches a maximum at k depending linearly on v. (This case corresponds to the unique other
case (modulo the Fourier transform and elementary transformations) of a divisor D such that
(D + Ca) ∩ Ca = ∅ and (D + Ca)
2 = 2, corresponding to a moduli space of line bundles on
genus 2 curves.)
Acknowledgements
The author would particularly like to thank P. Etingof both for asking the original seed question
(with an important assist from A. Okounkov!) and hosting the author’s sabbatical at MIT
(which the author would also like to thank, naturally) where much of the basic approach was
worked out, with great assistance from conversations with not only Etingof but also (regarding
various geometrical issues) B. Poonen. Thanks also go to T. Graber and E. Mantovan for helpful
and encouraging conversations regarding the constructions of Section 2 as well as various general
algebraic geometric questions, and to O. Chalykh for asking some fruitful questions about residue
conditions. The author’s work presented here was supported in part by grants from the National
Science Foundation, DMS-1001645 and DMS-1500806.
132 E.M. Rains
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1 Introduction
2 Line bundles on En and their sections
3 Coxeter group actions on abelian varieties
4 Elliptic analogues of affine Hecke algebras
5 Infinite groups
6 The (double) affine case
7 The CCn case
8 The (spherical) CCn Fourier transform
9 Deformations of Hilbert schemes
References
|
| id | nasplib_isofts_kiev_ua-123456789-211009 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T16:05:00Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Rains, Eric M. 2025-12-22T09:26:51Z 2020 Elliptic Double Affine Hecke Algebras. Eric M. Rains. SIGMA 16 (2020), 111, 133 pages 1815-0659 2020 Mathematics Subject Classification: 33D80; 39A70; 14A22 arXiv:1709.02989 https://nasplib.isofts.kiev.ua/handle/123456789/211009 https://doi.org/10.3842/SIGMA.2020.111 We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C~ₙ version of the construction to construct a flat noncommutative deformation of the nth symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points. The author would particularly like to thank P. Etingof both for asking the original seed question (with an important assist from A. Okounkov!) and hosting the authors sabbatical at MIT (which the author would also like to thank, naturally) where much of the basic approach was worked out, with great assistance from conversations with not only Etingof but also (regarding various geometrical issues) B. Poonen. Thanks also go to T. Graber and E. Mantovan for helpful and encouraging conversations regarding the constructions of Section 2, as well as various general algebraic geometric questions, and to O. Chalykh for asking some fruitful questions about residue conditions. The author's work presented here was supported in part by grants from the National Science Foundation, DMS-1001645 and DMS-1500806. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Double Affine Hecke Algebras Article published earlier |
| spellingShingle | Elliptic Double Affine Hecke Algebras Rains, Eric M. |
| title | Elliptic Double Affine Hecke Algebras |
| title_full | Elliptic Double Affine Hecke Algebras |
| title_fullStr | Elliptic Double Affine Hecke Algebras |
| title_full_unstemmed | Elliptic Double Affine Hecke Algebras |
| title_short | Elliptic Double Affine Hecke Algebras |
| title_sort | elliptic double affine hecke algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211009 |
| work_keys_str_mv | AT rainsericm ellipticdoubleaffineheckealgebras |