A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank
We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using R-matrices for Uq(slN). Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno the...
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| Cite this: | A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank / Y. Fu, S. Shelley-Abrahamson // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 15 назв. — англ. |
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| citation_txt | A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank / Y. Fu, S. Shelley-Abrahamson // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 15 назв. — англ. |
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| description | We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using R-matrices for Uq(slN). Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 055, 11 pages
A Family of Finite-Dimensional Representations
of Generalized Double Affine Hecke Algebras
of Higher Rank
Yuchen FU and Seth SHELLEY-ABRAHAMSON
Department of Mathematics, Massachusetts Institute of Technology,
182 Memorial Drive, Cambridge, MA 02139, USA
E-mail: yfu@mit.edu, sethsa@mit.edu
Received April 20, 2016, in final form June 11, 2016; Published online June 14, 2016
http://dx.doi.org/10.3842/SIGMA.2016.055
Abstract. We give explicit constructions of some finite-dimensional representations of gene-
ralized double affine Hecke algebras (GDAHA) of higher rank using R-matrices for Uq(slN ).
Our construction is motivated by an analogous construction of Silvia Montarani in the
rational case. Using the Drinfeld–Kohno theorem for Knizhnik–Zamolodchikov differential
equations, we prove that the explicit representations we produce correspond to Montarani’s
representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.
Key words: generalized double affine Hecke algebra; R-matrix; Drinfeld–Kohno theorem
2010 Mathematics Subject Classification: 20C08
1 Introduction
Generalized double affine Hecke algebras of higher rank (GDAHA) are a family of algebras that
generalize the well-known Cherednik algebras in representation theory and were first introduced
by Etingof, Oblomkov, and Rains in [5] (the rank 1 case) and [4] (the rank n case). They have
been related to del Pezzo surfaces in algebraic geometry and Calogero–Moser integrable systems.
A degenerate version of GDAHA, known as the rational GDAHA, was also introduced in [4].
In [14], Montarani introduces two constructions of finite-dimensional representations of rational
GDAHA, one using D-modules and the other using explicit Lie theoretic methods. The latter
of these methods, involving an isotypic subspace of a tensor product with a tensor power of the
vector representation of slN , is similar in spirit to the Arakawa–Suzuki functor from the BGG
category O of slN -modules to the category of finite-dimensional representations of degenerate
affine Hecke algebras of type A constructed in [1] and to the construction of representations of
affine braid groups given by Orellana and Ram in [15]. There has been much interest in and
effort devoted to developing similar techniques for the representation theory of related algebraic
objects, for instance by Calaque, Enriquez, and Etingof [2] for degenerate double affine Hecke
algebras and by Jordan [9] in the nondegenerate case, among others.
In this paper, we generalize the Montarani’s Lie theoretic construction to the non-degenerate
GDAHA using R-matrices for the quantum groups Uq(slN ). Furthermore, we show that the
explicit representations we produce are equivalent to the image of Montarani’s representations
under a monodromy functor constructed in [4] which introduces an action of a nondegenerate
GDAHA on a finite-dimensional representation of a rational GDAHA. In the same sense that
Jordan’s work [9] is a generalization of the work of Calaque, Enriquez, and Etingof [2] to the
nondegenerate case, this paper generalizes Montarani’s construction to the nondegenerate case.
The structure of this paper is as follows. In Section 2 we fix the notation we use for quantum
groups and review their basic properties and also introduce the definition of GDAHA. Following
mailto:yfu@mit.edu
mailto:sethsa@mit.edu
http://dx.doi.org/10.3842/SIGMA.2016.055
2 Y. Fu and S. Shelley-Abrahamson
that, our explicit construction of representations is given in Section 3, along with its proof. The
calculation that relates this representation with the monodromy representation given in [4] is
done in Section 4.
2 Background
2.1 Generalized double affine Hecke algebras of higher rank (GDAHA)
We recall the definition of GDAHA in [4]. Let D be a star-shaped finite graph that is not finite
Dynkin with m legs and leg lengths d1, . . . , dm (number of vertices on each leg, including the
center).
Definition 2.1 (GDAHA). Hn(D), the generalized double affine Hecke algebra of rank n
associated with graph D, is the associative algebra generated over C[u±11,1, . . . , u
±1
1d1
, u±12,1, . . . ,
u±1m,dm , t
±1] by invertible generators U1, . . . , Um, T1, . . . , Tn−1 and the following relations:
1. U1U2 · · ·UmT1T2 · · ·Tn−2Tn−1Tn−1Tn−2 · · ·T2T1 = 1;
2. TiTi+1Ti = Ti+1TiTi+1 for 1 ≤ i < n− 1;
3. [Ti, Tj ] = 0 for |i− j| > 1;
4. [Ui, Tj ] = 0 for 1 < i ≤ n− 1, 1 < j ≤ m;
5. [Ui, T1UiT1] = 0 for 1 ≤ i ≤ m;
6. [Ui, T
−1
1 UjT1] = 0 for 1 ≤ i < j ≤ m;
7.
dk∏
j=1
(Uk − uk,j) = 0 for 1 ≤ k ≤ m;
8. Ti − T−1i = t− t−1 for 1 ≤ i ≤ n− 1.
The rank-n GDAHA Hn,m can be seen as a quotient of the group algebra of the fundamental
group of the configuration space of n unordered points on the m-punctured sphere, where the
quotient is by the eigenvalue relations (7) and (8). From this perspective, Ui is represented by
a path in the configuration space in which one of the points loops around a missing point αi,
and Ti is represented by a typical braid group generator exchanging the positions of two points
in the configuration. Note that the generators Ti satisfy the relations of the finite-type Hecke
algebra of type An−1.
2.2 The quantum group Uq(slN)
Due to the large volume of literature and varying conventions regarding quantum groups, it
is necessary to fix the notations that we work with. We use conventions compatible with, for
example, [6, 8, 12]. Throughout this paper, it is assumed that q is a nonzero complex number
that is not a root of unity.
For every n ∈ Z, let [n]q denote the associated symmetrized q-number
[n]q :=
qn − q−n
q − q−1
.
In particular, we have
[n]q = qn−1 + qn−3 + · · ·+ q−n+3 + q−n+1
A Family of Finite-Dimensional Representations of GDAHA of Higher Rank 3
for n > 0 and [n]q = −[−n]q for all n. Define q-binomial coefficients by[
a
n
]
q
:=
[a]q[a− 1]q · · · [a− n+ 1]q
[1]q[2]q · · · [n]q
for all a ∈ Z, n ∈ N, so in particular
[
a
0
]
q
= 1 by the usual convention on empty products.
Define q-factorials [a]!q for a ∈ N by
[a]!q = [1]q[2]q · · · [a]q.
Note that [0]!q = 1.
Definition 2.2 (quantum group Uq(slN )). Let A = (aij) be the Cartan matrix for the Lie
algebra slN . The quantum group Uq(slN ) is the associative C-algebra with generators Ei, Fi,
Ki, K
−1
i (1 ≤ i < N) and relations:
[Ki,Kj ] = 0,
KiK
−1
i = 1 = K−1i Ki,
KiEj = qaijEjKi,
KiFj = q−aijFjKi,
[Ei, Fj ] = δij
Ki −K−1i
qi − q−1i
,
1−aij∑
r=0
(−1)r
[
1− aij
r
]
q
E
1−aij−r
i EjE
r
i = 0, and
1−aij∑
r=0
(−1)r
[
1− aij
r
]
q
F
1−aij−r
i FjF
r
i = 0 if i 6= j,
where δij is the Kronecker delta and [a, b] = ab− ba.
We will use the Hopf algebra structure on Uq(slN ) specified in the following standard propo-
sition:
Proposition 2.3 (Hopf algebra structure on Uq(slN )). Denote U = Uq(slN ). The assignments
below extend to unique algebra homomorphisms (∆: U → U ⊗U, ε : U → C, η : C→ U, S : U →
Uop) that give U a Hopf algebra structure:
∆(Ki) = Ki ⊗Ki, ∆(Ei) = Ei ⊗Ki + 1⊗ Ei, ∆(Fi) = Fi ⊗ 1 +K−1i ⊗ Fi,
ε(Ei) = ε(Fi) = 0, ε(Ki) = 1, S(Ki) = K−1i , S(Ei) = −EiK−1i ,
S(Fi) = −KiFi.
2.3 R-matrices
In this section we will fix our conventions for R-matrices for Uq(slN ). All statements below are
well-known and their proofs can be found in [12] or [10].
Definition 2.4. For a given pair (V, V ′) of finite-dimensional Uq(slN )-modules, define the linear
map f : V ⊗ V ′ → V ⊗ V ′ such that f(v ⊗ w) = q(λ,µ)(v ⊗ w) if v and w have weights λ and µ,
respectively, where (·, ·) is the standard pairing on the weight lattice.
4 Y. Fu and S. Shelley-Abrahamson
Proposition 2.5. There exists an element
R̃ ∈ 1 + Uq(sln)>0 ⊗̂Uq(sln)<0
in an appropriate completion Uq(slN )⊗̂Uq(slN ) such that the operators
R := f ◦ R̃, R := P ◦R = P ◦ f ◦ R̃
on tensor products of finite-dimensional Uq(slN )-modules satisfy, for any finite-dimensional mo-
dules V1, V2, V3:
• R : V1 ⊗ V2 → V2 ⊗ V1 is an isomorphism of Uq(sln)-modules,
• R12,3 = R13R23,
• R1,23 = R13R12,
where P denotes the operator exchanging tensor factors and the subscripts indicate the tensor
factor positions on which R acts.
For an explicit description of the element R̃, see [12, Section 8.3.3].
The operator R satisfies the quantum Yang–Baxter equation:
Proposition 2.6 (QYBE). R12R13R23 = R23R13R12.
Corollary 2.7. For 1 ≤ i 6= j < n, let Ri = Pi,i+1 ◦ Ri,i+1, then we have RiRi+1Ri =
Ri+1RiRi+1 and RiRj = RjRi for |i − j| > 1, and in particular the Ri operators yield a rep-
resentation of the braid group Bn on V ⊗n for any finite-dimensional Uq(slN )-representation V ,
and this representation is functorial in V .
Proposition 2.8. Let CN be the N -dimensional vector representation of Uq(slN ). Then Ri (1 ≤
i < n) act on the space
(
CN
)⊗n
. The operators q1/NRi on
(
CN
)⊗n
act with eigenvalues q, −q−1
and in particular define a representation of the Hecke algebra of type An−1 with parameter q.
Proof. This is Proposition 23 in Section 8.4.3 of [12]. �
2.4 Ribbon category structure on Uq(slN)-modf .d.
Recall that a type-1 representation of Uq(slN ) is a representation V such that V has a weight
decomposition V = ⊕λVλ where the direct sum is over the weight lattice for slN and Vλ is the
subspace
Vλ :=
{
v ∈ C : Kµ(v) = q(λ,µ)v for all roots µ
}
.
Let Uq(slN )-modf.d. denote the category of finite-dimensional type-1 representations of Uq(slN ).
We will only consider representations in this category.
The operator R, along with the Hopf algebra structure on Uq(slN ), gives the category
Uq(slN )-modf.d. of finite-dimensional type-1 representations of Uq(slN ) the structure of a braided
rigid tensor category. In fact, Uq(slN )-modf.d. has even richer structure – it is a ribbon category.
In particular, there exists an automorphism θ of the identity functor on Uq(slN )-modf.d. which
is given by multiplication by q(λ,λ+2ρ) (where, as usual, ρ is the half-sum of the positive roots)
on any irreducible representation with highest weight λ, and θ satisfies the compatibility
θV (λ)⊗V (µ) = R2 ◦
(
θV (λ) ⊗ θV (µ)
)
, (2.1)
where V (λ) and V (µ) are the irreducible highest weight representations with highest weights λ
and µ, respectively, and R is the R-matrix introduced above. An h-adic version of these state-
ments can be found in Proposition 21 in Section 8.4.3 of [12], and it is routine to translate the
result into the setting of numeric q.
A Family of Finite-Dimensional Representations of GDAHA of Higher Rank 5
3 Representations of GDAHA via R-matrices
3.1 The construction
Again, we assume q is a nonzero complex number which is not a root of unity, and all represen-
tations of Uq(slN ) we consider will be type-1 and finite-dimensional.
Let n ≥ 1 be an integer, let V1, . . . , Vm be irreducible finite-dimensional highest weight
modules for Uq(slN ), let Vm+1 = Vm+2 = · · · = Vm+n = CN be copies of its vector representation,
and let
V = V1 ⊗ · · · ⊗ Vm ⊗ Vm+1 ⊗ · · · ⊗ Vm+n.
Let E be the 0-isotypic component of V . For 1 ≤ i < m+n letRi denote id⊗i−1⊗R⊗id⊗m+n−i−1.
Theorem 3.1 (main theorem). Let λ1, . . . , λm ∈ C be any complex numbers and let c = (n +∑
k λk)/N . Then for t = q, some specific values of ukj, and some graph D, the following
formulas for 1 ≤ i < n and 1 ≤ k ≤ m define a representation of the GDAHA of rank n attached
to D on E:
• Ti = q1/NRm+i, and
• Uk = q2[(N−c)/m+λk/N ]RmRm−1 · · ·Rk+1R
2
kR
−1
k+1 · · ·R
−1
m−1R
−1
m .
Note that the Ti are endomorphisms of E because Vm+1, . . . , Vm+n are all copies of CN
and Rm+i acts as a Uq(slN )-module isomorphism. The Uk act on E for similar reasons. The
dependence of the graph D and the parameters ukj on the representations V1, . . . , Vm and the
scalars λ1, . . . , λm ∈ C is given later in equation (3.1).
3.2 Validity of the defining relations
We need to verify that the relations (1)–(8) defining GDAHA hold for the operators defined in
the theorem above. The relations (2), (3) and (8) among Ti are the defining relations of the
type An−1 Hecke algebra and hold by Proposition 2.8. Relation (4) holds trivially because the
operators act on different tensor factors. Relations (5) and (6) follow from routine calculations
using that the R-matrices Ri satisfy the braid relations, and so we need only check relations (1)
and (7). We begin with relation (7), which restricts the eigenvalues of the operators Ui.
3.2.1 Eigenvalues of Uk
Recall from Section 2 the functorial operator θ defined on any Uq(slN )-module and acting on
any copy of the highest weight module V (λ) by q(λ,λ+2ρ). From the compatibility between the R
matrix and θ given in (2.1) that R2 acts as a scalar on each highest weight submodule of the
tensor product of two highest weight modules. More precisely,
Lemma 3.2. Let V (µ) and V (µ′) be highest weight representations of U and let V (λ) ⊂ V (µ)⊗
V (µ′) be a copy of the simple highest weight module with highest weight λ. Then R2 acts on V (λ)
by the scalar q−(µ,µ+2ρ)−(µ′,µ′+2ρ)+(λ,λ+2ρ).
Proof. Immediate from θV (µ)⊗V (µ′) = R2(θV (µ) ⊗ θV (µ′)). �
For 1 ≤ k ≤ m, let Ũk := Rm · · ·Rk+1R
2
kR
−1
k+1 · · ·R
−1
m . Relation (7) requires that the eigenva-
lues of Uk counted with multiplicity are among uk,1, . . . , uk,dk , so it is equivalent to show that Ũk
has all its eigenvalues counted with multiplicity appearing in
q−2[(N−c)/m+λk/N ]uk,1, . . . , q−2[(N−c)/m+λk/N ]uk,dk .
6 Y. Fu and S. Shelley-Abrahamson
As Ũk is conjugate to R2
k via Rm · · ·Rk+1, we need only compute the eigenvalues of R2
k acting
on Vk ⊗ CN .
Suppose Vk is of highest weight µk. As CN is of highest weight ε1 := (1, 0, . . . , 0), it follows
from Lemma 3.2 that the eigenvalues of R2
k on Vk ⊗ CN are q−(µk,µk+2ρ)−(ε1,ε1+2ρ)+(η,η+2ρ),
where η ranges over the highest weights of the irreducible constituents appearing in the direct
sum decomposition of Vk ⊗CN . Let dk be the number of non-isomorphic irreducibles appearing
in Vk ⊗ CN , and suppose their distinct highest weights are ηk,1, . . . , ηk,dk . Then, setting
uk,j = q2[(N−c)/m+λk/N ]−(µk,µk+2ρ)+(ηk,j ,ηk,j+2ρ)−N+1/N (3.1)
provides the choice of d1, . . . , dm and ukj so that relation (7) holds (note that (ε1, ε1 + 2ρ) =
N − 1/N). In particular, the graph D is one with m legs and leg lengths d1, . . . , dm.
3.2.2 The first relation
If we expand relation (1) of the GDAHA, i.e., U1 · · ·UmT1 · · ·Tn−1Tn−1 · · ·T1 = 1, in terms of
the operators given in Theorem 3.1, we see that it simplifies to the following equality:
q2N−2/NRmRm−1 · · ·R1(R1R2 · · ·RmRm+1 · · ·Rm+n−1)Rm+n−1Rm+n−2 · · ·Rm+1 = idE .
Now, suppose we have the following identity:
R1R2 · · ·Rm+n−1Rm+n−1Rm+n−2 · · ·R2R1 = q−2N+2/N idE′ ,
where E′ is the 0-isotypic subspace of V ′ := Vm+1 ⊗ V1 ⊗ · · · ⊗ Vm ⊗ Vm+2 ⊗ · · · ⊗ Vm+n, then
one can write
R1R2 · · ·Rm+n−1 = q−2N+2/NR−11 R−12 · · ·R
−1
m+n−1
as maps from the 0-isotypic component of V1 ⊗ · · · ⊗ Vm ⊗ Vm+2 ⊗ · · · ⊗ Vm+n ⊗ Vm+1 to the
0-isotypic component E′ of V ′. Substituting this into the parenthesized part in the expression
above, we see it immediately proves relation (1). Thus we have reduced relation (1) to the
following lemma:
Lemma 3.3. The operator R1R2 · · ·Rm+n−1Rm+n−1Rm+n−2 · · ·R2R1 : V ′ → V ′ acts on the
zero isotypic component of V ′ by q−2N+2/N .
Proof. It follows from properties (2) and (3) of R-matrices in Proposition 2.5 that we have
R1 · · ·Rm+n−1 = R12···(m+n−1),m+n
and
Rm+n−1 · · ·R1 = R1,23···(m+n),
where the notations R12···(m+n−1),m+n and R1 = R1,23···(m+n) are as in Proposition 2.5, i.e.,
these represent the R-matrices with flip associated to the bracketings (•m+n−1)• and •(•m+n−1)
respectively. Let
V1 ⊗ · · · ⊗ Vm ⊗ Vm+2 ⊗ · · · ⊗ Vm+n =
⊕
i
Wi
be a decomposition of V1 ⊗ · · · ⊗ Vm ⊗ Vm+2 ⊗ · · · ⊗ Vm+n into irreducibles and similarly let
CN ⊗Wi =
⊕
j
Zij
A Family of Finite-Dimensional Representations of GDAHA of Higher Rank 7
be a decomposition of CN ⊗Wi into irreducibles. If Wi is of highest weight µi and if Zij is of
highest weight νij , by Lemma 3.2 and the first sentence of this proof, R1· · ·Rm+n−1Rm+n−1· · ·R1
acts on Zij by the scalar q−(ε1,ε1+2ρ)−(µi,µi+2ρ)+(νij ,νij+2ρ). However, we are only concerned with
those Zij with highest weight 0. By Pieri’s rule, V (0) can appear as a constituent of CN ⊗V (µi)
only if µi is the weight (1, . . . , 1, 0), i.e., when V (µi) is labeled by the Young diagram associated
to the partition 1N−1 [7]. In this case the power of q we just computed becomes q−2N+2/N , as
needed. The lemma, and hence Theorem 3.1, follow. �
4 Equivalence with the monodromy representation
4.1 The monodromy functor
In [4], the authors introduced a certain connection of Knizhnik–Zamolodchikov type whose mo-
nodromy defines a functor from the category of finite-dimensional representations of a rational
GDAHA (rGDAHA) Bn to the category of finite-dimensional representations of a corresponding
non-degenerate GDAHA Hn attached to the same diagram D with m legs of lengths d1, . . . , dm.
Recall that the rGDAHA Bn is the algebra generated over C[γ1,1, . . . , γm,dm , ν] by elements Yi,k
(1 ≤ i ≤ n, 1 ≤ k ≤ m) and the symmetric group Sn, such that the following hold for every
i, j, h ∈ [1, n] with i 6= j and every k, l ∈ [1,m]:
1. sijYi,k = Yj,ksij ;
2. sijYh,k = Yh,ksij if h 6= i, j;
3.
dk∏
j=1
(Yi,k − γk,j) = 0;
4.
m∑
j=1
Yi,j = ν
∑
j 6=i
sij ;
5. [Yi,k, Yj,k] = ν(Yi,k − Yj,k)sij ;
6. [Yi,k, Yj,l] = 0 if k 6= l.
Let us recall the monodromy functor from [4]. Fix a finite-dimensional Bn module M . For
convenience, let α1, . . . , αm ∈ C be distinct points defined by αi = −m − 1 + i and choose the
basepoint z0 = (1, . . . , n) in the ordered configuration space Confn(C\{α1, . . . , αm}) of n-points
in the m-punctured plane. In [4] the Knizhnik–Zamolodchikov-style connection
∇EGO := d−
n∑
i=1
m∑
k=1
Yi,k
zi − αk
−
∑
j 6=i
νsij
zi − zj
dzi (4.1)
is introduced on the trivial vector bundle EM with fiber M over Confn(C\{α1, . . . , αm}). It
follows readily from the defining relations of Bn and a calculation that the connection ∇ is
flat and has trivial residue at ∞. This connection is visibly Sn-equivariant and so descends
to a connection on the unordered configuration space UConfn(C\{α1, . . . , αm}). The residue
condition implies that the monodromy of this connection at z0 defines a representation
ρM : π1
(
UConfn
(
CP1\{α1, . . . , αm}
)
, z0
)
→ Aut(M).
The algebra Hn may be interpreted as the quotient of the group algebra
Cπ1
(
UConfn
(
CP1\{α1, . . . , αm}
)
, z0
)
by the eigenvalue relations (7) and (8) of Definition 2.1, where Ui and Ti are the generators
represented by the following loops:
8 Y. Fu and S. Shelley-Abrahamson
In [4, Section 4.2] it is shown that the monodromy operators ρM (Ti) and ρM (Uk) satisfy the
eigenvalue conditions (7) and (8) for t = e−πiν , uk,j = e2πiγk,j and therefore define a finite-
dimensional representation of Hn with these parameter values. This construction is clearly
functorial in M and defines a functor
F : Bn-modf.d. → Hn-modf.d.,
where as before -modf.d. denotes the category of finite-dimensional representations. Note that
this functor is the identity at the level of vector spaces.
4.2 Montarani’s rGDAHA representation
We now recall the construction of a family of finite-dimensional representations of the rGDAHA
Bn given in [14, Section 5]. Let V1, . . . , Vm be irreducible finite-dimensional representations
of glN , and let λ1, . . . , λm ∈ C be the scalars by which the identity matrix I ∈ glN acts,
respectively. Let χ be a character of glN , and denote the χ-isotypic subspace of V1⊗ · · · ⊗Vm⊗(
CN
)⊗n
by
En,χ :=
{
v ∈ V1 ⊗ · · · ⊗ Vm ⊗
(
CN
)⊗n |xv = χ′(x)v ∀x ∈ glN
}
.
Let c ∈ C be the scalar such that χ = cTrglN . Note that if En,χ 6= 0 then we have the relation
n+
m∑
j=1
λj = cN defining c in Theorem 3.1.
Let ΩglN ∈ glN ⊗ glN be the Casimir tensor of glN , and let Ω
glN
ij represent the tensor that
acts as ΩglN on the ith and jth tensor factors and as identity on other factors.
Theorem 4.1 ([14, Theorem 5.1]). For any choice of ν ∈ C, the assignments sij = Ω
glN
m+i,m+j,
and Yi,k = −ν
(
Ω
glN
k,m+i + N−c
m
)
define a representation of the rGDAHA Bn on En,χ′ for an
appropriate parameter value γ.
4.3 Equivalence of the representations
We may apply the monodromy functor of [4] to the representation of the rGDAHA Bn in the
previous theorem to produce a representation F (En,χ) of a corresponding GDAHA Hn. On
the other hand, given the representations V1, . . . , Vm+n of glN , we can extract the constants
λ1, . . . , λn as above. When ν /∈ Q, the associated parameter t = e−πiν is a nonzero complex
number which is not a root of unity. Let q = t. If V q
i denotes the irreducible representation
of Uq(slN ) with highest weight corresponding to the highest weight of Vi, then from the V q
i
and the λi we may produce, using Theorem 3.1, a finite-dimensional representation E of an
associated non-degenerate GDAHA H ′n.
Theorem 4.2. For ν /∈ Q, the parameters of the GDAHAs Hn and H ′n agree, and the repre-
sentations F (En,χ) and E are equivalent.
A Family of Finite-Dimensional Representations of GDAHA of Higher Rank 9
Proof. Let us first check that the eigenvalue parameters agree. The parameters t for Hn and H ′n
agree and equal e−πiν by definition. Let µk denote the highest weight of Vk, let dk be the number
of non-isomorphic irreducible subrepresentations in Vk ⊗ CN =
⊕
jWj , and let their highest
weights be ηk,1, . . . , ηk,dk . Then, by [14, Lemma 5.2], the graph D attached to the rGAHA Bn
in Theorem 4.1 has leg lengths d1, . . . , dm and parameter values γk,j = −ν(wj + (N − c)/m)
where wj is the eigenvalue of ΩglN on Wj . Therefore, with q = e−iπν , the ukj parameter for the
algebra Hn is given by
ukj := e2πiγk,j = q2(N−c)/m+(ηk,j ,ηk,j+2ρ)−(µk,µk+2ρ)−(N−1/N)+2λj/N ,
where the term 2λj/N comes from the discrepancy ΩglN = ΩslN + 1
N I ⊗ I between the Casimir
tensors for glN and slN . This agrees with the parameter values for H ′n obtained in Section 3.2.1.
The strategy for proving the equivalence statement in Theorem 4.2 is to relate the connection
∇EGO to the classical KZ connection and to use the Drinfeld–Kohno theorem to relate the
monodromy of the latter connection to R-matrices for Uq(slN ). Let V1, . . . , Vm+n be finite-
dimensional irreducible glN -representations as in Montarani’s construction, with Vm+i a copy of
the vector representation CN for 1 ≤ i ≤ n. We have the associated KZ-connection
∇KZ := d+ ν
∑
1≤j≤m+n, j 6=i
ΩslN
ij
zi − zj
dzi
on the trivial vector bundle EYm+n with fiber En,χ over the unordered configuration space
Ym+n :=
{
(z1, . . . , zm+n) ∈ Cm+n : zi 6= zj for all i 6= j
}
of m + n points in C. Here ΩslN
ij denotes the Casimir tensor for slN acting on the ith and jth
tensor factors and we view the glN representations as slN representations. It is well-known, and
easy to check, that this connection is flat. Observe that ∇KZ is Sn-equivariant, where Sn acts
on Ym+n by permuting the last n coordinates and on the fiber by permuting the last n tensor
factors.
Let Y := Confn(C\{α1, . . . , αm}) be the space on which ∇EGO is defined. There is a natural
Sn-equivariant map
r : Y → Ym+n
given by (z1, . . . , zn) 7→ (α1, . . . , αm, z1, . . . , zn). Pulling back the connection ∇KZ along r we
obtain the Sn-equivariant flat connection
r∗∇KZ = d+ ν
n∑
i=1
m∑
k=1
ΩslN
k,m+i
zi − αk
+
∑
1≤j 6=i≤n
ΩslN
m+i,m+j
zi − zj
dzi
on the trivial vector bundle EY with fiber En,χ over Y . On the other hand, inserting the
operators Yi,k and si,j on En,χ defined in Theorem 4.1 into the connection ∇EGO defined in
equation (4.1), we obtain
∇EGO = d+ ν
n∑
i=1
m∑
k=1
Ω
glN
k,m+i + N−c
m
zi − αk
+
∑
1≤j 6=i≤n
Ω
glN
m+i,m+j
zi − zj
dzi.
Both connections above are flat Sn-equivariant connections on the trivial vector bundle EY
over Y . To relate their monodromy, first recall that ΩslN and ΩglN are related by the equation
10 Y. Fu and S. Shelley-Abrahamson
ΩglN = ΩslN + 1
N I ⊗ I where I ∈ glN is the identity matrix. As I acts on Vi by λi for 1 ≤ i ≤ m
and as 1 for m+ 1 ≤ i ≤ m+ n, we have
∇EGO = r∗∇KZ + ν
n∑
i=1
m∑
k=1
λk/N + (N − c)/m
zi − αk
+
∑
1≤j 6=i≤n
1/N
zi − zj
dzi. (4.2)
Note that the connection
∇diff := d+ ν
n∑
i=1
m∑
k=1
λk/N + (N − c)/m
zi − αk
+
∑
1≤j 6=i≤n
1/N
zi − zj
dzi (4.3)
on EY is itself flat and scalar-valued, so it follows from equation (4.2) that the parallel transport
operator associated to a path γ in Y for the connection (EY ,∇EGO) is obtained by multiplying
the parallel transport operator associated to γ for the connection (EY , r
∗∇KZ) by the scalar-
valued parallel transport operator associated to γ for the connection (EY ,∇diff). By inspection
of the residues in ∇diff in equation (4.3), it follows that for the loops Uk and Ti about z0 in Y/Sn,
the monodromies µ∇EGO
and µ∇r∗KZ
for the connections ∇EGO and r∗∇KZ are related by
µ∇EGO
(Uk) = q2(λk/m+(N−c)/m)µ∇r∗KZ
(Uk) (4.4)
and
µ∇EGO
(Ti) = q1/Nµ∇r∗KZ
(Ti), (4.5)
where as before q is defined by q = e−πiν .
All that remains is to relate the monodromy operators µKZ(Uk) and µKZ(Ti) to the R-matrix
expressions appearing in Theorem 3.1 using the Drinfeld–Kohno theorem. The original formu-
lation of the Drinfeld–Kohno theorem as stated in [3] was for the ~-adic quantum group U~(g),
but here we need a version of this theorem for Uq(slN ) for q a nonzero complex number which
is not a root of unity (Drinfeld’s work in [3] was a generalization of previous results obtained
by Kohno [13] for slN ). Such a result was obtained in [11] and an exposition can be found,
for example, in [6, Theorem 8.6.4]. Similarly to [6, Corollary 8.6.5], this theorem immediately
implies in particular that for ν /∈ Q, the monodromy representation µ∇KZ
of π1(Ym+n/Sn) on
En,χ at z0 is equivalent to the representation given by R-matrix expressions in which the class of
a loop γ at z0 acts by the product of R-matrices Ri1 · · ·Ril whenever [π∗γ] factors as σi1 · · ·σil
in the braid group Bn+m := π1(Ym+n/Sm+n). Here σi is the standard ith generator of Bi (coun-
terclockwise half-loop around the hyperplane zi = zi+1), Ri is the R-matrix for Uq(slN ) with
q = e−πiν , and π : Ym+n/Sn → Ym+n/Sm+n is the natural projection. As µ∇r∗KZ
(γ) = µ∇KZ
(r∗γ)
for any loop γ, Theorem 4.2 now follows from equations (4.4) and (4.5) and the observation that
[π∗r∗Ti] = σm+i and [π∗r∗Uk] = σm · · ·σkσkσ−1k+1 · · ·σ
−1
m . �
Remark 4.3. A similar approach may be used for some ν ∈ Q using Part IV of [11], but the
analogous statements more complicated due to the failure of Uq(slN )-modf.d to be semisimple
in this case. If the representations V1, . . . , Vm are fixed, then there is an analogous construction
via R-matrices for ν ∈ Q as long as the denominator of ν is sufficiently large.
Acknowledgements
A portion of this work was conducted at the 2015 MIT Summer Program in Undergraduate
Research and the MIT Undergraduate Research Opportunities Program, for which the second
author was a graduate student mentor. We thank Pavel Etingof for many useful discussions
and for suggesting the direction of this work. We also thank the anonymous referees for their
valuable remarks and suggestions.
A Family of Finite-Dimensional Representations of GDAHA of Higher Rank 11
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1 Introduction
2 Background
2.1 Generalized double affine Hecke algebras of higher rank (GDAHA)
2.2 The quantum group Uq(slN)
2.3 R-matrices
2.4 Ribbon category structure on Uq(slN) -modf.d.
3 Representations of GDAHA via R-matrices
3.1 The construction
3.2 Validity of the defining relations
3.2.1 Eigenvalues of Uk
3.2.2 The first relation
4 Equivalence with the monodromy representation
4.1 The monodromy functor
4.2 Montarani's rGDAHA representation
4.3 Equivalence of the representations
References
|
| id | nasplib_isofts_kiev_ua-123456789-147751 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-28T12:14:54Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Fu, Y. Shelley-Abrahamson, S. 2019-02-15T19:09:31Z 2019-02-15T19:09:31Z 2016 A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank / Y. Fu, S. Shelley-Abrahamson // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20C08 DOI:10.3842/SIGMA.2016.055 https://nasplib.isofts.kiev.ua/handle/123456789/147751 We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using R-matrices for Uq(slN). Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov. A portion of this work was conducted at the 2015 MIT Summer Program in Undergraduate Research and the MIT Undergraduate Research Opportunities Program, for which the second author was a graduate student mentor. We thank Pavel Etingof for many useful discussions and for suggesting the direction of this work. We also thank the anonymous referees for their valuable remarks and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank Article published earlier |
| spellingShingle | A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank Fu, Y. Shelley-Abrahamson, S. |
| title | A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank |
| title_full | A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank |
| title_fullStr | A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank |
| title_full_unstemmed | A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank |
| title_short | A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank |
| title_sort | family of finite-dimensional representations of generalized double affine hecke algebras of higher rank |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147751 |
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