Quasi-Invariants in Characteristic and Twisted Quasi-Invariants
The spaces of quasi-invariant polynomials were introduced by Chalykh and Veselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series over fields of characteristic 0 was computed by Feigin and Veselov [Int. Math. Res. Not. 2002 (2002), 521-545]. In this paper, we show some partial results a...
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| description | The spaces of quasi-invariant polynomials were introduced by Chalykh and Veselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series over fields of characteristic 0 was computed by Feigin and Veselov [Int. Math. Res. Not. 2002 (2002), 521-545]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, Braverman, Etingof, and Finkelberg [arXiv:1611.10216] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 107, 13 pages
Quasi-Invariants in Characteristic p
and Twisted Quasi-Invariants
Michael REN † and Xiaomeng XU ‡
† Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
E-mail: mren36@mit.edu
‡ School of Mathematical Sciences and Beijing International Center for Mathematical Research,
Peking University, Beijing 100871, China
E-mail: xxu@bicmr.pku.edu.cn
Received July 10, 2020, in final form October 17, 2020; Published online October 27, 2020
https://doi.org/10.3842/SIGMA.2020.107
Abstract. The spaces of quasi-invariant polynomials were introduced by Chalykh and
Veselov [Comm. Math. Phys. 126 (1990), 597–611]. Their Hilbert series over fields of char-
acteristic 0 were computed by Feigin and Veselov [Int. Math. Res. Not. 2002 (2002), 521–
545]. In this paper, we show some partial results and make two conjectures on the Hilbert
series of these spaces over fields of positive characteristic. On the other hand, Braverman,
Etingof and Finkelberg [arXiv:1611.10216] introduced the spaces of quasi-invariant polyno-
mials twisted by a monomial. We extend some of their results to the spaces twisted by
a smooth function.
Key words: quasi-invariant polynomials; twisted quasi-invariants
2020 Mathematics Subject Classification: 81R12; 20C08
1 Introduction
A polynomial in variables x1, . . . , xn is symmetric if permuting the variables does not change it.
Another way to view symmetric polynomials is as invariant polynomials under the action of the
symmetric group. A natural generalization of symmetric polynomials then arises: if si,j is the
operator on polynomials that swaps the variables xi and xj , then we may consider polynomials F
such that F −si,j(F ) vanishes to some order at xi = xj . Notably, if F is symmetric in xi and xj ,
then F − si,j(F ) vanishes to infinite order. These polynomials may be viewed as quasi-invariant
polynomials of the symmetric group, and have been introduced by Chalykh and Veselov [4] in
the study of quantum Calogero–Moser systems.
Definition 1.1. Let k be a field, n be a positive integer, and m be a nonnegative integer. We
say that a polynomial F ∈ k[x1, . . . , xn] is m-quasi-invariant if
(xi − xj)
2m+1 |F (x1, . . . , xi, . . . , xj , . . . , xn)− F (x1, . . . , xj , . . . , xi, . . . , xn)
for all 1 ≤ i, j ≤ n. Denote by Qm(n) the set of all m-quasi-invariant polynomials over k in n
variables.
Here, we use the odd exponent 2m+1 because if the right-hand side is divisible by (xi−xj)2m,
then it is also divisible by (xi − xj)
2m+1. This follows by the anti-symmetry of the right-hand
side in xi and xj . Note that Qm(n) is a module over the ring of symmetric polynomials over k
in n variables. Also, as Qm(n) is a space of polynomials, it has a grading by degree. Thus, we
may define a Hilbert series and a Hilbert polynomial to encapsulate the structure of Qm(n).
mailto:mren36@mit.edu
mailto:xxu@bicmr.pku.edu.cn
https://doi.org/10.3842/SIGMA.2020.107
2 M. Ren and X. Xu
The motivation for studying quasiivariant polynomials arises from their relation with inte-
grable systems. In 1971, Calogero first solved the problem in mathematical physics of determin-
ing the energy spectrum of a one-dimensional system of quantum-mechanical particles with in-
versely quadratic potentials [3]. Moser later on connected the classical variant of his problem with
integrable Hamiltonian systems and showed that the classical analogue is indeed integrable [9].
These so-called Calogero–Moser systems have been of great interest to mathematicians as they
connect many different fields including algebraic geometry, representation theory, deformation
theory, homological algebra, and Poisson geometry. See, e.g., [6] and the references therein.
Quasi-invariant polynomials are deeply related with solutions of quantum Calogero–Moser
systems as well as representations of Cherednik algebras [7]. As such, the structure of Qm(n), in
particular freeness as a module, and its corresponding Hilbert series and polynomials have been
extensively investigated by mathematicians. Introduced by Feigin and Veselov in 2001, their
Hilbert series and lowest degree non-symmetric elements have subsequently been computed by
Felder and Veselov [8]. In 2010, Berest and Chalykh generalized the idea to quasi-invariant
polynomials over an arbitrary complex reflection group [1]. Recently in 2016, Braverman,
Etingof, and Finkelberg proved freeness results and computed the Hilbert series of a gener-
alization of Qm(n) twisted by monomial factors [2]. Our goal is to extend the investigation
of Qm(n) and its various generalizations.
In Section 2, we investigate quasi-invariant polynomials over finite fields. In particular, we
provide sufficient conditions for which the Hilbert series over characteristic p is greater than over
characteristic 0. We conjecture that our sufficient conditions are also necessary. We also make
conjectures about the properties of the Hilbert series over finite fields.
In Section 3, we investigate a generalization of the twisted quasi-invariants. In [2], Braverman,
Etingof and Finkelberg introduced the space of quasi-invariants twisted by monomial factors,
again a module over the ring of symmetric polynomials. They proved freeness results and
computed the corresponding Hilbert series. We generalize their work to the space of quasi-
invariants twisted by arbitrary smooth functions and determine the Hilbert series in certain
cases when there are two variables.
In Section 4, we discuss future directions for our research, in particular considering spaces of
polynomial differential operators and q-deformations.
2 Quasi-invariant polynomials over fields
of nonzero characteristic
Much of the previous research on quasi-invariant polynomials has been done over fields of char-
acteristic zero. The general approach is to use representations of spherical rational Cherednik
algebras [2]. In the case of fields of positive characteristic, we take a different approach.
Let k be Fp, and Qm(n) the set of all m-quasi-invariant polynomials over k in n variables.
To begin, we define the Hilbert series of Qm(n).
Definition 2.1. Let the Hilbert series of Qm(n) be
Hm(t) =
∑
d≥0
td · dimQm,d(n),
where Qm,d(n) is the k vector subspace of Qm(n) consisting of polynomials with degree d.
By the Hilbert basis theorem, Qm(n) is a finitely generated module over the ring of symmetric
polynomials. Thus, we may write
Hm(t) =
Gm(t)
n∏
i=1
(1− ti)
,
Quasi-Invariants in Characteristic p and Twisted Quasi-Invariants 3
where Gm(t) is the Hilbert polynomial associated with Hm(t) and the terms in the denomina-
tor correspond to the elementary symmetric polynomials that generate the ring of symmetric
polynomials in n variables.
We are mainly concerned with the difference between the Hilbert series of Qm(n) over charac-
teristic p and characteristic 0. The following proposition states that the Hilbert series of Qm(n)
is at least as large in the former case as in the latter case.
Proposition 2.2. dimQm,d(n) over Fp is at least as large as over C for each choice of m, n,
and d.
Proof. Suppose that F =
∑
i1+···+in=d ai1,...,inx
i1
1 · · ·xinn is in Qm,d(n). Then, either d < 2m+1,
in which case we must have F symmetric or (xi − xj)
2m+1 would divide a nonzero polynomial
with degree d for some choice of i and j, a contradiction. This means that the dimensions are
equal over either characteristic. Otherwise, we have that
F − si,jF = (xi − xj)
2m+1
∑
j1+···+jn=d−(2m+1)
bi,j,j1,...,jnx
j1
1 · · ·x
jn
n
for each pair i, j. These yield a system of linear equations in the undetermined coefficients
of F and
F−si,jF
(xi−xj)2m+1 , which is with integral coefficients we are considering. It then follows from
considering the null-space that the dimension of the solution space over a field of characteristic p
is at least the dimension over a field of characteristic 0. �
However, for each m, there are only finitely many primes for which the Hilbert series of Qm(n)
is strictly greater over Fp than over C.
Proposition 2.3. For any fixed m and n, there are only finitely many primes p for which the
Hilbert series of Qm(n) is greater over Fp than over C.
Proof. Let P = Z[x1, . . . , xn], Q =
⊕
1≤i<j≤n P/(xi − xj)
2m+1P , and h be the linear map
from P to Q defined as
h(F ) =
⊕
1≤i<j≤n
(1− si,j)F.
Note that Ker(h) coincides with Qm(n) by definition. Set M = Coker(h) as the cokernel of h
in Q and note that if Qm(n) over Fp has a higher dimension than Qm(n) over C for some degree
of the polynomials, then M must have p-torsion. To prove that there are only finitely many
such primes p, we use the following generic freeness lemma, see, e.g., [5, Theorem 14.4].
Lemma 2.4. For a Noetherian integral domain A, a finitely generated A-algebra B, and a fini-
tely generated B-module M , there exists a nonzero element r of A such that the localization Mr
is a free Ar module.
We apply this in the case where A = Z, B = Z[x1, . . . , xn]Sn and M = Coker(h). It is easy
to see that these satisfy the conditions for A, B, and M in the lemma. Thus there exists an
integer r ∈ Z \ {0} such that Mr is free over Z[1/r]. As M has no p-torsion for any p - r,
M has no p-torsion for all but finitely many primes p so the Hilbert series over Fp is the same
as over C. �
We now determine the primes for which Qm(n) is greater. First, we examine the case when
n = 2.
Proposition 2.5. When n = 2, the Hilbert series for Qm(2) over characteristic p coincides
with that of characteristic 0. It is 1+t2m+1
(1−t)(1−t2) over all fields.
4 M. Ren and X. Xu
Proof. We claim that the dimension of Qm,d(2) over C is equal to the dimension of Qm,d(2)
over Fp. By Proposition 2.2, it suffices to show that for each m and d, the dimension of Qm,d(2)
over C is at least the dimension of Qm,d(2) over Fp. Consider a basis f1, . . . , fk ∈ Fp[x, y]
of Qm,d(2) over Fp. We will show the existence of F1, . . . , Fk ∈ Z[x, y] of Qm,d(2) such that
Fi ≡ fi (mod p) for all i. This means that F1, . . . , Fk are linearly independent, as otherwise
there exist relatively prime integers n1, . . . , nk with n1F1 + · · ·+nkFk = 0. Taking the equation
modulo p yields n1f1 + · · ·+ nkfk ≡ 0 (mod p), a contradiction with f1, . . . , fk forming a basis
of Qm,d(2) as not all of n1, . . . , nk are divisible by p.
To show the existence of such F1, . . . , Fk, let f = fi for a fixed i and suppose that f(x, y)−
f(y, x) = (x − y)2m+1g(x, y) for some symmetric g(x, y) ∈ Fp[x, y]. Let us take a symmetric
G(x, y) ∈ Z[x, y] such that G ≡ g (mod p). Let f(x, y) =
∑d
i=0 aix
iyd−i and suppose that
G(x, y)(x − y)2m+1 =
∑d
i=1Bix
iyd−i with ai ∈ Fp and Bi ∈ Z. We have that ai − ad−i ≡ Bi
(mod p). Note that G(x, y) is symmetric, so G(x, y)(x−y)2m+1 is anti-symmetric, which implies
that Bi + Bd−i = 0 for all i. Now, define F (x, y) =
∑d
i=1Aix
iyd−i, where Ai ≡ ai (mod p) for
i ≤ d
2 and Ai = Ad−i + Bi for i > d
2 . Note that for i > d
2 , we have that Ai ≡ Ad−i + Bi ≡ ai
(mod p), so this F satisfies F ≡ f (mod p). It remains to check the quasi-invariance condition.
However, note that
F (x, y)− F (y, x) =
d∑
i=1
(Ai −Ad−i)x
iyd−i =
d∑
i=1
Bix
iyd−i = G(x, y)(x− y)2m+1
by definition, so we are done.
Hence, the dimension, and thus the series, is independent of p. It is known from [2] that the
series is 1+t2m+1
(1−t)(1−t2) , as desired. �
When n > 2, the series differs greatly for many primes. In this case, we have found a sufficient
condition for when the Hilbert series over characteristic p is greater.
Theorem 2.6. Let m ≥ 0 and n ≥ 3 be integers. Let p be a prime such that there exist integers
a ≥ 0 and k ≥ 0 with
mn(n− 2) +
(
n
2
)
n(n− 2)k +
(
n
2
)
− 1
≤ pa ≤ mn
nk + 1
.
Then the Hilbert series of Qm(n) with n variables over Fp is different from the Hilbert series
over C.
Proof. The following formula, due to [8], gives the Hilbert polynomial for Qm(n) over C:
n!tm(n2)
∑
Young diagrams
n∏
i=1
tm(`i−ai)+`i
1− ti
hi
(
1− thi
) .
Here, the sum is over Young diagrams with n boxes, ai denotes the number of boxes to the right
of the ith box, `i denotes the number of boxes below the ith box, and hi = ai + `i + 1. It is not
hard to see that the formula gives that the Hilbert polynomial is of the form 1+(n−1)tmn+1+· · · ,
where the exponents are sorted in ascending order. This is as the two terms of smallest degree
are contributed by the Young diagrams corresponding to the partitions (n) and (n− 1, 1). This
implies that all polynomials in Qm(n) with degree at most mn are symmetric, and Qm(n) as
a module over symmetric polynomials has a generator of degree mn+ 1. For any m, denote this
generator in Qm(n) by Pm. In the following construction, we will use the generator Pk of Qk(n)
for certain k < m.
Quasi-Invariants in Characteristic p and Twisted Quasi-Invariants 5
To show that the Hilbert series is different over Fp, we consider the following non-symmetric
polynomial:
F = P pa
k
∏
1≤i<j≤n
(xi − xj)
2b.
Here b = 2m+1−pa(2k+1)
2 , and a, k are integers such that the above inequalities are satisfied. So
degF = pa(nk + 1) + 2b
(
n
2
)
= pa(nk + 1) +
(
n
2
)
(2m + 1− pa(2k + 1))
=
(
n
2
)
(2m + 1) + pa
(
1−
(
n
2
)
− n(n− 2)k
)
≤
(
n
2
)
(2m + 1)−
(
mn(n− 2) +
(
n
2
))
= mn < degPm.
Hence, if we show that F ∈ Qm(n), then as degF < degPm we obtain a different Hilbert series
over Fp, in particular in the coefficient of tdegF . To do that, note that b is an integer when p is
odd and a half-integer when p = 2. Either way,
∏
(xi − xj)
2b is a symmetric polynomial in Fp,
so we have that (1 − si,j)
(
P pa
k
∏
(xi − xj)
2b
)
= ((1 − si,j)Pk)p
a∏
(xi − xj)
2b by the fact that
(u + v)p
a
= up
a
+ vp
a
in Fp. Hence, as (xi − xj)
2k+1 divides (1 − si,j)Pk by assumption, we
have that (xi−xj)
pa(2k+1)+2b = (xi−xj)
2m+1 divides (1− si,j)F . Hence, F is in Qm(n), so this
produces a generator of Qm(n) of lower degree in Fp and thus a different Hilbert series over Fp,
as desired. �
Remark 2.7. Let us write the inequalities in Theorem 2.6 in the form
k +
1
n
≤ m/pa ≤ k +
n + 1
2n
− n− 1
2(n− 2)pa
.
In this way, we can rewrite it as
1
n
≤
{
m
pa
}
≤ n + 1
2n
− n− 1
2(n− 2)pa
,
where {· · · } denotes fractional part, which eliminates k. Also from this form it is clear that a
cannot be zero, i.e., a ≥ 1.
Remark 2.8. Let k = 0 in the inequality in Theorem 2.6. Then, we see that all primes p
with a power pa between roughly 2m and mn satisfy the inequality. These primes p satisfy the
property that Qm(n) over Fp has a different Hilbert series than over C.
Conjecture 2.9. The sufficient condition we have given in Theorem 2.6 is also necessary. That
is if the Hilbert series of Qm(n) in Fp is different from the Hilbert series in C, then there exist
integers a ≥ 0 and k ≥ 0 such that
mn(n− 2) +
(
n
2
)
n(n− 2)k +
(
n
2
)
− 1
≤ pa ≤ mn
nk + 1
.
In particular, if p > mn, then the Hilbert series over Fp is the same as over C.
This is supported by computer calculations, especially in the case of n = 3, 4. They suggest
that the Hilbert series takes a form depending on the smallest non-symmetric element of Qm(n)
which is described by the proof of Theorem 2.6 and hence satisfies the conjecture. The following
table summarizes the results of our computer program verification for n = 3, m ≤ 15 and p ≤ 50.
6 M. Ren and X. Xu
m
p
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
0
1 1, 0
2 1, 0
3 3, 0 2, 0 1, 0
4 3, 0 2, 0 1, 0
5 2, 0 1, 0 1, 0
6 4, 0 1, 0 1, 0 1, 0
7 4, 0 1, 2 1, 1 1, 0 1, 0 1, 0
8 4, 0 1, 0 1, 0 1, 0
9 4, 0 3, 0 2, 0 1, 0 1, 0 1, 0
10 3, 0 2, 0 1, 1 1, 0 1, 0 1, 0 1, 0
11 5, 0 3, 0 2, 0 1, 0 1, 0 1, 0 1, 0
12 5, 0 3, 0 2, 0 1, 0 1, 0 1, 0
13 5, 0 3, 0 2, 0 1, 0 1, 0 1, 0 1, 0
14 5, 0 3, 0 2, 0 1, 0 1, 0 1, 0 1, 0 1, 0
15 5, 0 3, 0 2, 0 1, 1 1, 0 1, 0 1, 0 1, 0 1, 0
Each box in which the series is greater over Fp than over C is labeled with its integers a, k that
make the inequality hold.
Through our programs, we have found that when n = 3, the Hilbert series takes the form
1 + 2td + 2t6m+3−d + t6m+3
(1− t)(1− t2)(1− t3)
for small p, where d is the degree of the smallest non-symmetric generator of Qm in Fp. In
particular, this smallest non-symmetric polynomial in Qm is of the form P pa
k
∏
(xi−xj)
2b where
the Pk are as described in Theorem 2.6. Furthermore, we conjecture that Qm is a free module
over the ring of symmetric polynomials for n = 3 over any field.
In [8], the authors prove some properties of Hilbert series and polynomials over C, specifically
their maximal term and symmetry. We believe that similar results still hold over Fp, and this is
supported by our computer calculations for n = 3, 4.
Conjecture 2.10. The largest degree term in the Hilbert polynomial is always t(
n
2)(2m+1). Fur-
thermore, when p is an odd prime Qm is a free module over the ring of the symmetric polynomials
of rank n!, and the Hilbert polynomial is palindromic.
Remark 2.11. The condition that p is odd appears to be necessary. Indeed, a computer
calculation shows that when n = 4, m = 1 and p = 2, the Hilbert series is
1 + t + 2t2 + 3t3 + 8t4 + 9t5 + 15t6 + 23t7 + 38t8 + 50t9 + 71t10 + · · ·
=
1 + 3t4 + 3t7 + 5t8 + 3t9 − t10 + · · ·
(1− t)(1− t2)(1− t3)(1− t4)
,
and the negative coefficient implies that the module cannot be free. In particular, computing
the polynomial up to t18 also demonstrates that it is not symmetric.
3 Twisted quasi-invariants
3.1 A generalization of quasi-invariants
In [2], Braverman, Etingof and Finkelberg introduced quasi-invariants twisted by a monomial
xa11 · · ·xann , where a1 . . . , an ∈ C. We further generalize this by allowing the twist to be a product
Quasi-Invariants in Characteristic p and Twisted Quasi-Invariants 7
of general functions. To be more precise, let m be a nonnegative integer. Fix one-variable
meromorphic functions f1, f2, . . . , fn, and denote by D ⊂ Cn the domain where the product
f1(x1)f2(x2) · · · fn(xn) and its inverse are smooth.
Definition 3.1. We define Qm(f1, . . . , fn) to be the space of polynomials F ∈ C[x1, . . . , xn] for
which
(1− si,j) (f1(x1) · · · fn(xn)F (x1, . . . , xn))
(xi − xj)2m+1
is smooth on D for all 1 ≤ i < j ≤ n.
In the following, for simplicity when we say smooth functions we always mean smooth on D.
Remark 3.2. In [2], the authors studied the case fi(x) = xai for ai ∈ C, and denoted
Qm(f1, . . . , fn) by Qm(a1, . . . , an). In cases of unambiguous use, we will shorten this to Qm(n).
Similar to [2], we believe that Qm(f1, . . . , fn) is in general a free module.
Conjecture 3.3. For generic f1, . . . , fn, in particular when fi
fj
is not a monomial in x1, . . . , xn,
Qm(f1, . . . , fn) is a free module over the ring of symmetric polynomials in x1, . . . , xn.
3.2 Rationality of the logarithmic derivative
Note that for any f1, . . . , fn and i, j, any F ∈ (xi − xj)
2mC[x1, . . . , xn] has the property that
(1−si,j)(f1(x1)···fn(xn)F )
(xi−xj)2m+1 is smooth. This is a trivial case, thus we want to find out which choices
of the fi, fj yield polynomials in Qm(n) that are not in (xi − xj)
2mC[x1, . . . , xn].
Lemma 3.4. If F is divisible by xi−xj and
(1−si,j)(f1(x1)···fn(xn)F )
(xi−xj)2(k+1)+1 is smooth then (xi−xj)
2 |F
and
(1−si,j)
(
f1(x1)···fn(xn)
F
(xi−xj)
2
)
(xi−xj)2k+1 is smooth.
Proof. Let F (xi, xj) = (xi − xj)G(xi, xj) for some polynomial G. Here, we write F (xi, xj) for
F and G(xi, xj) for G to ease the notation, as we will only consider it as a function in the ith
and jth coordinates. Substituting, the condition becomes
fi(xi)fj(xj)G(xi, xj) + fj(xi)fi(xj)G(xj , xi) = (xi − xj)
2k+2g(xi, xj)
(here and for the rest of this section, g (with a possible subscript, e.g., ga, gb, gc) denotes
a function smooth on D) and setting xi = xj = x gives
fi(x)fj(x)G(x, x) = 0,
so G(x, x) = 0. This implies that (xi − xj) |G, so (xi − xj)
2 |F , as desired. The second part of
the proposition follows by definition as F
(xi−xj)2
is smooth. �
Proposition 3.5. Let hi,j = fi
fj
. If dlog(hi,j) is not a rational function, then Qm(n) ⊂ (xi −
xj)
2m+1C[x1, . . . , xn]. Here, dlog(f) = f ′
f denotes the logarithmic derivative of a function f .
Proof. The proposition is trivial for m = 0. For m > 0, note that F ∈ Qm(n) if and only if
fi(xi)fj(xj)F (xi, xj)− fj(xi)fi(xj)F (xj , xi) = (xi − xj)
2m+1ga(xi, xj),
hi,j(xi)F (xi, xj)− hi,j(xj)F (xj , xi) = (xi − xj)
2m+1gb(xi, xj).
8 M. Ren and X. Xu
Here, we treat the rest of the functions and variables as constants. Differentiating with respect
to xi, we have that
hi,j(xi)(dlog(hi,j)(xi)F (xi, xj) + F1(xi, xj))− hi,j(xj)F2(xj , xi) = (xi − xj)
2mgc(xi, xj),
where for a function F (x, y) we define F1 = ∂F
∂x and F2 = ∂F
∂y . Setting xi = xj = x, we have that
hi,j(x)(dlog(hi,j)(x)F (x, x) + F1(x, x))− hi,j(x)F2(x, x) = 0,
dlog(hi,j)(x)F (x, x) = F2(x, x)− F1(x, x),
which means that F (x, x) = 0. Otherwise, we would have
dlog(hi,j)(x) =
F2(x, x)− F1(x, x)
F (x, x)
,
which is a contradiction as the right hand side is a rational function. Hence, (xi− xj) |F . Now,
by Lemma 3.4 we have (xi − xj)
2 |F and F
(xi−xj)2
∈ Qm−1(n), which implies the desired result
by a straightforward induction. �
3.3 Hilbert series for n = 2
Let n = 2, x = x1, and y = x2. Note that scaling f1 and f2 by some smooth function does
not affect Qm(2). Hence, we may multiply them both by 1
f2
and let f = f1
f2
. For convenience,
we use Qm(f) to denote the space of quasi-invariants. Throughout this section, we will let
dlog(f(x)) = p(x)
q(x) for relatively prime p, q ∈ C[x], as we have from Section 3.2 that either
Qm = (x − y)2mC[x, y] or dlog(f(x)) is a rational function. For convenience, we will also set
F1 = ∂F
∂x , F2 = ∂F
∂y , and F12 = ∂2F
∂x∂y .
Lemma 3.6. If F (x, y) ∈ Qm(f), then
p(x)F2(x, y) + q(x)F12(x, y) ∈ Qm−1
(
f
q
)
,
−p(y)F1(x, y) + q(y)F12(x, y) ∈ Qm−1(fq).
Proof. We begin with our quasi-invariant condition, which in our case of n = 2 is
f(x)F (x, y)− f(y)F (y, x) = (x− y)2m+1ga(x, y).
Differentiating by x and then y, we obtain
f ′(x)F2(x, y) + f(x)F12(x, y)− f ′(y)F2(y, x)− f(y)F12(y, x) = (x− y)2m−1gb(x, y).
By the definition of p and q, this is equivalent to
f(x)
q(x)
(p(x)F2(x, y) + q(x)F12(x, y))− f(y)
q(y)
(p(y)F2(y, x) + q(y)F12(y, x))
= (x− y)2m−1gb(x, y),
which is exactly the quasi-invariant condition that is desired.
Dividing our quasi-invariant condition by f(x)f(y) gives
1
f(x)
F (y, x)− 1
f(y)
F (x, y) = (x− y)2m+1gc(x, y).
(Note that gc is a function smooth on D, because 1
f(x)f(y) is smooth on D by assumption.) Thus,
−p(x)F1(y, x) + q(x)F12(y, x) ∈ Qm−1
(
1
fq
)
by the above. Expanding the quasi-invariant condi-
tion and multiplying by f(x)f(y)q(x)q(y), we obtain the equivalent statement −p(y)F1(x, y) +
q(y)F12(x, y) ∈ Qm−1(fq), as desired. �
Quasi-Invariants in Characteristic p and Twisted Quasi-Invariants 9
Now, we specialize to the case in which f(x) =
∏k
i=1(x − ai)
bi for arbitrary complex num-
bers ai, bi. Note that in this case dlog(f) =
∑k
i=1
bi
x−ai .
Definition 3.7. For a nonnegative integer m and a complex number z, denote
dm(z) =
{
min(m, |z|) if z ∈ Z,
m otherwise,
and
dm(f) =
k∑
i=1
dm(bi),
where f(x) =
∏k
i=1(x− ai)
bi for a1, . . . , ak, b1, . . . , bk ∈ C with a1, . . . , ak pairwise distinct.
Lemma 3.8. We have that
k∏
i=1
(x− ai)
dm(bi) |F (x, x)
for any F ∈ Qm(f).
Proof. We proceed using induction, with the base case of m = 0 clearly true. Recall that
dlog(f(x)) = p(x)
q(x) . For f(x) =
∏k
i=1(x − ai)
bi , we have that p(x) =
∑k
i=1 bi
∏
j 6=i(x − ai)
and q(x) =
∏k
i=1(x − ai). It suffices to prove this divisibility for each (x − ai)
dm(bi). By the
inductive hypothesis and Lemma 3.6, we have that (x−ai)dm−1(bi−1) | p(x)F2(x, x)+q(x)F12(x, x)
and (x − ai)
dm−1(bi+1) | − p(x)F1(x, x) + q(x)F12(x, x). It is easy to see that dm(bi) − 1 ≤
dm−1(bi), dm−1(bi − 1), dm−1(bi + 1).
Thus, (x− ai)
dm(bi)−1 | (x− ai)
dm−1(bi) |F (x, x) as F is also in Qm−1(f). From the other two
divisibilities we also obtain (x − ai)
dm(bi)−1 | p(x)F2(x, x) + q(x)F12(x, x), and −p(x)F1(x, x) +
q(x)F12(x, x), so (x − ai)
dm(bi)−1 | p(x)(F1(x, x) + F2(x, x)) = p(x)dF (x,x)
dx (x, x). As p and q
are relatively prime, we must have (x − ai)
dm(bi)−1 | dF (x,x)
dx (x, x) which together with (x −
ai)
dm(bi)−1 |F (x, x) implies (x− ai)
dm(bi) |F (x, x), as desired. �
In fact, this lemma is sharp in the sense that there exists F such that the divisibility becomes
equality. To prove that, we utilize the following lemma:
Lemma 3.9. If F ∈ Qm(f) and G ∈ Qm(g), then FG ∈ Qm(fg).
Proof. We have that
f(x)g(x)F (x, y)G(x, y)− f(y)g(y)F (y, x)G(y, x)
(x− y)2m+1
= g(x)G(x, y)
f(x)F (x, y)− f(y)F (y, x)
(x− y)2m+1
+ f(y)F (y, x)
g(x)G(x, y)− g(y)G(y, x)
(x− y)2m+1
is smooth. �
Lemma 3.10. There exists Pm ∈ Qm(f) with
Pm(x, x) =
k∏
i=1
(x− ai)
dm(bi).
10 M. Ren and X. Xu
Proof. Note that by Lemma 3.9 it suffices to show this when k = 1 as for k > 1 we can take
the product of all such Pm in Qm
(
(x−a1)
dm(b1)
)
, . . . , Qm
(
(x−ak)dm(bk)
)
. Shifting, we may also
assume that a1 = 0. Now, note that if z = b1 is an integer less than m, we can simply take
Pm = yz. Otherwise, we claim that we can take
Pm(x, y) =
m∑
i=0
(
m−z
i
)(
m+z
m−i
)
xiym−i(
2m
m
) .
Indeed, note that we have that Pm(x, x) = xm by Vandermonde’s identity, so it suffices to show
that Pm, or equivalently the numerator of Pm, is in Qm. We proceed using induction, with the
base case of m = 0 obvious. For the inductive step, note that we wish to show that
H(x, y) :=
m∑
i=0
(
m− z
i
)(
m + z
m− i
)(
xi+zym−i − xm−iyi+z
)
vanishes at x = y to order 2m + 1.
It is easy to see that H(x, y) vanishes at x = y. Let us first show that H(x, y) vanishes at
x = y to order 2. Differentiating with respect to x and setting x = y, we would like to show
that
m∑
i=0
(
m− z
i
)(
m + z
m− i
)
(2i−m + z) = 0.
As we have
m∑
i=0
(
m− z
i
)(
m + z
m− i
)
i =
m∑
i=1
(m− z)
(
m− z − 1
i− 1
)(
m + z
m− i
)
= (m− z)
(
2m− 1
m− 1
)
and
m∑
i=0
(
m− z
i
)(
m + z
m− i
)
(i−m) = −
m−1∑
i=0
(m + z)
(
m− z
i
)(
m + z − 1
m− i− 1
)
= −(m + z)
(
2m− 1
m− 1
)
by Vandermonde’s identity, the expression reduces to
−2z
(
2m− 1
m− 1
)
+
m∑
i=0
z
(
m− z
i
)(
m + z
m− i
)
= −2z
(
2m− 1
m− 1
)
+ z
(
2m
m
)
= 0
as desired.
Secondly, let us show that ∂2H
∂y∂x vanishes at x = y to order 2m− 1. Differentiating by both x
and y, it suffices to show that
m∑
i=0
(
m− z
i
)(
m + z
m− i
)
(i + z)(m− i)
(
xi+z−1ym−i−1 − xm−i−1yi+z−1)
vanishes at x = y to order 2m− 1. But note that this expression is
m−1∑
i=0
(
m− z
i
)(
m + z − 1
m− 1− i
)
(i + z)(m + z)
(
xi+z−1ym−i−1 − xm−i−1yi+z−1)
Quasi-Invariants in Characteristic p and Twisted Quasi-Invariants 11
= (m + z)
m−1∑
i=0
(
m− z
i
)(
m + z − 2
m− 1− i
)
(m + z − 1)
(
xi+z−1ym−i−1 − xm−i−1yi+z−1)
= (m + z)(m + z − 1)
m−1∑
i=0
(
m− z
i
)(
m + z − 2
m− 1− i
)(
xi+z−1ym−i−1 − xm−i−1yi+z−1),
which vanishes at x = y to order 2m−1 by the inductive hypothesis on Qm−1(x
z−1), as desired.
Thus we have seen that H(x, y) vanishes at x = y to order 2 and
∂2H
∂y∂x
= (x− y)2m−1K(x, y)
for certain polynomial K(x, y). Since H(x, y) vanishes at x = y to order 2, the solution of
the above equation is unique. We can use the integration by parts to get an expression of the
polynomial H in terms of (x− y)k, k ≥ 2m + 1, and the derivatives of K(x, y) (the constant of
integration is chosen to be zero). In this way one checks that H(x, y) vanishes at x = y to order
2m + 1. �
Lemma 3.11. Let R denote the ring of symmetric polynomials in x and y. Then, for all m > 0
we have that
Qm = RPm + (x− y)2Qm−1.
Proof. Let F (x, y) be an element of Qm. By Lemma 3.8, Pm(x, x) |F (x, x), so there exists
a polynomial g ∈ C[x] with Pm(x, x)g(x) = F (x, x). Now, consider the polynomial
F ′(x, y) = F (x, y)− Pm(x, y)g
(
x + y
2
)
,
which is in Qm as F, Pm ∈ Qm and g
(x+y
2
)
∈ R. But now note that F ′(x, x) = F (x, x) −
Pm(x, x)g(x) = 0, so by Lemma 3.4, F ′ ∈ (x − y)2Qm−1, which immediately implies the desi-
red. �
Corollary 3.12. We have that
Qm = RPm + R(x− y)2Pm−1 + · · ·+ R(x− y)2m−2P1 + (x− y)2mQ0
for all m.
Now, we are finally ready to prove our main result of this section. Recall that dm(f) =∑k
i=1 dm(bi) where f(x) =
∏k
i=1(x − ai)
bi and dm(z) = min(m, |z|) if z ∈ Z and dm(z) = m
otherwise.
Theorem 3.13. The Hilbert series for Qm(f) is
t2m + t2m+1 +
∑m
i=1 t
2(m−i)+di(f) −
∑m
i=1 t
2(m−i)+di(f)+2
(1− t)(1− t2)
.
Proof. Note that Q0 = C[x, y], which is generated by P0 = 1 and x− y (as an R modules). By
the corollary of Lemma 3.11, Qm is generated by
Pm, (x− y)2Pm−1, . . . , (x− y)2m−2P1, (x− y)2m, (x− y)2m+1.
Let gm,i = (x − y)2(m−i)Pi and gm = (x − y)2m+1. We claim that Qm is generated by
gm, gm,0, . . . , gm,m and m independent relations of the form
(x− y)2gm,m = rm,m−1gm,m−1 + · · ·+ rm,0gm,0 + rmgm,
12 M. Ren and X. Xu
(x− y)2gm,m−1 = rm−1,m−2gm,m−2 + · · ·+ rm−1,0gm,0 + rm−1gm,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
(x− y)2gm,1 = r1,0gm,0 + r1gm
for some ri, ri,j ∈ R. We proceed using induction, noting that Q0 is generated by 1 and x−y with
no relations. For the inductive step, first note that as gm,m = Pm ∈ Qm ⊂ Qm−1, there exist
p, p0, . . . , pm−1 ∈ R with gm,m = pgm−1 + p0gm−1,0 + · · ·+ pm−1gm−1,m−1. This yields a relation
in the form of the first relation above by setting rm = p, rm,0 = p0, . . . , rm,m−1 = pm−1. This
equation is true as gm = (x−y)2gm−1, gm,0 = (x−y)2gm−1,0, . . . , gm,m−1 = (x−y)2gm−1,m−1 by
definition. Now, suppose that q, q0, . . . , qm ∈ R such that qgm+q0gm,0+· · ·+qmgm,m = 0. Then,
as (x− y)2 | gm, gm,0, . . . , gm,m−1 and (x− y)2 - gm,m, we must have that (x− y)2 | qm. Let qm =
(x− y)2q′m. Then, subtracting q′m times the first relation from qgm + q0gm,0 + · · ·+ qmgm,m = 0,
we obtain a relation of the form q′gm + q′0gm,0 + · · ·+ q′m−1gm,m−1 = 0 with q′, q′0, . . . , q
′
m−1 ∈ R.
Note that this relation is uniquely determined by the first generating relation we have so the
first generating relation is independent of the rest of the relations. Furthermore, this relation
is (x − y)2 times a relation among the generators of Qm−1. By the inductive hypothesis, such
a relation is generated by (x − y)2 times the m − 1 independent generating relations of Qm−1,
which are by definition the last m− 1 generating relations on our list. Hence, Qm is generated
by those m + 2 elements and m independent relations among those elements, as desired.
For the Hilbert polynomial, note that the generators have degrees
dm(f), 2 + dm−1(f), . . . , 2m− 2 + d1(f), 2m, 2m + 1
and that the independent relations have degrees
2 + dm(f), 4 + dm−1(f), . . . , 2m + d1(f),
which gives the Hilbert polynomial and series exactly as described in the theorem. �
4 Future prospects
It would be interesting to study Conjectures 2.9, 2.10 and 3.3. As with our current results,
we expect to make extensive use of computer programs to discover key properties of quasi-
invariant polynomials and their Hilbert series. We expect that resolving Conjecture 2.9 will
require studying the modular representation theory of Sn.
A possible approach to Conjecture 3.3 is to adapt the approach of the authors of [2], namely
to construct a Cherednik-like algebra related to f1, . . . , fn and the quasi-invariant polynomials.
Along the way, one may also find the formula of the Hilbert series.
Finally, it would be interesting to study q-deformations of the spaces of twisted quasi-invariant
polynomials. In [2], Braverman, Etingof and Finkelberg study q-deformations of their special
case and show that when Qm is free, its q-deformation is a flat deformation. They conjecture
that it is a flat deformation in general even when Qm is not a free module. Here, Qm,q(f1, . . . , fn)
is defined as the set of polynomials F for which
(1− si,j)(f1(x1) · · · fn(xn)F )
m∏
k=−m
(
xi − qkxj
)
is a smooth function for all 1 ≤ i < j ≤ n. It would be interesting to resolve this in the
case that Braverman, Etingof, and Finkelberg consider, as well as the general case we have
presented. We believe that q-analogues of some of our results hold. For example, the q-analogue
of Proposition 3.5 would be that Qm,q ⊂
∏m
k=−m
(
xi − qkxj
)
if
hi,j(qx)
hi,j(x)
is not rational.
Quasi-Invariants in Characteristic p and Twisted Quasi-Invariants 13
Acknowledgements
We would like to thank MIT PRIMES, specifically Pavel Etingof, for suggesting the project.
We would like to thank Eric Rains for very useful discussions. We also would like to thank the
referees for carefully reading our manuscript and for their valuable comments and suggestions
which substantially help to improve the readability and quality of the paper.
References
[1] Berest Yu., Chalykh O., Quasi-invariants of complex reflection groups, Compos. Math. 147 (2011), 965–1002,
arXiv:0912.4518.
[2] Braverman A., Etingof P., Finkelberg M., Cyclotomic double affine Hecke algebras (with an appendix by
Hiraku Nakajima and Daisuke Yamakawa), arXiv:1611.10216.
[3] Calogero F., Solution of the one-dimensional N -body problems with quadratic and/or inversely quadratic
pair potentials, J. Math. Phys. 12 (1971), 419–436.
[4] Chalykh O.A., Veselov A.P., Commutative rings of partial differential operators and Lie algebras, Comm.
Math. Phys. 126 (1990), 597–611.
[5] Eisenbud D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics,
Vol. 150, Springer-Verlag, New York, 1995.
[6] Etingof P., Lectures on Calogero–Moser systems, arXiv:math.QA/0606233.
[7] Feigin M., Veselov A.P., Quasi-invariants of Coxeter groups and m-harmonic polynomials, Int. Math. Res.
Not. 2002 (2002), 521–545, arXiv:math-ph/0105014.
[8] Felder G., Veselov A.P., Action of Coxeter groups on m-harmonic polynomials and Knizhnik–Zamolodchikov
equations, Mosc. Math. J. 3 (2003), 1269–1291, arXiv:math.QA/0108012.
[9] Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16
(1975), 197–220.
https://doi.org/10.1112/S0010437X10005063
https://arxiv.org/abs/0912.4518
https://arxiv.org/abs/1611.10216
https://doi.org/10.1063/1.1665604
https://doi.org/10.1007/BF02125702
https://doi.org/10.1007/BF02125702
https://doi.org/10.1007/978-1-4612-5350-1
https://arxiv.org/abs/math.QA/0606233
https://doi.org/10.1155/S1073792802106064
https://doi.org/10.1155/S1073792802106064
https://arxiv.org/abs/math-ph/0105014
https://doi.org/10.17323/1609-4514-2003-3-4-1269-1291
https://arxiv.org/abs/math.QA/0108012
https://doi.org/10.1016/0001-8708(75)90151-6
1 Introduction
2 Quasi-invariant polynomials over fields of nonzero characteristic
3 Twisted quasi-invariants
3.1 A generalization of quasi-invariants
3.2 Rationality of the logarithmic derivative
3.3 Hilbert series for n=2
4 Future prospects
References
|
| id | nasplib_isofts_kiev_ua-123456789-211013 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T20:34:05Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ren, Michael Xu, Xiaomeng 2025-12-22T09:28:21Z 2020 Quasi-Invariants in Characteristic and Twisted Quasi-Invariants. Michael Ren and Xiaomeng Xu. SIGMA 16 (2020), 107, 13 pages 1815-0659 2020 Mathematics Subject Classification: 81R12; 20C08 arXiv:1907.13417 https://nasplib.isofts.kiev.ua/handle/123456789/211013 https://doi.org/10.3842/SIGMA.2020.107 The spaces of quasi-invariant polynomials were introduced by Chalykh and Veselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series over fields of characteristic 0 was computed by Feigin and Veselov [Int. Math. Res. Not. 2002 (2002), 521-545]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, Braverman, Etingof, and Finkelberg [arXiv:1611.10216] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function. We would like to thank MIT PRIMES, specifically Pavel Etingof, for suggesting the project. We would like to thank Eric Rains for the very useful discussions. We also would like to thank the referees for carefully reading our manuscript and for their valuable comments and suggestions, which substantially helped to improve the readability and quality of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quasi-Invariants in Characteristic and Twisted Quasi-Invariants Article published earlier |
| spellingShingle | Quasi-Invariants in Characteristic and Twisted Quasi-Invariants Ren, Michael Xu, Xiaomeng |
| title | Quasi-Invariants in Characteristic and Twisted Quasi-Invariants |
| title_full | Quasi-Invariants in Characteristic and Twisted Quasi-Invariants |
| title_fullStr | Quasi-Invariants in Characteristic and Twisted Quasi-Invariants |
| title_full_unstemmed | Quasi-Invariants in Characteristic and Twisted Quasi-Invariants |
| title_short | Quasi-Invariants in Characteristic and Twisted Quasi-Invariants |
| title_sort | quasi-invariants in characteristic and twisted quasi-invariants |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211013 |
| work_keys_str_mv | AT renmichael quasiinvariantsincharacteristicandtwistedquasiinvariants AT xuxiaomeng quasiinvariantsincharacteristicandtwistedquasiinvariants |