Jack polynomials and the coinvariant ring of $G(r,p,n)$

We study the coinvariant ring of the complex reflection group $G(r,p,n)$ as a module for the corresponding rational Cherednik algebra $\HH$ and its generalized graded affine Hecke subalgebra $\mathcal{H}$. We construct a basis consisting of non-symmetric Jack polynomials, and using this basis decompose the coinvariant ring into irreducible modules for $\mathcal{H}$. The basis consists of certain non-symmetric Jack polynomials, whose leading terms are the ``descent monomials'' for $G(r,p,n)$ recently studied by Adin, Brenti, and Roichman and Bagno and Biagoli. The irreducible $\mathcal{H}$-submodules of the coinvariant ring are their ``colored descent representations''.


Introduction.
The aim of this paper is to understand the coinvariant ring for the complex reflection group G(r, p, n) as a module over the rational Cherednik algebra and its generalized graded affine Hecke subalgebra. As applications we construct a basis for the coinvariant ring consisting of certain of the non-symmetric Jack polynomials discovered in [6], and give a new realization of the "colored descent representations" studied in [2] as irreducible modules for a generalized graded affine Hecke algebra.
The classical formulas where S n is the group of permutations of {1, 2, . . . , n}, l(w) is the length of w and maj(w) is the major index of w may be obtained by computing the Hilbert series for the coinvariant ring of the symmetric group in three ways: the left hand side corresponds to the divided difference basis, the middle is the quotient of the Hilbert series for all polynomials by the Hilbert series for symmetric polynomials, and the right hand side corresponds to the descent basis.
In [1] Adin, Brenti and Roichman constructed "colored descent bases" and a corresponding "flag major index" for the coinvariant rings of the type B Weyl groups G(2, 1, n) (see [1]). As an application they decompose the coinvariant ring into "colored descent representations". Analagous results for the groups G(r, p, n) are contained in the paper [2] of Bagno and Biagoli. Our main theorem (5.2) constructs a new basis consisting of non-symmetric Jack polynomials by viewing the coinvariant ring as a module for the rational Cherednik algebra, and shows that upon restriction to the generalized graded affine Hecke algebra inside H, the coinvariant ring decomposes into irreducible submodules corresponding to "colored descent classes" of elements of G(r, p, n). These are the representations studied without the use of Hecke algebras in [2], and the leading terms of our basis elements are their "colored descent monomials". We suspect that a version of our results holds for Weyl groups, upon replacing the rational Cherednik algebra and Jack polynomials with the double affine Hecke algebra and Macdonald polynomials, and using the exponential coinvariant ring in place of the coinvariant ring.
We hope our results may eventually shed some light on the seeming intractibility of the corresponding problems for the exceptional complex reflection groups. Here the missing ingredient seems to be an analog of the generalized graded affine Hecke subalgebra of the rational Cherednik algebra.
Acknowledgements. Part of this paper is based on a thesis ( [11]) written at the University of Wisconsin under the direction of Arun Ram. I am greatly indebted to him for teaching me about affine Hecke algebras and for suggesting the problem that motivated this work.

Preliminaries and notation
Let h be a finite dimensional complex vector space. A reflection is an element s ∈ GL(h) such that codim(fix(s)) = 1. A complex reflection group is a finite subgroup W of GL(h) that is generated by the set of reflections it contains.
Let W be a complex reflection group, let T be the set of reflections in W , let κ be a variable, and let c s be a set of variables indexed by s ∈ T such that c wsw −1 = c s for all s ∈ T and w ∈ W . Let F be the field of rational functions with complex coefficients in the variables κ and c s , and by abuse of notation write h and h * for the vector spaces obtained by extension of scalars to F . Later we will specialize κ = 0; in what follows note that κ = 0 does not make any denominators vanish.
We write The semi-direct product of the tensor algebra T (h ⊕ h * ) and F W is and v, w ∈ W . From now on we will drop the tensor signs. The rational Cherednik algebra is where I is the ideal generated by the relations and (2.5) xy = yx for x, y ∈ h or x, y ∈ h * .
The PBW-theorem for H (see [4], [7], and [16]) asserts as a vector space, where S(h) and S(h * ) are the symmetric algebras of h and h * . It can be proved (for a field of any characteristic) by a straightforward adaptation of the standard proof of the PBW theorem for universal enveloping algebras. Given a F W -module V , define the Verma module M (V ) by where the set of positive degree polynomials S(h) >0 acts by 0 on V . The PBW theorem implies that as a complex vector space In particular, when V = 1 is the trivial representation of F W , we obtain the polynomial representation of H: for y ∈ h and f ∈ S(h * ), where ∂ y denotes the partial derivative in the direction y. These are the famous Dunkl operators. From our point of view, the fact that they commute is a consequence of the PBW theorem, though it is possible to prove the commutativity directly ( [6], for instance).
In Lemma 4.1 and Theorem 5.2 we will need the following notation: for µ ∈ Z n ≥0 let w µ ∈ S n be the shortest permutation such that w −1 µ .µ is a partition, and let v µ = w 0 w −1 µ be the longest permutation such that v µ .µ is an anti-partition.

The coinvariant ring of a complex reflection group
There is a useful Casimir element h in the algebra H that helps to distinguish between different lowest weight modules. This element is the analogue for H of the Euler vector field n i=1 x i ∂ ∂x i in the Weyl algebra. Fix dual bases x 1 , . . . , x n of h * and y 1 , . . . , y n of h. It is straightforward to check that the sum does not depend on the choice of dual bases. Let We have introduced the shift by c s in order to simply some formulas that occur later on. A calculation shows In the next proposition we use the fact that if V ∈ Irr(CW ) then M (V ) has a unique maximal proper graded submodule (even when κ = 0; otherwise the term "graded" may be omitted). The corresponding irreducible quotient is denoted L(V ).
For real reflection groups, the following proposition is a consequence of the results in [5]. Proof. In light of our assumption that κ = 0 and (2.9), the ideal I is H-stable and the coinvariant ring is an H-module. Let R be the (unique) maximal proper graded submodule of M (1). We must show that R ⊆ I. It suffices to prove that for all irreducible CW -modules V and all integers d that Suppose towards a contradiction that this is false, and choose d minimal so that it fails.
Thus c V = 0 and our hypothesis implies V = 1, contradiction.
Our strategy in the remainder of the paper is to construct a basis of the coinvariant ring for the groups G(r, p, n) that is particularly adapted to understanding its structure as an H-module.

Non-symmetric Jack polynomials
From now on we consider the case W = G(r, p, n), where G(r, p, n) is the group of n by n monomial matrices whose non-zero entries are rth roots of 1 and so that the product of the nonzero entries is an r/pth root of 1. We put ζ = e 2πi/r and let ζ i be the diagonal matrix with a ζ in the ith position and 1's elsewhere. As usual, s ij is the transposition matrix interchanging the ith and jth coordinates.
When n ≥ 3 the equations and s 1i ζ l i s −1 1i = ζ l 1 , for 1 ≤ i < j ≤ n, k = i, j, and 0 ≤ l ≤ r − 1, show that there are r/p conjugacy classes of reflections in G(r, p, n): (a) The reflections of order two: where ζ pl i and ζ pk j are conjugate if and only if k = l. Despite the fact that there are more conjugacy classes than described above when n = 2 and p > 1, the results in this paper go through without change.
The generalized degenerate affine Hecke algebra is the subalgebra H of H generated by CW and t. It was first constructed in [16], section 5, and in [3] it was observed that by the results of [6] it is a subalgebra of H. For W = G(1, 1, n), it is the usual graded affine Hecke algebra of the symmetric group.
Some formulas are simpler when written in terms of the following parameters: To efficiently describe the H-action on the basis f µ , we introduce the following operators: The operator σ i is well-defined on those t-weights spaces M α on which z i − z i+1 is invertible or π i acts as 0. We also define the intertwining operators Φ and Ψ by (4.12) Φ = x n t s n−1 ···s 1 and Ψ = y 1 t s 1 ···s n−1 .
The following lemma gives the action of the intertwiners on the basis f µ and is a special case of Lemma 5.2 of [12].
As a consequence of this lemma, the polynomials f µ are well-defined at κ = 0: they can be recursively constructed by using Φ and the σ i 's, which are well-defined on M (1) when κ = 0.

The coinvariant ring of G(r, p, n).
In this section we assume κ = 0 and that W = G(r, p, n). We will now obtain an eigenbasis for the coinvariant ring S/I for G(r, p, n) indexed by a certain subset of G(r, 1, n). First we need some definitions. We write elements of G(r, 1, n) as "colored permutations": The colored descent monomial corresponding to v is If w µ is the shortest permutation such that w −1 µ .µ is a partition, then we note that We will also need the formulas obtained by specializing (4.13) and (4.14) to κ = 0. Our proof our 5.2 requires the following combinatorial lemma: Then there exists a sequence s i 1 , . . . , s iq of simple transpositions so that des(s i j · · · s i 1 v) = des(s i j−1 . . . s i 1 v) for 1 ≤ j ≤ q and v ′ = s iq · · · s i 1 v.
Proof. Each descent class contains a unique v = wζ k 1 1 · · · ζ kn n so that if i is a descent of v with k i = k i+1 then w(i) = w(i + 1) + 1 and otherwise w(i) < w(i + 1). One checks that such a v is the unique element of its descent class with w of minimum length, and it is straightforward to check that for any other v ′ = w ′ ζ k 1 1 · · · ζ kn n in the same descent class, there is a simple reflection s i with l(s i w ′ ) < l(w ′ ) and des(s i v ′ ) = des(v ′ ).
The x λv generalize the descent monomials from [8] and [9], and recently appeared in [2]. The following theorem shows that they are the leading terms of a basis for the coinvariant ring consisting of certain κ = 0 specializations of non-symmetric Jack polynomials. It strengthens Theorem 8.8 of [13].
Theorem 5.2. Suppose κ = 0 and c s are generic. Then L(1) is the coinvariant ring for G(r, p, n) and has basis {f λv | v ∈ G(r, 1, n) p }, where G(r, 1, n) p = ζ k 1 w(1), . . . , ζ kn w(n) ∈ G(r, 1, n) | 0 ≤ k n ≤ r/p − 1 . The formulas (5.6) show that the t-eigenspaces on the span of f λv for v ∈ G(r, 1, n) p are all onedimensional. Therefore each irreducible H submodule of L(1) is spanned by the f µ 's it contains. Since H is generated by t and s 1 , . . . , s n , the C-span of a collection of f λv 's is an H-module exactly if it is stable under σ 1 , . . . , σ n , and is irreducible exactly if any two f λv 's can be connected by a sequence of invertible intertwiners. On the other hand, Lemma 4.1, Lemma 5.1, (5.5), and (5.7) can be combined once more to see that if des(v) = des(s i v) then σ i .f λv = cf λs i v with c = 0, and if v, v ′ ∈ G(r, 1, n) p then there is a sequence of invertible intertwiners σ i connecting f λv and f λ v ′ exactly if des(v) = des(v ′ ). The second assertion of the Theorem follows from this. Finally, the summands are pairwise non-isomorphic because their t-spectra are different.
The above theorem may help explain why the major index is difficult to define directly on the group G(r, p, n): it is the subset G(r, 1, n) p of G(r, 1, n) that naturally indexes the basis of Jack polynomials, not the group G(r, p, n).