We consider the polynomial representation S(V*) of the rational Cherednik algebra H_c(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and nonnegative integer m the space S(V*) contains a single copy of the reflection representation V of W spanned by the homogeneous singular polynomials of degree d-1+hm, where h is the Coxeter number of W; these polynomials generate an H_c(W) submodule with the parameter c=(d-1)/h+m. We express these singular polynomials through the Saito polynomials that are flat coordinates of the Saito metric on the orbit space V/W. We also show that this exhausts all the singular polynomials in the isotypic component of the reflection representation V for any constant parameter c.
In this paper we relate two remarkable constructions associated with a finite Coxeter group W . The first one is the Frobenius manifold structure on the space of orbits of W acting in its reflection representation V [1]. The key ingredient here is the Saito flat metric on the orbit space V /W [2]. This metric is defined as a Lie derivative of the standard contravariant (Arnold) metric. The flat coordinates form a distinguished basis in the ring of invariant polynomials S(V * ) W . This basis is now known explicitly for all irreducible groups W . All the cases except W of type E 7 , E 8 were covered in the original paper [2]. The flat coordinates in the latter two cases were found recently both in [3] and in [4].
The other famous construction associated with the group W is the rational Cherednik algebra H c (W ) [5]. It depends on the W -invariant function c on the set of reflections of W which we assume to be constant. The key ingredient here is the Dunkl operator ∇ ζ , ζ ∈ V , which acts in the ring of polynomials as a differential-reflection operator [6]. For particular values of c the polynomial representation S(V * ) has nontrivial submodules M. These values were completely determined by Dunkl, de Jeu and Opdam in [7] where it was shown that non-trivial submodules exist if and only if c is a non-integer number of the form c = l/d where d is one of the degrees of the Coxeter group W and l ∈ Z >0 . The lowest homogeneous component M 0 of M consists of so-called singular polynomials [7] which are annihilated by Dunkl operators ∇ ζ for any ζ ∈ V . All singular polynomials were found by Dunkl when W has type A [8,9]. Further, it was established in [10] that in this case any submodule M is generated by its lowest homogeneous component M 0 . In general the structure of submodules of S(V * ) and the corresponding singular polynomials are not known. Some singular polynomials for the classical groups W and for the icosahedral group were determined in [11] and [12] (see also [13]) respectively, while dihedral case was fully studied in [7] (see also [14]).
In the paper we study singular polynomials that belong to the isotypic component of the reflection representation V of the Coxeter group W . The existence of such singular polynomials is known for the Weyl groups when c = r/h where h is the Coxeter number of W and r is a positive integer coprime with h [15]. It appears that in general the corresponding parameter values have to be c = (d -1)/h + m, where d is one of the degrees of W and m ∈ Z ≥0 . We explain how to construct all the singular polynomials in the isotypic component of V in terms of the Saito polynomials that are flat coordinates of the Saito metric. We use theory of Frobenius manifolds and especially Dubrovin’s almost duality [17]. We show that singular polynomials under consideration correspond to the W -invariant polynomial twisted periods of the Frobenius manifold V /W , and we determine all such twisted periods.
Firstly we prove that the first order derivatives of the Saito polynomials are singular polynomials at appropriate values of the parameter c = (d -1)/h (Corollary 2.14). Then we explain how to construct further singular polynomials with parameter c shifted by an integer (Theorem 3.16). Then we show in Corollary 4.10 that this construction provides all the singular polynomials in the isotypic component of the reflection representation.
In the last section we present residue formulas for all the polynomial invariant twisted periods in the case of classical Coxeter groups W . Then we generalize them to get some singular polynomials for the complex reflection group W = S n ⋉ (Z/ℓZ) n .
Let V = C n with the standard constant metric g given by g(e i , e j ) = (e i , e j ) = δ ij where e i , i = 1, . . . , n, is the standard basis in V . Let (x 1 , . . . , x n ) be the corresponding orthogonal coordinates. Let W be an irreducible finite Coxeter group of rank n which acts in V by orthogonal transformations such that V is the complexified reflection representation of W . Let R ⊂ V be the Coxeter root system with the group W [16]. Let y 1 (x), . . . , y n (x) be a homogeneous basis in the ring of invariant polynomials
We assume the polynomials are ordered so that d 1 ≥ . . . ≥ d n ; d 1 = h is the Coxeter number of the group W . The polynomials y 1 , . . . , y n are coordinates on the orbit space M = V /W . The Euclidean coordinates x 1 , . . . , x n can also be viewed as local coordinates on M\Σ, where Σ is the discriminant set. Denote by S = {x ∈ V |(γ, x) = 0 for some γ ∈ R} the preimage of Σ in the space V .
The metric g is defined on M \ Σ due to its W -invariance. Let g αβ be the corresponding contravariant metric. Consider its Lie derivative η αβ (y) = ∂ y 1 g αβ (y). The metric η is called the Saito metric. It is correctly defined (up to proportionality), and it is flat. There exist homogeneous coordinates
such that η is constant and anti-diagonal:
where δ i j = δ ij is the Kroneck
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