Ju n 20 01 On m - quasiinvariants of a Coxeter group

Let W be a finite Coxeter group in a Euclidean vector space V , and m a W -invariant Z+-valued function on the set of reflections in W . Chalyh and Veselov introduced in [CV] an interesting algebra Qm, called the algebra of m-quasiinvariants for W , such that C[V ] W ⊆ Qm ⊆ C[V ], Q0 = C[V ], Qm ⊇ Qm′ if m ≤ m , and ∩mQm = C[V ] W . Namely, Qm is the algebra of quantum integrals of the rational Calogero-Moser system with coupling constants m. The algebra Qm has been studied in [CV], [VSC], [FeV] and [FV]. In particular, in [FV] Feigin and Veselov proposed a number of interesting conjectures concerning the structure of Qm, and verified them for dihedral groups and constant functions m. Our goal is to prove some of these conjectures in the general case. 1 Definitions and main results We recall some definitions from [FV]. Consider a real Euclidean space V of dimension n. We will often identify V and V ∗ using the inner product on V . Let W be a finite Coxeter group, i.e. a finite group generated by reflections of V . Let N = |W |. Let Σ be the set of reflections in W , and Πs be the reflection hyperplane for a reflection s. Let m : Σ → Z≥0, s 7→ ms, be a W -invariant function (called the multiplicity function). A complex polynomial q on V is said to be an m-quasiinvariant (under W ) if, for each s ∈ Σ, the function x 7→ q(x)− q(sx) vanishes up to order 2ms + 1 at the hyperplane Πs. Such polynomials form a graded subalgebra in the graded algebra C[V ] = ⊕ i≥0 C[V ]〈i〉, which will be denoted by Qm. It is obvious that Qm contains as a subalgebra the ring C[V ]W of invariant polynomials. We denote by Im the ideal in Qm generated by the augmentation ideal in C[V ]W . This is a graded ideal in Qm. The following two theorems, conjectured in [FV], are two of the main results of this paper. Let T be any graded complement of Im in Qm. Theorem 1.1. Qm is a free module over C[V ] W , of rank N . More specifically, the multiplication mapping defines a graded isomorphism C[V ]W ⊗ T → Qm. In particular, dim(Qm/Im) = codim(Im) = N . Consider now theN -dimensional graded algebraRm = Qm/Im. Let d = ∑ s∈Σ (2ms + 1). Theorem 1.2. (i) The space Rm〈d〉 is one dimensional. (ii) (Poincare duality). The multiplication mapping Rm〈j〉 × Rm〈d − j〉 → Rm〈d〉 is a nondegenerate pairing for any j. In particular, the Poincare polynomial PRm(t) is a palindromic polynomial of degree d (i.e. PRm(t −1) = tPRm(t)), and the algebra Rm is Gorenstein. (iii) The algebra Qm is Gorenstein.