Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups

Abstract

We determine the support of the irreducible spherical representation (i.e., the irreducible quotient of the polynomial representation) of the rational Cherednik algebra of a finite Coxeter group for any value of the parameter c. In particular, we determine for which values of c this representation is finite dimensional. This generalizes a result of Varagnolo and Vasserot, arXiv:0705.2691, who classified finite dimensional spherical representations in the case of Weyl groups and equal parameters (i.e., when c is a constant function). As an application, we compute the zero set of the kernel of the Macdonald pairing in the trigonometric case (for equal parameters). Our proof is based on the Macdonald-Mehta integral and the elementary theory of distributions.

12 pages, latex; in the revision there is a new appendix by Stephen Griffeth, proving directly and uniformly that the condition of Corollary 3.3 is equivalent to the requirement that m is an elliptic number, and also a new Section 5 on the trigonometric c