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

Abstract

In this paper 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 (2009) [20], who classified finite dimensional spherical representations in the case of Weyl groups and equal parameters (i.e., when c is a constant function). Our proof is based on the Macdonald–Mehta integral and the elementary theory of distributions.