Supports of Irreducible Spherical Representations of Rational Cherednik Algebras of Finite Coxeter Groups

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, [VV], 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. The organization of the paper is as follows. Section 2 contains preliminaries on Coxeter groups and Cherednik algebras. In Section 3 we state and prove the main result in the case of equal parameters. In Section 4 we deal with the remaining case of irreducible Coxeter groups with two conjugacy classes of reflections. Finally, in the appendix, written by Stephen Griffeth, it is shown by a uniform argument (using only the theory of finite reflection groups) that our classification of finite dimensional spherical representations of rational Cherednik algebras with equal parameters coincides with that of Varagnolo and Vasserot. Acknowledgements. It is my great pleasure to dedicate this paper to my father Ilya Etingof on his 80-th birthday. His selflessness and wisdom made him my main role model, and have guided me throughout my life. This work was partially supported by the NSF grants DMS-0504847 and DMS-0854764.