Finite dimensional representations of symplectic reflection algebras associated to wreath products

Abstract

In this paper we construct finite dimensional representations of the wreath product symplectic reflection algebra H(k,c,N,G) of rank N attached to a finite subgroup G of SL(2,C) (here k is a number and c a class function on the set of nontrivial elements of G). Specifically, we show that if W is an irreducible representation of S_N whose Young diagram is a rectangle, and Y an irreduible finite dimensional representation of H(c,1,G), then the representation M=W\otimes Y^N of H(0,c_0,N,G) can be deformed along a hyperplane in the space of parameters (k,c) passing through c_0. On the other hand, if Y is 1-dimensional and the Young diagram of W is not a rectangle, such a deformation does not exist.