Finite dimensional representations of sympletic reflection algebras for wreath products

Abstract

Symplectic reflection algebras are attached to any finite group G of automorphisms of a symplectic vector space V , and are a multi-parameter deformation of the smash product TV ?G, where TV is the tensor algebra. Their representations have been studied in connection with different subjects, such as symplectic quotient singularities, Hilbert scheme of points in the plane and combinatorics. Let ... SL(2,C) be a finite subgroup, and let Sn be the symmetric group on n letters. We study finite dimensional representations of the wreath product symplectic reflection algebra ... of rank n, attached to the wreath product group ... and to the parameters (k, c), where k is a complex number (occurring only for n > 1), and c a class function on the set of nontrivial elements of ... In particular, we construct, for the first time, families of irreducible finite dimensional modules when ... is not cyclic, n > 1, and (k, c) vary in some linear subspace of the space of parameters. The method is deformation theoretic and uses properties of the Hochschild cohomology of H1,k,c(...), and a Morita equivalence, established by Crawley-Boevey and Holland, between the rank one algebra H1, ... and the deformed preprojective algebra ?Q), where Q is the extended Dynkin quiver attached to ?? via the McKay correspondence. We carry out a similar construction for continuous wreath product symplectic reflection algebras, a generalization to the case when ... SL(2,C) is infinite reductive. This time the main tool is the definition of a continuous analog of the deformed preprojective algebras for the infinite affine Dynkin quivers corresponding to the infinite reductive subgroups of SL(2,C).