Reflection functors and representations for continuous wreath-product symplectic reflection algebras

Abstract

We introduce "continuous deformed preprojective algebras" attached to infinite affine Dynkin quivers of type A_{\infty}, A_{+\infty}, D_{\infty}. We define a one-parameter family of deformations of the wreath product of a symmetric group with these algebras and, using the generalized McKay correspondence for infinite reductive subgroups of SL(2,C), we establish a Morita equivalence to a corresponding continuous symplectic reflection algebra of wreath product type. We give applications of this Morita equivalence to the representation theory of continuous wreath product symplectic reflection algebras. In particular, we use Crawley-Boevey and Holland's results about the representation theory of deformed preprojective algebras to get a complete classification of the finite dimensional irreducible representations for the rank one continuous wreath product symplectic reflection algebras. In higher rank, we extend Gan's definition of the reflection functors to the continuous case and we use such functors to find an interesting class of finite dimensional representations.