Finite dimensional representations of symplectic reflection algebras associated to wreath products II

Details Finite dimensional representations of symplectic reflection algebras associated to wreath products II Finite dimensional representations of symplectic reflection algebras associated to wreath products II

2005-01-11

arxiv.org/abs/math/0501156v1

Mathematics

Representation Theory

10 pages, Latex

Scientific paper

This note extends some results of a previous paper (math.RT/0403250) about 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, let N'=(N_1,...,N_r) be a partition of N. Consider W=W_1\otimes >...\otimes W_r an irreducible representation of S_N'=S_{N_1}\times ...\times S_{N_r}\subset S_N s.t. W_i has rectangular Young diagram for any i and let {Y_i}, i=1,...,r be a collection of irreducible non isomorphic representation of the rank 1 algebra B=H(c_0,1,G) s.t. Ext^1_B(Y_i, Y_j)=0 for any i\neq j. Consider the module M'=W\otimes Y, where Y=Y_1^{N_1}\otimes ...\otimes Y_r^{N_r}, over the subalgebra S_N'#B^N \subset S_N#B^N= H(0,c_0,N,G) and form the induced module M over H(0,c_0,N,G). We show that M can be uniquely deformed along a linear subspace of codimension r in the space of the parameters (k,c) passing through c_0. This result implies the main result of math.RT/0403250 as a particular case, the case of the trivial partition N'=(N).

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