Skew Group Algebras, Invariants

The aim of this paper is two fold. We study rst nite groups G of automorphisms of the homogenized Weyl algebra Bn, the skew group algebra Bn G, the ring of invariants B G n , and the relations of these algebras with the Weyl algebra An, with the skew group algebra An G, and with the ring of invariants A G . Of particular interest is the case n=1. On the other hand, we consider the invariant ring C[X] G of the polynomial ring C[X] in n generators, where G is a nite subgroup of Gl(n,C) such that any element in G dierent from the identity does not have one as an eigenvalue. We study the relations between the category of nitely generated modules over C[X] G and the corresponding category over the skew group algebra C[X] G. We obtain a generalization of known results for n=2 and G a nite subgroup of Sl(2,C ). In the last part of the paper we extend the results for the polynomial algebra C[X] to the homogenized Weyl algebra Bn