Skew group algebras, invariants and Weyl Algebras

The aim of this paper is two fold: First to study finite groups $G$ of automorphisms of the homogenized Weyl algebra $B_{n}$, the skew group algebra $B_{n}\ast G$, the ring of invariants $B_{n}^{G}$, and the relations of these algebras with the Weyl algebra $A_{n}$, with the skew group algebra $A_{n}\ast G$, and with the ring of invariants $A_{n}^{G}$. Of particular interest is the case $n=1$. In the on the other hand, we consider the invariant ring $\QTR{sl}{C}[X]^{G}$ of the polynomial ring $K[X]$ in $n$ generators, where $G$ is a finite subgroup of $Gl(n,\QTR{sl}{C}$) such that any element in $G$ different from the identity does not have one as an eigenvalue. We study the relations between the category of finitely generated modules over $\QTR{sl}{C}[X]^{G}$ and the corresponding category over the skew group algebra $\QTR{sl}{C}% [X]\ast G$. We obtain a generalization of known results for $n=2$ and $G$ a finite 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 $B_{n}$.