  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 $B_{n}$.
