Some topological properties of quotients modulo semisimple algebraic groups

Abstract

We will prove a general result in Invariant Theory, viz. for a quotient $C^n//G,$ where $G$ is a connected complex semisimple algebraic group, the local first homology group at any point in the quotient $C^n//G$ is trivial and the local second homology group is finite. Using this we will prove that the completion of the local ring of any point in $C^n//G$ is a unique factorization domain (UFD).