Lifting automorphisms of generalized adjoint quotients

Abstract

Let G be a reductive algebraic group over an algebraically closed field K of characteristic zero. Let $\pi :{\mathfrak{g}^r} \to X = {\mathfrak{g}^r}//G$ be the categorical quotient where $\mathfrak{g}$ is the adjoint representation of G and r is a suitably large integer (in general r ≥ 5, but for many cases r ≥ 3 or even r ≥ 2 suffices). We show that every automorphism φ of X lifts to a map $\Phi :{\mathfrak{g}^r} \to {\mathfrak{g}^r}$ commuting with π. As an application we consider the action of φ on the Luna stratification of X.