Lifting automorphisms of generalized adjoint quotients

Let G be a reductive algebraic group over an algebraically closed field K of characteristic zero. Let p:\mathfrakg^r ? X = \mathfrakg^r//G \pi :{\mathfrak{g}^r} \to X = {\mathfrak{g}^r}//G be the categorical quotient where \mathfrakg \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 F:\mathfrakg^r ? \mathfrakg^r \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.