Reductive subgroups of reductive groups in nonzero characteristic

Let G be a (possibly nonconnected) reductive linear algebraic group over an algebraically closed field k, and let N is an element of N. The group G acts on G(N) by simultaneous conjugation. Let H be a reductive subgroup of G. We prove that if k has nonzero characteristic then the natural map of quotient varieties H-N/H --> G(N)/G is a finite morphism. We use methods introduced by Vinberg, who proved the same result in characteristic zero. As an application, we show that if Gamma is a finite group then the character variety C(Gamma, G) of closed conjugacy classes of representations from Gamma to G is finite. (C) 2003 Elsevier Science (USA). All rights reserved.