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 . The group G acts on GN 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 HN/H→GN/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 Γ is a finite group then the character variety C(Γ,G) of closed conjugacy classes of representations from Γ to G is finite.