Orbit closures and invariants

Abstract

Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let X^H denote the set of fixed points of H in X, and N_G(H) the normalizer of H in G. In this paper we study the natural map of quotient varieties psi _{X,H}:X^H/N_G(H) rightarrow X/G induced by the inclusion X^H subseteq X. We show that, given G and H, psi _{X,H} is a finite morphism for all affine G-varieties X if and only if H is a G-completely reducible subgroup of G (in the sense defined by Serre); this was proved in characteristic 0 by Luna in the 1970s. We discuss some applications and give a criterion for psi _{X,H} to be an isomorphism. We show how to extend some other results in Luna’s paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then the double coset HgK is closed for generic gin G if and only if Hcap gKg^{-1} is reductive for generic gin G.