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 XH denote the set of fixed points of H in X, and NG(H) the normalizer of H in G. In this paper we study the natural map of quotient varieties ψX,H: XH/ NG(H) → X/ G induced by the inclusion XH⊆ X. We show that, given G and H, ψX,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 ψX,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 g∈ G if and only if H∩ gKg- 1 is reductive for generic g∈ G.