Orbit Closures and Invariants

TL;DR

This paper investigates the conditions under which the natural map between quotient varieties induced by subgroup fixed points is finite, extending Luna's characteristic zero results to positive characteristic and exploring related geometric properties.

Contribution

It characterizes when the quotient map is finite based on G-complete reducibility of H, extends Luna's results to positive characteristic, and analyzes double coset closure conditions.

Findings

The quotient map is finite if and only if H is G-completely reducible.

Extension of Luna's results to positive characteristic.

A criterion for double coset closure involving reductiveness of intersections.

Abstract

Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let Y denote the set of fixed points of H in X, and N(H) the normalizer of H in G. In this paper we study the natural map from the quotient of Y by N(H) to the quotient of X by G induced by the inclusion of Y in X. We show that, given G and H, this map is a finite morphism for all G-varieties X if and only if H is G-completely reducible (in the sense defined by J-P. Serre); this was proved in characteristic zero by Luna in the 1970s. We discuss some applications and give a criterion for the map of quotients 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 in G if and only if the intersection of generic conjugates of H and K is reductive.