Closed H-orbits in G-varieties

TL;DR

This paper investigates the relationship between G-orbits and H-orbits in algebraic varieties, extending classical results to positive characteristic and providing criteria for orbit closure equivalences.

Contribution

It generalizes Luna's classical results on orbit closures from characteristic zero to arbitrary characteristic for reductive groups acting on affine varieties.

Findings

Established conditions for when G.x closed implies H.x closed.

Extended Luna's results to positive characteristic.

Provided new criteria for orbit closure in algebraic group actions.

Abstract

Let G be a linear algebraic group, H a subgroup of G and X a G-variety. This paper explores the connection between G-orbits and H-orbits in X, concentrating in particular on the question of when we have the implications G.x closed in X implies H.x closed in X, and vice versa. Some of the general results found in this paper are then used in the special case where G and H are reductive and X is affine to extend two classical results of Luna from characteristic zero to arbitrary characteristic.