Good Representations and Homogeneous Spaces

TL;DR

This paper investigates the properties of reductive group actions on homogeneous spaces, proving that generic orbits are closed, and applies this to representations and subgroup intersections.

Contribution

It establishes that generic orbits under reductive group actions on homogeneous spaces are closed, providing new insights into representation theory and subgroup intersections.

Findings

Generic H orbits on G/F are closed

Applications to G and H representations

Insights into reductive subgroup intersections

Abstract

Let G be a real or complex linear algebraic reductive group. Let H and F be reductive subgroups. We study the natural H action on G/F. The main theorem of this note shows that generic H orbits are closed. This theorem is then applied to study representations of G, representations of H which are induced from representations of G, and intersections of reductive subgroups of G.