Finite orbit decomposition of real flag manifolds

TL;DR

This paper proves that for a real semi-simple Lie group G, a subgroup H has finitely many orbits on the flag manifold G/P if and only if it has an open orbit, confirming a conjecture by T. Matsuki.

Contribution

It establishes a precise criterion linking open orbits and finite orbit decomposition for subgroups on flag manifolds, confirming a longstanding conjecture.

Findings

H has an open orbit on G/P if and only if it has finitely many orbits.

Confirms T. Matsuki's conjecture.

Provides a complete characterization of orbit structures on flag manifolds.

Abstract

Let G be a connected real semi-simple Lie group and H a closed connected subgroup. Let P be a minimal parabolic subgroup of G. It is shown that H has an open orbit on the flag manifold G/P if and only if it has finitely many orbits on G/P. This confirms a conjecture by T. Matsuki.