Equivariant formality of isotropy actions on homogeneous spaces defined by Lie group automorphisms

TL;DR

This paper proves that for certain homogeneous spaces formed by compact Lie groups with automorphism-defined subgroups, the isotropy action is equivariantly formal, and the pair forms a Cartan pair, advancing understanding in Lie group actions.

Contribution

It establishes the equivariant formality of isotropy actions on homogeneous spaces defined by automorphisms, and confirms that such pairs are Cartan pairs, a novel result in Lie theory.

Findings

Isotropy action is equivariantly formal for these spaces

The pair (G, K) forms a Cartan pair

Advances understanding of Lie group automorphism actions

Abstract

We show that the isotropy action of a homogeneous space $G/K$, where $G$ and $K$ are compact, connected Lie groups and $K$ is defined by an automorphism on $G$, is equivariantly formal and that $(G, K)$ is a Cartan pair.