Equivariant formality of isotropic torus actions

TL;DR

This paper investigates the conditions under which the left action of a connected Lie group on a homogeneous space is equivariantly formal, providing classifications and cohomology ring structures for such actions.

Contribution

It classifies pairs of compact Lie groups and embedded circles acting equivariantly formally on homogeneous spaces, and proves the structure of their cohomology rings.

Findings

Classification of pairs (G,S) with equivariantly formal actions

Proof of the cohomology ring structure of G/S

Reduction to the case where G is compact and simply-connected

Abstract

Considering the potential equivariant formality of the left action of a connected Lie group $K$ on the homogeneous space $G/K$, we arrive through a sequence of reductions at the case $G$ is compact and simply-connected and $K$ is a torus. We then classify all pairs $(G,S)$ such that $G$ is compact connected Lie and the embedded circular subgroup $S$ acts equivariantly formally on $G/S$. In the process we provide a proof of the structure (known to Leray and Koszul) of the cohomology rings $H^*(G/S;\mathbb Q)$.