Equivariant formality of isotropic torus actions

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 what seems to be the first published proof of the structure (known to Leray and Koszul) of the cohomology rings $H^*(G/S;\mathbb Q)$.Comment: Completely revised. Many proofs simplified, including reduction to toral isotropy and classification of reflected circles. An error in the reduction to the semisimple case is corrected. New: a reduction to the compact case; partial reductions if the groups are disconnected or compact but not Lie. Citations to literature improved. To be published in the Journal of Homotopy and Related Structure