Equivariant formality of isotropic torus actions, I

It is an open question for which pairs $(G,K)$ of compact Lie groups $G$ and closed, connected subgroups $K$ the left action of $K$ on the homogeneous space $G/K$ is equivariantly formal. In this work, we reduce this question to the case $K$ is a torus and $G$ is simply-connected by successively replacing members of the pair $(G,K)$ with simpler groups. The results, combined, essentially reduce equivariant formality of isotropy actions to the study of embeddings of tori in simply-connected groups. To illustrate the feasibility of this approach, we classify all pairs $(G,S^1)$ such that $G$ is compact connected Lie and the embedded $S^1$ subgroup of $G$ acts equivariantly formally on $G/S^1$. The computation depends only on the dimension of the cohomology ring $H^*(G/S^1)$ and the number of components of the normalizer $N_G(S^1)$.