Equivariant formality of isotropy actions

Abstract

Let $G$ be a compact connected Lie group and $K$ a connected Lie subgroup. In this paper, we collect an assortment of results on equivariant formality of the isotropy action of $K$ on $G/K$ and thus improving those from previous work. We show that if the isotropy action of $K$ on $G/K$ is equivariantly formal, then $G/K$ is formal in the sense of rational homotopy theory. This enables us to strengthen Shiga-Takahashi's theorem to a cohomological characterization of equivariant formality of isotropy actions. Using an analogue of equivariant formality in $K$-theory introduced by the second author and shown to be equivalent to equivariant formality in the usual sense, we provide a representation theoretic characterization of equivariant formality of isotropy actions, and give a new, uniform proof of equivariant formality for previously known examples of homogeneous spaces.