Equivariant formality of istropy 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 $G/K$. 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 a theorem of Shiga--Takahashi to a characterization of equivariant formality in this case. Using a K-theoretic analogue of equivariant formality introduced and shown by the second-named author to be equivalent to equivariant formality in the usual sense, we provide a representation-theoretic characterization for equivariant formality of the isotropy action and give a new, uniform proof of equivariant formality for some classes of homogeneous spaces for which it was previously known.