Equivariant formality of istropy actions

TL;DR

This paper investigates the conditions under which the isotropy action of a subgroup on a homogeneous space is equivariantly formal, linking it to formality in rational homotopy theory and providing new characterizations and proofs.

Contribution

It offers a representation-theoretic characterization of equivariant formality and extends previous results by strengthening theorems and providing a unified proof approach.

Findings

Equivariant formality implies rational homotopy formality of the space.

A K-theoretic analogue characterizes equivariant formality in terms of representation theory.

New proofs of equivariant formality for certain classes of homogeneous spaces.

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.