Equivariant formality of homogeneous spaces

Let $G$ be a compact connected Lie group and $K$ a connected Lie subgroup. In this paper, we continue the attempt of characterizing equivariant formality of the isotropy action of $K$ on $G/K$ explored in \cite{Sh}, \cite{ShTa} and \cite{Ca}. We introduce an analogue of equivariant formality in $K$-theory called rational $K$-theoretic equivariant formality (RKEF) and show that it is equivalent to equivariant formality in the usual sense. Using RKEF, we give a more uniform proof of the main results in \cite{Go} and \cite{GoNo} that the isotropy actions on (generalized) symmetric spaces are equivariantly formal, without appealing to the classification theorem of those spaces. We also give a representation theoretic condition which is equivalent to $G/K$ being formal and the isotropy action being equivariantly formal, and show how it provides a uniform proof of the fact that, if $(G, K)$ is an equal rank pair, cohomlogically surjective pair or (generalized) symmetric pair, $G/K$ is formal and the isotropy action is equivariantly formal.