For best-studied class of homogeneous spaces $G/H$ with $G > H$ compact, connected Lie groups, those for which rank($G$) = rank($H$), it is well known that the standard left ("isotropy") action of $H$ on $G/H$ is equivariantly formal, meaning every rational cohomology class on $G/H$ lifts to a class in Borel $H$-equivariant cohomology.

Moving to the case rank($G$) - rank($H$) = 1, we give a characterization of pairs $(G,H)$ such that the isotropy action is equivariantly formal, via a sequence of reductions ending with pairs such that $G/H$ has the rational homotopy type of a product of spheres. The irreducible such pairs are for the most part already classified in works of Kramer, Wolfrom, and Bletz-Siebert, which require only mild extension to handle the cases we are interested in and then reduce the entire problem to a verification of finitely many cases.

This work is joint with Chen He.