Equivariant formality of isotropy actions

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 prove a representation‐theoretic characterization for equivariant formality of the isotropy action and give a new, uniform proof of equivariant formality for some previously known classes of examples.