Equivariant formality and representation theory

Let G be a compact connected Lie group and K its connected Lie subgroup. Using the K-theoretic version of equivariant formality developed in [F2] which involves analysis of vector bundles over G/K, we find two characterizations of equivariant formality of the isotropy action of K on G/K. The first one equates equivariant formality with a "smoothness" condition of the restriction map of the representation ring of G to that of K. The second characterization asserts that equivariant formality amounts to the equivariant K-theoretic index pairing of two certain special K-theory classes being nontrivial in some sense. By applying these characterizations, we are able to give a representation theoretic criterion for equivariant formality of the isotropy action by a circle subgroup, as well as an invariant theory criterion for the case where K is a torus of dimension one less than the rank of G, culminating in a complete classification of equivariant formality where G is further assumed to be simple.