The fundamental representation-theoretic data associated to a continuous group action is the isotropy action on the normal bundle of a stratum, and the actions most amenable to algebraic study are the equivariantly formal actions, whose Borel cohomology determines their singular cohomology.

We characterize equivariant formality for an isotropy action and compute the equivariant formality of an equivariantly formal action, showing along the way that the homogeneous space underlying the action is rationally formal.

A rationally equivalent question in equivariant K-theory is surjectivity of the map to nonequivariant K-theory forgetting an action on a complex bundle, as shown by Fok. We use this correspondence and a map of Künneth spectral sequences to calculate the equivariant K-theory of an equivariantly formal isotropy action in the best-studied class of cases. We also show that in these cases, surjectivity of the forgetful map integrally is equivalent to its kernel being generated by virtual representations of dimension zero.

Some of this work is joint with Chi-Kwong Fok.