Localization in equivariant operational K-theory and applications

For a complete nonsingular variety with a torus action, the localization principle asserts, roughly speaking, that one can read-off the equivariant $K$-theory of the whole variety from that of fixed point subvariety, modulo certain relations given by the fixed loci of codimension-one subtori. For singular varieties, however,
such method quite often does not apply. The purpose of this talk is to show that in the setting of equivariant operational theories there is a version of the localization
principle that works perfectly well for both singular and nonsingular varieties. For instance, if $X$ is any complete variety where a torus acts with finitely many fixed points and invariant curves, then the equivariant operational $K$-theory of $X$ is a ring of piecewise exponential functions (a version of GKM theory). To illustrate the results, the case of projective embeddings of reductive groups is discussed in some detail.