Localization in equivariant operational K-theory and the Chang-Skjelbred property

We establish a localization theorem of Borel-Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne (Doc Math 20:357-399, 2015). Inspired by the work of Chang-Skjelbred and Goresky-Kottwitz-MacPherson, we establish a general form of GKM theory in this setting, applicable to singular schemes with torus action. Our results are deduced from those in the smooth case via Gillet-Kimura's technique of cohomological descent for equivariant envelopes. As an application, we extend Uma's description of the equivariant K-theory of smooth compactifications of reductive groups to the equivariant operational K-theory of all, possibly singular, projective group embeddings.