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.