A decomposition of equivariant K-theory in twisted equivariant K-theories

TL;DR

This paper presents a decomposition of G-equivariant K-theory for finite groups acting on spaces, expressing it as a sum of twisted theories based on the conjugation action on irreducible representations, revealing new structural insights.

Contribution

It introduces a novel decomposition of equivariant K-theory into twisted components parametrized by group actions on irreducible representations, with explicit cocycle twists.

Findings

Decomposition of G-equivariant K-theory into twisted components.

Identification of cocycle twists encoding representation lifting obstructions.

Structural understanding of K-theory in terms of group actions and representations.

Abstract

For G a finite group and X a G-space on which a normal subgroup A acts trivially, we show that the G-equivariant K-theory of X decomposes as a direct sum of twisted equivariant K-theories of X parametrized by the orbits of the conjugation action of G on the irreducible representations of A. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of A to the subgroup of G which fixes the isomorphism class of the irreducible representation.