Proper actions and decompositions in equivariant K-theory

TL;DR

This paper develops a new decomposition of G-equivariant K-theory for proper G-spaces, generalizing Mackey's theory to broader classes of groups, and applies it to spaces with a single isotropy type.

Contribution

It introduces a novel decomposition of equivariant K-theory applicable to discrete, linear, and almost connected groups, extending known results beyond compact Lie groups.

Findings

Decomposition applies to a wide class of groups including discrete and almost connected groups.

Provides examples demonstrating the decomposition's effectiveness and generality.

Studies equivariant K-homology using a configuration space model.

Abstract

In this paper we study a natural decomposition of $G$-equivariant $K$-theory of a proper $G$-space, when $G$ is a Lie group with a compact normal subgroup $A$ acting trivially. Our decomposition could be understood as a generalization of the theory known as Mackey machine under suitable hypotheses, since it decomposes $G$-equivariant K-theory in terms of twisted equivariant K-theory groups respect to some subgroups of $G/A$. Similar decompositions were known for the case of a compact Lie group acting on a space, but our main result applies to discrete, linear and almost connected groups. We also apply this decomposition to study equivariant $K$-theory of spaces with only one isotropy type. We provide a rich class of examples in order to expose the strength and generality of our results. We also study the decomposition for equivariant connective $K$-homology for actions of compact Lie groups using a suitable configuration space model, based on previous papers published by the third author.