Equivariant K-theory of compact Lie group actions with maximal rank isotropy

TL;DR

This paper proves that under certain conditions, the rationalized equivariant K-theory of a compact space with a Lie group action is a free module over the representation ring, enabling explicit computations.

Contribution

It establishes conditions under which equivariant K-theory is free over the representation ring for spaces with maximal rank isotropy subgroups.

Findings

Rationalized equivariant K-theory is free over R(G) under specified conditions.

Provides explicit computations for examples like commuting n-tuples in G.

Shows the role of fixed-point set homotopy type in K-theory freeness.

Abstract

Let G denote a compact connected Lie group with torsion-free fundamental group acting on a compact space X such that all the isotropy subgroups are connected subgroups of maximal rank. Let $T\subset G$ be a maximal torus with Weyl group W. If the fixed-point set $X^T$ has the homotopy type of a finite W-CW complex, we prove that the rationalized complex equivariant K-theory of X is a free module over the representation ring of G. Given additional conditions on the W-action on the fixed-point set $X^T$ we show that the equivariant K-theory of X is free over R(G). We use this to provide computations for a number of examples, including the ordered n-tuples of commuting elements in G with the conjugation action.