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

TL;DR

This paper computes twisted equivariant K-theory for compact Lie group actions with maximal rank isotropy, revealing a simplified spectral sequence and applications to the rational Verlinde algebra.

Contribution

It provides a new explicit description of the rational spectral sequence in twisted equivariant K-theory for specific group actions, connecting it to invariants under the Weyl group.

Findings

Spectral sequence simplifies to invariants under Weyl group

Explicit calculations for inertia stacks and cohomology

Recovery of the rational Verlinde algebra for a point

Abstract

We consider twisted equivariant K--theory for actions of a compact Lie group $G$ on a space $X$ where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational spectral sequence \`a la Segal has a simple $E_2$--term expressible as invariants under the Weyl group of $G$. Namely, if $T$ is a maximal torus of $G$, they are invariants of the $\pi_1(X^T)$-equivariant Bredon cohomology of the universal cover of $X^T$ with suitable coefficients. In the case of the inertia stack $\Lambda Y$ this term can be expressed using the cohomology of $Y^T$ and algebraic invariants associated to the Lie group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when $Y=\{*\}$.