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

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 $\mathrm{\Lambda }Y$ 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 $Y=\{\ast \}$.