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 sequenceà 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 π 1 (X T )-equivariant Bredon cohomology of the universal cover of X T with suitable coefficients. In the case of the inertia stack ΛY 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 = { * }.

In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p‐local finite group (S,F,L) in terms of representation rings of subgroups of S. We also prove a stable elements formula for generalized cohomological invariants of p‐local finite groups, which is used to show the existence of unitary embeddings of p‐local finite groups. Finally, we show that the augmentation C∗(|L|p∧;double-struckFpfalse)→double-struckFp is Gorenstein in the sense of Dwyer–Greenlees–Iyengar and obtain some consequences about the cohomology ring of false|scriptLfalse|p∧.