Classifying spaces of twisted loop groups

Abstract

We study the classifying space of a twisted loop group $L_{\sigma }G$, where $G$ is a compact Lie group and $\sigma$ is an automorphism of $G$ of finite order modulo inner automorphisms. Equivalently, we study the $\sigma$–twisted adjoint action of $G$ on itself. We derive a formula for the cohomology ring $H^{\ast }\left(B{L}_{\sigma }G\right)$ and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.