Let \(G\) be a compact connected Lie group, and let \(\sigma\) be an automorphism of \(G\). The twisted loop group \(L_\sigma G\) is the topological group of continuous paths \(\gamma: I\to G\) satisfying \(\gamma(1)=\sigma(\gamma(0))\), with pointwise multiplication and compact-open topology. The isomorphism type of \(L_\sigma G\) only depends on the outer automorphism \([\sigma]\) represented by \(\sigma\). If \(\sigma\) is the identity automorphism, then \(L_\sigma G\) equals the continuous loop group \(LG\).
The author studies the classifying space \(BL_\sigma G\). The main result is a formula for the cohomology ring \(H^\ast(BL_\sigma G)\): Assume \(G\) is semisimple. Let \(G^\sigma\) be the subgroup of elements of \(G\) fixed by \(\sigma\), and let \(G^\sigma_0\) be the identity component of \(G^\sigma\). Let \(T\) and \(T^\sigma\) denote the maximal tori of \(G\) and \(G^\sigma_0\), respectively. Let \(N_G(T^\sigma)\) denote the normalizer of \(T^\sigma\). Then \[ H^\ast(BL_\sigma G; F)\cong H^\ast(BLG^\sigma_0; F)^{W_\sigma}, \] where \(W_\sigma= N_G(T^\sigma)/T\), and \(F\) is a coefficient field of characteristic coprime to the order of the Weyl group of \(G\), to the number of path components of \(G^\sigma\), and to the order of \([\sigma]\).
The author explicitly carries out the calculation of \(H^\ast(BL_\sigma G;F)\) for all automorphisms of compact connected simple Lie groups. Moreover, he derives a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.