Twisted Weyl groups of Lie groups and nonabelian cohomology

Abstract

For a cyclic group $A$ and a connected Lie group $G$ with an $A$-module structure (with the additional conditions that $G$ is compact and the $A$-module structure on $G$ is 1-semisimple if $A\cong\ZZ$), we define the twisted Weyl group $W=W(G,A,T)$, which acts on $T$ and $H^1(A,T)$, where $T$ is a maximal compact torus of $G_0^A$, the identity component of the group of invariants $G^A$. We then prove that the natural map $W\backslash H^1(A,T)\to H^1(A,G)$ is a bijection, reducing the calculation of $H^1(A,G)$ to the calculation of the action of $W$ on $T$. We also prove some properties of the twisted Weyl group $W$, one of which is that $W$ is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.