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 assumptions that G is compact and the A-module structure on G is 1-semisimple if $$A\cong{\mathbb{Z}}$$ ), 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 $$W\backslash H^1(A,T)\rightarrow H^1(A,G)$$ 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.