Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology

Abstract

Given a compact connected Lie group G with an S 1 -module structure and a maximal compact torus T of G S 1 , we define twisted Weyl group W ( G , S 1 , T ) of G associated to S 1 -module and show that two elements of T are δ -conjugate if and only if they are in one W ( G , S 1 , T ) -orbit. Based on this, we prove that the natural map W ( G , S 1 , T ) \ H 1 ( S 1 , T ) → H 1 ( S 1 , G ) is bijective, which reduces the calculation for the nonabelian cohomology H 1 ( S 1 , G ) .