Nonabelian cohomology with coefficients in lie groups

In this paper we prove some properties of the nonabelian cohomology H-1(A, G) of a group A with coefficients in a connected Lie group G. When A is finite, we show that for every A-submodule K of G which is a maximal compact subgroup of G, the canonical map H-1(A, K) -> H-1(A, G) is bijective. In this case we also show that H-1(A, G) is always finite. When A = Z and G is compact, we show that for every maximal torus T of the identity component G(0)(Z) of the group of invariants G(Z), H-1(Z, T) -> H-1(Z, G) is surjective if and only if the Z-action on G is 1-semisimple, which is also equivalent to the fact that all fibers of H-1(Z, T) -> H-1(Z, G) are finite. When A = Z/nZ, we show that H-1(Z/nZ, T) -> H-1(Z/nZ, G) is always surjective, where T is a maximal compact torus of the identity component G(0)(Z/nZ) of G(0)(Z/nZ). When A is cyclic, we also interpret some properties of H-1(A, G) in terms of twisted conjugate actions of G.