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 $H^1(A,K)to H^1(A,G)$ is bijective. In this case we also show that $H^1(A,G)$ is always finite. When $A=ZZ$ and $G$ is compact, we show that for every maximal torus $T$ of the identity component $G_0^ZZ$ of the group of invariants $G^ZZ$, $H^1(ZZ,T)to H^1(ZZ,G)$ is surjective if and only if the $ZZ$-action on $G$ is 1-semisimple, which is also equivalent to that all fibers of $H^1(ZZ,T)to H^1(ZZ,G)$ are finite. When $A=Zn$, we show that $H^1(Zn,T)to H^1(Zn,G)$ is always surjective, where $T$ is a maximal compact torus of the identity component $G_0^{Zn}$ of $G^{Zn}$. When $A$ is cyclic, we also interpret some properties of $H^1(A,G)$ in terms of twisted conjugate actions of $G$.