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 $A=\text{ZZ}$ and $G$ is compact, we show that for every maximal torus $T$ of the identity component $G_0^{\text{ZZ}}$ of the group of invariants $G^{\text{ZZ}}$, $H^1(\ZZ,T)\to H^1(\ZZ,G)$issurjectiveifandonlyifthe$\ZZ$-actionon$G$ is 1-semisimple, which is also equivalent to that all fibers of $H^1(\ZZ,T)\to H^1(\ZZ,G)$arefinite\mathrm{.}When$A=\Zn$,weshowthat$H^1(\Zn,T)\to H^1(\Zn,G)$isalwayssurjective,where$T$ is a maximal compact torus of the identity component $G_0^{\text{Zn}}$ of $G^{\text{Zn}}$. When $A$ is cyclic, we also interpret some properties of $H^1(A,G)$ in terms of twisted conjugate actions of $G$.