Consider the complex Lie group $G=\text{sl}(2,\mathbb{C})$ and let us denote $\Omega$ the sheaf of top holomorphic forms of this group. Are the cohomology spaces $H^{*}(G,\Omega)$ known ? I am particularly interested in the third one.

Moreover, if $F$ is a closed subset, is there a strategy to compute $H^{*}_F(G,\Omega)$ ? (Cohomology supported by $F$) If there is, could it be coming from a general strategy for complex Lie groups ?

I am sorry if these questions are too elementary for the forum, I am beginner with complex Lie groups. Any help will be much appreciated.

Consider the complex Lie group $G=\text{SL}(2,\mathbb{C})$ and let us denote $\Omega$ the sheaf of top holomorphic forms of this group. Are the cohomology spaces $H^{*}(G,\Omega)$ known ? I am particularly interested in the third one.

Moreover, if $F$ is a closed subset, is there a strategy to compute $H^{*}_F(G,\Omega)$ ? (Cohomology supported by $F$) If there is, could it be coming from a general strategy for complex Lie groups ?

I am sorry if these questions are too elementary for the forum, I am beginner with complex Lie groups. Any help will be much appreciated.