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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.

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    $\begingroup$ This is the underlying complex analytic space of a complex affine variety, hence it is Stein. Therefore all of the higher cohomology groups of all coherent analytic sheaves are zero. Since $\Omega$ has a canonical left-invariant isomorphism with the sheaf of holomorphic functions into the one-dimensional complex vector space $\bigwedge^3_{\mathbb{C}} \mathfrak{sl}_2(\mathbb{C})$, the global sections of $\Omega$ are just the holomorphic functions on this complex Lie group. $\endgroup$
    – Jason Starr
    Commented Jun 28, 2017 at 14:22
  • $\begingroup$ Thank you for your comment. I was guessing that the first question was particularly trivial. Do you have any idea for the second one ? $\endgroup$
    – C. Dubussy
    Commented Jun 28, 2017 at 14:34
  • $\begingroup$ Since G is stein you can find compactly supported cohomology .See Theorem 3 of Serre's paper on Serre duality. $\endgroup$
    – Mohan Ramachandran
    Commented Jun 28, 2017 at 19:08
  • $\begingroup$ @Mohan Ramachandran : Could you tell me the exact title of the paper ? Also note that I'm not interested in $H^*_c$ but $H^*_F$. $\endgroup$
    – C. Dubussy
    Commented Jun 29, 2017 at 2:44
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    $\begingroup$ @C.Dubussy . Serre's paper is titled Un Theoreme de Dualite Commentarii Mathematici Helvetici vol 29 (1955) 9-26 . $\endgroup$
    – Mohan Ramachandran
    Commented Jun 29, 2017 at 20:26

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