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The Raghunathan's theorem states that if $G$ is a simple Lie group non-compact and $g:G\to GL_n(\mathbb{R})$ is a linear representation without fixed point and $\Gamma$ a discrete co-compact subgroup then, $H^1(\Gamma,\mathbb{R}^n)=0$ with $\Gamma$-module structure of $\mathbb{R}^n$ induced by $g$ except in few cases (which are $G\simeq SO(k,1)$ and the highest weight of $g$ is a multiple of the the weight of the canonical representation).

Is there a generalisation for $G$ semi-simple and $\Gamma$ only discrete subgroup, not co-compact ?

théo jamin
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  • What is a fixed point of $g$? – Arctic Char Oct 20 '21 at 10:29
  • I meant a vector in $\mathbb{R}^n$ which is globally stabilized by the action of $g(G)$. – théo jamin Oct 20 '21 at 10:35
  • You have to exclude $SO(2,1)$ and for nonuniform lattices you also have to exclude $SO(3,1)$ as well. Otherwise, I think it is the same theorem. – Moishe Kohan Oct 22 '21 at 12:25
  • I'm mainly interested in discrete subgroups of $SL_2(\mathbb{C})\times SL_2(\mathbb{C})$, do you have an idea in this case or a reference ? – théo jamin Oct 22 '21 at 15:49
  • I forgot to add that you also ned your discrete subgroup to be a lattice in such vanising theorems. In the product case you are interested in, you also need an irreducible lattice. I will try to find a reference for you. – Moishe Kohan Oct 23 '21 at 08:40

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