Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication.


What is known about group cohomology $H^\*(SL(2,{\mathbb R}),{\mathbb R}^2)$?


And about $$H^\*(\Gamma,{\mathbb R}^2)$$ for a cocompact lattice $$\Gamma\subset SL(2,{\mathbb R})?$$



Cohomology of SL(2,R) with coefficients given by linear action
It is zero. This is an application of the "centre kills" trick, which I will state in homology.


**Trick.** Let $M$ be a $G$-module for which there is an element $z$ in the centre of $G$ which acts as $-1$ on $M$. Then $2H\_\*(G;M)=0$.


In your situation the homology is a real vector space, so if multiplication by 2 kills it then it is already dead. You can then use Universal Coefficients to get the result in cohomology, or just work out the details of the trick in cohomology.


The proof is as follows. For any element $z \in G$, the map
$$m \vert g\_1 \vert g\_2 \vert \cdots \vert g\_n \longmapsto m \cdot z \vert g\_1^z \vert g\_2^z \vert \cdots \vert g\_n^z$$
on the bar complex is i) a chain map, and ii) chain homotopic to the identity, so induces the identity map on homology. But if $z$ has the properties described then this map is equal to $-1$.


