Cohomology of SL(2,R) with coefficients given by linear action 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 $$\Gamma\subset SL(2,{\mathbb R})?$$
 A: 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 $-1$.
A: I will answer only for cohomology in degree 1, as I am not sure for higher degrees. 
On the other hand I think that what I am saying remains true when R^2 is replaced by any finite dimensional representation of SL_2(R). 
For the first cohomology of a lattice with values in a finite dimensional representation, the key word is Eichler-Shimura isomorphism. These cohomology groups can be nonzero, see 
http://www.ams.org/journals/tran/1987-300-02/S0002-9947-1987-0876476-0/S0002-9947-1987-0876476-0.pdf 
or 
http://www.math.u-psud.fr/~labourie/preprints/pdf/propsurf.pdf 
and the (original) references there. For the cohomology group H^1(SL_2(R),V) with V a finite dimensional SL_2(R)-module, I think it is always zero. I don't have a reference on the top of my head. But if you understand the Eichler-Shimura isomorphism for lattices, it should give a proof of vanishing for SL_2(R).  
