# cohomology of finite groups of lie type with coefficients in the adjoint module

Let $\mathbb G$ be a connected, semisimple, split group over a finite field $\mathbb F_q$ and let $G = \mathbb G(\mathbb F_q)$. Let $\mathfrak g$ be its Lie algebra, an $\mathbb F_q$-vector space with the adjoint $G$-action.

I want to understand the group cohomology $H^*(G, \mathfrak g)$, and more generally $H^*(G, \bigwedge^n g)$.

I am aware there are some results about $H^1(G, \mathfrak g)$ (for example due to Cline, Parshall and Scott), as well as some conditions for the map $H^2(\mathbb G, \mathfrak g) \longrightarrow H^2 (G, \mathfrak g)$ to be an injection (due to the University of Georgia VIGRE algebra group - here the domain is the cohomology of the algebraic group $\mathbb G$).

Are there other known results? I am willing to assume that $p$ is as large as needed. I am also interested in special cases, such as $\mathbb G = \mathbf {SL}_2$.