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

Question

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 $\mathbb G = \mathbf {SL}_2$.

Answer 1

This area of the subject is somewhat frustrating, since there is a lot of literature but not many satisfactory results involving the entire cohomology ring. For what it's worth, I'll point you to a survey I wrote around 2005 (not up to date at present) here. See especially the short Chapter 14 and its many references. For groups of rank 1, see for example 14.7. [This survey was meant to be a supplement to Jantzen's large 1987 book Representations of Algebraic Groups, 2nd ed., AMS, 2003, but it got delayed many years.]

One person who can give a clearer view of the entire subject of cohomology for finite groups of Lie type is Dan Nakano at U. Georgia. But quite a few people have made contributions.