# Group cohomology of modular representations for finite groups of Lie type

$$GL_n(\mathbb F_q)$$ naturally acts on the vector space $$V=\mathbb F_q^n$$. As $$GL_n(\mathbb F_q)$$ is a finite group, the cohomology group $$H^i(GL_n(\mathbb F_q),V)$$ are all finite abelian groups. Can we compute those cohomology groups explicitly?

This is a baby example, and for odd $$p$$ one can use the trick in Cohomology of SL(2,R) with coefficients given by linear action. In general, Let $$G=\mathbb G(\mathbb F_q)$$ where $$\mathbb G$$ is a connected reductive group over $$\mathbb F_q$$ (or more generally a finite group of Lie type), $$V$$ be an irreducible algebraic representation of $$\mathbb G$$ defined over $$\mathbb F_{q^n}$$ (or more generally any irreducible equal characteristic modular representation), can we compute $$H^i(G,V)$$ explicitly or at least give some bounds?

• In the baby example you ask about in the first paragraph, the cohomology groups are zero whenever $q>2$, because there are nontrivial central elements of the group acting fixed-point-freely on the module. – Derek Holt Jan 6 at 22:44
• @DerekHolt Yes, this is exactly the trick in the link. How about the case $p=2$? – sawdada Jan 11 at 3:30
• You mean how about $q=2$. They are known for $i=1$ and $2$, but I am afraid that I don't know of any results for higher $i$ (although some smaller examples could be computed). – Derek Holt Jan 11 at 9:28
• @DerekHolt Thanks, can you give a reference for $i=1,2$? – sawdada Jan 11 at 19:23
• They are done in papers by G.W. Bell, On the cohomology of the special lienar groups I and II in Journal of Algebra, Volume 54, Issue 1, September 1978. That might not be the earliest reference for these specific results. – Derek Holt Jan 11 at 21:32

For example, SL$$_2$$ is simple in this sense and the associated finite group of Lie type has tricky cohomology to compute: see the old paper by Jon Carlson, which has not been improved on much, here . See also the many papers of Cline-Parshall-Scott, which lead to some bounds on dimension of cohomology and some other approaches to computation. A summary and references are given in my LMS Lecture Notes 326 (2006) here .