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$GL_n(\mathbb F_q)$ 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?

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    $\begingroup$ 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. $\endgroup$
    – Derek Holt
    Commented Jan 6, 2020 at 22:44
  • $\begingroup$ @DerekHolt Yes, this is exactly the trick in the link. How about the case $p=2$? $\endgroup$
    – Zhiyu
    Commented Jan 11, 2020 at 3:30
  • $\begingroup$ 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). $\endgroup$
    – Derek Holt
    Commented Jan 11, 2020 at 9:28
  • $\begingroup$ @DerekHolt Thanks, can you give a reference for $i=1,2$? $\endgroup$
    – Zhiyu
    Commented Jan 11, 2020 at 19:23
  • $\begingroup$ 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. $\endgroup$
    – Derek Holt
    Commented Jan 11, 2020 at 21:32

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As Derek Holt comments, cohomology has complications even for fimite general linear groups. Probably you are using the term "reductive" too casually and should replace it by "simple" or perhaps "semisimple" to get a finite group of Lie type: for example, an algebraic torus (direct product of copies of the multiplicative group of the field involved) is reductive but does not lead to a group of Lie type. Here "simple" refers to a connected algebraic group with no proper normal connected subgroups except the trivial group.

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 .

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  • $\begingroup$ Thank you for references, now I see this is a difficult problem in general. $\endgroup$
    – Zhiyu
    Commented Jan 7, 2020 at 21:58

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