TITLE: Group cohomology of modular representations for finite groups of Lie type
QUESTION [9 upvotes]: $GL_n(\mathbb F_q)$ 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?

REPLY [3 votes]: 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 .