Let $G$ be a reductive group over a finite field (i.e. finite groups over lie type). The case I am most interested in is $G=GL\_{n}(\mathbb{F}\_{q})$; other classical groups are also interesting I think. 


Deligne-Lusztig theory has a lot to say about the irreducible representations and characters of these groups. For $G=GL\_n(\mathbb{F}\_q)$, Green's paper from the 1940's gives the characters explicitly also. The following question I guess, is in part a reference request, since the question has probably been examined in the literature somewhere, but I am unable to find a reference. 


**Question:** Let $V$ and $W$ be two irreducible representations of $G$. What can be said about the decomposition of $V \otimes W$ into irreducibles? Specifically:


* Are there any special cases of $V$ for which the decomposition of $V \otimes W$ into irreducibles can always be explicitly determined? (for instance, with the symmetric group $S\_n$, there is some theory which does this for the regular representation of dimension $n-1$, and also I believe work which does this for representations corresponding to two-row partitions).
* Is there anything that can be said for the decomposition of $V \otimes V$ in general?
* What about, if $V$ and $W$ are not actually irreducibles, but instead representations obtained from $l$-adic cohomology; for instance the virtual representation $R\_{T, \theta}$ is defined as alternating sums of various cohomological representations. As an example, consider the representation of $G$ acting on the $i$-th cohomology of the Deligne-Lusztig variety $X\_{T}$ corresponding to a fixed torus $T$; if we tensor together two different cohomological representations corresponding to different tori, and cohomology for different values of $i$, what can we say? Since the $R\_{T, \theta}$ are defined as alternating sums of these, perhaps this question will help with our original problem.



Decomposing tensor products of irreducible representations of reductive groups over a finite field
Theorem 1.4.1 in [arxiv:0810.2076](http://arxiv.org/abs/0810.2076) answers some of your questions for generic semisimple irreducible representations. Emmanuel Letellier has hitherto unpublished results where he does answer your question for all generic irreducible representations in terms of intersection cohomology of certain quiver varieties. We did not know about other results on the representation ring of $GL\_n({\mathbb F}\_q)$. EDIT: but see Victor's answer for related results of Lusztig. EDIT 2 (added 16/03/11) Letellier's paper is now available at: <http://arxiv.org/abs/1103.2759>


