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; but I'm not really interested in exceptional groups of Lie type (for those, this problem should be a standard mindless computation). So if you want in order to answer the following questions, let $G=GL_n(\mathbb{F}_q)$ (this may be necessary for some of the following questions, where an explicit description of all the characters is I believe not known for all $G$). Everything here is characteristic $0$.

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? (of course this is the sum of the exterior and symmetric powers, I mean something more explicit than that..)
- 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.