If $G$ is an algebraic group over a field of positive characteristic $p$, we have in $G$ the two subgroups $G_r$ and $G(p^r)$ (the $r$'th Frobenius kernel and the finite group of Lie type given as the fixed points of the $r$'th Frobenius morphism). I wonder if the product $G_rG(p^r)$ (which is semidirect inside $G$) and its representations has ever been studied.

Or for that matter if products of the form $G_{r'}G(p^r)$ with $r$ not necessarily equal to $r'$.

We certainly know a bunch of irreducible representations of the group, as the two subgroups have the same irreducibles (these coming from representations of $G$).

Another topic, I would love some discussion on: Now that the Lusztig conjecture turns out to be false, what sort of implications does that have? What new interesting questions does it raise? What things were previously studied in specific ways because one expected this to give useful results once the conjecture was proven? How much of such things can we still hope to salvage (given that we know the conjecture is after all true for large enough primes)?

Good to see other in here. And of course, feel free to propose any other topics of discussion you feel like.

I know almost nothing about representation theory. Given the assortment of things out there that are called products which operate on groups, I wonder how to characterize those that can be decomposed easily, or satisfy some "nice" (interesting and not too hard to analyze) property.

Awesome!

I don't know anything about representation theory. But It's really really cool that this room is happening.