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If $G$ is a reductive algebraic group over an algebraically closed field of positive characteristic $p$, and $G$ is defined over the prime field, we have the Frobenius morphism $F: G\to G$, which for each positive number $r$ leads to two subgroups of $G$, namely $G_r$ which is the kernel of the $r$'th iteration of $F$, and $G^{F^r}$ which is the fixed points of the same.

Now, $G^{F^r}$ normalizes $G_r$ (since $G_r$ is normal in $G$), and they clearly intersect trivially. So in $G$ the subgroup $G^{F^r}G_r$ is a semidirect product.

My question is whether this subgroup and its representations has been studied. We certainly know some of its irreducible representations, as any irreducible representation of $G_r$ or $G^{F^r}$ extend to $G$ (and they even have the same irreducibles when these are seen as $G$-modules). So one of the main questions would be whether there are any other irreducible representations of this group.

Of course, one could also consider $G^{F^r}G_{r'}$ with $r\neq r'$ and ask the same questions.

The reason I am interested in this group is that it seems like it might provide a more direct link between the representation theories of $G_r$ and $G^{F^r}$, which are certainly very similar.

It might also provide an additional stepping stone for comparing the representation theories of $G_r$ or $G^{F^r}$ with that of $G$, since one of the ways one often does such comparison is via the induction from either $G_r$ or $G^{F^r}$ to $G$, and both of these factor through this subgroup.

(I asked this question in the representation theory chat room, trying to get the discussion going, but I realized it was focused enough that it might as well be an actual question here).

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At first sight your formulation looks rather confusing, since you attempt to mix finite groups with finite group schemes. Also, it is not so much irreducible modules which raise questions in these two settings, but rather projective modules and cohomol/ogy. Have you consulted local experts? –  Jim Humphreys Jul 11 '13 at 10:52
@JimHumphreys By subgroup I mean subgroup scheme, and I suppose you are right that the question of projectives might well be a lot more interesting. I asked my advisor (Henning Haahr Andersen), and he had not heard of any study of this subgroup. The question may well be too naive to lead anywhere. –  Tobias Kildetoft Jul 11 '13 at 10:59
Maybe what you mean by Frobenius is not the standard thing, as the standard Frobenius only maps $G$ to itself if $G$ is defined over the prime field. –  Felipe Voloch Jul 11 '13 at 15:11
@FelipeVoloch Sorry, I forgot to add the assumption that the group is defined over the prime field. –  Tobias Kildetoft Jul 12 '13 at 6:22
I definitely talked about this with Brian Parshall and Len Scott at one point. I think we even found a minor use for it. (Too minor to be worth following up.) They had someone in mind who'd worked with such things. I'll email them. As you say, the simples must be easy to work out. –  David Stewart Jul 12 '13 at 8:48
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