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Is there anything done on the (irreducible) $\mathbb{C}-$representations of (Zariski) connected, solvable groups over a finite field?

YCor
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João Dias
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    "Borel subgroup" doesn't mean much in isolation, so I guess you mean "soluble subgroup". A representation of a (Zariski) connected, soluble group is a representation of its unipotent radical, decorated with some additional data (the weights of a maximal torus), so it's probably no easier to understand them than to understand representations of unipotent groups—which is very hard: http://mathoverflow.net/questions/68207/irreducible-representations-of-the-unitriangular-group . – LSpice Nov 29 '16 at 20:00
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    The question is definitely incomplete and needs more context. As L Spice indicates, "borel subgroup" is unclear in isolation. Please clarify! – Jim Humphreys Nov 30 '16 at 15:03
  • Yes, sorry for the unclearness. I meant to say "connected, solvable groups over a finite field". I'm thinking in particular on the case of the (upper) triangular group. What's a good reference for representation of that groups (involving the weights of a maximal torus). And can you study the representations without the theory of weights. (What i'm concern is the relation of representations of the group and the unipotent radical). – João Dias Dec 02 '16 at 14:25
  • Even the group of rational points of the most standard unipotent group (upper triangular unipotent matrices) over a finite field is quite tricky to study. There has been a lot of work done by Drinfeld and others aimed at the representation theory of such groups, but I don't know the exact status at present. But weights of a torus action are not directly involved in ordinary representation theory of this finite group over the complex field.. – Jim Humphreys Dec 02 '16 at 17:05

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