What results are known about representations of reductive groups over finite rings in general? Here by finite rings I usually mean an algebra over $F_q$, I guess.

I know Lusztig has a paper generalizing Deligne-Lusztig theory to finite commutative rings of the form $F_q[x]/x^{\epsilon}$ for some integer $\epsilon > 0$ (so the quotient of that polynomial ring by that ideal), though even that theory (as best I understand) is not complete yet.

There would be no way of reducing representation theory of all commutative finite rings to the particular case of those rings (along with perhaps other "simple" rings, e.g. $\mathbb{Z} / p^{k} \mathbb{Z}$)?

Are any results known about representation theory of finite commutative rings that satisfy the Frobenius endomorphism condition, i.e. $t^q = t$ for all $ t \in R$? In that case, some aspects of Deligne-Lusztig theory can perhaps be salvaged.

What about non-commutative finite rings? In general, this seems to require something much more powerful than Deligne-Lusztig theory however, the Frobenius endomorphism breaks down.

  • $\begingroup$ For non-commutative rings (finite or otherwise), you do have one powerful simplification at your disposal: Morita equivalence. $\endgroup$ – Greg Kuperberg Nov 29 '09 at 22:48
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    $\begingroup$ Perhaps you could start by telling us the definition of a reductive group over a noncommutative ring. It's not clear to me that such a thing exists in general. $\endgroup$ – S. Carnahan Nov 29 '09 at 23:29
  • $\begingroup$ Sorry: for noncommutative rings, what I should have changed my definition. What I mean is $GL_{n}(R)$ where $R$ is the finite ring. I can't think of a way of defining $SL$ since determinant need not exist, let alone $Sp$ or $O$. How does Morita equivalence help in this case? $\endgroup$ – Vinoth Nov 30 '09 at 8:14

Lusztig's results mentioned above have been generalised to groups over arbitrary finite local rings, see Unramified representations of reductive groups over finite rings, Represent. Theory 13 (2009), 636-656.

There is a recent paper extending the above construction to certain ramified maximal tori, cf. Extended Deligne-Lusztig varieties for general and special linear groups, arXiv:0911.4593v1. It is shown here that the above "unramified" construction omits certain interesting representations. Some of these missing representations are connected with ramified maximal tori, and it is therefore desirable to have generalised Deligne-Lusztig varieties attached also to such tori.

In addition to the cohomological approaches to constructing representations, there are some partial purely algebraic constructions. There is a paper by G. Hill (Regular Elements and Regular Characters of $\text{GL}_n(\mathcal{O})$, which gives a construction of many so-called regular representations. These include most of the interesting representations, such as the strongly cuspidal ones. There is also an approach, due to U. Onn, which defines a new type of induction functor (called infinitesimal induction) which complements the classical parabolic induction. This construction considers general automorphism groups of finite modules over DVRs with finite residue field. So far, this approach has led to a classification of all the representations in the rank-2 case (which includes $\text{GL}_2(\mathcal{O}/\mathfrak{p}^r)$).

Since any finite commutative ring is the direct product of finite local rings, it is enough to study representations of groups over the latter. This is however a very hard problem, and one cannot expect an explicit list of all the representations in general. As in the representation theory of most infinite groups, the most fruitful approach seems to be to define a nice category of representations which is both possible to control, and at the same time is rich enough to include most (or all) the representations one is interested in. A candidate for such a category for $\text{GL}_n(\mathcal{O})$ is the one consisting of regular representations, mentioned above. One can ask interesting questions about the regular representations, such as which of them are given by the cohomological constructions, or which of them are types for supercuspidal representations of $\text{GL}_n(F)$, where $F$ is a local field with ring of integers $\mathcal{O}$.

  • $\begingroup$ Thank you very much. That was very helpful. I've seen your paper before, I'm reading it in detail now. The specific case I want to try is when $R$ is toric, and local. In small cases of $n$, I think I might want to attempt finding other representations not given by the constructions you outlined (since you mentioned those are not all of them), specific to toric rings. I don't know how feasible this will be. Do you think it will be interesting to examine finite local toric rings? $\endgroup$ – Vinoth Dec 17 '09 at 12:07
  • $\begingroup$ I think the first case to consider is that of finite local principal ideal rings (i.e., finite quotients of rings of integers in non-Archimedean local fields). If this is understood, one may move on to more general rings. The best thing, however, is to look for general structures which do not depend too much on specific rings. $\endgroup$ – A Stasinski Jan 6 '10 at 11:57

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