The defining characteristic representations of Lie type groups

Question

Let $\mathbf{G}$ be a connected reductive group over $\mathbb{F}_q$, let $G=\mathbf{G}(\mathbb{F}_q)$, and let $k$ be a large enough finite extension of $\mathbb{F}_q$.

If $\mathbf{G}=\mathrm{SL}_2$, then in page-109 of Bonnafe's book it is said that "It turns out that the simple $kG$-modules are the restrictions of simple "rational representations" of the algebraic group $\mathbf{G}_{\overline{k}}$."

Question 1: Where can I find a proof of this statement?

Question 2: Is this true for a general $\mathbf{G}$?

Any help would be appreciated.

Answer 1

This result is false for a general connected reductive algebraic group $\mathbf{G}$ but is true if $\mathbf{G}$ is simple and simply connected. This was proved by Steinberg in Theorem 1.3 of the following paper

• R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33--56.

It's also summarised in Theorem 13.1 of Steinberg's monograph "Endomorphisms of linear algebraic groups".

A parameterisation of the simple modules for general $\mathbf{G}$ was recently obtained by Brunat and Lübeck:

• O. Brunat and F. Lübeck, On defining characteristic representations of finite reductive groups, J. Algebra 395 (2013), 121--141.

A preprint version of this article is also available here.

Answer 2

Here is an extended comment on Jay Taylor's answer to both questions (in community-wiki style), with some other references added. Note however that the results for simple (or semisimple) algebraic groups and their relevant finite subgroups are well-known by now. Also, the results for the rank 1 groups treated by Bonnafe go back further to Steinberg's adviser Richard Brauer (in Toronto), so the history here is rather long.

1) In Steinberg's seminal 1963 paper linked by Jay (in a journal issue dedicated to Brauer), the approach relies on "lifting" irreducible representations of a (simple) Chevalley group to projective representations of the ambient algebraic group. This is then modified suitably for some twisted groups. At that time Chevalley's construction focused on adjoint algebraic groups, so there also needed to be some study of central extensions and covering groups (which Steinberg did basic work on).

Later on Steinberg, in his 1967-68 Yale lectures on "Chevalley groups" and the contemporary AMS Memoir cited by Jay, provided a more unified formulation of his earlier results. All of this depends on knowing how to characterize the irreducible representations of the simple (or semisimple) algebraic groups in the "highest weight" formulation of the 1956-58 Chevalley seminar, as well as Steinberg's generalization of the twisted tensor product theorem (worked out by Brauer-Nesbitt in rank 1) to arbitrary ranks. The main point is to reduce the study of irreducibles for finite groups of Lie type in the defining characteristic to the study of "restricted" highest weight representations for the algebraic groups. This is far from complete, however, in spite of many results involving Lusztig's ideas based on a characteristic $p$ analogue of Kazhdan-Lusztig theory.

Meanwhile, Jantzen's 1987 approach based partly on Kempf's ideas here were later exposed in my 2006 book here: see Chapter 2 along with the posted revisions. Jantzen's own 2003 AMS expanded edition of his 1987 text Representations of Algebraic Groups remains the best source for a thorough treatment of the overall theory for semisimple algebraic groups. But recent work by Achar, Riche, Williamson. and others shows the subtleties of the unsolved problems about weight space dimensions for all $p$. This is all relatively easy in rank 1, but not in general.

2) As Jay indicates, the other question here about reductive groups is addressed by the more recent work of Brunat-Lubeck. Recall that Borel and Tits explored their notion of "reductive" in arbitrary characteristic (in 1965) and showed that a connected reductive algebraic group is just the almost-direct product of a semisimple group and a torus. While the representation theory of semisimple groups reduces soon to the study of simple algebraic groups, the (algebraic) representations of a torus all come from dimension 1. So the main result of Brunat-Lubeck mainly involves assembling the pieces carefully, with attention to the fine points of the group structure.