# The defining characteristic representations of Lie type groups

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.

• This is false for a general connected reductive algebraic group but is true if the group is simple and simply connected. This is discussed in Steinberg's "Endomorphisms of linear algebraic groups". Parametrising the simple modules in the general case is disussed by Brunat and Lübeck in the following paper arxiv.org/pdf/1211.3692.pdf. – Jay Taylor Dec 7 '17 at 6:39
• @JayTaylor Thank you! This is very useful. Could you put this into an answer? – user148212 Dec 7 '17 at 8:54
• Note that your Question 1 has a familiar answer, though Question 2 was dealt with much more recently in a precise way. (In general, it's best here to formulate a single question.) – Jim Humphreys Dec 8 '17 at 0:03

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

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:

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.