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To fix notation, by $SU(n,p)$, I mean the subgroup of $SL_n(\mathbb F_{p^2})$ consisting of matrices $A$ which satisfy $\overline A^t A = 1$, where $\overline A$ is the matrix given by raising all the entries of $A$ to the power of $p$.

So my question is what are the irreducible algebraic representations of this group defined over $\overline {\mathbb{F_p}}$? In particular, are they the same as the representations of $SL(n,p)$? Mucking around with the MeatAxe in GAP, I've checked this is true for a few small primes and $n\leqslant 3$.

Is it true in general?

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up vote 7 down vote accepted

Yes, it's a theorem of Steinberg from his fundamental 1963 Nagoya Math. J. paper, in the pre-Meataxe era. This is treated in Chapter 2 (especially 2.11) of my LMS Lecture Note Series No. 326 Modular Representations of Finite Groups of Lie Type along with full references and discussion. The main point here is that the irreducible modular representations (in the defining characteristic) come from the ambient algebraic group (special linear group) by restriction to either of the two finite groups. Of course, Steinberg's theorem is not obvious.

By the way, the ordinary characters of the two groups are related in an interesting way as well, though of course the degrees can't match precisely because the group orders are somewhat different.

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many thanks for the reference! –  Max Flander Nov 8 '11 at 4:39
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