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I should preface this by saying that I am not a representation theorist, so I apologize if this can easily be found in standard sources (but sadly I cannot seem to extract it from any of the books I know of).

Let $p$ be a prime number. What are all the irreducible representations (over $\mathbb{C}$) of $\text{SL}(n,\mathbb{F}_p)$ and $\text{Sp}(2n,\mathbb{F}_p)$ whose dimensions are $p^k$ for some $k$? There are the Steinberg representations, but I assume that there are also others.

I'm particularly interested in the symplectic group.

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  • $\begingroup$ Try looking at Carter's Finite groups of Lie type - it discusses the Deligne-Lusztig theory for groups of Lie type in great detail. Email me if you want an e-copy. $\endgroup$
    – Nick Gill
    Commented Oct 29, 2014 at 15:16
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    $\begingroup$ Why on earth did someone down vote this question? I've certainly looked at books on Deligne-Lusztig theory, but they do things in so much generality that I cannot extract the answer to this simple-minded question. $\endgroup$
    – Sarah
    Commented Oct 29, 2014 at 15:24
  • $\begingroup$ @Sarah: Your question is straightforward-looking, but I don't think a full answer exists (yet) without heavy reliance on the Deligne-Lusztig theory and especially Lusztig's later work. The easiest thing to compute from the theory is the degree of a virtual Deligne-Lusztig character, but getting to the irreducible characters takes a lot more work. $\endgroup$
    – Jim Humphreys
    Commented Oct 29, 2014 at 17:52

1 Answer 1

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A full classification of such representations (and much more) can be found here:

Prime power degree representations of quasi-simple groups by Malle and Zalesskii

You can read this paper here. The main theorem implies that, apart from the Steinberg representation, there are no representations of this form.

Interestingly this is not the case if you allow quasisimple covers of such groups, for instance $2\cdot {\rm Sp}_6(2)$ has an irreducible complex representation of degree $2^9$.

As I mentioned in my comment, if you want to understand the theory of these representations then I would go to Finite groups of Lie type by Carter. I should also say that it's quite possible that the fact you seek can be proved more directly than via the full classification given by Malle and Zalesskii ... but I don't know enough of D-L theory to tell you how.

P.S. if anyone knows how to put the correct accent on Zalesskii's name, then I'd like to know how!

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  • $\begingroup$ This is very helpful. Thank you very much! $\endgroup$
    – Sarah
    Commented Oct 29, 2014 at 15:23
  • $\begingroup$ And the edit made it even more helpful. Thanks again! $\endgroup$
    – Sarah
    Commented Oct 29, 2014 at 15:36
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    $\begingroup$ Well, there's also the trivial representation if you don't specify that $k\geq 1$... $\endgroup$
    – Denis Chaperon de Lauzières
    Commented Oct 29, 2014 at 16:22
  • $\begingroup$ Re the PS: I believe that the spelling of Zalesskii's name has fairly recently been changed to Zalesski due to some new convention (not sure about the accent). $\endgroup$
    – Geoff Robinson
    Commented Oct 29, 2014 at 16:31
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    $\begingroup$ Jim, Thanks for your comments. I'm slightly surprised that this answer can't be answered more directly - I thought that, in my ignorance, I was using a sledgehammer to crack a nut but perhaps not. And, Geoff, thanks for the info on Zalesski's name. $\endgroup$
    – Nick Gill
    Commented Oct 29, 2014 at 18:26

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