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Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's Finite groups of Lie type:

Theorem (Steinberg): Let $G$ be a connected reductive group, $F$ a Frobenius endomorphism of level $q$, and $\Phi$ the root system associated to $G$. The number of $F$-stable maximal tori in $G$ is $q^{2N}$, where $N=|\Phi^+|$. This number can also be written $|G^F|_p^2$, where $|G^F|_P$ is the highest power of $p$ dividing $|G^F|$.

I'm interested in possible generalizations of this theorem. Let me restrict to the case where $G$ is simple of rank $r$, and let me observe that provided $q$ is big enough in terms of $r$, then the number of $F$-stable maximal tori in $G$ is equal to the number of maximal tori in $G^F$. So, in this restricted setting, we can think of Steinberg's theorem as counting maximal tori in $G^F$. Here's my question:

Question 1: Fix a positive integer $k<r$, and suppose that $q$ is big compared to $r$. Is there a nice formula for the number of tori in $G^F$ of dimension $k$ that are also the center of a Levi subgroup of $G^F$?

I guess the answer would be NO if I was just counting all tori of a given dimension, but if I restrict to Levi-centers, then it seems like there might be some hope that such a result exists.

Of course, you might just tell me to read Carter's proof to see if I can generalize it myself. To make sure I maximise my chances at succeeding with such a generalization, I have a second question:

Question 2 Do you know any proofs of Steinberg's theorem that are substantially different to the one given by Carter?

Edits in light of Jim's comments: Perhaps it would be more transparent to ask an alternative question to Question 1, as follows.

Question 3: Fix a positive integer $k\leq r$. Is there a nice formula for the number of $F$-stable Levi subgroups $L$ of $G$ for which $\dim(Z(L))=k$?

With this formulation, one can see that Steinberg's original result answers this question for $k=r$. Hopefully this also explains why I am asking for a formula for Levi's that have centre of a specified dimension.

I realise that I could (in theory) work through simple $G$'s on a case-by-case basis to answer this question. I'd like to know if a general result exists (a la Steinberg) that would do the lot all at once... Thanks in advance for any thoughts you might have.

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    $\begingroup$ There are different proofs; at least for GL_n, one is explained in Section 5.3 of arXiv:1309.6038 by Church, Ellenberg and Farb, and they give references to arguments by Srinivasan (using the Grothendieck-Lefschetz trace formula) and Lehrer. $\endgroup$ Aug 4, 2015 at 16:17
  • $\begingroup$ @DenisChaperondeLauzières, Thank you! I will look up the article you mention, and follow up the references there. $\endgroup$
    – Nick Gill
    Aug 6, 2015 at 13:34

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Concerning your Question 1, the formulation looks somewhat out of focus. Keep in mind that there is for each Lie type of simple algebraic group a well-organized classification of parabolic subgroups and their Levi subgroups (up to conjugacy). So the centers are in principle not impossible to find, and their dimensions in particular are limited. So I don't see what fixing $k$ in advance might do for you. Aside from that, I'm not sure what the concept of "dimension" has to do with tori in the finite group, so this language should be clarified.

Concerning Question 2, you should take a look at the much-cited 1992 paper by G.I. Lehrer here, and maybe also at the 1994 paper by J.M. Douglass here (if you have access to it). I'm not sure if Lehrer's approach will be suggestive to you, but it is substantially different from Steinberg's original method.

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  • $\begingroup$ Thanks Jim. I'll edit the original question to address some of your remarks. I was being a little sloppy with dimension, since if $q\gg r$ I believe I can transfer notions of dimension pretty straightforwardly to the finite case... But, anyway, I don't need this, so will edit accordingly. $\endgroup$
    – Nick Gill
    Aug 6, 2015 at 13:23
  • $\begingroup$ Also, ludicrously, I don't have access to MathSciNet... So, can I check that the Douglass article you mention is this - hilbert.math.unt.edu/downloads/papers/douglass/levi.pdf - and the Lehrer paper is entitled Rational Tori, semisimple orbits and the topology of hyperplane complements. $\endgroup$
    – Nick Gill
    Aug 6, 2015 at 13:25
  • $\begingroup$ @Nick: Yes, that's the paper by Matt Douglass. The Lehrer paper has open access if you follow the article link given and click on PDF at the lower right. (MathSciNet access shouldn't be needed for the two links I gave.) $\endgroup$ Aug 6, 2015 at 15:40

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