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 2Do 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.