# Maximal torus and parabolic subgroups in reductive groups over finite fields

Let $G$ be a reductive group, $B_0$ a $F$-stable Borel subgroup and $T_0$ a fixed $F$-stable maximal torus contained in $B_0$. ($F$ = Frobenius morphism). Let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$.

For $w$ in the Weyl group associated with $T_0$, let $x \in G$ such that $x^{-1} F(x) = w$ and define $T_w = x T_0 x^{-1}$.

It is then stated in the article I'm reading that :

If the maximal torus $T_w$ is not contained in any proper $F$-stable parabolic subgroup of $G$, it is well known that this implies $l(w) = r \pmod2$.

Does anyone have a reference for this?

Edit (following Jim's advices) : The article I'm currently working on is "Representations of finite Chevalley groups" (Google Books; MSN; errata) from Lusztig (1977) (CBMS Regional Conf. Series in Math.n°39). Even more specifically, I'm trying to understand many examples (3.10) that are not fully developed about unipotent representations. The previous quote is from the proof of the following claim:

Assume $q$ is greater than the Coxeter number of $G$. Let $\rho$ be an irreducible cuspidal $G^F$-submodule of $H^{i}_{c}(X_w)_{\mu}$ and let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$. Then all complex conjugates of $\mu$ have absolute value of the form $q^{\delta m /2}$ where $m$ is an integer congruent to $r$ modulo $2$.

• It would help to specify the specific source you are reading, which is presumably close to the work on the Deligne-Lusztig virtual characters of finite groups of Lie type (Annals of Mathematics, 1976) as developed in many later papers and presented in the books by Carter and Digne-Michel. Earlier foundations for the maximal tori relative to Frobenius action were laid by Steinberg. In your question, the fixed torus is probably meant to be of maximal split rank over the finite field, whereas the twisted torus is meant to be minisotropic''? – Jim Humphreys Jun 5 '11 at 22:29
• I edited my post, but concerning the fact that the fixed torus is probably of maximal split rank, I can't say it is in the setting of proof... but maybe I'm wrong. – th.ng Jun 6 '11 at 6:43

1) At the beginning it's important to specify that the Borel subgroup $B_0$ is $F$-stable. Lusztig (and others) try to deal simultaneously with split and quasisplit finite groups of Lie type such as $SL_n(\mathbb{F}_q)$ and $SU_n(\mathbb{F}_q)$ in the wider context of reductive groups such as $GL_n(\mathbb{F}_q)$ (and sometimes the groups of Suzuki and Ree are also included). In the split case, one is fixing a nice Borel subgroup such as the upper triangular matrices, along with a nice maximal torus such as the diagonal group, whereas other maximal tori defined over $\mathbb{F}_q$ need not be split and are determined by $F$-conjugacy classes in the Weyl group. In general one also has to distinguish the semisimple $\mathbb{F}_q$-rank, etc.

2) The moral is that it's helpful to work through the details first in a familiar split group as a guide to what Lusztig is doing in his increasingly sophisticated series of papers. The substantial but readable book by Carter and the smaller book by Digne-Michel (much less complete while in some ways more "modern") provide standard introductions to the Deligne-Lusztig theory. Lusztig's own papers are rewarding but not at all easy to read in isolation, especially the concise 1978 paper he wrote following his intense week of lectures at Madison in the summer of 1977 focused on the determination of all irreducible unipotent characters of these finite groups. To decipher this paper you already need substantial background from the 1976 Deligne-Lusztig paper (and more). It may be helpful to look at some of Lusztig's own comments on his papers here.

3) Closer to the question you raise, it's useful to compare Lusztig's 1976 paper on Coxeter tori and related characters. Twisting the given maximal torus by a Coxeter element provides an essential example (though not the only one in general) for the study of discrete series (= cuspidal) characters. These come from the twisted tori $T_w$ not contained in any proper $F$-stable parabolic. Note for instance that the length of a Coxeter element in the Weyl group relative to the fixed simple system (for a Borel subgroup) has the same parity as the semisimple rank of the group. Much of the motivation for the way characters of finite groups of Lie type are studied here goes back to this and other features of finite general linear groups, studied combinatorially by J.A. Green in his influential 1955 paper. But that case is only the tip of the iceberg in terms of theoretical complexity.

ADDED: To address your original question more directly, the expression "well known" basically means here that the fact quoted is lurking in the 1976 Deligne-Lusztig paper, especially Sections 7-8. It's easier to look at the 1985 text by R.W. Carter Finite Groups of Lie Type, which separates the algebraic group treatment somewhat from the etale cohomology framework in DL. Here you should study Chapter 7, especially 7.5. Carter writes $R_{T,\theta}$ for the virtual character DL attach to an $F$-stable maximal torus $T=T_w$ and a complex character $\theta$ of the finite group $T^F$.

The main notational complication throughout is that you deal with connected reductive groups $H$ such as $G$ and $T$, writing $\varepsilon_H = (-1)^r$ with $r$ the "relative rank" (= $\mathbb{F}_q$-rank). Now Carter's 7.3.5 and 7.5.1 develop a basic DL result: for $\theta$ in "general position", $\varepsilon_G \varepsilon_T R_{T,\theta}$ is an irreducible character of $G^F$. The proof of 7.5.1 separately treats the case when $T=T_w$ fails to lie in any proper $F$-stable parabolic, while 7.5.2 shows in general that $(-1)^{\ell(w)} = \det w = \varepsilon_G \varepsilon_T$. By unpacking the notation you get the asserted parity, as in the special case of a Coxeter element mentioned above. If $G$ is actually semisimple and split, the tori in question are anisotropic (so $\varepsilon_T =1$) while $\det w = \varepsilon_G$.

Thank you for adding more to the general context, pointing out what is important. As a matter of fact, you are completely right, the Borel $B_0$ is chosen $F$-stable. (I edited the post).

Closer to the question, since I met this morning my advisor (for my minor thesis) he sketched me a proof for what, however, did not seem to him as "well known". Just a warning though, the proof is not clear at the moment for me, but I'll try to state it anyway (for those interested by the answer and able to decipher what I'll be writing).

The case where $T_0$ is split is easier. One has to start with the claim that the condition for $T_w$ not to be contained in any $F$-stable parabolic subgroups implies that $1$ is not an eigenvalue for $w$ acting on $X(T)$. Then, as the eigenvalues for $w$ (in general) are $1$, $-1$ or pairs of conjugate complex numbers. So, by looking at the determinant of $w$, one should be able to conclude.