Let $G$ be a finite simple group of lie type. Let $s$ be a semisimple element lying in maximal torus $T_{w}$ for $w\in W$ where $W$ is the Weyl group of $G$. Can we say that $C_{G}(s)$ is Levi subgroup of a Parobolic containing $s$ by looking just conjugacy class of $w$?

I assume that $G'$ is simply connected.

  • $\begingroup$ G'=G? Why is your maximal torus indexed by W? Best, Marc $\endgroup$
    – Marc Palm
    Nov 23 '12 at 11:40
  • $\begingroup$ Please google: centralizer semisimple element. tons of material. $\endgroup$
    – Marc Palm
    Nov 23 '12 at 11:42
  • 1
    $\begingroup$ ıf $G=G_{0}^{F}$ be fixed points of a Frobenius map $F$ on $G_{0}$ which is an algebraic group over an algebraicly closed field with characteristic p> 0 Then it is known that the conjugacy class of maximal tori are corresponding to $F$ conjugacy class of Weyl group of $G$. G' denote the commutator subgroup of G and G' simply connected guarenties the connectedness of $C_{G}(s).$ $\endgroup$ Nov 23 '12 at 12:07
  • $\begingroup$ Yes and no, your question is up to conjugacy. Btw, you must fix a maximal tori apriori, and then you can index all others by W. Nevertheless, I think this question is better suited on another side like math exchange. I suggest you solve you question yourself. Start by defining $W_s$ of the subgroup of $W$ fixing $s$. Show that $C_G(s)= W_s T_w$ and then use something like the Theorem on pg. 184 in Humpreys' linear algebraic groups. $\endgroup$
    – Marc Palm
    Nov 23 '12 at 12:57
  • $\begingroup$ Carter's "Finite groups of Lie type" certainly has this in it. I don't have a copy with me to give a specific reference though. $\endgroup$
    – Nick Gill
    Nov 23 '12 at 14:27

In its present form the question is not clearly enough formulated to have a definite answer (as Marc Palm points out in his comments), so it's difficult to upvote. What I can do is point to some of the relatively old material which in principle should provide answers, though most of it is not readily available online.

1) It's natural to begin with centralizers of semisimple elements in a connected semisimple algebraic group $G$, then adapt to finite groups of Lie type. Here you can start with a connected reductive group, with no loss of generality, but in fact the centralizers depend just on the study of a semisimple group following Borel-Chevalley, Borel-Tits, Steinberg, Carter. Here the "simple" groups (in the sense of algebraic groups) are most crucial, and at some point you get involved with case-by-case study. For the finite groups, it's most efficient to work over an algebraic closure of a finite prime field, then study split and quasi-split groups along with groups of Suzuki and Ree.

2) Semisimple elements $s \in G$ are the best behaved ones, having reductive centralizers. But the centralizer need not be connected, unless $G$ is assumed to be simply connected (a difficult theorem proved by Steinberg with help from Springer and still awaiting a more transparent proof). Otherwise life gets more complicated, as seen in the format of character tables for the finite groups in the Atlas. (The Weyl group may contribute a little extra to the centralizer.)

3) Now the structure of $C_G(s)$ is transparent: it is connected and reductive, generated by a maximal torus containing $s$ along with root subgroups relative to this torus which belong to roots lying in a subsystem generated by roots corresponding to a proper subset of vertices of the extended Dynkin diagram (Borel de-Siebenthal theory, applied by Carter and his student Deriziotis). In particular, $C_G(s)$ may or may not be a Levi subgroup of some parabolic subgroup of $G$, though it is always a pseudo-Levi subgroup. For instance, in type $G_2$ you can have such a subgroup with derived group of Lie type $B_2$.

4) By the time all of this gets adapted carefully to rational points of groups which are split or quasi-split over a finite field (or groups of Suzuki, Ree types), the details multiply. Carter and Deriziotis organized much of this material in their papers and lectures, referenced in my 1995 AMS book Conjugacy Classes in Semisimple Algebraic Groups (with corrections posted on my homepage): see especially Chapters 2 and 8.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.