For which semisimple element $s$ in finite group of lie type centralizer $C_{G}(s)$ of $s$ is a Levi subgroup ?

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.

• G'=G? Why is your maximal torus indexed by W? Best, Marc Nov 23 '12 at 11:40
• Please google: centralizer semisimple element. tons of material. Nov 23 '12 at 11:42
• 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 $C_{G}(s).$ Nov 23 '12 at 12:07
• 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. Nov 23 '12 at 12:57
• 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. Nov 23 '12 at 14:27

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 $B_2$.