Questions a bit similar to this one have already appeared I think on the forum but I couldn't find the answer to my question using those answers. I must say from the beginning that my knowledge of group theory is very very basic so please adapt your answers.


Let $G$ be a reductive group over an algebraically closed field (say of reasonable characteristic). Fix a torus $T$ and let $w$ be an element of the Weyl group. Consider the subtorus of fixed points of $w$, say $S:=(T^w)^\circ$ and put $M=Z\_G(S)^\circ$ (I guess this is already connected if $G$ is simply-connected).


From the general theory we know that $M$ is reductive. 
Question: is $M$ always a Levi subgroup? If no, can we say when does this happen?
I'm only interested in the simply-connected case actually but the question makes sense for a general $G$.


Any references and comments are appreciated!



Centralizer of a subtorus in a reductive group is Levi?
Yes, it is true. You can look at the book of Digne-Michel Proposition 1.22 for a proof. 


