Centralisers of semi-simple elements are semidirect products

Abstract

Let $\bG$ be a connected reductive algebraic group over an algebraically closed field, and let $s\in\bG$ be a semisimple element. We show that the centraliser of $s$ is the semidirect product of its identity component by its group of components. We then look at the case where $\bG$ is defined over an algebraic closure of a finite field $\Fq$, and $F$ is an endomorphism such that some power is a Frobenius endomorphism attached to an $\Fq$-structure on $\bG$. We show that if the centraliser of $s$ is $F$-stable we have a semidirect product decomposition of its $F$-fixed points.