Centralisers of semi-simple elements are semidirect products

Abstract

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