Let $G$ be an connected reductive algebraic group over $k=\bar{\mathbb{F}_p}$. Suppose $G$ is defined over $\mathbb{F}_q$. Let $G^{F}$ be the corresponding finite group associated to $G$. Suppose $s\in G^{F}$ is a regular semisimple element. Now, $s$ is contained in a unique maximal Torus $T$, and $T$ is necessarily $F$-stable. Let $T^F$ denote the set of $F$-rational points of $T$.

It is clear that $T^F\subseteq C_{G^F}(s)$. My question is whether $T^F=C_{G^F}(s)$? In the case that this isn’t true is there a simple description of the quantity $C_{G^F}(s)$?

Thank you.

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