Galois group action and Jordan decomposition of characters of finite reductive groups with connected center

Abstract: Let $\mathbf{G}$ be a connected reductive group with connected center defined over $\mathbb{F}_q$, with Frobenius morphism F. Given an irreducible complex character $\chi$ of $\mathbf{G}^F$ with its Jordan decomposition, and a Galois automorphism $\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, we give the Jordan decomposition of the image ${^\sigma \chi}$ of $\chi$ under the action of $\sigma$ on its character values.