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Let $\mathbf{G}$ be a connected reductive group with connected center definedover $\mathbb{F}_q$, with Frobenius morphism F. Given an irreducible complexcharacter $\chi$ of $\mathbf{G}^F$ with its Jordan decomposition, and a Galoisautomorphism $\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, wegive the Jordan decomposition of the image ${^\sigma \chi}$ of $\chi$ under theaction of $\sigma$ on its character values.

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