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 $χ$ of $\mathbf{G}^F$ with its Jordan decomposition, and a Galois automorphism $σ\in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, we give the Jordan decomposition of the image ${^σχ}$ of $χ$ under the action of $σ$ on its character values.

It was pointed out to us by Gunter Malle that Lemma 3.3 in the previous version of this paper was incorrect. In the new version, Section 3.3 is re-written, and Section 4 has a new result in Proposition 4.1