Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer

We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne--Michel's unique Jordan decomposition in the connected center case.
Comment: v2, 19 pages, substantial revision, text reordered and shortened in places. Theorem 2.1 a stronger version of previous results, see (6) of the statement, with shorter proof. New results: Proposition 4.3, Lemma 6.1, Theorem 6.2 and Proposition 6.3