On Jordan decomposition of characters for SU(n,q)

As shown by Bonnafé, a step in proving a Jordan decomposition of characters of finite special linear groups is the parametrization of unipotent characters of centralizers of semi-simple elements in projective linear groups. We show the same kind of result in the case of finite special unitary groups. The proof leads to a mild adaptation of Bonnaféʼs methods expounded in [B99]. The outcome is a Jordan decomposition of characters compatible with Lusztigʼs twisted induction.