| Résume | John McKay's conjecture (1970) is an elementary-looking global/local statement relating the representation theory of a group to the normalizer of its Sylow subgroup. After the reduction theorem for this conjecture given by Isaacs-Malle-Navarro (2007), it is sufficient to check a strong version of this conjecture, the so-called inductive McKay condition for all finite quasi-simple groups.
We verify this condition for special linear groups and special unitary groups over finite fields. Key to the proofs is the study of (certain) generalized Gelfand-Graev representations. As a by-product we obtain some surprising statements about all characters of those groups, for example that as stabilizer of a character of $SL_n(q)$, only certain subgroups can occur. |
