  As a step to establish the McKay conjecture on character degrees of finite
groups, we verify the inductive McKay condition introduced by
Isaacs-Malle-Navarro for simple groups of Lie type $A_{n-1}$, split or twisted.
Key to the proofs is the study of certain characters of SL$_n(q)$ and SU$_n(q)$
related to generalized Gelfand-Graev representations. As a by-product we can
show that a Jordan decomposition for the characters of the latter groups is
equivariant under outer automorphisms. Many ideas seem applicable to other Lie
types.
