Action of Automorphisms on Irreducible Characters of Symplectic Groups

Assume $G$ is a finite symplectic group ${\mathrm{S}\mathrm{p}}_{2n}(q)$ over a finite field ${\mathbb{F}}_q$ of odd characteristic. We describe the action of the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ on the set $\mathrm{I}\mathrm{r}\mathrm{r}(G)$ of ordinary irreducible characters of $G$. This description relies on the equivariance of Deligne–Lusztig induction with respect to automorphisms. We state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding Levi subgroup; this may be of independent interest. As an application we prove that the global condition in Sp"ath’s criterion for the inductive McKay condition holds for the irreducible characters of ${\mathrm{S}\mathrm{p}}_{2n}(q)$.