Equivariance and extendibility in finite reductive groups with connected center

Abstract

We show that several character correspondences for finite reductive groups $$G$$ are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to $$G$$ has connected center. The correspondences we consider are the so-called Jordan decomposition of characters introduced by Lusztig and the generalized Harish-Chandra theory of unipotent characters due to Broue–Malle–Michel. In addition we consider a correspondence giving character extensions, due to the second author, in order to verify the inductive McKay condition from Isaacs–Malle–Navarro for the non-abelian finite simple groups of Lie types $$^3\mathsf{D }_4,\mathsf{E }_8,\mathsf{F }_4,^2\mathsf{F }_4$$ , and $$\mathsf{G }_2$$ .