Extensions of characters in type D and the inductive McKay condition, I

Abstract

This is a contribution to the study of $\operatorname {Irr}(G)$ as an $\operatorname {Aut}(G)$-set for $G$ a finite quasi-simple group. Focusing on the last open case of groups of Lie type $\mathrm D$ and $^2\mathrm D$, a crucial property is the so-called condition $A'(\infty)$ expressing that diagonal automorphisms and graph-field automorphisms of $G$ have transversal orbits in $\operatorname {Irr}(G)$. This is part of the stronger $A(\infty)$ condition introduced in the context of the reduction of the McKay conjecture to a question on quasi-simple groups. Our main theorem is that a minimal counter-example to condition $A(\infty)$ for groups of type $\mathrm D$ would still satisfy $A'(\infty)$. This will be used in a second paper to fully establish $A(\infty)$ for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of arbitrary standard Levi subgroups of $G={\mathrm D}_{ l,\mathrm{sc}}(q)$ extend to their stabilizers in the normalizer of that Levi subgroup. This allows to control the action of automorphisms on these extensions. From there Harish Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.