EXTENSIONS OF CHARACTERS IN TYPE D AND THE INDUCTIVE MCKAY CONDITION, I

Abstract

AbstractThis is a contribution to the study of$\mathrm {Irr}(G)$as an$\mathrm {Aut}(G)$-set forGa finite quasisimple 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$A'(\infty )$condition expressing that diagonal automorphisms and graph-field automorphisms ofGhave transversal orbits in$\mathrm {Irr}(G)$. This is part of the stronger$A(\infty )$condition introduced in the context of the reduction of the McKay conjecture to a question about quasisimple groups. Our main theorem is that a minimal counterexample 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 any standard Levi subgroup of$G=\mathrm {D}_{ l,\mathrm {sc}}(q)$extend to their stabilizers in the normalizer of that Levi subgroup. This allows us 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.