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

Author:

SPÄTH BRITTAORCID

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.

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Reference38 articles.

1. Inductive McKay condition for finite simple groups of type 𝖢

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3. A reduction theorem for Dade’s projective conjecture

4. [S7] Späth, B. , Extensions of characters in type D and the inductive McKay condition, II, preprint arXiv:2304.07373

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