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

Britta Späth

TL;DR

The paper addresses the action of Aut$(G)$ on Irr$(G)$ for quasisimple groups of Lie type $D$ and $^2D$, proving a weak form $A'( abla orall)$ that a minimal counterexample to $A( abla orall)$ would still satisfy. It develops an equivariant extension framework for cuspidal Levi-subgroup characters and uses Harish-Chandra theory to parametrize Irr$(G)$ via relative Weyl groups, controlling automorphism actions through detailed Clifford theory. A core contribution is a new, explicit extension-of-cuspidal-characters result for Levi subgroups (extending Geck–Lusztig’s theorem in a uniform way) and a thorough analysis of the relative inertia groups $W( ext{lambda})$. These advances enable a robust reduction toward the inductive McKay condition for type $D$, informing both transversal constructions and stabilizer computations for outer automorphisms. The methods pave the way for establishing $A( abla orall)$ in a follow-up work and have potential applications to broader character-theoretic conjectures via the transversal/extendibility framework.

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.

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

TL;DR

The paper addresses the action of Aut on Irr for quasisimple groups of Lie type and , proving a weak form that a minimal counterexample to would still satisfy. It develops an equivariant extension framework for cuspidal Levi-subgroup characters and uses Harish-Chandra theory to parametrize Irr via relative Weyl groups, controlling automorphism actions through detailed Clifford theory. A core contribution is a new, explicit extension-of-cuspidal-characters result for Levi subgroups (extending Geck–Lusztig’s theorem in a uniform way) and a thorough analysis of the relative inertia groups . These advances enable a robust reduction toward the inductive McKay condition for type , informing both transversal constructions and stabilizer computations for outer automorphisms. The methods pave the way for establishing in a follow-up work and have potential applications to broader character-theoretic conjectures via the transversal/extendibility framework.

Abstract

This is a contribution to the study of as an -set for a finite quasi-simple group. Focusing on the last open case of groups of Lie type and , a crucial property is the so-called condition expressing that diagonal automorphisms and graph-field automorphisms of have transversal orbits in . This is part of the stronger 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 for groups of type would still satisfy . This will be used in a second paper to fully establish 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 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.

Paper Structure

This paper contains 23 sections, 74 theorems, 68 equations, 1 table.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Basic Considerations
  4. Notation and Condition $A(\infty)$
  5. Action of $\mathrm{Aut}(G)$ on Harish-Chandra induced characters
  6. Action on characters of normalizers of Levi subgroups
  7. Reminder on cuspidal characters
  8. The Levi subgroup and its normalizer
  9. Subgroups of the Levi subgroup $L$
  10. The structure of $N/L$
  11. A supplement of $L$ in $\overline N$
  12. Extending cuspidal characters of Levi subgroups
  13. The inclusion $H_1 \lhd \overline V_{1}$
  14. The inclusion $K_{d}\lhd K_{d} \overline V_d$ for $d\geq 2$
  15. Consideration of $K_{-1}\lhd K_{-1}\overline V_{-1}$
  16. ...and 8 more sections

Key Result

Theorem A

Let $G= \mathrm{D}_{l,sc}(q)$ ($l\geq 4$, $q$ a power of an odd prime), let $\widetilde{G}$ and $E$ as above (see also not). If any $\mathrm{D}_{l',sc}(q)$ for $4\leq l'<l$ satisfies $A({\infty})$, then $G$ satisfies $A'({\infty})$.

Theorems & Definitions (149)

  • Theorem A
  • Theorem B
  • Lemma 2.4
  • proof
  • Theorem 2.7: Malle--Späth MS16
  • Theorem 2.8
  • Proposition 2.10
  • proof
  • proof : Proof of \ref{['thm_MS']}
  • Proposition 2.11
  • ...and 139 more