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Galois automorphisms and blocks covering unipotent blocks

L. Ruhstorfer, A. A. Schaeffer Fry

Abstract

In this paper we prove that a recent condition of Lyons--Martínez--Navarro--Tiep, regarding the field of values of extensions of characters in principal blocks, is satisfied for all finite simple groups, which when combined with their results gives a new characterization of finite groups with a normal $\ell$-complement for a prime $\ell$. This leads us to study the distribution of characters in unipotent blocks of disconnected reductive groups and show that this is well-behaved under a generalization of $d$-Harish-Chandra theory. We go on to study the blockwise Galois--McKay (also known as the Alperin--McKay--Navarro) conjecture for the blocks of almost (quasi-)simple groups above unipotent blocks.

Galois automorphisms and blocks covering unipotent blocks

Abstract

In this paper we prove that a recent condition of Lyons--Martínez--Navarro--Tiep, regarding the field of values of extensions of characters in principal blocks, is satisfied for all finite simple groups, which when combined with their results gives a new characterization of finite groups with a normal -complement for a prime . This leads us to study the distribution of characters in unipotent blocks of disconnected reductive groups and show that this is well-behaved under a generalization of -Harish-Chandra theory. We go on to study the blockwise Galois--McKay (also known as the Alperin--McKay--Navarro) conjecture for the blocks of almost (quasi-)simple groups above unipotent blocks.
Paper Structure (24 sections, 54 theorems, 59 equations)

This paper contains 24 sections, 54 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The conditions of Lyons--Martínez--Navarro--Tiep
  4. Initial observations on extensions
  5. The Alvis--Curtis duality and automorphisms
  6. Lifting the Cabanes--Rickard equivalence
  7. Comparison to Digne--Michel's lift
  8. Regular characters in the principal block
  9. Groups of Lie type
  10. Regular embeddings and regular characters
  11. Height zero characters in the principal block
  12. Some exceptional cases
  13. The generalized Steinberg character and the principal block
  14. Observations on the Dade group for twisted groups
  15. The principal block and generalized Harish-Chandra induction
  16. ...and 9 more sections

Key Result

Theorem A

Let $\ell$ be any odd prime number and let $G$ be a finite group with order divisible by $\ell$. Then $\ell$ divides $\chi(1)$ for all nonlinear ${\mathbb{Q}}_\ell$-valued $\chi\in\operatorname{Irr}(B_0(G))$ if and only if $G$ has a normal $\ell$-complement.

Theorems & Definitions (115)

  • Theorem A
  • Corollary B
  • Theorem C
  • Theorem D
  • Theorem E
  • Definition 2.1: Definition 3.1 of lmnt
  • Theorem 2.2: Theorem E of lmnt
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 105 more