Galois automorphisms and blocks covering unipotent blocks

TL;DR

This paper proves a key condition for all finite simple groups, leading to a new characterization of groups with a normal -complement, and studies the distribution of characters in unipotent blocks under generalized theory.

Contribution

It establishes the condition for all finite simple groups and advances understanding of character distribution in unipotent blocks using generalized -Harish-Chandra theory.

Findings

Condition satisfied for all finite simple groups.

Character distribution in unipotent blocks is well-behaved.

Progress on the blockwise Galois--McKay conjecture.

Abstract

In this paper we prove that a recent condition of Lyons--Mart\'inez--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.