Galois-equivariant McKay bijections for primes dividing $q-1$

TL;DR

This paper demonstrates that certain Galois-equivariant bijections for groups of Lie type can potentially satisfy the inductive Galois-McKay conditions, advancing the understanding of the McKay conjecture for prime 2.

Contribution

It proves Galois-equivariance of McKay bijections for most Lie type groups, supporting their use in inductive Galois-McKay conditions.

Findings

Galois-equivariance of McKay bijections established

Supports inductive Galois-McKay conditions for prime 2

Several Lie type groups satisfy McKay--Navarro conjecture

Abstract

We prove that for most groups of Lie type, the bijections used by Malle and Spaeth in the proof of Isaacs-Malle-Navarro's inductive McKay conditions for the prime 2 and odd primes dividing q - 1 are also equivariant with respect to certain Galois automorphisms. In particular, this shows that these bijections are candidates for proving Navarro-Spaeth-Vallejo's recently-posited inductive Galois-McKay conditions. On the way, we show that several simple groups of Lie type satisfy the McKay--Navarro conjecture for the prime 2.