The inductive McKay–Navarro conditions for the prime 2 and some groups of Lie type

Abstract

For a prime ℓ \ell , the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with ℓ ′ \ell ’ -degree and the corresponding set for the normalizer of a Sylow ℓ \ell -subgroup. Navarro’s refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro–Späth–Vallejo have reduced the McKay–Navarro conjecture to certain “inductive” conditions on finite simple groups. We prove that these inductive McKay–Navarro (also called the inductive Galois–McKay) conditions hold for the prime ℓ = 2 \ell =2 for several groups of Lie type, namely the untwisted groups without non-trivial graph automorphisms.

On the inductive McKay–Navarro condition for finite groups of Lie type in their defining characteristic

“…Since we need here that the automorphism extending G acts trivially on the Weyl group, we cannot use these methods to extend the characters to the exceptional graph automorphism. The same problem occurs using Harish-Chandra induction for disconnected groups as in [RSF22a].…”

“…So far, this has been done for groups of Lie type in their defining characteristic in [Ruh21] and [Joh22b] as well as for the Suzuki and Ree groups and all primes in [Joh21]. Ruhstorfer and Schaeffer Fry further showed in [RSF22b] that the inductive condition holds for all finite simple groups and the prime ℓ = 2 and thereby proved that the conjecture itself is true for ℓ = 2.…”