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.