Galois-equivariant McKay bijections for primes dividing q − 1

Abstract

We prove that for most groups of Lie type, the bijections used by Malle and Späth 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-Späth-Vallejo's recently-posited inductive Galois-McKay conditions.

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

“…We now prove the equivariance part of Condition 1 by working in the twisted setting. Note that for d = 1 this has already been shown in [SF22].…”

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

“…In our situation of Theorem A, the second author has proven in [SF21] that the groups M and bijections Ω from [Mal07,Mal08,MS16] for the inductive McKay conditions are indeed H-equivariant. Hence here we must study the behavior of the projective representations under the appropriate twists.…”

“…We remark that Theorem A gives the first result in which a series of groups of Lie type is shown to satisfy the inductive McKay-Navarro conditions for a non-defining prime. Given the results of [SF21], the bulk of what remains to prove Theorem A is to study the behavior of extensions of odd-degree characters of groups of Lie type to their inertia group in the automorphism group, and similarly for the local subgroup used in the inductive conditions. To do this, we utilize work by Digne and Michel [DM94] on Harish-Chandra theory for disconnected groups and work of Malle and Späth [MS16] that tells us that, in our situation, odd-degree characters lie in very specific Harish-Chandra series.…”

“…In [Ruh21], the first author proves that the simple groups of Lie type satisfy the inductive McKay-Navarro conditions for their defining prime, with a few lowrank exceptions that were settled in [Joh20]. In [SF21], the second author shows that the maps used in [Mal07,Mal08,MS16] to prove the original inductive McKay conditions for the prime = 2 are further H -equivariant, making them promising candidates for the inductive McKay-Navarro conditions.…”

“…Here we expand on the work of [SF21] to complete the proof that when = 2, the inductive McKay-Navarro (also called inductive Galois-McKay) conditions from [NSV20] are satisfied for the untwisted groups of Lie type that do not admit graph automorphisms.…”