Characters of odd degree

TL;DR

This paper proves the McKay conjecture for characters of odd degree by verifying the inductive McKay condition for certain groups of Lie type using a new equivariant Harish-Chandra induction approach.

Contribution

It introduces a novel equivariant Harish-Chandra induction method to verify the inductive McKay condition for specific Lie type groups, leading to the proof of the McKay conjecture for prime 2.

Findings

Proof of the McKay conjecture for characters of odd degree.

Verification of the inductive McKay condition for groups of Lie type.

Characters of odd degree lie in specific Harish-Chandra series.

Abstract

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely that they only lie in very particular Harish-Chandra series then allow us to deduce from it the McKay conjecture for the prime~$2$, hence for characters of odd degree.