On the Inductive Alperin-McKay Conditions in the Maximally Split Case

TL;DR

This paper advances understanding of the Alperin-McKay conjecture by analyzing characters of height zero in groups of Lie type over finite fields, and verifies the inductive conditions for certain groups and primes.

Contribution

It describes characters of height zero in specific blocks of Lie type groups and proves the inductive Alperin-McKay conditions for quasi-simple groups of type C over finite fields for primes dividing q-1.

Findings

Characters of height zero are characterized in these blocks.

Quasi-simple groups of type C satisfy the inductive conditions for primes ≥ 5 dividing q-1.

Methods from Malle--Späth are adapted for this analysis.

Abstract

The Alperin-McKay conjecture relates height zero characters of an $\ell$-block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin-McKay conditions about quasi-simple groups by the third author. Those conditions are still open for groups of Lie type. The present paper describes characters of height zero in $\ell$-blocks of groups of Lie type over a field with $q$ elements when $\ell$ divides $q-1$. We also give information about $\ell$-blocks and Brauer correspondents. As an application we show that quasi-simple groups of type $C$ over $\mathbb{F}_q$ satisfy the inductive Alperin-McKay conditions for primes $\ell\geq 5$ and dividing $q-1$. Some methods to that end are adapted from the work of Malle--Sp\"ath.