On the inductive Alperin–McKay conditions in the maximally split case

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. The validity of those conditions is 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 \(\mathsf C\) over \(\mathbb {F}_q\) satisfy the inductive Alperin–McKay conditions for primes \(\ell \ge 5\) and dividing \(q-1\). Some methods to that end are adapted from Malle and Späth (Ann. Math. (2) 184:869–908, 2016).