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

The Alperin-McKay conjecture relates height zero characters of an l-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 l-blocks of groups of Lie type over a field with q elements when l divides q- 1. We also give information about l-blocks and Brauer correspondents. As an application we show that quasi-simple groups of type C over Fq satisfy the inductive Alperin-McKay conditions for primes l = 5 and dividing q - 1. Some methods to that end are adapted from Malle and Spath (Ann. Math. (2) 184:869-908, 2016).