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

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 Spath (Ann. Math. (2) 184:869–908, 2016).