Inductive AM condition for the alternating groups in characteristic 2

The aim of this paper is to verify the inductive AM condition stated in [22] (Definition 7.2) for the simple alternating groups in characteristic 2. Such a condition, if checked for all simple groups in all characteristics would prove the Alperin–McKay conjecture (see [22]). We first check the Alperin–McKay conjecture for double cover of symmetric and alternating groups, S˜n and A˜n. The proof of this will extend known results about blocks of symmetric and alternating groups of a given weight.The first two parts are notations and basic results on blocks of symmetric, alternating groups and of their double covers that are recalled for convenience. In the third part we determine the number of height zero spin characters of a given block using certain height preserving bijections between blocks of a given weight. Then we determine the number of height zero spin characters in a given block of the normalizer of a defect group. We finish by showing that the inductive AM condition is true for the alternating groups in characteristic 2, the cases A7A7 and A6A6 being treated separately from the rest.I would like to thank Marc Cabanes and Britta Späth for taking the time to discuss all the mathematical issues I encountered while writing this paper.