Perfect generalized characters inducing the Alperin–McKay conjecture

Abstract

It is well known that the perfect isometries predicted in Broué's conjecture do not always exist when the defect groups are non-abelian, even when the blocks have equivalent Brauer categories. We consider perfect generalized characters which induce bijections between the sets of irreducible characters of height zero of a block and of its Brauer correspondent in the normalizer of a defect group, hence providing in these cases an ‘explanation’ for the numerical coincidence predicted in the Alperin–McKay conjecture. In this way the perfect isometries predicted in Broué's conjecture for blocks with abelian defect groups are generalized. Whilst such generalized characters do not exist in general, we show that they do exist when the defect groups are non-abelian trivial intersection subgroups of order p 3 , as well as for B 2 2 ( q ) for q a power of two and PSU 3 ( q ) for all q. Further, we show that these blocks satisfy a generalized version of an isotypy.