Unitriangular basic sets, Brauer characters and coprime actions

Abstract

We show that the decomposition matrix of a given groupGGis unitriangular, wheneverGGhas a normal subgroupNNsuch that the decomposition matrix ofNNis unitriangular,G/NG/Nis abelian and certain characters ofNNextend to their stabilizer inGG. Using the recent result by Brunat–Dudas–Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix whenever they are related via Bonnafé–Dat–Rouquier’s equivalence to a unipotent block. This is then applied to study the action of automorphisms on Brauer characters of finite quasi-simple groups. We use it to verify the so-calledinductive Brauer–Glauberman conditionthat aims to establish a Glauberman correspondence for Brauer characters, given a coprime action.