The Alperin Weight Conjecture and the Glauberman correspondence via character triples

Abstract

Recently, G. Navarro introduced a new conjecture that unifies the Alperin Weight Conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple groups and prove it for several classes of groups and blocks. Our reduction can be divided into two steps. First, we show that assuming the so-called Inductive (Blockwise) Alperin Weight Condition for finite simple groups, we obtain an analogous statement for arbitrary finite groups, that is, an automorphism-equivariant version of the Alperin Weight Conjecture inducing isomorphisms of modular character triples. Then, we show that the latter implies Navarro's conjecture for each finite group.