Weight parameterization of simple modules for \(p\)-solvable groups. (Q2859265)

Let \(G\) be a finite group, let \(p\) be a prime, and let \(k\) be an algebraically closed field of characteristic \(p\). \textit{J. L. Alperin} [in: Representations of finite groups, Proc. Conf., Arcata/Calif. 1986, Pt. 1, Proc. Symp. Pure Math. 47, 369-379 (1987; Zbl 0657.20013)] introduced weights for \(G\) with respect to \(p\) in order to formulate his celebrated weight conjecture. More precisely, a weight of \(G\) is a pair \((R,Y)\) where \(R\) is a \(p\)-subgroup of \(G\) and \(Y\) is an isomorphism class of simple \(kN_G(R)\)-modules with vertex \(R\). Alperin's Weight Conjecture says that the number of \(G\)-conjugacy classes of weights of \(G\) is the same as the number of isomorphism classes of simple \(kG\)-modules. In 1981, Okuyama proved a statement that implies Alperin's Weight Conjecture for \(p\)-solvable groups.NEWLINENEWLINE Suppose now that \(G\) is a \(p\)-solvable group. In the paper under review, the author exhibits a natural bijection between the set of isomorphism classes of simple \(kG\)-modules \(M\) and the set of \(G\)-conjugacy classes of weights \((R,Y)\), up to the choice of a polarization. Note that the bijection is compatible with the action of the group of outer automorphisms of \(G\). The author also determines the relationship between a multiplicity module of \(M\) and \(Y\). In an appendix, the author shows that the bijection defined by \textit{G. Navarro} [Math. Z. 212, No. 4, 535-544 (1993; Zbl 0795.20005)] for the groups of odd order coincides with his bijection for a particular choice of the polarization.