Brauer morphism between modular Hecke algebras
Let k be an algebraically closed field of prime characteristic p, let G be a finite group, and let M be a kG-module. For any p-subgroup Q of G, one has a \(kN_ G(Q)/Q\)-module M(Q) and a Brauer morphism \(Br_ Q: M^ Q\to M(Q)\). If M is a p-permutation module (also called a trivial source module) then \((End_ k(M))(Q)\) and \(End_ k(M(Q))\) are isomorphic so one obtains a Brauer morphism \(End_{kQ}(M)\to End_ k(M(Q))\). This Brauer morphism is studied in the case where G is a finite group with split BN-pair in characteristic p, \(U=O_ p(B)\), \(M=Ind^ G_ U(k)\), P is a parabolic subgroup of G and \(Q=O_ p(P)\); in this case \(N_ G(Q)/Q\) is isomorphic to a Levi subgroup of G. Along the way the author presents a more or less self-contained introduction into the representation theory of G (as developed by Curtis, Richen, Sawada, Green, Tinberg) simplifying proofs of some of the known results. One of the applications is the verification of Alperin's weight conjecture for finite groups with split BN-pair in characteristic p.
scientific article; zbMATH DE number 4071167
scientific article