Harish-chandra Vertices and Steinberg's Tensor Product Theorem for General Linear Groups in Non-describing Characteristic

Let G be a nite group of Lie type deened over some nite eld GF(q). Let k be a eld of positive characteristic p not dividing q. Hecke functors tie together the representation theory of kG and that of Hecke algebras associated with nite reeection groups. In DDu1] the theory of vertices and sources for such algebras was introduced in the case of Hecke algebras H of type A, (that is Hecke algebras associated with symmetric groups), and they were investigated involving q-Schur algebras. These come up as centralizer algebras of the action of H on tensor space, and they are the building blocks of the quantum coordinate algebra of general linear groups. One of the main results of DDu1] was a q-analogue for Steinberg's tensor product theorem for q-Schur algebras. This is produced by investigating Frobenius morphisms on quantum GL n (compare too Lu],,DS],,PW],,AW]). In DDu2] the notion of Harish-Chandra vertices and sources has been introduced and subsequently investigated for the special case of nite groups of Lie type. The main result there was that Harish-Chandra induction (and truncation) from conjugate Levi subgroups yields isomorphic functors. This result has been shown also by R. Howlett and G. Lehrer HL2]. In this paper, we shall present some applications which come from the combination of the results in DDu1] and DDu2] by relating them with Hecke functors, and in case of type A, by q-Schur functors. There are three main applications: First using Hecke functors, we show that vertices and sources of indecomposable modules for Hecke algebras H yield Harish-Chandra vertices and sources for the corresponding images under the Hecke functor. We have to impose some restrictions, but the result are valid for representations in the principal series for arbitrary nite groups