Representations of Hecke and $q$-Schur algebras are closely related to those of finite general linear groups $G$ in non-describing characteristics. Such a relationship can be described by certain functors. Using these functors, we determine the Harish-Chandra vertices and sources of certain indecomposable $G$-modules. The Green correspondence is investigated in this context. As a further application of our theory, we establish Steinberg's tensor product theorems for irreducible representations of $G$ in non-describing characteristics.
1991 Mathematics Subject Classification: 20C20, 20C33, 20G05, 20G40.
