This work is an account of the results originating in the work of James and the second author in the 1980s relating the representation theory of GL n(F q) over fields of characteristic coprime to q to the representation theory of quantum GL n at roots of unity. The new treatment allows us to extend the theory in several directions. First, we prove a precise functorial connection between the operations of tensor product in quantum GL n and Harish-Chandra induction in finite GL n. This allows us to obtain a version of the recent Morita theorem of Cline, Parshall and Scott valid in addition...