Quantum Linear Groups and Representations of Gln (Fq)Paperback, 1 January 2001

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 for p-singular classes. From that we obtain simplified treatments of various basic known facts, such as the computation of decomposition numbers and blocks of GL [n(F [q) from knowledge of the same for the quantum group, and the non-defining analogue of Steinberg's tensor product theorem. We also easily obtain a new double centralizer property between GL [n(F [[q) and quantum GL [n, generalizing a result of Takeuchi. Finally, we apply the theory to study the affine general linear group, following ideas of Zelevinsky in characteristic zero.