Quantum linear groups and representations of GL(n)(F-q)

We give a self-contained 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 representa tion theory of "quantum GL(n)" at roots of unity. The new treatment allows us to extend the theory in several directions. Fir st, we prove a precise functorial connection between the operations of tens or product in quantum GL(n) and Harish-Chandra induction in finite GL(n). T his allows us to obtain a version of the recent Morita theorem of Cline, Pa rshall and Scott valid in addition for p-singular classes. From that we obtain simplified treatments of various basic known facts, suc h 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 r esult of Takeuchi. Finally, we apply the theory to study the affine general linear group, foll owing ideas of Zelevinsky in characteristic zero. We prove results that car l be regarded as the modular analogues of Zelevinsky's and Thoma's branchin g rules. Using these, we obtain a new dimension formula for the irreducible cross-characteristic representations of GL(n)(F-q), expressing their dimen sions in terms of the characters of irreducible modules over the quantum gr oup.