Introduction This article is a contribution to the study of the modular representation theory of the finite general linear group GLn(¥q) over a field F of characteristic p coprime to q. We have attempted in the first place to give a self-contained account of the results originating in the work of James and the second author [Di, D2, DJ2, DJ3, Ji, J2] in the 1980s relating representation theory of GLn(¥q) to representation theory of "quantum GLn" at roots of unity. Since that time, there have been a number of conceptual simplifications to the theory, e.g. in [CPS3, D3, D4, DDui, DDU2, HL2, T], which we have incorporated in the present approach from the outset. We mention above all the centrally important Morita theorem of Cline, Parshall and Scott from [CPS3, §9]. We will reprove this Morita theorem in the present article in a self- contained way, thus leading us to a new and independent approach to the results of [Di, D2, DJ2, DJ3, Ji, J2] assumed in the Cline-Parshall-Scott argument. Along the way, we make many of these results more precise or more functorial, which is essential in order to prove the new results of the article described further below. Our point of view has been to deduce as much as possible of the modular theory from standard, often purely character theoretic results in the characteristic zero theory of GLn(¥q), combined with knowledge of the highest weight representation theory of quantum linear groups. For the former, we have adopted the point of view of the Deligne-Lusztig theory, as described for GLn{¥q) by Fong and Srinivasan [FS], supplemented by various other basic results most of which can be found in Carter's book [C] we also appeal to the result of [DDui, §5],[HL2] showing that Harish- Chandra induction is independent of the choice of parabolic subgroup, and the basic result of block theory proved in [BM] (also originally proved in [FS] though we do not use the full block classfication of Fong and Srinivasan). For quantum linear groups, we have followed the treatment by Parshall and Wang [PW] wherever possible, as well as [CI, JM, DDo, D07] for various additional results. We now summarize the main steps in the development, so that we can describe the new results of the article in more detail. We restrict our attention in this intro- duction to the case of unipotent representations, though there is no such restriction in the main body of the article. So let F be an algebraically closed field of char- acteristic p coprime to g, and let Gn = GLn(¥q). Let M denote the FGn-module arising from the permutation representation of Gn on the cosets of a Borel subgroup. Following the idea of Cline, Parshall and Scott, we introduce the cuspidal algebra,
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