On Quotients of Hom-functors and Representations of Nite General Linear Groups Ii

This is a second paper on quotients of Hom-functors and their applications to the representation theory of nite general linear groups in non-describing characteristic. After some general result on quotients of Hom-functors and their connection to Harish-Chandra theory these contructions are used to obtain a full classiication of thè-modular irreducible representations of GL n (q) for some prime power q which is not divisible by the primè and to explain some facts on their Harish-Chandra series and decomposition numbers. 0 Introduction This is a second paper dealing with representations of nite general linear groups G in non-describing characteristic and quotients of Hom-functors in fulllment of an anouncement in 9], promising new proofs for the classiication of the irreducible representations of G. In the rst three sections the general theory of those functors is further developed. In particular some errors of 9] are corrected. The relation between the functors for diierent projective resolutions is investigated as well as the image of the diierent right inverses of those. Our main applications of these functors concern representations of nite reductive groups G in nondescribing characteristic, (they have however been successfully applied to p-adic groups as well, see 46]). Quotients of Hom-functors arise here especially in connection with Harish-Chandra theory, which is used to subdivide the irreducible G-modules into Harish-Chandra series in terms of Hecke algebras associated with reeection groups. Partial results in this direction were obtained rst in 7], 8] (for general linear groups) and in 15], 16] for arbitrary nite reductive groups. In 28] Geck, Hiss and Malle established then This paper is a contribution to the DFG project on \Algorithmic number theory and algebra".