Green Theory for Hecke Algebras and Harish-chandra Philosophy

This paper can be considered as a continuation of a survey article on the representation theory of nite general linear groups in non describing characteristic D4]. The main theme there as well as in the present paper is the connection between the representation theory of quantum GL n and the non describing characteristic case for general linear groups. This connection is given through certain triangles of functors between the categories of representations of general linear groups, Hecke algebras of type A, and q-Schur algebras. The latter algebras were introduced in DJ3] and DJ4] and it was shown in DDo] that they are closely related to quantum GL n. All these connections were extensively discussed in D4]. Thus we shall give here only a very brief account of those in section one. In the present paper I want to concentrate on recent joint work with Jie Du, most of which was omitted in D4]. The main idea is to formulate a vertex theory for representations of Hecke algebras analogous to Green's vertex theory for representations of nite groups. This was carried out for the characteristic 0 case in Du1],,Du2] and Du3], and in general in DDu1]. Then we use the functors in our basic triangles to apply this to representations of q-Schur algebras and nite general linear groups. Two results of this methods are discussed. First one gets a tensor product theorem for q-Schur algebras, based on a Brauer homomorphism generalizing work of Scott Sc]. This was shown with diierent methods by Parshall and Wang in PW], and by Du and Scott in DS] for the case of elds of characteristic 0. For quantum enveloping algebras of type A it was shown by Lustzig Lu1] and in general by Andersen and Wen AW]. Using the functors connecting representations of q-Schur algebras and those of general linear groups one can derive a tensor product theorem for nite general linear groups in the non describing characetristic case. Its main features will be brieey discussed in section 3. The second result comes from working directly with the connection between Hecke algebras and general linear groups. It turns out that the vertex theory for Hecke algebras