Representations of Hecke Algebras at Roots of Unity

Hecke algebras arise naturally in the representation theory of nite or p-adic Cheval-ley groups, as endomorphism algebras of certain induced representations (see Carter 9], Lusztig 42], and the references there). They may also be viewed as quotients of group algebras of Artin{Tits braid groups, and then they can be used to construct invariants of knots and links (Jones 34]). Another point of view | which we take in this talk | is to regard them abstractly as deformations of group algebras of nite Coxeter groups, depending on a parameter u. If we specialize this parameter to a root of unity, we obtain in general a non-semisimple specialized algebra. The degree of non-semisimplicity is measured in terms of a corresponding decomposition matrix, which records in which way the irreducible representations of the generic algebra split up under specialization. The aim of this talk is to explain some of the main open problems about decomposition matrices of Hecke algebras, and to report on some signiicant recent advances in this area. For applications to the representation theory of a nite Chevalley group G over a eld of characteristic p, the most interesting specializations are those where u is mapped to a non-zero element in a eld of positive characteristic`6 = p. By Dipper's theory of Hom functors 13], a special case of which we describe in Section 1, the corresponding decomposition matrix is a submatrix of the usuaì-modular decomposition matrix of G. A general factorization result, which we explain in Section 2, shows that our decomposition matrices can be obtained in two steps: one step from u to a root of unity over Q, and another step from characteristic 0 to characteristic`. A conjecture which was rst formulated by James 30] for Hecke algebras associated to the symmetric group S n predicts that \nothing happens" in the second step, as long asìs not too small. (James actually considers an enlargement of that algebra, namely the q-Schur algebra introduced by Dipper and himself in 16]; he also gives a precise bound for`.) A rigorous formulation of this conjecture, for nite Coxeter groups of any type, will be given in Section 3. For Hecke algebras associated with S n , Lascoux, Leclerc and Thibon 37] have conjectured a solution of the rst step, by translating the original problem to that of computing