Representations of Coxeter groups and Hecke algebras

Abstract

here l(w) is the length of w. In the case where Wis a Weyl group and q is specialized to a fixed prime power, | ~ can be interpreted as the algebra of intertwining operators of the space of functions on the flag manifold of the corresponding finite Chevalley group G(Fq) (see [loc. cit., Ex. 24]). Therefore, the problem of decomposing this space of functions into irreducible representations of G(Fq) is equivalent to the problem of decomposing the regular representation o f ~ | (12. It is known that, in this case, | is isomorphic to the group algebra of W; however, in general, this isomorphism cannot be defined without introducing a square root of q (see [1]). It is therefore, natural to extend the ground ring of ~ as follows. For any Coxeter group (W, S) we define the Hecke algebra ~ to be J{' | A, where A is the ring of Laurent polynomials with integral coefficients in the indeterminate ql/2. Our purpose is to construct representations oL,Uf endowed with a special basis. They will be defined in terms of certain graphs. We define a W-graph to be a set of vertices X, with a set Y of edges (an edge is a subset of X consisting of two elements) together with two additional data: for each vertex xeX , we are given a subset I x of S and, for each ordered pair of vertices y, x such that {y, x} e Y, we are given an integer p(y, x) +0. These data are subject to the requirements (1.0.a), (1.0.b) below. Let E be