SOME GENERIC REPRESENTATIONS, W-GRAPHS, AND DUALITY

This paper begins by generalising the notion of a ''W-graph'' to show that the W-graph data determine not one but four closely related repre sentations of the generic Hecke algebra of an arbitrary Coxeter group. Canonical ''Kazhdan-Lusztig bases'' are then constructed for several families of ideals inside the Hecke algebra of a finite Coxeter system (W, S). In particular for each J subset-or-equal-to S we construct th e left cell module corresponding to the ''top'' left cell C(J) as a su bmodule of the Hecke algebra and give a precise description of its can onical basis. In the case of the symmetric group it is shown that ever y irreducible representation arises as a top cell representation. Fina lly, analogues of the representations considered are discussed for the case of an infinite Coxeter group. (C) 1994 Academic Press, Inc.