Abstract
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 representations 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 ⊆ S we construct the left cell module corresponding to the "top" left cell C J as a submodule of the Hecke algebra and give a precise description of its canonical basis. In the case of the symmetric group it is shown that every irreducible representation arises as a top cell representation. Finally, analogues of the representations considered are discussed for the case of an infinite Coxeter group.
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