Hecke Algebra with Unequal Parameters

Let (W,S) be a Coxeter system and H the associated Hecke algebra with unequal parameters. The Laurent polynomials Ms y,w and py,w for y, w ∈ W and s ∈ S play an important role in the representations of H. We study the properties of Ms y,w and py,w, the relations among them, as well as with the left, right and two-sided cells of W . In his book [5], Lusztig gave a systematic introduction to the Hecke algebras H associated to a Coxeter system (W,S) with unequal parameters, where the Laurent polynomials M y,w and py,w for y, w ∈ W and s ∈ S play an important role in the structure theory and the representation theory of H. However, owing to the lack of their explicit expressions, we know very little about the properties of M y,w’s and py,w’s. In the present paper, we give some closed investigation for those Laurent polynomials. We establish some criteria for the vanishing and the non-vanishing of M y,w. In particular, we generalize some results of Kazhdan and Lusztig in [2]. In [5. Corollary 6.5], Lusztig showed that for any y, w ∈ W with sy < y < w < sw and L(s) = 1, M y,w is equal to the coefficient of v −1 in py,w. In this paper, we generalize this result to unequal parameter case (see Proposition 3.1). We study the relation between