W-graph ideals and duality

This paper is the continuation of the work in [14]. In that paper we generalized the definition of W-graph ideal in the weighted Coxeter groups, and showed how to construct a W-graph from a given W-graph ideal in the case of unequal parameters.In this paper we study the duality and the full W-graph for a given W-graph ideal. We show that there are two modules associated with a given W-graph ideal, they are connected by a duality map. The full W-graph includes all the W-graph data determined by the dual and contragredient representations. Our construction closely parallels that of Kazhdan and Lusztig [6,10,11], which can be regarded as the special case J=∅. It also generalizes the work of Couillens [2], Deodhar [3,4], and Douglass [5], corresponding to the parabolic case.