W-graphs for Hecke algebras with unequal parameters(II)

This paper is the continuation of the work in~\cite{Yin}. 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 full $W$-graphs for a given $W$-graph ideal. We show that there exist a pair of dual modules associated with a given $W$-graph ideal, they are connected by a duality map. % and the dual $W$-graph bases can be established. For each of the dual modules, the associated full $W$-graphs can be constructed.% among them, another pair of dual bases are obtained by using %the inversions of the relative Kazhdan-Lusztig polynomials. Our construction closely parallels that of Kazhdan and Lusztig~\cite{KL, Lusztig1, Lusztig2}, which can be regarded as the special case $J=\emptyset$. It also generalizes the work of Couillens~\cite{C}, Deodhar~\cite{Deodhar, Deodhar2}, and Douglass \cite{MD}, corresponding to the parabolic cases.