Kazhdan-Lusztig cells in some weighted Coxeter groups

Let (W, S) be a Coxeter group with S = I aS"J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W (I) of W with w0 the longest element of W (I) . We describe all the left cells and two-sided cells of the weighted Coxeter group (W,S,L) that have non-empty intersection with W (J) , where the weight function L of (W,S) is in one of the following cases: (i) max{L(s) | s a J} < min{L(t) | t a I}; (ii) min{L(s) | s a J} a (c) 3/4 L(w0); (iii) there exists some t a I satisfying L(t) < L(s) for any s a I - {t} and L takes a constant value L (J) on J with L (J) in some subintervals of [1,L(w0) - 1]. The results in the case (iii) are obtained under a certain assumption on (W,W (I) ).