Fully commutative elements and Kazhdan{Lusztig cells in the and ane Coxeter groups

Abstract. Let W be an irreducible nite or ane Coxeter group and let Wc be the set of fully commutative elements in W. We prove that the set Wc is closed under the Kazhdan-Lusztig's preorder> LR if and only if Wc is a union of two-sided cells of W. Introduction. Let W = (W; S) be a Coxeter group with S the distinguished generator set. For any J = fs1;:::; srg S, denote by wJ or ws1s2:::sr the longest element in the subgroup WJ of W generated by J. The fully commutative elements of W were dened by Stembridge: w 2 W is fully commutative, if any two reduced expressions of w can be transforme