Fully commutative elements and Kazhdan–Lusztig cells in the finite and affine Coxeter groups

Abstract

The main goal of the paper is to show that the fully commutative elements in the affine Coxeter group C ~ n \widetilde {C}_{n} form a union of two-sided cells. Then we completely answer the question of when the fully commutative elements of W W form or do not form a union of two-sided cells in the case where W W is either a finite or an affine Coxeter group.