Left cells containing a fully commutative element

Let W be a finite or an affine Coxeter group and Wc the set of all the fully commutative elements in W. For any left cell L of W containing some fully commutative element, our main result of the paper is to prove that there exists a unique element (say wL) in L∩Wc such that any z∈L has the form z=xwL with ℓ(z)=ℓ(x)+ℓ(wL) for some x∈W. This implies that L is left connected, verifying a conjecture of Lusztig in our case.