Left cells containing a fully commutative element

Let W be a finite or an affine Coxeter group and W c 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 w L ) in L ∩ W c such that any z ∈ L has the form z = xw L with ℓ ( z ) = ℓ ( x ) + ℓ ( w L ) for some x ∈ W . This implies that L is left connected, verifying a conjecture of Lusztig in our case.