Coxeter Elements and Kazhdan-lusztig Cells

By the correspondence between Coxeter elements of a Coxeter system (W, S, Γ) and the acyclic orientations of the Coxeter graph Γ, we study some properties of elements in the set C0(W ). We show that when W is of finite, affine or hyperbolic type, any w ∈ C0(W ) satisfies w ∼ LR wJ with `(wJ ) = |J | = m(w) for some J ⊂ S. Now assume that W is of finite or affine type. We give an explicit description for all the distinguished involutions d of W with d ∼ L w for some w ∈ E(W ), which verifies a conjecture proposed in [12, Conjecture 8.10] in our case. We show that any left cell of W containing some element of C0(W ) is left-connected, which verifies a conjecture of Lusztig in [3] in our case.