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 with ℓ(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 for some w E(W), which verifies a conjecture proposed in [Jian-yi Shi, Adv. Sci. China, Math.3 (1990), 79–98, 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 [Ryoshi Hotta (Ed.), in “Problems from the Conference on Algebraic Groups and Representations held at Katata, August 29–September 3, 1983”] in our case.