COXETER ELEMENTS AND KAZHDAN-LUSZTIG CELLS Jian-yi Shi Fakultät für Mathematik Universität

Abstract. 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. Introduction. Let (W,S,Γ) be a Coxeter system: a Coxeter groupW with S the distinguished generator set and Γ the Coxeter graph. We always assume S finite in the present paper. A Coxete