Conjectures P1-P15 for Coxeter groups with complete graph

Abstract

We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph, using deceasing induction on a-values and a kind of decomposition formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a description of the left, right, and two-sided cells. In the appendix, we prove P1-P15 for right-angled Coxeter groups by the same methods.

Conjectures P1-P15 for hyperbolic Coxeter groups of rank 3

“…This paper is a subsequent work of [Xie19], where the second author proved P1-P15 for Coxeter groups with complete graph and right-angled Coxeter groups. The aim of this paper is to complete the proof of conjectures P1-P15 for all Coxeter groups of rank 3.…”

“…This paper takes similar methods. The main difficulty of this paper is proving some key properties stated in [Xie19]. In complete graph case, they can be proved easily.…”

“…Thus we focus on hyperbolic Coxeter groups of rank 3. We do not need to consider rank 3 Coxeter groups with complete graph and right-angled Coxeter groups by [Xie19]. Therefore, we only need to consider Coxeter groups (W, S) such that S = {r, s, t}, m rt = 2, 1 mrs + 1 mrs < 1 2 , and at least one of m rs , m st is not ∞.…”

“…The methods of [Xie19] are applying decreasing induction on a-functions and proving a kind of decomposition formula of some Kazhdan-Lusztig basis elements in the quotient algebra. Roughly speaking, the induction is proving P1-P15 for W ≥N under assuming P1-P15 for W >N (the set of elements of W with a-values bigger than N ).…”

“…But for our present cases, the situation becomes complicate, and all the sections 4-8 are devoted to prove them. Once we obtain these properties, we can repeat the procedures in [Xie19] to prove P1-P15. In fact we conjecture in section 3 that these properties should hold for all Coxeter groups such that finite parabolic groups are of at most rank 2 (Conjecture 3.1), which will imply P1-P15 (Theorem 3.2).…”