The boundedness of a weighted Coxeter group with non-3-edge-labeling graph

Abstract

Let [Formula: see text] be a weighted Coxeter group such that the order [Formula: see text] of the product [Formula: see text] is not 3 for any [Formula: see text] and that [Formula: see text], where [Formula: see text] is the longest element in the parabolic subgroup [Formula: see text] of [Formula: see text] generated by [Formula: see text]. We prove that [Formula: see text] is bounded with [Formula: see text] an upper bound in the sense of Lusztig in Sec. 13.2 of [Hecke Algebras with Unequal Parameters, arXiv:math/0208154 v2 [math.RT] 10 Jun 2014], verifying a conjecture of Lusztig in our case (see Conjecture 13.4 in loc. cite).