Conjectures P1-P15 for hyperbolic Coxeter groups of rank 3

Abstract

We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3 with any positive weight function. Combined with Guilhot and Parkinson's works on affine Weyl groups of type B2 , G2 , this completes the proof of P1-P15 for all rank 3 Coxeter groups. Our proof enables us to give a description of the a-function and right cells. At last, we determine when two different two-sided cells have the same a-value.