Lusztig's $a$-function in type $B_{n}$ in the asymptotic case

Abstract

In this paper, we study Lusztig's ${\mathbf{a}}$-function for a Coxeter group with unequal parameters. We determine that function explicitly in the "asymptotic case" in type $B_{n}$, where the left cells have been determined in terms of a generalized Robinson-Schensted correspondence by Bonnafé and the second author. As a consequence, we can show that all of Lusztig's conjectural properties (P1)--(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the "leading matrix coefficients" introduced by the first author. We also interprete the ideal structure defined by the two-sided cells in the associated Iwahori-Hecke algebra ${\mathcal{H}}_{n}$ in terms of the Dipper-James-Murphy basis of ${\mathcal{H}}_{n}$.