Lusztig's $a$-function in type $B_n$ in the asymptotic case

In this paper, we study Lusztig's $a$-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 Bonnaf\'e and the second author. As a consequence, we can also 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 $\bH_n$ in terms of the Dipper--james--Murphy basis of $\bH_n$.