  In this paper, we study the Kazhdan--Lusztig cells of a Coxeter group $W$ in
a ``relative'' setting, with respect to a parabolic subgroup $W_I \subseteq W$.
This relies on a factorization of the Kazhdan--Lusztig basis $\{C_w\}$ of the
corresponding (multi-parameter) Iwahori--Hecke algebra with respect to $W_I$.
We obtain two applications to the ``asymptotic case'' in type $B_n$, as
introduced by Bonnaf\'e--Iancu: we show that $\{C_w\}$ is a ``cellular basis''
in the sense of Graham--Lehrer, and we construct an analogue of Lusztig's
canonical isomorphism from the Iwahori--Hecke algebra to the group algebra of
the underlying Weyl group of type $B_n$.
