  We study the restrictions of simple modules of Ariki-Koike algebras
$\H_m(\v)$ with set of parameters $\v= (\zeta;\zeta^{v_0},...
,\zeta^{v_{l-1}})$, where $\zeta$ is an $n$th root of unity, to their
subalgebras $\H_{m-j}(\v)$. Using a theorem of Ariki and the crystal basis
theory of Kashiwara, we relate this problem to the calculation of tensor
product multiplicities of highest weight irreducible representations of the
affine Lie algebra $A_{n-1}^{(1)}$. These multiplicities have a combinatorial
description in terms of higher level paths or highest-lift multipartitions.
This enables us to solve the Jantzen-Seitz problem for Ariki-Koike algebras,
that is, to determine which irreducible representations of $\H_m(\v)$ restrict
to irreducible representations of $\H_{m-1}(\v)$. From a combinatorial point of
view, this problem is identical to that of computing the tensor product of an
$A_{n-1}^{(1)}$-module of level $l$ and one of level 1. We also consider
natural generalisations of the Jantzen-Seitz problem corresponding to the
product of a level $l$ module by a level $l'>1$ module, and from the
commutativity of tensor products, we deduce a remarkable symmetry between the
generalised Jantzen-Seitz conditions and the sets of parameters of the
Ariki-Koike algebras.
