  We study natural quantizations of branching coefficients corresponding to the
restrictions of the classical Lie groups to their Levi subgroups. We show that
they admit a stable limit which can be regarded as a $q$-analogue of a tensor
product multiplicity. According to a conjecture by Shimozono, the stable
one-dimensional sum for nonexceptional affine crystals are expected to occur as
special cases of these $q$-analogues.
