  Let X be the Dynkin diagram of a symmetrizable Kac-Moody algebra, and X_0 a
subgraph with all vertices of degree 1 or 2. Using the crystal structure on the
components of quiver varieties for X, we show that if we expand X by extending
X_0, the branching multiplicities and tensor product multiplicities stabilize,
provided the weights involved satisfy a condition which we call ``depth'' and
are supported outside $X_0$. This extends a theorem of Kleber and Viswanath.
  Furthermore, we show that the weight multiplicities of such representations
are polynomial in the length of X_0, generalizing the same result for A_\ell by
Benkart, et al.
