The Polynomial Behavior of Weight Multiplicities for the Affine   Kac-Moody Algebras $A^{(1)}_r$
  We prove that the multiplicity of an arbitrary dominant weight for an integrable highest weight representation of the affine Kac-Moody algebra $A_{r}^{(1)}$ is a polynomial in the rank $r$. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks. 
Georgia Benkart, Seok-Jin Kang, Hyeonmi Lee, Kailash C. Misra, Dong-Uy   Shin