Weight multiplicities for affine Lie algebras of type and Kostka numbers

In this paper, we prove the conjecture on the polynomial behavior of weight multiplicities for affine Lie algebras of type , which was originally due to Benkart and Kass. More precisely, we prove that the degree of the weight multiplicity function mλ(μ) is equal to the depth associated with dominant integral weights λ and μ, and compute the leading coefficient of mλ(μ) explicitly in terms of a Kostka number. As applications, we verify other conjectures on the leading coefficient of mλ(μ), and the polynomial behavior of weight multiplicities for classical Lie algebras of type Ar.