Weight multiplicities for affine Lie algebras of type Ar<SUP>(1)</SUP> and Kostka numbers

초록

In this paper, we prove the conjecture on the polynomial behavior of weight multiplicities for affine Lie algebras of type AM, which was originally due to Benkart and Kass. More precisely, we prove that the degree of the weight multiplicity function m(gimel) (mu) is equal to the depth associated with dominant integral weights gimel and mu, and compute the leading coefficient of m(gimel) (mu) explicitly in terms of a Kostka number. As applications, we verify other conjectures on the leading coefficient of m(gimel) (mu), and the polynomial behavior of weight multiplicities for classical Lie algebras of type A(r). (c) 2006 Elsevier Inc. All rights reserved.

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