WEIGHT MULTIPLICITY POLYNOMIALS FOR AFFINE KAC-MOODY ALGEBRAS OF TYPEA(R)((1))

For the affine Kac-Moody algebras X(tau)((1)) it has been conjectured by Benkart and Kass that for fixed dominant weights lambda, mu, the mu ltiplicity of the weight mu in the irreducible X(r)((1))-module L(lamb da) of highest weight lambda is a polynomial in r which depends on the type X of the algebra. In this paper we provide a precise conjecture for the degree of that polynomial for the algebras A(r)((1)). To offer evidence for this conjecture we prove it for all dominant weights lam bda and all weights mu of depth less than or equal to 2 by explicitly exhibiting the polynomials as expressions involving Kostka numbers.