Weight multiplicity polynomials for affne Kac-Moody algebras of type $A_n^{(1)}

Abstract

For the affine Kac--Moody algebras X_r^{(1)} it has been conjectured by Benkart and Kass that for fixe d dominant weights \lambda,\mu, the multiplicity of the weight \mu in the irreducible X_r^{(1)}-module L(\lambda) of highest wei ght \lambda is a polynomial in r which depends on the type X of the alg ebra. 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 p rove it for all dominant weights \lambda and all weights \mu of depth \leqslant 2 by explicitly exhibiting the polynomials as expressions involving Kostka numbers.