The polynomial behavior of weight multiplicities for the affine Kac-Moody algebras \(A_r^{(1)}\) (Q2721346)

In an earlier paper [Compos. Math. 104, No. 2, 153-187 (1996; Zbl 0862.17016)], \textit{G. M. Benkart, S.-J. Kang} and \textit{K. C. Misra} showed that the multiplicity of a weight in an irreducible highest weight module of the affine Kac-Moody algebra \(A_r^{(1)}\) is given by a polynomial in the rank \(r\), for weights of depth not more than 2. However, the techniques used there did not generalize for greater depths. NEWLINENEWLINENEWLINEIn this paper, using a different approach, the authors show that the result extends to arbitrary dominant weights of irreducible highest weight representations for algebras of sufficiently large rank. They give some explicit examples of these polynomials and show that the degree of the polynomial is bounded by the depth of the weight. In the examples given in the tables, this bound is always attained. NEWLINENEWLINENEWLINEA paper by \textit{G. M. Benkart, S.-J. Kang, H. Lee} and \textit{D. Shin} [Contemp. Math. 248, 1-29 (1999; Zbl 0948.17011)] treats the remaining classical affine cases, achieving parallel results. The fact that the Dynkin diagram of \(A_r^{(1)}\) is a cycle necessitated its separate treatment.