A non-recursive criterion for weights of a highest weight module for an affine Lie algebra

Abstract

Let $Λ$ be a dominant integral weight of level $k$ for the affine Lie algebra $\mathfrak g$ and let $α$ be a non-negative integral combination of simple roots. We address the question of whether the weight $η=Λ-α$ lies in the set $P(Λ)$ of weights in the irreducible highest-weight module with highest weight $Λ$. We give a non-recursive criterion in terms of the coefficients of $α$ modulo an integral lattice $kM$, where $M$ is the lattice parameterizing the abelian normal subgroup $T$ of the Weyl group. The criterion requires the preliminary computation of a set no larger than the fundamental region for $kM$, and we show how this set can be efficiently calculated.