Characters of Highest Weight Modules over Affine Lie Algebras Are Meromorphic Functions

We show that the characters of all highest weight modules over an affine Lie algebra with the highest weight away from the critical hyperplane are meromorphic functions in the positive half of the Cartan subalgebra, their singularities being at most simple poles at zeros of real roots. We obtain some information about these singularities. 0. Introduction 0.0.1. Let g be a simple finite-dimensional Lie algebra over C, and let g = g[t, t] ⊕ CK ⊕ CD be the associated non-twisted affine Kac-Moody algebra [K]. Recall that the commutation relations on g are: [at, bt] = [a, b]t +mδm,−n(a|b)K, [D, at ] = mat, [K, g] = 0, where a, b ∈ g, and (−,−) is a non-degenerate invariant symmetric bilinear form on g. Choosing a Cartan subalgebra h of g, and a triangular decomposition g = n−⊕h⊕n+, we have the associated Cartan subalgebra h = h⊕CK⊕CD and the triangular decomposition g = n− ⊕ h ⊕ n+, where n± = n± + g[t ]t. Let ∆ ⊃ ∆+ ⊃ Π = {α0, α1, . . . , αn} be the multiset of all roots, the multiset of positive roots and the set of simple roots of g, respectively. 0.1. Let M(λ) be the Verma module over g with highest weight λ ∈ h. Any quotient V (λ) of M(λ), called a highest weight module, has weight space decomposition V (λ) = ⊕μ∈h∗Vμ(λ), where dimVμ(λ) < ∞, hence we can define its character chV (λ)(h) = ∑