Classification of highest weight modules with meromorphic modular normalized characters

Establish that the only irreducible highest weight modules L(A) over an affine Lie algebra g whose normalized characters (as defined via the Kac–Wakimoto character formula) are meromorphic modular functions on the domain {h ∈ h | Re δ(h) > 0} are precisely the admissible modules, i.e., those with weights A satisfying the integrality conditions (A + ρ, α) ∈ ℤ for all positive real coroots α and whose set of integral real coroots equals the full set of positive real coroots.

Background

Admissible weights A for an affine Lie algebra g are defined by explicit integrality and maximality conditions on real coroots, and their characters satisfy a Kac–Wakimoto formula. Using this formula, normalized characters of admissible modules are known to be meromorphic modular functions on a half-plane.

The authors explicitly state a conjecture that admissible modules exhaust all irreducible highest weight modules with this meromorphic modularity property, providing a proposed classification criterion.