Classification of Irreducible Nonzero Level Modules with Finite-Dimensional Weight Spaces for Affine Lie Algebras

We classify the irreducible weight affine Lie algebra modules with finite-dimensional weight spaces on which the central element acts nontrivially. In particular, we show that any such module is a quotient of a generalized Verma module. The classification of such irreducible modules is reduced to the classification of irreducible torsion-free modules for algebras of type An and Cn which is due to O. Mathieu (forthcoming).