Abstract
This paper studies a class of simple integrable modules for an affine Lie algebras which are closely related to the finite-dimensional modules studied by V. Chari and A. Pressley, except that the Euler element is assumed to act. They are infinite-dimensional; but are shown to have finite-dimensional weight spaces. It is conjectured that any simple integrable module with a zero weight space belongs to this class and their classification is given. The main interest in studying such modules is that they may occur in the endomorphism rings of highest weight modules whilst those of Chari and Pressley in general do not. Their character theory is also more complicated.
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