Abstract
Let $\g$ be an arbitrary Kac-Moody algebra with a Cartan subalgebra $\h$. In this paper, we determine the category of $\g$-modules that are free $U(\h)$-modules of rank 1. More precisely, this category of $\g$-modules is not empty if and only if $\g$ is of type $A_l$ or $C_l$ for any positive integer $l$.
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