  We prove that for simple complex finite dimensional Lie algebras, affine
Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro
algebra, simple highest weight modules are characterized by the property that
all positive root elements act on these modules locally nilpotently. We also
show that this is not the case for higher rank Virasoro and for Heisenberg
algebras.
