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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.
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V. Bekkert, G. Benkart, V. Futorny, I. Kashuba; New Irreducible Modules for Heisenberg and Affine Lie Algebras. Preprint arxive:1107.0893
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