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I know that an irreducible infinite-dimensional weight module over the Virasoro algebra in which it has a non-zero finite-dimensional weight space, then all its weight spaces have finite dimension. Does this result extend to finite-dimensional simple Lie algebras over complex numbers?

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    $\begingroup$ Ptonsbly the question needs a more precise formulation, to express "this result" clearly. Then the answer is likely to be "yes". $\endgroup$ – Jim Humphreys Jul 12 at 18:51

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