Classification of Harish-Chandra Modules over the Higher Rank Virasoro Algebras

We classify the Harish-Chandra modules over the higher rank Virasoro and super-Virasoro algebras: It is proved that a Harish-Chandra module, i.e., an irreducible weight module with finite weight multiplicities, over a higher rank Virasoro or super-Virasoro algebra is a module of the intermediate series. As an application, it is also proved that an indecomposable weight module with finite weight multiplicities over a generalized Witt algebra is a uniformly bounded module (i.e., a module with weight multiplicities uniformly bounded), and all nonzero weights have the same multiplicity, as long as the generalized Witt algebra satisfies one minor condition.