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 either a module of the intermediate series, or a finitely-dense module. As an application, it is also proved that an indecomposable weight module with finite weight multiplicities over a generalized Witt algebra is either a uniformly bounded module (i.e., a module with weight multiplicities uniformly bounded) with all nonzero weights having the same multiplicity, or a finitely-dense module, as long as the generalized Witt algebra satisfies one minor condition.This work is supported by a NSF grant 10171064 of China and two grants ``Excellent Young Teacher Program' and ``Trans-Century Training Programme Foundation for the Talents' from Ministry of Education of China.