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In this paper, we completely classify the irreducible Harish-Chandra modulesover the full toroidal Lie algebra, which is a natural higher-dimensionalanalogue of the affine-Virasoro algebra. As a by-product, we also obtain theclassification of all possible irreducible Harish-Chandra modules over thehigher-dimensional Virasoro algebra. These directly generalize the well-knownresult of O. Mathieu [Invent. Math. 1992] for the classical Virasoro algebraand also the recent work of Billig-Futorny [J. Reine Angew. Math. 2016] for thehigher rank Witt algebra. In particular, we show that any irreducibleHarish-Chandra module over the higher-dimensional Virasoro algebra turns out tobe either a quotient of a module of tensor fields on a torus or a highestweight type module up to a twist of an automorphism, as conjectured by EswaraRao in 2004.

Classification of irreducible Harish-Chandra modules over full toroidal Lie algebras and higher-dimensional Virasoro algebras