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

Abstract

In this paper, we classify the irreducible Harish-Chandra modules over the full toroidal Lie algebra, which is a natural higher-dimensional analogue of the affine-Virasoro algebra. In particular, we complete the classification of irreducible bounded modules, which were studied by Billig for non-zero level modules [Int. Math. Res. Not. 2006]. As a by-product, we also obtain the classification of irreducible Harish-Chandra modules over the higher-dimensional Virasoro algebra, which was introduced by Rao-Moody [Comm. Math. Phys. 1994], thereby generalizing the well-known result of O. Mathieu [Invent. Math. 1992] for the classical Virasoro algebra. More precisely, we show that any irreducible Harish-Chandra module over the higher-dimensional Virasoro algebra turns out to be either a quotient of a module of tensor fields on a torus or a highest weight type module up to a twist of an automorphism, as conjectured by Eswara Rao in 2004.