The Virasoro algebra, which can be realized as a central extension of (complex) polynomial vector fields on the unit circle, plays a key role in the representation theory of affine Lie algebras, as it acts on almost every highest weight module for the affine Lie algebra. This remarkable phenomenon eventually led to constructing the affine-Virasoro algebra, which is a semi-direct product of the affine Lie algebra and the Virasoro algebra with a common extension. The representation theory of the affine-Virasoro algebra has been studied extensively and is an extremely well-developed classical object.
In this talk, we shall consider a natural higher-dimensional analogue of the affine-Virasoro algebra, popularly known as the full toroidal Lie algebra in the literature and henceforth classify the irreducible Harish-Chandra modules over this Lie algebra. As a by-product, we also obtain the classification of all possible irreducible Harish-Chandra modules over the higher-dimensional Virasoro algebra, thereby proving Eswara Rao’s conjecture (conjectured in 2004). These directly generalize the well-known result of O. Mathieu for the classical Virasoro algebra and also the recent work of Billig–Futorny for the higher rank Witt algebra.