Classification of simple weight Virasoro modules with a finite-dimensional weight space

We show that the support of a simple weight module over the Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite dimensional. As a corollary we obtain that every simple weight module over the Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module (and hence is either a simple highest or lowest weight module or a simple module from the intermediate series). This implies positive answers to two conjectures about simple pointed and simple mixed modules over the Virasoro algebra. 1 Description of the results The Virasoro algebra V over an algebraically closed field, k, of characteristic zero has a basis, consisting of a central element, c, and elements ei, i ∈ Z, with the Lie bracket defined for the basis elements as follows: [ei, ej] = (j − i)ei+j + δi,−j i − i 12 c. The linear span of c and e0 is called the Cartan subalgebra H of V and an H-diagonalizable V-module is usually called a weight module. If, additionally, all weight spaces of a weight V-module are finite-dimensional, the module is called a Harish-Chandra module, see for example [M]. All simple HarishChandra modules were classified in [MP1, MP2, M] and are exhausted by simple highest weight modules, simple lowest weight modules and simple modules from the so-called intermediate series (see e.g. [M] for definitions).