Extensions of “thickened” Verma modules of the Virasoro algebra

Let denote the Virasoro Lie algebra, its Cartan subalgebra, and the symmetric algebra on . In this paper we consider “thickened” Verma modules which are -bimodules satisfying where M(λ) is the usual Verma module with highest weight . We determine to be where φμ,λ is, up to a -algebra automorphism of , a product of irreducible factors of the determinant of the Shapovalov matrix. This result provides a conceptual explanation of the factorization of the Shapovalov determinant and implies that the inverse of the Shapovalov matrix has only simple poles.