Extensions of “thickened” Verma modules of the Virasoro algebra

Abstract

Let g denote the Virasoro Lie algebra, h its Cartan subalgebra, and S( h) the symmetric algebra on h . In this paper we consider “thickened” Verma modules M(λ) which are (U( g),S( h)) -bimodules satisfying M(λ)⊗ S( h) C≅M(λ) where M( λ) is the usual Verma module with highest weight λ∈ h ∗ . We determine Ext 1( M(μ), M(λ)) to be S( h)/φ μ,λS( h) where φ μ, λ is, up to a C -algebra automorphism of S( h) , 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.