Extensions of “thickened” Verma modules of the Virasoro algebra

Abstract

AbstractLet 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 Ext1(M(μ),M(λ)) to be S(h)/φμ,λS(h) where φμ,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