The Shapovalov determinant and translation functors for the Virasoro algebra

The Virasoro algebra Vir possesses a triangular decomposition; this decomposition allows us to define the category of modules O and to obtain a decomposition of O by blocks. We use the results of Feĭgin and Fuchs [B.L. Feĭgin, D.B. Fuchs, Representations of the Virasoro algebra, in: Representation of Lie Groups and Related Topics, in: Adv. Stud. Contemp. Math., vol. 7, Gordon and Breach, New York, 1990, pp. 465–554] to describe the blocks of O as group orbits. Also, following the work of Jantzen [J.C. Jantzen, Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, thesis, 1973] in the semisimple Lie algebra setting, we use the Shapovalov determinant to study the translation of Verma modules between blocks. In particular, for the translation (M(μ)⊗L(δ))[ν] of M(μ) to the [ν]-block, we compute certain error terms (μ+δ+j∈[ν]) and give a representation-theoretic interpretation for .