Category for truncated current Lie algebras

In this paper, we study an analogue of the Bernstein–Gelfand–Gelfand category O for truncated current Lie algebras gn attached to a complex semisimple Lie algebra. This category admits Verma modules and simple modules, each parametrized by the dual space of the truncated currents on a choice of Cartan subalgebra in g. Our main result describes an inductive procedure for computing composition multiplicities of simples inside Vermas for gn, in terms of similar composition multiplicities for ln−1 where l is a Levi subalgebra. As a consequence, these numbers are expressed as integral linear combinations of Kazhdan–Lusztig polynomials evaluated at 1. This generalizes recent work of the first author, where the case n=1 was treated.