Category $\mathcal{O}$ for truncated current Lie algebras

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