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Category
$\mathcal {O}$ for truncated current Lie algebras
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 5 / October 2024
- Published online by Cambridge University Press:
- 19 October 2023, pp. 1795-1821
- Print publication:
- October 2024
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- Article
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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 parametrized 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 generalizes recent work of the first author, where the case 
$n=1$ was treated.
