Geodesic
Category $\mathcal {O}$ for truncated current Lie algebras
Canadian journal of mathematics, Tome 76 (2024) no. 5, pp. 1795-1821

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DOI

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.
DOI : 10.4153/S0008414X23000664
Mots-clés : Lie algebras, representation theory, category O, truncated current Lie algebras 
Chaffe, Matthew; Topley, Lewis. Category $\mathcal {O}$ for truncated current Lie algebras. Canadian journal of mathematics, Tome 76 (2024) no. 5, pp. 1795-1821. doi: 10.4153/S0008414X23000664
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     title = {Category $\mathcal {O}$ for truncated current {Lie} algebras},
     journal = {Canadian journal of mathematics},
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     year = {2024},
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