Category $\mathcal{O}$ for truncated current Lie algebras
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2023
- DOI:
- 10.48550/arXiv.2304.09561
- arXiv:
- arXiv:2304.09561
- Bibcode:
- 2023arXiv230409561C
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Rings and Algebras;
- 17B10;
- 17B20;
- 17B45
- E-Print:
- 19 pages