Category O for Takiff Lie algebras

TL;DR

This paper investigates the structure of category O for Takiff Lie algebras, decomposing it into subcategories, establishing equivalences, and expressing composition multiplicities via Kazhdan-Lusztig polynomials.

Contribution

It introduces a decomposition of category O for Takiff Lie algebras and relates composition multiplicities to Kazhdan-Lusztig polynomials, extending known results.

Findings

Decomposition of category O into subcategories.

Equivalence of subcategories via induction and twisting functors.

Composition multiplicities expressed through Kazhdan-Lusztig polynomials.

Abstract

We study category $\mathcal{O}$ for Takiff Lie algebras $\mathfrak{g} \otimes \mathbb{C}[\epsilon]/(\epsilon^2)$ where $\mathfrak{g}$ is the Lie algebra of a reductive algebraic group over $\mathbb{C}$. We decompose this category as a direct sum of certain subcategories and use an analogue of parabolic induction functors and twisting functors for BGG category $\mathcal{O}$ to prove equivalences between these subcategories. We then use these equivalences to compute the composition multiplicities of the simple modules in the Verma modules in terms of composition multiplicities in the BGG category $\mathcal{O}$ for reductive subalgebras of $\mathfrak{g}$. We conclude that the composition multiplicities are given in terms of the Kazhdan-Lusztig polynomials.