# The category of weight modules for symplectic oscillator Lie algebras

## TL;DR

This paper classifies simple weight modules with finite-dimensional weight spaces over symplectic oscillator Lie algebras, establishing an equivalence with modules over symplectic Lie algebras when a certain parameter is non-zero.

## Contribution

It provides a classification of simple weight modules for symplectic oscillator Lie algebras and links their category to that of symplectic Lie algebras via an equivalence.

## Key findings

- Equivalence between categories of modules when z ≠ 0
- Classification of simple weight modules with finite-dimensional weight spaces
- Use of localization and highest weight techniques

## Abstract

The rank $n$ symplectic oscillator Lie algebra $\mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $\mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spaces over $\mathfrak{g}_n$. When $\dot z\neq 0$, it is shown that there is an equivalence between the full subcategory $\mathcal{O}_{\mathfrak{g}_n}[\dot z]$ of the BGG category $\mathcal{O}_{\mathfrak{g}_n}$ for $\mathfrak{g}_n$ and the BGG category $\mathcal{O}_{\mathfrak{sp}_{2n}}$ for $\mathfrak{sp}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over $\mathfrak{g}_n$ with finite-dimensional weight spaces.