The category of weight modules for symplectic oscillator Lie algebras

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.