  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.
