  Let $\mathbb{S}$ denote the oscillatory module over the complex symplectic
Lie algebra $\mathfrak{g}= \mathfrak{sp}(\mathbb{V}^{\mathbb{C}},\omega).$
Consider the $\mathfrak{g}$-module
$\mathbb{W}=\bigwedge^{\bullet}(\mathbb{V}^*)^{\mathbb{C}}\otimes \mathbb{S}$
of exterior forms with values in the oscillatory module. We prove that the
associative algebra $\hbox{End}_{\mathfrak{g}}(\mathbb{W})$ is generated by the
image of a certain representation of the ortho-symplectic Lie super algebra
$\mathfrak{osp}(1|2)$ and two distinguished projection operators. The space
$\bigwedge^{\bullet}(\mathbb{V}^*)^{\mathbb{C}}\otimes \mathbb{S}$ is
decomposed with respect to the joint action of $\mathfrak{g}$ and
$\mathfrak{osp}(1|2).$ This establishes a Howe type duality for
$\mathfrak{sp}(\mathbb{V}^{\mathbb{C}},\omega)$ acting on $\mathbb{W}.$
