Howe type duality for the metaplectic group acting on symplectic spinor valued forms

Abstract

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 $\mathbb{W}={\bigwedge }^{\bullet }({\mathbb{V}}^{\ast }{)}^{\mathbb{C}}\otimes \mathbb{S}$ of exterior forms with values in the oscillatory module. We prove that the associative algebra ${End}_{\mathfrak{g}}(\mathbb{W})$ 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 ${\bigwedge }^{\bullet }({\mathbb{V}}^{\ast }{)}^{\mathbb{C}}\otimes \mathbb{S}$ 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 $\mathbb{W}.$

16 pages