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

TL;DR

This paper establishes a Howe type duality for the symplectic Lie algebra acting on symplectic spinor-valued forms, revealing the algebraic structure and decomposition under joint symmetries.

Contribution

It proves that the endomorphism algebra is generated by an $ ext{osp}(1|2)$ representation and projections, providing a new duality framework for symplectic spinor modules.

Findings

Endomorphism algebra generated by $ ext{osp}(1|2)$ representation and projections

Decomposition of the module under joint $ ext{sp}$ and $ ext{osp}$ actions

Establishment of Howe type duality for symplectic group

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 $\mathbb{W}=\bigwedge^{\bullet}(\mathbb{V}^*)^{\mathbb{C}}\otimes \mathbb{S}$ 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 $\mathbb{W}.$