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

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

2012-02-12

arxiv.org/abs/1202.2541v1

Mathematics

Representation Theory

16 pages

Scientific paper

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 $\mathbb{W}.$

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