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

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

2008-05-19

arxiv.org/abs/0805.2904v1

Mathematics

Representation Theory

30 pages, 1 figure

Scientific paper

Let $\lambda: \tilde{G}\to G$ be the non-trivial double covering of the symplectic group $G=Sp(V,\omega)$ of the symplectic vector space $(V,\omega)$ by the metaplectic group $\tilde{G}=Mp(V,\omega).$ In this case, $\lambda$ is also a representation of $\tilde{G}$ on the vector space $V$ and thus, it gives rise to the representation of $\tilde{G}$ on the space of exterior forms $\bigwedge^{\bullet}V^*$ by taking wedge products. Let $S$ be the minimal globalization of the Harish-Chandra module of the complex Segal-Shale-Weil representation of the metaplectic group $\tilde{G}.$ We prove that the associative commutant algebra $\hbox{End}_{\tilde{G}}(\bigwedge^{\bullet}V^*\otimes S)$ of the metaplectic group $\tilde{G}$ acting on the $S$-valued exterior forms is generated by certain representation of the super ortho-symplectic Lie algebra $osp(1|2)$ and two distinguished operators. This establishes a Howe type duality between the metaplectic group and the super Lie algebra $\mathfrak{osp}(1|2).$ Also the space $\bigwedge^{\bullet}V^*\otimes S$ is decomposed wr. to the joint action of $Mp(V,\omega)$ and $osp(1|2).$

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