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

Abstract

Let $λ: \tilde{G}\to G$ be the non-trivial double covering of the symplectic group $G=Sp(V,ω)$ of the symplectic vector space $(V,ω)$ by the metaplectic group $\tilde{G}=Mp(V,ω).$ In this case, $λ$ 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 $Mp(V,ω)$ and $osp(1|2).$