Howe type duality for metaplectic group acting on symplectic spinor valued forms
نویسنده
چکیده
Let λ : G̃ → 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 G̃ = Mp(V, ω). In this case, λ is also a representation of G̃ on the vector space V and thus, it gives rise to the representation of G̃ on the space of exterior forms ∧• 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 G̃. We prove that the associative commutant algebra EndG̃( ∧• V ⊗ S) of the metaplectic group G̃ 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 osp(1|2). Also the space ∧• V ⊗ S is decomposed wr. to the joint action of Mp(V, ω) and osp(1|2). Math. Subj. Class.: 22E46, 22E47, 22E45, 70G45, 81S10
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