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Journal of Lie Theory 22 (2012), No. 4, 1049--1063
Copyright Heldermann Verlag 2012

Howe Duality for the Metaplectic Group Acting on Symplectic Spinor Valued Forms

Svatopluk Krýsl
Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Praha 8 - Karlín 186 75, Czechia


\def\g{{\frak g}} \def\o{{\frak o}} \def\p{{\frak p}} \def\s{{\frak s}} \def\C{{\Bbb C}} \def\SS{{\Bbb S}} \def\V{{\Bbb V}} \def\W{{\Bbb W}} Let $\SS$ denote the oscillatory module over the complex symplectic Lie algebra $\g= \s\p(\V^\C,\omega)$. Consider the $\g$-module $\W=\bigwedge^{\bullet}(\V^*)^\C\otimes\SS$ of forms with values in the oscillatory module. We prove that the associative commutant algebra $\hbox{\rm End}_\g(\W)$ is generated by the image of a certain representation of the ortho-symplectic Lie super algebra $\o\s\p(1|2)$ and two distinguished projection operators. The space $\W$ is then decomposed with respect to the joint action of $\g$ and $\o\s\p(1|2)$. This establishes a Howe type duality for $\s\p(\V^\C,\omega)$ acting on $\W$.

Keywords: Howe duality, symplectic spinors, Segal-Shale-Weil representation, Kostant spinor.

MSC: 17B10, 17B45, 22E46, 81R05

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