Journal of Lie Theory

no. 4
Howe Duality for the Metaplectic Group Acting on Symplectic Spinor Valued Forms
Journal of Lie Theory, Volume 22 (2012) no. 4, pp. 1049-1063
\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 $\W$.
DOI: 10.5802/jolt.704
Classification: 17B10, 17B45, 22E46, 81R05
Keywords: Howe duality, symplectic spinors, Segal-Shale-Weil representation, Kostant spinor 
@article{JOLT_2012_22_4_a5,
     author = {S. Kr\'ysl},
     title = {Howe {Duality} for the {Metaplectic} {Group} {Acting} on {Symplectic} {Spinor} {Valued} {Forms}},
     journal = {Journal of Lie Theory},
     pages = {1049--1063},
     year = {2012},
     volume = {22},
     number = {4},
     doi = {10.5802/jolt.704},
     zbl = {1275.17015},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.704/}
}
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S. Krýsl. Howe Duality for the Metaplectic Group Acting on Symplectic Spinor Valued Forms. Journal of Lie Theory, Volume 22 (2012) no. 4, pp. 1049-1063. doi: 10.5802/jolt.704

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