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

S. Krýsl

Journal of Lie theory, Tome 22 (2012) no. 4, pp. 1049-1063

Résumé

\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 $\W$.

Détail

Classification : 17B10, 17B45, 22E46, 81R05
Mots-clés : Howe duality, symplectic spinors, Segal-Shale-Weil representation, Kostant spinor