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Journal of Lie Theory 19 (2009), No. 4, 671--683
Copyright Heldermann Verlag 2009

A Note on Howe Duality Correspondence and Isotropy Representations for Unitary Lowest Weight Modules of Mp(n,R)

Noriyuki Abe
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153--8914, Japan

Hiroshi Yamashita
Department of Mathematics, Faculty of Science, Hokkaido University, N10 W8 Kita-ku, Sapporo 060-0810, Japan


We give a new proof of the Howe duality theorem for the reductive dual pair $({\rm Sp}(n,\mathbb{R}), {\rm O}(k))$ by using the isotropy representations for unitary lowest weight modules of the metaplectic group ${\rm Mp}(n,\mathbb{R})$. The irreducible representations of $O(k)$ appearing in the Howe duality correspondence are specified explicitly by means of the branching rule of the representations of O$(k)$ restricted to orthogonal groups of smaller size.

Keywords: Metaplectic group, lowest weight module, Howe duality theorem, branching rule, Harish Chandra modules.

MSC: 17B10, 22E45, 22E46

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