New proof demonstrates Howe duality, specifying O(k) representations in unitary modules of Mp(n,R), indicating relationships in representation theory.
We give a new proof of the Howe duality theorem for the reductive dual pair (Sp(n, R), O(k)) by using the isotropy representations for unitary lowest weight modules of the metaplectic group Mp(n, 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.
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Abe et al. (2009) studied this question.
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