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Journal of Lie Theory 12 (2002), No. 1, 081--112
Copyright Heldermann Verlag 2002

Mixed Models for Reductive Dual Pairs and Siegel Domains for Hermitian Symmetric Spaces

C. S. Leslie
Dept. of Computer Science, Columbia University, 1214 Amsterdam Avenue, New York, NY 10027-7003, U.S.A.


Let $(G,G^\prime)$ be the reductive dual pair $(Sp(n,\Bbb R), O(k))$ or $(U(p,q),U(k))$, and let $K$ be a maximal compact subgroup of the noncompact group $G$. Then for the representations $\pi$ of $\widetilde{G}$ which occur in the Howe duality correspondence for $(G, G^\prime)$, we construct explicit intertwining maps between mixed models of $\pi$ and spaces of holomorphic sections of vector bundles over the hermitian symmetric space $G/K$, where $G/K$ is embedded in its holomorphic tangent space as a type III Siegel domain. This result provides a link between the original construction of these representations using tube domain and type II Siegel domain realizations of $G/K$ and more recent constructions using the bounded domain realization of $G/K$.

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