Journal of Lie Theory

no. 1
Mixed Models for Reductive Dual Pairs and Siegel Domains for Hermitian Symmetric Spaces
Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 81-112
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 $G/K$.
DOI: 10.5802/jolt.254
Classification: 22E45, 32M15
Keywords: reductive dual pair, intertwining map, Siegel domain, Hermitian symmetric spaces 
@article{JOLT_2002_12_1_a5,
     author = {C. S. Leslie},
     title = {Mixed {Models} for {Reductive} {Dual} {Pairs} and {Siegel} {Domains} for {Hermitian} {Symmetric} {Spaces}},
     journal = {Journal of Lie Theory},
     pages = {81--112},
     year = {2002},
     volume = {12},
     number = {1},
     doi = {10.5802/jolt.254},
     zbl = {0991.22013},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.254/}
}
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C. S. Leslie. Mixed Models for Reductive Dual Pairs and Siegel Domains for Hermitian Symmetric Spaces. Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 81-112. doi: 10.5802/jolt.254

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