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A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras. (English) Zbl 1129.22005
The symplectic group \({\text{Sp}}(2n,{\mathbb{R}})\) may be realized as acting on the Siegel ball, a bounded domain \({\mathcal{D}}_n\) in the space of \(2n\times 2n\) complex matrices whilst the Heisenberg group can be realized as the standard definite hyperquadric in \({\mathbb{C}}^{n+1}\). In this article, the author combines these two realizations to write the semidirect product of these groups as acting on the product \({\mathbb{C}}^n\times{\mathcal{D}}_n\). This action preserves a Kähler metric and can be used to construct unitary representations of the group.

MSC:
22E30 Analysis on real and complex Lie groups
32M10 Homogeneous complex manifolds
22E70 Applications of Lie groups to the sciences; explicit representations
81R30 Coherent states
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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