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.


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