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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