A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras (Q855118)

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.