On the geometry of Siegel-Jacobi domains (Q2891196)

The \textit{Jacobi group} \(G^J\) is the semidirect product of a Zariski connected semisimple real algebraic group of Hermitian type \(G^s\) and the Heisenberg group \(H(V)\) associated with the symplectic \(\mathbb{R}\)-space \(V\). The Hermitian symmetric domain associated to \(G^s\) is \(\mathcal{D}=G^s/K^s\), where \(K^s\) is a maximal compact subgroup. The \textit{Jacobi-Siegel domain} associated to the Jacobi group \(G^J\) is \(\mathcal{D}^J=\mathcal{D}\times \mathbb{C}^N\), where dim \(V=2N\). The authors study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. The \textit{Siegel disk} \(\mathcal{D}_n\) of degree \(n\) consists of all the symmetric matrices \(W\in M_n(\mathbb{C})\) with \(I_n-W\overline{W}>0\). Explicit polynomial orthonormal bases of the Fock spaces based on the \textit{Siegel-Jacobi disk} \(\mathcal{D}_n^J=\mathcal{D}_n \times \mathbb{C}^n\) are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given.