On the geometry of Siegel-Jacobi domains

We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk 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.Comment: 15 pages, Latex, AMS fonts, paper presented at the the International Conference "Differential Geometry and Dynamical Systems", August 25-28, 2010, University Politehnica of Bucharest, Romani

A holomorphic representation of the Jacobi algebra

A representation of the Jacobi algebra $\mathfrak{h}_1\rtimes \mathfrak{su}(1,1)$ by first order differential operators with polynomial coefficients on the manifold $\mathbb{C}\times \mathcal{D}_1$ is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with polynomials coefficients act is constructed.Comment: 34 pages, corrected typos in accord with the printed version and the Errata in Rev. Math. Phys. Vol. 24, No. 10 (2012) 1292001 (2 pages) DOI: 10.1142/S0129055X12920018, references update

A convenient coordinatization of Siegel-Jacobi domains

We determine the homogeneous K\"ahler diffeomorphism $FC$ which expresses the K\"ahler two-form on the Siegel-Jacobi ball \mc{D}^J_n=\C^n\times \mc{D}_n as the sum of the K\"ahler two-form on \C^n and the one on the Siegel ball \mc{D}_n. The classical motion and quantum evolution on \mc{D}^J_n determined by a hermitian linear Hamiltonian in the generators of the Jacobi group G^J_n=H_n\rtimes\text{Sp}(n,\R)_{\C} are described by a matrix Riccati equation on \mc{D}_n and a linear first order differential equation in z\in\C^n, with coefficients depending also on W\in\mc{D}_n. $H_n$ denotes the $(2n+1)$-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on \mc{D}_n. When the transform $FC:(\eta,W)\rightarrow (z,W)$ is applied, the first order differential equation in the variable \eta=(\un-W\bar{W})^{-1}(z+W\bar{z})\in\C^n becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel-Jacobi upper half plane \mc{X}^J_n=\C^n\times\mc{X}_n, where \mc{X}_n denotes the Siegel upper half plane.Comment: 32 pages, corrected typos, Latex, amsart, AMS font