A convenient coordinatization of Siegel-Jacobi domains
arXiv:1204.5610 · doi:10.1142/S0129055X12500249
Abstract
We determine the homogeneous Kähler diffeomorphism which expresses the Kähler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the Kähler 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 $W\in\mc{D}_n$. denotes the -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 is applied, the first order differential equation in the variable $η=(\un-W\bar{W})^{-1}(z+W\bar{z})\in\C^n$ 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.
32 pages, corrected typos, Latex, amsart, AMS fonts