## A convenient coordinatization of Siegel-Jacobi domains    [PDF]

Stefan Berceanu
We determine the homogeneous K\"ahler diffeomorphism $FC$ which expresses the K\"ahler two-form on the Siegel-Jacobi ball $\mathcal{D}^J_n=\mathbb{C}^n\times\mathcal{D}_n$ as the sum of the K\"ahler two-form on $\mathbb{C}^n$ and the one on the Siegel ball $\mathcal{D}_n$. The classical motion and quantum evolution on $\mathcal{D}^J_n$ determined by a hermitian linear Hamiltonian in the generators of the Jacobi group $G^J_n=H_n\rtimes\text{Sp}(n,\mathbb{R})_{\mathbb{C}}$ are described by a matrix Riccati equation on $\mathcal{D}_n$ and a linear first order differential equation in $z\in\mathbb{C}^n$, with coefficients depending also on $W\in\mathcal{D}_n$. $H_n$ 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 $\mathcal{D}_n$. When the transform $FC:(\eta,W)\rightarrow (z,W)$ is applied, the first order differential equation in the variable $\eta=(1_n-W\bar{W})^{-1}(z+W\bar{z})\in\mathbb{C}^n$ decouples of the motion on the Siegel ball. Similar considerations are presented for the Siegel-Jacobi upper half-plane $\mathcal{X}^J_n=\mathbb{C}^n\times\mathcal{X}_n$, where $\mathcal{X}_n$ denotes the Siegel upper half-plane.
View original: http://arxiv.org/abs/1204.5610