Bergman representative coordinates on the Siegel-Jacobi disk

We underline some differences between the geometric aspect of Berezin’s approach to quantization on homogeneous K"ahler manifolds and Bergman’s construction for bounded domains in ${\mathbb{C}}^n$. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk ${\mathcal{D}}_1^J$, which is a partially bounded manifold whose points belong to $\mathbb{C}\times {\mathcal{D}}_1$, where ${\mathcal{D}}_1$ denotes the Siegel disk. The Bergman representative coordinates on ${\mathcal{D}}_1^J$ are globally defined, the Siegel-Jacobi disk is a normal K"ahler homogeneous Lu Qi-Keng manifold, whose representative manifold is the Siegel-Jacobi disk itself.