  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
$\mathcal{D}^J_1$, 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 $\mathcal{D}^J_1$ 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.
