Bergman representative coordinates on the Siegel-Jacobi disk

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We underline some differences between the geometric aspect of Berezin's approach to quantization on homogeneous Kähler manifolds and Bergman's construction for bounded domains in Cⁿ. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk Dʲ₁, which is a partially bounded manifold whose points belong to C₁, where D₁ denotes the Siegel disk. The Bergman representative coordinates on Dʲ₁ are globally defined, the Siegel-Jacobi disk is a normal Kähler homogeneous Lu Qi-Keng manifold, whose representative manifold is the Siegel-Jacobi disk itself.