Coherent states and geometry on the Siegel-Jacobi disk

The coherent state representation of the Jacobi group G(1)(J) is indexed with two parameters, mu(= 1/h), describing the part coming from the Heisenberg group, and k, characterizing the positive discrete series representation of SU(1, 1). The Ricci form, the scalar curvature and the geodesics of the Siegel-Jacobi disk D-1(J) are investigated. The significance in the language of coherent states of the transform which realizes the fundamental conjecture on the Siegel-Jacobi disk is emphasized. The Berezin kernel, Calabi's diastasis, the Kobayashi embedding and the Cauchy formula for the Siegel-Jacobi disk are presented.