Coherent States Associated to the Jacobi Group - a Variation on a Theme by Erich Kähler

Applying the methods developed in [3], in [4] we have constructed generalized CS attached to the Jacobi group G1 = H1 oSU(1, 1), based on the homogeneous Kähler manifold D 1 = H1/R × SU(1, 1)/U(1) = C 1 × D1. Here D1 denotes the unit disk D1 = {w ∈ C ; |w| < 1}, and Hn is the (2n + 1)-dimensional real Heisenberg-Weyl group with Lie algebra hn. In [4] we have also emphasized that, when expressed in appropriate coordinates on the manifold X J 1 = C ×H1, H1 = {v ∈ C ; Im(v) > 0}, the Kähler two-form ω1 is identical with the one considered by Kähler-Berndt [6, 7, 10–12].