A representation of the Jacobi algebra h 1 su(1, 1) by first-order differential operators with polynomial coefficients on the manifold C × D 1 is presented. The Hilbert space of holomorphic functions on which the holomorphic first-order differential operators with polynomials coefficients act is constructed.

The classical motion and the quantum evolution determined by linear Hamiltonians on coherent state manifolds of compact and noncompact Hermitian symmetric space structure are considered. A matrix Riccati equation, with opposite signs of the quadratic terms in the compact and noncompact cases, determines the classical motion and the quantum evolution. The geometric meaning of equations of motion as flow on a complex Grassmann manifold and his noncompact dual is emphasized. Possibilities of generalizing the results to flag manifolds are pointed out.

The real Jacobi group G J 1 (R), defined as the semi-direct product of the group SL(2, R) with the Heisenberg group H 1 , is embedded in a 4×4 matrix realisation of the group Sp(2, R). The left-invariant one-forms on G J 1 (R) and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates (x, y, θ, p, q, κ), and a left-invariant metric depending of 4 parameters (α, β, γ, δ) is obtained. An invariant metric depending of (α, β) in the variables (x, y, θ) on the Sasaki manifold SL(2, R) is presented. The well known Kähler balanced metric in the variables (x, y, p, q) of the four-dimensional Siegel-Jacobi upper half-is written down as sum of the squares of four invariant one-forms, where X 1 denotes the Siegel upper half-plane. The left-invariant metric in the variables (x, y, p, q, κ) depending on (α, γ, δ) of a five-dimensional manifoldFC −1 : η = z +zw 1 − |w| 2 , and FC : A(w, z) → dη − wdη.