Narrow Results

A representation of the Jacobi algebra 𝔥₁ ⋊ 𝔰𝔲(1, 1) by first-order differential operators with polynomial coefficients on the manifold is presented. The Hilbert space of holomorphic functions on which the holomorphic first-order differential operators with polynomials coefficients act is constructed.

We determine the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi ball as the sum of the Kähler two-form on ℂⁿ and the one on the Siegel ball . The classical motion and quantum evolution on determined by a hermitian linear Hamiltonian in the generators of the Jacobi group are described by a matrix Riccati equation on and a linear first-order differential equation in z ∈ ℂⁿ, with coefficients depending also on . H_n denotes the (2n+1)-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on . When the transform FC : (η, W) → (z, W) is applied, the first-order differential equation in the variable becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel–Jacobi upper half plane , where denotes the Siegel upper half plane.

We study the holomorphic unitary representations of the Jacobi group based on Siegel–Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel–Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel–Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given.

We find the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi domain as the sum of the Kähler two-form on ℂ and the one on the Siegel ball . The classical motion and quantum evolution on determined by a linear Hamiltonian in the generators of the Jacobi group is described by a Riccati equation on and a linear first-order differential equation in z ∈ ℂ, where H₁ denotes the three-dimensional Heisenberg group. When the transformation FC is applied, the first-order differential equation for the variable z ∈ ℂ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel–Jacobi space , where denotes the Siegel upper half-plane.

The coherent state representation of the Jacobi group is indexed with two parameters, , 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 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.