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Geometry and Arithmetic on the Siegel–Jacobi Space

Profile image of Jae-Hyun YangJae-Hyun Yang

2015, Progress in Mathematics

https://doi.org/10.1007/978-3-319-11523-8_10
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Abstract

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space. To the memory of my teacher, Professor Shoshichi Kobayashi Table of Contents 1. Introduction 2. Invariant Metrics and Laplacians on the Siegel-Jacobi Space 3. Invariant Differential Operators on the Siegel-Jacobi Space 4. The Partial Cayley Transform 5. Invariant Metrics and Laplacians on the Siegel-Jacobi Disk 6. A Fundamental Domain for the Siegel-Jacobi Space 7. Jacobi Forms 8. Singular Jacobi Forms 9. The Siegel-Jacobi Operator 10. Construction of Vector-Valued Modular Forms from Jacobi Forms 11. Maass-Jacobi Forms 12 The Schrödinger-Weil Representation 13. Final Remarks and Open Problems

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We determine the homogeneous Kähler diffeomorphism F C which expresses the Kähler two-form on the Siegel-Jacobi ball D J n = C n × D n as the sum of the Kähler two-form on C n and the one on the Siegel ball D n . The classical motion and quantum evolution on D J n determined by a hermitian linear Hamiltonian in the generators of the Jacobi group G J n = H n ⋊ Sp(n, R) C are described by a matrix Riccati equation on D n and a linear first order differential equation in z ∈ C n , with coefficients depending also on W ∈ D n . 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 D n . When the transform F C : (η, W ) → (z, W ) is applied, the first order differential equation in the variable η = (I n − WW ) −1 (z + Wz) ∈ C n becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel-Jacobi upper half plane X J n = C n × X n , where X n denotes the Siegel upper half plane.

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Related topics

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