Invariant metrics and Laplacians on Siegel-Jacobi Disk

arXiv: Number Theory, 2006

In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential operators. We present problems which are natural. We give some new partial solutions for these natural problems.

2012

Abstract. In this paper, we discuss the theory of the Siegel modular variety in the aspects of arithmetic and geometry. This article covers the theory of Siegel modular forms, the Hecke theory, a lifting of elliptic cusp forms, geometric properties of the Siegel modular variety, (hypothetical) motives attached to Siegel modular forms and a cohomology of the Siegel modular variety. To the memory of my mother

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.

2016

For two positive integers m and n, we let H n be the Siegel upper half plane of degree n and let C (m,n) be the set of all m × n complex matrices. In this article, we study differential operators on the Siegel-Jacobi space H n × C (m,n) that are invariant under the natural action of the Jacobi group Sp(n, R) ⋉ H (n,m) R on H n × C (m,n) , where H (n,m) R denotes the Heisenberg group. We give some explicit invariant differential operators. We present important problems which are natural. We give some partial solutions for these natural problems.

arXiv: Number Theory, 2011

For two positive integers m and n, we let Hn be the Siegel upper half plane of degree n and let C (m,n) be the set of all m × n complex matrices. In this article, we study differential operators on the Siegel-Jacobi space Hn ×C (m,n) that are invariant under the natural action of the Jacobi group Sp(n,R) ⋉ H (n,m) R on Hn × C (m,n) , where H (n,m) R denotes the Heisenberg group. We give some explicit invariant differential operators. We present important problems which are natural. We give some partial solutions for these natural problems.

International Journal of Geometric Methods in Modern Physics, 2011

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.

arXiv preprint arXiv:1110.5469, 2011

arXiv: Differential Geometry, 2019

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this short paper, we propose the basic problems in the geometry of the Siegel-Jacobi space.

arXiv preprint arXiv:1204.5610, 2012

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.

Bull. Korean Math. Soc. 50 (2013), No. 3, pp. 787-799

In this paper, we compute the sectional curvatures and the scalar curvature of the Siegel-Jacobi space H 1 × C of degree 1 and index 1 explicitly.