# On Siegel modular forms of half-integral weights and Jacobi forms

@article{Takase1999OnSM, title={On Siegel modular forms of half-integral weights and Jacobi forms}, author={Koichi Y. Takase}, journal={Transactions of the American Mathematical Society}, year={1999}, volume={351}, pages={735-780} }

We will establish a bijective correspondence between the space of the cuspidal Jacobi forms and the space of the half-integral weight Siegel cusp forms which is compatible with the action of the Hecke operators. This correspondence is based on a bijective correspondence between the irreducible unitary representations of a two-fold covering group of a symplectic group and a Jacobi group (that is, a semidirect product of a symplectic group and a Heisenberg group). The classical results due to…

## 13 Citations

The Schroedinger-Weil Representation and Jacobi Forms of Half-Integral Weight

- Mathematics
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In this paper, we define the concept of Jacobi forms of half-integral weight using Takase's automorohic factor of weight 1/2 for a two-fold covering group of the symplectic group on the Siegel upper…

ON THE GEOMETRY OF SIEGEL–JACOBI DOMAINS

- Mathematics
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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…

THE WEIL REPRESENTATIONS OF THE JACOBI GROUP

- Mathematics
- 2009

The Jacobi group is the semi-direct product of the symplectic group and the Heisenberg group. The Jacobi group is an important object in the frame- work of quantum mechanics, geometric quantization…

Consequences of the fundamental conjecture for the motion on the Siegel-Jacobi disk

- Mathematics
- 2011

We find the homogeneous Kahler diffeomorphism FC which expresses the Kahler two-form on the Siegel–Jacobi domain as the sum of the Kahler two-form on ℂ and the one on the Siegel ball . The classical…

A CONVENIENT COORDINATIZATION OF SIEGEL–JACOBI DOMAINS

- Mathematics
- 2012

We determine the homogeneous Kahler diffeomorphism FC which expresses the Kahler two-form on the Siegel–Jacobi ball as the sum of the Kahler two-form on ℂn and the one on the Siegel ball . The…

A Useful Parametrization of Siegel-Jacobi Manifolds

- Mathematics
- 2013

We determine the homogeneous Kahler diffeomorphism which expresses the Kahler two-form on the Siegel–Jacobi ball \(\mathcal{D}^J_n=\mathbb{C}^n \times \mathcal{D}_n\) as the sum of the Kahler…

Coherent states and geometry on the Siegel–Jacobi disk

- Mathematics
- 2014

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…

Geodesics on the extended Siegel-Jacobi upper half-plane

- Mathematics
- 2021

The semidirect product of the real Heisenberg group H1(R) with SL(2,R), called the real Jacobi groupGJ1 (R), admits a four-parameter invariant metric expressed in the S-coordinates. We determine the…

Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball

- Mathematics
- 2016

We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We…

Balanced Metric and Berezin Quantization on the Siegel –

- Mathematics
- 2016

We determine the matrix of the balanced metric of the Siegel–Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace–Beltrami operator of this manifold. We…

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