On the geometry of Siegel-Jacobi domains

TL;DR

This paper explores the structure of the Jacobi group on Siegel-Jacobi domains, constructing explicit orthonormal bases and analyzing holomorphic discrete series representations.

Contribution

It provides explicit polynomial orthonormal bases for Fock spaces on Siegel-Jacobi disks and constructs scalar holomorphic discrete series for the Jacobi group.

Findings

Explicit polynomial orthonormal bases for Fock spaces on Siegel-Jacobi disks

Construction of scalar holomorphic discrete series representations

Analysis of holomorphic unitary representations of the Jacobi group

Abstract

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.