On the geometry of Siegel-Jacobi domains
type: Item
instance of: scholarly article
zbMATH Open document ID: 1250.22010
DOI: 10.1142/S0219887811005920
author: Stefan Berceanu
author: A. Gheorghe
published in: International Journal of Geometric Methods in Modern Physics
publication date: 2012-06-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.3317
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 12E10
Mathematics Subject Classification ID: 81R30
zbMATH DE Number: 6046010
zbMATH Keywords: Jacobi group
zbMATH Keywords: Siegel-Jacobi domain
zbMATH Keywords: canonical automorphy factor
zbMATH Keywords: canonical kernel function
zbMATH Keywords: Fock representation
zbMATH Keywords: scalar holomorphic discrete series
MaRDI profile type: Publication
arXiv ID: 1011.3317
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OpenAlex ID: W3104202268
title: On the geometry of Siegel-Jacobi domains
review text: The \textit{Jacobi group} \(G^J\) is the semidirect product of a Zariski connected semisimple real algebraic group of Hermitian type \(G^s\) and the Heisenberg group \(H(V)\) associated with the symplectic \(\mathbb{R}\)-space \(V\). The Hermitian symmetric domain associated to \(G^s\) is \(\mathcal{D}=G^s/K^s\), where \(K^s\) is a maximal compact subgroup. The \textit{Jacobi-Siegel domain} associated to the Jacobi group \(G^J\) is \(\mathcal{D}^J=\mathcal{D}\times \mathbb{C}^N\), where dim \(V=2N\). The authors study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. The \textit{Siegel disk} \(\mathcal{D}_n\) of degree \(n\) consists of all the symmetric matrices \(W\in M_n(\mathbb{C})\) with \(I_n-W\overline{W}>0\). Explicit polynomial orthonormal bases of the Fock spaces based on the \textit{Siegel-Jacobi disk} \(\mathcal{D}_n^J=\mathcal{D}_n \times \mathbb{C}^n\) 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.
reviewed by: Nicolae Cotfas
description: scientific article; zbMATH DE number 6046010