Weighted Bergman spaces and the holomorphic discrete series of \(\widetilde{U(p,q)}\)
type: Item
instance of: scholarly article
title: Weighted Bergman spaces and the holomorphic discrete series of \(\widetilde{U(p,q)}\)
zbMATH Open document ID: 0957.43008
published in: Journal of Functional Analysis
publication date: 2000-08-27T00:00:00Z
review text: The symmetric space \(SU(n,n)/S(U(p,q) \times U(p,q))\) with \(n= p+q\) can be realized as a domain \({\mathcal D}_{p,q}\) in the complex \(n\times n\) matrices, defined by the condition that \(I_{p,q}-w* I_{p,q}w\) is positive definite, where \(I_{p,q}\) is a diagonal matrix with \(p\) entries 1 and then \(q\) entries \(-1\). Applying a Cayley transform to \({\mathcal D}_{p,q}\) one obtains a different realization \(T_\Omega\setminus \Sigma_{p,q}\). Here \(\Omega\) is the cone of hermitian positive definite \(n\times n\) matrices, \(T_\Omega\) is the tube domain \(V+i\Omega\), where \(V\) is the space of all hermitian \(n\times n\) matrices, and \(\Sigma_{p,q}\) is the complex hypersurface in \(T_\Omega\) defined by the equation \(\det(z+iI_{p,q}) =0\). The author gives an explicit construction of a covering \(\widetilde {{\mathcal D}_{p,q}}\) of \({\mathcal D}_{p,q}\) with infinitely many sheets and considers a family of weighted Bergman spaces \({\mathcal H}^2_\alpha(\widetilde {{\mathcal D}_{p,q}})\) on this covering. The weight functions indexed by \(\alpha\) are the powers \(\det(I_{p,q}- w^*I_{p,q} w)^{\alpha-2n}\). His first main result is an explicit isomorphism between \({\mathcal H}^2_\alpha (\widetilde {{\mathcal D}_{p,q}})\) and the classical weighted Bergman spaces \({\mathcal H}^2_\alpha (T_\Omega)\) with respect to the weight functions \(\det(\text{Im} z)^\alpha\). The weighted Bergman spaces carry a natural representation of the universal covering group \(\widetilde {U(n,n)}\) which decomposes as a direct sum of highest weight representations. The author gives an explicit description of this decomposition and the corresponding expansion of the reproducing kernel of the weighted Bergman spaces in terms of holomorphic discrete series characters.
reviewed by: Joachim Hilgert
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 32A10
zbMATH DE Number: 1462830
zbMATH Keywords: weighted Bergman spaces
zbMATH Keywords: highest weight representations
zbMATH Keywords: reproducing kernel
zbMATH Keywords: holomorphic discrete series
author: Salem Ben Said
MaRDI profile type: Publication
full work available at URL: jfan.1999.3553
OpenAlex ID: W2000708938
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DOI: 10.1006/JFAN.1999.3553
description: scientific article; zbMATH DE number 1462830