Operator-valued Bessel functions, holomorphic discrete series, and harmonic representations of \(U(p,q)\)
type: Item
instance of: scholarly article
title: Operator-valued Bessel functions, holomorphic discrete series, and harmonic representations of \(U(p,q)\)
zbMATH Open document ID: 0762.22010
DOI: 10.1016/0022-1236(92)90150-H
author: Hongming Ding
published in: Journal of Functional Analysis
publication date: 1993-01-16T00:00:00Z
review text: Let \(G=U(p,q)\) and let \(K\) be a maximal compact subgroup. Then the Hermitian symmetric space \(G/K\) can be realized as \(D\), a Siegel domain of type II. In this paper the author applies his theory of operator- valued Bessel functions and Schrödinger-Fock spaces to some problems in representation theory of \(G\). He gives a new and simpler characterization of the ``reduced'' Schrödinger-Fock space, and a representation theoretic proof of a Paley-Wiener theorem on \(D\). The main result is a decomposition theorem of the harmonic (or Segal- Shale-Weil) representation of \(G\) into a sum of holomorphic discrete series; square-integrability of the latter representations is proved by means of square-integrability of the ``reduced'' Bessel functions.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 32M15
zbMATH DE Number: 89207
zbMATH Keywords: harmonic representation
zbMATH Keywords: Segal-Shale-Weil representation
zbMATH Keywords: Siegel domain of type II
zbMATH Keywords: operator-valued Bessel functions
zbMATH Keywords: Schrödinger-Fock spaces
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: decomposition theorem
zbMATH Keywords: discrete series
zbMATH Keywords: square- integrability
reviewed by: François Rouvière
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 89207