Noyau de Cauchy-Szegö d'un espace symétrique de type Cayley. (Cauchy-Szegö kernel of a symmetric space of Cayley type.)
type: Item
instance of: scholarly article
title: Noyau de Cauchy-Szegö d'un espace symétrique de type Cayley. (Cauchy-Szegö kernel of a symmetric space of Cayley type.)
zbMATH Open document ID: 0920.43008
author: Mohammed Chadli
publication date: 1998-09-22T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75283
review text: The author studies the geometry and harmonic analysis of certain open domains \(\Xi\) in a complex symmetric space \(G_\mathbb{C}/H_\mathbb{C}\) whose Shilov boundary is a symmetric space \(G/H\) of Cayley type. The Hardy space \(H^2 (\Xi)\) then consists of holomorphic functions on \(\Xi\) with \(L^2\)-boundary values on \(G/H\). It carries a holomorphic representation of \(G\) which decomposes discretely into representations from the holomorphic discrete series of \(G/H\) as introduced by Ørsted and Ólafsson. The main achievement of this paper is to make the decomposition of \(H^2(\Xi)\) explicit in the language of Jordan algebras, which is particularly apt to describe the Cayley type situation, and in this way obtain an explicit formula for the Cauchy-Szegö kernel. On the way the author provides several remarkable results: (i) An explicit calculation of the \(L^2\)-norm for the highest weight vectors in the irreducible components of \(H^2(\Xi)\) which as a corollary yields the spectrum of \(H^2(\Xi)\). (ii) A map \(u\mapsto\psi_u\), which associates an \(H\)-invariant distribution vector to each element of a lowest \(K\)-type of an irreducible component of \(H^2 (\Xi)\), and which is explicit enough to identify summands in the general expansion formula for the Cauchy-Szegö kernel as quotients of well studied spherical functions by certain \(c\)-functions. In the last part the author derives an isomorphism between the Hardy space \(H^2(\Xi)\) and the classical Hardy space of the domain in which \(\Xi\) is realized as an open dense subset. This isomorphism gives rise to another explicit formula for the Cauchy-Szegö kernel.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 17C37
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1202018
zbMATH Keywords: Gelfand-Gindikin program
zbMATH Keywords: harmonic analysis
zbMATH Keywords: Cauchy-Szegö kernel
reviewed by: Joachim Hilgert
MaRDI profile type: Publication
OpenAlex ID: W2081275177
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DOI: 10.5802/AIF.1612
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1202018