Fourier transform of a space of holomorphic discrete series
type: Item
instance of: scholarly article
title: Fourier transform of a space of holomorphic discrete series
zbMATH Open document ID: 0611.43003
author: Takaaki Nomura
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: Essentially from the author's introduction: ''Let G be a connected non- compact real simple Lie group of matrices and K a maximal compact subgroup of G. Assume G/K is a Hermitian symmetric space. Then, G/K can be realized as a Siegel domain D of type II. Let \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak g}=Lie G\) contained in \({\mathfrak k}=Lie K\), \(\Delta\) the root system of (\({\mathfrak g}_{{\mathbb{C}}},{\mathfrak h}_{{\mathbb{C}}})\). We introduce an order in \(\Delta\) compatible with the complex structure of G/K. For each K-dominant K-integral linear form \(\Lambda\) on \({\mathfrak h}_{{\mathbb{C}}}\) satisfying \textit{Harish-Chandra}'s non-vanishing condition [Am. J. Math. 78, 564-628 (1956; Zbl 0072.017), p. 612], the holomorphic discrete series \(\Pi_{\Lambda}\) of G is realized on a Hilbert space \({\mathcal H}(\Lambda)\) of vector valued holomorphic functions on D. Let S(D) be the Shilov boundary of D. Then, one knows that S(D) is diffeomorphic to a certain nilpotent subgroup N(D) of the affine automorphisms of D. By identifying S(D) with N(D), the aim of this note is a description of the space \({\mathcal H}(\Lambda)\) by using the Fourier transform on N(D).'' When D is a tube domain the results are known from \textit{H. Rossi} and \textit{M. Vergne} [J. Funct. Anal. 13, 324-389 (1973; Zbl 0279.32019)]. The paper treats the non-tube domains.
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 3987935
zbMATH Keywords: Hermitian symmetric space
zbMATH Keywords: Siegel domain
zbMATH Keywords: holomorphic discrete series
zbMATH Keywords: Fourier transform
zbMATH Keywords: tube domain
reviewed by: Mogens Flensted-Jensen
MaRDI profile type: Publication
full work available at URL: pjaa.61.133
OpenAlex ID: W2021854918
cites work: Representations of Semisimple Lie Groups VI: Integrable and Square-Integrable Representations
cites work: Q3897665
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cites work: Harmonic analysis of a nilpotent group and function theory on Siegel domains of type II
cites work: Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group
cites work: Analytic continuation of the holomorphic discrete series of a semi-simple Lie group
DOI: 10.3792/PJAA.61.133
description: scientific article; zbMATH DE number 3987935