Conformal harmonic analysis on hyperboloids
type: Item
instance of: scholarly article
title: Conformal harmonic analysis on hyperboloids
zbMATH Open document ID: 0787.22014
DOI: 10.1016/0393-0440(93)90027-C
published in: Journal of Geometry and Physics
publication date: 1993-06-29T00:00:00Z
review text: The author studies the natural representation of \(SO(p,q)\), when \(p+q\) is even, in the space of functions on the isotropic cone of signature \((p,q)\), which are homogeneous of degree \(-(p+q)/2+1\). This representation is known to be unitarizable and to split into a direct sum of two irreducible representations [see for example \textit{R. Howe} and \textit{Eng-Chye Tan}, Bull. Am. Math. Soc., New Ser. 28, No. 1, 1-74 (1993) (case 00, (iv), p. 13)]. A direct proof of this fact is given; the spaces of the two irreducible representations are described using the distributions \(m_ \pm^ \lambda\) associated to a nonsingular quadratic form \(m\) of signature \((p-1,q-1)\). It is then shown, using former results of the author and \textit{G. M. Henkin} on the inversion of the Radon- Penrose transform of \(\overline{\partial}\)-cohomology, that these spaces are isomorphic to some Hardy spaces of \(\overline{\partial}\)-cohomology in pseudo-Hermitian symmetric manifolds of tube type.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 32A45
Mathematics Subject Classification ID: 32M10
zbMATH DE Number: 219772
zbMATH Keywords: \(\text{SO}(p,q)\)
zbMATH Keywords: irreducible representations
zbMATH Keywords: Hardy spaces of \(\overline{\partial}\)-cohomology
zbMATH Keywords: pseudo-Hermitian symmetric manifolds of tube type
author: S. G. Gindikin
reviewed by: Guy Roos
MaRDI profile type: Publication
cites work: Harmonic analysis on hyperboloids
cites work: Q5724171
cites work: Q3940444
cites work: Integral formulas and integral geometry for \({\bar\partial}\)-cohomologies in \({\mathbb{C}}{\mathbb{P}}^ n\)
full work available at URL: 0393-0440(93)90027-c
OpenAlex ID: W2024176815
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description: scientific article; zbMATH DE number 219772