On a global realization of a discrete series for SU(n,1) as applications of Szegö operator and limits of discrete series
type: Item
instance of: scholarly article
title: On a global realization of a discrete series for SU(n,1) as applications of Szegö operator and limits of discrete series
zbMATH Open document ID: 0699.22025
author: Takeshi Kawazoe
published in: Tokyo Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let G be a connected Lie group with finite center, having real rank one, and a discrete series. The aim of this paper is to give a global (not infinitesimal) realization of discrete series representations of G. For limit of discrete series, this can be achieved by means of the Szegö operator composed with a boundary value operator, relating this representation, in a G-equivariant way, to a reducible unitary principal series representation [\textit{B. Blank}, Pac. J. Math. 122, 299-318 (1986; Zbl 0595.22012)]. Here, the author shifts this realization from a limit of discrete series to a discrete series representation, by means of Zuckerman's technique (tensor product with a finite dimensional representation). His construction is valid for all rank one semisimple Lie groups. However, in the final step, he needs the extra assumption that the associated symmetric space is hermitian, in order to prove the square-integrability of the image of the Szegö operator. Thus, the results are complete for \(G=SU(n,1)\) only. The examples of SU(1,1) and SU(2,1) are considered with some detail at the end of the paper.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 4146301
zbMATH Keywords: connected Lie group
zbMATH Keywords: discrete series representations
zbMATH Keywords: unitary principal series representation
zbMATH Keywords: rank one semisimple Lie groups
zbMATH Keywords: square- integrability
zbMATH Keywords: Szegö operator
reviewed by: François Rouvière
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270133190
OpenAlex ID: W2067124405
DOI: 10.3836/TJM/1270133190
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description: scientific article; zbMATH DE number 4146301