Spherical functions on harmonic extensions of \(H\)-type groups
type: Item
instance of: scholarly article
title: Spherical functions on harmonic extensions of \(H\)-type groups
zbMATH Open document ID: 0957.22011
DOI: 10.1007/BF02921938
author: Anthony H. Dooley
author: Genkai Zhang
published in: The Journal of Geometric Analysis
publication date: 2000-12-11T00:00:00Z
review text: In the paper under review, the authors consider the harmonic extension \(AN\) of an \(H\)-type group \(N\) with Lie algebra \(\mu=\nu +\zeta\) with \([\nu,\nu ]=\zeta\), and characterize the positive definite spherical functions on \(AN\). The notion of \(H\)-type group was introduced by \textit{A. Kaplan} [Trans. Am. Math. Soc. 258, 147-153 (1980; Zbl 0416.35022)] as a natural generalization of the Iwasawa \(N\)-group of a rank one simple Lie group \(G\). \textit{J.-P. Anker, E. Damek} and {Ch. Yacoub} [Ann. Scuola Norm. Sup. Pisa, Cl. Sci (4) 23, 643-679 (1996; Zbl 0881.22008)] studied harmonic analysis on a solvable extension \(AN\) of an \(H\)-type group \(N\), where \(A={\mathbb R}^{+}\) acts on \(N\) by nonhomogeneous dilations and \(AN\) is the semi-direct product of \(A\) and \(N\). Analysis on the space \(AN\) has many similarities to that on a symmetric space of rank one. In particular, one can define the spherical functions \(\phi_\xi\) and they play the same role as in the study of a rank one symmetric space. An important problem in the study of rank one symmetric spaces is to determine all the positive spherical functions. First, the authors recall the basic definitions of \(H\)-type groups \(N\) and their harmonic extensions \(AN\), as well as the definition of a spherical function for \(AN\). The induced representations of \(AN\) and the Plancherel decomposition of \(L^2(N)\) under the group \(AN\) are studied and the Fourier transform of the kernel of the intertwining operator is calculated. The authors prove a necessary and sufficient condition for the spherical function \(\phi_\xi\) to be positive definite. This result was conjectured by Anker, Damek and Yacoub [op. cit.].
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A90
zbMATH DE Number: 1542094
zbMATH Keywords: Heisenberg-type group
zbMATH Keywords: spherical function
zbMATH Keywords: induced representation
zbMATH Keywords: Plancherel decomposition
zbMATH Keywords: Fourier transform
zbMATH Keywords: intertwining operator
reviewed by: Adela-Gabriela Mihai
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921938
OpenAlex ID: W1965221840
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description: scientific article; zbMATH DE number 1542094