Generating functions and integral representations for the spherical functions on some classical Gelfand pairs
type: Item
instance of: scholarly article
title: Generating functions and integral representations for the spherical functions on some classical Gelfand pairs
zbMATH Open document ID: 0801.43003
author: Shigeru Watanabe
published in: Journal of Mathematics of Kyoto University
publication date: 1994-11-21T00:00:00Z
review text: Let \(G\) be \(SO_ 0(1,n)\), the connected component of the identity element in the real case \(\mathbb{F}=\mathbb{R}\), \(SU(1,n)\) in the complex case \(\mathbb{F}=\mathbb{C}\) and \(U(1,n;\mathbb{H})=Sp(1,n)\) in the quaternion case \(\mathbb{F}=\mathbb{H}\). The group \(G\) acts on the unit ball \(B(\mathbb{F}^ n)\) and the unit sphere \(S(\mathbb{F}^ n)\). The isotropy group \(K\) of the point \(O\in B(\mathbb{F}^ n)\) in \(G\) is just a maximal compact subgroup and \(G/K=B(\mathbb{F}^ n)\), \(K\cong SO(n)\) in the real case, \(U(n)\) in the complex case and \(Sp(1)\times Sp(n)\) in the quaternion case. Let \(G=KAN\) be the Iwasawa decomposition and \(M\) be the centralizer of \(A\) in \(K\). Then \(M\) is the isotropy group of the point \(e_ 1={^ t}(1,0,\dots,0)\in S(\mathbb{F}^ n)\) in \(K\), \(M\cong SO(n-1)\) in the real case, \(U(n-1)\) in the complex case and \(Sp(1)\times Sp(n-1)\) in the quaternion case, and \(K/M\cong S(\mathbb{F}^ n)\). In the real case, \(SO(n-1)\cong S(\mathbb{R}^ n)\), it is well known that each zonal spherical function \(\varphi\) depends only on the first coordinate \(\eta_ 1\), \((\eta={^ t}(\eta_ 1,\dots,\eta_ n)\in S(\mathbb{R}^ n))\), \(\varphi(\eta)=\varphi(\eta_ 1)\) and there is a unique nonnegative integer \(p\) such that \(\varphi(\eta_ 1)=C^{(n-2)/2}_ p(\eta_ 1)/C_ p^{(n-2)/2}(1)\), where \(C^{(n-2)/2}_ p\) is the Gegenbauer polynomial, for which there is the well known generating function \[ (1-2tz+t^ 2)^{-(n-2)/2}=\sum_{p=0}^ \infty C_ p^{(n-2)/2}(z) t^ p,\quad -1\leq z\leq 1,\quad -1<t<1. \] In the paper under review the author finds generating functions for the zonal spherical functions in the complex and the quaternion cases. He also gives integral presentations for the zonal functions in these situations. In the complex case, \(L^ 2(\mathbb{S}(\mathbb{C}^ n))=\bigoplus_{p,q=0}^ \infty H_{p,q}^{(n)}\), where \(H^{(n)}_{p,q}\) denotes the space of restrictions to \(\mathbb{S}(\mathbb{C}^ n)\) of harmonic polynomials…
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 33D55
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 33D80
zbMATH DE Number: 584586
zbMATH Keywords: isotropy group
zbMATH Keywords: maximal compact subgroup
zbMATH Keywords: Iwasawa decomposition
zbMATH Keywords: Gegenbauer polynomial
zbMATH Keywords: generating function
zbMATH Keywords: zonal spherical functions
zbMATH Keywords: harmonic polynomials
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519133
OpenAlex ID: W1850641856
DOI: 10.1215/KJM/1250519133
description: scientific article; zbMATH DE number 584586