Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms
type: Item
instance of: scholarly article
title: Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms
zbMATH Open document ID: 0617.43005
DOI: 10.1016/0022-1236(87)90130-3
author: Henrik Schlichtkrull
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: One studies the eigenspaces \({\mathcal E}_{\lambda}\) of the Laplace- Beltrami operator \(\Delta\) of a hyperbolic space \(X=G/H\), \[ {\mathcal E}_{\lambda}=\{f\in C^{\infty}(X)| \quad \Delta f=(\lambda^ 2- \rho^ 2)f\}\quad, \] where \(G=U(p,q; {\mathbb{F}})\), \(H=U(1, {\mathbb{F}})\times U(p-1,q; {\mathbb{F}})\), \({\mathbb{F}}\) is one of the classical fields \({\mathbb{R}}\), \({\mathbb{C}}\) or \({\mathbb{H}}\), \(\rho =(dp+dq-2)\) \((d=\dim_{{\mathbb{R}}} {\mathbb{F}})\), (and also for the exceptional space \(F^ 1_ 4/Spin(1,8)\) for which one may say that \(p=2\), \(q=1\), and \({\mathbb{F}}={\mathbb{O}}\), the algebra of the Cayley numbers). One describes explicitly the closed C-invariant subspaces of \({\mathcal E}_{\lambda}\) in terms of the K-types occuring in such a subspace, where \(K=U(p; {\mathbb{F}})\times U(q; {\mathbb{F}})\), a maximal subgroup of G. In particular one determines for which \(\lambda\) the space \({\mathcal E}_{\lambda}\) is irreducible, and also the K-types occuring in the discrete series. The range of the Poisson transform \({\mathcal P}_{\lambda}\) is a subspace of \({\mathcal E}_{\lambda}\). One gives the values of \(\lambda\) for which this range is a proper subspace of \({\mathcal E}_{\lambda}\).
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 4000684
zbMATH Keywords: eigenspaces
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: hyperbolic space
zbMATH Keywords: Cayley numbers
zbMATH Keywords: K-types
zbMATH Keywords: maximal subgroup
zbMATH Keywords: discrete series
zbMATH Keywords: Poisson transform
reviewed by: Jacques Faraut
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90130-3
OpenAlex ID: W1976964009
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description: scientific article; zbMATH DE number 4000684