Eigenvalues of generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains
type: Item
instance of: scholarly article
title: Eigenvalues of generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains
zbMATH Open document ID: 1179.43008
author: Eisuke Imamura
author: Kiyosato Okamoto
author: Michiroh Tsukamoto
published in: Hiroshima Mathematical Journal
publication date: 2009-10-12T00:00:00Z
review text: The authors develop a group-theoretic method to generalize the Laplace-Beltrami operators on the classical domains. In [\textit{K. Okamoto}, Lect. Notes Math. 266, 255--271 (1972; Zbl 0234.43009)], inspired by \textit{S. Helgason}'s paper [Adv. Math. 5, 1--154 (1970; Zbl 0209.25403)], one of the authors defined the ``Poisson transforms'' for homogeneous vector bundles over symmetric spaces. In [\textit{K. Okamoto, M. Tsukamoto} and \textit{K. Yokota}, Jap. J. Math., New Ser. 26, No.~1, 51--103 (2000; Zbl 1019.32018)], the authors defined the generalized Poisson-Cauchy transforms for homogeneous holomorphic line bundles over hermitian symmetric spaces and computed explicitly the kernel functions for each type of the classical domains. In [\textit{E. Imamura, K. Okamoto, M. Tsukamoto} and \textit{A. Yamamori}, Proc. Japan Acad., Ser. A 82, No. 9, 167--172 (2006; Zbl 1154.43006)], making use of the Casimir operator, the authors defined the ``generalized Laplacians'' on homogeneous holomorphic line bundles over hermitian symmetric spaces and showed that the generalized Poisson-Cauchy transforms give rise to eigenfunctions of the ``generalized Laplacians''. In the paper under review, the authors use the canonical coordinates defined by the Harish-Chandra decomposition. This leads to a certain ``canonical Riemannian metric''. It turns out that Hua's differential operator is the Laplace-Beltrami operator defined by this metric. For each type of the classical domains, the authors carry out the direct computation to obtain the explicit formulas of line bundle valued invariant differential operators which are called the generalized Laplacians and compute their eigenvalues evaluated at the generalized Poisson-Cauchy kernel functions.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 43-06
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 32A26
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 5613990
zbMATH Keywords: harmonic analysis on symmetric spaces
zbMATH Keywords: Lie group representations
zbMATH Keywords: Laplace-Beltrami operators
zbMATH Keywords: Poisson-Cauchy transforms on classical domains
zbMATH Keywords: Casimir operator
zbMATH Keywords: eigenvalues of generalized Laplacians
zbMATH Keywords: Harish-Chandra decomposition
zbMATH Keywords: Hua's differential operator
zbMATH Keywords: homogeneous holomorphic line bundles over hermitian symmetric spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1249046339
OpenAlex ID: W1515665034
Wikidata QID: Q128870734
DOI: 10.32917/HMJ/1249046339
Recommended article: Generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains
Recommended article: Q3006804
Recommended article: Boundedness of the differential transforms for the generalized Poisson operators generated by Laplacian
Recommended article: Some classes of eigenvalues problems for generalized \(p\& q\)-Laplacian type operators on bounded domains
Recommended article: Generalized eigenvalues of the (P, 2)-Laplacian under a parametric boundary condition
Recommended article: Q4786343
Recommended article: Q4825247
Recommended article: Q4010094
Recommended article: Q3413856
Recommended article: Supersymmetry in quantum mechanics: an extended view
description: scientific article; zbMATH DE number 5613990