Martin boundaries for real semisimple Lie groups
type: Item
instance of: scholarly article
title: Martin boundaries for real semisimple Lie groups
zbMATH Open document ID: 0837.22010
published in: Journal of Functional Analysis
publication date: 1994-12-20T00:00:00Z
review text: The purpose of this paper is to construct the Martin boundaries for symmetric spaces of noncompact type \(X= G/K\), where \(G\) is a real semisimple Lie group with finite center. The author gives an explicit form of the Martin kernel \(K_y(x)\), which is expressed through the Green function \(v(x, y)\) of the Laplace-Beltrami operator on \(G\). The points of the Martin boundaries are in a correspondence with the limits \(\lim_{|y|\to \infty} K_y(x)\). Using the integral decomposition of the Green function over the zonal spherical functions, he calculates the asymptotics \(\lim_{|y|\to \infty} v(x, y)\) of the Green function from the corresponding asymptotics of the zonal spherical functions. He generalizes the \textit{Harish-Chandra} expansion [Am. J. Math. 80, 241-310, 553-613 (1958; Zbl 0093.128)] to derive asymptotic formulae that allow him to construct the Martin boundaries for \(X\). Then the author compares the Martin boundaries with other boundaries and shows that the Martin compactification is intermediate between the \textit{F. I. Karpelevich} [Trans. Mosc. Math. Soc. 14, 51-199 (1965; Zbl 0164.222)] and \textit{I. Satake} [Ann. Math., II. Ser. 72, 555-580 (1960; Zbl 0146.047)] compactifications. An application to the scattering problem of the quantum Calogero-Moser system is also discussed [cf. \textit{F. Calogero}, J. Math. Phys. 12, 419- 436 (1971) and \textit{J. Moser}, Adv. Math. 16, 1-23 (1975)].
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 704522
zbMATH Keywords: Martin boundaries
zbMATH Keywords: symmetric spaces
zbMATH Keywords: real semisimple Lie group
zbMATH Keywords: Martin kernel
zbMATH Keywords: Green function
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Martin compactification
zbMATH Keywords: scattering problem
zbMATH Keywords: quantum Calogero-Moser system
author: M. Olshanetsky
reviewed by: Hidenori Fujiwara
MaRDI profile type: Publication
full work available at URL: jfan.1994.1145
OpenAlex ID: W1976269662
DOI: 10.1006/JFAN.1994.1145
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description: scientific article; zbMATH DE number 704522