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Compactifications of symmetric spaces and positive eigenfunctions of the Laplacian. (English) Zbl 0855.58060
Fascicule de probabilités. Séminaire de probabilités, Université de Rennes, France, année 1994. Rennes: Université de Rennes, IRMAR, Publ. Inst. Rech. Math. Rennes. Exp. 2, 63 p. (1994).
The author considers symmetric spaces of the form \(X= G/K\), on which \(G\) acts by isometries, where \(G\) is a connected semi-simple real Lie group with a finite center and no compact factors, and \(K\) is an appropriate maximal compact subgroup of \(G\). Let \(L\) be the Laplace-Beltrami operator. By \(\partial m^\lambda\) one denotes the space of normalized \(\lambda\)-eigenfunctions of \(L\). Let \(m^\lambda= X\cup \partial m^\lambda\) be the Martin compactification of \(X\). The main problem is to identify the \(G\)-space \(m^\lambda\) in geometrical terms and to calculate the cocycle \(\theta^\lambda\) on \(G\times \partial m^\lambda\), defined by the Green function \(G_\lambda\) as a limit, \(\theta^\lambda (g, \mu)= \lim G_\lambda (g^{-1}.0,y)/G_\lambda (0, y)\), \(y\to \mu\), in terms of the geometry of \(X\) and its natural boundaries.
For the entire collection see [Zbl 0828.00015].

58J50 Spectral problems; spectral geometry; scattering theory on manifolds