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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].

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds