Martin boundaries for real semisimple Lie groups
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)].
scientific article; zbMATH DE number 704522