On Furstenberg's characterization of harmonic functions on symmetric spaces (Q1969005)

Let \(G=G_1\times \cdots\times G_n\) be a decomposition of a semisimple Lie group into simple components. Let \(\mu\) be a radial probability measure on \(G\). The author gives a short proof of the following theorem: Let \(f\) be a bounded function on \(D=G/K\), where \(K\) is a maximal compact subgroup of \(G\) which satisfies the equation \(f(g)=\int_G f(gg')d\mu(g')\) \(\forall g\in G\). Then \(f\) is harmonic, i.e. \(f(gK)=\int_K f(gkg'K)dk\) for any \(g,g'\in G\), if and only if the semigroup generated by the double cosets \(Kg'K\) in \(\text{Supp} (\mu)\) is equal to \(G\). This implies a classical result of Furstenberg which says that if \(\mu\) is radial and absolutely continuous, and if \(f\) is \(\mu\)-harmonic, then \(f\) is harmonic (see [\textit{H. Furstenberg}, Ann. Math. (2) 77, 335-386 (1963; Zbl 0192.12704)] and [\textit{H. Furstenberg}, Boundaries of Riemannian symmetric spaces, in: Symmetric spaces (W. M. Boothby and G. L. Weiss, eds), Pure Appl. Math. 8, 359-377 (1972; Zbl 0241.22028)]).