Equivariant maps and purely atomic spectrum (Q1340811)

Let \(G\) be a connected semisimple Lie group with no compact factors and finite center and \(\Gamma \subset G\) an irreducible lattice. If \((X_1, \mu_1)\) and \((X_2, \mu_2)\) are \(G\)-spaces with finite invariant measures and the action of \(G\) has purely atomic spectrum (and is essentially free or essentially transitive) it is proved that every measure preserving \(\Gamma\)-map \(\varphi : X_1 \to X_2\) is also a \(G\)-map.