Gelfand pairs and strong transitivity for Euclidean buildings

摘要

Let G be a locally compact group acting properly, by type-preserving automorphisms on a locally finite thick Euclidean building Delta, and K be the stabilizer of a special vertex in Delta. It is known that (G, K) is a Gelfand pair as soon as G acts strongly transitively on Delta; in particular, this is the case when G is a semi-simple algebraic group over a local field. We show a converse to this statement, namely that if (G, K) is a Gelfand pair and G acts cocompactly on Delta, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in G and their peculiar dynamics on the spherical building at infinity. Other equivalent formulations are also obtained, including the fact that G is strongly transitive on Delta if and only if it is strongly transitive on the spherical building at infinity.