Gelfand pairs and strong transitivity for Euclidean buildings. (English) Zbl 1355.37047
Summary: 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.
MSC:
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 22D15 | Group algebras of locally compact groups |
| 22D45 | Automorphism groups of locally compact groups |
| 22C05 | Compact groups |
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