Gelfand pairs and strong transitivity for Euclidean buildings

Caprace, Pierre-Emmanuel;Ciobotaru, Corina Gabriela
(2015) Ergodic Theory and Dynamical Systems — Vol. 35, n° 4, p. 1056-1078 (2015)

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Abstract
Let G be a locally compact group acting properly by type-preserving automorphisms on a locally finite thick Euclidean building X and K be the stabilizer of a special vertex in X. It is known that (G;K) is a Gelfand pair as soon as G acts strongly transitively on X; this is in particular the case when G is a semi-simple algebraic group over a local field. We show a converse of this statement, namely: if (G;K) is a Gelfand pair and G acts cocompactly on X, 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 innity. Other equivalent formulations are also obtained, including the fact that G is strongly transitive on X if and only if it is strongly transitive on the spherical building at infinity.
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Caprace, P.-E., & Ciobotaru, C. G. (2015). Gelfand pairs and strong transitivity for Euclidean buildings. Ergodic Theory and Dynamical Systems, 35(4), 1056-1078. https://doi.org/10.1017/etds.2013.102 (Original work published 2015)