Gelfand pairs and strong transitivity for Euclidean buildings

Pierre Emmanuel Caprace, Corina Ciobotaru

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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 δ , and K be the stabilizer of a special vertex in δ. It is known that (G, K) is a Gelfand pair as soon as G acts strongly transitively on δ ; 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 δ , 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 δ if and only if it is strongly transitive on the spherical building at infinity.

OriginalsprogEngelsk
TidsskriftErgodic Theory and Dynamical Systems
Vol/bind35
Nummer4
Sider (fra-til)1056-1078
Antal sider23
ISSN0143-3857
DOI
StatusUdgivet - 2015
Udgivet eksterntJa

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