Gelfand pairs and strong transitivity for Euclidean buildings

Pierre Emmanuel Caprace; Corina Ciobotaru

doi:10.1017/etds.2013.102

Gelfand pairs and strong transitivity for Euclidean buildings

Pierre Emmanuel Caprace, Corina Ciobotaru

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

16 Citationer (Scopus)

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.

Originalsprog	Engelsk
Tidsskrift	Ergodic Theory and Dynamical Systems
Vol/bind	35
Nummer	4
Sider (fra-til)	1056-1078
Antal sider	23
ISSN	0143-3857
DOI	https://doi.org/10.1017/etds.2013.102
Status	Udgivet - 2015
Udgivet eksternt	Ja

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title = "Gelfand pairs and strong transitivity for Euclidean buildings",

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.",

author = "Caprace, {Pierre Emmanuel} and Corina Ciobotaru",

note = "Publisher Copyright: {\textcopyright} 2014 Cambridge University Press.",

year = "2015",

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journal = "Ergodic Theory and Dynamical Systems",

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TY - JOUR

T1 - Gelfand pairs and strong transitivity for Euclidean buildings

AU - Caprace, Pierre Emmanuel

AU - Ciobotaru, Corina

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

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U2 - 10.1017/etds.2013.102

DO - 10.1017/etds.2013.102

M3 - Journal article

AN - SCOPUS:84929163034

SN - 0143-3857

VL - 35

SP - 1056

EP - 1078

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 4

ER -

Gelfand pairs and strong transitivity for Euclidean buildings

Abstract

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