TY - JOUR
T1 - Gelfand pairs and strong transitivity for Euclidean buildings
AU - Caprace, Pierre Emmanuel
AU - Ciobotaru, Corina
N1 - Publisher Copyright:
© 2014 Cambridge University Press.
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.
UR - http://www.scopus.com/inward/record.url?scp=84929163034&partnerID=8YFLogxK
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 -