Let k be a number field, G an algebraic group defined over k, and G(k) the group of k-rational points in G. We determine the set of functions on G(k) which are of positive type and conjugation invariant, under the assumption that G(k) is generated by its unipotent elements. An essential step in the proof is the classification of the G(k)-invariant ergodic probability measures on an adelic solenoid naturally associated to G(k). This last result is deduced from Ratner’s measure rigidity theorem for homogeneous spaces of S-adic Lie groups; this appears to be the first application of Ratner’s theorems in the context of operator algebras.
Classification :
22-XX, 20-XX
Mots-clés :
Algebraic groups, characters of discrete groups, von Neumann algebras, invariant random subgroups, Ratner’s theory
Affiliations des auteurs :
Bachir Bekka
1
;
Camille Francini
1
1
Université de Rennes 1, France
Bachir Bekka; Camille Francini. Characters of algebraic groups over number fields. Groups, geometry, and dynamics, Tome 16 (2022) no. 4, pp. 1119-1164. doi: 10.4171/ggd/678
@article{10_4171_ggd_678,
author = {Bachir Bekka and Camille Francini},
title = {Characters of algebraic groups over number fields},
journal = {Groups, geometry, and dynamics},
pages = {1119--1164},
year = {2022},
volume = {16},
number = {4},
doi = {10.4171/ggd/678},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/678/}
}
TY - JOUR
AU - Bachir Bekka
AU - Camille Francini
TI - Characters of algebraic groups over number fields
JO - Groups, geometry, and dynamics
PY - 2022
SP - 1119
EP - 1164
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IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/678/
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%J Groups, geometry, and dynamics
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%R 10.4171/ggd/678
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