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Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

Part of: Complex spaces with a group of automorphisms Automorphic functions Lie groups

Published online by Cambridge University Press:  18 October 2021

BRUNO DUCHESNE Open the ORCID record for BRUNO DUCHESNE [Opens in a new window] ,
JEAN LÉCUREUX  and
MARIA BEATRICE POZZETTI Open the ORCID record for MARIA BEATRICE POZZETTI [Opens in a new window]
BRUNO DUCHESNE*
Affiliation:
Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, F-54000 Nancy, France
JEAN LÉCUREUX
Affiliation:
Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France (e-mail: jean.lecureux@universite-paris-saclay.fr)
MARIA BEATRICE POZZETTI
Affiliation:
Mathematisches Institut, Universität Heidelberg, 69120 Heidelberg, Germany (e-mail: pozzetti@mathi.uni-heidelberg.de)
*
e-mail: bruno.duchesne@univ-lorraine.fr
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Abstract

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

Keywords

maximal representationsinfinite dimensional symmetric spacesboundary maps

MSC classification

Primary: 22E40: Discrete subgroups of Lie groups 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 1 , January 2023 , pp. 140 - 189
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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