Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces

Bruno Duchesne (1) , Jean Lecureux (2) , Maria Beatrice Pozzetti (3)

1 IECL - Institut Élie Cartan de Lorraine
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 Universität Heidelberg [Heidelberg] = Heidelberg University

We define a Toledo number for actions of surface groups and complex hyper-bolic 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, that 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.

Domaines: Géométrie différentielle [math.DG]; Théorie des groupes [math.GR]

Bruno Duchesne, Jean Lecureux, Maria Beatrice Pozzetti. Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces. 2018. ⟨hal-01903007v2⟩