Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces

TL;DR

This paper introduces a Toledo number for actions of surface groups on infinite dimensional Hermitian symmetric spaces, classifies maximal representations based on the space's type, and constructs examples of dense maximal representations in certain cases.

Contribution

It defines a Toledo number for infinite dimensional spaces, establishes non-existence of Zariski-dense maximal representations in non-tube type spaces, and constructs examples of geometrically dense maximal representations.

Findings

Maximal representations cannot be Zariski-dense in non-tube type spaces.

Boundary maps lead to preservation of finite-dimensional totally geodesic subspaces.

Constructed examples of geometrically dense maximal representations in tube type spaces.

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, 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.