Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric 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 maximal representations. When the target is not tube type, we show that there cannot be Zariski-dense representations, whenever existence boundary map can guaranteed, representation preserves finite-dimensional totally geodesic subspace action maximal. In opposite direction, construct examples geometrically dense in space type finite rank. Our approach based study maps, are able low ranks or under some suitable Zariski density assumption, circumventing lack local compactness setting.