Weakly maximal representations of surface groups

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. They are defined in terms of invariants in bounded cohomology and extend considerably the scope of maximal representations. We prove that weakly maximal representations are discrete and injective and describe the structure of the Zariski closure of the image. An interesting feature of these representations is that they admit an elementary topological characterization in terms of bi-invariant orderings. In particular if the target group is Hermitian of tube type, the ordering can be described in terms of the causal structure on the Shilov boundary.