  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.
