On order-preserving representations

In this article, we introduce order-preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order-preserving representations to weakly maximal representations, previously introduced by the authors, we show that order-preserving representations into Lie groups of Hermitian type are faithful with discrete image, and that the set of order-preserving representations is closed in the representation variety. For Lie groups of Hermitian type whose associated symmetric space is of tube type, we give a geometric characterization of these representations in terms of the causal structure on the Shilov boundary.