The hyperbolic 3-manifold associated to a Fuchsian
representation of a surface group admits a fibration over the surface with geodesic fibers that extends to a fibration of the conformal boundary. This also holds for almost Fuchsian representations, but not in general for quasi-Fuchsian representations.

I will present an analog of this picture for representations of surface groups into Sp(2n,R). Among these representations, there exists a union of connected components containing only discrete and faithful representations, called maximal representations. We will consider fibrations by projective codimension $2$ subspaces of a projective convex set containing the symmetric space of Sp(2n,R). These subspaces are described by pencils of quadrics, and we will see that one can characterize maximal representations by the existence of such a continuous fibration, satisfying some additional properties. The hyperbolic 3-manifold associated to a Fuchsian representation of a surface group admits a fibration over the surface with geodesic fibers that extends to a fibration of the conformal boundary. This also holds for almost Fuchsian representations, but not in general for quasi-Fuchsian representations.