Pseudo-Kähler geometry of Hitchin representations and convex projective structures

In this thesis we study the symplectic and pseudo-Riemannian geometry of the PSL(3,R)-Hitchin component associated with a closed orientable surface, using an approach coming from the theory of symplectic reduction in an infinite-dimensional context. In the case where the closed surface is homeomorphic to a torus, for each choice of a smooth real function with certain properties, we prove the existence of a pseudo-Kähler metric on the deformation space of properly convex projective structures. Moreover, we define a circle action and a SL(2,R)-action on the aforementioned space, which turn out to be Hamiltonian with respect to our symplectic form, and we give an explicit description of the moment maps. Then, we study the symplectic geometry of the deformation space as a completely integrable Hamiltonian system, and we find a geometric global Darboux frame for the symplectic form using the theory of complete Lagrangian fibrations. In the case of higher genus we define a mapping class group invariant pseudo-Kähler metric on the Hitchin component, by using a general construction of Donaldson. The complex structure is exactly the one coming from the identification with the holomorphic bundle of cubic differentials over Teichmüller space. In particular, we prove that Wang's equation for hyperbolic affine spheres has an interpretation as moment map for the action of an infinite-dimensional Lie group.