Maximal representations of complex hyperbolic lattices in infinite dimension.

Unlike lattices in higher rank, lattices of simple Lie groups in rank 1 are not rigid. This gives rise to the Teichmüller spaces for example.
For representations of lattices of the isometry group of the complex hyperbolic lattices in Hermitian Lie groups, the Kähler form yields a numerical invariant, the Toledo number. When this number is maximal, these representations are rigid when the dimension is at least 2.
We will focus on infinite dimensional representations of complex hyperbolic lattices that are not unitary but preserve a Hermitian form of finite index. This gives actions by isometries on infinite dimensional Hermitian symmetric spaces and one can define a Toledo number as well.
We will see that for surface groups, one can create maximal representations that do not preserve any finite dimensional subspace. Conversely, for complex hyperbolic lattices in dimension at least 2, these maximal representations factor through a finite dimensional Lie subgroup.