An important application of bounded cohomology is the theory of maximal representations: a class of homomorphisms of fundamental groups of Kähler manifolds (most notably fundamental groups of surfaces and finite volume manifolds covered by complex hyperbolic spaces) in Hermitian Lie groups (such as Sp(2n,R) or SU(p,q)). These representations have striking geometric properties and, in some cases, are even superrigid. In my talk I will discuss a joint work with Bruno Duchesne and Jean Lécureux in which we study generalizations to actions on infinite dimensional Hermitian symmetric spaces.
