Bounded Kähler class rigidity of actions on Hermitian symmetric spaces

We prove that the conjugacy class of a Zariski dense representation , q>p?1, of a finitely generated group Γ is completely determined by the pull-back via π of a bounded cohomology class in defined in terms of the Kähler form on the associated symmetric space. Under the assumption that is finite dimensional, we show that, up to equivalence, there is only a finite number of such representations for fixed q>p?1; moreover, under the hypothesis that injects into , we estimate the total number of such representations (for all q>p?1) to be bounded above by .