To sum up: We will define complementary series for hyperbolic groups and prove their irreducibility.
More precisely: The complementary series representations are a family of unitary representations that can be realized on the Gromov boundary of the hyperbolic group. They can be viewed as a one-parameter deformation of the quasi-regular representation arising on the boundary, "sometimes" approaching the trivial representation, in a certain sense. The starting point of this work is to find a suitable scalar product in order to unitarize the complementary series. Then, a spectral estimates combined with counting estimates enable us to prove an ergodic theorem à la Bader-Muchnik to achieve the irreducibility.
Joint work with Kevin Boucher and Jean-Claude Picaud.