Asymptotic Schur orthogonality in hyperbolic groups with application to   monotony
  We prove a generalization of Schur orthogonality relations for certain classes of representations of Gromov hyperbolic groups. We apply the obtained results to show that representations of non-abelian free groups associated to the Patterson-Sullivan measures corresponding to a wide class of invariant metrics on the group are monotonous in the sense introduced by Kuhn and Steger. This in particular includes representations associated to harmonic measures of a wide class of random walks. 
Adrien Boyer and {\L}ukasz Garncarek
math.GR math.RT