  We generalize an equidistribution theorem \`a la Bader-Muchnik for
operator-valued measures constructed from a family of boundary representations
associated with Gibbs measures in the context of convex cocompact discrete
group of isometries of a simply connected connected Riemannian manifold with
pinched negative curvature. We combine a functional analytic tool, namely the
property RD of hyperbolic groups, together with a dynamical tool: an
equidistribution theorem of Paulin, Pollicott and Schapira inspired by a result
of Roblin. In particular, we deduce irreducibility of these new classes of
boundary representations.
