abstract: Anantharam and Zelditch observed a remarkable connection between Wigner and Patterson-Sullivan distributions on compact hyperbolic surfaces. These are distributions associated with the eigenvalues of the Laplace-Beltrami operators and satisfy invariance properties under the geodesic flow. A key tool to establish this connection is a specific pseudodifferential calculus adapted to the symmetries of the situation. We reformulate these results in terms of group theory and generalize them to rank 1 symmetric spaces. This is joint work with M. Schroeder.