  We generalize parts of a special non-Euclidean calculus of pseudodifferential
operators, which was invented by S. Zelditch for hyperbolic surfaces, to
symmetric spaces $X=G/K$ of the noncompact type and their compact quotients
$Y=\Gamma\backslash G/K$. We sometimes restrict our results to the case of rank
one symmetric spcaes. The non-Euclidean setting extends the defintion of
so-called Patterson-Sullivan distributions, which were first defined by N.
Anantharaman and S. Zelditch for hyperbolic systems, in a natural way to
arbitrary symmetric spaces of the noncompact type. We find an explicit
intertwining operator mapping Patterson-Sullivan distributions into Wigner
distributions. We study the important invariance and equivariance properties of
these distributions. Finally, we describe asymptotic properties of these
distributions.
