There is a remarkable relation between two kinds of phase space distributions
associated to eigenfunctions of the Laplacian of a compact hyperbolic manifold:
It was observed in \cite{AZ} that for compact hyperbolic surfaces
$X_{\Gamma}=\Gamma\backslash\mathbb{H}$ Wigner distributions $\int_{S^*
X_{\Gamma}} a dW_{ir_j} = < Op(a)\phi_{ir_j},\phi_{ir_j}>_{L^2(X_{\Gamma})}$
and Patterson--Sullivan distributions $PS_{ir_j}$ are asymptotically equivalent
as $r_j\to\infty$. We will generalize the definitions of these distributions to
all rank one symmetric spaces of noncompact type and introduce off-diagonal
elements $PS_{\lambda_j,\lambda_k}$. We give explicit relations between
off-diagonal Patterson--Sullivan distributions and off-diagonal Wigner
distributions and generalize the asymptotic relation between these
distributions.