  For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and
an associated vector bundle $\mathbf V_\tau$ over the unit cosphere bundle
$S^\ast\mathcal M$, we give a precise description of those classical
(Pollicott-Ruelle) resonant states on $\mathbf V_\tau$ that vanish under
covariant derivatives in the Anosov-unstable directions of the chaotic geodesic
flow on $S^\ast\mathcal M$. In particular, we show that they are isomorphically
mapped by natural pushforwards into generalized common eigenspaces of the
algebra of invariant differential operators $D(G,\sigma)$ on compatible
associated vector bundles $\mathbf W_\sigma$ over $\mathcal M$. As a
consequence of this description, we obtain an exact band structure of the
Pollicott-Ruelle spectrum. Further, under some mild assumptions on the
representations $\tau$ and $\sigma$ defining the bundles $\mathbf V_\tau$ and
$\mathbf W_\sigma$, we obtain a very explicit description of the generalized
common eigenspaces. This allows us to relate classical Pollicott-Ruelle
resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of
sections of $\mathbf W_\sigma$. Our methods of proof are based on
representation theory and Lie theory.
