Quantum-classical correspondence on associated vector bundles over locally symmetric spaces

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.