Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces

For a compact Riemannian locally symmetric space M of rank 1 and an associated vector bundle V-tau over the unit cosphere bundle S*M, we give a precise description of those classical (Pollicott-Ruelle) resonant states on V-tau that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on S*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 W-sigma over 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 V-tau and 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 W-sigma. Our methods of proof are based on representation theory and Lie theory.