On quantum ergodicity for vector bundles

In the present paper we develop a framework in which questions of quantum ergodicity for operators acting on sections of hermitian vector bundles over Riemannian manifolds can be studied. We are particularly interested in the case of locally symmetric spaces. For locally symmetric spaces, we extend the recent construction of Silberman and Venkatesh (math.RT/0407413) of representation theoretic lifts to vector bundles.