  We study the ergodic properties of Schr\"odinger operators on a compact
connected Riemannian manifold $M$ without boundary in case that the underlying
Hamiltonian system possesses certain symmetries. More precisely, let $M$ carry
an isometric and effective action of a compact connected Lie group $G$. Relying
on an equivariant semiclassical Weyl law proved in Part I of this work, we
deduce an equivariant quantum ergodicity theorem under the assumption that the
symmetry-reduced Hamiltonian flow on the principal stratum of the singular
symplectic reduction of $M$ is ergodic. In particular, we obtain an equivariant
version of the Shnirelman-Zelditch-Colin-de-Verdi\`ere theorem, as well as a
representation theoretic equidistribution theorem. If $M/G$ is an orbifold,
similar results were recently obtained by Kordyukov. When $G$ is trivial, one
recovers the classical results.
