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

Semiclassical analysis and symmetry reduction II. Equivariant quantum ergodicity for invariant Schr\"odinger operators on compact manifolds