  In this article we examine the concentration and oscillation effects
developed by high-frequency eigenfunctions of the Laplace operator in a compact
Riemannian manifold. More precisely, we are interested in the structure of the
possible invariant semiclassical measures obtained as limits of Wigner measures
corresponding to eigenfunctions. These measures describe simultaneously the
concentration and oscillation effects developed by a sequence of
eigenfunctions. We present some results showing how to obtain invariant
semiclassical measures from eigenfunctions with prescribed symmetries. As an
application of these results, we give a simple proof of the fact that in a
manifold of constant positive sectional curvature, every measure which is
invariant by the geodesic flow is an invariant semiclassical measure.
