We consider a classical Hamiltonian H on R2d, invariant by a
Lie group of symmetry G, whose Weyl quantization H^ is a selfadjoint
operator on L2(Rd). If χ is an irreducible character of G,
we investigate the spectrum of its restriction H^_χ to the
symmetry subspace L2_χ(Rd) of L2(Rd) coming
from the decomposition of Peter-Weyl. We give semi-classical Weyl asymptotics
for the eigenvalues counting function of H^_χ in an interval of
R, and interpret it geometrically in terms of dynamics in the
reduced space R2d/G. Besides, oscillations of the spectral
density of H^_χ are described by a Gutzwiller trace formula
involving periodic orbits of the reduced space, corresponding to quasi-periodic
orbits of R2d.Comment: 23 page