  We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a
Lie group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint
operator on $L^2(\mathbb{R}^d)$. If $\chi$ is an irreducible character of $G$,
we investigate the spectrum of its restriction $\hat{H}\_{\chi}$ to the
symmetry subspace $L^2\_{\chi}(\mathbb{R}^d)$ of $L^2(\mathbb{R}^d)$ coming
from the decomposition of Peter-Weyl. We give semi-classical Weyl asymptotics
for the eigenvalues counting function of $\hat{H}\_{\chi}$ in an interval of
$\mathbb{R}$, and interpret it geometrically in terms of dynamics in the
reduced space $\mathbb{R}^{2d}/G$. Besides, oscillations of the spectral
density of $\hat{H}\_{\chi}$ are described by a Gutzwiller trace formula
involving periodic orbits of the reduced space, corresponding to quasi-periodic
orbits of $\mathbb{R}^{2d}$.
