  We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a
finite 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
reduced semi-classical asymptotics of a regularised spectral density describing
the spectrum of $\hat{H}\_\chi$ near a non critical energy $E\in\mathbb{R}$. If
$\Sigma\_E:=\{H=E \}$ is compact, assuming that periodic orbits are
non-degenerate in $\Sigma\_E/G$, we get a reduced Gutzwiller trace formula
which makes periodic orbits of the reduced space $\Sigma\_E/G$ appear. The
method is based upon the use of coherent states, whose propagation was given in
the work of M. Combescure and D. Robert.
