Reduced Gutzwiller formula with symmetry: case of a finite group

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.