Reduced Gutzwiller formula with symmetry: Case of a finite group

We consider a classical Hamiltonian H on R2d, invariant by a finite group of symmetry G, whose Weyl quantization H is a self-adjoint operator on L2(Rd). If χ is an irreducible character of G, we investigate the spectrum of its restriction Hχ to the symmetry subspace Lχ2(Rd) of L2(Rd) coming from the decomposition of Peter-Weyl. We give reduced semiclassical asymptotics of a regularized spectral density describing the spectrum of Hχ near a noncritical energy E set-membership sign R. If ΣE:={H=E} is compact, assuming that periodic orbits are nondegenerate in ΣE/G, we get a reduced Gutzwiller trace formula which makes periodic orbits of the reduced space ΣE/G appear. The method is based upon the use of coherent states, whose propagation was given in the work of Combescure and Robert