Wave front set at infinity and hyperbolic linear operators with multiple characteristics

We discuss a notion of wave front set which allows us to control the behaviour 'at infinity' of temperate distributions. We obtain the microlocality and microellipticity properties with respect to a class of global pseudodifferential operators and a propagation theorem for the corresponding class of Fourier Integral Operators. Through these results, we prove an adapted global version of the classical theorem concerning the singularities of solutions of hyperbolic Cauchy problems for linear operators with multiple characteristics of constant multiplicities. Finally, we make a comparison with the scattering wave front set introduced by Melrose.