La formule des traces pour les revêtements de groupes réductifs connexes. IV. Distributions invariantes

Abstract: We establish the invariant trace formula (`a la Arthur) for the ad\'elic covers of connected reductive groups over a number field, under the hypothesis that the trace Paley-Wiener theorem is verified for all Levi subgroups at the real archimedean places. For instance, this hypothesis can be verified for the metaplectic covers of GL(n), or the twofold metaplectic cover of Sp(2n). We also give simple trace formulae when the test function satisfies certain cuspidality properties.