Twisted index theory for foliations [Théorie de l'indice tordue pour des feuilletages]

For any Lie groupoid with a twisting, we define an analytic index morphism using the Connes tangent groupoid. This morphism agrees with the one of the Lie groupoid when the twisting is trivial. We discuss a longitudinal index theorem, geometric cycles, push-forward maps and Baum-Connes assembly maps for foliations with a twisting on the space of leaves