Résumé
We introduce and study the index morphism for leafwise G-transversally elliptic operators on smooth closed foliated manifolds. We prove the usual axioms of excision, multiplicativity and induction for closed subgroups. In the case of free actions, we relate our index class with the Connes–Skandalis index class of the corresponding leafwise elliptic operators on the quotient foliation. Finally we prove the compatibility of our index morphism with the Gysin morphism and reduce its computation to the case of tori actions. We also construct a topological candidate for an index theorem using the Kasparov–Dirac element for euclidean G-representations.