Résumé
We define and study the index map for families of G-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah-Singer results [1]. Finally, we compute the cup product of our index class against the K-homology class of an elliptic operator on the base. Our approach is based on the functorial properties of Kasparov's cup product, and relies on some constructions due to Connes-Skandalis and to Hilsum-Skandalis.