  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 Kasparov intersection 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 the intersection product, and relies
on some constructions due to Connes-Skandalis and to Hilsum-Skandalis.
