Bivariant Chern character and the longitudinal index theory

In this paper we consider a family of Dirac-type operators on fibration $P \to B$ equivariant with respect to an action of an etale groupoid. Such a family defines an element in the bivariant $K$ theory. We compute the action of the bivariant Chern character of this element on the image of Connes’ map $\mathrm{\Phi }$ in the cyclic cohomology. A particular case of this result is Connes’ index theorem for etale groupoids in the case of fibrations.