Bivariant Chern character and longitudinal index

In this paper we consider a family of Dirac-type operators on fibration P →B equivariant with respect to an action of an étale 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 Φ in the cyclic cohomology. A particular case of this result is Connes’ index theorem for étale groupoids [A. Connes, Noncommutative Geometry, Academic Press, 1994] in the case of fibrations. © 2006 Elsevier Inc. All rights reserved.