  We define the "localized index" of longitudinal elliptic operators on Lie
groupoids associated to Lie algebroid cohomology classes. We derive a
topological expression for these numbers using the algebraic index theorem for
Poisson manifolds on the dual of the Lie algebroid. Underlying the definition
and computation of the localized index, is an action of the Hopf algebroid of
jets around the unit space, and the characteristic map it induces on Lie
algebroid cohomology. This map can be globalized to differentiable groupoid
cohomology, giving a definition as well as a computation of the "global index".
The correspondence between the "global" and "localized" index is given by the
van Est map for Lie groupoids.
