The localized longitudinal index theorem for Lie groupoids and the van Est map

Abstract

We define the "localized index" of longitudinal elliptic operators on Lie groupoids associated with 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 of the "global index", that can be computed by localization. This correspondence between the "global" and "localized" index is given by the van Est map for Lie groupoids.

The index of geometric operators on Lie groupoids

“…In §1 we briefly review the set-up and the statement of the index theorem of [PPT1]. After that, in §2, we prove a Lie algebroid version of the Thom isomorphism.…”

“…The index theorem. In this section we explain the statement of the main index theorem of [PPT1] for Lie groupoids. The structure of this index theorem is the same as that of the original Atiyah-Singer index theorem: it gives an equality between two complex numbers, one defined analytically and the other topologically.…”

“…One of the mysterious looking features of the index theorem of [PPT1] is that the topological side involves integration over the dual of the Lie algebroid of cohomology classes of the pull-back of the Lie algebroid. This is analogous to the cohomological form of the Atiyah-Singer index theorem as an integral over the cotangent bundle of the original manifold.…”

“…The index theorem of [PPT1], recently proved by the authors, takes this chain of generalizations one step further: it gives a cohomological formula for the index, analytically defined by means of noncommutative geometry, of invariant elliptic operators along the fibers of a Lie groupoid over a compact base manifold. The original Atiyah-Singer index theorem is recovered by considering the so-called pair groupoid, whereas the family index theorem follows using the natural groupoid associated to a submersion of manifolds.…”

“…Preliminaries. The setting of the index theorem of [PPT1] is that of Lie groupoids. Here we shall describe the most basic definitions that we need.…”