Index theory for improper actions: localization at units

TL;DR

This paper develops local index theory for Fourier-integral operators associated with non-proper Lie groupoid actions, introducing geometric cocycles to represent cyclic cohomology classes and proving an equivariant index theorem for foliations.

Contribution

It introduces geometric cocycles for Lie groupoids to represent cyclic cohomology classes localized at units and computes their images under the excision map, advancing index theory for non-proper actions.

Findings

Defined geometric cocycles for Lie groupoids.

Computed images of cocycles under excision map.

Proved an equivariant longitudinal index theorem for foliations.

Abstract

We pursue the study of local index theory for operators of Fourier-integral type associated to non-proper and non-isometric actions of Lie groupoids, initiated in a previous work. We introduce the notion of geometric cocycles for Lie groupoids, which allow to represent fairly general cyclic cohomology classes of the convolution algebra of Lie groupoids localized at isotropic submanifolds. Then we compute the image of geometric cocycles localized at units under the excision map of the fundamental pseudodifferential extension. As an illustrative example, we prove an equivariant longitudinal index theorem for a codimension one foliation endowed with a transverse action of the group of real numbers.