Index theory for improper actions: localization at units

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.