  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.
