Adiabatic groupoid, crossed product by R+⁎ and pseudodifferential calculus

Let G be a Lie groupoid. Recall that the associated adiabatic groupoid Gad is the groupoid obtained by the deformation to the normal cone construction corresponding to the inclusion G(0)→G. In the present paper, we consider the crossed product Gga of the natural action of R+⁎ on the adiabatic groupoid Gad associated with any Lie groupoid G. We construct an explicit Morita equivalence between the exact sequence of order 0 pseudodifferential operators on G and (a restriction of) the natural exact sequence associated with Gga. As an important intermediate step, we express a pseudodifferential operator on G as an integral associated with a smoothing operator on the adiabatic groupoid Gad of G.