Adiabatic groupoid, crossed product by R + ⁎ and pseudodifferential calculus
Résumé
Let G be a Lie groupoid. Recall that the associated adiabatic groupoid G ad 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 G ga of the natural action of R * + on the adiabatic groupoid G ad 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 G ga. As an important intermediate step, we express a pseudodifferential operator on G as an integral associated to a smoothing operator on the adiabatic groupoid G ad of G.
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