Pseudodifferential operators on differential groupoids

We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincar'e-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual $A^{\ast }$ of a Lie algebroid.