Pseudodifferential operators on manifolds with a Lie structure at infinity

We define and study an algebra Ψ ∞ 1,0,V (M 0 ) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M 0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M 0 to a compact manifold with corners.We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M 0 ).We also consider the algebra Diff * V (M 0 ) of differential operators on M 0 generated by V and C ∞ (M ), and show thatOur construction solves a problem posed by Melrose in 1990.Finally, we introduce and study semi-classical and "suspended" versions of the algebra Ψ ∞ 1,0,V (M 0 ).