Pseudodifferential operators on manifolds with a Lie structure at infinity

We define and study an algebra ψ_1,0,ν ^∞(M₀) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M₀ whose geometry at infinity is described by a Lie algebra of vector fields ν on a compactification M of M₀ 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,ν ^∞(M₀). We also consider the algebra Diff _ν*(M₀) of differential operators on M₀ generated by ν and C^∞ (M), and show that ψ_1,0,ν^∞(M₀) is a microlocalization of Diff_ν*(M₀). Our 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,ν ^∞(M₀).