Pseudo differential operators on manifolds with a Lie structure at infinity

We define and study an algebra Psi(infinity)(1,0,nu)(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 Psi(infinity)(1,0,nu)(M-0) . We also consider the algebra Diff nu*(M-0) of differential operators on M-0 generated by nu and C infinity(M), and show that Psi(infinity)(1,0,nu)(M-0) is a microlocalization of Diff nu*(M-0). Our construction solves a prob lem posed by Melrose in 1990. Finally, we introduce and study semi-classical and "suspended" versions of the algebra Psi(infinity)(1,0,nu)(M-0).