Pseudodifferential operators on manifolds with a Lie structure at infinity

Several examples of non-compact manifolds $M_0$ whose geometry at infinity is described by Lie algebras of vector fields $V\subset \mathrm{\Gamma }(TM)$ (on a compactification of $M_0$ to a manifold with corners $M$) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of vector fields – baptised “manifolds with a Lie structure at infinity” there – was studied from an axiomatic point of view. In this paper, we define and study the algebra ${\mathrm{\Psi }}_{1,0,\text{VV}}^{\mathrm{\infty }}(M_0)$, which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at infinity $V\subset \mathrm{\Gamma }(TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to ${\mathrm{\Psi }}_{1,0,V}^{\mathrm{\infty }}(M_0)$. We also consider the algebra $\text{DiffV}\ast (M_0)$ of differential operators on $M_0$ generated by $V$ and $\text{CI}(M)$, and show that ${\mathrm{\Psi }}_{1,0,V}^{\mathrm{\infty }}(M_0)$ is a microlocalization'' of $\DiffV{*}(M_0)$. Finally, we introduce and study semi-classical and suspended’’ versions of the algebra ${\mathrm{\Psi }}_{1,0,V}^{\mathrm{\infty }}(M_0)$. Our construction solves a problem posed by Melrose in his talk at the ICM in Kyoto.