Fourier integral operators on Lie groupoids

As announced in [36], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study Lagrangian conic submanifolds of the symplectic groupoid T⁎G. This includes their product, transposition and inversion. We also study the relationship between these Lagrangian submanifolds and the equivariant families of Lagrangian submanifolds of T⁎Gx×T⁎Gx parametrized by the units x∈G(0) of G. This allows us to select a subclass of Lagrangian distributions on any Lie groupoid G that deserve the name of Fourier integral G-operators (G-FIOs). By construction, the class of G-FIOs contains the class of equivariant families of ordinary Fourier integral operators on the manifolds Gx, x∈G(0). We then develop for G-FIOs the first stages of the calculus in the spirit of Hormander's work. Finally, we illustrate this calculus in the case of manifolds with boundary.