A Lie group structure for Fourier integral operators (Q1089646)

On a compact manifold M, let \({\mathcal D}_{\theta}(T^*M\setminus 0)\) be the group of smooth diffeomorphisms of \(T^*M\setminus 0\) which preserve the canonical one-form \(\theta\). Let \((FIO)_*\) \((resp.(\psi DO)_*)\) denote the group of invertible formal Fourier integral operators (resp. pseudodifferential operators) on M. Then we have an exact sequence of groups \(I\to (\psi DO)_*\to^{j}(FIO)\to^{\pi}{\mathcal D}_{\theta}(T^*M\setminus 0)\to e\), where j is inclusion and \(\pi (A)=\eta\) if graph \(\eta \subset (T^*M\setminus 0)\times (T^*M\setminus 0)\) is the canonical relation associated to A. The authors define a Lie group structure on \((FIO)_*\) by doing this for \((\psi DO)_*\) and \({\mathcal D}_{\theta}(T^*M\setminus 0)\) and constructing a local section of the fibration \(\pi\) satisfying compatibility conditions that allow them to define charts on \((FIO)_*\) that make the group operations \(C^{\infty}\).