Getzler's symbol calculus and the composition of differential operators on contact Riemannian manifolds (Q6144557)

Having in mind the symbol calculus of pseudodifferential operators introduced on a spin manifold by \textit{E. Getzler} [Commun. Math. Phys. 92, 163--178 (1983; Zbl 0543.58026)], the author of the paper under review develops a symbol calculus of \(H-\)pseudodifferential operators on a manifold which is enhanced with a canonical Spin\(^C\) structure, namely on a contact Riemannian manifold with contact distribution \(H\), since the Clifford variables associated with the Spin\(^C\) structure furnishes a filtration of symbol space. Following \textit{M.-T. Benameur} and \textit{J. L. Heitsch}'s idea from [J. Noncommut. Geom. 11, No. 3, 1141--1194 (2017; Zbl 06802924)], but a different approach, the author obtains an explicit formula for the top grading part of the composite of polynomial symbols (the symbols of \(H-\)differential operators). The new strategy adopted in the reviewed paper can be applied to Getzler's and Benameur-Heitsch's situations from the two papers mentioned above.