A symbol calculus for foliations

Abstract

The classical Getzler rescaling theorem of [15] is extended to the transverse geometry of foliations. More precisely, a Getzler rescaling calculus, [15], as well as a Block–Fox calculus of asymptotic pseudodifferential operators (A‰$\Psi$DOs), [10], is constructed for all transversely spin foliations. This calculus applies to operators of degree $m$ globally times degree $\ell$ in the leaf directions, and is thus an appropriate tool for a better understanding of the index theory of transversely elliptic operators on foliations [13]. The main result is that the composition of A‰$\Psi$DOs is again an A$\Psi$‰DO, and includes a formula for the leading symbol. Our formula is more complicated due to its wide generality but its form is essentially the same, and it simplifies notably for Riemannian foliations. In short, we construct an asymptotic pseudodifferential calculus for the “leaf space” of any foliation. Applications will be derived in [5,6] where we give a Getzler-like proof of a local topological formula for the Connes–Chern character of the Connes–Moscovici spectral triple of [20], as well as the (semi-finite) spectral triple given in [5], yielding an extension of the seminal Atiyah–Singer $L^2$ covering index theorem, [2], to coverings of “leaf spaces” of foliations.