Index theory of hypoelliptic operators on Carnot manifolds

TL;DR

This paper develops a geometric $K$-theoretic framework to compute the index of hypoelliptic operators on Carnot manifolds, extending previous contact manifold results to more general regular polycontact cases.

Contribution

It extends Baum-van Erp's index theory to regular Carnot manifolds with flat coadjoint orbits, providing explicit $K$-homology solutions for Toeplitz and related operators.

Findings

Computed Fredholm index via operator $K$-theory under geometric conditions.

Extended index formulas from contact to polycontact Carnot manifolds.

Provided explicit $K$-homology solutions for specific hypoelliptic operators.

Abstract

We study the index theory of hypoelliptic operators on Carnot manifolds -- manifolds whose Lie algebra of vector fields is equipped with a filtration induced from sub-bundles of the tangent bundle. A Heisenberg pseudodifferential operator, elliptic in the calculus of van Erp-Yuncken, is hypoelliptic and Fredholm. Under some geometric conditions, we compute its Fredholm index by means of operator $K$-theory. These results extend the work of Baum-van Erp (Acta Mathematica '2014) for co-oriented contact manifolds to a methodology for solving this index problem geometrically on Carnot manifolds. Under the assumption that the Carnot manifold is regular, i.e. has isomorphic osculating Lie algebras in all fibres, and admits a flat coadjoint orbit, the methodology derived from Baum-van Erp's work is developed in full detail. In this case, we develope $K$-theoretical dualities computing the Fredholm index by means of geometric $K$-homology a la Baum-Douglas. The duality involves a Hilbert space bundle of flat orbit representations. Explicit solutions to the index problem for Toeplitz operators and operators of the form "$\Delta_H+\gamma T$" are computed in geometric $K$-homology, extending results of Boutet de Monvel and Baum-van Erp, respectively, from co-oriented contact manifolds to regular polycontact manifolds.