  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.
