The tangent grupoid of a Heisenberg manifold

Abstract

As a step towards proving an index theorem for hypoelliptic operators on Heisenberg manifolds, including for those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes? tangent groupoid of a manifold. As is well known for a Heisenberg manifold (M,H) the relevant notion of tangent bundle is rather that of a Lie group bundle of graded 2-step nilpotent Lie groups GM. We define the tangent groupoid of (M,H) as a differentiable groupoid gHM encoding the smooth deformation of M × M to GM. In particular, this construction makes a crucial use of a refined notion of privileged coordinates and of a tangent-approximation result for Heisenberg diffeomorphisms.