The tangent groupoid of a Heisenberg manifold

As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes’ tangent groupoid of a manifold $M$. As it is well known for a Heisenberg manifold $(M,H)$ the relevant notion of tangent is rather that of Lie group bundle of graded 2-step nilpotent Lie groups $GM$. We then construct the tangent groupoid of $(M,H)$ as a differentiable groupoid ${\text{cG}}_{H}M$ encoding the smooth deformation of $M\times M$to$GM$. In this construction a crucial use is made of a refined notion of privileged coordinates and of a tangent approximation result for Heisenberg diffeomorphisms.