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
the relevant
notion of tangent bundle is rather that of a Lie group bundle of graded 2-step nilpotent Lie groups
. We define the tangent
groupoid of
as a differentiable
groupoid
encoding the
smooth deformation of
to
.
In particular, this construction makes a crucial use of a refined notion of
privileged coordinates and of a tangent-approximation result for Heisenberg
diffeomorphisms.
