The tangent grupoid of a Heisenberg manifold

Abstract

RAPHAËL PONGEAs 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 G M. We define the tangent groupoid of (M, H) as a differentiable groupoid Ᏻ H M encoding the smooth deformation of M × M to G M. In particular, this construction makes a crucial use of a refined notion of privileged coordinates and of a tangent-approximation result for Heisenberg diffeomorphisms. IntroductionA somewhat long standing open question is the existence of an index theorem for geometric operators on contact and CR manifolds. In this context the operators are not elliptic, so we cannot apply the classical index theorem of Atiyah-Singer [1968a;1968b]. The natural pseudodifferential tool to deal with hypoelliptic operators on contact and CR manifolds is provided by the Heisenberg calculus of BealsGreiner [1988] and Taylor [1984]. The latter holds in full generality for Heisenberg manifolds, that is, manifolds M together with a distinguished hyperplane bundle H ⊂ T M. This definition includes that of CR and contact manifolds, as well as that of codimension one foliations and confoliations. Therefore, what we would like to have is an analogue of the Atiyah-Singer theorem for hypoelliptic operators on Heisenberg manifolds.There are various proofs of the Atiyah-Singer index theorem. A simple and fairly general proof is that of Connes [1994, Sect. II.5]. A salient feature in Connes' proof is the use of the tangent groupoid of a manifold, that is, the differentiable groupoid encoding the smooth deformation of M × M to T M (see [Connes 1994;Hilsum and Skandalis 1987]).In this paper, as a step towards proving an index theorem for hypoelliptic operators on Heisenberg manifolds, we construct an analogue for Heisenberg manifolds MSC2000: primary 58H05; secondary 53C10, 53D10, 32V05.

Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry

“…This isomorphism commutes with the dilations in (2.4) and it can be further shown that it gives rise to a Lie group isomorphism from GM onto GM (see [50]). For a ∈ U we let ψ a : R d+1 → R d+1 denote the unique affine change of variable such that ψ a (a) = 0 and (ψ a ) * X j (0) = ∂ ∂x j for j = 0, .…”

“…Next, the terminology Heisenberg manifold stems from the fact that the relevant tangent structure in this setting is that of a bundle GM of graded nilpotent Lie groups (see [1,3,19,23,27,50,56,64]). This tangent Lie group bundle can be described as follows.…”

“…In other words the class of [X, Y ](a) modulo H a depends only on the values X(a) and Y (a), not on the germs of X and Y near a (see [50]). This allows us to define the tangent Lie algebra bundle gM as the vector bundle (TM/H ) ⊕ H together with the grading and field of Lie brackets such that, for sections X 0 , Y 0 of TM/H and X , Y of H , we have…”

On the tangent groupoid of a filtered manifold

“…The Heisenberg calculus was developed by Folland-Stein [12] and Taylor [23] in the 1970s. An adaptation of the tangent groupoid to the Heisenberg calculus was developed in [8,21], which led to the index theorem for the Heisenberg calculus [9,10].…”

Lie groupoids, pseudodifferential calculus, and index theory

“…A deformation groupoid taking into account this inhomogeneous calculus was constructed independently in ( [98,25,27,26]) and [111,70,113]. Both these constructions are rather technical and based on higher jets.…”