C. R. Acad. Sci. Paris, t. 329, Série 1, p. 985-990, 1999 Géométrie différentielle/Differential Geometry

Differential Geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups

Géométrie différentielle sur les espaces de Carnot et exposants de Novikov-Shubin des groupes de Lie nilpotents

Michel Rumin

Abstract - We prove that the near cohomology [7] of a stratified nilpotent Lie group G concentrates on a sub-complex of the de Rham complex. This subcomplex is conjugated to a differential complex d_{c} acting on the Lie algebra cohomology of G. These constructions hold on any Carnot-Caratheodory space regular in some sense. The main result is that d_{c} exhibits hypoelliptic regularity. This allows to give bounds, and sometimes to compute some of the Novikov-Shubin numbers of G.

Résumé - Nous montrons que la presque cohomologie [7] d’un groupe de Lie nilpotent stratifié G se concentre sur un sous-complexe du complexe de de Rham. Celui-ci est lui même conjugué à un complexe d_{c} agissant sur la cohomologie de l’algèbre de Lie de G. Ces constructions s’appliquent à tous les espaces de Carnot-Caratheodory réguliers en un certain sens. Le résultat principal est que d_{c} possède une régularité hypoelliptique. Ceci permet de donner des bornes, et parfois de calculer, certains exposants de Novikov-Shubin de G.

Version française abrégée

Une structure de Carnot-Caratheodory équirégulière sur une variété M est la donnée d’une filtration de T M par des sous-fibrés H_{i} satisfaisant H_{i+1} = [H, H_{i}] et H_{r} = T M pour un certain entier r (cf [6] par ex). Le prototype (et le cône tangent) d’une telle structure s’obtient sur un groupe de Lie nilpotent stratifié G, c’est-à-dire un groupe dont l’algèbre de Lie 𝔤 est graduée et engendrée par les vecteurs de poids 1.

La filtration de T M induit une filtration H_{p}^{∗} de Λ^{∗}T M, et par dualité, une filtration descendante F_{p} = Ann(H_{p−1}) ⊂ Λ^{∗}T^{∗}M. Le fait que cette dernière soit stable par différentiation est dual à la propriété [H_{p}, H_{q}] ⊂ H_{p+q}. Dans l’esprit de la théorie adiabatique, la suite spectrale naturellement associée à ce complexe filtré, qui converge vers le gradué de la cohomologie de M, doit aussi jouer un rôle dans l’étude de certains problèmes spectraux asymptotiques sur M ([12]). Nous nous intéressons ici aux vitesses de décroissance en grand temps de la chaleur sur les formes d’un groupe stratifié G.

Soit G un tel groupe, muni d’une métrique pour laquelle les composantes homogènes de 𝔤 sont orthogonales. On suppose que G est rationnel, c’est-à-dire possède un sousgroupe discret cocompact Γ. D’après [7], le spectre du Laplacien près de 0 se décrit géométriquement par la presque cohomologie. C’est la famille de cônes de Hilbert C_{ε} = \{α ∈ L^{2}(Ω^{p}M)/\overline{im d}, ||dα|| ≤ ε||α||\}. En particulier la distribution spectrale de δd se lit dans C_{ε} par F_{δd}(ε^{2}) = dim_{Γ}(E_{δd}(ε^{2})) = sup_{L}(dim_{Γ}L), où L décrit les sous-Hilbert Γinvariants de C_{ε}. On retrouve ainsi les exposants de Novikov-Shubin α_{p} = lim_{ε→0} \frac{ln F_{δd}(ε)}{ln ε} donnant la vitesse de décroissance de la chaleur, sur les formes cofermées de Ω^{p}G ([7]).

Le lien avec la suite spectrale introduite ci-dessus est que celle-ci décrit, au sens des séries formelles en ε, la structure des formes satisfaisant d(h_{ε}^{∗}α_{ε}) = O(ε^{p}) à travers les dilatations h_{ε} de G. L’obstruction à produire de la presque cohomologie par dilatation provient de d_{0} = d agissant sur F_{p}/F_{p+1}. Sur une variété C-C générale, d_{0} en x_{0} est la différentielle de l’algèbre de Lie nilpotente tangente 𝔤_{x_{0}}. C’est un opérateur algébrique qui possède donc un inverse partiel d_{0}^{−1} = (δ_{0}d_{0})^{−1}δ_{0} pour une métrique donnée.

On pose alors E_{0} = ker d_{0}∩ker d_{0}^{−1} ≃ ker d_{0}/ im d_{0} = H^{∗}(𝔤_{x_{0}}). On dit qu’une structure C-C est E_{0}-régulière en x_{0} si E_{0} est de dimension localement constante au voisinage de x_{0}, condition génériquement satisfaite. La suite spectrale considérée agit sur E_{0}. L’algébricité de d_{0} permet de le faire plus géométriquement.

Théorème 1. - Soit M une variété C-C E_{0}-régulière. Il existe deux sous-complexes (E, d) et (F, d) du complexe de de Rham tels que Ω^{∗}M = E ⊕F, et tels que la projection sur E soit une équivalence d’homotopie de la forme Π_{E} = Id −Qd − dQ où Q est un opérateur différentiel. D’autre part, (E, d) est lui-même conjugué par Π_{E_{0}} à un complexe différentiel (E_{0}, d_{c}).

Ceci entraîne un résultat de concentration de presque-cohomologie. On sait, d’après [7], qu’une équivalence d’homotopie bornée entre deux complexes Hilbertiens (H, d) et (H′, d′) induit une équivalence des presque-cohomologies au sens qu’il existe des applications bornées f : C_{ε} → C′_{Kε} et g : C′_{ε} → C_{Kε} inversibles sur leurs images. Ainsi, les presque-cohomologies de (Ω^{∗}T^{∗}M, d) et du complexe de de Rham spectralement tronqué (E_{Δ([0,1])}, d) sont équivalentes. Sur M = G comme ci-dessus, on obtient alors :

Corollaire 2. - La presque-cohomologie C_{ε} de (E_{Δ([0,1])}, d) se contracte sur E (c’est-à-dire que ||Π_{F} ⌈C_{ε}|| ≤ Kε).

- Les complexes (E_{Δ([0,1])}, d), (E, d) et (E_{0}, d_{c}) ont des presque-cohomologies équivalentes.

Pour exploiter analytiquement cette construction, on introduit sur les variétés C-C une notion de régularité utilisant le calcul hypoelliptique développé dans [2]. Soit |∇| un opérateur scalaire pseudo-différentiel d’ordre 1 inversible dans ce calcul (voir [2] thm 6.1). On l’étend diagonalement à Ω^{∗}T^{∗}M et on note N le degré homogène induit par la filtration F_{p}. On dit qu’un opérateur P sur Ω^{∗}T^{∗}M est C-C elliptique si P^{∇} = |∇|^{−N}P |∇|^{N} a une paramétrix dans ce calcul.

Théorème 3. Les complexes d et d_{c} sont C-C elliptiques.

La preuve consiste à montrer que les complexes d^{∇} et d_{c}^{∇} sont exacts au niveau symbolique, c’est-à-dire à travers les représentations unitaires de G_{x_{0}} (voir ci-dessous).

Ce résultat permet d’utiliser le complexe d_{c} pour estimer certains exposants α_{p} sur les groupes considérés. On pose β_{p} = \frac{N(G)}{2α_{p}}, où N(G) = ∑ i dim(𝔤_{i}/𝔤_{i−1}) est la dimension homogène de G. L’espace E_{0}^{∗} = H^{∗}(𝔤) se décompose en composantes homogènes pour N. On dit que E_{0}^{k} est de poids pur p si N = p sur cet espace. E_{0}^{1} est toujours de poids pur 1 ici. Lorsque E_{0}^{k} est de poids pur p, on peut considérer les opérateurs tronqués d_{c}^{i} : E_{0}^{k} → E_{0}^{k+1}/F_{p+i}^{k+1}, et le plus petit entier i, noté r_{k}(d_{c}), pour lequel d_{c}^{i} + δ_{c} soit C-C elliptique. Cet entier s’interprète comme l’ordre de dégénérescence de la suite spectrale associée au complexe filtré d_{c} au niveau du calcul symbolique.

Théorème 4. Soit G un groupe nilpotent, stratifié, rationnel. Si E_{0}^{k} est de poids pur p, on a

sup(N_{min}(E_{0}^{k+1}) − p, 1) ≤ β_{k} ≤ r_{k}(d_{c}) ≤ N_{max}(E_{0}^{k+1}) − p.

En particulier, on a toujours 1 ≤ β_{1} ≤ r où r est le rang de nilpotence de G. Les groupes G à présentation quadratique (voir [3,4]) sont précisément ceux tels que E_{0}^{2} soit de poids pur 2 et ont donc β_{1} = 1. Il en est ainsi des groupes avec r = 2 et possédant un plan legendrien Ω-isotrope régulier (cf [6]). A l’opposé, les quotients des algèbres de Lie libres par un idéal engendré par des éléments de degré ≥ r + 1 ont r ≤ β_{1}. Enfin, il existe des groupes tels que β_{1} = r_{1}(d_{c}) = 1 et N_{max}(E_{0}^{2}) = 3.

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1. Filtered differential geometry on C-C spaces

An equiregular Carnot-Caratheodory structure on a manifold M consists in a filtration of T M by bundles H_{i} such that H_{i+1} = [H, H_{i}] and H_{r} = T M for some r (see [6]). The model for such a geometry is a stratified nilpotent Lie group G, whose Lie algebra is graded and generated by vectors of weight 1.

This filtration of T M induces a filtration H_{p}^{∗} = span{∧_{i=1}^{r}(∧^{k_{i}}H_{i}); ∑_{i=1}^{r} ik_{i} ≤ p} of Λ^{∗}T M and a dual descending filtration F_{p} = Ann(H_{p−1}) ⊂ Λ^{∗}T^{∗}M. This F_{p} is dstable as comes from the dual property [H_{p}, H_{q}] ⊂ H_{p+q}. As in the adiabatic theory, the spectral sequence induced by this filtered complex should play a role in some asymptotic spectral problems on M (see [12]). We are concerned here with the rate of decay of heat on forms at large time on G. The relation is that, on one hand, one knows (see [7]) that the heat decay is encoded in the near-cohomology of G, that is in the family of Hilbert cones C_{ε} = \{α ∈ L^{2}(Ω^{p}G)/\overline{im d}, ||dα|| ≤ ε||α||\}, and on the other hand, the above spectral sequence describes, in the sense of power series, the structure of forms α_{ε} such that d(h_{ε}^{∗}α_{ε}) = O(ε^{p}) through the dilations h_{ε} of G. The obstruction to producing near-cohomology by dilations is d_{0} = d acting on F_{p}/F_{p+1}. On a general C-C space, d_{0} at x_{0} is the differential of the tangent Lie algebra 𝔤_{x_{0}} whose structure comes from the bracket [ ,]_{0} : H_{p}/H_{p−1} × H_{q}/H_{q−1} → H_{p+q}/H_{p+q−1}. It is an algebraic operator so that a partial inverse d_{0}^{−1} = (δ_{0}d_{0})^{−1}δ_{0} can be chosen depending on a given metric.

Let E_{0} = ker d_{0} ∩ ker d_{0}^{−1} ≃ ker d_{0}/imd_{0} = H^{∗}(𝔤_{x_{0}}), and call a C-C structure E_{0}- regular at x_{0} if dimE_{0} is locally constant near x_{0}. The spectral sequence we consider works algebraically on E_{0} but the nature of d_{0} here allows one to achieve this more geometrically inside the de Rham complex. Namely, the operator r = Id −d_{0}^{−1}d − dd_{0}^{−1} induces a homotopical equivalence whose zero order term is r_{0} = Π_{E_{0}}. This map can be iterated in order to obtain representatives of forms with the smallest Π_{E_{0}^{⊥}} part.

Theorem 1. - Let M be a E_{0}-regular C-C space. The de Rham complex is the direct sum of two sub-complexes E = ker d_{0}^{−1} ∩ ker(d_{0}^{−1}d) and F = im d_{0}^{−1} + im(dd_{0}^{−1}). The projection Π_{E} on E along F is a homotopical equivalence of the form Π_{E} = Id −Qd−dQ, where Q is a differential operator. The maps r^{k} converge to Π_{E}.

- One has Π_{E_{0}}Π_{E}Π_{E_{0}} = Π_{E_{0}} and Π_{E}Π_{E_{0}}Π_{E} = Π_{E}, so that (E, d) is conjugated through Π_{E_{0}} and Π_{E} to the complex (E_{0}, d_{c}) with d_{c} = Π_{E_{0}}dΠ_{E}Π_{E_{0}}.

The main result we need is :

Lemma. The map d_{0}^{−1}d induces an isomorphism on im d_{0}^{−1}, whose inverse is a differential operator P. Define then Q = P d_{0}^{−1} for theorem 1.

Proofs. - On im d_{0}^{−1}, one can write d_{0}^{−1}d = Id +D where D = d_{0}^{−1}(d − d_{0}) is a nilpotent differential operator since D(F_{p}) ⊂ F_{p+1}. One has then P = (d_{0}^{−1}d)^{−1} = Id −D + D^{2} · · ·.

- Let Q = P d_{0}^{−1} and consider Π = Qd + dQ. By construction im Q ⊂ im P ⊂ im d_{0}^{−1}, so that, im Π ⊂ F = im(d_{0}^{−1} + dd_{0}^{−1}). Conversely, Qd_{0}^{−1} = P (d_{0}^{−1})^{2} = 0 and Qd = Id on im d_{0}^{−1} so that Π = Id on F. Regarding E = ker d_{0}^{−1} ∩ ker(d_{0}^{−1}d), one has Π = P d_{0}^{−1}d + dP d_{0}^{−1} = 0 on it. Moreover, d_{0}^{−1}(Id −Π) = d_{0}^{−1} − d_{0}^{−1}dP d_{0}^{−1} − d_{0}^{−1}Qd = 0 and d_{0}^{−1}d(Id −Π) = d_{0}^{−1}d − d_{0}^{−1}dQd = 0 so that im(Id −Π) ⊂ E. This shows that Π is the projection on F along E.

- The statement regarding r follows from the fact that first r = Id −d_{0}^{−1}d − dd_{0}^{−1} = Id on E. Also r preserves and induces a nilpotent operator on F. Namely, on im d_{0}^{−1}, one has r = 1 − d_{0}^{−1}d = −d_{0}^{−1}(d − d_{0}) = −D (the above nilpotent operator), whereas on im dd_{0}^{−1}, r(dd_{0}^{−1}) = (1 − dd_{0}^{−1})dd_{0}^{−1} = d(1 − d_{0}^{−1}(d_{0} + d − d_{0}))d_{0}^{−1} = −dd_{0}^{−1}(d − d_{0})d_{0}^{−1}. This shows that r(dd_{0}^{−1}F_{p}) ⊂ dd_{0}^{−1}(F_{p+1}) and the nilpotency holds.

- One has Π_{F} = Qd + dQ with, by construction, im Q ⊂ im d_{0}^{−1} ⊂ E_{0}^{⊥} and E_{0} ⊂ ker d_{0}^{−1} ⊂ ker Q, so that we obtain Π_{E_{0}}Π_{F} Π_{E_{0}} = 0. Also, Π_{E_{0}^{⊥}} = d_{0}^{−1}d_{0} + d_{0}d_{0}^{−1} with d_{0}^{−1}Π_{E} = Π_{E}d_{0}^{−1} = 0 since E ⊂ ker d_{0}^{−1} and im d_{0}^{−1} ⊂ F = ker Π_{E}, and therefore Π_{E}Π_{E_{0}^{⊥}}Π_{E} = 0.

Remarks. - Although this construction is obtained by elementary considerations, we have no doubt that it would also enter more general theories such as Spencer cohomology and related works of Bryant-Griffiths [1] and Vinogradov. We note also that in the case of two step distributions the spectral sequence we use turns out to be equivalent to the one defined by Julg in [9] using quite different methods. At last, we observe that the existence of d_{c} was suggested in [6] 4.11.C”.

We have derived two complexes (E, d) and (E_{0}, d_{c}) that still compute cohomology but now strictly increase the filtration weight. On M = G we can interpret them as nearcohomology complexes by showing that the de Rham near-cohomology concentrates in some sense on them. Namely, one knows by [7] that boundedly homotopically equivalent Hilbert complexes (H, d) and (H, d′) have equivalent near-cohomologies in the sense that there exists bounded maps f : C_{ε} → C′_{Kε} and g : C′_{ε} → C_{Kε} invertible on their images. In particular, the de Rham complex and the spectrally cut-off one (E_{Δ([0,1])}, d) have equivalent near-cohomologies. Therefore we obtain the following results :

Corollary 2. - The near-cohomology C_{ε} of (E_{Δ([0,1])}, d) contracts on E, that is

||Π_{F} ⌈C_{ε}|| ≤ Kε for some K.

- The complexes (E_{Δ([0,1])}, d), (E, d) and (E_{0}, d_{c}) have equivalent near-cohomologies.

2. Analytic properties

We now address regularity questions. The first fact about the two geometrically equivalent complexes (E, d) and (E_{0}, d_{c}) is that E is a space defined by differential equations whereas E_{0} is algebraic. Another feature of d_{c} is that it is Hodge ∗ self-dual. Indeed, one can show that ∗ preserves E_{0} and δ_{c} = (−1)^{k} ∗ d_{c}∗ on E_{0}^{k}. Proof. - The choice of a metric gives an orthogonal splitting of T M = ⊕V_{i} with H_{i+1} = H_{i}⊕^{⊥}V_{i}. This fixes a weight N on Λ^{∗}T^{∗}M such that F_{p} = F_{p+1}⊕^{⊥}G_{p} and G_{p} = ker(N − p). One has then ∗G_{p} = G_{N_{0}−p} with N_{0} = N(Λ^{max}T^{∗}M). Identifying F_{p}/F_{p+1} with G_{p}, we now get δ_{0} = (−1)^{k} ∗ d_{0}∗ as comes from d_{0}α ∧ ∗β + (−1)^{k}α ∧ d_{0} ∗ β = d_{0}(α ∧ ∗β) = 0, since d_{0}(α ∧ ∗β) is a top form of weight N_{0} − 1 here. This gives ∗E_{0} = E_{0}.

- To prove δ_{c} = (−1)^{k} ∗ d_{c}∗ we develop

\begin{aligned}d(Π_{E}α ∧ Π_{E} ∗ β) & = Π_{E}d_{c}α ∧ Π_{E} ∗ β + (−1)^{k}Π_{E}α ∧ Π_{E}d_{c} ∗ β \\ & = d_{c}α ∧ ∗β + (−1)^{k}α ∧ d_{c} ∗ β \end{aligned}

if we interpret these wedges as scalar products and observe that Π_{F} (E_{0}) ⊂ im d_{0}^{−1} = (ker d_{0})^{⊥}.

We consider now some regularity notion based on the pseudodifferential hypoelliptic calculus developed in [2]. Strictly speaking this calculus was developed there on groups but extends on equiregular C-C spaces (see [10]). This is because maximal hypoellipticity is a stable property so that it can be checked on the model space, the filtered Lie group here, the operators and the produced kernels of parabolic homogeneity being transplanted by Métivier’s approximation map available in equiregular C-C geometry (see [8 (7.2),10,11]). Let |∇| be a scalar first order peudodifferential operator invertible in this calculus. On a filtered group G, one can take |∇| = (Δ_{H})^{\frac{1}{2}} with Δ_{H} the Kohn Laplacian (see [2,5]). We locally extend |∇| as acting diagonally on Ω^{∗}T^{∗}M. An operator P on Ω^{∗}T^{∗}M is said C-C elliptic if P^{∇} = |∇|^{−N}P |∇|^{N} has a parametrice in the calculus, where N is the filtration weight.

Theorem 3. The complexes d and d_{c} are C-C elliptic on a E_{0}-regular C-C space.

Proof. - We have to show that the complexes d^{∇}, d_{c}^{∇} are acyclic at the symbolic level. Here the symbol consists in taking the images of the operator through all non-trivial unitary irreducible representations σ_{ξ} of G_{x_{0}}. They are parametrized by ξ ≠ 0 in 𝔤_{x_{0}}^{∗}. By nilpotency of 𝔤_{x_{0}}, one can find for each ξ a vector X ∈ i𝔤_{x_{0}} such that σ_{ξ}(X) = Id. Consider then L_{X} = X. Id +ad(X) on G_{x_{0}}. Its C-C symbol is σ_{ξ}(L_{X}^{∇}) = Id +σ_{ξ}(ad(X)^{∇}) with ad(X) nilpotent, so that we get its inverse P_{X} = ∑_{k≥0}(−1)^{k}σ_{ξ}(ad(X)^{∇})^{k}. One has [P, σ_{ξ}(d^{∇})] = [P, σ_{ξ}(i_{X}^{∇})] = 0 by [L_{X}, d] = [L_{X}, i_{X}] = 0. Cartan’s formula L_{X} = i_{X}d + di_{X} gives then

Id = (P_{X}σ(i_{X}^{∇}))σ_{ξ}(d^{∇}) + σ_{ξ}(d^{∇})(P_{X}σ(i_{X}^{∇}))

which is the acyclicity of σ_{ξ}(d^{∇}). The proof for d_{c} follows from the previous formula. We compose it on the left by σ_{ξ}(Π_{E_{0}}Π_{E})^{∇} and on the right by σ_{ξ}(Π_{E}Π_{E_{0}})^{∇} to get on E_{0}

Id = Q_{X}σ_{ξ}(d_{c}^{∇}) + σ_{ξ}(d_{c}^{∇})Q_{X}

where Q_{X} = (σ_{ξ}(Π_{E_{0}}Π_{E})^{∇})P_{X}σ(i_{X}^{∇})(σ_{ξ}(Π_{E}Π_{E_{0}})^{∇}).

3. Application to Novikov-Shubin numbers

We consider now nilpotent stratified Lie groups which moreover are rational, that is, possess a discrete cocompact subgroup Γ. Following [7], the spectral distribution of δd is interpreted as the Γ-linear thickness of the cones C_{ε}, which means that F_{δd}(ε^{2}) = dim_{Γ}(E_{δd}(ε^{2})) = sup_{L}(dim_{Γ}L), where L describes the closed Γ-invariant linear subspaces of C_{ε}. The Novikov-Shubin invariants of G are then α_{k} = lim_{ε→0} \frac{ln F_{δd}(ε)}{ln ε}, giving the heat decay exponents at large time, on coclosed k-forms of Ω^{∗}G ([7]). The space E_{0}^{∗} ≃ H^{∗}(G) naturally splits under the weight N. When E_{0}^{k} is of pure weight p, one can consider the truncated operators d_{c}^{i} : E_{0}^{k} → E_{0}^{k+1}/F_{p+i}^{k+1}, and the smallest integer i such that d_{c}^{i} + δ_{c} is C-C elliptic. This integer r_{k}(d_{c}) is the degeneracy order at the symbolic level of the spectral sequence associated to the filtered complex d_{c}. Define at last β_{k} = \frac{N_{0}}{2α_{k}} with N_{0} = ∑ i dim(𝔤_{i}/𝔤_{i−1}).

Theorem 4. Let G be a stratified nilpotent rational Lie group with E_{0}^{k} of pure weight p, then

sup(N_{min}(E_{0}^{k+1}) − p, 1) ≤ β_{k} ≤ r_{k}(d_{c}) ≤ N_{max}(E_{0}^{k+1}) − p.

As E_{0}^{1} is always of pure weight 1, one obtains 1 ≤ β_{1} ≤ r for r-steps groups. Quadratically presented groups (see [3,4]) are precisely those whose E_{0}^{2} is of pure weight 2, and thus have β_{1} = 1. It is the case of two step groups with a regular Ω-isotropic Legendrian plane (see [6]). In the opposite direction, quotients of free Lie algebras by ideal generated by elements of degree ≥ r + 1 satisfy r ≤ β_{1}. Furthermore, there exist groups with β_{1} = r_{1}(d_{c}) = 1 and N_{max}(E_{0}^{2}) = 3. The proof of theorem 4 consists mainly in exploiting homogeneity and C-C regularity with the shift weight N − p.

I thank Raphaël Ponge for his useful references.

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[12] Rumin M., Sub-Riemannian limit of the differential form spectrum of contact manifolds, GAFA (to appear).

———————— Michel Rumin, Mathématiques, Bât 425, Université de Paris-Sud, 91405 Orsay, France. tél : 01.69.15.79.55 E-mail : rumin@topo.math.u-psud.fr