Analytic Torsion of Generic Rank Two Distributions in Dimension Five

We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincaré duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds, we are able to show that this torsion coincides with the Ray–Singer analytic torsion, up to a constant.


Introduction
The Ray-Singer analytic torsion [77] of a closed smooth manifold is a zeta regularized graded determinant of its de Rham complex and computes the Reidemeister torsion [6,32,33,72]. Rumin and Seshadri [85] have recently introduced an analytic torsion for the Rumin complex of contact manifolds [79,80,82] and showed that it coincides with the Ray-Singer torsion for 3-dimensional CR Seifert manifolds equipped with a unitary representation [85,Theorem 4.2]. In this paper we propose an analytic torsion of the Rumin complex associated with another filtered geometry [81,83,84] known as generic rank two distributions in dimension five [29].
A generic rank two distribution on a 5-manifold M is a smooth rank two subbundle D in the tangent bundle T M such that Lie brackets of sections of D span a rank three subbundle [D, D] in T M and triple brackets span all of the tangent bundle. In other words, D is a bracket generating distribution with growth vector (2,3,5). Clearly, this is a C 2 -open condition on the 2-plane bundle D, whence the name generic.
If D is a generic rank two distribution, then the Lie bracket of vector fields induces an algebraic (Levi) bracket on the associated graded bundle  [D, D]. This turns tM into a bundle of graded nilpotent Lie algebras called the bundle of osculating algebras. Its fibers are all isomorphic to the graded nilpotent Lie algebra g = g −3 ⊕ g −2 ⊕ g −1 with graded basis X 1 , X 2 ∈ g −1 , X 3 ∈ g −2 , X 4 , X 5 ∈ g −3 and brackets A basic example of a generic rank two distribution is obtained by equipping the simply connected Lie group G corresponding to the Lie algebra g with the G-invariant 2-plane bundle obtained by translating g −1 .
The study of generic rank two distributions reaches back to Cartan's "five variables paper" [29] from 1910 where Cartan constructs a canonical connection and derives a non-trivial local invariant, the harmonic curvature tensor which is a section of S 4 D * . A generic rank two distribution is locally diffeomorphic to the invariant rank two distribution on the nilpotent Lie group G if and only if Cartan's curvature tensor vanishes.
In modern terminology, generic rank two distributions can equivalently be described as regular normal parabolic geometries [27] of type (G 2 , P), where P is a particular parabolic subgroup in the split real form of the exceptional Lie group G 2 , see also [88]. Their geometry has many intriguing features. The most symmetric example, the flat model, is the homogeneous space G 2 /P which is locally diffeomorphic to the nilpotent Lie group G and has an underlying manifold diffeomorphic to S 2 × S 3 , see [87]. In general, the symmetry group of a generic rank two distribution is a Lie group of dimension at most 14 and is subject to further restrictions [23,Theorem 7]. Nurowski [73] constructed a conformal metric of signature (2, 3) with conformal holonomy G 2 which is naturally associated with a generic rank two distribution, see also [24,54,[88][89][90]. Bryant and Hsu have shown that generic rank two distributions admit many rigid curves [18], i.e., curves which are isolated in the path space of curves tangent to D with fixed endpoints. In the same paper Bryant and Hsu also discuss the mechanical system of a surface rolling without slipping and twisting on another surface. This gives rise to a 5-dimensional configuration space equipped with a rank two distribution encoding the no slipping and twisting condition, which is generic iff the Gaussian curvatures are disjoint [18,Sect. 4.4]. If both surfaces are round spheres and the ratio of their radii is 3:1 then the universal covering of the configuration space is diffeomorphic to G 2 /P, see [8,29]. Let us also mention the following recent studies of generic rank two distributions [2,9,50,55,64].
By virtue of Gromov [37,Theorem 2] which open 5-manifolds admit globally defined generic rank two distributions. For closed manifolds the situation appears to be more subtle but the underlying topological problem has been settled, see [37,Theorem 1(b)]. The question to what extent generic rank two distributions abide by an h-principle [45,51] has served as a motivation for us to study the analytic torsion of the associated Rumin complex, cf. [10,30,31,41,44,48,97].
The Rumin complex [81,83,84] associated with a generic rank two distribution on a 5-manifold M is a natural complex of higher order differential operators, where E q = H q (tM) denotes the vector bundle obtained by taking the fiberwise Lie algebra cohomology of the bundle of osculating algebras tM. The ranks of the bundles E q are 1, 2, 3, 3, 2, 1 and the Heisenberg orders of the operators D q are 1, 3, 2, 3, 1, see [19,Sect. 5] or [39,Example 4.24]. There exist injective differential operators L q : ∞ (E q ) → q (M), embedding the Rumin complex as a subcomplex in the de Rham complex, whose image can be characterized using differential operators. The operators L q intertwine the Rumin differential with the de Rham differential, d L q = L q+1 D q , and induce isomorphisms in cohomology. In particular, the Rumin complex computes the cohomology of M.
Let us emphasize that the Rumin differentials D q are natural, i.e. independent of any further choices, and so is the map induced on cohomology by L q , see Lemma 3.2. The operators L q , however, depend on the choice of a sub-Riemannian metricg on D and a splitting of the filtration, S : tM → T M. By the latter we mean a filtration preserving isomorphism inducing the identity on the associated graded, that is, S| t −1 M = id, S(t −2 M) ⊆ [D, D], S| t −2 M = id mod D, and S| t −3 M = id mod [D, D]. Using parabolic geometry, one can construct a variant of the Rumin complex D q with splitting operators L q which are entirely natural. Indeed, the Rumin complex is isomorphic to the curved BGG sequence [25] associated with the trivial representation of G 2 . Writing down the BGG operators explicitly, however, requires the determination of (a good portion of) the canonical regular normal Cartan connection associated with the rank two distribution [29], a quite involved procedure, see [88]. While this more natural perspective might prove to be insightful when studying the sub-Riemannian limit, it does not appear to be helpful for the purpose of this paper, and we will therefore not adopt it here.
Suppose F is a flat complex vector bundle over M and let D F q denote the differential complex obtained by twisting the Rumin complex with F. The definition of the analytic torsion requires a sub-Riemannian metric [52,69], i.e., a fiberwise Euclidean inner productg on D, as well as a fiberwise Hermitian metric h on F. These choices give rise to fiberwise Hermitian metrics on the vector bundles E q and permit to define formal adjoints of the Rumin differentials, denoted by (D F q ) * . The essential part of the analytic torsion is a zeta regularized graded determinant which may be expressed in the form where the complex powers are understood to vanish on the kernels of the operators, see Remark 2.9. For our rigorous definition, however, we will proceed as in Ray-Singer [77] or Rumin-Seshadri [85] and rewrite this in terms of zeta functions associated with hypoelliptic Rumin-Seshadri type operators see Sect. 2.3. For appropriate choices of the numbers a q , the operators F q are Rockland of Heisenberg order 2κ and admit a parametrix in the Heisenberg calculus [39,66,96]. The zeta function tr ( F q ) −s converges for (s) > 10/2κ and extends to a meromorphic function on the entire complex plane which is holomorphic at zero, see [38,Theorem 2]. Correspondingly, as t → 0, the heat kernel admits an asymptotic expansion [38,Theorem 1] of the form where p F q, j ∈ ∞ end(H q (tM) ⊗ F) ⊗ | | are locally computable, and p F q, j = 0 for all odd j. Here | | denotes the line bundle of 1-densities over M.
Following [6] we incorporate the zero eigenspaces and consider the analytic torsion as a norm − sdet(H * (M;F)) D,g,h on the graded determinant line [62] sdet(H * (M; F)) := q det H q (M; F) where H q (M; F) denotes the de Rham cohomology of M with coefficients in the flat bundle F. This analytic torsion behaves as expected with respect to finite coverings and Poincaré duality, see Theorem 3.8. Moreover, this definition leads to a simple and local anomaly formula, describing the metric dependence, which is analogous to the corresponding formulas for the Ray-Singer torsion [6, Theorem 0.1] and the Rumin-Seshadri torsion [85,Corollary 3.7].
To state this formula, note that restriction provides natural isomorphisms Aut(tM) = Aut(D) and der(tM) = end(D), where Aut(tM) denotes the bundle of fiberwise graded Lie algebra automorphisms and der(tM) denotes the bundle of fiberwise derivations of degree zero. Hence, each A ∈ ∞ (Aut(D)) induces H q (A) ∈ ∞ (Aut(H q (tM))) and correspondingly, eacḣ A ∈ ∞ (end(D)) induces H q (Ȧ) ∈ ∞ (end(H q (tM))). u ∂ ∂u h u ∈ ∞ (end(F)), and p F q,u,10 ∈ ∞ end(H q (tM) ⊗ F) ⊗ | | denotes the constant term in the heat kernel expansion associated with the Rumin-Seshadri operator F u,q , cf. (1). Of course, the Ray-Singer torsion of a 5-manifold is independent of the Riemannian metric and the Hermitian metric on F, for there are no constant terms in the small time asymptotic expansion of the heat trace of the Hodge Laplacians in odd dimensions. The relevant analogue in the heat trace asymptotics for Rockland operators on filtered manifolds is the homogeneous dimension, which happens to be even (ten) for generic rank two distributions in dimension five. Hence, there is no apparent analytical reason, why the corresponding analytic torsion should be independent ofg and h. The situation is similar for the Rumin-Seshadri analytic torsion of contact manifolds which have even homogeneous dimension too. On contact 3-manifolds, by adding (potentially vanishing) local correction terms, Rumin and Seshadri were able to turn their torsion into a CR-invariant [85,Corollary 3.8(3)] which coincides with the Ray-Singer torsion according to [1]. We will not compute the local quantities appearing in the anomaly formula in Theorem 1.1 more explicitly in this paper.
In view of Gray's stability theorem [49, Theorem 2.2.2], a smooth deformation of a contact structure can always be absorbed by adjusting the sub-Riemannian metric. Hence, the infinitesimal change of the Rumin-Seshadri analytic torsion under a smooth deformation of the contact structure can be expressed as an integral over a local quantity. For generic rank two distributions the situation is fundamentally different due to the local invariant provided by Cartan's curvature tensor. Even a small perturbation of the 2-plane bundle will in general result in a distribution which is not even locally diffeomorphic to the original one. Nevertheless, the torsion changes by a local quantity. More precisely, we have group with Lie algebra g and let D denote a right invariant generic rank two distribution on G. We equip D with a right invariant sub-Riemannian metricg. Since the structure constants of g are rational, the group G admits lattices, i.e., discrete cocompact subgroups, [76,Theorem 2.12]. If is a lattice in G, then D andg descend to the nilmanifold G/ . The induced structures on G/ will be denoted by D andg , respectively. We have the following comparison with the Ray-Singer torsion [6,77] which will be denoted by − for every right invariant generic rank two distribution D on G, every lattice in G which is generated by two elements, and every flat line bundle F over G/ with parallel fiberwise Hermitian metric h.
Actually, we will establish a slightly stronger result in Theorem 4.7 below. Since the nilmanifold G/ admits a free circle action, its Ray-Singer torsion can be read off the corresponding Gysin sequence, see [65,Corollary 0.9].
To prove Theorem 1.3, we will consider the lattice 0 in G generated by two elements exp(X 1 ) and exp(X 2 ) where X 1 , X 2 is a basis of g −1 . The group of graded automorphisms Aut(g) ∼ = GL(2, R) acts on G by diffeomorphisms preserving D 0 , the right invariant rank two distribution obtained by translating g −1 . The subgroup preserving the lattice, Aut(g, log 0 ), is a subgroup of finite index in GL(2, Z) which acts on the nilmanifold G/ 0 by filtration preserving diffeomorphisms. Moreover, the space of unitary characters of 0 , that is, the space of flat line bundles over G/ 0 which admit a parallel Hermitian metric, can be covered by finitely many orbit closures of Aut(g, log 0 ). Exploiting this fact, and using several other results established here, including Theorems 1.1 and 1.2, we are able to give a proof of Theorem 1.3.
We expect that the statement in Theorem 1.3 remains true with c = 1 for arbitrary lattices and all flat bundles F of arbitrary rank with parallel Hermitian metric h. This expectation is motivated by a recent result of Albin and Quan [1,Corollary 3], asserting that the quotient of the Rumin-Seshadri torsion and the Ray-Singer torsion can be expressed in terms of an integral over a local quantity. If their analysis of the sub-Riemannian limit can be generalized to generic rank two distributions, the aforementioned strengthening of Theorem 1.3 would follow at once.
The remaining part of this paper is organized as follows. In Sect. 2 we define an analytic torsion for Rockland differential complexes over general closed filtered manifolds. We establish a metric anomaly formula for this torsion in Theorem 2.11 and describe the variation of the torsion through a deformation of the underlying filtration in Theorem 2.17. Some basic properties including compatibility with finite coverings and duality are collected in Sect. 2.6.
In Sect. 3 we apply this general framework to the Rumin complex associated with certain filtered manifolds. The resulting anomaly formulas are contained in Theorems 3.11 and 3.15, compatibility with Poincaré duality is formulated in Theorem 3.8. In order for the Rumin complex to be Rockland, it is necessary to assume that the osculating algebras have pure cohomology. This apparently very restrictive assumption will be discussed in Sect. 3.7. We only know of three types of filtered manifolds with this property: trivially filtered manifolds giving rise to the Ray-Singer torsion, contact manifolds giving rise to the Rumin-Seshadri torsion, and generic rank two distributions in dimension five giving rise to the torsion proposed in this paper.
In Sect. 4 we specialize to generic rank two distributions in dimension five and prove the results stated in the introduction.

Analytic Torsion of Rockland Complexes
To every Rockland complex of differential operators over a closed filtered manifold one can associate an analytic torsion. For the de Rham complex this specializes to the classical Ray-Singer torsion [6,77]. For the Rumin complex of a contact manifold it specializes to the Rumin-Seshadri analytic torsion [85].
The purpose of this section is to provide a rigorous definition of the analytic torsion of general Rockland complexes and to discuss some basic properties. We will show that this torsion is compatible with duality and finite coverings, see Sect. 2.6. Moreover, we will establish an anomaly formula which shows that the dependence on the metric is given by an integral over a local quantity, see Theorem 2.11. We will also study the variation of the torsion during a deformation of the underlying filtration, and show that in certain situations, this can be expressed in terms of local quantities also, cf. Theorem 2.17.
In Sects. 3 and 4 we will apply these results to Rumin complexes associated with certain filtered manifolds.

Differential Operators on Filtered Manifolds
Recall that a filtered manifold is a smooth manifold M together with a filtration of the tangent bundle T M by smooth subbundles, which is compatible with the Lie bracket of vector fields in the following sense: If X is a smooth section of T p M and Y is a smooth sections of T q M then the Lie bracket [X , Y ] is a section of T p+q M. Putting t p M := T p M/T p+1 M, the Lie bracket induces a vector bundle homomorphism t p M ⊗ t q M → t p+q M referred to as (generalized) Levi bracket [27,Definition 3.1.7]. This turns the associated graded vector bundle tM := p t p M into a bundle of graded nilpotent Lie algebras called the bundle of osculating algebras. The Lie algebra structure on the fiber t x M = p t p x M depends smoothly on the base point x ∈ M, but will in general not be locally trivial. In particular, the Lie algebras t x M might be non-isomorphic for different x ∈ M. The simply connected nilpotent Lie group with Lie algebra t x M is called osculating group at x and will be denoted by T x M.
Using negative degrees, we are following a convention common in parabolic geometry, see [27,70,71] for instance. The other convention, where everything is concentrated in positive degrees, is the one that has been adopted in [95,96].
The filtration on M gives rise to a Heisenberg filtration on differential operators which is compatible with composition and transposition. The basic idea is to consider differentiation along a vector field tangent to T −k M as an operator of Heisenberg order at most k. Suppose E and F are two vector bundles over M. A differential operator A : ∞ (E) → ∞ (F) of Heisenberg order at most k has a Heisenberg principal (co)symbol, encoding the highest order derivatives in the Heisenberg sense at x ∈ M, where U −k (t x M) denotes the degree −k part of the universal enveloping algebra of the graded nilpotent Lie algebra t x M = p t p x M. Equivalently, the Heisenberg principal symbol may be regarded as a left invariant differential operator on the osculating group T x M which is homogeneous of degree k with respect to the grading automorphism. The Heisenberg principal symbol is compatible with composition and transposition of differential operators. A differential operator A : ∞ (E) → ∞ (F) of Heisenberg order at most k is said to satisfy the Rockland condition [78] if is injective for every x ∈ M and every non-trivial irreducible unitary Hilbert space representation π of the osculating group T x M, where H x,∞ denotes the subspace of smooth vectors, cf. [61]. The Rockland theorem asserts that in this situation the operator admits a left parametrix which is of order −k in an appropriate Heisenberg calculus of pseudodifferential operators adapted to the filtration. In particular, Rockland operators are hypoelliptic. This result has a long history. For trivially filtered manifolds, that is if T M = T −1 M, it reduces to the classical, elliptic case. In this situation all irreducible unitary representations of the (abelian) osculating group are one dimensional, and the Rockland condition at x ∈ M becomes the familiar condition that the principal symbol of the operator is invertible at every 0 = ξ ∈ T * x M, provided rk(E) = rk(F). Helffer-Nourrigat [56][57][58] proved maximal hypoellipticity for left invariant scalar Rockland differential operators on graded nilpotent Lie groups, thus confirming a conjecture due to Rockland [78]. For contact and (more generally) Heisenberg manifolds, a pseudodifferential calculus has been developed independently by Beals-Greiner [3] and Taylor [93], see also [74]. Rockland theorems for Heisenberg manifolds can be found in [3,Theorem 8.4] or [74,Theorem 5.4.1]. These investigations can be traced back to the work of Kohn [63], Boutet de Monvel [11], and Folland-Stein [47] on CR manifolds, cf. the introduction of [3] for further historical comments. A pseudodifferential calculus for general filtered manifolds was first described by Melin [66]. In his unpublished manuscript Melin shows [66, Theorem 7.2] that a scalar Rockland differential operator admits a parametrix in his calculus. More recent constructions of pseudodifferential calculi on filtered manifolds are based on the Heisenberg tangent groupoid [34,53,68,95,96] and the idea of essential homogeneity introduced in [40]. In [35,Theorem 2.5(d)] one finds a scalar Rockland theorem for graded nilpotent Lie groups which suffices to study the flat models in parabolic geometry as well as topologically stable [75] structures like contact and Engel manifolds. For general systems of (pseudo)differential operators on arbitrary filtered manifolds the Rockland theorem may be found in [39,Theorem A].
Using the parametrix of the heat operator and the Heisenberg pseudodifferential calculus, one can extend the results on the structure of complex powers [4,67,74,91] to (formally) selfadjoint positive Rockland differential operators over general filtered manifolds, see [37,Corollary 2]. In particular, the zeta function of such an operator admits a meromorphic extension to the entire complex plane which is holomorphic at zero. Hence, its derivative at zero may be used to define regularized determinants for these operators.

Rockland Complexes
Consider a finite complex of differential operators over a closed filtered manifold M, which are of Heisenberg order k q ≥ 1, respectively. Here E q are smooth vector bundles over M, only finitely many of these vector bundles are non-zero, and D q D q−1 = 0 for all q. We assume the sequence is Rockland [39, Definition 2.14], i.e., the Heisenberg principal symbol sequence becomes exact in every non-trivial irreducible unitary representation of T x M. Fix a volume density μ on M and fiber wise Hermitian inner products h q on the bundles E q , and consider the associated L 2 inner product on ∞ (E q ), Fix natural numbers a q ∈ N such that k q−1 a q−1 = k q a q (5) for all q, and put Then the Rumin-Seshadri operators, are all of Heisenberg order 2κ. These operators generalize the classical Hodge Laplacians as well as the Laplacians associated with the Rumin complex in [85,Sect. 2.3] which are of Heisenberg order two or four. It will be convenient to consider the operators := q q , D = q D q and D * = q D * q acting on sections of the graded vector bundle E := q E q .

Lemma 2.1
The Rumin-Seshadri operator is formally selfadjoint and commutes with D and D * , i.e., we have the graded commutator relations Moreover, defining σ q : ∞ (E q ) → ∞ (E q−1 ) by and s q : we have the graded commutator relations Proof Clearly, * q = q . Using D q D q−1 = 0, we obtain D q q = q+1 D q and then q D * q = D * q q+1 by passing to the formal adjoint, whence (8). Moreover, q = D q−1 σ q + σ q+1 D q and thus q = s q−1 D * q−1 + D * q s q , whence (11). Lemma 2.18]) The Rumin-Seshadri operator is Rockland of Heisenberg order 2κ.
Using the spectral theorem [60, Sect. VI §5.3] we obtain a strongly differentiable semi-group e −t for t ≥ 0. Let n := − p p · rk(t p M) (12) denote the homogeneous dimension of M. Note that n ≥ 0 since with our convention the grading of tM = p t p M is concentrated in negative degrees. We will denote the line bundle of 1-densities on M by | |.
For A ∈ ∞ (end(E)) we thus obtain an asymptotic expansion as t → 0, where str denotes the graded trace. In particular, where LIM denotes the constant term in the asymptotic expansion.

Lemma 2.5
The graded heat trace str e −t is constant in t and Here χ(E, D) denotes the Euler characteristics of the complex in (2).

Proof
Moreover, with respect to the trace norm we have [38,Lemma 5] lim t→∞ e −t = P (17) where P denotes the spectral (orthogonal) projection onto the kernel of .

Analytic Torsion
For λ ≥ 0, let P λ denote the spectral projection of corresponding to the eigenvalues at most λ, and let Q λ := id −P λ denote the complementary projection. Fix numbers N q such that for all q. We let N ∈ (end(E)) denote the operator given by multiplication with N q on E q . For λ ≥ 0 and (s) > n/2κ, we put According to [38,Corollary 2] this zeta function admits a meromorphic continuation to the entire complex plane which is holomorphic at s = 0. Moreover, where LIM denotes the constant term in the asymptotic expansion, and Putting A = N in (14) and using (21) we obtain Recall that img(P λ ) is a finite dimensional subcomplex of ( ∞ (E), D). Moreover, the inclusion induces a canonical isomorphism in cohomology, The torsion of finite dimensional complexes, see [7,Sect. a] or [6, Sect. Ia], provides a canonical isomorphism of graded determinant lines [62] sdet(img(P λ )) = sdet H * img(P λ ), D .
Recall here that the graded determinant line of a finite dimensional graded vector space V = q V q is the line sdet(V ) := q (det V q ) (−1) q where det W := dim W W and the exponent −1 is used to denote the dual line. Combining the latter two identifications, we obtain a canonical isomorphism sdet(img(P λ )) = sdet(H * (E, D)).
The L 2 inner product on ∞ (E) restricts to a graded inner product on img(P λ ) and induces an inner product on the graded determinant line sdet(img(P λ )). Via (23) this corresponds to an inner product on the line sdet(H * (E, D)). We will denote the corresponding norm by − . Following [6,7], we define the analytic torsion of the Rockland complex (E, D) to be the norm on the graded determinant line A priori, the analytic torsion of a Rockland complex (E, D) depends on the graded fiber wise Hermitian inner product h on E, the volume density μ on M, the choice of numbers a q satisfying (5), the choice of numbers N q satisfying (19), and the choice of a spectral cutoff λ ≥ 0. It is fairly easy to see that the analytic torsion is actually independent of a q , N q and λ, see Lemmas 2.6, 2.7 and 2.8 below. The dependence on h q and μ will be discussed in Sect. 2.4, see Theorem 2.11 below.

Lemma 2.6
The analytic torsion of a Rockland complex, see (24), does not depend on the choice of the spectral cutoff λ ≥ 0.

Lemma 2.7
The analytic torsion of a Rockland complex, see (24), does not depend on the choice of numbers N q , see (19).
Proof IfÑ q is another choice such thatÑ q+1 −Ñ q = k q , then there exists a constant c such thatÑ q − N q = c for all q. Recall that the graded commutator relations = [D, σ ], see (11), and [D, Since lim t→∞ tr Q λ,q e −t q = 0, see (17), we conclude str Q λ e −t = 0 for all t > 0 and all λ ≥ 0. Hence, for (s) > n/2κ we obtain By analytic continuation, this remains true for all s, whence the lemma.

Lemma 2.8
The analytic torsion of a Rockland complex, see (24), does not depend on the choice of numbers a q , see (5).
Proof Supposeã q ∈ N is another choice of positive integers satisfying (5), that is, k q−1ãq−1 = k qãq for all q. Then there exist positive integers r ,r ∈ N such that ra q =rã q for all q. Indeed, we may use r :=ã q 0 andr := a q 0 for some fixed q 0 . W.l.o.g. we may assumer = 1 and thus a q = ra q for all q. Writingκ := k qãq we find, see (6), Denoting the corresponding Laplacian by˜ q : Using the spectral cutoffλ := λ r and denoting the spectral projections of˜ byQ, we havẽ Using ( r ) −s = −rs , (30) and (31) we find, see (20), for all s. Differentiating and using (29) we find In view of (31) we have img(Pλ) = img(P λ ) and thus Combining (32) with (33) and Lemma 2.6 we see that the analytic torsion with the choices a q andã q coincide, cf. (24).

Remark 2.9
The analytic torsion in (24) may be expressed in a way which does not require the choice of numbers a q as in (5) and N q as in (19). More precisely, for λ ≥ 0 we have Here str λ (D * D) −s denotes the graded trace, disregarding all eigenvalues ≤ λ, in particular, disregarding the eigenvalue zero which will have infinite multiplicity in general. To see this, note that D 2 = 0 gives, see (7) tr Using the latter two displayed equations, and proceeding by induction on q, one readily concludes, from the corresponding fact for tr(Q λ −s q ), that the functions tr λ (D * q D q ) −a q s and tr λ (D q−1 D * q−1 ) −a q s admit meromorphic extensions which are holomorphic at s = 0. Using (19) we obtain Using (20) and (6) this gives Similarly, Combining the latter two equations with the obvious equality obtained via analytic continuation, we arrive at the formula in (34).

Remark 2.10
The analytic torsion of a Rockland complex remains unchanged if the differentials D q are replaced with c q D q where c q ∈ C and |c q | = 1.

Dependence on the Metric
Suppose the volume density μ u on M and the graded Hermitian inner product h u on E depend smoothly on a real parameter u. Let D * u and u denote the corresponding family of operators. For real u and v, we let G v,u ∈ ∞ (Aut(E)), denote the unique vector bundle automorphism such that, cf. (3), for all φ, ψ ∈ ∞ (E). Clearly, G v,u preserves the decomposition E = q E q . Using (4) we immediately see that D * u varies by conjugation, i.e., The aim of this section is to establish the following variational formula generalizing well known results for the Ray-Singer torsion [6,77] and the Rumin-Seshadri torsion, see [85,Corollary 3.7].

Theorem 2.11 In this situation we have
u ∂ ∂u μ u , and p u,n ∈ ∞ (end(E) ⊗ | |) denotes the constant term in the heat kernel expansion associated with the Rumin-Seshadri operator u , see (13).

Remark 2.12
If the homogeneous dimension n is odd, then Theorem 2.11 implies that the analytic torsion of a Rockland complex does not depend on the choice of μ and h, for in this case p u,n = 0 according to Lemma 2.4. In the classical case this has already been observed by Ray and Singer [77].
where the middle equality is a consequence of the cocycle relation G w,v G v,u = G w,u , and the last equality follows by differentiating (35

Lemma 2.13 In this situation we have
and s u,q : Hence, from Lemma 2.1 we have graded commutator relations Using Duhamel's formula one shows Putσ u := ∂ ∂u σ u and let k denote the operator given by multiplication with k q on ∞ (E q ). Then we have graded commutator relations˙ u = [D,σ u ] and [N , D] = Dk, see (42) and (19), respectively. Combining this with [D, e −t u ] = 0 and the fact that the graded trace vanishes on graded commutators we obtain ,q ] = 0 and the trace vanishes on commutators this yields see also (6) and (41). Consequently, Differentiating (37) and using (38), (42), and using the fact that the graded trace vanishes on graded commutators we obtain Clearly, Combining equations (44), (45), (46), (47) and (48), we obtain (39).
Combining this with (52), we obtain the following analogue of (43) Proceeding exactly as in the proof of Lemma 2.13 and using the fact that Q u,λ commutes with u , D and D * u , one obtains Using (20) and (54) it is straight forward to derive (49). Using the fact that is holomorphic at s = 0, the equation (50) follows at once. Furthermore, we get Combining this with (14) we obtain (51).
The following can be proved as in the classical case, see for instance [20, p 56].

Remark 2.16
Twisting a Rockland complex as in (2) with a flat vector bundle F we obtain a new Rockland complex: Suppose h F is a parallel fiber wise Hermitian inner product on F and equip E q ⊗ F with the Hermitian metric h q ⊗h F . Locally, the flat bundle F and the parallel Hermitian metric h F can be trivialized simultaneously. Hence, locally, we have Using locality of the heat kernel expansion, we conclude the local quantities in the heat kernel expansions of the Rumin-Seshadri operators F and , respectively, see (13).

Deformation of the Filtration
In this section we consider a family of closed filtered manifolds M u smoothly depending on a real parameter u ∈ R. More explicitly, we assume the underlying manifold M remains fixed, and the filtration subbundles T p M u ⊆ T M depend smoothly on u. Suppose we have a smooth family of differential complexes D u acting between sections of vector bundles E q u which depend smoothly on u, We assume that each D u is a Rockland complex with respect to the filtration on M u and that the Heisenberg order k q of D q,u is independent of u. We consider graded Hermitian metrics h u on E u = q E q u and volume densities μ u on M depending smoothly on u, and aim at describing the dependence of the torsion − sdet(H * (E u ,D u )) h u ,μ u on the parameter u, cf. (24).
We will add the subscript u to the notation used in the preceding sections to indicate the dependence on this parameter. In particular, u denotes the Rumin-Seshadri operator associated with E u , D u , h u , μ u , see (7), and k t,u (x, y) will denote the (smooth) kernel of e −t u . We take the numbers a q satisfying (5) and the numbers N q as in (19) to be constant in u.
which depend smoothly on real parameters u and v such that We assume that A v,u induces an isomorphism in cohomology, and that A u,u = id. In this situation we have

Theorem 2.17 With respect to the isomorphism of determinant lines
induced by (56), we have In the remaining part of this section we will give a proof of this theorem. We begin by addressing the existence of the asymptotic expansion in (58).

Lemma 2.18
Suppose B is a differential operator, acting on sections of E u , which is of Heisenberg order at most r u with respect to the filtration on M u , and let (Bk t,u )(x, y) denote the (smooth) kernel of Be −t u . Then we have an asymptotic expansion, as t → 0, Proof According to [38,Sect. 4], the heat operator u + ∂ ∂t admits a parametrix Q in the Heisenberg calculus on M × R whose kernel k Q satisfies Here M × R is considered as a filtered manifold such that the vector field ∂ ∂t has Heisenberg order 2κ. Hence, B Q is a pseudodifferential operator of Heisenberg order r u − 2κ in the Heisenberg calculus on M × R. For its kernel k B Q we find Since B Q is in the Heisenberg calculus, its kernel k B Q admits an asymptotic expansion along the diagonal. Proceeding exactly as in [38,Sect. 4], this leads to the asymptotic expansion in (59).
Using parallel transport with respect to the auxiliary connection to identify the bundles E u with a single vector bundle, we may w.l.o.g. assume E u = E for all u.
whereḊ u := ∂ ∂u D u . By the chain rule, In view of Theorem 2.11 we may thus assume w.l.o.g. that the Hermitian metrics and the volume densities are independent of u, that is, h u = h and μ u = μ.
Proof From Lemma 2.1 we have the graded commutator relation Using Duhamel's formula one shows as in (53) Putσ u := ∂ ∂u σ u and let k denote the operator given by multiplication with k q on ∞ (E q ). Then we have graded commutator (62), (19) and (9), respectively. Combining Q λ,u ] and the fact that the graded trace vanishes on graded commutators, we obtain Clearly, see also (6). Consequently, Using (60), and the fact that the graded trace vanishes on graded commutators, we obtain str Clearly, Combining equations (63), (64), (65), (66) and (67), we obtain (61). Moreover, and ∂ ∂u wherep n,u is the constant term in the asymptotic expansion (58).
Proof Integrating (58), we obtain the asymptotic expansion as t → 0. Using Eq. (61) and proceeding exactly as in the second half of the proof of Lemma 2.14, we obtain the lemma.
Proof We consider the linear map This is a map of finite dimensional complexes inducing (56) on cohomology. Since α u,u = id, the map α v,u is an isomorphism of complexes, for v sufficiently close to u. Hence, for these v, we have Differentiating and using P λ,uṖλ,u P λ,u = 0, we obtain the lemma.

Elementary Properties of the Analytic Torsion
In this section we collect a few properties of the analytic torsion of Rockland complexes generalizing well known facts about the Ray-Singer and the Rumin-Seshadri analytic torsion. Throughout this section M denotes a closed filtered manifold, (E, D) is a Rockland complex over M, μ is a volume density on M and h is a graded fiberwise Hermitian inner product on E.
where the right hand side denotes the induced norm on the dual (inverse) line.
Proof Denoting the zeta function associated with the Rockland complex (Ẽ,D) bỹ ζ λ , see (20), we clearly haveζ λ (s) = −ζ λ (s), for any λ ≥ 0. Moreover, up to the canonical identification in (72) we have  Proof Since the Heisenberg orders of D q andD q coincide, D ⊕D is a Rockland complex and we may use the same numbers a q and N q to compute the analytic torsion of all three complexes. With theses choices, we clearly have In particular, for each λ ≥ 0, and we obtain From (74) we get a canonical isomorphism of finite dimensional complexes, It is well known that up to the canonical identification in (73)  Recall that the transposed of a differential operator A : where the parentheses (−, −) E denote the canonical pairing between sections of E and E := E * ⊗ | |. The following observation will be useful in the discussion of Poincaré duality in Sect. 3.3 below.
Then, up to the canonical identification induced by the canonical isomorphism H q (E , where the right hand side denotes the induced norm on the dual (inverse) line.
Dualizing, we obtain Note here that the vector bundle isomorphism h −q ⊗ μ :Ē −q → (E −q ) * ⊗ | | = (E ) q intertwines the Hermitian metrich onĒ with the Hermitian metric h on E , and thus, with respect to these Hermitian metrics, we have We obtain Denoting the Heisenberg order of (D t ) q by k q , we have Hence, we may use the numbers a q = a −q−1 to compute the analytic torsion of (E , D t ), cf. (5) and Lemma 2.8. With this choice, we find In particular, In view of (77), the numbers N q := −N −q also satisfy the relation (19), that is, Combining (78), (79) and (80), we obtain Recall here that according to Lemma 2.7, we may use any sequence of numbers satisfying (19) to compute this zeta function. Clearly, the adjoints of D with respect to h andh coincide. Hence, h,μ is real, we thus get Note that from (78) and (79) we also obtain Hence, the canonical weakly non-degenerate pairing ∞ ((E ) q ) × ∞ (E −q ) → C restricts to a (weakly) non-degenerate pairing The latter pairing induces an isomorphism between finite dimensional complexes, where the right hand side denotes the dual of the complex img P (E,D) λ , D with the grading reversed. Via this isomorphism, the inner product on the left hand side induced by the L 2 inner product on (E ) coincides with the inner product on the right hand side induced by the L 2 inner product on (E). Moreover, by the universal coefficient theorem, this identification induces isomorphisms in cohomology, H q (E , D t ) = H −q (E, D) * , which in turn induce the canonical identification (76). Hence, via (76) (81) and (82), the proposition follows at once, see (24).
With respect to finite coverings we have:
x ∈ M, and let h V denote the canonical parallel fiber wise Hermitian inner product on V.
Then, up to the canonical identification of determinant lines Proof This follows from Proposition 2.25. Indeed, via the canonical isomorphism of vector bundles π * π * E = E ⊗ V we have π * π * D = D V and π * π * h = h ⊗ h V .

Analytic Torsion of Rumin Complexes
The Rumin complex [81,83,84] associated with a filtered manifold is a complex of higher order differential operators which is conjugate to a subcomplex of the de Rham complex and computes the cohomology of the underlying manifold. Rumin has shown that these are Rockland complexes in a graded sense. Assuming that the cohomology of the osculating algebras is pure, we obtain a Rockland complex in the sense of Sect. 2. The aim of this section is to establish basic properties of the analytic torsion associated with these complexes. In particular, we will address Poincaré duality in Theorem 3.8, metric dependence in Theorem 3.11, and the dependence on the filtration in Theorem 3.15. For trivially filtered manifolds the Rumin complex coincides with the de Rham complex and gives rise to the classical Ray-Singer torsion [6,77]. For contact manifolds the Rumin complex [79,80,82] has been used by Rumin and Seshadri [85] to define an analytic torsion. Although these two classical torsions appear as special cases of our general construction, we will say nothing new about them. Cartan's (2, 3, 5) geometries [29] constitute another class of 5-dimensional filtered manifolds whose osculating algebras have pure cohomology. Their analytic torsion will be discussed in Sect. 4 below.
Our assumption that the osculating algebras have pure cohomology appears to be very restrictive. The three cases mentioned previously are the only types of filtered manifolds we know of which have this property. In Sect. 3.7 we present a necessary condition which restricts the dimensions of such Lie algebras tremendously.

Rumin Complexes
We continue to consider a closed filtered manifold M. Recall that tM denotes the bundle of osculating algebras and let ∂ q : q t * M → q+1 t * M denote the fiberwise Chevalley-Eilenberg codifferential. We assume that the dimension of the Lie algebra cohomology Letg be a graded fiberwise Euclidean inner product on tM = p t p M. We will denote the induced fiberwise Euclidean inner product on q t * M byg −1 . Let ∂ * q : q+1 t * M → q t * M denote the corresponding fiberwise adjoint of ∂ q . Fiberwise finite dimensional Hodge theory provides an orthogonal decomposition of vector bundles where We fix a splitting of the filtration S : tM → T M, i.e., a filtration preserving vector bundle isomorphism inducing the identity on the associated graded. Then (S t ) −1 : t * M → T * M is a splitting for the dual filtration which will be denoted by S too. We extend it to a splitting S : * t * M → * T * M characterized by for α, β ∈ * t * x M. Hence δ q : q T * M → q−1 T * M, is a filtration preserving vector bundle homomorphism inducing gr(δ) = ∂ * on the associated graded. Actually, δ is a Kostant type codifferential [39, Definition 4.8 and Remark 4.14] for the de Rham complex * (M). Kostant's box operator = δd + dδ is filtration preserving and induces gr( ) = := ∂ * ∂ + ∂∂ * on the associated graded. If 0 = z ∈ C is sufficiently close to zero, then the vector bundle map z −˜ is invertible. This implies that z − is invertible too and its inverse is again a filtration preserving differential operator. Indeed, the inverse can be expressed using a finite geometric series, cf. [ is a filtration preserving differential projector, 2 = , inducing gr( ) =˜ on the associated graded, where˜ denotes the orthogonal projection onto the subbundle ker˜ = ker ∂ * ∩ ker ∂ of * t * M. Since commutes with d and δ, the same is true for , that is, d = d, δ = δ. The operator coincides with Rumin's projector is filtration preserving and induces the identity on the associated graded, gr(L) = id. Hence, L is invertible and its inverse is a filtration preserving differential operator. Since L = L˜ and d = d , the differential operator L −1 dL commutes with˜ . Hence, L −1 dL decouples into a sum of two differential operators,  [81,83]. Indeed, ∂ * B + B∂ * is invertible for it is filtration preserving and the induced homomorphism on the associated graded, gr(∂ * B+B∂ * ) = ∂ * ∂ + ∂∂ * , is invertible on ker˜ . One can show that the complex B is actually conjugate, via an invertible filtration preserving differential operator, to the acyclic tensorial complex ∂| ∞ (ker˜ ) , see [39,Proposition 4.5(c)]. We use the decomposition in (83) to identify H(tM) = ker ∂/ img ∂ with img˜ = ker ∂ * ∩ ker ∂. More precisely, we let ι : H(tM) → img˜ ⊆ ker ∂ denote the corresponding isomorphism of vector bundles, splitting the canonical projection ker ∂ → ker ∂/ img ∂ = H(tM). Putting D := ι −1 Dι, we obtain a complex of differential operators we will refer to as Rumin complex: The latter three equations follow from L = Lι, L˜ = L, (87), and D = ι −1 Dι. The construction of the Rumin complex presented here is motivated by the construction of natural differential operators [22,25,28,39] in parabolic geometry.
To obtain a Rockland complex, we assume from now on that the osculating algebras have pure cohomology [81,83], that is, H q (t x M) = p H q (t x M) p is concentrated in a single degree for each q. In other words, we assume that the grading automorphism φ t given by multiplication with t p on t p M, t > 0, acts as a scalar in each cohomology group. Hence, there exist numbers p q such that It follows from Lemma 3.21 below that H q (tM) = 0 for all 0 ≤ q ≤ dim M and that the numbers p q are mutually different. In fact, the Betti numbers of any nilpotent Lie algebra are all nonzero [43,Théorème 2]. Clearly, p 0 = 0. The Rumin differential D q has Heisenberg order We expect p q+1 > p q , but are unable to prove this in general. If this inequality failed, then D q would have to vanish. Subsequently, we will assume p q+1 > p q .

Lemma 3.2 If the osculating algebras have pure cohomology, then the Rumin complex (88) does not depend on the choice of a graded fiberwise Euclidean inner productg on tM or the choice of a splitting S : tM → T M. Moreover, the isomorphism in cohomology induced by L is independent of these choices too.
Proof Suppose the Euclidean inner productg u and the splitting S u depend on a (discrete) parameter u. Let˜ u , L u , L u , ι u , and D u denote the associated operators.
Combining (89) and (90), we obtain It remains to show that ι −1 v˜ v L −1 v L u = id, for any two parameters u and v. Note that each factor is filtration preserving and the composition induces on the associated graded. Indeed, gr(L) = id, ι takes values in ker ∂, and ι −1˜ | ker ∂ coincides with the canonical projection ker ∂ → ker ∂/ img ∂ = H(tM). Since the filtration on H q (tM) was assumed to be trivial, this implies ι −1 v˜ v L −1 v L u = id on ∞ (H q (tM)), for each q. Composing with L v from the left, the latter equation also yields v L u = L v . Hence, as v induces the identity on cohomology, L u and L v induced the same isomorphism in cohomology.

Analytic Torsion
We continue to consider a closed filtered manifold M whose osculating algebras have pure cohomology of locally constant dimension. Twisting the Rumin complex in (88) with a flat vector bundle F, we obtain a Rockland complex computing the cohomology of M with coefficients in the flat bundle F. More explicitly, the twisted operator L F : ∞ (H(tM) ⊗ F) → (M; F) provides a chain map, which induces an isomorphism on cohomology, Letg be a graded fiberwise Euclidean inner product on tM and letg −1 denote the induced fiberwise Euclidean inner product on q t * M. Via the orthogonal decomposition in (83) we obtain a fiberwise Euclidean inner product on H q (tM) that will be denoted byg −1 too. If h is a fiberwise Hermitian inner product on F, we obtain a fiberwise Hermitian inner product on H q (tM) ⊗ F we will denote byg −1 ⊗ h. Using the volume density μg on M induced from the Euclidean inner productg −1 via the canonical identification m t * M = m T * M where m = dim M, we obtain formal adjoints (D F ) * and Rumin-Seshadri operators, see (7), which will be denoted by These are Rockland operators of Heisenberg order 2κ, see (6), which also depend on the choice of numbers a q as in (5), but they are independent of the splitting S according to Lemma 3.2. According to Lemma 2.5, the graded heat trace of F g,h is constant in t and cf. (24). The subscript F indicates the dependence on the filtration on M. We will refer to this as the analytic torsion of the filtered manifold M with coefficients in the flat bundle F. It is defined for closed filtered manifolds whose osculating algebras have pure cohomology of locally constant dimension.
The analytic torsion defined above is a common generalization of the Ray-Singer torsion [6,77] and the Rumin-Seshadri [85] analytic torsion. In Sect. 4 we will discuss another instance of this torsion associated with a certain five dimensional geometry. We are not aware of further filtered manifolds whose osculating algebras have pure cohomology, cf. Sect. 3.7 below.
Let us spell out the following immediate consequence of Proposition 2.23:

Proposition 3.3 Suppose F 1 and F 2 are two flat vector bundles with Hermitian inner products h 1 and h 2 , respectively. Then, up to the canonical identification of determinant lines
With respect to finite coverings we have Proof This follows from Proposition 2.25. Indeed, since the covering map is a local diffeomorphism of filtered manifolds, we have canonical isomorphisms of vector bundles π * (H q (tM) ⊗F) = H q (tM) ⊗ π * F intertwining π * DF q with D π * F q , cf. (95), and intertwining the Hermitian inner product π * (π * g−1 ⊗h) with the Hermitian inner productg −1 ⊗ π * h . Moreover, π * μg = μ π * g .

Poincaré Duality
Rumin observed that his complex is Hodge self-dual, see [81,Sect. 2] or [83,Proposition 2.8]. In this section we recall this duality and discuss implications for the analytic torsion.
Let us denote the dimension of M by Recall that the wedge product provides a fiberwise non-degenerate bilinear pairing The corresponding vector bundle isomorphism is an isometry with respect to the Euclidean metricg −1 on the left hand side and the induced Euclidean metricg ⊗g −1 on the right hand side. Hence, Since the Chevalley-Eilenberg codifferential ∂ : * t * x M → * +1 t * x M is a graded derivation, the wedge product induces a fiberwise bilinear pairing where we regard H(tM) = ker ∂/ img ∂. By Poincaré duality for the nilpotent Lie algebra t x M, this pairing is fiberwise non-degenerate. In particular, the codifferential in top degree, ∂ : m−1 t * M → m t * M vanishes. Hence, we have canonical identifications Note that the vector bundle map π : ker δ → H(tM) from Remark 3.1 coincides with the canonical identification (104) in top degree. Moreover, in top degree L is algebraic, inducing the canonical identification in (104) inverse to π , cf. (91).
Proof Since ∂ is a graded derivation that vanishes in top degree, we have (105). Using the vector bundle isomorphism in (101), this equation may be written in the form Combining (106) with (85) and (84), we obtain (107).
For every flat vector bundle F, the wedge product in (103) induces a fiberwise non-degenerate bilinear pairing Here O denotes the orientation bundle of M and we are using the canonical identification of line bundles From (109) we obtain Proof By the Leibniz rule for the de Rham differential, (96) and (112) we find: In view of (112), the isomorphisms induced on cohomology by L F * ⊗O and L F , see (97), intertwine this pairing with the Poincaré duality pairing In particular, the pairing in (113) is non-degenerate. .

Here h −1 denotes the fiberwise Hermitian inner product on F * ⊗ O induced by the fiberwise Hermitian inner product h on F.
Proof From the preceding lemma and Stokes' theorem we obtain the following commutative diagram: Here the vertical arrows denote the vector bundle isomorphisms Moreover, from Proposition 2.24 we obtain Combining the preceding to equations, we obtain the theorem.
Duality can also be understood in terms of an analogue of the Hodge star operator, cf. [81,Sect. 2] or [83,Proposition 2.8]. To this end, we let for α, β ∈ q t * x M where we are using the canonical identification given by the second and third equality in (111). Recall that the star operator is isometric, i.e., * = −1 .

The star restricts to vector bundle isomorphisms
cf. (83). From Lemma 3.7 we obtain [81,83] Using (115) this also yields Hence, Since we have a q = a m−q−1 , see (6). Hence, we may use the same numbers a q for the Rumin-Seshadri operator associated with flat bundle F * ⊗ O, and obtain (7) In particular, In view of (116), the numbersÑ q := −N m−q also satisfy the relation (19), that is, Combining (117), (118) and (119), we obtain cf. (20) and (82). Recall here that according to Lemma 2.7, we may use any sequence of numbers satisfying (19) to compute this zeta function. The relation (120) immediately leads to a slightly different proof of Theorem 3.8. Let us spell out two more consequences.

Proposition 3.9 If M is an orientable manifold of even dimension and h is a parallel
where p F q, j ∈ ∞ end(H q (tM) ⊗ F) ⊗ | | denote the degree q part of the local quantities in the heat kernel asymptotics associated with the Rumin-Seshadri operator F g,h , cf. (13). Using locality, this gives

Proposition 3.10 If the dimension of M is odd and if h is a parallel Hermitian metric on F, then str(
Proof Since h is parallel, it provides an isomorphism of flat vector bundles h :F → F * which maps the Hermitian metrich onF to the Hermitian metric h −1 on F * . Hence, pF q, j = p F * q, j . Combining this with (121), we see that the summands in str( p F j ) cancel pairwise if the dimension is odd.

Variation of the Metrics
In this section we apply Theorem 2.11 to determine how the analytic torsion of a Rumin complex depends on the fiberwise inner productsg on tM and h on F.
If A ∈ ∞ (Aut(tM)) is a fiberwise automorphism of graded (nilpotent) Lie algebras, we let H(A) ∈ ∞ (Aut(H(tM))) denote the automorphism in fiberwise cohomology induced by the transpose inverse Theorem 3.11 Suppose the graded fiberwise Euclidean inner productg u on tM and the fiberwise Hermitian inner product h u on F depend smoothly on a real parameter u such thatg −1 vg u ∈ ∞ (Aut(tM)) is a fiberwise automorphism of graded (nilpotent) Lie algebras, for all u and v. Then , and p F u,n ∈ ∞ end(H(tM) ⊗ F) ⊗ | | denotes the constant term in the heat kernel expansion associated with the Rumin-Seshadri operator F g u ,h u .
Proof Recall that we are using the fiberwise Hermitian inner productsg −1 u ⊗ h u on the Rumin complex H(tM)⊗ F, whereg −1 u denotes the fiberwise Euclidean inner product on H(tM) obtained by restriction via the fiberwise Hodge decomposition (83) from the induced Euclidean inner product on * t * M which is also denoted byg −1 u . Put A v,u =g −1 vg u ∈ ∞ (Aut(tM)) and let A t v,u =g ug −1 v ∈ ∞ (Aut(t * M)) denote the fiberwise dual automorphism. Extend this to an automorphism of the exterior bundle, for α, β ∈ * t * x M. Note that the equality holds on * t * M for we are using the induced Euclidean inner products on the exterior bundle. Since A v,u is a fiberwise automorphism of Lie algebras, A t v,u commutes with fiberwise Chevalley-Eilenberg codifferential ∂, the fiberwise adjoint ∂ * does not depend on u, and A t v,u commutes with ∂ * too. Hence, the decomposition in (83) is independent of u and invariant under A t v,u . Hence, Combining (122) and (123), we conclude Differentiating, we find Furthermore, we clearly have μg u = det 1/2 (g −1 vg u )μg v and thus The statement thus follows from Theorem 2.11.
Recall that φ t denotes the grading automorphism of t x M = p t p x M given by multiplication with t p on the summand t p x M, t > 0. For a positive smooth function f on M, we let φ f ∈ ∞ (Aut(tM)) denote the fiberwise Lie algebra automorphism acting by φ f (x) on the fiber t x M.

Corollary 3.12 Suppose f is a real valued smooth function on M and consider a family of Euclidean inner products of the formg u
where p F u,n ∈ ∞ end(H(tM) ⊗ F) ⊗ | | denotes the constant term in the heat kernel expansion associated with the Rumin-Seshadri operator F g u ,h .
Proof Sinceġ u =g −1 u ∂ ∂ug u acts by p f on t p M, we have cf. (12) tr(ġ u ) = −n f .

Remark 3.13
Conformal invariance of the analytic torsion, that is, independence under scaling of the Euclidean inner product on tM as in Corollary 3.12, thus is equivalent to the pointwise vanishing of the local quantity str (N − N 0 − n 2 ) p F n . Using Lemma 2.5 and (22), we find In view of (50) the integral in (124) is independent ofg and h, hence this is a smooth invariant of the filtered manifold M and the flat bundle F. Actually, this invariant only depends on F and the homotopy class of the underlying filtration, see Remark 3.17 below for a more precise statement. In the trivially filtered case, this vanishes according to [6,Theorem 7.10]. A discussion of the contact case can be found at the end of Sect. 3.2 in [85]. Whether (124) vanishes in general is unclear. If m is odd and h is parallel then str( p F n ) = 0 according to Proposition 3.10.

Remark 3.
14 If h u are parallel Hermitian inner products on F, then the integrands in the preceding statements can be simplified somewhat: str H(ġ u ) p F u,n = rk(F) str H(ġ u ) p u,n str ḣ u p F u,n = tr(ḣ u ) str( p u,n ) str p F u,n = rk(F) str( p u,n ) str N p F u,n = rk(F) str N p u,n Here p u,n ∈ ∞ end(H(tM)) ⊗ | | denotes the constant term in the heat kernel expansion for the corresponding Rumin-Seshadri operator associated with the trivial flat line bundle over M equipped with the canonical Hermitian metric. This follows from Remark 2.16.

Deformation of the Filtration
In this section we consider a family of closed filtered manifolds M u smoothly depending on a real parameter u. More explicitly, we assume the underlying manifold M remains fixed, and the filtration subbundles T p M u ⊆ T M depend smoothly on u.
As before, we assume that the osculating algebras of M u have pure cohomology of locally constant dimension. Moreover, supposeg u is a smooth family of graded fiberwise Euclidean metrics on tM u . For any flat bundle F with Hermitian metric h, we obtain an analytic torsion − sdet(H * (M;F)) on the determinant line sdet(H * (M; F)). Below, we will describe qualitatively how this torsion depends on the parameter u.
Clearly, the bundle of osculating algebras tM u depends smoothly on u also. For simplicity, we will assume that these are isomorphic as bundles of graded Lie algebras. Hence, there exist isomorphisms of graded vector bundlesψ v,u : tM u → tM v which intertwine the fiberwise Lie algebra structures and depend smoothly on the parameters u and v. W.l.o.g. we may assumeψ w,vψv,u =ψ w,u and in particularψ u,u = id. Let S u : tM u → T M u be splittings of the filtration which depend smoothly on u and define ψ v,u ∈ ∞ (Aut(T M)) by the equation Then ψ v,u : T M u → T M v is a filtration preserving isomorphism, ψ v,u (T p M u ) = T p M v , which inducesψ v,u : tM u → tM v on the associated graded. Moreover, ψ w,v ψ v,u = ψ w,u and ψ u,u = id. Since we have already discussed the dependence on the metric before, we will consider a family of fiberwise Euclidean metricsg u on wherep F n,u ∈ ∞ end(H(tM u ) ⊗ F) ⊗ | | is the constant term in the asymptotic expansion, as t → 0, HereȦ F u is the differential operator given by the expression (139) below,ψ u := ∂ ∂v | v=u ψ v,u ∈ ∞ (end(T M)), and k F t,u denotes the heat kernel of the Rumin-Seshadri operator F g u ,h . In particular,p F n,u is locally computable. More precisely,p F n,u (x) can be computed from the germ of (M, F u ,g u , F, h, S u ,ψ u ) at x. Furthermore,p F n,u = 0 if the homogeneous dimension n is odd.
Proof The differential operators A v,u : ∞ (H(tM u )) → ∞ (H(tM v )) (89) and (90). Twisting with the flat bundle F, we get differential operators Clearly, A F v,u induces the identity on H * (M; F), up to the canonical identifications in (97). In order to apply Theorem 2.17, we have to analyzeȦ F u . It will be convenient to rewrite A v,u in the form, see (92), We let ψ v,u : * T * M u → * T * M v andψ v,u : * t * M u → * t * M v also denote the isomorphisms induced by ψ v,u : T M u → T M v andψ v,u : tM u → tM v , respectively. Then Eq. (125) remains true on * t * M u if S u : * t * M u → * T * M u denotes the splitting induced by S u : tM u → T M u as in (84). Furthermore, (126) yields whereg −1 u denotes the induced Euclidean metric on * t * M u as before. We havẽ sinceψ v,u is an isomorphism of graded Lie algebra bundles. By (129) this gives and combination with (125) also yields see (85). Moreover, When applying Theorem 2.17 we use an auxiliary linear connection on the bundle whereψ u = ∂ ∂v | v=u ψ v,u ∈ ∞ (der( * T * M)) denotes the fiberwise graded derivation given by contraction withψ u ∈ ∞ (end(T M)).
and thus Combining this with (86), we find Combining this with [δ u , u ] = 0, δ u L u = 0 and˜ u S −1 u δ u = 0, we conclude that the first summand in (135) vanishes. Hence, using (92) we obtaiṅ In particular, the coefficients of the differential operatorȦ F u at x can be expressed in terms of the germ of (M, Moreover, (129) and (133) give The statement thus follows from Theorem 2.17.

Remark 3.16
Note thatȦ F u , tr(ψ u ) andp F j,u in Theorem 3.15 remain unchanged if ψ v,u is replaced by any other filtration preserving family ψ v,u : T M u → T M v satisfying ψ w,v ψ v,u = ψ w,u and inducing the same automorphism tM u → tM v on the associated graded, that is, gr(ψ v,u ) = gr(ψ v,u ) =ψ v,u . Indeed, in this case there exists a smooth family of filtration preserving automorphism B v,u of T M u such that ψ v,u = ψ v,u B v,u and gr(B v,u ) = id. Differentiating, we obtainψ u =ψ u +Ḃ u whereḂ u := ∂ ∂v | v=u B v,u is a filtration preserving endomorphism of T M u inducing gr(Ḃ u ) = 0 on tM u . In particular, tr(Ḃ u ) = 0 and thus tr(ψ u ) = tr(ψ u ). Moreover, gr(π u uḂu L u ) = gr(π u u ) gr(Ḃ u ) gr(L u ) = 0 since each factor is filtration preserving. As tM u has pure cohomology, we conclude π u uḂu L u = 0, cf. the proof of Lemma 3.2. We obtain π u uψ u L u = π u uψu L u and thusȦ F u remains unchanged, see (139). Clearly this implies thatp F j,u remains unchanged as well, see (127).

Remark 3.17
Using (69), we conclude that the quantity in (124) remains unchanged if the filtration on M is smoothly deformed as above, i.e., in such a way that the bundles of osculating algebras remain isomorphic. Hence, this quantity only depends on F and the corresponding homotopy class of the filtration.

Comparison with the Ray-Singer Torsion
We continue to use the notation set up above. Moreover, we consider a Riemannian metric g on M.
Comparing the torsion of the Rumin complex constructed above, see (99), with the Ray-Singer torsion [6,77], we obtain a positive real number For trivially filtered manifolds the analytic torsion constructed above coincides, by definition, with the Ray-Singer torsion. We will add the decoration RS to our notation if an object is associated to the underlying trivially filtered manifold. For contact manifolds, Albin and Quan [1,Corollary 3] have shown that the quotient (140) can be expressed as an integral over local quantities. Possibly, their analysis of the sub-Riemannian limit can be extended to filtered manifolds whose osculating algebras have pure cohomology. We will not pursue this here though. Returning to the general case we have. Proof Let u = F ũ g,h denote the Rumin-Seshadri operator (98) and let RS,u = F u RS,g,h denote the Hodge Laplacian. Given u 0 , we choose λ > 0 such that all nonzero eigenvalues of u 0 and all non-zero eigenvalues of RS,u 0 are strictly larger than λ. By definition, see (24) and (99), we have If u is sufficiently close to u 0 , then the spectra of u and RS,u do not contain λ. For these u, the spectral projectors P λ,u and P RS,λ,u depend smoothly on u, and so do ζ λ,u (0) and ζ RS,λ,u (0), cf. (20). Hence, numerator and denominator of the first factor in (141) depend smoothly on u, provided u remains sufficiently close to u 0 . To analyze the second factor in (141), we consider the chain map, cf. (96), u : img(P λ,u ), D F u → img(P RS,λ,u ), d F u , u := P RS,λ,u • L F u | img(P λ,u ) .
Similarly, d F u 0 vanishes on img(P RS,λ,u 0 ). Hence, u 0 is an isomorphism of (trivial) chain complexes. We conclude that u is an isomorphism of chain complexes, provided u is sufficiently close to u 0 . In this situation, we have the well-known formula where for these u, the right hand side clearly depends smoothly on u.

(b) If π :M → M is a finite covering of filtered manifolds andF is a flat vector bundle overM with fiberwise Hermitian inner producth, then
(c) If π :M → M is a finite covering of filtered manifolds, then R π * F,π * h π * g ;π * g (M, π * F) = R F⊗V,h⊗h Ṽ g;g (M, F).
(d) With respect to Poincaré duality, we have and, in particular, the analytic torsion on the left hand side is independent of the metric g. However, besides the trivially filtered case, we do not know of any even dimensional filtered manifold whose osculating algebras have pure cohomology: contact manifolds are odd dimensional and so are the five dimensional geometries discussed in Sect. 4 below.
We expect that the analytic torsion proposed in this paper can be refined to a complex bilinear form on the determinant line sdet(H * (M; F)) such that the comparison with the refined complex valued Ray-Singer torsion [12-16, 20, 21, 92] analogous to (140) yields a ratio that depends holomorphically on the flat connection.

Nilpotent Lie Algebras with Pure Cohomology
The analytic torsion discussed above is only defined for filtered manifolds which have osculating algebras with pure cohomology. This assumption appears to be very restrictive. The only examples we are aware of are: trivially filtered manifolds corresponding to abelian osculating algebras giving rise to the classical Ray-Singer torsion; contact manifolds corresponding to Heisenberg algebras giving rise to the Rumin-Seshadri torsion; and the five dimensional geometries discussed in Sect. 4 below.
The aim of this section is to present a necessary condition which the dimensions of the grading components of a finite dimensional graded nilpotent Lie algebra g = g −r ⊕ · · · ⊕ g −2 ⊕ g −1 have to satisfy, if the cohomology of g is pure.
To this end let φ t denote the grading automorphism acting by multiplication with t p on the summand g p and consider the polynomial [42] P(t) := str φ t |H * (g) .
If g has pure cohomology, then there exist numbers p q ∈ N 0 such that φ t acts by the scalar t p q on H q (g). Hence, in this case the polynomial has at most d + 1 non-trivial coefficients, where d := dim g.
Clearly, str(φ t |H * (g)) = str(φ t | * g * ) = p str(φ t | * g * p ) and thus This is a polynomial of degree n := p pn p . Moreover, where the (positive, integral) coefficients a l are defined by n−d l=0 a l t l := (1 + t) n 2 1 + t + t 2 n 3 · · · 1 + t + t 2 + · · · + t r −1 n r and we are using the convention d m = 0 if m < 0 or m > d. Using which is valid for all integers 0 ≤ i ≤ n and 0 ≤ l ≤ n − d, we obtain are polynomials of degree n − d in i. Hence, the polynomial P(t) has at least d + 1 non-trivial coefficients. Moreover, we obtain Lemma 3.21 If g has pure cohomology, then P(t) has precisely d + 1 non-trivial coefficients and c(i) has n − d mutually different integral zeros located in the range {0, . . . , n}. More explicitly, where P := { p q : q = 0, . . . , d} and P := {0, . . . , n} \ P.

Example 3.22
If g = g −2 ⊕ g −1 with n 2 = 2, then a l = 1, 2, 1 and Hence, if g has pure cohomology, then n must be a square according to Lemma 3.21. The case n = 4 corresponds to n 1 = 0 and is realized by an abelian Lie algebra concentrated in degree 2. The case n = 9 corresponds to n 1 = 5. It is not hard to see that this cannot be realized by a nilpotent Lie algebra though. We do no know if larger squares n can be realized by nilpotent Lie algebras. Example 3. 23 If g = g −2 ⊕ g −1 with n 2 = 3, then a l = 1, 3, 3, 1 and Hence, if g has pure cohomology, then n must be even and 3n − 2 must be a square according to Lemma 3.21. The case n = 6 corresponds to n 1 = 0 and is realized by an abelian Lie algebra concentrated in degree 2. The general n consistent with Lemma 3.21 is of the form n = (m 2 + 2)/3 where m ≥ 4 is an integer congruent to 2 or 4 mod 6. Which of these can be realized by nilpotent Lie algebras is unclear.

Example 3.24
If g = g −2 ⊕ g −1 with n 2 = 4, then a l = 1, 4, 6, 4, 1 and Hence, if g has pure cohomology, then the four roots of c(r ), must all be integral according to Lemma 3.21. The case n = 8 corresponds to n 1 = 0 and is realized by an abelian Lie algebra concentrated in degree 2. We do not know of any other n for which this happens. Using a computer, we ruled out all 8 < n ≤ 10000. For some n, e.g. n = 17, n = 66, or n = 1521, two roots are integral, but never all four, in this range.

Example 3.25
If g = g −2 ⊕ g −1 with n 2 = 5, then a l = 1, 5, 10, 10, 5, 1 and Hence, if g has pure cohomology, then the root i = n/2 must be integral, hence n must be even and the other four roots, must all be integral according to Lemma 3.21. The case n = 10 corresponds to n 1 = 0 and is realized by an abelian Lie algebra concentrated in degree 2. We do not know of any other n for which this happens. Using a computer, we ruled out all 10 < n ≤ 10000. For some n, e.g. n = 17, n = 36, or n = 289, two roots are integral, but never all five. For n = 67 four roots are integral, but this one is odd.
Using computer algebra, we also ruled out Lie algebras of the form g = g −2 ⊕ g −1 in the range 1 ≤ n 1 ≤ 1000 and 6 ≤ n 2 ≤ 11 -non of these dimensions are consistent with the condition in Lemma 3.21.

Example 3.26
If g = g −3 ⊕ g −2 ⊕ g −1 with n 2 = n 3 = 1, then a l = 1, 2, 2, 1 and c(i) = (n − 2i) 3 4 (n − 2i) 2 + n 2 4 − 3n + 2 Hence, if g has pure cohomology, then the root i = n/2 must be integral, hence n must be even and the other two roots, must be integral according to Lemma 3.21. All n ≥ 12 can be ruled out since in this case these roots are complex. Hence, n = 10 is the only case consistent with Lemma 3.21. However, one readily shows that this can not be realized by a nilpotent Lie algebra.

Example 3.27
If g = g −3 ⊕ g −2 ⊕ g −1 with n 2 = 2 and n 3 = 1, then we have a l = 1, 3, 4, 3, 1 and If this polynomial has four real zeros, then it must take a positive value at i = n/2, hence we must have n 2 − 14n + 24 < 0. This rules out any n ≥ 12. The remaining finitely many n can easily be excluded. Hence, no nilpotent Lie algebra of this form is consistent with the condition in Lemma 3.21.
If g has pure cohomology, then the zero i = n/2 must be integral, hence n must be even. Moreover, in order to obtain five real zeros, the polynomial c(i) must have a local maximum at i = n/2. Hence we must have 6n 2 − 132n + 252 < 0, which rules out any n ≥ 20. Checking the remaining n by computer, we see that the only case consistent with Lemma 3.21 is n = 10 corresponding to n 1 = 2. The latter can indeed be realized by a nilpotent Lie algebra, see Sect. 4 below.

Generic Rank Two Distributions in Dimension Five
Recall that a generic rank two distribution on a 5-manifold M is a smooth rank two subbundle D in the tangent bundle T M such that Lie brackets of sections of D span a rank three subbundle [D, D] We will denote this filtration by F D .
The osculating algebras x M of such a filtration are all isomorphic, as graded Lie algebras, to the five dimensional graded Lie algebra with graded basis X 1 , X 2 ∈ g −1 , X 3 ∈ g −2 , X 4 , X 5 ∈ g −3 and non-trivial brackets see [29], [ . Then there exists a unique (in general ungraded) Lie algebra automorphism ϕ : g → g such that ϕ(X 1 ) = Y 1 and ϕ(X 2 ) = Y 2 .
Proof Define a linear map ϕ : g → g by ϕ( Using the relations in (145) one readily checks that ϕ is a homomorphism of Lie algebras. With respect to the decomposition g = g −3 ⊕ g −2 ⊕ g −1 the linear map ϕ has upper triangular block form with invertible diagonal blocks. Hence, ϕ is invertible.
To obtain an analytic torsion, it suffices to choose a sub-Riemannian metricg on the rank two bundle D = T −1 M = t −1 M. We can use the fiberwise Lie algebra structure to extend this to a graded Euclidean inner product on tM, which will be denoted bỹ g too. Indeed, in view of (145), the fiberwise (Levi) bracket on tM induces canonical isomorphisms of vector bundles More explicitly, using a graded basis as in (145), we extendg such that For the purpose of this paper, any other pair of positive constants in (147) and (148) would work equally well. We will not discuss here to what extent the torsion depends on the choice of these constants. The following will be crucial though: Ifg u is a family of sub-Riemannian metrics on D, then their extensions differ by a graded automorphism of the bundle of osculating algebras tM, that is,g −1 vg u ∈ ∞ (Aut(tM)), see (146). Moreover, tr tM (ġ u ) = 5 tr D (ġ u ). Hence, Theorem 3.11 is applicable and specializes Theorem 1.1.

Proof of Theorem 1.2
We begin by noting that a sub-Riemannian metricg on a generic rank two distribution D induces a splitting of the filtration S D,g : tM → T M. Indeed, the sub-Riemannian metric induces a nowhere vanishing section of the line bundle | 2 D * | that provides a scale which is closely related to the generalized contact forms considered in [ [29], see also [26]. In particular, it provides a splitting of the filtration such that S D,g (x) only depends on the germ of (M, D,g) at x. Using these splittings, the coefficients at x of the differential operators L : ∞ (H(tM)) → (M) and π : (M) → ∞ (H(tM)) only depend on the germ of (M, D,g) at x.
We denote the filtered manifold corresponding to the generic rank two distribution D u by M u . Letψ u : tM → tM u denote the unique isomorphism of graded Lie algebra bundles which restricts to u | D : D → D u . Clearly, (ψ u ) * g =g u , cf. (126), whereg andg u denote the graded Euclidean metrics on tM and tM u which are induced from the sub-Riemannian metrics on D and D u as indicated in (147) and (148). We use the splittings discussed above to define ψ u ∈ ∞ (Aut(T M)) by ψ u S D,g = S D u ,g uψ u , cf. (125). Putψ = ∂ ∂u | u=0 ψ u and consider the differential operator, cf. (139), We will now show that tr(ψ)(x) and the coefficients ofȦ at x only depend on the germ of ( Clearly gr(ψ u ) =ψ u and ψ 0 = id. Differentiating, we see thatψ (x) = ∂ ∂u | u=0 ψ u (x) only depends on the germ of (M, D,˙ | D , X , Y ) at x. According to Remark 3.16 we have tr(ψ)| U = tr(ψ ) andȦ| U = −π ψ L| U . We conclude that tr(ψ)(x) and the coefficients ofȦ at x only depend on the germ of (M, D,g,˙ | D ) at x.
From the preceding paragraph we see that the coefficients at x of the differential operatorȦ F + (Ȧ F ) * − tr(ψ) only depend on the germ of (M, D,g, F, h,˙ | D ) at x. Hence, in the asymptotic expansion the termsp F j (x) can be computed from the germ of (M, D,g, F, h,˙ | D ) at x. Hence, in view of Theorem 3.15, the density α = str(p F 10 ) has the desired property.

Computation for Nilmanifolds
Let G denote the simply connected nilpotent Lie group with Lie algebra (144). Since the structure constants are rational, this group admits lattices, i.e., discrete cocompact subgroups, [76,Theorem 2.12]. Suppose ⊆ G is a lattice and let ρ : → U (k) be a finite dimensional unitary representation of . Then F ρ := G × C k is a flat vector bundle over the nilmanifold G/ which comes with a parallel Hermitian metric h ρ . Let D be a right G-invariant generic rank two distribution on G and let D denote the induced generic rank two distribution on G/ . Note that a 2-dimensional subspace in g gives rise to a right invariant generic rank two distribution on G iff it intersects [g, g] = g −3 ⊕ g −2 trivially, cf. Lemma 4.1. We consider a right invariant sub-Riemannian metricg on D and letg denote the induced sub-Riemannian metric on D . does not depend on the right invariant generic rank two distribution D on G nor does it depend on the right invariant sub-Riemannian metricg on D.
Proof We start by showing independence ofg. By convexity, any two right invariant sub-Riemannian metrics on D can be connected by a straight path of right invariant sub-Riemannian metrics on D. In view of Theorem 1.1, it therefore suffices to show that the constant term in the asymptotic expansion of the heat kernel of the Rumin-Seshadri operator F ρ D ,g ,h ρ vanishes. Since this asymptotic expansion can be computed locally, we may work on the universal covering p : G → G/ . Over G, the flat bundle F ρ may be (canonically) trivialized and the Hermitian metric h ρ becomes constant in this trivialization. In view of Lemma 4.1 there exists a group isomorphism φ : G → G such that φ * D = D 0 , the right invariant standard distribution obtained by translating g −1 . Moreover,g 0 := φ * g is a right invariant sub-Riemannian metric on D 0 . Hence, by naturality, where h 0 denotes the standard (constant) Hermitian metric on the trivial flat bundle φ * p * F ρ = G × C rk(ρ) . Note that D 0 andg 0 are also homogeneous with respect to the grading automorphism of G. Hence, C rk(ρ) D 0 ,g 0 ,h 0 is right invariant and homogeneous of degree 2κ. The same is true for the heat operator ∂ t + C rk(ρ) D 0 ,g 0 ,h 0 on R × G, after assigning degree 2κ to the time direction R. Consequently, this heat operator admits a parametrix which is homogeneous of degree −2κ. Combining this with (152), we see that all but the leading term in the asymptotic expansion of the heat kernel of To show independence of D, we proceed analogously using Theorem 1.2. Since the space of 2-dimensional subspaces in g intersecting g −3 ⊕ g −2 trivially is connected, any two right invariant generic rank two distributions on G can be connected by a smooth family of rank two distributions of the same type. According to Theorem 1.2 and the formula in (151) it thus suffices to show that for every differential operator B of Heisenberg order at most r on G/ we have an asymptotic expansion of the form, as t → 0, and, in particular, the constant termp 10 vanishes. This in turn follows from (152).
Combining these observations we will now show The theorem thus follows form Lemma 4.6.
Let us finally give a proof of Theorem 1.3 formulated in the introduction. To this end, recall that every flat line bundle F with parallel Hermitian metric h over G/ is of the form F = F ρ with h = h ρ for some unitary character ρ : → U (1). Hence, it suffices to show that R (ρ) is constant in and ρ, cf. (153). Since was assumed to be generated by two elements, there exists an automorphism φ of G mapping 0 onto , cf. Lemma 4.1. Using Lemma 4.3(e) and applying Lemma 4.6 with the trivial character ρ 0 = 1 and χ 0 = ρφ, we obtain R (ρ) = R 0 (ρφ) = R 0 (1).
Hence, the constant c = R 0 (1) has the desired property.