Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds

We prove quantitative equidistribution results for actions of Abelian subgroups of the $2g+1$ dimensional Heisenberg group acting on compact $2g+1$-dimensional homogeneous nilmanifolds. The results are based on the study of the $C^\infty$-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalisation method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in $g$ variables, generalizing the classical results of Hardy and Littlewood \cite{MR1555099, MR1555214} and the optimal result of Fiedler, Jurkat and K\"orner \cite{MR0563894} to higher dimension.


INTRODUCTION
In the analysis of the time evolution of a dynamical system many problems reduce to the study of the cohomological equation; in the case, for example, of a smooth vector field X on a connected compact manifold M this means finding a function u on M that is a solution of the equation where f is a given function on M .
For a detailed discussion of the cohomological equation for flows and transformations in ergodic theory the reader may consult [28].
In the 2006 paper [14], Forni and the second author used renormalization techniques coupled with the study of the cohomological equation to derive the equidistribution speed of nilflows on Heisenberg three-manifolds. This approach had initially been used by Forni for the study of flows on translation surfaces and subsequently by Forni and the second author [13] for the study of horocycle flows, where precise asymptotics of the equidistribution of these flows were obtained (see also [3]). Renormalization fails for homogeneous flows on higher-step nilmanifolds as, in general, the automorphism group of the underlying nilpotent group is rather poor, lacking semi-simple elements. In a recent paper [16] Forni and the second author developed a novel "rescaling technique" to overcome this difficulty in higher-step nilmanifold; as a consequence they obtained non-trivial estimates on Weyl sums, estimates which have recently been improved independently by Wooley [50].
The present paper moves in a different direction: the study of higher-rank Abelian actions, a theme of research that has been the subject of several investigations, primarily by A. Katok and co-authors (e.g., [29,35,11,34,11,33]). In fact, homogeneous actions of Abelian subgroups of higher-dimensional Heisenberg groups provide a setting where renormalization methods may still be applied, yielding precise quantitative estimates of the rate of equidistribution of the orbits once an in-depth analysis of the cohomological equations is carried out. Thus, an important part of this work is devoted to the study of the full cohomology of the actions of these groups; our attention has been focused on obtaining tame estimates for the solutions of cohomological equations with minimal loss of smoothness, a result that has its own interest in view of future applications to the study of some perturbations of these actions.
An immediate consequence of the quantitative estimates of the rate of equidistribution are bounds on exponential sums for quadratic forms in terms of certain diophantine properties of the form. To our knowledge these bounds, which generalise the classical results of Hardy and Littlewood [25,26] and the optimal result of Fiedler, Jurkat, and Körner [17], are new.
Cohomology in Heisenberg manifolds. In this article we study the cohomology of the action of an abelian subgroup P of the (2g + 1)-dimensional Heisenberg group H g on the algebra of smooth functions on a homogeneous manifold H g /Γ. The linearity of the problem and the fact that the unitary dual of H g is classical knowledge make the use of harmonic analysis particularly suitable to our goal, as it was the case in the works of L. Flaminio and G. Forni [13,14,15].
As a consequence, our results on the cohomology of P also apply to more general H g -modules, those for which the action of the center of H g has a spectral gap.
Before stating our results, let us fix some notation.
Let G be a connected Lie group of Lie algebra g, and let M = G/Γ be a compact homogeneous space of G. Then G acts by left translations on C ∞ (M) via (1. 2) Let H g be the Heisenberg group of dimension 2g + 1. Any compact homogeneous space M = H g /Γ is a circle bundle p : M → H g /(Γ Z(H g )) over the 2gdimensional torus T 2g = H g /(Γ Z(H g )), with fibers given by the orbits of the center Z(H g ) of H g . The space of C ∞ functions on M splits as a direct sum of the H g -invariant subspace p * (C ∞ (T 2g )) and the H g -invariant subspace F 0 = C ∞ 0 (M) formed by the smooth functions on M having zero average on the fibers of the fibration p. The following theorem is a particular case of Theorem 3.16 below. DEFINITION 1.4. A connected Abelian subgroup of H g without central elements will be called an isotropic subgroup of H g . A Legendrian subgroup of H g is an isotropic subgroup of H g of maximal dimension g . associating to each ω ∈ B k ( p,F 0 ) a primitive of ω and satisfying tame estimates of degree (k + 1)/2 + ε for any ε > 0.
We have H k ( p,F 0 ) = 0 for k < d ; in degree d , we have that The p-module F 0 has tame splitting in all degrees: for k = 0, . . . , d and for any ε > 0 there exist a constant C and linear maps such that the restriction of M k to B k (p, F 0 ) is the identity map, and the following estimates hold: where w = (k + 3)/2 + ε, if k < d and w = d /2 + ε if k = d .
Let P < H g be a subgroup as in the theorem above and letP be the group obtained by projecting P on H g /Z(H g ) ≈ R 2g . As before we set T 2g = H g /(Γ Z(H g )).
It should be considered as folklore that the cohomology of the action of a sub-groupP on a torus depends on the Diophantine properties ofP, considered as a vector space. The Diophantine conditionp ∈ DC τ (Γ) mentioned in the theorem below will be made precise in Section 3.1.
Equidistribution of isotropic subgroups on Heisenberg manifolds. We denote by M = H g /Γ the standard Heisenberg nilmanifold (see Section 2 for details on the definitions and notations). Let (X 1 , . . . , X g , Ξ 1 , . . . , Ξ g , T ) be a fixed rational basis of h g = Lie(H g ) satisfying the canonical commutation relations. Then the symplectic group Sp 2g (R) acts on H g by automorphisms 1 . For 1 ≤ d ≤ g , let P d be the subgroup generated by (X 1 , . . . , X d ) and, for any α ∈ Sp 2g (R), set We define a parametrization of the subgroup α −1 (P d ) according to Given a Jordan region U ⊂ R d and a point m ∈ M, we define a d -dimensional p-current P d ,α U m by It is well-known that the Diophantine properties of a real number may be formulated in terms of the speed of excursion, into the cusp of the modular surface, of a geodesic ray having that number as limit point on the boundary of hyperbolic space. This observation allows us to define the Diophantine properties of the subgroup P d ,α in terms of bounds on the height of the projection, in the Siegel modular variety Σ g = K g \Sp 2g (R)/Sp 2g (Z), of the orbit of α under the action of some one-parameter semi-group of the Cartan subgroup of Sp 2g (R) (here K g denotes the maximal compact subgroup of Sp 2g (R)). We refer to Section 4.4 for the definition of height function.
Let {exp t δ(d )} t ∈R be the Cartan subgroup of Sp 2g (R) defined by the formula exp(t δ(d ))X i = e t X i , for i = 1, . . . , d and exp(t δ(d ))X i = X i , for i = d + 1, . . . , g . Roughly, the Definition 4.10 states that α ∈ Sp 2g (R) satisfies a δ(d )-Diophantine condition of type σ if the height of the projection of exp(−t δ(d ))α in the Siegel modular variety Σ g is bounded by e 2t d (1−σ) ; if, for any ε > 0, the height considered above is bounded by e 2t d ε , then we say that α ∈ Sp 2g (R) satisfies a δ(d )-Roth condition; finally we say that α is of bounded type if the height of exp(− δ)α stays bounded as δ ranges in a positive cone a + in the Cartan algebra of diagonal symplectic matrices (see Definition 4.10).
As the height function is defined on the Siegel modular variety Σ g , the Diophantine properties of α depend only on its class [α] in the quotient space M g = Sp 2g (R)/Sp 2g (Z). 1 By acting on the left on the components of elements of h g in the given basis.
The definitions above agree with the usual definitions in the g = 1 case. Several authors (Lagarias [36], Dani [10], Kleinbock and Margulis [32], Chevallier [7]) proposed, in different contexts, various generalizations of the g = 1 case. We postpone to Remark 4.11 the discussion of these generalizations.
We may now state our main equidistribution result.
For any s > 1 4 d (d + 11) + g + 1/2 and any ε > 0 there exists a constant C = C (P, α, s, g , ε) > 0 such that, for all T 1 and all test p-forms The exponent of the logarithmic factor in the first case is certainly not optimal. When d = 1, a more precise result is stated in Proposition 5.9, which coincides with the optimal classical result for d = g = 1 (Fiedler, Jurkat, and Körner [17]).
The method of proof is, to our knowledge, the first generalization of the methods of renormalization of Forni [20] and of Flaminio and Forni [14,15] to actions of higher dimensional Lie groups. A different direction is the one taken by Flaminio and Forni in [16], where equidistribution of nilflows on higher step nilmanifolds requires a subtler rescaling technique, due to the lack of a renormalization flow.
A drawback of the inductive scheme that we adopted is that we are limited to consider averages on cubes Q(T ) (the generalization to pluri-rectangles is however feasible, but more cumbersome to state). For more general regions, growing by homotheties, we can obtain weak estimates where the power T d /2 is replaced by T d −1 . However, N. Shah's ideas [45] suggest that equidistributions estimates as strong as those stated above are valid for general regions with smooth boundary.
Application to higher-dimensional Theta sums. In their fundamental 1914 paper [25], Hardy and Littlewood introduced a renormalization formula to study the exponential sums N n=0 e(n 2 x/2+ξn), usually called finite theta sums, where N ∈ N and e(t ) := exp(2πi t ). Their algorithm provided optimal bounds for these sums when x is of bounded type. Since then, Hardy and Littlewood's renormalization method has been applied or improved by several authors obtaining finer estimates on finite theta sums (Berry and Goldberg [4], Coutsias and Kazarinoff [8], Fedotov and Klopp [18]). Optimal estimates have been obtained by Fiedler, Jurkat, and Körner [17]. Differently from the previously quoted authors, who relied heavily on the continued fractions properties of the real number x, Fiedler, Jurkat, and Körner's method was based on an approximation of x by rational numbers with denominators bounded by 4N .
In this paper we consider the g -dimensional generalization, the finite theta sums where Q[x] := x Qx is the quadratic form defined by a symmetric g × g real matrix Q, and (x) := x is the linear form defined by a vector ∈ R g . In the spirit of Flaminio and Forni [14], our method consists of reducing the sum (1.4) to a Birkhoff sum along an orbit (depending on ) of some Legendrian subgroup (depending on Q) in the standard (2g + 1)-dimensional Heisenberg nilmanifold.
The occurrence of Heisenberg nilmanifolds is not a surprise; in fact the connection between the Heisenberg group and the theta series is well known and very much exploited [1,2,48,14,42,43].
The application to g -dimensional finite theta sums (1.4) is the following corollary of Theorem 5.11. • There exists a full measure set Ω g ⊂ M g such that if [α] ∈ Ω g and ε > 0 then The Diophantine conditions in terms of the symmetric matrix Q are discussed in Remark 4.11.
Geometrical methods, similar to ours, to estimate finite theta sums are also used by Griffin and Marklof [23] and Cellarosi and Marklof [9]. They focus on the distributions of these sums as Q and are uniformly distributed in the g = 1 case. As they are only interested in theta sums, they may consider a single irreducible representation ρ of the Heisenberg group and a single intertwining operator between ρ and L 2 (M). The other more technical difference is that as Q and vary, it is more convenient to generalize the ergodic sums (1.3) to the case when ω is transverse current.
Estimates of theta sums are also crucial in the paper of Götze and Margulis [22], which focuses on the finer aspects of the "quantitative Oppenheim conjecture". There, it is a matter of estimating the error terms when counting the number of integer lattice points of given size for which an indefinite irrational quadratic form takes values in a given interval. This is clearly a subtler problem than the one considered here.
Article organization. In Section 2, we introduce the necessary background on the Heisenberg and symplectic groups. In Section 3 we prove the results about the cohomology of isotropic subgroups of the Heisenberg groups. Section 4 deals with the relation between Diophantine properties and dynamics on the Siegel modular variety. Finally, in Section 5 we prove the main equidistribution result and the applications to finite theta sums.
Applications to the rigidity problem of higher-rank Abelian actions on Heisenberg nilmanifolds, as a consequence of the tame estimates for these actions, will be the subject of further works.
2. HEISENBERG GROUP AND SIEGEL SYMPLECTIC GEOMETRY 2.1. The Heisenberg group and the Schrödinger representation.
The Heisenberg group and Lie algebra. Let ω denote the canonical symplectic form on R 2g ≈ R g × R g , i.e., the non-degenerate alternate bilinear form ω ((x, ξ), (x , ξ )) = ξ · x − ξ · x, where we use the notations (x, ξ) ∈ R g × R g and ξ · x := ξ 1 x 1 + · · · + ξ g x g . The Heisenberg group over R g (or the real (2g It is a central extension of R 2g by R, as we have an exact sequence The Lie algebra of H g is the vector space h g = R g × R g × R equipped with the commutator (q, p, t ), (q , p , t ) = (0, 0, p · q − p · q).
JOURNAL OF MODERN DYNAMICS VOLUME 9, 2015, 305-353 Let T = (0, 0, 1) ∈ Z (h g ). If (X i ) is a basis of R g , and (Ξ i ) the symplectic dual basis, we obtain a basis (X i , Ξ j , T ) of h g satifying the canonical commutation relations: A basis (X i , Ξ j , T ) of h g satisfying the relations (2.2) will be called a Heisenberg basis of h g . The Heisenberg basis (X 0 i , Ξ 0 j , T ) where X 0 i and Ξ 0 j are the standard bases of R g , will be called the standard Heisenberg basis.
Given a Lagrangian subspace l ⊂ R g × R g , there exists a Heisenberg basis (X i , Ξ j , T ) such that (X i ) spans l; in this case the span l = Ξ j is also Lagrangian and we say that the basis (X i , Ξ j , T ) is adapted to the splitting l×l × Z (h g ) of h g .
Standard lattices and quotients. The set Γ := Z g × Z g × 1 2 Z is a discrete and cocompact subgroup of the Heisenberg group H g , which we shall call the standard lattice of H g . The quotient is a smooth manifold that will be called the standard Heisenberg nilmanifold. The natural projection map maps M onto a 2g -dimensional torus T 2g := R 2g /Z 2g . All lattices of H g were described by Tolimieri in [48]. Henceforth we will limit ourselves to consider only a standard Heisenberg nilmanifold, our results extending trivially to the general case. Observe that exp T is the element of Z(H g ) generating Γ ∩ Z(H g ).
Unitary H g -modules and Schrödinger representation. The Schrödinger representation is a unitary representation of ρ : H g → U (L 2 (R g , dy)) of the Heisenberg group into the group of unitary operators on L 2 (R g , dy); it is explicitly given by (see [19]). Composing the Schrödinger representation with the automorphism (x, ξ, t ) → (|h| 1/2 x, |h| 1/2 ξ, ht ) of H g , where h = 0 and = sign(h) = ±1, we obtain the Schrödinger representation with parameters h: for all ϕ ∈ L 2 (R g , dy) According to the Stone-von Neumann theorem [37], the unitary irreducible representations π : H g → U (H ) of the Heisenberg group on a Hilbert space H are • either trivial on the center; then they are equivalent to a one-dimensional representation of the quotient group Z(H g )\H g , i.e., equivalent to a character of R 2g , • or infinite dimensional and unitarily equivalent to a Schrödinger representation with some parameter h = 0.
Infinitesimal Schrödinger representation. The space of smooth vectors of the Schrödinger representation ρ h : H g → U (L 2 (R g , dy)) is the space of Schwartz functions S (R g ) ⊂ L 2 (R g , dy) [44]. By differentiating the Schrödinger representation ρ h we obtain a representation of the Lie algebra h g on S (R g ) by essentially skew-adjoint operators on L 2 (R g , dy); this representation is called the infinitesimal Schrödinger representation with parameter h. With an obvious abuse of notation, we denote it by the same symbol ρ h ; the action of X ∈ h g on a function f will be denoted ρ h (X ) f or X . f when no ambiguity can arise. Differentiating the formulas (2.4) we see that, for all k = 1, 2, . . . , g , we have where (y i ) are the coordinates in R g relative to the standard basis (X 0 i ) and = sign(h). More generally, by the Stone-von Neumann theorem quoted above, given any Heisenberg basis (X i , Ξ j , T ) of h g , the formulas define, via the exponential map, a Schrödinger representation ρ h with parameter h on L 2 (R g , dy) such that  Symplectic group and moduli space. Let Sp 2g (R) be the group of symplectic automorphisms of the standard symplectic space (R 2g , ω). The group of those automorphisms of H g that are trivial on the center is the semi-direct product Aut 0 (H g ) = Sp 2g (R) R 2g of the symplectic group with the group of inner auto- The group of automorphisms of H g acts simply transitively on the set of Heisenberg bases, hence we may identify the set of Heisenberg bases of h g with the group of automorphisms of H g . However since we are interested in the action of subgroups defined in terms of a choice of a Heisenberg basis and since the dynamical properties of such action are invariant under inner automorphisms, we may restrict our attention to bases which are obtained by applying automorphisms α ∈ Sp 2g (R) to the standard Heisenberg basis.
Explicitly, the symplectic matrix written in block form α = A B C D ∈ Sp 2g (R), with the g ×g real matrices Siegel symplectic geometry. The stabilizer of the standard lattice Γ < H g inside Sp 2g (R) is exactly the group Sp 2g (Z). We call M g = Sp 2g (R)/Sp 2g (Z) the moduli space of the standard Heisenberg manifold. We may regard Sp 2g (R) as the deformation (or Teichmüller) space of the standard Heisenberg manifold M = H g /Γ and M g as the moduli space of the standard nilmanifold, in analogy with the 2-torus case. The Siegel modular variety, the moduli space of principally polarized abelian varieties of dimension g , is the double coset space where K g is the maximal compact subgroup Sp 2g (R) ∩ SO 2g (R) of Sp 2g (R), isomorphic to the unitary group U g (C). Thus, M g fibers over Σ g with compact fibers K g .
The quotient space K g \Sp 2g (R)/±1 2g may be identified with Siegel upper half-space in the following way. Recall that the Siegel upper half-space of degree/genus g [46] is the complex manifold This action leaves invariant the Riemannian metric d s 2 As the the kernel of this action is given by ±1 2g and the stabilizer of the point i := i 1 g ∈ H g coincides with K g , the map α ∈ Sp 2g (R) → α −1 (i ) ∈ H g induces an identification K g \Sp 2g (R)/±1 2g ≈ H g and consequently an identification of the Siegel modular variety Σ g ≈ Sp 2g (Z)\H g .

COHOMOLOGY WITH VALUES IN H g -MODULES
Here we discuss the cohomology of the action of a subgroup P ⊂ H g on a Fréchet H g -module F , that is to say the Lie algebra cohomology of p = Lie(P) with values in the H g -module F . We assume that P is a connected Abelian Lie subgroup of H g contained in a Legendrian subgroup L.
The modules interesting for us are, in particular, those arising from the regular representation of H g on the space C ∞ (M) of smooth functions on a (standard) nilmanifold M := H g /Γ. As mentioned in the introduction, the fact that H g acts on M by left translations, implies that the space F = C ∞ (M) is a p-module: in fact for all V ∈ p and f ∈ F one defines (cf. formula (1.2)) , (m ∈ M).
As P is an Abelian group, the differential on the cochain complex A * (p, F ) = Λ * p ⊗ F of F -valued alternating forms on p is given, in degree k, by the usual

NOTATION 3.1.
When F is the space of C ∞ -vectors of a representation π of H g we may denote the complex A * (p, F ) also by the symbol A * (p, π ∞ ).
In order to study the cohomology of the complex A * (p,C ∞ (M)), it is convenient to observe that the projection p of M onto the quotient torus T 2g (see (2.3)) yields a H g -invariant decomposition of all the interesting function spaces on M into functions with zero average along the fibers of p -we denote such function spaces with a suffix 0 -and functions that are constant along such fibers; these latter functions can be thought of as pull-backs of functions defined on the quotient torus T; hence we write, for example, and we have similar decompositions for L 2 (M) and -when a suitable Laplacian is used to define them -for the L 2 -Sobolev spaces W s (M).
If we denote byP the projection of P into T 2g and byp its Lie algebra, we obtain that we may split the complex A * (p,C ∞ (M)) into the sum of A * (p,C ∞ 0 (M)) and A * (p, p * (C ∞ (T 2g ))) ≈ A * (p,C ∞ (T 2g )). The action ofP on T 2g being linear, the computation of the cohomology of this latter complex is elementary and folklore when dimP = 1. For lack of references we review it in the next Section 3.1 for any dimP. In Section 3.2 we shall consider the cohomology of C * (p,C ∞ 0 (M)).

REMARK 3.2. To define the norm of the Hilbert Sobolev spaces
·〉 is the ordinary L 2 Hermitian product. This has the advantage that for Currents. Let F be any tame Fréchet h g -module, graded by increasing norms The space of continuous linear functionals on A k (p, F ) = Λ k p⊗F will be called the space of currents of dimension k and will be denoted A k (p, F ), where F is JOURNAL OF MODERN DYNAMICS VOLUME 9, 2015, 305-353 the strong dual of F ; the notation is justified by the fact that the natural pairing (Λ k p,Λ k p) between k vectors and k-forms allows us to write A k (p, F ) ≈ Λ k p⊗F .
Endowed with the strong topology, A k (p, F ) is the inductive limit of the spaces The boundary operators ∂ : A k (p, F ) → A k−1 (p, F ) are, as usual, the adjoint of the differentials d; hence they are defined by 〈∂T, ω〉 = 〈T, dω〉. A closed current T is one such that ∂T = 0. We denote by Z k (p, F ) the space of closed currents of dimension k and by Z k (p, (W s ) ) the space of closed currents with coefficients in (W s ) .
3.1. Cohomology of a linear R d action on a torus. Let Λ be a lattice subgroup of R and let R act on the torus T = R /Λ by translations. We consider the restriction of this action to a subgroup Q < R isomorphic to R d , with Lie algebra q. Then the Fréchet space C ∞ (T ) is a q-module. In this section we consider the cohomology of the associated complex A * (q,C ∞ (T )).
Let Λ ⊥ = λ ∈ (R ) λ · n = Z ∀ n ∈ Λ denote the dual lattice of Λ. We say that the subspace q satisfies a Diophantine condition of exponent τ > 0 with respect to the lattice Λ, and we write q ∈ DC τ (Λ), if

REMARK 3.3.
The Diophantine condition considered here is dual to the Diophantine condition on subspaces of (R ) ≈ R considered by Moser in [41]. In fact, if we set q ⊥ = λ ∈ (R ) ker λ ⊃ q , the condition (3.2) is equivalent to Thus, by Theorem 2.1 of [41], the inequalities (3.2) are possible only if τ ≥ /d −1, and the set of subspaces q ⊥ with µ(q, Λ) = /d − 1 has full Lebesgue measure in the Grassmannian Gr(R d ; R ).
We say that q is resonant (with respect to Λ) if, for some λ ∈ Λ ⊥ {0}, we have q ⊂ ker λ; in this case the closure of the orbits of Q on R /Λ are contained in lower dimensional tori, the orbits of the rational subspace ker λ, and we may understand this case by considering a lower dimensional ambient space R with < .
Thus we may limit ourselves to non-resonant q; in this case, if q is not Diophantine, we have µ(q, Λ) = +∞ and we say that q is Liouvillean (with respect to Λ).
the cohomology classes being represented by forms with constant coefficients. Furthermore, the q-module C ∞ (T ) is tamely cohomologically C ∞ -stable and has tame splitting in all degrees.
Proof. Without loss of generality we may assume Λ = Z . The s-Sobolev norm of a function f ∈ C ∞ (T ) with Fourier series representation is given by We have a direct sum decomposition Consequently, the cohomology H * (q,C ∞ (T )) splits into the sum of cohomology classes represented by forms with constant coefficients and H * (q,C ∞ 0 (T )). We now show that, under the assumption (3.2) on q, we have hence, for ω ∈ Λ k q ⊗ C n and V 0 , . . . ,V k ∈ q, Let X 1 , X 2 , . . . , X d be a basis of q, and define the co-differential d * by We conclude that the map d −1 := H −1 d * is a right inverse of d on the space Z k (q, C n ) of closed forms. From the definitions of the maps d * and H we obtain the estimate It is easily seen that the Diophantine condition (3.2) is equivalent to the existence of a constant C > 0 such that d m=1 |n · X m | 2 > C n −2τ for all n ∈ Z .
Hence, for some constant C > 0 we have d −1 ω 0 ≤ C −1 n τ ω 0 , and therefore Since the Sobolev space (W s 0 (T ), · s ) is equal to the Hilbert direct sum n =0 (C n , · s ), the map d −1 extends to a tame map ) satisfying a tame estimate of degree τ with base 0 and associating a primitive to each closed form.
Combining these results with the previous remark on constant coefficient forms, we conclude that under the Diophantine assumption (3.2) the q-module C ∞ (T ) is tamely cohomologically C ∞ -stable and has a tame splitting in all degrees.
The "only if" part of the statement may be proved as in the case dim Q = 1 (see Katok [28, page 71]). Thus let P be an isotropic subgroup of H g of dimension d . Fix a Legendrian subgroup L such that P ≤ L < H g . Let |h| > h 0 > 0.

Cohomology with values in C
Since the group of automorphisms of H g acts transitively on Heisenberg bases, we may assume that we have fixed a Heisenberg basis (X i , Ξ j , T ) of h g such that (X 1 , . . . , X d ) forms a basis of p and (X 1 , . . . , X g ) is a basis of Lie(L). This yields isomorphisms L ≈ R g and P ≈ R d , with the latter group embedded in R g via the first d coordinates. With these assumptions, the formulas yielding the representation ρ h on L 2 (R g ) are given by the equations (2.6). The space ρ ∞ h of C ∞ vectors for the representation ρ h is identified with S (R g ), on which h g acts by the formulas (2.5).
Homogeneous Sobolev norms. The infinitesimal representation extends to a representation of the enveloping algebra U(h g ) of h g ; this allows us to define the "sub-Laplacian" as the image via ρ h of the element JOURNAL OF MODERN DYNAMICS VOLUME 9, 2015, 305-353 Since H g is a positive operator with (discrete) spectrum bounded below by g |h|, we define the space W s (ρ h , R g ) of functions of Sobolev order s as the Hilbert space of vectors ϕ of finite homogeneous Sobolev norm This makes explicit the fact that the space ρ ∞ h of C ∞ vectors for the representation ρ h coincides with S (R g ).
The homogeneous Sobolev norms (3.4) are not the standard ones (later on we shall make a comparison with standard Sobolev norms). They have, however, the advantage that the norm on W s (ρ h , R g ) is obtained by rescaling by the factor |h| s/2 the norm on W s (ρ 1 , R g ). For this reason we can limit ourselves to studying the case h = 1; later we shall consider the appropriate rescaling. Thus we denote ρ = ρ 1 and, to simplify, we write H g for ρ(H g ) and W s (R g ) for W s (ρ 1 , R g ); we also set The cochain complex A * (p, ρ ∞ ). It will be convenient to use the identification for a function ϕ defined on R g . We also write dx = dx 1 · · · dx d . Then, by the formula (2.4), the group element q ∈ P ≈ R d acts on ϕ ∈ S (R g ) according to Thus the complex A * (p, ρ ∞ ) is identified with the complex of differential forms on p ≈ R d with coefficients in S (R g ). It will be also convenient to define the S (R g −d ), respectively; they may be also considered as operators on S (R g ), and then Then, for any s > g /2, I g extends to a bounded linear functional on W s (R g ), that is I g ∈ W −s (R g ).
Proof. Using Cauchy-Schwartz inequality we have As g +|x| 2 ≤ 2H g , the second integral is bounded by a constant times f 2 s , and the result follows.
For the next lemma we adopt the convention R 0 = {0} and S (R 0 ) = W s (R 0 ) = C with the usual norm.
We consider S (R g ) and S (R g −d ) as H g and H g −d -modules, respectively, with parameter h = 1. Then, for any ε > 0 and s ≥ 0, the map I d ,g extends to a bounded linear map from W s+d /2+ε (R g ) to W s (R g −d ), i.e., for some constant C = C (s, ε, d , g ). In particular this proves the inclusion Proof. For d = g we have I g ,g = I g and the result is a restating of the previous lemma. Now suppose d < g . The operators H d and H g −d , considered as operators on their joint spectral measure on L 2 (R g ) is the product of the spectral measures on L 2 (R d ) and L 2 (R g −d ) respectively. Clearly H g ≥ H d and Let (v m ) and (w n ) be orthonormal bases of L 2 (R d ) and L 2 (R g −d ) of eigenvectors of H d and H g −d with eigenvalues (λ m ) and (µ n ), respectively. We may choose these bases so that {v m } ⊂ S (R d ) and {w n } ⊂ S (R g −d ) . Writing The first term in this product equals I d 2 −(d /2+ε) , which is bounded by Lemma 3.5; the second term is majorated by The proof of the following corollary is immediate. COROLLARY 3.7. We use the notation of the previous lemma. Suppose d < g . For all t ≥ 0 and all We consider S (R g ) and S (R g −d ) as H g and H g −d -modules, respectively, with Proof. Consider H d = and H g −d as operators on S (R g ). For all integers n, from the binomial identity for ( where for the last inequality we used H d ≥ 1 and H g −d ≥ 1. This proves the lemma for integer s; the general claim follows by interpolation. For all t ≥ 0 and all ε > 0 there exists a constant C = C (t , ε) such that In particular this proves that P (S (R g )) ⊂ S (R g −d ).
Proof. When g = 1, the lemma is a variation on the statement of Lemma 6.1 in [14], which can be easily proved by use of the Cauchy-Schwartz inequality as in Lemma 3.5.
Suppose now that g > 1 and consider the decomposition Using the result for the case g = 1 and the definition of the norm · 0 we have for all t ≥ 0 and all ε > 0 VOLUME 9, 2015, 305-353 For integer values of the Sobolev order, using the above inequality and the binomial formula, we may write, for any ε > 0 and n ∈ N, The general inequality follows by interpolation of the family of norms · n .
Sobolev cocycles and coboundaries. Having fixed a Euclidean product on h g , we obtain, by restriction, a Euclidean product on p ⊂ h g and, by duality and extension to the exterior algebra, a Euclidean product on Λ k p . The spaces of cochains of degree k are endowed with the Hermitian products obtained as tensor product of the Euclidean product on Λ k p and the Hermitian products · s or · s on S (R g ). Completing with respect to these norms, we define the Sobolev spaces Λ k p ⊗W s (R g ) of cochains of degree k and use the same notations for the norms. It is clear that, for k < d , the cohomology groups are H k (p, S (R g )) = 0. Here we estimate the Sobolev norm of a primitive Ω ∈ A k−1 (p, S (R g )) of a coboundary ω = dΩ ∈ B k (p, S (R g )) = Z k (p, S (R g )) in terms of the Sobolev norm of ω.
By Lemma 3.6 we obtain that for any t ≥ 0 and ε > 0 we have It follows from this inequality that the image of I lies in ) is obvious, and by Lemma 3.8 we have, for any s ≥ 0, From (3.9) and (3.10) it follows that, for any s ≥ 0, The maps I and E commute with the differential d. It is well known that I and E are homotopy inverses of each other. In fact, it is clear that I E is the identity.
We claim that the usual homotopy operator (3.12) Then by Lemma 3.9 and (3.11) we have that for all s ≥ 0 This proves the claim.
Assume, by induction, that the proposition is true for all g ≥ 1, all d ≤ g and all k ≤ min{n, d }−1. Let ω ∈ A n (R d , S (R g )), with n < d , be closed. Then the (n − 1)-form I ω ∈ A n−1 (R d −1 , S (R g −1 )) is also closed. By the induction assumption, I ω = dη for a primitive η ∈ A n−2 (R d −1 , S (R g −1 )) satisfying the estimate η s ≤ C I ω s+n/2+ε . (3.15) Since E I ω = E dη and E commutes with d, we obtain that a primitive of ω is given by d −1 ω := Ω := K ω + E η. Therefore, from Lemma 3.6 and the estimates (3.9), (3.10), (3.13), and (3.15), we have, for some constants C 's which only depend on s ≥ 0 and ε > 0, Thus the estimate (3.6) holds also for k = n. This concludes the proof.
We are left to consider the space H k (p, S (R g )) when k = d := dim p.
The map I d ,g extends to a map associating to every ω ∈ ker I d ,g a primitive Ω of ω satisfying the estimate Proof. The "only if" part of the statement is obvious. For d = 1 and any g ≥ 1, this is Lemma 3.9. Indeed, a primitive of the 1-form ω = f (x, y) dx is the 0-form Ω := (P f )(x, y), and the estimate for the norms comes from (3.5).
Assume, by recurrence, that the Proposition is true for all g < g and all 1 )), where I is the operator defined in the previous proof (see (3.7)). It is clear from the definitions that I d ,g (ω) = 0 implies I d −1,g −1 I ω = 0. By recurrence, I ω = dη for a primitive η ∈ A k−1 (R k , S (R g )) satisfying the estimate η s ≤ C I ω s+d /2+ε . (3.19) As in the previous proof, one verifies that the form d −1 ω := Ω : ) is a primitive of ω (where the operators E and K are defined in the previous proof, see (3.8) and (3.12)). Therefore, from Lemma 3.6 and the estimates (3.9), (3.10), (3.13), and (3.19), we have, for some constants C 's which only depend on s ≥ 0 and ε > 0, The proof is complete.
satisfying the following properties: • the restriction of M k to B k (p, S (R g )) is the identity map; • the map M k satisfies, for any ε > 0, tame estimates of degree Lemmas 3.6 and 3.8 show that M d is a linear tame map of degree d /2 + ε for every ε > 0. Clearly for ω ∈ B d (p, S (R g )) we have M d (ω) = ω. Since the map M d maps A d (p, S (R g )) into B d (p, S (R g )), we have proved that B d (p, S (R g )) is a direct summand of A d (p, S (R g )). Now consider the case where k < d . We have B k (p, S (R g )) = Z k (p, S (R g )).
The map M k is a linear tame map of degree (k + 3)/2 + ε for every ε > 0. Clearly for ω ∈ Z k (p, S (R g )) we have M (ω) = ω. Furthermore d • M = 0. Thus the map M k sends A k (p, S (R g )) into Z k (p, S (R g )). We have proved that Z d (p, S (R g )) is a direct summand of A d (p, S (R g )).

P-invariant currents of dimension dim P.
Recall that the space of currents of dimension k is the space A k (p, S (R g )) of continuous linear functionals on A k (p, S (R g )) and that A k (p, S (R g )) is identified with Λ k p ⊗ S (R g ). For any s ≥ 0, the space Λ k p⊗W −s (R g ) is identified with the space of currents of dimension k and Sobolev order s.
It is clear, from Lemma 3.5, that I g = I g ,g ∈ W −s (R g ) for any s > g /2, i.e., it is a closed current of dimension g and Sobolev order g /2 + ε, for any ε > 0.
For d < g and t > 0, consider the currents D • I d ,g with D ∈ W −t (R g −d ). It follows from Lemma 3.6 that such currents belong to Λ d p ⊗ W −s (R g ) for any It is also easily seen that they are closed.
JOURNAL OF MODERN DYNAMICS VOLUME 9, 2015, 305-353 We have the following proposition, whose proof follows immediately from Lemma 3.6 and Proposition 3.11. • an infinite-dimensional space generated by Then ω admits a primitive Ω if and only if T (ω) = 0 for all T ∈ I d (p, S (R g )); under this hypothesis we may have

Bounds uniform in the parameter h.
Here we observe that the estimates in Propositions 3.10 and 3.11 are uniform in the Planck constant h, provided that this constant is bounded away from zero. PROPOSITION 3.14. Let s ≥ 0 and 1 ≤ k ≤ d ≤ g , and consider the H g -module ε, g , d , h 0 ) and a linear map associating to every ω ∈ B a primitive Ω = d −1 ω ∈ A k−1 (p, S (R g )) satisfying the estimate Furthermore, for any ε > 0 there exists a constant C = C (s, ε, g , d , h 0 ) > 0 such that the splitting linear maps of Proposition 3.12 for some C depending also on h 0 . The second statement is proved in an analogous manner.
Comparison with the usual Sobolev norms. The standard Sobolev norms associated with a Heisenberg basis (X i , Ξ j , T ) of h g were defined in Remark 3.2.
For a H g -module S (R g ) with parameter h, the image of the Laplacian −(X 2 1 + · · · + X 2 g + Ξ 2 1 + · · · + Ξ 2 Here we claim that the uniform bound as in Proposition 3.14 continues to hold with respect to the usual Sobolev norms. This is a consequence of the following easy lemma which applies to S (R g ) but also to any tensor product of S (R g ) with some finite dimensional Euclidean space.
Then for every s ≥ 0 we have Proof. For integer s = n, using the binomial formula, we get For non integer s the lemma follows by interpolation.
3.3. Proofs of Theorems 1.5 and 1.6. We are now in a position to integrate over Schrödinger representations and obtain our main result on the cohomology of P < H g with values in Fréchet H g -modules.
We have H k (p, F ∞ ) = 0 for k < d ; in degree d , we have that H d (p, F ∞ ) is finite dimensional only if d = g and the measure d α has finite support.
For any k = 0, . . . , d and any ε > 0, there exist a constant C and a linear map such that the restriction of M k to B k (p, F ∞ ) is the identity map and the following estimate holds: (The hypotheses 1 and 2 of the above theorem could be stated more briefly by saying that F satisfies the following property: any non-trivial unitary H g -module weakly contained in F is infinite dimensional.) Proof. Let F ∞ be the Fréchet space of C ∞ -vectors of a unitary H g -module (ρ, F ).
Let F = F α dα be the direct integral decomposition of F into irreducible submodules (ρ α , F α ). The hypotheses of Theorem 3.16 imply that there exists h 0 > 0 such that for almost every α the H g -module F α is unitarily equivalent to a Schrödinger module with parameter h satisfying |h| ≥ h 0 . For any s ∈ R, we also have a decomposition of the Sobolev spaces W s (F, ρ) as direct integrals W s (F α , ρ α )dα; this is because the operator 1 + ∆ g defining the Sobolev norms is an element of the enveloping algebra U(h g ) and because the spaces F α are U(h g )-invariant. It follows that any form ω ∈ A k (p, F ∞ ) has a decomposition ω = ω α dα with ω a ∈ A k (p, F ∞ α ) and For the same reason mentioned above, we have dω = (dω α ) dα. (3.24) Hence ω is closed if and only if ω α is closed for almost all α, that is, are the tame maps defined, for each α, as in (3.17). By Proposition 3.14 and Lemma 3.15, we have a constant C = C (s, ε, g , d , h 0 ) and, for each α, a linear map The above estimate shows that it is possible for one to define a linear map This shows that d −1 is a tame map of degree (k + 1)/2 + ε for all ε > 0 associating to each ω ∈ B k a primitive of ω. Thus , hence the top degree cohomology is infinite dimensional if d < g and one-dimensional if d = g . This shows that H d (p, F ∞ ) is finite dimensional if and only if d = g and the measure dα has finite support.
Finally for each α, we have tame maps M k α given by Proposition 3.12. Setting M k = M k α dα we obtain maps M k satisfying the Theorem's conclusion. Proof of Theorem 1.6. The theorem follows from the theorem above and the "folklore" Theorem 3.4, as explained at the beginning of Section 3.

Sobolev bundles.
Sobolev spaces. The group Sp 2g (R) < Aut(H g ) ≈ Aut(h g ) acts (on the right) on the enveloping algebra U h g in the following way: we identify U h g with the algebra of right invariant differential operators on H g ; if V ∈ U h g and α ∈ Sp 2g (R), the action of α on V yields the differential operator V α defined by Let ∆ = −(X 2 1 +· · ·+X 2 g +Ξ 2 1 +· · ·+Ξ 2 g +T 2 ) ∈ U(h g ) denote the Laplacian on H g defined via the "standard" basis (X i , Ξ j , T ) (cf. sect. 2.1). Then ∆ α = −((α −1 X 1 ) 2 + · · · + (α −1 Ξ g ) 2 + T 2 ), that is, ∆ α is the Laplacian on H g defined by the basis Let Γ be any lattice of H g and M := H g /Γ the corresponding nilmanifold. which are Hilbert spaces equipped with the inner product It is immediate that the pull-back map α * : since α * preserves the volume, we obtain an isometry α * : W s (M α ) → W s α (M). Observe that, as topological vector spaces, the spaces W s α (M), with α∈Sp 2g (R), are all isomorphic to W s (M). Only their Hilbert structure varies as α ranges in Sp 2g (R). In fact we have the following lemma, whose proof is omitted.

Best Sobolev constant.
The best Sobolev constant. The Sobolev embedding theorem implies that for any α ∈ Sp 2g (R) and any s > g + 1/2 there exists a constant B s (α) > 0 such that any f ∈ W s α (M) has a continuous representative such that For any Sobolev order s > g + 1/2, the best Sobolev constant is defined as the function on the group of automorphisms Sp 2g (R) given by For α = δ 0 0 δ −1 ∈ A + , where δ = diag(δ 1 , . . . , δ g ), we define

PROPOSITION 4.5.
For any order s > g + 1/2 and any α ∈ A + there exists a con- We fix the fundamental domain F = [0, 1] g × [0, 1] g × [0, 1/2] for the action of the lattice Γ on H g . By the standard Sobolev embedding theorem, for any where I = (0, 0, 0) is the identity of H g and dx is the Haar measure assigning volume 1 to F . Since left and right translation commute and since (1 + ∆) operates on the left, for every f ∈ W s loc (H g ) and every h ∈ H g we have It easy to see that, for any h ∈ H g , the set F h is also a fundamental domain for Γ. Furthermore, if we let p α : h ∈ H g → hΓ α ∈ M α denote the natural projection, the projection p α ((F h) o ) of the interior of F h covers each point of M α −1 at most times.
Given any f ∈ W s (M α ), letf = f • p α . Then, for any h ∈ H g and any integer . We deduce, by interpolation and by (4.6), that for any s ≥ g + 1/2 there exists a constant C such that Let F g ⊂ H g denote the Siegel fundamental domain for the action of Sp 2g (Z) on H g (see [31]). We define the height function Hgt : Σ g → R + to be the maximal height of a Sp 2g (Z)-orbit (which is attained by Proposition 1 of [5]), or, equivalently, the height of the unique representative of an orbit inside is a symmetric real matrix, W = (w i j ) is an upper triangular real matrix with ones on the diagonal, and D = diag(δ 1 , . . . , δ g ) is a diagonal positive matrix. The coordinates (x i j ) 1≤i ≤ j ≤g , (w i j ) 1≤i < j ≤g , and (δ i ) 1≤i ≤g thus defined are called Iwasawa coordinates on the Siegel upper half-space. For t > 0, define S g (t ) ⊂ H g as the set of those Z = X + iW DW ∈ H g such that For all t sufficiently large, S g (t ) is a "fundamental open set" for the action of Sp 2g (Z) on H g , containing the Siegel fundamental domain F g (see [5] or [31]).
We will need the following Lemma, which is an easy consequence of the expression for the Siegel metric in Iwasawa coordinates, where Y = W DW .

LEMMA 4.6. Any point Z = X + iW DW inside a Siegel fundamental open set S g (t ) is at a bounded distance from the point i D.
Proof. Let Z = X + iW DW , with W and D as explained above, be a point in S g (t ). In the sequel of the proof we denote by C 1 , C 2 etc., positive constants depending only on t and the dimension g .
We first observe that (4.12) says that the entries of the matrices W and W are bounded by t . Since these matrices are unipotent, their inverses are also bounded by a constant C 1 . Consider the path Z (τ) = X + iW (τ) D W (τ), with W (τ) := τW and τ ∈ [0, 1]. The entries of (W ) −1 dW D dW W −1 D −1 along this path are all proportional to C 2 (δ i /δ j )(dτ) 2 , where j > i . Since δ i /δ j < t j −i by (4.13), it follows from (4.14) that the length of the path is bounded by a constant C 3 . Thus, the arbitrary point Z = X + iW DW ∈ S g (t ) is within a bounded distance from X + i D.
But X + i D is within a bounded distance from i D. Indeed, fixed any pair of indices 1 ≤ i ≤ j ≤ g , we may consider the path Z (i j ) (τ) = X (i j ) (τ)+i D, (τ ∈ [0, 1]), where X (i j ) (τ) is the symmetric matrix with entries x i j (τ) = x j i (τ) = τx i j and all other entries constant and equal to those of X . It follows from (4.14) that the length of any such path is which is bounded by some constant C 4 because of (4.11) and (4.13). The claim follows by choosing successively all pair of indices, thus constructing a sequence of paths joining X + i D to i D.  2 . More precisely, for any τ 0 Proof. A change of variable as in page 67 of [31] shows that this volume is within a bounded ratio of  Proof. We recall that Hgt is the maximal hgt of a Sp 2g (Z) orbit. Therefore, we may take the representative β = αγ, with γ ∈ Sp 2g (Z), such that (e −t δ β) −1 (i ) ∈ H g realizes the maximal height, that is, and prove the inequality for the function hgt, namely hgt((e −t δ β) −1 (i )) ≤ (det e t δ ) 2 hgt(β −1 (i )), since then hgt(β −1 (i )) ≤ Hgt([[α]]). By the Iwasawa decomposition, any symplectic matrix β ∈ Sp 2g (R) sending the base point i := i 1 g into the point β −1 (i ) = X + iW DW may be written as and κ ∈ K g . By the formula (4.16), hgt(νηκ(Z )) = hgt(ηκ(Z )) = (det D) hgt(κ(Z )) (because detW = 1) for all Z ∈ H g . Therefore, since hgt(κ(i )) = 1, we only need to prove hgt(κe t δ (i )) ≤ det e 2t δ .
Let κ = A B −B A ∈ K g , i.e., with A A + B B = 1 g and A B symmetric. Since e t δ (i ) = i e 2t δ , using formula (4.16), the above inequality is equivalent to that is, to | det(A − i B e 2t δ )| 2 ≥ 1, and therefore to | det(A A + B e 4t δ B )| ≥ 1.
But, by our hypothesis on δ and t , the norm of e 2t δ is e 2t δ ≥ 1, and therefore for any vector x ∈ R g . Hence, all the eigenvalues of the symmetric matrix A A + B e 4t δ B are ≥ 1, and the same occurs for the determinant. Let δ = diag(δ 1 , . . . , δ g ) be a non-negative diagonal matrix, and δ = δ 0 0 −δ ∈ a + ⊂ sp 2g . We say that an automorphism α ∈ Sp 2g (R), or, equivalently, a point [α] ∈ M g in the moduli space, • is δ-Diophantine of type σ if there exists a σ > 0 and a constant C > 0 such that • satisfies a δ-Roth condition if for any ε > 0 there exists a constant C > 0 such that for all δ ∈ a + and all t ≥ 0. For such δ, the Diophantine properties of an automorphism α ∈ Sp 2g (R) only depend on the right T class of α −1 , where T ⊂ Sp 2g (R) is the subgroup of blocktriangular symplectic matrices of the form A B 0 (A ) −1 . In particular, those α in the full measure set of those automorphisms such that α −1 = A B C D with A ∈ GL g (R) are in the same Diophantine class of β = I 0 −X I , where X is the symmetric matrix X = C A −1 . For such lower-triangular block matrices β, the Height in the Diophantine conditions above is (see (4.16)) the maximum being over all N M P Q ∈ Sp 2g (Z). When g = 1, we recover the classical relation between Diophantine properties of a real number X and geodesic excursion into the cusp of the modular orbifold Σ 1 , or the behaviour of a certain flow in the space M 1 = SL 2 (R)/SL 2 (Z) of unimodular lattices in the plane.
Indeed, our (4.20) coincides with the function δ(Λ t ) = max v∈Λ t \{0} v −2 2 , where Λ t is the unimodular lattice made of e t 0 0 e −t 1 X 0 1 P Q , with P,Q ∈ Z. The maximizers, for increasing time t , define a sequence of relatively prime integers P n and Q n which give best approximants P n /Q n to X in the sense of continued fractions. In particular, our definitions of Diophantine, Roth, and bounded type coincide with the classical notions.
This same function δ(Λ t ), extended to the space SL n (R)/SL n (Z) of unimodular lattices in R n , has been used by Lagarias [36], or, more recently, by Chevallier [6] to understand simultaneous Diophantine approximations. A similar function, ∆(Λ t ) = max v∈Λ t \{0} log(1/ v ∞ ), has been considered by Dani [10] in his correspondence between Diophantine properties of systems of linear forms and certain flows on the space SL n (R)/SL n (Z), or more recently by Kleinbock and Margulis [32] to prove a "higher-dimensional multiplicative Khinchin theorem".

Khinchin-Sullivan-Kleinbock-Margulis logarithm law.
A stronger control on the best Sobolev constant comes from the following generalization of the Kinchin-Sullivan logarithm law for geodesic excursion [47], due to Kleinbock and Margulis [32].
Let X = G/Λ be a homogeneous space, equipped with the probability Haar measure µ. A function φ : X → R is said to be k-DL (for "distance-like") for some exponent k > 0 if it is uniformly continuous and if there exist constants c ± > 0 such that In particular, any such [α] satisfies a δ-Roth condition.

EQUIDISTRIBUTION
In According to (4.1), the group Sp 2g (R) acts on the right on the enveloping algebra U (h g ) and in particular for V ∈ h g , V α = α −1 (V ). For simplicity we set, Then p d ,α := α −1 (p d ,0 ) and P d ,α = α −1 (P d ,0 ) are respectively the algebra and the subgroup generated by (X α i , Ξ α j , T ). Every isotropic subgroup of H g is obtained in this way, i.e., given by some P d ,α defined as above.
It is immediate that for every α, β ∈ Sp 2g (R) we have in particular, if β belongs to the diagonal Cartan subgroup A, then P d ,βα = P d ,α .
We define a parametrization of P d ,α , hence a R d -action on M subordinate to α, by setting In the following, it will be convenient to set ω d ,α = d X α 1 ∧· · ·∧d X α d and to identify top-dimensional currents D with distributions by setting D, f := D, f ω d ,α .
Given a Jordan region U ⊂ R d and a point m ∈ M, we define a top-dimensional p-current P d ,α U m as the Birkhoff sums given by integration along the chain x m x ∈ U . Explicitly, if ω = f dX α 1 ∧· · ·∧dX α d is a top-dimensional JOURNAL OF MODERN DYNAMICS VOLUME 9, 2015, 305-353 p-form, then Our goal is to understand the asymptotic of these distributions as U R d in a Følner sense. A particular case is obtained when We remark that the Birkhoff sums satisfy the following covariance property: Left multiplication by the one parameter group (r t i ) yields a flow on Sp 2g (R) that projects to the moduli space M g according to [α] → r t Above this flow, we consider its horizontal lift to the bundles A j (p d , W s ) and for α ∈ Sp 2g (R) and ω ∈ A j (p d ,α , W s ) or D ∈ A j (p d ,α , W −s ). This is well defined since, as we remarked before, p d ,α = p d ,r t i α .
Consequently, denoting by (e −t 1 , . . . , e −t d )U the obvious diagonal automorphism of R d applied to the region U , the Birkhoff sums satisfy the identities Without loss of generality we may assume that D belongs to the space A d (p d ,α ,W −s (ρ h )), where ρ h is an irreducible Schrödinger representation in which the basis (X α i , Ξ α , T ) acts according to (2.5). LetL α = (ρ h ) * L α and L r t d α = (ρ h ) * L r t d α be the push-forward to L 2 (R g ) of the operators defining the norms · s,α and · s,r t d α . By Proposition 3.13, the space of closed currents of dimension d is spanned by I g if d = g and by the dense set of currents D = D y • I d ,g with D y in , intertwines the differential operator L α with the operator L r α , i.e., U t ( L α f ) = L r α U t f for any smooth f . Thus  with "boundary term" Z −s α (D) ∈ Z d (p d ,α ,W −s α (M)) and "remainder term" We will also need an estimate for the distortion of the Sobolev norms along the renormalization flow. Below, |t | denotes the sup norm of a vector t ∈ R d .
There exists a constant C = C (s) such that if |τt | is sufficiently small then the orthogonal projection Proof. As in the proof of Proposition 5.2, we may restrict to a fixed Schrödinger representation ρ h in which the basis (X α i , Ξ α i , T ) acts according to (2.5). It is also clear from Lemma 3.15 that we may use the homogeneous Sobolev norm defined in (3.4). If H = (ρ h ) * L α denotes the sub-Laplacian inducing the Sobolev structure of W −s α (R g ), then the Sobolev structure of W −s r τ α (R g ) is induced by H τ = U −τ HU τ where U τ = U τt is the one-parameter group of unitary operators of L 2 (R g ) defined according to (5.5). We denote by φ, ψ −s,τ = φ, H −s τ ψ the inner product in W −s r τ α (R g ). A computation shows that the infinitesimal generator of U τ is i times the self-adjoint operator A = (ρ h ) * d k=1 t k (1/2 − X k Ξ k ) . Moreover, using the Hermite basis, one can show that there exists a constant C such that Aψ ≤ C |t | H ψ for ψ in the domain of A. Now, let R ∈ W −s+2 α (R g ) be a distribution (we identify top-dimensional currents with distributions as explained in 5.1) which is orthogonal to the subspace Z of closed distributions when τ = 0, i.e., such that 〈R, D〉 −s,0 = R, H −s D = 0 for all D ∈ Z . In order to bound the norm of its projection to Z w.r.t. the Sobolev structure at τ we must bound the absolute values of the scalar products 〈R, D〉 −s,τ for all D in Z . Now, If R is in the domain of A, we may write According to Proposition 5.2, the group U τ preserves Z . Therefore, since R is orthogonal to U τ D for all τ, we may write Since s > (d + 1)/2 and since, by definition, T, ω s Proof. For simplicity we set r t = r t 1 . To start, we observe that, according to (5.4) and Lemma 5.6, we have  If we take first T = 1, then rename e t := T ≥ 1, we finally get The reminder term in the decomposition (5.7) is estimated as at the beginning of the proof, using Lemma 5.6, Proposition 4.8 and Lemma 4.9, and is bounded by The theorem follows.
The next result follows immediately from the above Theorem 5.8 and the Kleinbock-Margulis logarithm law, i.e., from Proposition 4.13. PROPOSITION 5.9. Let the notation as in Theorem 5.8. There exists a full mea-  To estimate the term I I , we start observing that, provided s < s −2−(d +1)/2, using (5.4) and Lemma 5.6, we have If s > s d −1 (and therefore s > s d −1 + (d + 1)/2 + 2 = s d ), denoting by P d −1,r −u α the generic summand of ∂(P d ,r −u α U d (t −u) ), we may estimate the norm of each such boundary term using the inductive hypothesis (5.8). For the j -face we obtain From (5.9) and (5.11) we obtain the following estimate for the term I I : (5.12) Applying the change of variable u j = t − u, majorizing the integrals t −u 0 with integrals t 0 and observing that there are at most k + 1 integer intervals ]i t , i t +1 [ in which the integer j in the above sum may land, we obtain The remainder term R −s [α, P d ,α U d (t ) ] in the decomposition (5.7) is estimated using Lemma 5.6, Proposition 4.8 and Lemma 4.9. We have: producing one more term like (5.10). The theorem follows from the estimates (5.10) and (5.13) for the terms I and II and from (5.14) for the remainder.  (0, ξ, t ) ξ ∈ R g , t ∈ R/ 1 2 Z is a normal subgroup of H g red . The quotient H g red /N is isomorphic to the Legendrian subgroup P = (x, 0, 0) | x ∈ R g , and we have an exact sequence 0 → N → H g red → P → 0. Therefore H g red ≈ P N, and in particular any (x, ξ, t ) ∈ H g red may be uniquely written as the product (x, ξ, t ) = exp(x 1 X 1 + · · · + x g X g ) · (0, ξ, t ) = (x, 0, 0) · (0, ξ, t ) .