Higher cohomology of parabolic actions on certain homogeneous spaces
aa r X i v : . [ m a t h . D S ] D ec Higher cohomology of parabolicactions on certain homogeneousspaces
Felipe A. Ram´ırez ∗ University of York
Abstract
We show that for a parabolic R d -action on PSL(2 , R ) d / Γ, the co-homologies in degrees 1 through d − d coboundary equation, along withbounds on Sobolev norms of primitives. In previous papers we haveestablished these results for certain Anosov systems. The present workextends the methods of those papers to systems that are not Anosov.The main new idea is in §
4, where we define special elements of rep-resentation spaces that allow us to modify the arguments from theprevious papers. In § R d -systems coming from a product of Lie groups, like in thesystems we have here. ∗ email: [email protected] . A. Ram´ırez Higher cohomology of parabolic actions
Contents
This article is a complement to [Ram13a], where we studied the smooth co-homology of Anosov R d -actions on homogeneous spaces of the d -fold productSL(2 , R ) d = SL(2 , R ) × · · · × SL(2 , R ). The main results there are a descrip-tion of top-degree (degree d ) cohomology and a vanishing statement for thelower degrees (degrees 1 , . . . , d − unipotent R d -actions onirreducible compact quotients of PSL(2 , R ) d , and prove similar statements tothose we proved in the Anosov case. Unlike in the Anosov case, these resultsare not expected to hold for all unipotent systems. In fact, they are in a senseoptimal for our examples. We comment on this after stating Theorem 1.1. Cohomology in dynamics
Cohomology is a fundamental tool in the study of rigidity properties of dy-namical systems. For some background on this, we recommend [Fur81, Kat01,KN11]. Let us briefly review some definitions that are relevant to this paper.(The initiated reader may skip to the “Past work” section.)For a flow on a manifold M along a vector field V , the (degree- )coboundary equation is the familiar V g = f , asking us to determinewhether a given smooth function (or 1 -cocycle ) f ∈ C ∞ ( M ) is the deriva-tive of some other smooth function g ∈ C ∞ ( M ) in the flow direction. Thereis a host of literature, even on this seemingly modest question. For now weonly mention the famous Livshitz Theorem, which gives a beautiful answer . A. Ram´ırez Higher cohomology of parabolic actions in the case of an Anosov flow on a compact manifold. In words, the theoremstates that we only need to check that f has integral 0 around every periodicorbit of the flow [Liv72, GK80a, GK80b, dlLMM86]. (Though we refer to itas the Livshitz Theorem, the full statement, with regularity of solutions andall, is really the culmination of work by several authors.)For an R d -action generated by commuting vector fields V , . . . , V d on amanifold M , one can define cohomology in degrees 1 , . . . , d . The degree- d coboundary equation is V g + · · · + V d g d = f. (1)This corresponds to asking whether a closed orbit-leafwise differential d -form (or d -cocycle , the one determined by f ∈ C ∞ ( M )) is a coboundary,in the sense of having a ( d − -primitive defined by d smooth functions g , . . . , g d ∈ C ∞ ( M ). Accordingly, there are coboundary equations in all de-grees, corresponding to the usual coboundary equation d η = ω for leafwisedifferential forms, asking us to find an ( n − η ∈ Ω n − whose leafwiseexterior derivative is the given leafwise n -cocycle ω ∈ Ω n . The leafwise exte-rior derivative d is defined by the formulad ω ( V i , . . . , V i n +1 ) = n +1 X j =1 ( − j +1 V i j ω ( V i , . . . , c V i j , . . . , V i n +1 ) , where 1 ≤ i < · · · < i n +1 ≤ d and c V i j means that V i j is omitted. So in degrees1 , . . . , d −
1, the coboundary equation is a system of (cid:0) dn (cid:1) partial differentialequations, instead of just one (1 = (cid:0) dd (cid:1) ) differential equation as in (1). Past work
There is a conjectural generalization of the Livshitz Theorem to higher-rankAnosov actions, due to A. and S. Katok [KK95]. They predict that in orderfor the degree- d coboundary equation to have a smooth solution, one shouldonly have to check that f integrates to 0 over closed orbits of the Anosov R d -action. Furthermore, when d ≥
2, all lower cohomologies should trivialize.(It is already known by work of A. Katok and R. Spatzier that the first coho-mology does [KS94].) Katok and Katok proved their conjecture for partiallyhyperbolic Z d -actions on tori [KK95, KK05]. . A. Ram´ırez Higher cohomology of parabolic actions
This problem was the motivation for our previous papers on higher coho-mology [Ram13a, Ram13b]. There, we studied certain families of
Weyl cham-ber flows , and proved that indeed the lower cohomologies trivialize for thosesystems, and that in degree d —the top degree—one only has to deal withobstructions coming from action-invariant distributions. Integration over aclosed orbit is itself an invariant distribution, so our results in top degreeare a priori weaker than the expected statements, but they are a step in theright direction. The present work
In this article we work with unipotent systems, where there may not even beany closed orbits. Yet, we know from work of L. Flaminio and G. Forni [FF03]that for the horocycle flow of a hyperbolic surface, there are always obstruc-tions to the degree-1 coboundary equation, coming from flow-invariant dis-tributions. Therefore, a statement like the Katok–Katok Conjecture cannotpossibly be true in the unipotent case, at least not the top-degree part.On the other hand, it is natural to expect that the obstructions to the top-degree coboundary equation for a unipotent R d -action come from invariantdistributions, as in the work of Flaminio–Forni. Indeed, we prove the followingresult for the cohomology of unipotent R d -actions on homogeneous spaces ofthe d -fold product PSL(2 , R ) d . Theorem 1.1.
Let G = G × · · · × G d with G i ∼ = PSL(2 , R ) for i = 1 , . . . , d ,and let Γ ⊂ G be an irreducible cocompact lattice. Consider the maximalunipotent R d -action on G/ Γ . For any smooth member of the kernel of all R d -invariant distributions there is a smooth solution to the degree- d coboundaryequation. Also, the cohomologies in degrees , . . . , d − trivialize, meaningthat any smooth cocycle is smoothly cohomologous to one defined by constantfunctions on G/ Γ .Remark. We in fact prove versions of Theorem 1.1 that also give bounds onSobolev norms of solutions to coboundary equations, listed here as Theo-rems 2.2 and 2.3.
Expectations
Notice that since there are no closed orbits in the systems of Theorem 1.1,a statement of this type is the best one can hope for in terms of describ- . A. Ram´ırez Higher cohomology of parabolic actions ing the obstructions to solving the top-degree coboundary equation. On theother hand, not even the lower -degree part of Theorem 1.1 holds for everyunipotent system: Already, there are non-vanishing obstructions (to trivial-ization of first cohomology) for unipotent R -actions on homogeneous spacesof SL(2 , C ) found by Mieczkowski [Mie07]; Wang [Wan12] has shown thesame for unipotent R -actions on homogeneous spaces of SL(3 , R ); and, inan ongoing project, L. Flaminio and the author show that there are non-vanishing obstructions to cohomology of the horospheric action in infinitelymany irreducible unitary representations of SO ◦ (1 , N ), for any N ≥ Methods
The arguments used here are adaptations of the representation-theoreticmethods used in [Ram13a, Ram13b]. The main new idea here is in defin-ing “ ϕ ’s” (see § , R ) that facilitate aninductive argument in the proof of the main theorem. After reinterpretation,one can say that similar special elements were also present in the previouspapers, but were “hidden,” so went without notice. By drawing attention tothem we are now able to modify the procedures from our previous articles(which were for Anosov systems) so that they work for the unipotent systemsin this article. Our § Theorem 1.2.
Let G = G × · · · × G d with each G i ∼ = PSL(2 , R ) and let Γ ⊂ G be an irreducible cocompact lattice. Consider an R d -action on G/ Γ whoseprojection to each factor corresponds to either the geodesic or horocycle flow.Then for any smooth member of the kernel of all R d -invariant distributionsthere is a smooth solution to the degree- d coboundary equation. Also, thecohomologies in degrees , . . . , d − trivialize. Our main results are stated for unitary representations of PSL(2 , R ) d thatsatisfy the following assumption, which in particular holds for the regular . A. Ram´ırez Higher cohomology of parabolic actions representation of PSL(2 , R ) d on L (PSL(2 , R ) d / Γ) where Γ ⊂ PSL(2 , R ) d isan irreducible cocompact lattice. Assumption 2.1.
For the unitary representation
PSL(2 , R ) d → U ( H ) thereexists ǫ > and direct integral decomposition H = Z ⊕ R H λ ds ( λ ) such that for ds -almost every λ , H λ = H ν ( λ ) ⊗ · · · ⊗ H ν d ( λ ) where ν j ( λ ) / ∈ B (0 , ǫ ) \{ } for all j = 1 , . . . , d . Colloquially, “none of the ν j ’s accumulate to zero.” The following two theorems are the main results of this paper. Theyconstitute a version of Theorem 1.1 for unitary representations of PSL(2 , R ) d that satisfy Assumption 2.1 and have spectral gap. The statements includeestimates on Sobolev norms. Theorem 2.2 (Version of Theorem 1.1 for unitary representations with spec-tral gap; top degree) . Let H be the Hilbert space of a unitary representa-tion of PSL(2 , R ) d with a spectral gap and satisfying Assumption 2.1. Sup-pose f ∈ C ∞ ( H ) lies in the kernel of all U , . . . , U d -invariant distributions.Then there exists a solution g , . . . , g d ∈ C ∞ ( H ) to the degree- d coboundaryequation for f that satisfies the Sobolev estimates k g i k t ≪ ν ,t k f k σ d , where σ d := σ d ( t ) is some increasing function of t > . Theorem 2.3 (Version of Theorem 1.1 for unitary representations with spec-tral gap; lower degrees) . Let H be the Hilbert space of a unitary representationof PSL(2 , R ) d with spectral gap and satisfying Assumption 2.1, and supposethat almost every irreducible representation appearing in its direct decompo-sition has no trivial factor. Then for ≤ n ≤ d − , any smooth n -cocycle ω ∈ Ω n R d ( C ∞ ( H )) is the coboundary d η = ω of some η ∈ Ω n − R d ( C ∞ ( H )) and k η k t ≪ ν ,t k ω k ς d where ς d := ς d ( t ) is an increasing function of t > . Let U = ( ) ⊂ sl (2 , R ) be the generator of the horocycle flow. We will takeour unipotent R d -action to be that generated by U i := U coming from eachfactor of the d -fold sum sl (2 , R ) ⊕ · · · ⊕ sl (2 , R ). . A. Ram´ırez Higher cohomology of parabolic actions
Irreducible unitary representations
We work with irreducible unitary representations π ν : PSL(2 , R ) → U ( H ν )and π ν : PSL(2 , R ) d → U ( H ν ), where ν is a parameter taking the values ν ∈ i R principal series representations( − , \{ } complementary series representations2 N − H ν := H ν ⊗ · · · ⊗ H ν d . Each H ν has a basis { u ( k ) } k ∈Z ν , where Z ν = ( Z if H ν is from the principal or complementary series Z ≥ + n if ν = 2 n − H ν from discrete series.Any element f ∈ H ν can therefore be written in terms of its coefficients f = P k ∈ Z ν f ( k ) u ( k ). The basis { u ( k ) } consists of eigenvectors for the Laplacian,so Sobolev norms can be easily computed in terms of coefficients. We have f ∈ W t ( H ν ) if and only if k f k t = X k ∈Z ν (1 + µ + 2 k ) t | f ( k ) | k u ( k ) k < ∞ , where µ is the eigenvalue of the Casimir operator (cid:3) corresponding to ν through ν = 1 − µ . For convenience, we put Q ν ( k ) = µ + 2 k .We will use the following lemma. Lemma 3.1.
Let
PSL(2 , R ) → U ( H ) be a unitary representation with directdecomposition H = Z ⊕ R H λ ds ( λ ) . For any f ∈ W t +1 ( H ) with decomposition f = Z ⊕ R f λ ds ( λ ) we have k U f λ k t ≪ k f λ k t +1 for ds -almost every λ . . A. Ram´ırez Higher cohomology of parabolic actionsProof.
Notice that from the point of view of another standard definition ofSobolev norm, namely, k f k t +1 = X { V i ,...,V it +1 }⊂ sl (2 , R ) (cid:13)(cid:13) V i V i . . . V i t +1 f (cid:13)(cid:13) , this lemma is obvious, and in fact k U f k t ≤ k f k t +1 . This other norm isequivalent to the norm we are using, meaning that the two are asymptotic inthe sense of “ ≍ ”. In particular, k U f k t ≪ k f k t +1 , and this continues to holdin almost every component of the direct integral decomposition of H . Irreducible unitary representations of products
As hinted above, an irreducible unitary representation of the d -fold productPSL(2 , R ) d is a d -fold tensor product H ν = H ⊗ · · · ⊗ H d , where each H j := H ν j is an i.u.r. of PSL(2 , R ). We now have a basis { u ( k ) } k ∈ Z ν = { u (1) ( k ) ⊗ · · · ⊗ u ( d ) ( k d ) } ( k ,...,k d ) ∈Z ν ×···×Z νd . Again, we have nice expressions for Sobolev norms. Namely, k f k t = X k ∈ Z ν (1 + µ + · · · + µ d + 2 | k | ) t | f ( k ) | k u ( k ) k . It is convenient to define projected versions of elements of H ν , for example( f | k j ,...,k d ) = j − X i =1 X k i ∈Z νi f ( k , . . . , k d ) k u ( k j ) k . . . k u ( k d ) k u ( k ) ⊗ · · · ⊗ u ( k j − )is f projected to H ν ⊗ · · · ⊗ H ν j − by fixing k j , . . . , k d . Easy calculations ofSobolev norms show that k ( f | k j ,...,k d ) k τ ≤ k f k τ (2)and X k ∈Z ν (1 + Q ν ( k )) τ k ( f | k ) k σ ≤ k f k τ + σ . (3)We use (2) and (3) repeatedly. . A. Ram´ırez Higher cohomology of parabolic actions
Invariant distributions
Let PSL(2 , R ) → U ( H ) be a unitary representation. A distribution is anelement D ∈ E ′ ( H ) := ( C ∞ ( H )) ∗ of the dual to the space of smooth vectors.The distribution is U -invariant if L U D = 0. That is, if D ( U v ) = 0 for every v ∈ C ∞ ( H ). Similarly, we define distributions of order s to be elements of W − s ( H ), the dual to the Sobolev space of order s .Flaminio and Forni [FF03] find all the U -invariant distributions in ir-reducible unitary representations of PSL(2 , R ). First, there is D + , definedby D + ( u ( k )) = 1 ∀ k ∈ Z ν , regardless of ν ’s value. For discrete series representations, there are no otherindependent invariant distributions. But for principal and complementaryseries, we also have D − ( u ( k )) = | k | Y i =1 i − − ν i − ν if ν = 0 | k | X i =1 i − ν = 0 , where empty products are by convention 1 and empty sums are by convention0. Lemma 3.2.
There is a number c > such that X ± X m ∈Z (1 + Q ν ( m )) − t |D ± ( u ( m )) | k u ( m ) k ≪ ν ,t µ ) t − c princ., comp. µ + 2 n ) t − c disc.Proof. The proof is a calculation in each of the three families of irreducibleunitary representations of PSL(2 , R ). For principal series representations, where ν ∈ i R , X ± X m ∈Z (1 + Q ν ( m )) − t |D ± ( u ( m )) | k u ( m ) k = X m ∈Z (1 + Q ν ( m )) − t (cid:0) |D + ( u ( m )) | + |D − ( u ( m )) | (cid:1) . (4) . A. Ram´ırez Higher cohomology of parabolic actions If ν = 0, this is rewritten X m ∈Z (1 + Q ν ( m )) − t | m | Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) i − − ν i − ν (cid:12)(cid:12)(cid:12)(cid:12) = X m ∈Z (1 + Q ν ( m )) − t | m | Y i =1 (2 i − + | ν | (2 i − + | ν | , since ν is purely imaginary. Hence we only need to bound X m ∈Z (1 + Q ν ( m )) − t = X m ∈Z (1 + µ + 2 m ) − t , which for t >
0, can be compared to the integral R dx (1+ µ +2 x ) t to obtain thedesired bound ≪ t (1 + µ ) / − t .Now, if ν = 0, (4) is rewritten X m ∈Z (1 + Q ( m )) − t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | m | X i =1 i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Comparing to an integral, we bound this by ≤ X m ∈Z (1 + Q ( m )) − t " (cid:12)(cid:12)(cid:12)(cid:12) m − (cid:12)(cid:12)(cid:12)(cid:12) = X m ∈Z (1 + 1 / m ) − t " (cid:12)(cid:12)(cid:12)(cid:12) m − (cid:12)(cid:12)(cid:12)(cid:12) which is also bounded by ≪ t (1 + µ ) / − t . For complementary series representations, where ν ∈ ( − , \{ } wecan rewrite (4) as X m ∈Z (1 + Q ν ( m )) − t | m | Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) i − − ν i − ν (cid:12)(cid:12)(cid:12)(cid:12) = X m ∈Z (1 + Q ν ( m )) − t (cid:2) k u ( m ) k (cid:3) , . A. Ram´ırez Higher cohomology of parabolic actions which, applying [FF03, Lemma 2.1], is bounded by ≪ X m ∈Z (1 + Q ν ( m )) − t " (cid:18) − ν ν (cid:19) (1 + m ) − ν , which is in turn bounded ≪ ν ,t (1 + µ ) / − t where | ν | ≤ ν < For discrete series representations, where ν = 2 n −
1, the expressionto bound becomes X m ∈Z (1 + Q ν ( m )) − t |D + ( u ( m )) | k u ( m ) k = X m ∈ N (1 + Q ν ( n + m )) − t k u ( n + m ) k − . Again, we appeal to [FF03, Lemma 2.1] to bound this by ≪ X m ∈ N (1 + Q ν ( n + m )) − t (cid:18) m + 1 (cid:19) − ν , which is bounded ≪ (1 + µ + 2 n ) c − t . The strategy for the top-degree part of Theorem 1.1 is to work in an irre-ducible unitary representation H ⊗ · · · ⊗ H d := H ⊗ ⊗ H d and write f as asum f ⊗ + f d , in such a way that ( f ⊗ | ℓ ) ∈ H ⊗ is always in the kernel of all U , . . . , U d − -invariant distributions, and ( f d | k ) ∈ H d is always in the kernelof every U d -invariant distribution. This will facilitate an induction on d , withbase case given by [FF03, Theorem 4 . H µ , choose ϕ ± ∈ W s ( H µ ) so that ϕ + ∈ ker D − µ and ϕ − ∈ ker D + µ D + ( ϕ + ) = 1 and D − ( ϕ − ) = 1 . (5) . A. Ram´ırez Higher cohomology of parabolic actions
For example, one can easily check that the following choices satisfy (5). If ν = 0, and π ν is from the principal or complementary series, ϕ + ( k ) = ν − ν if k = 0 ν +12 ν if k = 10 otherwise ϕ − ( k ) = ν +12 ν if k = 0 − ν +12 ν if k = 10 otherwise. (6)If ν = 0, ϕ + ( k ) = ( k = 00 otherwise ϕ − ( k ) = − k = 01 if k = 10 otherwise. (7)And if π ν is from the discrete series, there is only one invariant distribution, D + , so we only choose ϕ + as ϕ + ( k ) = ( k = 00 otherwise. (8)and set ϕ − ≡ ϕ ± defined as in (6), (7), (8),but other choices would work as well. See § ϕ ’s. Lemma 4.1. X ± k ϕ ± k t ≪ ǫ ,ν ( (1 + µ ) t in principal and complementary series (1 + µ + 2 n ) t in discrete seriesProof. Let t >
0, and compute X ± k ϕ ± k t = X k ∈Z ν (1 + Q ν ( k )) t (cid:0) | ϕ + ( k ) | + | ϕ − ( k ) | (cid:1) k u ( k ) k = (1 + Q ν (0)) t (cid:0) | ϕ + (0) | + | ϕ − (0) | (cid:1) k u (0) k + (1 + Q ν (1)) t (cid:0) | ϕ + (1) | + | ϕ − (1) | (cid:1) k u (1) k . (9)Now we just bound in all possible cases. . A. Ram´ırez Higher cohomology of parabolic actions
For principal series representations, where ν = 0, the expression (9)becomes(1 + µ ) t (cid:12)(cid:12)(cid:12)(cid:12) ν − ν (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ν + 12 ν (cid:12)(cid:12)(cid:12)(cid:12) ! + (3 + µ ) t (cid:12)(cid:12)(cid:12)(cid:12) ν + 12 ν (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) − ν + 12 ν (cid:12)(cid:12)(cid:12)(cid:12) ! , and since ν is purely imaginary we can bound by ≪ (1 + µ ) t | ν | + 1 | ν | ≪ ǫ (1 + µ ) t . This is what Assumption 2.1 was for.On the other hand, if ν = 0, (9) becomes(1 + µ ) t (cid:0) | | + |− | (cid:1) + (3 + µ ) t (cid:0) | | + | | (cid:1) , so the bound is obvious. For complementary series representations, where ν = 0, the expres-sion (9) becomes(1 + µ ) t (cid:12)(cid:12)(cid:12)(cid:12) ν − ν (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ν + 12 ν (cid:12)(cid:12)(cid:12)(cid:12) ! + (3 + µ ) t (cid:12)(cid:12)(cid:12)(cid:12) ν + 12 ν (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) − ν + 12 ν (cid:12)(cid:12)(cid:12)(cid:12) ! (cid:12)(cid:12)(cid:12)(cid:12) − ν ν (cid:12)(cid:12)(cid:12)(cid:12) , which we can bound on | ν | ∈ ( ǫ , ν ) ⊂ (0 ,
1) by ≪ ǫ ,ν (1 + µ ) t . Again, Assumption 2.1 was made for this.
For discrete series representations, where ν = 2 n − n , theexpression (9) becomes (1 + Q ν (0)) t = (1 + µ + 2 n ) t which is exactly what we want to bound by. . A. Ram´ırez Higher cohomology of parabolic actions
We now define for ( k , ℓ ) ∈ Z × × Z d f ⊗ ( k , ℓ ) = X ± ϕ ± ( ℓ ) X m ∈Z d f ( k , m ) D ± ( u d ( m )) (10)and put f d = f − f ⊗ . Let us prove that f ⊗ (and therefore f d also) retainssome of f ’s Sobolev regularity. Lemma 4.2.
There is a function L : R + → R + such that k f ⊗ k t ≪ ν ,ǫ ,t k f k L ( t ) for all t > .Proof. We compute k f ⊗ k t = X k ,ℓ (1 + Q + ( k ) + Q d ( ℓ )) t | f ⊗ ( k , ℓ ) | k u ⊗ ( k ) ⊗ u d ( ℓ ) k = X k ,ℓ (1+ Q + ( k )+ Q d ( ℓ )) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X { + , −} ϕ ± ( ℓ ) X m ∈Z d f ( k , m ) D ± ( u d ( m )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k u ⊗ ( k ) ⊗ u d ( ℓ ) k and by repeated use of the Cauchy–Schwartz Inequality, ≤ X k ,ℓ (1 + Q + ( k ) + Q d ( ℓ )) t X ± | ϕ ± ( ℓ ) | k u d ( ℓ ) k × X ± X m ∈Z d (1+ Q d ( m )) t ′ | f ( k , m ) | k u ⊗ ( k ) ⊗ u d ( m ) k X m ∈Z d (1+ Q d ( m )) − t ′ |D ± ( u d ( m )) | k u d ( m ) k . Re-writing this using (3), we obtain ≤ k f k t + t ′ X ± k ϕ ± k t X ± X m ∈Z d (1 + Q d ( m )) − t ′ |D ± ( u d ( m )) | k u d ( m ) k . If we choose t ′ = t + c , Lemmas 4.1 and 3.2 imply ≪ ν ,ǫ ,t k f k t + c , which is the lemma, with L ( t ) = 2 t + c . . A. Ram´ırez Higher cohomology of parabolic actions
Lemma 4.3.
Suppose f ∈ ker I U ,...,U d ( H ⊗ ⊗ H d ) . Then for every ℓ ∈ Z d , wehave ( f ⊗ | ℓ ) ∈ ker I U ,...,U d − ( H ⊗ ) and for every k ∈ Z × , we have ( f d | k ) ∈ ker I U d ( H d ) .Proof. This is essentially [Ram13a, Lemma 4.3] and [Ram13b, Lemma 13.3],but since our f ⊗ is now defined differently, it is worth reproducing the prooffor this new scenario.We calculate, for • a multi-index of length d − − ’s, D • ( f ⊗ | ℓ ) = X k ∈ Z × f ⊗ ( k , ℓ ) D • ( u ⊗ ( k ))= X k ∈ Z × "X ± ϕ ± ( ℓ ) X m ∈Z d f ( k , m ) D ± ( u d ( m )) D • ( u ⊗ ( k ))= X ± ϕ ± ( ℓ ) (cid:0) D • , + ( f ) + D • , − ( f ) (cid:1) = 0 , and for • ∈ { + , −} , D • ( f d | k ) = X ℓ ∈Z d [ f ( k , ℓ ) − f ⊗ ( k , ℓ )] D • ( u d ( ℓ ))= D • ( f | k ) − X ± D • ( ϕ ± ) X m ∈Z d f ( k , m ) D ± ( u d ( m ))= 0 , because D • ( ϕ ± ) = δ • , ± . Theorem 4.4 (Version of Theorem 2.2 for i.u.r.s) . Suppose f ∈ C ∞ ( H ⊗· · ·⊗H d ) lies in the kernel of all U , . . . , U d -invariant distributions. Then thereexists a solution g , . . . , g d ∈ C ∞ ( H ⊗ · · · ⊗ H d ) to the degree- d coboundaryequation for f that satisfies the Sobolev estimates k g i k t ≪ ν ,ǫ ,t k f k σ d , where σ d := σ d ( t ) is some increasing function of t > .Proof. The proof is an induction on d , with the base case being [FF03, The-orem 4.1].By Lemma 4.3 we have that ( f ⊗ | ℓ ) ∈ ker I U ,...,U d − ( H ⊗ ) for all ℓ ∈ Z d and ( f d | k ) ∈ ker I U d ( H d ) for every k ∈ Z × , so the inductive assumptionprovides g ,ℓ , . . . , g d − ,ℓ ∈ C ∞ ( H ⊗ ) and g d, k ∈ C ∞ ( H d ) satisfying U g ,ℓ + . A. Ram´ırez Higher cohomology of parabolic actions · · · + U d − g d − ,ℓ = ( f ⊗ | ℓ ) and U d g d, k = ( f d | k ) and the bounds k g i,ℓ k t ≪ ν ,t k ( f ⊗ | ℓ ) k σ d − and k g d, k k t ≪ ν ,t k ( f d | k ) k σ . By putting g i ( k , ℓ ) := g i,ℓ ( k ) for i = 1 , . . . , d − g d ( k , ℓ ) := g d, k ( ℓ ), we define a solution g , . . . , g d ∈ C ∞ ( H ⊗ ⊗ H d ) to the degree- d coboundary equation for f . To see the boundson Sobolev norms, we calculate k g i k t = X k ,ℓ (1 + Q + ( k ) + Q d ( ℓ )) t | g i ( k , ℓ ) | k u ⊗ ( k ) ⊗ u d ( ℓ ) k ≤ X k ,ℓ (1 + Q + ( k )) t (1 + Q d ( ℓ )) t | g i ( k , ℓ ) | k u ⊗ ( k ) k k u d ( ℓ ) k = X ℓ (1 + Q d ( ℓ )) t k g i,ℓ k t k u d ( ℓ ) k ≪ ν ,t X ℓ (1 + Q d ( ℓ )) t k ( f ⊗ | ℓ ) k σ d − k u d ( ℓ ) k ≪ ν ,t k f ⊗ k σ d − + t ≪ ν ,ǫ ,t k f k L ( σ d − + t ) , by Lemma 4.2. A very similar computation holds for Sobolev norms of g d .The theorem is proved by setting σ d ( t ) = L ( σ d − ( t ) + t ). Let Ω n R d ( W s ( H ⊗ · · · ⊗ H d )) denote the set of leafwise n -forms defined byelements of Sobolev order s .For an n -form ω ∈ Ω n R d ( W s ( H ⊗· · ·⊗H d )), we define ω ℓ for ℓ ∈ { , . . . , d } to be the part of ω that forgets ℓ : ω ℓ ( U i , . . . , U i n ) = ω ( U i , . . . , U i n )where i < · · · < i n ⊂ { , . . . , b ℓ, . . . , d } . If we now fix some basis element u ℓ ( k ) ∈ H ℓ , we can define a restriction( ω ℓ | k ) ∈ Ω n R d − ( W s ( H ⊗ · · · ⊗ c H ℓ ⊗ · · · ⊗ H d ))by setting ( ω ℓ | k )( U i , . . . , U i n ) = ( ω ( U i , . . . , U i n )) | k for i < · · · < i n ⊂ { , . . . , b ℓ, . . . , d } . . A. Ram´ırez Higher cohomology of parabolic actions
Lemma 5.1.
Let ω ∈ Ω n R d ( W s ( H ⊗ · · · ⊗ H d )) with d ω = 0 . Then for any ℓ = 1 , . . . , d , we have that d( ω ℓ | k ) = 0 for all k ∈ Z ℓ .Proof. This is [Ram13b, Lemma 14.1].
Proposition 5.2.
Let ω ∈ Ω d − R d ( W s ( H ⊗· · ·⊗H d )) be a closed ( d − -form,and ℓ = 1 , . . . , d . Then for every k ∈ Z ℓ , we have that ( ω ℓ | k )( U , . . . , b U ℓ , . . . , U d ) ∈ ker I sU ,..., c U ℓ ,...,U d ( H ⊗ , b ℓ ) . Proof. [Ram13a, Lemma 5.2]
Theorem 5.3 (Version of Theorem 2.3 for i.u.r.s) . Let H = H ⊗ · · · ⊗ H d be an irreducible representation of PSL(2 , R ) d with no trivial factor, and let ≤ n ≤ d − . Then any smooth n -cocycle ω ∈ Ω n R d ( C ∞ ( H )) is a coboundary d η = ω for some η ∈ Ω n − R d ( C ∞ ( H )) and k η k t ≪ ν ,ǫ ,t k ω k ς d where ς d := ς d ( t ) is some increasing function of t > .Proof. This proof is a version of the proof of [Ram13b, Theorem 14.6], whichin turn is adapted from an induction in [KK95, p. 25]. For us, the base caseis [Mie07, Theorem 17], for 1-cocycles an R -action. The inductive strategyis to suppose we have the result for R p -actions, whenever 2 ≤ p ≤ d −
1. Let t > k ∈ Z , ( ω | k ) is a closed n -form over the R d − ∼ = h U , . . . , U d i -action on H ⊗ , b , by Lemma 5.1.If n < d −
1, then our induction assumption produces an ( n − η ,k ∈ Ω n R d − ( C ∞ ( H ⊗ , b )) for ( ω | k ) satisfying k η ,k k τ ≪ ν ,ǫ ,τ k ( ω | k ) k ς d − ( τ ) (11)for any τ >
0. On the other hand, if n = d −
1, then ( ω | k ) is a top-degreeform for the R d − ∼ = h U , . . . , U d i -action, and our Proposition 5.2 tells us that( ω | k )( U , . . . , U d ) ∈ ker U ,...,U d ( H ⊗ , b ). Theorem 4.4 now tells us that there isan ( n − ω | k ) which we will again call η ,k , and that satisfies k η ,k k τ ≪ ν ,ǫ ,τ k ( ω | k ) k σ d − ( τ ) (12)for any τ >
0. Now we just define η by the requirement that ( η | k ) = η ,k for every k ∈ Z . . A. Ram´ırez Higher cohomology of parabolic actions
It is left to find a primitive for the components of ω which contain theindex 1. Let θ ( U i , . . . , U i n ) = ω ( U , U i , . . . , U i n ) − U η ( U i , . . . , U i n )and notice that θ ∈ Ω n − R d ( C ∞ ( H )) and that k θ ( U i , . . . , U i n ) k τ ≤ k ω ( U , U i , . . . , U i n ) k τ + k U η ( U i , . . . , U i n ) k τ . By Lemma 3.1, k θ ( U i , . . . , U i n ) k τ ≤ k ω ( U , U i , . . . , U i n ) k τ + k η ( U i , . . . , U i n ) k τ +1 ≪ ν ,τ k ω k ς d ( τ +1) and this in turn implies that k θ k τ ≪ ν ,τ k ω k ς d ( τ +1) . (13)The next calculation shows that( θ | k ) = ( θ | k ) ∈ Ω n − R d − ( C ∞ ( H ⊗ , b ))is a closed form for any k ∈ Z . For 1 < i < · · · < i n ≤ d ,d( θ | k )( U i , . . . , U i n ) = n X j =1 ( − j +1 U i j ( θ | k )( U i , . . . , b U i j , . . . , U i n )= n X j =1 ( − j +1 U i j (cid:16) ω ( U , U i , . . . , b U i j , . . . , U i n ) | k (cid:17) − n X j =1 ( − j +1 U i j (cid:16) U η ( U i , . . . , b U i j , . . . , U i n ) | k (cid:17) = U ( ω | k )( U i , . . . , U i n ) (because d ω = 0) − U n X j =1 ( − j +1 U i j ( η | k )( U i , . . . , b U i j , . . . , U i n )= 0 . (because d( η | k ) = ( ω | k ))Therefore, our induction implies that there exists a primitive ζ k ∈ Ω n − R d − ( C ∞ ( H ⊗ , b ))for ( θ | k ) satisfying k ζ k k τ ≪ ν ,ǫ ,τ k ( θ | k ) k ς d − ( τ ) (14) . A. Ram´ırez Higher cohomology of parabolic actions for every τ >
0. Finally, we define η by( η | k )( U i , . . . , U i n − ) = ( ζ k ( U i , . . . , U i n − ) if i = 1( η | k )( U i , . . . , U i n − ) if i > . Then d η = ω , so we have found our primitive. Let us check the Sobolevestimates by calculating k η k t = X ≤ i < ···
On the other hand, if i >
1, then k η ( U i , . . . , U i n − ) k t = X ( k, ℓ ) ∈Z × Z × (1 + Q ( k ) + Q + ( ℓ )) t × | η ( U i , . . . , U i n − )( k, ℓ ) | k u ( k ) ⊗ v ( ℓ ) k ≤ X k ∈Z (1 + Q ( k )) t X ℓ ∈ Z × (1 + Q + ( ℓ )) t × | η ( U i , . . . , U i n − )( k, ℓ ) | k u ( k ) k k v ( ℓ ) k = X k ∈Z (1 + Q ( k )) t k ( η | k )( U i , . . . , U i n − ) k t ≪ ν ,ǫ ,t X k ∈Z (1 + Q ( k )) t k ( ω | k ) k { ς d − ( t ) ,σ d − ( t ) } (by (11) and (12)) ≤ k ω k { ς d − ( t ) ,σ d − ( t ) } + t ≤ k ω k { ς d − ( t ) ,σ d − ( t ) } + t These two calculations imply that there is some increasing function ς d ( t ) suchthat k η k t ≪ ν ,ǫ ,t k ω k ς d ( t ) holds. (For example, ς d ( t ) = max ς d − ( ς d − ( t ) + t + 1) ,ς d − ( t ) + tσ d − ( t ) + t , works.) The proofs of Theorems 2.2 and 2.3 now follow from Theorems 4.4 and 5.3by arguments identical to the proofs of [Ram13b, Theorems 10.1 and 10.2].Theorem 1.1 follows trivially from Theorems 2.2 and 2.3. Theorem 1.2 isproved by mixing the arguments in this note with those in [Ram13a].
Proof of Theorem 2.2.
Let H be a unitary representation of SL(2 , R ) d withspectral gap and satisfying Assumption 2.1. This means we can choose ν , ǫ uniformly over the direct integral decomposition H = Z ⊕ R H ν λ ds ( λ ) (15) . A. Ram´ırez Higher cohomology of parabolic actions where ds -almost every H λ := H ν λ is an irreducible unitary representation.(Sobolev spaces W s ( H ) also decompose accordingly.) Now f ∈ ker I U ,...,U d ( H )decomposes as f = Z ⊕ R f λ ds ( λ )where f λ ∈ ker I U ( H λ ) for ds -almost every λ . Therefore, Theorem 4.4 guar-antees that for ds -almost every λ there are g ,λ , . . . , g d,λ ∈ C ∞ ( H λ ) satisfying U g ,λ + · · · + U d g d,λ = f and the estimates k g i,λ k t ≪ ν ,ǫ ,t k f λ k σ d ( t ) , where ν i,λ ≤ ν < ν λ ∈ C appearing in the decomposition (15). Setting g i = Z ⊕ R g i,λ ds ( λ ) , we have a solution to the degree- d coboundary equation for f , satisfying theestimate k g i k t ≪ ν ,ǫ ,t k f k σ d ( t ) , proving the theorem. Proof of Theorem 2.3.
Let H , ν , ǫ , t be as in the theorem statement, and1 ≤ n ≤ d −
1. Let ω ∈ Ω n R d ( C ∞ ( H )) with d ω = 0 . Again, we have a direct integral decompostion C ∞ ( H ) = Z ⊕ R C ∞ ( H ν λ ) ds ( λ )where ds -almost every H ν λ := H λ is irreducible and without trivial factors(by assumption), and ω decomposes ω ( U i , . . . , U i n ) = Z ⊕ R ω λ ( U i , . . . , U i n ) ds ( λ )such that ds -almost every ω λ is a cocycle in Ω n R d ( C ∞ ( H λ )).For these λ , Theorem 5.3 supplies η λ := η ν λ ∈ Ω n − R d ( C ∞ ( H ν λ )) withd η λ = ω λ and k η λ k t ≪ ν ,ǫ ,t k ω k ς d ( t ) . Defining η ( U i , . . . , U i n − ) := Z ⊕ R η λ ( U i , . . . , U i n − ) ds ( λ ) , gives a solution to the coboundary equation d η = ω satisfying the bound k η k t ≪ ν ,ǫ ,t k ω k ς d ( t ) . . A. Ram´ırez Higher cohomology of parabolic actions
This is the third paper where this general strategy has been implemented,each time with significant adjustments. However, we can interpret the previ-ous efforts in terms of the procedure used here. The strategies in [Ram13a],where we treated Anosov R d -actions onSL(2 , R ) d / Γ = SL(2 , R ) × · · · × SL(2 , R ) / Γ , and [Ram13b], where we considered Weyl chamber flows associated to d -foldproducts SO ◦ ( N, d = SO ◦ ( N, × · · · × SO ◦ ( N, , correspond to making particular choices of ϕ ’s in their respective situations.Therefore, the innovation in this note has been the observation that thearguments from those articles work if we re-define f ⊗ in terms of these ϕ ’s,as we did here in (10).In principle, the recipe should apply more generally for R d -actions onirreducible homogeneous spaces of G × · · · × G d where G i are (semisimple)Lie groups with the following ingredients: A nice description of the invariant distributions in irreducible uni-tary representations of G , with spanning set {D w } w ∈W indexed bysome set W . • In the SL(2 , R ) × · · · × SL(2 , R ) case, we used work of Mieczkowski ongeodesic flows of hyperbolic surfaces [Mie06]. He showed that in anyirreducible unitary representation of PSL(2 , R ), the space of (cid:16) / − / (cid:17) -invariant distributions is at most two-dimensional, and explicitly gave aspanning set {D , D } in each irreducible. The methods followed thoseof Flaminio–Forni for horocycle flows [FF03]. • For hyperbolic manifolds of arbitrary dimension, we proved [Ram13b,Theorem 1.2], a result showing that contrary to the surfaces case,the invariant distributions for the geodesic flow element form an in-finite -dimensional space in any irreducible unitary representation ofSO ◦ ( N, N ≥
3. Like in Mieczkowski and Flaminio–Forni’swork, the proof proceeded by looking at the action of the geodesic flowelement on K -types in irreducible unitary representations, and relating . A. Ram´ırez Higher cohomology of parabolic actions invariant distributions to a (partial) difference equation with a combi-natorial flavor. We found a spanning set {D m ,λ } ( m ,λ ) ∈ M × Λ , indexed bycertain Gelfand–Cejtlin arrays. (See [Ram13b, Figure 1].) • In this article, we use the seminal work of Flaminio–Forni [FF03], wherethe space of horocycle flow-invariant distributions is studied. As inthe case of geodesic flows of surfaces, it turns out that the space ofinvariant distributions is at most two-dimensional in any irreducibleunitary representation of PSL(2 , R ). A way to choose ϕ ’s that satisfy (5) and some sort of controllingstatement like Lemma 4.1. • Mieczkowski gives the geodesic flow-invariant distributions in any ir-reducible unitary representation of PSL(2 , R ). This is a space of (atmost) two dimensions, and he labels a spanning set by {D , D } . If wewere to follow the strategy used in this paper, the results in [Ram13a]could be proved by defining elements ϕ and ϕ satisfying (5). Actually,we can interpret the strategy in that paper as a family of choices of ϕ ’s and ϕ ’s, which reflects the fact that primitives to higher-degreecoboundary equations are not unique, as they are for the first degree. • For geodesic flows of higher-dimensional hyperbolic manifolds, since wehave an infinite-dimensional space of invariant distributions in every ir-reducible unitary representation of SO ◦ ( N, { ϕ m ,λ } where m , λ correspond to certain Gelfand–Cejtlin arrays. Part of thechallenge therefore is in coping with this infinite-dimensionality. Car-rying the arguments from the SL(2 , R ) situation directly over to thiscase is impossible. Therefore, instead of making the “family” of choicesthat we had made in the previous paper, we simply chose the “easiest” ϕ ’s: Each ϕ m ,λ was a basis element of the irreducible unitary represen-tation, where the basis { u m ,λ } was exactly what had been used to findour basis {D m ,λ } of invariant distributions in the first place. • In this paper, ϕ ± are indexed by { + , −} , the indices for the spanning set {D + , D − } of invariant distributions found in [FF03], and are engineeredto satisfy (5). . A. Ram´ırez Higher cohomology of parabolic actions
A “base case” for the induction telling us that we indeed have acomplete set of obstructions to the degree- coboundary equationand giving us control over the Sobolev norms of primitives. • For Anosov R d -actions on SL(2 , R ) d / Γ we used Mieczkowski’s [Mie06,Theorem 4.3], a theorem stating that one can solve the coboundaryequation for the geodesic flow as long as the given function is in thekernel of all geodesic flow-invariant distributions, and giving estimateson the Sobolev norms of primitives. (Actually, Mieczkowski proved thisfor PSL(2 , R ), but the extension to SL(2 , R ) is not hard.) • For Weyl chamber flows of SO ◦ ( N, d , we had to establish a basecase [Ram13b, Theorem 1.1] for the geodesic flow of a hyperbolic man-ifold. The result was a statement analogous to Mieczkowski’s. This wasthe main concern of Part I of that article. • In fact, both of the above mentioned theorems were inspired by thework of Flaminio and Forni, where a similar analysis was carried outfor horocycle flows of surfaces, resulting in [FF03, Theorem 4.1], whichhas been our “base case” here.Once these ingredients are present, it should be possible to carry out aninduction like the one used here for the top-degree theorem (see the proof ofTheorem 4.4). The lower-degree statement follows comparatively easily afterthe top-degree part is settled, by using the induction used here in the proofof Theorem 5.3, which is itself adapted from [KK95].
Acknowledgments
This work was completed while the author was employed at the University ofBristol and supported by ERC. Much of it was written in October–December2012, while visiting the Department of Mathematics at the PennsylvaniaState University and enjoying their hospitality.
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