Cocycle rigidity of abelian partially hyperbolic actions
aa r X i v : . [ m a t h . D S ] M a r COCYCLE RIGIDITY OF ABELIAN PARTIALLYHYPERBOLIC ACTIONS
ZHENQI JENNY WANG Abstract.
Suppose G is a higher-rank connected semisimple Lie groupwith finite center and without compact factors. Let G = G or G = G ⋉ V , where V is a finite dimensional vector space V . For any unitaryrepresentation ( π, H ) of G , we study the twisted cohomological equation π ( a ) f − λf = g for partially hyperbolic element a ∈ G and λ ∈ U (1),as well as the twisted cocycle equation π ( a ) f − λ f = π ( a ) g − λ g forcommuting partially hyperbolic elements a , a ∈ G . We characterizethe obstructions to solving these equations, construct smooth solutionsand obtain tame Sobolev estimates for the solutions. These results canbe extended to partially hyperbolic flows parallelly.As an application, we prove cocycle rigidity for any abelian higher-rank partially hyperbolic algebraic actions. This is the first paper ex-ploring rigidity properties of partially hyperbolic that the hyperbolicdirections don’t generate the whole tangent space. The result can beviewed as a first step toward the application of KAM method in obtain-ing differential rigidity for these actions in future works. Introduction
Various abelian algebraic actions.
We define Z k × R ℓ , k + ℓ ≥ H be a connected Lie group, A ⊆ H a closedabelian subgroup which is isomorphic to Z k × R ℓ , L a compact subgroup ofthe centralizer Z ( A ) of A , and Γ a torsion free lattice in H . Then A acts byleft translation on the compact space M = L \ H/ Γ. Denote this action by α A . The three specific types of examples discussed below correspond to: • for the symmetric space examples take H a semisimple Lie group ofthe non-compact type. • for the twisted symmetric space examples take H = G ⋉ ρ R m or H = G ⋉ ρ N , a semidirect product of a reductive Lie group G withsemisimple factor of the non-compact type with R m or a simplyconnected nilpotent group N . • for the parabolic action examples, take H a semisimple Lie groupof the non-compact type and A a subgroup of a maximal abelianunipotent subgroup in H . Key words and phrases: Higher rank Abelian group actions, cocycle rigidity, partiallyhyperbolic dynamical systems. ) Based on research supported by NSF grant DMS-1346876.
History and method.
In contrast to the classical rank-one actionswhere Livsic showed that there is an infinite-dimensional space of obstruc-tions to solving the cohomological equation for a hyperbolic action by R or Z , in the past two decades various rigidity phenomena for (partially) hyper-bolic actions have been well understood. Significant progresses have beenmade in the case of cocycle rigidity for higher rank (partially) hyperbolicalgebraic actions (see [4], [17], [18], [19] and [28]) obtained from symmetricand twisted symmetric space examples. In these papers, the higher rankproperty is used to show the existence of a distributional or continuoustransfer function; then the smoothness of the transfer function follows fromthe fact that it is smooth along stable and unstable directions and thatthose generate the tangent space at every point. Hence all actions consid-ered in previous papers satisfy the following property which is essential forobtaining smooth rigidity:( B ) The stable directions of various action elements generate the tangentspace as a Lie algebra.
In [18] and [19] the proofs are based on harmonic analysis of semisim-ple Lie groups, specifically, on exponential decay of matrix coefficients ofpartially hyperbolic elements. In [4], [17] and [28] the main geometric ingre-dient is the accessibility of stable and unstable foliations, which enables theconstruction of continuous transfer function. The natural difficulty in ex-tending the rigidity results to general partially hyperbolic algebraic actionscomes from three aspects: firstly, how to obtain exponential decay of ma-trix coefficients in general twisted spaces. The method used in [18] requiresthat individual acting element acts ergodicly on the torus bundle. But thiscondition fails once 0 weight appears. Secondly, for general partially hy-perbolic actions, the stable and unstable foliations are no longer accessible.This means geometric method (the method in [4], [17] and [28]) can’t beadapted to general cases. Thirdly, the smoothness of the solution to the co-homological equation followed from subelliptic regularity theorem. But thiscomes with three disadvantages: firstly, this requires that that the actionstaken into account should satisfy condition ( B ); secondly, the solution ofthe cohomological equation loses at least half of regularity. Tame estimates(finite loss of regularity) for the solution is important in dynamics, since it isclosely related to obtain smooth action rigidity in dynamics, see [6] and [5];thirdly, subelliptic regularity theorem fails for general Hilbert spaces. Forexample, the methods in previous papers all fail if projection of the actinggroup to one simple factor of the semisimple part is trivial.In this paper, we study the cohomological equation for general partiallyhyperbolic acting elements and build up cocycle rigidity results for gen-eral higher-rank partially hyperbolic algebraic actions. We characterizethe obstructions to solving the (twisted) cohomological equation, constructsmooth solution and obtain the tame Sobolev estimates for the solution, i.e,there is finite loss of regularity (with respect to Sobolev norms) between OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 3 the coboundary and the solution. As an application, we prove the smooth(twisted) cocycle rigidity for any higher rank partially hyperbolic actionsover G . To prove these results, we introduce new ingredients from repre-sentation theory and obtain more elaborate information about estimates inneutral directions. These results are of independent interest and have wideapplicability.1.3. Motivation.
So far an effective approach to local differentiable rigidityis the “geometric” method first introduced in [4, 6] to prove local smoothrigidity for generic restrictions in SL ( n, R ) / Γ and SL ( n, C ) / Γ. This ap-proach is based on geometry and combinatorics of invariant foliations andusing insights from algebraic K -theory as an essential tool. The approachwas further employed in [30], [31], [32] and [28] for extending cocycle rigidityand differentiable rigidity to most higher rank actions for symmetric spaceand twisted symmetric space examples satisfying the following genuinelyhigher rank condition: the projection of the acting group to each simplefactor of the semisimple part contains a Z subgroup. The genuinely higherrank condition is necessary for the application of geometric method. In manysituations of interest, however this condition is not present (for example, forthe homogeneous space SL (2 , k ) n / Γ, where k = R or C ).One important application of the results in the present paper is that theseresults open a prospect of proving a version of local differentiable rigidity for general partially hyperbolic actions. This should work as follows: bylinearization of the conjugacy equation, we get the corresponding linearizedequation: Ad( α )Ω − Ω ◦ α = R (1.1)where α is an A -algebraic action (the unperturbed action) and R is the errorbetween α and its perturbation ˜ α . If Ω is a solution for the linearized equa-tion, or at least an approximate solution, i.e., it solves the above equationwith a small error with respect to R , then one may expect that the newperturbation ˜ α (1) defined by is much closer to α than ˜ α . Carrying out theiteration process, one may produce a smooth conjugacy between α and ˜ α .This method first appeared in [5] to prove the differentiable rigidity ofpartially hyperbolic but not hyperbolic actions on torus. A scheme similarto that of [5] applies to certain parabolic cases, i.e. homogeneous actions ofunipotent abelian groups in [7]. To carry out the above scheme, the first taskis to precisely describe the solution to the equation (1.1), which is studiedin Section 6.6. Note that during the iteration process, the acting groupsare not fixed, but vary in a small neighbourhood. So we need to obtainuniform estimates for these actions. Hence the results in the present paperare essential for successful application of the scheme to general partiallyhyperbolic actions in the future work, see [33]. COCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS Background, definition, and statement of results
Preliminaries on cocycles.
Let α : A × M → M be an action of atopological group A on a (compact) manifold M by diffeomorphisms. Fora topological group ( Y, ∗ ) and a homomorphism ψ : A → Aut( Y ), a ( ψ -twisted)-cocycle over α is a continuous function β : A × E → Y satisfying: β ( ab, x ) = β ( a, α ( b, x )) ∗ ψ ( a ) β ( b, x )(2.1)for any a, b ∈ A . A (twisted) cocycle is cohomologous to a constant twistedcocycle (cocycle not depending on x ) if there exists a constant (twisted)cocycle s : A → Y and a continuous transfer map H : M → Y such that forall a ∈ A β ( a, x ) = H ( α ( a, x )) ∗ s ( a ) ∗ ( ψ ( a ) H ( x )) − (2.2)(2.2) is called the cohomology equation.In particular, a cocycle is a coboundary if it is cohomologous to the trivialcocycle π ( a ) = id Y , a ∈ A , i.e. if for all a ∈ A the following equation holds: β ( a, x ) = H ( α ( a, x )) ∗ ( ψ ( a ) H ( x )) − . (2.3)For more detailed information on cocycles adapted to the present setting see[4] and [15].In this paper we will only consider smooth C k -valued cocycles over alge-braic partially hyperbolic actions on smooth manifolds. By taking compo-nent functions we may always assume that β is valued on C . Further, bytaking real and imaginary parts, we can extend the results for real valuedcocycles as well. Adapted to the settings in this paper, A is isomorphicto Z k or R k and the space X = G / Γ, if G = G and X = G / Γ ⋉ Z N if G = G ⋉ R N , where Γ is an irreducible torsion free lattice in G . A cocy-cle is called H r if the map β ( a, · ) ∈ H r ( L ( G / Γ)) for any a ∈ A , where H r ( L ( G / Γ)) is Sobolev space of order r for the left regular representationof G on L ( G / Γ). We can also define β to be of class C r . We also note thatif the cocycle β is cohomologous to a constant cocycle, then the constantcocycle is given by s ( a ) = R G / Γ β ( a, x ) dx. In what follows, C will denote any constant that depends only on thegiven group G . C x,y,z, ··· will denote any constant that in addition to theabove depends also on parameters x, y, z, · · · .2.2. Statement of the results.
In this paper, G denotes a higher-rankconnected semisimple Lie group with finite center and without compactfactors. Fix a maximal compact subgroup K of G and a right invariant,bi- K -invariant metric d on G . Let A be the R -split Cartan subgroup of G admitting the Cartan decomposition G = KA K .For a finite dimensional vector space V , a continuous representation ρ : G → GL ( V ) is called excellent if ρ ( G i )-fixed points in V are { } for eachsimple factor G i of G . Set G = G or G ⋉ ρ V , where ρ is excellent. The OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 5 multiplication of elements in G ⋉ ρ V is defined by( g , r ) · ( g , r ) = ( g g , ρ ( g − ) r + r ) . (2.4)For any s ∈ G ⋉ ρ V , we have the decomposition s = ( g s , v s ), where g s ∈ G and v s ∈ V . Then for any s = ( g s , v s ) ∈ G , we haveAd( s )( X ) = Ad( g s )( X ) − ρ ( g s ) dρ ( X ) v s , andAd( s )( v ) = ρ ( g s ) v. (2.5)for any X ∈ Lie( G ) and v ∈ V .Set d (cid:0) ( g, v ) , e (cid:1) = d ( g, e ) + d ( v, e ), where d is a metric on V . Fix aninner product on G = Lie( G ) (determined by d or d ). Let G be the set ofunit vectors in G . Definition 2.1.
An algebraic flow φ R on G (resp. an element a ∈ G ) iscalled partially hyperbolic if the spectrum of the group Ad( φ t ) (resp. Ad( a ))acting on G is not contained in U (1).Our first two results characterize the obstructions to solving the cohomo-logical equation and obtain Sobolev estimates for the solution. The nexttheorem shows that the ( π ( a ) − λI )-invariant distributions are the only ob-structions to solving the cohomological equation π ( a ) f − λf = h where a ∈ G is partially hyperbolic and λ ∈ U (1). Theorem 2.2.
Suppose ( π, H ) is a unitary representation of G such thatthe restriction of π to any simple factor of G is isolated from the trivialrepresentation (in the Fell topology) if G = G ; or π contains no non-trivial V -fixed vectors if G = G ⋉ ρ V . For any partially hyperbolic element a and λ ∈ U (1) , (1) if f ∈ H is a solution of the equation: π ( a ) f − λf = h , then P + ∞ j = −∞ λ − ( j +1) π ( a j ) h = 0 as a distribution. (2) there exist constants m > σ > (only depending on G ) such thatfor any m ≥ m , there exists δ ( m ) > , such that for any b ∈ G with d ( a, b ) < δ , if h ∈ H m satisfying P + ∞ j = −∞ λ − ( j +1) π ( b j ) h = 0 asa distribution, then the equation π ( b ) f − λf = h (2.6) has a solution f ∈ H m − σ and the following estimate holds k f k m − σ ≤ C m,a k h k m . (3) if h ∈ H ∞ and D ( h ) = 0 for any ( π ( a ) − λI ) -invariant distribution D ∈ H − , then the cohomological equation π ( a ) f − λf = h , has asolution f ∈ H ∞ . The next result is about the existence of common solution to the coho-mological equations. Suppose a, b ∈ G are partially hyperbolic and linearlyindependent and m , σ as in Theorem 2.2. COCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS
Theorem 2.3.
Suppose ( π, H ) is a unitary representation of G such thatthe restriction of π to any simple factor of G is isolated from the trivialrepresentation if G = G ; or π contains no non-trivial V -fixed vectors if G = G ⋉ ρ V . For any m ≥ m + σ , there exists δ ( m ) > , such that for any a , b ∈ G with d ( a, a ) + d ( b, b ) < δ and a b = b a , if f, h ∈ H m andsatisfy the cocycle equation π ( a ) f − λ f = π ( b ) h − λ h where λ , λ ∈ U (1) , then the equations π ( b ) p − λ p = f, π ( a ) p − λ p = h have a common solution p ∈ H m − σ satisfying the Sobolev estimate k p k m − σ ≤ C m,a,b max {k h k m , k f k m } . Remark 2.4.
Results in Theorem 2.2 and 2.3 can be extended to partiallyhyperbolic flows correspondingly. See Corollary 4.6 and 5.4.As an application, for the symmetric space examples and the twistedsymmetric space examples we prove locally cocycle rigidity for any higher-rank partially hyperbolic action. All relevant definitions appear in Section6.3.
Theorem 2.5.
Let α A on L \ G / Γ be an abelian higher-rank partially hyper-bolic algebraic action of symmetric space examples or of the twisted symmet-ric space examples. N ⊂ G is the neutral distribution of α A on G / Γ . Thenthere exist p > s > such that for any m ≥ p : (1) any H m -cocycle β over α A is cohomologous to a constant cocycle viaa H m − s -transfer map; (2) if β is a H m -Ad-twisted cocycle taking values on N over α A , then β is cohomologous to a constant twisted cocycle via a H m − s -transfermap. Acknowledgements.
I would like to thank Roger Howe for discussionof matrix coefficients decay on twisted symmetric spaces. Livio Flaminiosuggested a method of obtaining tame estimates in the centralizer directionin a different setting that inspired our arguments on that topic.3.
Preliminaries on unitary representation theory
Sobolev space and elliptic regularity theorem.
Let π be a unitaryrepresentation of a Lie group S with Lie algebra s on a Hilbert space H = H ( π ). Definition 3.1.
For k ∈ N , H k ( π ) consists of all v ∈ H ( π ) such that the H -valued function s → π ( s ) v is of class C k ( H = H ). For X ∈ s , dπ ( X )denotes the infinitesimal generator of the one-parameter group of operators t → π (exp tX ), which acts on H as an essentially skew-adjoint operator. Forany v ∈ H , we also write Xv := dπ ( X ) v . OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 7
We shall call H k = H k ( π ) the space of k -times differentiable vectors for π or the Sobolev space of order k . The following basic properties of thesespaces can be found, e.g., in [25] and [8]:(1) H k = T m ≤ k D ( dπ ( Y j ) · · · dπ ( Y j m )), where { Y j } is a basis for s , and D ( T ) denotes the domain of an operator on H .(2) H k is a Hilbert space, relative to the inner product h v , v i S,k : = X ≤ m ≤ k h Y j · · · Y j m v , Y j · · · Y j m v i + h v , v i (3) The spaces H k coincide with the completion of the subspace H ∞ ⊂ H of infinitely differentiable vectors with respect to the norm k v k S,k = (cid:8) k v k + X ≤ m ≤ k k Y j · · · Y j m v k (cid:9) . induced by the inner product in (2). The subspace H ∞ coincideswith the intersection of the spaces H k for all k ≥ H − k , defined as the Hilbert space duals of the spaces H k , are sub-spaces of the space H −∞ of distributions, defined as the dual spaceof H ∞ .We write k v k k := k v k S,k and h v , v i k := h v , v i S,k if there is no confusion.Otherwise, we use subscripts to emphasize that the regularity is measuredwith respect to S .3.2. Elliptic regularity theorem.
We list the well-known elliptic regular-ity theorem which will be frequently used in this paper (see [26, Chapter I,Corollary 6.5 and 6.6]):
Theorem 3.2.
Fix a basis { Y j } for s and set L m = P Y mj , m ∈ N . If L m v ∈ H , then v ∈ H m with Sobolev estimate k v k m ≤ C m ( k L m v k + k v k ) , ∀ m ∈ N where C m is a constant only dependent on m and { Y j } . Suppose Γ is a torsion-free cocompact lattice in S . Denote by Υ theregular representation of S on H (Υ) = L ( S/ Γ). Then we have the followingsubelliptic regularity theorem (see [19]):
Theorem 3.3.
Fix { Y j } in s such that commutators of Y j of length at most r span s . Also set L m = P Y mj , m ∈ N . Suppose f ∈ H (Υ) or f ∈ H −∞ .If L m f ∈ H (Υ) for any m ∈ N , then f ∈ H ∞ (Υ) and satisfies k f k mr − ≤ C m ( k L m f k + k f k ) , ∀ m ∈ N (3.1) where C m is a constant only dependent on m and { Y j } . COCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS
Remark 3.4.
The elliptic regularity theorem is a general property, while thesubelliptic regularity theorem only has local versions on manifolds (see [27]).More precisely, the proof of the above theorem is based on the followinggeneral subelliptic regularity theorem: for any x ∈ S/ Γ there is an openneighbourhood V containing x such that if f and L m f are both in L ( V ),then k f k mr − ≤ C m,V ( k L m f k + k f k ) , ∀ m ∈ N where C m,V is a constant only dependent on m , { Y j } and V . In otherwords, we can only get a local version of (3.1) on the manifold S/ Γ. Thenthe compactness of S/ Γ is essential to the existence of a uniform constant C m for the global Sobolev estimates.3.3. Exponential matrix coefficients decay.Definition 3.5.
Let π be a unitary representation of S on a Hilbert space H . Say that a vector v ∈ H is δ -Lipschitz if δ = sup g ∈ G −{ e } k π ( g ) v − v k dist( e, g ) < ∞ ;we will refer to the number δ as to the δ -Lipschitz coefficient of v , and saythat the vector v is δ -Lipschitz.If S is semisimple without compact factors and with finite center, Klein-bock and Margulis (see [21, appendix]) extended the matrix coefficient decayresult about smooth vectors in [18] to Lipschitz vectors. Fix a maximal com-pact subgroup K of S and a Riemannian metric d on S which is bi-invariantwith respect to K . Theorem 3.6 (Kleinbock and Margulis) . Let ( π, H ) be a unitary repre-sentation of S such that the restriction of π to any simple factor of S isisolated from the trivial representation. Then there exist constants γ, E > ,dependent only on S such that if v i ∈ H , i = 1 , , be δ i -Lipschitz vectorsthen for any g ∈ S |h π ( g ) v , v i| ≤ ( E k v kk v k + δ k v k + δ k v k + δ δ ) e − γd ( e,g ) . Remark 3.7. If G = G ⋉ ρ V , then it also follows from Theorem 1 . π of G without V -fixed vectors, its re-striction to any simple factor of G is isolated from the trivial representation.The implies that the above theorem applies for the restriction of π to G .4. Solution of the twisted coboundary
Throughout this part, ( π, H ) always denotes a unitary representation of G such that the restriction of π to any simple factor of G is isolated from thetrivial representation if G = G ; or π contains no non-trivial V -fixed vectorsif G = G ⋉ ρ V . OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 9
The subsequent discussion will be devoted to the proof of Lemma 4.4.Typically differences between the cases G = G and G = G ⋉ ρ V are minimal,and usually appear at the level of notations. However, the case G = G ⋉ ρ V requires a separate argument in order to obtain exponential decay of matrixcoefficients.First we define obstructions to solvability of the twisted coboundary equa-tion, for a single element and show that vanishing of those obstructions im-plies solvability of the equation with tame estimates with respect to Sobolevnorms. The latter property is an instance of cohomological stability , thenotion first defined in [16]. The scheme of the proof is as follows:(1) We note that f is the solution of the equation (2.6) if and only if π ( s ) f is the solution of the equation π ( sbs − ) π ( s ) f − λπ ( s ) f = π ( s ) h ;(4.1) and P + ∞ j = −∞ λ − ( j +1) π ( b j ) h = 0 as a distribution if and only if + ∞ X j = −∞ λ − ( j +1) π (( sbs − ) j )( π ( s ) h ) = 0as a distribution. Then instead of solving the equation (2.6), wesolve the equation (4.1) for a well chosen s (see Section 4.2).(2) The crucial step in proving Theorem 2.2 is Lemma 4.4. Since gen-erally, condition ( B ) fails, we study the restricted representation π ′ of π on G ′ (see Section 4.1) instead of π . Note that condition ( B )holds on G ′ .(3) Decay estimates for matrix coefficients imply existence of two distri-bution solutions obtained by iteration in positive and negative direc-tions: one of those solutions is differentiable along stable directionsand the other along unstable directions.(4) Vanishing of the obstructions implies that those distribution solu-tions coincide. Since solution along the stable and unstable di-rections is given by explicit exponentially converging “telescopingsums”, they can be differentiated without loss of regularity. Upto this point the proof follows the same general scheme as in [18]although we obtain more elaborate information about estimates inother directions.(5) Remaining directions in G ′ with Ad eigenvalues sufficiently close to1 for the acting element; hence derivatives of all orders in thosedirection have every slow increasing speed. Tame estimates followfrom that and from the fact that those vector-fields can be expressedas polynomial of hyperbolic ones, i.e. from condition ( B ) on G ′ .(6) Note that the derivatives of all orders in directions outside G ′ arestill distributions for π ′ and satisfy the solvability condition. Thenestimates for these directions follow from previous steps. Spectral space decomposition. If G = G , for any s ∈ G under theadjoint representation of G , Ad( s ) has the decomposition:Ad( s ) = exp( Z s ) exp( X s ) exp( Y s )(4.2)for 3 commuting elements, where Z s is compact, X s is R -semisimple, Y s isnilpotent (see, eg, [10, Proposition 2]).The decomposition (4.2) also implies that we have the corresponding de-composition for a : s = k s x s n s (4.3)where x a = exp( X a ) ∈ G is R -semisimple, k a = exp( Z s ) ∈ G is compactand n s = exp( Y a ) ∈ G is nilpotent. It is clear that x a and n a commuteand k s x s = x s k s z and k s n s = n s k s z , where z , z ∈ Z ( G ). Since Z ( G )is finite, there exists n ∈ N such that k s x ns k − s = x ns , which implies thatAd( k s )( X s ) = X s . This shows that k s and x s commute. Similarly, we get k s n s = n s k s .If s is partially hyperbolic, x s is non-trivial. If G = G ⋉ ρ V , For any s = ( g s , v s ) ∈ G , if s is partially hyperbolic, x g s is non-trivial.For any partially hyperbolic element s , the Lie algebra G of G has theeigenspace decomposition for Ad( x s ) or Ad( x g s ): G = X µ ∈ ∆( s ) g µ ( s )(4.4)where ∆( s ) is the set of eigenvalues and g µ ( s ) is the eigenspace for eigenvalue µ . We note that the eigenvalues of Ad( s ) are determined by those of Ad( x s )or Ad( x g s ) up to some elements in U (1).Let g be the subalgebra generated by all g µ , µ = 1. Then: Lemma 4.1. (1) g is an ideal in G . (2) If G = G ⋉ V , then V ⊂ g .Proof. (1) follows directly form the fact that [ g µ , g µ ] ⊆ g µ µ . Let G ′ ( s ) bethe connected subgroup with Lie algebra g . The semisimple part of G ′ ( s )is an almost direct product of simple factors of the semisimple part of G .If G = G ⋉ V , by complete reducibility for representations of semisimplegroups, there is a decomposition of V = M i ∈ I V i , for restricted representation ρ on G ′ ( s ) such that ρ is irreducible on each V i . Since ρ is excellent (see Section 2.2), the restricted representation isnon-trivial on each V i . By (2.5) we see that each V i is contained in g . Thenwe get (2) immediately. (cid:3) Remark 4.2.
For a fixed partially hyperbolic element a ∈ G , let g be thesubalgebra generated by all g µ , µ = 1. Let G ′ = G ′ ( a ) be the connectedsubgroup with Lie algebra g . G ′ is dependent on a ; furthermore, even if b is OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 11 close to a , G ′ ( b ) is not necessarily to be equal to G ′ ( a ). For example, if G issemisimple and a is inside a simple factor G of G , then for b close enoughto a , G ′ ( b ) can be any product of simple factors of G containing G . But ifwe consider the subalgebra generated by the g µ ( b ), for those µ sufficientlyclose to ∆( a ) \
1, then it is still g (see Lemma 4.3).Next, we study the spectral space decomposition of elements sufficientlyclose to a . Suppose b is sufficiently close to a . We will need to consider theeigenspaces for Ad( x b ) or Ad( x g b ). In next section, we want to study therestricted representation π on G ′ . Since b is probably not contained in G ′ as explained in previous part, we need to consider the decomposition of x b or x g b instead. We have a direct sum decomposition ofLie( G ) = g ⊕ g . where g is the Lie algebra of the semisimple part of G ′ . Then we have thedecomposition: x b = x b, x b, or x g b = x g b , x g b , , where x b,i and x g b ,i , i = 1 , g i . Note that x b,i or x g b ,i , i = 1 , x a or x g a respectively. Itis clear that x b, or x g b , are sufficiently close to identity if b is sufficientlyclose to a .We have the eigenspace decomposition for Ad( x b, ) or Ad( x g b , ): G = X µ ∈ ∆ ′ ( b ) l µ . where ∆ ′ ( b ) is the set of eigenvalues of Ad( x b, ) or Ad( x g b , ) and l µ is theeigenspace for eigenvalue µ . Let ∆ ′′ ( b ) = { µ ∈ ∆ ′ ( b ) : µ ∈ ( ν − ǫ, ν + ǫ ) where ν ∈ ∆( a ) \ } for sufficiently small ǫ . In fact, ∆ ′′ ( b ) excludes thoseeigenvalues sufficiently close to 1. Then: Lemma 4.3.
Let g ′ be the subalgebra generated by all l µ , µ ∈ ∆ ′′ ( b ) . Then g ′ is an ideal and g ′ = g .Proof. To prove that g ′ is an ideal it suffices to shows that for any µ ∈ ∆ ′′ ( b )and µ ∈ ∆ ′ ( b ) \ ∆ ′′ ( b ), [ g µ , g µ ] ⊆ g µ ′ for some µ ′ ∈ ∆ ′′ ( b ). We notethat [ g µ , g µ ] ⊆ g µ µ . By assumption, µ µ ∈ ∆ ′′ ( b ) since µ µ is “faraway” from 1 by assumption. If G = G , then it is clear that g ′ = g . If G = G ⋉ V , g ′ = g have the same semisimple part by previous arguments.Lemma 4.1 shows that both g ′ and g contain V . Then the result followsimmediately. (cid:3) Hence we have the decomposition: G = g ⊕ g = g ⊕ X µ ∈ ∆ ′′′ ( b ) l µ , (4.5)where ∆ ′′′ ( b ) ⊂ ∆ ′ ( b ) \ ∆ ′′ ( b ). Note that elements in ∆ ′′′ ( b ) are sufficientlyclose to 1. For each µ ∈ ∆ ′ ( b ), set X µ , · · · , X µ dim l µ ∈ G to be a basis of l µ . Abovelemma shows that there exists r ( a ) > b sufficiently closeto a , X µj , where µ ∈ ∆ ′′ ( b ), 1 ≤ j ≤ dim l µ as well as their commutators oflength no larger than r ( a ) span g .Let p j be polynomials with degree no greater than r ( a ) such that thelinear span of the set (cid:8) p j ( X µ l (1) j (1) , · · · , X µ l ( i ) j ( i ) ) , X µt (cid:9) (4.6)where µ l ( k ) ∈ ∆ ′′ ( b ) for all 1 ≤ k ≤ i and 1 ≤ t ≤ dim l µ , µ ∈ ∆ ′′ ( b ), generate g . Here we note that after substituting elements in g , these p j take valuesin the universal enveloping algebra U ( g ).4.2. Conjugation in semidirect product. If G = G ⋉ ρ V , we note that I − ρ ( g b ) restricted on the subspace W = P µ ∈ ∆ ′′ ( b ) g µ T V is invertible.Denote by ( I − ρ ( g b )) − | W the inverse map on W . Then we have a decom-position: v b = X µ ∈ ∆ ′′ ( b ) v b,µ | {z } v b (1) + X µ ∈ ∆ ′ ( b ) \ ∆ ′′ ( b ) v b,µ | {z } v b (2) where v b,µ ∈ g µ T V . Then u = ρ ( g b )( v b (1)) ∈ W . Set u b = − ( I − ρ ( g b )) − | W ( ρ ( g b ) u ) and s = ( g b , u b ). Then by easy computation we have sbs − = (cid:0) g b , ρ ( g b ) v ′ b (cid:1) , where ρ ( g b ) v ′ b ∈ P µ ∈ ∆ ′ ( b ) \ ∆ ′′ ( b ) v b,µ T V .Note that the norms of such s are uniformly bounded for b sufficiently closeto a . By discussion at the beginning of Section 4 (we see that conjugationby s doesn’t affect the conclusions in Theorem 2.2), we can just assume that b has the decomposition: b = ( g b , v b ), v b ∈ P µ ∈ ∆ ′ ( b ) \ ∆ ′′ ( b ) v b,µ . Then wecan write b = ( x g b , , x g b , k g b n g b , v b ). Since the eigenvalues of Ad( x g b , ) on∆ ′ ( b ) \ ∆ ′′ ( b ) are sufficiently close to 1, we have k Ad( x jg b , )( v b ) k ≤ C (1 + ǫ ) | j | k v b k , ∀ j ∈ Z . (4.7)Using (2.5), we have b j = (cid:0) g jb , j − X i =0 ρ ( g − ib ) v b (cid:1) = ( x jg b , , (cid:0) x jg b , k jg b n jg b , j − X i =0 ρ ( g − ib ) v b (cid:1) = ( x jg b , , (cid:0) x jg b , k jg b n jg b , j − X i =0 Ad( g − ib ) v b (cid:1) (4.8)Let y b,j = x jb, k jb n jb if G = G or y b,j = (cid:0) x jg b , k jg b n jg b , P j − i =0 Ad( g − ib ) v b (cid:1) if G = G ⋉ ρ V . If there is no confusion, we abuse x b, and x g b , for simplicity. OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 13
Twisted cohomological stability.
Fix a partially hyperbolic element a ∈ G . It is clear that π is also a unitary representation for G ′ ( a ). We use( π ′ , H ) to denote the restricted representation. Define r to be the minimalpositive integer satisfying: χ · φ r > φ > , and χ · φ r < φ < χ, φ ∈ ∆( a ).Let m = max { r , r ( a ) } , where r ( a ) is defined (4.6). We list the followinglemma which is very important for the sequel. Lemma 4.4.
Suppose b ∈ G is sufficiently close to a . Also suppose λ ∈ C with | λ | sufficiently close to . Then: (1) For any j ∈ Z , v ∈ H ( π ) and u ∈ H ( π ′ ) we have |h π ( b j ) v, u i| ≤ C (1 + ǫ ) | j | ( | j | + 1) dim G (cid:0) k v kk u k + k u kk v k G , + k v kk u k G ′ , + k u k G ′ , k v k G , (cid:1) e − γ | j | l ( b ) , where l ( b ) = P µ ∈ ∆( b ) | log µ | > , γ > is a constant only depen-dent on G and ǫ ≥ is sufficiently small. (2) Suppose v = { v j } , j ∈ Z is a sequence in H ( π ) satisfying k v j k G , ≤ P ( | j | ) , where P a polynomial, then D ( + − ) v ( b ) = (cid:18) − + (cid:19) X ( j ≥ j ≤− ) λ − ( j +1) π ( b j ) v j are distributions in π ′ . (3) Suppose m > m + 3 and suppose v = { v j } , j ∈ Z is a sequence in H m ( π ) satisfying k v j k G ,s ≤ P s ( | j | ) k h k G ,s , where P s are polynomials, ≤ s ≤ m and h is a vector in H m ( π ) . There exists δ ( m ) > ,such that for any b ∈ G with d ( a, b ) < δ , if f v def = D + v ( b ) = D − v ( b ) as distributions in π ′ . Then f v ∈ H m − m − ( π ′ ) and the followingestimate holds k f v k G ′ ,m − m − ≤ C m,p , ··· ,P m k h k G ,m . Remark 4.5.
In (3), m is dependent on the closeness of b and a for generalpartially hyperbolic element a . If G = G and a is regular, i.e., χ ( a ) = 0 forany roots of G , then m is independent on the closeness. Proof.
Let G ′ denote the semisimple part of G ′ . If G = G , then it is clearthat the restriction of π ′ to any simple factor of G ′ is isolated from thetrivial representation; if G = G ⋉ ρ V , then it also follows from Theorem 1 . π ′ to any simple factor of G ′ is isolated fromthe trivial representation. The above arguments justify the application ofTheorem 3.6 for the restricted representation of ( π ′ , H ) on G ′ . Proof of (1). Recall notations in Section 4.2. By Theorem 3.6 for any v ∈ H ( π ) and u ∈ H ( π ′ ) we have, |h π ( b j ) v, u i| = |h π ′ ( x jb, )( π ( y b,j ) v ) , u i|≤ C ( k π ( y b,j ) v kk u k + δ j k u k + δ k π ( y b,j ) v j k + δ j δ ) e − γ d ( e,x jb, ) = C ( k v kk u k + δ j k u k + δ k v k + δ j δ ) e − γ d ( e,x jb, ) (4.10)where δ = k u k G ′ , , δ j = k π ( y b,j ) v k G ′ , and γ > G .Since x b, is partially hyperbolic and conjugated to an element in A , then d ( e, x jb, ) ≥ C | j | l ( b )(4.11)where l ( b ) = P µ ∈ ∆( b ) | log µ | >
0. Here we use ∆( b ) instead of ∆ ′ ( b ) since x b, is sufficiently close to x b .If G = G , since k b is compact, n b is unipotent, x b is sufficiently close toidentity and they are commuting, then for any Y ∈ G we have k Ad( y b,j ) Y k ≤ C (1 + ǫ ) | j | ( | j | + 1) dim G k Y k , ∀ j ∈ Z , (4.12)where ǫ ≥ G = G ⋉ ρ V , first, we note that in the expression of y b,j (see (4.8)),Ad( g − ib ) v b = Ad( x − ig b , k − ig b n − ig b x − ig b , ) v b . Then by using (4.7) and noting that Ad( x g b , ) is sufficiently close to identity,we see that estimates in (4.12) still hold. Note that in (4.11) and (4.12) wecan take uniform C for all b sufficiently close to a .Also note that for any Y ∈ G Y π ( y b,j ) v = π ( y b,j ) (cid:0) Ad( y − b,j ) Y (cid:1) v. (4.13)Then it follows that δ j ≤ C (1 + ǫ ) | j | ( | j | + 1) dim G k v k G , . (4.14)Then the estimate for |h π ( b j ) v, u i| follows directly from (4.10), (4.11) and(4.14). OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 15
Proof of (2). It follows from previous result that ∞ X j = −∞ |h λ − ( j +1) π ( b j ) v j , u i| = ∞ X j = −∞ | λ | − ( j +1) · |h π ( b j ) v j , u i|≤ C ∞ X j = −∞ | λ | − ( j +1) (1 + ǫ ) | j | ( k v j kk u k + δ j k u k + δ k v j k + δ j δ ) e − γd ( e,x jb, ) ≤ C ∞ X j = −∞ | λ | − ( j +1) (1 + ǫ ) | j | ( | j | + 1) dim G k v j k G , k u k G ′ , e − Cγl ( b ) | j | ≤ C ∞ X j = −∞ | λ | − ( j +1) (1 + ǫ ) | j | ( | j | + 1) dim G P ( | j | ) k u k G ′ , e − Cγl ( b ) | j | < + ∞ This shows that D ( + − ) v ( b ) are distributions in π ′ . Proof of (3) We will show differentiability of f v by using both of its forms.First, we show differentiability of f v in X µi , µ ∈ ∆ ′′ ( b ).For any X µi , 1 ≤ i ≤ dim l µ , if µ >
1, we may use the D + v form to obtainthe following bound on s ’th derivative ∞ X j =0 ( X µi ) s (cid:0) λ − ( j +1) π ( b j ) h (cid:1) = ∞ X j =0 λ − ( j +1) µ − js π ( x jb, )( X µi ) s (cid:0) π ( y b,j ) h (cid:1) = ∞ X j =0 λ − ( j +1) µ − js π ( x jb, ) (cid:0) π ( y b,j )( Z j ) s h (cid:1) (4.15)where Z j = Ad( y − b,j )( X µi ) for all 1 ≤ s ≤ m .By using (4.12) we see that the left-hand side of (4.15) converges abso-lutely in H with estimates k ( X µi ) s D + v k ≤ ∞ X j =0 µ − js k λ − ( j +1) π ( x jb, ) (cid:0) π ( y b,j )( Z j ) s v j (cid:1) k = ∞ X j =0 µ − js | λ | − ( j +1) k ( Z j ) s v j k≤ ∞ X j =0 Cµ − js | λ | − ( j +1) (1 + ǫ ) | j | ( | j | + 1) dim G P s ( | j | ) k h k G ,s ≤ C s,P s k h k G ,s . (4.16)Similarly, if µ < f v = D − v , the estimates k ( X µi ) s D − v k ≤ C s,P s k h k G ,s . (4.17)holds if 1 ≤ s ≤ m . Before we show differentiability in other directions, we obtain tame esti-mates for X µi f v , µ ∈ ∆ ′′ ( b ) instead, which is important for that purpose.Fix X µi , µ ∈ ∆ ′′ ( b ), 1 ≤ i ≤ dim g µ . (4.16) and (4.17) show that k ( X µi ) f v k ≤ C P k h k G , . (4.18)If µ >
1, for any X χj with χ ∈ ∆ ′′ ( b ) and χ > χ ∈ ∆ ′ ( b ) \ ∆ ′′ ( b ), we have( X χj ) s ( X µi D + v ) = − ∞ X ℓ =0 λ − ( ℓ +1) ( χ s µ ) − ℓ π ( x ℓb, ) (cid:0) π ( y b,ℓ )( Z ℓ ) s Y ℓ v ℓ (cid:1) where Z ℓ = Ad( y − b,ℓ )( X χj ) and Y ℓ = Ad( y − b,ℓ )( X µi ) for all 1 ≤ s ≤ m − χ is succinctly close to 1 if χ ∈ ∆ ′ ( b ) \ ∆ ′′ ( b ), this together with(4.12) shows that the estimates k ( X χj ) s ( X µi D + v ) k ≤ C s,P s +1 k h k G ,s +1 . (4.19)holds if 1 ≤ s ≤ m − X χj with χ > χ ∈ ∆ ′ b \ ∆ ′′ ( b ).For any X χj , χ ∈ ∆ ′′ ( b ) with χ <
1, since b is sufficiently to a , similar to(4.9) we also have χ s µ < r ≤ s . Then we have( X χj ) s ( X µi D − v ) = −∞ X ℓ = − λ − ( ℓ +1) ( χ s µ ) − ℓ π ( x ℓb, ) (cid:0) π ( y b,ℓ )( Z ℓ ) s Y ℓ v ℓ (cid:1) , for r ≤ s ≤ m −
1, where Z ℓ = Ad( y − b,ℓ )( X χj ) and Y ℓ = Ad( y − b,ℓ )( X µi ).Together with (4.12) it follows that k ( X χj ) s ( X µi D − v ) k ≤ C s,P s +1 k h k G ,s +1 . (4.20)if r ≤ s ≤ m − X χj with χ ∈ ∆ ′′ ( b ) and χ < X µi D v ∈ H m − of π withestimates: k X µi f v k G ,s ≤ C s,P ,P s +2 k h k G ,s +2 . (4.21)for any r ≤ s ≤ m − µ <
1, by using the form D − v we see that (4.19) also holds if 1 ≤ s ≤ m − X χj with χ ∈ ∆ ′′ ( b ) and χ < χ ∈ ∆ ′ ( b ) \ ∆ ′′ ( b ). Furthermore,by using the form D + v we see that (4.20) also holds if 1 ≤ s ≤ m − X χj with χ ∈ ∆ ′′ ( b ) and χ >
1. Hence, (4.21) also follows for the case of µ < f v in other directions of g . Let Y j = p j ( X µ l (1) j (1) , · · · , X µ l ( i ) j ( i ) ), 1 ≤ j ≤ r ( a ) (see (4.6)). We note that Y sj f v = Y s − j p j ( X µ l (1) j (1) , · · · , X µ l ( i ) j ( i ) ) f v , and µ l ( k ) = 1 for any 1 ≤ k ≤ i . Then by using (4.21) we obtain: k Y sj f v k ≤ C s,P ,P s + m k h k G ,s + m +1 (4.22)for any 1 ≤ s ≤ m − m − OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 17
Using (4.16), (4.17) and (4.22) it follows from Theorem 3 . f v ∈ H m − m − in π ′ ; furthermore, Theorem 3.2 shows that k f v k G ′ ,s ≤ C s,P , ··· ,P s + m k h k G ,s + m +2 + C s k f v k . (4.23)If 1 ≤ s ≤ m − m − s = 1 we have k f v k G ′ , ≤ C P , ··· ,P m k h k G ,m +3 + C k f v k . (4.24)It it fairly clear what one has to do now. We show how to get the estimateof k f v k . By using (4.10) and (4.12) we have k f v k = (cid:12)(cid:12) + ∞ X j =0 λ − ( j +1) π ( b j ) v j , f v i (cid:12)(cid:12) ≤ + ∞ X j =0 (cid:12)(cid:12) h π ( b j ) v j , f v i (cid:12)(cid:12) ≤ C P k h k G , ( k f v k + k f v k G ′ , ) . Together with (4.24) we get k f v k ≤ C P , ··· ,P m k h k G ,m +3 . Combined with (4.23), we obtain k f v k G ′ ,s ≤ C s,P , ··· ,P s + m k h k G ,s + m +2 + C s,P , ··· ,P m k h k G ,m +3 if 1 ≤ s ≤ m − m −
2. Hence we finish the proof. (cid:3)
We are now in a position to proceed with the proof of Theorem 2.2.4.4.
Proof of Theorem 2.2.
Proof of (1). It clear that h ∈ H byassumption. From (2) of Lemma 4.4 we see that P + ∞ j = −∞ λ − ( j +1) π ( a j ) h is adistribution in π . By using h = π ( a ) f − λf we have + ∞ X j = −∞ λ − ( j +1) π ( a j ) h = + ∞ X j = −∞ λ − ( j +1) π ( a j ) (cid:0) π ( a ) f − λf (cid:1) (1) = + ∞ X j = −∞ λ − ( j +1) π ( a j +1 ) f − + ∞ X j = −∞ λ − j π ( a j ) f = 0 , where (1) follows from the fact that P + ∞ j = −∞ λ − ( j +1) π ( a j ) f is also a distri-bution in π . Then we finish the proof. Proof of (2). The assumption that P + ∞ j = −∞ λ − ( j +1) π ( b j ) h = 0 as a dis-tribution in π implies that P + ∞ j = −∞ λ − ( j +1) π ( b j ) h = 0 as a distribution in π ′ (since H ∞ ( π ) ⊂ H ∞ ( π ′ )). Then the equation ρ ( b ) f − λf = h has twodistributional solutions f = D ( + − ) h = (cid:18) − + (cid:19) X ( j ≥ j ≤− ) λ − ( j +1) π ( b j ) h in π ′ by (2) of Lemma 4.4.From (3) of Lemma 4.4 we see that f ∈ H m − m − of π ′ with the followingestimates k f k G ′ ,m − m − ≤ C m k h k G ,m . (4.25)Next, we will show how to obtain differentiability along directions outside g . For any X µi , µ ∈ ∆ ′′′ ( b ) (see (4.5)), 1 ≤ i ≤ dim g µ and 1 ≤ s ≤ m − Y j = Ad( y − b,j )( X µi ), j ∈ Z . From (4.12) we have k ( Y j ) s h k G ,t ≤ C (1 + ǫ ) | j | ( | j | + 1) s dim G k h k G ,s + t . all 0 ≤ s + t ≤ m − X µi ) s f = − ∞ X j =0 λ − ( λµ s ) − j π ( b j )(( Y j ) s h ) = −∞ X j = − λ − ( λµ s ) − j π ( b j )(( Y j ) s h )are distributions in π ′ for any 0 ≤ s ≤ m − µ is sufficiently close to 1,i.e., b is sufficiently close to a . Furthermore, (3) of Lemma 4.4 implies that( X µi ) s f ∈ H with the estimate k ( X µi ) s f k ≤ C a,s k h k s + m +3 . for any s ≤ m − m − f ∈ H m − m − with the estimate k f k G ,m − m − ≤ C a,m k h k m from Theorem 3.2. Hence we finish the proof. Proof of (3). We just need to show that if D ( h ) = 0 for any ( ρ ( a ) − λI )-invariant distribution D ∈ H − , then P + ∞ j = −∞ λ − ( j +1) π ( a j ) h = 0 as adistribution. Note that for any f ∈ H ∞ , D f := P + ∞ j = −∞ ¯ λ − ( j +1) π ( a − j ) f ∈H − by (2) of Lemma 4.4. Furthermore, for any g ∈ H ∞ X j = −∞ (cid:10) π ( a ) g − λg, ¯ λ − ( j +1) π ( a − j ) f (cid:11) = + ∞ X j = −∞ (cid:10) λ − ( j +1) π ( a j +1 ) g − λ − j π ( a j ) g, f (cid:11) = 0 . This shows that D f ∈ H − is ( ρ ( a ) − λI )-invariant. On the other hand, if D f ( h ) = 0 for any f ∈ H ∞ , then0 = + ∞ X j = −∞ (cid:10) h, ¯ λ − ( j +1) π ( a − j ) f (cid:11) = + ∞ X j = −∞ (cid:10) λ − ( j +1) π ( a − ( j +1) ) h, f (cid:11) . This shows that P + ∞ j = −∞ λ − ( j +1) π ( a j ) h = 0 as a distribution. This provesthe result. OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 19
Cohomological equation for partially hyperbolic flow.
Recallthat for a C cocycle β , if A = R k the infinitesimal generator of β is definedby ϑ ( ν, β ) = ddt β (exp tν ) (cid:12)(cid:12) t =0 (4.26)The cocycle identity and commutativity of A imply that ϑ is a closed 1-formon the A -orbits in X . We can also recover β from ϑ by β (exp X ) = Z ϑ ( X, β ) · exp tXdt Thus, if A = R k we can restrict our attention to infinitesimal version of thecohomology equation ϑ = η − dH , where η is another infinitesimal generatorof a smooth cocycle and H is the transfer function. Therefore a cocycle β is cohomologous to a constant cocycle if the associated 1-form ϑ is exactand the problem of finding which cocycle is cohomologous to a trivial oneboils down to the problem of determining which closed 1-form on the orbitfoliation is exact. In fact, this point of view is the most useful for ourpurposes.For the cohomological equation v f = h where v ∈ G is a partially hy-perbolic element, i.e., the spectrum of Ad(exp( t v )) on G is not contained in U (1), we get results similar to the discrete-action cases. Corollary 4.6.
Suppose ( π, H ) is a unitary representation of G such thatthe restriction of π to any simple factor of G is isolated from the trivialrepresentation (in the Fell topology) if G = G ; or π contains no non-trivial V -fixed vectors if G = G ⋉ ρ V . If v ∈ G is partially hyperbolic, then: (1) if f ∈ H is a solution of the equation: v f = h , then Z ∞−∞ π (exp( t v )) hdt = 0 as a distribution. (2) there exist constants m > σ > (only depending on G ) such thatfor any m ≥ m , exist δ ( m ) > , such that for any u ∈ G with k v − u k < δ , if h ∈ H m satisfying R ∞−∞ π (exp( t u )) hdt = 0 as adistribution, then the equation u f = h have a solution f ∈ H m − σ and the following estimate holds k f k m − σ ≤ C m k h k m . (3) if h ∈ H ∞ and D ( h ) = 0 for any v -invariant distribution D ∈ H − ,then the cohomological equation v f = h , has a solution f ∈ H ∞ .Proof. If G = G , for any u ∈ G , it has the Iwasawa decomposition u = k u + x u + n u for 3 commuting elements, where k u is compact, x u is in a R -split Cartanalgebra, n u is nilpotent.For partially hyperbolic u , x u is non-trivial. If G = G ⋉ ρ V for any u ∈ G ,we have the decomposition u = g u + v u , where g u ∈ Lie( G ) and v u ∈ V . Forpartially hyperbolic u , x g u is non-trivial.For partially hyperbolic u , similar to (4.4) we consider the eigenspacedecomposition of G for Ad(exp( x u )) or Ad(exp( x g u )): G = X µ ∈ ∆( u ) g µ . If G = G ⋉ ρ V , for u = g u + v u , we have a decomposition of v u = P µ ∈ ∆( a ) v u ,µ ,where v u ,µ ∈ g µ T V . Set u = P = µ ∈ ∆( u ) v u ,µ . We note that dρ ( g u ) re-stricted on the subspace W = P = µ ∈ ∆( a ) g µ T V is invertible. Set v =( dρ ( g u )) − ( P = µ ∈ ∆( u ) v u ,µ ). By using (2.5) we get v exp( u ) v − = (cid:0) g u , v u , (cid:1) . Hence we can just assume that v u ∈ g . Then we can write u = x u + ( k u + n u + v u )as 2 commuting elements. Set y u = k u + n u if G = G or y u = k u + n u + v u if G = G ⋉ ρ V . Then we haveexp( t u ) = exp( tx u ) · exp( ty u ) , ∀ t ∈ R . We note that the formal solutions to the equation u f = h are: f ( + − ) = (cid:18) − + (cid:19) Z ( t ≥ t ≤ ) π (exp( t u )) hdt. Then we can follow the proof scheme of Lemma 4.4 and Theorem 2.2 toobtain the results. (cid:3) Twisted cocycle rigidity
Higher rank trick and trivialization of cohomology.
Now wewill show that in the higher rank case obstructions to solving the cocycleequation: π ( a ) f − λ f = π ( b ) h − λ h (5.1)vanish. Here we assume that λ , λ ∈ U (1) and a and b are commut-ing partially hyperbolic elements. The reason for that is the commutationrelation (5.1) means that the pair f, h form a twisted cocycle over the ho-mogeneous action generated by a and b . Joint solvability of the cocycleequations for commuting elements means that this cocycles is a coboundary,hence corresponding twisted first cohomology is trivial. The key ingredient OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 21 in the proof is the “higher rank trick” that proves vanishing of the obstruc-tions in both equations: π ( a ) p − λ p = hπ ( b ) p − λ p = f if the pair pair f, h satisfy condition (5.1). It appears in virtually identicalform in all proofs of cocycle and differentiable rigidity for actions of higherrank abelian groups that use some form of dual, i.e. harmonic analysisarguments. For its earliest appearance see Lemmas 4.3, 4.6 and 4.7 in [18].Recall notations in Section 4.1. Lemma 5.1. If ab = ba in G , then: (1) if G = G , then a m b j = x ma x jb k ma n ma k jb n jb , for any m, j ∈ Z . (2) if G = G ⋉ ρ V and v a is in the -weight space for Ad ( x g a ) , then v b is also in the -weight space for Ad ( x g b ) ; and a m b j = x mg a x jg b ( k g a n g a , v a ) m ( k g b n g b , v b ) n , ∀ m, j ∈ Z . (3) suppose x a and x b (resp. x g a and x g b ) are linear independent and λ , λ ∈ U (1) . Also suppose h = { h n,j } , n, j ∈ Z is a sequence in H ( π ) satisfying k h n,j k G , ≤ P ( | j | , | n | ) , where P a polynomial, then ∞ X n = −∞ ∞ X j = −∞ λ − ( j +1)1 λ − ( n +1)2 π ( b n a j ) h n,j (5.2) is a distributionProof. Proof of (1): Let c stand for a or b . The adjoint map Ad( c ) de-composes as Ad( c ) = Ad( k c )Ad( x c )Ad( n c ) , where Ad( k c ) is compact, Ad( x c ) is in a R -semisimple, Ad( n c ) is unipotent.We claim that Ad( x a ) commute with Ad( k b ), Ad( x b ) and Ad( n b ). Indeed,we can consider the decomposition of G into generalized eigenspaces for theaction of Ad( a ). By assumption Ad( x a ) is a scalar multiple of identity oneach generalized eigenspace of Ad( a ). Since b commutes with a , Ad( x b )preserves each generalized eigenspace of Ad( a ) and acts by a scalar multiplein each, which implies the claim.It is clear that x a n b = n b x a , x a x b = x b x a and x a k b = k b x a z , where z ∈ Z ( G ), the center of G . Their exists l ∈ N such that k b x la k − b = x la bynoting that Z ( G ) is finite. This shows that Ad( k b ) log x a = log x a . Then itfollows that x a and k b commute. Then we finish the proof. Proof of (2): By Proposition 4 . s ∈ G such that scs − = ( g c , v ′ c ) , c = a, or b where v ′ c is in the 1-weight spaces both for Ad( x g a ) and Ad( x g b ). This andprevious result imply the conclusion immediately. Proof of (3): If G = G it follows from (1) that d ( b n a j , e ) ≥ C X µ ∈ ∆( b n a j ) | log µ | , ∀ j, n ∈ Z . For any j, n ∈ Z we note that X µ ∈ ∆( b n a j ) | log µ | = X µ l ∈ ∆( a ) ,λ l ∈ ∆( b ) | j log µ l + n log λ l | where µ l and λ l are eigenvalues of Ad( x a ) and Ad( x b ) on G respectively forthe same eigenvectors. Write X µ l ∈ ∆( a ) ,λ l ∈ ∆( b ) | j log µ l + n log λ l | = ( | j | + | n | ) X µ l ∈ ∆( a ) ,λ l ∈ ∆( b ) | j log µ l + j log λ l | where j = j | j | + | n | and j = n | j | + | n | .Since x a and x b are linearly independent elements, c = min | r | + | r | =1( r ,r ) ∈ R X µ l ∈ ∆( a ) ,λ l ∈ ∆( b ) | r log µ l + r log λ l | > . Hence we have d ( b n a j , e ) ≥ Cc ( | j | + | n | ) . If G = G , for any u ∈ H ∞ by Theorem 3.6, we have ∞ X n = −∞ ∞ X j = −∞ (cid:12)(cid:12) h λ − ( j +1)1 λ − ( n +1)2 π ( b n a j ) h n,j , u i (cid:12)(cid:12) = ∞ X n = −∞ ∞ X j = −∞ |h π ( b n a j ) h n,j , u i|≤ ∞ X n = −∞ ∞ X j = −∞ C k h n,j k k u k e − γc ( | n | + | j | ) ≤ ∞ X n = −∞ ∞ X j = −∞ CP ( | j | , | n | ) k u k e − γc ( | n | + | j | ) < ∞ , where γ is a constant only dependent on G . This shows that (5.3) is adistribution.If G = G ⋉ ρ V , denote by y c = ( k g c n g c , v c ); and for simplicity, denote x g c by x c , where c stands for a or b . By arguments at the beginning of the proof OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 23 of Lemma 4.4 and Theorem 3.6 for any u ∈ H ∞ we have |h π ( b n a j ) h, u i| (1) = (cid:12)(cid:12) h π ( x nb x ja )( π ( y nb y ja ) h n,j ) , u i (cid:12)(cid:12) ≤ C (cid:0) k π ( y nb y ja ) h n,j kk u k + k u kk π ( y nb y ja ) h n,j k + k π ( y nb y ja ) h n,j kk u k + k u k k π ( y nb y ja ) h n,j k (cid:1) e − γ d ( x nb x ja ,e ) = C (cid:0) k h n,j kk u k + k u kk π ( y nb y ja ) h n,j k + k h kk u k + k u k k π ( y nb y ja ) h n,j k (cid:1) e − γd ( x nb x ja ,e ) , where γ is a constant only dependent on G . Here in (1) we used (2) ofLemma 5.1.By using (2.5) it follows that for any Y ∈ G we have k Ad( y nb y ja ) Y k ≤ C ( | j | + 1) dim G ( | n | + 1) dim G k Y k , ∀ j ∈ Z . Also note that for any Y ∈ G Y π ( y nb y ja ) h n,j = π ( y nb y ja ) (cid:0) Ad( y − ja y − nb ) Y (cid:1) h n,j . Hence we get k π ( y nb y ja ) h n,j k ≤ C ( | j | + 1) dim G ( | n | + 1) dim G k h n,j k . Then the estimate for |h π ( a j ) v, u i| follows directly from (4.10), (4.11) and(4.14).By arguments similar to the case of G = G , there is c > d ( x nb x ja , e ) ≥ c ( | j | + | n | ) . Then it follows that for any n, j ∈ Z |h π ( b n a j ) h, u i| ≤ C ( | j | + 1) dim G ( | n | + 1) dim G (cid:0) k h n,j kk u k + k u kk h n,j k + k h n,j kk u k + k u k k h k (cid:1) e − cγ ( | j | + | n | ) ≤ C ( | j | + 1) dim G ( | n | + 1) dim G P ( | j | , | n | ) k u k e − cγ ( | j | + | n | ) The above estimates imply that (5.3) is a distribution. Hence we finish theproof. (cid:3)
To prove Theorem 2.3, we also need the following result:
Fact 5.2. If b ∈ G is partially hyperbolic and λ ∈ U (1) and the equation π ( b ) f − λf = 0 has a solution f ∈ H ( π ) then f = 0. Proof.
Since π ( b j ) f = λ j f for any j ∈ Z , by using estimates obtained in (1)of Lemma 4.4, for any u ∈ H ( π ) we get h f, u i = lim j →∞ h λ − j π ( b j ) f, u i = 0Then f = 0 follows immediately. (cid:3) Proof of Theorem 2.3.
From the equation (5.1) we get j = n X j = − n λ − ( j +1)1 π ( b a j ) h − j = n X j = − n λ λ − ( j +1)1 π ( a j ) h = λ − ( n +1)1 π ( a n +11 ) f − λ n π ( a − n ) f. By (1) of Lemma 4.4, the right-hand converges to 0 as a distribution (in π )when n → ∞ . Hence we see that λ −
12 + ∞ X j = −∞ λ − ( j +1)1 π ( b a j ) h = + ∞ X j = −∞ λ − ( j +1)1 π ( a j ) h (5.3)as distributions. This shows that λ − k ∞ X j = −∞ λ − ( j +1)1 π ( b k a j ) h = + ∞ X j = −∞ λ − ( j +1)1 π ( a j ) h as distributions for any k ∈ Z .On the other hand, by iterating equation (5.3) we obtain: + ∞ X n = −∞ + ∞ X j = −∞ λ − ( n +1)2 λ − ( j +1)1 π ( b n a j ) h = + ∞ X n = −∞ λ −
12 + ∞ X j = −∞ λ − n λ − ( j +1)1 π ( b n a j ) h = + ∞ X n = −∞ λ −
12 + ∞ X j = −∞ λ − ( j +1)1 π ( a j ) h Since the series in the left hand side of above equation is a distribution by(3) of Lemma 5.1, it forces P + ∞ j = −∞ λ − ( j +1)1 π ( a j ) h to be a 0 distribution.This is the “higher rank trick”!By (2) of Theorem 2.2 each equation of π ( a ) p − λ p = hπ ( b ) p − λ p = f (5.4)has a H m − σ solution. Moreover, we will show that they coincide. In thefollowing proof, to simply notation, for any g ∈ G and λ ∈ C , we define thelinear operator F ( g, λ ) on H : F ( g, λ ) v = π ( g ) v − λv, ∀ v ∈ H . If p solves the first equation, i.e. F ( a , λ ) p = h then by equation (5.1) wehave F ( b , λ ) ◦ F ( a , λ ) p = F ( b , λ ) h = F ( a , λ ) f Since operators F ( a , λ ) and F ( b , λ ) commute this implies F ( a , λ ) (cid:0) F ( b , λ ) p − f (cid:1) = 0 OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 25
By Fact 5.2 F ( b , λ ) p = f by noting that p ∈ H , which is implied byassumption m ≥ m + σ and m >
1. Therefore π ( b ) p − λ p = f i.e., p solves the second equation as well.5.3. Cocycle rigidity for partially hyperbolic flow.
Similar to Fact5.2, we have the following result for partially hyperbolic flow:
Fact 5.3. If X ∈ G is partially hyperbolic and the equation Xf = 0 has asolution f ∈ H ( π ) then f = 0. Proof.
The assumption Xf = 0 implies that f is invariant under π (exp( tX )),i.e., π (exp( tX )) f = f for any t ∈ R . Then the result follows directly fromFact 5.2 immediately. (cid:3) Then by arguments in the proof of Corollary 4.6, we can follow the proofline of Lemma 5.1 and Theorem 2.3 to obtain the cocycle rigidity for R partially hyperbolic actions.Suppose X, Y ∈ G are commuting partially hyperbolic and linearly inde-pendent and m , σ as in Theorem 2.2. Corollary 5.4.
Suppose ( π, H ) is a unitary representation of G such thatthe restriction of π to any simple factor of G is isolated from the trivialrepresentation if G = G ; or π contains no non-trivial V -fixed vectors if G = G ⋉ ρ V . For any m ≥ m + σ , exist δ ( m ) > , such that for any X , Y ∈ G with k X − X k + k Y − Y k < δ and X Y = Y X , if f, h ∈ H m and satisfy the cocycle equation X f = Y h, then the equations Y p = f, X p = h have a common solution p ∈ H m − σ satisfying the Sobolev estimate k p k m − σ ≤ C m,X,Y max {k h k m , k f k m } . Application to algebraic partially hyperbolic actions
Recall that G denotes a real semisimple connected Lie group of R -rank ≥ G . Definition 6.1.
Coarse Lyapunov distributions are defined as minimal non-trivial intersections of stable distributions of various action elements.In the setting of present paper those are homogeneous distributions ortheir perturbations, that integrate to homogeneous foliations called coarseLyapunov foliations ; see [4, Section 2] and [14] for detailed discussion ingreater generality.
Symmetric space examples.
For any abelian set A ⊂ G there existsa R -split Cartan subgroup A such that x a ∈ A (see (4.3)) for any a ∈ A (see Proposition 4.2 of [28]). Set A ′ = { x a , a ∈ A } . Then A ′ is a subgroupof A . We consider the decomposition of g with respect to the adjointrepresentation of A ′ and the resulting root system ∆ A is called the restrictedroot system with respect to A . Then we get the decomposition of the Liealgebra g of G : g = g + X µ ∈ ∆ A g µ (6.1)where g µ is the root space of µ and g is the Lie algebra of the centralizer C G ( p ( A )) of A ′ .Elements of { a ∈ A : x a / ∈ S φ ∈ ∆ A ker( φ ) } are regular elements for α A .Connected components of the set of regular elements are Weyl chambers . Forany µ ∈ ∆ A let g ( µ ) = P k> g kµ and U [ µ ] be the corresponding subgroupof G . Then these subalgebra g ( µ ) form coarse Lyapunov distributions and(double) cosets of these subgroups U [ µ ] form coarse Lyapunov foliations of α A , which coincide with those of α p ( A ) (see Proposition 4.2 of [28]).If A ′ = A , the left translations of A on G/ Γ is sometimes referred to as full Cartan action (see [18]). If the coarse Lyapunov foliations of α A coincidewith those of α A , then A ′ is in a generic position (see [4]) and the action of A on G/ Γ is called a generic restriction . Remark 6.2. If A ′ = A , i.e., ∆ A is the standard root system. Calling ∆ A a restricted root system is somewhat abusive. Indeed, ∆ A does not carrythe usual structures of a (reduced) root system, such as a canonical innerproduct and associated Weyl group. For more details, see Section 1.1 of [28].6.2. Twisted symmetric space examples.
Let ρ : Γ → SL ( N, Z ) bea representation of Γ which admits no invariant subspace on which Γ actstrivially. We also assume that the Zariski closure of ρ (Γ) has no compactfactors.Then Γ acts on the N -torus T N via ρ and hence on G × T N via γ ( g, t ) = ( gγ − , ρ ( γ ) t ) . Let M = G × T N / Γ be the quotient of this action. A acts on the product G × T N given by a ( g, t ) = ( ag, t ) and since, the action of A and Γ commuteit induces an action of A on M , which is the suspension of Γ-action on T N .We can assume G has the following property: every Lie algebra homo-morphism g → sl ( N, R ) is the derivative of a Lie group homomorphism G → SL ( N, R ). Otherwise we pass to some finite cover of G .By Margulis’ superrigidity theorem [23], semisimplicity of the algebraichull H of ρ (Γ) and the non-compactness of ρ (Γ) the representation ρ of Γextends to a rational homomorphism G → H ad over R where H ad is theadjoint group of H . Note that ρ (Γ) has finite center Z (which follows, eg,from Margulis’ finiteness theorem [23]), then G acts on the orbifold R N /Z OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 27 via ρ , which can be lifted to a representation of G on R N , which we denoteby ˆ ρ . Then ρ ( γ )ˆ ρ ( γ ) − ∈ Z for any γ ∈ Γ. Then, by passing to a finite indexsubgroup Γ of Γ, we get ρ ( γ ) = ˆ ρ ( γ ) for any γ ∈ Γ . Then G × T N / Γ is a finite cover of M . For simplicity, we still use ρ to denote the liftedrepresentation of G on R N .We can build the associated semi-direct product G ρ = G ⋉ ρ R N . Themultiplication of elements in G ρ is given by (2.4), which shows that G × T N / Γ = G ρ / Γ ⋉ Z N . The A action on G × T N / Γ is isomorphic to theaction of A as a homogeneous flow on G ρ / Γ ⋉ Z N . We can view G × T N / Γ as a torus bundle over G/ Γ . Remark 6.3.
Firstly, if the Zariski closure of ρ (Γ) has compact factors, wecan pass to a suspension space (see [29]).Secondly, passing to finite covers of the homogeneous actions will notaffect the local rigidity results. Indeed, the constriction of the transfer maprelies on vanishing of the obstructions (see Section 6.6). If the obstructionsvanishes on a finite cover, then it also vanishes on the original space (seeLemma 6.9). This allows us to assume that Γ = Γ. We denote by Γ ρ =Γ ⋉ Z N .For any abelian set A ⊂ G ρ and set A ′ = { x g a : a ∈ A } (see below (2.4)),there exists an element s ∈ G ρ such that the coarse Lyapunov distributionsfor the action of α sAs − is the same as those for α A ′ (see Proposition 4.4 of[28]). Let Φ A denote the restricted weights of G with respect to A ′ . Thenthe Lie algebra g ρ of G ρ decomposes g ρ = g + X r ∈ ∆ A g r + X µ ∈ Φ A e µ where e µ is the weight space of µ and g is the Lie algebra of the centralizer C G ρ ( A ′ ) of A ′ in g ρ .Note that if A ′ = A , Φ A is the standard weights. For any r ∈ ∆ A ∪ Φ A let g ( r ) = P k ∈ R + ( g kr + e kr ) and U [ r ] be the corresponding subgroupof G ρ . Note that r or µ may appear in the set of both restricted rootsand weights. These subalgebra g ( r ) form coarse Lyapunov distributions and(double) cosets of these subgroups U [ r ] form coarse Lyapunov foliations of α A , which coincide with those of α A ′ (see Proposition 4.2 of [28]).Elements of { a ∈ A : x g a / ∈ S φ ∈ ∆ A ∪ Φ A \{ } ker( φ ) } are regular for α A and connected components of the set of regular elements Weyl chambers . If A = A , the left translations of A on G ρ / Γ ρ is sometimes referred to as fullCartan action .Similarly to the symmetric space setting we will consider actions of higherrank subgroups of A by left translations on double coset space L \ G ρ / Γ ρ where L is a compact subgroup commuting with A .6.3. Higher rank restrictions and standard perturbations.
Let X bea double coset space L \ G/ Γ as in symmetric space examples or L \ G ρ / Γ ρ as in twisted symmetric space examples; and let ¯ X be a coset space G/ Γas in symmetric space examples or G ρ / Γ ρ as in twisted symmetric spaceexamples. We consider the action α A on both X and ¯ X .Since A is the image of an embedding i : Z k × R ℓ → A , one can naturallyconsider the action α i of A = Z k × R l on X (or on ¯ X ) given by α i ( a, x ) = i ( a ) · x (6.2)Then we will say that A action α i generates A action α A since α i is α A with a fixed system of coordinates. Note that A can be obtained as theimage of different embeddings; corresponding actions of A differ by a timechange. It is immediately obvious that if α i is cocycle rigid then the sameis true for any time change obtained by an automorphism of A ; hence thenotion of cocycle rigidity for α A depends only on the subgroup A . Definition 6.4. α A is called a higher-rank partially action, if the set ofcoarse Lyapunov distributions of α A is not generated by a rank-one sub-group.It is clear that if α A is higher-rank, then we can choose regular elements a and b of A such that x a and x b (resp. x g a and x g b ) are linearly independent. a and b will be referred to as regular generators .Let L be the Lie algebra of the group L . Let N and N denote the neutraldistributions of α A on X and on ¯ X respectively (neutral distribution is thesubspace spanned by Lyapunov distributions with 0 Lyapunov exponents).Then: • For the symmetric space examples N is L \ N where N = g ; • for the twisted symmetric space examples N is L \ N where N = g + e . Remark 6.5.
Notice that the neutral distribution for α A coincides with thehomogeneous distribution into cosets of the centralizer of A , or its factor by L in the case of actions on double coset spaces.6.4. Exponential matrix coefficients decay on homogeneous space.
Here we review some relevant facts concerning the preservation of spectralgaps when restricting to the subgroups in the example we consider.
Theorem 6.6.
Let S = S × · · · × S k be a product of noncompact simpleLie groups with finite center, Γ an irreducible lattice in S , and let ρ standfor the regular representation of S on the subspace of L ( S/ Γ) orthogonal toconstant functions. Then the restriction of ρ to any simple factor of S isisolated (in the Fell topology) from the trivial representation. If k = 1 and the rank of S is at least 2 then S has property T and theresult follows directly from [2]. If k = 1 and the rank of S is 1, the spectralgap is already known. If k ≥
2, in the case of nonuniform Γ, this was provedby Kleinbock and Margulis and appeared as Theorem 1.12 in [22]. L. Clozel[1] extended this result to congruence lattices discussed in [1]. By Margulis
OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 29 [23], Γ is arithmetic and hence is commensurable with a congruence latticeof the type in [1], the result holds for all lattices.We assume notations in Section 3.1 for the regular representation ( π, H )of G or G ρ , where H = L ( G/ Γ) or L ( G ρ / Γ ρ ) correspondingly. It is clearthat given any a ∈ G (resp. a ∈ G ρ )( π ( a ) f )( x ) = f ( a − x ) , ∀ x ∈ X, f ∈ H . For simplicity, we denote π ( a ) f by f ( a − ) or by f ◦ a − . Set H to be vectorsin H orthogonal to constants. The following result is essential for later part: Corollary 6.7.
There exist constants γ, E > , dependent only on G , Γ and ρ such that if h, f ∈ H , i = 1 , orthogonal to the constants in H thenfor any a ∈ G : |h h ( a ) , f i| ≤ E ( k h kk f k + k h k k f k + k f k k h k + k h k k f k ) e − γ dist( e,a ) (6.3) where h· , ·i denotes the inner product in H with respect to the Haar measure.Proof. Let G ′ denote the direct product of simple factors of G and let p bethe projection of G ′ to G . Then G ′ /p − (Γ) is isomorphic to G/ Γ. Since G has finite center then p − (Γ) is also an irreducible lattice. This impliesthat (6.3) follows directly from Theorem 6.6 and Theorem 3.6 for vectors in H = L ( G/ Γ) orthogonal to constants.For H = L ( G ρ / Γ ρ ), we note that G ρ / Γ ρ can be viewed as a torus bundleover G/ Γ. Set V = { f ( g, t ) ∈ H : f ( g, t ) = R T N f ( g, t ) dt } . Then it is clearthat V is the set of R N -invariant vectors in H . For any u ∈ H , write u ( g, t ) = u ( g, t ) − Z T N u ( g, t ) dt | {z } u o ( g,t ) + Z T N u ( g, t ) dt | {z } u ( g ) . (6.4)Then f ∈ V ⊥ and f ∈ V . Note that both V ⊥ and V are closed andinvariant under G ρ . Hence we get a direct decomposition of H invariantunder G ⋉ R N . Then we have h h ( a ) , f i = h h o ( a ) , f o i + h h ( a ) , f i . It follows from Theorem 1 . π to any simplefactor of G is isolated from the trivial representation. Then Theorem 3.6implies that |h h o ( a ) , f o i| ≤ E ( k h o kk f o k + k h o k k f o k + k f o k k h o k + k h o k k f o k ) e − γ dist ( e,a ) ≤ E ( k h kk f k + k h k k f k + k f k k h k + k h k k f k ) e − γ dist ( e,a ) , (6.5)where γ , E >
0, dependent only on G and ¯ X . Note that f , h can be viewed as functions in L ( G/ Γ) orthogonal toconstants. The arguments at the beginning of the proof show that |h h ( a ) , f i| ≤ E ( k h kk f k + k h k k f k + k f k k h k + k h k k f k ) e − γ dist ( e,a ) ≤ E ( k h kk f k + k h k k f k + k f k k h k + k h k k f k ) e − γ dist ( e,a ) , (6.6)where γ , E >
0, dependent only on G and X .Hence (6.3) follows from (6.5) and (6.6) immediately. (cid:3) Remark 6.8.
Corollary 6.7 shows that for the restricted regular represen-tation ( π, H ) of G or G ρ , its restriction to any simple factor of G is isolatedfrom the trivial representation.6.5. H s and H s space on X . Fix a basis { Y i } ≤ i ≤ q of L \ g or L \ g ρ . Let H mX to be the subspace of L ( X ) such that f and Y ji ( f ), 1 ≤ j ≤ m exist as L functions for 1 ≤ i ≤ q . Define k f k m def = ( q X i =1 m X j =1 k Y ji f k + k f k ) / . Let H r ,X def = { f ∈ H rX | R X f = 0 } . We use subscripts to emphasize that thespaces H mX and H r ,X are different from the Hilbert spaces H m and H r forthe regular representation π .6.6. Twisted cohomological stability for homogeneous space.
For amap F with coordinate functions f i , 1 ≤ i ≤ n and −∞ ≤ s ≤ ∞ , we write F ∈ H sX if f i ∈ H sX , 1 ≤ i ≤ n ; and define kF k s = max ≤ i ≤ n k f i k s . F ∈ H s ,X is defined similarly. For two maps F , G define kF , Gk s =max {kF k s , kGk s } . Write R X F = ( R X f dµ, · · · , R X f n dµ ) where µ is theHaar measure.In this part, we show the solvability condition for the existence of a so-lution to equation (6.7). This argument is essentially the reduction of thevector values equation (6.7) and solvability condition in (2) to scalar equa-tions. After showing the vanishing of obstructions, the tame estimates forthe solution of equation (6.7) follow from Theorem 2.2.We use G to denote G or G ρ . For any partially hyperbolic element z ∈ G ,we use N z to denote the neutral distribution of Ad( z ). Lemma 6.9.
For any partially hyperbolic element z ∈ G : (1) If F : X → N z and F ∈ H ,X , then Λ( + − ) = (cid:18) − + (cid:19) X ( j ≥ j ≤− ) Ad( z ) − ( j +1) ( F ◦ z j ) are distributions. OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 31 (2)
There exist constants m > σ > (only depending on G ) such thatfor any l ≥ m , if F : X → N z , F ∈ H l ,X and + ∞ X j = −∞ Ad( z ) − ( j +1) ( F ◦ z j ) = 0 as a distribution, then the equation Λ ◦ z − Ad( z )Λ = F (6.7) have a solution Λ ∈ H l − σ ,X with estimates k Λ k l − σ ≤ C l,z kF k l . (3) If F : X → N z and the equation Λ ◦ z − Ad( z )Λ = F has a solution Λ ∈ H ,X , then Λ is unique. (4) There exist constants m > σ > (only depending on G ) such thatfor any l ≥ m , there exists δ ( l ) > , such that for any b ∈ G with d ( z, b ) < δ , if F : X → N z with Ad( b ) neutral and invariant on N z and F ∈ H l ,X satisfying + ∞ X j = −∞ Ad( b ) − ( j +1) ( F ◦ z j ) = 0 as a distribution, then the equation Λ ◦ b − Ad( b )Λ = F (6.8) has a solution Λ ∈ H l − σ ,X and the following estimate holds for Λ : k Λ k l − σ ≤ C l,z kF k l . Proof.
Lift F from X to ¯ X , which we denote by ˜ F . Then ˜ F ∈ H l if F ∈ H l ,X for any l .(1): It is clear that if we can show Λ( + − ) are distributions for ˜ F , then Λ( + − )are also distributions for F . Let ( a i,jn ) denote the matrix of Ad( z ) n on N z .Since the eigenvalues of Ad( z ) | N z are all in U (1), we have k ( a i,jn ) k ≤ C z ( | n | + 1) dim N z , ∀ n ∈ Z . (6.9)Denote by Λ i − and Λ i + the i -th coordinates of Λ − and Λ + respectively. Wehave Λ i ( + − ) = (cid:18) − + (cid:19) X ( j ≥ j ≤− ) dim N z X k =1 a i,k − ( j +1) f k ◦ z j Then the conclusion follows directly from (2) of Lemma 4.4.(2): If we can show the solution of lifted equation 6.7 on ¯ X is left- L invariantwith appropriate estimates (which is obvious from the expression of Λ( + − ) since ˜ F is left L invariant ) then it descends to a map in H ,X on X , whichimplies equation 6.7 has a solution on X with desired estimates.Let N C be the complexification of the subalgebra N z . There exists a basisin N C such that in the basis Ad( z ) | N z has its Jordan normal form. As usual,this basis may be chosen to consists of several real vectors and several pairsof complex conjugate vectors. Let J = ( q i,j ) be an m × m matrix whichconsists of blocks of Ad( z ) | N z corresponding to the eigenvalue λ ; i.e., let q i,i = λ for all 1 ≤ i ≤ m and q i,i +1 = 0 for all 1 ≤ i ≤ m − q i,i +1 = 1for all 1 ≤ i ≤ m − J the equation (6.7) has the form:Λ ◦ z − J Λ = Θ(6.10)and solvability condition splits as + ∞ X j = −∞ J − ( j +1) Θ ◦ z j = 0(6.11)in this block.(1) The semisimple case . Assume that J is diagonalizable, i.e., q i,i +1 = 0for all 1 ≤ i ≤ m −
1. Then equations (6.10) and condition (6.11) split intofinitely many equations of the form ω ◦ z − λω = ϕ (6.12)and + ∞ X j = −∞ λ − ( j +1) ϕ ◦ z j = 0(6.13)where ϕ is a H l function and λ ∈ U (1) is the corresponding eigenvalue ofAd( z ) | N z . Then the conclusion follows directly from Theorem 2.2.(2) The non-semisimple case . Assume that q i,i +1 = 1 for all 1 ≤ i ≤ m − + − ) = (cid:18) − + (cid:19) X ( j ≥ j ≤− ) J − ( j +1) Θ ◦ z j (6.14)are in fact H l − r − solutions. Let the coordinate functions of Θ be ϑ i ,1 ≤ i ≤ m . The m -th equation of (6.10) becomes: ω m ◦ z − λω m = ϑ m (6.15)and the condition (6.11) splits as + ∞ X j = −∞ λ − ( j +1) ϑ m ◦ z j = 0 . OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 33
Then the existence of a solution follows from Theorem 2.2. Moreover, theestimate: k ω m k l − σ ≤ C l k ϑ m k l ≤ C l k Θ k m holds.Now we proceed by induction. Fix i between 1 and m − ≤ k ≤ m − i , we have obtained a solution ω i + k with the appropriateestimate, i.e., for every 1 ≤ k ≤ m − i we have a H l − ( m − i − k +1) σ function ω k + i which solves the k + i -th equation: ω k + i ◦ z − λω k + i = ϑ k + i + ω k + i +1 and that the following estimates k ω k + i k l − ( m − i − k +1) σ ≤ C l k ϑ k + i + ω k + i +1 k l − ( m − i − k ) σ ≤ C l k Θ k l hold every 1 ≤ k ≤ m − i .We wish to find ω i that solves the i -th equation of (6.10): ω i ◦ z − λω i = ϑ i + ω i +1 (6.16)providing Λ i + − Λ i − is a 0 distribution.The i -th coordinate function of J − ( j +1) Θ ◦ z j is: λ − ( j +1) ϑ i ◦ z j + m − i X k =1 C j,k ϑ k + i ◦ z j , where C j,k = ( − k k ! λ − ( j +1+ k ) ( j + 1) · · · ( j + k ); and it follows thatΛ i − − Λ i + = + ∞ X j = −∞ λ − ( j +1) ϑ i ◦ z j + + ∞ X j = −∞ m − i X k =1 C j,k ϑ k + i ◦ z j . Note that both Λ i − and Λ i + are distributions by (2) of Lemma 4.4. Bysubstituting ϑ k + i = ω k + i ◦ z − λω k + i − ω k + i +1 for all 1 ≤ k ≤ m − i into P m − ik =1 C j,k ϑ k + i ◦ z j we get m − i X k =1 C j,k ϑ k + i ◦ z j = m − i − X k =1 C j,k ( ω k + i ◦ z − λω k + i − ω k + i +1 ) ◦ z j + C j,m − i ( ω m ◦ z − λω m ) ◦ z j . By noting that + ∞ X j = −∞ C j,k ( ω k + i ◦ z − λω k + i ) ◦ z j = + ∞ X j = −∞ C j,k − ω k + i ◦ z j it follows that:Λ i − − Λ i + = + ∞ X j = −∞ λ − ( j +1) ϑ i ◦ z j + + ∞ X j = −∞ λ − ( j +1) ω i +1 ◦ z j . By assumption (6.11) the left side is a 0 distribution. Thus the equation(6.16) satisfies the solvability and use Theorem 2.2 again to conclude thatthere exists a H l − ( m − i − σ solution ω i with estimate k ω i k l − ( m − i − σ ≤ C l k ϑ k + i + ω k + i +1 k l − ( m − i − k ) σ ≤ C l k Θ k l Since k is an arbitrary integer between 1 and m − z ) | N z . Sincethe maximal size of a Jordan block is bounded by dim N z , we obtain thefollowing estimates for the solution Λ: k Λ k l − σ ≤ C l kF k l . (3): It suffices to show that if the lifted equation ˜Λ ◦ z − Ad( z ) ˜Λ = ˜ F , if˜ F = 0 then ˜Λ = 0. We assume notations in (1). Since ˜Λ = Ad( z − j ) ˜Λ( z j )for any j ∈ Z , we getΛ i = dim N z X k =1 a i,k − j Λ k ◦ z j , ∀ j ∈ Z , where Λ i is the i -th coordinates of ˜Λ.By using (6.9) and estimates obtained in (1) of Lemma 4.4, for any u ∈ H we get h Λ i , u i = lim j →∞ h dim N z X k =1 a i,k − j Λ k ◦ z j , u i = 0Then Λ i = 0 follows immediately for any i . This shows that Λ = 0.(4): By using arguments in the proof of (2), we can assume that F is on ¯ X .If Ad( b ) is invariant on N z , there exists a basis in N C such that under thisbasis Ad( b ) has Jordan block form and each element in the basis has length1. As usual, this basis may be chosen to consists of several real vectors andseveral pairs of complex conjugate vectors. Let J = ( q i,j ) be an m × m matrix which consists of blocks of Ad( b ) corresponding to the eigenvalue λ ; i.e., let q i,i = λ for all 1 ≤ i ≤ m ; and q i,j = 0 if j = i + 1 for all1 ≤ i ≤ m −
1. Note that since elements in the basis are of length 1, then q i,i +1 is not necessarily 0 or 1. It is clear that the norms of all blocks ofAd( b ) | N z are uniformly bounded from above for any b sufficiently close to z . If b is sufficiently close to z , b is partially hyperbolic. It follows from (2)and (3) that the equation (6.8) has a unique solution Λ ∈ H l − σ . Let thecoordinate functions of Θ and Λ be ϑ i and ω i respectively, 1 ≤ i ≤ m . The m -th equation of (6.10) becomes: ω m ◦ b − λω m = ϑ m (6.17) OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 35 and for 1 ≤ k ≤ m − k -th equation is ω k ◦ b − λω k = ϑ k + q k,k +1 ω k +1 (6.18)The estimates of ω m follow directly from equation (6.17) and Theorem 2.2: k ω m k l − σ ≤ C l k ϑ m k l − σ ≤ C l k Θ k l . Inductively, the estimates of ω i follow directly from equations (6.17) and(6.18) for k ≤ i ≤ m and Theorem 2.2: k ω i k l − ( m − i − σ ≤ C l k ϑ i + q i,i +1 ω i +1 k l − ( m − i ) σ ≤ C l k Θ k l . Repeated the above process for all corresponding blocks Ad( b ) | N z . Weobtain the following estimates for the solution Λ: k Λ k l − σ ≤ C l kF k l . (cid:3) Proof of Theorem 2.5
Reduction to finding a solution for a single cocycle equation.Lemma 7.1.
Suppose α A is a higher rank partially hyperbolic action on X and N is the neutral distributions of α A on ¯ X . If R is a H Ad-twistedcocycle over α A valued on N and Ω ∈ H ,X solves the equation Ω ◦ z − Ad( z )Ω = R z + c (7.1) for some e = z ∈ A and c ∈ N (Here we use R z := R ( z, · ) to denote themap from X to N ). Then Ω solves (7.1) for all the elements of the cocyclei.e. there exists a homomorphism c : A → N such that for all d ∈ A we have Ω ◦ d − Ad d Ω = R d + c ( d ) . Proof.
From (7.1) we see that c = − R X R z dµ (note that Ω ∈ H ,X ). Set R ′ d = R d − R X R d dµ for any d ∈ A . Then R ′ ∈ H ,X . It is easy to check that R ′ is also an Ad-twisted cocycle over α A . By the twisted cocycle condition R ′ d ◦ z − Ad( z ) R ′ d = R ′ z ◦ d − Ad( d ) R z we have T d R ′ z = T z R ′ d , ∀ d ∈ A, where T d f = f ◦ d − Ad( d ) f for any f ∈ H X .Substituting (7.1) into the above equation we have T d ( T z Ω) = T z R ′ d , ∀ d ∈ A. Since operators T d and T z commute this implies T z ( T d Ω − R ′ d ) = 0 . By (3) of Lemma 6.9 it follows that T d Ω − R ′ d , i.e., Ω solvesΩ ◦ d − Ad( d )Ω = R d + c ( d )as well. (cid:3) Remark 7.2.
The result still holds if we change R to a cocycle over α A ,since we just need to show that the operator D d f = f ◦ d − f is injective,which is obvious from Fact 5.2.The first part of Theorem 2.5 follows directly from Theorem 2.3 and theabove remark. Next, we prove the second part. Lemma 7.1 shows thatobtaining a tame solution of cocycle R for one regular generator suffices forthe proofs of Theorem 2.5. Hence to prove Theorem 2.5, it is equivalent toprove the following lemma: Lemma 7.3.
Suppose m and σ are as defined in Lemma 6.9. Also suppose a and b commute and are regular generators for α A on X . If F , G : X → N satisfying F ◦ b − Ad( b ) F = G ◦ a − Ad( a ) G (7.2) and F , G ∈ H m ,X , m ≥ m , then the equations Ω ◦ a − Ad( a )Ω = F Ω ◦ b − Ad( b )Ω = G (7.3) have a common solution Ω ∈ H m − σ ,X with the following estimate k Ω k m − σ ≤ C m kF , Gk m . (7.4) Proof.
By arguments in the proof of (2) of Lemma 6.9, we can assume F , G : ¯ X → N . Next, we show that the obstructions to solving equations7.3 vanish. From the equation (7.2) we get j = n X j = − n Ad( a ) − ( j +1) F ◦ ( b a j ) − j = n X j = − n Ad( b ) Ad( a ) − ( j +1) F ◦ a j = Ad( a ) − ( n +1) G ◦ a n +11 − Ad( a ) n G ◦ a − n . From proof of (1) of Lemma 6.9 and (2) of Lemma 4.4 we see that the right-hand converges to 0 as a distribution (in π ) when n → ∞ . Hence we seethat Ad( b ) − ∞ X j = −∞ Ad( a ) − ( j +1) F ◦ ( b a j ) = + ∞ X j = −∞ Ad( a ) − ( j +1) F ◦ a j (7.5)as distributions. This shows thatAd( b ) − k + ∞ X j = −∞ Ad( a ) − ( j +1) F ◦ ( b k a j ) = + ∞ X j = −∞ Ad( a ) − ( j +1) F ◦ a j as distributions for any k ∈ Z .If we can show that + ∞ X n = −∞ + ∞ X j = −∞ Ad( b ) − ( n +1) Ad( a ) − ( j +1) F ◦ ( b n a j )(7.6) OCYCLE RIGIDITY OF ABELIAN PARTIALLY HYPERBOLIC ACTIONS 37 is a distribution, then by iterating equation (7.5) we obtain: + ∞ X n = −∞ + ∞ X j = −∞ Ad( b ) − ( n +1) Ad( a ) − ( j +1) F ◦ ( b n a j )= + ∞ X n = −∞ Ad( b ) − ∞ X j = −∞ Ad( b ) − n Ad( a ) − ( j +1) F ◦ ( b n a j )= + ∞ X n = −∞ Ad( b ) − ∞ X j = −∞ Ad( a ) − ( j +1) F ◦ a j Since the series in the left hand side of above equation is a distribution by(1) of Lemma 6.9, it forces P + ∞ j = −∞ Ad( a ) − ( j +1) F ◦ a j to be a 0 distribution.Then the result follows from Lemma 6.9.Let ( a k,ln,j ) denote the matrix of Ad( a ) n Ad( b ) j on N . Since the eigen-values of Ad( a ) and Ad( b ) on N are all in U (1), we have k ( a k,ln,j ) k ≤ C ( | n | + 1) dim N ( | j | + 1) dim N , ∀ n, j ∈ Z . (7.7)The i -th coordinate of (7.6) is: X j ∈ Z X i ∈ Z dim N X k =1 a i,k − ( j +1) , − ( n +1) F k ◦ ( b n a j )where F k is the i -th coordinate of F .By using (7.7) it follows from (5.3) of Lemma 5.1 that (7.6) is a distribu-tion. (cid:3) References [1] L.Clozel. D´ e monstration de la conjecture τ . Invent. Math. 151 (2003), no. 2, 297-328.[2] M. Cowling, Sur le coefficients des reprsesentations unitaries des groupes de Lie sim-ple, Lecture Notes in Math., vol. 739, Springer-Verlag, Berlin and New York, 1979,pp. 132-178.[3] M. Cowling, U. Haggerup and R. E. Howe, Almost L matrix coefficients, J. ReinerAngew. Math 387 (1988), 97-110.[4] D. Damjanovic and A. Katok, Periodic cycle functionals and cocycle rigidity forcertain partially hyperbolic R k actions, Discr. Cont. Dyn. Syst. (2005), 985–1005.[5] D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions. I. KAMmethod and Z k actions on the torus, Annals of Mathematics (2010), 1805–1858.[6] D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions.II. Thegeometric method and restrictions of Weyl Chamber flows on SL ( n, R ) / Γ, Int. Math.Res. Notes , 2010.[7] D. Damjanovic and A. Katok, Local rigidity of homogeneous parabolic actions: I. Amodel case,
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Department of Mathematics, Michigan State University, East Lansing, MI48824,USA
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