A Haar component for quantum limits on locally symmetric spaces
aa r X i v : . [ m a t h . A P ] S e p A HAAR COMPONENT FOR QUANTUM LIMITS ON LOCALLYSYMMETRIC SPACES
NALINI ANANTHARAMAN AND LIOR SILBERMAN
Abstract.
We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive cur-vature. In the case of certain compact quotients of the space of positive definite n × n matrices (any quotient for n = 3 , quotients associated to inner forms in general), measureclassification results then show that the limit measures must have a Lebesgue component.This is consistent with the conjecture that the limit measures are absolutely continuous. Contents
1. Introduction 12. Background and notation regarding semisimple Lie groups 103. Quantization and pseudodifferential operators 144. The WKB Ansatz 195. The Cotlar–Stein argument. 246. Measure Rigidity 35References 381.
Introduction
Background and motivations.
The study of high-energy Laplacian eigenfunctionson negatively curved manifolds has progressed considerably in recent years. In the so-called “arithmetic” case, Elon Lindenstrauss has proved the Quantum Unique Ergodicityconjecture for Hecke eigenfunctions on congruence quotients of the hyperbolic plane [16].In the “general case” (variable negative curvature, with no arithmetic structure), the firstauthor has proved that semiclassical limits of eigenfunctions have positive Kolmogorov-Sinai entropy, in a joint work with Stéphane Nonnenmacher [1, 3, 4].The two approaches are very different, but have in common the central role of thenotion of entropy. In Lindenstrauss’ work, an entropy bound is obtained from arithmeticconsiderations [5], and then combined with the measure rigidity phenomenon to proveQuantum Unique Ergodicity.It is very natural to ask about a possible generalization of these results to locally sym-metric spaces of higher rank and nonpositive curvature. In this case the Laplacian will be
N. Anantharaman wishes to acknowledge the support of Agence Nationale de la Recherche, under thegrants ANR-09-JCJC-0099-01 and ANR-07-BLAN-0361. replaced by the entire algebra of translation-invariant differential operators, as proposedby Silberman and Venkatesh in [23]. A generalization of the entropic bound of [5] hasbeen worked out by these authors in the adelic case, and as a result they could prove aform of Arithmetic Quantum Unique Ergodicity in the case of the locally symmetric space Γ \ SL n ( R ) , when n is prime and Γ is derived from a division algebra over Q [24]. The goalof this paper is to generalize the “non-arithmetic” approach of [3, 4] in this context – thatis to say, prove an entropy bound without using the Hecke operators or other arithmetictechniques. Doing so, we will not require some of the assumptions used in [24]: we willwork with an arbitrary connected semisimple Lie group with finite center G , Γ will be any cocompact lattice in G , and we will not use the Hecke operators. Combining the entropybound with the measure classification results of [8, 9, 17], in the case of G = SL ( R ) , Γ arbitrary, or G = SL n ( R ) , n arbitrary but Γ derived from a division algebra over Q , wewill prove a weakened form of Quantum Unique Ergodicity : any semiclassical measurehas the Haar measure as an ergodic component .In addition to the intrinsic interest of locally symmetric spaces, there is yet anothermotivation to study these models. So far, the entropic bound of [3, 4] is not satisfactory formanifolds of variable negative curvature ([1] proves that the entropy is positive, but withoutgiving an explicit bound). Gabriel Rivière has been able to treat the case of surfaces [19, 20];he is even able to work in nonpositive curvature, but the case of higher dimensions remainsopen. The problem comes from the existence of several distinct Lyapunov exponents at eachpoint. Locally symmetric spaces are an attempt to make some progress in this direction :we will deal with flows that have distinct Lyapunov exponents, some of which may evenvanish. Still, considerable simplifications arise from the fact that they are homogeneousspaces, and that the stable and unstable foliations are smooth. It would be extremelyinteresting to extend the techniques of [3, 4, 19, 20] to systems that are not uniformlyhyperbolic (euclidean billiards would be the ultimate goal).Let G be a connected semisimple Lie group with finite center, K < G be a maximalcompact subgroup, Γ < G a uniform lattice. We will work on the symmetric space S = G/K , the compact quotient Y = Γ \ G/K , and the homogeneous space X = Γ \ G . We willendow G with its Killing metric, yielding a G -invariant Riemannian metric on G/K , withnonpositive curvature.Call D the algebra of G -invariant differential operators on S ; it follows from the structureof semisimple Lie algebras that this algebra is commutative and finitely generated [11, Ch.II §4.1, §5.2]. The number of generators, to be denoted r , coincides with the real rank of S (that is the dimension of a maximal flat totally geodesic submanifold), and, in a morealgebraic fashion, with the dimension of a , a maximal abelian semisimple subalgebra of Unfortunately, we are not able to extend the method to the case of
Γ = SL n ( Z ) , which is not cocompact– unless we input the extra assumption that there is no escape of mass to infinity, or that the mass escapesvery fast. We do not assume that Γ is torsion free. When speaking of smooth functions on Y , we have in mindsmooth functions on S that are Γ -invariant. We shall denote g the Lie algebra of G , k the Lie algebra of K , and so on. UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 3 g orthogonal to k . More background and notations concerning Lie groups are given inSection 2. Remark 1.1.
The algebra D always contains the Laplacian. If the symmetric space S hasrank r = 1 , then D is generated by the Laplacian. Example 1.2.
The case G = SO o ( d, yields the d -dimensional hyperbolic space S = H d (of rank ), already dealt with in [3, 4].We will focus on the example of G = SL n ( R ) , K = SO ( n, R ) . In that case, g is the setof matrices with trace , k the antisymmetric matrices, and one can take a to be the set ofdiagonal matrices with trace . The connected group generated by a is denoted A , in thisexample it is the set of diagonal matrices of determinant and with positive entries. Therank is r = n − .We will be interested in Γ -invariant joint eigenfunctions of D ; in other words, eigen-functions of D that go to the quotient Γ \ G/K . If we choose a set of generators of D ,the collection of eigenvalues can be represented as an element of R r . We will recall inSection 2.2 that it is more natural to parametrize the eigenvalue by an element ν ∈ a ∗ C ,the complexified dual of a . More precisely, ν ∈ a ∗ C /W where W is the Weyl group of G , afinite group given by M ′ /M where M ′ is the normalizer of A , and M the centralizer of A ,in K .1.2. Semiclassical limit.
Silberman and Venkatesh suggested to study the L -normalizedeigenfunctions ( ψ ) in the limit k ν k −→ + ∞ , as a variant of the very popular questionof understanding high-energy eigenfunctions of the Laplacian. The question of “quantumergodicity” is to understand the asymptotic behaviour of the family of probability measures d ¯ µ ψ ( y ) = | ψ ( y ) | dy on Y = Γ \ G/K . They considered the case where ν k ν k has a limit ν ∞ ∈ a ∗ C /W , with the sequence ν satisfying a certain number of additional assumptions thatwe shall recall later. For the moment, we just note that the real parts ℜ e ( ν ) are uniformlybounded, so that ℜ e ( ν ∞ ) = 0 ([23, Thm. 2.7 (3)]). We will denote Λ ∞ = ℑ m ( ν ∞ ) = − iν ∞ .1.3. Symplectic lift vs. representation-theoretic lift.
The locally symmetric space Y should be thought of as the configuration space of our dynamical system. To properlyanalyze the dynamics it is necessary to move to an appropriate phase space . Once we lift the eigenfunctions there, the measures become approximately invariant under the dynamicsand we can apply the tools of ergodic theory. Two different kinds of lifts have beenconsidered thus far: the microlocal lift (we also call it the symplectic lift ) lifts the measure ¯ µ ψ to a distribution ˜ µ ψ on the cotangent bundle T ∗ Y = Γ \ T ∗ ( G/K ) , taking advantageof its symplectic structure. This construction applies in great generality, for examplewhen Y is any compact Riemannian manifold. The representation theoretic lift used in[26, 16, 23, 24, 6], specific to locally symmetric spaces, lifts the measure ¯ µ ψ to a measure µ ψ defined on X = Γ \ G , taking advantage of the homogeneous space structure of G/K .The two lifts are very natural, and closely related. In our proofs we will use a lot thesymplectic point of view, as we will use the Helgason-Fourier transform of L functions,and interprete it geometrically as a decomposition into lagrangian states. But we will also N. ANANTHARAMAN AND L. SILBERMAN need to translate our results in terms of the representation theoretic lift, in order to applysome measure classification results from [8, 9].In the symplectic point of view, the dynamics is defined as follows. On T ∗ ( G/K ) ,consider the algebra H of smooth G -invariant Hamiltonians, that are polynomial in thefibers of the projection T ∗ ( G/K ) −→ G/K . This algebra is isomorphic to the algebra of W -invariant polynomials on a ∗ (consider the restriction on a ∗ ⊂ T ∗ o ( G/K ) ). The structuretheory of semisimple Lie algebras shows that H is isomorphic to a polynomial ring in r generators. Moreover, the elements of H commute under the Poisson bracket. Thus, wehave on T ∗ ( G/K ) a family of r independent commuting Hamiltonian flows H , ..., H r . TheKilling metric, seen as a function on T ∗ ( G/K ) , always belongs to H , and its symplecticgradient generates the geodesic flow. Of course, since all these flows are G -equivariant,they descend to the quotient T ∗ Y .Joint energy layers of H are naturally parametrized by elements Λ ∈ a ∗ /W . This is easyto explain geometrically: fix a point in G/K , say the origin o = eK . Consider the flattotally geodesic submanifold A.o ⊂ G/K going through o . It is isometric to R r , and thecotangent space T ∗ o ( A.o ) is naturally isomorphic to a ∗ . If E ⊂ T ∗ ( G/K ) is a joint energylayer of H (or equivalently a G -orbit in T ∗ ( G/K ) ), then there exists Λ ∈ a ∗ such that E ∩ T ∗ o ( A.o ) = W. Λ . See [13] for details. We will denote E Λ the energy layer of parameter Λ .In Section 3 we will use a quantization procedure to associate to every Γ -invariant eigen-function ψ a distribution ˜ µ ψ on T ∗ Y , called its microlocal lift. This distribution projectsto ¯ µ ψ on Y . This is a very standard construction, and so is the following theorem, whichis an avatar of propagation of singularities for solutions of partial differential equations: Theorem 1.3.
Assume that k ν k −→ + ∞ , and that ν k ν k has a limit ν ∞ . Denote Λ ∞ = − iν ∞ ∈ a ∗ /W . Any limit (in the distribution sense) of the sequence ˜ µ ψ is a probabil-ity measure on T ∗ Y , carried by the energy layer E Λ ∞ , and invariant under the family ofHamiltonian flows generated by H . In order to transport this statement to get an A -invariant measure on Γ \ G , we mustnow make some assumptions on Λ ∞ . Silberman and Venkatesh assume that ν ∞ is a regularelement of a ∗ C , in the sense that it is not fixed by any non-trivial element of W , andthey show that it implies ℜ e ( ν n ) = 0 for all but a finite number of ν n s in the sequence.The element ν ∞ being regular is, of course, equivalent to Λ ∞ being regular; and this isalso equivalent to the energy layer Λ ∞ being regular, in the sense that the differentials dH , ..., dH r are independent there [13].There is a surjective map π : G/M × a ∗ −→ T ∗ ( G/K ) (1.1) ( gM, λ ) ( gK, g.λ ) . (1.2)Remember that M is the centralizer of A in K . The image of G/M × { λ } under π is theenergy layer E λ . The map π λ : G/M × { λ } −→ E λ is a diffeomorphism if and only if λ isregular (otherwise π λ is not injective). Under π − λ , the action of the Hamiltonian flow Φ tH UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 5 generated by H ∈ H on E λ is conjugate to gM g exp( t dH ( λ )) M. The same statements hold after quotienting on the left by Γ . Since H is a function on a ∗ , the differential dH ( λ ) is an element of a . Denoting R ( e tX ) the one–parameter flow on G/M generated by X ∈ a (acting by multiplication on the right), we can rephrase this bywriting π ◦ R ( e tdH ( λ ) ) = Φ tH ◦ π on E λ . If λ is regular, the elements dH ( λ ) can be shown to span a as H varies over H . Otherwise,we have [13](1.3) { dH ( λ ) , H ∈ H} = { X ∈ a , ∀ α ∈ ∆ , ( h α, λ i = 0 = ⇒ α ( X ) = 0) } , where ∆ ⊂ a ∗ is the set of roots.Thus, Theorem 1.3 may be rephrased as follows: Theorem 1.4.
Assume Λ ∞ is regular. Then any limit (in the distribution sense) of thesequence ˜ µ ψ yields a probability measure on Γ \ G/M , invariant under the right action of A by multiplication. This theorem was proved in [23, Thm. 1.6 (3)] using the representation-theoretic lift;the equivariance of that lift shows that the construction is compatible with the Heckeoperators on Γ \ G . It is also shown there that the symplectic lift ˜ µ ψ and the representationtheoretic lift µ ψ have the same asymptotic behaviour as ν tends to infinity, and if weidentify E Λ ∞ ⊂ Γ \ T ∗ ( G/K ) with Γ \ G/M . Definition 1.5.
We will call any limit point of the sequence ˜ µ ψ (or µ ψ ) a semiclassicalmeasure in the direction Λ ∞ .Semiclassical measures in a regular direction are, equivalently, positive measures on T ∗ (Γ \ G/K ) (carried by a regular energy layer), positive measures on Γ \ G/M , or positivemeasures on Γ \ G (which are M -invariant).1.4. Entropy bounds.
Our main result is a non-trivial lower bound on the entropy ofsemiclassical measures. We fix H ∈ H , and we consider the corresponding Hamiltonianflow Φ tH on E Λ ∞ , which has Lyapunov exponents − χ J ( H ) ≤ · · · ≤ − χ ( H ) ≤ ≤ χ ( H ) ≤ · · · ≤ χ J ( H ) . In addition, the Lyapunov exponent appears trivially with multiplicity r , as a consequenceof the existence of r integrals of motion. The dimension of E Λ ∞ is r + 2 J . The integer J , the rank r and the dimension d of G/K are related by d = J + r . In general, theLyapunov exponents are measurable functions on the phase space, but here, because of thehomogeneous structure, the Lyapunov exponents are constants.In the following theorem we will denote χ max ( H ) = χ J ( H ) , the largest Lyapunov expo-nent. We denote h KS ( µ, H ) the Kolmogorov-Sinai entropy of a (Φ tH ) -invariant probability N. ANANTHARAMAN AND L. SILBERMAN measure µ . We recall the Ruelle-Pesin inequality, h KS ( µ, H ) ≤ X j χ j ( H ) , which holds for any (Φ tH ) -invariant probability measure µ . Theorem 1.6. (Symplectic version) Let µ be a semiclassical measure in the direction Λ ∞ .Assume that Λ ∞ is regular.For H ∈ H , we consider the corresponding Hamiltonian flow Φ tH on E Λ ∞ . Then (1.4) h KS ( µ, H ) ≥ X j : χ j ( H ) ≥ χ max( H )2 (cid:18) χ j ( H ) − χ max ( H )2 (cid:19) . Continuing with the assumption that Λ ∞ is regular, we can transport the theorem to Γ \ G/M . If we fix a 1-parameter subgroup ( e tX ) of A (with X ∈ a ), it is well known thatthe (non trivial) Lyapunov exponents of the flow ( e tX ) acting on X/M are the real numbers ( α ( X )) , where α ∈ a ∗ run over the set of roots ∆ (see Section 2 for background related toLie groups). If α is a root then so is − α (one of the two will be called positive , the other negative ). The notion of positivity is explained in detail later. For now it suffices to notethat we may assume that α ( X ) ≥ for positive roots α . We write α max ( X ) for max α α ( X ) (this is the largest Lyapunov exponent of the associated Hamiltonian flow). Each rootoccurs with multiplicity m α , which must be taken into account in the statements below(the corresponding Lyapunov exponent α ( X ) would be counted repeatedly, m α times). Theorem 1.7. (Group-theoretic version) Let µ be a semiclassical measure in the direction Λ ∞ . Assume that Λ ∞ is regular.Let ( e tX ) ( X ∈ a ) be a one parameter subgroup of A such that α ( X ) ≥ for all positiveroots α .Let h KS ( µ, X ) be the entropy of µ with respect to the flow ( e tX ) . Then (1.5) h KS ( µ, X ) ≥ X α : α ( X ) ≥ α max( X )2 m α (cid:18) α ( X ) − α max ( X )2 (cid:19) . Our lower bound is positive for all non-zero X , in fact greater than α max ( X )2 . In [1, 3],the first author and S. Nonnenmacher had conjectured the following stronger bound h KS ( µ, H ) ≥ X j χ j ( H ) or equivalently(1.6) h KS ( µ, X ) ≥ X α> m α · α ( X ) . We are still unable to prove it, except in one case: when all the positive Lyapunov exponentsare equal to each other, so that formula (1.5) reduces to (1.6). One case is that of hyperbolic d -space ( G = SO ( d, ) alluded to above. Another, the main focus of the present paper, is UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 7 of the “extremely irregular” elements of the torus in G = SL n ( R ) . These are the elementsconjugate under the Weyl group to X = diag( n − , − , ..., − . Application: towards Quantum Unique Ergodicity on locally symmetricspaces.
In Section 6 we combine our entropy bounds with measure classification results.Let n ≥ , G = SL n ( R ) , Γ < G a cocompact lattice. Let µ be a semiclassical measure on Γ \ G in the regular direction Λ ∞ .The measure µ can be written uniquely as a sum of an absolutely continuous measureand a singular measure (with respect to Lebesgue or Haar measure). Since µ is invariantunder the action of A , the same holds for both components. Because the Haar measureis known to be ergodic for the action of A , the absolutely continuous part of µ is, in fact,proportional to Haar measure. We call this the Haar component of µ . Its total mass is the weight of this component. Theorem 1.8.
Let n = 3 . Then µ has a Haar component of weight ≥ . Theorem 1.9.
Let n = 4 . Then either µ has a Haar component, or each ergodic componentis the Haar measure on a closed orbit of the group ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
00 0 0 ∗ (or one of its 4 imagesunder the Weyl group), and the components invariant by each of these 4 subgroups havetotal weight . In fact, the result is slightly stronger: if some “extremely irregular” element acts on µ with entropy strictly larger than half of its entropy w.r.t. Haar measure, then there is aHaar component.It does not seem to be possible to push this technique beyond SL . The problem is thatthere are large subgroups (in the style of those occuring in Theorem 1.9) whose closed orbitssupport measures of large entropy. For particular lattices, however, these large subgroupsdo not have closed orbits, so the only possible non-Haar components have small entropyand cannot account for all the entropy. For co-compact lattices this occurs, for example,when Γ is the set of elements of reduced norm of an order in a central division algebraover Q , or more generally for any lattice commensurable with one obtained this way (wesay that Γ is associated to the division algebra). Such lattices are said to be of “inner type”since they correspond to inner forms of SL n over Q (there also exist non-uniform latticesof inner type, corresponding to central simple Q -algebras which are not division algebras).For a brief description of the construction and references see Section 6. Theorem 1.10.
For n ≥ let Γ < SL n ( R ) be a lattice associated to a division algebraover Q , and let µ be a semiclassical measure on Γ \ SL n ( R ) in a regular direction. Then µ has a Haar component of weight ≥ n +12 − tn − t > where t is the largest proper divisor of n . N. ANANTHARAMAN AND L. SILBERMAN
It is not surprising that strongest implication is for n prime (so that there are few inter-mediate algebraic measures). Indeed, setting t = 1 we find w ∆ ≥ in that case. Howeverfor n prime Silberman-Venkatesh [24] show that the semiclassical measures associated to Hecke eigenfunctions are equal to Haar measure. The main impact of Theorem 1.10 is thuswhen the n is composite, where previous methods only showed that semiclassical measuresare convex combinations of algebraic measures but could not establish that Haar measureoccurs in the combination. Remark 1.11.
We compare here our result with that of [24]. That paper studies thecase of lattices in G = P GL n ( R ) associated to division algebras of prime degree n andjoint eigenfunctions of D and of the Hecke operators. It is then shown that any ergodiccomponent of a semiclassical measure µ has positive entropy; it follows that µ must be theHaar measure. Our result is neither stronger nor weaker: • We cannot prove that all ergodic components of µ have positive entropy, only thatthe total entropy of µ is positive. Hence, we are not able to exclude components ofzero entropy; • On the other hand, our lower bound on the total entropy ( / of the maximalentropy) is explicit and quite strong. This allows to detect the presence of a Haarcomponent in a variety of cases; • In particular, for n = 3 we do not need any assumption on the cocompact lattice Γ ; and for Γ associated to a division algebra, our result holds for all n . • The Hecke-operator method applies more naturally to adelic quotients G ( Q ) \ G ( A ) /K ∞ K f .When G is a form of SL n there is no distinction, but when G = P GL n the adelicquotients are typically disjoint unions of quotients Γ \ G . Even when the quotient iscompact, G -invariance of the limit measure does not show that all components havethe same proportion of the mass. Our result applies to each connected componentseparately. • We do not assume that our eigenfunctions are also eigenfunctions of the Heckeoperators: this means that multiplicity of eigenvalues is not an issue in this work. • The methods of Silberman-Venkatesh apply to non-cocompact lattices as well.1.6.
Hyperbolic dispersive estimate.
The proof of Theorem 1.6 (and 1.7) follows themain ideas of [3], with a major difference which lies in an improvement of the “hyperbolicdispersive estimate” : [1, Thm. 1.3.3] and [3, Thm. 2.7]. If we applied directly the resultof [3], we would get h KS ( µ, H ) ≥ X k (cid:18) χ k ( H ) − χ max ( H )2 (cid:19) . This inequality is often trivial (the right-hand term being negative) whereas in (1.4) wemanaged to get rid of the negative terms (cid:16) χ k ( H ) − χ max ( H )2 (cid:17) .Since the “hyperbolic dispersive estimate” has an intrinsic interest, and is the core of thispaper, we state it here as one of our main results. We fix a quantization procedure, set atscale ~ = k ν k − , that associates to any reasonable function a on T ∗ Y an operator Op ~ ( a ) on L ( Y ) . An explicit construction is given in Section 3. In particular, it is useful to know UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 9 that Op ~ can be defined so that, if H ∈ H is real valued, Op ~ ( H ) is a self-adjoint operatorbelonging to D . More explicitly, Op ~ ( H ) is defined so that Op ~ ( H ) ψ ν = H ( − i ~ ν ) ψ ν forany D -eigenfunction ψ ν , with spectral parameter ν (hence the choice of the normalisation ~ = k ν k − ).Let ( P k ) k =1 ,...,K be a family of smooth real functions on Y , such that(1.7) ∀ x ∈ Y , K X k =1 P k ( x ) = 1 . We assume that the diameter of the supports of the functions P k is small enough. We willalso denote P k the operator of multiplication by P k ( x ) on the Hilbert space L ( Y ) .We denote U t = exp( i ~ − t Op ~ ( H )) the propagator of the “Schrödinger equation” gener-ated by the Hamiltonian H . This is a unitary Fourier Integral Operator associated with theclassical Hamiltonian flow Φ − tH . The ~ -dependence of U will be implicit in our notations.We fix a small discrete time step η .Throughout the paper we will use the notation b A ( t ) = U − tη b AU tη for the quantumevolution at time tη of an operator b A . For each integer T ∈ N and any sequence of labels ω = ( ω − T , · · · , ω − , ω , · · · ω T − ) , ω i ∈ [1 , K ] (we say that the sequence ω is of length | ω | = 2 T ), we define the operators P ω = P ω T − ( T − P ω T − ( T − . . . P ω P ω − ( − . . . P ω − T ( − T ) . (1.8)We fix a smooth, compactly supported function χ on T ∗ Y , supported in a tubularneighbourhood of size ǫ of the energy layer E Λ ∞ (which is assumed to be regular); and wedefine P χω = P ω T − ( T − P ω T − ( T − . . . P / ω Op( χ ) P / ω P ω − ( − . . . P ω − T ( − T ) . (1.9)The operator P χω should be thought of as P ω restricted to a spectral window around theenergy layer E Λ ∞ . Theorem 1.12.
Fix H ∈ H , and a time step η , small enough. Let K > be fixed,arbitrary. Let χ ∈ C ∞ ( T ∗ Y ) , supported in a tubular neighbourhood of size ǫ of the regularenergy layer E Λ ∞ . Assume that ǫ , as well as the diameters of the supports of each P k , aresmall enough.Then, there exists ~ K > such that, for all ~ ∈ (0 , ~ K ) , for T = ⌊ K| log ~ | η ⌋ , and for everysequence ω of length T , (1.10) k P χω k ≤ C ~ − cǫ Y k, χ k ( H ) ≥ K e − T η χ k ( H ) ~ / where the χ k ( H ) denote the Lyapunov exponents of Φ tH on the energy layer E Λ ∞ . Theconstant C does not depend on K nor on H , whereas c does. The method used in [3] only yielded the upper bound:(1.11) k P χω k ≤ C ~ − cǫ Y k e − T η χ k ( H ) ~ / This is clearly not optimal when Φ tH has some neutral, or slowly expanding directions. Forinstance, if H = 0 then Φ tH = I has only neutral directions. In this case, (1.11) reads(1.12) k P χω k ≤ C ~ − d − cǫ , where d is the dimension of Y , which is obviously much worse (for any T ) than the trivialbound(1.13) k P χω k ≤ . On the other hand, if some of the χ k ( H ) are (strictly) positive, then (1.11) is muchbetter than the trivial bound (1.13), for very large T η . The bound given by Theorem 1.12interpolates between the two, for
T η ∼ K| log ~ | .The proof of the hyperbolic dispersion estimates is quite technical, and occupies Sections3, 4, 5. It uses a version of the pseudodifferential calculus adapted to the geometry of locallysymmetric spaces, based on Helgason’s version of the Fourier transform for this spaces, andinspired by the work of Zelditch in the case of G = SL (2 , R ) [27]. We point out the fact thatan alternative proof of Theorem 1.12 is given in [2], based on more conventional Fourieranalysis. The reader might prefer to read [2] instead of Sections 3, 4, 5, however we feelthat the two techniques have an interest of their own.We will not repeat here the argument that leads from Theorem 1.12 to the entropybound Theorem 1.6; it would be an exact repetition of the argument given in [3, §2].Let us just make one comment : in this argument, we are limited to K = χ max ( H ) (thetime T E = | log ~ | χ max ( H ) is sometimes called the Ehrenfest time for the Hamiltonian H , andcorresponds to the time where the approximation of the quantum flow U t by the classicalflow Φ tH breaks down). This means that we eventually keep the Lyapunov exponents suchthat χ k ( H ) ≥ χ max ( H )2 , and explains why this restriction appears in (1.4).2. Background and notation regarding semisimple Lie groups
Our terminology follows Knapp [15].2.1.
Structure.
Let G denote a non-compact connected simple Lie group with finite cen-ter . We choose a Cartan involution Θ for G , and let K < G be the Θ -fixed maximal If G is semisimple our discussion remains valid, but one can even do something finer, as remarkedin [23, §5.1]. After decomposing g into simple factors ⊕ g ( j ) , and assuming that the Cartan involution, thesubalgebra a , etc. are compatible with this decomposition, one can decompose the spectral parameter ν into its components ν ( j ) ∈ a ( j ) ∗ . Instead of assuming that k ν k −→ + ∞ and ν k ν k has a regular limit ν ∞ ,one can assume the same independently for each component ν ( j ) . This means that we do not have toassume that all the norms k ν ( j ) k go to infinity at the same speed. UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 11 compact subgroup. Let g = Lie ( G ) , and let θ denote the differential of Θ , giving the Car-tan decomposition g = k ⊕ p with k = Lie ( K ) . Let S = G/K be the symmetric space, with o = eK ∈ S the point with stabilizer K . We fix a G -invariant metric on G/K : observethat the tangent space at the point o is naturally identified with p , and endow it with theKilling form. For a lattice Γ < G we write X = Γ \ G and Y = Γ \ G/K , the latter being alocally symmetric space of non-positive curvature. In this paper, we shall always assumethat X and Y are compact.Fix now a maximal abelian subalgebra a ⊂ p .We denote by a C the complexification a ⊗ C . We denote by a ∗ (resp. a ∗ C ) the real dual(resp. the complex dual) of a . For ν ∈ a ∗ C , we define ℜ e ( ν ) , ℑ m ( ν ) ∈ a ∗ to be the realand imaginary parts of ν , respectively. For α ∈ a ∗ , set g α = { X ∈ g , ∀ H ∈ a : ad ( H ) X = α ( H ) X } , ∆ = ∆( a : g ) = { α ∈ a ∗ \ { } , g α = { }} and call the latter the (restricted) rootsof g with respect to a . The subalgebra g is θ -invariant, and hence g = ( g ∩ p ) ⊕ ( g ∩ k ) .By the maximality of a in p , we must then have g = a ⊕ m where m = Z k ( a ) , the centralizerof a in k .The Killing form of g induces a standard inner product h ., . i on p , and by duality on p ∗ . By restriction we get an inner product on a ∗ with respect to which ∆( a : g ) ⊂ a ∗ isa root system. The associated Weyl group, generated by the root reflections s α , will bedenoted W = W ( a : g ) . This group is also canonically isomorphic to N K ( a ) /Z K ( a ) . Inwhat follows we will represent any element w of the Weyl group by a representative in N K ( a ) ⊂ K (taking care to only make statements that do not depend on the choice of arepresentative), and the action of w ∈ W ( a : g ) on a or a ∗ will be given by the adjointrepresentation Ad( w ) . The fixed-point set of any s α is a hyperplane in a ∗ , called a wall.The connected components of the complement of the union of the walls are cones, calledthe (open) Weyl chambers. A subset Π ⊂ ∆( a : g ) will be called a system of simple rootsif every root can be uniquely expressed as an integral combination of elements of Π witheither all coefficients non-negative or all coefficients non-positive. For a simple system Π ,the open cone C Π = { ν ∈ a ∗ , ∀ α ∈ Π : h ν, α i > } is an (open) Weyl chamber. Theclosure of an open chamber will be called a closed chamber; we will denote in particular C Π = { ν ∈ a ∗ , ∀ α ∈ Π : h ν, α i ≥ } . The Weyl group acts simply transitively on thechambers and simple systems. The action of W ( a : g ) on a ∗ extends in the complex-linearway to an action on a ∗ C preserving i a ∗ ⊂ a ∗ C , and we call an element ν ∈ a ∗ C regular if itis fixed by no non-trivial element of W ( a : g ) . Since − C Π ⊂ a ∗ is a chamber, there is aunique w ℓ ∈ W ( a : g ) , called the “long element”, such that Ad( w ℓ ) .C Π = − C Π . Note that w ℓ C Π = C Π and hence w ℓ = e . Also, w ℓ depends on the choice of Π but we suppress thisfrom the notation.Fixing a simple system Π we get a notion of positivity. We will denote by ∆ + the set ofpositive roots, by ∆ − = − ∆ + the set of negative roots. We use ρ = P α> (dim g α ) α ∈ a ∗ to denote half the sum of the positive roots. For n = ⊕ α> g α and ¯ n = Θ n = ⊕ α< g α wehave g = n ⊕ a ⊕ m ⊕ ¯ n . Note that ¯ n = Ad( w ℓ ) . n . We also have (“Iwasawa decomposition”) g = n ⊕ a ⊕ k . We can therefore uniquely write every X ∈ g in the form X = X n + X a + X k .We also write H ( X ) for X a . Let
N, A, N < G be the connected subgroups corresponding to the subalgebras n , a , ¯ n ⊂ g respectively, and let M = Z K ( a ) . Then m = Lie ( M ) , though M is not necessarilyconnected. Moreover P = N AM is a minimal parabolic subgroup of G , with the map N × A × M −→ P being a diffeomorphism. The map N × A × K −→ G is a (surjective)diffeomorphism (Iwasawa decomposition), so for g ∈ G there exists a unique H ( g ) ∈ a such that g = n exp( H ( g )) k for some n ∈ N , k ∈ K . The map H : G −→ a is continuous;restricted to A , it is the inverse of the exponential map.We will use the G -equivariant identification between G/M and
G/K × G/P , given by gM ( gK, gP ) . The quotient G/P can also be identified with K/M .Starting from H we define a “Busemann function” B on G/K × G/P ∼ G/M :(2.1) B ( gK, g P ) = H ( k − g ) , where k is the K -part in the KAN decomposition of g (if g is defined modulo P , then k is defined modulo M ). Equivalently, if gM ∈ G/M , we have B ( gM ) = a , where g = kna is the KN A decomposition of g (if g is defined modulo M , then a is uniquely defined and k is defined modulo M ).In G/K , a “flat” is a maximal flat totally geodesic submanifold. Every flat is of the form { gaK, a ∈ A } for some g ∈ G . The space of flats can be naturally identified with G/M A ,or with an open dense subset of
G/P × G/ ¯ P , via the G -equivariant map gM A ( gP , g ¯ P ) where ¯ P = M AN = w ℓ P w − ℓ . We will also use the following injective map from
G/M A into
G/P × G/P , gM A ( gP , gw ℓ P ) . Its image is an open dense subset of
G/P × G/P , namely { ( g P , g P ) , g − g ∈ P w ℓ P } .Finally we recall the Bruhat decomposition G = ⊔ w ∈ W ( a : g ) P wP , with P w ℓ P being anopen dense subset (the “big cell”).2.2. The universal enveloping algebra; Harish-Chandra isomorphisms.
We ana-lyze the structure of D by comparing it with other algebras of differential operators. Fora Lie algebra s we write s C for its complexification s ⊗ R C . In particular, g C is a complexsemisimple Lie algebra. We fix a maximal abelian subalgebra b ⊂ m and let h = a ⊕ b .Then h C is a Cartan subalgebra of g C , with an associated root system ∆( h C : g C ) satisfying ∆( a : g ) = { α | a } α ∈ ∆( h C : g C ) \ { } .If s C is a complex Lie algebra, we denote by U ( s C ) its universal enveloping algebra; U ( g C ) is isomorphic to the algebra of left- G -invariant differential operators on G with complexcoefficients [10].There is an isomorphism, called the Harish-Chandra isomorphism, between the algebra D of G -invariant differential operators on G/K and the algebra D W ( A ) of A - and W -invariant differential operators on A ∼ R r . The latter is obviously isomorphic to U ( a C ) W ,the subalgebra of U ( a C ) formed of W -invariant elements. Since a C is abelian, U ( a C ) is canbe identified to the space of polynomial functions on a ∗ with complex coefficients. UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 13
The Harish-Chandra isomorphism
Γ :
D −→ D W ( A ) can be realized in a geometric wayas follows [11, Cor. II.5.19]. Consider the flat subspace A.o ⊂ G/K , naturally identifiedwith A . Fixing D ∈ D , let ∆ N ( D ) be the translation-invariant differential operator on A (that is, an element of U ( a ) ) given by [∆ N ( D ) f ]( a ) = D ˜ f ( a.o ) , for a ∈ A , f ∈ C ∞ ( A.o ) , and where ˜ f stands with the unique N -invariant function on G/K that coincides with f on A.o . Then, we define
Γ : D e − ρ ◦ ∆ N ( D ) ◦ e ρ , remembering that ρ is half the sum of positive roots and thus can be seen as a function on A . Note that e − ρ ◦ ∆ N ( D ) ◦ e ρ = τ ρ . ∆ N ( D ) , where τ ρ is the automorphism of U ( a ) defined by putting τ ρ ( X ) = X + ρ ( X ) for every X ∈ a .In what follows, we denote by Z ( g C ) the center of U ( g C ) . Thus, Z ( g C ) is the algebraof G -bi-invariant operators. Differentiating the action of G on S gives a map Z ( g C ) → D .For the next lemma we shall compare the isomorphism Γ with an isomorphism ω HC : Z ( g C ) −→ U ( h C ) W ( h C : g C ) , also called the Harish-Chandra isomorphism . Lemma 2.1.
Assume that the restriction from h C to a induces a surjection from U ( h C ) W ( h C : g C ) to U ( a C ) W (thought of as functions on the respective linear spaces).Let D ∈ D , of degree ¯ d . Then there exists Z ∈ Z ( g C ) such that Z and D coincide on(right-) K -invariant functions, and such that Z − τ − ρ Γ( D ) ∈ U ( n C ) U ( a C ) ¯ d − + U ( g C ) k C . Remark 2.2.
The assumption is automatically satisfied when G is split. It is also satisfiedwhen G/K is a classical symmetric space, that is when G is a classical group [11, p. 341].In fact the lemma itself is Proposition II.5.32 of [11], with the difference of degree between Z and τ − ρ Γ( D ) made precise. Proof.
Let D ∈ D be of degree ¯ d , so that Γ( D ) ∈ U ( a C ) W is a polynomial of degree ≤ ¯ d . Byassumption, we can extend Γ( D ) to an element of U ( h C ) W ( h C : g C ) . Consider Z = ω − HC Γ( D ) . It is shown in [23, Cor. 4.4] that Z − τ − ρ Γ( D ) ∈ U ( n C ) U ( a C ) ¯ d − + U ( g C ) k C . It is not completely clear that Z and D coincide on K -invariant functions, but the aboveformula shows that Γ( Z ) − Γ( D ) is of degree ≤ ¯ d − , and hence that Z − D has degreeat most ¯ d − .By descending induction on the degree of Γ( Z ) − Γ( D ) , we see that we can thus construct Z ∈ Z ( g C ) such that Z − τ − ρ Γ( D ) ∈ U ( n C ) U ( a C ) ¯ d − + U ( g C ) k C This is the isomorphism denoted by γ HC in [23], and defined by γ HC ( z ) = τ ρ h pr( z ) , where pr( z ) ∈ U ( h C ) is such that z − pr( z ) ∈ U ( n C ) U ( a C ) + U ( g C ) k C . and such that Γ( Z ) − Γ( D ) = 0 (which precisely means that Z and D coincide on right- K -invariant functions). (cid:3) The Helgason-Fourier transform.
For any θ ∈ G/P , ν ∈ a ∗ C , the function e ν,θ : x ∈ G/K e ( ρ + ν ) B ( x,θ ) is a joint eigenfunction of D , and one can verify easily (for instance in the case θ = eM )that De ν,θ = [Γ( D )]( ν ) e ν,θ , for every D ∈ D . Here we have seen Γ( D ) as a W -invariant polynomial on a ∗ C . In fact for any joint eigenfunction ψ of D there exists ν ∈ a ∗ C such that Dψ = [Γ( D )]( ν ) ψ for every D ∈ D [11, Ch. II Thm. 5.18, Ch. III Lem. 3.11]. The parameter ν is called the“spectral parameter” of ψ ; it is uniquely determined up to the action of W .The Helgason–Fourier transform gives the spectral decomposition of a function u ∈ C ∞ c ( S ) on the “basis” ( e ν,θ ) of eigenfunctions of D . It is defined as(2.2) e u ( λ, θ ) = Z S u ( x ) e − iλ,θ ( x ) dx, ( λ ∈ a ∗ , θ ∈ G/P ). It has an inversion formula: u ( x ) = Z θ ∈ G/P ,λ ∈ C Π e u ( λ, θ ) e iλ,θ ( x ) dθ | c ( λ ) | − dλ. Here dθ denotes the normalized K -invariant measure on G/P ∼ K/M . The function c is the so-called Harish-Chandra function, given by the Gindikin-Karpelevic formula [11,Thm. 6.14, p. 447].The Plancherel formula reads k u k L ( S ) = Z θ ∈ G/P ,λ ∈ C Π | e u ( λ, θ ) | dθ | c ( λ ) | − dλ. Remark 2.3.
For D ∈ D , D acts on u by Du ( x ) = Z θ ∈ G/P ,λ ∈ C Π [Γ( D )]( iλ ) e u ( λ, θ ) e iλ,θ ( x ) dθ | c ( λ ) | − dλ Quantization and pseudodifferential operators
In this section we develop a pseudodifferential calculus for S , inspired by the work ofZelditch [27]. We do not push the analysis as far as in [27] (a more detailed analysis is donein Michael Schröder’s thesis [22]). For us, the most important feature of this quantizationis that it is based on the Helgason-Fourier transform, in other words, on the spectraldecomposition of the algebra D . UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 15
Semiclassical Helgason transform.
We now introduce a parameter ~ . In the se-quel it will tend to at the same speed as k ν k − ; the reader may identify the two. Theparameter will be assumed to go to infinity in the conditions of §1.2, the limit ν ∞ assumedto be regular.From now on we rescale the parameter space a ∗ of the Helgason–Fourier transform by ~ .We define the semiclassical Fourier transform, b u ~ ( λ, θ ) = e u ( ~ − λ, θ ) . Thus, for u ∈ C ∞ c ( S ) ,we rewrite equation (2.2) as: b u ~ ( λ, θ ) = Z S u ( x ) e − i ~ − λ,θ ( x ) dx ( λ ∈ C Π , θ ∈ G/P ). The inversion formula now reads u ( x ) = Z θ ∈ G/P ,λ ∈ C Π b u ~ ( λ, θ ) e i ~ − λ,θ ( x ) dθ | c ~ ( λ ) | − dλ, with the “semiclassical Harish-Chandra c -function”, | c ~ ( λ ) | − = ~ − r | c ( ~ − λ ) | − . Remark 3.1.
By the Gindikin-Karpelevic formula, we have | c ( ~ − λ ) | − ≍ ~ − dim n uniformly for λ in a compact subset of C Π , and thus | c ~ ( λ ) | − ≍ ~ − d where d = dim a + dim n = dim( G/K ) .We also adjust the Plancherel formula to k u k L ( S ) = Z | b u ~ ( λ, θ ) | dθ | c ~ ( λ ) | − dλ. In the sequel we will always use the semiclassical Fourier transform, and will in generaldenote b u instead of b u ~ .3.2. Pseudodifferential calculus on Y . We identify the functions on the quotient Y =Γ \ G/K (respectively T ∗ Y ) with the Γ –invariant functions on S = G/K (resp. T ∗ ( G/K ) ).If Γ has torsion, we shall use “smooth function on Y ” to mean a Γ -invariant smooth functionon S . For a compactly supported function χ on S , we denote Π Γ χ ( x ) = P γ χ ( γ.x ) . Thissum is finite for any x ∈ S , and hence defines a function on Y .On S , we fix once and for all a positive, smooth and compactly supported function φ such that P γ ∈ Γ φ ( γ.x ) ≡ . We call such a function a “smooth fundamental cutoff” or a“smooth fundamental domain”. Here we have used the assumption that Y is compact. Wealso introduce ˜ φ ∈ C ∞ c ( S ) which is identically on the support of φ . We note that for any D ∈ D and for any smooth Γ -invariant u on S we have(3.1) Π Γ (cid:16) ˜ φD ( φu ) (cid:17) = Π Γ D ( φu ) = D Π Γ φu = Du.
The analogue of left-quantization on R n in our setting associates to a function a on G/K × G/P × C Π the operator which acts on u ∈ C ∞ c ( G/K ) by(3.2) Op L ~ ( a ) u ( x ) = Z θ ∈ G/P ,λ ∈ C Π a ( x, θ, λ ) b u ( λ, θ ) e i ~ − λ,θ ( x ) dθ | c ~ ( λ ) | − dλ . A similar formula was introduced by Zelditch in [27] (with ~ = 1 ) in the case G = SL (2 , R ) ;it is shown there that a Op L ~ ( a ) is G -equivariant. The operator Op L ~ ( a ) can be definedif a belongs to a nice class of functions (possibly depending on ~ ). If a is smooth enoughand has reasonable growth, it will be a pseudodifferential operator. We give the regularityassumptions on a below. In any case, we shall always require a to be of the form b ◦ π , where b is a symbol on T ∗ ( G/K ) and π was defined in (1.1); besides, we will assume that b issupported away from the singular G -orbits in T ∗ ( G/K ) (which means that a is supportedaway from the walls in C Π ). This allows to identify a in a natural way with a functiondefined on (a subset of) T ∗ ( G/K ) .Let us define symbols of order m on T ∗ ( G/K ) (independent of ~ ) in the usual fashion : S m ( G/K ) := (cid:8) a ∈ C ∞ ( T ∗ ( G/K )) / for every compact F ⊂ G/K, for every α, β, there exists C such that | D αz D βξ a ( x, ξ )) | ≤ C (1 + | ξ | ) m −| β | for all ( x, ξ ) ∈ T ∗ ( G/K ) , x ∈ F (cid:9) . We also define semiclassical symbols of order m and degree l — thus called because theydepend on a parameter ~ :(3.3) S m,l ( G/K ) = { a ~ ( x, ξ ) = ~ l ∞ X j =0 ~ j a j ( x, ξ ) , a j ∈ S m − j } . This means that a ~ ( x, ξ ) has an asymptotic expansion in powers of ~ , in the sense that a − ~ l N − X j =0 ~ j a j ∈ ~ l + N S m − N for all N , uniformly in ~ . In this context, we denote S −∞ , + ∞ = ∩ m ≥ S − m,m . Remark 3.2.
As indicated above, we define symbols on
G/K × G/P × C Π by trans-porting the standard definition on T ∗ ( G/K ) through the map π (1.1). We will exclu-sively consider the case where a vanishes outside a fixed neighbourhood of the singular G -orbits in T ∗ ( G/K ) . In other words, a can be identified (through (1.1)) with a functionon G/K × G/P × C Π , that vanishes in a neighbourhood of G/K × G/P × ∂C Π . Defining agood pseudodifferential calculus using formula (3.2) for symbols supported near the wallsof C Π raises delicate issues about the behaviour of the c -function near the walls, and wedo not address this problem here. This is one among several reasons why we assume that Λ ∞ is regular in our main theorem.We now project this construction down to functions on Y , which we identify with Γ -invariant functions on S . Here we do not follow Zelditch, who defined the action of Op ~ ( a ) UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 17 on Γ -invariant functions in a global manner, using the Helgason-Fourier decomposition ofsuch functions. We continue to work locally, which is sufficient for our purposes.For us, the quantization of a ∈ S m,k ∩ C ∞ ( T ∗ Y ) (supported away from singular G -orbits)is defined to act on u ∈ C ∞ ( Y ) by:(3.4) Op ~ ( a ) u = Π Γ ˜ φ Op L ~ ( a ) φu ∈ C ∞ ( Y ) . Note that (3.1) and Remark 2.3 imply that Op ~ ( H ) = Γ − [ H ( − i ~ • )] for H ∈ H .The image of S m,k by this quantization will be denoted Ψ m,k ( Y ) . This quantizationprocedure depends on the fundamental cutoff φ and on ˜ φ . However, this dependence onlyappears at second order in ~ . The space Ψ m,k ( Y ) itself is perfectly well defined modulo Ψ −∞ , + ∞ ( Y ) = ∩ k ′ ,m ′ Ψ m ′ ,k ′ ( Y ) . Moreover, it coincides with the more usual definition ofpseudodifferential operators, defined using the euclidean Fourier transform in local coordi-nates .3.3. Action of Op ~ ( H ) on WKB states. Fix a Hamiltonian H ∈ H .The letter H will stand for several different objects which are canonically related: afunction H on T ∗ ( G/K ) , a W -invariant polynomial function on a ∗ , and an element of U ( a ) W . As such, we can also let H act as a left- G -invariant differential operator on G or G/M .In the following lemma, all functions on
G/K and
G/M are lifted to functions on G ,and in that sense we can apply to them any differential operator on G . If b is a functiondefined on G/M = G/K × G/P , and θ is an element of G/P , we denote b θ the functiondefined on G/K by b θ ( x ) = b ( x, θ ) . Lemma 3.3.
Let H ∈ H be of degree ¯ d , and let b be a smooth function on G/M . Fix λ ∈ a ∗ . Then, there exist D k ∈ U ( n C ) U ( a C ) of degree ≤ k (depending on λ and on H )such that for any θ ∈ G/P , for any x ∈ G/K , Op ~ ( H )[ b θ .e i ~ − λ,θ ]( x ) = H ( λ ) b ( x, θ ) − i ~ [ dH ( λ ) .b ]( x, θ ) + ¯ d X k =2 ~ k D k b ( x, θ ) ! e i ~ − λ,θ ( x ) . On the right H is seen as a function on a ∗ , so its differential dH ( λ ) is an element of a ,and it acts as a differential operator of order on G/M . Each operator D k actually definesa differential operator on G/M .Proof.
By linearity, it is enough to treat the case where H ∈ U ( a ) W is homogeneous ofdegree ¯ d . In this case, we have Op ~ ( H ) = ~ ¯ d Op ( H ) = ~ ¯ d Γ − [ H ( − i • )] . This could be checked by testing the action of Op ~ ( a ) on a local plane wave of the form φ ( x ) e iξ.x ~ inlocal euclidean coordinates. One then uses the stationary phase method and the facts that the complexphase of e ~ − λ,θ is ~ − λB ( x, θ ) , and that the covector ( x, d x λB ( x, θ )) ∈ T ∗ x ( G/K ) corresponds preciselyto ( x, θ, λ ) under the identification (1.1). We have also introduced the differential operator Op ( H ) = Γ − [ H ( − i • )] acting on G/K . These are not the same objects, but [23, Cor. 4.4] relates the two.
Consider the operator Z related to D = Op ( H ) by Lemma 2.1. We have Op ( H )[ b θ .e i ~ − λ,θ ]( x ) = Z [ b θ .e i ~ − λ,θ ]( x ) . In what follows we consider the point ( x, θ ) ∈ G/K × G/P . We choose a representativeof θ in K ( θ is then defined modulo M , but the calculations do not depend on the choiceof this representative). We write x = θnaK . This means that ( x, θ ) represents the point θnaM ∈ G/M . All functions on
G/K and
G/M are lifted to functions on G , and in thatsense we can apply to them any differential operator on G .By Lemma 2.1, we have Z [ b θ .e i ~ − λ,θ ]( x ) = Z [ b θ .e i ~ − λ,θ ]( θna ) = τ − ρ H ( − i • ) . [ b θ .e i ~ − λ,θ ]( θna )+ D [ b θ .e i ~ − λ,θ ]( θna ) where D ∈ U ( n C ) U ( a C ) ¯ d − .Because of the identity e i ~ − λ,θ ( θnag ) = e ( ρ + i ~ − λ ) B ( θna ) e ( ρ + i ~ − λ ) H ( g ) , (valid for any g ∈ N A ) we see that, for any D ∈ U ( n C ) U ( a C ) , the term D [ e i ~ − λ,θ ]( θna ) isof the form Ce i ~ − λ,θ ( θna ) , where the constant C depends on D and ~ − λ . This constant C is in fact polynomial in ~ − λ .This results in an expression : Z [ b θ .e i ~ − λ,θ ]( x ) = Z [ b θ .e i ~ − λ,θ ]( θna ) = τ − ρ H ( − i • ) . [ b θ .e i ~ − λ,θ ]( θna )+ " ¯ d − X k =0 ~ − k D ¯ d − k b ( θna ) e i ~ − λ,θ ( θna ) where D ¯ d − k ∈ U ( n C ) U ( a C ) depends only on λ and H .A term in ~ − k can only arise if e i ~ − λ,θ is differentiated k times; but Z being of degree ¯ d ,we see then that D ¯ d − k can be of order ¯ d − k at most. The last term, when multiplied by ~ d , becomes P ¯ dk =2 ~ k D k b . We do not know a priori if the function D ¯ d − k b (defined on G ) is M -invariant, but the sum P ¯ d − k =0 ~ − k D ¯ d − k b necessarily defines an M -invariant function on G , since all the other terms do. Since ~ is arbitrary, we see that each D k must necessarilysend an M -invariant function to an M -invariant function.Finally, we write τ − ρ H ( − i • ) . [ b θ .e i ~ − λ,θ ]( θnaM ) = H ( − i • )[ b θ .e i ~ − λ − ρ,θ ] .e ,θ ( θnaM )= [ τ i ~ − λ H ( − i • ) .b θ ] .e i ~ − λ,θ ( θnaM ) . When multiplying by ~ ¯ d , and using the Taylor expansion of H at λ , we have ~ ¯ d τ i ~ − λ H ( − i • ) = H ( λ ) − i ~ dH ( λ ) + ¯ d X k =2 ( − i ~ ) k k ! d ( k ) H ( λ ) . (cid:3) We will refer to a function of the form x b θ ( x ) e i ~ − λ,θ ( x ) as a WKB state , using thelanguage of semiclassical analysis.
UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 19
Symplectic lift.
Let ψ be a D -eigenfunction, of spectral parameter ν . We let ~ = k ν k − (the choice of the norm here is arbitrary, one can take the Killing norm forinstance). We sometimes write ψ = ψ ν to indicate the spectral parameter, but this nota-tion is imprecise in that ψ may not be uniquely determined by ν .To ψ ν we attach a distribution ˜ µ ψ (sometimes denoted ˜ µ ν ) on T ∗ Y : for a ∈ C ∞ c ( T ∗ Y ) set ˜ µ ψ ( a ) = h ψ, Op ~ ( a ) ψ i L ( Y ) As described in Section 1 we are trying to classify weak-* limits of the distibutions ˜ µ ν in the limit ν → ∞ . We fix such a limit (“semiclassical measure”) µ and a sequence ( ψ j ) j ∈ N = ( ψ ν j ) j ∈ N of eigenfunctions such that the corresponding sequence (˜ µ ν j ) convergesweak-* to µ . In the sequel we write ν for ν j . We assume that ν goes to infinity in theconditions of paragraph 1.2, the limit ν ∞ assumed to be regular. We let ~ = k ν k − . Writing
Λ = Λ ν = ~ ℑ m ( ν ) we have Λ −→ Λ ∞ = ℑ m ( ν ∞ ) = − iν ∞ . Note that ℜ e ( ν ) is bounded[15, §16.5(7) & Thm. 16.6]), so that ~ ν = iλ ν + O ( ~ ) . Necessarily ν ∞ is purely imaginary.With the notations of Section 2.2, the state ψ ν satisfies(3.5) Op ~ ( H ) .ψ ν = H ( − i ~ ν ) ψ ν for all H ∈ H . From now on, we fix a Hamiltonian H ∈ H . The letter H will stand for twodifferent objects that are canonically related: a function H on T ∗ ( G/K ) ( G -invariant andpolynomial in the fibers of the projection T ∗ ( G/K ) −→ G/K ), a W -invariant polynomialfunction on a ∗ , an element of U ( a ) W .We denote X Λ = dH (Λ) ∈ a . Since Λ is only defined up to an element of W , so is X Λ .One can assume that α ( X Λ ∞ ) ≥ for all α ∈ ∆ + . For simplicity (and without loss ofgenerality), we will also assume that Λ ∞ belongs to the Weyl chamber C Π . Other miscellaneous notations: d is the dimension of G/K , r is the rank, and J the dimension of N (so that d = r + J ). We call ˜ J the number of roots. We index thepositive roots α , . . . , α ˜ J in such a way that α ( X Λ ∞ ) ≤ α ( X Λ ∞ ) ≤ . . . ≤ α ˜ J ( X Λ ∞ ) (withour previous notations, we have α ˜ J ( X Λ ∞ ) = χ max ( H ) ). We fix K as in Theorem 1.12, andwe denote j = j ( X Λ ∞ ) the largest index j such that α j ( X Λ ∞ ) < K .With w ℓ ∈ W the long element, we set: n fast = ⊕ j>j g α j , n slow = ⊕ j ≤ j g α j , ¯ n fast = ⊕ j>j g w ℓ .α j , ¯ n slow = ⊕ j ≤ j g w ℓ .α j J = dim n slow = P j ≤ j m α j . The spaces n fast and ¯ n fast aresubalgebras, in fact ideals, in n , ¯ n respectively; they generate subgroups N fast , N fast thatare normal in N, N respectively. 4.
The WKB Ansatz
We now start the proof of Theorem 1.12. We first describe how the operator P χω acts onWKB states. In Section 5, we will use the fact that these states form a kind of basis toestimate the norm of the operator.4.1. Goal of this section.
Fix a sequence ω = ( ω − T , · · · , ω − , ω , · · · ω T − ) , of length T chosen so that T η ≤ K| log ~ | . Theorem 1.12 requires us to estimate the norm of the operator P χω acting on L ( Y ) (for a suitable choice of the time step η ). This operator isthe same as U − ( T − η P where P = P ω T − U η . . . U η P / ω Op ~ ( χ ) P / ω U η . . . P ω − T +1 U η P ω − T , where we recall that U t = exp( i ~ − t Op ~ ( H )) . On the “energy layer” E λ , U t quantizes the action of e − tX λ , in other words the time − t ofthe Hamiltonian flow generated by H . Under the action of e − tX λ for t ≥ , elements of n are expanded and elements of ¯ n are contracted (the vector X Λ may be singular, so thatthese stable or unstable spaces can also contain neutral directions).In what follows we estimate the norm of P . To do so, we will first describe how P actson our Fourier basis e i ~ − λ,θ , using the technique of WKB expansion (§4.2). Then, we willuse the Cotlar-Stein lemma (§5) to estimate as precisely as possible the norm of P .The sequence ω − T , . . . , ω T − is fixed throughout this section. Instead of working withfunctions on Y we work with functions on G/K that are Γ -invariant. For instance, P ω is themultiplication operator by the Γ –invariant function P ω . We assume that each connectedcomponent of the support of P ω has very small diameter (say ǫ ). We will fix Q ω , a functionin C ∞ c ( S ) such that Π Γ Q ω = P ω and such that the support of Q ω has diameter ǫ . We alsodenote Q ω the corresponding multiplication operator. Finally we need to introduce Q ′ ω in C ∞ c ( S ) which is identically on the support of Q ω and supported in a set of diameter ǫ .We decompose(4.1) P = S ∗ U χ where U χ = Op( χ ) P / ω U η P ω − . . . U η b P ω − T +1 U η P ω − T and S = P / ω . . . U − η P ω T − U − η P ω T − . The WKB Ansatz for the Schrödinger propagator .
We recall some standardcalculations, already done in [3], with some additional simplifications coming from the factthat the functions e i ~ − λ,θ are eigenfunctions of Op ~ ( H ) .On S , let us try to solve − i ~ ∂ ˜ u∂t = Op ~ ( H )˜ u, in other words ˜ u ( t ) = U t ˜ u (0) , with initial condition the WKB state ˜ u (0 , x ) = a ~ (0 , x ) e i ~ − λ,θ ( x ) . We only consider t ≥ .We assume that a ~ is compactly supported and has an asymptotic expansion in all C l norms as a ~ ∼ P k ≥ ~ k a k . We look for approximate solution up to order ~ M , in the form u ( t, x ) = e itH ( λ ) ~ e i ~ − λ,θ ( x ) a ~ ( t, x ) = e itH ( λ ) ~ e i ~ − λ,θ ( x ) M − X k =0 ~ k a k ( t, x ) . UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 21
Let us denote(4.2) u ( t, x ) = e itH ( λ ) ~ e i ~ − λ,θ ( x ) a ~ ( t, x, θ, λ ) = e itH ( λ ) ~ e i ~ − λ,θ ( x ) M − X k =0 ~ k a k ( t, x, θ, λ ) to keep track of the dependence on θ and λ ; the pair ( x, θ ) then represents an elementof G/K × G/P = G/M . Identifying powers of ~ , and using Lemma 3.3, we find theconditions:(4.3) ∂a ∂t ( x, θ ) = [ dH ( λ ) .a ]( x, θ ) ( -th transport equation) ∂a k ∂t ( x, θ ) = [ dH ( λ ) .a k ]( x, θ ) + i P ¯ dl =2 P l + m = k +1 D l a m ( x, θ ) ( k -th transport equation) . The equations (4.3) can be solved explicitly by a ( t, ( x, θ ) , λ ) = a (0 , ( x, θ ) e tX λ , λ ) , in other words a ( t ) = R ( e tX λ ) a (0) , where R here denotes the action of A on functions on G/M by right translation; and a k ( t ) = R ( e tX λ ) a k (0) + Z t R ( e ( t − s ) X λ ) i ¯ d X l =2 X l + m = k +1 D l a m ( s, x, θ ) ! ds. If we now define u by (4.2), u solves − i ~ ∂ ˜ u∂t = Op ~ ( H )˜ u − e itH ( λ ) ~ e i ~ − λ,θ " ¯ d X l =2 M − X k = M +1 − l ~ k + l D l a k and thus k u ( t ) − U t u (0) k L ( S ) ≤ Z t " ¯ d X l =2 M − X k = M +1 − l ~ k + l − k D l a k ( s ) k L ( S ) ds ≤ te (2 M + ¯ d − t max α ∈ ∆+ α ( X λ ) − " ¯ d X l =2 M − X k = M +1 − l ~ k + l − k X j =0 k a k − j (0) k C j + l ≤ Ct ~ M e (2 M + ¯ d − t max α ∈ ∆+ α ( X λ ) − " M − X k =0 k a k (0) k C M − k )+ ¯ d − . Since D k belongs to U ( n C ) U ( a C ) , in the co-ordinates ( x, θ ) it only involves differentiationwith respect to x . We also recall that D k is of order k . We have used the following estimateon the flow R ( e tX λ ) (for t ≥ ) : k d N dx N a (( x, θ ) e tX λ ) k ≤ e − tN min α ∈ ∆+ α ( X λ ) k d N dx N a (( x, θ ) k and we have denoted x − = max( − x, . Remark 4.1.
In what follows we will always have λ ∈ supp( χ ) , where by assumption χ issupported on a tubular neighbourhood of size ǫ of E Λ ∞ , and α (Λ ∞ ) ≥ for α ∈ ∆ + . Forsuch λ we have α ( X λ ) ≥ − ǫ for all α ∈ ∆ + . We see that our approximation method makessense if t is restricted by ~ M e (2 M + ¯ d − tǫ ≪ . Since ǫ can be chosen arbitrarily small, wecan assume that the WKB approximation is good for t ≤ K| log ~ | . Remark 4.2.
On the quotient Y = Γ \ S , the same method applies to find an approximatesolution of U t Π Γ u (0) in the form Π Γ u ( t ) , with the same bound(4.4) k Π Γ u ( t ) − U t Π Γ u (0) k L ( Y ) ≤ Ct ~ M e ǫt (2 M + ¯ d − " M − X k =0 k a k (0) k C M − k )+ ¯ d − , provided that the projection S −→ Y is bijective when restricted to the support of a ~ ( t ) .If λ stays in a compact set and if the support of a ~ (0) has small enough diameter ǫ , thiscondition will be satisfied in a time interval t ∈ [0 , T ] . In the applications below, we mayand will always assume that η < T .We can iterate the previous WKB construction T times to get the following descriptionof the action of U χ on Π Γ Q ′ ω − T e i ~ − λ,θ (the induction argument to control the remaindersat each step is the same as in [3] and we won’t repeat it here): Proposition 4.3. (4.5) U χ (Π Γ Q ′ ω − T e i ~ − λ,θ ) = Π Γ h e iTηH ( λ ) ~ e i ~ − λ,θ A ( T ) M ( • , θ, λ ) i + O L ( Y ) ( ~ M ) k Q ′ ω − T e i ~ − λ,θ k L ( S ) where A ( T ) M ( x, θ, λ ) = M − X k =0 ~ k a ( T ) k ( x, θ, λ ) . The function a ( T )0 ( x, θ, λ ) is equal to a ( T )0 ( x, θ, λ ) = χ ( λ ) P / ω ( x ) P ω − (( x, θ ) e ηX λ ) P ω − (( x, θ ) e ηX λ ) . . . Q ω − T (( x, θ ) e T ηX λ ) , where we have lifted the functions P ω (originally defined on G/K ) to
G/M = G/K × G/P .The functions a ( T ) k have the same support as a ( T )0 . Moreover, if we consider a ( T ) k as afunction of ( x, θ ) , that is, as a function on G/M , we have the following bound k Z mα a ( T ) k k ≤ P k,m,Z α ( T ) sup j =0 ,...T { e − ( m +2 k ) jη α ( X λ ) } if Z α belongs to g α ( P k,m,Z α ( T ) is polynomial in T ). In particular, for α ∈ ∆ + , k Z mα a ( T ) k k ≤ P k,m,Z α ( T ) e ( m +2 k ) T η ǫ
The energy parameter λ will always stay ǫ -close to Λ ∞ . Recall that we denote by thesame letter ǫ the diameter of the support of each Q ω . We choose ǫ and η (the time step) UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 23 small enough to ensure the following: there exists γ = γ ω − T ,...,ω ∈ Γ (independent of θ or λ ) such that(4.6) a ( T )0 ( x, θ, λ ) = χ ( λ ) Q / ω ◦ γ − ( x ) P ω − (( x, θ ) e ηX λ ) P ω − (( x, θ ) e ηX λ ) . . . Q ω − T (( x, θ ) e T ηX λ ) . This means that the function a ( T )0 ( • , θ, λ ) is supported in a single connected component ofthe support of P / ω .We will also use the following variant: Proposition 4.4.
Let γ = γ ω − T ,...,ω . U χ ( Q ′ ω − T ◦ γ e i ~ − λ,θ ) = e iTηH ( λ ) ~ e i ~ − λ,θ ( x ) A ( T ) M ◦ γ ( x, θ, λ ) + O ( ~ M ) k Q ′ ω − T ◦ γ e i ~ − λ,θ k where A ( T ) M ( x, θ, λ ) = M − X k =0 ~ k a ( T ) k ( x, θ, λ ) . Remark 4.5.
For the operator S , analogous results can be obtained if we replace every-where λ by w ℓ .λ , − t by + t , and the label ω − j by ω + j . Remark 4.6.
Let u, v ∈ L ( Y ) . We explain how the previous Ansatz can be used toestimate the scalar product h v, U χ u i L ( Y ) (up to a small error). This is done by decomposing u and v , locally, into a combination of the functions e i ~ − λ,θ (using the Helgason-Fouriertransform), and inputting our Ansatz into this decomposition.In more detail, we note that P ω − T = P ω − T Π Γ Q ′ ω − T , so that U χ u = U χ Π Γ Q ′ ω − T u . We usethe Fourier decomposition to write Q ′ ω − T u ( x ) = Q ′ ω − T ( x ) Z θ ∈ G/P ,λ ∈ C Π \ Q ′ ω − T u ( λ, θ ) e i ~ − λ,θ ( x ) dθ | c ~ ( λ ) | − dλ. By Cauchy-Schwarz and the asymptotics of the c -function (Remark 3.1), we note that Z χ ( λ ) =0 | \ Q ′ ω − T u ( λ, θ ) | dθ | c ~ ( λ ) | − dλ = O ( ~ − d/ ) k u k L ( Y ) , and write(4.7) h v, U χ u i L ( Y ) = D v, U χ Π Γ Q ′ ω − T u E L ( Y ) = Z χ ( λ ) =0 \ Q ′ ω − T u ( λ, θ ) D v, U χ Π Γ Q ′ ω − T e i ~ − λ,θ E dθ | c ~ ( λ ) | − dλ + O ( ~ ∞ ) k u k L ( Y ) k v k L ( Y ) . We now use Proposition 4.3 to replace U χ by the Ansatz, D v, U χ Π Γ Q ′ ω − T e i ~ − λ,θ E L ( Y ) = D v, e iTηH ( λ ) ~ e i ~ − λ,θ A ( T ) M ( • , θ, λ ) E L ( S ) + O ( ~ M ) k v k L ( Y ) = D Q ′ ω ◦ γ − . v, e iTηH ( λ ) ~ e i ~ − λ,θ A ( T ) M ( • , θ, λ ) E L ( S ) + O ( ~ M ) k v k L ( Y ) = D Q ′ ω v, e iTηH ( λ ) ~ e i ~ − λ,θ ◦ γ A ( T ) M ( γ • , θ, λ ) E L ( S ) + O ( ~ M ) k v k L ( Y ) for γ = γ ω − T ,...,ω defined above. Thus,(4.8) h v, U χ u i L ( Y ) = Z χ ( λ ) =0 \ Q ′ ω − T u ( λ, θ ) D Q ′ ω v, e iTηH ( λ ) ~ e i ~ − λ,θ ◦ γ A ( T ) M ( γ • , θ, λ ) E L ( S ) dθ | c ~ ( λ ) | − dλ + O ( ~ M − d/ ) k v k L ( Y ) k u k L ( Y ) . In this last line we see that replacing the exact expression of U χ by the Ansatz inducesan error of O ( ~ M − d/ ) k v k L ( Y ) k u k L ( Y ) . We will take M very large, depending on theconstant K in Theorem 1.12, so that the error O ( ~ M − d/ ) is negligible compared to thebound announced in the theorem.5. The Cotlar–Stein argument.
We now use the previous approximations of U χ and S to estimate the norm of P . Thisis done in a much finer, and more technical manner, than in [1, 3], because we want toeliminate the slowly expanding/contracting directions.5.1. The Cotlar-Stein lemma.Lemma 5.1.
Let
E, F be two Hilbert spaces. Let ( A α ) ∈ L ( E, F ) be a countable family ofbounded linear operators from E to F . Assume that for some R > we have sup α X β k A ∗ α A β k ≤ R and sup α X β k A α A ∗ β k ≤ R Then A = P α A α converges strongly and A is a bounded operator with k A k ≤ R . We refer for instance to [7] for the proof.5.2.
A non-stationary phase lemma.
The following lemma is just a version of integra-tion by parts.
Lemma 5.2.
Let Ω be an open set in a smooth manifold. Let Z be a vector field on Ω and µ be a measure on Ω with the property that R ( Zf ) dµ = R f J dµ for every smooth function f and for some smooth J .Let S ∈ C ∞ (Ω , R ) and a ∈ C ∞ c (Ω) . Assume that ZS does not vanish. Consider theintegral (5.1) I ~ = Z e iS ( x ) ~ a ( x ) dµ ( x ) . Then we have I ~ = i ~ R e iS ( x ) ~ D Z a ( x ) dµ ( x ) , where the operator D Z is defined by D Z a = Z (cid:16) aZS (cid:17) − aJZS . UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 25
If we iterate this formula n times we get I ~ = ( i ~ ) n Z e iS ( x ) ~ D nZ a ( x ) dµ ( x ) and D nZ has the form D nZ a = X m ≥ n,k + m ≤ n, P l j ≤ n f k, ( l j ) ,m Z k aZ l S . . . Z l r S ( ZS ) m where the f k, ( l j ) ,m ( x ) are smooth functions that do not depend on a nor S . Study of several phase functions.
Sum of two Helgason phase functions.
Proposition 5.3. (i) Let g P , g P ∈ G/P be two points on the boundary. Let λ, µ ∈ C Π be two elements of the closed nonnegative Weyl chamber. Consider the function on G/K , (5.2) gK λ.H ( g − gK ) + µ.H ( g − gK ) . Then, this map has critical points if and only if µ = − Ad( w ℓ ) .λ .(ii) Let λ, µ ∈ C Π be two (regular) elements of the positive Weyl chamber. Let g P , g P ∈ G/P be two points on the boundary, and assume that g − g ∈ P w ℓ P (we don’t assumehere that the conclusion of (i) is satisfied). Write g − g = b w ℓ b with b , b ∈ P .Then, the set of critical points for variations of the form t λ.H ( e tX g − gK ) + µ.H ( e tX g − gK ) , with X ∈ n is precisely { gK, g ∈ g b A } . Moreover, these critical points are non-degenerate. Remark 5.4.
The set of critical points is { gK, g ∈ g P , gw ℓ ∈ g P } , that is, the flat in G/K determined by the two boundary points g P , g P . Proof. (i) It is enough to consider the case g = e . By the Bruhat decomposition, we knowthat there exists a unique w ∈ W such that g ∈ BwB , that is, g = b wb for some b , b ∈ B . The map (5.2) has the same critical points as the map(5.3) gK λ.H ( gK ) + µ.H ( w − b − gK ) , and those are the image under gK b gK of the critical points of(5.4) gK λ.H ( gK ) + µ.H ( w − gK ) . For X ∈ a the derivative at t = 0 of(5.5) t λ.H ( e tX gK ) + µ.H ( w − e tX gK ) is λ ( X ) + µ (Ad( w − ) X ) . Thus, for the map (5.4) to have critical points, we must have λ ( X ) + µ (Ad( w − ) X ) = 0 for every X ∈ a . Letting X vary over the dual basis to a positive basis of a ∗ , we see that µ = − Ad( w ) .λ is nonnegative, and this is only possible if µ = − Ad( w ℓ ) .λ (this does notnecessarily mean that w = w ℓ if λ is not regular). (ii) Here we assume that µ and λ are regular, and that we are in the “generic” case where g − g ∈ P w ℓ P . Starting from (5.4), we now consider variations of the form(5.6) t λ.H ( e tX gK ) + µ.H ( w − ℓ e tX gK ) for X ∈ n . The term λ.H ( e tX gK ) is constant, and it remains to deal with µ.H ( w − ℓ e tX gK ) .Write g = w ℓ anK , n ∈ N, a ∈ A , and denote Y = Ad( w ℓ ) .X ∈ ¯ n , Y ′ = Ad( a − ) Y . Wehave µ.H ( w − ℓ e tX gK ) = µ.H ( e tY anK ) = µ ( a ) + µ.H ( e tY ′ nK ) = µ ( a ) + µ.H ( n − e tY ′ nK ) . Hence ddt µ.H ( e tY anK ) = µ.H (Ad( n − ) Y ′ ) . We see that the set of critical points of (5.6) is the set of those points gK , with g = w ℓ anK such that n satisfies µ.H (Ad( n − ) Y ′ ) = 0 for all Y ′ ∈ ¯ n . Since µ is regular, one can checkthat this implies n = e . This proves the first assertion of (ii).Finally, assume that we are at a critical point, that is, gK = aK in (5.6). We calculatethe second derivative at t = 0 of t µ.H ( w − ℓ e tX aK ) when X ∈ n . We keep the samenotation as above for Y and Y ′ .Let U = Y ′ − θ ( Y ′ ) ∈ k . By the Baker-Campbell-Hausdorff formula, we have(5.7) e tY ′ = e tθ ( Y ′ )+ t [ Y ′ ,θ ( Y ′ )]+ O ( t ) e tU = e tθ ( Y ′ ) e t [ Y ′ ,θ ( Y ′ )]+ O ( t ) e tU . Remember that θ ( Y ′ ) ∈ n , and that H is left- N -invariant. This calculation shows thatthe second derivative of t µ.H ( w − ℓ e tX aK ) is the quadratic form X µ ([ Y ′ , θ ( Y ′ )]) , where Y ′ = Ad( a − ) Ad( w ℓ ) .X . This is a non-degenerate quadratic form if µ is regular. (cid:3) Variations with respect to N . In this section we need the decomposition g = n ⊕ a ⊕ m ⊕ ¯ n . We will denote π n , π a , π ¯ n the corresponding projections. We note that π a = H ,since ¯ n ⊂ n + k . Lemma 5.5.
Fix n ∈ N and a ∈ A . Then there exist two neighbourhoods V , V of in ¯ n ,and a diffeomorphism Ψ = Ψ na : V −→ V such that e − Y nae Y ∈ N A, Y ∈ V , Y ∈ V ⇐⇒ Y = Ψ( Y ) . Moreover, the differential at of Ψ (denoted Ψ ′ ) preserves the subalgebra ¯ n slow . Finally, ifwe write e − Y nae Ψ( Y ) = n ( Y ) a ( Y ) , we have a ′ .Y = π a [Ad( na )Ψ ′ ( Y )] . Proof.
We apply the implicit function theorem. For Y = 0 , the differential of Y nae Y ( na ) − at Y = 0 is Y Ad( na ) .Y . What we need to check is the equivalence of UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 27 π ¯ n [Ad( na ) .Y ] = 0 and Y = 0 , which is the case since Ad( na ) preserves n ⊕ a ⊕ m . So theexistence of Ψ is proved, in addition the differential Ψ ′ is defined by Y = π ¯ n [Ad( na ) . Ψ ′ .Y ] for Y ∈ ¯ n . Since Ad( na ) preserves the space n ⊕ a ⊕ m ⊕ ¯ n slow (without preserving thedecomposition, of course), Ψ ′ .Y must belong to ¯ n slow if Y does.The last formula is simply obtained by differentiating e − Y nae Ψ( Y ) = n ( Y ) a ( Y ) . (cid:3) For the next lemma we need to recall our two decompositions ¯ n = P k ≤ j g w ℓ .α k ⊕ P j>j g w ℓ .α k = ¯ n slow ⊕ ¯ n fast and n = P k ≤ j g α k ⊕ P j>j g α k = n slow ⊕ n fast . The space n fast is an ideal of n , and we call N fast the associated (normal) subgroup. Lemma 5.6. (i) The set { n ∈ N, H (Ad( n ) Y ) = 0 ∀ Y ∈ ¯ n slow } is, near identity, a submanifold of N , tangent to n fast .(ii) If n is close enough to identity, if we write n = e P α T α with T α ∈ g α ( α ∈ ∆ + ), andif µ ∈ a ∗ is a regular element, we have (cid:12)(cid:12)(cid:12)(cid:12) µ.H (cid:18) Ad( n ) θ ( T β ) k θ ( T β ) k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C µ k T β k with C µ > .Proof. The differential of n H (Ad( n ) Y ) is Z H ([ Z, Y ]) ( Z ∈ n ). Write Z = Z slow + Z fast , Z slow = P Z α , with Z α ∈ g α , and take Y = θ ( Z β ) for some β . We have H ([ Z, Y ]) = −h Z β , Z β i H β where H β ∈ a is the coroot [14, Ch. VI §5, Prop. 6.52]. Notethat µ ( H β ) = h µ, β i , hence µ ( H β ) = 0 if µ is regular. This proves the lemma. (cid:3) First decomposition of P . We want to use the Cotlar-Stein lemma to estimate thenorm of the operator P , defined in (4.1). To do so, we will decompose P into many pieces.Our first decomposition of P is obtained by covering the boundary G/P by a finite numberof small sets Ω , . . . , Ω M described below. We use the fact that there is a neighbourhood Ω of eP in G/P that is diffeomorphic to a neighbourhood of e in N , via the map N −→ G/P ¯ n ¯ nP . Using compactness, we can find an open cover of
G/P by a finite number of open sets Ω , . . . , Ω M such that, for every m , there exists g m ∈ G with Ω m ⊂ g m Ω ⊂ g m N P .Introduce a family of smooth functions χ Ω m on G/P such that χ Ω m is supported inside Ω m and P m χ Ω m ≡ . We then define the pseudodifferential operators Q m u ( x ) = Z b u ( w ℓ .λ, k ) Q ′ ω ( x ) χ Ω m ( k ) e i ~ − w ℓ .λ,k dk | c ~ ( λ ) | − dλ, and P m u = Π Γ S ∗ Q ∗ m U χ u Obviously, P = P m P m . The sum over m is finite, and we now fix m . The variable k stays in g m N P . Remark 5.7.
Let γ = γ ω T − ,...,ω defined as in (4.6). Proposition (4.4) (and Remark 4.5)can be generalized to(5.8) Q m S (cid:16) Q ′ ω T − ◦ γ e i ~ − w ℓ .µ,k (cid:17) = e − iTηH ( µ ) ~ e i ~ − w ℓ .µ,k B ( T ) M ◦ γ ( x, k, w ℓ .µ )+ O L ( S ) ( ~ M ) k Q ′ ω T ◦ γ e i ~ − w ℓ .µ,k k L ( S ) , where now B ( T ) M ( x, k, w ℓ .µ ) = M − X k =0 ~ k b ( T ) k ( x, k, w ℓ .µ ) , (5.9) b ( T )0 ( x, k, w ℓ .µ ) = χ Ω m ( k ) P / ω ( x ) P ω (( x, k ) e − ηX wℓ.µ ) P ω (( x, k ) e − ηX wℓ.µ ) . . . Q ω T − (( x, k ) e − ( T − ηX wℓ.µ )= χ Ω m ( k ) Q / ω ◦ γ − ( x ) P ω (( x, θ ) e − ηX wℓ.µ ) P ω (( x, k ) e − ηX wℓ.µ ) . . . Q ω T − (( x, k ) e − ( T − ηX wℓ.µ ) and the next terms have the same support as the leading one (their derivatives are boundedthe same way as in Proposition 4.3).In the next paragraphs we will concentrate our attention on brackets of the form: D Q ′ ω T − ◦ γ e i ~ − w ℓ .µ,k , S ∗ Q ∗ m U χ Q ′ ω − T ◦ γ e i ~ − λ,θ E L ( S ) , for λ, µ ∈ C Π , θ, k ∈ G/P . We take γ = γ ω − T ,...,ω and γ = γ ω T − ,...,ω as defined in (4.6).These are none other than the matrix elements of the operator P m in the Fourier basis e i ~ − λ,θ .5.5. Second decomposition of P . The index m being fixed, we will apply the Cotlar-Stein lemma to bound the norm of P m . We decompose P m as a sum of countably manyoperators, and this decomposition is much more involved.We have assumed that we have a diffeomorphism from a relatively compact subset of N to Ω m : ¯ n g m ¯ n P . We can write the Haar measure on Ω m as dk = Jac(¯ n ) d ¯ n , where Jac is a smooth function on N (we suppress from the notation its dependence on g m ).An element ( x, k ) ∈ G/K × Ω m corresponding to the point g m ¯ n n a M ∈ G/M can alsobe represented as ( g m ¯ n n a K, g m ¯ n ) ∈ G/K × g m N . Accordingly we now write denote e i ~ − w ℓ .µ,g m ¯ n for e i ~ − w ℓ .µ,k . Let us look at a scalar product D Q m S Q ′ ω T − ◦ γ e i ~ − w ℓ .µ ,g m ¯ n P , U χ Q ′ ω − T ◦ γ e i ~ − λ,θ E .We only need to consider the generic case where θ ∈ g m ¯ n P w ℓ P , that is, θ is of the form g m ¯ n n w ℓ P (with n ∈ N ). In addition, we always assume that λ and µ are regular.Proposition 5.3 (ii) tells us that the stationary points of the phase function gK λ.H ( θ − gK ) − ( w ℓ .µ ) .H ( k − gK ) , k = g m ¯ n P with respect to variations ( g m ¯ n n ) e tX ( g m ¯ n n ) − gK, X ∈ n , UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 29 are the points of the form gK = g m ¯ n n a K with a ∈ A . Thus the set of critical pointsis of codimension J . The stationary phase method then gives:(5.10) D Q m S Q ′ ω T − ◦ γ e i ~ − w ℓ .µ,g m ¯ n P , U χ Q ′ ω − T ◦ γ e i ~ − λ,θ E = ~ J/ Z a ∈ A d ( λ, a ) C ~ ( g m ¯ n n a M, λ, w ℓ .µ ) ¯ e i ~ − w ℓ .µ,g m ¯ n P ( g m ¯ n n a K ) e i ~ − λ,g m ¯ n n w ℓ P ( g m ¯ n n a K ) da where C ~ ( g m ¯ n n a M, λ, w ℓ .µ ) ∼ P ~ k c k ( g m ¯ n n a M, λ, w ℓ .µ ) and c ( g m ¯ n n a M, λ, w ℓ .µ ) = (cid:16) A ( T ) M ◦ γ ( g m ¯ n n a w ℓ M, λ ) (cid:17) (cid:16) ¯ B ( T ) M ◦ γ ( g m ¯ n n a M, w ℓ .µ ) (cid:17) . (and the next terms have the same support as the leading one). The term d ( λ, a ) is theprefactor involving the hessian of the phase function in the application of the method ofstationary phase, it is a smooth function. So the asymptotics of our scalar product onlytakes into account the elements g m ¯ n n a M with A ( T ) M ◦ γ ( g m ¯ n n a w ℓ M, λ ) ¯ B ( T ) M ◦ γ ( g m ¯ n n a M, w ℓ .µ ) = 0 . Lemma 5.8.
Assume that the diameter of Ω and of supp Q ω is smaller than ǫ . Thenthere exist n ∈ N and a in A such that B ( T ) M ◦ γ ( g m ¯ n n a M, w ℓ .µ ) = 0 implies n a = n a g , where g ∈ N A is ǫ -close to identity.Proof. Just note from the expression of B ( T ) M ◦ γ that, if it is not , we must have g m ¯ n n a ∈ supp Q ω . The element g m varies in a finite set and ¯ n varies over Ω which is of diameter ≤ ǫ . Wealso assume that supp Q ω is of diameter ≤ ǫ , so that n (and a ) must both vary in setsof diameter ≤ ǫ . (cid:3) It follows that ¯ n n a M itself is ǫ -close to n a M in G/M . From now on we write g m ¯ n n a M = g m n a gM , where gM ∈ G/M varies in a neighbourhood of eM of diameter ≤ ǫ . We will always choose a representative g ∈ exp( n ⊕ a ⊕ ¯ n ) . By G -equivariance wemay assume g m n a = 1 , which we do from now on. Proposition 5.9. (Contracting and expanding foliations) (1)
Let µ be such that α k ( X µ ) > for all α k ∈ ∆ + with k > j (this is of coursethe case if µ is close enough to Λ ∞ ). Suppose we have gM and g ′ M both ǫ -closeto eM such that B ( T ) M ◦ γ ( gM, w ℓ .µ ) = 0 and B ( T ) M ◦ γ ( g ′ M, w ℓ .µ ) = 0 , then g ′ − g = exp( X + P α ∈ ∆ + Y α + P α ∈ ∆ + Y w ℓ .α ) with X ∈ a , Y α ∈ g α , k X k , k Y α k ≤ ǫ ,and k Y w ℓ .α k k ≤ ǫe − T η ( w ℓ .α k )( X wℓ.µ ) = ǫe − T ηα k ( X µ ) for k > j . (2) Similarly, assume that α k ( X λ ) > for all α k ∈ ∆ + with k > j . Suppose wehave gM and g ′ M both ǫ -close to eM such that A ( T ) M ◦ γ ( gw ℓ M, λ ) = 0 and A ( T ) M ◦ γ ( g ′ w ℓ M, λ ) = 0 . Then g ′ − g = exp( X + P α Y α ) with X ∈ a , Y α ∈ g α , k X k , k Y α k ≤ ǫ , k Y α k k ≤ ǫe − T ηα k ( X λ ) for k > j . Actually, the claim holds for all k (not only for k > j ), but we will only use it for k > j .For the other indices, there is something more optimal to do. Proof.
Assume that the term B ( T ) M ◦ γ ( gM, w ℓ .µ ) does not vanish. The evolution equation(5.9) shows that we must have • ge − ( T − ηX wℓ.µ M ∈ γ − . supp Q ω T − ; • gM ∈ supp Q ω .If gM and g ′ M both satisfy the two conditions above, then we see that g ′− g must be ǫ -close to identity. For ǫ small enough we can write this element using the co-ordinatesdescribed in part (1) of the claim. Also, e ( T − ηX wℓ.µ g ′− ge − ( T − ηX wℓ.µ must stay in thefixed compact set M [supp Q ω T − ] − supp Q ω T − M ⊂ G. Writing the action of A in the co-ordinate system gives the claim. The proof of the secondpart is similar. (cid:3) Finally we write gM = ¯ nnaM with ¯ n ∈ N , n ∈ N, a ∈ A all ǫ -close to . We decompose n = e Y n fast , and ¯ n = e ¯ Y ¯ n fast , Y ∈ n slow ≃ R J , Y ∈ ¯ n slow ≃ R J both ǫ -close to (we fix avector space isomorphism that sends the root spaces to the coordinate axes of R J ); and n fast ∈ N fast , ¯ n fast ∈ N fast both ǫ -close to . The quantity ǫ is fixed, but can be chosenas small as we wish. Note that the previous Proposition restricts n fast and ¯ n fast to sets ofmeasure Q k>j ǫe − T ηm αk α k ( X λ ) and Q k>j ǫe − T ηm αk α k ( X µ ) , respectively.We will now break P m into countably many pieces, P m = X (¯ y,y,t,λ ) ∈ Z J × Z r P m, (¯ y,y,t,λ ) to which we will apply the Cotlar-Stein lemma.For j = J and j = r choose a smooth nonnegative compactly supported function χ j on R j such that(5.11) X y ∈ Z j χ j ( Y − y ) ≡ and such that χ j ( Y ) .χ j ( Y + 2 y ) = 0 for all Y ∈ R j and y ∈ Z j \ { } . Let (¯ y, y ) ∈ Z J and let ( t, λ ) ∈ Z r . Denote + a fixed real number > . Define χ ~ (¯ y,y ) ( Y , Y ) = χ J ( ~ − / + Y − ¯ y ) χ J ( ~ − / + Y − y ) ; and χ ~ λ ( λ ) = χ r ( ~ − / + λ − λ ) and χ ~ t ( a ) = χ r ( ~ − / + a − t ) . Also define χ ~ (¯ y,y,t ) ( gM ) = χ ~ (¯ y,y,t ) ( Y , Y ) χ ~ t ( a ) if gM is an elementof G/M that can be decomposed as gM = e Y ¯ n fast e Y n fast aM , as described above. Here the Q ω are treated as functions on G/M that factor through
G/K . UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 31
We define a bounded operator S m, (¯ y,y,t,λ ) : L ( G/K ) −→ L ( G/K ) by S m, (¯ y,y,t,λ ) [ e i ~ − w ℓ .µ,k ] ( x ) def = e − iTηH ( µ ) ~ e i ~ − w ℓ .µ,k ( x ) χ ~ (¯ y,y,t ) ( x, k ) χ ~ λ ( µ ) B ( T ) M ◦ γ ( x, k, w ℓ .µ ) . We then define P m, (¯ y,y,t,λ ) def = Π Γ Q ′ ω T − ◦ γ S ∗ m, (¯ y,y,t,λ ) U χ Q ′ ω − T ◦ γ . It can be checked that kP m − X (¯ y,y,t,λ ) ∈ Z ko +2 r P m, (¯ y,y,t,λ ) k L ( Y ) −→ L ( Y ) = O ( ~ M − d/ ) , by noting that the sum P (¯ y,y,t,λ ) ∈ Z ko +2 r S m, (¯ y,y,t,λ ) gives back our Ansatz (5.8) for Q m S ,and by arguing as in (4.7) that the difference between Q m S and the Ansatz is of order O ( ~ M − d/ ) . Again we choose M large enough so that the error O ( ~ M − d/ ) is negligiblecompared to the bound announced in Theorem 1.12.Let us now look at a scalar product D S m, (¯ y,y,t,λ ) e i ~ − w ℓ .µ, ¯ nP , U χ Q ′ ω − T ◦ γ e i ~ − λ,θ E . Weneed only consider the generic case where θ ∈ ¯ nP w ℓ P , that is, θ is of the form θ = ¯ nnw ℓ P (with n ∈ N ). From the previous discussions, it follows that this scalar product is non-negligible only if ¯ n and n stay in some sets of diameters ≤ ǫ ; and, without loss of generality,we have assumed they are both ǫ -close to . As in (5.10), we have by the stationary phasemethod(5.12) D S m, (¯ y,y,t,λ ) e i ~ − w ℓ .µ, ¯ nP , U χ Q ′ ω − T ◦ γ e i ~ − λ,θ E = ~ J/ Z a ∈ A d ( λ, a ) C (¯ y,y,t,λ ) ~ (¯ nnaM, λ, w ℓ .µ ) ¯ e i ~ − w ℓ .µ, ¯ nP (¯ nnaK ) e i ~ − λ, ¯ nnwP (¯ nnaK ) da = ~ J/ Z a ∈ A d ( λ, a ) C (¯ y,y,t,λ ) ~ (¯ nnaM, λ, w ℓ .µ ) ¯ e i ~ − w ℓ .µ, ¯ nP (¯ nnaK ) e i ~ − λ, ¯ nnw ℓ P (¯ nnaK ) da. where C (¯ y,y,t,λ ) ~ (¯ nnaM, λ, w ℓ .µ ) = P ~ k c k (¯ nnaM, λ, w ℓ .µ ) and c (¯ nnaK, ¯ nP , ¯ nnw ℓ P , λ, w ℓ .µ ) = A ( T ) M ◦ γ (¯ nnaw ℓ M, λ ) ¯ B ( T ) M ◦ γ (¯ nnaM, w ℓ .µ ) × χ ~ (¯ y,y,t ) (¯ nnaM ) χ ~ λ ( µ ) (the next terms have the same support as the leading one). Remember the notation ¯ n = e Y ¯ n fast , n = e Y n fast . By Proposition 5.9, and by definition of the cut-off functions χ J , χ r , our scalar product is non-negligible only if Y , Y stay in a set of measure ~ J / + ,and n fast , ¯ n fast stay in a set of measure Q k>j e − T ηm αk α k ( X λ ) and Q k>j e − T ηm αk α k ( X µ ) , re-spectively.5.6. Norm of P ∗ m, (¯ x,x,s,µ ) P m, (¯ y,y,t,λ ) . We are now ready to check the first assumption ofthe Cotlar-Stein lemma, that is, to bound from above the norm of P ∗ m, (¯ x,x,s,µ ) P m, (¯ y,y,t,λ ) . Let u, v ∈ L (Γ \ G/K ) . We write (cid:10) P m, (¯ x,x,s,µ ) v, P m, (¯ y,y,t,λ ) u (cid:11) Γ \ G/K = D Q ′ ω T − ◦ γ S ∗ m, (¯ x,x,s,µ ) U χ Q ′ ω − T ◦ γ v, Q ′ ω T − ◦ γ S ∗ m, (¯ y,y,t,λ ) U χ Q ′ ω − T ◦ γ u E G/K = D S ∗ m, (¯ x,x,s,µ ) U χ Q ′ ω − T ◦ γ v, S ∗ m, (¯ y,y,t,λ ) U χ Q ′ ω − T ◦ γ u E G/K + O ( ~ ∞ ) k u kk v k . We develop fully this scalar product using the Fourier transform.(5.13) D S ∗ m, (¯ x,x,s,µ ) U χ Q ′ ω − T ◦ γ v, S ∗ m, (¯ y,y,t,λ ) U χ Q ′ ω − T ◦ γ u E G/K = Z dθdθ ′ | c ~ ( λ ) | − dλ | c ~ ( λ ′ ) | − dλ ′ \ Q ′ ω − T ◦ γ u ( λ, θ ) \ Q ′ ω − T ◦ γ v ( λ ′ , θ ′ ) D S ∗ m, (¯ x,x,s,µ ) U χ Q ′ ω − T ◦ γ e i ~ − λ ′ ,θ ′ , S ∗ m, (¯ y,y,t,λ ) U χ Q ′ ω − T ◦ γ e i ~ − λ,θ E G/K = Z dθdθ ′ dk | c ~ ( λ ) | − dλ | c ~ ( λ ′ ) | − dλ ′ | c ~ ( µ ) | − dµ \ Q ′ ω − T ◦ γ u ( λ, θ ) \ Q ′ ω − T ◦ γ v ( λ ′ , θ ′ ) D U χ Q ′ ω − T ◦ γ e i ~ − λ ′ ,θ ′ , S m, (¯ x,x,s,µ ) e i ~ − w ℓ .µ,k E D S m, (¯ y,y,t,λ ) e i ~ − w ℓ .µ,k U χ Q ′ ω − T ◦ γ e i ~ − λ,θ E G/K = Z dθdθ ′ Jac(¯ n ) d ¯ n | c ~ ( λ ) | − dλ | c ~ ( λ ′ ) | − dλ ′ | c ~ ( µ ) | − dµ \ Q ′ ω − T ◦ γ u ( λ, θ ) \ Q ′ ω − T ◦ γ v ( λ ′ , θ ′ ) D U χ Q ′ ω − T ◦ γ e i ~ − λ ′ ,θ ′ , S m, (¯ x,x,s,µ ) e i ~ − w ℓ .µ, ¯ nP E D S m, (¯ y,y,t,λ ) e i ~ − w ℓ .µ, ¯ nP , U χ Q ′ ω − T ◦ γ e i ~ − λ,θ E G/K
Finally, in equation (5.13), we write θ = ¯ nnw ℓ P and θ ′ = ¯ nn ′ w ℓ P (we can do so on aset of full measure). We have shown in (5.12) that(5.14) D U χ Q ′ ω − T ◦ γ e i ~ − λ ′ ,θ ′ , S m, (¯ x,x,s,µ ) e i ~ − w ℓ .µ, ¯ nP E D S m, (¯ y,y,t,λ ) e i ~ − w ℓ .µ, ¯ nP , U χ Q ′ ω − T ◦ γ e i ~ − λ,θ E G/K = ~ J Z a ∈ A d ( λ, a ) C (¯ y,y,t,λ ) ~ (¯ nnaM, λ, w ℓ .µ ) ¯ e i ~ − w ℓ .µ, ¯ n (¯ nnaK ) e i ~ − λ, ¯ nnw (¯ nnaK ) da Z a ′ ∈ A d ( λ ′ , a ′ ) ¯ C (¯ x,x,s,µ ) ~ (¯ nn ′ a ′ M, λ ′ , w ℓ .µ ) e i ~ − w ℓ .µ, ¯ n (¯ nn ′ a ′ K )¯ e i ~ − λ ′ , ¯ nn ′ w (¯ nn ′ a ′ K ) da ′ . Already we can note that C (¯ y,y,t,λ ) ~ (¯ nnaM, λ, w ℓ .µ ) ¯ C (¯ x,x,s,µ ) ~ (¯ nn ′ a ′ M, λ ′ , w ℓ .µ ) can onlybe non zero if χ ~ (¯ y,y,t ) (¯ nnaM ) χ ~ (¯ x,x,s ) (¯ nn ′ a ′ M ) = 0 , and from the way we chose χ J this canhappen only for k ¯ x − ¯ y k ≤ . For the same reason, it can only be non zero if k µ − λ k ≤ .Now we try to show that (5.13) decays fast when k x − y k gets large. Under the lastintegral in (5.13) we have a function of the pair (¯ nna, ¯ nn ′ a ′ ) . We have an oscillatory integralof the form (5.1), with a phase S (¯ nna, ¯ nn ′ a ′ ) = λ.B (¯ nnaw ℓ M ) + ( w ℓ .µ )[ B (¯ nn ′ a ′ ) − B (¯ nna )] − λ ′ .B (¯ nn ′ a ′ w ℓ M )= λ.B (¯ nnaw ℓ M ) + ( w ℓ .µ )[ a ′ − a ] − λ ′ .B (¯ nn ′ a ′ w ℓ M ) , UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 33 where B is the function defined in (2.1). We want to do “integration by parts with respectto ¯ n ” (as in Lemma 5.2). However, because the derivatives of S with respect to ¯ n are trickyto compute, it is preferable to use a vector field Z whose definition is a bit delicate butwith the property that Z.B (¯ nnaw ℓ M ) = 0 and Z.B (¯ nn ′ a ′ w ℓ M ) = 0 .Consider a variation of the form Ψ τ : (¯ nna, ¯ nn ′ a ′ ) (¯ nne τY a, ¯ nn ′ a ′ a − e Ψ( τY ) a ) = ¯ nn ( e τY a, n − n ′ a ′ a − e Ψ( τY ) a ) , for Y ∈ ¯ n , and Ψ = Ψ n − n ′ a ′ a − defined in lemma 5.5. By definition of Ψ , the two elements ¯ nne τY a and ¯ nn ′ a ′ a − e Ψ( τY ) a are in the same N A orbit, for all τ . Such a variation preservesthe terms B (¯ nn ′ a ′ w ℓ M ) and B (¯ nnaw ℓ M ) . We call Z the vector field d Ψ τ dτ | τ =0 . We take Y ∈ g w ℓ .α k with ≤ k ≤ j . We note that each term of the product \ Q ′ ω − T ◦ γ u ( λ, ¯ nnw ℓ P ) \ Q ′ ω − T ◦ γ v ( λ ′ , ¯ nn ′ w ℓ P ) e i ~ − λ, ¯ nnw ℓ (¯ nnaK )¯ e i ~ − λ ′ , ¯ nn ′ w ℓ (¯ nn ′ a ′ K ) is invariant under Ψ τ . The function C (¯ y,y,t,λ ) ~ (¯ nnaM ) ¯ C (¯ x,x,s,µ ) ~ (¯ nn ′ aM ) satisfies k Z m C (¯ y,y,t,λ ) ~ (¯ nnaM ) ¯ C (¯ x,x,s,µ ) ~ (¯ nn ′ a ′ M ) k ≤ C ( m ) ~ − m/ + just by the definition of C (¯ y,y,t,λ ) ~ and C (¯ x,x,s,µ ) ~ . Now we want to apply the nonstationaryphase lemma 5.2, so we need to understand ZS = Z [( w ℓ .µ )[ B (¯ nn ′ a ′ ) − B (¯ nna )]] . Lemmas 5.5 and 5.6 tell us that if we write n − n ′ = exp( T ) with T = P α T α ǫ -close to , choose β among the slow exponents so that the norm of T β is comparable to the normof log( n − n ′ ) slow and take Y ∈ ¯ n slow of norm such that Ψ ′ ( Y ) = θ ( T β ) then(5.15) | ZS (¯ nna, ¯ nn ′ a ′ ) | ≥ C k log( n − n ′ ) slow k . Note that we have k log( n − n ′ ) slow k ≥ ~ / + ( k x − y k− if C (¯ y,y,t,λ ) ~ (¯ nnaM ) ¯ C (¯ x,x,s,µ ) ~ (¯ nn ′ aM ) =0 . We now apply Lemma 5.2 to the last expression of integral (5.13), integrating by parts ˜ M -times using the vector field Z . This yields that (cid:10) P m, (¯ x,x,s,µ ) v, P m, (¯ y,y,t,λ ) u (cid:11) Y is boundedfrom above by C ( M ) ~ ˜ M (1 − / + ) max(16 , k x − y k ) ˜ M ~ J Z Jac( n ) dn Jac( n ′ ) dn ′ Jac(¯ n ) d ¯ nda da ′ χ ~ (¯ y,y,t ) (¯ nnaM ) χ ~ (¯ x,x,s ) (¯ nn ′ a ′ M ) | c ~ ( λ ) | − dλ | c ~ ( λ ′ ) | − dλ ′ | c ~ ( µ ) | − dµχ ~ λ ( µ ) χ ~ µ ( µ ) | \ Q ′ ω − T ◦ γ u ( λ, ¯ nnw ℓ P ) \ Q ′ ω − T ◦ γ v ( λ ′ , ¯ nn ′ w ℓ P ) | for an arbitrarily large integer ˜ M . For any ¯ n, n, n ′ , we have Z da da ′ χ ~ (¯ y,y,t ) (¯ nnaM ) χ ~ (¯ x,x,s ) (¯ nn ′ a ′ M ) = O ( ~ r/ + ) , so the previous bound becomes ~ ˜ M (1 − / + ) max(16 , k x − y k ) ˜ M ~ J ~ r/ + Z Jac( n ) dn Jac( n ′ ) dn ′ Jac(¯ n ) d ¯ n | c ~ ( λ ) | − dλ | c ~ ( λ ′ ) | − dλ ′ | c ~ ( µ ) | − dµχ ~ λ ( µ ) χ ~ µ ( µ ) | \ Q ′ ω − T ◦ γ u ( λ, ¯ nnw ℓ P ) \ Q ′ ω − T ◦ γ v ( λ ′ , ¯ nn ′ w ℓ P ) | Similarly, ˜ M integrations by parts in (5.13) with respect to the variable µ allows togain a factor ~ ˜ M (1 − / k a − a ′ k ˜ M ≤ ~ ˜ M (1 − / k t − s k ˜ M if k t − s k is large enough. Integrations by parts withrespect to a allow to gain a factor ~ ˜ M (1 − / k λ − µ k ˜ M ; and integrations by parts with respect to a ′ allow to gain a factor ~ ˜ M (1 − / k λ ′ − µ k ˜ M . In particular, the contribution to (5.13) of those λ, λ ′ , µ with k λ ′ − µ k ≥ ~ / or k λ − µ k ≥ h / is O ( ~ ∞ ) . In these cases the application of thenon-stationary phase lemma 5.2 is made simpler by the fact that the phase S is linear in µ , a and a ′ .We find that (cid:10) P m, (¯ x,x,s,µ ) v, P m, (¯ y,y,t,λ ) u (cid:11) Y is bounded from above by(5.16) , k x − y k ) ˜ M , k t − s k ) ˜ M ~ J ~ r/ + Z Jac( n ) dn Jac( n ′ ) dn ′ Jac(¯ n ) d ¯ n | c ~ ( λ ) | − dλ | c ~ ( λ ′ ) | − dλ ′ | c ~ ( µ ) | − dµχ ~ λ ( µ ) χ ~ µ ( µ ) | \ Q ′ ω − T ◦ γ u ( λ, ¯ nnw ℓ P ) \ Q ′ ω − T ◦ γ v ( λ ′ , ¯ nn ′ w ℓ P ) | . In this integral, λ ′ , λ, µ are all ǫ -close to Λ ∞ , and each of them runs over a set of volume ~ r/ + ; ¯ n runs over a set of measure ~ J / + Q k>j e − T ηm αk α k ( X µ ) , n runs over a set of measure ~ J / + Q k>j e − T ηm αk α k ( X λ ) , and n ′ runs over a set of measure ~ J / + Q k>j e − T ηm αk α k ( X λ ′ ) .Using Cauchy-Schwarz and the Plancherel formula we find that the integral Z Jac( n ) dn Jac( n ′ ) dn ′ | c ~ ( λ ) | − dλ | c ~ ( λ ′ ) | − dλ ′ | \ Q ′ ω − T ◦ γ u ( λ, ¯ nnw ℓ P ) \ Q ′ ω − T ◦ γ v ( λ ′ , ¯ nn ′ w ℓ P ) | is bounded by ~ − d ~ J / + ~ r/ + Q k>j e − T ηm αk α k ( X Λ ∞ ) ~ − J K ǫ k u k L ( Y ) k v k L ( Y ) .The integral R Jac(¯ n ) d ¯ n | c ~ ( µ ) | − dµ adds another factor ~ − d ~ J / + ~ r/ + Q k>j e − T ηm αk α k ( X Λ ∞ ) ~ − J K ǫ .Overall we find that kP ∗ m, (¯ x,x,s,µ ) P m, (¯ y,y,t,λ ) k≤ , k x − y k ) ˜ M , k t − s k ) ˜ M ~ J +4 r/ + − d +2 J / + Y k>j e − T ηm αk α k ( X Λ ∞ ) ~ − J K ǫ UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 35 and vanishes for k ¯ x − ¯ y k > or k µ − λ k > .Choosing ˜ M large enough, we can sum over all (¯ y, y, t, λ ) , and we find X (¯ y,y,t,λ ) ∈ Z J r kP ∗ m, (¯ x,x,s,µ ) P m, (¯ y,y,t,λ ) k / ≤ ~ J/ r/ + − d + J / + Y k>j e − T ηm αk α k ( X Λ ∞ ) ~ − J K ǫ . Remembering that J = d − r and that + could be chosen arbitrarily close to , we get X (¯ y,y,t,λ ) ∈ Z J r kP ∗ m, (¯ x,x,s,µ ) P m, (¯ y,y,t,λ ) k / ≤ ~ J − J − cǫ Y k>j e − T ηm αk α k ( X Λ ∞ ) with a constant c that depends on K .5.7. Norm of P m, (¯ x,x,s,µ ) P ∗ m, (¯ y,y,t,λ ) . Using a similar calculation reversing the roles of N and N , we get the same bound, X (¯ y,y,t,λ ) ∈ Z J r kP m, (¯ x,x,s,µ ) P ∗ m, (¯ y,y,t,λ ) k / ≤ ~ J − J − cǫ Y k>j e − T ηm αk α k ( X Λ ∞ ) . Using the Cotlar-Stein lemma and the fact that the α ( X Λ ∞ ) coincide with the Lyapunovexponents χ ( H ) on the energy layer E Λ ∞ , we get Theorem 1.12.6. Measure Rigidity
In this section we prove Theorems 1.8, 1.9 and 1.10. The proofs combine our entropybounds with the measure classification results of [8, 9] and the orbit classification resultsof [17, 25] which give information about A -invariant and ergodic measures that have a largeentropy. Proposition 6.1. (Measure rigidity theory) Let G be a split group, and let µ be an ergodic A -invariant measure on X = Γ \ G .(1) [8, Lem. 6.2] there exist constants s α ( µ ) ∈ [0 , associated to the roots α ∈ ∆ , suchthat for any a ∈ A , h KS ( µ, a ) = X α ∈ ∆ s α ( µ ) (log α ( a )) + . Here t + = max { , t } for t ∈ R . Furthermore, s α ( µ ) = 1 if and only if µ is invariant by theroot subgroup U α .(2) [8, Prop. 7.1] Assume that s α ( µ ) , s β ( µ ) > for two roots α, β ∈ ∆ such that α + β ∈ ∆ . Then s α + β ( µ ) = 1 .(3) [8, Thm. 4.1(iv)] If G is locally isomorphic to SL n and s α ( µ ) > for all α , then µ is G -invariant.(4) [9, Cor. 3.4] In the case G = SL n , we have s α ( µ ) = s − α ( µ ) for all roots α . We do not know if (4) holds in general.Now let µ be an A -invariant probability measure with ergodic decomposition µ = R X µ x dµ ( x ) . For each subset R ⊂ ∆ let X R be the set of ergodic components µ x such that { α, s α ( µ x ) > } = R . Write w R = µ ( X R ) and if w R > , let µ R = w R R X R µ x dµ ( x ) , sothat µ = P R w R µ R . From Proposition 6.1(1) we have for a ∈ Ah KS ( µ R , a ) ≤ X α ∈ R (log α ( a )) + , (this is in fact an avatar of the Ruelle-Pesin inequality) and hence h KS ( µ, a ) ≤ X R w R X α ∈ R (log α ( a )) + . By Proposition 6.1(2) it is enough to consider those R that are closed under the additionof roots. In the case G = SL n , parts (3) and (4) show, respectively, that it is enough toconsider those R which are symmetric and that also µ ∆ = µ Haar . Proposition 6.2.
Let G = SL ( R ) , Γ a lattice in G , and µ an A -invariant probabilitymeasure on Γ \ G , such that h KS ( µ, a ) ≥ h KS ( µ Haar , a ) for a = e X , X = diag(2 , − , − , diag( − , − , and diag( − , − , . Then w ∆ ≥ , that is, the Haar component has weightat least .Proof. The possible sets R are ∆ , ∅ , { α, − α } . In the case of SL n the roots are indexed by { ij, ≤ i, j ≤ n, i = j } : α ij is defined by α ij ( X ) = X ii − X jj . Consider a = diag( e , e , e − ) .Then h KS ( µ Haar , a ) = 6 (since s α = 1 for all α ), h KS ( µ ∅ , a ) = 0 , h KS ( µ , a ) = 0 , h KS ( µ , a ) ≤ , h KS ( µ , a ) ≤ . Thus,(6.1) ≤ h KS ( µ, a ) ≤ w + 3 w + 6 w ∆ . This implies w ∆ − w ≥ − ( w ∆ + w + w + w ) ≥ . By symmetry it follows that w ∆ ≥ w and w ∆ ≥ w . Returning to (6.1), it follows that ≤ w ∆ .In fact, if h KS ( µ, a ) ≥ (cid:0) + ǫ (cid:1) h KS ( µ Haar , a ) for a = e X , X = diag(2 , − , − , diag( − , − , or diag( − , − , , then w ∆ ≥ ǫ. (cid:3) Putting together Theorem 1.7 and Proposition 6.2 gives Theorem 1.8.For SL the analogue of Proposition 6.2 is given below. Theorem 1.9 is an immediatecorollary. Proposition 6.3.
Let G = SL ( R ) , µ an A -invariant probability measure on Γ \ G , suchthat h KS ( µ, a ) ≥ (cid:0) + ǫ (cid:1) h KS ( µ Haar , a ) for a = e X , X in the Weyl orbit of diag(3 , − , − , − .Then w ∆ ≥ ǫ . If ǫ = 0 and there is no Haar component, then each ergodic component isthe Haar measure on a closed orbit of the group ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
00 0 0 ∗ (or one of its 4 imagesunder the Weyl group), and the components invariant by any of these 4 subgroups havetotal weight . UANTUM LIMITS ON LOCALLY SYMMETRIC SPACES 37
Theorem 1.8 and its analogue for G = SL apply to any lattice Γ . On the other handfor G = SL n with n large some quotients Γ \ G support ergodic invariant measures of largeentropy other than Haar measure, so our entropy bound is not strong enough to obtain aHaar component. However, for some lattices Γ there are further restrictions on the set ofergodic components, so that non-Haar measures have much smaller entropy. This is thecase where Γ is a lattice associated to a divison algebra.We give here a quick outline of the construction, refering the reader to [25] and itsreferences (or [18]) for a detailed discussion. Let F be a central simple algebra of degree n over Q and assume that F splits over R , that is that F ⊗ Q R ≃ M n ( R ) . Next, let O ⊂ F be an order , that is a subring whose additive group is generated by a basis for F over Q . Finally, let O ⊂ SL n ( R ) denote the subgroup of elements of O with determinant (“reduced norm ”). Such O are in fact lattices; any lattice Γ < SL n ( R ) commensurablewith some O is said to be of inner type . We simply say that they are associated to thealgebra F . Our Theorem 1.7 applies when the lattice is co-compact, which is the case ifand only if F is a division algebra.We shall need the fact that those measure rigidity results of [9] which are stated specif-ically for SL n ( Z ) apply, in fact, to any lattice of inner type, since the proof of Lemma 5.2of that paper carries over to the more general situation. We give the easy argument here: Lemma 6.4.
Let Γ < SL n ( R ) be a lattice of inner type. Then there is no γ ∈ Γ , diag-onalizable in SL n ( R ) , such that ± are not eigenvalues of γ and all eigenvalues of γ aresimple except for precisely one which occurs with multiplicity two.Proof. Say that Γ is associated to the central simple algebra F , and let O be an order in F such that Γ ∩ O has finite index in Γ .Assume by contradiction that there exists γ as in the statement, and choose r so that γ r ∈ O . Since O is a ring with a finitely generated additive group, the Cayley-HamiltonTheorem shows that γ r is integral over Z . It follows that every eigenvalue of γ r , hence of γ ,is an algebraic integer. The fact that det( γ ) = 1 now shows that the rational eigenvaluesof γ must be integral divisors of , so by assumption all eigenvalues of γ are irrational. Let f ( x ) ∈ R [ x ] be the characteristic polynomial of γ , when γ is thought of as an element of SL n ( R ) . We will show f ( x ) ∈ Q [ x ] . Then the multiplicity the eigenvalues of f would beGalois invariant giving the desired contradiction. For the last claim extend scalars to C and note that the usual proof that the reduced trace and norm belong to Q applies to theentire characteristic polynomial. (cid:3) Proposition 6.5.
Let n ≥ and let t be the largest proper divisor of n . Let G = SL n ( R ) and let Γ < G be a lattice of inner type. Let µ be an A -invariant probability measure on Γ \ G such that h KS ( µ, a ) ≥ h KS ( µ Haar , a ) for a = e X , X a Weyl conjugate of diag( n − , − , · · · , − . Then w ∆ ≥ ( n +1)2 − tn − t > . In other words, µ must contain an ergodiccomponent proprtional to Haar measure. Theorem 1.10 follows.
Proof.
As above, let µ x be an ergodic component of µ that has positive entropy with respectto e X . By [9, Thm. 1.3] (replacing Lemma 5.2 of that paper with Lemma 6.4 above) µ x must be algebraic : there exists a closed subgroup H containing A , and a closed orbit zH in Γ \ G , such that µ x is the H-invariant measure on zH . By [17] (the arguments are containedin the proof of Lemma 4.1 and Lemma 6.2) and [25] (see Thm 1.2 and §4.2), H must bereductive, and conjugate to the connected component of GL k ( R ) l ∩ SL n ( R ) ; where n = kl and GL k ( R ) l denotes the block-diagonal embedding of l copies of GL k ( R ) into GL n ( R ) .By the discussion following Proposition 6.1 we see that for such lattices Γ the possiblesets R are obtained by partitioning n into l subsets B , B , . . . , B l of equal size k , andletting R = { α ij , ≤ i, j ≤ n, ∃ u such that i ∈ B u and j ∈ B u } . Consider a = diag( e n − , e − , . . . , e − ) . Then h KS ( µ Haar , a ) = n ( n − , and for everysubset R defined as above by a non-trivial partition, we have h KS ( µ R ) ≤ n ( t − . Theinequality h KS ( µ, a ) ≥ h KS ( µ Haar , a ) now shows that w ∆ ( n −
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