Patterson--Sullivan distributions in higher rank
aa r X i v : . [ m a t h . SP ] M a y Patterson–Sullivan distributions in higher rank
S. Hansen · J. Hilgert ⋆ · M. Schr¨oder ⋆⋆ Version: May 29, 2011
Abstract
For a compact locally symmetric space X Γ of non-positive curva-ture, we consider sequences of normalized joint eigenfunctions which belong tothe principal spectrum of the algebra of invariant differential operators. Usingan h -pseudo-differential calculus on X Γ , we define and study lifted quantumlimits as weak ∗ -limit points of Wigner distributions. The Helgason boundaryvalues of the eigenfunctions allow us to construct Patterson–Sullivan distri-butions on the space of Weyl chambers. These distributions are asymptoticto lifted quantum limits and satisfy additional invariance properties, whichmakes them useful in the context of quantum ergodicity. Our results gener-alize results for compact hyperbolic surfaces obtained by Anantharaman andZelditch. Keywords
Patterson–Sullivan distributions · Wigner distributions · quantumergodicity · lifted quantum limits · locally symmetric spaces · geometricpseudo-differential analysis · Weyl chamber flow
Mathematics Subject Classification (2000) · · For a locally symmetric space X Γ of non-positive curvature, we consider se-quences, ( ϕ h ) h ⊂ L ( X Γ ), of normalized joint eigenfunctions which belong to ⋆ Part of this research was done at the Hausdorff Research Institute for Mathematics inthe context of the trimester program “Interaction of Representation Theory with Geometryand Combinatorics” ⋆⋆ Partially supported by the DFG-IRTG 1133 “Geometry and Analysis of Symmetries”
S¨onke Hansen · Joachim Hilgert · Michael Schr¨oder [email protected] · [email protected] · [email protected] Institut f¨ur Mathematik, Universit¨at Paderborn, Warburger Str. 100, 33098Paderborn, Germany. the principal spectrum of the algebra of invariant differential operators. Usinga h -pseudo-differential calculus on X Γ , we define and study lifted quantumlimits or microlocal lifts as weak ∗ -limit points of Wigner distributions W h : a (cid:0) Op h ( a ) ϕ h | ϕ h (cid:1) L ( X Γ ) . Here, h − is the norm of a spectral parameter associated with ϕ h , and h ↓ ϕ h , we construct Patterson–Sullivan distri-butions on the space of Weyl chambers. In the context of quantum ergodicity,Patterson–Sullivan distributions are important because they are asymptoticto lifted quantum limits and satisfy invariance properties.For compact hyperbolic surfaces X Γ = Γ \ H , the asymptotic equivalence oflifted Wigner distributions and Patterson–Sullivan distributions was observedby Anantharaman and Zelditch [2]. While it was known from earlier work (see[31,28]) that lifted quantum limits on compact hyperbolic surfaces are invari-ant under geodesic flows it turned out that Patterson–Sullivan distributionsare themselves invariant under the geodesic flow. Moreover, in [2] it is shownthat they have an interpretation in terms of dynamical zeta functions whichcan be defined completely in terms of the geodesic flow.Although lifted quantum limits do not depend on the specific pseudo-differential calculus chosen for their definition, it is useful, for establishinginvariance properties, to have an equivariant calculus. For hyperbolic surfaces,based on the non-euclidean Fourier analysis and closely following the euclideanmodel, such a calculus was provided by Zelditch [29]. In [20] this calculus wasextended to rank one symmetric spaces. Using this calculus the constructionof the Patterson–Sullivan distributions and the proof of the asymptotic equiv-alence from [2] has been generalized in [15]. However, due to singularitiesarising from Weyl group invariance, it is difficult to construct an equivariantnon-euclidean pseudo-differential calculus in higher rank; see [20]. Silbermanand Venkatesh [22], generalizing work of Zelditch and Wolpert for surfaces tocompact locally symmetric spaces, introduced a representation theoretic liftas a replacement for a microlocal lift. They sketch, in [22, Remark 1.7(4) and § X = G/K denote a Riemanniansymmetric space of noncompact type, where G is a connected semisimple Liegroup with finite center and K a maximal compact subgroup of G . Further, let Γ be a co-compact and torsion free discrete subgroup of G . Then we obtain alocally symmetric space X Γ as the quotient Γ \ X , i.e., the double coset space Γ \ G/K . Let G = KAN be a corresponding Iwasawa decomposition of G andlet M denote the centralizer of A in K . The Furstenberg boundary of X canbe identified with the flag manifold B := K/M . Denote by P = M AN theminimal parabolic associated with the Iwasawa decomposition. Identifying B with G/P we define a G -action on B . Under the diagonal action, there isa unique open G -orbit B (2) ∼ = G/M A in B × B . For rank 1 spaces B (2) isthe set of pairs of distinct boundary points. In this case each geodesic of X has a unique forward limit point and a unique backward limit point in B . Inparticular, one can identify B (2) with the space of geodesics. In higher rank thegeometric interpretation is more complicated. It involves the Weyl chamberflow rather than the geodesic flow.Joint eigenfunctions come with a spectral parameter λ ∈ a ∗ C , where a isthe Lie algebra of A . The spectral parameters are unique up to the action ofthe Weyl group W associated with the Iwasawa decomposition. The principalpart of the spectrum comes from the purely imaginary spectral parameters.We assume that the spectral parameter of ϕ h is iν h /h ∈ i a ∗ , | ν h | = 1. ThePatterson–Sullivan distribution P S Γh ∈ D ′ ( Γ \ G/M ) associated with ϕ h isconstructed as follows. The Poisson–Helgason transform allows us to write ϕ h ( x ) = Z B e ( iν h /h + ρ ) A ( x,b ) T h (d b ) , x ∈ X, where T h ∈ D ′ ( B ) is the boundary value of ϕ h . Here, 2 ρ is the sum of positiverestricted roots counted according to multiplicities, and A : X × B → a isthe horocycle bracket. For dealing with non-real ϕ h , it is important that theconjugate of ϕ h also is a unique transform, ϕ h ( x ) = Z B e ( − iw · ν h /h + ρ ) A ( x,b ) ˜ T h (d b ) , x ∈ X, where ˜ T h ∈ D ′ ( B ). Here w is the longest element of W . The weighted Radontransform R h : C ∞ c ( G/M ) → C ∞ c ( G/M A ) is defined by( R h f )( gM A ) = Z A d h ( gaM, ν h ) f ( gaM ) d a with a weight function related to the the horocycle bracket. Denote by R ′ h : D ′ ( B × B ) → D ′ ( G/M ) the dual of R h . The Patterson–Sullivan distribution P S Γh ∈ D ′ ( Γ \ G/M ) is defined as the Γ -average of R ′ h ( T h ⊗ ˜ T h ).Let ω = lim h W h ∈ D ′ ( T ∗ X Γ ) be a lifted quantum limit which, afterpassing to a subsequence if necessary, has a regular direction θ = lim h ν h . Inaddition, assume ν h = θ + O ( h ) as h ↓ ω to the sequence ( P S Γh ) h of Patterson–Sullivan distributions, we makeuse of a natural G -equivariant map Φ : G/M × a ∗ → T ∗ X . For regular θ ∈ a ∗ this induces a push-forward of distributions, Φ ( · , θ ) ∗ : D ′ ( Γ \ G/M ) → D ′ ( T ∗ X Γ ) . Our main result (Theorem 7.4) can now be stated as follows: ω = κ ( w · θ ) lim h ↓ (2 πh ) dim N/ Φ ( · , θ ) ∗ P S Γh in D ′ ( T ∗ X Γ ). (1.1)Here κ is a normalizing function defined in terms of structural data of X . Wepoint out that Theorem 7.4 is more general. It also describes the situationarising from off-diagonal Wigner distributions (cid:0) Op Γ,h ( a ) ϕ h | ϕ ′ h (cid:1) L ( X Γ ) .If one had a formula interwining Patterson–Sullivan distributions P S Γh intolifted Wigner distributions W h , one might be able to deduce (1.1) as a corollary.Presumably, an intertwining formula holds only for special pseudo-differentialcalculi.The paper is organized as follows. In Section 2 we collect various geometricfacts needed to construct the lifted quantum limits and the Patterson-Sullivandistributions. In particular we discuss the function Φ and the G -orbit B (2) .In Section 3 we recall the Helgason-Poisson transform and prove a regularitytheorem of Γ -invariant boundary values which is instrumental in proving ourmain result but also of independent interest (see Theorem 3.13). In Section 4we give the details of the construction of the Patterson-Sullivan distributionsand observe its natural A -invariance properties (Remark 4.11). Section 5 pro-vides the technical results on oscillatory integrals which are instrumental inestablishing our asymptotic results. In Section 6 we describe the lifted quan-tum limits constructed via the geometric pseudo-differential calculus and de-rive their invariance under the Weyl chamber flow (Theorem 6.6). In the finalSection 7 we put things together and prove Theorem 7.4. Let g the Lie algebra of G , and h , i the Killing form of g . Let θ be a Cartaninvolution of g such that the form ( X, Y ) ( X, Y ) θ := −h X, θY i is positivedefinite on g × g . Let g = k + p be the decomposition of g into eigenspaces of θ and K the analytic subgroup of G with Lie algebra k . We choose a maximalabelian subspace a of p and denote by a ∗ its dual and a ∗ C the complexificationof a ∗ . Let A = exp a denote the corresponding analytic subgroup of G and letlog denote the inverse of the map exp : a → A .Given λ ∈ a ∗ , put g λ = { X ∈ g | ( ∀ H ∈ a )[ H, X ] = λ ( H ) X } . If λ = 0 and g λ = { } , then λ is called a (restricted) root and m λ = dim( g λ ) is called its multiplicity . Let g C denote the complexification of g and if s is any subspaceof g let s C denote the complex subspace of g C spanned by s .For λ ∈ a ∗ let H λ ∈ a be determined by λ ( H ) = h H λ , H i for all H ∈ a . For λ, µ ∈ a ∗ we put h λ, µ i := h H λ , H µ i . Since h , i is positive definite on p × p weset | λ | := h λ, λ i / for λ ∈ a ∗ and | X | := h X, X i / for X ∈ p . The C -bilinearextension of h , i to a ∗ C will be denoted by the same symbol.Let a ′ be the open subset of a where all restricted roots are = 0. Theelements of a ′ are called regular , and the components of a ′ are called Weylchambers . We fix a Weyl chamber a + and call a root α positive ( >
0) if it is positive on a + . Let a ∗ + denote the corresponding Weyl chamber in a ∗ , thatis the preimage of a + under the mapping λ H λ . Let Σ denote the setof restricted roots, Σ + the set of positive roots and Σ − := − Σ + the set ofnegative roots.Let Σ = (cid:8) α ∈ Σ : α / ∈ Σ (cid:9) be the set of indivisible roots, and put Σ +0 = Σ + ∩ Σ , Σ − = Σ − ∩ Σ . We set ρ := Σ α ∈ Σ + m α α and let N denote theanalytic subgroup of G with Lie algebra n := Σ α> g α . Then n = θ ( n ) = Σ α< g α . The involutive automorphism θ of g extends to an analytic involutiveautomorphism of G , also denoted by θ , whose differential at the identity e ∈ G is the original θ . It thus makes sense to define N = θN . The Lie algebra of N is θ ( n ).Let G = KAN be the Iwasawa decomposition of G corresponding to thechoice of a positive system in Σ . Writing g = k ( g ) exp H ( g ) n ( g ) , (2.1)where k ( g ) ∈ K , H ( g ) ∈ a , n ( g ) ∈ N , the functions k, H, n are called the Iwasawa projections . By M we denote the centralizer of A in K . Then P := M AN is a minimal parabolic subgroup of G and G/P is the
Furstenbergboundary of X := G/K . In view of the Iwasawa decomposition, it can beidentified with the flag manifold B := K/M . The group G acts on G/P via g · xP = gxP and K/M → G/P, kM kP is a diffeomorphism ([14], p. 407)inverted by gP k ( g ) M . Hence this map intertwines the G -action on G/P with the action on
K/M defined by g · kM = k ( gk ) M . These spaces are thusequivalent for the study of B = K/M = G/P .Let o := K ∈ G/K denote the origin of the symmetric space X and b + := M ∈ K/M the canonical base point in B . Then the diagonal action of G on X × B = G/K × G/P = G/K × K/M is transitive and the stabilizer of( o, b + ) is K ∩ P = M , so we can identify X × B with the space G/M of Weylchambers as a G -space.Let M ′ be the normalizer of A in K . Then W := M ′ /M is the correspond-ing Weyl group . It acts on Σ and contains unique element w ∈ W exchanging Σ + and Σ − . This element is called the longest element of W and by abuse ofnotation we will sometimes also denote a representative of w in M ′ by w .Further, we set b − := w · b + = w M ∈ K/M = B .2.1 The Horocycle BracketThe horocycle bracket is defined by X × B → a , ( gK, kM ) A ( gK, kM ) := − H ( g − k ) . (2.2)Each ( x, b ) ∈ X × B is of the form ( gK, kM ) and it is easy to see that (2.2) iswell-defined. The horocycle bracket is often denoted by h x, b i = h gK, kM i = − H ( g − k ). In order to avoid confusion with the Killing form we prefer touse the notation A ( x, b ) over h x, b i as in [12]. For details on the geometricinterpretation of the horocycle bracket we refer to [12], Ch. II. Proposition 2.1
The horocycle bracket A : X × B → a is invariant under thediagonal action of K on X × B . Lemma 2.2
Let g , g ∈ G , k ∈ K . Then H ( g g k ) = H ( g k ( g k )) + H ( g k ) .Proof Decompose g k = ˜ k ˜ a ˜ n and g ˜ k = k ′ a ′ n ′ . Then H ( g g k ) = H ( k ′ a ′ n ′ ˜ a ˜ n ) = H ( a ′ n ′ ˜ a ) . Since A normalizes N this equals log( a ′ ) + log(˜ a ). ⊓⊔ Lemma 2.3
Let x = hK ∈ G/K , b = kM ∈ K/M , g ∈ G . Then (i) A ( g · x, g · b ) = A ( x, b ) + A ( g · o, g · b ) . (ii) A ( g − · o, b ) = − A ( g · o, g · b ) .Proof By definition, A ( g · x, g · b ) = − H ( h − g − k ( gk )). Then by Lemma 2.2applied to g = h − g − and g = g this equals − H ( h − g − gk ) + H ( gk ) = − H ( h − k ) + H ( gk ) . For h = e we obtain A ( g · o, g · b ) = − H ( k ) + H ( gk ) = H ( gk ). Hence A ( g · x, g · b ) − A ( g · o, g · b ) = − H ( h − k ) = A ( hK, kM ) = A ( x, b ) , which implies (i). For (ii) we use (i) to calculate0 = A ( o, g · b ) = A ( g · ( g − · o ) , g · b ) = A ( g − · o, b ) + A ( g · o, g · b ) . ⊓⊔ Lemma 2.4
Let γ, g ∈ G . Then (i) A ( g · o, g · b + ) = H ( g ) = − A ( g · o, b + ) . (ii) A ( g · o, g · b − ) = H ( gw ) = − A ( g − · o, b − ) . (iii) H ( γg ) = H ( g ) + A ( γ · o, γg · b + ) and H ( γgw ) = H ( gw ) + A ( γ · o, γg · b − ) .Proof Parts (i) and (ii) are direct computations. The second part of (iii) followsfrom the first part applied to gw instead of g . For this assertion, let z = g · o .Then by (i) H ( γg ) = A ( γg · o, γg · b + ) = A ( γ · z, γg · b +) , which by Lemma 2.3 equals A ( z, g · b + ) + A ( γ · o, γg · b + ) = H ( g ) + A ( γ · o, γg · b + ) . ⊓⊔ T ∗ ( X ) can be found in [16]. We onlyrecall a few facts we will need later on. The G -action on X lifts to an action T ( X ) by taking derivatives and then to an action on T ∗ ( X ) by duality. T ∗ ( X ) is a G -homogenous vector bundle. In fact, it can be written as G × K p ∗ , where K acts on p ∗ via the coadjoint representation. Using theKilling form on p = T o ( X ), i.e. the invariant Riemannian metric defined bythe Killing form, one can identify T ( X ) and T ∗ ( X ). Under this identificationadjoint and coadjoint action of K on p and p ∗ get identified.Let L g : G/K → G/K be the left translation by g ∈ G . Then map Φ : G/M × a → T ( X ) = G × K p , ( gM, X ) dL g ( o ) X = [ g, X ] (2.3)is G -equivariant and surjective, but not a covering unless one restricts it tothe set a ′ of regular elements in a . If one wants to keep p and p ∗ apart, thefunction Φ is written Φ : G/M × a ∗ → T ∗ ( X ) = G × K p ∗ , ( gM, θ ) [ g, θ ] . (2.4)The fibers of Φ can be described as follows: Φ ( gM, θ ) = Φ ( g ′ M, θ ′ ) if and onlyif there exists a k ∈ K such that g ′ = gk and k · θ = Ad ∗ ( k ) θ = θ ′ . This means Φ − ([ g, θ ]) = { (˜ gM, ˜ θ ) ∈ G/M × a ∗ | ∃ k ∈ K : gk = ˜ g, k · θ = ˜ θ } . If θ is regular, then such a k has to be in M ′ . Therefore, ˜ gM = gM · w and˜ θ = w · θ , where w = kM is in the Weyl group W = M ′ /M .Note that a continuous function f : G/M × a ∗ → C that factors through Φ will have to satisfy f ( gM · w, w · θ ) = f ( gM, θ ) for all w ∈ W . But eventhough the regular elements in a ∗ are dense in a ∗ , this condition does notautomatically guarantee that f factors through Φ since the Φ -fibers over thesingular points have positive dimension and W -invariance cannot guaranteethat the function is constant on those fibers as well.The map Φ : G/M × a ∗ → T ∗ ( X ) can also be written in terms of theIwasawa projection (cf. [1], § Proposition 2.5
Consider the function F : X × B × a ∗ → R defined by F ( x, b, θ ) = θ (cid:0) A ( x, b ) (cid:1) . Then Φ ( x, b, θ ) = dF x ( x, b, θ ) ∈ T ∗ x ( X ) .Proof Identifying X × B with G/M the map Φ can be written Φ ( gM, θ ) = dL g ( o ) −⊤ θ ∈ T ∗ g · o ( X ). Note that the embedding of a ∗ ֒ → p ∗ is given viaextension by 0 on the orthogonal complement of a in p . Thus for v = [ x, ξ ] = dL g ( o ) ξ ∈ T x ( X ) we have Φ ( gM, θ )( v ) = θ ( ξ ). Therefore it suffices to showthat ddt (cid:12)(cid:12)(cid:12) t =0 θ (cid:16) A ( g exp tξ · o, b ) (cid:17) = θ ( ξ ) (2.5)for x = g · o , b = g · b + ∈ B , and ξ ∈ p . To prove this, note first the identity(Lemma 2.3) A ( g exp tξ · o, b ) = A (exp tξ · o, b + ) + A ( g · o, b ) = H (exp tξ ) + A ( g · o, b ) . We claim that lim t → H (exp tξ ) t = p a ( ξ ) , (2.6)for all ξ ∈ p , where p a : p → a is the orthogonal projection with respect to theKilling form (cf. [11], proof of Theorem 2). Since θ ( ξ ′ ) = θ (cid:0) p a ( ξ ′ ) (cid:1) equation(2.6) proves (2.5).To prove (2.6) it suffices to consider a spanning subset of p . If ξ ∈ a , thenthe claim is clear. If ξ ∈ a ⊥ , then we have ξ = η + θη with η ∈ n , and one hasto show. lim t → H (exp tξ ) t = 0 . Writing θη + η = ( θη − η ) + 2 η ∈ k + n and using the Campbell–Hausdorffmultiplication one calculates H (cid:0) exp t ( θη + η ) (cid:1) = H (cid:0) exp t (cid:0) ( θη − η ) + 2 η (cid:1)(cid:1) = H (cid:0) exp (cid:0) t ( θη − η ) ∗ t η + O ( t ) (cid:1)(cid:1) = H (cid:0) (exp t ( θη − η ))(exp t η ) g t (cid:1) = H (cid:0) exp t ( θη − η ) exp t η (cid:1) + O ( t )= O ( t ) , where g t is a group element differing from the identity by O ( t ). ⊓⊔ We introduce the involutive algebra of functions on T ∗ X which are thesymbols of invariant differential operators on X . Definition 2.6
Denote by A the algebra of G -invariant real valued functionsin C ∞ ( T ∗ X ) which restrict to polynomials on p ∗ = T o ( X ).According to [16], Theorem 1.1, A is finitely generated and its joint levelsets are precisely the G -orbits in T ∗ ( X ). In fact, the proof of that theo-rem shows that the restriction to a ∗ induces an isomorphism between A andthe algebra I ( a ∗ ) of Weyl group invariant polynomials on a ∗ (see also [13],Cor. II.5.12). Note that A is also closed under the Poisson bracket { f, h } .The Weyl chamber flow on G/M is the right A -action given by gM · a := gaM . If X is of rank one, i.e. if dim R a = 1, it reduces to the geodesic flow onthe sphere bundle on X .Given a G -invariant function f ∈ C ∞ ( T ∗ X ) G , let h ∈ C ∞ ( p ∗ ) be therestriction to T ∗ o X ∼ = p ∗ . In [16], §
1, it is shown that the hamiltonian flow R × T ∗ ( X ) → T ∗ ( X ) , ( t, ω ) Φ tf ( ω ) associated with f is given by Φ tf ([ g, ξ ]) = (cid:2) g exp (cid:0) t grad h ( ξ ) (cid:1) , ξ (cid:3) , where the gradient of a function on p ∗ is taken with respect to the innerproduct coming from the Killing form. Moreover, considering the restrictionof h to a one obtains the following relation between the Weyl chamber flowand the function Φ from (2.4) Φ ( gM · e t grad h ( ξ ) , ξ ) = Φ tf ◦ Φ ( gM, ξ ) ∀ ξ ∈ a ∗ , gM ∈ G/M. (2.7)
Here it should be noted that (grad h ) | a ∗ = grad( h | a ∗ ).In order to see which Weyl chamber actions ( gM, ξ ) ( gaM, ξ ) we obtainfrom (2.7), we recall from loc. cit. that { grad p ( ξ ) ∈ a ∗ | p ∈ I ( a ∗ ) } = a ∗ if ξ is regular. Note here that the calculations in [16] are done in T ( X ) ratherthan T ∗ ( X ), but identifying the two bundles via the invariant metric gives theresults mentioned above.We call an element in ( gM, ξ ) ∈ G/M × a ∗ regular if ξ ∈ a ∗ is regular.Thus the Hamilton flows associated with functions in A preserve the regularelements and produce the entire Weyl chamber flow on the regular elements.2.3 Open CellsThe Bruhat decomposition says that G is the disjoint union of the doublecosets P wP with w ∈ W , or more precisely, with representatives of the Weylgroup elements in M ′ . Moreover, w P is open in G and this is the only opendouble coset. In particular P w P ⊆ G is dense. Recall b − = w M ∈ K/M in B and note that b − does not depend on the choice of the representative w in M ′ . Proposition 2.7
The orbit B (2) := G · ( b + , b − ) in B × B under the diagonalaction is open and dense. The stabilizer of ( b + , b − ) is M A .Proof
We claim that G · ( b + , b − ) = { ( h P, h P ) ∈ B × B | h − h ∈ P w − P } . Since
P w P is dense and open in G and for U running through a basis of neigh-borhoods of the identity in G , the sets h − U h form a basis of neighborhoodsof h − h , this set is dense and open in B × B . Moreover g · ( b + , b − ) = ( b + , b − )if and only if g ∈ P and gw ∈ w P , which is equivalent to g ∈ P ∩ w P w − = P ∩ θP = M A . Thus it only remains to prove the claim. The inclusion “ ⊆ ”is clear, so assume that h − h = p w − p . Then h = h p − w p − implies h P = h p − w P , whence( h P, h P ) = ( h p − P, h p − w P ) = h p − · ( b + , b − ) , which proves the claim. ⊓⊔ Remark 2.8 (a) The Weyl group W := M ′ /M acts from the right on G/M A via gM A · wM := gwM A and the induced W -action on B (2) is( g · b + , g · b − ) · wM = ( gw · b + , gw · b − ) = ( g · ( w · b + ) , g · ( w · b − )) . In particular, we have ( b , b ) · w M = ( b , b ) since w · b ± = b ∓ . (b) The Weyl group W = M ′ /M acts from the right on G/M via gM · wM := gwM and the induced W -action on X × B is( g · o, g · b + ) · wM = ( gw · o, gw · b + ) = ( g · o, k ( gw ) · b + ) . (c) W acts also on G/M and
K/M from the right such that
K/M → G/M → G/M A are W -equivariant. It is also possible to view the W -action on G/M = X × B as follows: Given ( z, b ) ∈ X × B ∼ = G/M one finds acorresponding element g ( z, b ) M of G/M and defines b · z w = g ( z, b ) w · b + .Then ( z, b ) · w = ( z, b · z w ) , i.e., the W -action on X × B is a twisted version of the W -action on thefibers of X × B → X .(d) The argument from the proof of Proposition 2.7 works for any w ∈ W andproves G · ( b + , w · b + ) = { ( h P, h P ) ∈ B × B | h − h ∈ P w − P } . Thus the Bruhat decomposition implies that each element ( b, b ′ ) ∈ B isof the form g · ( b + , w · b + ) for some w ∈ W . Remark 2.9
It will turn out to be useful to have a smooth section σ : G/M A → G/M for the canonical projection
G/M → G/M A . To construct σ we use theIwasawa decomposition G = KN A to define a smooth map ˜ σ : G → G/M, g = kna knM . Then knama ′ = km ( m − nm ) a ′ for m ∈ M and a ′ ∈ A showsthat ˜ σ factors through the canonical projection π : G → G/M A . Since π isa submersion and ˜ σ is smooth, the universal property of submersions impliesthat the resulting map σ : G/M A → G/M, knaM A knM is indeed smooth.Using the identifications G/M A = B (2) and G/M = X × B from Lemma 2.7and Remark 2.8, we write σ ( b, b ′ ) = σ ( g · ( b + , b − ) = kn · ( o, b + ) = ( kn · o, kn · b + ) = ( z b,b ′ , b ) , where ( b, b ′ ) z b,b ′ is defined as the composition of σ with the canonicalprojection G/M → G/K .The space
G/M can also be interpreted in terms of B (2) as the followingproposition shows. Proposition 2.10
The map Ψ : G/M → B (2) × A = G/M A × Akan ( kan · b + , kan · b − , a ) = ( gM A, a ) is a diffeomorphism.Proof Using the properties of the Iwasawa decomposition, it is elementary tocheck that Ψ is bijective. Moreover, it is clear that K × N × A → G/M A × A, ( k, n, a ) ( knM A, a ) is a submersion. So G → G/M A × A, g = kan ( k ( ana − ) M A, a ) is a submersion. Thus Ψ is a submersion as well, whence itis a diffeomorphism. ⊓⊔ If we compose Ψ with the canonical embedding B (2) × A ֒ → B × A , wefind an embedding G/M ֒ → B × B × A . A , a and a ∗ . For l = dim( A ) we multiply these measures by (2 π ) − l/ and obtain invariant measures d a, d H and d λ on A, a and a ∗ . This normaliza-tion has the advantage that the Euclidean Fourier transform of A is invertedwithout a multiplicative constant. We normalize the Haar measures dk and dm on the compact groups K and M such that the total measure is 1. If U isa Lie group and L a closed subgroup, with left invariant measures d u and d l ,the U -invariant measure d u L = d( uL ) on U/L (if it exists) will be normalizedby Z U f ( u ) d u = Z U/L (cid:18)Z L f ( ul ) d l (cid:19) d u L . (2.8)This measure exists in particular if L is a compact subgroup of U . In particular,we have a K -invariant measure d k M = d( kM ) on K/M of total measure 1.We also have a G -invariant measure d x = d g K = d ( gK ) on X = G/K . Byuniqueness, d x is a constant multiple of the measure on X induced by theRiemannian structure on X given by the Killing form. The Haar measures dn and d n on the nilpotent groups N and N are normalized such that θ (d n ) = d n, Z N e − ρ ( H ( n )) d n = 1 . (2.9)As for X one can also for N consider the Riemannian volume d n Riem on N given by the left-invariant Riemannian structure on N derived from the Killingform. Then d n and d n Riem are proportional and we define the constant C N via d n = C N d n Riem . (2.10) Proposition 2.11
Set η ( n ) = w nw − . Then η (d n ) = d n .Proof Since η is an automorphism of G , η (d n ) is a Haar measure on η ( N ) = θN = N . Therefore η (d n ) = c · d n for some constant c >
0. We claim that c = 1. In view of the normalizations (2.9) the constant equals Z η ( N ) e − ρ ( H ( η ( n ))) d( ηn ) = Z N e − ρ ( H ( η ( n ))) d n = Z N e − ρ ( H ( nw − )) d n and we have Z N e − ρ ( H ( θn )) d n = Z θN e − ρ ( H ( θn )) d( θn ) = Z N e − ρ ( H ( n )) d n = 1 . Let c w be the conjugation by w on G . Since w ∈ K and K is the fixedpoint set of θ , we have θ ◦ c w = c w ◦ θ . Thus κ := θ ◦ c w is an involutive automorphism of G , which fixes N . This implies κ ( dn ) = d n , since κ (d n ) = d d n with d > d = 1. Usingd n = κ (d n ) = θ ( c w (d n ))we find θ (d n ) = c w (d n ) and calculate Z N e − ρ ( H ( nw − )) d n = Z N e − ρ ( H ( c w n )) d n = Z c w ( N ) e − ρ ( H ( c w n )) d( c w n )= Z θN e − ρ ( H ( θn )) d( θn )= 1 . ⊓⊔ The Haar measure on G can ([13], Ch. I, §
5) be normalized such that Z G f ( g ) d g = Z KAN f ( kan ) e ρ (log a ) d k d a d n (2.11)= Z NAK f ( nak ) e − ρ (log a ) d n d a d k (2.12)for all f ∈ C c ( G ). Let f ∈ C c ( AN ), f ∈ C c ( G ), a ∈ A . Then ([13], pp. 182) Z N f ( na ) d n = e ρ (log( a )) Z N f ( an ) d n (2.13)and Z G f ( g ) d g = Z KNA f ( kna ) d k d n d a = Z ANK f ( ank ) d a d n d k. (2.14)Let f ∈ C c ( X ). It follows from (2.14) that Z X f ( x ) d x = Z AN f ( an · o ) d a d n. (2.15) γ : D ( X ) → I ( a ∗ ) which associatesa Weyl group invariant polynomial on a with each invariant differential oper-ator on X = G/K . The formula χ λ ( D ) = γ ( D )( λ ) defines a homomorphism χ λ : D ( X ) → C for each λ ∈ a ∗ C . In this way one obtains the joint eigenspace E λ ( X ) = (cid:8) f ∈ E ( X ) (cid:12)(cid:12) (cid:0) ∀ D ∈ D ( X ) (cid:1) Df = χ λ ( D ) f (cid:9) . Since χ λ = χ λ ′ if and only if there exists a w ∈ W with λ = w · λ ′ , we seethat this is equivalent also to E λ ( X ) = E λ ′ ( X ).Let A ( B ) denote the vector space of analytic functions on B = K/M ,topologized as in [12], § V.6.1. The analytic functionals are (loc. cit.) the func-tionals in the dual space A ′ ( B ) of A ( B ). Fix λ ∈ a C ∗ and recall the set Σ of restricted roots. For α ∈ Σ we write α := α/ h α, α i . We will need Harish-Chandra’s e -functions ([12], p. 163; note that Helgason uses a slightly differentnotation), defined by e − s ( λ ) := Y α ∈ Σ + s Γ (cid:18) m α h λ, α i (cid:19) Γ (cid:18) m α m α h λ, α i (cid:19) , (3.1)where s ∈ W , Σ + s := Σ +0 ∩ s − · Σ − and where Γ denotes the classical Gamma-function. Note that Σ + w = Σ +0 for the longest Weyl group element w . Thenthe fundamental result ([18], see also [19], § Theorem 3.1
The Poisson–Helgason transform P λ : A ′ ( B ) → E λ ( X ) givenby P λ ( T )( x ) := Z B e ( λ + ρ ) A ( x,b ) T (d b ) (3.2) is a bijection if and only if e w ( λ ) = 0 . Since χ λ = χ wλ for w ∈ W , one can always assume Re λ ∈ a ∗ + , so that e w ( λ ) = 0. Thus each joint eigenfunction is the Poisson integral of an analyticfunctional (see [12], Theorem V.6.6 and [19], Corollary 5.5.4).One also has a characterization of the class of joint eigenfunctions havingdistributional boundary values: Let d X denote the distance function on X anddefine the space E ∗ ( X ) of smooth functions of exponential growth by E ∗ ( X ) := n f ∈ E ( X ) | ( ∃ C > ∀ x ∈ X : | f ( x ) | ≤ Ce Cd X ( o,x ) o . (3.3)Put E ∗ λ ( X ) := E ∗ ( X ) ∩ E λ ( X ). Then one has (cf. [5], Theorem 12.2): Theorem 3.2
Suppose that λ ∈ a ∗ C is contained in the set Λ := (cid:26) λ ∈ a ∗ C (cid:12)(cid:12)(cid:12) h λ, α ih α, α i 6∈ − N (cid:27) . Then P λ : D ′ ( B ) → E ∗ λ ( X ) is a topological isomorphism. For λ ∈ Λ and ϕ ∈ E ∗ λ ( X ) we denote the unique distribution T ∈ D ′ ( B )with P λ ( T ) = ϕ by T λ,ϕ . We call T λ,ϕ the λ -boundary values of ϕ . Note that T λ,ϕ actually depends on λ , since P λ and P λ ′ in general differ even if λ ∈ W · λ ′ .The space C ∞ ( X ) has a natural real structure given by the real valuedfunctions. This real structure induces a real structure on the space D ( X ) ofinvariant differential operators. Here the space D R ( X ) of real invariant dif-ferential operators is given as the set of operators in D ( X ) commuting withthe complex conjugation on the function spaces. Equivalently, D R ( X ) is thesubspace of operators preserving the space of real valued smooth functions. Proposition 3.3 D ( X ) is spanned by D R ( X ) .Proof According to Theorem II.4.9 in [13] the Harish-Chandra homomorphismmaps the algebra D ( X ) isomorphically onto the algebra I ( a ) of W -invariantpolynomial functions on a . The Harish-Chandra homomorphism is a compo-sition of operations (e.g. taking radial parts) preserving real valued maps (seethe arguments leading up to Theorem II.5.18 in [13]). Therefore D R ( X ) getsmapped to the space I R ( a ) of real valued W -invariants. Since I R ( a ) spans I ( a ),this implies the claim. ⊓⊔ Note that the complex conjugation D of D ∈ D ( X ) is defined by D ( f ) := D ( f ). Similarly the complex conjugate of a character of D ( X ) is defined by χ ( D ) := χ ( D ). Therefore, D ( f ) = χ ( D ) f implies D ( f ) = D ( f ) = χ ( D ) f = χ ( D ) f = χ ( D ) f . (3.4)Since the Harish-Chandra homomorphism commutes with complex conjuga-tion, we have χ λ ( D ) = γ ( D )( λ ) = γ ( D )( λ ) = γ ( D )( λ ) = χ λ ( D ) = χ λ ( D ) . Together, we have proved the first part of the following proposition.
Proposition 3.4 D ( f ) = χ λ ( D ) f implies D ( f ) = χ λ ( D ) f . (i) If λ ∈ a ∗ is real, then E λ ( X ) is invariant under taking real and imaginaryparts. Moreover, χ λ ( D ) is real for D ∈ D R ( X ) and E λ ( X ) is spanned byits real valued elements. (ii) If w = − id , so that γ ( D )( − λ ) = γ ( D )( w λ ) = γ ( D )( λ ) , then we have χ λ = χ λ also for iν = λ ∈ i a ∗ . In particular, E λ ( X ) is again invariantunder taking real and imaginary parts. Finally, χ λ ( D ) is real for D ∈ D R ( X ) and E λ ( X ) is spanned by its real valued elements. (iii) Conversely, suppose that there exists a real valued joint eigenvector ϕ ∈E λ ( X ) with λ ∈ i a ∗ + . Then λ is contained in the subspace ker( w +id) ⊆ a ∗ ,which is proper if w = − id .Proof To show (ii) we calculate χ λ ( D ) = χ − λ ( D ) = γ ( D )( − λ ) = γ ( D )( λ ) = χ λ ( D ) . For (iii) we note that ϕ = ϕ ∈ E λ ( X ) implies χ λ = χ λ , whence there exists a w ∈ W with − λ = λ = w · λ .If λ = iν is regular, then ν belongs to an open Weyl chamber in a ∗ .Since W acts simply transitively on the set of Weyl chambers, we can find aunique s ∈ W such that s · ν ∈ a ∗ + . But then sw · ν = − s · ν ∈ − a ∗ + so that sws − ( s · ν ) ∈ − a ∗ + . Since w is the unique element in W sending a ∗ + to − a ∗ + ,this implies sws − = w . In particular, if ν ∈ a ∗ + , i.e. s = id, we find w = w ,and the claim follows. ⊓⊔ Recall that complex conjugation on distributions is defined by T ( f ) := T ( f ). Remark 3.5
Let λ ∈ Λ . Since Λ is invariant under complex conjugation, also λ ∈ Λ . By Proposition 3.4, ϕ ∈ E ∗ λ ( X ) implies ϕ ∈ E ∗ λ ( X ) and we can write ϕ and ϕ as Poisson integrals of uniquely determined distributions T λ,ϕ and T λ,ϕ : ϕ ( x ) = P λ ( T λ,ϕ )( x ) = Z B e ( λ + ρ ) A ( x,b ) T λ,ϕ (d b )and ϕ ( x ) = P λ ( T λ,ϕ )( x ) = Z B e ( λ + ρ ) A ( x,b ) T λ,ϕ (d b ) . On the other hand, taking complex conjugates we find ϕ ( x ) = T λ,ϕ (cid:0) e ( λ + ρ ) A ( x, · ) (cid:1) = T λ,ϕ (cid:16) e ( λ + ρ ) A ( x, · ) (cid:17) (3.5)= Z B e ( λ + ρ ) A ( x,b ) T λ,ϕ (d b ) = P λ (cid:0) T λ,ϕ (cid:1) ( x ) . From (3.5) we deduce T λ,ϕ = T λ,ϕ .The following immediate consequence of Remark 3.5 will allow us to dealwith non-real eigenfunctions (cf. [3], where a special case is used). Lemma 3.6
Let λ ∈ Λ . If w ∈ W satisfies w · λ ∈ Λ , then ϕ ( x ) = P w · λ ( T w · λ,ϕ )( x ) = Z B e ( w · λ + ρ ) A ( x,b ) T w · λ,ϕ (d b ) . principal series representations of G .Following [12] and [27], let ν ∈ a ∗ and consider the representation σ ν ( man ) = e ( iν + ρ ) log( a ) of P = M AN on C . We denote the induced representation on G by π ν = Ind GP ( σ ν ). The induced picture of this representation is constructedas follows: A dense subspace of the representation space is H ∞ ν := n f ∈ C ∞ ( G ) : f ( gman ) = e − ( iν + ρ ) log( a ) f ( g ) o with inner product( f | f ) = Z K/M f ( k ) f ( k ) d k = ( f | K | f | K ) L ( K/M ) and corresponding norm k f k = R K/M | f ( k ) | d k . The group action of G isgiven by ( π ν ( g ) f )( x ) = f ( g − x ). The actual Hilbert space, which we denote by H ν , and the representation on H ν , which we also denote by π ν , is obtained by completion (cf. [27], Ch. 9). The representations π ν ( ν ∈ a ) form the sphericalprincipal series of G . The representation ( π ν , H ν ) is a unitary ([12], p. 528)and irreducible (loc. cit. p. 530) Hilbert space representation.Given f ∈ C ∞ ( K/M ) we may extend it to a function on G by ˜ f ( g ) = e − ( iν + ρ ) H ( g ) f ( k ( g )). A direct computation shows that ˜ f ∈ H ∞ ν . On the otherhand, if f ∈ H ∞ ν , then the restriction f | K of f to K is an element of C ∞ ( K/M ).Moreover, if f ∈ C ∞ ( K/M ) and if ˜ f is as above, then ˜ f | K = f . The mapping f ˜ f described above is isometric with respect to the L ( K/M )-norm. Wemay hence identify C ∞ ( K/M ) ∼ = H ∞ ν . The advantage is that the represen-tation space is independent of ν . The group action on C ∞ ( K/M ) is realizedby ( π ν ( g ) f )( kM ) = f ( k ( g − k ) M ) e − ( iν + ρ ) H ( g − k ) . (3.6)This is called the compact picture of the (spherical) principal series. Notice thatfor g ∈ K the group action (3.6) simplifies to the left-regular representationof the compact group K on K/M .Let ν ∈ a ∗ . It follows from( π ν ( g )1)( k ) = e − ( iν + ρ ) H ( g − k ) = e ( iν + ρ ) A ( gK,kM ) (3.7)that the Poisson transform P iν ( T ) : G/K → C of T ∈ D ′ ( B ) is given by P iν ( T )( gK ) = T ( π ν ( g ) · . (3.8)A smooth vector f ∈ L ( K/M ) is a smooth function on
K/M . This followsfrom the Sobolev lemma, since f and all its derivatives are in L ( K/M ).3.3 Regularity of Γ -invariant Boundary ValuesIn this subsection we prove a regularity statement for distribution boundaryvalues of joint eigenfunctions on a compact quotient X Γ := Γ \ X of X , where Γ is a a co-compact, torsion free discrete subgroup of G . Choose a G -invariantmeasure ν on Γ \ G such that Z G f ( x ) d x = Z Γ \ G X γ f ( γx ) ! d ν ( Γ x ) (3.9)for f ∈ C c ( G ). We will denote the Hilbert space L ( Γ \ G, ν ) simply by L ( Γ \ G ).The G -invariance of ν implies that the equation( R Γ ( g ) f )( Γ x ) = f ( Γ xg )( g, x ∈ G , f ∈ L ( Γ \ G )) defines a unitary representation R Γ of G on L ( Γ \ G ),which is called the right-regular representation of G on Γ \ G .The action of G on B induces an action on D ′ ( B ) by push-forward: Given T ∈ D ′ ( B ), a test function f ∈ E ( B ) and g ∈ G , this action is ( gT )( f ) = T ( f ◦ g − ). When we denote the pairing between distributions and test functions byan integral, we also write T (d γb ) for ( γT )(d b ). Remark 3.7
A joint eigenfunction in ϕ ∈ L ( X Γ ) is automatically smooth,since the Laplace-Beltrami operator is elliptic. Thus we can view it as Γ -invariant joint eigenfunction ϕ ∈ E λ ( X ) which is automatically contained in E ∗ λ ( X ) since Γ \ G is compact. According to [13], formula (7) in § IV.5, theeigenvalues of the Laplacian − ∆ X Γ are non-negative and of the form h iλ, iλ i + | ρ | . Thus, either λ ∈ i a ∗ or else λ ∈ a ∗ with | λ | ≤ | ρ | . In the first case λ clearly is contained in A . In the second case this cannot be guaranteed. Thespectral parameters λ in i a ∗ are called the principal part of the spectrum of L ( X Γ ). Thus, for a joint eigenfunction in ϕ ∈ L ( X Γ ) with spectral parameterbelonging to the principal part, we have a unique boundary value distribution T iν,ϕ . Proposition 3.8
Let ϕ ∈ L ( X Γ ) be a joint eigenfunction with spectral pa-rameter λ = iν belonging to the principal part of the spectrum. Then theboundary value T iν,ϕ satisfies the invariance condition e π ν ( γ ) T iν,ϕ = T iν,ϕ ∀ γ ∈ Γ, (3.10) where e π ν denotes the dual representation on D ′ ( B ) corresponding to the prin-cipal series π ν acting on H ∞ λ = C ∞ ( B ) .Conversely, if a distribution T ∈ D ′ ( B ) is invariant under e π ν ( γ ) , then P iν ( T ) is invariant under π ν ( γ ) .Proof The equality ϕ ( γx ) = ϕ ( x ) for all γ and x implies (recall A ( g · x, g · b ) = A ( x, b ) + A ( g · o, g · b ) from Lemma 2.3) ϕ ( x ) = Z B e ( iν + ρ ) A ( γ · x,b ) T iν,ϕ (d b ) = Z B e ( iν + ρ ) A ( γ · x,γ · b ) T iν,ϕ (cid:0) d( γ · b ) (cid:1) = Z B e ( iν + ρ ) A ( x,b ) e ( iν + ρ ) A ( γ · o,γ · b ) T iν,ϕ (cid:0) d( γ · b ) (cid:1) . By the uniqueness of the boundary value, we obtain T iν,ϕ (cid:0) d( γ · b ) (cid:1) = e − ( iν + ρ ) A ( γ · o,γ · b ) T iν,ϕ (d b ) . (3.11)Now (3.11) and (3.8) imply the claim. ⊓⊔ In the situation of Proposition 3.7 we consider the mapping Φ ϕ : H ∞ ν → C ∞ ( Γ \ G ) , Φ ϕ ( f )( Γ g ) = T iν,ϕ ( π ν ( g ) f ) . Lemma 3.9 Φ ϕ is an isometry w.r.t. the norms of L ( K/M ) and L ( Γ \ G ) .Proof The operator Φ ϕ is equivariant with respect to the actions π ν on H ∞ ν andthe right regular representation of G on L ( Γ \ G ). We pull-back the L ( Γ \ G )inner product onto the ( g , K )-module H ∞ ν,K of K -finite and smooth vectors(which is dense in H ∞ ν , [25], p. 81):( f | f ) := ( Φ ϕ ( f ) | Φ ϕ ( f )) L ( Γ \ G ) . Let f ∈ H ∞ ν,K . Then A f : H ∞ ν,K → C , f ( f | f ) is a conjugate-linear, K -finite functional on the ( g , K )-module H ∞ ν,K . This module is irreducible andadmissible, since H ν is unitary and irreducible ([25], Theorems 3.4.10 and3.4.11). As A f is K -finite it is nonzero on at most finitely many K -isotypiccomponents. It follows that there is a linear map A : H ∞ ν,K → H ∞ ν,K such thatfor each f ∈ H ∞ ν,K the functional A f equals f ( Af | f ) L ( K/M ) . Theequivariance of Φ ϕ and the unitarity of π ν imply that A is ( g , K )-equivariant.Using Schur’s lemma for irreducible ( g , K )-modules ([25], p. 80), we deducethat A is a constant multiple of the identity and hence ( · | · ) is a constantmultiple of the original L ( K/M )-inner product on H ∞ ν,K . This constant is 1:First, Φ ϕ (1) = P iν ( T iν,ϕ ) = ϕ is the K -invariant lift of ϕ to L ( Γ \ G ). Then k Φ ϕ (1) k L ( Γ \ G ) = 1 = k k L ( K/M ) . ⊓⊔ Let ( y j ) and ( x j ) be bases for k and p , respectively, such that h y j , y i i = − δ ij , h x j , x i i = δ ij , where h , i as before denotes the Killing form. The Casimiroperator of k is Ω k = P i y i and the Casimir operator of g is Ω g = − X j x j + Ω k ∈ Z ( g ) , where Z ( g ) is the center of the universal enveloping algebra U ( g ) of g .It follows from T iν,ϕ ( f ) = Φ ϕ ( f )( Γ e ) that | T iν,ϕ ( f ) | ≤ k Φ ϕ ( f ) k ∞ . (3.12)We may now estimate this by a convenient Sobolev norm on L ( Γ \ G ). Let e ∆ denote the Laplace operator of Γ \ G . Then we have e ∆ = − Ω g + 2 Ω k . Definition 3.10
Let s ∈ R . The Sobolev space W ,s ( Γ \ G ) is (cf. [24], p. 22)the space of functions f on Γ \ G satisfying (1 + e ∆ ) s/ ( f ) ∈ L ( Γ \ G ) withnorm k f k W ,s ( Γ \ G ) = k (1 + e ∆ ) s/ ( f ) k L ( Γ \ G ) . Let m = dim( Γ \ G ) = dim( G ), and let s > m/
2. The Sobolev imbed-ding theorem for the compact space Γ \ G ([24], p. 19) states that the identity W ,s ( Γ \ G ) ֒ → C ( Γ \ G ) is a continuous inclusion ( C ( Γ \ G ) is equipped withthe usual sup-norm k · k ∞ ). It follows that there exists a C > k Φ ϕ ( f ) k ∞ ≤ C k Φ ϕ ( f ) k W ,s ( Γ \ G ) ∀ f ∈ C ∞ ( K/M ) . (3.13)Now we derive the announced regularity estimate for the boundary values:First, by increasing the Sobolev order, we may assume s/ ∈ N , so(1 + e ∆ ) s/ = (1 − Ω g + 2 Ω k ) s/ ∈ U ( g ) . Hence (1 + e ∆ ) s/ commutes with each G -equivariant mapping. Let f ∈ H ∞ ν .Then k Φ ϕ ( f ) k W ,s ( Γ \ G ) = (cid:13)(cid:13)(cid:13) (1 + e ∆ ) s/ Φ ϕ ( f ) (cid:13)(cid:13)(cid:13) L ( Γ \ G ) = (cid:13)(cid:13)(cid:13) Φ ϕ ((1 − Ω g + 2 Ω k ) s/ ( f )) (cid:13)(cid:13)(cid:13) L ( Γ \ G ) = (cid:13)(cid:13)(cid:13) (1 − Ω g + 2 Ω k ) s/ ( f ) (cid:13)(cid:13)(cid:13) L ( K/M ) . (3.14)Recall π ν ( Ω k ) = ∆ K/M and Ω g ∈ Z ( g ). Then (3.14) equals (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s/ X k =0 (cid:18) s/ k (cid:19) (1 + 2 ∆ K/M ) k ( − Ω g ) s/ − k ( f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( K/M ) ≤ s/ X k =0 (cid:18) s/ k (cid:19) (cid:13)(cid:13)(cid:13) (1 + 2 ∆ K/M ) k ( − Ω g ) s/ − k ( f ) (cid:13)(cid:13)(cid:13) L ( K/M ) . (3.15)Assume f ∈ H ∞ ν.K and recall that Ω g acts on the irreducible U ( g )-module H ∞ ν,K by multiplication with the scalar − ( h ν, ν i + h ρ, ρ i ) (cf. [27], p. 163), that is Ω g | H ∞ ν,K = − ( h ν, ν i + h ρ, ρ i ) id H ∞ ν,K . Then (3.15) equals s/ X k =0 (cid:18) s/ k (cid:19) (cid:13)(cid:13)(cid:13) (1 + 2 ∆ K/M ) k ( | ν | + | ρ | ) s/ − k ( f ) (cid:13)(cid:13)(cid:13) L ( K/M ) . (3.16)But (cid:0) | ν | + | ρ | (cid:1) − k ≤ | ρ | − s =: C ′ (0 ≤ k ≤ s/ C ′ (cid:0) | ν | + | ρ | (cid:1) s/ s/ X k =0 (cid:18) s/ k (cid:19) (cid:13)(cid:13) (1 + 2 ∆ K/M ) k ( f ) (cid:13)(cid:13) L ( K/M ) . (3.17)Since H ∞ ν.K is dense in H ∞ ν , this bound holds for all f ∈ H ∞ ν . Using (3.12)-(3.17) we get | T iν,ϕ ( f ) | ≤ C ′ (cid:0) | ν | + | ρ | (cid:1) s/ s/ X k =0 (cid:18) s/ k (cid:19) (cid:13)(cid:13) (1 + 2 ∆ K/M ) k ( f ) (cid:13)(cid:13) L ( K/M ) (3.18)for all f ∈ H ∞ ν and hence for all f ∈ C ∞ ( K/M ). We set k f k ( s ) := C ′ s/ X k =0 (cid:18) s/ k (cid:19) (cid:13)(cid:13) (1 + 2 ∆ K/M ) k ( f ) (cid:13)(cid:13) L ( K/M ) and note that it is a continuous C ∞ ( K/M )-seminorm independent of ϕ and ν . Since W leaves the norm on a ∗ C invariant, (3.18) yields: Proposition 3.11
Let s > dim( G ) such that s/ ∈ N . Then | T iν,ϕ ( f ) | ≤ (1 + | ν | ) s k f k ( s ) ∀ f ∈ C ∞ ( K/M ) (3.19) for the distribution boundary values T iν,ϕ corresponding to a Γ -invariant jointeigenfunction ϕ ∈ E iν ( X ) . For ν ∈ a ∗ , let D ′ ( B ) Γ denote the space of distributions T on B whichsatisfy e π ν ( γ ) T = T for all γ ∈ Γ . By Proposition 3.8, the Poisson transform P ν ( T ) of a distribution T ∈ D ′ ( B ) Γ is a function on the quotient X Γ . We mayhence also define D ′ ( B ) (1) Γ := n T ∈ D ′ ( B ) Γ (cid:12)(cid:12)(cid:12) k P ν ( T ) k L ( X Γ ) = 1 o . (3.20)Fix s as in Proposition 3.11. Then with D ′ ( B ) ν := n T ∈ D ′ ( B ) (cid:12)(cid:12)(cid:12) | T ( f ) | ≤ (1 + | ν | ) s k f k ( s ) ∀ f ∈ C ∞ ( K/M ) o (3.21)the above observations imply: Lemma 3.12 D ′ ( B ) (1) Γ ⊆ D ′ ( B ) ν . In other words: There exist s > and acontinuous norm k · k ( s ) on C ∞ ( B × B ) such that for any Γ -invariant jointeigenfunction ϕ ∈ E iν ( X ) with spectral parameters ν ∈ a ∗ C with real part in a ∗ + ,we have | T iν,ϕ ( f ) | ≤ (1 + | ν | ) s k f k ∀ f ∈ C ∞ ( B ) . The constant s > and the norm k · k ( s ) are independent of ϕ and ν . Each f ∈ C ∞ ( B ) ⊗ C ∞ ( B ) has the form f = P i,j c i,j f i ⊗ f j . We define across-norm k · k on C ∞ ( B ) ⊗ C ∞ ( B ) by k f k := inf n X i,j | c i,j |k f i k ( s ) k f j k ( s ) (cid:12)(cid:12)(cid:12) f = X i,j c i,j f i ⊗ f j o . This norm induces a continuous seminorm on the projective tensor product C ∞ ( B ) b ⊗ π C ∞ ( B ) (cf. [23], p. 435). Let ψ ∈ E iµ ( X ) denote another Γ -invariantjoint eigenfunction with distribution boundary values T iµ,ψ ∈ D ′ ( B ) and spec-tral parameter µ ∈ a ∗ . Given f = P i,j c i,j f i ⊗ f j ∈ C ∞ ( B ) ⊗ C ∞ ( B ) weobtain | ( T iν,ϕ ⊗ T iµ,ψ )( f ) | ≤ X i,j | c i,j | · | T iν,ϕ ( f i ) | · | T iµ,ψ ( f j ) |≤ (1 + | ν | ) s (1 + | µ | ) s X i,j | c i,j | · k f i k ( s ) · k f j k ( s ) , (3.22)which implies (by taking the infimum) | ( T iν,ϕ ⊗ T iµ,ψ )( f ) | ≤ (1 + | ν | ) s (1 + | µ ) s k f k (3.23)for all f ∈ C ∞ ( B ) ⊗ C ∞ ( B ). But C ∞ ( B × B ) ∼ = C ∞ ( B ) b ⊗ π C ∞ ( B ) (cf. [23],p. 530) implies that (3.23) holds for all f ∈ C ∞ ( B × B ).Summarizing we obtain the main result of this section: Theorem 3.13
There exist s > and a continuous norm k · k on C ∞ ( B × B ) such that for any two Γ -invariant joint eigenfunctions ϕ ∈ E iν ( X ) and ψ ∈E iµ ( X ) with spectral parameters ν, µ ∈ a ∗ C with real part in a ∗ + , we have | ( T iν,ϕ ⊗ T iµ,ψ )( f ) | ≤ (1 + | ν | ) s (1 + | µ ) s k f k ∀ f ∈ C ∞ ( B × B ) . The constant s > and the norm k · k are independent of ϕ, ψ, ν, µ . Definition 4.1
Given ν, ν ′ ∈ a ∗ C , we define d ν,ν ′ : G/M → C by d ν,ν ′ ( gM ) := e ( iν + ρ ) H ( g ) e ( iν ′ + ρ ) H ( gw ) (4.1) Lemma 4.2
Let γ, g ∈ G and a ∈ A . Then (i) d ν,ν ′ ( γgM ) = e ( iν + ρ ) A ( γ · o,γg · b + ) e ( iν ′ + ρ ) A ( γ · o,γg · b − ) d ν,ν ′ ( gM ) . (ii) d ν,ν ′ ( gaM ) = e i ( ν + w · ν ′ ) log a d ν,ν ′ ( gM ) . Proof
Part (i) follows from Lemma 2.4 and for (ii) we recall that w · ρ = − ρ to calculate d ν,ν ′ ( gaM ) = e ( iν + ρ ) H ( ga ) e ( iν ′ + ρ ) H ( gaw ) = e ( iν + ρ )( H ( g )+log a ) e ( iν ′ + ρ )( H ( gw )+log( w − aw )) = d ν,ν ′ ( gM ) e ( iν + ρ ) log a e ( iν ′ + ρ ) log( w − aw ) = d ν,ν ′ ( gM ) e iν log a e iw · ν ′ log( a ) . ⊓⊔ Definition 4.3
For functions f on G/M , the weighted Radon transform R ν,ν ′ on G/M is given by( R ν,ν ′ f )( g ) := Z A d ν,ν ′ ( ga ) f ( ga ) d a, (4.2)whenever this integral exists.If R ν,ν ′ ( f ) exists, then it is a right- A -invariant function on G/M and hencea function on
G/M A ∼ = B (2) (cf. Lemma 2.7). Lemma 4.4
Let f ∈ C ∞ c ( G/M ) . Then R ν,ν ′ ( f ) ∈ C ∞ c ( G/M A ) = C ∞ c ( B (2) ) .Proof Projecting the support of f to G/M A we can find a compact subset C of G/M A such that f a ( gM ) := f ( gaM ) = 0for all a ∈ A , whenever gM A / ∈ C . For these g we have R ν,ν ′ ( f )( g ) = 0. ⊓⊔ Remark 4.5 (i) Identifying
G/M A with B (2) we see that elements of C ∞ c ( G/M A )can be extended by zero to yield elements of C ∞ c ( B ). In particular we mayinterpret R λ,ν ′ also as an integral transform C ∞ c ( G/M ) → C ∞ c ( B ), wherewe view C ∞ c ( B (2) ) as a subset of C ∞ c ( B ), extending all functions by zeroon B \ B (2) .(ii) Lemma 4.2 implies that for ν, ν ′ ∈ a ∗ C we have R ν,ν ′ ( f a ) = e − i ( ν + w · ν ′ ) log a R ν,ν ′ ( f ) . In particular, R ν, − w · ν is A -invariant. Proposition 4.6
Let ν, ν ′ ∈ a ∗ C and f ∈ C ∞ c ( G/M ) . For γ ∈ G set f γ ( gM ) := f ( γ − gM ) . Then the following equivariance property holds for ( b, b ′ ) ∈ B × B . ( R ν,ν ′ f γ )( b, b ′ ) = e ( iν + ρ ) A ( γ · o,b ) e ( iν ′ + ρ ) A ( γ · o,b ′ ) ( R ν,ν ′ f )( γ − · b, γ − · b ′ ) . Proof
By Remark 4.5 it suffices to prove the claim for ( b, b ′ ) = ( g · b + , g · b − )in B (2) , where gM A is determined uniquely by ( b, b ′ ) (see Proposition 2.7).Using first Lemma 4.2 and then Lemma 2.3 we can calculate( R ν,ν ′ f γ )( gM A ) = Z A d ν,ν ′ ( gaM ) f ( γ − gaM ) da = Z A d ν,ν ′ ( γ − gaM ) f ( γ − gaM ) e − ( iν + ρ ) A ( γ − · o,γ − g · b + ) × e − ( iν ′ + ρ ) A ( γ − · o,γ − g · b − ) da = Z A d ν,ν ′ ( γ − gaM ) f ( γ − gaM ) e ( iν + ρ ) A ( γ · o,g · b + ) × e ( iν ′ + ρ ) A ( γ · o,g · b − ) da = e ( iν + ρ ) A ( γ · o,g · b + ) e ( iν ′ + ρ ) A ( γ · o,g · b − ) ( R ν,ν ′ f )( γ − gM A ) . ⊓⊔ If one considers ν, ν ′ ∈ a ∗ , then it is clear from Definition 4.1 that d ν,ν ′ as well as its derivatives are of polynomial growth in the spectral parameters.Hence Proposition 4.7
Let χ ∈ C ∞ c ( G/M ) . For each continuous seminorm k · k on C ∞ ( B ) there is K > and a continuous seminorm k · k on C ∞ ( G/M ) such that for all f ∈ C ∞ ( G/M ) and all ( ν, ν ′ ) ∈ ( a ∗ ) the estimate kR ν,ν ′ ( χf ) k ≤ ((1 + | ν | ) · (1 + | ν ′ | )) K k χf k (4.3) holds. G/M and Γ \ G/M
Definition 4.8
Fix ν, ν ′ ∈ a ∗ + and ϕ ∈ E ∗ iν ( X ) , ϕ ′ ∈ E ∗ iν ′ ( X ). Let T iν,ϕ and T iν ′ ,ϕ ′ denote their respective boundary values. The Patterson-Sullivan distri-bution
P S ϕ,ϕ ′ on G/M associated to ϕ and ϕ ′ is defined by h f, P S ϕ,ϕ ′ i G/M := Z B (2) R ν, − w · ν ′ ( f )( b, b ′ ) T iν,ϕ (d b ) T − iw · ν ′ ,ϕ ′ (d b ′ ) , (4.4)where f ∈ C ∞ c ( G/M ) is a test function. Note that this makes sense since B (2) is open in B , so the distribution T iν,ϕ (d b ) ⊗ T − iw · ν ′ ,ϕ ′ (d b ′ ) on B canbe restricted to B (2) , and R ν, − w · ν ′ ( f ) is compactly supported in B (2) byLemma 4.4. More precisely, we obtain h f, P S ϕ,ϕ ′ i G/M = Z B × B ( R ν, − w · ν ′ f )( b, b ′ ) T iν,ϕ (d b ) ⊗ T − iw · ν ′ ,ϕ ′ (d b ′ ) . (4.5)Since boundary values of Γ -invariant and L ( X Γ )-normalized eigenfunc-tions also have polynomial bounds in the eigenvalue parameters, Proposition4.7 and Theorem 3.13 imply the following estimate: Proposition 4.9
Let χ ∈ C ∞ c ( G/M ) . Then there exists K > and a semi-norm k·k on C ∞ c ( G/M ) such that following estimate holds for all f ∈ C ∞ ( G/M ) ,all ν, ν ′ ∈ a ∗ + , and all joint eigenfunctions ϕ ∈ E ∗ iν ( X ) and ϕ ′ ∈ E ∗ iν ′ ( X ) , whichare Γ -invariant and L ( X Γ ) -normalized: | P S ϕ,ϕ ′ ( χf ) | ≤ ((1 + | ν | ) · (1 + | ν ′ | )) K k χf k . (4.6) Proof
By Theorem 3.13 and by Proposition 4.7, we have, for f ∈ C ∞ ( G/M ), | P S ϕ,ϕ ′ ( χf ) | = | ( T iν,ϕ ⊗ T − iw · ν ′ ,ϕ ′ )( R ν, − w · ν ′ ( χf )) |≤ ((1 + | ν | ) · (+ | ν ′ | )) s kR ν, − w · ν ′ ( χf ) k ′ ≤ ((1 + | ν | ) · (1 + | ν ′ | )) s + K k χf k , where k · k ′ is the fixed seminorm on C ∞ ( B × B ) constructed in Theorem 3.13.The constants s and K are independent of f , since k · k ′ is fixed. ⊓⊔ The following proposition will allow us to define Patterson–Sullivan distri-butions also on the quotient Γ \ G/M . Proposition 4.10
Suppose that ϕ and ϕ ′ are Γ -invariant joint eigenfunctionswith spectral parameters iν and iν ′ in i a ∗ + . Then the distribution P S ϕ,ϕ ′ on G/M is Γ -invariant. Proof
For f ∈ C ∞ c ( G/M ) we calculate, using first (3.11) and then Proposi-tion 4.6, h f γ , P S ϕ,ϕ ′ i G/M = Z B × B (cid:0) R ν, − w · ν ′ f γ (cid:1) ( b, b ′ ) T iν,ϕ (d b ) ⊗ T − iw · ν ′ ,ϕ ′ (d b ′ )= Z B × B (cid:0) R ν, − w · ν ′ f γ (cid:1) ( γ · ( b, b ′ )) e − ( iν + ρ ) A ( γ · o,γ · b ) × e − ( − iw · ν ′ + ρ ) A ( γ · o,γ · b ′ ) T iν,ϕ (d b ) ⊗ T − iw · ν ′ ,ϕ ′ (d b ′ )= Z B × B (cid:0) R ν, − w · ν ′ f (cid:1) ( b, b ′ ) T iν,ϕ (d b ) ⊗ T − iw · ν ′ ,ϕ ′ (d b ′ )= h f, P S ϕ,ϕ ′ i G/M . ⊓⊔ Remark 4.11
Let ϕ ∈ E ∗ iν and ϕ ′ ∈ E ∗ iν ′ be Γ -invariant eigenfunctions. Thenby Remark 4.5 we see that h f a , P S ϕ,ϕ ′ i G/M = e − i ( ν − ν ′ ) log( a ) h f, P S ϕ,ϕ ′ i G/M . (4.7)In other words, the P S ϕ,ϕ ′ are eigendistributions for the action of A on G/M (given by right-translation). In particular, if ν − ν ′ = 0, then the associatedPatterson–Sullivan distribution is invariant under right-translation by A .Since B is compact, we can (by using partition of unity) also choose a cutoff χ ∈ C ∞ c ( X × B ) such that P γ ∈ Γ χ ( γ · ( z, b )) = 1. Such a function we call a smooth fundamental domain cutoff for Γ . Let T ∈ D ′ ( X × B ) be a Γ -invariantdistribution and f a Γ -invariant smooth function on X × B . Suppose there is f ∈ C ∞ c ( X × B ) such that P γ ∈ Γ f ( γ · ( z, b )) = f ( z, b ). Then h f , T i X × B = Z X × B n X γ ∈ Γ χ ( γ · ( z, b )) o f ( z, b ) T (d z, d b )= Z X × B X γ ∈ Γ χ ( z, b ) f ( γ · ( z, b )) T (d z, d b ) . By the invariance of T this equals R X × B χ ( z, b ) f ( z, b ) T (d z, d b ). We thus have Proposition 4.12
Let T ∈ D ′ ( G/M ) be a Γ -invariant distribution. Let f be a Γ -invariant smooth function on G/M . Then for any f , f ∈ C ∞ c ( G/M ) suchthat P γ ∈ Γ f j ( γ · ( z, b )) = f ( z, b ) ( j = 1 , ) we have h f , T i G/M = h f , T i G/M . This proposition implies that the following definition of Patterson–Sullivandistributions on Γ \ G/M is independent of the choice of a smooth fundamentaldomain cutoff.
Definition 4.13
Let ν, ν ′ ∈ a ∗ + . Suppose that ϕ ∈ E ∗ iν ( X ) and ϕ ′ ∈ E ∗ iν ′ ( X )are Γ -invariant joint eigenfunctions. Since P S ϕ,ϕ ′ is a Γ -invariant distributionon G/M , the definition descends to the quotient Γ \ G/M via h f, P S Γϕ,ϕ ′ i Γ \ G/M := h χf, P S Γϕ,ϕ ′ i G/M , (4.8)where χ is a smooth fundamental domain cutoff. We deal with the asymptotic behavior of oscillatory integrals Z X f h ( x, y ) e iψ ( x,y ) /h d x as h ↓ y = ( b, b ′ , ν, ν ′ ) ranges in B × ( a ∗ ) , and the phase functionarises from non-euclidean plane waves, ψ ( x, b, b ′ , ν, ν ′ ) = νA ( x, b ) − ( w · ν ′ ) A ( x, b ′ ) . (5.1)5.1 Phase FunctionsWe rewrite (5.1) as follows: ψ ( x, b, b ′ , ν, ν ′ ) = νA ( gan · o, g · b + ) − ( w · ν ′ ) A ( gan · o, gw · b + ) . Here we used Remark 2.8(d) to write ( b, b ′ ) = g · ( b + , w · b + ) with g ∈ G and w ∈ W , and we defined a ∈ A and n ∈ N through x = gan · o . Lemma 2.3 andLemma 2.4 give A ( gan · o, gw · b + ) = A ( n · o, w · b + ) + A ( ga · o, gaw · b + )= − H ( n − w ) + H ( gaw )= − H ( n − w ) + H ( gw ) + log( w − aw )= H ( gw ) + w − · log a − H ( n − w ) . In particular, A ( gan · o, g · b + ) = H ( ga ) = H ( g ) + log a . It follows that ψ ( x, b, b ′ , ν, ν ′ ) = νH ( g ) − ( w · ν ′ ) H ( gw )+ ( ν − ww · ν ′ ) log a + ( w · ν ′ ) H ( n − w ) . (5.2)We impose assumptions which will imply that stationary points of the phasefunction ψ only arise from the last term. In that context the following set willbe important. a ∗ (2) := { ( ν, ν ′ ) ∈ ( a ∗ ) | ∀ = w ∈ W, ν = w · ν ′ } . (5.3)Notice that ( ν, ν ) ∈ a ∗ (2) iff ν is regular, i.e. ν ∈ a ∗ reg . Moreover, ( a ∗ + ) ⊆ a ∗ (2) .We start with a standard observation: Proposition 5.1
The derivative of the Iwasawa projection H : G → a is givenby d nak H ( nak )( X, Y, Z ) = ˜ n · k − · a − · X + ˜ n · k − · Y + ˜ n · Z, where nak = ˜ k ˜ a ˜ n ∈ KAN . Now we consider the map ϕ µw given by ϕ µw ( n ) = µ (cid:0) H ( nw ) (cid:1) = h H µ , H ( nw ) i for H µ ∈ a . Then (5.4) impliesd ϕ µw ( n )( X ) = h H µ , ˜ nw − · X i = h w · (˜ n − · H µ ) , X i for X ∈ n and nw = ˜ k ˜ a ˜ n . In order to have a clean description of the criticalpoints of ϕ µw we introduce Σ w, ± := { α ∈ Σ + | w · α ∈ Σ ± } and set N w := exp( n w ), where n w := P α ∈ Σ w, + g α . Note that N w = { e } . Lemma 5.2
For w ∈ W and µ ∈ a ∗ reg the set of critical points of the map ϕ µw : N → R is N w .Proof Writing ˜ n − = exp Y we obtain w · (˜ n − · H µ ) = w · H µ + w · ( e ad Y − id) H µ , so that d ϕ µw ( n ) vanishes if and only if the part of ( e ad Y − id) H µ ∈ n whichgets mapped into θ n by w is zero.Write Y = P α ∈ Σ w, + Y α + P β ∈ Σ w, − Y β . and let β be the minimal ele-ment β ∈ Σ w, − with Y β = 0 and note that ( e ad Y − id) H µ is a finite linearcombination of iterated Lie brackets of Y α ’s and Y β ’s. Such an element be-long to the root space given by the sum of all the involved α ’s and β ’s. Theminimality condition shows that this root cannot be β . In fact, if it where,no β ’s could occur in the sum, but a sum of roots in Σ w, + is again in Σ w, + .Therefore ( e ad Y − id) H µ contains a summand of the form −h µ, β i Y β , and if h µ, β i 6 = 0, then n cannot be a critical point of ϕ w . Thus, if n is a criticalpoint, then Y ∈ n w and ˜ n = exp( − Y ) ∈ N w . This implies w ˜ nw − ∈ N , andtogether with nw = ˜ k ˜ a ˜ n , also n = w ˜ nw − ∈ N w ⊆ N ∩ wN w − , ˜ a = 1, and˜ k = w .Conversely, assume that n ∈ N w . Then H ( nw ) = H ( w ˜ n ) = 0, so thatd ϕ µw ( n )( X ) = h w · H λ , w ˜ nw − · X i = h w · H µ , n · X i = 0for all X ∈ n , since n · X ∈ n and w · H µ ∈ a . ⊓⊔ Proposition 5.3 ([8])
For µ ∈ a ∗ reg the function ψ µ : N → R , ψ µ ( n ) = µH ( n − w ) , has n = e as its only critical point. The Hessian S ( µ ) := ∇ ψ µ ( e ) is symmetricand non-degenerate. Its signature and determinant are sgn( S ( µ )) = X α ∈ Σ + sign( h µ, α i ) dim( g α ) , (5.4) (cid:12)(cid:12) det S ( µ ) (cid:12)(cid:12) = Y α ∈ Σ + (cid:12)(cid:12) h µ, α i (cid:12)(cid:12) dim( g α ) . (5.5) Proof
By [8, Corollary 5.2], the differential of g µH ( g ) equals Y
7→ h
Y, n ( g ) − · H µ i at g ∈ KAn ( g ) ⊂ G . A calculation shows that the differential of the em-bedding ι : N → G , n n − w , is d ι ( n ) : X w − n · ( − X ). It followsthat d ψ µ ( n ) : X
7→ −h w − n · X, n ( n − w ) − · H µ i . In particular, d ψ µ ( e ) : X
7→ −h w · X, H µ i = 0 because n = w · n is orthogonalto a . That e is the only critical point of ψ µ follows from Lemma 5.2, appliedto w ∈ W .By [8, Lemma 6.1], the Hessian form g × g → R at g = e of g µH ( g )equals ( Y, Z ) X α ∈ Σ + h µ, α i (cid:10) p α Y − θp − α Y, p − α Z (cid:11) . (5.6)Here p α is the projection g → g α corresponding to the direct sum decompo-sition g = m ⊕ a ⊕ α ∈ Σ g α . Composing (5.6) with d ι ( e ) : X
7→ − w · X , wededuce ∇ ψ µ ( e )( w · X, w · Y ) = X α ∈ Σ + h µ, α i (cid:10) − θp − α X, p − α Y (cid:11) , X, Y ∈ n . (5.7)By the regularity of µ , we have h µ, α i 6 = 0. Since ( X, Y )
7→ h− θX, Y i is aninner product, the non-degeneracy of the Hessian and the formulae for thesignature and the determinant are seen after choosing a suitable orthonormalbasis of n = ⊕ α ∈ Σ + g − α . ⊓⊔ Lemma 5.4
Assume ( ν, ν ′ ) ∈ a ∗ (2) . Then d x ψ ( x, b, b ′ , ν, ν ′ ) = 0 iff ν ′ = ν , ( b, b ′ ) = g · ( b + , b − ) ∈ B (2) , and x ∈ gA · o .Proof Suppose d x ψ ( x, b, b ′ , ν, ν ′ ) = 0. Since log is a diffeomorphism, it followsthat ν − ww · ν ′ = 0 in (5.2). Therefore, in view of the assumption, w = w = w − , ( b, b ′ ) = g · ( b + , w · b + ) ∈ B (2) , and ν = ν ′ . With these parameters (5.2)reduces to ψ ( x, b, b ′ , ν, ν ′ ) = νH ( g ) − ( w · ν ′ ) H ( gw ) + ( w · ν ′ ) H ( n − w ) . (5.8)The remaining assertions follow from Proposition 5.3.5.2 AsymptoticsIt is convenient to have notation for describing asymptotic behavior. In general,for a given locally convex space E , we denote by h − k E the locally convex spaceof functions f : I → E , h f h , such that h k f h is uniformly bounded in E .In particular, h E denotes the space of bounded functions I → E . Here I isa bounded set of positive reals, having 0 as a limit point. The seminorms are f sup h ∈ I h k k f h k , where k · k runs through the seminorms of E . Asymptoticexpansions are defined with respect to the scale (cid:0) h j − k E (cid:1) ≤ j ∈ Z . The locallyconvex space E = C ∞ c ( Ω ) is a regular inductive limit for any second countable smooth manifold Ω . Therefore, ( f h ) ∈ h − k C ∞ c ( Ω ) iff ( f h ) ∈ h − k C ∞ c ( K ) forsome compact K ⊂ Ω .Lemma 5.4 and the principle of non-stationary phase imply the followingresult. Lemma 5.5
Let f h ∈ h C ∞ c ( X × B × a ∗ (2) ) and compact sets S ⊂ X , S B ⊆ B , such that S × S B contains the projections to X × B of the supports of f h . Assume that g · ( b + , b − ) ∈ S B implies ( gA · o ) ∩ S = ∅ . Then Z X f h ( x, b, b ′ , ν, ν ′ ) e iψ ( x,b,b ′ ,ν,ν ′ ) /h d x ∈ h ∞ C ∞ c ( B × a ∗ (2) ) . (5.9) Remark 5.6
Lemma 5.4 states in particular that the phase function ψ doesnot have a critical point if ( ν, ν ′ ) ∈ a ∗ (2) and ν = ν ′ . Therefore, (5.9) alsoholds if the a ∗ (2) -component of the supports of f h is contained in a compactsubset disjoint to the diagonal.We shall be interested in the asymptotic behavior of oscillatory integrals F h ( b, b ′ , ν, ν ′ ) = Z X f h ( x, b, ν, ν ′ ) e ψ ( x,b,b ′ ,ρ,ρ ) e iψ ( x,b,b ′ ,ν,ν ′ ) /h d x. (5.10)Lemma 5.5 implies that F h ( b, b ′ , ν, ν ′ ) ∈ h ∞ C ∞ c ( B × a ∗ (2) ) if f h ∈ h C ∞ c ( X × B × a ∗ (2) ).The following construction gives a function useful for cutting off the inte-grand near the stationary points. Lemma 5.7
Let S ⊂ X compact. There exists β ∈ C ∞ c ( B (2) ) ⊂ C ∞ ( B ) such that ( gA · o ) ∩ S = ∅ implies that ( g · M, g · w M ) is in the interiorof the support of − β . Moreover, if we view β ∈ C ∞ c ( G/M A ) , then the A -invariant lift ˆ β ∈ C ∞ ( G/M ) of β is well-defined. If S A ⊂ A is compact, thenthe projection of KS A N to G/M intersects the support of ˆ β in a compact set.Proof In view of the smooth Urysohn lemma, to prove the existence of β , itsuffices to show that the set of all gM A ∈ G/M A ∼ = B (2) for which gAK/K intersects S is compact. If S ′ is the preimage of S in G under the canonicalprojection G → G/K , then this amounts to the observation that S ′ A/M A iscompact.An A -invariant lift ˆ β satisfies ˆ β ( gaM ) = β ( gM A ). The existence anduniqueness of ˆ β is clear. The support of ˆ β , when viewed as a M A -invariantfunction on G , is contained in KAS N for some compact S N ⊂ N . The asser-tion about the compactness of the intersection follows. ⊓⊔ We introduce a notation for the ordinary
Radon transform R : C ∞ c ( G/M ) → C ∞ c ( G/M A ) , R f ( gM A ) = Z A f ( gaM ) d a (5.11)and note that, in the situation of Lemma 5.7, we have β · R ( f ) = R ( ˆ βf ) forall f ∈ C ∞ c ( G/M ). For µ ∈ a ∗ , we set κ ( µ ) = C N (cid:18) Y α ∈ Σ + (cid:12)(cid:12) h µ, α i (cid:12)(cid:12) dim( g α ) (cid:19) − / e πis/ , (5.12)where C N is defined in (2.10) and the signature s = P α ∈ Σ + sign( h µ, α i ) dim( g α )is, as a function of µ , constant in each Weyl chamber.Fix f h ∈ h C ∞ c (cid:0) X × B × a ∗ (2) (cid:1) and suppose that ( b, b ′ ) = g · ( b + , b − ) = gM A . Then (5.2) holds with w = w = w − , and we have, setting x = an · oψ ( g · x, b, b ′ , ν, ν ′ ) = νH ( g ) − ( w · ν ′ ) H ( gw ) + ( ν − ν ′ ) log a + ( w · ν ′ ) H ( n − w ) ,ψ ( g · x, b, b ′ , ρ, ρ ) = ψ ( g · x, b, b ′ , ρ ) = ρ ( H ( g ) + H ( gw )) − ρH ( n − w ) . Here we also used ρ = − w · ρ . Furthermore, f h ( g · x, b, ν, ν ′ ) = f h ( gan · o, gan · b + , ν, ν ′ ) = f h ( ganM, ν, ν ′ ) . Using the weight function d h ( gM, ν, ν ′ ) := d ν/h, − w · ν ′ /h ( gM ) = e ( ih ν + ρ ) H ( g ) e ( − ih w · ν ′ + ρ ) H ( gw ) (5.13)(5.10), Lemma 4.2(ii), and d − n = e − ρH ( n − w ) d n yield F h ( b, b ′ , ν, ν ′ ) = Z A Z N f h ( ganM, ν, ν ′ ) e ih ( w · ν ′ ) H ( n − w ) d ν/h, − w · ν ′ /h ( gM ) · e ih (cid:0) ( ν − ν ′ ) log a (cid:1) d − n d a = Z A d h ( gaM, ν, ν ′ ) Z N f h ( ganM, ν, ν ′ ) e ih ( w · ν ′ ) H ( n − w ) d − n d a. Let S ⊂ X be a compact set which contains the X -projections of thesupports of f h . Then consider I h ( g, ν, ν ′ ) := ˆ β ( gM ) Z N f h ( gnM, ν, ν ′ ) e i ( w · ν ′ ) H ( n − w ) /h d − n, (5.14)where β is chosen as in Lemma 5.7, and ˆ β denotes the A -invariant lift of β to G/M . We have I h ( g, ν, ν ′ ) = I h ( gm, ν, ν ′ ) for m ∈ M since the weightedmeasure d − n is M -invariant. By Lemma 5.7, Proposition 5.3 and the methodof stationary phase applied to (5.14) we get I h ∈ h dim N/ C ∞ c (cid:0) G/M × a ∗ (2) (cid:1) and an asymptotic expansion I h ( gM, ν, ν ′ ) = κ ( w · ν ′ ) (2 πh ) dim N/ (cid:0) f h ( gM, ν, ν ′ ) + O ( h ) (cid:1) . (5.15)Here κ is defined by (5.12).The calculation above shows β ( gM A ) F h ( gM A, ν, ν ′ ) = Z A d h ( gaM, ν, ν ′ ) I h ( gaM, ν, ν ′ ) d a (5.16)= R (cid:0) d h I h ( · , ν, ν ′ ) (cid:1) ( gM A ) . On the other hand, Lemma 5.5 implies (1 − β ) F h ∈ h ∞ C ∞ (cid:0) B × a ∗ (2) (cid:1) . To-gether, we obtain F h − R ( d h I h ) ∈ h ∞ C ∞ c (cid:0) B × a ∗ (2) (cid:1) . (5.17)We collect these results in the following proposition: Proposition 5.8
Let f h ∈ h C ∞ c (cid:0) X × B × a ∗ (2) (cid:1) . Let S ⊂ X be a compactset which contains the X -projections of the supports of f h . Choose β as inLemma 5.7, and denote by ˆ β the A -invariant lift of β to G/M . Then I h ∈ h dim N/ C ∞ c ( G/M × a ∗ (2) ) has the asymptotic expansion I h ( gM, ν, ν ′ ) = κ ( w · ν ′ ) h dim N/ (cid:0) f h ( gM, ν, ν ′ ) + O ( h ) (cid:1) and the oscillatory integral (5.10) satisfies F h − R ( d h I h ) ∈ h ∞ C ∞ c (cid:0) B × a ∗ (2) (cid:1) . The definition of quantum limits of Wigner measures lifted to the cotangentbundle, also called semi-classical defect measures, and the study of their prop-erties is based on semi-classical microlocal analysis. It is convenient to use a ge-ometric h -pseudo-differential calculus. Refer to [6], [9] for h -pseudo-differentialoperators and to [21] and [10, Appendix] for geometric pseudo-differential cal-culi. The results in [6] and [9] are stated for the Weyl quantization. However,operator classes and principal symbols of operators do not depend on thechosen quantization.6.1 Geometric Pseudo-Differential CalculusLet X be a Riemannian manifold. Denote by exp x : T x X → X the exponentialmap of its Levi-Civita connection. With a symbol a h = a ( · ; h ) depending ona small parameter h > h ( a h ),Op h ( a h ) u ( x ) = Z T ∗ x X Z T x X e − i h ξ,v i /h χ ( x, v ) a h ( x, ξ ) u (exp x v ) d v d − ξ, (6.1) x ∈ X . Here d − ξ = (2 πh ) − dim X d ξ , and χ ∈ C ∞ ( T X ) is chosen such that χ = 1 holds in a neighborhood of the zero section and that its supportis contained in a bounded open neighborhood of the zero section where theexponential map is injective. In our applications, the x -support of the symbolsis compact.The symbols belong to symbol spaces S m,k ( T ∗ X ) = h − k S m ( T ∗ X ). Often a h ∈ S m,k ( T ∗ X ) has an asymptotic expansion in powers of h , a h ( x, ξ ) ∼ X j ≥ h − k + j a m − j ( x, ξ ) , a ℓ ∈ S ℓ ( T ∗ X ) . We shall always assume that a h has a principal symbol h − k a , i.e., a h − h − k a ∈ S m,k − ( T ∗ X ) with a ∈ S m ( T ∗ X ) necessarily uniquely determined.If X = exp x ( B r ) is a geodesic ball, then we trivialize the cotangent bundle, B r × T ∗ x X → T ∗ X, ( v, ξ ) ( y, τ T ∗ X [ y ← x ] ξ ) , y = exp x v. Here [ y ← x ] denotes the unique geodesic from x to y , and τ T ∗ Xγ the paralleltransport in the cotangent bundle T ∗ X along a curve γ in X . Using the triv-ialization, the quantization (6.1) is, after a change of variables, expressed asfollows,Op h ( a h ) u ( x ) = Z Z T ∗ X e − i h ξ, log x y i /h ψ ( x, y ) a h ( x, ξ ) u ( y ) d y d − η. (6.2)Here ξ = τ T ∗ X [ x ← y ] η , ψ ( x, exp x v ) = χ ( x, v ) /J ( x, v ), log x = exp − x , and J ( x, v ) isthe determinant of the differential of exp x at v . Observe that the phase functionin (6.2) is linear in ξ and generates the conormal bundle of the diagonal in X × X . Applying (6.2) with X replaced by convex charts, one deduces thatthe definition (6.1) leads to known classes Ψ m,k ( X ) of h -dependent pseudo-differential operators, [9, Section 8]. Notice that the cutoff χ in (6.1) insuresthat the operators Op h ( a h ) are properly supported. Remark 6.1
If the restriction of a h to each fiber of T ∗ X is a polynomial,then Op h ( a h ) has its Schwartz kernel supported in the diagonal and thus is adifferential operator.Modulo residual operators in Ψ −∞ , −∞ ( X ) the quantization map given by(6.1) is independent of the choice of χ . The symbol isomorphism of the geo-metric pseudo-differential calculus, S m,k ( T ∗ X ) /S −∞ , −∞ ( T ∗ X ) ∼ = Ψ m,k ( X ) /Ψ −∞ , −∞ ( X ) , is given by the quantization map Op h and inverted by a symbol homomorphism σ h . On the principal symbol level the rules for compositions and adjoints agreewith those of the Weyl calculus and other quantizations.The geometric calculus behaves nicely under pullback by isometries. Let ϕ : X → X a bijective isometry. Denote ϕ ∗ : D ′ ( X ) → D ′ ( X ), u u ◦ ϕ , thepullback operator, and ϕ −∗ its inverse. Denoted ϕ −⊤ : T ∗ X → T ∗ X, ( x, d ϕ ( x ) ⊤ η ) ( ϕ ( x ) , η ) , the symplectic map induced by ϕ . Lemma 6.2
For a h ∈ S m,k , ϕ ∗ Op h ( a h ) ϕ −∗ ≡ Op h ( a h ◦ d ϕ −⊤ ) mod Ψ −∞ , −∞ ( X ) . (6.3) Equality holds for differential operators, and if the cutoff in (6.1) satisfies χ ◦ d ϕ = χ . Proof
Let a h ∈ S −∞ ,k , u ∈ C ∞ c ( X ). Then (6.1) is an absolutely convergentintegral. Set A = Op h ( a h ) and B = Op h ( a h ◦ d ϕ −⊤ ). Fix x ∈ X , and set y = ϕ ( x ), S = d ϕ ( x ). Since ϕ is an isometry, ϕ (exp x v ) = exp y w if w = Sv .Using the linear symplectic change of variables ( w, η ) ( v, ξ ), w = Sv and ξ = S ⊤ η , we obtain Bϕ ∗ u ( x ) = Z Z T ∗ x × T x e − i h ξ,v i /h χ ( x, v ) a h (cid:0) y, S −⊤ ξ (cid:1) u ( ϕ (exp x v )) d v d − ξ = Z Z T ∗ y × T y e − i h η,w i /h χ ( y, w ) a h ( y, η ) u (exp y w ) d w d − η. Here χ ( y, w ) = χ ( x, S − w ). Hence Bϕ ∗ = ϕ ∗ A + ϕ ∗ R , where Ru ( y ) = Z Z T ∗ y × T y e − i h η,w i /h ( χ − χ )( y, w ) a h ( y, η ) u (exp y w ) d w d − η. Extending by density and continuity to a ∈ S m,k we obtain Bϕ ∗ = ϕ ∗ A + ϕ ∗ R with R ∈ Ψ −∞ , −∞ ( X ). Formula (6.3) follows. Obviously, R = 0 if χ = χ .To complete the proof we observe that are no non-zero differential operatorsin Ψ −∞ , −∞ ( X ). ⊓⊔ X = G/K be a symmetric space of noncompact type as in Section 2. Thegroup G acts on X by left translations which are isometries. For every x ∈ X ,the exponential map exp x : T x X → X is a diffeomorphism. Therefore, wedefine h -pseudo-differential operators on X by (6.2).The following lemma relates the geometric pseudo-differential calculus toFourier analysis on X . Set e λ,b ( x ) = e ( λ + ρ ) A ( x,b ) for x ∈ X , b ∈ B , λ ∈ a C ∗ .We associate a non-euclidean symbol ˜ a h with a symbol a h . Recall the map Φ : ( x, b, θ ) d x θA ( x, b ) from Proposition 2.5. Lemma 6.3
Let a h ∈ S m, ( T ∗ X ) . Define ˜ a h by Op h ( a h ) e iθ/h,b = ˜ a h ( · , b, θ ) e iθ/h,b . (6.4) Then ˜ a h ∈ h C ∞ ( X × B × a ∗ ) . Moreover, there exists r h ∈ h C ∞ ( X × B × a ∗ ) such that ˜ a h ( x, b, θ ) = a h ( ξ ) + ih ( D (2) a h )( ξ ) + r h ( x, b, θ ) , (6.5) ξ = d x θA ( x, b ) ∈ T ∗ x X . Here D (2) is a second order differential operator on T ∗ X with real coefficients.Proof Using (6.2) we write ˜ a h as an oscillatory integral over ( y, η ) ∈ T ∗ X .The phase function is ϕ ( x, y, b, θ, η ) = −h ξ, log x y i + θ ( A ( y, b ) − A ( x, b )) . We determine the stationary points of ϕ as a function of y and η . First, ϕ ′ η := d η ϕ = − τ T X [ y ← x ] log x y . It follows that y = x at a stationary point.Moreover, ϕ ′′ ηη = 0 and ϕ ′′ ηy = − I at y = x . Furthermore, ϕ ′ y = 0 at y = x implies ϕ ′ x ( x, b, θ ) = η . Hence for given x, b, θ the phase ϕ has the unique sta-tionary point ( y, η ) = ( x, ϕ ′ x ( x, b, θ )) which is non-degenerate. The signatureof the Hessian is zero, and the modulus of its determinant is unity. Recall thedefinition of d − η , and apply the method of stationary phase. ⊓⊔ Recall the Definition 2.6 of the algebra
A ⊂ S ∞ ( T ∗ X ) and the homomor-phisms χ λ from Section 3.1. Lemma 6.4 If p ∈ A , then Op h ( p ) ∈ D ( X ) . If P h = Op h ( p h ) ∈ D ( X ) , p h ∈ S m, , with principal symbol p ∈ A , then χ iν/h ( P h ) = p ( ν ) + O ( h ) as h ↓ , (6.6) uniformly as ν stays bounded, ν ∈ a ∗ ⊂ T ∗ o X . If P ∗ h = P h and χ iν/h ( P h ) isreal, then (6.6) holds with O ( h ) replaced by O ( h ) .Proof Left translation by an element of G acts as an isometry on X . The firstassertion follows from Remark 6.1. Let P h = Op h ( p h ) ∈ D ( X ) with principalsymbol p ∈ A . For ν ∈ a ∗ , h >
0, and ( x, b ) ∈ X × B , we have P h e iν/h,b = χ iν/h ( P h ) e iν/h,b = ˜ p h ( x, b, ν ) e iν/h,b , where we used (6.4). Hence χ iν/h ( P h ) = ˜ p h ( x, b, ν ) = p ( ν ) + O ( h ) , by (6.5) and (2.5).If χ iν/h ( P h ) is real, then χ iν/h ( P h ) = Re ˜ p h ( x, b, ν ) = p ( ν ) + O ( h ). Since P h is formally self-adjoint, p is real and the subprincipal symbol of P h is purelyimaginary. Therefore, the second term of the stationary phase expansion (6.5)for ˜ p h is also purely imaginary. This proves the last assertion. ⊓⊔ Let Γ be a co-compact, torsion-free discrete subgroup of G . The locallysymmetric space X Γ = Γ \ X is a Riemannian manifold. We denote the quan-tization map of (6.1) by Op Γh , if X is replaced by X Γ . The notation Op h ( a h )continues to denote pseudo-differential operators on X . We identify functions(distributions) on X Γ with Γ -invariant functions (distributions) on X . Opera-tors on D ′ ( X ) which are Γ -invariant restrict to operators on D ′ ( X Γ ). In (6.1)the cutoff χ ∈ C ∞ ( T X ) is assumed to equal unity in a neighbourhood of thezero section. In addition, we assume that χ is Γ -invariant, and is supportedin a sufficiently small neighbourhood of the zero section where the exponentialmap of X Γ is a diffeomorphism. By Lemma 6.2, we then haveOp Γh ( a h ) u = Op h ( a h ) u for a h ∈ S m,kΓ , u ∈ D ′ ( X Γ ). (6.7) Here, S m,kΓ denotes the subspace of symbols in S m,k ⊂ C ∞ ( T ∗ X ) which are Γ -invariant. Denote by Ψ m,kΓ ( X ) := Op Γh ( S m,kΓ ) the corresponding space ofpseudo-differential operators on X Γ .Denote by B ( H ) the algebra, equipped with the operator norm, of boundedoperators on a Hilbert space H . Since X Γ is compact, we have Ψ , Γ ( X ) ⊂ B ( L ( X Γ ), uniformly bounded in h . This follows from standard L -continuityproperties of pseudo-differential operators. Moreover, by H¨ormanders’s proof[17, Theorem 18.1.11] of L -continuity we have, for given ε > h >
0, the estimate k Op Γh ( a h ) k B ( L ( X Γ )) ≤ (1 + ε ) sup T ∗ X | a | + O ( √ h ) , (6.8)where a is the principal symbol of a h ∈ S , . Let a ∈ S Γ , a ≥
0. The sharpG˚arding inequality gives that there exists c > (cid:0) Op Γh ( a ) u | u (cid:1) L ( X Γ ) ≥ − ch k u k (6.9)for u ∈ C ∞ c ( X Γ ). For a proof see [6, Theorem 7.12], and [9, Theorem 5.3].6.3 Lifted Quantum LimitsEvery bounded sequence of distributions has a weak*-convergent subsequence. Lemma 6.5
Let ( ϕ j ) j , ( ϕ ′ j ) j be bounded sequences in L ( X Γ ) , < h j → .Set W j ( a ) = (cid:0) Op Γh j ( a ) ϕ j | ϕ ′ j (cid:1) L ( X Γ ) , a ∈ C ∞ c ( T ∗ X Γ ) . Then ( W j ) j is a bounded sequence in D ′ ( T ∗ X Γ ) . Assume that ω = lim j W j in D ′ ( T ∗ X Γ ) as j → ∞ . Then ω is a Radon measure on T ∗ X Γ of finite totalvariation, and Z T ∗ X Γ a d ω = lim j →∞ (cid:0) Op Γh j ( a h j ) ϕ j | ϕ ′ j (cid:1) L ( X Γ ) (6.10) if a h j ∈ S , Γ has principal symbol a ∈ S Γ . If k ϕ j k L ( X Γ ) = 1 and ϕ ′ j = ϕ j ,then ω is a probability measure.Proof Since Op Γh maps S , Γ continuously into B ( L ( X Γ )), the boundednessof ( W j ) j follows. Now assume lim j →∞ W j = ω . Let M ≥ sup j ( k ϕ j k , k ϕ ′ j k ). Itfollows from (6.8) that lim sup j | W j ( a ) | ≤ M sup T ∗ X | a | , implying that ω isa Radon measure of total variation ≤ M . Now assume k ϕ j k L ( X Γ ) = 1 and ϕ ′ j = ϕ j . Thus, we can choose M = 1. If 0 ≤ a ∈ C ∞ c ( T ∗ X Γ ), then it followsfrom the sharp G˚arding inequality (6.9) and Im (cid:0) Op h ( a ) u | u (cid:1) = O ( h ) that ω ( a ) ≥
0. Notice ω (1) = 1. Thus ω is a probability measure. ⊓⊔ Let ( ϕ j ) j ⊂ L ( X Γ ) be a sequence of normalized joint eigenfunctions of thealgebra D ( X ) of invariant differential operators on X with associated spectralparameters λ j ∈ a C ∗ , Dϕ j = χ λ j ( D ) ϕ j if D ∈ D ( X ). Let ∆ = ∆ X Γ ∈ D ( X )denote the Laplacian on X Γ . The eigenvalues χ λ j ( − ∆ ) = −h λ j , λ j i + | ρ | arenon-negative. We restrict attention to the principal spectrum, [7]. Therefore,assume that λ j ∈ i a ∗ . Set λ j = iν j /h j with unit vectors ν j ∈ a ∗ , h j = | λ j | − .We say that ( ϕ j ) j has lifted quantum limit ω if the sequence of distributions W j ∈ D ′ ( T ∗ X Γ ), W j ( a ) = (cid:0) Op Γh j ( a ) ϕ j | ϕ j (cid:1) L ( X Γ ) , a ∈ C ∞ c ( T ∗ X Γ ) , converges, ω = lim j →∞ W j . Passing to a subsequence, we can assume that θ = lim j →∞ ν j ∈ a ∗ C exists. Following [1] we then say that ω is the liftedquantum limit in the direction θ . The distributions W j are lifts of the Wignermeasures w j = | ϕ j | d x under the canonical projection π : T ∗ X Γ → X Γ , π ∗ W j = w j .In addition, we assume that the sequence ( h j ) j is strictly decreasing. Wecan then use h as a subscript, removing j from the notation. In particular, wedenote the spectral parameters iν h /h , and we write Z T ∗ X Γ a d ω = lim h ↓ (cid:0) Op Γh ( a ) ϕ h | ϕ h (cid:1) L ( X Γ ) . (6.11)Using the metric tensor we regard the unit sphere bundle S ∗ X Γ as a sub-set of the cotangent bundle T ∗ X Γ . Then, in view of the results recalled inSubsection 2.2, propagation of singularities and Lemma 6.4 allow us to provethe following invariance properties of lifted quantum limits. Theorem 6.6 ([22, Theorem 1.6(3)], [1, Theorem 1.3])
Assume that ( ϕ h ) h has the lifted quantum limit ω . Then supp( ω ) ⊂ S ∗ X Γ , and ω is invari-ant under the geodesic flow. Moreover, supp ω is contained in a joint level setof A , i.e. in a G -orbit in S ∗ X Γ . Moreover, for every p ∈ A , ω is invariantunder the Hamilton flow generated by p . If the direction θ ∈ a ∗ of ω is regular,then ω is A -invariant.Proof We can assume that ω is a lifted quantum limit in the direction θ =lim h ↓ ν h ∈ a ∗ ⊂ T ∗ o X .Note that − h ∆ X = Op h ( g ), where g ∈ A is the metric form, g ( ξ ) = | ξ | , ξ ∈ T ∗ X . Since | ν h | = 1 and ν h is real, χ iν h /h ( − h ∆ ) = 1 + h | ρ | . Hence k h ∆ X Γ ϕ h + ϕ h k L = O ( h ) . It follows from [9, Theorem 5.4] that the support of ω is contained in S ∗ X Γ = g − (1) because this is the characteristic variety of the h -differential operator h ∆ X Γ + 1. The invariance under the geodesic flow follows from [9, Theo-rem 5.5]. Let p ∈ A . Set P h = Op h ( p ) ∈ D ( X ). Choose an integer m such that theorder of P h is < m . Define the h -differential operator Q h = Op h ( p − p ( ξ ) + g m − ∈ D ( X ), 0 < h <
1. By Lemma 6.4 we have k Q h ϕ h k L = O ( h ) as h → ω ) is contained in the characteristicvariety of Q h . The latter intersected with the unit sphere bundle is containedin the level set p − ( p ( θ )). This proves that supp( ω ) is contained in a joint levelset.We prove that ω is invariant under the Hamilton flow generated by p .Adding a constant to p if necessary, we may assume that p = 1 on supp( ω ).By selfadjointness, the eigenvalues of P ∗ h P h + ( − h ∆ ) m are real so by the lastassertion of Lemma 6.4 we have k (cid:0) P ∗ h P h + ( − h ∆ ) m − (cid:1) ϕ h k L = O ( h ) . By [9, Theorem 5.5] we have, for every a ∈ C ∞ c ( T ∗ X ),0 = Z { p + g m , a } d ω = 2 Z { p, a } d ω. Here we used the invariance of ω under the geodesic flow and p = 1 on supp ω .This proves the invariance of ω under the Hamilton flow generated by p .Recall from Subsection 2.2 that (2.7) intertwines the Weyl chamber flowwith certain Hamilton flows. Indeed the last statement of that subsection saysthat because θ is regular, each one-parameter subgroup of the Weyl chamberflow can be realized as a Hamilton flow associated with a function in A . Thus,the A -invariance follows. ⊓⊔ We study the asymptotic behavior of the principal spectrum of X Γ whichcorrespnds to spectral parameters λ ∈ i a ∗ ; see [7]. Let ( ϕ h ) h , ( ϕ ′ h ) h ⊂ L ( X Γ )be sequences of normalized joint eigenfunctions, with purely imaginary spectralparameters iν h /h ∈ i a ∗ , iν ′ h /h ∈ i a ∗ . The Poisson–Helgason transform (3.2)gives unique representations, ϕ h ( x ) = Z B e ( iν h /h + ρ ) A ( x,b ) T iν h /h,ϕ h (d b ) , x ∈ X. (7.1)We use Lemma 3.6 to pick a suitable representation of ϕ ′ h as a Poisson integral: ϕ ′ h ( x ) = Z B e ( − iw · ν ′ h /h + ρ ) A ( x,b ′ ) T − iw · ν ′ h /h,ϕ ′ h (d b ′ ) , x ∈ X. (7.2)This reduces to (7.1) if w = − id and if ϕ h = ϕ ′ h is real valued. To simplifyour notation we write T h and ˜ T h for T iν h /h,ϕ h and T − iw · ν ′ h /h,ϕ ′ h , respectively. Lemma 7.1
Let χ ∈ C ∞ c ( X ) real-valued, and a h ∈ S , ( T ∗ X ) . Then (cid:0) Op h ( a h ) ϕ h | χϕ ′ h (cid:1) L ( X ) = Z B F h ( b, b ′ , ν h , ν ′ h ) T h (d b ) ⊗ ˜ T h (d b ′ ) , (7.3) where F h ( b, b ′ , ν, ν ′ ) = Z X χ ( x )˜ a h ( x, b, ν ) e ψ ( x,b,b ′ ,ρ,ρ ) e iψ ( x,b,b ′ ,ν,ν ′ ) /h d x. (7.4) Here, ˜ a h ∈ h C ∞ ( X × B × a ∗ ) is the non-euclidean symbol of Op h ( a h ) definedin Lemma 6.3, and ψ is the phase function of (5.2) .Proof We apply Op h ( a h ) to (7.1). The rules for composing Schwartz kernelsjustify interchanging the operator Op h ( a h ) with the integral (duality bracket).In the notation of Lemma 6.3, we getOp h ( a h ) ϕ h ( x ) = Z B ˜ a h ( x, b, ν h ) e iν h /h,b ( x ) T h (d b ) . Using the tensor product of distributions, we derive (cid:0) Op h ( a h ) ϕ h | χϕ ′ h (cid:1) L ( X ) = Z X Z B χ ( x )˜ a h ( x, b, ν h ) e iν h /h,b ( x ) e − w · iν ′ h /h,b ′ ( x ) T h (d b ) ⊗ ˜ T h (d b ′ ) d x. We interchange the integral over X with the duality bracket of distributionson B , (cid:0) Op h ( a h ) ϕ h | χϕ ′ h (cid:1) L ( X ) = Z B F h ( b, b ′ , ν h , ν ′ h ) T h (d b ) ⊗ ˜ T h (d b ′ ) . Here we used w − = w , and − w · ρ = ρ . ⊓⊔ Consider the weight function d h ( gM, ν, ν ′ ) := d ν/h, − w o · ν ′ /h ( gM ) = e ( iν/h + ρ ) H ( g ) e ( − iw · ν/h + ρ ) H ( gw ) . Following (5.11), (5.13), and (4.2), we have the weighted Radon transform R h : C ∞ c ( G/M ) → C ∞ c ( G/M A ) ⊂ C ∞ ( B ) , ( R h f )( gM A ) = Z A d h ( gaM, ν h , ν ′ h ) f ( gaM ) d a, and its dual R ′ h : D ′ ( B ) → D ′ ( G/M ). Further, (4.8) suggests to define
P S Γh := P S Γϕ h ,ϕ ′ h ∈ D ′ ( Γ \ G/M ), so that h P S Γh , f i Γ \ G/M = h P S Γϕ h ,ϕ ′ h , f i Γ \ G/M = hR ′ h ( T h ⊗ ˜ T h ) , χf i G/M for f ∈ C ∞ c ( Γ \ G/M ), where χ ∈ C ∞ c ( G/M ) is a smooth fundamental domaincutoff. Given χ ∈ C ∞ c ( G/M ) and χ a ∈ C ∞ c ( a ∗ (2) ), we define I h = I h,χ : h C ∞ c ( T ∗ X Γ ) → h dim N/ C ∞ c ( G/M × a ∗ (2) )as follows. For S = supp χ , choose β ∈ C ∞ c ( B (2) ) ⊂ C ∞ ( B ) as in Lemma 5.7.Denote by ˆ β the A -invariant lift of β to G/M . Recall the definition of thenon-euclidean symbol ˜ a h ∈ h C ∞ ( G/M × a ∗ ) of an operator Op h ( a h ) fromLemma 6.3. Following (5.14) we set( I h a h )( gM, ν, ν ′ ):= ˆ β ( gM ) χ a ( ν, ν ′ ) Z N χ ( gnM )˜ a h ( gnM, ν ) e i ( w · ν ′ ) H ( n − w ) /h d − n. (7.5)We relate lifted quantum limits to Patterson–Sullivan distributions. Lemma 7.2
Set W h ( a ) = (cid:0) Op Γ,h ( a ) ϕ h | ϕ ′ h (cid:1) L ( X Γ ) , a ∈ C ∞ c ( T ∗ X Γ ) . Assume that ω = lim h W h in D ′ ( T ∗ X Γ ) as h → . Assume further that lim h → ν h = θ and lim h → ν ′ h = θ ′ with ( θ, θ ′ ) ∈ a ∗ (2) . Suppose χ is smoothfundamental domain cutoff, and χ a = 1 in a neighborhood of ( θ, θ ′ ) . Let a h ∈ S , ( T ∗ X ) with principal symbol a = lim h ↓ a h ∈ C ∞ c ( T ∗ X Γ ) . Then,with I h = I h,χ , Z X a d ω = lim h ↓ hR ′ h ( T h ⊗ ˜ T h ) , ( I h a h )( · , ν h , ν ′ h ) i G/M . (7.6) Proof
Combine Proposition 5.8, Lemma 6.5, and Lemma 7.1. ⊓⊔ Remark 7.3
Observe that for any χ ′ ∈ C ∞ c ( G/M ),lim h ↓ (cid:0) Op h ( a h ) ϕ h | χ ′ ϕ ′ h (cid:1) L ( X ) = 0if a h ∈ S , − ( T ∗ X ). This observation will allow us to add terms to (7.3)without changing the limit as h ↓ G -equivariant map Φ : G/M × a ∗ → T ∗ X . If θ ∈ a ∗ is regular, then Φ ( · , θ ) : G/M → T ∗ X is an imbedding having a joint level setas its range, [16, Lemma 1.6]. Since this map is proper, the push-forward ofdistributions, Φ ( · , θ ) ∗ : D ′ ( Γ \ G/M ) → D ′ ( T ∗ X Γ ) , is well-defined. Moreover, we can define an extension operator E θ : C ∞ c ( G/M ) → C ∞ c ( T ∗ X ) , ( E θ u )( Φ ( gM, θ )) = u ( gM ) . Theorem 7.4
Let ( ϕ h ) h , ( ϕ ′ h ) h ⊂ L ( X Γ ) be sequences of normalized jointeigenfunctions, with purely imaginary spectral parameters iν h /h, iν ′ h /h ∈ i a ∗ .Assume that ω = lim h W h in D ′ ( T ∗ X Γ ) as h → . Assume further that lim h → ν h = θ and lim h → ν ′ h = θ ′ with ( θ, θ ′ ) ∈ ( a ∗ + ) such that ν h = θ + O ( h ) , ν ′ h = θ ′ + O ( h ) as h ↓ . (7.7) Then, with κ defined in (5.12) , ω = κ ( w · θ ′ ) lim h ↓ (2 πh ) dim N/ Φ ( · , θ ) ∗ P S Γh in D ′ ( T ∗ X Γ ) . (7.8) Proof
Let a ∈ C ∞ c ( T ∗ X Γ ). Let a h ∈ S , Γ with principal symbol a = lim h ↓ a h .Applying Proposition 5.8, we obtain, with χ now a smooth fundamental do-main cutoff,( I h a h )( gM, ν, ν ′ ) = κ ( w · ν ′ )(2 πh ) dim N/ (cid:0) χ ( gM )˜ a h ( gM, ν ) + O ( h ) (cid:1) = κ ( w · θ ′ )(2 πh ) dim N/ · (cid:0) χ ( gM )˜ a h ( gM, θ ) + O ( | ν − θ | ) + O ( | ν ′ − θ ′ | ) + O ( h ) (cid:1) , in h dim N/ C ∞ c (cid:0) G/M × a ∗ (2) (cid:1) . Here we used Taylor expansion around θ for˜ a h ( gM, ν ) and Taylor expansion around θ ′ for κ ( w · ν ′ ). Setting ν = ν h and ν ′ = ν ′ h , and using the assumption (7.7), we have, as h ↓ I h a h )( gM, ν h , ν ′ h ) = κ ( w · θ ′ )(2 πh ) dim N/ (cid:0) χ ( gM )˜ a h ( gM, θ ) + O ( h ) (cid:1) = κ ( w · θ ′ )(2 πh ) dim N/ (cid:0) χ ( gM ) a h ( Φ ( gM, θ )) + O ( h ) (cid:1) = κ ( w · θ ′ )(2 πh ) dim N/ (cid:0) χ ( gM ) a ( Φ ( gM, θ )) + O ( h ) (cid:1) . The second equation follows from (6.5).For ℓ > a h by lower order terms, i.e.,terms in h C ∞ c ( T ∗ X Γ ), such that the above error term O ( h ) gets replaced by O ( h ℓ ). This will, in view of Proposition 4.9 and Lemma 7.2, imply Z X a d ω = lim h ↓ hR ′ h ( T h ⊗ ˜ T h ) , κ ( w · θ ′ )(2 πh ) dim N/ χΦ ( · , θ ) ∗ a i , and hence the theorem.Set r h ( gM ) = ( I h a h )( gM, ν h , ν ′ h ) − κ ( w · θ ′ )(2 πh ) dim N/ χ ( gM ) a ( Φ ( gM, θ )).By the computation above, r h ∈ h ℓ +dim N/ C ∞ c ( G/M ) (7.9)with ℓ = 1. Define a ′ h = (2 πh ) − dim N/ E θ r h ∈ h ℓ C ∞ c ( T ∗ X ). Choose χ ′ ∈ C ∞ c ( X ) such that χ ′ = 1 on the support of r h . The computations above with a h replaced by a ′ h give( I h,χ ′ a ′ h )( gM, ν h , ν ′ h ) = κ ( w · θ ′ ) h dim N/ (cid:0) χ ′ ( gM ) a ′ h ( Φ ( gM, θ )) + O ( h ℓ ) (cid:1) = κ ( w · θ ′ ) (cid:0) r h ( gM ) + O ( h ℓ ) (cid:1) . We replace, in (7.6), I h a h by I h a h − κ − I h,χ ′ a ′ h . By Remark 7.3, the formula(7.6) remains true. In addition, by the arguments above, we have a new re-mainder r h which satisfies (7.9) with ℓ replaced by ℓ + 1. Arguing by inductionover ℓ , the proof follows. ⊓⊔ Remark 7.5 (i) If θ = θ ′ , then Remark 5.6, combined with the method of non-stationary phase and Proposition 4.9 imply that ω = 0. Thus, in this casealso the right hand side of (7.8) vanishes(ii) Combining Theorem 7.4 for ϕ ′ h = ϕ h with Remark 4.11 yields yet anotherproof of the A -invariance of the lifted quantum limits (see Theorem 6.6,where | ν h | = 1). Acknowledgements
We thank J. M¨ollers, A. Pasquale, and in particular N. Ananthara-man and S. Zelditch for helpful discussions. Special thanks go to M. Olbrich for providingthe idea of a proof of Proposition 3.11.
References
1. N. Anantharaman, L. Silberman,
A Haar Component for Quantum Limits in LocallySymmetric Spaces , preprint, 2010, arXiv:1009.492.2. N. Anantharaman, S. Zelditch,
Patterson–Sullivan Distributions and Quantum Ergodic-ity , Ann. Henri Poincar´e (2007), 361–426.3. N. Anantharaman, S. Zelditch, Intertwining the geodesic flow and the Schr¨odinger groupon hyperbolic surfaces , preprint, 2010, arXiv:1010.0867.4. J.-P. Anker,
A Basic Inequality for Scattering Theory of Riemannian Symmetric Spacesof the Noncompact Type . Amer. J. Math.. (1991), 391–398.5. E. van den Ban, H. Schlichtkrull,
Asymptotic expansions and boundary values of eigen-functions on Riemannian symmetric spaces , J. reine angew. Math. (1987), 108–165.6. M. Dimassi and J. Sj¨ostrand,
Spectral Asymptotics in the Semi-Classical Limit , Cam-bridge University Press, Cambridge (1999).7. J. J. Duistermaat, J. A. C. Kolk, V. S. Varadarajan,
Spectra of compact locally symmetricmanifolds of negative curvature , Invent. Math. (1979), 27–93.8. J. J. Duistermaat, J. A. C. Kolk, V. S. Varadarajan, Functions, Flows and oscillatory in-tegrals on flag manifolds and conjugacy classes in real semisimple Lie groups , CompositioMath. (1983), 309–398.9. L.E. Evans, M. Zworski Lectures on Semiclassical Analysis , UC Berkeley.10. S. Hansen,
Rayleigh-type surface quasimodes in general linear elasticity , preprint 2010,arXiv:1008.2930.11. G. Heckman,
Projection of Orbits and Asymptotic Behaviour of Multiplicities for Com-pact Lie Groups , Dissertation, Univ. Utrecht, 1980.12. S. Helgason,
Geometric Analysis on Symmetric Spaces , Mathematical surveys andmonographs, American Mathematical Society, Providence, RI (1994).13. S. Helgason,
Groups and Geometric Analysis , Mathematical surveys and monographs,American Mathematical Society, Providence, RI (2000).14. S. Helgason,
Differential Geometry, Lie Groups, and Symmetric Spaces , Graduate Stud-ies in Mathematics, American Mathematical Society, Providence, RI (2001).15. J. Hilgert, M. Schr¨oder,
Patterson–Sullivan Distributions for Rank One SymmetricSpcaces of the Non-Compact Type . arXiv0909.214216. J. Hilgert,
An Ergodic Arnold–Liouville Theorem for Locally Symmetric Spaces ,“Twenty Years of Bialowieza: A Mathematical Anthology”. S.T. Ali et al. eds., WorldScientific, Singapore (2005).17. L. H¨ormander,
The Analysis of Linear Partial Differential Operators , volumes I-IV,Springer-Verlag, Berlin and New York (1983–1985).118. M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. ¯Oshima, and M. Tanaka.
Eigenfunctions of invariant differential operators on a symmetric space , Ann. of Math. (1978), 1–39.19. H. Schlichtkrull,
Hyperfunctions and Harmonic Analysis on Symmetric Spaces , Progressin Math. , Birkh¨auser, 1984.20. M. Schr¨oder, Patterson-Sullivan distributions for rank one symmetric spaces of thenoncompact type , Dissertation, Univ. Paderborn, 2010.21. V.A. Sharafutdinov,
Geometric symbol calculus for pseudo differential operators I andII . Siber. Adv. Math. , no.3 (2005), 81–125, and no.4 (2005), 71–95.22. L. Silberman, A. Venkatesh, On quantum unique ergodicity for locally symmetric spaces ,Geom. Funct. Anal. (2007), 960–998.23. F. Treves, Topological Vector Spaces, Distributions and Kernels , Acad. Press, (1967)24. M. E. Taylor,
Pseudodifferential Operators , Princeton Univerity Press, Princeton, NewJersey, (1981).25. N. R. Wallach,
Real reductive groups 1 , Academic Press, Pure and Applied Mathematics,San Diego (1988).26. H. Widom,
A complete symbolic calculus for pseudodifferential operators.
Bull. Sci.Math (2) (1980), 19–63.27. F. L. Williams,
Lectures on the spectrum of L ( Γ \ G ), Pitman Research Notes in Math-ematics Series 242, Essex (1991).28. S. Wolpert, The Modulus of Continuity for Γ ( m ) Semi-Classical Limits , Comm. Math.Phys. (2001), 313–323.29. S. Zelditch,