aa r X i v : . [ m a t h . DG ] S e p Intrinsic sound of anti-de Sitter manifolds
Toshiyuki Kobayashi
Abstract
As is well-known for compact Riemann surfaces, eigenvalues of theLaplacian are distributed discretely and most of eigenvalues vary viewed as func-tions on the Teichm¨uller space. We discuss a new feature in the Lorentzian geom-etry, or more generally, in pseudo-Riemannian geometry. One of the distinguishedfeatures is that L -eigenvalues of the Laplacian may be distributed densely in R inpseudo-Riemannian geometry. For three-dimensional anti-de Sitter manifolds, wealso explain another feature proved in joint with F. Kassel [Adv. Math. 2016] thatthere exist countably many L -eigenvalues of the Laplacian that are stable under anysmall deformation of anti-de Sitter structure. Partially supported by Grant-in-Aid forScientific Research (A) (25247006), Japan Society for the Promotion of Science. Keywords and phrases:
Laplacian, locally symmetric space, Lorentzian manifold,spectral analysis, Clifford–Klein form, reductive group, discontinuous group
Primary 22E40, 22E46, 58J50: Secondary 11F72, 53C35
Our “common sense” for music instruments says:“shorter strings produce a higher pitch than longer strings”,“thinner strings produce a higher pitch than thicker strings”.Let us try to “hear the sound of pseudo-Riemannian locally symmetric spaces”.Contrary to our “common sense” in the Riemannian world, we find a phenomenonthat compact three-dimensional anti-de Sitter manifolds have “intrinsic sound”
Toshiyuki KobayashiKavli IPMU and Graduate School of Mathematical Sciences, The University of Tokyo, e-mail: [email protected] which is stable under any small deformation. This is formulated in the frame-work of spectral analysis of anti-de Sitter manifolds, or more generally, of pseudo-Riemannian locally symmetric spaces X G . In this article, we give a flavor of this newtopic by comparing it with the flat case and the Riemannian case.To explain briefly the subject, let X be a pseudo-Riemannian manifold, and G adiscrete isometry group acting properly discontinuously and freely on X . Then thequotient space X G : = G \ X carries a pseudo-Riemannian manifold structure suchthat the covering map X → X G is isometric. We are particularly interested in thecase where X G is a pseudo-Riemannian locally symmetric space, see Section 3.2.Problems we have in mind are symbolized in the following diagram: existence problem deformation v.s. rigidityGeometry Does cocompact G exist? Higher Teichm¨uller theory v.s. rigidity theorem(Section 4.1) (Section 4.2)Analysis Does L -spectrum exist? Whether L -eigenvalues vary or not(Problem A) (Problem B) In [5, 6, 12] we initiated the study of “spectral analysis on pseudo-Riemannian lo-cally symmetric spaces” with focus on the following two problems:
Problem A
Construct eigenfunctions of the Laplacian D X G on X G . Does there exista nonzero L -eigenfunction? Problem B
Understand the behaviour of L -eigenvalues of the Laplacian D X G onX G under small deformation of G inside G. Even when X G is compact, the existence of countably many L -eigenvalues isalready nontrivial because the Laplacian D X G is not elliptic in our setting. We shalldiscuss in Section 2.2 for further difficulties concerning Problems A and B when X G is non-Riemannian.We may extend these problems by considering joint eigenfunctions for “invariantdifferential operators” on X G rather than the single operator D X G . Here by “invariantdifferential operators on X G ” we mean differential operators that are induced from G -invariant ones on X = G / H . In Section 7, we discuss Problems A and B in thisgeneral formulation based on the recent joint work [6, 7] with F. Kassel. ntrinsic sound of anti-de Sitter manifolds 3 Spectral analysis on a pseudo-Riemannian locally symmetric space X G = G \ X = G \ G / H is already deep and difficult in the following special cases:1) (noncommutative harmonic analysis on G / H ) G = { e } .In this case, the group G acts unitarily on the Hilbert space L ( X G ) = L ( X ) by translation f ( · ) f ( g − · ) , and the irreducible decomposition of L ( X ) ( Plancherel-type formula ) is essentially equivalent to the spectral analysis of G -invariant differential operators when X is a semisimple symmetric space.Noncommutative harmonic analysis on semisimple symmetric spaces X hasbeen developed extensively by the work of Helgason, Flensted-Jensen, Matsuki–Oshima–Sekiguchi, Delorme, van den Ban–Schlichtkrull among others as a gen-eralization of Harish-Chandra’s earlier work on the regular representation L ( G ) for group manifolds.2) (automorphic forms) H is compact and G is arithmetic.If H is a maximal compact subgroup of G , then X G = G \ G / H is a Riemannianlocally symmetric space and the Laplacian D X G is an elliptic differential operator.Then there exist infinitely many L -eigenvalues of D X G if X G is compact by thegeneral theory for compact Riemannian manifolds (see Fact 1). If furthermore G is irreducible, then Weil’s local rigidity theorem [18] states that nontrivial defor-mations exist only when X is the hyperbolic plane SL ( , R ) / SO ( ) , in which casecompact quotients X G have a classically-known deformation space modulo con-jugation, i.e. , their Teichm¨uller space. Viewed as a function on the Teichm¨ullerspace, L -eigenvalues vary analytically [1, 20], see Fact 11.Spectral analysis on X G is closely related to the theory of automorphic forms inthe Archimedian place if G is an arithmetic subgroup.3) (abelian case) G = R p + q with H = { } and G = Z p + q .We equip X = G / H with the standard flat pseudo-Riemannian structure of sig-nature ( p , q ) (see Example 1). In this case, G is abelian, but X = G / H is non-Riemannian. This is seemingly easy, however, spectral analysis on the ( p + q ) -torus R p + q / Z p + q is much involved, as we shall observe a connection with Op-penheim’s conjecture (see Section 5.2). If we try to attack a problem of spectral analysis on G \ G / H in the more generalcase where H is noncompact and G is infinite, then new difficulties may arise fromseveral points of view:(1) Geometry. The G -invariant pseudo-Riemannian structure on X = G / H is notRiemannian anymore, and discrete groups of isometries of X do not always actproperly discontinuously on such X . Toshiyuki Kobayashi (2) Analysis. The Laplacian D X on X G is not an elliptic differential operator. Fur-thermore, it is not clear if D X has a self-adjoint extension on L ( X G ) .(3) Representation theory. If G acts properly discontinuously on X = G / H with H noncompact, then the volume of G \ G is infinite, and the regular represen-tation L ( G \ G ) may have infinite multiplicities. In turn, the group G may nothave a good control of functions on G \ G . Moreover L ( X G ) is not a subspaceof L ( G \ G ) because H is noncompact. All these observations suggest that anapplication of the representation theory of L ( G \ G ) to spectral analysis on X G israther limited when H is noncompact.Point (1) creates some underlying difficulty to Problem B: we need to considerlocally symmetric spaces X G for which proper discontinuity of the action of G on X is preserved under small deformations of G in G . This is nontrivial. This questionwas first studied by the author [9, 11]. See [4] for further study. An interestingaspect of the case of noncompact H is that there are more examples where nontrivialdeformations of compact quotients exist than for compact H ( cf . Weil’s local rigiditytheorem [18]). Perspectives from Point (1) will be discussed in Section 4.Point (2) makes Problem A nontrivial. It is not clear if the following well-knownproperties in the Riemannian case holds in our setting in the pseudo-Riemannian case.
Fact 1
Suppose M is a compact Riemannian manifold. (1)
The Laplacian D M extends to a self-adjoint operator on L ( M ) . (2) There exist infinitely many L -eigenvalues of D M . (3) An eigenfunction of D M is infinitely differentiable. (4) Each eigenspace of D M is finite-dimensional. (5) The set of L -eigenvalues is discrete in R .Remark 1. We shall see that the third to fifth properties of Fact 1 may fail in thepseudo-Riemannian case, e.g. , Example 6 for (3) and (4), and M = R , / Z (Theo-rem 7) for (5).In spite of these difficulties, we wish to reveal a mystery of spectral analysis ofpseudo-Riemannian locally homogeneous spaces X G = G \ G / H . We shall discussself-adjoint extension of the Laplacian in the pseudo-Riemannian setting in Theo-rem 13, and the existence of countable many L -eigenvalues in Theorems 8, 12 and13. A pseudo-Riemannian manifold M is a smooth manifold endowed with a smooth,nondegenerate, symmetric bilinear tensor g of signature ( p , q ) for some p , q ∈ N . ntrinsic sound of anti-de Sitter manifolds 5 ( M , g ) is a Riemannian manifold if q =
0, and is a Lorentzian manifold if q =
1. Themetric tensor g induces a Radon measure d m on X , and the divergence div. Then theLaplacian D M : = div grad , is a differential operator of second order which is a symmetric operator on theHilbert space L ( X , d m ) . Example 1.
Let ( M , g ) be the standard flat pseudo-Riemannian manifold: R p , q : = ( R p + q , dx + · · · + dx p − dx p + − · · · − dx p + q ) . Then the Laplacian takes the form D R p , q = ¶ ¶ x + · · · + ¶ ¶ x p − ¶ ¶ x p + − · · · − ¶ ¶ x p + q . In general, D M is an elliptic differential operator if ( M , g ) is Riemannian, and is ahyperbolic operator if ( M , g ) is Lorentzian. A typical example of pseudo-Riemannian manifolds X with “large” isometry groupsis semisimple symmetric spaces, for which the infinitesimal classification was ac-complished by M. Berger in 1950s. In this case, X is given as a homogeneous space G / H where G is a semisimple Lie group and H is an open subgroup of the fixedpoint group G s = { g ∈ G : s g = g } for some involutive automorphism s of G . Inparticular, G ⊃ H are a pair of reductive Lie groups.More generally, we say G / H is a reductive homogeneous space if G ⊃ H are apair of real reductive algebraic groups. Then we have the following: Proposition 1.
Any reductive homogeneous space X = G / H carries a pseudo-Riemannian structure such that G acts on X by isometries.Proof.
By a theorem of Mostow, we can take a Cartan involution q of G such that q H = H . Then K : = G q is a maximal compact subgroup of G , and H ∩ K is thatof H . Let g = k + p be the corresponding Cartan decomposition of the Lie algebra g of G . Take an Ad ( G ) -invariant nondegenerate symmetric bilinear form h , i on g such that h , i| k × k is negative definite, h , i| p × p is positive definite, and k and p areorthogonal to each other. (If G is semisimple, then we may take h , i to be the Killingform of g . )Since q H = H , the Lie algebra h of H is decomposed into a direct sum h = ( h ∩ k ) + ( h ∩ p ) , and therefore the bilinear form h , i is non-degenerate when restrictedto h . Then h , i induces an Ad ( H ) -invariant nondegenerate symmetric bilinear form h , i g / h on the quotient space g / h , with which we identify the tangent space T o ( G / H ) Toshiyuki Kobayashi at the origin o = eH ∈ G / H . Since the bilinear form h , i g / h is Ad ( H ) -invariant, theleft translation of this form is well-defined and gives a pseudo-Riemannian structure g on G / H of signature ( dim p / h ∩ p , dim k / h ∩ k ) . By the construction, the group G acts on the pseudo-Riemannian manifold ( G / H , g ) by isometries. ⊓⊔ Let Q p , q ( x ) : = x + · · · + x p − x p + − · · · − x p + q be a quadratic form on R p + q of sig-nature ( p , q ) , and we denote by O ( p , q ) the indefinite orthogonal group preservingthe form Q p , q . We define two hypersurfaces M p , q ± in R p + q by M p , q ± : = { x ∈ R p + q : Q p , q ( x ) = ± } . By switching p and q , we have an obvious diffeomorphism M p , q + ≃ M q , p − . The flat pseudo-Riemannian structure R p , q (Example 1) induces a pseudo-Riemannianstructure on the hypersurface M p , q + of signature ( p − , q ) with constant curvature 1,and that on M p , q − of signature ( p , q − ) with constant curvature − O ( p , q ) on R p , q induces an isometric and transi-tive action on the hypersurfaces M p , q ± , and thus they are expressed as homogeneousspaces: M p , q + ≃ O ( p , q ) / O ( p − , q ) , M p , q − ≃ O ( p , q ) / O ( p , q − ) , giving examples of pseudo-Riemannian homogeneous spaces as in Proposition 1.The anti-de Sitter space AdS n = M n − , − is a model space for n -dimensionalLorentzian manifolds of constant negative sectional curvature, or anti-de Sitter n-manifolds . This is a Lorentzian analogue of the real hyperbolic space H n . For theconvenience of the reader, we list model spaces of Riemannian and Lorentzian man-ifolds with constant positive, zero, and negative curvatures.Riemannian manifolds with constant curvature: S n = M n + , + ≃ O ( n + ) / O ( n ) : standard sphere , R n : Euclidean space , H n = M n , − ≃ O ( , n ) / O ( n ) : hyperbolic space , Lorentzian manifolds with constant curvature: ntrinsic sound of anti-de Sitter manifolds 7 dS n = M n , + ≃ O ( n , ) / O ( n − , ) : de Sitter space , R n − , : Minkowski space , AdS n = M n − , − ≃ O ( , n − ) / O ( , n − ) : anti-de Sitter space , Let H be a closed subgroup of a Lie group G , and X = G / H , and G a discretesubgroup of G . If H is compact, then the double coset space G \ G / H becomes a C ¥ -manifold for any torsion-free discrete subgroup G of G . However, we have to becareful for noncompact H , because not all discrete subgroups acts properly discon-tinuously on G / H , and G \ G / H may not be Hausdorff in the quotient topology. Weillustrate this feature by two general results: Fact 2 (1) (Moore’s ergodicity theorem [15])
Let G be a simple Lie group, and G a lattice. Then G acts ergodically on G / H for any noncompact closed subgroupH. In particular, G \ G / H is non-Hausdorff. (2) (Calabi–Markus phenomenon ([2, 8]))
Let G be a reductive Lie group, and G an infinite discrete subgroup. Then G \ G / H is non-Hausdorff for any reductivesubgroup H with rank R G = rank R H . In fact, determining which groups act properly discontinuously on reductive ho-mogeneous spaces G / H is a delicate problem, which was first considered in fullgenerality by the author; we refer to [13, Section 3.2] for a survey.Suppose now a discrete subgroup G acts properly discontinuously and freely on X = G / H . Then the quotient space X G : = G \ X ≃ G \ G / H carries a C ¥ -manifold structure such that the quotient map p : X → X G is a covering,through which X G inherits any G -invariant local geometric structure on X . We say G is a discontinuous group for X and X G is a Clifford–Klein form of X = G / H . Example 2. (1) If X = G / H is a reductive homogeneous space, then any Clifford–Klein form X G carries a pseudo-Riemannian structure by Proposition 1.(2) If X = G / H is a semisimple symmetric space, then any Clifford–Klein form X G = G \ G / H is a pseudo-Riemannian locally symmetric space, namely, the (local)geodesic symmetry at every p ∈ X G with respect to the Levi-Civita connection islocally isometric.By space forms , we mean pseudo-Riemannian manifolds of constant sectionalcurvature. They are examples of pseudo-Riemannian locally symmetric spaces. Forsimplicity, we shall assume that they are geodesically complete. Toshiyuki Kobayashi
Example 3.
Clifford–Klein forms of M p + , q + = O ( p + , q ) / O ( p , q ) (respectively, M p , q + − = O ( p , q + ) / O ( p , q ) ) are pseudo-Riemannian space forms of signature ( p , q ) with positive (respectively, negative) curvature. Conversely, any (geodesicallycomplete) pseudo-Riemannian space form of signature ( p , q ) is of this form as faras p = q = G / H is: Question 1.
Does compact Clifford–Klein forms of G / H exist?or equivalently, Question 2.
Does there exist a discrete subgroup G of G acting cocompactly andproperly discontinuously on G / H ?This question has an affirmative answer if H is compact by a theorem of Borel.In the general setting where H is noncompact, the question relates with a “globaltheory” of pseudo-Riemannian geometry: how local pseudo-Riemannian homoge-neous structure affects the global nature of manifolds? A classic example is spaceform problem which asks the global properties ( e.g. compactness, volume, funda-mental groups, etc. ) of a pseudo-Riemannian manifold of constant curvature (localproperty). The study of discontinuous groups for M p + , q + and M p , q + − shows the fol-lowing results in pseudo-Riemannian space forms of signature ( p , q ) : Fact 3
Space forms of positive curvature are (1) always closed if q = , i.e., sphere geometry in the Riemannian case; (2) never closed if p ≥ q > , in particular, if q = (de Sitter geometry in theLorentzian case [2]). The phenomenon in the second statement is called the
Calabi–Markus phenomenon (see Fact 2 (2) in the general setting).
Fact 4
Compact space forms of negative curvature exist (1) for all dimensions if q = , i.e., hyperbolic geometry in the Riemannian case; (2) for odd dimensions if q = , i.e., anti-de Sitter geometry in the Lorentzian case; (3) for ( p , q ) = ( m , ) ( m ∈ N ) or ( , ) . See [13, Section 4] for the survey of the space form problem in pseudo-Riemanniangeometry and also of Question 1 for more general G / H .A large and important class of Clifford–Klein forms X G of a reductive homoge-neous space X = G / H is constructed as follows (see [8]). Definition 1.
A quotient X G = G \ X of X by a discrete subgroup G of G is called standard if G is contained in some reductive subgroup L of G acting properly on X .If a subgroup L acts properly on G / H , then any discrete subgroup of G acts prop-erly discontinuously on G / H . A handy criterion for the triple ( G , H , L ) of reductivegroups such that L acts properly on G / H is proved in [8], as we shall recall below. ntrinsic sound of anti-de Sitter manifolds 9 Let G = K exp a + K be a Cartan decomposition, where a is a maximal abelian sub-space of p and a + is the dominant Weyl chamber with respect to a fixed positivesystem S + ( g , a ) . This defines a map m : G → a + ( Cartan projection ) by m ( k e X k ) = X for k , k ∈ K and X ∈ a . It is continuous, proper and surjective. If H is a reductive subgroup, then there exists g ∈ G such that m ( gHg − ) is given by the intersection of a + with a subspace ofdimension rank R H . By an abuse of notation, we use the same H instead of gHg − .With this convention, we have: Properness Criterion 5 ([8])
L acts properly on G / H if and only if m ( L ) ∩ m ( H ) = { } . By taking a lattice G of such L , we found a family of pseudo-Riemannian lo-cally symmetric spaces X G in [8, 13]. The list of symmetric spaces admitting stan-dard Clifford–Klein forms of finite volume (or compact forms) include M p , q + − = O ( p , q + ) / O ( p , q ) with ( p , q ) satisfying the conditions in Fact 4. Further, by ap-plying Properness Criterion 5, Okuda [16] gave examples of pseudo-Riemannianlocally symmetric spaces G \ G / H of infinite volume where G is isomorphic to thefundamental group p ( S g ) of a compact Riemann surface S g with g ≥ X G (see Theorem 10 and Theorem 12(2) below), we introduced in [6, Section 1.6] the following concept: Definition 2.
A discrete subgroup G of G acts strongly properly discontinuously (or sharply ) on X = G / H if there exists C , C ′ > g ∈ G , d ( m ( g ) , m ( H )) ≥ C k m ( g ) k − C ′ . Here d ( · , · ) is a distance in a given by a Euclidean norm k ·k which is invariant underthe Weyl group of the restricted root system S ( g , a ) . We say the positive number C is the first sharpness constant for G .If a reductive subgroup L acts properly on a reductive homogeneous space G / H ,then the action of a discrete subgroup G of L is strongly properly discontinuous ([6,Example 4.10]). Let G be a Lie group and G a finitely generated group. We denote by Hom ( G , G ) the set of all homomorphisms of G to G topologized by pointwise convergence. Bytaking a finite set { g , · · · , g k } of generators of G , we can identify Hom ( G , G ) as asubset of the direct product G × · · · × G by the inclusion:Hom ( G , G ) ֒ → G × · · · × G , j ( j ( g ) , · · · , j ( g k )) . (1) If G is finitely presentable, then Hom ( G , G ) is realized as a real analytic variety via(1).Suppose G acts continuously on a manifold X . We shall take X = G / H withnoncompact closed subgroup H later. Then not all discrete subgroups act properlydiscontinuously on X in this general setting. The main difference of the followingdefinition of the author [9] in the general case from that of Weil [18] is a requirementof proper discontinuity. R ( G , G ; X ) : = { j ∈ Hom ( G , G ) : j is injective, (2)and j ( G ) acts properly discontinuously and freely on G / H } . Suppose now X = G / H for a closed subgroup H . Then the double coset space j ( G ) \ G / H forms a family of manifolds that are locally modelled on G / H withparameter j ∈ R ( G , G ; X ) . To be more precise on “parameter”, we note that theconjugation by an element of G induces an automorphism of Hom ( G , G ) whichleaves R ( G , G ; X ) invariant. Taking these unessential deformations into account, wedefine the deformation space ( generalized Teichm¨uller space ) as the quotient set T ( G , G ; X ) : = R ( G , G ; X ) / G . Example 4. (1) Let G be the surface group p ( S g ) of genus g ≥ G = PSL ( , R ) , X = H (two-dimensional hyperbolic space). Then T ( G , G ; X ) is the classicalTeichm¨uller space, which is of dimension 6 g − G = R n , X = R n , G = Z n . Then T ( G , G ; X ) ≃ GL ( n , R ) (see (4) below).(3) G = SO ( , ) , X = AdS , and G = p ( S g ) . Then T ( G , G ; X ) is of dimension12 g −
12 (see [6, Section 9.2] and references therein).
Remark 2.
There is a natural isometry between X j ( G ) and X j ( g G g − ) . Hence, the setSpec d ( X j ( G ) ) of L -eigenvalues is independent of the conjugation of j ∈ R ( G , G ; X ) by an element of G . By an abuse of notation we shall write Spec d ( X j ( G ) ) for j ∈ T ( G , G ; X ) when we deal with Problem B of Section 2. R p , q / Z p + q and Oppenheim conjecture This section gives an elementary but inspiring observation of spectrum on flatpseudo-Riemannian manifolds. R p , q / j ( Z p + q ) Let G = R n and G = Z n . Then the group homomorphism j : G → G is uniquelydetermined by the image j ( e j ) (1 ≤ j ≤ n ) where e , · · · , e n ∈ Z n are the standard ntrinsic sound of anti-de Sitter manifolds 11 basis, and thus we have a bijectionHom ( G , G ) ∼ ← M ( n , R ) , j g g (3)by j g ( m ) : = g m for m ∈ Z n , or equivalently, by g = ( j g ( e ) , · · · , j g ( e n )) . Let s ∈ Aut ( G ) be defined by s ( x ) : = − x . Then H : = G s = { } and X : = G / H ≃ R n is a symmetric space. The discrete group G acts properly discontinu-ously on X via j g if and only if g ∈ GL ( n , R ) . Moreover, since G is abelian, G acts trivially on Hom ( G , G ) by conjugation, and therefore the deformation space T ( G , G ; X ) identifies with R ( G , G ; X ) . Hence we have a natural bijection betweenthe two subsets of (3): T ( G , G ; X ) ∼ ← GL ( n , R ) . (4)Fix p , q ∈ N such that p + q = n , and we endow X ≃ R n with the standardflat indefinite metric R p , q (see Example 1). Let us determine Spec d ( X j g ( G ) ) ≃ Spec d ( R p , q / j g ( Z n )) for g ∈ GL ( n , R ) ≃ T ( G , G ; X ) .For this, we define a function on X = R n by f m ( x ) : = exp ( p √− t m g − x ) ( x ∈ R n ) for each m ∈ Z n where x and m are regarded as column vectors. Clearly, f m is j g ( G ) -periodic and defines a real analytic function on X j g ( G ) . Furthermore, f m isan eigenfunction of the Laplacian D R p , q : D R p , q f m = − p Q g − I p , qt g − ( m ) f m , where, for a symmetric matrix S ∈ M ( n , R ) , Q S denotes the quadratic form on R n given by Q S ( y ) : = t y S y for y ∈ R n . Since { f m : m ∈ Z n } spans a dense subspace of L ( X j g ( G ) ) , we have shown: Proposition 2.
For any g ∈ GL ( n , R ) ≃ T ( G , G ; X ) , Spec d ( X j g ( G ) ) = {− p Q g − I p , qt g − ( m ) : m ∈ Z n } . Here are some observation in the n = , Example 5.
Let n = ( p , q ) = ( , ) . Then Spec d ( X j g ( G ) ) = {− p m / g : m ∈ Z } for g ∈ R × ≃ GL ( , R ) by Proposition 2. Thus the smaller the period | g | is, thelarger the absolute value of the eigenvalue | − p m / g | becomes for each fixed m ∈ Z \ { } . This is thought of as a mathematical model of a music instrument forwhich shorter strings produce a higher pitch than longer strings (see Introduction). Example 6.
Let n = ( p , q ) = ( , ) . Take g = I , so that j g ( G ) = Z is thestandard lattice. Then the L -eigenspace of the Laplacian D R , / Z for zero eigen-value contains W : = { y ( x − y ) : y ∈ L ( R / Z ) } . Since W is infinite-dimensionaland W C ¥ ( R / Z ) , the third and fourth statements of Fact 1 fail in this pseudo-Riemannian setting. By the explicit description of Spec d ( X j ( G ) ) for all j ∈ T ( G , G ; X ) in Proposi-tion 2, we can also tell the behaviour of Spec d ( X j ( G ) ) under deformation of G by j . Obviously, any constant function on X j ( G ) is an eigenfunction of the Laplacian D X j ( G ) = D R p , q / j ( Z p + q ) with eigenvalue zero. We see that this is the unique stable L -eigenvalue in the flat compact manifold: Corollary 1 (non-existence of stable eigenvalues).
Let n = p + q with p , q ∈ N . Forany open subset V of T ( G , G ; X ) , \ j ∈ V Spec d ( X j ( G ) ) = { } . In 1929, Oppenheim [17] raised a question about the distribution of an indefinitequadratic forms at integral points. The following theorem, referred to as Oppen-heim’s conjecture, was proved by Margulis (see [14] and references therein).
Fact 6 (Oppenheim’s conjecture)
Suppose n ≥ and Q is a real nondegenerateindefinite quadratic form in n variables. Then either Q is proportional to a formwith integer coefficients (and thus Q ( Z n ) is discrete in R ), or Q ( Z n ) is dense in R . Combining this with Proposition 2, we get the following.
Theorem 7.
Let p + q = n, p ≥ , q ≥ , G = R n , X = R p , q and G = Z n . We definean open dense subset U of T ( G , G ; X ) ≃ GL ( n , R ) byU : = { g ∈ GL ( n , R ) : g − I p , qt g − is not proportional to an element of M ( n , Z ) . } Then the set
Spec d ( X j ( G ) ) of L -eigenvalues of the Laplacian is dense in R if andonly if j ∈ U. Thus the fifth statement of Fact 1 for compact Riemannian manifolds do fail in thepseudo-Riemannian case.
In general, it is not clear whether the Laplacian D M admits infinitely many L -eigenvalues for compact pseudo-Riemannian manifolds. For anti-de Sitter 3-manifolds,we proved in [6, Theorem 1.1]: ntrinsic sound of anti-de Sitter manifolds 13 Theorem 8.
For any compact anti-de Sitter 3-manifold M, there exist infinitelymany L -eigenvalues of the Laplacian D M . In the abelian case, it is easy to see that compactness of X G is necessary for theexistence of L -eigenvalues: Proposition 3.
Let G = R p + q , X = R p , q , G = Z k , and j ∈ R ( G , G ; X ) . Then Spec d ( X j ( G ) ) = /0 if and only if X j ( G ) is compact, or equivalently, k = p + q. However, anti-de Sitter 3-manifolds M admit infinitely many L -eigenvalues evenwhen M is of infinite-volume (see [6, Theorem 9.9]): Theorem 9.
For any finitely generated discrete subgroup G of G = SO ( , ) actingproperly discontinuously and freely on X = AdS , Spec d ( X G ) ⊃ { l ( l − ) : l ∈ N , l ≥ C − } where C ≡ C ( G ) is the first sharpness constant of G . The above L -eigenvalues are stable in the following sense: Theorem 10 (stable L -eigenvalues). Suppose that G ⊂ G = SO ( , ) and M = G \ AdS is a compact standard anti-de Sitter 3-manifold. Then there exists a neigh-bourhood U ⊂ Hom ( G , G ) of the natural inclusion with the following two proper-ties: U ⊂ R ( G , G ; AdS ) , (5) ( \ j ∈ U Spec d ( X G )) = ¥ . (6)The first geometric property (5) asserts that a small deformation of G keeps properdiscontinuity, which was conjectured by Goldman [3] in the AdS setting, andproved affirmatively in [11]. Theorem 10 was proved in [6, Corollary 9.10] in astronger form ( e.g. , without assuming “standard” condition).Figuratively speaking, Theorem 10 says that compact anti-de Sitter manifoldshave “intrinsic sound” which is stable under any small deformation of the anti-deSitter structure. This is a new phenomenon which should be in sharp contrast to theabelian case (Corollary 1) and the Riemannian case below: Fact 11 (see [20, Theorem 5.14])
For a compact hyperbolic surface, no eigenvalueof the Laplacian above is constant on the Teichm¨uller space. We end this section by raising the following question in connection with the flatcase (Theorem 7):
Question 3.
Suppose M is a compact anti-de Sitter 3-manifold. Find a geometriccondition on M such that Spec d ( M ) is discrete. The results in the previous section for anti-de Sitter 3-manifolds can be extended tomore general pseudo-Riemannian locally symmetric spaces of higher dimension:
Theorem 12 ([6, Theorem 1.5]).
Let X G be a standard Clifford–Klein form of asemisimple symmetric space X = G / H satisfying the rank condition rank G / H = rank K / H ∩ K . (7) Then the following holds. (1)
There exists an explicit infinite subset I of joint L -eigenvalues for all the differ-ential operators on X G that are induced from G-invariant differential operatorson X. (2) (stable spectrum) If G is contained in a simple Lie group L of real rank oneacting properly on X = G / H, then there is a neighbourhood V ⊂ Hom ( G , G ) ofthe natural inclusion such that for any j ∈ V , the action j ( G ) on X is properlydiscontinuous and the set of joint L -eigenvalues on X j ( G ) contains the infiniteset I.Remark 3. We do not require X G to be of finite volume in Theorem 12. Remark 4.
It is plausible that for a general locally symmetric space G \ G / H with G reductive, no nonzero L -eigenvalue is stable under nontrivial small deformationunless the rank condition (7) is satisfied. For instance, suppose G = p ( S g ) with g ≥ R ( G , G ; X ) = /0. (Such semisimple symmetric space X = G / H was re-cently classified in [16].) Then we expect the rank condition (7) is equivalent to theexistence of an open subset U in R ( G , G ; X ) such that ( \ j ∈ U Spec d ( X j ( G ) )) = ¥ . It should be noted that not all L -eigenvalues of compact anti-de Sitter manifoldsare stable under small deformation of anti-de Sitter structure. In fact, we provedin [7] that there exist also countably many negative L -eigenvalues that are NOTstable under deformation, whereas the countably many stable L -eigenvalues thatwe constructed in Theorem 9 are all positive. More generally, we prove in [7] thefollowing theorem that include both stable and unstable L -eigenvalues: Theorem 13.
Let G be a reductive homogeneous space and L a reductive sub-group of G such that H ∩ L is compact. Assume that the complexification X C isL C -spherical. Then for any torsion-free discrete subgroup G of L, we have: (1) the Laplacian D X G extends to a self-adjoint operator on L ( X G ) ; (2) d ( X G ) = ¥ if X G is compact. ntrinsic sound of anti-de Sitter manifolds 15 By “ L C -spherical” we mean that a Borel subgroup L C has an open orbit in X C . Inthis case, a reductive subgroup L acts transitively on X by [10, Lemma 5.1].Here are some examples of the setting of Theorem 13, taken from [13, Corollary3.3.7]. Table 1 G H L (i) SO ( n , ) SO ( n , ) U ( n , ) (ii) SO ( n , ) U ( n , ) SO ( n , ) (iii) SU ( n , ) U ( n , ) Sp ( n , ) (iv) SU ( n , ) Sp ( n , ) U ( n , ) (v) SO ( n , ) SO ( n , ) Sp ( ) × Sp ( n , ) (vi) SO ( , ) SO ( , ) Spin ( , ) (vii) SO ( , C ) SO ( , C ) Spin ( , ) (viii) SO ( , ) Spin ( , ) SO ( , ) × SO ( ) (ix) SO ( , ) G ( R ) SO ( , ) × SO ( ) Examples for Theorem 13 include Table 1 (ii) for all n ∈ N , whereas we need n ∈ N in Theorem 12 for the rank condition (7).The idea of the proof for Theorem 12 is to take an average of a (nonperi-odic) eigenfunction on X with rapid decay at infinity over G -orbits as a general-ization of Poincar´e series. Geometric ingredients of the convergence (respectively,nonzeroness) of the generalized Poincar´e series include “counting G -orbits” statedin Lemma 1 below (respectively, the Kazhdan–Margulis theorem, cf . [6, Proposi-tion 8.14]). Let B ( o , R ) be a “pseudo-ball” of radius R > o = eH ∈ X = G / H , and we set N ( x , R ) : = { g ∈ G : g · x ∈ B ( o , R ) } . Lemma 1 ([6, Corollary 4.7]). (1) If G acts properly discontinuously on X, then N ( x , R ) < ¥ for all x ∈ X and R > . (2) If G acts strongly properly discontinuously on X, then there exists A x > suchthat N ( x , R ) ≤ A x exp ( RC ) for all R > , where C is the first sharpness constant of G . The key idea of Theorem 13 is to bring branching laws to spectral analysis [10,12], namely, we consider the restriction of irreducible representations of G that arerealized in the space of functions on the homogeneous space X = G / H and analyzethe G -representations when restricted to the subgroup L . Details will be given in [7]. Acknowledgements
This article is based on the talk that the author delivered at the eleventhInternational Workshop: Lie Theory and its Applications in Physics in Varna, Bulgaria, 15-21,June, 2015. The author is grateful to Professor Vladimir Dobrev for his warm hospitality.
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