Intrinsic sound of anti-de Sitter manifolds
Toshiyuki Kobayashi

Abstract As is well-known for compact Riemann surfaces, eigenvalues of the
Laplacian are distributed discretely and most of eigenvalues vary viewed as functions on the Teichmüller space. We discuss a new feature in the Lorentzian geometry, or more generally, in pseudo-Riemannian geometry. One of the distinguishedfeatures is that L2 -eigenvalues of the Laplacian may be distributed densely in R in
pseudo-Riemannian geometry. For three-dimensional anti-de Sitter manifolds, we
also explain another feature proved in joint with F. Kassel [Adv. Math. 2016] that
there exist countably many L2 -eigenvalues of the Laplacian that are stable under any
small deformation of anti-de Sitter structure. Partially supported by Grant-in-Aid for
Scientific 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 2010
MSC: Primary 22E40, 22E46, 58J50: Secondary 11F72, 53C35

1 Introduction
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 phenomenon
that compact three-dimensional anti-de Sitter manifolds have “intrinsic sound”
Toshiyuki Kobayashi
Kavli IPMU and Graduate School of Mathematical Sciences, The University of Tokyo, e-mail:toshi@ms.u-tokyo.ac.jp

1

2

Toshiyuki Kobayashi

which is stable under any small deformation. This is formulated in the framework of spectral analysis of anti-de Sitter manifolds, or more generally, of pseudoRiemannian locally symmetric spaces XΓ . In this article, we give a flavor of this new
topic by comparing it with the flat case and the Riemannian case.
To explain briefly the subject, let X be a pseudo-Riemannian manifold, and Γ adiscrete isometry group acting properly discontinuously and freely on X. Then thequotient space XΓ := Γ \X carries a pseudo-Riemannian manifold structure suchthat the covering map X → XΓ is isometric. We are particularly interested in thecase where XΓ 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. rigidity
Geometry Does cocompact Γ exist? Higher Teichmüller theory v.s. rigidity theorem(Section 4.1)
(Section 4.2)
Analysis Does L2 -spectrum exist?
Whether L2 -eigenvalues vary or not(Problem A)(Problem B)

2 A program
In [5, 6, 12] we initiated the study of “spectral analysis on pseudo-Riemannian locally symmetric spaces” with focus on the following two problems:
Problem A Construct eigenfunctions of the Laplacian ∆XΓ on XΓ . Does there exist
a nonzero L2 -eigenfunction?
Problem B Understand the behaviour of L2 -eigenvalues of the Laplacian ∆XΓ on
XΓ under small deformation of Γ inside G.
Even when XΓ is compact, the existence of countably many L2 -eigenvalues isalready nontrivial because the Laplacian ∆XΓ is not elliptic in our setting. We shalldiscuss in Section 2.2 for further difficulties concerning Problems A and B when XΓ
is non-Riemannian.
We may extend these problems by considering joint eigenfunctions for “invariant
differential operators” on XΓ rather than the single operator ∆XΓ . Here by “invariant
differential operators on XΓ ” 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.

Intrinsic sound of anti-de Sitter manifolds

3

2.1 Known results
Spectral analysis on a pseudo-Riemannian locally symmetric space XΓ = Γ \X =
Γ \G/H is already deep and difficult in the following special cases:
1) (noncommutative harmonic analysis on G/H) Γ = {e}.
In this case, the group G acts unitarily on the Hilbert space L2 (XΓ ) = L2 (X)by translation f (·) 7→ f (g−1 ·), and the irreducible decomposition of L2 (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 has
been developed extensively by the work of Helgason, Flensted-Jensen, Matsuki–
Oshima–Sekiguchi, Delorme, van den Ban–Schlichtkrull among others as a generalization of Harish-Chandra’s earlier work on the regular representation L2 (G)for group manifolds.
2) (automorphic forms) H is compact and Γ is arithmetic.
If H is a maximal compact subgroup of G, then XΓ = Γ \G/H is a Riemannian
locally symmetric space and the Laplacian ∆XΓ is an elliptic differential operator.
Then there exist infinitely many L2 -eigenvalues of ∆XΓ if XΓ is compact by thegeneral theory for compact Riemannian manifolds (see Fact 1). If furthermore Γis irreducible, then Weil’s local rigidity theorem [18] states that nontrivial deformations exist only when X is the hyperbolic plane SL(2, R)/SO(2), in which casecompact quotients XΓ have a classically-known deformation space modulo conjugation, i.e., their Teichmüller space. Viewed as a function on the Teichmüller
space, L2 -eigenvalues vary analytically [1, 20], see Fact 11.
Spectral analysis on XΓ is closely related to the theory of automorphic forms in
the Archimedian place if Γ is an arithmetic subgroup.
3) (abelian case) G = R p+q with H = {0} and Γ = Z p+q .
We equip X = G/H with the standard flat pseudo-Riemannian structure of signature (p, q) (see Example 1). In this case, G is abelian, but X = G/H is nonRiemannian. 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 Oppenheim’s conjecture (see Section 5.2).

2.2 Difficulties in the new settings
If we try to attack a problem of spectral analysis on Γ \G/H in the more generalcase where H is noncompact and Γ is infinite, then new difficulties may arise fromseveral points of view:(1) Geometry. The G-invariant pseudo-Riemannian structure on X = G/H is not
Riemannian anymore, and discrete groups of isometries of X do not always act
properly discontinuously on such X.

4Toshiyuki Kobayashi(2) Analysis. The Laplacian ∆X on XΓ is not an elliptic differential operator. Furthermore, it is not clear if ∆X has a self-adjoint extension on L2 (XΓ ).
(3) Representation theory. If Γ acts properly discontinuously on X = G/H with
H noncompact, then the volume of Γ \G is infinite, and the regular representation L2 (Γ \G) may have infinite multiplicities. In turn, the group G may nothave a good control of functions on Γ \G. Moreover L2 (XΓ ) is not a subspaceof L2 (Γ \G) because H is noncompact. All these observations suggest that an
application of the representation theory of L2 (Γ \G) to spectral analysis on XΓ israther limited when H is noncompact.
Point (1) creates some underlying difficulty to Problem B: we need to consider
locally symmetric spaces XΓ for which proper discontinuity of the action of Γ on X
is preserved under small deformations of Γ in G. This is nontrivial. This question
was first studied by the author [9, 11]. See [4] for further study. An interesting
aspect of the case of noncompact H is that there are more examples where nontrivial
deformations of compact quotients exist than for compact H (cf . Weil’s local rigidity
theorem [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-known
properties in the Riemannian case holds in our setting in the pseudo-Riemanniancase.
Fact 1 Suppose M is a compact Riemannian manifold.
(1) The Laplacian ∆M extends to a self-adjoint operator on L2 (M).
(2) There exist infinitely many L2 -eigenvalues of ∆M .
(3) An eigenfunction of ∆M is infinitely differentiable.
(4) Each eigenspace of ∆M is finite-dimensional.
(5) The set of L2 -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 = R2,1 /Z3 (Theorem 7) for (5).
In spite of these difficulties, we wish to reveal a mystery of spectral analysis of
pseudo-Riemannian locally homogeneous spaces XΓ = Γ \G/H. We shall discuss
self-adjoint extension of the Laplacian in the pseudo-Riemannian setting in Theorem 13, and the existence of countable many L2 -eigenvalues in Theorems 8, 12 and13.

3 Pseudo-Riemannian manifold
3.1 Laplacian on pseudo-Riemannian manifolds
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.

Intrinsic sound of anti-de Sitter manifolds

5(M, g) is a Riemannian manifold if q = 0, and is a Lorentzian manifold if q = 1. The
metric tensor g induces a Radon measure d µ on X, and the divergence div. Then the
Laplacian∆M := div grad,is a differential operator of second order which is a symmetric operator on theHilbert space L2 (X, d µ ).
Example 1. Let (M, g) be the standard flat pseudo-Riemannian manifold:R p,q := (R p+q , dx12 + · · · + dx2p − dx2p+1 − · · · − dx2p+q ).
Then the Laplacian takes the form

∆R p,q =

∂2
∂2
∂2
∂2
−
+
·
·
·
+
−
·
·
·
−
.
∂ x2p ∂ x2p+1∂ x12∂ x2p+q

In general, ∆M is an elliptic differential operator if (M, g) is Riemannian, and is a
hyperbolic operator if (M, g) is Lorentzian.

3.2 Homogeneous pseudo-Riemannian manifolds
A typical example of pseudo-Riemannian manifolds X with “large” isometry groups
is semisimple symmetric spaces, for which the infinitesimal classification was accomplished 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 fixed
point group Gσ = {g ∈ G : σ g = g} for some involutive automorphism σ of G. In
particular, 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 pseudoRiemannian structure such that G acts on X by isometries.
Proof. By a theorem of Mostow, we can take a Cartan involution θ of G such that
θ H = H. Then K := Gθ 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 ⟨ , ⟩ on gsuch that ⟨ , ⟩|k×k is negative definite, ⟨ , ⟩|p×p is positive definite, and k and p are
orthogonal to each other. (If G is semisimple, then we may take ⟨ , ⟩ to be the Killing
form of g. )Since θ H = H, the Lie algebra h of H is decomposed into a direct sum h = (h ∩k) + (h ∩ p), and therefore the bilinear form ⟨ , ⟩ is non-degenerate when restricted
to h. Then ⟨ , ⟩ induces an Ad(H)-invariant nondegenerate symmetric bilinear form
⟨ , ⟩g/h on the quotient space g/h, with which we identify the tangent space To (G/H)

6Toshiyuki Kobayashi

at the origin o = eH ∈ G/H. Since the bilinear form ⟨ , ⟩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. ⊓⊔3.3 Pseudo-Riemannian manifolds with constant curvature,
Anti-de Sitter manifolds
Let Q p,q (x) := x12 + · · · + x2p − x2p+1 − · · · − x2p+q be a quadratic form on R p+q of signature (p, q), and we denote by O(p, q) the indefinite orthogonal group preserving
the 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) = ±1}.
By switching p and q, we have an obvious diffeomorphismq,pM+p,q ≃ M−.

The flat pseudo-Riemannian structure R p,q (Example 1) induces a pseudo-Riemannian
structure on the hypersurface M+p,q of signature (p − 1, q) with constant curvature 1,
and that on M−p,q of signature (p, q − 1) with constant curvature −1.
The natural action of the group O(p, q) on R p,q induces an isometric and transitive action on the hypersurfaces M±p,q , and thus they are expressed as homogeneous
spaces:M+p,q ≃ O(p, q)/O(p − 1, q),M−p,q ≃ O(p, q)/O(p, q − 1),

giving examples of pseudo-Riemannian homogeneous spaces as in Proposition 1.
n−1,2
The anti-de Sitter space AdSn = M−
is a model space for n-dimensional
Lorentzian manifolds of constant negative sectional curvature, or anti-de Sitter nmanifolds. 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 manifolds with constant positive, zero, and negative curvatures.
Riemannian manifolds with constant curvature:n+1,0
Sn = M+≃O(n + 1)/O(n)Rn: standard sphere,: Euclidean space,n,1
H n = M−
≃ O(1, n)/O(n): hyperbolic space,

Lorentzian manifolds with constant curvature:

Intrinsic sound of anti-de Sitter manifolds

7

n,1≃ O(n, 1)/O(n − 1, 1)
dSn = M+: de Sitter space,n−1,1: Minkowski space,

R

AdS

n

n−1,2
= M−
≃O(2, n − 1)/O(1, n − 1) : anti-de Sitter space,

4 Discontinuous groups for pseudo-Riemannian manifolds
4.1 Existence problem of compact Clifford–Klein forms
Let H be a closed subgroup of a Lie group G, and X = G/H, and Γ a discrete
subgroup of G. If H is compact, then the double coset space Γ \G/H becomes a
C∞ -manifold for any torsion-free discrete subgroup Γ of G. However, we have to becareful for noncompact H, because not all discrete subgroups acts properly discontinuously on G/H, and Γ \G/H may not be Hausdorff in the quotient topology. We
illustrate this feature by two general results:
Fact 2(1) (Moore’s ergodicity theorem [15]) Let G be a simple Lie group, and Γa lattice. Then Γ acts ergodically on G/H for any noncompact closed subgroup
H. In particular, Γ \G/H is non-Hausdorff.
(2) (Calabi–Markus phenomenon ([2, 8])) Let G be a reductive Lie group, and Γ
an infinite discrete subgroup. Then Γ \G/H is non-Hausdorff for any reductive
subgroup H with rankR G = rankR H.
In fact, determining which groups act properly discontinuously on reductive homogeneous 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 Γ acts properly discontinuously and freely on
X = G/H. Then the quotient spaceXΓ := Γ \X ≃ Γ \G/H
carries a C∞ -manifold structure such that the quotient map p : X → XΓ is a covering,
through which XΓ inherits any G-invariant local geometric structure on X. We sayΓ is a discontinuous group for X and XΓ 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 XΓ carries a pseudo-Riemannian structure by Proposition 1.
(2) If X = G/H is a semisimple symmetric space, then any Clifford–Klein form XΓ =
Γ \G/H is a pseudo-Riemannian locally symmetric space, namely, the (local)
geodesic symmetry at every p ∈ XΓ with respect to the Levi-Civita connection is
locally isometric.
By space forms, we mean pseudo-Riemannian manifolds of constant sectional
curvature. They are examples of pseudo-Riemannian locally symmetric spaces. For
simplicity, we shall assume that they are geodesically complete.

8Toshiyuki Kobayashi

Example 3. Clifford–Klein forms of M+p+1,q = O(p + 1, q)/O(p, q) (respectively,
M−p,q+1 = O(p, q + 1)/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 far
as p ̸= 1 for positive curvature or q ̸= 1 for negative curvature.
A general question for reductive homogeneous spaces G/H is:
Question 1. Does compact Clifford–Klein forms of G/H exist?
or equivalently,
Question 2. Does there exist a discrete subgroup Γ 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 “global
theory” of pseudo-Riemannian geometry: how local pseudo-Riemannian homogeneous structure affects the global nature of manifolds? A classic example is space
form problem which asks the global properties (e.g. compactness, volume, fundamental groups, etc.) of a pseudo-Riemannian manifold of constant curvature (local
property). The study of discontinuous groups for M+p+1,q and M−p,q+1 shows the following results in pseudo-Riemannian space forms of signature (p, q):
Fact 3 Space forms of positive curvature are(1) always closed if q = 0, i.e., sphere geometry in the Riemannian case;(2) never closed if p ≥ q > 0, in particular, if q = 1 (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 = 0, i.e., hyperbolic geometry in the Riemannian case;(2) for odd dimensions if q = 1, i.e., anti-de Sitter geometry in the Lorentzian case;(3) for (p, q) = (4m, 3) (m ∈ N) or (8, 7).
See [13, Section 4] for the survey of the space form problem in pseudo-Riemannian
geometry and also of Question 1 for more general G/H.
A large and important class of Clifford–Klein forms XΓ of a reductive homogeneous space X = G/H is constructed as follows (see [8]).
Definition 1. A quotient XΓ = Γ \X of X by a discrete subgroup Γ of G is called
standard if Γ 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 Γ acts properly discontinuously on G/H. A handy criterion for the triple (G, H, L) of reductive
groups such that L acts properly on G/H is proved in [8], as we shall recall below.

Intrinsic sound of anti-de Sitter manifolds

9

Let G = K exp a+ K be a Cartan decomposition, where a is a maximal abelian subspace of p and a+ is the dominant Weyl chamber with respect to a fixed positive
system Σ + (g, a). This defines a map µ : G → a+ (Cartan projection) byµ (k1 eX k2 ) = X

for k1 , k2 ∈ K and X ∈ a.

It is continuous, proper and surjective. If H is a reductive subgroup, then there existsg ∈ G such that µ (gHg−1 ) is given by the intersection of a+ with a subspace ofdimension rankR H. By an abuse of notation, we use the same H instead of gHg−1 .
With this convention, we have:
Properness Criterion 5 ([8]) L acts properly on G/H if and only if µ (L) ∩ µ (H) ={0}.
By taking a lattice Γ of such L, we found a family of pseudo-Riemannian locally symmetric spaces XΓ in [8, 13]. The list of symmetric spaces admitting standard Clifford–Klein forms of finite volume (or compact forms) include M−p,q+1 =O(p, q + 1)/O(p, q) with (p, q) satisfying the conditions in Fact 4. Further, by applying Properness Criterion 5, Okuda [16] gave examples of pseudo-Riemannian
locally symmetric spaces Γ \G/H of infinite volume where Γ is isomorphic to thefundamental group π1 (Σg ) of a compact Riemann surface Σg with g ≥ 2.
For the construction of stable spectrum on XΓ (see Theorem 10 and Theorem 12(2) below), we introduced in [6, Section 1.6] the following concept:
Definition 2. A discrete subgroup Γ of G acts strongly properly discontinuously (orsharply) on X = G/H if there exists C, C′ > 0 such that for all γ ∈ Γ ,d(µ (γ ), µ (H)) ≥ C∥µ (γ )∥ −C′ .
Here d(·, ·) is a distance in a given by a Euclidean norm ∥·∥ which is invariant under
the Weyl group of the restricted root system Σ (g, a). We say the positive number C
is the first sharpness constant for Γ .
If a reductive subgroup L acts properly on a reductive homogeneous space G/H,
then the action of a discrete subgroup Γ of L is strongly properly discontinuous ([6,Example 4.10]).

4.2 Deformation of Clifford–Klein forms
Let G be a Lie group and Γ a finitely generated group. We denote by Hom(Γ , G)
the set of all homomorphisms of Γ to G topologized by pointwise convergence. Bytaking a finite set {γ1 , · · · , γk } of generators of Γ , we can identify Hom(Γ , G) as asubset of the direct product G × · · · × G by the inclusion:
Hom(Γ , G) ,→ G × · · · × G,

ϕ 7→ (ϕ (γ1 ), · · · , ϕ (γk )).

(1)

10

Toshiyuki Kobayashi

If Γ is finitely presentable, then Hom(Γ , G) is realized as a real analytic variety via(1).
Suppose G acts continuously on a manifold X. We shall take X = G/H with
noncompact closed subgroup H later. Then not all discrete subgroups act properly
discontinuously on X in this general setting. The main difference of the following
definition of the author [9] in the general case from that of Weil [18] is a requirementof proper discontinuity.
R(Γ , G; X) := {ϕ ∈ Hom(Γ , G) : ϕ is injective,(2)and ϕ (Γ ) acts properly discontinuously and freely on G/H}.
Suppose now X = G/H for a closed subgroup H. Then the double coset spaceϕ (Γ )\G/H forms a family of manifolds that are locally modelled on G/H withparameter ϕ ∈ R(Γ , G; X). To be more precise on “parameter”, we note that the
conjugation by an element of G induces an automorphism of Hom(Γ , G) whichleaves R(Γ , G; X) invariant. Taking these unessential deformations into account, we
define the deformation space (generalized Teichmüller space) as the quotient set
T (Γ , G; X) := R(Γ , G; X)/G.
Example 4.(1) Let Γ be the surface group π1 (Σg ) of genus g ≥ 2, G = PSL(2, R),
X = H 2 (two-dimensional hyperbolic space). Then T (Γ , G; X) is the classicalTeichmüller space, which is of dimension 6g − 6.
(2) G = Rn , X = Rn , Γ = Zn . Then T (Γ , G; X) ≃ GL(n, R) (see (4) below).
(3) G = SO(2, 2), X = AdS3 , and Γ = π1 (Σg ). Then T (Γ , G; X) is of dimension
12g − 12 (see [6, Section 9.2] and references therein).
Remark 2. There is a natural isometry between Xϕ (Γ ) and Xϕ (gΓ g−1 ) . Hence, the set
Specd (Xϕ (Γ ) ) of L2 -eigenvalues is independent of the conjugation of ϕ ∈ R(Γ , G; X)
by an element of G. By an abuse of notation we shall write Specd (Xϕ (Γ ) ) for ϕ ∈T (Γ , G; X) when we deal with Problem B of Section 2.

5 Spectrum on R p,q /Z p+q and Oppenheim conjectureThis section gives an elementary but inspiring observation of spectrum on flat
pseudo-Riemannian manifolds.

5.1 Spectrum of R p,q /ϕ (Z p+q )Let G = Rn and Γ = Zn . Then the group homomorphism ϕ : Γ → G is uniquelydetermined by the image ϕ (e j ) (1 ≤ j ≤ n) where e1 , · · · , en ∈ Zn are the standard

Intrinsic sound of anti-de Sitter manifolds

11basis, and thus we have a bijection∼

Hom(Γ , G) ← M(n, R),

ϕg →g7(3)by ϕg (m) := gm for m ∈ Zn , or equivalently, by g = (ϕg (e1 ), · · · , ϕg (en )).
Let σ ∈ Aut(G) be defined by σ (x) := −x. Then H := Gσ = {0} and X :=G/H ≃ Rn is a symmetric space. The discrete group Γ acts properly discontinuously on X via ϕg if and only if g ∈ GL(n, R). Moreover, since G is abelian, G
acts trivially on Hom(Γ , G) by conjugation, and therefore the deformation space
T (Γ , G; X) identifies with R(Γ , G; X). Hence we have a natural bijection between
the two subsets of (3):∼T (Γ , G; X) ← GL(n, R).
(4)
Fix p, q ∈ N such that p + q = n, and we endow X ≃ Rn with the standard
flat indefinite metric R p,q (see Example 1). Let us determine Specd (Xϕg (Γ ) ) ≃Specd (R p,q /ϕg (Zn )) for g ∈ GL(n, R) ≃ T (Γ , G; X).
For this, we define a function on X = Rn by√
fm (x) := exp(2π −1 tmg−1 x)(x ∈ Rn )for each m ∈ Zn where x and m are regarded as column vectors. Clearly, fm is
ϕg (Γ )-periodic and defines a real analytic function on Xϕg (Γ ) . Furthermore, fm isan eigenfunction of the Laplacian ∆R p,q :∆R p,q fm = −4π 2 Qg−1 Ip,q tg−1 (m) fm ,where, for a symmetric matrix S ∈ M(n, R), QS denotes the quadratic form on Rn
given by
QS (y) := tySy
for y ∈ Rn .
Since { fm : m ∈ Zn } spans a dense subspace of L2 (Xϕg (Γ ) ), we have shown:
Proposition 2. For any g ∈ GL(n, R) ≃ T (Γ , G; X),
Specd (Xϕg (Γ ) ) = {−4π 2 Qg−1 Ip,q tg−1 (m) : m ∈ Zn }.
Here are some observation in the n = 1, 2 cases.
Example 5. Let n = 1 and (p, q) = (1, 0). Then Specd (Xϕg (Γ ) ) = {−4π 2 m2 /g2 : m ∈Z} for g ∈ R× ≃ GL(1, R) by Proposition 2. Thus the smaller the period |g| is, thelarger the absolute value of the eigenvalue | − 4π 2 m2 /g2 | becomes for each fixedm ∈ Z \ {0}. This is thought of as a mathematical model of a music instrument for
which shorter strings produce a higher pitch than longer strings (see Introduction).
Example 6. Let n = 2 and (p, q) = (1, 1). Take g = I2 , so that ϕg (Γ ) = Z2 is thestandard lattice. Then the L2 -eigenspace of the Laplacian ∆R1,1 /Z2 for zero eigenvalue contains W := {ψ (x − y) : ψ ∈ L2 (R/Z)}. Since W is infinite-dimensionaland W ̸⊂ C∞ (R2 /Z2 ), the third and fourth statements of Fact 1 fail in this pseudoRiemannian setting.

12

Toshiyuki Kobayashi

By the explicit description of Specd (Xϕ (Γ ) ) for all ϕ ∈ T (Γ , G; X) in Proposition 2, we can also tell the behaviour of Specd (Xϕ (Γ ) ) under deformation of Γ byϕ . Obviously, any constant function on Xϕ (Γ ) is an eigenfunction of the Laplacian∆Xϕ (Γ ) = ∆R p,q /ϕ (Z p+q ) with eigenvalue zero. We see that this is the unique stableL2 -eigenvalue in the flat compact manifold:
Corollary 1 (non-existence of stable eigenvalues). Let n = p +q with p, q ∈ N. For
any open subset V of T (Γ , G; X),∩ϕ ∈VSpecd (Xϕ (Γ ) ) = {0}.

5.2 Oppenheim’s conjecture and stability of spectrum
In 1929, Oppenheim [17] raised a question about the distribution of an indefinite
quadratic forms at integral points. The following theorem, referred to as Oppenheim’s conjecture, was proved by Margulis (see [14] and references therein).
Fact 6 (Oppenheim’s conjecture) Suppose n ≥ 3 and Q is a real nondegenerate
indefinite quadratic form in n variables. Then either Q is proportional to a form
with integer coefficients (and thus Q(Zn ) is discrete in R), or Q(Zn ) is dense in R.
Combining this with Proposition 2, we get the following.
Theorem 7. Let p + q = n, p ≥ 2, q ≥ 1, G = Rn , X = R p,q and Γ = Zn . We define
an open dense subset U of T (Γ , G; X) ≃ GL(n, R) by
U := {g ∈ GL(n, R) : g−1 I p,q tg−1 is not proportional to an element of M(n, Z).}
Then the set Specd (Xϕ (Γ ) ) of L2 -eigenvalues of the Laplacian is dense in R if andonly if ϕ ∈ U.
Thus the fifth statement of Fact 1 for compact Riemannian manifolds do fail in thepseudo-Riemannian case.

6 Main results—sound of anti-de Sitter manifolds
6.1 Intrinsic sound of anti-de Sitter manifolds
In general, it is not clear whether the Laplacian ∆M admits infinitely many L2 eigenvalues for compact pseudo-Riemannian manifolds. For anti-de Sitter 3-manifolds,
we proved in [6, Theorem 1.1]:

Intrinsic sound of anti-de Sitter manifolds

13

Theorem 8. For any compact anti-de Sitter 3-manifold M, there exist infinitely
many L2 -eigenvalues of the Laplacian ∆M .
In the abelian case, it is easy to see that compactness of XΓ is necessary for the
existence of L2 -eigenvalues:
Proposition 3. Let G = R p+q , X = R p,q , Γ = Zk , and ϕ ∈ R(Γ , G; X). Then Specd (Xϕ (Γ ) ) ̸=
0/ if and only if Xϕ (Γ ) is compact, or equivalently, k = p + q.
However, anti-de Sitter 3-manifolds M admit infinitely many L2 -eigenvalues evenwhen M is of infinite-volume (see [6, Theorem 9.9]):
Theorem 9. For any finitely generated discrete subgroup Γ of G = SO(2, 2) acting
properly discontinuously and freely on X = AdS3 ,Specd (XΓ ) ⊃ {l(l − 2) : l ∈ N, l ≥ 10C−3 }where C ≡ C(Γ ) is the first sharpness constant of Γ .
The above L2 -eigenvalues are stable in the following sense:
Theorem 10 (stable L2 -eigenvalues). Suppose that Γ ⊂ G = SO(2, 2) and M =Γ \ AdS3 is a compact standard anti-de Sitter 3-manifold. Then there exists a neighbourhood U ⊂ Hom(Γ , G) of the natural inclusion with the following two properties:
U ⊂ R(Γ , G; AdS3 ),
(5)
#(

∩

ϕ ∈U

Specd (XΓ )) = ∞.

(6)

The first geometric property (5) asserts that a small deformation of Γ keeps proper
discontinuity, which was conjectured by Goldman [3] in the AdS3 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 manifolds
have “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 14 is constant on the Teichmüller 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 Specd (M) is discrete.

14

Toshiyuki Kobayashi

7 Perspectives and sketch of proof
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 XΓ 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 L2 -eigenvalues for all the differential operators on XΓ that are induced from G-invariant differential operatorson X.
(2) (stable spectrum) If Γ is contained in a simple Lie group L of real rank one
acting properly on X = G/H, then there is a neighbourhood V ⊂ Hom(Γ , G) of
the natural inclusion such that for any ϕ ∈ V , the action ϕ (Γ ) on X is properly
discontinuous and the set of joint L2 -eigenvalues on Xϕ (Γ ) contains the infinite
set I.
Remark 3. We do not require XΓ to be of finite volume in Theorem 12.
Remark 4. It is plausible that for a general locally symmetric space Γ \G/H with
G reductive, no nonzero L2 -eigenvalue is stable under nontrivial small deformation
unless the rank condition (7) is satisfied. For instance, suppose Γ = π1 (Σg ) withg ≥ 2 and R(Γ , G; X) ̸= 0.
/ (Such semisimple symmetric space X = G/H was recently classified in [16].) Then we expect the rank condition (7) is equivalent to theexistence of an open subset U in R(Γ , G; X) such that
#(

∩

ϕ ∈U

Specd (Xϕ (Γ ) )) = ∞.

It should be noted that not all L2 -eigenvalues of compact anti-de Sitter manifolds
are stable under small deformation of anti-de Sitter structure. In fact, we provedin [7] that there exist also countably many negative L2 -eigenvalues that are NOT
stable under deformation, whereas the countably many stable L2 -eigenvalues that
we constructed in Theorem 9 are all positive. More generally, we prove in [7] thefollowing theorem that include both stable and unstable L2 -eigenvalues:
Theorem 13. Let G be a reductive homogeneous space and L a reductive subgroup of G such that H ∩ L is compact. Assume that the complexification XC is
LC -spherical. Then for any torsion-free discrete subgroup Γ of L, we have:(1) the Laplacian ∆XΓ extends to a self-adjoint operator on L2 (XΓ );(2) # Specd (XΓ ) = ∞ if XΓ is compact.

Intrinsic sound of anti-de Sitter manifolds

15

By “LC -spherical” we mean that a Borel subgroup LC has an open orbit in XC . In
this 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, Corollary
3.3.7].
Table 1 1G
HL(i) SO(2n, 2) SO(2n, 1)U(n, 1)(ii) SO(2n, 2) U(n, 1)SO(2n, 1)Sp(n, 1)(iii) SU(2n, 2) U(2n, 1)(iv) SU(2n, 2) Sp(n, 1)U(2n, 1)(v) SO(4n, 4) SO(4n, 3) Sp(1) × Sp(n, 1)(vi) SO(8, 8) SO(8, 7)Spin(8, 1)(vii) SO(8, C) SO(7, C)
Spin(7, 1)(viii) SO(4, 4) Spin(4, 3) SO(4, 1) × SO(3)(ix) SO(4, 3) G2 (R) SO(4, 1) × SO(2)Examples for Theorem 13 include Table 1 (ii) for all n ∈ N, whereas we needn ∈ 2N in Theorem 12 for the rank condition (7).
The idea of the proof for Theorem 12 is to take an average of a (nonperiodic) eigenfunction on X with rapid decay at infinity over Γ -orbits as a generalization of Poincaré series. Geometric ingredients of the convergence (respectively,
nonzeroness) of the generalized Poincaré series include “counting Γ -orbits” statedin Lemma 1 below (respectively, the Kazhdan–Margulis theorem, cf . [6, Proposition 8.14]). Let B(o, R) be a “pseudo-ball” of radius R > 0 centered at the origino = eH ∈ X = G/H, and we set
N(x, R) := #{γ ∈ Γ : γ · x ∈ B(o, R)}.

Lemma 1 ([6, Corollary 4.7]).
(1) If Γ acts properly discontinuously on X, then N(x, R) < ∞ for all x ∈ X and R > 0.
(2) If Γ acts strongly properly discontinuously on X, then there exists Ax > 0 suchthatR
N(x, R) ≤ Ax exp( ) for all R > 0,C
where C is the first sharpness constant of Γ .
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 are
realized in the space of functions on the homogeneous space X = G/H and analyze
the G-representations when restricted to the subgroup L. Details will be given in [7].

16

Toshiyuki Kobayashi

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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