Scattering matrices and scattering geodesics of locally symmetric spaces

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4 série, t. 34, 2001, p. 441 à 469.

SCATTERING MATRICES AND SCATTERING GEODESICS OF LOCALLY SYMMETRIC SPACES

B

Y

L

IZHEN

JI

AND

M

ACIEJ

ZWORSKI

ABSTRACT. – Let Γ/X be aQ-rank one locally symmetric space. We describe the frequencies of oscillation of scattering matrices onΓ/X in the energy variable in terms of sojourn times of scattering geodesics. Scattering geodesics are the geodesics which move to infinity in both directions and are distance minimizing near both infinities. The sojourn time of a scattering geodesic is the time it spends in a fixed compact region. The frequencies of oscillation come from the singularities of the Fourier transforms of scattering matrices and we show that these occur at sojourn times of scattering geodesics on the locally symmetric space. This generalizes a result of Guillemin obtained in the case of finite volume noncompact Riemann surfaces.2001 Éditions scientifiques et médicales Elsevier SAS

RÉSUMÉ. – SoitΓ/Xun espace localement symétrique de rang 1 surQ. Nous décrivons les fréquences d’oscillations de la matrice de scattering deΓ/X dans les variables d’énergie. Définissons pour cela les géodésiques de scattering de l’espace localement symétrique comme les géodésiques atteignant l’infini à leurs deux extrémités, et qui sont minimisantes au voisinage de l’infini. Appelons temps de séjour d’une telle géodésique le temps qu’elle passe dans un domaine compact fixé. Nous prouvons que les singularités de la transformée de Fourier (relativement au paramètre d’énergie) de la matrice de scattering sont contenues dans l’ensemble des temps de séjour, et décrivons la nature de ces singularités. Cela généralise un résultat de Guillemin obtenu dans le cas des surfaces de Riemann non compactes de volume fini._2001 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

The purpose of this paper is to describe the frequencies of oscillation (in the energy variable) of scattering matrices onQ-rank one locally symmetric spacesΓ\X. In other words, we describe the singularities of the Fourier transforms (with respect to the energy parameter) of scattering matrices and show that they occur at sojourn times of scattering geodesics on the locally symmetric space. This generalizes a result of Guillemin [9] obtained in the case of finite volume noncompact Riemann surfaces. In the case whenQ-rank is greater than one, the method of this paper would also give some results (see Remarks at the end of Section 3) but we restrict ourselves to a simpler case at this early stage.

Scattering geodesics are the geodesics which move to infinity in both directions and are distance minimizing near both infinities – see Proposition 2.5 and Definition 2.4 for a more direct geometric description. The sojourn time of a scattering geodesic is, roughly speaking, the time it spends in a fixed compact region.

LetX=G/K be a symmetric space of noncompact type, andΓ⊂Gan arithmetic discrete subgroup of finite covolume. ThenΓ\X is a locally symmetric space of finite volume. When Γ\X is noncompact, it has both discrete and continuous spectra. The continuous spectrum is

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given by the Eisenstein series, and the constant terms of the Eisenstein series along rational parabolic subgroups are described by the scattering matrices. For generalΓ\X, the Eisenstein series and scattering matrices were studied by Langlands in [18] (see Lemma 3.5 and (3.3) below). Roughly speaking, the scattering matrices measure the density of the continuous spectrum, and their analytic properties and functional equations play an important role in the Langlands’ program (see [19]).

The scattering matrices in theQ-rank one case take a particularly simple form. Specifically, in theQ-rank case, there is a one-to-one correspondence between the set of ends ofΓ\X and the set ofΓ-conjugacy classes of rational parabolic subgroups. Topologically, each end is a cylinder.

For each end of the space, let us denote the associated parabolic subgroup byP. We associate to it a compact locally symmetric space,ΓXP\XP, of smaller dimension (see Proposition 2.3).

Then, a section of the topological end associated withP is a fiber bundle overΓXP\XP, with fiber equal toΓNP\NP, a compact quotient of a unipotent groupNP. LetP1, . . . , Pnbe a set of representatives ofΓ-conjugacy classes of rational parabolic subgroups. For eachµ∈Rwe then consider the direct sum of theµ-eigenspaces of the Laplacians onΓX_Pi\XP_i,i= 1, . . . , n. The scattering matrixS^µ (also denoted bycµ below) is a unitary operator on that space. Forµ= 0, the dimension of the space is equal to the number of the ends ofΓ\X, and it is non-zero only for a discrete set ofµ’s inn

i=1Spec(ΓPi\XPi). For a particular orthonormal basis of eigenvectors corresponding toµ,

φ^µ₁, . . . , φ^µ_K(µ), φ^µ_j ∈L²(ΓX_Pk(j)\XP_k(j)), (1.1)

where K(µ)is the dimension of the direct sum of the eigenspaces, and k(j)∈ {1, . . . , n} is the unique index such that φ^µ_j is an eigenfunction on ΓX_Pk(j)\XP_k(j), we have the following expression for the scattering matrixS^µ(λ):

S^µ(λ) =

s^µ_ij(−iλ)

1i,jK(µ).

The operator S^µ(λ) is meromorphic in λ∈C (we are in the Q-rank one case) and we are interested in values ofλ∈R, corresponding to the continuous spectrum.

Using this notation we can now state our results. The detailed definitions of all other objects appearing in the statement will be given in Section 2.

THEOREM 1. – LetΓ\Xbe aQ-rank one locally symmetric space. Forl, m∈ {1, . . . , n}, let Tlm⊂Rbe the set of sojourn times of scattering geodesics between the ends corresponding to parabolic subgroupsPlandPm. ThenTlmis discrete and for eachT∈ Tlmthe corresponding scattering geodesics form a smooth family parametrized by a common finite cover XT,ml of ΓX_Pl\XP_landΓX_Pm\XP_m with projections

XT,ml

πl,T πm,T

ΓX_Pl\XP_l ΓX_Pm\XPm

(1.2)

defined in (2.4).

If we think of the geometric Theorem 1 as a classical statement, the next theorem provides the corresponding quantum property of locally symmetric spaces. It partially answers [16, Question 13.17.2].

To formulate it, we need to discuss the structure ofΓ\Xmore – it will be carefully reviewed in Section 2. As recalled above, to each infinite end we associate a parabolic subgroupP, which

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admits the Langlands decompositionNPMPAP. LetΦ⁺(P, AP)be the set of the roots ina^∗_P of the adjoint action ofAP onNP. We then define

2ρ= 2ρP=sum of the roots inΦ⁺(P, AP), (1.3)

where we add with multiplicities equal to the dimensions of the root spaces. In theQ-rank one case, dimAP = 1, anda^∗_P can be identified withR such that the norm ona^∗_P defined by the Killing form is equal to the standard Euclidean norm ofRand the roots inΦ⁺(P, AP)and hence ρP are identified with positive numbers. Then ρis the same positive number for all rational parabolic subgroupsP. The groupNPcan be identified with the Riemannian submanifoldNPx0

inX, wherex0=K∈X=G/K. We then have a natural quotientΓNP\NP,ΓNP = Γ∩NP, also equipped with a Riemannian density.

THEOREM 2. – Let Γ\X be a Q-rank 1 locally symmetric space, and s^µ_ij(λ) be the components of the scattering matrix for the eigenvalueµwith respect to the orthonormal basis φ^µ₁, . . . , φ^µ_K(µ)as above. Letl=k(i), m=k(j)∈ {1, . . . , n}be defined as in 1.1, andTlm the set of sojourn times between the ends ofPlandPmas in Theorem 1. Then

sing supp ˆs^µ_ij⊂ Tlm, and, more precisely,sˆ^µ_ij(t)is equal to

T∈Tlm

e⁻^ρT(2π)⁻^e^ml^T ^/2+1 vmvl

X_{T ,ml}

πl,T^∗ φiπm,T^∗ φj

±

(t−T±i0)⁻^1/2⁻^e^ml^T ^/2

1 +gT^ml(t) ,

whereg_T^mlare smooth in a neighbourhood ofT,g^ml_T (T) = 0, e^ml_T = dimXT,ml,

vp is the volume ofΓN_Pp\NP_p with respect to the induced Riemannian metric,p=l, m,ρis given by (1.3), and the integration overXT,mlis with respect to the Riemannian density.

The statement of the theorem can be transformed into a statement about the frequencies of oscillations of the scattering matrix. Suppose thatφ∈ Cc^∞(R). Then

s^µ_ij∗φ(λ) =ˆ

T∈suppφ∩Tlm

a^ml_T,φ(λ)e⁻^{iT λ}+Oφ

λ^−∞ ,

a^ml_T,φ(λ) =

k0

a^ml_T,φ,kλ^e^ml^T ^/2⁻^1/2⁻^k, (1.4)

with coefficients easily determined from the leading coefficient in Theorem 2 and the Taylor expansion ofg_T^mlatT. It is possible (see (1.5) below) that a finer analysis based on less general methods will give an expansion without the need for a cut-offφ.

As mentioned above, when X=SL(2,R)/SO(2) andΓ is a co-finite discrete subgroup of SL(2,R), the result is due to Guillemin [9], [10]. In that case the only contribution comes from µ= 0, the spacesXT,mlreduce to points andm, lrange from one to the number of cusp ends.

The argument of [9] was a direct computation, and in a more general case of a non-constant curvature, an application of the method coming from Euclidean scattering. It was based on the

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construction of wave operators using the theory of Fourier Integral Operators. A nice feature of the Riemann surface case is that the same theorem holds if a compactly supported metric perturbation is introduced. In Theorem 2 that is not the case. The spectral decomposition changes dramatically when the structure of a locally symmetric space is perturbed and we do not know a proper analogue of the quantum statement. Similarly, the structure of scattering geodesics will also change dramatically under perturbations: for example, the set of sojourn times may not be discrete anymore.

To compare with (1.4), we recall Guillemin’s explicit expansion in the case ofΓ\H²:

sij(λ) =c(λ)

T∈Tij

e⁻^{T /2}e⁻^iλT, c(λ) = ∞

−∞

dq (1 +q²)^1/2+iλ, (1.5)

where now we only haveµ= 0andk(i) =i,k(j) =j, index the cusps.

In this paper, rather than to follow [9] and construct the wave operators or to study Eisenstein series explicitly, we adapt an observation of Zelditch [25]. It was in turn inspired by Kuznecov sum formulæ: the real part of the scattering matrix can be given as an integral of the wave group over horocycles (see Proposition 4.1). That gives an easy way for understanding the singularities of the Fourier transform of the real part of the scattering matrix [25, (2.10),(2.12)]. We observe (Lemma 5.1) that we can also obtain singularities of the Fourier transform of the scattering matrix itself that way. Once the geometry is understood (Theorem 1 and Section 2), the same method applies in the case studied here.

We also point out that in the Euclidean setting, sojourn times are an old object of classical (and semi-classical) scattering. For that, and for pointers to recent work, we refer to [22].

2. Geometry of locally symmetric spaces

In this section, we define arithmetic subgroups, locally symmetric spaces and their geometry at infinity. The geometry at infinity is needed to understand scattering geodesics and hence their sojourn time, and is also crucial to understanding the continuous spectrum and the spectral decomposition. Though we will mainly study locally symmetric spaces ofQ-rank 1 in §4 and

§5, we formulate some results in §2 and §3 without this restriction. An important class ofQ-rank 1 locally symmetric spaces consists of Hilbert modular varieties (see [7]).

We begin by recalling some notions from algebraic group theory. The references are [1–4].

LetG=G(C)be a connected semisimple linear algebraic group defined overQ,G=G(R) the real locus ofG, which is a semisimple Lie group with finitely many connected components.

LetK⊂Gbe a maximal compact subgroup. The Killing form of the Lie algebra ofGdefines aG-invariant Riemannian metric onX =G/K, andX is a Riemannian symmetric space of noncompact type.

DEFINITION 2.1. – A subgroupΓ of the rational locusG(Q) ofGis called an arithmetic subgroup if under an embeddingG→GL(n,C)which is defined overQ,Γis commensurable toG∩GL(n,Z).

The embeddingG→GL(n,C)exists becauseG is by definition a linear algebraic group defined over Q. The class of arithmetic subgroups is well-defined and does not depend on the embeddingG→GL(n,C). Furthermore, a result of Selberg [24, Lemma 8] says that any arithmetic subgroup has a subgroup of finite index which is torsion free, and which is also clearly arithmetic.

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Any torsion free arithmetic subgroupΓacts properly discontinuously and fixed point freely onX, and the quotientΓ\X is a locally symmetric space of finite volume. In the following, we assume thatΓis torsion free, andΓ\X is noncompact . A good example to keep in mind while reading this paper isΓ = Γn={g∈SL(n,Z)|g≡Id mod n}, andX is the space of positive definite unimodular matrices.

The first natural question is to understand the geometry ofΓ\Xnear infinity, and hence to see how geodesics and points ofΓ\Xgo to infinity. For this purpose, we need to recall the reduction theory, in particular, rational parabolic subgroups and their Langlands decomposition.

A closed subgroupPofGis called a parabolic subgroup ifPcontains a maximal connected solvable subgroup, i.e., a Borel subgroup, ofG. This condition is equivalent to the condition that the quotientP\Gis compact (or a projective variety, to be precise). IfPis defined overQ,Pis called a rational parabolic subgroup.

For any rational parabolic subgroupP, let NP be the greatest unipotent normal subgroup ofP, which is called the unipotent radical ofPand is defined overQ. WhenG=SL(n), the subgroup of upper triangular matrices is a rational parabolic subgroupP, andNPis the subgroup of upper triangular matrices with 1 on the diagonal, i.e., unipotent upper triangular matrices. The quotientNP\Pis called the Levi quotient ofPand is a reductive algebraic group defined over Q, denoted byLP.

To decomposeP, we need to liftLPto a subgroup ofP(see [4, §1]). Letx0=K∈X=G/K be a fixed basepoint. Then there is a unique lift ix0:LP →P such that ix0(LP)is a closed subgroup ofGstable under the Cartan involutionθassociated withK. In the following,ix0 is also denoted byi0for simplicity.

Let NP =NP(R), P =P(R), L=L(R). Let SP be the maximal torus in LP split over Q, i.e., isomorphic toC^× over Q, and let AP =SP(R)⁰, the identity component of the real locusSP(R). Then there is a complementary subgroupMP ofLP defined overQsuch that LP=MPAPMP×AP, whereMP =MP(R).

Under the lifti0:L_P→P, we identityAP, MPwith their imagesi0(AP), i0(MP)inP. Then we have the following (rational) Langlands decomposition ofP:

P=NPMPAPNP×MP×AP, (2.1)

i.e., the map(n, m, a)→nmais a diffeomorphism fromNP×AP×MP toP. The dimension ofAP is called the split rank ofP, and theQ-rank ofGis the maximum of the split rank of all rational parabolic subgroupsPofG.

When the Q-rank of Gis equal to 1, Γ\X is called aQ-rank 1 locally symmetric space.

More generally, the Q-rank of G is also called the Q-rank of Γ\X. For n2, the space SL(n,Z)\SL(n,R)/SO(n)hasQ-rankn−1, and hence is ofQ-rank 1 if and only ifn= 2.

An important classΓ\XofQ-rank 1 consists of Hilbert modular varieties. See [7] for details.

SinceG=P K, the parabolic subgroupP acts transitively onX=G/K, and the Langlands decomposition ofP induces the following horospherical decomposition ofX:

X∼=NP×MP/(K∩P)×AP, (2.2)

where the map is given by(n, a, m(K∩P), a)→namK∈X. Note that this map is well-defined sinceAP, MP commute. In the above identification, we used the fact thatK∩P=K∩MP is a maximal compact subgroup ofMP. DenoteK∩MP byKP, andMP/KP byXP, called the boundary symmetric space associated with the rational parabolic subgroup P. Then the horospherical decomposition can be written as

X=NP×XP×AP. (2.3)

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This is a basic decomposition in this paper and plays an important role in describing the geometry at infinity ofΓ\Xand the continuous spectrum ofΓ\X. WhenX=SL(2,R)/SO(2) is identified with the upper half plane andPconsists of upper triangular matrices,XP reduces to a point,NPcorresponds to thex-coordinates, andAPto they-coordinates. For a generalQ-rank 1 spaceΓ\X,XP is nontrivial. For example, for the Hilbert modular varieties in [7],XP is an Euclidean space.

In the following, the coordinates of a point x∈X in this horospherical decomposition are denoted by

x=

n(x), m(x), a(x) ∈NP×XP×AP.

LetaPbe the Lie algebra ofAP, thenH(x) = loga(x)∈aP, andxis also written as x=

n(x), m(x),e^H(x) .

The arithmetic subgroupΓ induces several subgroups. LetΓNP = Γ∩NP. ThenΓNP is a cocompact discrete subgroup inNP. Under the projectionP→LP, the subgroupΓP= Γ∩P is mapped to an arithmetic subgroup of L_P, which turns out to be contained in M_P. Denote this image inMP byΓX_P and identify it with its lift inP underi0. ThenΓX_P acts properly discontinuously onXPwith a quotient of finite volume. In general,ΓX_P is not torsion free even ifΓis. On the other hand, if we assumeΓto be neat, i.e., every element inΓrealized as a matrix via a linear embedding ofGhas no root of unity except 1 as an eigenvalue, thenΓis torsion free, and furthermore,ΓX_P is also torsion free. It is known that any arithmetic subgroup contains a neat subgroup of finite index. In fact, the neat condition is motivated by this property. (See [1,

§17].) For convenience, we assume in the rest of the paper thatΓis neat. The quotientΓX_P\XP

is a smooth locally symmetric space and called the boundary component ofΓ\Xassociated with the rational parabolic subgroupP.

LetΦ⁺(P, AP)be the set of roots of the adjoint action ofAP onNP. For any real numberr, define

AP,r=

a∈AP|α(loga)> r, for allα∈Φ⁺(P, AP) ,

in particular, when r= 0, AP,r is the positive chamber A⁺_P. The corresponding chamber in aP is denoted by a⁺_P = logA⁺_P. For any bounded set w in NP ×XP, the set w×AP,r

in NP ×XP ×AP X is called a Siegel set of X associated with the rational parabolic subgroupP.

The main result in the reduction theory of arithmetic subgroups can be summarized as follows (see [3],[23]).

PROPOSITION 2.2. – There are only finitely manyΓ-conjugacy classes of (proper) rational parabolic subgroups ofG. LetP1, . . . , Pnbe a set of representatives of these conjugacy classes.

For every r0, and eachPi, there is a bounded set wi =wi(r) such that wi is mapped injectively to a compact set in ΓP\NP ×XP andwi×APi,r is injectively intoΓ\X under the mapπ:X=NP×XP×AP→Γ\X, and there is also a compact setw0=w0(r)inΓ\X such that the following disjoint decomposition holds:

Γ\X=w0∪ n

i=1

π(wi×APi,r).

For convenience, we often identifywi×AP_i,rwith its imageπ(wi×AP_i,r)inΓ\X. Basically, the reduction theory says that the noncompactness ofΓ\Xcomes from the Siegel sets of rational

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parabolic subgroups. For the case of a Riemann surface Γ\H, each Siegel set π(wi×APi,r) corresponds to a cusp neighborhood ofΓ\H. So, in general, we can say that there is a one-to- one correspondence between the “cusps” ofΓ\X andΓ-conjugacy classes of rational parabolic subgroups ofG.

In theQ-rank 1 case, the shape ofΓ\Xnear infinity and the reduction theory can be described more explicitly as follows. In this case, all rational parabolic subgroupsPihave split rank 1, i.e., dimAPi= 1, andΓX_Pi\XPiis a compact locally symmetric space. SinceΓPi,ΓN_Pi andΓX_Pi

fit into an exact sequence0→ΓN_Pi →ΓPi→ΓX_Pi →0, andΓN_Pi is a normal subgroup ofΓPi, the quotientΓPi\(NPi×XPi)is a fiber bundle overΓX_Pi\XPi with fiber equal toΓN_Pi\NPi, which is a compact nilmanifold. The bundleΓP_i\(NP_i×XP_i)admits a flat connection whose (horizontal) sections are images of {n} ×XP_i. Such sections will appear later as parameter spaces for scattering geodesics.

PROPOSITION 2.3. – Assume G is of Q-rank 1, i.e., the Q-rank of Γ\X is equal to 1.

When r0, two points inNP_i×XP_i ×AP_i,r ⊂X areΓ-equivalent if and only if they are ΓP_i-equivalent, and henceΓP_i\NP_i×XP_i×AP_i,r is mapped injectively intoΓ\X under the projectionΓP_i\X→Γ\X. Each subsetΓP_i\NP_i×XP_i×AP_i,r is a topological cylinder with a section equal toΓP_i\NP_i×XP_i, and is an end ofΓ\X. Furthermore, all ends ofΓ\Xare of this form, and hence there exists a compact subsetw0⊂Γ\X such thatΓ\X admits a disjoint decomposition:

Γ\X=w0∪ n

i=1

ΓPi\NPi×XPi×APi,r.

Briefly, the above proposition says that in the Q-rank 1 case, all the ends of Γ\X are topological cylinders, with one end corresponding to oneΓ-conjugacy class of rational parabolic subgroups. On the other hand, we would like to emphasize that each end is not a metric cylinder which is defined to be an isometric product R0×B, where B is a compact Riemannian manifold. In fact, in the decomposition

ΓPi\(NPi×XPi×APi,r) = (ΓPi\NPi×XPi)×APi,r,

when the AP_i,r-component goes to the positive infinity, the fibers ΓN_Pi\NP_i in the bundle ΓP_i\NP_i×XP_ishrink exponentially while the baseΓX_Pi\XP_istays fixed. In a certain sense, an end of aQ-rank 1 locally symmetric spaceΓ\Xis combination of a cusp of a Riemann surface and a (metric) cylindrical end, where the shrinking factorΓN_Pi\NPicorresponds to the horocycle of a cusp, andΓX_Pi\XPicorresponds to a section of a cylindrical end. This combination will also be reflected in the description of the spectral decomposition ofΓ\X, in particular, the continuous spectrum.

Next we study geodesics in Γ\X which run from one “cusp” to another one, the so-called scattering geodesics.

DEFINITION 2.4. – A geodesicγ: (−∞,∞)→Γ\X is called a scattering geodesic if there exist two numbers t1< t2 and two rational parabolic subgroups Pi, Pj such that (1) for all tt2,γ(t)∈wi×AP_i,r in Proposition 2.2 for some larger, and in the decompositionγ(t) = (n, m, a(t)), the componentsn, m are independent of t fortt2, andloga(t) =tH+H0, whereH0∈aP,H∈a⁺_P. (2) The same conditions are satisfied byγ(t)fortt1with respect to the rational parabolic subgroupPj, andloga(t) =−tH+H0. (Note that the minus sign in front ofsis needed sincet→ −∞in this case.) When the above conditions are satisfied,γis called a scattering geodesics between the ends associated withPi, Pj, or a scattering geodesic fromPi

toPj.

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In theQ-rank 1 case, a scattering geodesicγruns from the infinity of one end to the infinity of another end, and inside each end, the coordinate on the sectionΓPi\NPi×XPi is constant.

This latter condition is automatically satisfied for Riemann surfaces. On the other hand, in the generalQ-rank 1 case, there are geodesics going out to infinity of an end which do not satisfy this condition, i.e., the component ofΓX_Pi\XP_i is not constant and they spiral out to infinity.

Therefore, in the generalQ-rank 1 case, scattering geodesics are not exactly the geodesics going from one end to another. But the additional condition imposed in the above definition is natural in view of the following result (see [16]).

PROPOSITION 2.5. – A geodesicγinΓ\X is a scattering geodesic if and only if there exist two numbers t1< t2 such that both rays γ(t), tt2;γ(t), tt1 are distance minimizing in Γ\X, i.e.,γis eventually distance minimizing in both directions.

More importantly, the condition on the scattering geodesics is also natural from the point of view of microlocal analysis, in particular, the wave equation as seen below. In the following, we only consider scattering geodesics between parabolic subgroups of rank 1.

For each parabolic subgroupPof rank 1, i.e.,dimAP = 1, and for every sufficiently large heightr, letYP,r= ΓP\NP×XP× {r}, a section at heightr of the end associated withP.

ThenYP,r is a codimension 1 submanifold inΓ\X. LetN YP,r be the normal bundle ofYP,r. The complement of the zero section inN YP,r has two connected components. Denote the one containing the positive directionA⁺_P byN⁺YP,r, and the other component byN⁻YP,r. Then we have the following characterization of scattering geodesics.

PROPOSITION 2.6. – A geodesicγ(t)is a scattering geodesic from a parabolic subgroupPi

to another parabolic subgroupPj if and only if for every sufficiently larger, there exist t1, t2

such thatγ(t1)∈YPi,r,γ(t1)∈N⁺YPi,r, andγ(t2)∈YPj,r,γ(t2)∈N⁻YPj,r.

Proof. – If γ(t) is a scattering geodesic, it is clear that it satisfies the conditions in the proposition. On the other hand, if γ(t)satisfies the above condition, then a lift γ˜ ofγ(t) in the universal covering space X is of the form ˜γ(t) = (ni, mi, ai(t))∈NP_i ×XP_i ×AP_i in the horospherical decomposition ofX with respect toPi, whereni, mi are independent oft, andlogai(t)is linear int. The reason for these expressions is that every geodesic inX is of the form(n, m,exp(H0+tH))∈NP×XP ×AP for a unique real parabolic subgroupP of G, wheren∈NP,m∈XP, andH∈a⁺_P. Since theAPi component of˜γ(t)goes to+∞when t→+∞, it is clear thatAP_i⊂APandP⊂Pi. The conditionsγ(t˜ 1)∈YP_i,r=NP_i×XP_i×{r}

andγ˜(t1)∈N⁺YPi,r imply that AP =APi and henceP=Pi. By considering the direction t→ −∞, we get a similar expression for another lift˜γ(t)in the horospherical decomposition determined by Pj. These expressions imply that γ(t) is a scattering geodesic as defined in Definition 2.4. ✷

An important invariant of a scattering geodesics is its sojourn time. In the following, all geodesics have unit speed and are directed.

DEFINITION 2.7. – Letrbe a sufficiently large height, as in Proposition 2.2. For a scattering geodesic γ(t) as in Definition 2.4, let t2(r) be the largest numbert such that γ(t)∈YP_i,r

the first time as tdecreases from+∞. Similarly, lett1(r)be the smallest numbert such that γ(t)∈YP_j,r for the first time astincreases from−∞. Then the sojourn time ofγis defined to bet2(r)−t1(r)−2r, and will be denoted byT(γ).

We observe thatt2(r)−t2(r) =t1(r)−t1(r) =r−r and hence t2(r)−t1(r)−2r is independent of the heightr. The sojourn times defined with this modification are not necessarily positive, though they are uniformly bounded from below by−2r.

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In the case of Riemann surfaces, this sojourn time was first introduced by Guillemin [9]. In theQ-rank 1 case, the sojourn time of a scattering geodesic is the length of the geodesic segment between two sectionsYP_i,r, PP_j,r of the ends ofPi, Pj at the heightrmodified by the heightr so that it is independent ofr.

PROPOSITION 2.8. – If X has strictly negative sectional curvature, i.e., the rank of X (or the R-rank ofG)is equal to 1, then there are countably infinitely many scattering geodesics between every pair of ends, and their sojourn times form a discrete sequence of numbers with finite multiplicity inR.

Proof. – Two unit speed geodesicsγ(t), γ(t)inXare defined to be equivalent if

t→lim+∞supd

γ(t), γ(t) <+∞.

LetX(∞)be the set of equivalence classes of geodesics inX, called the sphere at infinity, and X ∪X(∞)the usual geodesic compactification. Since the rank ofX is equal to 1, there is a one-to-one correspondence between real parabolic subgroups ofGand the points inX(∞), and furthermore, for any two distinct points p , qin X(∞), there is a unique geodesic γ(t)in X such thatlimt→+∞γ(t) =p, andlimt→−∞γ(t) =q. For each parabolic subgroupP, denote the corresponding point inX(∞)by[P].

Fix two representativesPi, Pj ofΓ-conjugacy classes of rational parabolic subgroups as in Proposition 2.2. For any two rational parabolic subgroups P_i, P_j which are Γ-equivalent to Pi, Pj, letγ(t)be the unique geodesic connecting the points[P_i],[P_j]as above. Then the image of γ in Γ\X is a scattering geodesic from the end ofPi to the end ofPj. Conversely, any scattering geodesic from the end ofPito that ofPj is of this form.

To show that the sojourn times form a discrete sequence, we denote the unique geodesicγ(t)in X connecting[P_i],[P_j]byP_i, P_j. Then every scattering geodesic connecting the ends ofPi, Pj

is the image inΓ\Xof one of the geodesics in the union

γ∈Γ_Pi\Γ

γ⁻¹Piγ, Pj.

Asγruns over the cosets and off to infinity, it can be shown that the sojourn time of the geodesic γ⁻¹Piγ, Pjgoes to infinity. The reason is that the sojourn time ofγ⁻¹Piγ, Pjis greater than the norm of theaP_i-component ofγin the Langlands decomposition, up to a constant independent ofγ. By the proof of absolute convergence of Eisenstein series for large parameters (see [11,

§2]), these norms form a discrete sequence. This implies that the sojourn times of the scattering geodesics between the ends ofPiandPj form a discrete sequence. SinceΓ\X has only finitely many ends, the discreteness of the spectrum of the sojourn times ofΓ\Xfollows. ✷

An immediate corollary of the above proposition is that if theR-rank ofGis equal to 1, the scattering geodesics ofΓ\X are isolated. On the other hand, if theR-rank ofGis greater than one, scattering geodesics are often embedded in smooth, non-discrete families.

PROPOSITION 2.9. – Assume that theQ-rank of Gis equal to 1. Let γ(t) be a scattering geodesic inΓ\Xbetween the ends associated with two rational parabolic subgroupsPiandPj. Thenγ(t)lies in a smooth family of scattering geodesics of the same sojourn time parametrized by a common finite covering spaceXT,ij of the boundary locally symmetric spacesΓX_Pi\XPi

andΓX_Pj\XPj.

Proof. – First, we assume that a liftγ(t)˜ inX ofγ(t)is of the form˜γ(t) = (id, m, a(t))∈ NP_i×XP_i×AP_i,r, wherem∈XP_i,a(t) = expH0+tH, H∈a⁺_P

i. LetP_i⁻ be the opposite

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parabolic subgroup of Pi with respect to the split componentAPi, i.e.,P_i⁻=N_P⁻_iMPiAPi, where the Lie algebra of N_P⁻_i is equal to sum of the root spaces of the roots −α, for α∈Φ⁺(P, APi). Since theNPi component ofγ(t)˜ is trivial, the horospherical coordinates of

˜

γ(t)forPiare also the horospherical coordinates forP_i⁻. Therefore,γ(t)˜ is a scattering geodesic betweenPiandP_i⁻. By assumption,γ(t)˜ is a scattering geodesic betweenPiand aΓ-conjugate of Pj. This implies thatP_i⁻ is a rational parabolic subgroup, hence the lifted Levi quotient i0(MP_iAP_i), which is also denoted byMP_iAP_i, is given byMP_iAP_i=Pi∩P_i⁻and hence is a rational subgroup. In particular, the lifted subgroupMP_i=ix₀(MP_i)is defined overQ, and the split componentAPiis also rational.

The above argument shows that in the equation˜γ(t) = (id, m, a(t)), the componentmcan be changed to another point inXPi andγ(t)˜ is still a scattering geodesic betweenPiandP_i⁻. It is clear that the sojourn time of γ(t)˜ does not depend onm. The projections of two such geodesicsγ˜1(t) = (id, m1, a(t)),˜γ2(t) = (id, m2, a(t))inΓ\Xdefine the same geodesic if and only if there exists an elementγ∈Γsuch thatγm1=m2. This implies thatγ∈MP_iand hence γ∈Γ∩MP_i= ΓP_i∩MP_i. Therefore, the projectionγ(t)of˜γ(t) = (id, m, a(t))inΓ\X only depends on the image ofmin the quotientΓP_i∩MP_i\XP_i, and hence the scattering geodesic γ(t)belongs to a smooth family parametrized byΓP_i∩MP_i\XP_i. The spaceΓP_i∩MP_i\XP_i

can be identified with the image of{id} ×XPi in the bundleΓPi\NPi×XPi and hence is a horizontal section of this bundle (see the comments before Proposition 2.3). This implies that ΓPi∩MPi\XPiis a covering space of the baseΓX_Pi\XPiof this bundle. SinceΓis torsion free, ΓPi∩MPi is also torsion free, and henceΓPi∩MPi\XPi is a smooth manifold which we will denote byXT,ij.

As mentioned earlier, MP_i is a rational subgroup ofPi. This implies that ΓP_i ∩MP_i is an arithmetic subgroup of MP_i and hence a subgroup of finite index of ΓX_Pi, which is defined earlier as the image ofΓP_iinMP_iunder the projection defined by the Langlands decomposition.

Therefore,ΓP_i∩MP_i\XP_i is a finite covering space ofΓX_Pi\XP_i. SincePj isΓ-conjugate to P_i⁻andM_P−

i =MP_i, it follows thatXP_j=XP_i andΓP_j∩MP_j= Γ∩MP_j = ΓP_i∩MP_i, and ΓPj ∩MPj\XPj can be identified withΓPi∩MPi\XPi. This implies thatΓPi∩MPi\XPi is a finite common covering space ofΓX_Pi\XPi andΓX_Pj\XPj. The proposition is proved in this case.

On the other hand, suppose that a lift ˜γ(t) has a non-trivial NP_i-component n, γ(t) =˜ (n, m, a(t)). Let x1=nx0 be a new basepoint. Then with respect to this basepoint x1, the Langlands decomposition ofPibecomes

Pi=NPi×

nMPin⁻¹ ×

nAPin⁻¹ . In the induced horospherical decomposition ofX:

X=NP_i×nMP_in⁻¹/

nKP_in⁻¹ ×nAP_in⁻¹, the components of˜γ(t)are given by

˜ γ(t) =

id,nmn⁻¹

nKP_in⁻¹ ,na(t)n⁻¹ .

Since the NPi-component is trivial, by the previous argument,˜γ(t) is a scattering geodesic betweenPiand opposite parabolic subgroup ofPiwith respect to this split componentnAPin⁻¹. If we denote the opposite parabolic subgroup of Pi with respect to the fixed split component AP_i as above byP_i⁻, then this opposite parabolic subgroup with respect to the split component nAP_in⁻¹ is equal to nP_i⁻n⁻¹. By assumption, γ(t) is a scattering geodesic between two

e ◦

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rational parabolic subgroups Pi and Pj. This implies that nP_i⁻n⁻¹ is also rational and Γ- conjugate to Pj. This in turn implies that the split componentnAPin⁻¹ is rational, and the liftix₁(MP_i) =nMP_in⁻¹is also defined overQ.

As in the previous case, theMP_i component ofγ(t)˜ can be moved inXP_i andγ(t)˜ is still a scattering geodesic of the same sojourn time. Two such geodesics(n, m1, a(t))and(n, m2, a(s)) inX project to the same geodesic inΓ\Xif and only if there exists an elementγ∈Γsuch that γnm1=nm2. This implies that n⁻¹γn∈n⁻¹Γn∩MP_i. Therefore, the image in Γ\X of a geodesic(n, m, a(t))inX only depends on the image ofminn⁻¹Γn∩MP_i\XP_i. This shows thatγ(t)belongs to a smooth family of scattering geodesics parametrized byn⁻¹Γn∩MPi\XPi. As above, sinceΓ is torsion free,n⁻¹Γn∩MPi is also torsion free and hence the parameter space n⁻¹Γn∩MPi\XPi is a smooth manifold. This parameter space can be identified with the image of the horizontal section{n} ×XPi in the bundleΓPi\NPi×XPi and hence is a covering space of the baseΓX_Pi\XP_i. In fact it is a finite covering space ofΓX_Pi\XP_i. First note that n⁻¹Γn∩MP_i\XP_i can be identified with Γ∩nMP_in⁻¹\nMP_in⁻¹/(nKP_in⁻¹).

SincenMP_in⁻¹is a rational subgroup, the intersectionΓ∩nMP_in⁻¹is an arithmetic subgroup in nMP_in⁻¹, and hence n⁻¹Γn∩MP_i is a cofinite discrete subgroup of MP_i. This proves the finiteness of the covering n⁻¹Γn∩MP_i\XP_i →ΓX_Pi\XP_i. The corresponding smooth manifold will be denoted byXT,ij.

SincePj isΓ-conjugate to nP_i⁻n⁻¹andn⁻¹Γn∩MPi\XPi∼= Γ∩nMPin⁻¹\XPi can be identified withΓ∩MPj\XPj, it follows that the parameter spacen⁻¹Γn∩MPi\XPi is also a common finite covering space of ΓX_Pi\XPi andΓX_Pj\XPj. This completes the proof of this proposition. ✷

Remark. – If a sojourn time has a multiplicity, then we will have different manifoldsXT,ijfor each different family of geodesics with the same sojourn time. To fix an identification ofXT,ij, we will identify it with the quotient ofXP_ias in the proof of the proposition.

Remark. – In this paper, we fixed a basepointx0=K∈X and identified the subgroupMP in the Levi quotient of a rational parabolic subgroupPwith its liftix0(MP). In general,ix0(MP) is not a rational subgroup. But for any rational parabolic subgroup P, we can always find a basepointx1such thatix₁(MP)is rational. The existence of such a basepoint can be seen from the proof of the above proposition.

Suppose thatix₀(MP)is rational. Then the set of connected components of the continuous families of scattering geodesics coming out of the parabolic subgroupP are parametrized by ΓN_P\NP(Q). Precisely, letm0=KP be a basepoint inXP=MP/KP. LetH be the unique unit vector ina⁺_P. Then the proof of the above proposition shows that for everyn∈NP(Q), the geodesic(n, m0,exptH)inX projects to a scattering geodesic inΓ\X coming fromP. Two such scattering geodesics belong to one connected family if and only if theirNPcomponents are in the sameΓNP orbit. Furthermore, every scattering geodesic coming out ofP is of this form.

This parametrization of scattering geodesics coming out of a cusp can be seen clearly in the case of Riemann surfaces.

Each connected family of scattering geodesics between two rational parabolic subgroups or two ends ofΓ\Xhas a common sojourn time. All these sojourn times form a spectrum of sojourn times. In the following, we say a geodesic inXis a scattering geodesic if its image inΓ\X is a scattering geodesic, and its sojourn time is equal to the sojourn time of the image.

PROPOSITION 2.10. – Assume thatQ-rank ofΓ\X is equal to1as above. Then for any two (not necessarily different) ends ofΓ\X, there are countably infinitely many smooth families of scattering geodesics between them, and the spectrum of sojourn times of all scattering geodesics forms a discrete sequence of points inRof finite multiplicities.