Document [original]

Universität Paderborn

Fakultät für Elektrotechnik, Informatik und Mathematik

Université Paul Verlaine Metz

École Doctorale, IAEM Lorraine

Patterson-Sullivan distributions

for symmetric spaces

of the noncompact type

Dissertation

Thèse de doctorat

vorgelegt von

Michael Schröder

27. Mai 2010

Abstract

There is a curious relation between two kinds of phase space distributions asso-

ciated to Laplace-eigenfunctions ϕλkon a compact hyperbolic manifold Y.

Given a pseudodifferential operator quantization Op :C∞(S∗Y)→B(L2(Y)),

that is an assignment of bounded operators to smooth zero order symbols aon

the unit (co-)tangent bundle S∗Y, the functionals ρλj,λk(A) = hAϕλj, ϕλkiL2(Y)

on the space of zero-order pseudodifferential operators give rise to Wigner distri-

butions Wλj,λk(a) = ρλj,λk(Op(a)) on S∗Y, which are the key objects in quantum

ergodicity. One studies the oscillation and concentration properties of the eigen-

functions through the so-called large energy limits of the distributions Wλj,λk,

that is one investigates their behaviour when the eigenvalues tend to infinity.

If Yis a symmetric space of the noncompact type, the Laplace operator

is replaced by the corresponding algebra of translation invariant differential

operators. Given moderate eigenfunctions ϕand ψ, their distributional boudary

values in the sense of Helgason give rise to the Patterson-Sullivan distribution

PSϕ,ψ on S∗Y.

In the case of compact hyperbolic surfaces Y= Γ\Hit was observed by N.

Anantharaman an S. Zelditch that there is an exact and an asymptotic relation

between these phase space distributions.

We generalize parts of a special non-Euclidean calculus of pseudodifferential

operators, which was invented by S. Zelditch for hyperbolic surfaces, to sym-

metric spaces X=G/K of the noncompact type and their compact quotients

Y= Γ\G/K. We sometimes restrict our results to the case of rank one sym-

metric spcaes. The non-Euclidean setting extends the defintion of Patterson-

Sullivan distributions in a natural way to arbitrary symmetric spaces of the

noncompact type. Generalizing the exact formula given by Zelditch and Anan-

tharaman, we find an explicit intertwining operator mapping Patterson-Sullivan

distributions into Wigner distributions. We study the important invariance and

equivariance properties of these distributions. Finally, we describe asymptotic

properties of these distributions.

Zusammenfassung

Es gibt eine interessante Beziehung zwischen zwei Familien von Distributionen,

welche zu Eigenfunktionen ϕλkdes Laplace-Operators einer kompakten hyper-

bolischen Mannigfaltigkeit Yassoziiert werden:

Gegeben eine Pseudodifferentialoperatoren-Quantisierung, d. h. eine Vorschrift

Op :C∞(S∗Y)→B(L2(Y)), die Symbolen ader Ordnung 0auf dem Kosphären-

bündel L2-beschränkte Operatoren auf Yzuweist, so erhält man aus den Funk-

tionalen ρλj,λk(A) = hAϕλj, ϕλkiL2auf den Raum der Pseudodifferentialope-

ratoren nullter Ordnung die Wigner-Distributionen Wλj,λk(a) = ρλj,λk(Op(a))

auf dem Kosphärenbündel S∗Y. Diese sind die Schlüsselobjekte der Quanten-

Ergodizität: Man studiert die Schwingungs- und Konzentrationseigenschaften

der Eigenfunktionen, indem man das Hochfrequenzverhalten der Distributionen

Wλj,λkuntersucht, d.h. wenn die Eigenwerte gegen unendlich streben.

Falls Yein symmetrischer Raum nichtkompakten Typs ist, so wird der Laplace-

Operator durch die gesamte Algebra der invarianten Differentialoperatoren er-

setzt. Gegeben moderate Eigenfunktionen ϕund ψauf Y, so liefern ihre Helgason-

Randwerte sogenannte Patterson-Sullivan Distributionen PSϕ,ψ auf S∗Y.

Im Falle kompakter hyperbolischer Flächen Y= Γ\Hbeobachteten N. Anan-

tharaman und S. Zelditch eine exakte und eine asymptotische Beziehung zwi-

schen diesen Distributionen.

Wir verallgemeinern Teile eines speziellen nicht-euklidischen Kalküls von Pseu-

dodifferentialoperatoren, welcher zuerst von S. Zelditch für hyperbolische Flä-

chen eingeführt wurde, auf symmetrische Räume X=G/K nichtkompakten

Typs und ihre kompakten Quotienten Y= Γ\G/K. Wir werden uns bei einigen

Resultaten auf den Fall von Räumen vom Rang eins beschränken. Das nicht-

euklidische Setting erweitert die Definitionen der Patterson-Sullivan Distribu-

tionen auf natürliche Weise auf symmetrische Räume nichtkompakten Typs.

Wir verallgemeinern die exakte Beziehung zwischen diesen und den Wigner-

Distributionen und studieren die wichtigen Eigenschaften der Patterson-Sullivan

Distributionen. Schließlich beschreiben wir asymptotische Verbindungen zwi-

schen verschiedenen Arten von Distributionen.

Acknowledgements

First of all I wish to express my sincere thanks to my supervisor Joachim Hilgert.

I thank him for leaving to me this interesting and challenging topic, and for giv-

ing me the chance to be a part of the IRTG, as which I was able to look beyond

the scenes of mathematics. I thank him for his enduring help, his patience, and

for making me a member of his friendly working group. I express my thanks

to Angela Pasquale for her help and for being a kind and helpful advisor, and

for a very pleasant stay in Metz. Many thanks go to my longstanding men-

tor Sönke Hansen for always having sympathetic ear and an open door for me.

Special thanks go to Martin Olbrich for his expertise and experience in very

difficult questions. In particular, I want to thank Nalini Anantharaman and

Steve Zelditch for helpful discussions on further issues, problems, and research

projects. I thank my colleagues from the IRTG and the Department of Mathe-

matics of the University of Paderborn and last but not least, I thank my friends

in Paderborn for their insight and for their support.

Contents

1 Introduction 8

2 Preliminaries 15

2.1 Symmetric spaces and real semisimple Lie groups ........ 16

2.2 Geodesics, horocycles, and the boundary at infinity ....... 26

2.3 Invariant differential operators ................... 35

2.4 The classical examples ....................... 42

3 Component computations 49

3.1 Some integral formulas ....................... 49

3.2 Derivatives corresponding to the Iwasawa decomposition . . . . 52

3.3 Critical sets and Hessian forms .................. 56

4 Equivariant pseudodifferential operators on symmetric spaces 66

4.1 Non-Euclidean Fourier analysis .................. 67

4.2 Invariance and equivariance properties .............. 69

4.3 Classes of symbols ......................... 72

4.4 The Kohn-Nirenberg operator ................... 75

4.5 Conjugation by a wave group-type operator ........... 87

5 Helgason boundary values 91

5.1 Poisson transform and principal series representations ...... 93

5.2 Regularity of distributional boundary values ........... 95

5.3 Tensor products of distributional boundary values ........ 99

6 Patterson-Sullivan distributions 102

6.1 Intermediate values ......................... 102

6.2 Definitions and invariance properties ............... 108

6.3 The Knapp-Stein intertwining operators ............. 113

6.4 An integral formula ......................... 116

6.5 Eigenfunctions on a compact quotient ............... 118

6.6 The spectral order principle .................... 122

Bibliography 127

Index 131

1 Introduction

Quantum ergodicity is a subfield of mathematics combining dynamical systems

and microlocal analysis to investigate the global topography of eigenfunctions

of the Laplace-Beltrami operator on Riemannian manifolds.

We begin by describing how the questions of quantum ergodicity are inte-

grated in the greater picture of science. Then we give a brief summary of the

basic definitions which are important in quantum ergodicity, and we list a cou-

ple of simple properties of the objects we want to investigate. It is important

to collect these things in this introduction to motivate the concrete results of

this work. We use definitions from the overview articles [Zel05], [Zel09a] and

[Zel09b], and we also follow the descriptions in [Zel87], [BO05] and [SV].

Background

Let (M, g)denote a (compact) Riemannian manifold with metric g. We denote

by ∆ = ∆gthe corresponding positive Laplace-Beltrami operator

∆ = −1

p|det gij|

i,j=1

∂

∂xigijq|det gij|∂

∂xj,

where gij =g(∂

∂xi,∂

∂xj)and where gij is the inverse matrix to gij. The starting

point is the eigenvalue problem

∆ϕλ=λ2ϕλ, λ ∈R.(1.1)

In the compact case, the spectrum of ∆is discrete and we arrange the eigen-

values in non-decreasing order λ0≤λ1≤λ2≤. . . → ∞. We denote by ϕλj

an orthonormal basis of real-valued eigenfunctions with respect to the inner

product hϕλj, ϕλki=RMϕλj(x)ϕλk(x)dx, where dx denotes the volume density.

The eigenvalue problem on Mis dual under the Fourier transform to the wave

equation. We denote the eigenvalues by λ2, which saves us from writing a few

square root signs. We will later (in the other chapters) often consider the usual

defintion of the Laplace-Beltrami operator, that is we will consider −∆instead

of ∆.

Eigenfunctions of Laplace operators arise in physics as modes of periodic vi-

bration of drums and membranes. They can also represent stationary states of

a free quantum particle on a Riemannian manifold. More generally, eigenfunc-

tions of Schrödinger operators represent stationary energy states of atoms and

molecules in quantum mechanics.

In mathematics, studies of eigenfunctions tend to fall into two categories:

•The analysis of ground states, i.e. ϕ0or ϕ1. An eigenfunction is always the

ground state Dirichlet eigenfunction in any of its nodal domains. Other

questions in the spectral theory of the Laplacian concern estimates for the

least positive eigenvalue (for example, see [U80]).

1 Introduction

•The analysis of high frequency limits (semi-classical limits) of eigenfunc-

tions, i.e. the limit as the eigenvalue tends to infinity.

Our emphasis is on the high frequency behavior of eigenfunctions. Studies of

high frequency behavior eigenfunctions also fall into two categories:

•Local results, which often hold for any solution of (1.1) on a (small) ball

Br(x), irrespective of whether the eigenfunction extends to a global eigen-

function.

•Global results for eigenfunctions that extend to M. A typical global as-

sumption is that the eigenfunctions are also eigenfunctions of the wave

group.

We are interested in global properties of eigenfunctions. These generally re-

flect the relation of the wave group and geodesic flow, particularly the long time

behavior of waves and geodesics on the manifold.

The general approach to understand the global behavior of eigenfunctions is

to do a phase space analysis, where the phase space is the co-tangent bundle

T∗Mor an energy surface S∗M. We often identify T∗Mand TM using the

metric. For example, one often wishes to construct highly localized eigenfunc-

tions or approximate eigenfunctions (quasi-modes) of ∆or to prove that they

do not exist. To obtain global phase space results relating the behavior of eigen-

functions to the behavior of geodesics, it is necessary to use microlocal analysis,

i.e. the calculus of pseudo-differential operators. Microlocal analysis is a math-

ematically precise formulation of the semi-classical limit in quantum mechanics.

Pseudo-differential operators are quantizations Op(a)of functions on the phase

space T∗M: The classical pseudodifferential operators Op(a)on Rnare defined

by the action on exponentials:

Op(a)eihx,ξi=a(x, ξ)eihx,ξi.

The symbol a(x, ξ)has order m∈Rif supK(1 + |ξ|)j−m|Dα

xDβ

ξa(x, ξ)|<∞for

all compact sets Kand all α, β, j. Symbol classes can also be defined locally and

the definition of pseudodifferential operators can be extended to manifolds. A

symbol is called polyhomogeneous if it admits a classical asymptotic expansion

a(x, ξ)∼∞

j=0

am−j(x, ξ),

where the alare homogeneous in |ξ| ≥ 1of order l. We call the leading term

σOp(a):= amthe principal symbol of Op(a). By Ψmwe denote the space of

classical pseudodifferential operators on Mof order m. We have the exact

sequence of algebras 0→Ψ−1→Ψ0σ

→C∞(SM)→0, where σis the principal

symbol map. A right-inverse of σmapping homogeneous symbols of order 0into

1 Introduction

L2-bounded operators is called quantization or operator convention. Functions

on S∗Mare also called observables.

The cotangent bundle is equipped with the symplectic form Pidxi∧dξi. The

metric defines the Hamiltonian vector field

H(x, ξ) = |ξ|g=v

i,j=1

gij(x)ξiξj

on T∗M. The classical evolution is given by the geodesic flow of (M, g), i.e.

the Hamiltonian flow gtof Hon T∗M: By definition, gt(x, ξ) = (xt, ξt), where

(xt, ξt)is the terminal tangent vector at time tof the unit speed geodesic starting

at xin the direction ξ. The Liouville measure µLon S∗Mis by definition the

measure dµL=dxdξ

dH induced by the Hamitonian and the symplectic volume

element dx dξ on T∗M. The geodesic flow preserves the Liouville measure. We

can thus define a unitary operator Vton L2(S∗M, dµL)by

Vt(a) := a◦gt.

The operator Vtis called the translation operator associated to the geodesic

flow. The geodesic flow is called ergodic, if Vthas no invariant L2-functions

besides the constants. Equivalently, the geodesic flow is called ergodic, if any

invariant set E⊆S∗Mhas either zero measure or full measure.

The quantization of the Hamiltonian is the square root √∆of the positive

Laplacian. Quantum evolution is given by the wave group

Ut=eit√∆.

It is generated by the pseudodifferential operator √∆as defined by the spectral

theorem: It has the same eigenfunctions as ∆, but to the eigenvalues λ.

Evolution of observables is known in physics as the ’Heisenberg picture’. It

is defined by

αt(A) = UtAU−t, A ∈Ψm.

Egorov’s theorem yields a correspondence to the classical evolution Vt(a) =

a◦gt. It says that αtis an order preserving automorphism on the space of

pseudodifferential operators, that is αt(A)∈Ψmfor all A∈Ψmand that

σαt(A)(x, ξ) = σA(gt(x, ξ)) = Vt(σA).

This formula is almost universally taken to be the definition of quantization of

a flow or map in the physics literature.

In quantum ergodicity, one studies the concentration and oscillation proper-

ties of eigenfunctions through the linear functionals

ρλj(A) = hAϕλj, ϕλji

1 Introduction

on the space of zeroth order pseudo-differential operators A. The possible limits

of the family ρλjare called quantum limits or microlocal defect measures. The

diagonal elements ρλj(A)are interpreted in quantum mechanics as the expected

value of the observable Op(a)in the energy state λj. The off-diagonal matrix

elements

ρλj,λk(A) = hAϕλj, ϕλki

are interpreted as transition amplitudes between states. We fix a quantization

a→Op(a). The matrix elements are then also called Wigner distributions:

Wλj,λk(a) = ρλj,λk(Op(a)).

We first observe that ρλj,λk(I) = δj,k (Kronecker-Delta), since the eigenfunctions

are orthonormal in L2(M). In the diagonal case, the functionals ρλkare positive

in the sense that for any operator Awe have ρλk(A∗A)≥0. This can be seen by

moving A∗to the right side in the L2-inner product. Writing out ρλj,λk(UtAU−t)

and moving Utto the right side we find

ρλj,λk(UtAU−t) = eit(λk−λj)ρλj,λk(A).

These properties are summarized by saying that ρλjis an invariant state on (the

closure in the operator norm of) the algebra Ψ0.

Let Qdenote the set of possible quantum limits. Any orthonormal basis such

as {ϕλk}tends to 0weakly in L2. Hence {Kϕλk}tends to 0weakly in L2for

each compact operator K. Then, the diagonal elements ρλj(K)tend to 0for

all compact K. Given two pseudodifferential operators on Mwith the same

principal symbol of order zero, their difference is an operator of negative order

and thus compact. It follows that Qis independent of the choice of quantization.

Using standard estimates on pseudodifferential operators one shows ([Zel09a],

§6) that any weak limit is continuous on C(S∗M). It is a positive functional

since each ρλkis and hence any limit is a probability measure.

By the invariance of the ρλkunder the automorphisms αton Ψ0and by

Egorov’s theorem we find that any limit of ρλk(A)is a limit of ρλk(Op(σA◦gt)),

and hence the limit is invariant under the geodesic flow gt.

It follows from ρλj,λk(UtAU−t) = eit(λk−λj)ρλj,λk(A)that the off-diagonal

matrix elements can only have a limit for subsequences {λjn}and {λkn}of

eigenvalue-parameters such that the spectral gap |λjn−λkn|tends to a limit

τ∈R. In that case, each limit µis an eigenmeasure for the geodesic flow:

µ(a◦gt) = eitτ µ(a).

A measure is called invariant under time-reversal, if it is invariant under the

anti-symplectic involution (x, ξ)→(x, −ξ)on T∗M. Since the eigenfunctions

are (by our assumption) real-valued and hence complex-conjugation invariant,

it follows that any quantum limit is invariant under time-reversal.

1 Introduction

From the mathematical point of view, one would like to know the behavior

of the diagonal matrix elements and the off diagonal matrix elements, when the

eigenvalue tends to infinity. One of the principal problems is:

Problem 1.1. Determine the set Qof quantum limits.

As a motivating example, suppose that for a subsequence kjthe functionals

ρkjtend to the Liouville measure µL. Let E⊆Mdenote a measureable set

whose boundary has measure zero. Testing against multiplication operators

(with symbols given by smoothed versions of the characteristic function of E)

yields ([Zel09b], p. 19)

vol(M)ZE|ϕkj(x)|2dx →vol(E)

vol(M).

We interprete |ϕkj(x)|2dx as the probability density of finding a particle of en-

ergy λ2

kjin E. Then this sequence of probabilities tends to uniform measure

and the eigenfunctions become uniformly distributed on M. However, the as-

sumption ρkj→µLis much stronger, since then

hOp(1E)ϕkj, ϕkji → µL(π−1(E))

µL(S∗M),

where π:S∗M→Mis the natural projection. The Laplacian of (M, g)is

said to be QUE (quantum uniquely ergodic) if the only quantum limit for any

orthonormal basis of eigenfunctions is the Liouville measure. The following

conjecture was first stated by Rudnick-Sarnak ([RS94]):

Conjecture 1.2. Let (M, g)be a negatively curved manifold. Then ∆is QUE.

Off-diagonal matrix elements are also important as transition amplitudes be-

tween states. As described above, a sequence of such matrix elements cannot

have a weak limit unless the spectral gap tends to a limit τ. We denote the

corresponding set of limits by Qτ. Then we can also formulate:

Problem 1.3. Determine the set Qτof off-diagonal quantum limits.

For examples of possible quantum limits we refer to the overview articles

[Zel09a] and [Zel09b], which also describe recent developments of mathematical

quantum chaos such as mixing properties of eigenfunctions, boundary quantum

ergodicity, converse quantum ergodicity, and other problems.

Outline and statement of results

Let X=G/K denote a symmetric space of the noncompact type, where Gis

a connected semisimple Lie group with finite center and Ka maximal compact

subgroup of G. In Section 2we recall basic definitions concerning symmetric

spaces and we give detailed descriptions of their geometry. Our setting is as

follows: Let G=KAN be an Iwasawa decomposition of Gand let Mdenote

the centralizer of Ain K. The geodesic boundary of Xcan be identified with

the flag manifold B:= K/M. Let o:= K∈G/K denote the origin of the

symmetric space X. We fix a cocompact and torsion free discrete subgroup Γ

of G. Let ∆, resp. ∆Γ, denote the Laplace operator of X, resp. XΓ.

In [Zel86], S. Zelditch introduced a natural pseudodifferential operator con-

vention for G/K, when G=PSU(1,1),K=PSO(2). In Section 4we gener-

alize this calculus to symmetric spaces of the noncompact type. We sometimes

restrict our results to rank one spaces. The interesting aspect of this calculus

is its G-equivariance: Let SX denote the unit tangent bundle of X=G/K. If

a∈C∞(SX)is Γ-invariant under the natural action of Gon SX, then it yields

a pseudodifferential operator on the quotient XΓ:= Γ\G/K. We can hence use

the G-equivariant non-Euclidean pseudodifferential calculus to define Wigner

distributions on the quotient XΓ= Γ\G/K.

If Yis a manifold, ua distribution or hyperfunction on Yand ϕa test func-

tion, then we denote the pairing hϕ, uiYby RYϕ(y)u(dy). The starting point of

all following observations is Helgason’s representation theorem for joint eigen-

functions of the algebra D(G/K)of translation invariant differential operators:

Given a joint eigenfunction ϕ∈Eλ(X)(see Section 5), then there is a linear

functional Tϕon the space of analytic functions on Bsuch that ϕis given by

the Poisson-Helgason-transform ϕ(z) = he(iλ+ρ)hz,bi, TiB=RBe(iλ+ρ)hz,biT(db).

Here, the function eλ,b := e(iλ+ρ)hz,bidenotes a generalized Poisson kernel (see

Section 2.3).

In Section 5we describe the theory of Helgason boundary values. In partic-

ular, we describe their regularity as a function of the spectral parameter λ∈a,

where ais the Lie algebra of A.

Wigner distributions tend to measures with certain invariance properties. The

question arises whether there exist distributions constructed from eigenfunc-

tions which are related to the Wigner-distributions and which already possess

these invariance properties. For hyperbolic surfaces, such distributions were

constucted by N. Anantharaman and S. Zeldirch in [AZ07]. These distributions

were termed Patterson-Sullivan distributions by analogy with their construction

of boundary measures associated to ground states on infinite volume hyperbolic

manifolds ([Sul79]): The Patterson-Sullivan distribution associated to a real

eigenfunction ϕir corresponding to the eigenvalue 1/4 + r2and with associated

boundary values Tir is the distribution on B(2) (the space consisting of distinct

1 Introduction

boundary points b, b0∈B) defined by

psir(db, db0) := Tir(db)Tir(db0)

|b−b0|1+2ir .(1.2)

The interesting aspect of quotients XΓlies in the study of Γ-invariant eigenfunc-

tions on the original symmetric space: If the eigenfunction is Γ-invariant, then

the corresponding Patterson-Sullivan distribution is Γ-invariant and invariant

under time reversal. To obtain a geodesic flow invariant distribution PSir on

SX, Anantharaman and Zelditch tensor with dt. They also define normalized

Patterson-Sullivan distributions by dividing by the integral against 1. The result

is a geodesic flow invariant distribution c

PSir constructed as a quadratic expres-

sion in the eigenfunctions. Anantharaman and Zelditch then proved that there

is an explicit intertwining operator Lir mapping Patterson-Sullivan distributions

into Wigner distributions.

We explain how to generalize these definitions to symmetric spaces of the

noncompact type: Following [Eber96] we say that two distinct boundary points

b, b0∈Bcan be joint at infinity if there is a geodesic in Xwith forward endpoint

band backwards endpoint b0. We describe in Section 2the open dense subset

B(2) of distinct boundary points that can be joint at infinity. It turns out

that this space is invariant under the action of Gon Band identifies with the

homogeneous space G/MA. We introduce functions dλon B(2) and a geodesic

Radon transform R:C∞

c(SX)→C∞

c(B(2))such that the expression

ha, PSλiSX := ZB(2)

dλ(b, b0)R(a)(b, b0)Tλ(db)Tλ(db0)(1.3)

defines a Γ-invariant distribution on SX, and this is the generalized Patterson-

Sullivan distribution associated to the eigenfunction ϕ∈Eλ(Sec. 2.3). The

PSλare invariant under the geodesic flow and under time reversal. The weight

functions dλwill be called intermediate values because they satisfy a certain

equivariance property, which generalizes a so-called intermediate values formula

for hyperbolic surfaces (Sec. 6).

As was pointed out in the introduction of [AZ07] it is of interest to also have

analogous definitions for off-diagonal matrix entries. We will in fact also consider

these off-diagonal elements and off-diagonal Patterson-Sullivan distributions:

In Section 6we use off-diagonal intermediate values dλ,µ on B(2). Given joint

eigenfunctions ϕand ψwe then introduce general off-diagonal Patteson-Sullivan

distributions on SX.

The point is that all Patterson-Sullivan distributions we consider are Γ-

invariant. We show how this lets the definitions descend to quotients XΓ. In

order to generalize the above mentioned results for hyperbolic surfaces, we will

find an explicit intertwining operator that maps off-diagonal Patterson-Sullivan

distributions into non-Euclidean Wigner distributions.

2 Preliminaries

A Riemannian manifold Xis a called a homogeneous space if its group of Rie-

mannian isometries acts transitively on X. We consider a point xof a connected

Riemannian manifold X. Let Udenote a symmetric neighborhood of Min the

tangent space of xsuch that the exponential map is well-defined on Uand a

diffeomorphism onto its image V. The symmetry u7→ −uof Uinduces a map

sxon V, which we call the local geodesic symmetry centered at x. We say that

Xis a Riemannian locally symmetric space if for any xin Xthe corresponding

local symmetry at xis a local isometry of X. We say that Xis globally sym-

metric for any xthis isometry may be extended uniquely to X. A complete

simply connected locally symmetric space is globally symmetric. In this sense,

globally symmetric spaces are complete spaces which possess a very large group

of isometries. In particular, their group of isometries acts transitively. We recall

material from [DS] and [Eber96] for some background.

A globally symmetric space Xis the Cartesian Riemannian product of three

globally symmetric spaces X=Rn×D×T(de Rham decomposition), where

Dhas nonpositive curvature, where Thas nonnegative curvature, and where D

and Tmay not be written as a product of Rwith another Riemannian manifold.

We say that Dis of noncompact type and Tis of compact type. We will be

interested in symmetric spaces of the noncompact type.

The structure of Riemannian symmetric spaces is intrinsically linked with

the theory of Lie groups: Let Gdenote the isometry group of the connected

Riemannian manifold X. For a compact subset Cof Xand an open subset Uof

Xput W(C, U) := {g∈G:g·C⊂U}. The compact open topology is defined

as the smallest topology on Gfor which all the sets W(C, U)are open. For

this topology, Gis Hausdorff, separable, locally compact and second countable.

If Xis globally symmetric, Gcan be proved to carry a structure of Lie group

compatible with this topology. Let G0denote the identity component of G,

select a point p∈Xand denote by Kthe subbgroup of G0which stabilizes p.

Then Kis a maximal compact subgroup of G0and G0/K is isometric to X. On

the other hand, given a connected Lie group G0and a closed subgroup Kof G0,

we call (G0, K)a Riemannian symmetric pair if the group AdG0(K)is compact

and if there exists an involutive smooth automorphism σof G0, which is not

the identity, such that (Kσ)0⊂K⊂Kσ, where Kσis the set of fixed point of σ

in G0and where (Kσ)0is its identity component. Then there is a Riemannian

metric on G0/K such that G0/K is a Riemannian symmetric space. We now

explain these constructions for symmetric spaces of the noncompact type.

Call a Lie algebra semisimple if it is a direct sum of simple (non-abelian) Lie

algebras without proper ideals. A (connected) Lie group is said to be semisimple

if its Lie algebra is semisimple, that is it has no non-trivial abelian connected

normal closed subgroup. We denote the Lie algebra of Gby gand let Tr denote

the trace of a vector space endomorphism. We consider the symmetric bilinear

form B(X, Y ) = Tr(ad Xad Y)on g×gand call B(·,·)the Killing form of g.

2 Preliminaries

A Lie algebra gover a field of characteristic 0is semisimple if and only if its

Killing form Bof gis non-degenerate.

Let Xbe a symmetric space of the noncompact type. If pis any point of X,

its stabilizer is a maximal compact subgroup of G0. If Kis a maximal compact

subgroup of G0, then there is a unique point pin Xsuch that Kis the stabilizer

of p. Any two maximal compact subgroups of G0are conjugate by an element of

G0. If kis the Lie algebra of K, the Killing form of gis strictly negative definite

on k. The group G0acts transitively on X. It is a semisimple Lie group with

finite center. Fix a point p∈Xand let Kdenote its stabilizer in G0. Consider

the coset space G0/K and the diffeomorphism ϕ:G0/K →X, ϕ(gK) = g(p)

for g∈G0. Denote by h,ithe metric on G0/K obtained by pulling back

the metric of Xby ϕ. Then ϕis an isometry and the metric h,iis left G0-

invariant, that is left translations on G0/K by elements of G0are isometries

of the metric space (G0/K, h,i). Hence each globally symmetric space of the

noncompact type can be written in the form G0/K as above. These observations

are summarized by

Theorem 2.1 (E. Cartan).The Riemannian globally symmetric spaces of the

noncompact type are the spaces of the form G/K equipped with a G-invariant

metric, where Gis a connected semisimple Lie group with finite center and K

a maximal compact subgroup of G.

2.1 Symmetric spaces and real semisimple Lie groups

Definition 2.2. ARiemannian symmetric space of the noncompact type is a

homogeneous space X=G/K, where Gis a real connected semisimple Lie

group with finite center and Kis a maximal compact subgroup of G.

Let Gdenote a connected Lie group with Lie algebra gand let Hbe a closed

subgroup of Gwith Lie algebra h. By G/H we denote the quotient space

consisting of left cosets gH,g∈G. Let π:G→G/H denote the natural

projection. Choose a complementary subspace mof gsuch that g=h⊕m.

Let X1, ..., Xrand Xr+1, ..., Xnbe bases of mand h, respectively. If g∈G, the

mapping

(x1, ..., xr)7→ π(gexp(x1X1+... +xrXr)) (2.1)

is a diffeomorphism of a neighborhood of 0∈monto a neighborhood of the

point π(g) = gH ∈G/H. The inverse of (2.1) is a local coordinate system near

gH, turning a neighborhood of each π(g)and hence G/H into a manifold.

The Lie algebra gis naturally identified with the tangent space TeGof Gat

the identity e∈G. We list basic results and definitions about semisimple Lie

groups. Details can be found in standard sources ([DS]).

For each X∈g∼

=TeGthere is a unique homomorphism γX: (R,+) →G

such that γ0

X(0) = X. The image of γXis called a one parameter subgroup of

2 Preliminaries

G. The mapping exp : g→G, X 7→ exp(X) := γX(1) is called the exponential

map of G. We have etX =γX(t)for all t∈R.

Each g∈Gdefines an inner automorphism Cg:G→Gby Cg(h) = ghg−1of

the group G. Taking the derivative we define a Lie algebra automorphism

Ad(g) = dCg:g→g.

The map Ad : G→Aut(g)is called the adjoint representation of G. We will

often denote the corresponding group action of Gon gby g·X(X∈g). For

X∈gwe define a linear transformation

ad X:g→g,(ad X)(Y) = [X, Y ],

where [,]denotes the Lie bracket of vector fields on G.

If σis an automorphism of gthen ad(σX) = σ◦ad X◦σ−1so by Tr(AB) =

Tr(BA)we have B(σX, σY ) = B(X, Y )and B([X, Y ], Z) + B(Y, [X, Z]) = 0.

If ais an ideal in g, then the Killing form of acoincides with the restriction of

Bto a×a.

The space G/H is called reductive, if mas above can be chosen such that

g=h⊕m,AdG(h)m⊂m(h∈H).(2.2)

If Ad(H)is compact, then G/H is reductive: In fact, gwill then admit a positive

definite quadratic form invariant under AdG(H)and mcan be chosen to be the

orthogonal complement (w.r.t. this quadratic form) of hin g([GGA], p. 284).

2.1.1 Tangent spaces and Cartan decomposition

For the descriptions of the geometric structure of a symmetric space X=G/K

in terms of algebraic data given by the semisimple Lie group Gwe orient our-

selves on [Eber96].

We write o:= K∈G/K and call othe origin of the symmetric space X=

G/K. Define an involution σ:G→Gby σ(g) = s◦g◦s([Eber96], p.

71), where sdenotes the geodesic symmetry at o. The differential of σat eis

θ=dσ :g→g, which is also characterized by the equation σ(etX ) = etθ(X)for

all X∈g. Since θ2= idgwe obtain the Cartan decomposition

g=k⊕p,

where k={X∈g:θX =X}and p={X∈g:θX =−X}are the eigenspaces

corresponding to the eigenvalues +1 and −1. The Lie algebra automorphism θ

preserves Lie brackets, so we have

[k,k]⊆k,[k,p]⊆p,[p,p]⊆k.(2.3)

We consider the map π:G→Xgiven by g→g·o. Taking the differential

we obtain a linear map dπ :g→g, whose kernel is precisely k. Moreover, kis

2 Preliminaries

the Lie algebra of the maximal compact subgroup K={g∈G:g·o=o}of

G. The restriction dπ :p→gis a monomorphism and we use it to identify

ToX=p. Although we restricted the above constructions to the particular

point o, these results can be obtained at for each p∈X.

It also follows that Ad(K)leaves pinvariant. Moreover, the elements of

Ad(K)are orthogonal transformations on pwith respect to the restriction to p

of the Killing form Bof g. The spaces pand kare orthogonal with respect to

the Killing form Bof g.

For X, Y ∈gwe set Bθ(X, Y ) = −B(θX, Y ). Then Bθis a positive definite

bilinear form on g. We can therefore call |X|2=Bθ(X, X)the norm on g

induced by the Killing form. The restriction of Bθto pequals the Killing form

of grestricted to p.

2.1.2 Rank of Symmetric Spaces

A totally geodesic submanifold of a globally symmetric space Xis necessar-

ily itself a globally symmetric space. If Xis of the noncompact type, totally

geodesic submanifolds have nonpositive curvature and thus don’t have compact

type factors ([Eber96], Ch. 2). We say that Xhas rank kif it contains a

flat totally geodesic submanifold of dimension kand if every other flat totally

geodesic submanifold has dimension ≤k. As Xcontains geodesics, its rank is

≥1. A symmetric space has rank one if and only if it has negative sectional cur-

vature, that is its sectional curvature (as a function on the Grassmanian bundle

of tangent 2-planes of X) is everywhere negative.

As usual let X=G/K and g=k+pbe a Cartan decomposition. The rank

of Xmay also be defined by the dimension of a maximal abelian subspace a

of p. It does neither depend on the choice of anor p([Eber96], p. 76). These

definitions of rank are equivalent ([Eber96], 1.12.12 and (2.15.4)).

2.1.3 Root space decompositions

Fix a maximal abelian subspace aof p. Let a∗be the real dual space of aand

let a∗

Cbe its complexification. The operators ad Xand ad Ycommute in End(g)

for all X, Y ∈a. Let a∗be the real dual of aand let α∈a∗. Then define

gα={X∈g: (ad H)(X) = α(H)Xfor all H∈a}.

An element 06=α∈a∗is called a restricted root if gα6={0}. It also follows

that ad(a)is a commuting family of linear transformations of g. We denote the

set of roots determined by aby Σ. Then Σ⊂g∗is a nonempty finite set. We

have the Bθ-orthogonal direct sum decomposition

g=g0+X

α∈Σ

gα

2 Preliminaries

([DS], p. 263 or [Eber96], p. 78). This is called the root space decomposition

of gdetermined by a. For X∈glet Z(X)denote the centralizer of Xin g.

An element X∈pis called regular if Z(X)∩pis a maximal abelian subspace

of p. Otherwise Xis called singular. An element X6= 0 is regular if and

only if α(X)6= 0 for every nonzero root α∈Σthat occurs in the root space

decomposition of gdetermined by a. Let a0=R(a)denote the set of regular

elements. This set is the complement in aof the union of the finite collection of

hyperplanes

aα={X∈a:α(X) = 0}, α ∈Σ.

We write

H∼H0⇐⇒ α(H)α(H0)>0∀α∈Σ.

This ∼defines an equivalence relation in R(a). The corresponding equivalence

classes are called Weyl chambers. We fix a Weyl chamber a+and call it the

positive Weyl chamber. We call a root αpositive and write α > 0or α∈Σ+if

αhas positive values on a+. A root αis called simple if it is not the sum of two

positive roots. Then a+is given by

a+={H∈a:α1(H), ..., αl(H)>0},

where {α1, ..., αl}is the set of simple roots. The set of simple roots is a basis

of a∗([Eber96], p. 81). Let the real dual space a∗be ordered lexicographically

with respect to this basis ([DS], p 173).

2.1.4 The Weyl group

Let h,idenote the Killing form. The Riesz representation theorem states that

for each α∈Σthere is a unique root vector Hα∈asuch that α(H) = hH, Hαi

for all H∈a. Given a root α, we consider the reflection Sαin the hyperplane

aαof athat is orthogonal to Hα(the kernel of α). This reflection is concretely

given by the Householder transformation

Sα(H) = H−2hHα, Hi

hHα, HαiHα.

The Sαpermute the root vectors ([Eber96], p. 81).

The Weyl group W=W(g,a)is defined as the group Wof isometries of a

generated by the Sα. We write

ZK(a) = {k∈K: Ad(k)(H) = H∀H∈a}

for the centralizer of ain Kand

NK(a) = {k∈K: Ad(k)a=a}

2 Preliminaries

for the normalizer ain K. Then NK(a)normalizes ZK(a). Both groups are

compact and have the same Lie algebra, namely

m=g0∩k={X∈k: [X, H] = 0 for all H∈a}.

The restriction of the exponential map of Gto ais an analytic diffeomorphism

onto the abelian subgroup A:= exp(a). The inverse diffeomorphism is denoted

by log. We can also set NK(A)and ZK(A)for the normalizer and the centralizer

of Ain K, respectively. The Weyl group Wis isomorphic to the finite group

NK(A)/ZK(A)([Eber96], p. 82). Write

M:= {k∈K:ka =ak ∀a∈A}

for the centralizer of Ain Kand

M0:= k∈K:kAk−1=A

for the normalizer of Ain K. Then W∼

=M0/M ∼

=N(A)/Z(A), where N(A)

and Z(A)denote the normalizer and the centralizer of Ain G, respectively. We

always consider Wto be the group W=M0/M. The Weyl group is acting

simply transitively on the collection of Weyl chambers of a([Eber96], p. 83).

Its action extends to a∗by duality, to Avia the exponential map, and to the

complexifications of aand a∗by complex linearity. The Weyl group permutes

the root vectors and it permutes the root spaces.

2.1.5 Decomposition theorems

Let Gbe a semisimple Lie group and carry over the algebraic data g,θ,k,K,

p,a,A,a+from the preceding paragraphs. Write A+:= exp a+and let A+

denote the closure of A+in G. The real rank of Gis the dimension dim a(it

is independent on the choice of a⊂p). We need the following decomposition

theorems for G([DS], Ch. IX).

Theorem 2.3 (Cartan decomposition).Each g∈Gcan be written g=k1ak2,

where k1, k2∈K. The element a=a+(g)∈A+is uniquely determined by g.

Thus G=KA+K.

Recall that we denote the set of positive roots by Σ+. Let

n=X

λ∈Σ+

gλ.

Then nis a subalgebra of g. Let Nbe the corresponding connected subgroup

of G. Then nand Nare nilpotent ([DS], Ch. VI, Thm. 3.4, Ch. IX, Lemma

1.6) and a+nis a solvable Lie algebra. Each element a∈Anormalizes N, that

is we have aN =Na for all a∈A. In particular, AN =NA is a subgroup of G.

2 Preliminaries

Theorem 2.4 (Iwasawa decomposition).We have g=k+a+n(direct vector

space sum) and G=KAN. The mapping (k, a, n)→kan is a diffeomorphism

of K×A×Nonto G.

We fix some notation: If g∈G, we will always write

g=k(g) exp H(g)n(g),

where k(g)∈K,H(g)∈aand n(g)∈N. The corresponding projections onto

the K,aand Nare called Iwasawa projections. We can also decompose

g=n(g) exp A(g)k(g)

corresponding to G=NAK, where A(g)∈a. Clearly A(g) = −H(g−1).

Remark 2.5. Each point p∈Xgives rise to another Cartan involution and

another Cartan decomposition. Let θpbe the Cartan involution and g=kp+pp

be the Cartan decomposition determined by p∈X. If q∈Xdetermines θqand

g=kq+pq, then kq=g·kpand pq=g·ppwhenever g·p=q([Eber96], §2.3, §2.8,

[DS], Ch. III, Thm. 7.2). It follows that all Cartan decompositions of gare

conjugate in G. By [DS], Ch. V, Lem. 6.3 (or [Eber96], §2.8), any two maximal

abelian subspaces a1and a2of ppare conjugate by an element k∈K. Since

the Weyl group acts simply transitively on the Weyl chambers ([DS], Ch. VII,

Theorem 2.12) we deduce that for another choice a1resp. A1the corresponding

Iwasawa decomposition components AN and A1N1are conjugate by an element

of K. It follows that all Iwasawa decompositions are conjugate in G.

Note that Ad(m)(m∈M) leaves apointwise fixed, so it maps a root space α

into itself. Hence Mnormalizes N, so MN =NM is a group. Then P=MAN

is a closed subgroup of G. For s∈W=M0/M we fix a representative ms∈M0.

Theorem 2.6 (Bruhat decomposition).Let Gbe any noncompact semisimple

Lie group. Then Gdecomposes into double cosets of P=MAN, that is

G=[

s∈W

PmsP(disjoint union).

We can also write S= exp p. Then (cf. [DS], Ch. VI)

Theorem 2.7. G=K·S=S·K. The indicated decomposition of an element

of Gis unique. The mapping (X, k)7→ (exp X)kis a diffeomorphism of p×K

onto G. Write π:G→G/K. Then the mapping π◦exp is a diffeomorphism

of ponto the globally symmetric space X=G/K.

Definition 2.8. For α∈Σ+we call mα= dim gαthe multiplicity of α. Once

for all we define the parameter ρ∈a∗by

ρ=1

α∈Σ+

mαα.

2 Preliminaries

We finish this subsection with a few remarks on the nilpotent subgroup N.

Let ·denote the adjoint action of Gon g.

Remark 2.9. (1) Let H∈a0(regular). The mapping n7→ n·H−Hdefines

a diffeomorphism of Nonto n([DS], p. 403).

(2) Assume H∈a0(i.e. His regular) such that α(H)>0for all α∈Σ+.

Then ([DS], p. 278)

N=ng∈G: lim

t→∞exp(−tH)gexp tH =eo.

(3) For X∈p, let ZN(X)denote the cantralizer of Xin Nand let Zn(X) =

{X∈n: [X, X] = 0}denote the centralizer of Xin n. Let H∈a. Then

ZN(H) = exp(Zn(H)). Each X∈nis of the form X=Pα∈Σ+Xα,

where Xα∈gα. By definition we thus have [H, X] = Pα∈Σ+α(H)Xα.

Now assume [H, X]=0. Then α(H)Xα= 0 for all α. Then X= 0,

hence Zn(H) = {0}and ZN(H) = {e}. In general, for X∈pwe have

ZH(X) = {e}if and only if Xis regular.

2.1.6 Measure theoretic preliminaries

We establish some conventions about the normalization of invariant measures

on the groups and homogeneous spaces we work with. We follow the standard

source [GASS], Ch. II.

If Yis any manifold we denote by C(Y)the space of real- or complex-valued

continuous functions on Y. By Cc(Y)we denote the subspace of C(Y)consisting

of functions with compact support.

The Killing form induces Euclidean measures on A, its Lie algebra aand the

dual space a∗. If l= dim(A), we multiply these measures by the factor (2π)−l/2

and thereby obtain invariant measures da, dH and dλ on A, aand on a∗. This

normalization has the advantadge that the Euclidean Fourier transform on A

is inverted without a multiplicative constant. We normalize the Haar measures

dk and dm on the compact groups Kand Msuch that the total measure is 1.

In general, if Uis a Lie group and Pa closed subgroup, with left invariant

measures du and dp, the U-invariant measure duP=d(uP)on U/P (when it

exists) will be normalized by

f(u)du =ZU/P ZP

f(up)dpduP.(2.4)

This measure exists if Uis unimodular and Pis a compact subgroup of U

([GGA], Ch. I, Thm. 1.9). In particular, we have a K-invariant measure

dkM=d(kM)on K/M of total measure 1. We also use the notation

dx =dgK=d(gK), dξ =dgMN =d(gMN)(2.5)

2 Preliminaries

for the invariant measures on X=G/K and Ξ = G/MN. By uniqueness, dx is

a constant multiple of the measure on Xinduced by the Riemannian structure

on Xgiven by the Killing form B.

The involutive automorphism θof ginduces a unique ([DS], Ch. IV, Prop.

3.5) analytic involutive automorphism, also denoted by θ, of Gwhose differential

at e∈Gis the original θ. ([DS], Ch. VI, Thm. 1.1). It thus makes sense to

define N=θN. The mapping (n, m, a, n)7→ nman is a bijection of N×M×

A×Nonto the open submanifold NMAN of G, whose complement is a null-set

for the Haar measure of G([DS], Ch. IX, §1). In the Iwasawa decomposition

notation, the mapping N→K/M,n7→ k(n)M, is a diffeomorphism of Nonto

an open subset of K/M whose complement is a null set for the invariant measure

d(kM)on K/M.

The Haar measures dn and dn on the nilpotent groups Nand Ncan be

normalized ([GGA], Ch. IV, §6) such that

θ(dn) = dn, ZN

e−2ρ(H(n))dn = 1.

By loc. cit., Ch. I, §5, we can then normalize the Haar measure on Gsuch

that for all f∈Cc(G)

f(g)dg =ZKAN

f(kan)e2ρ(log a)dkdadn (2.6)

=ZNAK

f(nak)e−2ρ(log a)dndadk. (2.7)

Recall that each m∈Mleaves apointwise fixed, so mmaps a root space into

and onto itself. Hence n7→ mnm−1is an automorphism of Nmapping dn into a

multiple of dn. Since Mis compact, dn is preserved. It follows that the product

measure dmdn is a bi-invariant measure on MN =NM. Let m∗∈M0denote

any representative of the the Weyl group element mapping the positive Weyl

chamber a+onto −a+. Then the mapping n7→ (m∗)−1nm∗is a diffeomorphism

between Nand N=θ(N)([GASS], p. 102).

We will also need the following integral formulas ([GGA], Ch. I).

Lemma 2.10. (1) Let f∈Cc(AN)and a∈A. Then

f(na)dn =e2ρ(log(a)) ZN

f(an)dn. (2.8)

(2) Let f∈Cc(G). Then

f(g)dg =ZKNA

f(kna)dk dn da =ZANK

f(ank)da dn dk. (2.9)

(3) Let f∈Cc(X). Then

f(x)dx =ZAN

f(an ·o)da dn. (2.10)

2 Preliminaries

2.1.7 Special functions and the Plancherel density

Recall that we denote by Σ+the system of positive roots. The set of all (re-

stricted) roots is the disjoint union of Σ+and −Σ+. We write Σ−:= −Σ+. A

root α∈Σis called indivisible if α/2/∈Σ. For the sets of indivisible, respec-

tively positive indivisible roots, we write Σ0and Σ+

0, respectively. We can then

define

Σ+

0:= Σ+∩Σ0and Σ−

0:= Σ−∩Σ0.(2.11)

Also recall that the Cartan-Killing form B(·,·)is positive definite on p×p,

so hX, Y i:= B(X, Y )defines a Euclidean structure in pand in a⊂p. Given

γ∈a∗, there is a unique Hγ∈asuch that γ(H) = hHγ, Hifor all H∈a.

We can thus extend h·,·i to a∗by duality, that is we set hλ, µi=hHλ, Hµifor

λ, µ ∈a∗. Finally we denote the C-bilinear extension of h·,·i to a∗

Cby the same

symbol. Given α∈Σand λ∈a∗

Cwe write

λα:= hλ, αi

hα, αi.(2.12)

Let Γdenote the classical Γ-function. Here and in the following we adopt the

convention that m2α= 0 if 2αis not a root. Harish-Chandra’s c-function is the

meromorphic function on a∗

Cgiven by the Gindikin-Karpelevich product formula

c(λ) = c0Y

α∈Σ+

cα(λ)(2.13)

where

cα(λ) = 2−iλαΓ(iλα)

Γ(iλα

2+mα

4+1

2) Γ(iλα

2+mα

4+m2α

2),(2.14)

and where the constant c0is defined by c(−iρ)=1. Note that the function

|c(λ)|−2=c(λ)c(−λ) = c(sλ)c(−sλ)∀s∈W(2.15)

is Weyl group invariant ([GGA], p. 451). The singularities of the Plancherel

density

c(λ)c(−λ)=1

α∈Σ+

cα(λ)cα(−λ)(2.16)

can be explicitly written down. We recall some formulas given in [HP09]. Note

that if both αand 2αare roots, then mαis even and m2αis odd ([DS], p. 530).

For α∈Σ+

0, the singularities of

cα(λ)cα(−λ)(2.17)

are described by distinguishing the following four cases:

2 Preliminaries

(a) mαeven, m2α= 0,

(b) mαodd, m2α= 0,

(d) mα/2odd, m2αodd.

It follows from simple identities for the Γ-function that

cα(λ)cα(−λ)=Cαλαpα(λ)qα(λ),(2.18)

where Cαis a positive constant depending on αand on the multiplicities, where

pαis a polynomial, and where qαis a function. We make the convention that a

product taken over the empty set is equal to one. Then the explicit expressions

for pαand qαin the four cases listed above are ([HP09], p. 501)

(a) pα(λ) = λαQmα/2−1

k=1 (λ2

α+k2),

qα(λ) = 1,

(b) pα(λ) = Q(mα−3)/2

k=0 [λ2

α+ (k+1

2)2],

qα(λ) = tanh(πλα),

k=0 [(λα/2)2+(k+1

2)2]·Qmα/4+(m2α−1)/2−1

l=0 [(λα/2)2+(l+1

2)2],

qα(λ) = tanh(πλα/2),

(d) pα(λ) = Q(mα−2)/4

k=0 [(λα/2)2+k2]·Q(mα+2m2α)/4−1

l=1 [(λα/2)2+l2],

qα(λ) = coth(πλα/2),

Note that in each of the above cases the degree of the polynomial λαpα(λ)

equals mα, and hence the dimension of the root subspace gα. Given λ∈a∗

we sometimes write λ→ ∞ and mean that λ(H)→ ∞ for all H∈a+. Recall

that tanh ∼1and coth ∼1to all orders. Hence if asymptotics λ→ ∞ are

involved, we can replace the factor qα(λ)by 1, and the Plancherel density is

asymptotically a polynomial of degree dim(N).

For any (restricted) root αwe can also write α0:= α/hα, αi. We will later

need Harish-Chandra’s e-functions ([GASS], p. 163)

es(λ) = Y

α∈Σ+

Γmα

4+1

2+hiλ, α0i

2Γmα

4+m2α

2+hiλ, α0i

2,(2.19)

where s∈Wand Σ+

s= Σ+

0∩s−1Σ−

2 Preliminaries

2.2 Geodesics, horocycles, and the boundary at infinity

Let Xbe a symmetric space of the noncompact type, hence X=G/K, where

Gis a noncompact connected semisimple Lie group with finite center and where

Kis a maximal compact subgroup of G. We carry over the notations from the

previous section.The origin of Xis the identity coset o:= K∈G/K. A basic

remark which follows from Theorem 2.7 is that the geodesics through the origin

are ([Eber96], p. 74) the curves

γX:t7→ etX ·o, (X∈p)(2.20)

As Xis a simply connected manifold of nonpositive sectional curvature, for each

points p6=qin Xthere exists a unique unit speed geodesic γp,q :R→Xwith

γp,q(0) = pand γp,q(a) = q, where d(p, q) = a, and where ddenotes the distance

function on X(loc. cit, p. 20).

Definition 2.11. Two unit speed geodesics γand σof Xare asymptotes or

asymptotically equivalent if there exists C≥0such that the d(γ(t), σ(t)) ≤

Cfor all t≥0. Two unit vectors v, w ∈SX are said to be asymptotes or

asymptotically equivalent if the corresponding geodesics γvresp. γwwith initial

velocity vand whave this property.

The asymptote relation is an equivalence relation on the unit speed geodesics

of Xand on the unit vectors of SX.

Definition 2.12. Apoint at infinity for Xis an equivalence class of asymptotic

geodesics of X([Eber96], p. 27). The set of all points at infinity for Xis denoted

by X(∞). The equivalence class represented by a geodesic γis denoted by

γ(∞)and the equivalence class represented by the oppositely oriented geodesic

γ−1:t7→ γ(−t)is denoted by γ(−∞).

If γis any geodesic of the complete, simply connected space Xwith nonpos-

itive curvature, then for each p∈Xthere exists a unique geodesic σof Xsuch

that σ(0) = pand σis asymptotic to γ([Eber96], p. 28).

Definition 2.13. We say that points x6=yin X(∞)can be joined by a geodesic

of Xif there exists a geodesic γof Xwith γ(∞) = xand γ(−∞) = y. The

geodesic γis said to join xand y.

Throughout this work we will mainly be interested in points at infinity that

can be joined by a geodesic. We first recall a basis result ([EO], Proposition

4.4):

Theorem 2.14. Let Xhave rank one. The sectional curvature of Xis strictly

negative. Any two distinct points x, y ∈X(∞)can be joined by a geodesic of X.

To motivate this setting, we will now describe the geometry of a rank one

space in detail. The group theoretical aspects can then be generalized to higher

rank spaces.

2 Preliminaries

2.2.1 The boundary at infinity

Let X=G/K have rank one. We call B=X(∞)the boundary at infinity.

For X∈plet γX=etX ·odenote the geodesic through the origin o∈Xwith

inital direction X. We introduce an action of Gon B. For b= limt→∞ γX(t)

and g∈G, define

g·b:= g·lim

t→∞γX(t) = lim

t→∞γg·X(t)∈B.

(Here, g·Xdenotes the adjoint action.) Since G/K has rank one, we define

once and for all Hto be the unique unit vector (w.r.t. the norm induced by

the Killing form) in the positive Weyl chamber a+. We write S(p)for the unit

sphere of p. Let b∞∈Bdenote the boundary point limt→∞ γH(t). Let b−∞ ∈B

denote the boundary point limt→−∞ γH(t) = limt→∞ γ−H(t).

The only orthogonal transformations of the one-dimensional space aare ±id.

It follows that (in the rank one case) the Weyl group has exactly two elements.

Let w∈M0denote any representative of the nontrivial Weyl group element. The

adjoint action of won ais −id, so w·H=−H. It follows that w·b∞=b−∞

and vice versa.

For b∈Bthere exists X∈S(p)such that b=γX(∞)for γX(t) = etX ·o.

Since Kacts transitively on S(p), there is k∈Ksuch that k·H=X. Hence

k·b=b∞. In particular Kacts transitively on B. The stabilizer of b∞is by

definition the stabilzer Mof H. The action of Kon Bis continuous ([Eber96],

Ch. 3) and since it is transitive, Bis compact. Hence under the mapping

AH:K/M →B, kM 7→ lim

t→∞γk·H(t),(2.21)

Bis in a natural way homeomorphic to the compact space K/M. We make

Ba smooth manifold by giving it the differentiable structure that makes AHa

diffeomorphism ([Eber96], Ch. 3.8). The natural Lie topology of K/M agrees

with the compact open topology of the homeomorphism group of B, so B=

K/M as homogeneous spaces.

2.2.2 The real flag manifold

We drop the rank one assumption and let X=G/K be a general symmetric

space of the noncompact type. Each g∈Gcan be written g=k(g)a(g)n(g)

corresponding to G=KAN. We introduce the map

G×K→K, (g, k)7→ g·k:= Tg(k) := k(gk)(2.22)

Then Tgis a group action of Gon K. In particular, Tgis inverted by Tg−1

and defines a diffeomorphism of Konto itself. This can easily be verified using

the Iwasawa decomposition. For g∈G,k∈Kand m∈Mwe clearly have

k(gkm) = k(gk)m, since mnormalizes Nand centralizes A. Hence k7→ k(gk)is

2 Preliminaries

right-M-equivariant, so (2.22) descends to an action of Gon the quotient K/M.

We write Tg:K/M →K/M, kM 7→ k(gk)Mfor this action.

Let man ∈P=MAN. Then man·M=k(man)M=M. Thus MAN is the

centralizer in Gof M∈K/M. The group Gacts naturally (by left-translations)

on G/P. The mapping ϕ:K/M →G/P, kM 7→ kP, is a bijection of K/M onto

G/P which is regular at the origin, hence everywhere, so it is a diffeomorphism

([DS], p. 407). The identification ϕ:K/M →G/P intertwines the actions of

Gon K/M and the natural group action of Gon G/P:

ϕ(g·kM) = ϕ(k(gk)M) = k(gk)MAN =gkMAN =g·ϕ(kM).

The spaces K/M and G/P are thus equivalent from this group theoretical point

of view. We will write B:= K/M =G/P. We also recall the following useful

lemma ([DS], p. 407):

Lemma 2.15. The mapping n7→ k(n)Mis a diffeomorphism of Nonto an

open submanifold of K/M whose complement consists of finitely many disjoint

manifolds of lower dimension.

Remark 2.16. AHadamard manifold is a simply connected complete Rieman-

nian manifold of nonpositive curvature and arbitrary dimension. We say that

a Hadamard manifold Xsatisfies the visibility axiom, if any two points of the

geodesic boundary ([EO]) can be joined by a geodesic X. A Hadamard manifold

may or may not satisfy the visibility axiom. The extreme cases are as follows:

(a) The sectional curvature is zero. Then asymptoticity of geodesics coincides

with ordinary parallelism, hence the visibility axiom is not satisfied.

(b) The sectional curvature is negative and bounded away from zero. In this

case the behaviour of geodesic rays is qualitatively the same as in hyper-

bolic geometry, the visibility axiom is satisfied, and the geodesic joining

two given boundary points is unique ([EO], Cor. 5.2).

A special class of Hadamard manifolds consists of Riemannian symmetric spaces

of the noncompact type. If the symmetric space has rank one, then its sectional

curvature is bounded between two negative constants (and thus the space falls

into category (b) from above), so the visibility axiom is satisfied. On the other

hand, higher rank spaces are characterized by the existence of totally geodesic

flat subspaces, in which the visibility axiom fails, and hence it fails the ambient

space as well ([Hof81]).

The description of the geodesic boundary of a higher rank space X=G/K

differs from the rank one case. For details we refer to [Eber96]. If X∪X(∞)is

given the so-called cone topology (loc. cit., p. 28), then isometries and geodesic

symmetries of Xextend to the boundary X(∞)(loc. cit. p. 30).

2 Preliminaries

Remark 2.17. Given a boundary point x∈X(∞), let Gx⊂Gdenote its

stabilizer. Then Gxacts transitively on X=G/K (loc. cit., p.101). Suppose

that another point yat infinity can be joined with xby a geodesic. Then the

set of points to which xcan be joined is the orbit Gx(y)(loc. cit., p. 151). If

Xhas rank one, then Gxacts transitively on X(∞)\{x}. This fails whenever

the rank of Xis ≥2.

Irrespective from the geometric point of view, many group theoretical as-

pects generalize to the higher rank case. We take the preceding remark as a

motivation.

Definition 2.18. A subgroup of P∗of Gis parabolic if there exists a point

b∈Bsuch that P∗=Gb={g∈G:gb =b}is the stabilizer of bin G.

Remark 2.19. (1) Our definition of a parabolic subgroup follows [Eber96]

and does only consider the minimal parabolic subgroups of G.

(2) Unlike the subgroups of Gthat fix a point in X, the parabolic subgroups

are noncompact.

(3) The parabolic subgroup fixing b=M3K/M is P=MAN (M∈K/M

corresponds to P∈G/P).

(4) Let b=hP ∈G/P (h∈G). Then g·b=b⇔g∈hPh−1, so all parabolic

subgroups of Gare conjugate to each other.

(5) AN acts transitively on X, so the same holds for P=MAN. It follows

that all parabolic subgroups act transitively on X.

2.2.3 The rank one case

Let X=G/K have rank one. The Weyl group W=M0/M has exactly two

elements. Let w∈M0denote any representative of the nontrivial Weyl group

element. As before, let Hdenote the unit vector in a+. We also write

at:= exp(tH)∈A. (2.23)

We consider the geodesic t7→ at·o. Its forward limit point is b∞and it identifies

with M∈K/M (that is P∈G/P). Its backward limit point b−∞ identifies

with wM ∈K/M (that is wP ∈G/P).

Since wM 6=Min K/M, the geodesic t7→ at·ois the unique (up to parameter

translation and time reversal) geodesic of Xthat joins the boundary points

M∈K/M and wM ∈K/M at infinity ([Quint06]).

We consider the homogeneous space G/M. The group Mis the stabilizer

in Kof the unit vector tangent at oto the geodesic t7→ at·o. As Kacts

transitively on the set of unit vectors in ToX∼

=p, the unit tangent bundle of X

identifies G-equivariantly with G/M and the geodesic flow reads as the action

of Aby right translations on G/M.

2 Preliminaries

Lemma 2.20. Let b∈B. Then Gbacts transitively on B\{b}. In particular,

Pacts transitively on B\{b∞}.

Proof. Since all parabolic subgroups are conjugate, it suffices to prove the as-

sertion for Gb∞=P. Recall the Bruhat decomposition

G=P∪PwP (disjoint union),

Let b∈B\{b∞}and select g∈Gsuch that b=g·b∞. Note that p·b∞=b∞

for each p∈P. Thus g=p1wp2(p1, p2∈P). Hence b=p1wP =p1·b−∞,

which shows that b∈P·b−∞, as desired.

Definition 2.21. Let ∆ = {(b, b)∈B×B}denote the diagonal of B×B. Let

B(2) := (B×B)\∆denote the set of distinct boundary points.

We study the space of geodesics and the geodesic connections in the rank one

case and describe the map that assigns to a geodesic its forward and backward

limit points. We consider the diagonal action of Gon B(2) given by

G×B(2) →B(2), g ·(b1, b2)=(g·b1, g ·b2).(2.24)

Note that g·b1=g·b2implies b1=b2, so (2.24) is well-defined.

Lemma 2.22. Gacts transitively on B(2). The stabilizer of (b∞, b−∞)is MA.

In particular, B(2) =G/MA as a homogeneous space.

Proof. Let b16=b2be points in B. Since Kacts transitive on B, we find k∈K

such that k·b1=b∞. Since Pacts transitively on B\{b∞}, we find p∈Psuch

that p·k·b2=b−∞. Let g=pk. Then g·(b1, b2)=(b∞, b−∞), so the group

action is transitive.

It remains to show that g·(b∞, b−∞) = (b∞, b−∞)⇔g∈MA. Clearly an

element ma ∈MA fixes both M∈K/M and wM ∈K/M, since M0normalizes

both Aand M.

Conversely assume that g·(b∞, b−∞) = (b∞, b−∞). Then g·b∞=b∞, hence

g=man ∈MAN. It suffices to prove that n=e. By the assumption we have

n∈Gb∞∩Gb−∞ =MAN ∩wMANw−1⊂θ(N). Hence n∈N∩θN ={e}.

(Recall that gis the direct vector space sum of the root-subspaces gα.)

Remark 2.23. (1) The unit tangent bundle SX ∼

=G/M identifies with the

set of pointed oriented complete geodesics of X.

(2) B(2) ∼

=G/MA is the set of oriented geodesics up to parameter translation.

We can also write SX ∼

=B(2) ×R.

(3) One could also prove Lemma 2.22 by using that the flats nA ·oand A·o

([GASS]) coincide if and only if n=e.

2 Preliminaries

(4) Lemma 2.22 is false for G/K of rank ≥2. This follows from the Bruhat

decomposition, too. We will later see which subspace of B×Bidentifies

with the homogeneous space G/MA.

We can now give group-theoretical proof of Theorem 2.14. See also [Quint06].

Theorem 2.24. Each geodesic σof G/K has two distinct limit points in B.

For (b1, b2)∈B(2) there exists up to parameter translation a unique geodesic σ

with limit points b1and b2. For (x, b)∈X×Bthere is a unique geodesic through

xwith limit point b.

Proof. The first point is true for the geodesic t7→ at·oand therefore for a general

geodesic as Gacts transitively on the set of geodesics, since it acts transitively

on Xand Kacts transitively on the unit sphere of ToX. The third point is true

for x=oas Kacts transitively on Band therefore for any x, as for every b∈B

its stabilizer Gbin Gacts transitively on X. The second point is true for b∞

and b−∞, hence by transitivity of Gon B(2) for all pairs of limit points.

Remark 2.25. B\ {b∞}∼

=Nas homogeneous spaces. In fact, the action of

Non B\{b∞}is already transitive, since the action of P=MAN is and MA

fixes b−∞. It follows from 2.9 that the stabilizer in Nof b−∞ is {e}.

2.2.4 The general case

We drop the rank one assumption and let X=G/K be a general symmetric

space of the noncompact type. Consider the diagonal action of Gon G/K×G/P

given by

γ·(gK, hP) := (γgK, γhP), γ, g, h ∈G. (2.25)

Note that in the customary sense, P=MAN is still a minimal parabolic sub-

group of G(we do not describe this concept here). The action (2.25) yields the

useful identification G/M ∼

=X×B(as homogeneous spaces). To describe

this identification, we use simple Iwasawa decomposition arguments. First,

let b0denote the identity coset of K/M ∼

=G/P. For γ∈Gwe observe

γ·(o, b0)=(o, b0)⇔γ∈K∩P=M. It follows that M⊂Gis the sta-

bilizer of (o, b0)∈X×B. For a proof of X×B∼

=G/M it remains to show that

the diagonal action of Gon X×Bis transitive. We say that cosets γP ∈G/P

and hK ∈G/K are incident, if as subsets of Gthey are not disjoint.

Lemma 2.26. Let g∈G. Then gK ∈G/K and P∈G/P are incident. Let

h∈G. Then gK and hP ∈G/P are incident.

Proof. Write g=nak. Then gK =naK ⊂Gcontains p=na ∈MAN =P.

For general hP ∈G/P select p∈h−1gK ∩P. Then p=h−1gk for some k∈K,

so hP 3hp =gk ∈gK.

2 Preliminaries

Corollary 2.27. Gacts transitively on G/K ×G/P.

Proof. First, given (gK, hP)∈G/K ×G/P, we apply Lemma 2.26 and write

gk =hp, where k∈Kand p∈P. Then gk ·(o, b0) = (gk ·o, gk ·b0) =

(g·o, h ·b0).

Corollary 2.28. Each element (gK, kM)∈G/K ×K/M can be written in the

form (kanK, kM). If (z, b)∈X×B, then there is g∈Gsuch that g·(o, b0) =

(z, b). The element g∈Gis uniquely determined modulo M.

If H∈a, the geodesic t7→ exp(tH)·oin Xis said to be regular if the vector

His regular. A general geodesic γin Xis said to be regular if its stabilizer

{g∈G:g·γ=γ}in Ghas minimum dimension ([GASS], p.82). A flat in Xis

a totally geodesic submanifold of Xwhose curvature tensor vanishes identically.

The maximal flats in Xare of the form gA ·o(g∈G) ([DS], Ch. V, §6).

Recall the Bruhat decomposition

G=[

s∈W

PmsP(disjoint union),

where for s∈W(Weyl group) we picked a representative ms∈M0. Exactly

one of the above sets PmsPis open and dense in G, namely PwP, where wis

the longest Weyl group element. The other summands have lower dimension.

Recall N=wNw−1(conjugation by wis not necessarily θ|N, the restriction of

θto N). It follows that the manifold NMAN is open and dense in G. Thus the

space of flats can be naturally identified with G/MA, or a dense open subset of

G/P ×G/P, where P:= MAN, via the G-equivariant map

G/MA 3gMA 7→ (gP, gwPw−1)∈G/P ×G/P.

We also consider the G-equivariant map

G/MA 3gMA 7→ (gP, gwP)∈G/P ×G/P =B×B.

It follows from the Bruhat decomposition that its image is an open and dense

subset of G/P×G/P =B×B, namely {(gP, hP)∈G/P ×G/P :h−1g∈PwP}.

This open and dense subset of B×Bis the G-orbit of (P, wP)in B×B. We

will from now on write B(2) :∼

=G/MA for this G-orbit. If Xhas rank one, then

B(2) = (B×B)\∆, where ∆denotes the diagonal of B×B.

2.2.5 The space of horocycles

Definition 2.29. Ahorocycle ξin Xis any orbit ξ=N0·x, where x∈Xand

N0=g−1Ng is a subgroup of Gconjugate to N. In particular, we define ξ0to

be the horocycle N·o.

2 Preliminaries

The choice of Iwasawa-decomposition is immaterial since all such decomposi-

tions are conjugate ([Eber96], p.105). We note that each horocycle is a closed

submanifold of X. The group Gacts transitively on the set of horocycles. The

subgroup of Gwhich maps the horocycle ξ0into itself equals MN ([GASS], Ch.

II, §1).

The set of horocycles in Xwith the differentiable structure of G/MN is called

the dual space of Xand will be denoted by Ξ. We write Ξ = G/MN. Then each

ξ∈Ξcan be written in the form ξ=gMN, where g∈G. Decompose g=kan

corresponding to the Iwasawa decomposition. Then ξ=kanMN =kaMN,

since Mnormalizes N. Let hbe another representative of ξ, that is hMN =

gMN, so h=gmn0, since Mnormalizes N. Then ξ=hMN =kanmn0MN =

kaMN =kmaMN. It follows that each horocycle ξ∈Ξcan be written in the

form kaMN, where kM ∈K/M and a∈Aare unique.

Definition 2.30. If ξ=kaMN is any horocycle, then b=kM ∈B=K/M is

said to be normal to ξ.

Lemma 2.31. Each horocycle ξ=gNg−1·x(g∈G, x ∈X) can be written in

the form ξ=ka ·ξ0, where kM ∈K/M and a∈Aare unique.

Proof. Write g=kan and g−1·x= ˜n˜a·ocorresponding to the Iwasawa decompo-

sition. Since Anormalizes Nwe obtain ξ=gNg−1·x=kaN ˜n˜aK =ka1NK =

ka1·ξ0. The uniqueness follows from the fact that MN is the stabilizer of the

horocycle ξ0.

Definition 2.32. Let ξ=kaMN ∈Ξbe any horocycle. We call log(a)the

composite distance from oto ξ. In general, for x=g1K∈Xand ξ=g2MN ∈Ξ

we call hx, ξi=H(g−1

1g2)the composite distance from xto ξ.

Recall that H:G→ais left-K-invariant and right-MN-invariant, so hx, ξiis

well-defined and invariant under the natural diagonal action of Gon the product

space X×Ξ∼

=G/K ×G/MN. We also state the following uniqueness result

([GASS], p. 81).

Lemma 2.33. Given x∈X,b∈B, there exists a unique horocycle passing

through xwith normal b. For x=gK ∈G/K and b=kM ∈K/M,

ξ=ξ(x, b) = kexp(−H(g−1k))ξ0(2.26)

is the unique horocycle in question.

2.2.6 Horocycles brackets and the Iwasawa-projection

For x∈Xand b∈Blet ξ(x, b)denote the unique horocycle passing through

the point x∈Xwith normal b∈B=K/M. We denote by hx, ξi ∈ athe

composite distance from the origin oto the horocycle ξ(x, b). This vector-valued

inner product has a simple expression in terms of the Iwasawa decomposition

2 Preliminaries

G=KAN =NAK. Therefore recall the projections H:KAN →aand

A:G=NAK →a. In view of (2.26) we define A:X×B→avia

(x, b)7→ A(x, b) = hx, bi=hgK, kMi:= A(k−1g) = −H(g−1k).

We will mostly use the notation h,ifor this inner product and call it the

horocycle bracket. Sometimes, when this horocycle bracket is needed in one

equation with the Killing form, we use the notation A(x, b), which is also used

in [GASS]. We clearly have

Lemma 2.34. h·,·i is invariant under the diagonal action of Kon X×B.

Recall that g∈Gacts on Kby g·k=k(gk), where k:G=KAN →K

denotes the Iwasawa projection. By the right-M-equivariance of this projection

the action descends to an action of Gon K/M.

Lemma 2.35. Let g1, g2∈G,k∈K. Then H(g1g2k) = H(g1k(g2k))+H(g2k).

Proof. Decompose g2k=˜

k˜a˜nand g1˜

k=k0a0n0. Then

H(g1g2k) = H(k0a0n0˜a˜n) = H(a0n0˜a).

Since Anormalizes Nthis equals log(a0) + log(˜a).

Lemma 2.36. Let x=hK ∈G/K,b=kM ∈K/M,g∈G. Then

hg·x, g ·bi=hx, bi+hg·o, g ·bi.(2.27)

Proof. By definition, hg·x, g ·bi=−H(h−1g−1k(gk)). Then by Lemma 2.35

with g1=h−1g−1and g2=gthis equals

−H(h−1g−1gk) + H(gk) = −H(h−1k) + H(gk).

Also hg·o, g ·bi=−H(k) + H(gk) = H(gk)as above for h=e. Hence

hg·x, g ·bi−hg·o, g ·bi= [−H(h−1k) + H(gk)] −[−H(k) + H(gk)],

and the right hand side equals −H(h−1k) = hhK, kMi=hx, bi.

Corollary 2.37. hg−1·o, bi=−hg·o, g ·bi.

Proof. 0 = ho, g ·bi=hg·g−1·o, g ·bi=hg−1·o, bi+hg·o, g ·bi, since the

distance to the origin of a horocycle passing through the origin is 0.

We go on using the Iwasawa decomposition and easily derive

Lemma 2.38. (i) hg−1z, Mi=hz, g ·Mi−hg·o, g ·Mi,

(ii) hg−1z, g−1bi=hz, bi−hg·o, bi.

2 Preliminaries

Lemma 2.39. Let g∈G. Then hg·o, g ·Mi=H(g).

Proof. Write g=kan corresponding to the Iwasawa decomposition. Then

g−1k=n−1a−1=a−1˜n, so hkan ·o, kan ·Mi=−H(g−1k) = log(a) = H(g).

Note that one could also prove (2.27) using Lemma 2.39. We will need some

more component computations for later reference. Under X×B∼

=G/M, each

(z, b)∈X×Bcan be written (g·o, g ·M). Then hz, bi=H(g)follows from

Lemma (2.39). We go on using the Iwasawa decomposition and easily derive

Corollary 2.40. Given z, w ∈X,b, b0∈B, let (z, b)∈X×Bcorrespond to

gM ∈G/M and let (w, b0) = (hK, h ·M)∈X×Bcorrespond to hM ∈G/M,

respectiveley. Then

(1) hz, bi=H(g),

(2) hz, b0i=−H(g−1k(h)) = −H(g−1h) + H(h),

(3) hw, bi=−H(h−1k(g)) = −H(h−1g) + H(g),

(4) hw, b0i=H(h).

2.3 Invariant differential operators

We recall the theory of invariant differential operators to put results concerning

the Laplacian of a symmetric space into a general context. We will need to recall

relations between invariant differential operators and invariant polynomials for

the Weyl group. We recall the definition of the Laplace-Beltrami operator and

give the explicit and important formula (2.57) for the so-called complete symbol

of this invariant differential operator. The material is taken mostly from [GGA].

If Vis an open subset of Rnwe let E(V) = C∞(V)denote the set of smooth

functions on Vand D(V)denote the space of functions in E(V)with compact

support contained in V. Let ∂jdenote partial differentiation with respect to xj,

where x= (x1, . . . , xn)∈Rn. If α= (α1, ..., αn)∈Nn

0, put

Dα=∂α1

1···∂αn

n, xα=xα1

1···xαn

n,(2.28)

|α|=α1+···+αn, α! = α1!···αn!.(2.29)

If Sis any subset of the open set Vand m∈N0we put

kfkS

m=X

|α|≤m

sup

x∈S|Dαf(x)|.(2.30)

Adifferential operator on Vis a linear mapping D:D(V)→D(V)such that

for each relatively compact open set U⊂Vsuch that U⊂V(closure in Rn),

there exists a finite famliy of functions aα∈E(U),α∈Nn

0, such that

Dϕ =X

aαDαϕ, ϕ ∈D(U).(2.31)

2 Preliminaries

Differential operators decrease supports:

supp(Dϕ)⊂supp(ϕ).(2.32)

Conversely, Peetre’s theorem states that any linear mapping D:D(V)→D(V)

decreasing supports is a differential operator ([GGA], p. 236).

Let Mbe a manifold. A differential operator Don Mis a linear mapping of

C∞

c(M)into itself which decreases supports:

supp(Df)⊂supp(f), f ∈C∞

c(M).

The definition of a differential operator extends naturally to C∞(M)if one puts

(Df)(x) = (Dϕ)(x), where ϕ∈C∞

cequals f∈C∞in a neighborhood of x∈M.

To describe the function and distribution spaces we work with, we follow

[GGA], Ch. II. Let Msatisfy the second axiom of countability, that is the

topology of Madmits a countable base for the open sets. If (U, ϕ)is a local

coordinate system on M, the mapping

Dϕ:F7→ (D(F◦ϕ)) ◦ϕ−1, F ∈C∞

c(ϕ(U)),

is support-decreasing. It follows that for each open relatively compact set W

such that W⊂Uthere are finitely many aα∈C∞(W)such that

Df =X

aα(Dα(f◦ϕ−1)) ◦ϕ, f ∈C∞

c(W).

Just as for open sets in Rnthe definition of differential operators extends to

C∞(M). We write

D(M) = C∞

c(M)and E(M) = C∞(M).

If Kis a compact subset of M, we denote by DK(M)the subset of functions in

D(M)with support in K.

For an open set Vof Rnthe spaces E(V)are topologized by the seminorms

kfkC

m, as Cruns through the compact subsets of Vand kruns through N0. If

(U, ϕ)runs through all local coordinate systems on M, this gives a topology of

E(U)with the property that a sequence fnin E(U)converges to 0if and only if

for each differential operator Don U, the sequence Dfn→0uniformly on each

compact subset of U. It follows that the topology of E(U)is independent of the

coordinate system.

The space E(M)is provided with the weakest topology for which the restric-

tions f7→ f|U, when (U, ϕ)runs through the local coordinate systems of M,

are continuous. By the countability assumption, we may restrict the (U, ϕ)to

a countable family of charts (Uj, ϕj). It follows that E(M)is a Fréchet space

and again the topology is described by uniform convergence (of all derivatives)

2 Preliminaries

on compact subsets. Since Mis the union of an increasing sequence of compact

subsets, this implies that D(M)is dense in E(M).

When Kis a compact subset of M, the space DKis given the topology induced

by E(M). As a closed subspace of E(M)it is a Frechet space.

A linear functional Ton D(M)is called a distribution if for any compact

subset K⊂Mthe restriction of Ton DK(M)is continuous. The set of distri-

butions is denoted by D0(M). We often write RMf(m)dT(m)instead of T(f).

The space D(M)is given the inductive limit topology of the spaces DK(M)

by taking as a fundamental system of neighborhoods of 0the convex sets W

such that for each compact subset K⊂Mthe space set W∩DK(M)is a

neighborhood of 0in DK(M). It follows that D0(M)is the dual space of D(M).

A distribution Tis said to vanish on an open set V⊂Mif T(f) = 0 for all

f∈D(V). The support of Tis the complement of the largest open subset of M

on which Tvanishes. Let E0(M)denote the set of distributions with compact

support. The restriction of a functional from E(M)to D(M)identifies the dual

of E(M)with E0(M)(cf. [GGA], p. 240).

If Nis another manifold and ϕis a diffeomorphism of Mand Nand if

f∈D(N),g∈E(N),T∈D0(M),D∈E(M), we write

gϕ−1=g◦ϕ, Tϕ=T(fϕ−1), Dϕ(g)=(D(gϕ−1))ϕ.

If ϕis a diffeomorphism of Monto itself, then Dis said to be invariant under

ϕ, if Dϕ=D, that is

Dg = (D(g◦ϕ)) ◦ϕ−1for all g∈E(M).

Given a measure µon M, the space E(M)is imbedded in D0(M)associating

with f∈D(M)the distribution

f7→ If:= g7→ ZM

f g dµ(2.33)

on M. We call this the canonical imbedding of functions into distributions.

2.3.1 The Laplace-Beltrami operator

Let Mbe a pseudo-Riemannian manifold with pseudo-Riemannian structure g

and let ϕ:q7→ (x1(q), ..., xn(q)) be a coordinate system valid on an open subset

U⊂M. As customary we define the functions gij,gij and gon Uby

gij =g(∂

∂xi

,∂

∂xj

),X

gijgjk =δik, g =|det(gij)|.(2.34)

In this section we write h|iin place of gand extend it C-bilinearly to complex

vector fields. Each f∈C∞(M)gives rise to the vector field grad f(gradient of

f) defined by

hgrad f|Xi=Xf (2.35)

2 Preliminaries

for each vector field X.

On the other hand, if Xis a vector field on M, the divergence of Xis the

function on Mwhich on Uis given by

div(X) = 1

√gX

∂i(pgXi),(2.36)

if X=PiXi(∂/∂xi)on U. Then div(X)is well-defined ([GGA], p.243) and

independent of the coordinate system.

The Laplace-Beltrami operator on Mis defined by

Lf = div grad f, f ∈E(M).(2.37)

In terms of local coordinates one has (loc. cit., p.245)

Lf =1

√gX

∂k X

gikpg∂if!,(2.38)

so Lis a differential operator on M. The Laplace-Beltrami operator Lof a

pseudo-Riemannian manifold Mis symmetric:

u(x)(Lv)(x)dx =ZM

(Lu)(x)v(x)dx, u ∈D(M), v ∈E(M),(2.39)

where dx is the Riemannian measure on M. If Φis a diffeomorphism of M, then

Φleaves the Laplace-Beltrami operator invariant if and only if it is an isometry.

Let Mbe an m-dimensional Riemannian manifold and pa point in M. Given

normal coordinates (x1, . . . , xm)around psuch that (∂/∂xi)p(1≤j≤m) is an

orthonormal basis of the tangent space at p, then the Laplace-Beltrami operator

Lof Mis given at pby ([DS], p. 330)

(Lf)(p) = X

∂2f

∂x2

(p), f ∈E(M).(2.40)

Suppose that Mis a compact Riemannian manifold of dimension m≥2. Let

ddenote the distance function on Mand write

(f1|f2) = ZM

f1(x)f2(x)dx, f1, f2∈L2(M),(2.41)

for the customary L2-product of M. Given λ∈C, define the eigenspace Eλby

Eλ={u∈E(M) : Lu =λu}(2.42)

and Λthe spectrum

Λ = {λ∈C:Eλ6= 0}.(2.43)

Then ([War70], Chapter 6)

2 Preliminaries

(a) Λis a discrete subset of Cand λ≤0for each λ∈Λ.

(b) Each eigenspace Eλis finite-dimensional: dim Eλ<∞for each λ.

in L2(M)such that each Eλis spanned by some of the ϕi. Then, if

f∈L2(M),

f=∞

0hf, ϕniϕn,(2.44)

where the sum converges in L2(M).

(d) If f∈E(M), the expansion in (c) converges absolutely and uniformly.

2.3.2 Harish-Chandra’s isomorphism and radial parts

Suppose His a closed subgroup of Gwith Lie algebra h. Let D(G/H)be the

algebra of differential operators on G/H which are invariant under the trans-

lations τ(g) : xH 7→ gxH (g∈G) of G/H onto itself. We write D(G)instead

of D(G/ {e}). For g∈G, let ρgdenote the right-translation by gin G. Then

define

DH(G) = {D∈D(G) : Dρh=Dfor all h∈H}.(2.45)

Write π:G→G/H. If fis a function on G/H, we put e

f=f◦π, so that e

fis

a function on G. Given u∈DK(G)and f∈E(G/K), let Du∈D(G/K)denote

the operator defined by (Duf)∼=ue

f. Then we have ([GGA], p. 285):

Theorem 2.41. The mapping µ:u7→ Duis a homomorphism of DK(G)onto

D(G/K). The kernel of µis DK(G)∩D(G)k.

Recall the Iwasawa decomposition G=KAN. Let D(A)denote the algebra of

translation-invariant differential operators (with constant coefficients) on Aand

let DW(A)⊂D(A)denote the subalgebra consisting of W-invariant differential

operators on A. If D∈D(G), there is ([GGA], p. 302) a unique element

Da∈D(A)such that

D−Da∈nD(G) + D(G)k.(2.46)

If νis a linear function on awe denote by eν:A→Cthe function a7→ eν(log(a)).

Let ◦denote the composition of differential operators. The mapping

γ:D7→ e−ρDa◦eρ

is a surjective homomorphism of DK(G)onto DW(A)with kernel DK(G)∩D(G)k

([GGA], 304). The next theorem ([GGA], 306) involves Harish-Chandra’s iso-

morphism Γ:

2 Preliminaries

Theorem 2.42. Let µdenote the isomorphism from Theorem 2.41. Consider

the diagram

DK(G)

yyr

rγ

D(G/K)Γ//DW(A).

Then γfactors through µto yield an isomorphism Γof algebras

Γ : D(G/K)→DW(A),(2.47)

given by Γ(µ(D)) = γ(D)for D∈DK(G).

When the nilpotent subgroup Nof Gacts on the symmetric space G/K, the

orbits are transversal (in the sense of [GGA], Ch. II, §3 (29)) to the submanifold

A·o([GGA], p. 266). Thus if Dis a differential operator on X, it follows

from [DS], Ch. II, Theorem 3.6 that there is a uniquely determined differential

operator ∆N(D)on A·osuch that for each N-invariant function on G/K

(Df)(a·o) = (∆N(D)f|A·o)(a·o),(2.48)

where f|A·odenotes the restriction of fto A·o. The operator ∆N(D)is called

the radial part of D. The isomorphism (2.47) is then given by ([GGA], p. 306)

Γ(D) = e−ρ∆N(D)◦eρ.(2.49)

In particular (loc. cit.), for the Laplacian LXon X=G/K we have

Γ(LX) = LA−hρ, ρi,(2.50)

where LAdenotes the Laplace operator of the submanifold A·oof G/K.

2.3.3 Joint eigenfunctions and joint eigenspaces

If Vis a finite-dimensional vector space over R, the symmetric algebra S(V)

over Vis defined as the algebra of complex-valued polynomial functions on the

dual space V∗([GGA], p. 280). If X1, ..., Xnis a basis of V, then S(V)can be

identified with the commutative algebra of polynomials

ak1...knXk1

1...Xkn

n.(2.51)

Let Ube any Lie group with Lie algebra u. Consider the exponential mapping

exp : u→U, which maps a line RXthrough 0in uonto the one-parameter

subgroup t7→ exp(tX)of U. As usual, if X∈u, let e

X∈D(U)([GGA], p.280)

denote the vector field of Ugiven by

Xf)(g) = X(f◦λg) = d

dtf(gexp tX)t=0

, f ∈E(G).(2.52)

The relation between S(u)and D(U)is as follows ([DS], p. 281):

2 Preliminaries

Theorem 2.43. There exists a unique linear bijection λ:S(u)→D(U)such

that λ(Xm) = e

Xmfor all X∈uand all m∈N0.

Theorem 2.43 states that the algebra D(U)of translation invariant differen-

tial operators on Uis generated by the e

X(X∈u). The mapping λis called

symmetrization and identifies the commutative algebras S(a)and D(A). Fur-

ther, it identifies the set S(a)Wof W-invariants in S(a)with the set DW(A)of

W-invariant differential operators on A·owith constant coefficients.

Given a homomorphism χ:D(G/K)→Cwe introduce the joint eigenspace

Eχ(X) = {f∈E(G/K) : Df =χ(D)ffor all D∈D(G/K)}.

We know from (2.47) that D(G/K)∼

=S(a)W. Since D(A)is a commutative

polynomial ring, each ν∈a∗

Cextends uniquely to a homomorphism of D(A)into

C, denoted by D7→ D(ν). We then have ([DS], Chapter III, Lemma 3.11):

Lemma 2.44. The homomorphisms of S(a)Winto Care precisely

χµ:P7→ P(µ),

where µis an element of a∗

It follows that the characters of D(G/K)(and hence the joint eigenspaces)

are parameterized by the orbits of Win a∗

C: Given λ∈a∗

Cwe define

Eλ(X) = {f∈E(X) : Df = Γ(D)(iλ)ffor all D∈D(X)}.(2.53)

Lemma 2.44 implies that each Eχ(X)is given by a suitably chosen Eλ(X).

Definition 2.45. Let λ∈a∗

Cand b∈B. We define

eλ,b :X→C, z 7→ e(iλ+ρ)hz,bi.(2.54)

The exponential functions eλ,b are called non-Euclidean plane waves.

Recall our notation b∞=eM ∈K/M. Let λ∈a∗

C. The function

eλ,b∞:G/K →C, gK 7→ e−(iλ+ρ)H(g−1)(2.55)

is N-invariant and its restriction to A·ois given by eλ,b∞|A·o(a·o) = e(iλ+ρ)(log a).

By (2.48) and (2.49), if D∈D(G/K),

(Deλ,b∞)|A·o= ∆N(D)(eλ,b∞)|A·o

= (eρΓ(D)◦e−ρ)(eλ,bo)|A·o

= Γ(D)(iλ)(eλ,bo)|A·o.

2 Preliminaries

Hence Deλ,b∞= Γ(D)(iλ)eλ,b∞, since both sides are N-invariant. In general,

when b=kM ∈K/M is arbitrary, then eλ,b(x) = eλ,b∞(k−1·x), so the K-

invariance of Dimplies

Deλ,b = Γ(D)(iλ)eλ,b (2.56)

for all λ∈a∗

C,b∈Band D∈D(G/K). It follows that each eλ,b is a joint

eigenfunction and belongs to Eλ(X). Moreover, (2.56) explains why one takes

iλ instead of λin the definition (2.53) of the Eλ(X). Finally, (2.50) implies

LXeλ,b = Γ(LX)(iλ)eλ,b =−(hλ, λi+hρ, ρi)eλ,b.(2.57)

This explicit formula for the eigenvalues of the Laplacian is of particular impor-

tance and will be applied a couple of times in the following sections.

Remark 2.46. A Riemannian manifold Xwith distance function dis called

two-point homogeneous if whenever d(p, q) = d(p0, q0), then there is an isometry

gof Xsuch that g(p) = p0and g(q) = q0. A Riemannian symmetric space of the

noncompact type is two-point homogeneous if and only if its real rank is one.

If X∼

=G/K is a two-point homogeneous space, then D(G/K)is generated by

the Laplacian, that is the algebra of invariant differential operators consists of

the polynomials in the Laplace-Beltrami operator ([DS], p. 288).

2.4 The classical examples

It is always useful to have concrete examples in mind. The classification of

globally symmetric spaces of noncompact type is the same as the classification

of semisimple Lie groups. As often in Lie group theory, the classification contains

a finite number of infinite lists (as the one of special linear groups SLn(R)for

n≥2), the so-called classical groups, and a finite set of “exceptional” examples.

2.4.1 Hyperbolic spaces and their realizations

For rank one symmetric spaces, there are three lists of classical spaces: Real,

complex, and quaternionic hyperbolic spaces. There is only one exceptional one,

the Cayley hyperbolic plane. For the latter we refer to the standard literature

on exceptional Lie groups and Lie algebras, for example [D78]. In this Section

we describe the realizations of the classical hyperbolic spaces. We follow [DH97].

Let F∈ {R,C,H}denote the field of real numbers, complex numbers, or the

quaternions. On Fn+1, regarded as a right-vector space over F, we consider the

Hermitian form

[x, y] = y0x0−y1x1−···−ynxn.

Let G=U(1, b;F)be the group of (n+ 1) ×(n+ 1) matrices with coefficients

in Fwhich preserve this Hermitian form. The group Gacts on the projective

2 Preliminaries

space Pn(F)and the stabilizer of the line generated by the vactor (1,0,...,0)

is the group K=U(1; F)×U(n;F), which is compact. We call X=G/K a

hyperbolic space. Xis a Riemannian symmetric space of the noncompact type

of rank one. By πwe denote the natural projection map

π:Fn+1 \{o} → Pn(F).

The hyperbolic space Xis then the image under πof the open set

x∈Fn+1 : [x, x]>0.

On Fnwe have the inner product (x, y) = Pjyjxjwith norm kxk= (x, x)1/2.

Let B(Fn)denote the unit ball in Fn. Then the space Xcan also be realized as

the unit ball in Fn. In fact, the map

x∈Fn+1 : [x, x]>0→Fn

given by x7→ y, where yp=xpx−1

0, defines, after going to the quotient space,

a real analytic bijection of Xonto B(Fn)and Gacts transitively by fractional

linear transformations ([DH97]).

Let ddenote the dimension of Fover R, so d= 1,2or 4respectively. On

{x∈Fn: [x, x]>0}we consider the Riemannian metric

ds2=−[dx, dx]

[x, x].

This metric is invariant under x7→ xλ (λ∈F\{0})and thus defines a Rieman-

nian metric on X, which is invariant under G, of signature (dn, 0).

We can now describe the group theoretical deompositions of G. Let Jbe the

(n+ 1) ×(n+ 1) diagonal-matrix

J:= 





−1

1...





.

it will turn out that this matrix is a representative in M0for longest Weyl group

element w∈W. For any (n+ 1) ×(n+ 1)-matrix Xwith coefficients in Fwe

set X∗:= JXtrJ. The Lie algebra gof Gconsists of matrices Xwhich satisfy

X+X∗= 0. These are the matrices of the form

X=Z1Z2

tr Z3,

where Z1and Z3are anti-Hermitian and Z2is arbitrary. The involutive auto-

morphism θof gis given by

θ(X) := JXJ.

2 Preliminaries

This θis the Cartan involution with the usual decomposition g=k+pinto

eigenspaces to the eigenvalues +1 and −1. The space kis the Lie algebra of the

subgroup K=U(1; F)×U(n;F).

Let Lbe the element

L:= 

0 0 1

0 0n−10

1 0 0

∈g.

Then L∈pand a:= RLis a maximal abelian subspace of p. The centralizer

of Lin kis

m:= 





u0 0

0v0

0 0 u

:u∈F, u +u= 0, v ∈u(n−1; F)



,

where u(n−1; F)denotes the Lie algebra of U(n−1,F). Let α:= 1 The nonzero

eigenvalues of Lare ±αif F=Rand ±α, ±αif F=Cor F=H. The root-space

gαconsists of the matrices

X=

0z∗0

z0n−1−z

0z∗0

,

where zis an (n−1) ×1-matrix with coefficients in F, and where z∗=−ztr.

We have mα:= dim(gα) = d(n−1). The space g2αconsists of matrices of the

form

X=

w0−w

0 0n−10

w0−w

,

where w∈Fwith w+w= 0. Then m2α:= dim(g2α) = d−1. We have

g=g−2α+g−α+a+m+gα+g2α.

The subgroup A= exp(a)of Gis given by the matrices

at:= 

cosh t0 sinh t

0 idn−10

sinh t0 cosh t

,

where t∈R. The centralizer of Ain Kis the subgroup Mof matrices



u0 0

0v0

0 0 u

,

2 Preliminaries

where u∈F,|u|= 1,v∈U(n−1; F). The Lie algebra of Mis m. The subspace

n=gα+g2αis a nilpotent subalgebra of gand the Lie algebra of the analytic

subgroup Nof Ggiven by the matrices

n(w, z) := 

1 + w−1

2[z, z]z∗−w+1

2[z, z]

zidn−1−z

w−1

2[z, z]z∗1−w+1

2[z, z]

,

where w∈F,w+w= 0, where zis an (n−1) ×1-matrix with coefficients in

Fand with z∗=−ztr. If

z=









, z0=









,

then [z, z0] = −z0

2z2−···−z0

nzn. The composition law in Nis

n(w, z)·n(w0, z0) = n(w+w0+Im[z, z0], z +z0).

In particular, since [z, z]is real, the inverse of n(w, z)is n(−w, −z). The sub-

group Anormalizes N:

atn(w, z)a−t=n(e2tw, etz).

The parameter ρis given by ρ=1

2(mα+m2α). The Iwasawa decomposition

reads G=KAN =NAK. Each g∈Gcan be written g=kexp H(g)n, where

k∈K,n∈N, and H(g)∈a. Let |·|denote the norm in F.

Lemma 2.47. Let g= (gi,j)with i, j = 0,1, ..., n be an element in G=

U(1, n;F). Then H(g) = tL, and t= ln |g0,0+g0,n|.

Proof. Set f(g) := ln |g0,0+g0,n|. Let bt:= exp(tL)∈A. Then

ln |(bt)00 + (bt)0n|= log |cosh t+ sinh t|

= ln(et)

=t.

Hence H(g) = f(g). Moreover, f(g)is left-K-invariant, since k∈K=

U(1; F)×U(n;F)) is unitary, and right N-invariant (this follows from the explicit

expression of n(w, z)). Hence f(g) = H(g)for all g∈G.

An explicit computation shows the following: If g=n(w, z), then

|(gw)00 + (gw)0n|2=|1−2w+ [z, z]|2

= (1 + [z, z]2)2+ 4w2,

since [z, z]is real and wis purely imaginary. This formula is even in zand w,

so considering n(−w, −z) = n(w, z)−1instead of n(z, w)gives the same result.

2 Preliminaries

Corollary 2.48. Let X=G/K be a hyperbolic space. Then H(nw) = H(n−1w),

where n∈N,G=KAN, and where w∈Wdenotes the longest Weyl group

element.

Remark 2.49. The formula H(nw) = H(n−1w)can alternatively be shown as

follows: The set of positive roots P+consists of αand possibly 2α. Recall

that mαand m2αdenote the multiplicities of these roots. We write B(·,·)for

the Killing form and put |Z|2=−B(Z, θZ)for Z∈g. If n∈Nwe write

n= exp(X+Y), where X∈g−αand Y∈g−2α. Set

c−1:= 4(mα+ 4m2α).

Then by [GASS], p. 180, we have

eρH(n)= [(1 + c|X|2)2+ 4c|Y|2]1

4(mα+2m2α).

We always have wNw−1=N, although conjugation with wdoes not have to

coincide with the involution θin all cases (it is true for the classical hyperbolic

spaces). It follows that the inverse of n= exp(X+Y)is given by n= exp(−X−

Y). In particular, the formula for eρH(n)is even in Xand Y, so H(n) = H(n−1)

for all n∈N. This implies H(nw) = H(n−1w)for all n∈N, since H(·)is

left-K-invariant.

2.4.2 The special linear groups

The groups G=SLn(R)are generic examples for higher rank spaces. In partic-

ular, if K=SOn(R), then G/K is a Riemannian symmetric space of the non-

compact type of rank n−1. We will briefly recall the Iwasawa-decomposition

components of this group and give a counterexample for the formula H(nw) =

H(n−1w)we already analyzed in the case of rank one spaces. The interest of

the function n7→ H(nw)arises in the fact that it is the phase function of sev-

eral integrals, such as the Harish-Chandra’s c-function, and another family of

operators we will consider in Section 6.

Let G=SLn(R). The subgroup Aarising in the Iwasawa decomposition

consists of the n×n-diagonal matrices

a:= 





a1...





,

where a1···an= 1 and aj>0for all 1≤j≤n. The nilpotent subgroups N

and Nare given by upper, respectively lower, triangular matrices with 10sin

the main diagonal. The subgroup M0of Kis generated by the subgroup Mand

2 Preliminaries

by the diagonal-matrices

si:=







1...







where the matrix

s:= 0 1

−1 0

is placed in the i−th and (i+ 1) −th rows. The Weyl group W(imbedded

into the subgroup M0) is generated by the matrices si. The action of Won

Ais defined by the formula w0·a:= w0aw0−1(w0∈W,a∈A). The group

Wcoincides with the symmetric group Snand therefore has n!elements. The

matrix wwith all zero entries, except for the entries (w)k,n−k+1 =±1, is the

longest element in W. It permutes the entries akand an−k+1 (k= 1,2, . . . , n)

of the matrices a=diag(a1, a2, . . . , an∈A. Moreover, we have N=wNw−1.

Let G=SL3(R). We will now find an n∈Nsuch that H(nw)6=H(n−1w).

An element a∈Ahas the form

a=

es0 0

0et0

0 0 e−s−t

,

where s, t ∈R. The longest Weyl group element w∈Wis

w=

001

0−1 0

100

.

We fix an element n∈N. Then there are d, e, f ∈Rsuch that

n=

1d e

0 1 f

0 0 1

.

Multiplying out we find

nw =

e−d1

f−1 0

1 0 0

and n−1w=

df −e d 1

−f−1 0

1 0 0

.

2 Preliminaries

Now suppose that nw =˜

k˜a˜nis written corresponding to the Iwasawa decom-

position, where ˜a=a(s, t)as above. Then ˜

k=nw˜n−1˜a−1∈SO3(R)yields

e2+f2+ 1 = e2s.(2.58)

Similarly, if n−1wis Iwasawa decomposed with A-part a(s0, t0), then

(df −e)2+f2+ 1 = e2s0.(2.59)

If we now assume that H(nw) = H(n−1w)then in particular s=s0. The

equations (2.58) and (2.59) have solutions for suitable chosen d, e, f and s, but

surely not for all choices. For example, the equations contradict if d=e=f= 1,

which shows that H(nw) = H(n−1w)is not a general property in SL3(R). The

method used here can be extended to all special linear groups SLn(R)for n≥3.

We always have H(nw) = H(n−1w)in the group SL2(R)(see Section 6.4).

3 Component computations

For later reference, we outhouse long and technical computations.

3.1 Some integral formulas

If Uis a Lie group with closed subgroup Vand with a left-invariant positive

measure on Vwe put

F(uV ) := ZV

F(uv)dv, F ∈Cc(U).(3.1)

Note that this factorization e

Fis not the same as the lift F◦πfrom the preceding

sections. The mapping F7→ e

Fis a linear and surjective mapping of Cc(U)onto

Cc(U/V )([DS], p. 91). In what follows, we will often use the following integral

formula due to Harish-Chandra ([DS], p. 197).

Lemma 3.1. Let g∈G. Then

f(k(g−1k))dk =ZK

f(k)e−2ρH(gk)dk, f ∈C(K).(3.2)

Hence (Tg)∗(dk) = e−2ρ(H(gk))dk, where (Tg)∗(dk)denotes the pull-back mea-

sure corresponding to the G-action on K. We write dk(gk)

dk =e−2ρ(H(gk)) to express

the Jacobian |det dTg(k)|. We will need a similar formula for the quotient K/M.

Therefore first observe that

(F◦T−1

g)(km)dm =e

F(k(g−1k)M)

F(T−1

g(kM))

F◦T−1

g(kM).

Hence

(F◦T−1

g)∼(kM) = e

F◦T−1

g(kM).(3.3)

Recall that the Iwasawa projection g7→ H(g)is M-bi-invariant. It follows that

the Jacobian e−2ρH(gk)of the action of gon Kis a function on K/M.

Corollary 3.2. The Jacobian of Tg:K/M →K/M, kM 7→ k(gk)M, is

|det dTg(kM)|=e−2ρH(gk).

Proof. We need to show that for each f∈C(K/M)

ZK/M

(f◦T−1

g)(kM)dkM =ZK/M

f(kM)e−2ρH(gk)d(kM).(3.4)

3 Component computations

Select F∈C(K)such that f=e

F. Then by 2.4 and the M-equivariance of Tg

ZK/M |det dTg(k)|f(kM)d(kM) = ZK/M |det dTg(k)|e

F(kM)d(kM)

=ZK/M |det dTg(k)|ZM

F(km)dmd(kM).

(Recall RMdm = 1.) Then the last expression equals

F(k)|det dTg(k)|dk =ZK

(F◦Tg−1)(k)dk

=ZK/M ZM

F◦Tg−1(km)dmd(kM),

and by (3.3) the last term equals RK/M (e

F◦T−1

g)(kM)d(kM), as desired.

Remark 3.3. The measure dp =dmdadn (in the notation of 2.1.6) is a left-

invariant measure on P=MAN. Let db denote the normalized K-invariant

measure on K/M =G/P. Using (2.6) we get ([GASS], p. 512) for f∈Cc(G)

f(g)e−2ρ(H(g))dg =ZG/P

db(gP)ZP

f(gp)dp. (3.5)

Corollary 3.2 states that dk(gk)M

d(kM)=e−2ρ(H(gk)). Given b=kM we use 2.37 to

find

d(g·b)

db =e−2ρ(H(gk)) =e+2ρ(hg−1K,kMi)(3.6)

=e−2ρ(hgK,g·kMi)=e−2ρ(hg·o,g·bi).(3.7)

It follows for f∈C(B)that

f(g·b)e2ρ(hg·o,g·bi)d(gb) = ZB

f(b)db =ZK

f(kM)dk. (3.8)

Remark 3.4. Let Cc(G)Mdenote the right-M-invariant functions in Cc(G).

Then Cc(G/M)∼

=Cc(G)Mvia (3.1), so M-invariant functions on Gare functions

on G/M and vice versa. Under G/K ×K/M ∼

=G/M a function g7→ f(gM)

on G/M becomes a function (gK, g ·B)7→ f(gM)on X×B.

Lemma 3.5. Let g∈Gand z∈X. Then

e−2ρH(gk)dk = 1 and ZB

e2ρhz,bidb = 1.(3.9)

3 Component computations

Proof. Apply Harish-Chandra’s formula (3.2) to f(k) = 1:

1 = ZK

f(k)dk =ZK

f(k(g−1k)dk =ZK

f(k)·e−2ρH(gk)dk.

Given z=g·o, where g∈G, we then find

e2ρhz,bidb =ZK

e−2ρH(g−1k)dk = 1,

as desired.

Recall the formulas d(g·b)

db =e−2ρ(hg·o,g·bi)and hg·z, g ·bi=hz, bi+hg·o, g ·bi.

Let f∈Cc(X×B). The G-invariance of dz then yields

ZX×B

f(z, b)e2ρhz,bidz db =ZX×B

f(g·z, g ·b)e2ρhg·z,g·bie−2ρhg·o,g·bidz db

=ZX×B

f(g·z, g ·b)e2ρhz,bidz db.

This proves:

Proposition 3.6. e2ρhz,bidz db is a G-invariant measure on X×B.

Hence by uniqueness, under the inverse of the G-equivariant diffeomorphism

G/M →X×B,gM 7→ (g·o, g ·M), the measure e2ρhz,bidz db is mapped into

a scalar multiple of d(gM), the G-invariant measure on G/M. To compute this

scalar c, select f(z)∈C∞

c(X)such that RXf(z)dz = 1. Lift f(g) := f(g·o)to

aK-invariant function on G. Then RGf(g)dg = 1. Also lift f(z, b) := f(z)to a

function on X×B, which is independent of b. Then f(g) = f(g·o, g ·M), so

c=cZG

f(g·o, g ·M)dg

=ZX×B

f(z, b)e2ρhz,bidz db

=ZX

f(z)ZB

e2ρhz,bidb dz. (3.10)

But RBe2ρhz,bidb = 1 and hence (3.10) equals RXf(z)dz = 1. Thus c= 1.

Corollary 3.7. Let f∈Cc(X×B). Then

ZG/M

f(g·o, g ·M)dg =ZX×B

f(z, b)e2ρhx,bidz db. (3.11)

3 Component computations

Given (z, b)∈X×Bwe can find g∈Gsuch that (z, b) = g·(o, M). Then

hg·o, g ·Mi=H(g). If we replace f(g·o, g ·M)by f(g·o, g ·M)e−2ρH(g)it

follows from (3.11) that

ZX×B

f(z, b)dz db =ZG/M

f(g·o, g ·M)e−2ρH(g)d(gM).(3.12)

One can directly prove (3.12) by using the G-invariance of dz and the integral

formulas (2.6) and (2.10):

ZX×B

f(z, b)db, dz =ZKZX

f(k·z, kM)dz dk

=ZKAN

f(kan ·o, kan ·M)dk da dn

=ZG/M

f(g·o, g ·M)e−2ρH(g)d(gM).

3.2 Derivatives corresponding to the Iwasawa

decomposition

We begin this subsection by recalling some material from [DKV83] concerning

derivatives of the Iwasawa projection. We will later apply these derivatives to

functions defined by the Iwasawa decomposition.

Let g, h ∈G. We write hg=ghg−1. Let U(g)be the universal enveloping

algebra of the complexification of g. The adjoint representation of Gon g

extends to a representation of Gon U(g)by automorphisms. We write ug=

Ad(g)u, if u∈U(g). Then we have

ugh = (uh)g,(uv)g=ugvg,(g, h ∈G, u, v ∈U(g)).

We shall view elements of U(g)as left invariant differential operators acting on

functions on G. To explain this interpretation, we now specify how an element

u=X1···Xr(Xj∈g), acts as a differential operator. Let f:G→Cbe a

function on Gand define

∂(u)f(g) := f(g;u) := ∂r

∂t1···∂tr|t1=...=tr=0

f(gexp t1X1···exp trXr).

If u∈U(g)is a complex number c∈C, then f(g;c) = cf(g).

The Iwasawa decomposition g=k⊕a⊕ngives rise to the decomposition

U(g) = (kU(g) + U(g)n)⊕U(a).

Therefore it makes sense to speak of the projection

Ea:U(g)→U(a).

3 Component computations

It is clear that this projection preserves the degree filtrations on both sides. Let

ε:U(g)→C

be a homomorphism that sends all elements of gto 0. We call ε(u)the constant

term of u∈U(g). If uhas zero constant term, then the same is true for Ea(u).

Since ais abelian, U(a)is canonically isomorphic to the symmetric algebra

(see Subsection 2.3.3) over a. Thus, on U(a)the degree filtration arises from

a grading. So in U(a)we may speak of the homogeneous components of an

element.

We can now give the main calculation on the derivatives of the Iwasawa

projection

H:G→a, kan 7→ log(a).

We will later use these formulas several times in applications of the method of

stationary phase.

Lemma 3.8. Let g∈G,b∈U(g). Then we have the formula

H(g;b) = ε(b)H(g) + Ea(bt(g))1,

where the subscript 1means the homogeneous component of degree 1, and t(g) =

a(g)n(g)is the “triangular part“ of the KAN Iwasawa decomposition of g∈G.

Proof. We copy the proof given in [DKV83], p. 337 to fix some notation. Since

His left-invariant under Kand right-invariant under Nwe have

H(1; u)=0 ∀u∈kU(g) + U(g)n.

Let g, h ∈Gand Iwasawa decompose g=kan. Then

H(gh) = H(kanh) = H(anh) = H(ht(g)t(g)) = H(ht(g)a(g)),(3.13)

where a(g) = a. The right hand side of (3.13) equals H(ht(g)) + H(a(g)) =

H(ht(g)) + H(g)and hence

H(g;b) = H(1; bt(g)) = ε(b)H(g) + H(1; Ea(bt(g))).(3.14)

But as H(exp X1···exp Xr) = X1+···+Xrfor Xj∈ait follows that for any

c∈U(a)we have H(1; c) = c1.

Lemma 3.9. Let h·,·i denote the Killing form of gand let H∈a. Let ϕdenote

the function

ϕ:G→R, g 7→ hH(g), Hi.

Let g∈Gand X∈g. Then

ϕ(g;X) = hXt(g), Hi=hX, Hn(g)−1i.(3.15)

3 Component computations

Proof. X∈ghas constant term 0, so H(g;X) = Ea(Xt(g)). The linear func-

tional λ(Y) = hY, Hi(Y∈a) has derivative λ(Y)and from the chain rule we

obtain for ϕ=λ◦Hthat

ϕ(g;X) = hEa(Xt(g)), Hi=hXt(g), Hi,

since ais orthogonal to k⊕nwith respect to the Killing form, while Ht(g)−1=

Hn(g)−1a(g)−1=Hn(g)−1, since a∈Afixes H, since ais abelian.

Given any Lie group G, we denote by Lgthe left translation by a group element

g∈G. The tangent vector to the curve t7→ gexp tX at gis dLg(X). Suppose

gis a direct sum g=u⊕v, where uand vare subalgebras of g(not necessarily

ideals). Let Uand Vbe the analytic subgroups of Gwith Lie algebras uand v.

Let α:U×V→Gdenote the mapping (u, v)7→ uv. We identify Uand Vwith

the subgroups (U, e)and (e, V )of the product group U×Vand we also identify

the tangent space T(u,v)(U×V)with the direct sum TuU+TvV(u∈U, v ∈V).

Let g·X(g∈G, X ∈g) denote the adjoint action. Let Y∈u, Z ∈v. We then

have

α(uexp tY, v) = uv exp(t v−1·Y), t ∈R

and

α(u, v exp tZ) = uv exp tZ, t ∈R.

It follows that the differential of αat (u, v)∈U×Vis given by

dα(u,v)(dLuY, dLvZ) = dLuv(v−1·Y+Z).(3.16)

Identifying TuU=uand TvV=vwe will from now on denote the differential

dα =α0of the product map αby

α0(u, v)(X, Y ) = v−1·X+Y, (3.17)

where u∈U, v ∈V, X ∈u, Y ∈v.

Corollary 3.10. The mapping αfrom above is everywhere regular.

Proof. h−1·Y+Z= 0 ⇔Y=−h·Z∈u∩v={0} ⇔ Y=Z= 0.

Assume that Gis a semisimple Lie group with Iwasawa decomposition G=

NAK. Then NA is a group, since Anormalizes N. We consider the following

mappings:

(i) σ1:N×A→NA,(n, a)7→ na,

(ii) σ2:NA ×K→NAK =G,(na, k)7→ nak,

(iii) σ3:N×A×K→NAK =G,(n, a, k)7→ nak,

3 Component computations

(iv) σ4:A×N→AN,(a, n)7→ an,

(v) σ5:A×N×K→AN ×K,(an, k)7→ ank,

(vi) σ6:A×N×K→ANK =G,(a, n, k)7→ ank.

Then σ3=σ2◦(σ1×idK). It follows from the chain rule that

σ0

3(n, a, k) : n×a×k→g

is given by

σ0

3(n, a, k)(X, Y, Z) = Ad(k−1)(Ad(a−1)X+Y) + Z,

where X∈n, Y ∈a, Z ∈k. Then

σ0

3(n, a, k)(X, Y, Z) = k−1a−1·X+k−1·Y+Z. (3.18)

Similarly, we obtain

σ0

6(a, n, k)(X, Y, Z) = k−1n−1·X+k−1·Y+Z, (3.19)

for (a, n, k)∈A×N×Kand (X, Y, Z)∈a×n×k.

Fix H∈a,H6= 0 and let h·,·i denote the Killing form. We introduce the

C∞-functions

(i) ϕ1:N×A×K→R, ϕ1(n, a, k) = hH(nak), Hi,

(ii) ϕ2:A×N×K→R, ϕ3(a, n, k) = hH(ank), Hi.

We factorize ϕ1in the following way: As above, let σ3:N×A×K→Gdenote

the map (n, a, k)7→ nak and let λ0denote the linear functional X7→ hX, Hion

a. Then ϕ1=λ0◦H◦σ3. For the differential of ϕ1we obtain from the chain

rule

ϕ0

1(n, a, k) = λ0

0(H(σ3(n, a, k))) ◦H0(σ3(n, a, k)) ◦σ0

3(n, a, k).

Now replace Xin (3.15) by k−1a−1·X+k−1·Y+Zfrom (3.18). Then ϕ0

1(n, a, k)

is a map

ϕ0

1(n, a, k) : T(n,a,k)(N×A×K) = n×a×k→TnakG=g→a→R

given by

(X, Y, Z)7→ hk−1a−1·X+k−1·Y+Z, Hn(nak)−1i.

We can now write nak =˜

k˜a˜ncorresponding to the Iwasawa decomposition.

Then

ϕ0

1(n, a, k)(X, Y, Z) = hk−1a−1·X+k−1·Y+Z, Hn(nak)−1i(3.20)

=h˜n·k−1·a−1·X, Hi+h˜n·k−1·Y, Hi+h˜n·Z, Hi.

For the derivatives of ϕ2, write ank =˜

k˜a˜n. Then (3.19) yields

ϕ0

2(a, n, k)(X, Y, Z)=h˜n·k−1·n−1·X, Hi+h˜n·k−1·Y, Hi+h˜n·Z, Hi.(3.21)

3 Component computations

3.3 Critical sets and Hessian forms

Let h·,·i denote the Killing form and let H∈a+with kHk= 1. We investigate

the critical set of the phase function ψ:a∗

+×N×A×K→R,

(µ, n, a, k)7→ µ(log(a)) −hH(nak), Hi,(3.22)

arising in Section 4for an oscillatory integral named Ua. We analyze the critical

set of ψand write it down explicitly in the case when X=G/K has rank one.

Viewed as a function on a∗

+×N×A×K/M, the critical set will then consist

of one single point. We then prove the non-degeneracy of the Hessian form of

ψat this critical point.

Note that in order to determine the critical set of ψ, we have to solve

dψ(µ, n, a, k) = 0.(3.23)

Written out, (3.23) is equivalent to the equations

(a) ∂ψ

∂µ (µ, n, a, k) = 0,

(b) ∂ψ

∂s |s=0(µ, n exp sX, a, k) = 0 for all X∈n,

∂t |t=0(µ, n, a exp tY, k) = 0 for all Y∈a,

(d) ∂ψ

∂θ |θ=0(µ, n, a, k exp θZ)=0 for all Z∈k.

Lemma 3.11. Let n⊥denote the orthogonal complement (w.r.t. the Killing

form) of nin g. Then n⊥∩p=a.

Proof. Let Z∈n⊥∩p. Write Z=Za+Zqcorresponding to the orthogonal

decomposition g=k+a+q. For Y∈nwe then have

0 = hZ, Y i=hZa+Zq, Y i=hZq, Y i,(3.24)

since a⊥n. It follows that Zq⊥g, so Zq= 0, so Z∈a. Conversely, if Z∈a,

then Z∈pand Z⊥n.

Lemma 3.12. Let X=G/K have rank one. If µ6= 1 or kM 6=M, then the

phase function ψgiven in (3.22)has no critical points in {µ}×A×N×{k}.

Proof. Suppose that (µ, n, a, k)is a critical point for A×N. Write nak =˜

k˜a˜n

corresponding to the Iwasawa decomposition. We rewrite the A-derivative given

in (3.20) as follows:

ϕ0

1(n, a, k)(0, H, 0) = h˜nk−1·H, Hi

=h˜

k˜a˜nk−1·H, ˜

k·Hi

=hnakk−1·H, ˜

k·Hi

=hn·H, ˜

k·Hi.

3 Component computations

Similarly we find for X∈n

ϕ0

1(n, a, k)(X, 0,0) = hn·X, ˜

k·Hi.

The assumption that (µ, n, a, k)is critical is then equivalent to the conditions

(a’) hn·H, ˜

k·Hi=µ,

(b’) hn·X, ˜

k·Hi= 0 ∀X∈n.

It follows from (b’) that ˜

k·H⊥n. But since also ˜

k·H∈p, Lemma 3.11 yields

k·H∈a. Hence ˜

k∈M0. (In higher rank, the same argument applies for H

regular.) Now equation (a’) yields

0< µ =hn·H, ˜

k·Hi=±hn·H, Hi=±1,

since n·H−H∈n. It follows that µ= 1 and that ˜

k=m∈M. Finally,

nak =m˜a˜nyields (by uniqueness of the Iwasawa decomposition) that k∈M,

and the lemma is proven.

3.3.1 Critical points

For H∈a, let ZN(H)denote the centralizer of Hin N. Recall that g=k+p,

where pdenotes the orthogonal complement (with respect to the Killing form)

of kin g.

Lemma 3.13. Let H∈a,n∈N. Then

n·H∈p⇐⇒ n∈ZN(H).

Proof. n·H∈pis satisfied if and only if hZ, n ·Hi= 0 ∀Z∈k. But a⊆p

yields hZ, Hi= 0 ∀Z∈kand since n∈Nwe obtain n·H∈H+n. Thus

0 = hZ, n ·Hi ∀Z∈k⇐⇒ 0 = hZ, n ·H−Hi ∀Z∈k

⇐⇒ n·H−H∈p∩n.

We may now use p∩n={0}, which follows from the fact that the elements of

pare semisimple, while the elements of nare nilpotent. Thus n·H=H, as

desired.

Assume that (µ, n, a, k)is a critical point in all variables. It follows from (a)

that log(a)=0, that is

a=e. (3.25)

We use the notation of (3.20) and Iwasawa decompose

nak =˜

k˜a˜n. (3.26)

Condition (d) yields

0 = −∂ψ

∂θ |θ=0(µ, n, a, k exp θZ) = h˜n·Z, Hi=hZ, H˜n−1i ∀Z∈k.(3.27)

It follows from Lemma 3.13 that ˜n∈NH=ZN(H), the centralizer of Hin N.

3 Component computations

Remark 3.14. (1) It is sufficient for (3.27) to be satisfied only for all Z∈m⊥,

the orthogonal complement of min k. This can be seen as follows: If X=

˜n·H−H=Xp+Xm∈(p+m)∩n, then 2Xm=X+θ(X)∈m∩m⊥={0},

so X∈p∩n={0}, so ˜n∈NH=ZN(H).

(2) However, given ˜n∈Nand Z∈m, set X= ˜n−1·H−H∈nand

Y=X+θ(X)∈m⊥. Since m⊥awe have 2h˜n·Z, Hi=hZ, ˜n−1·H−Hi+

hZ, ˜n−1·H−Hi=hZ, Xi+hZ, θ(X)i=hZ, Y i= 0, so (3.27) holds for

all Z∈mand ˜n∈N.

Next, recall a=eand also note that ˜n·H=His equivalent to ˜n−1·H=H.

We may then plug (3.20) into equation (b) above and obtain the condition

0 = −∂ψ

∂s |s=0(µ, n exp sX, a, k) = h˜n·k−1·a−1·X, Hi=hX, k ·Hi ∀X∈n.

It is immediate from Lemma 3.11 that

Lemma 3.15. Let 06=H∈a,k∈K. The following assertions are equivalent:

(i) hX, k ·Hi= 0 ∀X∈n,

(ii) k·H∈a.

3.3.2 Regular elements

Let from now on H∈a+be regular. Let λH∈a∗

+denote the linear functional

on agiven by λ0(X) = hX, Hifor X∈a(Killing form). Then λ0∈a∗

the dual positive Weyl chamber. Also H=Hλ0in the notation of the Riesz

representation (Section 2.1.4). As before, we study the critical set of

ψ:a∗

+×N×A×K→R,(µ, n, a, k)7→ µ(log(a)) −hH(nak), Hλ0i,

Let (µ, n, a, k)be a critical point of ψ. We already know a=e. Lemma 3.15

states k·H∈a. For regular elements we have the following refinement:

Lemma 3.16. Let 06=H∈a,k∈K. The following assertions are equivalent:

(i) hX, k ·Hi= 0 ∀X∈n,

(ii) k=m0∈M0, where M0is the normalizer of Ain K.

Since His regular, it follows that k=m0∈M0.

Next, we Iwasawa decompose nak =˜

k˜a˜n. Then by the above observations we

have ˜n∈NH. But since His regular, Remark 2.9 implies ˜n=e. Using a=e

and k=m0we then observe

nm0=nak =˜

k˜a˜n=˜

k˜a, (3.28)

3 Component computations

which implies (by uniqueness of the Iwasawa decomposition)

n=˜

k˜a(m0)−1=˜

km0−1˜

˜a∈N∩KA ={e},(3.29)

so n=e,˜

k=m0and ˜a=e.

Condition (c) above and (3.20) yield

0 = ∂ψ

∂t |t=0(µ, n, a exp tY, k) = µ(Y)−h˜n·k−1·Y, Hi.(3.30)

Evaluating this at the critical point (µ, n, a, k), where ˜n=eand k=m0, we get

µ(Y) = hY, m0·Hi ∀Y∈a.(3.31)

Recall that H∈a+induces the linear function λ0(Y) = hY, Hi(Y∈a) on a. It

follows that µ∈a∗

+is in the W-orbit of λ0∈a∗

+. Hence k=m0=m∈Mand

µ=λ0. We summarize this as follows:

Proposition 3.17. Let H∈a+be regular. Write λ0(Y) = hY, Hi(Y∈a).

The critical points (µ, n, a, k)of

ψ:a∗

+×N×A×K→R,(µ, n, a, k)7→ µ(log(a)) −hH(nak), Hλ0i,

are precisely

(µ, n, a, k) = (λ0, e, e, m), m ∈M. (3.32)

On the quotient a∗

+×N×A×K/M, the phase function ψhas exactly one critical

point, namely (µ, n, a, k) = (λ0, e, e, M).

Proof. We have seen that each critical point has this form. In the K-variable, ψ

is M-invariant, since H:KAN →ais invariant. The proposition follows.

3.3.3 The Hessian form

Let Xhave rank one. Then a=RH, where H∈a+is the unique vector such

that kHk= 1 (the norm on ainduced by the Killing form). Let λ0∈a∗

+denote

the linear functional on agiven by λ0(X) = hX, Hifor X∈a(Killing form).

Then a∗=Rλ0. Then λ0∈a∗

+, the dual positive Weyl chamber. Also H=Hλ0.

We compute the Hessian form of ψfor the rank one case. First, we note that

the second order derivatives of ψare clear if they contain at least one derivative

in direction µ. We will now also compute the Hessian matrix of the C∞-function

ϕ1:N×A×K→R, ϕ1(n, a, k) = hH(nak), Hi(3.33)

at the points (e, e, m)∈N×A×Kand conclude that as a function on the

quotient a∗

+×N×R×K/M the phase function ψhas a non-degenerate Hessian

3 Component computations

form at the critical point (λ0, e, e, M). Note that under a∼

=Rwe identify λ0

with 1∈R.

Recall that H:KAN →ais a smooth mapping and hence ϕ1is smooth as

well. Derivatives of ϕ1are given by

ϕ0

1(n, a, k)(X, Y, Z)=h˜n·k−1·a−1·X, Hi+h˜n·k−1·Y, Hi+h˜n·Z, Hi,(3.34)

where (X, Y, Z)∈n×a×kand where nak =˜

k˜a˜nis written corresponding to

the Iwasawa decomposition.

The Hessian form is bilinear, hence we must prove its non-degenerateness

only with respect to a certain basis, which we will later construct. We will

now successively fill up the following 3×3-matrix of question marks, where

each row and each column corresponds to the Lie algebra direction in which we

differentiate:







µ n atk

µ0 0 1 0

n0 ? ? ?

at1 ? ? ?

k0 ? ? ?





(3.35)

Because of symmetry we only have to consider the following 6cases:

(1) X, X0∈n. Then

XX0(hH(nak), Hi)|n=e,a=e,k=m

dt|t=0hn(nexp(tX)ak)k−1a−1·X0, Hi|n=e,a=e,k=m

dt|t=0 0 = 0,

since at the critical points we have n=e,k=m∈Mand a=e, so the left

vector in the Killing form is an element of nfor each tin a neighborhood of

0∈Rand n⊥awith respect to the Killing form.

(2) X∈n,Y∈a. Then

dt|t=0hn(na exp(tY )k)k−1a−1·X, Hi|n=e,a=e,k=m= 0

as above.

(3) Y, Y 0∈a. Then

dt|t=0hn(na exp(tY 0)k)·Y, Hi|n=e,a=e,k=m=d

dt|t=0hY, Hi= 0.

(4) Y∈a,Z∈k. Then

dt|t=0hn(na exp(tY )k)·Z, Hi|n=e,a=e,k=m= 0,

3 Component computations

since the left vector in the Killing form is an element of kand k⊥awith respect

to the Killing form (recall that Mand Acommute elementwise).

(5) Z, Z0∈k. Then

dt|t=0hn(nak exp(tZ0)) ·Z, Hi|n=e,a=e,k=m= 0,

since k⊥a.

(6) X∈n,Z∈k. Then

dt|t=0hn(nak exp(tZ)) exp(−tZ)k−1a−1·X, Hi|n=e,a=e,k=m

dt|t=0hexp(−tZ)m−1·X, Hi

dt|t=0he

X, exp(tZ)·Hi(e

X=m−1·X∈n)

=he

X, [Z, H]i,(3.36)

since Mnormalizes N. This vanishes for Z∈m, since then [Z, H] = 0.

We now analyse the last expression with respect to the transversal direction

m⊥. If α > 0is a positive root, we find vectors Xα∈gαsuch that

Z=X

α>0

(Xα+θXα).(3.37)

Plugging (3.37) into the commutator-bracket of gwe obtain

[Z, H] = "X

α>0

(Xα+θXα), H#

α>0−α(H)Xα+α(H)θXα

α>0

α(H) (θXα−Xα)∈p.(3.38)

Next, we write e

X∈nas a sum

X=X

α>0e

Xαe

Xα∈gα.(3.39)

Plugging (3.38) and (3.39) into the Killing form (3.36) we obtain

X, [Z, H]i=*X

α>0

α(H) (θXα−Xα),X

β>0e

Xβ+(3.40)

α>0

α(H)DθXα,e

XαE,(3.41)

3 Component computations

since hgα,gβi 6= 0 ⇔α+β= 0 and since θXα∈g−α.

Now let {X1, ..., Xs}be a basis for nconsisting of root vectors such that

hXj, θXii=δij. Then

{X1+θX1, ..., Xs+θXs}(3.42)

is a basis of m⊥. Hence e

Xand Z∈m⊥are linear combinations

X=X

ajXj, Z =X

bj(Xj+θXj).(3.43)

It follows that

X, [Z, H]i=X

αj(H)bjaj,(3.44)

where αj=αif Xj∈gα. Hence in this basis, for n×m⊥the second derivatives

X, [Z, H]iat the critical points are given by the invertible diagonal matrix

Q0:= 





α1(H)...

αs(H)





.(3.45)

Finally, with respect to this basis, the second derivatives of ψare

Q:=







sntm m⊥

s0 0 1 0 0

n0 0 0 0 −Q0

t1 0 0 0 0

m0 0 0 0 0

m⊥0−Q00 0 0







.(3.46)

We drop the m-rows and columns, which describe the stable direction.

Theorem 3.18. The phase function ψhas a non-degenerate Hessian form at

its critical point (λ0, e, e, M).

3.3.4 Another phase function

Let X=G/K have rank one and as usual, denote by H∈a+the unique unit

vector. We also need to determine the critical points of ψt:R+×A×N×K→R,

(µ, n, a, k)7→ t(µ−1) −hlog(a), Hi−µH(a−1n−1k).

First, dψt(µ, n, a, k) = 0 is equivalent to

(a) ∂ψt

∂µ (µ, n, a, k) = 0,

3 Component computations

(b) ∂ψt

∂t |t=0(µ, n, a exp tY, k)=0 for all Y∈a,

∂s |s=0(µ, n exp sX, a, k) = 0 for all X∈n,

(d) ∂ψt

∂θ |θ=0(µ, n, a, k exp θZ) = 0 for all Z∈k.

We may first consider the mapping ϕ2:A×N×K→R,

ϕ2(a, n, k) = hH(ank), Hi.

Given (X, Y, Z)∈a×n×k, the differential of ϕ2at (a, n, k)is (cf. Sec. 3.2)

ϕ0

2(a, n, k)(X, Y, Z)=h˜n·k−1·n−1·X, Hi+h˜n·k−1·Y, Hi+h˜n·Z, Hi.(3.47)

Assume that (µ, n, a, k)is a critical point of ϕ2and Iwasawa decompose ank =

k˜a˜n. Then by (3.47)

h˜n·Z, Hi= 0 for all Z∈k.

It follows follows from Lemma 3.13 that ˜n∈NH, where NHdenotes the cen-

tralizer of Hin N. Since G/K has rank one this yields ˜n=e. Again by (3.47)

we have that

h˜n·k−1·Y, Hi= 0 for all Y∈n.

It follows from Lemma 3.11 that k=m0∈M0, where M0is the normalizer of A

in K. Then

anm0=ank =˜

k˜a˜n=˜

k˜a,

and by uniqueness of the Iwasawa decomposition this implies

k=anm0˜a−1=⇒˜

k=m0.

Again by uniqueness of the Iwasawa decomposition we find

anm0=m0˜a=⇒n=e.

Now assume that (µ, n, a, k)is a critical point of ψt. Since the first two

summands in the definition of ψtare independent of kand n, it follows that

for the critical point of ψtwe have n=eand k=m0as well. Then by the

assumption, we have for the derivatives of ψtwith respect to aand µ(given by

equations (a) and (b) above)

(i) ∂ψt

∂µ =t−hH(a−1n−1k), Hi= 0,

(ii) ψa=−1 + µ·hm0−1·X, Hi= 0.

3 Component computations

Recall that we identify the unit vector H∈a+with the real number 1∈R+.

Condition (ii) yields

0<1/µ =hH, m0·Hi=±1,

since M0is acting by orthogonal transformations on a. It follows from µ > 0

that µ= 1 and m0=m∈M, where Mis the centralizer of Ain K. Evaluating

(i) at the critical point we obtain

t−hH(a−1m), Hi= 0 =⇒log(a) = −t.

Summarizing we have proven:

Proposition 3.19. The critical points (µ, n, a, k)of ψtare precisely

(µ, n, a, k) = (1, e, a−t, m), m ∈M.

On the quotient R+×N×A×K/M, the phase function ψthas exactly one

critical point, namely (µ, n, a, k) = (1, e, a−t, M).

The Hessian matrix of ψtat the critical points is then given by







µn a m m⊥

µ0 0 1 0 0

n0 0 0 0 −Q0

a1 0 0 0 0

m0 0 0 0 0

m⊥0−Q00 0 f(t)







,(3.48)

where Q0is as in (3.46) and all other computations are exactly as in Subsection

The Hessian form of this ongoing section, and where f(t)is the matrix

f(t) := ∂2

∂θ1∂θ2ha−t·o, kθ1Z,θ2Z0Mi|θ1=θ2=0,(3.49)

where Z, Z0∈m⊥and kθZ = exp θZ for small tand Z∈k(we can restrict this

to the critical point m=e, since all functions involved are M-invariant). We

can also rewrite f(t)in terms of (3.47) and the Killing form to obtain

f(t) = d

dθ|θ=0hn(atkθZ0)·Z, Hi.(3.50)

Example 3.20. If G=PSU(1,1) and K=PSO(2), then G/K is identifiesd

with the open unit disk D. Then ([Zel86], p. 103)

f(t) = −2 tanh(t) tanh(t+ 1)−26= 0 ∀t6= 0.(3.51)

3 Component computations

Note that given a∈Aand k∈K, the horocycle bracket ha·o, kMiequals the

Iwasawa projection −H(a−1k). It seems not to be easy to give a short derivation

for an explicit formula for the matrix f(t), but we observe the following: The

matrix coefficients of the principal series of representations of G(see Section 5)

may be expressed as integrals of the form

eλ(H(ak)) ϕ(k)dk. (3.52)

Here a∈Aand ϕis an analytic function on Kexpressed in terms of matrix

coefficients of representations of K, and the eigenvalue parameter is λ∈a∗

We can keep Re(λ) = νfixed and absorb the factor eν(H(ak)) into the amplitude

ϕ. We write ξ=Im(λ)∈a∗and denote by H=Hξ∈athe vector satisfying

ξ(Y) = hY, Hξiwith respect to the Killing form. Then (3.52) becomes

eihH(ak),Hiϕ(k)dk. (3.53)

If we replace Hby τH and let τ→ ∞, then the principle of stationary phase

states that the main contributions to the asymptotic expansion of (3.53) come

from the critical points of the phase function Fa,H on Kdefined by

Fa,H(k) = hH(ak), Hi,(k∈K).(3.54)

These functions have been studied in [DKV83] (for proofs see Sections 5 and 6

loc. cit.). Let Ka, respectively KH, denote the centralizer of ain K, respectively

of Hin K. The study of the critical points of Fa,H reveals that the critical

set of Fa,H is equal (for Xbeing of arbitrary rank) to the disjoint union of

smooth manifolds KawKH, where wruns through the Weyl group. Note that the

notation wKHmakes sense, as always M⊆KHfor all H∈a. The Hessians of

the Fa,H are tranversally non-degenerate to the critical manifolds. In particular,

if Xhas rank one, then the subgroup M=ZK(A)is a critical manifold for Fa,H

and its Hessian is non-degenerate in transversal direction. Since our matrix f(t)

equals the Hessian form of Fat,H we can summarize:

Theorem 3.21. The Hessian form ψtis non-degenerate at the critical point.

4 Equivariant pseudodifferential operators on

symmetric spaces

The Euclidean Fourier transform of a sufficiently regular function on Rnis

f(ξ) = (2π)−nZf(x)e−ix·ξdx. (4.1)

Writing Dj=−i(∂/∂xj), we differentiate the Fourier inversion formula

f(x) = Zb

f(ξ)eix·ξdξ

and get

Dαf(x) = Zξαb

f(ξ)eix·ξdξ,

where α∈Nn

0. Hence for a differential operator p(x, D) = P|α|≤kaα(x)Dα.

p(x, D)f(x) = Zp(x, ξ)b

f(ξ)eix·ξdξ;

The function

p(x, ξ) = X

α≤k

aα(x)ξα

is called the full symbol of the operator p(x, D). These observations lead to

the Euclidean version of pseudodifferential operators on the Euclidean space

Rn([Tay81], [Hor85]). As described in the introduction, pseudodifferential

operators can be very useful in determining the asymptotic behaviour of the

eigenvalues and eigenfunctions of the Laplace operator. In 1986, Steve Zelditch

([Zel86]) presented a calculus of pseudodifferential operators that in that case

of the unit disk Dand a corresponding compact hyperbolic surface XΓ= Γ\D,

where Γ⊂PSU(1,1) is a cocompact discrete subgroup, is best adapted for this

purpose. The main idea is to use Helgason’s non-Euclidean Fourier analysis in

place of the local Euclidean Fourier analysis in manifolds. An advantage of this

calculus lies in its equivariance and invariance properties: Γ-invariant symbols

define Γ-invariant operators on T∗D. Other objects of interest in ΨDO-theory,

such as lower terms in asymptotic expansions, are invariantly defined in this

calculus, too.

In this section we generalize parts of this calculus to symmetric spaces of the

noncompact type. Eventually we will have to restrict some results to the case

of rank one symmetric spaces.

4 Equivariant pseudodifferential operators on symmetric spaces

4.1 Non-Euclidean Fourier analysis

The non-Euclidean Fourier transform F([GASS]) converts sufficiently regular

functions fon X(e.g. f∈C∞

c(X)) into functions Ff =˜

fon a∗

C×K/M.

This integral transform was introduced by S. Helgason in 1965 ([Helg65]) and

shows a lot of analogies with the Euclidean Fourier-transform ([Hor83]). There

is an inversion formula, a Plancherel formula, and a non-Euclidean Paley-Wiener

theorem. Let fbe a complex valued function on X. Its non-Euclidean Fourier

transform Ff =˜

fis defined by

Ff(λ, b) := ˜

f(λ, b) := ZX

f(x)e(−iλ+ρ)hx,bidx (4.2)

for all λ∈a∗

C,b∈B, for which the integral exists.

Proposition 4.1. Let u∈C∞

c(X). Then ˜u(λ, b)is rapidly decreasing in λ.

Proof. We use (2.57) and iterate integration by parts via the Laplace operator

LX(see Section 2.3):

˜u(λ, b) = ZX

e(−iλ+ρ)A(z,b)u(z)dz

=ZX−1

hλ, λi+hρ, ρiLXe(−iλ+ρ)A(z,b)u(z)dz

=ZX−1

hλ, λi+hρ, ρie(−iλ+ρ)A(z,b)LXu(z)dz

=ZX−1

hλ, λi+hρ, ρik

e(−iλ+ρ)A(z,b)Lk

Xu(z)dz.

This proves the proposition.

As usual, we denote Harish-Chandra’s c-function by c(λ). Explicit formulas

for the Plancherel density |c(λ)|−2∈C∞(a)can be found in Section 2.1.7. We

introduce the notation

¯dλ =|c(λ)|−2dλ. (4.3)

Let w=|W|denote the order of the Weyl group. In analogy with the inversion

formula for the Euclidean Fourier transform we have ([GASS], pp. 225-226):

Theorem 4.2 (Fourier inversion formula).For each f∈D(X)the Fourier

transform is inverted by the formula

f(x) = w−1Za∗ZB

e(iλ+ρ)hx,bi˜

f(λ, b) ¯dλdb, x ∈X. (4.4)

4 Equivariant pseudodifferential operators on symmetric spaces

Let B(·,·)denote the restriction to aof the Killing form of g. Given λ∈a∗,

we denote by Hλ∈athe uniquely determined element such that

B(Hλ, H) = λ(H)∀H∈a(4.5)

Recall that we denote the dual positive Weyl chamber, that is the preimage

(under the mapping λ7→ Hλ)of the positive Weyl chamber a+, by

a∗

+=λ∈a∗:Hλ∈a+.(4.6)

The following theorem ([GASS], p. 227) is the symmetric space analog of the

Plancherel formula for the Euclidean Fourier transform.

Theorem 4.3 (Plancherel formula).The Fourier transform f(x)7→ ˜

f(λ, b)

extends to an isometry of L2(X)onto L2(a∗

+×B, |c(λ)|−2dλ db). For f∈L2(X),

the Plancherel formula reads

f1(x)f2(x)dx =w−1Za∗×B

f1(λ, b)˜

f2(λ, b)|c(λ)|−2dλdb (4.7)

Given λ∈a∗

C, we can find ξ, µ ∈a∗such that λ=ξ+iµ, where i=√−1.

We employ the notation Imλ=µand |λ|= (|ξ|2+|µ|2)1/2. A C∞-function

ψ(λ, b)on a∗

C×B, holomorphic in λ, is called a holomorphic function of uniform

exponential type if there exists a constant R≥0such that for each N∈N

sup

λ∈a∗

C, b∈B

e−R|Imλ|(1 + |λ|)N|ψ(λ, b)|<∞.(4.8)

We denote the space of ψsatisfying (4.8) by HR(a∗

C×B)and define

H(a∗

C×B) := [

R>0

HR(a∗

C×B).(4.9)

By H(a∗

C×B)Wwe denote the space of functions ψ∈H(a∗

C×B)satisfying

e(isλ+ρ)(A(x,b))ψ(sλ, b)db =ZB

e(iλ+ρ)(A(x,b))ψ(λ, b)db (4.10)

for all s∈W,λ∈a∗

Cand x∈X.

The following theorems ([GASS], Ch. III, Theorem 5.1 and [GASS], Ch. III,

Corollary 5.9) are the symmetric space versions of the Paley-Wiener theorems

for the Fourier transform and answers the questions concerning the range of the

Fourier transform.

Theorem 4.4. The Fourier transform f(x)7→ ˜

f(λ, b)is a bijection of D(X)

onto H(a∗

C×B)W.

4 Equivariant pseudodifferential operators on symmetric spaces

AC∞-function ψon a∗

C×B, holomorphic in λ, is called a holomorphic function

of uniform exponential type and slow growth if there exist constants R, C ≥0

and N∈Nsuch that

|ψ(λ, b)| ≤ C(1 + |λ|)NeR|Imλ|(4.11)

for all λ∈a∗

Cand b∈B. Given R≥0, let KR(a∗

C×B)denote the space of

these ψsatisfying (4.11) for some Nand C. We then define

K(a∗

C×B) := [

R≥0

KR(a∗

C×B).(4.12)

Let K(a∗

C×B)Wdenote the space of functions in K(a∗

C×B)satisfying 4.10.

Theorem 4.5. The distributional Fourier transform T7→ e

T, where

T(λ, b) = ZX

e(−iλ+ρ)hx,bidT(x),

is a bijection of E0(X)onto the space K(a∗

C×B)W.

4.2 Invariance and equivariance properties

In this section we describe important invariance properties of operators defined

using the non-Euclidean Fourier transform.

The group action of Gon Xinduces a translation of functions on X: Given

g∈Gand a function fon X, we denote by Tgfthe function Tgf(z) = f(gz). A

function a(z, λ, b)on X×a×Bis called invariant under translation (on X×B)

by gif and only if

a(gz, λ, gb) = a(z, λ, b)for all (z, λ, b).(4.13)

Functions on X×Bare identified with functions on G/M and we call a function

aon G/M invariant under translation by gif and only if a(ghM) = a(hM)for

all g, h ∈G. Let fbe a function on X×X. For g, h ∈Gwe define Tg,hfby

(Tg,hf)(z, w) := f(gz, hw). A function fon X×Xis called invariant under

g∈Gif and only if Tg,gf=f.

Let for a moment h·,·i denote the duality bracket of C∞

c(X). Given a distri-

bution uon Xwe define the distribution Tguon Xvia duality by

hTgu, vi:= hu, Tg−1vi, v ∈C∞

c(X).(4.14)

This definition is consistent with the imbedding (2.33)C∞

c(X),→E0(X): Given

a function u∈C∞

c(X)one has

hTgu, vi=ZX

u(gz)v(z)dz =ZX

u(z)v(g−1z)dz =hu, Tg−1vi,(4.15)

4 Equivariant pseudodifferential operators on symmetric spaces

since dz is G-invariant. If uis a distribution on the product space X×X,

we define the distribution Tg,huon X×Xvia duality on the algebraic tensor

product by defining it on the tensor products ϕ⊗ψ∈C∞

c(X×X), where

ϕ, ψ ∈C∞

c(X), by

hTg,hu, ϕ ⊗ψi:= hu, Tg−1ϕ⊗Th−1ψi.(4.16)

This definition is again consistent with the imbedding of functions into distri-

butions.

Definition 4.6. (1) Let Abe an operator with Schwartz kernel kA. We say

that kAis properly supported if the projections of X×Xto each factor

when restricted to the support of the kernel are proper mappings.

(2) We say that an operator Ais properly supported provided A, A∗:C∞

c(X)→

C∞

c(X), hence A, A∗:C∞(X)→C∞(X), where A∗is the adjoint of A

with respect to the L2(X)-inner product. Ais properly supported if and

only if its kernel is.

Lemma 4.7. Let A:C∞(X)→C∞(X)denote a linear and continuous op-

erator with properly supported Schwartz kernel kA, viewed as a distribution on

X×X. Then Tgcommutes with A(i.e. TgAu(z) = ATgu(z)) if and only if kA

is invariant under the action of g.

Proof. Let h·,·i denote the pairing of distributions and test functions. Then

hTgAu, vi=hAu, Tg−1vi

=hkA, Tg−1,e(v⊗u)i

=hTg,ekA, v ⊗ui

and

hATgu, vi=hkA, v ⊗(Tgu)i.

The algebraic tensor product C∞

c(X)⊗C∞

c(X)is dense in the test functions of

X⊗X([Treves67], p. 530) and hence we obtain

TgA=ATg⇐⇒ Tg,ekA=Te,g−1kA⇐⇒ Tg,gkA=kA.

This proves the lemma.

Recall the notion of non-Euclidean plane waves (2.54): Given λ∈a∗,b∈B,

the functions eλ,b :X→Care defined by

eλ,b :x7→ e(iλ+ρ)hx,bi.(4.17)

4 Equivariant pseudodifferential operators on symmetric spaces

Definition 4.8. Given a linear operator A:C∞(X)→C∞(X), we define the

complete symbol (full symbol)a(z, λ, b)∈C∞(X×a∗

+×B)of Aby

(Aeλ,b) (z) = a(z, λ, b)eλ,b(z).(4.18)

The complete symbol is defined if A:C∞(X)→C∞(X). We will later see for

which classes of operators this condition is satisfied.

Let u∈C∞

c(X). We will now use the Fourier inversion formula to represent

Au by an integral. The following observations have to be understood formally.

We will later define concrete classes of symbols a(z, λ, b)for which these com-

putations make sense. We write

Au(z) = Za∗

+ZB

e(iλ+ρ)hz,bia(z, λ, b)˜u(λ, b) ¯dλdb

=ZXZa∗

+ZB

e(iλ+ρ)hz,bie(−iλ+ρ)hw,bia(z, λ, b)u(w) ¯dλ db dw.

On the level of distributions we then have for the Schwartz kernel

kA(z, w) = Za∗

+ZB

e(iλ+ρ)hz,bie(−iλ+ρ)hw,bia(z, λ, b) ¯dλ db (4.19)

in the sense that

hAu, vi=hkA, v ⊗ui(4.20)

=ZXZXZa∗

+ZB

e(iλ+ρ)hz,bie(−iλ+ρ)hw,bia(z, λ, b)u(w)v(z) ¯dλ db dw dz.

By the Fourier inversion formula, for properly supported kernels kA(z, w)

we can then reconstruct the full symbol of Aby using the Helgason-Fourier

transform of the kernel:

a(z, λ, b) = ZX

e(iλ+ρ)(hw,bi−hz,bi)kA(z, w)dw. (4.21)

We observe

hTg,gkA, v ⊗ui=Ze(iλ+ρ)hz,bie(−iλ+ρ)hw,bia(z, λ, b)u(g−1w)v(g−1z) ¯dλ db dw dz

=Ze(iλ+ρ)hgz,bie(−iλ+ρ)hgw,bia(gz, λ, b)u(w)v(z) ¯dλ db dw dz. (4.22)

Recall the equation hg·z, g ·bi=hz, bi+hg·o, g ·bi(cf. (2.27)), which implies

hg·z, bi=hz, g−1bi+hg·o, bi. Similarly we obtain hg·w, bi=hw, g−1bi+hg·o, bi.

Hence the integral (4.22) becomes

Ze(iλ+ρ)hz,g−1bie(−iλ+ρ)hw,g−1bia(gz, λ, b)u(w)v(z)e+2ρhg·o,bi¯dλ db dw dz.

4 Equivariant pseudodifferential operators on symmetric spaces

Also recall the formula dg·b

db =e−2ρhg·o,g·bifrom Subsection 2.2.6 and change g−1·b

into b. This yields

Ze(iλ+ρ)hz,bie(−iλ+ρ)hw,bia(gz, λ, g ·b)u(w)v(z) ¯dλ db dw dz. (4.23)

Proposition 4.9. Let A:C∞(X)→C∞(X)have properly supported kernel

kA. The following assertions are equivalent:

(1) Tgcommutes with A.

(2) The symbol aof Ais invariant under the action of g.

(3) kAis invariant under the action of g.

Proof. It follows from the equivariance property (2.27) for the horocycle bracket

that eλ,b(gz) = eλ,g·b(g·o)eλ,g−1·b(z). Using this we compute

(TgAeλ,b)(z) = a(gz, λ, b)eλ,b(gz)

=a(gz, λ, b)eλ,g·b(g·o)eλ,g−1·b(z)

and

(ATgeλ,b)(z) = A(eλ,g·b(g·o)·eλ,g−1·b)(z)

=eλ,g·b(g·o)Aeλ,g−1·b(z)

=eλ,g·b(g·o)a(z, λ, g−1·b)eλ,g−1·b(z).

(1) ⇒(2): Assume TgA=ATg. Then amust be invariant in the sense of (4.13).

(2) ⇒(3): Assume a(gz, λ, g ·b) = a(z, λ, b)for all (z, λ, b). Then the integral

(4.23) equals hkA, v ⊗ui, which proves the invariance of kA.

(1) ⇔(3): This is proven in Lemma 4.7.

4.3 Classes of symbols

Let Γdenote a cocompact discrete subgroup of Gand let XΓdenote the cor-

responding compact quotient Γ\X. We now define classes of symbols Sm

cl and

cl,Γand establish C∞-continuities for corresponding classes of operators. If X

has rank one, the properly supported operators (in Lm

cl and Lm

cl,Γ) are closed

under composition and adjoints, and properly supported operators of order 0

are L2-continuous. In the beginning of this section we let the rank r:= dim(A)

of Xbe arbitrary.

Definition 4.10. Let a∗

+denote the closure in a∗of the positive Weyl chamber.

A function a∈C∞(X×a∗

+×B)is a symbol of order m∈Rif for all β∈Nr

0, for

each differential operator Don X×B, and for each compact subset C⊂⊂ X

it satisfies

k∂β

λD a(z, λ, b)k ≤ Cβ,D(C)(1 + |λ|)m−|β|∀z∈C. (4.24)

By Smwe denote the space of symbols of order m.

4 Equivariant pseudodifferential operators on symmetric spaces

Remark 4.11. Suppose that Xhas rank one. Then a=RH, where we choose

Has a generator of aas the unique unit vector in the positive Weyl chamber.

Then a∗

+=Rλ0, where λ0is the linear functional λ0(X) = hX, Hi,X∈a. We

always identify R=aand R=a∗. It follows that the multi-index β∈Nr

0in

(4.24) is an integer k∈N0and (4.24) becomes

k∂k

λD a(z, λ, b)k ≤ Ck,D(C)(1 + |λ|)m−k∀z∈C. (4.25)

Definition 4.12. A symbol a(z, λ, b)is homogeneous of degree m∈Rif for all

t≥1and |λ| ≥ 1it satisfies

a(z, tλ, b) = tma(z, λ, b).(4.26)

A symbol a∈Smis classical if it has an asymptotic expansion

a(z, λ, b)∼∞

j=0

aj(z, λ, b),(4.27)

where the ajare symbols, homogeneous of degree sj, such that sj→ −∞,

s0=m. Asymptotics is here denoted by ∼and means that for all N≥1

a−

N−1

j=0

aj!∈Sm−N.(4.28)

The space of classical symbols of order mis denoted by Sm

cl . The set of symbols

which are invariant under the action of Γon X×B(see 4.13) is denoted by Sm

Γ.

By Sm

cl,Γwe denote the space of Γ-invariant classical symbols. We will in most

cases replace aj(z, λ, b)by |λ|m−jaj(z, λ/|λ|, b).

Proposition 4.13. (i) Suppose a(z, λ, b)is homogeneous of degree min λ

and ϕis a smooth cutoff-function such that ϕ(λ)=0for |λ| ≤ C1and

ϕ(λ) = 1 for |λ| ≥ C2> C1, then ϕ(λ)a(z, λ, b)is a symbol of order m.

(ii) If a(z, λ, b)is a symbol of order m, then a k-th order derivative of awith

respect to λhas order m−k.

(iii) Let aand bbe symbols of order mand k, respectively. Then the symbol ab

defined by ab =a(z, λ, b)b(z, λ, b)has order m+k.

(iv) Let a∈Smsuch that 1/a ≤C(1 + |λ|)−m. Then 1/a ∈S−m.

(v) Let a∈Sm

Γ,cl such that a∼P∞

j=0 aj(z, λ, b). Then aj∈Sm−j

Γfor all

j∈N0.

Proof. (i)-(iv) follow from the chain rule ([Tay81], p. 37). To prove (v), we note

that the terms ajare uniquely determined by a:

a0(z, λ/|λ|, b) = lim

λ→∞|λ|−ma(z, λ, b).

4 Equivariant pseudodifferential operators on symmetric spaces

The other terms ajcan be successively recovered. Then

∞

j=0 |λ|m−jaj(z, λ/|λ|, b)∼a(z, λ, b) = a(γz, λ, γb)∼∞

j=0 |λ|m−jaj(γz, λ/|λ|, γb),

so aj(γz, λ/|λ|, γb) = aj(z, λ/|λ|, b)for each jand γ.

Definition 4.14. Given a symbol a(z, λ, b)we define the corresponding pseu-

dodifferential operator A:= Op(a) := a(z, D)by

a(z, D)u(z) = ZXZa∗ZB

e(iλ+ρ)hz,bie(−iλ+ρ)hw,bia(z, λ, b)u(w)db ¯dλdw

=ZXZa∗ZB

eiλ(hz,bi−hw,bi)eρ(hz,bi+hw,bi)a(z, λ, b)u(w)db ¯dλdw.

Then A=Op(a) = a(z, D)acts on functions uon X, for which the integral

exists. We write OPSm=Op(Sm).

Theorem 4.15. Let a∈Sm. Then A=Op(a) = a(z, D)

(i) is a continuous operator A:C∞

c(X)→C∞(X).

(ii) is a continuous operator A:E0(X)→D0(X).

Proof. For (i), let u∈C∞

c(X). The Fourier transform ˜u(λ, b)is rapidly de-

creasing (Prop. 4.1). Hence Au(z)and all of its derivatives are absolutely and

uniformly convergent integrals. For (ii), let u∈E0(X). Then by Theorem 4.5

we have |˜u(λ, b)| ≤ C(1 + |λ|)nfor some C > 0and n > 0. Then for v∈D(X)

hAu, vi=ZX×a∗

+×B

e(iλ+ρ)A(z,b)v(z)a(z, λ, b)˜u(λ, b) ¯dλdbdz

=Za∗

+×B

av(λ, b)˜u(λ, b) ¯dλdb,

where (using integration by parts via LXas in the proof of Prop. 4.1)

av(λ, b) = ZX

e(iλ+ρ)A(z,b)v(z)a(z, λ, b)dz

=+1

hλ, λi+hρ, ρikZX

e(iλ+ρ)A(z,b)Lk

X(va)dz.

Thus |av(λ, b)| ≤ Ck(v, a) (hλ, λi+hρ, ρi)m−kfor any k∈N0, where Ck(v, a)

depends on the C2k

supp v-norm of v(2.30). The order of the Plancherel density is

s:= dim N. Choose klarge enough to finish the proof.

Definition 4.16. (1) Let Lm,Lm

Γ, denote the properly supported operators

with symbols in their respective symbol spaces.

(2) Let dX(z, w)denote the non-Euclidean distance from z∈Xto w∈X. We

say that A∈Lmis uniformly properly supported if there exists a constant

d0>0such that kA(z, w) = 0 for all zand wwith d(z, w)≥d0.

4 Equivariant pseudodifferential operators on symmetric spaces

4.4 The Kohn-Nirenberg operator

For G=SU(1,1) ∼

=SL(2,R), it is proven in [Zel86] that the non-Euclidean

operator classes (4.14) are contained in the classical space of pseudodifferential

operators ([Hor85]). Proofs of these facts are based on the equivalence of phase

functions and amplitudes in the definitions of operators. We note that equiva-

lence of phase functions generalizes to arbitrary symmetric spaces (see [Zel86]

for references, similar results are announced by N. Anantharaman and L. Silber-

man). The problem is to show, at least in the case of rank one spaces, that the

symplectic volume element of T∗(G/K), if expressed in (z, λ, b)-coordinates, is

asymptotically equivalent to the measure e2ρhz,bi¯dλdbdz. This is an open prob-

lem to me, and I will not go into any more detail at this point. In this section, we

build up the analysis of the operator U:C∞

c(X×a∗

+×B)→C∞(X×a∗

+×B),

Ua(z, λ, b) := (4.29)

e−(iλ+ρ)hz,biZXZa∗

+ZB

e(iµ+ρ)hz,b0ie(iλ+ρ)hw,bie(−iµ+ρ)hw,b0ia(w, µ, b0) ¯dµdwdb0.

In the non-Euclidean calculus of pseudodifferential operators, proofs of many

facts are based on the properties of this Kohn-Nirenberg operator, which is the

composition of the quantization map a7→ Op(a)and the symbol map.

Lemma 4.17. Uis an isometry of L2(X×a∗

+×B, e2ρhz,bidz ¯dλdb).

Proof. The Fourier inversion formula (4.4) says that each sufficiently regular

function fon Xsatisfies

(1) f(z) = Ra∗

+RBRXe(iλ+ρ)hz,bie(−iλ+ρ)hw,bif(w)dw ¯dλ db,

(2) ˜

f(λ, b) = RXRa∗

+RBe(−iλ+ρ)hz,bie(iµ+ρ)hz,˜

bi˜

f(µ,˜

b) ¯dµ d˜

b dz.

Let a∈L2(X×a∗

+×B, e2ρhz,bidzdb ¯dλ)and for the moment, let h|idenote the

L2-inner product. Let the overline denote complex conjugation. Then hUa|Uai

is the ninefold integral

hUa|Uai=Ze(iµ+ρ)hz,b1ie(iλ+ρ)hw1,bie(−iµ1+ρ)hw1,b1ia(w1, µ1, b1)

e(−iµ+ρ)hz,b2ie(−iλ+ρ)hw2,bie(iµ1+ρ)hw2,b2ia(w2, µ2, b2)

¯dλdbdz ¯dµ1db1dw1dµ2db2dw2,

where integration is over (X×a∗

+×B)×(X×a∗

+×B).

If we do the dz ¯dµ2db2integral first, it follows from formula (2) above that

hUa|Uai=Ze(−iµ1+ρ)hw1,b1ie(iλ+ρ)hw1,bie(−iλ+ρ)hw2,bie(iµ1+ρ)hw2,b1i

a(w1, b1, µ1)a(w2, b1, µ1) ¯dλdb ¯dµ1db1dw1dw2.

4 Equivariant pseudodifferential operators on symmetric spaces

Doing the dz ¯dλ db integral next, formula (1) above yields

Ze(−iµ1+ρ)hw1,b1ie(iµ1+ρ)hw1,b1i|a(w1, µ1, b1)|2dw1dµ1db1=ha|ai,

and the lemma is proven.

Remark 4.18. Consider the operator

Fa(z, λ, b) = e−ρhz,biZXZa∗

+ZB

e(−iλ+ρ)hw,bie−(iµ+ρ)hz,b0i

×eρhw,b0ia(w, µ, b0)db0¯dµ dw.

Using the Fourier inversion formula as above, one checks that e

Fis an isometry

of L2(X×a∗

+×B, e2ρhz,bidzdb ¯dλ). Then, e

Fis inverted by

Ga(z, λ, b) = e−ρhz,biZXZa∗

+ZB

e(iλ+ρ)hw,bie(iµ+ρ)hz,b0i

×eρhw,b0ia(w, µ, b0)db0¯dµ dw.

By definition we have U=e−iλhz,bie

Ge−iµhw,b0iand U−1=eiλhz,bie

Feiµhw,b0i, which

shows that Uis invertible.

Proposition 4.19. Uis a unitary operator on L2(X×a∗

+×B, e2ρhz,bidz ¯dλdb)

and commutes with each g∈G, that is UTg=TgUin the notation of (4.13).

Proof. Uis unitary on L2(X×a∗

+×B, e2ρhz,bidz ¯dλdb)by Lemma 4.17 and

Remark 4.18. For a proof of UTg=TgUnote that dw is G-invariant, so

Ua(gz, λ, gb) =

e−(iλ+ρ)hgz,gbiZXZa∗

+ZB

e(iµ+ρ)hgz,b0ie(iλ+ρ)hgw,gbie(−iµ+ρ)hgw,b0ia(gw, µ, b0) ¯dµdwdb0.

If we substitute b07→ g·b0and use hgz, gb0i=hz, b0i+hg·o, b0iand dg·b0

db0=

e−2ρhg·o,g·b0i, the integral becomes

e−(iλ+ρ)hz,biZXZa∗

+ZB

e(iµ+ρ)hz,b0ie(iλ+ρ)hw,bie(−iµ+ρ)hw,b0ia(gw, µ, gb0) ¯dµdwdb0

=U(a◦g)(z, λ, b),

where (f◦g)(z, λ, b) = f(gz, λ, gb). The proposition follows.

4 Equivariant pseudodifferential operators on symmetric spaces

4.4.1 A convolution formula

Given two functions aand bon G, at least one with compact support, their

convolution product a∗bis defined by

(a∗b)(h) := ZG

a(g)b(g−1h)dg, h ∈G. (4.30)

Since Gis locally compact and unimodular we may substitute g7→ hg and then

change ginto g−1. The unimodularity and the G-invariance of dg yield

(a∗b)(h) = ZG

a(hg−1)b(g)dg. (4.31)

This convolution descends to convolution of M-invariant functions on G, which

we also denote by ∗: If πdenotes the projection G→G/M and if fis a function

on G/M, then, f◦πis an M-right-invariant function on G. Convolution on

G/M is then defined via

(a∗b)◦π:= (a◦π)∗(b◦π),

where aand bdenote functions on G/M. Written out, this means

(a∗b)(hM) = ZG

a(gM)b(g−1hM)dg. (4.32)

To see this is well-defined, let aand bbe M-invariant functions on G, such that

the convolution integral a∗bexists. Given g∈G,m∈M, observe

(a∗b)(gm) = ZG

a(h)b(h−1gm)dh =ZG

a(h)b(h−1g)dh = (a∗b)(g).(4.33)

It follows that a∗bis invariant and thus the convolution product is well-defined.

We identify functions on X×Band functions on G/M. The non-Euclidean

Fourier analysis is written in X×B-terms (for example using horocycle bracket),

but it is often more convenient to work with the space G/M (and Iwasawa pro-

jections) instead. We then observe that under G/M ∼

=X×B(4.32) corresponds

to the convolution on X×Bdefined by

(a∗b)(z, b) = ZG

a(g·(o, M))b(g−1·(z, b))dg, (z, b)∈X×B, (4.34)

where ·denotes the action of Gon X×B. The integral exists whenever at least

one of the functions aand bhas compact support.

Given µ, λ ∈a∗

+, we write

Eµ,λ :X×B→C,(z, b)7→ e(iµ+ρ)hz,Mie−(iλ+ρ)hz,bi.(4.35)

4 Equivariant pseudodifferential operators on symmetric spaces

In order to rewrite Eµ,λ in terms of G/M, recall that (z, b)=(gK, g ·M)∈

X×Bcorresponds to gM ∈G/M. We have hz, bi=−H(g−1k(g)) = H(g)and

hz, Mi=−H(g−1)by Corollary 2.40. Hence

Eµ,λ :G/M →C, gM 7→ e−(iµ+ρ)H(g−1)e−(iλ+ρ)H(g).(4.36)

Note that (4.36) is well-defined since the Iwasawa projection His M-biinvariant.

Proposition 4.20. Let a∈C∞

c(X×a∗

+×B). Then

Ua(z, λ, b) = Za∗

(a(·, µ, ·)∗Eµ,λ)(z, b) ¯dµ (4.37)

Proof. Note that we sometimes write a(z, b, λ)instead of a(z, λ, b)for simplicity

of notation (when a group action is involved). Consider the integral

(a(·, µ, ·)∗Eµ,λ)(z, b) = ZG

Eµ,λ(g−1·(z, b))a(g·(o, M), µ)dg

=ZG

e(iµ+ρ)hg−1z,Mie−(iλ+ρ)hg−1z,g−1·bia(g·(o, M), µ)dg.

We fix z∈X,λ, µ ∈a∗

+,b∈Band write

f(g) = e(iµ+ρ)hg−1z,Mie−(iλ+ρ)hg−1z,g−1·bia(g·(o, M), µ).

We claim that fis M-invariant and hence a function on G/M. The action of m

on X×Bleaves (o, M)∈X×Bfixed. Recall that hz, biis invariant under the

diagonal action of Kon X×B. Thus hm−1g−1z, m−1g−1·bi=hg−1z, g−1·bi

and hm−1g−1z, Mi=hm−1g−1z, m−1Mi=hg−1z, Mi, and hence

f(gm) = e(iµ+ρ)hm−1g−1z,Mie−(iλ+ρ)hm−1g−1z,m−1g−1·bia(gm ·(o, M), µ)

=e(iµ+ρ)hg−1z,Mie−(iλ+ρ)hg−1z,g−1·bia(g·(o, M), µ) = f(g).

We have hg−1·z, Mi=hz, g·Mi−hg·o, g·Miand hg−1z, g−1bi=hz, bi−hg·o, bi

by Lemma 2.38 and thus

f(g) = e(iµ+ρ)hg−1z,Mie−(iλ+ρ)hg−1z,g−1·bia(g·(o, M), µ)

=e(iµ+ρ)(hz,g·Mi−hg·o,g·Mi)e−(iλ+ρ)(hz,bi−hg·o,bi)a(g·o, µ, g ·M).

Then by Corollary 3.11

(a(·, µ, ·)∗Eµ,λ)(z, b) = ZG

f(g)dg

=ZXZB

e(iµ+ρ)(hz,b0i−hw,b0i)e−(iλ+ρ)(hz,bi−hw,bi)a(w, µ, b0)e2ρhw,b0idw db0

=e−(iλ+ρ)hz,biZXZB

e(iµ+ρ)hz,b0ie(iλ+ρ)hw,bie(−iµ+ρ)hw,b0ia(w, µ, b0)dw db0,

4 Equivariant pseudodifferential operators on symmetric spaces

and integrating against µ∈a∗

+yields

Za∗

(a(·, µ, ·)∗Eµ,λ)(z, b) ¯dµ

=e−(iλ+ρ)hz,biZa∗

+ZXZB

e(iµ+ρ)hz,b0ie(iλ+ρ)hw,bie(−iµ+ρ)hw,b0ia(w, µ, b0) ¯dµ dw db0

=Ua(z, λ, b),

as desired.

4.4.2 Asymptotic expansions in the rank one case

Let X=G/K have rank one. We identify aand a∗with Rby means of the

Killing form and make no difference between a∗

+and the positive real numbers

R+: The unit vector H∈a+is identified with the real number 1. Let a(z, λ, b)∈

C∞

c(X×B×a∗

+). Recall the definition

Ua(z, λ, b) =

ZX×B×a∗

e−(iλ+ρ)hz,bie(iµ+ρ)hz,b0ie(iλ+ρ)hw,bie(−iµ+ρ)hw,b0ia(w, µ, b0)|c(µ)|−2dµ db0dw.

We collect the λ-terms and the ρ-terms in the integral defining Ua, change

variables from µto λµ and factor out λfrom the phase function to find

Ua(z, λ, b) = ZX×B×a∗

eiλ[hw,bi−hz,bi+µ(hz,b0i−hw,b0i)] eρ[hw,bi+hw,b0i+hz,b0i−hz,bi]

×a(w, λµ, b0)λ

|c(λµ)|2dw db0dµ. (4.38)

Hence we have an oscillatory integral Ua =Reiλψαdx with phase function

ψz,b(w, µ, b0) = hw, bi−hz, bi+µ(hz, b0i−hw, b0i).(4.39)

Let (z, b)=(g·o, g ·M),(w, b0)=(h·o, h ·M)∈X×Bcorrespond to

gM ∈G/M and hM ∈G/M, respectively. Then by Corollary 2.40

(1) hz, bi=H(g),

(2) hz, b0i=−H(g−1k(h)) = −H(g−1h) + H(h),

(3) hw, bi=−H(h−1k(g)) = −H(h−1g) + H(g),

(4) hw, b0i=H(h).

4 Equivariant pseudodifferential operators on symmetric spaces

It follows that in terms of G/M the function ψz,b has the form

ψg(h, µ) = −H(h−1g)−µ(H(g−1h)).(4.40)

Note that ψg(h, µ)is right-M-invariant in both gand h. Writing h=nak, we

get for gM =eM

ψeM (n, a, k, µ) = log(a)−µH(nak).

Writing h−1=nak, we get for gM =eM

ψeM (n, a, k, µ) = µlog(a)−H(nak).

These functions are defined on N×A×K/M ×R+. As proven in Subsection 3.3,

the unique critical point of ψeM is (n, a, kM, µ) = (e, e, eM, 1) (and the Hessian

form at the critical point is non-degenerate). Under the natural diffeomorphisms

N×A×K/M ∼

=X×B∼

=G/M,

the critical point corresponds to (hM, µ)=(eM, 1) in G/M ×R+, so if ψeM =ψ

is as in (4.40) and gM =eM, the critical point is (hM, µ) = (eM, 1). But

ψgM (h, µ) = ψeM (g−1h, µ)has the critical set {g−1h∈M}, so the critical point

of ψgM (hM, µ)is (hM, µ)=(gM, 1) and corresponds to (z, 1, b)in X×R+×B.

This proves

Lemma 4.21. If ψz,b(w, µ, b0)is as in (4.39), then its unique critical point is

(w, µ, b0)=(z, 1, b)and the Hessian form at this point is non-degenerate.

Theorem 4.22. Let a(z, λ, b)∈Sm

cl be compactly supported in z(uniformly in

the other variables). Then there exist eak(z, λ, b), homogeneous of order m−k

for λ≥1, such that

Ua −

N−1

k=0 eak≤CN(1 + λ)m−N.(4.41)

Proof. Let ∇=∇wdenote the gradient taken w.r.t. w∈X. Then the vector

∇hw, bihas norm one for all b∈B, since it is a unit vector pointing along a

geodesic orthogonal to level sets of hw, bitowards b([Eber96], Prop. 1.10.2). It

follows that ∇ψ6= 0 for all µ > 1. We choose a cutoff χ(µ)∈C∞

c(R+)such

that χ(µ) = 1 in [0,2] and χ(µ)=0in [3,∞), and write

Ua(z, λ, b) = Ia(z, λ, b) + IIa(z, λ, b)(4.42)

corresponding to 1 = χ(µ) + [1 −χ(µ)]. Then

IIa(z, λ, b) = ZX×B×a∗

[1 −χ(µ)] eiλ[hw,bi−hz,bi+µ(hz,b0i−hw,b0i)]

×eρ[hw,bi+hw,b0i+hz,b0i−hz,bi]a(w, λµ, b0)λ

|c(λµ)|2dw db0dµ.

4 Equivariant pseudodifferential operators on symmetric spaces

Since ∇ψ6= 0 for µ > 1, the operator L:= 1

iλ(|∇ψ|2)−1∇ψ·∇ is defined in the

support of the integrand. Then Leiλψ =eiλψ, so we can apply the transpose

Ltof Lto the amplitude. The order of the Plancherel density is s:= dim(N).

Each (|∇ψ|2)−1is at least O(µ−1). Thus |(Lt)k(α)| ≤ Ck(λµ)m+1+s−kat each

point. Since αis compactly supported in X×B, we have absolute and uniform

convergence of IIa(z, λ, b). Thus IIa(z, λ, b) = O(λ−∞).

Recall that g·(z, b)=(g·z, g ·b)denotes the diagonal action of Gon X×B.

A function f(z, b)on X×Bis pulled-back to an M-invariant function on Gvia

f(g) = f(g·o, g·M). We denote by f◦gthe function (z, b)7→ f(g·z, g·b). Recall

that Ucommutes with translation by elements g∈G, that is U(a◦g)=(Ua)◦g.

We write (z, b) = g·(o, M)corresponding to X×B∼

=G/M. The equivariance

still holds if we insert χ(µ)into the R+-integral:

Ia(g, λ) = Ia(z, λ, b) = I(a◦g)(o, λ, M)(4.43)

ZX×B×a∗

χ(µ)eiλ[hw,Mi−µhw,b0i]eρ[hw,Mi+hw,b0i]λ a(g·(w, b0), λµ)λ

|c(λµ)|2dw db0dµ.

The phase function ψo,M (w, µ, b0) = hw, Mi− µhw, b0iis non-degenerate at its

critical point (w, µ, b0) = (o, 1, M). We can further assume (by using another

cutoff around the critical point) that the integrand is supported in a coordinate

patch around the critical point. All remainder integrals will again be O(λ−∞),

which follows from the standard principle of non-stationary phase for compactly

supported amplitudes.

We use coordinates x= (x1, . . . , xd, µ), where d:= dim(X×B), around the

critical point (w, b, µ)=(o, M, 1). In these coordinates, (0,1) ∈Rd

(w,b)×R+

corresponds to (o, M, 1). Let D= (∂x1, . . . , ∂xd, ∂µ)and let HD

0denote the

Hessian operator of ψ=ψo,M at this point. The Taylor expansion of ψaround

(0,1) is ψ(x, µ) = Q(x, µ)+h(x, µ), where hvanishes up to order 3in (0,1) and

Q(x, µ) = 1

2hHD

0(x, µ)|(x, µ)i(the customary inner product on Rd+1). Then

Ia(g, λ) = ZeiλQ χαeiλhdx dµ +O(λ−∞).(4.44)

Set s:= dim(N) = dim(B). Since tanh ∼1and coth ∼1to all orders, the

Plancherel density is asymptotically a polynomial of degree s(cf. (2.13)). For

the asymptotics we can hence replace |c(ν)|−2by a polynomial p(ν) = Ps

j=1 cjνj

of degree s(without constant term). We split Ua =PjUja,Ia =PjIjaand

α=Pjαjinto the corresponding ssummands.

We start by assuming that ais homogeneous of degree m. Then a(z, λ, b)∼

λma(z, 1, b)(for λ→ ∞), so we can assume that the amplitude αjin each Ij(a)

is homogeneous.

By (3.46) we can choose coordinates such that sign(HD

0)=0. We thus set

C0=(2π)s+1

√det HD

and R=1

2(HD

0)−1D, D. The method of stationary phase yields

Uja(g, λ)∼C0

λs+1

∞

k=0 i

λk1

k!Rk(αjeiλh)|(x,µ)=(0,1) (4.45)

4 Equivariant pseudodifferential operators on symmetric spaces

in the sense that |Uj(a)−PN−1

j=0 eαj,k| ≤ Cjλ−Nkαjk, where kαjkis a seminorm

of the amplitude (and still influences the order in λ, see [Hor85], Sec. 7.7.). We

rearrange (4.45) to provide a classical asymptotic expansion: Differentiations in

µpreserve the order in λ, hence all differentiations do. Thus only derivatives

of eiλh affect the order of a term. But hvanishes of order 3, and so one needs

three derivatives to bring down one λ. It follows that the k-th term has an order

≤m−[k/3] −(s−j). Hence for each lthere are only finitely many ksuch that

the k-th term has an order ≥l. After rearrangement into homogeneous terms,

Uja(z, λ, b)∼∞

k=0

(i/λ)ke

Λj,k(a)(g, o, λ, M),(4.46)

where e

Λj,k(a)(g, o, λ, M)is homogeneous of degree m, and where e

Λj,k is a dif-

ferential operator on X×B×R+of order 2kwith coefficients in g. The

Λj,k(a)(g, o, λ, M)are left-G-invariant and right-M-invariant in g, since each

Ujis invariant. They also decrease supports, so by Peetre’s theorem they define

differential operators on G/M ×[1,∞)∼

=X×B×[1,∞). Hence

Uja(z, λ, b)∼∞

k=0

(i/λ)kλmΛj,k(a)(z, 1, b),(4.47)

where Λj,k is a left-invariant differential operator of order 2k. Summing up we

find U(a)(z, λ, b)∼PkΛk(a)(z, λ, b), where Λkis a differential operator of order

2k, and the sum can be rearranged into homogeneous summands. If ais not

homogeneous, then a∼Pak, and we again rearrange to provide Ua ∼Peak,

where the order of eakis m−k, and Ua −PN−1

k=0 eak≤CN(1 + λ)m−N.

Remark 4.23. The expansion

U(a)∼∞

k=0 i

λk

Λk(a)(4.48)

can be obtained directly from the method of stationary phase with parameters:

We write Ua(z;z, λ, b) = Reiλψz,b α dx as in (4.38), where the phase function is

ψz,b(w, µ, b0) = hw, bi−hz, bi+µ(hz, b0i−hw, b0i).

The critical point (z, 1, b)of ψz,b is given by the parameter (z, b)(Lemma 4.21).

Then (4.48) follows from the method of stationary phase. If ais a classical

symbol of order m, the sum (4.48) can be rearranged as above in homogeneous

summands Ua ∼Peaj, where the order of eajis m−j. The Λkin the expansion

are left-G-invariant, since Uis left-invariant.

4 Equivariant pseudodifferential operators on symmetric spaces

Remark 4.24. (1) Given a function a(z, w, λ, b)of two spatial variables we

sometimes write a(z;w, λ, b)to emphasize the special role of z. The oper-

ator Ustill operates in X×R+×Band we write

Ua(z;z, λ, b) =

ZX×B×a∗

e−(iλ+ρ)hz,bie(iµ+ρ)hz,b0ie(iλ+ρ)hw,bie(−iµ+ρ)hw,b0ia(z;w, µ, b0)|c(µ)|−2dw db0dµ.

(2) Let m << 0be so small such that for a∈Sm

cl the integral U(a)makes

sense. We can write Ua ∼P∞

k=0(i/λ)kΛk(a)and expand out U∗U(a) = a.

In particular, the principal symbol of U(a)equals c·σa, where σadenotes

the principal symbol of a, and where cis a constant with |c|= 1. Since

c > 0by the MSP-formula, we find that principal symbol of Ua equals the

principal symbol of a.

Definition 4.25. (1) Let Lm

1,0,0denote the space of properly supported oper-

ators in OPSm

1,0,0=Op(Sm

1,0,0), where Sm

1,0,0⊂C∞(X×X×R+×B)is the

space of functions a(z, w, λ, b)satisfying

(∂/∂λ)kD a(z, w, λ, b)≤CD,k(C)(1 + |λ|)m−k∀(z, w)∈C, (4.49)

for all k∈N0, all compact subsets Cof X×X, and all differential operators

on X×X×B. We call a∈Sm

1,0,0classical (a∈Sm

1,0,0,cl) if for all N∈N0

a(z, w, λ, b)∼∞

j=0

λm−jaj(z, w, b) (λ→ ∞).(4.50)

Asymptotics here means a−PN−1

j=0 aj∈Sm−N

1,0,0for all N∈N0. By trans-

lation on X×X×Bwe mean g·(z, w, b) = (g·z, g ·w, g ·b). Let Sm

1,0,0,Γ

denote the set of symbols, which are Γ-invariant:

a(γz, γw, λ, γ ·b) = a(z, w, λ, b)∀z, w ∈X, λ ∈R+, b ∈B, γ ∈Γ.(4.51)

(2) An operator Op(a) = a(z, z, D)∈OPSm

1,0,0operates according to the

formula

a(z, z, D)u(z) :=

ZXZBZR+

e(iλ+ρ)(hz,bi−hw,bi)a(z, w, λ, b)u(w)e2ρhw,bidw db ¯dλ.

Corollary 4.26. Ua ∼P∞

k=0(i/λ)kΛk(a)in symbol norm asymptotics.

Proof. First note that an expansion for Ua(z;z, λ, b)is obtained by the method

of stationary phase with parameters as above, where the parameter is (z, b). For

the eajand eathat arise in Remark 4.23, we need that

∂

∂λk

D(Ua −ea)(z, w, λ, b)≤Ck,D,N (C)(1 + λ)−N(4.52)

4 Equivariant pseudodifferential operators on symmetric spaces

for each N∈N, any compact subset Cof X×X, all (z, w)∈C, and each

differential operator Don X×X×B. Recall that a sequence {aj}of symbols,

where the order of ajis m−j, can be asymptotically summed by setting ea=

P∞

j=0 ϕ(εjλ)aj(z, w, λ, b), where ϕ= 0 for λ≤1/2,ϕ= 1 for λ≥1and where

the εjare chosen appropriately ([Tay81], p. 41). If the ajare Γ-invariant, then

so is ea. We claim that this holds if derivatives of Ua have at most polynomial

growth, that is

|(∂/∂λ)kD U(a)| ≤ Ck,D(C)(1 + λ)σ,(4.53)

for (z, w)∈Cand σ=σ(k, α). It suffices to prove (4.52) for an open coordinate

patch Vof X×X×Band for D=∂

∂xj. By [Tay81], Prop. 3.3,

kDak2

∞≤4kak∞D2a∞.(4.54)

So if p−qis rapidly decreasing in λand if D2(p−q)has at most polynomial

growth,

sup |D(p−q)|2≤c1sup |p−q|sup |D2(p−q)| ≤ Ck,N (1 + λ)−N(4.55)

for any N, since the first factor is rapidly decreasing and the second is at most

polynomially growing. Here sup means supVand c1is a constant. λ-derivatives

can be handled similarly (loc. cit., p. 41). We thus have Ua ∼P∞

k=0(i/λ)kΛk(a)

in symbol norm asymptotics if derivatives of U(a)have polynomial growth.

Therefore write U(a)in the form I(a) + II(a)as before, integrate by parts in

II(a)as before, pass derivatives under the integral and see that the result is

O(λ−∞)uniformly in compact subsets. I(a)is a compactly supported integral,

and derivatives can be estimated by a constant times a suitable symbol norm

of atimes a convenient power of 1 + |λ|.

Proposition 4.27. Let A∈Lm

1,0,0,cl. Then A∈Lm

cl and a complete symbol of

Ais given by

a(z, λ, b) = Ua(z;z, λ, b)∼∞

k=0

(i/λ)kΛka(z;w, λ, b)|w=z.

Proof. Since Ais properly supported, a(z, λ, b) := e−(iλ+ρ)hz,biAe(iλ+ρ)hz,biis well-

defined and yields a complete symbol for A. Written out,

a(z, λ, b) :=

e(−iλ+ρ)hz,biZX×R+×B

e(iµ+ρ)hz,b0ie(−iµ+ρ)hw,b0ia(z;w, µ, b0)e(iλ+ρ)hw,bi¯dµ dw db0

=Ua(z;·,·,·)|(z,λ,b)∼∞

k=0

(i/λ)kΛka(z;w, λ, b)|w=z,

where the Λkoperate in the variables (w, λ, b). The ∼holds in the sense of

Corollary 4.26.

4 Equivariant pseudodifferential operators on symmetric spaces

If ais classical, we write U(a)∼Pajin homogeneous terms and expand

out U∗U= id. In particular, the principal symbol of U(a)equals the principal

symbol of a. The proofs given in [Zel86] for the adjoint of properly supported

operators (Theorem 2.8, loc. cit.) is formal enough to cover the case of all rank

one spaces: It is proven there that if a(z, λ, b)is an amplitude of order 0, then

the adjoint Op(a)∗has amplitude a(w, λ, b), and the principal symbol of Op(a)∗

is a0(z, 1, b), so that the principal symbol of Op(a)∗Op(a)is |a(z, b)|2(Thm. 2.9

loc. cit.). In particular, it is shown in Theorem 2. 11 loc. cit. that if Γis

cocompact, then properly supported zero order pseudodifferential operator are

continuous on L2(Γ\X).

Remark 4.28. The computations of the critical set, the Hessian form, and

the application of the method of stationary phase generalize to higher rank

spaces, if the spectral parameters are assumed to be regular (cf. Subsection

3.3). More integral formulas for the integral operator U(a)(z, λ, b)are given in

the following subsection. It seems reasonable to believe that the proofs given

here and in [Zel86] can be generalized to higher rank spaces with only slight

modifications.

4.4.3 Some integral formulas for the Kohn-Nirenberg operator

We list a few possibilities to write Ua as an oscillatory integral. These represen-

tations of the Kohn-Nirenberg operator may be useful to approach a formula for

the Hessian operator in the asymptotic expansion for Ua(z, λ, b), which would

yield a commutator formula for the non-Euclidean calculus of pseudodifferential

operators. First, let G/K have rank one. Write h=h(w, b0)corresponding to

G/M ∼

=X×B: By 3.12 we have

Ua(g, λ) = Ua(z, λ, b)

=Za∗

+ZG

eiλ[−H(h−1g)−µH(g−1h)]e−ρ(H(g−1h)+H(h−1g))a(h, λµ)λ|c(λµ)|−2dh dµ.

The integral is actually taken over G/M, since all terms in the integrand are

M-invariant. From now on we will work in G. First, substitute h7→ gh. Then

Ua(g, λ) = Za∗

+ZG

eiλ[−H(h−1)−µH(h)]e−ρ(H(h)+H(h−1))a(gh, λµ)λ|c(λµ)|−2dh dµ.

Now by (2.6) the integral equals

Za∗

+ZNAK

eiλ[log(a)−µH(nak)] eρ[log(a)−H(nak)] a(gnak)λ

|c(λµ)|2dn da dk dµ. (4.56)

4 Equivariant pseudodifferential operators on symmetric spaces

We could have also changed hto h−1(Gis unimodular). Then

Ua(g, λ)(4.57)

=Za∗

+ZG

eiλ[−H(h)−µH(h−1)]e−ρ(H(h−1)+H(h))a(gh−1, λµ)λ|c(λµ)|−2dh dµ

=Za∗

+ZNAK

eiλ(µlog(a)−H(nak))e−ρ(log(a)+H(nak))a(g(nak)−1, λµ)λ

|c(λµ)|2dn da dk dµ.

In higher rank, the same computations are possible: Given 06=λ∈a∗

+, write

λ=τλ0, where |λ0|= 1 (the norm on a∗induced by the Killing form). Set

Ua(g, τ) := Ua(g, τλ0). Then

Ua(g, τ) = Za∗

+ZNAK

ei(µlog(a)−τλ0H(nak))e−ρ(log(a)+H(nak))

×a(g(nak)−1, µ)dn da dk ¯dµ

=Za∗

+ZNAK

eiτ(µlog(a)−λ0H(nak))e−ρ(log(a)+H(nak))

×a(g(nak)−1, τµ)τdim(A)

|c(τµ)|2

wdn da dk dµ.

where we factored out τfrom the phase function and substituted µ7→ µ/τ.

Recall that Ucommutes with translation by g∈G. We rewrite Ua(z, λ, b)as

given in (4.38) corresponding to X×B∼

=AN ×K/M, evaluate the integral at

(o, λ, M), and finally choose g∈Gsuch that g·(o, M) = (z, b)to insert g∈G

in the amplitude. Then with |λ0|= 1 and λ > 0,

Ua(z, λλ0, b) = U(a◦g)(o, λλ0, M)(4.58)

=ZAN×K/M×R+

eiλλ0[han·o,Mi−µ(han·o,kMi)] eρ[han·o,Mi+han·o,kMi](4.59)

×a(g·an ·o, λµ, g ·kM)λdim(A)

|c(λµ)|2dw db0dµ

=ZAN×K/M×R+

eiλλ0[−log(a−1)+µ(H(n−1a−1k))] eρ[−log(a−1)−H(n−1a−1k)] (4.60)

×a(g·an ·o, λµ, g ·kM)λdim(A)

|c(λµ)|2dw db0dµ

=ZAN×K/M×R+

eiλλ0[−log(a)+µ(H(nak))] e−ρ[log(a)+H(nak)] (4.61)

×a(g·a−1n−1·o, λµ, g ·kM)λdim(A)

|c(λµ)|2dw db0dµ.

The phase function µ(H(nak))−λ0(log(a)) has the critical point (µ, n, a, kM) =

(λ0, e, e, eM)and the Hessian at this point is non-degenerate. The method of

stationary phase can be applied to all these integral exactly as before.

4 Equivariant pseudodifferential operators on symmetric spaces

4.5 Conjugation by a wave group-type operator

Let Xhave rank one. We identify a=Rby means of the Killing form: λ0

denotes the functional on agiven by λ0(X) = hX, Hi, where His the unit

vector in a+. Then λ0∈a+. We identify λ∈awith the real number λsuch

that λ=λλ0.

We denote by Gtthe geodesic flow on X×B. The latter space identifies with

G/M and hence the geodesic flow on X×Breads by right-translations with

elements a∈A, that is Gt(g·o, g ·M) = (gat·o, gat·M) = (gat·o, b)for g∈G,

at= exp(tH)∈A. Right-translation on X×Bis well-defined, since Mand A

commute elementwise. The point b∈Bis not moved under Gt. Recall that if Γ

is a cocompact subgroup of G, the geodesic flow on SXΓ= Γ\G/M also reads

by right-A-translation.

Let A=a(x, D)∈Lm

cl . We denote by σAthe principal symbol of A, that is

the highest order term in the asymptotic sum (4.27). Let λ∈a∗and b∈B.

Recall the character of the Laplace operator (cf. (2.57)):

∆e(iλ+ρ)hz,bi=−(hλ, λi+hρ, ρi)e(iλ+ρ)hz,bi.

Using functional calculus (the spectral theorem), we define R:= p−(∆ + |ρ|2)

and the group of operators eitR by its action on the non-Euclidean plance waves

eλ,b(z) = e(iλ+ρ)hz,bi, that is

eitRe(iλ+ρ)hz,bi=eitλe(iλ+ρ)hz,bi.

Given t∈R, we write

At:= eitRAe−itR.(4.62)

4.5.1 The complete symbol after conjugation

The complete symbol of Atis

Ut(a) := at(z, λ, b) = e−(iλ+ρ)hz,bieitRAe−itRe(iλ+ρ)hz,bi

=e−(iλ+ρ)hz,bieitRAe−itλe(iλ+ρ)hz,bi

=e−itλe−(iλ+ρ)hz,bieitR a(z, λ, b)e(iλ+ρ)hz,bi.(4.63)

Recall the Fourier inversion formula (4.4), which states that each sufficiently

regular function fon Xsatisfies

f(z) = Za∗

+ZBZX

e(iλ+ρ)hz,bie(−iλ+ρ)hw,bif(w)dw ¯dλ db.

It follows that

a(z, λ, b)e(iλ+ρ)hz,bi=Ze(iµ+ρ)hz,b0ie(−iµ+ρ)hw,b0ie(iλ+ρ)hw,bia(w, λ, b)dw ¯dµ db0,

4 Equivariant pseudodifferential operators on symmetric spaces

and hence

at(z, λ, b)

=Ze−itλe−(iλ+ρ)hz,bieitRe(iµ+ρ)hz,b0ie(−iµ+ρ)hw,b0ie(iλ+ρ)hw,bia(w, λ, b)dw ¯dµ db0

=Zeit(µ−λ)e−(iλ+ρ)(hz,bi−hw,bi)e(iµ+ρ)(hz,b0i−hw,b0i)e2ρhw,b0ia(w, λ, b)dw ¯dµ db0

=Zeit(µ−λ)e−(iλ+ρ)(hz,bi−hw,bi)e(iµ+ρ)(hz,b0i−hw,b0i)a(w, λ, b)e2ρhw,b0idw db ¯dµ.

Corollary 4.29. For sufficiently regular functions a(w, µ, b0)on X×a∗

+×B,

Ut(a) = Zeit(µ−λ)e−(iλ+ρ)(hz,bi−hw,bi)e(iµ+ρ)(hz,b0i−hw,b0i)a(w, λ, b)e2ρhw,b0idw db ¯dµ.

Proposition 4.30. The Utare a one-parameter group of unitary operators on

L2(G/M ×R+, dg ¯dµ) = L2(X×B×a+, e2ρhw,b0idw db ¯dµ).

Proof. Let h·,·i denote the L2-inner product. Then

hUta, Utai=Zeitµ1e−itµ2e(iµ1+ρ)(hz,b1i−hw1,b1i)

×e−(iλ+ρ)(hz,bi−hw1,bi)a(w1, b, λ)e(−iµ2+ρ)(hz,b2i−hw2,b2i)

×e(iλ−ρ)(hz,bi−hw2,bia(w2, b, λ)e2ρhw2,b2ie2ρhw1,b1i

×e2ρhz,bi¯dλ db dz ¯dµ1db1dw1¯dµ2db2dw2.

The Fourier inversion formula (4.4) states for sufficiently regular f:X→C

(1) f(z) = Ra∗

+RBRXe(iλ+ρ)hz,bie(−iλ+ρ)hw,bif(w)dw ¯dλ db,

(2) ˜

f(λ, b) = RXRa∗

+RBe(−iλ+ρ)hz,bie(iµ+ρ)hz,˜

bi˜

f(µ,˜

b) ¯dµ d˜

b dz.

As in the proof of Proposition 4.19 we use these formulae to find hUta, Utai=

ha, ai. (The SU(1,1)-proof given in [Zel86], p. 100, generalizes completely).

Recall X×B∼

=G/M. We change variables to w=g·o,b0=g·Mand prove

exactly as in Subsection 4.4 that

Ut(a)(z, λ, b) = ZR+ZG

eit(µ−λ)hg−1z,Mie(iµ+ρ)hg−1z,g−1bie−(iλ+ρ)a(g·o, b, λ)dg ¯dµ

=ZR+

(a(·, b, λ)∗Eµ,λ(z, b)) eit(µ−λ)¯dµ.

Lemma 4.31. Utcommutes with translations Tgby elements g∈G.

4 Equivariant pseudodifferential operators on symmetric spaces

Proof. Let g∈G. Then by (4.63) we have

(TgUt)(a)(z, λ, b) = at(gz, λ, gb)

=e−itλe−(iλ+ρ)hgz,gbieitR a(gz, λ, gb)e(iλ+ρ)hgz,gbi.

Since hgz, gbi=hz, bi+hgo, gbithis equals

e−itλe−(iλ+ρ)hz,bieitR a(gz, λ, gb)e(iλ+ρ)hz,bi

= (a◦g)t(z, λ, b)

= (UtTg)(a)(z, λ, b),

where (a◦g)(z, λ, b) = a(gz, λ, gb). This proves UtTg=TgUt.

Recall from Corollary 4.29 that

Ut(a) = Zeit(µ−λ)e−(iλ+ρ)(hz,bi−hw,bi)e(iµ+ρ)(hz,b0i−hw,b0i)a(w, λ, b)e2ρhw,b0idw db ¯dµ,

where the integration space is X×B×R+. We factor out λfrom the phase,

change variables to eµ=µ/λ, and drop the tilde. Then

Ut(a)(z, λ, b) = Zeiλ[t(µ−1)+hw,bi−hz,bi+µ(hz,b0i−hw,b0i)]

×eρ[hw,bi−hz,bi+hz,b0i−hw,b0i]λ a(w, λ, b)e2ρhw,b0i|c(λµ)|−2dw db dµ.

Writing (z, λ, b) = (g·o, λ, g ·M)and using Ua(z, λ, b) = U(a◦g)(o, λ, M), we

find (also note that g·M=b∈B)

Uta(g, λ) = Ut(a)(z, λ, b)(4.64)

=Zeiλ[t(µ−1)+hw,Mi−µhw,b0i]eρ[hw,Mi−hw,b0i]λ a(g·w, λ, b)e2ρhw,b0i|c(λµ)|−2dw db dµ.

We change variables (w, b0) = h·(o, M)corresponding to X×B∼

=G/M. Then

hw, Mi=−H(h−1)and hw, b0i=H(h), so by 3.11

Uta(g, λ) = Ut(a)(z, λ, b)

=Zeiλ[t(µ−1)−H(h−1)−µH(h)] e−ρ[H(h−1)+H(h)] λ a(gh ·o, λ, b)|c(λµ)|−2dh dµ.

Next, write h=a−1n−1kcorresponding to G=ANK. Then by (2.6) and since

Aand Nare unimodular we obtain

Uta(g, λ) = Ut(a)(z, λ, b)

=Zeiλ[t(µ−1)−log(a)−µH(a−1n−1k)]

×e−ρ[log(a)+H(a−1n−1k)] λ a(g(na)−1·o, λ, b)|c(λµ)|−2da dn dk dµ.

4 Equivariant pseudodifferential operators on symmetric spaces

4.5.2 An Egorov-type formula

The classical Egorov theorem states that conjugation by the wave group defines

an order preserving automorphism on the space of pseudodifferential operators.

We will now be able to prove the following version:

Theorem 4.32. Let a(z, λ, b)∈Sm

cl be compactly supported in z(uniformly in

the other variables). Write A=Op(a)and At:= eitRAe−itR. Then Athas

complete symbol Ut(a)∈Sm

cl and σAt=ctσA(Gt(z, b), λ), where ctis a constant.

Proof. The phase function of the symbol Ut(a)(z, λ, b)on A×N×K/M ×R+

is given by

ψt(µ, n, a, kM) = t(µ−1) −log(a)−µH(a−1n−1k).

As proven in Subs. 3.3, the phase function ψthas the critical point (µ, n, a, kM) =

(1, e, a−t, eM), and the Hessian form of ψtat the critical point is non-degenerate.

Under X×B∼

=G/M ∼

=A−1N−1×K/M the critical point (a−t, e, eM)cor-

responds to (w, b0) = (at·o, M) = Gt(o, M). Given (z, b) = (g·o, g ·M)we

then have (g·a−1n−1·o, λ, b)a=a−t,n=e= (Gt(z, b), λ). As before, the princi-

ple of non-stationary phase yields that Ut(a)is uniquely determined modulo

S−∞ := ∩mSmby a compactly supported cutoff of the integrand. The method

of stationary phase is applied to this cutoff of Ut(a)(z, λ, b)exactly as in Subs.

4.4.2, only the critical point of the phase function is different. The MSP-formula

yields an expansion for Ut(a)(z, λ, b)which can be rearranged in homogeneous

terms, so Ut(a)∈Sm

cl . In particular, the principal symbol of Ut(a)is given

by a constant times an evaluation at the critical point of the principal symbol

of a(all other terms in the MSP-formula have lower order). It follows that

σAt=ctσA(Gt(z, b), λ), so the theorem is proven.

Remark 4.33. (1) It seems reasonable to conjecture that ct= 1 for all t. In

fact, the operators in the MSP-formula are left-G-invariant, so the theo-

rem descends to a compact quotient XΓ. Write 1(z, λ, b)for the constant

function f(z, λ, b) = 1 on XΓ×R+×B. Then 1(Gt(z, b), λ) = 1 and we

can use diagonal matrix elements ρλj(Op(a)) as in the introduction to see

that 1 = ρλj(Op(1)) ∼ct(when j→ ∞).

(2) One should caution that conjugation eitRAe−itR is not equivalent to con-

jugation by the wave group. If one uses [BO05], Lemma 2.2, to compute

(for the standard quantization) the infinitesimal action of the wave group

on a symbol of a pseudodifferential operator, one finds that shifting the

Laplace operator influences the velocity of the (geodesic) flow defining the

symbol in an Egorov theorem.

5 Helgason boundary values

We start by recalling some fundamental relations first proven by Helgason

([Helg70]) between joint eigenfunctions of the algebra of invariant differential

operators with hyperfunctions and distributions on the real flag manifold of a

symmetric space. These relations are described by means of the Poisson trans-

form. In Subsection 5.2 we prove a regularity statement (Lemma 5.12) for

boundary values of certain eigenfunctions, which seems to be a new result.

Recall that D(G/K)denotes the algebra of differential operators on X=

G/K, which are invariant under left-translations by elements of G. Given a

homomorphism χ:D(G/K)→C, let χ(D)(D∈D(G/K)) denote the corre-

sponding system of eigenvalues. The space

Eχ(X) = {f∈E(X) : Df =χ(D)ffor all D∈D(G/K)}

is called a joint eigenspace of D(G/K). We also know that the homomorphisms

χas above can be parameterized by the orbits of the Weyl group in a∗, that is

each χis of the form χλ, where λ∈a∗. As in Section 2.3.3 we write

Eλ(X) = {f∈E(X) : Df = Γ(D)(iλ)ffor all D∈D(X)}.

A smooth function f∈E(G/K)is called joint eigenfunction if it belongs to one

of the spaces Eλ(X).

Definition 5.1. Let Ldenote the Laplace-Beltrami operator of B. Let A(B)

denote the vector space of analytic functions on B=K/M. For T > 0put

|F|T= sup

k∈Z+1

2k!TkLkF,

where k·kis the L2-norm on B, and

AT(B) = {F∈E(B) : |F|T<∞}.

Then AT(B)is a Banach space, A(B)is the union of the spaces AT(B)and is

accordingly given the inductive limit topology. The analytic functionals (hyper-

functions) are the functionals in the dual space A0(B)of A(B)(cf. [LM63]).

We use the integral notation for distributions or hyperfunctions and test func-

tions: For any space Ywe denote the pairing of distributions uand test functions

ϕon Yby RYϕ(y)u(dy) = hϕ, uiY.

The Poisson kernel P(x, b) = e2ρhx,biand its powers eλ,b(x) = e(iλ+ρ)hx,bi, where

λ∈a∗

C, are analytic functions ([GASS], p. 119).

Definition 5.2. Given a function, distribution or hyperfunction Ton Band

λ∈a∗

Cwe define the Poisson transform Pλ:A0(B)→Eλ(X)by

Pλ(T)(z) := ZB

e(iλ+ρ)hz,biT(db).(5.1)

5 Helgason boundary values

As a consequence of [Rou63], p. 167, the function Pλ(T)(z)is analytic and its

derivatives can be computed under the integral sign. It follows from (2.57) that

z7→ Pλ(T)(z)is a joint eigenfunction and belongs to Eλ(X).

If the functional Tabove is actually a function fon B, then T(db) = f(b)db.

Now suppose that ψis a function on a∗×B. Writing ψλ(b) = ψ(λ, b)we see

that (4.10) can be written in the form

Psλ(ψsλ) = Pλ(ψλ), s ∈W. (5.2)

The following fundamental theorem ([GASS], p. 507, Theorem 6.5) relates

eigenfunctions with hyperfunctions:

Theorem 5.3. The joint eigenfunctions of D(G/K)are the functions

f(x) = ZB

e(iλ+ρ)hz,bidT(b),

where λ∈a∗

Cand T∈A0(B).

Given a joint eigenfunction ϕof D(G/K), we call the unique functional T=Tϕ

given by Theorem 5.3 the boundary values (Helgason boundary values) of ϕ.

We will consider the following special class of eigenfunctions: Let ddenote the

distance function on X. We define the subspace E∗(X)of E(X)of functions of

exponential growth by

E∗(X) = f∈E(X) : ∃C > 0 : |f(x)| ≤ CeCdX(o,x)∀x∈X(5.3)

and we put E∗

λ(X) = E∗(X)∩Eλ(X). Denote by wthe longest Weyl group

element and recall Harish-Chandra’s e-functions (Subsection 2.1.7). It turns

out that eigenfunctions with exponential growth have distributional boundary

values (cf. [GASS], p. 508):

Theorem 5.4. Let λ∈a∗

Cbe such that ew(λ)6= 0. Then Pλ(D0(B)) = E∗

λ(X).

We will always consider eigenfunctions with unique and distributional bound-

ary values as in Theorem 5.4.

Fix any subgroup Γof Gand let ϕ∈E∗

λ(X)(λ∈a∗

C) denote a Γ-invariant

eigenfunction with unique and distributional boundary values Tλ. Then

ϕ(z) = ZB

e(iλ+ρ)hz,biTλ(db).

The group Gacts on the boundary Bof X(cf. Section 2.2.1). Hence Gacts

on D0(B)by push-forward: Given a distribution Ton B, a test function ϕ∈

E(B) = D(B)and g∈G, this action is given by

(gT)(ϕ) := T(ϕ◦g−1).(5.4)

5 Helgason boundary values

When we denote the pairing of distributions and functions by an integral, we

also write T(dgb)instead of (gT)(db)(g∈G). One might expect that Tas well

is invariant under the pull-back action of Γ. But in fact, since ϕ(γz) = ϕ(z)for

all γ∈Γand z∈X, we observe (recall (2.27), that is hg·x, g ·bi=hx, bi+

hg·o, g ·bi)

ϕ(z) = ϕ(γz) = ZB

e(iλ+ρ)hγz,biTλ(db)

=ZB

e(iλ+ρ)hγz,γbiTλ(dγb) = ZB

e(iλ+ρ)hz,bie(iλ+ρ)hγo,γbiTλ(dγb).

By uniqueness of the boundary values (Theorem 5.3) this implies

Tλ(db) = e(iλ+ρ)hγo,γbiTλ(dγb),

or equivalently

Tλ(dγb) = e−(iλ+ρ)hγo,γbiTλ(db).(5.5)

Definition 5.5. Let ϕand Tbe as above. We define eλ∈D0(X×B)as the

distribution on X×B=G/M given by

hf, eλi:= ZX×B

e(iλ+ρ)hz,bif(z, b)T(db)dz, f ∈D(X×B).(5.6)

The action of Gon distributions on X×Bis defined by pulling back the

action of Gon X×B: Given a distribution uand a test function fon X×B,

we write (g·u)(f) := u(f◦g−1). Let γ∈Γ. Then by the invariance of dz, by

(2.27) and (5.5) we obtain

hf, eλi=ZX×B

e(iλ+ρ)hγz,γbif(γz, γb)T(dγb)dz

=ZX×B

e(iλ+ρ)hz,bif(γz, γb)T(db)dz

=hf◦γ, eλi=hf, γ−1·eλi.

Corollary 5.6. eλis a Γ-invariant distribution on X×B.

5.1 Poisson transform and principal series representations

We recall some facts concerning the principal series representations of G. We

follow [GASS] and [Wil91]. Let λ∈aand consider the representation

σλ(man) = e(iλ+ρ) log(a)

5 Helgason boundary values

of P=MAN on C. We denote the induced representation on Gby πλ=

IndG

P(σλ). The induced picture of this representation is constructed as follows:

A dense subspace of the representation space is

H∞

λ:= f∈C∞(G) : f(gman) = e−(iλ+ρ) log(a)f(g).

We define an inner product on H∞

λby

(f1, f2) = ZK/M

f1(k)f2(k)dk =hf1|K, f2|KiL2(K/M)

and denote the corresponding norm by kfk2=RK/M |f(k)|2dk. The group

action of Gis given by

(πλ(g)f)(x) = f(g−1x).

The actual Hilbert space, which we denote by Hλ, and the representation on

Hλ, which we also denote by πλ, is obtained by completion (cf. [Wil91], Ch. 9).

The representations πλ(λ∈a) form the spherical principal series of G. The

representation (πλ, Hλ)is a unitary ([GASS], p. 528) and irreducible (loc. cit.

p. 530) Hilbert space representation.

Given f∈C∞(K/M), we extend it to a function on Gby putting

f(g) = e−(iλ+ρ)H(g)f(k(g)),(5.7)

where g=k(g) exp H(g)n(g)according to the Iwasawa decomposition.

Proposition 5.7. (i) For f∈C∞(K/M)let e

fas in (5.7). Then e

f∈H∞

λ.

(ii) Let e

f∈H∞

λand denote restriction to Kby e

f|K. Then e

f|K∈C∞(K/M)

(iii) Let f∈C∞(K/M)and e

fas in (5.7). Then e

f|K=f.

(iv) The mapping f7→ e

fis isometric with respect to the L2(K/M)-norm. It

intertwines the representation πλand the representation (which we also

denote by πλ) on C∞(K/M)defined by

(πλ(g)f)(kM) = f(k(g−1k)M)e−(iλ+ρ)H(g−1k).(5.8)

Proof. All assertions are clear.

In view of Proposition 5.7 we identify C∞(K/M)∼

=H∞

λ. The advantage of

C∞(K/M)is that the representation space is independent of λ. The representa-

tions (5.8) are called the compact picture (compact realization) of the (spherical)

principal series. Notice that for g∈Kthe group action (5.8) simplifies to the

left-regular representation of the compact group Kon K/M.

5 Helgason boundary values

Let λ∈aand denote by 1the constant function k7→ 1on K/M. In the

compact picture we observe

(πλ(g)1)(k) = e−(iλ+ρ)H(g−1k)=e(iλ+ρ)hgK,kMi,(5.9)

and it follows that the Poisson transform

Pλ(T) : G/K →C(5.10)

of T∈D0(B)is given by

Pλ(T)(gK) = T(πλ(g)·1).(5.11)

It follows that the Poisson transform Pλintertwines the dual spherical prin-

cipal series representation eπλand the translation on G/K. Now suppose that

ϕ∈Eλ(X)is a Γ-invariant joint eigenfunction of D(G/K)with boundary val-

ues Tϕ∈D0(B)such that ϕ=Pλ(Tϕ). Since ϕis invariant, it follows from

(5.11) and the uniqueness of the boundary values that Tϕis invariant under all

eπλ(γ), for γ∈Γ. Vice versa, if Tis a Γ-invariant distribution, then Pλ(T)is a

Γ-invariant eigenfunction.

5.2 Regularity of distributional boundary values

Before we start with our investigation on the regularity of distributional bound-

ary values for eigenfunctions in general symmetric spaces, we motivate this sec-

tion by recalling some results proven by Otal for compact hyperbolic surfaces.

We use the notation of [Otal98] and [AZ07].

Definition 5.8. For 0≤δ≤1we say that a 2π-periodic function F:R→Cis

δ-Hölder if there exists C≥0such that |F(x)−F(y)| ≤ C|x−y|δ. The smallest

constant Cis denoted by kFkδ. The Banach space of δ-Hölder functions with

norm kFkδis denoted by Λδ.

Theorem 5.9 ([Otal98], Proposition 4).Suppose that s=1

2+ir with Re(s)≥

0, and that ϕis an eigenfunction of the Laplace operator of HΓsatisfying

k∇ϕkL∞<0. Then its Helgason boundary value Tϕis the derivative of a Re(s)-

Hölder function.

Since outside a finite number of small eigenvalues sof HΓbelonging to the

complementary series we always have Re(s) = 1

2(for eigenvalues s=1

2+ir of

the Laplacian on HΓ), it follows that almost all boundary values associated to

eigenfunctions and eigenvalues belonging to the discrete spectrum of HΓ, are

derivatives of certain 1

2-continuous Hölder functions. To be more precise, the

boundary values are not literally the derivative of a periodic function, but the

derivative of a function Fon Rsatisfying F(x+ 2π) = F(x) + Cfor all x∈R.

5 Helgason boundary values

As described in [AZ07], it follows from Otal’s regularity statement, that given

an eigenfunction ϕto the eigenvalue s=1

2+ir with r∈R, then the Hölder

norm of the corresponding boundary values Tϕ,r is bounded by a power of r.

As noted in [GO05], there seems to be no straightforward generalization of

these concepts, not even in the case of the real hyperbolic spaces. However,

related approaches can be found, for example, in [GO05].

In this subsection we give a representation theoretic approach to describe the

regularity of distributional boundary values and its dependence on the spectral

parameter λand we prove a regularity statement for the boundary values cor-

responding to joint eigenfunctions with real eigenvalue parameter λ∈a∗on a

compact quotient XΓ. These estimates may not be the sharpest possible, but

they are sufficient for our purposes.

Given λ∈a∗

C, let D0(B)Γdenote the space of distribuions on Bwhich are

invariant under all actions eπλ(γ)(γ∈Γ). As described in the preceding sub-

section, if T∈D0(B)Γ, then the Poisson transform Pλ(T)is a function on the

quotient XΓ. We may hence also define

D0(B)(1)

Γ:= nT∈D0(B)Γ:kPλ(T)kL2(XΓ)= 1o.(5.12)

Now fix λ∈a∗and a Γ-invariant joint eigenfunction ϕ∈Eλ(X)of D(G/K)(it

has automatically exponential growth, since it is Γ-invariant). We also assume

that ϕis normalized with respect to the customary L2(XΓ)-norm. Let Tϕ∈

D0(B)(1)

Γdenote be the (unique) preimage (under the Poisson transform) of ϕ.

Under the identification H∞

λ∼

=C∞(K/M)we view Tϕas a functional on H∞

λ:

For f∈H∞

λlet Tϕ(f)be defined by Tϕ(f|K). Then Tϕis a continuous linear

functional on H∞

λ, invariant under eπλ(γ). As proven in [CG89], Theorem A.1.4,

if fis a smooth vector for the principal series representation, then f∈H∞

λis a

smooth function on G. We consider the mapping

Φϕ:H∞

λ→C∞(Γ\G),Φϕ(f)(Γg) = Tϕ(πλ(g)f).

Lemma 5.10. Φϕis an isometry w.r.t. the norms of L2(K/M)and L2(Γ\G).

Proof. The operator Φϕis equivariant with respect to the actions πλon H∞

λand

the right regular representation of Gon L2(Γ\G). We pull-back the L2(Γ\G)in-

ner product onto the (g, K)-module H∞

λ,K of K-finite and smooth vectors (which

is dense in H∞

λ, [Wal88], p. 81):

hf1, f2i2:= hΦϕ(f1),Φϕ(f2)iL2(Γ\G).

Let f1∈H∞

λ,K. Then

Af1:H∞

λ,K →C, f27→ hf1, f2i2

is a conjugate-linear, K-finite functional on the (g, K)-module H∞

λ,K. This mod-

ule is irreducible and admissible, since Hλis unitary and irreducible ([Wal88],

5 Helgason boundary values

Thm. 3.4.10, Thm. 3.4.11). As Af1is K-finite it is nonzero on at most

finitely many K-isotypic components. It follows that there is a linear map

A:H∞

λ,K →H∞

λ,K such that for each f1∈H∞

λ,K the functional Af1equals

f27→ hAf1, f2iL2(K/M). The equivariance of Φϕand the unitarity of πλimply

that Ais (g, K)-equivariant. Using Schur’s lemma for irreducible (g, K)-modules

([Wal88], p. 80), we deduce that Ais a constant multiple of the identity and

hence h·,·i2is a constant multiple of the original L2(K/M)-inner product on

H∞

λ,K. This constant is 1: First, Φϕ(1) = Pλ(Tϕ) = ϕis the K-invariant lift of

ϕto L2(Γ\G). Then kΦϕ(1)kL2(Γ\G)=1=k1kL2(K/M).

Let (yj)and (xj)be bases for kand p, respectively, such that hyj, yii=−δij,

hxj, xii=δij, where h,idenotes the Killing form. The Casimir operator of kis

Ωk=Piy2

iand the Casimir operator of gis

Ωg=−X

j+ Ωk∈Z(g),

where Z(g)is the center of the universal enveloping algebra U(g)of g.

It follows from Tϕ(f) = Φϕ(f)(Γe)that

|Tϕ(f)|≤kΦϕ(f)k∞.(5.13)

We may now estimate this by a convenient Sobolev norm on L2(Γ\G). Let e

∆

denote the Laplace operator of Γ\G. Then we have

∆ = −Ωg+ 2Ωk,

where Ωgand Ωkare the Casimir operators on Gand K, respectively.

Definition 5.11. Let s∈R. The Sobolev space W2,s(Γ\G)is (cf. [Tay81], p.

22) the space of functions fon Γ\Gsatisfying (1 + e

∆)s/2(f)∈L2(Γ\G)with

norm

kfkW2,s(Γ\G)=k(1 + e

∆)s/2(f)kL2(Γ\G).

Let m= dim(Γ\G) = dim(G), and let s > m/2. The Sobolev imbedding

theorem for the compact space Γ\G([Tay81], p. 19) states that the identity

W2,s(Γ\G),→C0(Γ\G)is a continuous inclusion (C0(Γ\G)is equipped with

the usual sup-norm k·k∞). It follows that there exists a C > 0such that

kΦϕ(f)k∞≤CkΦϕ(f)kW2,s(Γ\G)∀f∈C∞(K/M).(5.14)

Now we derive the announced regularity estimate for the boundary values:

First, by increasing the Sobolev order, we may assume s/2∈N, so

(1 + e

∆)s/2= (1 −Ωg+ 2Ωk)s/2∈U(g).

5 Helgason boundary values

Hence (1 + e

∆)s/2commutes with each G-equivariant mapping. Let f∈H∞

λ.

Then

kΦϕ(f)kW2,s(Γ\G)=(1 + e

∆)s/2Φϕ(f)L2(Γ\G)

=Φϕ((1 −Ωg+ 2Ωk)s/2(f))L2(Γ\G)

=(1 −Ωg+ 2Ωk)s/2(f)L2(K/M).(5.15)

Recall πλ(Ωk) = ∆K/M and Ωg∈Z(g). Then (5.15) equals



s/2

k=0 s/2

k(1 + 2∆K/M )k(−Ωg)s/2−k(f)L2(K/M)

≤

s/2

k=0 s/2

k(1 + 2∆K/M )k(−Ωg)s/2−k(f)L2(K/M).(5.16)

Assume f∈H∞

λ.K and recall that Ωgacts on the irreducible U(g)-module H∞

λ,K

by multiplication with the scalar −(hλ, λi+hρ, ρi)(cf. [Wil91], p. 163), that is

Ωg|H∞

λ,K =−(hλ, λi+hρ, ρi) idH∞

λ,K .

Then (5.16) equals

s/2

k=0 s/2

k(1 + 2∆K/M )k(|λ|2+|ρ|2)s/2−k(f)L2(K/M).(5.17)

But (|λ|2+|ρ|2)−k≤1 + |ρ|−s=: C0(0≤k≤s/2), so (5.17) is bounded by

C0|λ|2+|ρ|2s/2

s/2

k=0 s/2

k(1 + 2∆K/M )k(f)L2(K/M).(5.18)

Since H∞

λ.K is dense in H∞

λ, this bound holds for all f∈H∞

λ. Using (5.13)-(5.18)

we get

|Tϕ(f)| ≤ C0|λ|2+|ρ|2s/2

s/2

k=0 s/2

k(1 + 2∆K/M )k(f)L2(K/M).(5.19)

for all f∈H∞

λand hence for all f∈C∞(K/M). We estimate the sum in

(5.19) by the continuous C∞(K/M)-seminorm (recall that K/M has normalized

volume)

kfk0:=

s/2

k=0 s/2

ksup

K/M |(1 + 2∆K/M )k(f)|, f ∈E(B),(5.20)

5 Helgason boundary values

(where 2s > dim(G)is arbitrary, but fixed) and define

D0(B)λ:= {T∈D0(B) : |T(f)| ≤ (1 + |λ|)skfk0∀f∈C∞(K/M)}.(5.21)

(Note that D0(B)λdepends on the number s > dim(G)/2). We summarize

these obervations as follows:

Lemma 5.12. D0(B)(1)

Γ⊆D0(B)λ.

5.3 Tensor products of distributional boundary values

We need to recall some background concerning tensor products of distributions,

which is naturally based on the tensor product of the underlying test function

spaces and their completions. We assume that the reader is familiar with the

definitions of the customary algebraic tensor product of general vector spaces.

We are mainly interested in the compatibility of the tensor product for distribu-

tions with the embedding f7→ If(2.33) of functions into distributions and the

tensor product for functions. The material is taken from [Treves67] and [BB02].

If Ωjare non-empty open subsets of Rnjand ϕj∈D(Ωj)are test functions,

their tensor product is the function ϕ1⊗ϕ2∈D(Ω1×Ω2)defined by

ϕ1⊗ϕ2(x1, x2) = ϕ(x1)ϕ(x2) (xj∈Ωj).

The vector space spanned by all these tensors is denoted by D(Ω1)⊗D(Ω2). A

general element in D(Ω1)⊗D(Ω2)is a finite sum Pjϕj⊗ψj, where ϕj∈D(Ω1),

ψj∈D(Ω2). Then

D(Ω1)⊗D(Ω2)⊂D(Ω1×Ω2).

This tensor product space is dense in the test function space D(Ω1×Ω2).

On the algebraic tensor product E⊗Fof two Hausdorff locally convex topo-

logical spaces Eand Fover the same field one can define the projective tensor

product as follows. Let Pand Qdenote the respective filtering systems of semi-

norms defining the topology of the respective spaces Eand F. A general element

χ∈E⊗Fis of the form

χ=

j=1

ej⊗fj(ej∈E, fj∈F).

This representation as a finite sum is not unique. Given semi-norms p∈P, q ∈Q

we define the projective tensor product by

p⊗πq:= inf (m

j=1

p(ej)q(fj) : χ=

j=1

ej⊗fj).

5 Helgason boundary values

Then p⊗πqdefines a semi-norm on E⊗F, and the system

P⊗πQ:= {p⊗πq:p∈P, q ∈Q}

is a filtering and thus defines a locally convex topology on E⊗F, called the

projective tensor product topology. The vector space equipped with this topology

is denoted by

E⊗πF

and is called the projective tensor product of the spaces Eand F·

In particular, if X, Y ⊂Rnare both open or compact, then the completion

D(X)b

⊗πD(Y)of the projective tensor product D(X)⊗πD(Y)is equal to the

test function space D(X×Y)over the product X×Y([Treves67], p. 530):

D(X)b

⊗πD(Y) = D(X×Y).

Let Ωjbe as above. For ϕ∈D(Ω1⊗Ω2)and T∈D0(Ω1)we define a function

ψon Ω2by ψ(y) = hT, ϕyi, where ϕ(y)(x) := ϕ(x, y). Then ψ∈D(Ω2), and

F(T, ϕ) := ψdefines a bilinear map F:D0(Ω1)×D(Ω1×Ω2)→D(Ω2). This

yields the existence of the tensor product for distributions:

For Tj∈D0(Ωj)there is exactly one distribution T∈D0(Ω1×Ω2), called the

tensor product of T1and T2, such that ([BB02], Ch. 6.2)

hT, ϕ1⊗ϕ2i=hT1, ϕ1ihT2, ϕ2i.

Recall the embedding of functions into distributions as given in (2.33). If

f, g ∈L1

loc(Ω), a direct computation shows (loc. cit.)

hIf⊗Ig, ϕ ⊗ψi=hIf, ϕihIg, ψi,

so the tensor product of distributions is consistent with the tensor product of

functions.

For convenience, if Tis a distribution and fa test function on a space Y, we

sometimes write hT(y), f(y)ifor the pairing between Tand finstead of hT, fi

to point out the active variables.

The tensor product of T1and T2is a continuous linear functional on Ω1×Ω2

and it satisfies Fubini’s theorem for distributions: For every Tj∈D(Ωj)and for

every χ∈D(Ω1×Ω2)one has (loc. cit.)

hT1⊗T2, χi=h(T1⊗T2)(x, y), χ(x, y)i

=hT1(x),hT2(y)χ(x, y)ii

=hT1(y),hT2(x)χ(x, y)ii.

We can now apply the definitions given above to tensor products of distribu-

tional boundary values. As usual, let B=K/M denote the real flag manifold of

belonging to the Riemannian symmetric space X=G/K of noncompact type.

100

5 Helgason boundary values

In the notation of Section 5.2, there is a continuous seminorm k·k0on C∞(B)

and a constant Ksuch that for all distributional boundary values Tϕ,λ cor-

responding under the Poisson transform to a Γ-invariant joint eigenfunction

ϕ∈E∗

λ(X)with kϕkL2(XΓ)= 1 we have

|T(f)| ≤ (1 + |λ|)Kkfk0∀f∈C∞(B).

Each f∈C∞(B)⊗C∞(B)has the form

f=X

i,j

ci,jfi⊗fj.

We define a cross-(semi-)norm k·k00 on the customary algebraic tensor product

C∞(B)⊗C∞(B)by

kfk00 = inf (X

i,j |ci,j|kfik0kfjk0:f=X

i,j

ci,jfi⊗fj).

Then by [Treves67], p. 435, this norm induces a continuous seminorm on the

projective tensor product C∞(B)b

⊗πC∞(B).

Let ϕ∈E∗

λ(X)and ψ∈E∗

µ(X)denote Γ-invariant and L2(XΓ)-normalized

eigenfunction with distributional boundary values Tϕ, Tψ∈D0(B)and eigen-

value parameter µ∈a∗. Given

f=X

i,j

ci,jfi⊗fj∈C∞(B)⊗C∞(B)

we obtain

|(Tϕ⊗Tψ)(f)| ≤ X

i,j |ci,j|·|Tϕ(fi)|·|Tψ(fj)|

≤(1 + |λ|)s(1 + |µ|)sX

i,j |ci,j|·kfik0·kfjk0,

which implies (by taking the infimum)

|(Tϕ⊗Tψ)(f)| ≤ (1 + |λ|)s(1 + |µ|)skfk00 (5.22)

for all f∈C∞(B)⊗C∞(B). But

C∞(B×B)∼

=C∞(B)b

⊗πC∞(B)

([Treves67], p. 530) implies that (5.22) holds for all f∈C∞(B×B).

101

6 Patterson-Sullivan distributions

In this Section, we introduce Patterson-Sullivan distributions for symmetric

spaces of the noncompact type and establish a couple of invariance properties.

It will then turn out how these phase space distributions are related to the

questions of quantum ergodicity.

We carry over the notation from the preceding chapters. Gdenotes a non-

compact semisimple Lie group with finite center and Iwasawa decomposition

G=KAN. By X=G/K we denote the corresponding symmetric space of

the noncompact type. By B=K/M we denote the (Fürstenberg) boundary of

X. Given a cocompact torsion free discrete subgroup Γof G, we denote by XΓ

the corresponding locally symmetric compact manifold of nonpositive sectional

curvature. At this point, we make no restriction on the rank of X. In general,

a (diagonal) Patterson-Sullivan distribution psλ=psϕ,λ will be associated to a

joint eigenfunction ϕ∈E∗

λ(X), where λ∈a∗

In Subsection 6.1 we build up a concept of functions, which we call inter-

mediate values. The intermediate values depend on the spectral parameter λ.

Invariance properties of the Patterson-Sullivan distributions arise from equiv-

ariance properties of the intermediate values. Tensoring the psλ-distributions

with an appropriate Radon transform, one obtains A-invariant distributions

PSϕ,λ. In Subsection 6.2.1 we generalize the constructions given in [AZ07] to

symmetric spaces of the noncompact type. We will explain that these special

constructions are only possible for eigenvalue-parameters that satisfy a certain

condition (see Lemma 6.10). It is not possible to generalize these definitions to

a larger class of eigenfunctions and eigenvalues. Eigenvalues of the Laplacian

of a rank one space satisfy this condition. In Subsection 6.2.3 we introduce

off-diagonal Patterson-Sullivan distributions PSϕ,λ,ψ,µ, which are associated to

two eigenfunctions ϕ∈E∗

λ(X)and ψ∈E∗

µ(X). These distributions exist for

all symmetric spaces of the noncompact type. If ϕ=ψ, they coincide with the

PSϕ,λ for the special cases considered in Subsection 6.2.1.

6.1 Intermediate values

Let Hnbe the real hyperbolic space of dimension n, that is, the complete and

simply connected Riemannian manifold of constant curvature −1. Using the

Poincaré model, we identify Hnwith the unit ball of Rnand its (geodesic)

boundary at infinity ∂Hnwith the unit sphere Sn−1of Rn.

For z∈Hn, let γbe an isometry of Hnsuch that z=γ−1·0, where 0∈Rn

is the origin of Hn. Then |γ0(ξ)|=P(z, ξ), where Pis the Poisson kernel of

Hnand where |γ0(ξ)|is the conformal factor of the derivative of γat the point

ξ∈Sn−1.

Given two points ξ, ξ0∈Sn−1, we denote their chordal (Euclidean) distance

by |ξ−ξ0|= 2 sin(θ/2), where θis the spherical distance between ξand ξ0. One

6 Patterson-Sullivan distributions

has the intermediate value formula (cf. [Sul79])

|γ(ξ0)−γ(ξ)|2=|γ0(ξ0)||γ(ξ)||ξ0−ξ|2.(6.1)

The derivatives in (6.1) are (cf. (3.6)) given by d(γ·b)

db =e−2ρhγ·o,γ·bi. Suppose

that G=SU(1,1) and

K=eiθ 0

0e−iθ, θ ∈R.

Then the non-Euclidean disk Didentifies with the symmetric space G/K. Writ-

ing ρ=1

2we find

|γb −γb0|=e−(hγ·o,γ·bi+hγ·o,γ·b0i)|b−b0|(6.2)

for b, b0∈∂D. Caution that the horocycle bracket hz, biwe use is written 1

2hz, bi

in [AZ07], [Nich89] etc., because the hyperbolic metric on Dis often defined

to be a multiple of the metric used in [DS], [GGA], [GASS]. (Sometimes the

abelian subgroup A=atof Gis parameterized by t/2instead of t, that is

at=diag(et/2, e−t/2)∈G.) Raising (6.2) to the power 1

2+ir we obtain

|γb −γb0|1

2+ir =e−(1

2+ir)·(hγ·o,γ·bi+hγ·o,γ·b0i)|b−b0|1

2+ir.(6.3)

In this setting it is standard ([AZ07]) to parameterize the eigenvalue parameters

corresponding to the eigenvalues of ∆on compact hyperbolic surfaces by λj=

2+irj. In the disk model we have (b∞, b−∞) = (M, wM)∈B×Bis (1,−1) ∈

∂D×∂D. Writing (b, b0) = (γ·M, γ ·wM), then (6.3) yields

|b−b0|1

2+ir = 21

2+ire−(1

2+ir)·(hγ·o,γ·1i+hγ·o,γ·(−1)i).(6.4)

For a general symmetric space X=G/K with real flag manifold B=K/M

we can neither use a Poincaré ball model nor Euclidean distances. We will now

see how to generalize equation (6.3) in group-theoretical terms.

6.1.1 Generalized intermediate values

As usual, let Hdenote the Iwasawa projection KAN →a. We denote the

longest Weyl group element and (by abuse of notation) a representative of it in

M0by w, where M0is the normalizer of Ain K. Let λ, µ ∈a∗

C. We introduce

the function dλ,µ :G→C,

dλ,µ(g) = e(iλ+ρ)H(g)e(iµ+ρ)H(gw).(6.5)

Definition 6.1. We call the functions dλ,µ off-diagonal intermediate values. In

the case when λ=µwe define diagonal intermediate values dλ:= dλ,λ.

103

6 Patterson-Sullivan distributions

Recall that the action of Won a∗is defined via duality by

(s·ν)(X) := ν(s−1·X),

where s∈W,ν∈a∗,X∈a, and where ·denotes the adjoint action. We have

s·X∈a∗

C, since M0(hence s∈W) normalizes Aand a∗

C. The action is extended

to a∗

Cby complex linearity.

Lemma 6.2. Let g∈G,m∈M,a∈A. Then

dλ,µ(gam) = dλ,µ(g)ei(λ+w·µ) log(a).(6.6)

Proof. Recall that the Iwasawa-projection His M-invariant and that M0(hence

w∈W) normalizes M. Then

dλ,µ(gam) = e(iλ+ρ)H(gam)e(iµ+ρ)H(gamw)

=e(iλ+ρ)H(ga)e(iµ+ρ)H(gww−1aw)

=e(iλ+ρ)(H(g)+log(a))e(iµ+ρ)(H(gw)+log(w−1aw))

=e(iλ+ρ)H(g)e(iµ+ρ)H(gw)e(iµ+ρ)(log(w−1aw))e(iλ+ρ) log(a)

=dλ,µ(g)e(iµ+ρ)(log(w−1aw))e(iλ+ρ) log(a).

Also recall log(w−1aw) = w−1·log(a), since exp and log intertwine AdG(w)with

the conjugation by won A. It follows from w·ρ=−ρthat the last line equals

dλ,µ(g)ei(λ+w·µ) log(a),

and the lemma is proven.

Remark 6.3. (1) The functions dλ,µ are right-M-invariant. Thus

dλ,µ :G/M →C.

(2) Suppose that w·λ=−λ. This is satisfied if the longest Weyl group

element satisfies AdG(w) = −ida∗, which is for example true if G/K has

rank one. Then the diagonal intermediate values function dλis invariant

under right-translation by elements a∈Aand hence a function on G/MA.

In all other cases, dλis not a function on G/MA. We will see in (6.27)

how to circumvent this problem.

(3) Let G/K have rank one. If m0∈M0, then dλ(gm0) = dλ(g), so dλis a

function on G/M0A.

Recall (from 2.2.4) that B(2) ∼

=G/MA: The group Gacts transitively on

B(2). The closed subgroup of Gfixing (M, wM)∈B(2) is MA. Thus each pair

of distinct boundary points (b, b0)may be written in the form g·(M, wM), where

g(b, b0)MA =gMA ∈G/MA is unique.

104

6 Patterson-Sullivan distributions

Definition 6.4. Time reversal refers to the involution on the unit cosphere

bundle defined by ι(x, ξ)=(x, −ξ). Suppose that G/K has rank one. Under

Γ\G/M =S∗XΓthe time reversal map takes the form Γg7→ Γwg. We say that

a distribution Tis time-reversible if ι∗T=T. Let (b, b0)=(g·M, g·wM)∈B(2),

where g∈Gand gMA ∈G/MA is unique. Recall w2∈M. Then time reversal

means

(b, b0) = (g·M, g ·wM)7→ (gw ·M, g ·w2M) = (b0, b),

that is the interchanging (b, b0)↔(b0, b). We call a function or distribution on

B2time reversal invariant if it is invariant under (b, b0)↔(b0, b).

Corollary 6.5. Let G/K have rank one. The functions dλare time reversal

invariant.

For the rest of this subsection suppose that AdG(w) = −ida∗. Under the

identification B(2) ∼

=G/MA the function dλcorresponds to a function on B(2)

which we also denote by dλ. If g=g(b, b0), then dλ:B(2) →C,

dλ(b, b0) = dλ(g·M, g ·wM) = e(iλ+ρ)(H(g)+H(gw)).

Recall the horocycle bracket h·,·i on G/K ×K/M. Let g∈G. We have

shown in Lemma 2.39 that hg·o, g ·Mi=H(g)and hg·o, g ·wMi=H(gw).

Corollary 6.6. Let h·,·i denote the horocycle-bracket. Then

dλ(g·M, g ·wM) = e(iλ+ρ)(hg·o,g·Mi+hg·o,g·wMi).(6.7)

Proof. This follows from hg·o, g ·Mi=H(g)and hg·o, g ·wMi=H(gw).

Lemma 6.7. Let γ, g ∈G. Then

dλ,µ(γg) = e(iλ+ρ)hγ·o,γg·Mie(iµ+ρ)hγ·o,γg·wMidλ,µ(g).(6.8)

Proof. Let z=g·o∈G/K. By (2.27) and by Lemma 2.39 we find

H(γg) = hγg ·o, γg ·Mi

=hγ·z, γg ·Mi

=hz, g ·Mi+hγ·o, γg ·Mi

=H(g) + hγ·o, γg ·Mi.

Similarly we compute

H(γgw) = hγg ·o, γg ·wMi

=hγ·z, γg ·wMi

=hz, g ·wMi+hγ·o, γg ·wMi

=H(gw) + hγ·o, γg ·wMi.

Summing up we obtain the assertion.

105

6 Patterson-Sullivan distributions

Corollary 6.8. dλ(γg) = e(iλ+ρ)(hγ·o,γg·Mi+hγ·o,γg·wMi)dλ,µ(g).

Lemma 6.9. Let (b, b0)∈B(2) and γ∈G. Then

(dλ◦γ)(b, b0) = dλ(γ·b, γ ·b0) = e(iλ+ρ)(hγ·o,γ·bi+hγ·o,γ·b0i)dλ(b, b0).(6.9)

Proof. Let g∈Gsuch that (b, b0)=(g·M, g·wM). Then dλ(γ·b, γ·b0) = dλ(γg),

so the Lemma follows from Corollary 6.8.

6.1.2 An equivariance property

Recall from Section 5that in case of Γ-invariant joint eigenfunctions ϕλthe

corresponding distribution boundary values satisfy

Tλ(dγb)⊗Tλ(dγb0) = e−(iλ+ρ)(hγo,γbi+hγo,γb0i)Tλ(db)⊗Tλ(db0).

To obtain Γ-invariant distributions we multiply with so-called intermediate val-

ues dλ(b, b0)which satisfy the inverse equivariance property

dλ(γ·b, γ ·b0) = e(iλ+ρ)(hγ·o,γ·bi+hγ·o,γ·b0i)dλ(b, b0).(6.10)

The result of this subsection is very interesting: We prove in the following that

the existence of a non-trivial function satisfying (6.10) is equivalent to a certain

condition on the eigenvalue parameter.

The idea is that the function dλis independent of the concrete subgroup Γ,

so we suppose (6.10) to be satisfied for all g, γ ∈G. Let w∈Wdenote the

longest Weyl group element. We identify wwith a representative in M0.

Lemma 6.10. Suppose that there exists a function dλ:G/MA →Csatisfying

(6.10)for all γ∈Gand all (b, b0)∈B(2). Then w·λ=−λ.

Proof. Given (b, b0)∈B(2), there is g∈Gsuch that under G/MA ∼

=B(2) we

can write b=g·Mand b0=g·wM. Then (6.10) for a function on B(2) is

equivalent to the existence of a function dλon G/MA satisfying

dλ(γg) = e(iλ+ρ)(hγ·o,γ·g·Mi+hγ·o,γ·g·wMi)dλ(g)∀γ, g ∈G. (6.11)

Let a∈A, n ∈N. We first have

dλ(n) = e(iλ+ρ)(hn·o,Mi+hn·o,nw·Midλ(e) = e(iλ+ρ)H(nw)dλ(e).(6.12)

Since ana−1∈Nthe assumed MA-invariance then yields

dλ(an) = dλ(ana−1) = e(iλ+ρ)H(ana−1w)dλ(e).(6.13)

Combining (6.11) and (6.12) we also find

dλ(an) = e(iλ+ρ)(ha·o,a·n·Mi+ha·o,a·n·wMi)dλ(n)

=e(iλ+ρ)(log(a)+ha·o,an·wMi+H(nw))dλ(e).(6.14)

106

6 Patterson-Sullivan distributions

Comparing (6.13) with (6.14) and assuming dλ(e)6= 0 (otherwise dλ= 0

everywhere by the transitivity of the G-action on G/MA) we get

(iλ +ρ)H(ana−1w) = (iλ +ρ)[log(a) + ha·o, an ·wMi+H(nw)].(6.15)

On the the left hand side of (6.15) we have

H(ana−1w) = H(anww−1a−1w).(6.16)

Note that w−1a−1w∈A, since Wnormalizes A. Thus (6.16) equals

H(anw) + log(w−1a−1w).(6.17)

For the right hand side of (6.15) recall that

ha·o, an ·wMi=−H(a−1k(anw)).

If anw =˜

k˜a˜n, then a−1k(anw) = nw˜n−1˜a−1, so

ha·o, an ·wMi=−H(a−1k(anw))

=−H(nw˜n−1˜a−1)

=−H(nw) + log(˜a).

Thus on the right hand side of (6.15) we have

log(a) + ha·o, an ·wMi+H(nw) = log(a)−H(nw) + log(˜a) + H(nw)

= log(a) + log(˜a)

= log(a) + H(anw).(6.18)

If we now compare (6.17) with (6.18) we see that (6.10) implies

(iλ +ρ) log(a)=(iλ +ρ) log(w−1a−1w)

for all a∈A. But

ρ(log(w−1a−1w)) = (w·ρ) log(a−1) = −ρlog(a−1) = ρlog(a),

since w·ρ=−ρ, since wmaps positive roots into negative roots. Moreover,

λlog(w−1a−1w) = λ(w−1·log(a−1))) = (w·λ)(−log(a)),

so our final condition is w·λ=−λ, as desired.

Remark 6.11. Note that equation (6.11) can be satisfied by a function dλ

defined on G/M. We will later see how to circumvent the problem of missing

A-invariance.

107

6 Patterson-Sullivan distributions

6.2 Definitions and invariance properties

We now build up the theory of Patterson-Sullivan distributions. We start by

generalizing the definitions given in [AZ07], which is possible if Ad(w) = −ida

(recall that by wwe denote the longest Weyl group element). Later we see how

to define Patterson-Sullivan distributions for general symmetric spaces. We also

study interesting invariance properties of these distributions.

6.2.1 Diagonal Patterson-Sullivan distributions

In this Section we fix λ∈a∗

Cand suppose that w·λ=−λ. We fix an eigenfunc-

tion ϕ∈E∗

λ(X). At this point, we do not assume that ϕis real-valued. Let Tϕ

denote the boundary values of ϕ. The assumption on λis satisfied if the longest

Weyl group element wsatisfies AdG(w∗)|a=−ida. This is the case for all rank

one spaces. Recall the concept of intermediate values (Section 6.1)

dλ(b, b0) = dλ(g·M, g ·wM) = e(iλ+ρ)(H(g)+H(gw)),

where g=g(b, b0)corresponding to B(2) ∼

=G/MA. We have proven in Subsec-

tion 6.1.2 that this function exists if and only if w·λ=−λ.

Definition 6.12. The Patterson-Sullivan distribution psϕ,λ(db, db0)associated

to ϕ∈Eλis the distribution on C∞

c(B(2))defined by

psϕ,λ(db, db0) := dλ(b, b0)·Tϕ(db)⊗Tϕ(db0).(6.19)

The same definition (6.19) extends psλto a linear functional on the larger space

dλ(b, b0)−1·C∞(B×B). If ϕ∈E∗

λ(X)is fixed we write for simplicity psλinstead

of psϕ,λ. Moreover, we often write Tϕ(db)Tϕ(db0)instead of Tϕ(db)⊗Tϕ(db0).

Proposition 6.13. Let ϕ∈E∗

λ(X)be a Γ-invariant eigenfunction of D(G/K).

Let Tϕdenote its boundary values. Then psλ(db, db0)is Γ-invariant.

Proof. Given a test function f∈C∞

c(B(2))and γ∈Γ, we observe

psλ(f◦γ−1) = (Tϕ⊗Tϕ)(dλ·(f◦γ−1)) = (γTϕ⊗γTϕ)((dλ◦γ)·f).

It follows from (5.5) that

Tϕ(dγb)Tϕ(dγb0) = e−(iλ+ρ)hγ·o,γ·bie−(iλ+ρ)hγ·o,γ·b0iTϕ(db)Tϕ(db0).

By (6.9), the dλ(b, b0)have the inverse equivariance property, so multiplying

with (6.9) yields (Tϕ⊗Tϕ)(dλ·f) = psλ(f)and completes the proof of Γ-

invariance.

Recall the time reversal map b↔b0. Then by Corollary 6.5:

Proposition 6.14. Suppose that ϕ∈E∗

λ(X)is Γ-invariant. Then the distribu-

tion psϕ,λ(db, db0)is time reversal invariant.

108

6 Patterson-Sullivan distributions

We now construct A-invariant distributions. Recall that under the identifica-

tion G/MA ∼

=B(2) we write g(b, b0)∈Gif g(b, b0)·(M, wM)=(b, b0)∈B(2).

The element g(b, b0)is uniquely determined modulo MA.

Definition 6.15. For functions fon G/M, the Radon transform Ron G/M is

given by

Rf(b, b0) = ZA

f(g(b, b0)aM)da, (6.20)

whenever this integral exists. Then Rf(b, b0)is a function on B(2). By unimod-

ularity of Awe find that (6.20) does not depend on the choice of g(b, b0).

Lemma 6.16. The Radon transform maps R:Cc(G/M)→Cc(B(2)).

Proof. Recall B(2) ∼

=G/MA as homogeneous spaces. Given f∈Cc(G/M)we

define ˜

f∈Cc(G)by ˜

f(g) = f(gM). Then

Rf(gMA) = ZA

f(ga)da =ZMA

f(gam)dadm.

It follows from (3.1) and its subsequent remark applied to MA that Rfhas

compact support.

6.2.2 Patterson-Sullivan distributions on the compact quotient

We keep the assumption that w·λ=−λ. (w∈Wis the longest Weyl group

element, λ∈a∗

C).

Definition 6.17. Let Fdenote a bounded fundamental domain for Γin X.

Following [AZ07], pp. 380-381, we say that χ∈C∞

c(X)is a smooth fundamental

domain cutoff function if it satisfies

γ∈Γ

χ(γz) = 1 ∀z∈X. (6.21)

Such a function can for example be constructed by taking ν∈C∞

c(X),ν= 1

on F, and putting χ(z) = ν(z)·(Pγ∈Γν(γz))−1. If χsatisfies (6.21), then

f dz =ZX

χf dz, f ∈C(XΓ).(6.22)

Since Bis compact, we can (by using partition of unity) also choose a cutoff

χ∈C∞

c(X×B)such that Pγ∈Γχ(γ·(z, b)) = 1. Let T∈D0(X×B)be a

Γ-invariant distribution and aaΓ-invariant smooth function on X×B. Suppose

there is a1∈D(X×B)such that Pγ∈Γa1(γ·(z, b)) = a(z, b). Then

ha1, TiX×B=ZX×B(X

γ∈Γ

χ(γ·(z, b)))a1(z, b)T(dz, db)

=ZX×BX

γ∈Γ

χ(z, b)a1(γ·(z, b)) T(dz, db).

109

6 Patterson-Sullivan distributions

By the invariance of Tthis equals RX×Bχ(z, b)a(z, b)T(dz, db). We thus have

Proposition 6.18. Let T∈D0(X×B)be a Γ-invariant distribution. Let abe

aΓ-invariant smooth function on X×B. Then for any a1, a2∈D(X×B)such

that Pγ∈Γaj(γ·(z, b)) = a(z, b)(j= 1,2) we have ha1, Ti=ha2, Ti.

Given Tand aas in Proposition 6.18 and if moreover χj(j= 1,2) are

smooth fundamental domain cutoffs, then aj=χjasatisfy the assumptions of

the proposition. Hence ha, TiΓ\G/M := hχa, TiG/M defines a distribution on the

quotient Γ\G/M and this definition is independent of the choice of χ.

Definition 6.19. Let λ∈a∗

Cand ϕ∈E∗

λ(X)denote a Γ-invariant joint eigen-

function. The Patterson-Sullivan distribution PSλ=PSϕ,λ associated to ϕis

defined by

ha, PSλiG/M =ZB(2)

(Ra)(b, b0)psϕ,λ(db, db0).(6.23)

On the quotient Γ\G/M, we define the Patterson-Sullivan distributions by

ha, PSλiΓ\G/M := hχa, PSλiG/M .(6.24)

We define normalized Patterson-Sullivan distributions by

PSλ=1

h1, PSλiΓ\G/M

PSλ.(6.25)

In view of Proposition 6.18 these definitions do not depend on χ.

We look at the expression

ha, PSλi=ZB(2)

dλ(b, b0)R(a)(b, b0)Tϕ(db)Tϕ(db0).(6.26)

It follows that PSλ(a)is well-defined if (dλ·Ra)(b, b0)∈C∞(B×B), which is

the case for a∈C∞

c(G/M): In fact, then Ra∈C∞

c(B(2)), so

dλ(b, b0)R(a)(b, b0)∈C∞

c(B(2))⊂C∞

c(B×B) = C∞(B×B).

As a consequence of Proposition 6.18 we obtain (recall that w·λ=−λ):

Proposition 6.20. PSϕ,λ is an A-invariant and Γ-invariant distribution on

G/M. On the quotient Γ\G/M, the distribution PSϕ,λ is still A-invariant.

110

6 Patterson-Sullivan distributions

6.2.3 Off-diagonal Patterson-Sullivan distributions

In this Subsection, we drop the assumption that wa=−id. Let λ, µ ∈a∗

Cand

fix ϕ∈E∗

λ(X)and ψ∈E∗

µ(X). At this point, we do not assume that these

eigenfunctions are real-valued. Let Tϕand Tψdenote the respective boundary

values. Recall the off-diagonal intermediate values (Section 6.1)

dλ,µ(g) = e(iλ+ρ)H(g)e(iµ+ρ)H(gw).

Definition 6.21. For functions fon G/M, the weighted Radon transform Rλ,µ

on G/M is by definition the Radon transform (6.20) of dλ,µf, that is

Rλ,µf(g) := ZA

dλ,µ(ga)f(ga)da, (6.27)

whenever this integral exists.

It is clear that Rλ,µ(f)is an A-invariant function on G/M (right-A-invariant),

that is a function on G/MA ∼

=B(2). Note that by integrating dλ,µ with respect

to a∈Awe circumvent the problem that dλ,µ alone is not a function on G/MA

(see (6.6) and its subsequent remark).

Exactly as in Lemma 6.16 we find

Lemma 6.22. Let f∈C∞

c(G/M). Then Rλ,µ(f)∈C∞

c(G/MA).

Definition 6.23. As usual, let g(b, b0)∈Gbe a representative for the element

g(b, b0)MA ∈G/MA that corresponds to (b, b0)∈B(2). Let f∈C∞

c(G/M). We

pull-back the Radon transform (6.27) to B(2) and define

Rλ,µf(b, b0) = Rλ,µf(g(b, b0)).

Then Rλ,µf∈C∞

c(B(2)). This definition is independent of the choice of repre-

sentative g(b, b0), since Rλ,µ(f)is invariant.

Let f∈C∞

c(B(2))⊂C∞

c(B×B)⊂C∞(B×B). We interpret Rλ,µfas a

function on B×Bwith compact support contained in B(2).

Definition 6.24. Let ϕ∈E∗

λ(X)and ψ∈E∗

µ(X)have boundary values Tϕand

Tψ. The off-diagonal Patterson-Sullivan distribution PSλ,µ associated to ϕand

ψon G/M is defined by

hf, PSλ,µi=ZB(2)

Rλ,µf(b, b0)Tϕ(db)Tψ(db0).(6.28)

It follows that PSλ,µ(f)is well-defined if Rλ,µf(b, b0)∈C∞(B×B). A simple

case is when f∈C∞

c(G/M): Then Rλ,µ ∈C∞

c(B(2)), so

Rλ,µ(f)(b, b0)∈C∞

c(B(2))⊂C∞

c(B×B) = C∞(B×B).

111

6 Patterson-Sullivan distributions

Proposition 6.25. Suppose that ϕ∈E∗

λ(X)and ϕ∈E∗

µ(X)are Γ-invariant

eigenfunctions. Then the distribution PSλ,µ on G/M is Γ-invariant.

Proof. Let f∈C∞

c(G/M)and let fγdenote the translation f◦γ−1. Then

hfγ, PSλ,µi=ZB(2) ZA

dλ,µ(g(b, b0)a)f(γ−1g(b, b0)a)da Tλ(db)Tµ(db0),

where (b, b0) = (g·M, g ·wM)for g=g(b, b0). By (5.5) this equals

ZB(2) ZA

dλ,µ(g(γ·(b, b0))a)f(γ−1g(γ(b, b0))a)

×e−(iλ+ρ)hγ·o,γ·bie−(iµ+ρ)hγ·o,γ·b0ida Tλ(db)Tµ(db0).

Recall that a∈Aacts trivially on (M, wM). Using this and (6.8) we observe

dλ,µ(γga) = e(iλ+ρ)hγ·o,γ·bie(iµ+ρ)hγ·o,γ·b0idλ,µ(ga).

We also have g(γ·(b, b0)) = γg(b, b0)), since (b, b0)7→ g(b, b0)∈G/MA is G-

equivariant. Hence γ−1g(γ·(b, b0)) = g(b, b0). Thus we have

hfγ, PSλ,µi=ZB(2) ZA

dλ,µ(g(b, b0)a)f(g(b, b0)a)da Tλ(db)Tµ(db0)

=ZB(2)

Rλ,µf(b, b0)Tλ(db)Tµ(db0) = hf, PSλ,µi,

and the proposition follows.

Remark 6.26. Let (b, b0)∈B(2),g=g(b, b0)and suppose w·λ=−λ. Then

Rλ,λ(f)(g) = ZA

dλ,λ(ga)f(ga)da =dλ(g(b, b0))(Rf)(b, b0).(6.29)

Let ϕ∈E∗

λ(X)and consider the distributions PSλ,λ and PSϕ,λ associated to ϕ.

By (6.29) we have PSλ,λ =PSλ. If ϕ=ψand λ=µ, it follows as in Subsection

6.2.1 that the PSλ,λ are invariant under time-reversal and right-translation by

A. Vice versa, if Tϕ6=Tψ, then PSϕ,ψ needs not to be invariant under b↔b0.

Remark 6.27. Given ˜a∈Awe write f˜a:= f◦˜a−1. Then

Rλ,µ(f˜a)(g) = ZA

dλ,µ(ga˜a)f(ga)da =ei(λ+w·µ) log(˜a)Rλ,µ(f)(g),(6.30)

which follows from

dλ,µ(g˜a) = ei(λ+w·µ) log(˜a)dλ,µ(g)(6.31)

(cf. (6.6)). Given eigenfunctions ϕ, ψ we thus have

hf˜a, PSλ,µi=ei(λ+w·µ) log(˜a)hf, PSλ,µi.(6.32)

In other words, the PSλ,µ are eigendistributions for the action of Aon G/M

(given by right-translation). In particular, if λ+w·µ= 0, then the associated

Patterson-Sullivan distribution is invariant under right-translation by A. This

is for example the case when ϕ=ψ,λ=µ, and w·λ=−λ.

112

6 Patterson-Sullivan distributions

Definition 6.28. Suppose that ϕ∈E∗

λ(X)and ϕ∈E∗

µ(X)are Γ-invariant joint

eigenfunctions. Since PSλ,µ is a Γ-invariant distribution on G/M, the definition

descends to the quotient Γ\G/M via

ha, PSλ,µiΓ\G/M := hχa, PSλ,µiG/M ,(6.33)

where χis a smooth fundamental domain cutoff. We normalize these distribu-

tions by setting

PSλ,µ := 1

h1, PSµ,µiΓ\G/M

PSλ,µ.(6.34)

In view of Proposition 6.18 these definitions do not depend on χ.

6.3 The Knapp-Stein intertwining operators

In this Section we introduce the Knapp-Stein intertwiners. We will later see

how these operators yield an explicit relation between the Patterson-Sullivan

distributions and the Wigner distributions (6.5.2). For background on similar

intertwining operators see [Knapp86]. Let λ∈a∗

Cand define

Lλa(g) := ZN

e−(iλ+ρ)(H(n−1w)) a(gn)dn, a ∈C(G),(6.35)

whenever the integral exists. The integrals Lλa(g)may be viewed as a weighted

horocyclic Radon transform.

Remark 6.29. Each AdG( ˜m),˜m∈M, fixes the elements of aand hence the

root subspaces. Thus Mnormalizes N, that is ˜mN =N˜mfor all ˜m∈M.

Hence n7→ ˜m−1n˜mdefines an automporphism of Nwhich by uniqueness of

Haar-measures maps dn into a multiple of dn. Since Mis compact, dn is

preserved.

It is a basic remark that Lλpreserves M-invariance:

Lemma 6.30. Lλ:C∞

c(G/M)→C∞(G/M).

Proof. Suppose that a∈C∞

c(G/M)and let g∈G,n∈N,m∈M. Then

a(gmn) = a(gmnm−1)and by 6.29 we know that n7→ ˜n:= mnm−1∈Npre-

serves dn. Moreover, H(n−1w) = H(mn−1m−1w)by invariance of the Iwasawa

projection and since wnormalizes M. Thus

Lλa(gm) = ZN

e−(iλ+ρ)(H(n−1w)) a(gmn)dn =ZN

e−(iλ+ρ)(H(˜n−1w)) a(g˜n)dn

=ZN

e−(iλ+ρ)(H(n−1w)) a(gn)dn =Lλa(g).

113

6 Patterson-Sullivan distributions

6.3.1 Harish-Chandra’s phase function

We absorb the term e−ρH(n−1w)in (6.35) into the amplitude, so that the phase

function is

ψ(n) = −H(n−1w).

By uniqueness of the longest element of a Coxeter group, we have w−1=w∈W.

Thus w−1=wm (m∈M)as elements in M0, so H(n−1w) = H(wn−1w−1)by

invariance of H(kan) = log(a). We write

θ:N→N, n 7→ wnw−1.(6.36)

Then e

θ(dn) = dn (cf. Subsection 2.1.6), since M0is compact, so since Nis

unimodular

Lµ(a)(g) = ZN

e−iµ(H(n)) e−ρ(H(n)) a(gw−1n−1w)dn.

Given 06=µ∈a∗

C, we identify the ray R+µ⊂a∗

Cwith R+by means of the Killing

form: First, we denote by Hµthe unique element in aCsuch that µ(X) = hX, Hµi

for all X∈a∗

C. Then

µ(X) = |µ|hX, Hµ/|µ|i, X ∈a∗

C,|µ| ∈ R+.(6.37)

We can now fix µ∈a∗and H:= Hµ/|µ|∈a∗. Using these identifications we

make from now on no difference between |µ|and µ. We rewrite the integrals

(6.35) in the form (note that ρ“remains” an element of a∗)

Lµ(a)(g) = ZN

e−iµhH(n),Hie−ρH(n)a(gwn−1w−1)dn, µ ∈R.

We choose a smooth fundamental domain cutoff function χ. Then Lµ(χa)(g)is

an oscillatory integral with real-valued phase function

ψH:N→R, n 7→ hH(n), Hi.(6.38)

We would be able to compute the critical points and the Hessian form of n→

H(n−1w)as we did for the other phase functions in Subsection 3.3. However,

we do not have to: The point is that ψHis the phase in the integral

c(λ) = ZN

e−(iλ+ρ)H(n)dn, Re(iλ)∈a∗

+,(6.39)

defining Harish-Chandra’s c-function. The calculations concerning the critical

points and Hessians of the ψHwere for example carried out in [Cohn74], §19.

The following proposition taken from [DKV83], Section 7, gives the complete

description of facts concerning ψH. Recall that NHdenotes the centralizer of

114

6 Patterson-Sullivan distributions

H∈ain N. For a root β, let Rβdenote the orthogonal projection g→gβ.

If g∈Gis decomposed g=kan corresponding to the Iwasawa decomposition,

then we denote its triangular part by t(g) = an ∈AN. Writing, as usual, h·,·i

for the Killing form, we denote in the next Proposition by (·,·)the inner product

Z, Z07→ −hZ, θZ0ion g×g.

Proposition 6.31. Let H∈a. The critical set of ψHis equal to NH. For the

Hessian of ψHat the critical points we have the formula

Hessn(Y , Y 0) = −X

α∈∆+

α(H)(θRα(Yt(n))−R−α(Yt(n)), R−α(Y0t(n)),

valid for n∈NHand Y , Y 0∈n. The index of the Hessian Hessnat any point

of NHis X

α∈∆+, α(H)<0

dim(gα).

Let nHdenote the Lie algebra of the closed subgroup NHof N. Write nλfor the

eigenspace of ad(H)in nfor the eigenvalue λ∈R. Then, with respect to the

Lie algebra decomposition n=nH⊕⊕λ6=0nλ(cf. [DKV83] Corollary 7.3), the

matrix Hessnis diagonal and ψHis clean.

Remark 6.32. It is clear that Proposition 6.31 still holds if H∈a∗

C: The case

of complex His dealt by passing to the real and imaginary part of ψH, since by

uniqueness of real and imaginary parts a point is critical for ψHif and only if it

is critical for both ψRe(H)and ψIm(H). In this way we could also handle complex

µin (6.37) with no extra work. However, in view of our results of Section 5,

we only consider real eigenvalue parameters. Anyway, the mehod of stationary

phase only applies for phase functions with non-negative imaginary part.

6.3.2 Asymptotic expansions for the Knapp-Stein intertwiner

Recall that an element X∈pis called regular, if Z(X)∩pis a maximal abelian

subspace of p. We call an element µ∈a∗regular, if Hµis regular, where Hµ

is the vector in asuch that µ(X) = hX, Hµi(Killing form) for all X∈a. The

centralizer of a regular element X∈ain N(resp. N) is the trivial subgroup

{e}of G. If G/K has rank one, then all nonzero elements of a(resp. a∗) are

regular.

We fix a regular µ∈a∗and write H=Hµ∈aand ψ=ψHµ. Then

ψ(e)=0and for the amplitude α(n) = e−ρH(n)χa(gwn−1w−1)we have α(e) =

χa(g). Let s= dim(N) = dim(N). It follows from Proposition 6.31 that after

the coordinate change (6.36) the function n7→ hH(n−1w), Hµihas the unique

critical point n=eand its Hessian form at n=eis non-degenerate. The

method of stationary phase ([Hor83]) yields

Lµ(χa)(g)∼C·(2π/µ)s/2∞

n=0

µ−nR2n(χa)(g),(6.40)

115

6 Patterson-Sullivan distributions

where R2nis a differential operator on Gof order 2nand R0is the identity. If

Qdenotes the Hessian matrix at the critical point, then

C=|det Q|−1/2eπi sign(Q)/4.(6.41)

One could also show C·(2π/µ)s/2∼c(µ)for the factor in (6.40) by applying the

method of stationary phase to the integrals (6.39) and using Proposition (6.31).

Lemma 6.33. For each n∈N, the operator R2narising in the expansion (6.40)

is a left-invariant differential operator on G/M.

Proof. We can replace χa in (6.40) by an arbitrary a∈C∞

c(G/M). The coef-

ficients R2n(a)(g)are independent of µand hence uniquely determined. Since

Lµ(a)is M-invariant, it follows that R2n(a)(g) = R2n(a)(gm)for all n∈N,

g∈G,m∈M,a∈C∞

c(G/M). Hence

R2n:C∞

c(G/M)→C∞

c(G/M)

is a linear operator. To see that R2nis a local operator, take a∈C∞

c(G/M).

Then K:= supp(a)⊂⊂ G/M is compact. Write π:G→G/M and set

V=π−1(K). Then suppG(R2n(a)) ⊂V, since R2nis a differential operator

on G. Thus supp R2n(a)⊂K. It follows that R2n:C∞

c(G/M)→C∞

c(G/M)

decreases supports, so by Peetre’s theorem it is a differential operator on G/M.

The same reasoning shows that the R2nare left-invariant.

6.4 An integral formula

In this subsection we prove an important integral formula involving the Radon

transform, intermediate values and the intertwining operators.

Lemma 6.34. Let a∈C∞

c(G/M) = C∞

c(X×B)and (b, b0)∈B(2). Then

Rλ,µ(Lµa)(b, b0) = ZX

a(z, b)e(iλ+ρ)hz,bie(iµ+ρ)hz,b0idz. (6.42)

Proof. Let g∈Gsuch that (b, b0) = (g·M, g ·wM). We manipulate the right

side of (6.42): First note that since dz is G-invariant we obtain

a(z, b)e(iλ+ρ)hz,bie(iµ+ρ)hz,b0idz =ZX

a(g·z, b)e(iλ+ρ)hg·z,bie(iµ+ρ)hg·z,b0idz.(6.43)

We consider aas a function on G/M ∼

=X×B. Then since b=g·oit follows

that a(gan ·o, g ·M) = a(gan ·o, gan ·M) = a(ganM). Recall that P=MAN

fixes b∞=M∈K/M. By (2.10) we find that (6.43) equals

ZAN

a(ganM)e(iλ+ρ)hgan·o,g·Mie(iµ+ρ)hgan·o,g·wMidn da. (6.44)

116

6 Patterson-Sullivan distributions

We first have

hgan ·o, g ·Mi=hgan ·o, gan ·Mi=H(gan) = H(ga).(6.45)

Next, by (2.27), by the definition of hz, biand since a·wM =wM for all a∈A,

hgan ·o, g ·wMi=hga ·n·o, ga ·wMi

=hn·o, wMi+hga ·o, ga ·wMi

=−H(n−1w) + H(gaw).(6.46)

It follows that (6.44) equals

ZAN

a(ganM)e(iλ+ρ)H(ga)e(iµ+ρ)H(gaw)e−(iµ+ρ)H(n−1w)dn da

=ZA

dλ,µ(gaM)ZN

a(ganM)e−(iµ+ρ)H(n−1w)dn da (6.47)

=ZA

dλ,µ(gaM)Lµa(gaM)da

=Rλ,µ(Lµa)(b, b0).

Note that Rλ,µ(Lµa)(b, b0) = Rλ,µ(Lµa)(g)is defined if ahas compact support.

This follows from the Fubini theorem and the often used formula given in (2.10).

The lemma is proven.

Remark 6.35. If g˜m˜ais another representative of gMA ∈G/MA, then in

(6.47)

dλ,µ(g˜m˜aaM)ZN

a(g˜m˜aanM)e−(iµ+ρ)H(n−1w)dnda. (6.48)

Since Ais unimodular we get rid of ˜a. Moreover, H(n−1w)is preserved under

n−17→ ˜m−1n−1˜m, since H(kan) = log(a)is M-bi-invariant and wnormalizes M.

Then by Remark 6.29 we find that (6.47) and (6.48) coincide. Hence the proof

of Lemma 6.34 does not depend on the choice of representative of g(b, b0)MA.

Remark 6.36. (1) If Xhas rank one can show that H(n−1w) = H(nw)for

all n∈N. This follows from [GASS], Ch. II, §6, Thm. 6.1). Hence

in these cases we obtain a slight simplification of the formulae above. In

general, the formula H(n−1w) = H(nw)is not correct. It is easy to find

counterexamples for example in SL(3,R), where AdG(w)|a6=−ida(see

Section 2.4.2).

(2) In the notation of [AZ07], we identify iλ +ρ=1

2+ir. Then dλ(b, b0)and

|b−b0|−1

2−ir satisfy the same equivariance property. By the transitivity of

the G-action on B(2) these function are constant multiples of each other.

This explains the factor 21

2+ir in [AZ07]. It appears because |1−(−1)|= 2

in the disk model, whereas we defined dλsuch that dλ(M, wM)=1.

117

6 Patterson-Sullivan distributions

(2) The intertwining operator

Lra(g) = ZR

a(gnu)(1 + u2)−(1

2+ir)du

introduced in [AZ07] is generalized by our intertwiner

Lλa(g) = ZN

a(gn)e−(iλ+ρ)(H(n−1w))dn.

In the notation of [AZ07], we always identify iλ +ρ=1

2+ir. The group

PSL(2,R)has the following Iwasawa decomposition components:

kα=cos(α)−sin(α)

sin(α) cos(α)at=et/20

0e−t/2

nu=1u

0 1w=0 1

−1 0

It suffices to prove H(n−1

uw) = ln(1 + u2)for all u∈R. Writing out nuw

gives

nuw=−u1

−1 0.

We have the following cases: u= 0,u < 0and u > 0.

(i) u= 0. Then nuw=w∈K, so the formula is obvious.

(ii) u < 0. Let t= ln(1 + u2),α=−arcsin( 1

et/2). Then let nvbe the

element nv:= a−1

tk−1

αnuw. Multiplying out shows that nvis of the

form

nv=1v

0 1.

Then kαatnv=n−1

uw, so H(n−1

uw) = t= ln(1 + u2).

(iii) u > 0. This case is very similar to the preceding case (ii). The

formula also follows from H(nuw) = H(n−1

uw) = H(n−uw), since in

this example G/K has rank one.

6.5 Eigenfunctions on a compact quotient

As before, let X=G/K denote a symmetric space of the noncompact type with

Laplace-Beltrami operator LX. Let Γdenote a cocompact, discrete and torsion

free subgroup Γof Gand let XΓ:= Γ\G/K be given the quotient metric. Then

XΓis a compact hyperbolic manifold and a locally symmetric space. We write

∆for the Laplace operator of XΓ.

118

6 Patterson-Sullivan distributions

Let 0 = c0< c1< c2< . . . denote the discrete spectrum of −∆on XΓ

(cf. Subsection 2.3.1). We choose a corresponding complete Hilbert space basis

(ϕj)of L2(XΓ)consisting of normalized (with respect to the L2-norm of XΓ)

eigenfunctions of ∆. Then

∆ϕj=−cjϕjfor all j∈N0.(6.49)

Let πdenote the natural projection of Xonto XΓ. Then πis a local isometry

and since the Laplace operator is isometry-invariant, πintertwines the Laplace

operators LXof Xand ∆of XΓ. It follows that an eigenfunction on XΓ(for the

Laplacian of XΓ) is a Γ-invariant eigenfunction on X(for the Laplacian of X).

AΓ-invariant eigenfunction of LXis called an automorphic eigenfunction.

Thus, (6.49) corresponds to the automorphic eigenvalue problem

LXϕ=−c ϕ,

ϕ(γz) = ϕ(z)for all γ∈Γand for all z∈G/K.

The rank of an algebra is defined as the maximal number of pairwise commut-

ing generators of the algebra. The rank of the algebra D(G/K)of translation

invariant differential operators equals the real rank of G/K, that is the number

dim(A), where G=KAN is an Iwasawa decomposition, or equivalently the

dimension of a maximal flat subspace of G/K. It follows that if Xhas higher

rank ≥2, the ϕjchosen above may not necessarily be joint eigenfunctions of

D(G/K). However, if Xhas rank one, then this is true (Remark 2.46). In

particular, if Xhas rank one, the joint eigenspaces are given by (h·,·i denotes

the extension of the Killing form to a∗

Eλ(X) = {f∈E(X) : LXf=−(hλ, λi+hρ, ρi)f}.

Suppose that ϕ∈E∗

λ, where λ∈a∗

C, is a Γ-invariant joint eigenfunction of

G/K. Then (|·|denotes the norm on a∗induced by the Killing form of g)

Dϕ =−(hλ, λi+|ρ|2)for all D∈D(G/K).

6.5.1 The rank one case

Recall the situation when the symmetric space has rank one: We only consider

joint eigenfunctions with exponential growth. Given such a ϕj, it follows that

there is λj∈a∗

Csuch that cj=−(hλj, λji+hρ, ρi). Then

∆ϕj=−(hλj, λji+hρ, ρi)ϕj.(6.50)

We can then fix the eigenvalue parameters λjcorresponding to the spectrum

0 = c0< c1< c2< . . .

119

6 Patterson-Sullivan distributions

It follows from hρ, ρi ∈ Rthat hλj, λji→∞(j→ ∞). Suppose that Xhas

rank one. Then for all j∈N0we must have λj∈a∗∪ia∗, where i=√−1. We

can hence for at most finitely many jhave λj∈ia∗, that is only finitely many

λjare contained in the so-called complementary series. All remaining λjare

contained in the unitary principal series, which we have studied in Section 5.

We will in this context sometimes also write λ→ ∞, which means λ(H)→ ∞

for each Hin the positive Weyl chamber a+.

6.5.2 Wigner distributions

Given a joint eigenfunction ϕ∈E∗

λ(X), we denote the corresponding (uniquely

determined) distributional boundary values by Tϕ∈D(B)(Theorem 5.3). Then

ϕ(z) = ZB

e(iλ+ρ)hz,biTϕ(db), z ∈X.

Recall that given λ∈a∗

Cand b∈B, the functions

eλ,b :X→C, z 7→ e(iλ+ρ)hz,bi.

are called non-Euclidean plane waves. The symmetric space calculus of pseu-

dodifferential operators (Chapter 4) is defined by

(Op(a)eλ,b) (z) = a(z, λ, b)eλ,b(z).(6.51)

Non-Euclidean Fourier analysis extends this definition to C∞

c(X). We always

assume that the symbol a:X×B×a→Cof Op(a)is a polyhomogeneous

function in λin the classical sense defined in (4.26). We know from Section 4.2

that Op(a)commutes with the action of γ∈Γif and only if ais invariant under

the diagonal action of Γon X×B=G/M. We will from now on always assume

that Op(a)is properly supported. In the non-Euclidean calculus we then have

Op(a)ϕ(z) = ZB

a(z, λ, b)e(iλ+ρ)hz,biTϕ(db).(6.52)

Definition 6.37. Let λ, µ ∈a∗

Cand suppose that ϕ∈E∗

λ(X)and ψ∈E∗

µ(X)

are L2(XΓ)-normalized and Γ-invariant joint eigenfunctions of D(G/K). We

define the Wigner distributions Wϕ,ψ associated to ϕand ψon C∞(Γ\G/M)by

Wϕ,ψ(a) := hOp(a)ϕ, ψiL2(XΓ).(6.53)

We view a∈C∞(Γ\G/M)as a symbol a∈S0, which is is independent of λ.

Note that Wϕ,ψ is a well-defined distribution: Using the boundary values, we

express (as we will do in (6.56)) the L2-inner product by means of the Poisson

transform and obtain the distribution

Wϕ,ψ =e(iλ+ρ)hz,bie(iµ+ρ)hz,b0idz Tϕ(db)Tψ(db0).(6.54)

120

6 Patterson-Sullivan distributions

Hence Wϕ,ψ(a)is bounded by a continuous C∞(Γ\G/M)-seminorm of a. In the

special case when ϕ=ψwe write Wϕ:= Wϕ,ϕ.

Let Xhave rank one. Recall from 6.5.1 the fixed basis (ϕj)of eigenfunctions

of ∆. We denote the corressponding boundary values by Tj. Then ϕj=Pλj(Tj)

by means of the Poisson-Helgason transform, where λjis as in Subsection 6.5.

We will then write Wj,k := Wϕj,ϕk.

Remark 6.38. Let ϕ∈E∗

λ(X)and λ∈abe real valued. The distributions Wϕ

are quantum time reversible in the following sense: Let Cf =fdenote complex

conjugation and write Ca(z, λ, b) = a(z, −λ, b). We have COp(a)C=Op(Ca)

by a direct computation. Hence hCOp(a)Cϕ, ϕi=hOp(a)ϕ, ϕi, so C∗Wλ=Wλ.

6.5.3 An intertwining formula

Asymptotic properties of Wigner distributions only concern principal symbols.

We hence assume symbols a(z, λ, b)of order 0to be independent of λ. Recall

that if χis a smooth fundamental domain cutoff function, then Wϕ,ψ(a) =

hOp(χa)ϕ, ψiL2(X).

Remark 6.39. In what follows we need a certain amount of regularity for

the boundary values we work with. From now on, we will always work with

distributional boundary values which are actually functions, that is Tϕ∈L1(B),

the space of integrable functions on B.

Theorem 6.40. Let ϕ∈E∗

λ(X)and ψ∈E∗

µ(X)be Γ-invariant joint eigenfunc-

tions with respective boundary values Tϕ∈L1(B)and Tψ∈L1(B). Let ψbe

real-valued. Then for a∈C∞(Γ\G/M)we have

Wϕ,ψ(a) = hLµ(χa), PSλ,µi.(6.55)

Proof. We express this L2(X)-inner product by means of the Poisson-Helgason

transform formula (5.1):

hOp(χa)ϕ, ψiL2(X)=ZX

(Op(χa)ϕ)(z)ψ(z)dz

=ZB×BZX

(χa)(z, b)e(iλ+ρ)hz,bie(iµ+ρ)hz,b0idzTϕ(db)Tψ(db0).(6.56)

It follows from Lemma 6.34 that Rλ,µ(Lµχa)(b, b0)extends to a smooth function

on B×B, which is given by the inner X-integral above. Then (6.56) equals

hRλ,µ(Lµχa), Tϕ⊗TψiB×B=hLµ(χa), PSλ,µi,(6.57)

and the theorem is proven.

Remark 6.41. If ϕ=ψand λ=µ, then (6.57) shows

Wϕ(a) = hdλR(Lλχa), Tϕ⊗TϕiB×B

=hR(Lλχa), psλiG/M

=hLλ(χa), PSλiG/M .

121

6 Patterson-Sullivan distributions

6.6 The spectral order principle

Let X=G/K have rank one. As usual, we identify aand a∗with Rby means

of the Killing form h·,·i: The unit vector (w.r.t. the Killing form) H∈a+and

the linear functional λ0∈a∗given by λ0(X) = hX, Hiare identified with the

real number 1.

In this section we introduce an idea which we call the spectral order principle.

This principle is geared to explain asymptotic relations between phase space

distributions and Wigner-distributions. To be as general as possible, we let

Op :C∞(SXΓ)→B(L2(SXΓ)) denote an arbitrary operator convention.

Let {ϕλ}denote a family of Γ-invariant joint eigenfunctions ϕλ∈E∗

λto spec-

tral parameters λ∈a∗

C, which are all normalized w.r.t. the norm of L2(XΓ).

Recall that Γ-invariant distributions on SX descend to distributions on SXΓ

by using smooth fundamental domain cutoff functions. We fix a smooth funda-

mental domain cutoff function χ.

Definition 6.42 (Intertwining operator).We say a family {Tλ,µ} ⊂ D0(SX)of

Γ-invariant distributions is intertwined with the Wigner distributions Wϕλ,ϕµif

for each µthere is a linear operator Lµ:C∞

c(SX)→C∞

c(SX)such that

Wϕλ,ϕµ(a) = Tλ,µ(Lµ(χa)) ∀a∈C∞

c(SXΓ).(6.58)

The operators Lµare called intertwining operators.

Definition 6.43 (Spectral order of a distribution).Let {Tλ,µ} ⊂ D0(SX)denote

a family of distributions. We say that {Tλ,µ} ⊂ D0(SX)has spectral order

K∈Rif there is a continuous seminorm k·k on C∞

c(SX)such that for all λ, µ

|Tλ,µ(f)| ≤ (1 + |λ|)K(1 + |µ|)K·kfk ∀f∈C∞

c(SX).(6.59)

Definition 6.44 (Left-invariant asymptotic expansion).Let Lµ:C∞

c(SX)→

C∞

c(SX)be a family of intertwining operators (in the sense of 6.42. Suppose

that there is an aymptotic expansion

Lµ(a)(gM)∼∞

j=0

µ−j−s/2Rj(a)(gM)(6.60)

in the sense that |Lµ(a)−PN−1

j=0 µ−j−sRj(a)| ≤ CN(1 + |µ|)−N, where s∈Ris

a constant and where the Rj:C∞

c(SX)→C∞

c(SX)are differential operators

on SX. We say that (6.60) is a left-invariant asymptotic expansion, if the Rj

are left-invariant differential operators.

Suppose that Tλ,µ ∈D0(SXΓ)is a distribution depending on two spectral

parameters, with Tµ,µ(1) 6= 0. We denote by b

Tλ,µ ∈D0(SXΓ)the normalized

distribution

Tλ,µ, fi:= hT, fi

hTµ,µ,1i.(6.61)

122

6 Patterson-Sullivan distributions

Theorem 6.45. Suppose that {Tλ,µ}is a family of distributions of spectral

order Kwhich is intertwined with the Wigner distributions Wλby the uniformly

continuous (in µ) intertwining operators Lµ. Let the Lµhave an asymptotic

expansion with left-invariant coefficients. Suppose O(|λ|−1) = O(|µ|−1). Let

a∈C∞(SXΓ). Then we have the asymptotic equivalence

Wλ,µ(a) = b

Tλ,µ(a) + O(µ−1).(6.62)

The constant in the O-term is a C∞(SXΓ)-seminorm of a.

Proof. We copy the asymptotic argument given in [AZ07]. First, integrating

(6.60) with respect to Tλ,µ and comparing with (6.58) we get an asymptotic

expansion (in the sense of (6.63))

hOp(a)ϕλ, ϕµiSXΓ∼X

n≤0

µ−n−s/2hRn(χa), Tλ,µiSX.

Note that the coefficients of this expansion depend on the spectral parameters.

By left-invariance, each distribution

f7→ hRn(f), Tλ,µiSX

is Γ-invariant, so by Proposition 6.18, the functional

a7→ hRn(χa), Tλ,µiSX

defines a distribution on SXΓand the first term (for n= 0) is Tλ,µ. Then

hOp(a)ϕλ, ϕµiSX =hLµ(χa), Tλ,µiSX

n=0

µ−n−s/2hRn(χa), Tλ,µi+O(µ−N−1+2K).(6.63)

We choose N > 2K. Since R0is the identity, the operator L(N)

µ=PN

nµ−nRn

can be inverted up to O(µ−N−1), i.e. one finds differential operators M(N)

µ=

n=0 µ−nMn, where M0= id, and R(N)

µ, such that

L(N)

µM(N)

µ= id +µ−N−1R(N)

µ.

We apply 6.58 to M(N)

µ(a)and find

hOp(M(N)

µa)ϕλ, ϕµiSXΓ=hL(N)

µχM(N)

µa, Tλ,µiSX +O(µ−N−1+2K)

=hL(N)

µM(N)

µχa, Tλ,µiSX +O(µ−N−1+2K)

=ha, Tλ,µiSXΓ+O(µ−N−1+2K).

The second line is a consequence of Proposition 6.18. But

M(N)

µ(a) = a+µ−1M1+. . . +µ−N+1M2(a),

123

6 Patterson-Sullivan distributions

so by the L2-continuity of zero-order pseudodifferential operators,

hOp(M(N)

µ(a))ϕλ, ϕµiL2(XΓ)=hOp(a)ϕλ, ϕµiL2(XΓ)+O(1/µ).

This proves

ha, Tλ,µiSXΓ=hOp(a)ϕλ, ϕµiSXΓ+O(1/µ).(6.64)

Putting ha, Tλ,µi=h1, Tµ,µiha, b

Tλ,µiinto (6.64) we obtain

h1, Tµ,µi·ha, b

Tλ,µi=ha, Wλ,µi+O(1/µ).(6.65)

In particular, for a= 1, we get

h1, Tµ,µiSXΓ= 1 + O(1/µ).

Together with (6.65) this yields

(1 + O(1/µ)) ·ha, b

Tλ,µi=ha, Wλ,µi+O(1/µ).(6.66)

The Wigner distributions and hence by (6.66) the ha, b

Tλ,µiare uniformly bounded.

It follows that the left side of (6.66) is asymptotically the same as ha, b

Tλ,µi.

Remark 6.46. One can weaken some assumptions of the above principle (Theo-

rem 6.45). For example, it is not really neccessary to claim O(|λ|−1) = O(|µ|−1).

The condition O(|λ|−1)≤O(|µ|−L)for an L≥1will still be sufficient: We

can then choose N > 2LK in the above asymptotic expansions. Moreover, the

condition that the intertwiners Lµpreserve compact supports is not neccessary,

if the expression Tλ,µ(Lµ(χa)) still makes sense for a∈C∞(SXΓ), and if for

f=Lµ(χa)the spectral estimate (6.59) is still satisfied.

The problem is to show that the spectral order principle (or a version with

weaker assumptions) can be applied to the intertwining formula 6.40 for the

non-Euclidean Wigner distributions, the Patterson-Sullivan distributions, and

the Knapp-Stein intertwiners. I will now describe what the concrete problems

are and restrict these considerations to the case of diagonal elements (ϕ=ψ,

λ=µ). Let f∈C∞

c(G/M). The values |dλ(b, b0)|are independent of λand all

derivatives of dλhave polynomial growth in λ. It follows that given a continuous

seminorm k·k1on C∞(B×B)there exist K1>0and a continuous seminorm

k·k2on C∞

c(G/M)such that

kdλ(b, b0)R(f)(b, b0)k1≤(1 + |λ|)K1kfk2.(6.67)

Note that k·k2may depend on the support of f. Assume dλ(b, b0)R(f)(b, b0)∈

C∞(B×B). Then PSλ(f)is well-defined. A simple example is when f∈

C∞

c(SX) = C∞

c(SX). Let λ∈a∗. In this case, it follows from (5.22), (6.28)

124

6 Patterson-Sullivan distributions

and (6.67) that there exist K > 0and a continuous seminorm k·k2on C∞

c(SX)

(possibly depending on the support of f) such that

|PSλ(f)| ≤ (1 + |λ|)Kkfk2.(6.68)

It is stated in [AZ07] (equation (3.14) there) that there is a seminorm inde-

pendent of the function f. I cannot find such an estimate. However, even if

we would have this equation for f∈C∞

c(SX), another problem would occur

in the well-definedness of the intertwining formula from Theorem 6.40: The

Knapp-Stein intertwiners do not preserve compact supports, so the intertwin-

ing formula can only be understood formally in the sense of continuation from

B(2) to B×B(Lemma 6.34). The problem is that for the psλ-distributions

there is no spectral order estimate in the sense of (6.59) for the enlarged do-

main dλ(b, b0)−1·C∞(B×B). For the PS-distributions, the constant Kand the

seminorm k·k2cannot be used in a proof of 6.45, since the remainder terms in

the asymptotic expansion (6.40) are not compactly supported.

For a∈C∞(Γ\G/M), let fa,λ,µ(b, b0)∈C∞(B×B)denote the inner X-

integral in (6.56). The intertwining formula in Theorem 6.40 is understood in

the sense of hLµ(χa), PSλ,µiG/M =hfa,λ,µ, Tλ⊗TλiB×B. In this sense, (5.22)

yields |hLµ(χa), PSλ,µi| ≤ (1 + |λ|)K(1 + |µ|)Kkχak, where k·kis a seminorm

on C∞(G/M)and only depends on the support of χ.

6.6.1 Further remarks and some open questions

(1) Recall that the intertwining formula is the same in each case. One could

conjecture that the asymptotic argument given in the proof of the spectral

order principle can be generalized to all symmetric spaces of the noncom-

pact type. It should be conjectured that most limits of Wigner distri-

butions are A-invariant (see [SV]. Similar results are announced by L.

Silberman and N. Anantharaman). In view of Remark 6.27, we see that

limits of Patterson-Sullivan distributions, as defined via B(2) will not al-

ways be A-invariant.

(2) It is in some cases possible to modify the definitions and to obtain off-

diagonal psλ,µ-distributions: For simplicity, let G/K have rank one, so

that the function dλ(b, b0)exists. Recall

Rλ,µf(b, b0) = ZA

dλ,µ(g(b, b0)a)f(g(b, b0)a)da. (6.69)

The choice of g=g(b, b0)was immaterial (modulo MA), so if we as-

sume H(g) = 0, then dλ,µ(g) = e(iλ+ρ)H(g)e(iµ+ρ)H(gw)=dµ(b, b0). One

can then define the distributions psλ,µ(db, db0) = dµ(b, b0)Tλ(db)Tµ(db0)

and f

PSλ,µ(f) = psλ,µ(R(f)). However, psλ,µ is not Γ-invariant in the

off-diagonal case.

125

6 Patterson-Sullivan distributions

(3) It is possible to express the normalized version of the Patterson-Sullivan

distributions by means of Harish-Chandra’s c-function. Therefore, a gen-

eralization of Lemma 6.4 in [AZ07] is needed, which does not a priori make

sense in G/M, since there is no horocycle flow on G/M. However, some of

the formulas given in Theorem 1.2 of [AZ07], in particular the one for the

normalization of the PS-distributions, generalize to arbitrary symmetric

spaces.

(4) It is still an open question if there is a purely classical dynamical interpre-

tation of the Patterson-Sullivan distributions in terms of closed geodesics

(see [AZ07]).

Details concerning these open questions are in progress and will eventually ap-

pear later.

126

References

[AZ07] N. Anantharaman, S. Zelditch, Patterson-Sullivan Distributions and

Quantum Ergodicity, Ann. Henri Poincaré 8 (2007), pp. 361-426, Birkhäuser

Verlag Basel, Switzerland, 2007.

[AZ2] N. Anantharaman, S. Zelditch, Patterson-Sullivan Distributions and

Quantum Ergodicity II - Preliminary version.

[BB02] P. Blanchard, E. Brüning, Mathematical Methods in Physics - Distribu-

tions, Hilbert Space Operators, and Variational Methods, Progress in Mathe-

matical Physics Vol. 26. Birkhäuser 2002.

[BO05] U. Bunke, M. Olbrich, On Quantum Ergodicity for Vector Bundles,

Springer Science + Business Media B.V. 2006.

[Cohn74] L. Cohn, Analytic Theory of the Harish-Chandra C-function. Lecture

Notes in Mathematics 429, Berlin: Springer-Verlag 1974.

[CG89] L. Corwin, F. P. Greenleaf, Representations of nilpotent Lie groups and

their applications, Part I, Cambridge Studies in Adv. Math., No 18, Cam-

bridge Univ. Press (1989).

[DH97] G. van Dijk, S. C. Hille, Canonical Representations Related to Hyper-

bolic Spaces, journal of functional analysis 147, 109-139 (1997).

[D78] D. Drucker, Exceptional Lie algebras and the structure of hermitian sym-

metric spaces, Memoirs of the AMS No. 208, 1978.

[DKV83] J. J. Duistermaat, J. A. C. Kolk and V. S. Varadarajan, Functions,

Flows and oscillatory integrals on flag manifolds and conjugacy classes in

real semisimple Lie groups, Compositio Mathematica 49 (1983) pp. 309-398,

Martinus Nijhoff Publishers, The Hague. Printed in the Netherlands.

[EO] P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973),

45-109.

[Eber79] P. Eberlein, Surfaces of nonpositive curvature, American Mathemati-

cal Society, 1979, No. 218: Vol 20.

[Eber96] P. B. Eberlein, Geometry of Nonpositiveley Curved Manifolds, Lectures

in Mathematic Series, The University of Chicago Press, Chicago an London

1996.

[GO05] S. Grellier and J.-P. Otal, Bounded Eigenfunctions in the Real Hyper-

bolic Space, Int. Math. Res. Not. 2005, no. 62, 3867-3897.

[GS77] V. Guillemin, S. Sternberg, Geometric Asymptotics, Amer. Math. Soc.,

Providence, Rhode Island, 1977.

References

[HC58] Harish-Chandra, Spherical functions on a semisimple Lie group, II.,

Amer. J. Math. 80 (1958) 553-613.

[Helg65] S. Helgason, Radon-Fourier transforms on symmetric spaces and re-

lated group representations, Bull. Amer. Math. Soc. 71 (1965), pp. 757-763

[Helg70] S. Helgason, A Duality for Symmetric Spaces with Applications to

Group Representations, Advances in Mathematics 5, 1-154 (1970), Mas-

sachusetts Institute of Technology, Cambridge.

[Helg74] S. Helgason, Eigenspaces of the Laplacian; Integral Representations

and Irreducibility, Journal of Functional Analysis 17, 328-353 (1974), Mas-

sachusetts Institute of Technology, Cambridge.

[GASS] S. Helgason, Geometric Analysis on symmetric spaces, Mathematical

surveys and monographs, American Mathematical Society, Providence, Rhode

Island (1994), ISSN 0076-5376; v. 39.

[GGA] S. Helgason, Groups and geometric analysis: integral geometry, in-

variant differential opsrators, and spherical functions, Mathematical surveys

and monographs, American Mathematical Society, Providence, Rhode Island

(2000).

[DS] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces,

Graduate Studies in Mathematics, American Mathematical Society, Provi-

dence, Rhode Island (2001).

[H05] J. Hilgert, An Ergodic Arnold-Liouville Theorem for Locally Symmetric

Spaces, Twenty Years of Bialowieza: A Mathematical Anthology. S.T. Ali et

al. eds., World Scientific, Singapore, 2005.

[HP09] J. Hilgert, A. Pasquale, Resonances and residue operators for symmetric

spaces of rank one, J. Math. Pures Appl. (9) 91 (2009), no. 5, 495-507.

[HS09] J. Hilgert, M. Schröder, Patterson-Sullivan distributions for rank one

symmetric spaces, preprint, arXiv:0909.2142v2.

[Hor65] L. Hörmander, Pseudo-differential operators, Comm. Pure Appl. Math.

18 (1965), p. 501-517.

[Hor83] L. Hörmander, The Analysis of Linear Partial Differential Operators I,

Springer-Verlag, Berlin 1983.

[Hor85] L. Hörmander, The Analysis of Linear Partial Differential Operators

III, Springer-Verlag, Berlin Heidelberg 1985.

[Hof81] H. C. Im Hof, Visibility, Horocycles, and the Bruhat decomposition, in

Global Differential Geometry and Global Analysis, Lecture Notes in Mathe-

matics Volume 838/1981, p. 149-153, Springer Berlin / Heidelberg.

128

References

[Knapp86] A. W. Knapp, Representation Theory of Semisimple Lie Groups,

Princeton University Press, Princeton (1986).

[LM63] J. L. Lions, E. Magenes, Problèmes aux limites non homogènes (VII)

Ann. Mat. Pura Appl. 4 63 (1963), 201-224.

[Nich89] P. J. Nicholls, The Ergodic Theory of Discrete Groups, London Math.

Soc. Lect. Notes Series 143, Cambridge Univ. Press, Cambridge 1989.

[Otal98] J.-P. Otal, Sur les fonctions propres du Laplacien du disque hyper-

bolique, C. R. Acad. Sci. Paris, t. 327, Série I, p. 161-166, 1998.

[Quint06] J.-F. Quint, An overview of Patterson-Sullivan theory, Notes de cours,

An overview of Patterson-Sullivan theory, Workshop The barycenter method,

FIM, Zurich, May 2006. http://www-math.univ-paris13.fr/ quint/

[Rou63] C. Roumieu, Ultra-distributions définies sur Rnet sur certaines classes

de variétés differentiables, J. Analyse Math. X (1962-63), 153-192.

[Rez04] A. Reznikov, Microlocal lifts of eigenfunctions on hyperbolic surfaces

and trilinear invariant functionals, preprint, arXiv:math/0404294v1.

[RS94] Z. Rudnick, P. Sarnak, The behaviour of eigenstates of arithmetic hy-

perbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195-213.

[Schwartz66] L. Schwartz, Théorie des Distributions, 2nd e., Hermann, Paris,

1966.

[SV] L. Silberman and A. Venkatesh, On quantum unique ergodicity for locally

symmetric spaces I, Geom. Func. Anal. 17 (2007), no. 3, 960-998.

[Sul79] D. Sullivan, The density at infinity of a discrete group of hyperbolic

motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), p. 171-202.

[Tay81] M. E. Taylor, Pseudodifferential Operators, Princeton Univerity Press,

Princeton, New Jersey 1981.

[Treves67] F. Treves, Topological Vector Spaces, Distributions and Kernels,

Acad. Press (1967)

[U80] H. Urakawa, On the least positive eigenvalue of the Laplacian for the

compact quotient of a certain Riemannian symmetric space, Nagoya Math. J.

Vol. 78 (1980), 137-152.

[Var] V. S. Varadarajan, Oscillatory integrals and their applications to harmonic

analysis on semisimple Lie groups, preprint.

129

References

[Var97] V. S. Varadarajan, The Method of Stationary Phase and Applications to

Geometry and Analysis on Lie Groups, Perspectives in Mathematics (1997)

pp. 167-242, Academic Press.

[Wal73] N. R. Wallach, Harmonic Analysis on Homogeneous Spaces, in Pure

and Applied Mathematics, No. 19., Marcel Dekker, Inc., New York (1973).

[Wal88] N. R. Wallach, Real reductive groups 1, Academic Press, Pure and Ap-

plied Mathematics, San Diego (1988)

[War70] F. Warner, Foundations of Differentiable Manifolds and Lie Groups,

Scott Foresman, Glenview, Illinois, 1970.

[Wil91] F. L. Williams, Lectures on the spectrum of L2(Γ\G), Pitman Research

Notes in Mathematics Series 242, Essex, England (1991).

[Zel86] S. Zelditch, Pseudo-differential Analysis on Hyperbolic surfaces, Journal

of Functional Analysis 68, pp. 72-105 (1986); Department of Mathematics,

Columbia University, New York 1986.

[Zel87] S. Zelditch, Uniform distribution of eigenfunctions on compact hyper-

bolic surfaces, Duke Math. J., Vol. 55, No. 4 (1987)

[Zel05] S. Zelditch, Quantum Ergodicity and Mixing, Expository article for the

Encyclopedia of Mathematical Physics on the subject of the title, arXiv:math-

ph/0503026v1.

[Zel09a] S. Zelditch, Local and global analysis of eigenfunctions on Riemannian

manifolds, preprint, arXiv:0903.3420v1.

[Zel09b] S. Zelditch, Recent developments in mathematical quantum chaos - Pre-

liminary version/ comments appreciated, preprint, arXiv:0911.4312v1.

130

Index

A(g), Iwasawa projection NAK →

a,21

AH, homeomorphism, 27

B, boundary at infinity, 27

B(·,·), Killing form, 15

Bθ, positive definie bilinear form on

g,18

Dj, differential operator, 66

Da, projection of D∈D(G)onto

D(A),39

Ea, projection, 52

Eµ,λ, function, 77

G, real semisimple Lie group with fi-

nite center, 16

G, semisimple Lie group, 13

H(g), Iwasawa projection KAN →

a,21

Hα∈a, root vector, 19

Hγ, representation in aof γ∈a∗,24

H∞

λ, representation space, 94

K, maximal compact subgroup of G,

13,16

L1(B), space of integrable functions

on B,121

LA, Laplace operator on A·o,40

Lg, left translation, 54

Lλ, Knapp-Stein intertwiner, 113

M, centralizer of Ain K,20

M0, normalizer of Ain K,20

NK(a), normalizer of ain K,20

Op, pseudodifferential operator quan-

tization, 74

Op, pseudodifrerential operator quan-

tization, 83

Op(a), psudodifferential operator, 9

P(x, b) = e2ρhx,bi, Poisson kernel, 91

Sm, space of symbols, 72

1,0,0,Γ, space of Γ-invariant symbols,

1,0,0, space of classical symbols, 83

1,0,0, space of symbols of order m,

Γ, space of Γ-invariant symbols, 73

cl,Γ, space of Γ-invariant classical

symbols, 73

S−∞ =∩mSm, smoothing symbols,

Sα, reflection in the hyperplane aα,

cl , space of classical symbols, 73

Tg, group action, 27

Tg, translation, 69

Tg,h, translation, 69

U, Kohn-Nirenberg operator, 75

Ut, wave group, 10

Vt, translation operator associated

to the geodesic flow, 10

W, Weyl group, 20

Wλj, Wigner distribution, 11

X(∞), set of points at infinity, 26

X=G/K, symmetric space, 13

X=G/K, symmetric space of the

noncompact type, 16

XΓ, compact hyperbolic manifold, 13

Z(X), centralizer of X,19

ZK(a), centralizer of ain K,19

ZN(H), the centralizer of Hin N,57

ZN(X), centralizer of Xin N,22

Ad, adjoint representation, 17

∆N, radial part, 40

Γ, Harish-Chandra’s isomorphism, 39

Ψm(M), space of classical pseudod-

ifferential operators on Mof

order m,9

Σ+, set of positive roots, 20

Σ+

s= Σ+

0∩s−1Σ−

0, partial positive

roots, 25

ad, adjoint representation, 17

∗, convolution product, 77

χµ, homomorphism, 41

D(A), algebra of translation-invariant

differential operators on A,

131

Index

D(G), algebra of translation-invariant

differential operators on G,

D(G/H), algebra of translation-invariant

differential operators on G/H,

DW(A), Weyl group invariant differ-

ential operators on A,39

γ(−∞), equivalence class of a geodesic,

γ(∞), equivalence class of a geodesic,

γX, geodesic, 27

λ, symmetrization map, 41

λα, normalization of λ∈a∗

C,24

h,i, Killing form, 19

h,i, metric, 16

hx, ξi, composite distance from a point

xto a horocycle ξ,33

n⊥, orthogonal complement of n,56

log, inverse of the exponential map,

D(G/K), algebra of translation invariant-

differential operators on G/K,

AT(B), Banach space, 91

D0(B)(1)

Γ, space of distributions, 96

D0(B)(1)

Γ, subspace of D0(B),96

D0(B)λ, subspace of D0(B),99

D(M) = C∞

c(M), space of compactly

supported smooth function on

M,36

D(V), compactly supported smooth

functions on Van open set

V,35

DK(M), space of functions supported

with compact support in K,

E(M) = C∞(M), space of smooth

functions on M,36

E(V), smooth functions on an open

set V,35

E∗(X), subspace of E(X),92

E∗

λ(X), subspace of Eλ(X),92

Eχ(X), joint eigenspace, 91

Eλ(X), joint eigenspace, 91

E∗

λ(X), joint eigenspace, 92

H(a∗

C×B), space of functions, 68

H(a∗

C×B)W, space of functions, 68

HR(a∗

C×B), space of functions, 68

K(a∗

C×B)W, space of functions, 69

K(a∗

C×B), space of functions, 69

KR(a∗

C×B), space of functions, 69

Lm, properly supported operators in

OPSm,74

1,0,0, properly supported operators

in OPSm

1,0,0,83

Γ, properly supported operators in

OPSm

Γ,74

R, Radon transform, 109

R, weighted Radon transform, 111

a, abelian Lie algebra, 18

a+, positive Weyl chamber, 19

gα, root subspace, 18

n, nilpotent Lie algebra, 20

p, orthogonal complement of k,17

µL, Liouville measure, 10

Tg, group action, 28

ρ=1

2Pα∈Σ+mαα, parameter, 21

ρg, right-translation by gin G,39

σOp(a), principal symbol of a pseu-

dodifferential operator Op(a),

τ(g), translation on G/H,39

ε, homomorphism, 53

at, one-dimensional parameterization

of A,29

b−∞, boundary point, 27

b∞, boundary point, 27

es(λ), Harish-chandra’s e-functions,

eλ,b, non-Euclidean plane waves, 41

hg, conjugation in G,52

mα, multiplicity of a simple root α,

ug, action of g∈Gon u∈U(g),52

132

Index

analytic functionals (hyperfunctions),

asymptotic expansion, 73

asymptotically equivalent geodesics,

boundary values, 92

Cartan decomposition, 17

centralizer of Ain N,22

centralizer of Xin n,22

classical symbol, 73

complete symbol, 71

composite distance, 33

constant term of u∈U(g),53

convolution product, 77

differential operator on Rn,35

differential operator on a manifold,

distance function, 38

distribution, 37

distributional Schwartz-kernel of an

operator, 71

divergence, 38

dual positive Weyl chamber, 68

dual space, 33

eigenspace, 38

Euclidean Fourier inversion formula,

Euclidean Fourier transform, 66

exponential growth, 92

exponential map, 17

Fourier transform, 67

full symbol, 66,71

geodesic symmetry, 15

gradient, 37

Helgason boundary values, 92

homogeneous symbol, 73

horocycle, 32

horocycle bracket, 34

imbedding of functions into distribu-

tions, 37

incident, 31

inductive limit topology, 37

invariant differential operator, 37

invariant function, 69

invariant state, 11

invariant under a diffeomorphism, 37

Iwasawa projections, 21

joined by a geodesic, 26

joint eigenfunction, 91

joint eigenspace, 41,91

Killing form, 15

Laplace-Beltrami operator, 38

left-invariant asymptotic expansion,

122

microlocal defect measure, 11

non-Euclidean Fourier transform, 67

norm induced by the Killing form,

normal, 33

one-parameter subgroup, 40

OPSm, pseudodifferential operators

in Op(Sm),74

origin, 26

Paley-Wiener estimates, 68

parabolic, 29

Patterson-Sullivan distribution, 108

Plancherel formula, 68

point at infinity, 26

Poisson transform, 91

positive root, 19

positive Weyl chamber, 19

preimage of the positive Weyl cham-

ber, 68

principal series, 93

principal series representations, 65

principal symbol, 87

properly supported operator, 70

133

Index

quantum limit, 11

radial part of a differential operator,

Radon transform, 109

rank of a symmetric space, 18

real rank, 20

reductive, 17

regular, 19

root, 18

root space decomposition, 19

root vector, 19

second countable manifold, 36

semisimple, 16

simple root, 19

spectral order principle, 122

spectrum, 38

support, 36

support of a distribution, 37

symbol asymptotics, 73

symbol estimates, 72

symbol of order m∈R,72

symmetric algebra, 40

symmetrization, 41

trace, 15

triangular part of the Iwasawa de-

composition, 53,115

two-point homogeneous space, 42

uniform exponential type, 68

uniformly properly supported, 74

visibility axiom, 28

weighted Radon transform, 111

Weyl chambers, 19

Weyl group, 19

134