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DECAY ON HOMOGENEOUS SPACES OFREDUCTIVE TYPE

BERNHARD KROTZ, EITAN SAYAG AND HENRIK SCHLICHTKRULL

Abstract. In this paper we explore homogeneous spaces Z =G/H of a a real reductive Lie group G with a closed connectedsubgroup H. The investigation concerns the decay at infinity ofsmooth functions on Z, and Lp-integrability of matrix coefficients.These results are used in a study of the asymptotic density of latticepoints on Z. Explicit examples are given of spaces for which resultsare new.

Date: June 6, 2011.2000 Mathematics Subject Classification. 22F30, 22E46, 53C35, 22E40.

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1. Introduction

The question of decay of matrix coefficients of unitary representa-tions plays central role in the applications of harmonic analysis andrepresentation theory to geometry, probability and number theory. Inparticular, the fundamental theorem of Howe-Moore, that matrix coef-ficients of infinite dimensional unitary representations decay to zero atinfinity, has been an important ingredient in studying lattice countingon symmetric spaces [20], [22], [9].

The present paper establishes basic results concerning decay and in-tegrability of generalized matrix coefficients, and more general smoothfunctions, in the context of unimodular homogeneous spaces.

The paper contains three types of results. We first establish a quali-tative property of decay, valid for functions on general reductive homo-geneous spaces. Next a more quantitative property of decay is estab-lished on spaces with refined structure and for matrix coefficients only.In particular, the decay allows one to conclude Lp-integrability. Finallythe results on integrability, combined with the abstract Plancherel theo-rem, are used to study decay of periods of automorphic forms. Buildingon the work of [20], these results are applied to the asymptotic countingof lattice points on reductive homogeneous spaces.

1.1. Qualitative results. Let G be a real reductive group and Z =G/H a unimodular homogeneous G space with a closed connected sub-group H. Qualitative bounds are developed for the decay of functionson Z. These bounds are expressed through the property VAI of Van-ishing At Infinity. The space Z admits VAI if and only if for each1 ≤ p < ∞, the smooth functions on Z, all of whose derivatives belongto Lp(Z, µZ), vanish at infinity.

Our main result in regards to this property is the following classifi-cation:

Theorem A. A unimodular homogeneous G-space Z = G/H admitsVAI if and only if H ⊂ G is reductive.

The direction ‘if’ is proved in Proposition 4.7. If Z is of reductivetype and B ⊂ G is a compact ball we provide essentially sharp lowerand upper bounds for volZ(Bz) where z ∈ Z moves off to infinity (Sec-tion 4.2). These results generalize and simplify previous approaches in[36] and [31]. The lower bounds in particular imply that Z has VAI.

The converse implication is established in Proposition A.1 (see Ap-pendix A). The main lemma shows that in the non-reductive case thevolume of the above mentioned sets Bz can be made exponentiallysmall.

DECAY ON HOMOGENEOUS SPACES 3

1.2. Quantitative results. We impose on Z = G/H the conditionthat there exists an open H orbit on a flag manifold G/Pmin of G,with Pmin ⊂ G a minimal parabolic subgroup. Spaces that admit thisproperty are said to be of spherical type.

Restricting analysis to spherical reductive homogenous spaces, westudy the decay of matrix coefficients. More precisely, for a unitaryirreducible representation π of G we consider the generalized matrixcoefficient mv,η(g) := 〈v, π(g)η〉 on G/H, with η an H-fixed distributionvector and v a smooth vector.

Assuming reductive spherical type, we obtain in Theorem 6.4 an up-per bound for mv,η with a spectrally defined rate of decay. To formu-late the result we denote by A the torus in the Iwasawa decompositionG = KAN , by A+ its positive part and by Λπ the exponent associatedto π in [46] 4.3.5.

Theorem B. Suppose for the minimal parabolic subgroup P ⊂ Gthat PH is open in G. Let V be a Harish-Chandra module for G andη ∈ (V −∞)H . Then there exists d ∈ N0 such that for all v ∈ V there isa constant Cv > 0

(1.1) |mv,η(a)| ≤ CvaΛπ(1 + ‖ log a‖)d (a ∈ A+) .

Theorem B is a generalization of [3], Thm. 6.1, and we view it as abasis for analysis on spherical spaces Z. We emphasize that TheoremB is applicable for general (non-unitary) Harish-Chandra modules. Inthe unitarizable case, the theorem of Howe-Moore implies that Λπ is asum of negative roots. Rather than generalizing the approach of [16] aswas done in [3], we rest our arguments of [46] adapted to our situation.This yields a significantly shorter argument.

In many applications the Lp-integrability of the generalized matrixcoefficients mv,η associated with unitary representations plays an im-portant role. For example, the importance of matrix coefficient inte-grability was seen in [18], where the equivalence of an Lp-property withKazhdan’s Property (T) is shown.

In our case we express the desired integrability by a condition onZ, called Property (I), which asserts that for each irreducible unitaryrepresentation π of G there exists p < ∞ such that mv,η ∈ Lp(G/Hη)for all K-fixed vectors v and all H-fixed distribution vectors η. HereHη is the stabilizer of η, often equal to H, but possibly strictly larger.

We believe that all reductive spherical type spaces admits this prop-erty (see Appendix D for our conjecture). The result we prove is weaker.To formulate it we introduce a class of reductive spaces, called strong

4 BERNHARD KROTZ, EITAN SAYAG AND HENRIK SCHLICHTKRULL

spherical type (see definition 7.7). These spaces admit polar decompo-sition G = KAH where K is maximal compact and A is a non-compacttorus. Symmetric spaces are known to be of both polar and sphericaltype, and they satisfy the conditions of strong sphericality. It is ourbelief that all spaces of reductive spherical type are strongly spherical,a result which would make the stronger notion essentially superfluous.We provide a list of strongly spherical spaces below (the list is notexhaustive).

Our main result on integrability of generalized matrix coefficients ofunitary representations is the following (see Theorem 8.5):

Theorem C. Suppose Z is of strong spherical type. Then Z admitsProperty (I).

The polar decomposition and the decay along A provided by Theo-rem B, yields the required integrability.

1.3. Lattice counting. In the last part of the paper we relate Prop-erty (I) to the problem of lattice counting problems which are situatedat the crossroad of analysis, geometry and number theory.

Let z0 ∈ Z denote the base point. A lattice on G/H is the orbit Γ.z0

of a lattice Γ in G for which ΓH = Γ ∩H is a lattice in H. The latticecounting problem on Z consists of the determination of the asymptoticbehavior of the density of Γ.z0 in K-invariant balls BR ⊂ Z, as theradius R →∞. We note that the Gauss circle problem in the Euclideanplane is an instance of this problem and that Selberg developed his traceformula to establish lattice counting in the hyperbolic plane (see [39] forreferences). In the setting of semisimple symmetric spaces the problemwas initiated by Duke, Rudnick and Sarnak in [44] and extensivelystudied in [20].

With proper normalizations of invariant measures, the Main TermCounting is the statement that the asymptotic density is 1. Moreprecisely, with

NR(Γ, Z) := #γ ∈ Γ/ΓH | γ.z0 ∈ BRand |BR| := vol(BR) we have

(1.2) NR(Γ, Z) ∼ |BR| (R →∞),

This was established in [20] for lattices on G/H for which H/ΓH iscompact. In subsequent work Eskin and McMullen [22] removed theobstruction and presented an ergodic approach. Later Eskin, Mozesand Shah [23] refined the ergodic methods and discovered that mainterm counting holds for a wider class of reductive spaces: It is validwhenever H is a maximal torus or a maximal reductive subgroup.


Our approach to the lattice counting problem is close in spirit to[20]. In proposition 10.2 we reformulate their approach in the form ofa criterion. This criterion guarantees that main term counting holdsas long as vanishing at infinity can be established for enough functionson Z, obtained as periods of automorphic functions on Γ\G.

Based on (I), Plancherel decomposition and VAI we prove main termcounting in Theorem 10.1 for symmetric spaces as well as for the spacesin the list below. We emphasize that in those cases H is not a maximalsubgroup of G and main term counting is new.

Below we state a general result concerning main term counting whichis applicable for any space admitting property (I). The statement,which is a simplification of 10.1 in the body of the paper, requiresthat the balls BR factorize well, a simple notion introduced in Section9.3 and related to the notion of focusing in [23].

These assumptions are verified for symmetric spaces and for an im-portant class of geometric balls in all the examples listed below.

Theorem D. Let G be reductive and H a closed subgroup such thatZ = G/H is of reductive type. Suppose that Γ ⊂ G is co-compact andZ admits (I). Suppose that (BR)R>0 factorizes well, then Main TermCounting holds.

Being of spectral nature, our approach extends and simplifies themethod used in [20]. We believe that the co-compactness assumptioncan be removed, but this will require further results on regularizationof periods of Eisenstein series that presently are not available in theliterature (see comments in Appendix D).

In the end of Section 10 we consider a space for which Eskin andMcMullen showed that main term counting fails to hold ([22]). It turnsout that main term counting does hold for our family of geometric balls.

1.4. Error Terms. The problem of determining the error term incounting problems is notoriously difficult and in many cases relies ondeep arithmetic information. Sometimes, like in the Gauss circle prob-lem, some error term is easy to establish but getting an optimal errorterm is a very difficult problem.

We restrict ourselves to the cases where the cycle H/ΓH is com-pact, and where Z is either symmetric or a triple product space asSL2(R)3/∆(SL2(R)), where ∆ stands for the diagonal embedding.

The error we study is measure theoretic in nature, and will be de-noted here as err(R, Γ). Thus, err(R, Γ) measures the deviation of twomeasures on Y = Γ\G, the counting measure arising from lattice pointsin a ball of radius R, and the invariant measure µY . It is easy to compare


this error term with the pointwise error errpt(R, Γ) = |NR(Γ, Z)−|BR||,see Remark 11.1.

To formulate our result we introduce the exponent pH(Γ), which mea-sures the worst Lp-behavior of any generalized matrix coefficient associ-ated with a spherical unitary representation π, which is H-distinguishedand occurs in the automorphic spectrum of L2(Γ\G) (see definition11.1). A related notion was studied in recent work of [30].

We first state our result for the non-symmetric case of triple productspaces, which is Theorem 12.4 from the body of the paper.

Theorem E. Let Z = G30/ diag(G0) for G0 = SOe(1, n) and assume

that H/ΓH is compact. For all p > pH(Γ) there exists a C = C(p) > 0such that

err(R, Γ) ≤ C|BR|−1

(6n+3)p

for all R ≥ 1.

To the best of our knowledge this is the first error term obtainedfor a non-symmetric space. The crux of the proof is locally uniformcomparison between Lp and L∞ norms of generalized matrix coefficientswhich is achieved by applying the model of [12] for the triple productfunctional.

Let us now go back to the symmetric case. We will assume thatΓH\H is compact. The existence of a non-quantitative error term forsymmetric spaces was established in [9] and improved in [25].

Under these general assumptions we obtain error term estimateswhich in a considerable number of situations improve the bounds ob-tained in [25]. The next result is Theorem 11.8 in the body of thepaper.

Theorem F. Let Z be symmetric. Assume

• ΓH = H ∩ Γ is co-compact in H.• p > pH(Γ)

• k > rank(G/K)+12

dim(G/K) + 1

Then, there exists a constant C = C(p, k) > 0 such that


(2k+1)p

for all R ≥ 1.Moreover, if Y = Γ\G is compact one can replace the third condition

by k > dim(G/K) + 1.

The key estimate for the proof of Theorem F is Lemma 11.5 which isa quantitative version of property (I). To be more precise we compare ina uniform way the Lp and L∞ norms. The proof found in Appendix C


relies, among other things, on the Harish-Chandra-Gangolli recursionrelations satisfied by spherical matrix coefficients.

We note that in case of the hyperbolic plane our error term is stillfar from the quality of the bound of A. Selberg. This is because weonly use a week version of the trace formula, namely Weyl’s law, anduse simple soft Sobolev bounds between eigenfunctions on Y .

1.5. List of examples. To summarize, in this paper we prove thatthe following spaces are strongly spherical and hence, by Theorems Cand D, admit Property (I) and Main Term Counting for the geometricballs.

(1) symmetric spaces(2) some almost symmetric examples

• G = GL(n + 1,R) and H = diag GL(n,R)• G = SL(n + 1,R) and H = SL(n,R)• G = Sp(n + 1,R) and H = Sp(n,R)× R• G = U(p, q + 1,C) and H = U(p, q,C) with p + q ≥ 2.

(3) Some G2 related examples• G = G2(C) and H = diag SL(3,C)• G = G2(R) and H = diag SL(3,R)• G = SO(7,C) and H = G2(C)• G = SO(8,C) and H = G2(C)

(4) Gross-Prasad type spaces• G = GL(n + 1,R)×GL(n,R) and H = diag GL(n,R)• G = U(p, q + 1,F)× U(p, q,F) and H = diag U(p, q,F)

Here F = R or C.(5) Triple product spaces

• G = G0 ×G0 ×G0 and H = diag G0

where G0 = SOe(1, n) and n ≥ 2.(6) Some non-maximal subgroups that are non-symmetric.

• G = SL(2n + 1,R) and H = Sp(n,R)• G = SU(n, n + 1) and H = Sp(k, k) with n = 2k.• G = SO(n, n + 1) and H = U(k, k) with n = 2k.• G = SU(n, n + 1) and H = Sp(k, k + 1) with n = 2k + 1.• G = SO(n, n + 1) and H = U(k, k + 1) with n = 2k + 1.

Acknowledgment: The authors would like to thank Joseph Bernstein,Heiko Gimperlein and Erez Lapid for several enlightening discussionson various aspects of the paper.


2. Vanishing at infinity

Let G be a real Lie group and H ⊂ G a closed subgroup. Considerthe homogenous space Z = G/H and assume that it is unimodular,that is, it carries a G-invariant measure µZ . Note that such a measureis unique up to a scalar multiple.

For a Banach representation (π, E) of G let us denote by E∞ thespace of smooth vectors. In the special case for the left regular rep-resentation of G on E = Lp(Z) with 1 ≤ p < ∞, it follows from thelocal Sobolev lemma that E∞ ⊂ C∞(Z). Let C∞

0 (Z) be the space ofsmooth functions on Z that vanish at infinity. Motivated by the decayof eigenfunctions on symmetric spaces ([43]), the following definitionwas taken in [34]:

Definition 2.1. We say Z has the property VAI (vanishing at infinity)if for all 1 ≤ p < ∞ we have

Lp(Z)∞ ⊂ C∞0 (Z).

By a result of [40], Z = G has the VAI property for G unimodular andH = 1. The main result of [34] establishes that all reductive sym-metric spaces admit VAI. On the other hand, if H is a non-cocompactlattice in G then Z = G/H is not VAI.

Let G be a real reductive group (see [46]). We say that H is a reduc-tive subgroup of G and that Z is of reductive type (or just reductive),if H is real reductive and the adjoint representation of H in the Liealgebra g of G is completely reducible. Note that Z is unimodular inthis case.

Theorem 2.2. Let G be a real reductive group and H ⊂ G a closedconnected subgroup such that Z = G/H is unimodular. Then VAI holdsfor Z if and only if it is of reductive type.

3. The invariant measure

In this section we provide a suitable framework for a discussion of theinvariant measure on Z. Throughout G is a connected real reductivegroup and H ⊂ G is a closed connected subgroup such that Z := G/His unimodular.

Let g and h be the Lie algebras of G and H. We fix a Levi-decomposition h = r o s of h and choose a subalgebra l ⊃ s whichis reductive in g and contained in h, and which is maximal with theseproperties. Then Z = G/H is of reductive type if and only if h = l.Let L be the analytic subgroup of H corresponding to the subalgebra l.


Using [38], Ch. 6, Thm. 3.6, we fix a Cartan involution of G whichpreserves L, and such that the restriction to L is a Cartan involutionof L. The derived involution g → g will also be called θ. We may andshall assume that θ(s) = s.

The fixed point set of θ determines a maximal compact subgroupK of G whose Lie algebra will be denoted k. Let p denote the −1-eigenspace of θ on g, then g = k ⊕ p. Let κ be a non-degenerateinvariant symmetric bilinear form on g such that

• κ|p > 0• κ|k < 0• k ⊥ p with respect to κ.

Having defined κ we define an inner product on g by 〈X,Y 〉 =−κ(θ(X), Y ). Let q be the orthogonal complement of h in g.

Remark 3.1. Let Z be of reductive type. Then [h, q] ⊂ q. Moreoverone has [q, q] ⊂ h if and only if the pair (g, h) is symmetric, that is, ifand only if h = X ∈ g | σ(X) = X for an involution σ of g. Thenq = X ∈ g | σ(X) = −X3.1. Construction of the invariant measure. The differential geo-metric way to obtain an invariant measure on Z is by defining aninvariant differential form of top degree. Let us briefly recall this con-struction.

For every g ∈ G we denote by

τg : Z → Z, xH 7→ gxH

the diffeomorphic left displacement by g on Z. Let z0 = H ∈ Z be thebase point. Given g ∈ G we shall identify the tangent space Tgz0Z ofZ at the point gz0 with g/h via the map

(3.1) g/h → Tgz0Z, X + h 7→ dτg(z0)X .

Let us emphasize that if gz0 = g′z0, then g = g′h for some h ∈ Hand the two identifications differ by the automorphism Ad(h)|g/h. Theassumption that an invariant measure exists on Z implies that thedeterminant of this automorphism is 1.

Let Y1, ..., Ys be a basis of g/h and ω1, ..., ωs the corresponding dualbasis in (g/h)∗ ⊂ g∗. We define the H-invariant volume form on g/hby

ω = ω1 ∧ ω2 ∧ ... ∧ ωs ∈s∧

(g/h)∗ .

As ω is Ad(H)-invariant we can extend ω to a G-invariant volume formωZ on Z. The measure µZ corresponding to ωZ is then a Haar measureon Z.


4. Reductive Spaces are VAI

In this section Z = G/H is of reductive type. Our goal is to establishuniform bounds for the invariant measure and deduce VAI for thesespaces. As Z is of reductive type we can and will identify g/h withq in an H-equivariant way. Note that q is θ-stable and in particularq = q ∩ k + q ∩ p. We denote by prq : g → q the orthogonal projection.

The characterization of infinity on Z is obtained by the polar de-composition which asserts that the polar map

(4.1) π : K ×H∩K (q ∩ p) → Z, [k, Y ] 7→ k exp(Y )z0

is a homeomorphism (see [37]). Then a function f ∈ C(Z) vanishes atinfinity if and only if

limY 7→∞Y ∈q∩p

supk∈K

|f(π(k, Y ))| = 0 .

4.1. Local coordinates. The objective of this subsection is to providesome useful local coordinates on Z and to give a uniform estimate ofthe invariant measure in terms of these local coordinates.

Let UR = X ∈ g : ||X|| < R for R > 0 and UR,q = UR ∩ q. Notethat when R is sufficiently small then exp |UR

is diffeomorphic onto itsimage in G. We define for all R > 0 a ‘ball’ in G by BR,g = exp(UR).Likewise we define BR,q = exp(UR,q) ⊂ G.

Let g ∈ G and define a map φg by

φg : UR,q → Z, Y 7→ exp(Y )gz0 .

Observe that

volZ(BR,ggz0) ≥ volZ(BR,qgz0) =

∫

UR,q

φ∗gωZ

with the last equality holding if φg is diffeomorphic onto its image.We will now show that φg is a coordinate chart with a Jacobian

uniformly bounded from below provided g = exp(X) with X ∈ p ∩ qsufficiently large. We shall identify Texp(Y )gz0Z with q as in (3.1). Astandard computation yields for all Y ∈ Uq,R:

(4.2)dφg(Y ) : X 7→ prq(Ad(g−1)

(1− e− ad Y

ad YX

)),

q → Texp(Y )gz0Z = q

For Y ∈ Uq,R we shall denote by

Jg(Y ) = | det dφg(Y )|the Jacobian of φg at Y .


Lemma 4.1. There exists a neighborhood U ⊂ q of 0 and constantsC, d > 0 such that

(4.3) Jexp(X)(Y ) ≥ d

for all X ∈ p∩q with ‖X‖ ≥ C, and all Y ∈ U . In particular, the map

φexp X : U → Z

is then diffeomorphic onto its image.

Proof. We may assume that the basis Y1, . . . , Ys of g/h is an orthonor-mal basis of q. It then follows from (4.2) that

(4.4) Jexp X(Y ) = | det

(〈e− ad X

(1− e− ad Y

ad Y

)Yi, Yj〉1≤i,j≤s

)|.

Since θ(X) = −X we can rewrite the matrix elements in (4.4) as

(4.5) 〈(

1− e− ad Y

ad Y

)Yi, e

− ad X Yj〉 .

Observe that ad X is real semisimple and let V1, . . . , Vn be an orthonor-mal basis for g of eigenvectors, with corresponding real eigenvaluesλ1, ..., λn. Let

bik = 〈(

1− e− ad Y

ad Y

)Yi, Vk〉 , ckj = 〈Vk, Yj〉,

then (4.5) equals∑n

k=1 bikckje−λk . The determinant in (4.4) is a sum

of products of such expressions.We replace X by tX for t ∈ R and set

p(t) = pX,Y (t) := det

(〈(

1− e− ad Y

ad Y

)Yi, e

− ad tX Yj〉1≤i,j≤s

).

Then it follows from the reasoning above that p is a linear combinationof exponential functions e−λt with exponents λ ∈ R which are sumsof eigenvalues −λk. We observe that the exponents depend on X in away which can be arranged to be locally uniform, and likewise for thedependence of the coefficients on X and Y .

For Y = 0 we have pX,0(−t) = pX,0(t) for all t and all X, sinceθ(X) = −X and q is θ-invariant. Thus eλt and e−λt will occur withthe same coefficients in the expansion of pX,0. If we denote by λX themaximal exponent λ such that eλt occurs in pX,0(t) with a non-zerocoefficient, then we conclude that λX ≥ 0. By compactness and localuniformity it follows that there exists a compact neighborhood U ⊂ qof 0 such that eλX t occurs in the expansion of pX,Y (t) with non-zerocoefficient for all Y ∈ U and any unit vector X ∈ p ∩ q.


In the expansion of p(t) the term with maximal exponent λ willdominate the others when t →∞. As we have just seen, this maximalexponent is ≥ 0 for all Y ∈ U and all X. Hence there exist constantsC, d > 0 such that |p(t)| ≥ d for t > C. Again by compactness,these constants can be chosen independently of Y ∈ U and X with‖X‖ = 1. ¤

In general, the constant lower bound (4.3) is sharp. However, inmany cases one can improve to an exponential lower bound. A par-ticularly simple case is obtained when (g, h) is a symmetric pair. Letprs : q → q ∩ [g, g] denote the orthogonal projection of q to its semi-simple part, and let Xs = prs(X) for X ∈ q.

Lemma 4.2. Assume G/H is a symmetric space. Then there exists aneighborhood U ⊂ q of 0 and constants C, d, δ > 0 such that

Jexp(X)(Y ) ≥ deδ‖Xs‖

for all X ∈ p ∩ q with ‖X‖ ≥ C, and all Y ∈ U .

Proof. We denote by σ the involution of g associated with h. Let abe a maximal abelian subspace of p containing X and let Σ be theassociated system of restricted roots. Root spaces in g are denoted gα,where α ∈ Σ. Let

δX =∑

α∈Σ,α(X)>0

α(X)

(roots counted with multiplicities), then δX is independent of the choiceof subspace a, and δX ≥ δ‖Xs‖ for some constant δ > 0, independentof X. We claim that there exist U , C and d as above such that

(4.6) Jexp(X)(Y ) ≥ deδX

for all Y ∈ U and ‖X‖ ≥ C. Obviously this will imply the lemma.Notice that the ad X-eigenspace for the eigenvalue λ ∈ R is given by

gλX = ⊕α∈Σ,α(X)=λg

α. Note also that V ∈ gλX implies σ(V ) ∈ g−λ

X , sinceσ(X) = −X.

We follow the proof of Lemma 4.1. It suffices to prove that if λX isthe maximal exponent in this proof then λXt = δtX for t > 0. It followsfrom the preceding paragraph that we can choose the orthonormal ba-sis (Vk)1,...,n for g such that basis vectors with non-zero eigenvalues ±λfor ad X are mutually paired by σ, and that each root with α(X) > 0 isrepresented by such pairs according to its multiplicity. The orthonor-mal basis (Yj)1,...,s for q can then be chosen to consist of normalizedmultiples of the Vk − σ(Vk) for each such pair, and additional vectorscommuting with X. An elementary computation now shows that

pX,0(t) = Πα∈Σ,α(X)>0 cosh(α(tX)),


from which the expression for the maximal exponent follows. ¤4.2. Volume bounds. We record the following corollaries.

Corollary 4.3. There exist a neighborhood U ⊂ q of 0 and constantsC, d > 0 such that the following holds for all X ∈ q∩ p with ‖X‖ > C:

For each R > 0 with UR,q ⊂ U ,

volZ(BR,q exp(X)z0) ≥ d volq(UR,q).

When G/H is symmetric, the lower volume bound can be improvedto deδ‖Xs‖ volq(UR,q) with δ > 0 independent of X and R.

Remark 4.4. The proof of Lemma 4.1 also provides the followingupper volume bound. There exist constants D, λ > 0 such that

volZ(BR,q exp(X)z0) ≤ Deλ‖X‖ volq(UR,q)

for all X ∈ q ∩ p. See [31] for such a bound in the literature.

Corollary 4.5. There exist constants C, d, R0 > 0 such that

volZ(BR,gz) ≥ dRdim q

for all R ≤ R0 and all z = k exp(X)z0 ∈ Z with k ∈ K, X ∈ p∩ q and‖X‖ > C. Furthermore, in the symmetric case we have

volZ(BR,gz) ≥ deδ‖Xs‖Rdim q

with δ > 0 independent of R and z.

Remark 4.6. For fixed R > 0 and G,H semisimple it was shown in[36] that there exists a constant c > 0 such that

volZ(BR,gz) > c

for all z ∈ Z. The corollary above sharpens this bound.

4.3. Vanishing at infinity.

Proposition 4.7. Let Z be a homogeneous space of reductive type.Then Z has the property VAI. Moreover, the inclusion

Lp(Z)∞ ⊂ C∞0 (Z)

is continuous.

Proof. By applying the Sobolev inequality in local coordinates, we ob-tain the following for 1 ≤ p < ∞ and for each compact neighborhoodB of e in G (see [40] for details). There exist finitely many elementsvi ∈ U(g) (of degree up to the smallest integer > dim Z/p) in theenveloping algebra of g, and for each z ∈ Z a constant D > 0 such that

(4.7) |f(z)| ≤ D max ‖(Lvif)1Bz‖p


for all f ∈ Lp(Z)∞. Here 1Bz denotes the characteristic function ofBz ⊂ Z. The constant D is locally uniform with respect to z.

Based on Lemma 4.1, we can improve the local estimate (4.7), suchthat for G/H of reductive type it holds with D independent of z. LetU ⊂ q and C, d > 0 be as in Lemma 4.1, and fix R > 0 such thatUR,q ⊂ U . It follows that

(4.8) ‖(f φexp X)1UR,q‖p ≤ d−1/p‖f 1BR,q exp(X)z0‖p

for f ∈ Lp(Z) and X ∈ p∩q with ‖X‖ > C. By the Sobolev inequalityfor Rdim q, the value |f φexp X(0)| is estimated above by the p-norms,over any neighborhood of 0, of the derivatives of f φexp X . Hence iff ∈ Lp(Z)∞ and ‖X‖ > C, we obtain an upper bound

|f(exp(X)z0)| ≤ D max ‖(Lvf)1BR,q exp(X)z0‖p

with derivatives as before by finitely many elements in U(g), and witha constant D independent of f and X. After conjugation by k ∈ K weconclude that (4.7) holds at z = k exp(X)z0, with B = BR,g and with auniform constant D. As the set of elements k exp(X)z0 with ‖X‖ ≤ Cis compact, the inequality is finally obtained for all z ∈ Z.

The proposition is a straightforward consequence of the uniform ver-sion of (4.7). ¤

For G/H symmetric we obtain a better result by replacing the useof Lemma 4.1 by Lemma 4.2 in the estimate (4.8) while following thepreceding proof:

Proposition 4.8. Let Z = G/H be symmetric. There exists a constantδ > 0 with the following property. Let f ∈ Lp(Z)∞ where 1 ≤ p < ∞.Then for each ε > 0 there exists C > 0 such that

|f(k exp(X)z0)| ≤ εe−δ‖Xs‖

for all z = k exp(X)z0, where X ∈ p ∩ q, ‖X‖ > C and k ∈ K.

5. Homogeneous spaces of polar type

Recall that all homogeneous spaces of reductive type admit the de-composition (4.1). A stronger decomposition property can be definedas follows.

Definition 5.1. Let Z = G/H be a homogeneous space of reductivetype. A polar decomposition of Z consists of an abelian subspace a ⊂ pand a surjective proper map

K × A 3 (k, a) 7→ kaH ∈ Z,


where A = exp a. We say that Z is of polar type if such a polardecomposition exists.

Notice that we do not require a ⊂ q in Definition 5.1. This is notpossible for example in the spaces considered in 5.1.2 and 5.1.3.

5.1. Examples. We provide some examples of homogeneous spaces ofpolar type. Further examples will be given later (see Section 6.1)

5.1.1. Symmetric spaces. Symmetric spaces are of polar type with a ⊂q. In fact (see [45] p. 117) suppose that Z is symmetric and let aq ⊂p ∩ q be a maximal abelian subspace (it is unique up to conjugationby K ∩ H). Then G = KAqH where Aq = exp(aq). For this case werecall that more structure is known. Associated to aq is a root systemΣ(g, aq) ⊂ a∗q and hence a notion of positivity. For each associatedWeyl chamber a+

q the map

K ×M a+q → Z, [k,X] 7→ k exp(X)z0

is a diffeomorphism onto an open set. Here M = ZK∩H(aq). Further-more, the union of the sets KA+

q z0, over all K ∩ H-conjugacy classesof positive chambers A+

q = exp(a+q ), is disjoint and dense in Z.

Remark 5.2. Although not subject proper of this paper it is quiteuseful to compare the situation in the p-adic setup: Let G be a p-adicreductive group G, K < G a maximal compact subgroup and H < Ga symmetric subgroup. Then, in general, there exists no torus A suchthat G = KAH. However, there exists a torus A < G and a finitesubset F such that G = KAFH, see [19].

5.1.2. Triple spaces. Let G0 be a real reductive group and define

G = G30 = G0 ×G0 ×G0

and H = diag(G0) = (g, g, g) | g ∈ G0. Then Z = G/H is ahomogeneous space of reductive type.

Let K0 < G0 be a maximal compact subgroup. We fix an Iwasawadecomposition G0 = K0A0N0 and let P0 = M0A0N0 be the associatedminimal parabolic subgroup. Set K = K0 ×K0 ×K0.

Proposition 5.3. Suppose that B0 ⊂ G0 is a subset such that

G0 = A0M0B0K0.

Then, for A = A0 × A0 ×B0, one has G = KAH.

Proof. Let (g1, g2, g3) ∈ G. From the KAH-decomposition of the sym-metric space G0 ×G0/ diag(G0) we obtain

(g1, g2) = (g, g)(a1, a2)(k1, k2)


for some g ∈ G0, a1, a2 ∈ A0 and k1, k2 ∈ K0. Now choose m ∈ M0,a0 ∈ A0, b0 ∈ B0 and k0 ∈ K0 such that g−1g3 = a0m0b0k0. Then

(g1, g2, g3) = (ga0m0, ga0m0, ga0m0)(a−10 a1, a

−10 a2, b0)(m

−10 k1,m

−10 k2, k0)

as asserted. ¤

The proposition applies in the following cases:

• G0 = SOe(1, n) and B0 = exp(RX) for some 0 6= X ∈ p0 ∩ a⊥0 .

Note that it also applies to B0 = N0 for general G0,

Corollary 5.4. Let G0 = SOe(1, n) for n ≥ 2 and Z = G30/ diag(G0).

Then Z is of polar type.

5.1.3. Gross-Prasad spaces. We let G0 be a reductive group and H0 <G0 be a reductive subgroup. Set G = G0 × H0 and H = diag(H0).Note that Z = G/H ' G0. However in this isomorphism G0 is viewedas a homogeneous G0 ×H0 space.

We consider the following choices for G0 and H0, with which we referto Z as a Gross-Prasad space (cf. [26]):

• G0 = GL(n + 1,R) and H0 = GL(n,R) for n ≥ 0.• G0 = U(p, q + 1,F) and H0 = U(p, q,F) for p + q ≥ 2.

Here F = R, C or H.

Lemma 5.5. Gross-Prasad spaces are of polar type.

Proof. We first treat the case (G0, H0) = (GL(n+1,R), GL(n,R)). Letus embed H0 in G0 as the lower right corner.

We define a two-dimensional non-compact torus of GL(2,R) by

B = SOe(1, 1) · R+1 .

In GL(2k,R) we define a 2k-dimensional non-compact torus A2k by kblock matrices of form B along the diagonal. In GL(2k+1,R) we defineA2k+1 to consist of similar blocks together with a positive number inthe last diagonal entry. Finally we let

A = An+1 × An ⊂ G = GL(n + 1,R)×GL(n,R).

With K = O(n + 1,R)×O(n,R) we claim that

G = KAH ,

or, equivalently,

GL(n + 1,R) = O(n + 1,R)An+1An O(n,R) .


We proceed by induction on n. The case n = 0 is clear. We shall usethe known polar decomposition for the almost symmetric pair (GL(n+1,R), GL(n,R)):

GL(n + 1,R) = O(n + 1,R)B1 GL(n,R)

where B1 is the two-dimensional torus of form B located in in the upperleft corner. Now insert for GL(n,R) by induction, but in opposite order:

GL(n,R) = O(n− 1,R)An−1An O(n,R)

and observe that O(n− 1,R) commutes with B1.The case with G0 = U(p, q + 1,F) is similar. Choose non-compact

Cartan subspaces for g0 and h0 along antidiagonals, and note that theoverlap between these, as subspaces of g0, is trivial. Now proceed byinduction as before. ¤Remark 5.6. Note that in Corollary 5.4 and Lemma 5.5 all polardecompositions G = KAH are with a a Cartan subspace in p.

5.2. Some structure theory. Let Z = G/H be a homogeneous spaceof reductive type. We assume that the Cartan involution θ is chosensuch that θ(H) = H.

Lemma 5.7. There exists a finite dimensional representation (π, V )of G and a vector vh ∈ V such that h = X ∈ g|dπ(X)vh = 0.Proof. Follows from Sect. 5.6, Th. 3 in [2]. ¤

Let a ⊂ p be an abelian subspace with a∩ h = 0. Let A = exp(a).

Lemma 5.8. The set AH is closed in G and (a, h) 7→ ah is properA×H → AH.

Proof. We argue by contradiction. Suppose that AH were not closedin G or that (a, h) 7→ ah were not proper. Then there would existsequences (an)n∈N in A and (hn)n∈N in H, both leaving every compactsubset, such that p = limn→∞ anhn exists in G.

Let (π, V ) be a finite dimensional representation as in Lemma 5.7.Then the limit limn→∞ π(an)vh exists. Passing to a subsequence wemay assume that

X := limn→∞

log an

‖ log an‖ ∈ a

exists. This implies that vh is a sum of dπ(X)-eigenvectors with non-positive eigenvalues. Likewise (θ(anhn))n∈N converges and we obtainthat vh is fixed under dπ(X). But this contradicts the assumption thata ∩ h = 0. ¤


Corollary 5.9. The set KAH is closed in G and (k, a, h) 7→ kah isproper K × A×H → KAH.

6. Spaces of spherical type

Recall (see [15]) that a complex homogeneous space GC/HC is saidto be spherical if there exists a Borel subgroup BC such that BCHC isopen in GC. The following definition is analogous. Let Z = G/H be areductive homogeneous space.

Definition 6.1. The space Z is of spherical type if there exists a min-imal parabolic subgroup P such that PH is open in G. If in additiondim(P ∩H) = 0 then we say that Z is of pure spherical type.

Note that the main intention with the concept in [15] is the clas-sification of Gel’fand pairs. Having that intention one should add toDefinition 6.1 the condition that (M, M ∩H) is a Gel’fand pair. How-ever this is not our purpose. The non-symmetric space Sp(n, 1)/ Sp(n),for example, is of spherical type but fails the Gel’fand pair condition.

It will be convenient to consider also non-minimal parabolic sub-groups. If P ⊂ G is a parabolic subgroup, we denote by P = MP AP NP

its Langlands decomposition.

Definition 6.2. Let P ⊂ G be a parabolic subgroup. The pair (P,H)is called spherical if

(1) MP /(MP ∩H) is compact,(2) PH is open in G.

Lemma 6.3. If there exists a spherical pair then Z is spherical.

Proof. Let (P, H) be a spherical pair. It follows from condition (1)that mP ∩ p ⊂ h. Hence all non-compact ideals of mP belong to h. LetP ∗ = M∗A∗N∗ be a minimal parabolic in MP , then A∗N∗ ⊂ H, andhence P0 := AP NP P ∗ is a minimal parabolic in G such that P0H isopen. ¤6.1. Examples. In a symmetric space Z, the minimal σθ-stable par-abolic subgroups P satisfy (1) and (2), see [4], hence Z is of sphericaltype. In this case we have in addition that P ∩H ⊂ MP AP , and thatthe modular function of P is trivial on P ∩H.

6.1.1. Triple spaces, Gross-Prasad spaces. It is easily seen that thetriple spaces in Corollary 5.4 and the Gross-Prasad spaces with G0 =GL(n + 1,R) are of pure spherical type. Indeed, for the chosen Cartansubspaces (see Remark 5.6) one obtains h ⊕ Lie(P ) = g with P aminimal parabolic and suitable choices of positive systems.


6.1.2. Complex spherical spaces. Let GC/HC be a complex sphericalspace with open Borel orbit BCHC. When we regard the complexgroups as real Lie groups, GC/HC is of spherical type and (BC, HC) is aspherical pair. The complex spherical spaces have been classified (seethe lists in [33] and [15]). For example, the triple space of G0 = SL(2,C)is a complex spherical space ([33] p. 152).

6.1.3. Real forms of spherical spaces. Let GC, HC, BC be as above,and assume that G is a quasisplit real form of GC. Then BC is thecomplexification of a minimal parabolic P in G, for which PH is open.Hence G/H is of spherical type. The triple space with G0 = SL(2,R)is obtained in this fashion.

Notice that for n > 3 the triple spaces with G0 = SOe(n, 1) do notcorrespond to any spaces in 6.1.2 or 6.1.3.

6.1.4. SL(2n+1,C)/ Sp(n,C) and SO(2n+1,C)/ GL(n,C). Accordingto [33] p. 143, these are complex spherical spaces. Dimension countshows they are pure. The split or quasisplit real forms are

SL(2n + 1,R)/ Sp(n,R)

SU(n, n + 1)/ Sp(k, k), n = 2k

SO(n, n + 1)/ U(k, k), n = 2k

SU(n, n + 1)/ Sp(k, k + 1), n = 2k + 1

SO(n, n + 1)/ U(k, k + 1), n = 2k + 1.

None of these subgroups are maximal.

6.2. Bounds for generalized matrix coefficients. In this para-graph we prove a fundamental bound for generalized matrix coefficientsfor spaces of spherical type.

To begin with we need to recall a few notions from basic represen-tation theory. Let (π, E) be a Banach representation of G. Let usdenote by p a defining norm of E. As in [10], Sect. 2, we topologizethe Frechet space of smooth vectors E∞ by Sobolev norms Sps of p.These Sobolev norms are defined for any order s ∈ R. We note thatSps1 ≤ Sps2 for s1 ≤ s2. We denote by Es the Banach-completion of(E∞, Sps). If E−∞ denotes the continuous dual of E∞, then we record

E∞ =⋂s>0

Es = lim←−

Es and E−∞ =⋃

s∈REs = lim

−→Es .

In case s ∈ 2Z is an even integer the Sobolev norms Sps are equiv-alent to norms ps which are G-continuous, that is the completion of


(E∞, ps) defines a Banach-module for G. Hence for k ∈ N and η ∈ E−2k

and v ∈ E2k we have

(6.1) |η(π(g)v)| ≤ p−2k(η)p2k(π(g)v)

and further

(6.2) p2k(π(g)v) ≤ C‖g‖lp2k(v)

for some constants C, l > 0. Here ‖ · ‖ is a norm on G in the sense of[46], Sect. 2.

If V is a Harish-Chandra module for (g, K), , then we call a Banachrepresentation (π, E) of G a Banach globalization of V provided thatthe K-finite vectors of E are isomorphic to V as (g, K)-modules. TheCasselman-Wallach theorem asserts that E∞ does not depend on theparticular globalization (π, E) of V and thus we may define V ∞ := E∞.

Let Z = G/H be a reductive homogeneous space and V a Harish-Chandra module. For v ∈ V ∞ and η ∈ (V −∞)H an H-fixed distributionvector we denote by

(6.3) mv,η(gH) := η(π(g)−1v), (g ∈ G)

the corresponding generalized matrix coefficient. It is a smooth func-tion on Z.

Let a ⊂ p be maximal abelian. In this situation we fix an Iwasawadecomposition G = KAN and denote by Π ⊂ a∗ the set of simpleroots. Let P = MAN denote the corresponding minimal parabolicsubgroup, and P = θ(P ) its opposite.

In [46] 4.3.5 one associates to V an exponent Λ ∈ a∗. If g is simpleand π is non-trivial unitary, then it follows from the Howe-Moore-Theorem (see [43], p. 447), that

(6.4) Λ ∈ ∑

α∈Π

cαα | cα < 0

.

Theorem 6.4. Suppose for the minimal parabolic subgroup P ⊂ G thatPH is open in G. Let V be a Harish-Chandra module and η ∈ (V −∞)H .Then there exists d ∈ N0 such that for all v ∈ V there is a constantCv > 0

(6.5) |mv,η(a)| ≤ CvaΛ(1 + ‖ log a‖)d (a ∈ A+) .

Proof. The proof will be an adaption of the proof of Thm. 4.3.5 in[46]. Note that we have mv,η(a) = η(π(a)−1v) whereas [46] considersµ(π(a)v). The linear form η will be fixed throughout the proof.

We confine ourselves to providing the key step. Our starting point isthe following estimate, which follows from (6.1) and (6.2). Let (π, E)


be a Banach globalization of V , Suppose that η ∈ E−l for l ∈ 2N0.Then there exists δ ∈ a∗ such that

(6.6) |mv,η(a)| ≤ aδ‖η‖−l‖v‖l (a ∈ A+)

where we write ‖.‖k in place of pk. If δ happens to be ≤ Λ on a+ weare done. Otherwise we need to improve (6.6). The key ingredient isas follows: Suppose that v ∈ V is of the form v = dπ(X)u for somenormalized positive root vector X ∈ gα ⊂ n (this corresponds to theassumption µ ∈ nF V ∼ in [46] p.116). As PH is open in G we can writeX = X1 + X2 with X1 ∈ h and X2 ∈ a + m + n. Now observe that

mv,η(a) = η(π(a)−1dπ(X)u) = a−αη(dπ(X)π(a)−1u)

= a−αη(dπ(X2)π(a)−1u) = a−αη(π(a)−1dπ(Ad(a)X2)u) .

As Ad(a) is contractive on a + m + n we can write dπ(Ad(a)X2)u as alinear combination of elements from V with a-dependent coefficients,which are bounded. We thus obtain with (6.6) an improved bound

|mv,η(a)| ≤ Cvaδ−α (a ∈ A+)

Having seen that, the rest of the proof follows [46] p.117–118. ¤

Remark 6.5. (a) The theorem applies to symmetric spaces. Withnotation from Example 5.1.1 we let aq ⊂ a where a ⊂ p is maximalabelian. Then a = aq ⊕ ah = (a ∩ q) ⊕ (a ∩ h). A proper choice

of a positive chamber for a yields a+q ⊂ a+. Then PH, and hence

PH, is open for the corresponding minimal parabolic P . Thus (6.5)applies to all a ∈ A+

q . In this situation fundamental bounds as (6.5)have previously been established in [24] and [5], Thm. 14.1, and furtherexplored in [43].(b) We expect that there is a more quantitative version of the upperbound (6.5), namely

(6.7) |mv,η(a)| ≤ p(v)aΛ(1 + ‖ log a‖)d (a ∈ A+) .

for some G-continuous norm p on V .

7. Spherical type implies polar type

It is our belief that all spherical spaces are of polar type. In thissection we prove this implication under certain additional hypotheseson Z.


7.1. Algebraic type. We will say that Z is of algebraic type (or al-gebraic) if there exists a reductive algebraic group G, defined over R,and a reductive subgroup H < G such that G = G(R) and H = H(R).We will also write GC = G(C) and likewise we declare KC and HC.

The crucial property that we need of an algebraic type space is theexistence of a faithful finite dimensional algebraic G-module (π, V ) withan H-fixed vector vH ∈ V such that

(7.1) Z → V, gH 7→ π(g)vH

is a closed embedding (see Sect. 5.6, Th. 3 in [2]).Assumption: From now on we assume

(7.2) Z is of algebraic type.

7.2. Main theorem. The main theorem in this section is

Theorem 7.1. Let Z = G/H be a reductive space of spherical type,and assume that there exists a parabolic subgroup P ⊂ G such that

(1) (P, H) is spherical,(2) L := P ∩H ⊂ MP .

Then Z is of polar type.

We shall say that the pair (P,H) is strongly unimodular if (1)-(2)holds. Obviously, pure spherical spaces (see Definition 6.1) are stronglyunimodular. In particular it follows that the triple spaces and Gross-Prasad spaces of Example 6.1.1, as well as the spaces in 6.1.4, are allstrongly spherical. Further examples are given below.

It should however be emphasized that the existence of such a paircannot be attained for spherical spaces in general.

7.3. Spherical unimodular type. We prepare for the proof of The-orem 7.1 with a simple implication of Condition (2).

Definition 7.2. Let (P, H) be a spherical pair and let L := P ∩ H.The pair (P,H) is said to be unimodular if the homogeneous spacesH/L and P/L are both unimodular.

If such pairs exist, we say that Z is of spherical unimodular type.Not all spherical pairs (P, H) are unimodular (see Example 7.4.1). Thefollowing is clear, since the modular function of P is trivial on MP .

Lemma 7.3. A strongly unimodular spherical pair is unimodular.

For a symmetric space and P a minimal σθ-stable parabolic, the pair(P,H) is unimodular (see Example 6.1), but not necessarily stronglyunimodular.


7.4. Examples. The first example shows that it is important to allowthat P is not minimal.

7.4.1. A case where (P, H) is not unimodular for P minimal. Let

G/H = SL(n + 1,R)/ SL(n,R).

This is an almost symmetric space. Let H = SL(n,R) be embeddedinto the right lower corner of G = SL(n + 1,R), and let P < G bea minimal parabolic. Then dim G/P = n(n + 1)/2. Specifically wechoose P to be the stabilizer of the complete flag

〈e1 + en+1〉 ⊂ 〈e1 + en+1, e2〉 ⊂ 〈e1 + en+1, e2, e3〉 ⊂ . . . ⊂ Rn+1

from which we readily identify L = P ∩ H with the upper triangularsubgroup of SL(n − 1,R), where the latter group is embedded in theupper left corner of H. Thus dim H/L = n(n + 1)/2 and HP is openby dimension count. However H/L is not unimodular. This can beremedied by enlarging P .

If we take P to be the maximal parabolic which is the stabilizer ofthe line 〈e1 +en+1〉, then L ' SL(n−1,R)nRn−1 ⊂ MP NP and (P,H)is unimodular.

If instead we define P with the intermediate flag

〈e1 + en+1〉 ⊂ 〈e1 + en+1, e2, . . . , en〉 ⊂ Rn+1

then L = SL(n− 1,R) ⊂ MP and (P,H) is strongly unimodular.The non-symmetric space

G/H = Sp(n + 1,R)/ Sp(n,R)× Rsimilarly admits a strongly unimodular pair (P, H) with a parabolicsubgroup P of rank two.

7.4.2. Gross-Prasad space with G0 = U(p, q+1,F). (See Section 5.1.3).A similar procedure as for Lemma 5.5 shows that this space admits astrongly unimodular pair with a minimal parabolic subgroup.

7.4.3. Examples involving the exceptional group G2. In all what followsthe symbol G2(C) denotes the simply connected complex group of typeG2. We denote by G2(R), resp. U2(R), the non-compact, resp. com-pact, real form of G2(C). All complex cases G/H below are taken fromKramer’s list [33] and regarded over R as in Example 6.1.2.

1) G/H = G2(C)/ SL(3,C). In this case h is maximal but non-symmetric. According to Kramer, G/H is spherical and thus thereexists a Borel subgroup P = BC of G2(C) such that HP is open. Thenh ∩ Lie(P ) = 0 by dimension count and (P,H) is a pure sphericalpair.


It is quite instructive to have a concrete model for Z. For that letVR ' R7 be the space of trace zero octonions and recall that U2(R) isthe automorphism group of VR. If we endow VR with an U2(R)-invariantinner product, then U2(R) acts transitively on the unit sphere SR ⊂ VRand SR ' U2(R)/SU(2) as follows from the proof of [1], Th. 5.5. Inparticular, if SC is the complex quadric and complexification of SR inVC, then SC ' G2(C)/ SL(3,C) = Z. It is known that G2(R) ∩ H =SL(3,R) and we obtain another pure spherical space G2(R)/ SL(3,R).

2) G/H = SO(7,C)/G2(C). In this case h is maximal, non-symmetric.Let us first recall a standard model for Z. Consider the real 8-di-mensional spin representation VR of SO(7,R) and endow VR with anSO(7,R)-invariant inner product (, ). We extend (, ) to a complex bi-linear form (, )C on VC. In VC we choose a highest weight vector ufor G and note that u is isotropic, i.e. (u, u)C = 0. The nullconeQC := [v] = Cv ∈ P(VC) | (v, v)C = 0 is isomorphic to G/P where Pis the parabolic which fixes the line [u] = Cu. Then the Levi subgroupMP AP of P is isomorphic to GL(3,C). Further note that dimCG = 21and dimC P = 15. Now let v be a non-zero vector in VR. According to[1], Th. 5.5, the stabilizer of v in G is isomorphic to H = G2(C). Bydimension count we have dimC L ≥ 8. We claim that there is a specificchoice of v such that this intersection is isomorphic to M = SL(3,C);in particular, HP is open. For that let u = u1 + iu2 the decompositionaccording to VC = VR + iVR. The fact that u is isotropic means that(u1, u1) = (u2, u2) > 0 and (u1, u2) = 0. Take now v = u1. ThenL is the stabilizer of u1 and u2. As u1 and u2 are orthogonal, thereis an orthogonal transformation moving u1 to u2. In particular L isreductive. It is a proper reductive subgroup of G2(C) and thus at most8-dimensional. Our claim follows. To summarize, we constructed astrongly unimodular spherical pair (P,H) such that P ∩H = M . Thisproperty inherits to the real form SO(4, 3)/G2(R) of Z.

3) G/H = SO(8,C)/G2(C). In this case h is not symmetric and notmaximal. We argue as in 2) and use a real 8-dimensional spin rep-resentation VR of SO(8,R). The stabilizer of a highest weight vectoru ∈ VC is then a parabolic P < G with dim P = 22 and Levi subgroupMA = GL(3,C) × GL(1,C) in 10 dimensions. As before we arguethat H ∩ P is reductive and thus L = M . In particular, Z is stronglyunimodular. Likewise the real form SO(4, 4)/G2(R) of Z is stronglyunimodular as it admits a spherical pair (P, H) with L = M .

7.5. Proof of Theorem 7.1. We first study the polar decompositionon the Lie algebra level.


Lemma 7.4. Let (P, H) be a spherical pair with Langlands decompo-sition P = MAN of P . Then there exists a ∈ A such that

(7.3) ka + a + h = g

where ka := Ad(a−1)(k).

Note that in view of Sard’s theorem an equivalent formulation of theconclusion is that KAH has an interior point.

Proof. Otherwise L(a) := Ad(a)k + a + h is a proper subspace of gfor all a ∈ A. For t 7→ at a ray tending to infinity in A+ we note thatlimt→∞ L(at) = m∩k+a+n+h in the Grassmann variety of all subspacesof g. By (1) and (2) in Definition 6.1 we obtain limt→∞ L(at) = g,but as the set of subspaces of positive codimension is closed in theGrassmannian, this is a contradiction. ¤

Strong unimodularity of (P,H) implies the stronger decomposition

(7.4) a⊕ (ka + h) = g.

This results from the following lemma, where we replace L ⊂ M bythe weaker assumption L ⊂ MA.

Lemma 7.5. Assume L ⊂ MA and let a ∈ A satisfy (7.3). Then

a ∩ (ka + h) = a ∩ h and ka ∩ h = m ∩ k ∩ h.

Proof. Since (P, H) is a spherical pair we have

(7.5) h + (m⊕ a⊕ n) = g

As m∩p ⊂ h we have l = (m∩p)⊕ (m∩ k∩h)⊕ (a∩h). Comparing theexcess dimensions in (7.3) and (7.5) and using that dim k = dim(m ∩k) + dim n we infer

dim[a ∩ (ka + h)] + dim(ka ∩ h) = dim(m ∩ k ∩ h) + dim(a ∩ h).

Since a centralizes m we have m∩ k∩ h ⊂ ka ∩ h, and we are done. ¤

Lemma 7.6. Assume that (P,H) is a strongly unimodular sphericalpair. Then G = KAH.

Proof. Given in Appendix B. ¤

Theorem 7.1 now follows from Corollary 5.9.


7.6. Strongly spherical. To facilitate the exposition in the next sec-tion we need the following concept, which combines polar and sphericaltypes compatibly.

Definition 7.7. A reductive homogeneous space Z = G/H is of strongspherical type if the following holds. There exists a ⊂ p maximalabelian and minimal parabolic subgroups P1, . . . , Pl ⊃ A such that

(1) Z is of polar type and G = KAH;(2) Z is of spherical type with PjH is open for all 1 ≤ j ≤ l;

(3)⋃l

j=1 A+(Pj)(A ∩H) = A.

Symmetric spaces are strongly spherical, see Remark 6.5. We believethat

(7.6) spherical type ⇒ strong spherical type.

but again we shall only prove this under the additional assumption ofstrong unimodularity.

Example 7.8. Consider G/H with

(1) G = GL(n + 1,R) and H = GL(n,R) for n ≥ 0.(2) G = U(p, q + 1,C) and H = U(p, q,C) for p + q ≥ 2.

These spaces are (rank one) symmetric modulo the center of G. Itfollows easily that they are strongly spherical.

Corollary 7.9. If Z is of strongly unimodular spherical type, then itis strongly spherical.

Proof. Let (P,H) be a strongly unimodular spherical pair. We haveseen in Lemma 7.6 that G = KAP H.

Let P ∗ be a minimal parabolic in MP and define a minimal parabolicP0 in G by P0 = AP NP P ∗ as in the proof of Lemma 6.3. Then P ⊃ P0

and (P0, H) is a spherical pair. Let a0 ⊂ p be the corresponding Cartansubspace.

Note that the sphericality of (P0, H) is an open condition for P0. Thisimplies that (kP0k

−1, H) is spherical for generic k ∈ K. By choosing ksufficiently generic we can thus ensure that (kwP0w

−1k−1, H) is spher-ical for all Weyl group elements w of a0. We now replace P by itsk-conjugate kPk−1, while noting that in view of Lemma B.2, this con-jugate is again unimodular. Likewise we replace P0 and A0 by theirk-conjugates, and obtain thus that (Q,H) is spherical for all minimalparabolics Q with AQ = A0,

Let Q1, . . . , Ql be an enumeration of the parabolic subgroups Q forwhich AQ = AP . Each Qj contains one or more minimal parabolics


with split part a0, of which we choose one and denote it by Pj. ThenPjH is open for each j. The result follows. ¤

8. Property (I)

Let Z = G/H be of reductive type. We introduce an integrabilitycondition for matrix coefficients on Z. It has some similarity withKazdan’s Property (T).

We denote by G the unitary dual of G and by Gs the subset of K-spherical representations. With a ⊂ p a maximal abelian subspace, and

W the corresponding Weyl group, we can identify Gs with a subset ofa∗C/W .

Definition 8.1. We say that a reductive homogeneous space Z = G/H

has Property (I) if for all π ∈ Gs and η ∈ (H−∞π )H the stabilizer Hη of

η is reductive in G and there exists 1 ≤ p < ∞ such that

(8.1) mv,η ∈ Lp(G/Hη),

for all v ∈ H∞π .

The following lemma shows that it suffices to have (8.1) for v ∈ HKπ .

Lemma 8.2. Let (π,Hπ) be irreducible unitary, and let η ∈ (H−∞π )H

and 1 ≤ p < ∞. The following statements are equivalent:

(1) mv,η ∈ Lp(Z) for all v ∈ H∞π .

(2) mv,η ∈ Lp(Z) for all K-finite vectors in Hπ.(3) mv,η ∈ Lp(Z) for some K-finite vector v 6= 0.

Proof. Let V be the Harish-Chandra module of (π,Hπ), i.e. the spaceof K-finite vectors. According to Harish-Chandra, V is an irreducible(g, K)-module. The map v 7→ mv,η is equivariant V → C∞(Z).

We first establish “(3) ⇒ (2)”. Let v ∈ V be non-zero with mv,η ∈Lp(Z), then v generates V , and (2) is equivalent with the statementthat mv,η ∈ Lp(Z)∞.

Let E be the closed G-invariant subspace of Lp(Z) generated by mv,η.As the left action on Lp(Z) is a Banach representation, the same holdsfor E. The Casimir element C acts by a scalar on V , hence it acts (inthe distribution sense) on E by the same scalar. It follows that all K-finite vectors in E are smooth for the Laplacian ∆ associated to C (see(11.2) below). Thus any K-finite vector of E belongs to E∞ ⊂ Lp(Z)∞

by [10], Prop. 2.13.Finally “(2) ⇒ (1)” follows from the Casselman-Wallach global-

ization theorem (see [10]), which implies that the map v 7→ mv,η,V → Lp(Z), extends to H∞

π → Lp(Z)∞ . ¤


In the definition of property (I) we have to take into account thatthe stabilizer Hη inflates H. To discuss this efficiently it is useful tohave appropriate notion of factorization for Z.

8.1. Factorization. Given a reductive homogeneous space Z = G/H.If there exists a reductive subgroup H∗ ⊃ G such that h ( h∗ ( g,then we call Z factorizable, set Z∗ = G/H∗ and call

Z 7→ Z∗, gH 7→ gH∗

a factorization of Z.

Example 8.3. 1) Irreducible symmetric spaces are not factorizable.In fact, if (g, h) is a symmetric pair and h ( h∗ ( g a factorization,then g = g1× g2, h = h1× h2 and h∗ = h1× g2 for some decompositionof g.2) Suppose that H < G is a proper reductive subgroup. Then Z =G×H/ diag(H) is factorizable with H∗ = H ×H.3) Let Gn := G × · · · × G denote the direct product of n copies of G.Suppose that G is simple. The homogeneous space Z = Gn/ diag(G)with the diagonal subgroup diag(G) = (g, . . . , g)|g ∈ G is factor-izable if and only if n > 2. For example, for n = 3, we can takeH∗ = (g1, g2, g2) | g1, g2 ∈ G ' G×G and permutations thereof.4) Let G/H = SO(8,C)/G2(C) as in Example 7.4.3.3. The symmetricspace G/H∗ = SO(8,C)/ SO(7,C) is a factorization.

Having the notion of factorization in the back of our mind we resumeour discussion of property (I) in the context of these examples.

Example 8.4. 1) Suppose that Z = G ×H/ diag(H) with G simple.Irreducible spherical unitary representations of G×H are of the form

π = π1 ⊗ π2 where π1 ∈ Gs and π2 ∈ Hs. Moreover π is H-spherical ifand only if π∗2 is contained in π1|H . If π2 was the trivial representation,then the stabilizer of η inflates to Hη = H ×H.2) Suppose that Z = G× G× G/ diag(H) with G simple. Irreduciblespherical representations of G × G × G are tensor products π = π1 ⊗π2 ⊗ π3. If π is non-trivial and H-spherical and if one constituent, sayπ1, is trivial, then Hη = (g1, g2, g2) | g1, g2 ∈ G.8.2. Main Theorem on Property (I). We can now state one of themain results of the paper. Recall that a space with the properties inDefinition 7.7 is called strongly spherical.

Theorem 8.5. Let Z = G/H be reductive and suppose that Z isstrongly spherical. Then Z has Property (I).


Proof. Note that any factorization Z∗ of Z will be strongly spherical,once Z is strongly spherical. Proceeding by induction, we may thus as-sume that all factorizations Z∗ have property (I). We may also assumethat h contains no non-trivial ideal of g.

Let π ∈ Gs be non-trivial and let f = mv,η with η ∈ (H−∞π )H and

v a K-fixed vector. In view of Lemma 8.2 it is sufficient to show thatf ∈ Lp(G/Hη) for some 1 ≤ p < ∞.

Since G = KAH the invariant measure µZ on Z can be expressed inK × A-coordinates as∫

Z

φ(z)dµZ(z) =

∫

K

∫

A/(A∩H)

φ(ka · z0)J(a) da dk (φ ∈ Cc(Z)) .

The Jacobian is readily verified to be the absolute value of a polynomialin aα, α ∈ Σ. If the exponent Λ of π satisfies (6.4), it follows fromTheorem 6.4 that |f |p becomes integrable over A+(Pj)/(A ∩ H), forsufficiently large p. Hence f ∈ Lp(Z). If the exponent Λ does notsatisfy (6.4), then G is not simple and π is 1 on some non-trivial normalsubgroup in G. This subgroup is contained in Hη but not in H, henceZ → G/Hη is a factorization. Now the assertion follows from ourinductive hypothesis. ¤

Corollary 8.6. Let Z be a homogeneous space of reductive type. As-sume that Z is symmetric or one of the following spherical spaces

(1) triple spaces G3/ diag(G) with G = SOe(1, n) and n ≥ 2,(2) Gross-Prasad spaces G×H/ diag(H), see Section 5.1.3,(3) the spaces discussed in Examples 6.1.4, 7.4.1, 7.4.3, and 7.8.

Then Z has Property (I).

We emphasize that this is by no means an exhaustive list of spaceswhich can be treated by the present methods.

9. Counting lattice points, I: Preliminaries

In this section we let G/H be a homogeneous space of reductivetype. We assume that there exists a lattice (a discrete subgroup withfinite covolume) Γ ⊂ G such that ΓH := Γ ∩ H is a lattice in H. Wenormalize Haar measures on G and H such that:

• vol(G/Γ) = 1.• vol(H/ΓH) = 1.

To simplify the presentation we assume that G is linear semi-simple.Let us emphasize that this is just a matter of convenience because itallows us to skip integration over a possibly non-compact center.


Our concern is with the double fibration

Z := G/H ← G/ΓH → Y := G/Γ .

Fibrewise integration yields transfer maps from functions on Z to func-tions on Y and vice versa. In more precision,

(9.1) L∞(Y ) → L∞(Z), φ 7→ φH ; φH(gH) :=

∫

H/ΓH

φ(ghΓ) d(hΓH)

and we record that this map is contractive, i.e

(9.2) ‖φH‖∞ ≤ ‖φ‖∞ (φ ∈ L∞(Y )) .

Likewise we have

(9.3) L1(Z) → L1(Y ), f 7→ fΓ; fΓ(gΓ) :=∑

γ∈Γ/ΓH

f(gγH) ,

which is contractive, i.e

(9.4) ‖fΓ‖1 ≤ ‖f‖1 (f ∈ L1(Z)) .

Unfolding with respect to the double fibration yields, in view of ournormalization of measures, the following adjointness relation:

(9.5) 〈fΓ, φ〉L2(Y ) = 〈f, φH〉L2(Z)

for all φ ∈ L∞(Y ) and f ∈ L1(Z). Let us note that (9.5) applied to |f |and φ = 1Y readily yields (9.4).

9.1. Distance function. We recall that on G one defines a K-bi-invariant distance function g 7→ ‖g‖G by ‖k exp X‖G = ‖X‖ for k ∈ Kand X ∈ p. In geometric terms, ‖g‖G is the Riemannian distance ofKg ∈ K\G from the origin. It is well known that

‖xy‖G ≤ ‖x‖G + ‖y‖G

for all x, y ∈ G. The norm ‖g‖ used in (6.2) relates as log ‖g‖ = c‖g‖G

with c > 0.On Z we recall the polar decomposition (4.1) and define similarly

‖[k, X]‖Z = ‖X‖ for k ∈ K and X ∈ q ∩ p. We record the following

Lemma 9.1. Let z = gH ∈ Z. Then ‖z‖Z = infh∈H ‖gh‖G.

Proof. (cf [31], Lemma 5.4). It suffices to prove that ‖ exp(X)h‖G ≥‖X‖ for X ∈ q ∩ p, h ∈ H, and by Cartan decomposition of H, wemay assume h = exp(T ) with T ∈ h ∩ p. Thus we have reduced to thestatement that

‖ exp(X) exp(T )‖G ≥ ‖ exp(X)‖G

for X ⊥ T in p. This follows from the fact that the sectional curvaturesof K\G are ≤ 0 (see [28], p. 73). ¤


In particular, it follows that

(9.6) ‖gz‖Z ≤ ‖z‖Z + ‖g‖G (z ∈ Z, g ∈ G).

9.2. Generalized balls. The problem of counting lattice points in Zleads to a natural question of exhibiting families of compact subsetsthat exhaust Z. We introduce here two families of generalized balls.

We define the intrinsic ball of radius R > 0 on Z by

BR := z ∈ Z | ‖z‖Z < R .

Note that the notation BR was used differently in Section 4. It fol-lows from (9.6) that for each r > 0 there exists a K × K-invariantneighborhood U of 1 in G such that

(9.7) gBR ⊂ BR+r

for all g ∈ U and all R > 0.We are interested in the volume of BR, and shall write it |BR|. It

follows easily from the lower bounds in Section 4 that |BR| ∞ asR → ∞ (in fact |BR| ≥ CRdim p∩q). On the other hand, we have thefollowing upper bound.

Lemma 9.2. There exists a constant c > 0 such that

(9.8) |BR+r| ≤ ecr|BR|for all R ≥ 1, r ≥ 0.

Proof. It follows from the formula for the invariant integral with respectto (4.1) that

|BR| =∫

X∈q∩p,‖X‖<R

J(X) dX

where J(X) = J1(X).Hence it suffices to prove that there exists c > 0 such that

∫ R+r

0

J(tX)tl−1 dt ≤ ecr

∫ R

0

J(tX)tl−1 dt

for all X ∈ q ∩ p with ‖X‖ = 1. Here l = dim q ∩ p. Equivalently, thefunction

R 7→ e−cR

∫ R

0

J(tX)tl−1 dt

is decreasing, or by differentiation,

J(RX)Rl−1 ≤ c

∫ R

0

J(tX)tl−1 dt

for all R. The latter inequality is established in [23, Lemma A.3] withc independent of X. ¤


We write 1R ∈ L1(Z) for the characteristic function of BR and deducefrom the definitions and (9.5):

• 1ΓR(eΓ) = NR(Γ, Z) := #γ ∈ Γ/ΓH | γ.z0 ∈ BR.

• ‖1ΓR‖L1(G/Γ) = |BR|.

9.2.1. Other balls. A second family of generalized balls can be con-structed by embedding ι : Z → V in a vector space V by choosing afinite dimensional representation (ρ, V ) of G with H the stabilizer of avector in V (see Lemma 5.7). We fix a K-invariant norm ρ on V anddefine

BR,ρ = z ∈ Z : ρ(ι(z)) < Rto be the intersection of the ball of radius R in V with our subvarietyZ. We write |BR,ρ| for the volumes of these balls in Z, and note thatproperties of these functions of R are established in [23, Appendix 1].

Formulas similar to the bulleted hold for the corresponding charac-teristic functions 1R,ρ.

9.3. Factorization of balls. Let us call Z = G/H rigid if any fac-torization is compact modulo the center Z of G, i.e. if Z → Z∗ is afactorization of Z, then H∗Z/HZ is compact. Note that all irreduciblesymmetric spaces are rigid.

In case Z is not rigid, then, with regard to the lattice count, we limitour consideration to balls which satisfy natural additional properties.In the sequel UR either stands for BR or BR,ρ.

Let Z be non-rigid and F → Z → Z∗ be a factorization with fiberspace F := H∗/H. We write U∗

R for the image of UR under the factor-ization map, i.e. U∗

R = URH∗/H∗. Further we set UFR := F ∩ UR.

For a compactly supported bounded measurable function φ on Z wedefine the fiberwise integral

φF (gH∗) :=

∫

H∗/H

φ(gh∗) d(h∗H)

and recall the integration formula

(9.9)

∫

Z

φ(gH) d(gH) =

∫

Z∗φF (gH∗) d(gH∗)

under appropriate normalization of measures. Consider the character-istic function 1R of UR and note that its fiber average 1F

R is supported inthe compact ball U∗

R. We say that the family of balls (UR)R>0 factorizeswell provided for all compact subsets Q ⊂ G

(9.10) limR→∞

supg∈Q 1FR(gH∗)

|UR| = 0


holds for all factorizations. For our balls UR = BR, BR,ρ one readilyobtains for all compact subsets Q an R0 = R0(Q) > 0 such that

supg∈Q

1FR(gH∗) ≤ |UF

R+R0| .

by (9.7). Thus the balls UR factorize well provided

limR→∞

|UFR+R0

||UR| = 0 .

for all R0 > 0. In practice, this appears always to be satisfied.

Example 9.3. (a) Let us consider geometric balls in a triple spaceZ = G×G×G/∆(G). Write BG

R for the geometric ball in G. Then

BR = (BGR ×BG

R ×BGR)∆(G) ⊂ Z

by Lemma 9.1. We identify Z with G×G via the coordinate map

G×G → Z, (g1, g2) 7→ (g1, g2,1)∆(G)

and within this identification we have BR = (BGR×BG

R)∆(BGR) ⊂ G×G.

Consider H∗ = (g, h, h) | g, h ∈ G and note that H∗/H identifieswith G× 1 ⊂ Z. In particular BF

R = BG2R. Thus (BR)R>0 factorizes

well provided

limR→∞

volG BG2R+R0

volG×G(BGR ×BG

R)∆(BGR)

= 0

for all R0 > 0. With Fubini the denominator is volG BG3R and thus the

balls factorize well.(b) More generally, all geometric balls for the spaces listed in the

introduction factorize well. These cases fit in a more general pattern.Suppose the factorization H∗/H → G/H → G/H∗ has the followingproperties:

(1) There exist a strongly unimodular spherical pair (P, H) for Gsuch that G = KAH.

(2) With A∗ = A ∩ H∗, M∗ = M ∩ H∗, N∗ = N ∩ H∗ one getsa spherical pair (P ∗, H) for H∗ with P ∗ = M∗A∗N∗ such thatH∗ = K∗A∗H.

Under this conditions the geometric balls BR factorize well. Indeed letus denote by J : a → R≥0 the Jacobian of the polar map K × A → Zand likewise we define J∗. Then |BR| =

∫‖X‖≤R J(X) dX and an

according formula for the volume of the fibered balls BFR = BR ∩ H∗.

Now for X ∈ a+ away from the walls one readily gets the asymptotic


behavior J(X) ∼ e2ρn(X) and likewise J∗(X) ∼ e2ρn∗ (X). The assertionfollows.

9.4. Weak asymptotics. Under the assumption that Z = G/H issymmetric and H/ΓH is compact the main result in [20] reads as

(MTρ) NR,ρ(Γ, Z) ∼ |BR,ρ| (R →∞).

The compactness assumption was later dropped in [22] where an er-godic proof was presented.

For spaces with property (I) and Y compact we show in the followingsection that (MTρ), as well as the corresponding result for the intrinsicballs BR,

(MT) NR(Γ, Z) ∼ |BR| (R →∞),

can be deduced from VAI, Theorem 2.2.With notation from (9.3) we set

F ΓR :=

1

|BR|1ΓR, and F Γ

R,ρ :=1

|BR,ρ|1ΓR,ρ (R > 0) .

We shall concentrate in verifying the following limits of weak type:

(wMTρ) 〈F ΓR,ρ, φ〉 →

∫

Y

φ dµY (R →∞), (∀φ ∈ C0(Y ))

(wMT) 〈F ΓR , φ〉 →

∫

Y

φ dµY (R →∞), (∀φ ∈ C0(Y )) .

Lemma 9.4. (wMTρ) ⇒ (MTρ) and (wMT) ⇒ (MT).

Proof. The first implication is shown in [20] Lemma 2.3 for Z a sym-metric space. The proof is based on a property of the action of G onthe family of balls under consideration, derived in an appendix of [20].The same property is derived for general reductive homogenous spacesin [23, Appendix 1] (in [22] the relevant property is denominated aswell-roundedness). It follows that (wMTρ) implies (MTρ) in general.The second implication is proved similarly from (9.7) and (9.8). ¤

10. Counting lattice points, II: Main term analysis

In this section we will establish main term counting under the man-date of property (I) and Y being compact.

In Section 9.3 we defined the notions of non-rigid spaces and wellfactorizable balls therein. To have a uniform notation, we understandthat both families (BR)R>0 and (BR,ρ)R>0 are well factorizable in caseZ is rigid.


10.1. Main Theorem on main term counting.

Theorem 10.1. Let G be reductive and H a closed subgroup such thatZ = G/H is of reductive type. Suppose that Y is compact and Z admits(I). Then the following assertion hold true:

(1) If (BR)R>0 factorizes well, then (wMT) and (MT) hold.(2) If (BR,ρ)R>0 factorizes well, then (wMTρ) and (MTρ) hold.

By combining with Theorem 8.5 one obtains main term counting forsymmetric spaces, as in [20], but also for the geometric balls for thespaces listed in the introduction.

The proof is based on the following lemma. For a function spaceF(Y ) consisting of integrable functions on Y we denote by F(Y )o thesubspace of functions with vanishing integral.

Proposition 10.2. Let Z = G/H be of reductive type. Assume thatthere exists a dense subspace A(Y ) ⊂ Cb(Y )K

o such that

(10.1) φH ∈ C0(Z) for all φ ∈ A(Y ) .

Then (wMT) and (wMTρ) hold.

Proof. We concentrate on (wMT) and establish it for φ ∈ Cb(Y ). AsCb(Y ) = Cb(Y )o⊕C1Y , and (wMT) is trivial for φ a constant, it sufficesto establish

(10.2) 〈F ΓR , φ〉 → 0 (φ ∈ Cb(Y )o) .

We will show (10.2) is valid for φ ∈ A(Y ). By density, as F ΓR is K-

invariant and has norm one in L1(Y ), this will finish the proof.Let φ ∈ A(Y ) and let ε > 0. By the unfolding identity (9.5) we have

(10.3) 〈F ΓR , φ〉L2(Y ) =

1

|BR|〈1R, φH〉L2(Z).

Using (10.1) we choose Kε ⊂ Z compact such that |φH(z)| < ε outsideof Kε. Then

1

|BR| 〈1R, φH〉L2(Z) =

∫

Kε

+

∫

Z−Kε

1R(z)

|BR| φH(z) dµZ(z) .

By (9.2), the first term is bounded by |Kε|||φ||∞|BR| , which is ≤ ε for R

sufficiently large. As the second term is bounded by ε for all R, weobtain (10.2). Hence (wMT) holds. Exactly the same argument gives(wMTρ). ¤


Remark 10.3. It is possible to replace (10.1) by a weaker requirement:Suppose thatA(Y ) =

∑j∈J A(Y )j whereA(Y )j is a subspace for which

the φH ∈ A(Y )j factorize to an invariant function φH∗j . Suppose that

(10.4) φH∗j ∈ C0(Z

∗j ) (φ ∈ A(Y )j)

holds for all j ∈ J . Then the conclusion in Lemma 10.2 is still valid,provided the relevant balls BR or BR,ρ factorize well. In fact, using(9.9) the last part of the proof modifies to (with H∗ = H∗

j to simplifynotation):

1

|BR| 〈1R, φH〉L2(Z) =1

|BR| 〈1FR, φH∗〉L2(Z∗) =

=

∫

K∗ε

+

∫

Z∗−K∗ε

1FR(z)

|BR| φH∗(z) dµZ(z) .

As ‖1FR‖L1(Z∗) = |BR|, the second term is bounded by ε for all R, and

the first term we get as small as we wish with (9.10)

10.2. The space A(Y ). We now construct a specific subspace A(Y ) ⊂Cb(Y )K

0 for and verify condition (10.1).As Y is compact, the abstract Plancherel-theorem reads:

L2(G/Γ)K '⊕

π∈Gs

Vπ,Γ

where Vπ,Γ ⊂ (H−∞π )Γ is a subspace of finite dimension accounting for

multiplicities. If we denote the Fourier transform by f 7→ f∧ then thecorresponding inversion formula is given by

(10.5) f =∑

π

mvπ,f∧(π)

with vπ ∈ Hπ normalized K-fixed and f∧(π) ∈ Vπ,Γ. The matrix coef-ficients for Y are defined as in (6.3), and the sum in (10.5) is requiredto include multiplicities.

Note that L2(Y ) = L2(Y )o ⊕ C · 1Y . We define A(Y ) ⊂ L2(Y )Ko to

be the dense subspace of functions with finite Fourier support. ThenA(Y ) ⊂ L2(Y )∞ is dense and since C∞(Y ) and L2(Y )∞ are topologi-cally isomorphic, it follows that A(Y ) is dense in C(Y )K

o as required.

Lemma 10.4. Assume that Y is compact and Z has (I), and defineA(Y ) as above. Then condition (10.1) holds if Z is rigid, and otherwise(10.4).

Proof. Assume first that Z is rigid. It is no loss of generality to assumethat the center of G is compact. The map φ 7→ φH corresponds on the


spectral side to a map (H−∞π )Γ → (H−∞

π )H , which can be constructedas follows.

As H/ΓH is compact, we can define

(10.6) Λπ : (H−∞π )Γ → (H−∞

π )H , Λπ(η) =

∫

H/ΓH

η π(h−1) d(hΓH)

by H−∞π -valued integration: the defining integral is understood as in-

tegration over a compact fundamental domain F ⊂ H with respect tothe Haar measure on H; as the integrand is continuous and H−∞

π is acomplete locally convex space, the integral converges in H−∞

π .Let φ ∈ A(Y ), then it follows from (10.5) that

(10.7) φH =∑

π 6=1

mvπ,Λπ(φ∧(π)) .

As Z has property (I), the finite sum in (10.7) belongs to Lp(G/H) forsome 1 ≤ p < ∞ (to be precise modulo the center). We conclude thatφH ∈ Lp(G/H), and from Lemma 8.2 that φH ∈ Lp(G/H)∞. In viewof VAI (Theorem 2.2), the claim now follows.

In case Z is not rigid we let J denote the set of all factorizations Z∗ →Z (including also Z∗ = Z) and define A(Y )j for j ∈ J accordingly toaccommodate summands in (10.7) for which Hη = H∗

j . ¤10.2.1. The example of Eskin-McMullen. We end this section by treat-ing another class of non-symmetric spaces Z = G/H for which thelattice density along the intrinsic balls satisfies (MT). The theory in-cludes the spaces which were introduced by Eskin and McMullen [22]as counterexamples to (MTρ), see Example 10.6 below.

We are interested in spaces Z = G/H for which the normalizerNK(H) of H in K is not contained in H. In this case

Z ′ := G/H ′, H ′ := HNK(H) ,

is a factorization of Z.

Proposition 10.5. Let Z ′ = G/H ′ be as above. If Y is compact, Z ′

admits (I), and the balls BR in Z factorize well, then (wMT) and (MT)hold for Z = G/H.

Proof. Letting NK(H) act on Z = G/H from the right we obtain anaction of the extended group G = G×NK(H) on Z such that

Z ' G/H, H = (hk, k) | h ∈ H, k ∈ NK(H).The G-invariant measure on Z agrees with the G-invariant, by unique-ness of the latter. Furthermore, as q ∩ p and its norm are preserved inthe adjoint action of NK(H), the intrinsic balls BR of Z for G and G


are identical (note that in contrast the other type of balls BR,ρ are not

defined for G, unless the H-fixed vector of ρ is also fixed by NK(H)).We claim that Z ′ = G/H ′ admits (I) if and only if Z = G/H admits

it. Let K = K × NK(H), then an irreducible unitary representationπ ⊗ δ of G has a non-trivial K-fixed vector only if δ is trivial. Thisreduces the verification of (I) for G/H to representations π of G with(H−∞

π )H′ 6= 0, and the claim follows. The proposition now follows fromTheorem 10.1 ¤

We end this section by providing some examples where Proposition10.5 can be applied.

Example 10.6. Consider the case of G = SL(2,C) and H < G thesubgroup of real diagonal matrices. Hence H ′ = HNK(H) is the sub-group of all diagonal matrices, so that G/H ′ is a symmetric space

The lattice Γ is a conjugate of SL(2,Z[i]) so that H/ΓH is compact.In view of Proposition 10.5 and the result of the next section, it followsthat (MT) holds.

This example was considered in [22] and [23], where it is shown that(MTρ) fails for a particular finite dimensional representation ρ. Thisshows that (wMTρ) is false in general for the situation in Theorem10.5.

Example 10.7. Let G be a real reductive group and let H = e.Then Z = G is a homogeneous space for the left action of G. In thiscase NK(H) = K and Z ′ = G/K is symmetric, hence admits (I). Hence(MT) holds for every cocompact lattice in G.

11. Counting lattice points, III: Error term analysis

In this section we assume that Z is symmetric. However, we wishto point out that this assumption enters only in the key Lemma 11.5.For simplicity we also assume that G is simple. In addition we requestthat the cycle H/ΓH ⊂ Y is compact. This technical condition ensuresthat the integration map

(11.1) Λπ : (H−∞π )Γ → (H−∞

π )H , Λπ(η) =

∫

H/ΓH

η π(h−1) d(hΓH)

considered in (10.6) is defined. In the sequel we use the Planchereltheorem (see [27])

L2(G/Γ)K '∫ ⊕

Gs

Vπ,Γ dµ(π) ,


where Vπ,Γ ⊂ (H−∞π )Γ is a finite dimensional subspace and of constant

dimension on each connected component in the continuous spectrum(parametrization by Eisenstein series), and where the Plancherel mea-sure µ has support

Gµ := supp(µ) ⊂ Gs.

The first error term for (wMT) can be expressed by

err(R, Γ) := supφ∈Cb(Y )

‖φ‖∞≤1

|〈F ΓR − 1Y , φ〉| (R > 0),

and our goal is to give an upper bound for err(R, Γ) as a function ofR.

Remark 11.1. In the literature results are sometimes stated with re-spect to the pointwise error term errpt(R, Γ) = |F Γ

R(1) − |BR||. Notethat

errpt(R, Γ) ≤ supφ∈L1(Y )‖φ‖1≤1

|〈F ΓR − 1Y , φ〉| (R > 0).

The Sobolev estimate ‖φ‖∞ ≤ C‖φ‖1,k, for K-invariant functions φ onY and with k = dim Y/K the Sobolev shift, then relates these errorterms.

According to the decomposition Cb(Y ) = Cb(Y )o ⊕ C1Y we decom-pose functions as φ = φo + φ1 and obtain

err(R, Γ) = supφ∈Cb(Y )

‖φ‖∞≤1

|〈F ΓR , φo〉| = sup

φ∈Cb(Y )

‖φ‖∞≤1

|〈1R, φHo 〉|

|BR| .

Further, from ‖φo‖∞ ≤ 2‖φ‖∞ we obtain that err(R, Γ) ≤ 2 err1(R, Γ)with

err1(R, Γ) := supφ∈Cb(Y )o‖φ‖∞≤1

|〈1R, φH〉||BR| .

11.1. Lp bounds on generalized matrix coefficients of H-distinguishedrepresentations. Let (π,Hπ) be a non-trivial unitary irreducible spher-ical representation of G. Then by Lemma 8.5 it follows that there existsa smallest index 1 ≤ pH(π) < ∞ such that all K-finite generalized ma-trix coefficients mv,η with η ∈ (H−∞

π )H (see (6.3)) belong to Lp(Z) forany p > pH(π).

The representation π is said to be H-distinguished if (H−∞π )H 6= 0.

Note that if π is not H-distinguished then pH(π) = 1. We say that πis H-tempered if pH(π) = 2.

Given a lattice Γ ⊂ G we define

pH(Γ) := suppH(π) : π ∈ Gµ, π 6= 1


and record the following.

Lemma 11.2. pH(Γ) < ∞.

Proof. This follows from [18] in the cases where G has property (T).The remaining cases are SOe(n, 1) and SU(n, 1) (up to covering), forwhich the result is well known. ¤Remark 11.3. (a) It is not difficult to relate the finiteness of pH(Γ)to the notion of “spectral gap” which is common in ergodic theory (see[21] or [30]).(b) In many cases one expects a Ramanujan-Selberg property, see [44].In particular one expects pH(Γ) = 2 in these cases.

Recall the Cartan-Killing form κ on g = k + p and choose a basisX1, . . . , Xl of k and X ′

1, . . . , X′s of p such that κ(Xi, Xj) = −δij and

κ(X ′i, X

′j) = δij. With that data we form the standard Casimir element

C := −l∑

j=1

X2j +

s∑j=1

(X ′j)

2 ∈ U(g) .

Set ∆K :=∑l

j=1 X2j ∈ U(k) and obtain the commonly used Laplace

element

(11.2) ∆ = C + 2∆K ∈ U(g)

which acts on Y = G/Γ from the left.Let d ∈ N. For 1 ≤ p ≤ ∞, it follows from [10], Section 2, that

Sobolev norms on Lp(Y )∞ ⊂ C∞(Y ) can be defined by

||f ||2p,2d =d∑

j=0

||∆jf ||2p .

Basic spectral theory allows to declare ‖ · ‖p,d for any d ≥ 0.Let us define

s := dim p = dim G/K = dim Γ\G/K

andr := dim a = rankR(G/K) ,

where a ⊂ p is maximal abelian.

Proposition 11.4. Let Z = G/H and p > pH(Γ). The map

AvH : C∞b (Y )K

o → Lp(Z)K ; AvH(φ) = φH

is continuous. More precisely, for all

(1) k > s + 1 if Y is compact.


(2) k > r+12

s + 1 if Y is non-compact and Γ is arithmetic

there exists a constant C = C(p, k) > 0 such that

‖φH‖p ≤ C‖φ‖∞,k (φ ∈ C∞b (Y )K

o )

Before we prove the proposition we need a fiberwise estimate. Given

π ∈ Gµ we consider the functions φ = φπ ∈ C∞b (Y )K which are gener-

alized matrix coefficients

(11.3) φπ(gΓ) := mv,η(gΓ) = η(π(g−1)v), (g ∈ G)

with v ∈ H∞π and η ∈ (H−∞

π )Γ.

Let 1 ≤ p < ∞. Let us say that a subset Λ ⊂ Gs is Lp(Z)-boundedprovided that mv,η ∈ Lp(Z) for all π ∈ Λ and v ∈ H∞

π , η ∈ (H−∞π )H .

The proof of the following Lemma is postponed to Appendix C.

Lemma 11.5. Suppose that Λ ⊂ Gs is Lp(Z)-bounded for some 1 ≤p < ∞. Then there exists C > 0 such that

‖mv,η‖p ≤ C‖mv,η‖∞for all π ∈ Λ, η ∈ (H−∞

π )H and v ∈ HKπ .

As a consequence we obtain that:

Lemma 11.6. Let p > pH(Γ). Then there exists C > 0 such that

‖φHπ ‖p ≤ C‖φπ‖∞

for all π ∈ Gµ, v ∈ HKπ , η ∈ (H−∞

π )Γ, with φπ given by (11.3).

Proof. Recall from (9.2), that integration is a bounded operator fromL∞(Y ) → L∞(Z). Hence the assertion follows from the previouslemma. ¤Proof. (Proposition 11.4) For all π ∈ G the operator dπ(C) acts as ascalar which we denote by

|π| := dπ(C) .

Let φ ∈ C∞b (Y )K

o and write φ = φd+φc for its decomposition in discreteand continuous Plancherel parts. We assume first that φ = φd.

In case Y is compact we have Weyl’s law: There is a constant cY > 0such that ∑

|π|≤R

m(π) ∼ cY Rs/2 (R →∞) .

Here m(π) = dimVπ,Γ. We conclude that

(11.4)∑

π

m(π)(1 + |π|)−k < ∞


for all k > s/2 + 1. In case Y is non-compact, we let Gµ,d be thethe discrete support of the Plancherel measure. Then assuming Γ isarithmetic, the upper bound in [29] reads:

∑

π∈Gµ,d|π|≤R

m(π) ≤ cY Rrs/2 (R > 0) .

For k > rs/2 + 1 we obtain (11.4) as before.As φ is in the discrete spectrum we decompose it as φ =

∑π φπ and

obtain with Lemma 11.6

‖φH‖p ≤∑

π

‖φHπ ‖p ≤ Cp

∑π

‖φπ‖∞ .

In these sums representations occur according to their multiplicitiesm(π). The last sum we estimate as follows:

∑π

‖φπ‖∞ =∑

π

(1 + |π|)−k/2(1 + |π|)k/2‖φπ‖∞

≤ C∑

π

(1 + |π|)−k/2‖φπ‖∞,k

with C > 0 a constant depending only on k. Applying the Cauchy-Schwartz inequality combined with (11.4) we obtain

‖φH‖p ≤ C( ∑

π

‖φπ‖2∞,k

) 12

with C > 0 (we allow universal positive constants to change from lineto line).

To finish the proof we apply the Sobolev lemma on K\G. HereSobolev norms are defined by the central operator C, whose actionagrees with the left action of ∆. It follows that ‖f‖∞ ≤ C‖f‖2,k1 withk1 > s

2for K-invariant functions f on G. This gives

‖φH‖p ≤ C(∑

π

||φπ||22,k+k1)

12 = C||φ||2,k+k1 ≤ C||φ||∞,k+k1

which proves the proposition for the discrete spectrum.If φ = φc belongs to the continuous spectrum, where multiplicities

are bounded, the proof is simpler. Let µc be the restriction of thePlancherel measure to the continuous spectrum. As this is just Eu-clidean measure on r-dimensional space we have

(11.5)

∫

Gs

(1 + |π|)−k dµc(π) < ∞


if k > r/2. We assume for simplicity in what follows that m(π) = 1 forall π ∈ supp µc. As supπ∈supp µc

m(π) < ∞ the proof is easily adaptedto the general case.

Let

φ =

∫

Gs

φπ dµc(π).

As ‖φH‖∞ ≤ ‖φ‖∞ we conclude with Lemma 11.6, (11.5) and Fubini’stheorem that

φH =

∫

Gs

φHπ dµc(π)

and, by the similar chain of inequalities as in the discrete case

‖φH‖p ≤ C‖φ‖∞,k+k1

with k > r2

and k1 > s2. This concludes the proof. ¤

11.2. Smooth versus non-smooth counting. Like in the classicalGauss circle problem one obtains much better estimates for the remain-der term if one uses a smooth cutoff. Let α ∈ C∞

c (G) be a non-negativetest function with normalized integral. Set 1R,α := α ∗ 1R and define

errα(R, Γ) := supφ∈Cb(Y )Ko‖φ‖∞≤1

|〈1R,α, φH〉||BR| .

Lemma 11.7. Let k > s+1 if Y is compact and k > r+12

s+1 otherwise.Let p > pH(Γ). Then there exists C > 0 such that

errα(R, Γ) ≤ C‖α‖1,k|BR|−1p

for all R ≥ 1 and all α ∈ C∞c (G).

Proof. First note that

〈1R,α, φH〉 = 〈1R,α, (−1 + ∆)k/2(−1 + ∆)−k/2φH〉 .

With ψ = (−1 + ∆)−k/2φ we have ‖ψ‖∞,k ≤ C‖φ‖∞ for some C > 0.We thus obtain

errα(R, Γ) ≤ C supψ∈Cb(Y )Ko‖ψ‖∞,k≤1

|〈1R,α, (−1 + ∆)k/2ψH〉||BR| .


Moving (−1+∆)k/2 to the other side we get with Proposition 11.4 andHolder’s inequality that

errα(R, Γ) ≤ C supψ∈Cb(Y )Ko‖ψ‖∞,k≤1

|〈(−1 + ∆)k/2α ∗ 1R, ψH〉||BR|

≤ C‖(−1 + ∆)k/2α ∗ 1R||q

|BR|where q is the conjugate exponent satisfying 1

p+ 1

q= 1. Finally,

‖(−1 + ∆)k/2α ∗ 1R‖q ≤ C‖α‖1,k‖1R‖q

and the Lemma follows. ¤

Finally we have to compare err1(R, Γ) with errα(R, Γ). For that wenote that

| err1(R, Γ)− errα(R, Γ)| ≤ supφ∈Cb(Y )Ko‖φ‖∞≤1

|〈1ΓR,α − 1Γ

R, φ〉||BR| .

Suppose that supp α ⊂ BGε for some ε > 0 with the superscript G

indicating that we take the balls in the reductive space “G”. Then(9.7) implies that 1R,α is supported in BR+ε, and hence

|〈1ΓR,α − 1Γ

R, φ〉| ≤ ‖1ΓR,α − 1Γ

R‖1

≤ ‖1R,α − 1R‖1

≤ |BR+ε| 12‖1R,α − 1R‖2

≤ |BR+ε| 12 |BR+ε\BR| 12 .

From (9.8) we have

|BR+ε\BR| ≤ Cε|BR| (R ≥ 1, ε < 1) .

Thus we obtain that

| err1(R, Γ)− errα(R, Γ)| ≤ Cε12 .

Combining this with the estimate in Lemma 11.7 we arrive at theexistence of C > 0 such that

err1(R, Γ) ≤ C(ε−k|BR|−1p + ε

12 )

for all R ≥ 1 and all 0 < ε < 1. The minimum of the function

ε 7→ ε−kc + ε1/2 is attained at ε = (2kc)2

2k+1 and thus we get:


Theorem 11.8. The first error term err(R, Γ) for the lattice countingproblem on Z = G/H can be estimated as follows: for all p > pH(Γ)and k > s+1 for Y compact, resp. k > r+1

2s+1 otherwise, there exists

a constant C = C(p, k) > 0 such that


(2k+1)p

for all R ≥ 1.

Remark 11.9. The point where we loose essential information is inthe estimate (11.4) where we used Weyl’s law. In the moment point-wise multiplicity bounds are available the estimate would improve. Tocompare the results with Selberg on the hyperbolic disc, let us as-sume that pH(Γ) = 2. Then with r = 1 and s = 2 our bound is

err(R, Γ) ≤ Cε|BR|− 114

+ε while Selberg showed err(R, Γ) ≤ Cε|BR|− 13+ε.

12. Counting lattice points IV: Triple spaces

In this final section we reveal an explicit study of the triple caseZ = PSl(2,R)3/ diag(PSl(2,R)). We present an alternative approachtowards (I) and establish the key Lemma 11.5, which implies error termcounting.

Let us define subgroups of G0 = PSl(2,R) ' SOe(2, 1):

A0 =

at =

(t 00 1

t

)| t > 0

,

B0 =

bs =

(cosh s sinh ssinh s cosh s

)| s ∈ R

,

N0 = (

1 x0 1

)| x ∈ R

,

and P0 = A0N0. We note that G/H is known to be a multiplicity onespace, that is, the space (H−∞)H of H-fixed distribution vectors is at

most one-dimensional for all π ∈ G (see [41]). This property is closelyrelated to the fact that there is only one open P = P 3

0 -orbit on Z.Indeed, if

w0 =

(0 −11 0

)and s0 =

1√2

(1 −11 1

)

and z0 = (1, w0, s0), then

Pz0H

is open and dense in G: to see that, observe that P\G = P1(R)3 andthat H acts transitively on all transversal triples.


This suggests the following definition

A = A0 × A0 ×B0

and, indeed, we have already established in Proposition 5.3 that G =KAH.

12.1. The invariant measure on Z. For a general semisimple groupG0 we note that the map

(12.1) G0 ×G0 → Z, (g1, g2) 7→ (g1, g2,1)H

is a G0 × G0-equivariant diffeomorphism. Accordingly the invariantmeasure on Z identifies with the Haar measure on G0 ×G0.

For G0 = PSl(2,R) we can actually compute the Haar-measure interms of the KAH-coordinates. Let us identify A with R3 via the map

(t1, t2, s) 7→ (at1 , at2 , bs) .

Then

(12.2) dµZ(gH) = J(t1, t2, s) dk dt1 dt2 ds .

Lemma 12.1. J(t1, t2, s) = | sinh(2(t1 − t2)) cosh(2s)|Proof. On G0×G0 we use the formula for integration in HAK coordi-nates for the symmetric space diag(G0)\G0 ×G0 = G0: The map

G0 × A0 ×K0 ×K0 → G0 ×G0

defined by(g, at, k1, k2) 7→ (g, g)(at/2, a−t/2)(k1, k2)

is a parametrization and the Haar measure on G0 ×G0 writes as

| sinh(2t)|dg dat dk1 dk2 .

Note that the “half” is swallowed in the identification (x, y) 7→ x−1y ofdiag(G0)\G0 ×G0 with G0.

Further we decompose G0 = diag(G0) by means of the KAH coor-dinates for the symmetric space G0/A0: the map

K0 ×B0 × A0 → G0, (k, bs, a) 7→ kbsa

givesdg = cosh(2s) dk dbs da .

Combining, we have the coordinates (kbsau+t/2k1, kbsau−t/2k2) on G0×G0, with Jacobian | sinh(2(t1 − t2)) cosh(2s)| where we have writtent1 = u + t/2 and t2 = u− t/2.

Finally we identify G0 ×G0 with diag(G0)\G0 ×G0 ×G0 by

(g1, g2) 7→ diag(G0).(g1, g2,1)


so that the above coordinates correspond to

diag(G0).(at1 , at2 , b−s)(k1, k2, k−1)

¤12.2. The key lemma. As H is not maximal we first need a modified

notion for an Lp-bounded subset Λ ⊂ Gs\1 (compare the definitiongiven above Lemma 11.5): we need to request that mv,η ∈ Lp(G/Hη)instead of mv,η ∈ Lp(G/H). Having said that, the key lemma reads:

Lemma 12.2. Let Z = G30/ diag(G0) and G0 = PSl(2,R). Suppose

that Λ ⊂ Gs is Lp-bounded for some 1 ≤ p < ∞. Then there existsC > 0 such that

‖mv,η‖p ≤ C‖mv,η‖∞for all π ∈ Λ, η ∈ (H−∞


Proof. We write Λ0 := Λ ∩ [G0s\1]3 and Λ1 := Λ\Λ0.Let (π,Hπ) be a non-trivial K-spherical unitary representation of G.

Then π = π1 ⊗ π2 ⊗ π3 with each factor a K0-spherical unitary repre-sentation of G0. We assume that π has non-trivial H-fixed distributionvectors, then at least two of the factors πi are non-trivial.

Let vi be normalized K0-fixed vectors of πi and set v = v1⊗ v2⊗ v3.Since Z is a multiplicity one space, the functional I ∈ (H−∞

π )H isunique up to scalars. Our concern is with the Lp-integrability of thegeneralized matrix coefficient fπ := mv,I :

fπ(g1, g2, g3) := I(π1(g1)−1v1 ⊗ π2(g2)

−1v2 ⊗ π3(g3)−1v3) ,

when π belongs to Λ. Consider first the case where π ∈ Λ1, i.e. one πi

is trivial, say π1. Then π2 = π∗3. Viewed as a function on G × G ' Zby means of (12.1), fπ becomes

fπ(g1, g2) = 〈π2(g2)v2, v2〉 .This function is constant on the first factor and by assumption Lp-integrable on the second. As the key lemma is satisfied for the sym-metric space G0/K0, the assertion follows for π ∈ Λ1.

Suppose now that π ∈ Λ0, i.e. all πi are non-trivial. In order toanalyze fπ we use G = KAH and thus assume that g = a = (a1, a2, b) ∈A. We work in the compact model of Hπi

= L2(S1) and use the explicitmodel for I in [11]: for f1, f2, f3 smooth functions on the circle one has

I(f1 ⊗ f2 ⊗ f3) =1

(2π)3

∫ 2π

0

∫ 2π

0

∫ 2π

0

f1(θ1)f2(θ2)f3(θ3)··K(θ1, θ2, θ3) dθ1dθ2dθ3 ,


where

K(θ1, θ2, θ3) = | sin(θ2−θ3)|(α−1)/2| sin(θ1−θ3)|(β−1)/2| sin(θ1−θ2)|(γ−1)/2 .

In this formula one has α = λ1 − λ2 − λ3, β = −λ1 + λ2 − λ3 andγ = −λ1 − λ2 + λ3 where λi ∈ iR ∪ (−1/2, 1/2) are the standardrepresentation parameters of πi. We note that when the λi are real, wehave a free choice of their sign, so that it always can be arranged thatK is integrable.

Returning to our analysis of fπ we now take f1(θ1) = [π1(at1)v1](θ1),f2(θ2) = [π2(at2)v2](θ2) and f3 = [π3(bs)v3](θ3). Then

f1(θ1) =1

(t21 + sin2 θ1(1t21− t21))

12(1+λ1)

and likewise formulas for f2 and f3.A simple computation then yields the existence of constants ci =

ci(π) > 0, C = C(π) > 0 depending on π only through the distance ofRe λi to the trivial representation, such that

|fπ(at1 , at2 , bs)| ≤ C1

[cosh log t1]c1 · [cosh log t2]c2 · [cosh s]c3.

This bound is essentially sharp. Hence, in view of the explicit form(12.2) of the invariant measure in KAH-coordinates, it follows that

infπ∈Λ0

ci(π) > 0 and infπ∈Λ0

C(π) > 0 .

In particular we get that

supπ∈Λ0

‖fπ‖p < ∞ .

On the other hand for g = 1 = (1,1,1), the value fπ(1) is obtained byapplying I to the constant function 1 = 1⊗1⊗1. This value has beencomputed explicitly by Bernstein and Reznikov in [12]. The result isthat I(1) is quotient of Γ-functions which, after Stirling approximation,yields a lower bound for fπ(1):

infπ∈Λ0

|fπ(1)| > 0 .

As ‖fπ‖∞ ≥ |fπ(1)| the proof is finished. ¤Remark 12.3. Let us emphasize that the same proof is applicable tothe case where G0 = SOe(1, n) (see [17] for the generalization of [12]).

With the key Lemma 12.2 we obtain as in Section 11 (see Theorem11.8) the following error term bound:


Theorem 12.4. Let Z = G30/ diag(G0) for G0 = SOe(1, n) and assume

that H/ΓH is compact. Then the first error term err(R, Γ) for thelattice counting problem on Z = G/H can be estimated as follows: forall p > pH(Γ) there exists a C = C(p) > 0 such that


(6n+3)p

for all R ≥ 1.

Appendix A. Proof of Theorem 2.2

In this appendix we prove that VAI does not hold on any homo-geneous space Z = G/H of G, which is not of reductive type. Wemaintain the assumption that G is a real reductive Lie group and es-tablish the following result.

Proposition A.1. Assume that H ⊂ G is a closed connected subgroupsuch that Z = G/H is unimodular and not of reductive type. Then forall 1 ≤ p < ∞ there exists an unbounded function f ∈ Lp(Z)∞. Inparticular, VAI does not hold.

The idea is to show that there is a compact ball B ⊂ G and asequence (gn)n∈N such that

• Bgnz0 ∩Bgmz0 = ∅ for n 6= m.• volZ(Bgnz0) ≤ e−n for all n ∈ N.

Out of these data it is straightforward to construct a smooth Lp-function which does not vanish at infinity.

Before we give a general proof we first discuss the case of unipotentsubgroups. The argument in the general case, although more technical,will be modeled after that.

A.1. Unipotent subgroups. Let H = N be a unipotent subgroup,that is, n := h is an ad-nilpotent subalgebra of [g, g]. Now, the situationwhere n is normalized by a particular semi-simple element is fairlystraightforward and we shall begin with a discussion of that case.

If X ∈ g is a real semi-simple element, i.e., ad X is semi-simplewith real spectrum, then we denote by gλ

X ⊂ g its eigenspace for theeigenvalue λ ∈ R, and by g±X the sum of these eigenspaces for λ posi-tive/negative. We record the triangular decomposition

g = g+X + zg(X) + g−X .

Here zg(X) =: g0X is the centralizer of X in g.

Lemma A.2. Assume that n is normalized by a non-zero real semisim-ple element X ∈ g such that n ⊂ g+

X . Set at := exp(tX) for all t ∈ R.


Let B ⊂ G be a compact ball around 1. Then there exists c > 0 andγ > 0 such that

volZ(Batz0) = c · etγ (t ∈ R)

Proof. Let A = expRX and note that A normalizes N . Thus for alla ∈ A the prescription

µZ,a(Bz0) := µZ(Baz0) (B ⊂ G measurable)

defines a G-invariant measure on Z. By the uniqueness of the Haarmeasure we obtain that

µZ,a = J(a)µZ

where J : A → R+0 is the group homomorphism J(a) = det Ad(a)|n.

All assertions follow. ¤Having obtained this volume bound we can proceed as follows. Let

us denote by χk the characteristic function of Ba−kz0 ⊂ Z. We claimthat the non-negative function

(A.1) χ :=∑

k∈Nkχk

lies in Lp(G/H). In fact

‖χ‖p ≤∑

k∈Nk‖χk‖p ≤ c

∑

k∈Nke−γk/p .

Finally we have to smoothen χ: For that let φ ∈ Cc(G)∞ with φ ≥ 0,∫G

φ = 1 and supp φ ⊂ B. Then χ := φ ∗χ ∈ Lp(Z)∞ with χ(a−kz0) ≥k. Hence χ is unbounded.

In general, given a unipotent subalgebra n, there does not necessarilyexist a semisimple element which normalizes n. For example if U ∈ g =sl(5,C) is a principal nilpotent element, then n = spanU,U2 + U3 isa 2-dimensional abelian unipotent subalgebra which is not normalizedby any semi-simple element of g. The next lemma offers a remedy outof this situation by finding an ideal n1 C n which is normalized by areal semisimple element X with n ⊂ g+

X + g0X .

Lemma A.3. Let n ⊂ [g, g] be an ad-nilpotent subalgebra and let 0 6=U ∈ z(n). Then there exists a real semi-simple element X ∈ g suchthat [X, U ] = 2U and n ⊂ g+

X + g0X .

Proof. According to the Jacobson-Morozov theorem one finds elementsX, V ∈ g such that X,U, V form an sl2-triple, i.e. satisfy the com-mutator relations [X,U ] = 2U , [X,V ] = −2V , [U, V ] = X. Note that


n ⊂ zg(U) and that zg(U) is ad X-stable. It is known and in fact easyto see that zg(U) ⊂ g+

X + g0X . All assertions follow. ¤

Within the notation of Lemma A.3 we set n1 = RU and N1 =exp(n1). Furthermore we set Z1 = G/N1 and consider the contractiveaveraging map

L1(Z1) → L1(Z), f 7→ f ; f(gN) =

∫

N/N1

f(gnN1) d(nN1).

Let B ⊂ G be a compact ball around 1, of sufficiently large size to bedetermined later, and let B1 = B · B ⊂ G. Let χ be the function onZ1 constructed as in (A.1), using the element X from Lemma A.3 andthe compact set B1. Let χ ∈ L1(Z) be the average of χ. We claimthat χ(Ba−kz0) ≥ k for all k. In fact let Q ⊂ N/N1 be a compactneighborhood of 1 in N/N1 with volN/N1(Q) = 1. Then for B largeenough we have a−kQak ⊂ B for all k (Lemma A.3). Hence for b ∈ B,

χ(ba−kz0) ≥∫

Q

χ(ba−knN1) d(nN1) ≥ k,

proving our claim.

To continue we conclude that fp := (χ)1p ∈ Lp(Z) is a function

with fp(Ba−kz0) ≥ k for all k. Finally we smoothen fp as before andconclude that VAI does not hold true.

A.2. The general case of a non-reductive unimodular space.Finally we shall prove Proposition A.1 in the general situation where His a non-reductive closed and connected subgroup for which Z = G/His unimodular.

Proof. We will argue by induction on dim g. Suppose first that h iscontained in a proper subalgebra h of g, which is reductive in g. Thenh is not reductive in h ([13], §6.6 Cor. 2). By induction H/H is notVAI, in the strong sense that for every 1 ≤ p < ∞ there exists anunbounded function f ∈ Lp(H/H)∞. We claim that G/H is not VAIin the same strong sense. Let q ⊂ g be the orthogonal complement toh in g. Then for a small neighborhood V ⊂ q of 0 the tubular map

V × H → G, (X, h) 7→ exp(X)h

is diffeomorphic. The Haar measure on G is expressed by J(X)dXdhwith J > 0 a bounded positive function. Since H normalizes q, thisallows us to extend smooth Lp-functions from H/H to G/H and wesee that G/H is not VAI in the strict sense.

We assume from now on that h is not contained in any reductiveproper subalgebra h.


To proceed we recall the characterization of maximal subalgebrasof g (see [14], Ch. 8, §10, Cor. 1). A maximal subalgebra is either amaximal parabolic subalgebra or it is a maximal reductive subalgebra.In the present case it follows that h is contained in a maximal parabolicsubalgebra p0. We write p0 = n0 o l0 where n0 is the unipotent radicalof p0. Note that l0 is reductive in g and hence h is not contained in l0.In addition we may assume that s ⊂ l0 ([13] §6.8 Cor. 1).

In the next step we claim that z(l0) is not contained in h. In factas G/H is unimodular, | det Ad(h)|h| = 1 for h ∈ H. If in additionh ∈ Z(l0), then h centralizes s and it follows that | det Ad(h)|r| = 1.Hence z(l0) ∩ h centralizes r as well. If z(l0) would be contained in h,then this would force h ⊂ l0, since l0 is the centralizer of its center. Wewould thus arrive at the already excluded case h ⊂ l0.

We fix 0 6= X ∈ z(l0) \ h such that n0 ⊂ g+X . As before we set

at := exp(tX) and observe that atz0 → ∞ in Z for |t| → ∞ (this isbecause at[L0, L0]N0 goes to infinity in G/[L0, L0]N0.)

Next we define a subspace l1 ⊂ l0 by

l1 = prl0(h)⊥

where prl0 : p0 → l0 is the projection along n0. Then l1⊕ (h+n0) = p0.In addition, we construct an ad X-invariant subspace n1 of n0 suchthat h + n0 = h ⊕ n1, as follows. If n0 ⊂ h, then n1 = 0. Other-wise we choose an ad X-eigenvector, say Y1, in n0 with largest possibleeigenvalue, such that h + RY1 is a direct sum. If this sum containsn0, we set n1 = RY1. Otherwise we continue that procedure until acomplementary subspace is reached. Now l1 ⊕ h⊕ n1 = p0.

Let us first exclude the case where n1 = n0, i.e. h ∩ n0 = 0. Inthis situation the projection map prl0 |h : h → l0, which is a Lie-algebrahomomorphism, is injective. Write h0 for the homomorphic image of hin l0. The analysis will be separated in two cases, the first being thath0 is reductive in l0.

Assume h0 is reductive in l0. In particular it is a reductive Lie alge-bra, hence so is h. In the Levi decomposition h = s ⊕ r we now knowthat r is the center of h. Let u be the subalgebra of g generated by rand θ(r), then s + u is a direct Lie algebra sum. Moreover, s + u isθ-invariant, hence reductive in g, and hence in fact = g by our previ-ous assumption on h. Thus s is an ideal in g which we may as wellassume is 0. Now h = r is an abelian subalgebra which together withθ(r) generates g. We shall reduce to the case where r is nilpotent in g,which we already treated. Every element X ∈ r has a Jordan decom-position Xn +Xs (in g), and we let o1, o2 be the subalgebras generatedby the Xn’s and Xs’s, respectively. Then o = o1 ⊕ o2 is abelian and o2


consists of semisimple elements. The centralizer of o2 is reductive in gand contains r, hence equal to g. Hence o2 is central in g, and we mayassume that it is θ-stable. Let g1 be the subalgebra of g generated byo1 and θ(o1). It is reductive in g, and (g1, o1) is of the type alreadytreated, hence not VAI. Since g = g1 + o2 we can now conclude that(g, r) = (g, h) is not VAI either.

We now assume that h0 is not reductive in g. Let H0 and L0 bethe connected subgroups of G corresponding to h0 and l0. As G/H isunimodular and H is homomorphic to H0, it follows that G/H0 andthus L0/H0 is unimodular. By induction we find for every 1 ≤ p < ∞an unbounded function f ∈ Lp(L0/H0)

∞. As before in the case of H/Hwe extend f to a smooth vector in Lp(G/H) (note that P0/H → L0/H0

is a fibre bundle, and we first extend f to a function on P0/H and thento a function on G/H).

In the sequel we assume that n1 is a proper subspace of n0.Let n0 be the nilradical of the parabolic opposite to p0 and consider

the ad X-invariant vector space

v := n0 × l1 × n1 ⊂ g

which is complementary to h.For fixed t ∈ R we consider the differentiable map

Φ = Φt : v = n0 × l1 × n1 → Z,

(Y −, Y 0, Y +) 7→ exp(Y −) exp(Y 0) exp(Y +)atz0 .

With y± = exp(Y ±) and likewise y0 = exp(Y 0) we get for the differen-tial of Φ:

dΦ(Y −, Y 0, Y +)(X−, X0, X+) = dτy−y0y+at(z0) Ad(at)

−1

(

Ad(y0y+)−11− e− ad Y −

ad Y − (X−) + Ad(y+)−11− e− ad Y 0

ad Y 0(X0) +

+1− e− ad Y +

ad Y +(X+) + h

).

In order to estimate the Jacobian of dΦ we will identify Ty−y0y+atz0Z

with v via the map

Ty−y0y+atz0Z → v, dτy−y0y+at

(z0)(X + h) 7→ πv(X + h)


where πv : g → v is the projection along h. Within this notation weobtain

dΦ(Y −, Y 0, Y +)(X−, X0, X+) = πv Ad(at)−1

(

Ad(y0y+)−11− e− ad Y −

ad Y − (X−) + Ad(y+)−11− e− ad Y 0

ad Y 0(X0) +

+1− e− ad Y +

ad Y +(X+)

).

Let Y = (Y −, Y 0, Y +) ∈ v. It follows that there exists a linear mapL(Y ) : v → g such that

dΦt(Y ) = πv Ad(at)−1(1v + L(Y ))

for all t, and that ‖L(Y )‖ → 0 for Y → 0. We rewrite as

(A.2) dΦ(Y ) = Ad(at)−1(1v + Ad(at)πv Ad(at)

−1L(Y ))

In order to control the remainder term, we define

Mt := supU∈g,‖U‖=1

‖Ad(at)πv Ad(at)−1U‖

and claim that Mt stays bounded for t → −∞. For U ∈ v we haveAd(at)πv Ad(at)

−1U = U , hence we may assume U ∈ h. Since h ⊂ p0

we can write U as a combination of an element Y0 ∈ l0 and possiblysome ad X-eigenvectors Yλ with eigenvalues λ > 0. Then

Ad(at)−1U = Y0 +

∑e−λtYλ = U +

∑(e−λt − 1)Yλ

(possibly with an empty sum). If Yλ ∈ n1 then

Ad(at)πv(e−λt − 1)Yλ = (1− eλt)Yλ → Yλ

as t → −∞. On the other hand it follows from the definition of n1,that if Yλ is not in n1 then either it belongs to h or it is a sum of anelement from h and some eigenvectors Vµ ∈ n1 with eigenvalues µ ≥ λ.Then

Ad(at)πv(e−λt − 1)Yλ =

∑eµt(e−λt − 1)Vµ

(possibly with an empty sum), which stays bounded for t → −∞. Ourclaim is thus established, and it follows that Ad(at)

−11v dominates(A.2), for Y ∈ v sufficiently small.

Combining our reasoning, and using that n1 is proper in n0, we ob-tain for every small enough compact neighborhood Q ⊂ v of 0 someconstants cQ, CQ > 0 such that

cQetγ ≤ supY ∈Q

| det dΦt(Y )| ≤ CQetγ (t ≤ 0)

for some γ > 0 independent of Q. In particular Φt|Q is a chart.


Fix now such a compact neighborhood Q, and let ψ ∈ C∞c (Q) be a

function with 0 ≤ χ ≤ 1 and ψ(0) = 1. For all t < 0 define χt ∈ C∞c (Z)

byχt(z) = ψ(Φ−1

t (z)) (z ∈ Z),

then χt ∈ Lp(Z) for all 1 ≤ p < ∞ and t ≤ 0, with ‖χt‖p ≤ Cetγ/p forsome C > 0 not depending on t (but possibly on p). Finally we set

χ :=∑

n∈Nnχ−n ,

Then χ ∈ Lp(Z) for all 1 ≤ p < ∞. It is also clear that χ ∈ C∞(Z)and that χ is unbounded. It remains to be seen that χ ∈ Lp(Z)∞.

Let U ∈ g and consider the left derivatives L(U)χt. At z = Φt(Y ) =y−y0y

+atz0 these are given by

L(U)χt(z) = d/ds|s=0 χt(exp(sU)z).

For Y in a compact set, we may as well consider the derivatives of

χt(y−y0y

+ exp(sU)atz0).

Notice that exp(sU)z0 ' exp(sπvU)z0 for s → 0 and rewrite the deriv-ative as

d/ds|s=0 χt(y−y0y

+ exp(s Ad(at)−1πv Ad(at)U)atz0).

We conclude that this is a linear combination of derivatives of ψ onQ, with coefficients that depend smoothly on Y and are bounded fort < 0. Here the previously attained bound on Mt is used. As beforewe conclude L(U)χt ∈ Lp(Z) for all t ≤ 0, with exponentially decayingp-norms. It follows that L(U)χ ∈ Lp(Z). By repeating the argumentfor higher derivatives we finally see that χ ∈ Lp(Z)∞. ¤

Appendix B. Proof of Lemma 7.6

We assume that Z is algebraic, that P = MAN is a parabolic sub-group such that PH is open, that M/M ∩ H is compact and thatL := P ∩H ⊂ M . According to (7.4)

(B.1) ∃a ∈ A : a⊕ (ka + h) = g.

The statement in the lemma is that G = KAH.We first claim that we may assume that PH is open for all all choices

of parabolics with AP = A. Note that both (B.1) and the openess ofPH are open conditions. This implies that if we replace A by itsconjugate by an element k ∈ K, then both conditions hold for generick. It follows that in each neighborhood of 1 in K we can find anelement k such that after replacing A by k−1Ak, then (B.1) holds andPH is open for all P ⊃ A. But if we know that KAkH = G for all k in


a sequence k → 1, then KAH = G. This proves the claim and allowsus to make the asserted assumption. Notice however that conjugatingP possibly destroys that L ⊂ M for L = P ∩ H. However, LemmaB.2 applied to (R,S) = (P, H), (H, P ) implies that H/L and P/L stayunimodular homogeneous spaces. We claim that L ⊂ MN . In fact ifP = Pk := kPk−1 and Lk := Pk ∩ H then Lemma B.1 implies thatLk = (H ∩ (Mk)

n)(A∩ (Mk)n)(H ∩Nk). Note that n ∈ Nk depends on

k but n → 1 for k → 1. The Lie algebras lk converge to l ⊂ m, andthus Lk = H ∩ (Mk)

n ⊂ MkNk as was to be shown.Note that KCACHC being the image of the algebraic morphism

KC × AC ×HC → GC, (k, a, h) 7→ kah

is an affine subvariety of GC which is of maximal dimension as it issubmersive at a point. It follows that KCACHC contains a Zariskiopen subset of GC and G ∩KCACHC has dense interior in G.

Let CC ⊂ (KCACHC)o be a connected component (with respect to

the Zariski topology). We consider the sets C := CC∩G. Since these arereal algebraic varieties, every component that intersects non-triviallywith KAH is necessarily contained in KAH. Hence it suffices to provethat C ∩KAH is non-empty for each component

C ⊂ G ∩ (KCACHC)o.

Observe first that C has unbounded AC-part, that is there exists a se-quence pn = knxnhn ∈ C with kn ∈ KC, hn ∈ HC and with xn ∈ ACtending to infinity. Otherwise the projection of C in the categori-cal quotient QC := GC//HC × KC (see [32], Sect. II.3.2) would bebounded. Hence the image of C would be a bounded subvariety of QRwith interior points, and this is not possible. To finish the proof, weshall show that eventually pn ∈ KAH.

Let T := exp(ia) and note that AC = AT . Further we set AR =AC ∩ G and note that AR = AF for F ⊂ T ∩K a finite 2-group. Wewrite xn = antn with an ∈ A and tn ∈ T . Passing to a subsequence ifnecessary we may assume that an lies in the closure of a fixed chamberA+. We claim that tn ∈ F for sufficiently large n. This is the crucialpart of the proof.

It is here that we need analytic arguments. We shall provide a suffi-cient supply of analytic K × H-invariant functions on G. We workwith generalized principal series: each λ ∈ a∗C defines a characterχλ = 1⊗ eλ ⊗ 1 : P → C∗ and defines a line bundle G×P Cλ → G/P .The smooth sections of this line bundle we denote by Eλ. As a topolog-ical vector space we identify Eλ with C∞(K/KM) in a K-equivariantway (compact model). Here KM = K ∩M .


To continue we recall that L = H ∩ P ⊂ MN . We claim that

ηλ : Eλ → C, f 7→∫

H/L

f(h) dh

defines an invariant functional provided Re λ ∈ a∗ is large and lies ina∗,+. Our concern is with convergence only. Recall that G/P ' K/KM

and write O ⊂ K/KM for the open subset corresponding to HP/P ⊂G/P . In the sequel we use

H ×L M × A×N 3 (h,m, a, n) 7→ hman ∈ HP ⊂ G

as coordinates on the open set HP ⊂ G and note that in HP 3 g =h(g)m(g)a(g)n(g) the entry a(g) ∈ A is defined whereas h(g) is definedmod L. In particular, kKM 7→ h(k)L is defined on O. For a measurablefunction φ on H/L, the integration formula in Lemma B.3 gives

∫

H/L

φ(hL) d(hL) =

∫

Oφ(h(k))a(k)−2ρ dk .

In particular, for f ∈ Eλ it follows that∫

H/L

f(hL) d(hL) =

∫

Of(k)a(k)λ−2ρ dk .

Let P0 = M0A0N0 ⊂ P be a minimal parabolic, as in the proof ofLemma 6.3, such that P0H is open. Recall the closed embedding (7.1).Let µ ∈ a∗0 be the highest weight of the corresponding finite dimensionalrepresentation π of G whose highest weight ray is fixed by P0. Thenthe fact that P0H is open allows the conclusion that 〈vµ, vH〉 6= 0, andin particular that µ = 0 on a0 ∩ h. Furthermore the identity

a(g)µ〈vµ, vH〉 = 〈vµ, π(g−1)vH〉allows us to conclude that aµ is continuous on HP .

Note that with µ1 and µ2 being highest weights of finite dimensionalH-spherical representations, the same holds for µ1 + µ2 (take tensorproducts). Thus the H-spherical highest weights form a semi-group.By (B.1) we see that the categorical quotient GC//HC × KC has di-mension equal to dim a. It follows that the elements of the semi-groupspan a∗. Let us denote by C ⊂ a∗ the convex hull of all µ′s. Hence ηλ

is defined and non-zero for all λ ∈ Cρ := 2ρ + C.For λ ∈ Cρ the matrix coefficient ηλ(π(g)−1vK), where vK = 1K/M ∈

EKλ , is a smooth K ×H-invariant real function on G. We define

fλ(g) := ηλ(πλ(g)−1vK), hλ(g) := Re[cfλ(g)], (g ∈ G)

with c = c(λ) ∈ C a non-zero constant to be determined.


We assume first that an ∈ A+, uniformly away from walls. Note thatexp−1

AC(1) =: Λ defines a lattice in t and that we can take F = exp(i1

2Λ).

We let Ω ⊂ t be the interior of the standard fundamental domain forthe action of 1

2Λ and set T ′ := exp(iΩ) so that D := A+T ′ ⊂ AC

is simply connected. As we are free to replace xn by an appropriaterepresentative in xnF we may (after possibly deforming Ω slightly) infact assume that xn ∈ D.

Our proof of Th. 6.4 yields, as in [46], Sect. 4.4, an asymptoticexpansion:

fλ(a) ∼∑w∈W

∑

ν∈N0[Σ+]

cw,νawλ−ρ−ν (a ∈ A+) .

In particular, c1,0 6= 0, so that we can fix c := c−11,0. Then

hλ(a) ∼ aλ−ρ +∑

wλ−ρ−ν 6=0

Re[ccw,ν ]awλ−ν .

The expansion converges for a ∈ D and allows to define hλ(a) fora ∈ D. In these formulas we suppressed logarithmic terms which can beavoided under suitable regularity assumptions. We have thus extendedhλ to a real analytic function on KCDHC which is KC ×HC-invariant.As pn ∈ G we obtain

R 3 hλ(pn) = hλ(antn) ' (antn)λ

for n large and λ ∈ Cρ generic. Thus tλn is close to the real axis if n issufficiently large. We claim that tn = 1 provided n is sufficiently large.For that we write tn = exp(iXn) with Xn ∈ Ω. The fact that tλn isclose to R means more precisely that for each ε > 0 and each λ ∈ Cρ

there exists Nε(λ) ∈ N such that

λ(Xn) ∈ πZ+ (−ε, ε) .

for n ≥ Nε(λ). As Ω is bounded, we have in addition

|λ(Xn)| ≤ C‖λ‖.Let us fix a basis µ1, . . . , µr of a∗ which is contained in Cρ and

introduce coordinates on Ω, say Xj := µj(X) for X ∈ Ω. With regardto our basis we express λ ∈ Cρ as λ =

∑rj=1 λjµj with λj ∈ R. Let

d > 1 be an irrational number. Then by taking λ equal to each ofthe finitely many elements µ1, . . . , µr, dµ1, . . . , dµr in Cρ we obtain aconstant C > 0 and for every ε > 0 an Nε ∈ N such that

Xnj ∈ [π Z+ (−ε, ε)] ∩ [

π

dZ+ (−ε, ε)] ∩ [−C, C]


for all n ≥ Nε and 1 ≤ j ≤ r. Clearly this set has only the element 0provided ε > 0 is small enough. This proves our claim.

The case where an does not lie in A+, only in A+, is treated via theseries regrouping argument in [16], Sect. 5, which allows control alongthe walls.

As F ⊂ K we conclude that our component C contains a pointp = kah with k ∈ KC and h ∈ HC and a ∈ A. As C is left K-invariant,we may assume that k = exp(iY ) for Y ∈ k (use that KC = K exp(ik)).

Write τ : GC → GC for the associated complex conjugation whichhas G as the set of real points. As p ∈ G we get that kah = τ(kth) =k−1aτ(h) and thus

exp(2i Ad(a−1)Y ) = a−1k2a = τ(h)h−1 ∈ HC

As 2i Ad(a−1)Y ∈ gC is a semi-simple element with real spectrum itfollows that i Ad(a−1)Y ∈ hC. Hence p = kah = ah′h for some h′ ∈ HCand as p lies in G we get that p ∈ AH. This completes the proof. ¤

Lemma B.1. Let H be an algebraic subgroup of G and P < G aparabolic subgroup with Langlands decomposition P = MAN . Thenthere exists n ∈ N such that

H ∩ P = (H ∩Mn)(H ∩ An)(H ∩N)

where Mn = nMn−1 and An = nAn−1.

Proof. The algebraic group S := H ∩ P has a Levi decompositionS = LU with U the unipotent radical. As N is the unique maximalunipotent subgroup of P it follows that U ⊂ N . Further, any Levisubgroup of P is conjugate under N to MA. As Levi subgroups of Pcoincide with maximal reductive subgroups of P , the assertion of thelemma follows. ¤

Lemma B.2. Let G be a connected real linear algebraic group and R, Salgebraic subgroups of G. Suppose that RS ⊂ G is open and R/R ∩ Sis unimodular. Then all open R-orbits in G/S are unimodular, i.e. ifRxS is open for some x ∈ G, then R/R ∩ xSx−1 is unimodular.

Proof. As RS is open, the same holds for RCSC in GC. Note that RCSCis Zariski open in GC. It follows that there exists only one open RC-orbit in GC/SC. This RC-orbit is unimodular and as any open R-orbitin G/S has the same complexification, the assertion follows. ¤

Lemma B.3. Let Z = G/H be of reductive type and (P, H) is a uni-modular spherical pair such that L := H ∩ P is reductive in G. Then,


up to normalization of measures,∫

H/L

φ(hL) d(hL) =

∫

K∩HP

φ(h(k))a(k)−2ρ dk

for φ ∈ L1(H/L). In this formula one has HP 3 g = h(g)m(g)a(g)n(g)according to P = MAN .

Proof. Let φ be a test function on H/L. Choose a test function Φ onH such that

φ(hL) = ΦL(hL) :=

∫

L

Φ(hl) dl .

Next choose a test function ψ on P with∫

Pψ(p) drp = 1 where drp

is a right Haar measure on P . Define a function F on H × P byF (h, p) = Φ(h)ψ(p). We let L act on H × P as (h, p) 7→ (hl, l−1p) andwrite f := FL for the corresponding fiber integral. Up to normalizationof the Haar measure dg of G we then have∫

HP

f(g) dg =

∫

H/L

φ(hL) d(hL) .

In fact ∫

HP

f(g)dg =

∫

H

∫

P

F (h, p) drp dh

=

∫

H

∫

P

Φ(h)ψ(p) drp dh

=

∫

H

Φ(h) dh =

∫

H/L

φ(hL) d(hL) .

On the other hand we have

∫

HP

f(g)dg =

∫

K

∫

P

f(kp) drp dk

=

∫

K

∫

P

f(h(k)p(k)p) drp dk

=

∫

K

∫

P

f(h(k)p) a(k)−2ρ drp dk

=

∫

K

∫

L

∫

P

F (h(k)l, l−1p) a(k)−2ρ drp dl dk

=

∫

K

∫

L

Φ(h(k)l) a(k)−2ρ dl dk

=

∫

K

φ(h(k)) a(k)−2ρ dk


where from line 4 to line 5 we use the fact that a(l)−2ρ = 1 which holdstrue as Ad(L) acts measure preserving on P . ¤

Appendix C. Proof of Lemma 11.5

Let 1 ≤ p < ∞ and suppose that Λ ⊂ Gs is Lp(Z)-bounded. Theassertion is that there exists C > 0 such that

(C.1) ‖mv,η‖p ≤ C‖mv,η‖∞for all π ∈ Λ, η ∈ (H−∞


We recall the polar decomposition G = KA+q H of of Z (see Subsec-

tion 5.1.1). Let ρ ∈ a∗q be the associated Weyl half sum and recall thatthe Haar measure on Z, expressed in K/M × a+

q coordinates is givenby

(C.2) J(X) dX d(kM)

with the Jacobian J(X) satisfying the bound

(C.3) J(X) ≤ e2ρ(X) .

It follows from [3] that a subset Λ ⊂ Gs is Lp(Z)-bounded if andonly if there exists a λ ∈ a∗q such that λ |a+

q \0> 0 and

• For all π, v and η as above there exists C > 0 such that

|mv,η(a)| ≤ Caλ (a ∈ A+q ) .

• ∫a+

qepλ(X)J(X) dX < ∞.

Denote in this proof by Σ = Σ(aq, g) ⊂ a∗q the root system of aq

and by W the corresponding Weyl-group. We embed aq in a maximalabelian subspace a ⊂ p, of which the corresponding Weyl group is thendenoted by W . Let λπ ∈ a∗C be a spherical principal series parameter(unique up to W -conjugacy) of the representation π. Note that thereal part of λπ is uniformly bounded.

In the sequel we will say that λπ is generic if the following twoconditions are satisfied:

(1) 〈ν − 2w.λπ, ν〉 6= 0 for all ν ∈ N0[Σ+]\0 and w ∈ W .

(2) λπ is regular, i.e. if w.λπ − λπ ∈ ZΣ+, then w = 1.

The first condition is related to the recurrence relation in [8], Prop. 5.2,and we note that it is automatically satisfied if λπ is purely imaginary.The second condition is related to the linear independence of the basicfunctions Φλπ .


In the sequel we let f = mv,η|A+q. If λπ is generic then we have an

asymptotic expansion of the form (see [5], Section 17):

f(a) =∑w∈W

∑

ν∈N0[Σ+]

cw,νa−ρ+w.λπ−ν (a ∈ A+

q ) .

A similar expansion holds in the non-generic case, but with additionallogarithmic terms of bounded degree.

We first work under the simplifying assumption that all elements ofΛ are generic. In addition we will first assume that Λ ⊂ ia∗. It followsthat there exists a constant c > 0 such that

(C.4) |〈ν − 2w.λπ, ν〉| ≥ c

for all π ∈ Λ, ν ∈ N0[Σ+]\0 and w ∈ W . Then, in view of [7],

Th. 7.4, it follows that there exist constants C, κ > 0, independent ofπ ∈ Λ, such that:

(C.5) |cw,ν | ≤ C(1 + ‖ν‖)κ|cw,0|for all w ∈ W .

Let us call Lf(a) := a−ρ∑

w∈W cw,0awλπ the leading term of f and

put Lf(a) = aρL(f)(a).To normalize matters we request that

(C.6) maxw∈W

|cw,0| = 1 .

Fix X0 ∈ a+q and choose a small compact ball U around 0 in aq such

that X0 + U ⊂ a+q . We let CU := R+U be the convex cone generated

by U . For R > 0 we define a compact ball in A+q by

BR := exp(X ∈ CU | ‖X‖ ≤ R) .

Further we let SR ⊂ BR be the subset obtained by elements X with‖X‖ = R.

To begin with we suppose that for all t > 1 there exists a constantct > 0 such that for all R ≥ 1 one has

(C.7) ‖L(f)‖∞,BtR\BR≥ ct

for all π ∈ Λ and all f subject to the normalization (C.6). Note that

‖f‖∞ ≥ ‖f‖∞,BtR\BR≥ ‖L(f)‖∞,BtR\BR

− ‖L(f)− f‖∞,BtR\BR.

For all R we set cR := mina∈BRa−ρ = mina∈SR

a−ρ and note thatcR = e−γ0R for some γ0 > 0. Likewise we set dR := maxa∈SR

a−ρ andnote that dR = e−γ1R. By choosing the ”opening angle” U of CU smallenough we can make the difference γ0 − γ1 arbitrarily small.

Now observe that ‖L(f)‖∞,BtR\BR≥ e−γ0tRct. Moreover, from (C.5)

we obtain that f is approximated well by its leading term outside a


ball of sufficiently large radius: there exists ε > 0, C > 0 such that forsufficiently large R and all t > 1 one has

‖L(f)− f‖∞,BtR\BR≤ Ce−(γ1+ε)R

for all π ∈ Λ. Putting matters together, there exists a choice of CU

such that there exists a constant c′ > 0 such that

(C.8) ‖f‖∞ ≥ c′

for all f subject to the normalization (C.6).For any R > 0 observe that

‖f‖p ≤ vol(BR)1p‖f‖∞,BR

+ ‖f‖p,BcR

.

Here vol stands for the volume with respect to the measure J(X)dX.We claim, given δ > 0 we find R > 0 such that ‖f‖p,Bc

R< δ uniformly.

For a ∈ A+q and the Lp-norm of functions restricted to KaA+

q .z0 thisfollows from (C.5). Along the walls we use the power series regroupingprocedure from [16], Sect. 6, (adapted to the symmetric case in [3],Sect. 5) to conclude that the contributions along the walls stay uni-formly small. Combining the claim with (C.8) we obtain a constantc′′ > 0 such that

‖f‖p ≤ c′′‖f‖∞ .

This proves (C.1) subject to all the simplifying assumptions. Step bystep we now drop those.

To begin with let us analyze what happens if (C.7) fails. It is instruc-tive to see the rank one case, i.e. dim a = 1. We choose coordinates xon a = R such that a−ρ = e−x and note that

L(f)(x) = e−x(ceλx + de−λx) .

Here λ = λπ is a complex number with |Re λ| < 1. We implement ournormalization (C.6) by the request c = 1 and |d| ≤ 1. We see that (C.7)holds provided λ is bounded away from 0. The worst case scenario is forλ = is small and imaginary and d = −1: then L(f)(x) = 2ie−x sin(sx).In this case however we note that

f(x) = 2ie−x sin(sx) + e−x

∞∑n=1

e−nx(eisxcn + e−isxdn) .

The precise form of the recurrence relation in [7], Prop. 5.2, impliesthat

|e−nx(eisxcn + e−isxdn)| ≤ Ce(−n+1/2)x|s|for a uniform constant C > 0. Again ‖f‖∞ and ‖f‖p are controlled bythe leading term as before and we are done.


Let us now analyze in general the case where (C.7) starts to fail.From the Lemma below we conclude that (C.7) holds true providedthere exists an ε > 0 such that

infw∈W\1

‖w.λπ − λπ‖ ≥ ε

for all π ∈ Λ. Suppose now that this conditions starts to fail for somew ∈ W\1. If w is the longest element, then this means that λπ → 0and, in view of the explicit recurrence relation in [7], we can argue asin the rank one case above. In general let us write Π ⊂ Σ+ for the setof simple roots. Then we find a non-empty subset F ⊂ Π such that〈λπ, α〉 → 0 for α ∈ F whereas |〈λπ, α〉| ≥ ε > 0 for all α ∈ Π\F . Thisreduces in essence the situation to a lower rank situation, i.e. in thelimit we group

∑w∈W/WF

cwaw.λπ with cw =∑

s∈WFcws and normalize

with regard maxw∈W/WF|cw| = 1.

We assumed that Λ ⊂ ia∗ and wish to drop this assumption. In factthe weaker condition (C.4) is enough and we analyze now when thisstarts to fail. Again it is instructive to understand the rank one casenear a point where (1) fails. Say (1) fails near ν = nα, n ∈ N and αthe simple root. Instead of using the leading term L(f) we use now then-th term to normalize matters, i.e.

Ln(f)(x) = e−nx(cneiλx + dne−iλx) .

We then assume max|cn|, |dn| = 1 and proceed as before. The higherrank case works in a similar way.

Finally we drop the genericity assumptions (1) and (2). As we saidlogarithmic terms appear now in the expansion. But since the boundholds for the dense set if generic parameters we obtain the bound forall parameters by continuity. ¤

Lemma C.1. Let c = (c1, . . . , cn) ∈ Cn, λ = (λ1, . . . , λn) ∈ Rn andδ > 0. For the the function

fλ,c(t) :=n∑

j=1

cjeiλjt (t ∈ R) .

the following inequality holds true:

[sup|t|≤δ

|fλ,c(t)|]2 ≥ ‖c‖22 − n(n− 1)/2‖c‖2

∞ supj 6=k

1

δ|λj − λk| .

Proof. From ∫ δ

−δ

|fλ,c(t)|2 dt ≤ 2δ[sup|t|≤δ

|fλ,c(t)|]2


and ∫ δ

−δ

|fλ,c(t)|2 dt =n∑

j,k=1

cjck2sin(λj − λk)δ

λj − λk

the assertion follows. ¤

Appendix D. Concluding Remarks

D.1. Remarks on VAI. We investigated (VAI) with regard to uni-modular homogeneous spaces Z = G/H for which G was reductive andH < G was a closed subgroup with finitely many connected compo-nents.

We did not address here the cases where H is not connected or Gis not reductive. Without any further assumption let us assume thatG is a connected Lie group and H ⊂ G is a closed subgroup such thatZ = G/H is unimodular. In case G is infinitesimally simple and Z isnot compact one might suspect that Z has VAI if and only if the Zariskiclosure of H is a proper reductive subgroup. For G and H algebraicand G reductive, H is reductive in G if and only if it is reductive. Onemight then suspect for G and H algebraic and G general, that Z hasVAI if and only if the nilradical of H is contained in the nilradical of G.

Initially we wanted to prove the converse implication in Theorem 2.2via a temperedness result for invariant measures. To be more specific,assume G and H < G to be algebraic groups and Z = G/H to be uni-modular and quasi-affine. Under these assumptions we conjecture thatthere is a rational G-module V , and an embedding Z → V such thatthe invariant measure µZ , via pull-back, defines a tempered distributionon V . Note that if Z is of reductive type, then there exists a V suchthat the image of Z → V is closed, and hence µZ defines a tempereddistribution on V . If Z is not of reductive type, then all images Z → Vare non-closed and our conjecture would imply that VAI does not hold.Our conjecture is supported by a result of Deligne, established in [42],which asserts that for a reductive group G and X ∈ g := Lie(G) theinvariant measure on the adjoint orbit Z := Ad(G)(X) ⊂ g defines atempered distribution on g. Various particular results in the theoryof prehomogeneous vector spaces provide additional support for ourconjecture (see [12]).

D.2. Spaces of spherical type, polar type and property (I).In the context of polar type and Property (I) we state the followingconjecture. All the conditions below are known to hold in case Z issymmetric, and also in the (non-reductive) case H = N of Iwasawadecomposition. In case Z is algebraic we showed (Theorem 7.9 and


Theorem 8.5) that (3) ⇒ (1)∧(2) under the additional hypothesis ofunimodular spherical type.

Conjecture D.1. Let G be linear simple group with finite center.(a) For a unimodular homogeneous space Z = G/H we expect the fol-lowing properties to be equivalent:

(1) Z has property (I);(2) Z is of polar type;(3) Z is of spherical type;(4) For every irreducible admissible moderate growth smooth Frechet

representation (π,E) of G, the space of H-invariant functionalsis of finite dimension.

If in addition G is split, then the above should be equivalent to:

(5) ZC = GC/HC is a spherical variety ([15]).

(b) Let Z(g) denote the center of the universal enveloping algebra U(g).If Z is of polar type with decomposition G = KAH, then all K-finiteZ(g)-eigenfunctions on Z admit asymptotic expansions along A.

D.3. Main term counting for non-compact Y . When Y = G/Γ isnon-compact, the Fourier inversion (10.5) involves continuous spectrumand it takes more care to define an appropriate space A(Y ). Onepossibility is to use pseudo-Eisenstein series as in [20] or simply takeCc(Y )K

o .So let f ∈ A(Y ) and

f =

∫

Gs

fπ dµ(π)

its Plancherel-decomposition: here fπ = mvπ,f∧(π) as in (10.5) before.In the first step we would like to define fH

π . In case the cycle H/ΓH ⊂Y is compact, this causes no difficulties. In general, this goes underthe term “regularization (of Eisenstein series)” which was establishedin a variety of cases (see [35] for some results). However, we wouldlike to point out that all the technical tools (Paley-Wiener-theory) areavailable to establish regularization, but this will not be subject of thispaper.

After we have defined fHπ , main term analysis needs a more quantita-

tive version of (I). To be more precise one needs to exhibit a 1 ≤ p < ∞and establish uniform quantitative control of ‖mH

v,η‖p for all π ∈ Gs\1which appear in in L2(Y )K . In case Z is symmetric harmonic analysisis sufficiently developed to cope with this problem (see the key- Lemma11.5 in the Section 11). For more general spaces, one needs to studyasymptotic expansion in sufficient detail.


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