The Harmonic Analysis of Lattice Counting

on Real Spherical Spaces

Bernhard Krötz^{1}, Eitan Sayag^{2}, Henrik Schlichtkrull

Received: August 28, 2014 Revised: December 17, 2015

Communicated by Patrick Delorme

Abstract. By the collective name of lattice counting we refer to a setup introduced in [10] that aims to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in [10].

2010 Mathematics Subject Classification: 22E40, 22E46, 43A85 Keywords and Phrases: Lattice counting, homogeneous spaces, real spherical spaces, spectral analysis

1 Introduction

1.1 Lattice counting

Let us recall from Duke, Rudnick and Sarnak [10] the setup of lattice counting on a homogeneous space Z = G/H. Here G is an algebraic real reductive group and H < G an algebraic subgroup such that Z carries an invariant measure. Further we are given a lattice Γ < G such that its trace Γ_{H} := Γ ∩ H in H is a lattice in H. Attached to invariant measures dh and dg on H and G we obtain an invariant measure d(gH) on Z via Weil-integration:

∫_{Z} ( ∫_{H} f(gh)dh) d(gH) = ∫_{G} f(g) dg (f ∈ C_{c}(G)) .

1Supported by ERC Advanced Investigators Grant HARG 268105 2Partially supported by ISF 1138/10 and ERC 291612

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Likewise the measures dg and dh give invariant measures d(gΓ) and d(hΓ_{H}) on Y := G/Γ and Y_{H} := H/Γ_{H}. We pin down the measures dg and dh and hence d(gH) by the request that Y and Y_{H} have volume one. Further we are given a family B of “balls” B_{R} ⊂ Z depending on a parameter R ≥ 0. At this point we are rather imprecise about the structure of these balls and content us with the property that they constitute an exhausting family of compact sets as R → ∞. Let z_{0} = H ∈ Z be the standard base point. The lattice counting problem for B consists of the determination of the asymptotic behavior of the density of Γ · z_{0} in balls B_{R} ⊂ Z, as the radius R → ∞. By main term counting for B we understand the statement that the asymptotic density is 1. More precisely, with

N_{R}(Γ, Z) := #{γ ∈ Γ/Γ_{H} | γ · z_{0} ∈ B_{R}}

and |B_{R}| := vol_{Z}(B_{R}) we say that main term counting holds if

N_{R}(Γ, Z) ∼ |B_{R}| (R → ∞).  (1.1)

1.2 Relevant previous works

The main term counting was established in [10] for symmetric spaces G/H and certain families of balls, for lattices with Y_{H} compact. Furthermore, the main term counting in the case where Y_{H} is non-compact was proven using a hypothesis on regularization of periods of Eisenstein series, whose proof remains unpublished. In subsequent work Eskin and McMullen [11] removed the obstruction that Y_{H} is compact and presented an ergodic approach. Later Eskin, Mozes and Shah [12] refined the ergodic methods and discovered that main term counting holds for a wider class of reductive spaces: For reductive algebraic groups G, H defined over ℚ and arithmetic lattices Γ < G(ℚ) it is enough to request that the identity component of H is not contained in a proper parabolic subgroup of G which is defined over ℚ and that the balls B_{R} satisfy a certain condition of non-focusing. In these works the balls B_{R} are constructed as follows. All spaces considered are affine in the sense that there exists a G-equivariant embedding of Z into the representation module V of a rational representation of G. For any such embedding and any norm on the vector space V, one then obtains a family of balls B_{R} on Z by intersection with the metric balls in V. For symmetric spaces all families of balls produced this way are suitable for the lattice counting, but in general one needs to assume non-focusing in addition. In particular all maximal reductive subgroups satisfy all the conditions and hence fulfill the main term counting.

1.3 Real spherical spaces

In this paper we investigate the lattice counting for a real spherical space Z, that is, it is requested that the action of a minimal parabolic subgroups P < G on Z admits an open orbit. In addition we assume that H is reductive and remark that with our standing assumption that Z is unimodular this is automatically satisfied for a spherical space when the Lie algebra 𝔥 of H is self-normalizing (see [17], Cor. 9.10). Our approach is based on spectral theory and is a natural continuation to [10]. We consider a particular type of balls which are intrinsically defined by the geometry of Z (and thus not related to a particular representation V as before).

1.3.1 Factorization of spherical spaces

In the spectral approach it is of relevance to get a control over intermediate subgroups H < H^{⋆} < G which arise in the following way: Given a unitary representation (π, H) one looks at the smooth vectors H^{∞} and its continuous dual H^{−∞}, the distribution vectors. The space (H^{−∞})^{H} of H-invariant distribution vectors is of fundamental importance. For all pairs (v, η) ∈ H^{∞} ×(H^{−∞})^{H} one obtains a smooth function on Z, a generalized matrix-coefficient, via

m_{v,η}(z) = η(g^{−1} · v)  (z = gH ∈ Z) .  (1.2)

The functions (1.2) are the building blocks for the harmonic analysis on Z. The stabilizer H_{η} in G of η ∈ (H^{−∞})^{H} is a closed subgroup which contains H, but in general it can be larger than H even if π is non-trivial. Let us call Z^{⋆} = G/H^{⋆} a factorization of Z if H < H^{⋆} and Z^{⋆} is unimodular. For a general real spherical space Z the homogeneous spaces Z_{η} = G/H_{η} can happen to be non-unimodular (see [19] for H the Iwasawa N-subgroup). However there is a large subclass of real spherical spaces which behave well under factorization. Let us call a factorization co-compact if H^{⋆}/H is compact and basic if (up to connected components) H^{⋆} is of the form H_{I} := HI for a normal subgroup I ◁ G. Finally we call a factorization weakly basic if it is obtained by a composition of a basic and a co-compact factorization.

1.3.2 Wavefront spherical spaces

A real spherical space is called wavefront if the attached compression cone is a quotient of a closed Weyl-chamber. The relevant definitions will be recalled in Section 3. Many real spherical spaces are wavefront: all symmetric spaces and all Gross-Prasad type spaces G × H/H (see (3.2) - (3.4)) are wavefront.^{3} The terminology wavefront originates from [24] because wavefront real spherical spaces satisfy the “wavefront lemma” of Eskin-McMullen (see [11], [18]) which is fundamental in the approach of [11] to lattice counting. On the geometric side wavefront real spherical spaces enjoy the following property from [19]: All Z_{η} are unimodular and the factorizations of the type Z_{η} are precisely the weakly basic factorizations of Z.

3Also, if Z is complex, then of the 78 cases in the list of [4], the non-wavefront cases are (11), (24), (25), (27), (39-50), (60), (61)

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On the spectral level wavefront real spherical spaces are distinguished by the following integrability property, also from [19]: The generalized matrix coefficients m_{v,η} of (1.2) belong to L^{p}(Z_{η}) for some 1 ≤ p < ∞ only depending on π and η.

1.3.3 Main term counting

In the theorem below we assume that Z is a wavefront real spherical space of reductive type. For simplicity we also assume that all compact normal subgroups of G are finite. Using soft techniques from harmonic analysis and a general property of decay from [21], our first result (see Section 5) is:

Theorem A. Let Z = G/H be as above, and assume that Y = G/Γ is compact. Then main term counting (1.1) holds.

Since wavefront real spherical spaces satisfy the wavefront lemma by [18], Section 6, this theorem could also be derived with the ergodic method of [11]. In the current context the main point is thus the proof by harmonic analysis. To remove the assumption that Y is compact and to obtain error term bounds for the lattice counting problem we need to apply more sophisticated tools from harmonic analysis. This will be discussed in the next paragraph with some extra assumptions on G/H.

1.4 Error Terms

The problem of determining the error term in counting problems is notoriously difficult and in many cases relies on deep arithmetic information. Sometimes, like in the Gauss circle problem, some error term is easy to establish but getting an optimal error term is a very difficult problem. We restrict ourselves to the cases where the cycle H/Γ_{H} is compact.^{4} To simplify the exposition here we assume in addition that Γ < G is irreducible, i.e. there do not exist non-trivial normal subgroups G_{1}, G_{2} of G and lattices Γ_{i} < G_{i} such that Γ_{1}Γ_{2} has finite index in Γ. The error we study is measure theoretic in nature, and will be denoted here as err(R, Γ). Thus, err(R, Γ) measures the deviation of two measures on Y = Γ\G, the counting measure arising from lattice points in a ball of radius R, and the invariant measure dμ_{Y} on Y. More precisely, with 1_{R} denoting the characteristic function of B_{R} we consider the densities

F_{R}^{Γ}(gΓ) := \frac{∑_{γ∈Γ/Γ_{H}} 1_{R}(gγH)}{|B_{R}|}.

Then,

err(R, Γ) = ||F_{R}^{Γ} − dμ_{Y} ||_{1},

drop this assumption.

4After a theory for regularization of H-periods of Eisenstein series is developed, one can

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where || · ||_{1} denotes the total variation of the signed measure. Notice that |F_{R}^{Γ}(eΓ) − 1| = \frac{|N_{R}(Γ,Z)−|B_{R}||}{|B_{R}|} is essentially the error term for the pointwise count (1.1). Our results on the error term err(R, Γ) allows us to deduce results toward the error term in the smooth counting problem, a classical problem that studies the quantity

err_{pt,α}(R, Γ) = |B_{R}||F_{α,R}^{Γ}(eΓ) − 1|

where α ∈ C_{c}^{∞}(G) is a positive smooth function of compact support (with integral one) and F_{α,R}^{Γ} = α∗F_{R}^{Γ}. See Remark 7.2 for the comparison of err(R, Γ) with err_{pt,α}(R, Γ). To formulate our result we introduce the exponent p_{H}(Γ) (see (6.2)), which measures the worst L^{p}-behavior of any generalized matrix coefficient associated with a spherical unitary representation π, which is H-distinguished and occurs in the automorphic spectrum of L^{2}(Γ\G). We first state our result for the non-symmetric case of triple product spaces, which is Theorem 8.2 from the body of the paper.

Theorem B. Let Z = G_{0}^{3}/ diag(G_{0}) for G_{0} = SO_{e}(1, n) and assume that H/Γ_{H} is compact. For all p > p_{H}(Γ) there exists a C = C(p) > 0 such that

err(R, Γ) ≤ C|B_{R}|^{− \frac{1}{(6n+3)p}}

for all R ≥ 1. (In particular, main term counting holds in this case). Furthermore, in regards to smooth counting, for any α ∈ C_{c}^{∞}(G) and for all p > p_{H}(Γ) there exists a C = C(p, α) > 0 such that

err_{pt,α}(R, Γ) ≤ C|B_{R}|^{1− \frac{1}{(6n+3)p}}

for all R ≥ 1.

To the best of our knowledge this is the first error term obtained for a nonsymmetric space. The crux of the proof is locally uniform comparison between L^{p} and L^{∞} norms of generalized matrix coefficients m_{v,η} which is achieved by applying the model of [3] and [9] for the triple product functional η in spherical principal series. It is possible to obtain error term bounds under a certain technical hypothesis introduced in Section 6 and refered to as Hypothesis A. This hypothesis in turn is implied by a conjecture on the analytic structure of families of Harish-Chandra modules which we explain in Section 9.1. The conjecture and hence the hypothesis appear to be true for symmetric spaces but requires quite a technical tour de force. In general, the techniques currently available do not allow for an elegant and efficient solution. Under this hypothesis we show that:

Theorem C. Let Z be wavefront real spherical space for which Hypothesis A is valid. Assume also

• G is semisimple with no compact factors • Γ is arithmetic and irreducible

• Γ_{H} = H ∩ Γ is co-compact in H.

• p > p_{H}(Γ)

• k > \frac{rank(G/K)+1}{2} dim(G/K) + 1

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

err(R, Γ) ≤ C|B_{R}|^{− \frac{1}{(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 existence of a non-quantitative error term for symmetric spaces was established in [1] and improved in [14]. We note that in case of the hyperbolic plane our error term is still far from the quality of the bound of A. Selberg. This is because we only use a weak version of the trace formula, namely Weyl’s law, and use simple soft Sobolev bounds between eigenfunctions on Y.

2 Reductive homogeneous spaces

In this section we review a few facts on reductive homogeneous spaces: the Mostow decomposition, the associated geometric balls and their factorizations. We use the convention that real Lie groups are denoted by upper case Latin letters, e.g A, B, C, and their Lie algebras by the corresponding lower case German letter 𝔞, 𝔟, 𝔠. Throughout this paper G will denote an algebraic real reductive group and H < G is an algebraic subgroup. We form the homogeneous space Z = G/H and write z_{0} = H for the standard base point. Furthermore, unless otherwise mentioned we assume that H is reductive in G, that is, the adjoint representation of H on 𝔤 is completely reducible. In this case we say that G/H is of reductive type. Let us fix a maximal compact subgroup K < G for which we assume that the associated Cartan involution θ leaves H invariant (see the references to [21], Lemma 2.1). Attached to θ is the infinitesimal Cartan decomposition 𝔤 = 𝔨 + 𝔰 where 𝔰 = 𝔨^{⊥} is the orthogonal complement with respect to a non-degenerate invariant bilinear form κ on 𝔤 which is positive definite on 𝔰 (if 𝔤 is semi-simple, then we can take for κ the Cartan-Killing form). Further we set 𝔮 := 𝔥^{⊥}.

2.1 Mostow decomposition

We recall Mostow’s polar decomposition:

K ×_{H∩K} 𝔮 ∩ 𝔰 → Z, [k, X] ↦ k exp(X) · z_{0}  (2.1)

which is a homeomorphism. With that we define

∥k exp(X) · z_{0}∥_{Z} = ∥X∥ := κ(X, X)^{\frac{1}{2}}

for k ∈ K and X ∈ 𝔮 ∩ 𝔰.

2.2 Geometric balls

The problem of lattice counting in Z leads to a question of exhibiting natural exhausting families of compact subsets. We use balls which are intrinsically defined by the geometry of Z. We define the intrinsic ball of radius R > 0 on Z by

B_{R} := {z ∈ Z | ∥z∥_{Z} < R} .

Write B_{R}^{G} for the intrinsic ball of Z = G, that is, if g = k exp(X) with k ∈ K and X ∈ 𝔰, then we put ∥g∥_{G} = ∥X∥ and define B_{R}^{G} accordingly. Our first interest is the growth of the volume |B_{R}| for R → ∞. We have the following upper bound.

Lemma 2.1. There exists a constant c > 0 such that:

|B_{R+r}| ≤ e^{cr}|B_{R}|

for all R ≥ 1, r ≥ 0.

Proof. Recall the integral formula

∫_{Z} f(z) dz = ∫_{K} ∫_{𝔮∩𝔰} f(k exp(X).z_{0})δ(X) dX dk,  (2.2)

for f ∈ C_{c}(Z), where δ(Y) is the Jacobian at (k, Y) of the map (2.1). It is independent of k because dz is invariant. Then

|B_{R}| = ∫_{X∈𝔮∩𝔰,∥X∥<R} δ(X) dX .

Hence it suffices to prove that there exists c > 0 such that

∫_{0}^{R+r} δ(tX)t^{l−1} dt ≤ e^{cr} ∫_{0}^{R} δ(tX)t^{l−1} dt

for all X ∈ 𝔮 ∩ 𝔰 with ∥X∥ = 1. Here l = dim 𝔮 ∩ 𝔰. Equivalently, the function

R ↦ e^{−cR} ∫_{0}^{R} δ(tX)t^{l−1} dt

is decreasing, or by differentiation,

δ(RX)R^{l−1} ≤ c ∫_{0}^{R} δ(tX)t^{l−1} dt

for all R. The latter inequality is established in [12, Lemma A.3] with c independent of X.

Further we are interested how the volume behaves under distortion by elements from G.

Lemma 2.2. For all r, R > 0 one has B_{r}^{G}B_{R} ⊂ B_{R+r}.

To prove the lemma we first record that:

Lemma 2.3. Let z = gH ∈ Z. Then ∥z∥_{Z} = inf_{h∈H} ∥gh∥_{G}.

Proof. It suffices to prove that ∥ exp(X)h∥_{G} ≥ ∥X∥ for X ∈ 𝔮 ∩ 𝔰, h ∈ H, and by Cartan decomposition of H, we may assume h = exp(T) with T ∈ 𝔥 ∩ 𝔰. Thus we have reduced to the statement that

∥ exp(X) exp(T)∥_{G} ≥ ∥ exp(X)∥_{G}

for X ⊥ T in 𝔰. In order to see this, we note that for each g ∈ G the norm ∥g∥_{G} is the length of the geodesic in K\G which joins the origin x_{0} to x_{0}g. More generally the geodesic between x_{0}g_{1} and x_{0}g_{2} has length ∥g_{2}g_{1}^{−1}∥_{G}. Hence c = ∥ exp(X) exp(T)∥_{G} is the distance from A = x_{0} exp(−T) to B = x_{0} exp(X). As X ⊥ T the points A and B form a right triangle with C = x_{0}. The hypotenuse has length c and the leg CB has length a = ∥ exp(X)∥. As the sectional curvatures are non-positive we have a^{2} + b^{2} ≤ c^{2}. In particular a ≤ c.

In particular, it follows that

∥gz∥_{Z} ≤ ∥z∥_{Z} + ∥g∥_{G} (z ∈ Z, g ∈ G)  (2.3)

and Lemma 2.2 follows.

Remark 2.4. Observe that the norm ∥ · ∥_{G} on G depends on the chosen Cartan decomposition θ. However, by applying (2.3) with Z = G one sees that the norm obtained with a conjugate θ′ of θ will satisfy

∥g∥′_{G} ≤ ∥g∥_{G} + c,  ∥g∥_{G} ≤ ∥g∥′_{G} + c′  (2.4)

for all g ∈ G with some constants c, c′ ≥ 0. For the definition of ∥ · ∥_{Z} we assumed that θ leaves H invariant. If instead we use the identity in Lemma 2.3 as the definition of ∥ · ∥_{Z} then this assumption can be avoided. In any case, it follows that the norms on Z obtained from two different Cartan involutions will satisfy similar inequalities as (2.4). The corresponding families of balls are then also compatible,

B_{R} ⊂ B′_{R+c}, B′_{R} ⊂ B_{R+c′},

for all R > 0.

2.3 Factorization

By a (reductive) factorization of Z = G/H we understand a homogeneous space Z^{⋆} = G/H^{⋆} with H^{⋆} an algebraic subgroup of G such that

• H^{⋆} is reductive.

• H ⊂ H^{⋆}.

A factorization is called compact if Z^{⋆} is compact, and co-compact if the fiber space F := H^{⋆}/H is compact. It is called proper if dim H < dim H^{⋆} < dim G.

Lemma 2.5. Let Z = G/H → Z^{⋆} = G/H^{⋆} be a factorization. Then the following assertions are equivalent:

1. Z → Z^{⋆} is co-compact.

2. There exist a compact subgroup K^{⋆} < H^{⋆} such that K^{⋆}H = H^{⋆}.

3. There exists a compact subalgebra 𝔨^{⋆} < 𝔥^{⋆} such that 𝔥^{⋆} = 𝔨^{⋆} + 𝔥 and exp(𝔨^{⋆}) < H^{⋆} compact.

Proof. First (1) implies (2) by the Mostow decomposition of the reductive homogeneous space H^{⋆}/H. Clearly (2) implies (3) as the multiplication map K^{⋆} × H → H^{⋆} needs to be submersive by Sard’s theorem. Finally, for (3) implies (1) we observe that H^{⋆}/H has finitely many components and exp(𝔨^{⋆})H is compact and open in there.

Let F → Z → Z^{⋆} be a factorization of Z. We write B_{R}^{⋆} and B_{R}^{F} for the intrinsic balls in Z^{⋆} and F, respectively.

Lemma 2.6. We have B_{R}^{⋆} = B_{R}H^{⋆}/H^{⋆} and B_{R}^{F} = B_{R} ∩ F.

Proof. Follows from Lemma 2.3.

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

ϕ^{F}(gH^{⋆}) := ∫_{H^{⋆}/H} ϕ(gh^{⋆}) d(h^{⋆}H)

and recall the integration formula

∫_{Z} ϕ(gH) d(gH) = ∫_{Z^{⋆}} ϕ^{F}(gH^{⋆}) d(gH^{⋆})  (2.5)

under appropriate normalization of measures. Consider the characteristic function 1_{R} of B_{R} and note that its fiber average 1_{R}^{F} is supported in the compact ball B_{R}^{⋆}. We say that the family of balls (B_{R})_{R>0} factorizes well to Z^{⋆} provided for all compact subsets Q ⊂ G

lim_{R→∞} \frac{sup_{g∈Q} 1_{R}^{F}(gH^{⋆})}{|B_{R}|} = 0 .  (2.6)

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Observe that for all compact subsets Q there exists an R_{0} = R_{0}(Q) > 0 such that

sup_{g∈Q} 1_{R}^{F}(gH^{⋆}) ≤ |B_{R+R_{0}}^{F}|

by Lemma 2.2. Thus the balls B_{R} factorize well provided

lim_{R→∞} \frac{|B_{R+R_{0}}^{F}|}{|B_{R}|} = 0 .  (2.7)

for all R_{0} > 0.

Remark 2.7. The condition that the balls B_{R} factorize well is closely related to the non-focusing condition (Definition 1.14 in [12]). Thus, in the case of semi-simple connected H, the non-focusing condition of the intrinsic balls is implied by the condition that they factorize well to all factorizations.

2.4 Basic factorizations

There is a special class of factorizations with which we are dealing with in the sequel. ¿From now on we assume that 𝔤 is semi-simple and write

𝔤 = 𝔤_{1} ⊕ … ⊕ 𝔤_{m}

for the decomposition into simple ideals. For a reductive subalgebra 𝔥 < 𝔤 and a subset I ⊂ {1, …, m} we define the reductive subalgebra

𝔥_{I} := 𝔥 + 𝔤_{I} = 𝔥 + ⨁_{i∈I} 𝔤_{i} .  (2.8)

We say that the factorization is basic provided that 𝔥^{∗} = 𝔥_{I} for some I. Finally we call a factorization weakly basic if it is built from consecutive basic and cocompact factorizations, that is, there exists a sequence

𝔥^{⋆} = 𝔥^{k} ⊃ ⋯ ⊃ 𝔥^{0} = 𝔥  (2.9)

of reductive subalgebras such that for each i we have 𝔥^{i} = (𝔥^{i−1})_{I} for some I or 𝔥^{i}/𝔥^{i−1} is compact. The following lemma shows that in fact it suffices with k ≤ 2.

Lemma 2.8. Let Z → Z^{⋆} be a weakly basic factorization. Then there exists an intermediate factorization Z → Z_{b} → Z^{⋆} such that Z → Z_{b} is basic and Z_{b} → Z^{⋆} co-compact.

Proof. Let a sequence (2.9) of factorizations which are consecutively basic or compact be given. We first observe that two consecutive basic factorizations make up for a single basic factorization, and likewise two consecutive compact factorizations yield a single compact factorization by Lemma 2.5. Hence it suffices to prove that we can modify a string

𝔥^{i+2} ⊃ 𝔥^{i+1} ⊃ 𝔥^{i}

with 𝔥^{i+2}/𝔥^{i+1} basic and 𝔥^{i+1}/𝔥^{i} compact to

𝔥^{i+2} ⊃ 𝔥_{b}^{i+1} ⊃ 𝔥^{i}

with 𝔥^{i+2}/𝔥_{b}^{i+1} compact and 𝔥_{b}^{i+1}/𝔥^{i} basic. We have 𝔥^{i+2} = 𝔥^{i+1} + 𝔤_{I} for some I, and by Lemma 2.5 that 𝔥^{i+1} = 𝔥^{i} + 𝔠 with 𝔠 compact. Then 𝔥_{b}^{i+1} := 𝔥^{i} + 𝔤_{I} is a reductive subalgebra and a basic factorization of 𝔥^{i}. Furthermore 𝔥^{i+2} = 𝔥_{b}^{i+1} + 𝔠. This establishes the lemma

3 Wavefront real spherical spaces

We assume that Z is real spherical, i.e. a minimal parabolic subgroup P < G has an open orbit on Z. It is no loss of generality to assume that P H ⊂ G is open, or equivalently that 𝔤 = 𝔥 + 𝔭. If L is a real algebraic group, then we write L_{n} for the normal subgroup of L which is generated by all unipotent element. In case L is reductive we observe that 𝔩_{n} is the sum of all non-compact simple ideals of 𝔩. According to [20] there is a unique parabolic subgroup Q ⊃ P with the following two properties:

• QH = P H.

• There is a Levi decomposition Q = LU with L_{n} ⊂ Q ∩ H ⊂ L.

Following [20] we call Q a Z-adapted parabolic subgroup. Having fixed L we let L = K_{L}A_{L}N_{L} be an Iwasawa decomposition of L. We choose an Iwasawa decomposition G = KAN which inflates the one of L, i.e. K_{L} < K, A_{L} = A and N_{L} < N. Further we may assume that N is the unipotent radical of the minimal parabolic P.

Remark 3.1. It should be noted that the assumption on the Cartan decomposition θ, which was demanded in Section 2.2, may be overruled by the above requirement to K. However, it follows from Remark 2.4 that the balls B_{R} can still be defined, and that the difference does not disturb the lattice counting on Z.

Set A_{H} := A ∩ H and put A_{Z} = A/A_{H}. We recall that dim A_{Z} is an invariant of the real spherical space, called the real rank (see [20]). In [18], Section 6, we defined the notion of wavefront for a real spherical space, which we quickly recall. Attached to Z is a geometric invariant, the so-called compression cone which is a closed and convex subcone 𝔞_{Z}^{−} of 𝔞_{Z}. It is defined as follows. Write Σ_{𝔲} for the space of 𝔞-weights of the 𝔞-module 𝔲 and let \overline{𝔲} denote the corresponding sum of root spaces for −Σ_{𝔲}. According to [20] there exists a linear map

T : ⊕_{α∈Σ_{𝔲}}𝔤^{−α} = \overline{𝔲} → 𝔩_{H}^{⊥} ⊕ 𝔲 ⊂ ⊕_{β∈\{0\}∪Σ_{𝔲}}𝔤^{β}  (3.1)

such that 𝔥 = 𝔩∩𝔥+\{\overline{X} +T (\overline{X}) | \overline{X} ∈ \overline{𝔲}\}. Here 𝔩_{H}^{⊥} denotes the orthocomplement of 𝔩 ∩ 𝔥 in 𝔩. For each pair α, β we denote by

T_{α,β} : 𝔤^{−α} → 𝔤^{β}

the map obtained from T by restriction to 𝔤^{−α} and projection to 𝔤^{β}. Then T = ∑_{α,β} T_{α,β} and by definition

𝔞_{Z}^{−} = {X ∈ 𝔞 | (α + β)(X) ≥ 0, ∀α, β with T_{α,β} ≠ 0}.

It follows from (3.1) that α + β vanishes on 𝔞_{H} if T_{α,β} ≠ 0. Hence 𝔞_{Z}^{−} ⊂ 𝔞_{Z}. If one denotes by 𝔞^{−} ⊂ 𝔞 the closure of the negative Weyl chamber, then 𝔞^{−} + 𝔞_{H} ⊂ 𝔞_{Z}^{−} and by definition Z is wavefront if

𝔞^{−} + 𝔞_{H} = 𝔞_{Z}^{−}.

Let us mention that many real spherical spaces are wavefront; for example all symmetric spaces and all Gross-Prasad type spaces Z = G × H/H with (G, H) one of the following

(GL_{n+1}(ℂ), GL_{n}(ℂ)), (GL_{n+1}(ℝ), GL_{n}(ℝ)),  (3.2)
(GL_{n+1}(ℍ), GL_{n}(ℍ)), (U(p + 1, q), U(p, q)),  (3.3)
(SO(n + 1, ℂ), SO(n, ℂ)), (SO(p + 1, q), SO(p, q)) .  (3.4)

We recall from [18] the polar decomposition for real spherical spaces

Z = ΩA_{Z}^{−}F · z_{0}  (3.5)

where

• Ω is a compact set of the type F′K with F′ ⊂ G a finite set.

• F ⊂ G is a finite set with the property that F · z_{0} = T · z_{0} ∩ Z where T = exp(i𝔞) and the intersection is taken in Z_{ℂ} = G_{ℂ}/H_{ℂ}.

3.1 Volume growth

Define ρ_{Q} ∈ 𝔞^{∗} by ρ_{Q}(X) = \frac{1}{2}tr(ad_{𝔲} X), X ∈ 𝔞. It follows from the unimodularity of Z and the local structure theorem that ρ_{Q}|_{𝔞_{H}} = 0, i.e. ρ_{Q} ∈ 𝔞_{Z}^{∗} = 𝔞_{H}^{⊥}.

Lemma 3.2. Let Z = G/H be a wavefront real spherical space. Then

|B_{R}| ≍ sup_{{X∈𝔞 \atop ∥X∥≤R}} e^{2ρ_{Q}(X)} = sup_{{X∈𝔞{− \atop Z} \atop ∥X∥≤R}} e^{−2ρ_{Q}(X)} .  (3.6)

Here the expression f(R) ≍ g(R) signifies that the ratio \frac{f(R)}{g(R)} remains bounded below and above as R tends to infinity.

Proof. First note that the equality in (3.6) is immediate from the wavefront assumption. Let us first show the lower bound, i.e. there exists a C > 0 such that for all R > 0 one has

|B_{R}| ≥ C sup_{{X∈𝔞 \atop ∥X∥≤R}} e^{2ρ_{Q}(X)} .

For that we recall the volume bound from [19], Prop. 4.2: for all compact subsets B ⊂ G with non-empty interior there exists a constant C > 0 such that vol_{Z}(Ba · z_{0}) ≥ Ca^{2ρ_{Q}} for all a ∈ A_{Z}^{−}. Together with the polar decomposition (3.5) this gives us the lower bound. As for the upper bound let

𝔞_{R}^{−} := {X ∈ 𝔞^{−} | ∥X∥ ≤ R} .

Observe that B_{R} ⊂ B′_{R} := KA_{R}^{−}K · z_{0}. In the sequel it is convenient to realize A_{Z} as a subgroup of A (and not as quotient): we identify A_{Z} with A_{H}^{⊥} ⊂ A. The upper bound will follow if we can show that

|B′_{R}| ≤ C sup_{{X∈𝔞 \atop ∥X∥≤R}} e^{2ρ_{Q}(X)} (R > 0) .

for some constant C > 0. This in turn will follow from the argument for the upper bound in the proof of Prop. 4.2 in [19]: in this proof we considered for a ∈ A_{Z}^{−} the map

Φ_{a} : K × Ω_{A} × Ξ → G, (k, b, X) ↦ kb exp(Ad(a)X)

where Ω_{A} ⊂ A is a compact neighborhood of 1 and Ξ ⊂ 𝔥 is a compact neighborhood of 0. It was shown that the Jacobian of Φ_{a}, that is \sqrt{det(dΦ_{a}dΦ_{a}^{t})}, is bounded by Ca^{−2ρ_{Q}}. Now this bounds holds as well for the right K-distorted map

Ψ_{a} : K × Ω_{A} × K × Ξ → G, (k, b, k′, X) ↦ kb exp(Ad(ak′)X) .

The reason for that comes from an inspection of the proof; all what is needed is the following fact: let d := dim 𝔥 and consider the action of Ad(a) on V = ⋀^{d} 𝔤. Then for a ∈ A^{−} we have

a^{−2ρ} ≥ sup_{{v∈V, \atop ∥v∥=1}}⟨Ad(a)v, v⟩ .

We deduce an upper bound

vol_{Z}(KΩ_{A}aK · z_{0}) ≤ Ca^{−2ρ} .  (3.7)

We need to improve that bound from ρ to ρ_{Q} on the right hand side of (3.7). For that let W_{L} be the Weyl group of the reductive pair (𝔩, 𝔞). Note that ρ_{Q} = \frac{1}{|W_{L}|} ∑_{w∈W_{L}} w · ρ. Further, the local structure theorem implies that L_{n} ⊂ H and hence W_{L} can be realized as a subgroup of W_{H∩K} := N_{H∩K}(𝔞)/Z_{H∩K}(𝔞). We choose Ω_{A} to be invariant under N_{H∩K}(𝔞) and observe that a ∈ A_{Z} is fixed under W_{H∩K}. Thus using the N_{H∩K}(𝔞)-symmetry in the a-variable we refine (3.7) to

vol_{Z}(KΩ_{A}aK · z_{0}) ≤ Ca^{−2ρ_{Q}} .

The desired bound then follows.

Corollary 3.3. Let Z = G/H be a wavefront real spherical space of reductive type. Let Z → Z^{⋆} be a basic factorization such that Z^{⋆} is not compact. Then the geometric balls B_{R} factorize well to Z^{⋆}.

Proof. As Z → Z^{⋆} is basic we may assume (ignoring connected components) that H^{⋆} = G_{I}H for some I. Note that F = H^{⋆}/H ≃ G_{I}/G_{I} ∩ H is real spherical. Let Q be the Z-adapted parabolic subgroup attached to P. Let P_{I} = P ∩G_{I} and G_{I} ⊃ Q_{I} ⊃ P_{I} be the F-adapted parabolic above P_{I} and note that Q_{I} = Q∩G_{I}. With Lemma 3.2 we then get

|B_{R}^{F}| ≍ sup_{{X∈𝔞_{I} \atop ∥X∥≤R}} e^{2ρ_{Q_{I}} (X)} ,

which we are going to compare with (3.6). Let 𝔲_{I} be the Lie algebra of the unipotent radical of Q_{I}. Note that 𝔲_{I} ⊂ 𝔲 and that this inclusion is strict since G/H^{⋆} is not compact. The corollary now follows from (2.7).

3.2 Property I

We briefly recall some results from [19]. Let (π, H_{π}) be a unitary irreducible representation of G. We denote by H_{π}^{∞} the G-Fréchet module of smooth vectors and by H_{π}^{−∞} its strong dual. One calls H_{π}^{−∞} the G-module of distribution vectors; it is a DNF-space with continuous G-action. Let η ∈ (H_{π}^{−∞})^{H} be an H-fixed element and H_{η} < G the stabilizer of η. Note that H < H_{η} and set Z_{η} := G/H_{η}. With regard to η and v ∈ H^{∞} we form the generalized matrix-coefficient

m_{v,η}(gH) := η(π(g^{−1})v) (g ∈ G)

which is a smooth function on Z_{η}. We recall the following facts from [19] Thm. 7.6 and Prop. 7.7:

Proposition 3.4. Let Z be a wavefront real spherical space of reductive type. Then the following assertions hold:

1. Every generalized matrix coefficient m_{v,η} as above is bounded.

2. Let H < H^{⋆} < G be a closed subgroup such that Z^{⋆} is unimodular. Then Z^{⋆} is a weakly basic factorization.

3. Let (π, H) be a unitary irreducible representation of G and let η ∈ (H_{π}^{−∞})^{H}. Then:

(a) Z → Z_{η} is a weakly basic factorization.

(b) Z_{η} is unimodular and there exists 1 ≤ p < ∞ such that m_{v,η} ∈ L^{p}(Z_{η}) for all v ∈ H_{π}^{∞}.

The property of Z = G/H that (3b) is valid for all π and η as above is denoted Property (I) in [19]. Note that (1) and (3b) together imply m_{v,η} ∈ L^{q}(Z_{η}) for q > p. Assuming Property (I) we can then make the following notation.

Definition 3.5. Given π as above, define p_{H}(π) as the smallest index ≥ 1 such that all K-finite generalized matrix coefficients m_{v,η} with η ∈ (H_{π}^{−∞})^{H} belong to L^{p}(Z_{η}) for any p > p_{H}(π).

Notice that m_{v,η} belongs to L^{p}(Z_{η}) for all K-finite vectors v once that this is the case for some non-trivial such vector v, see [19] Lemma 7.2. For example, this could be the trivial K-type, if it exists in π. It follows from finite dimensionality of (H_{π}^{−∞})^{H} (see [23]) that p_{H}(π) < ∞. We say that π is H-tempered if p_{H}(π) = 2. The representation π is said to be H-distinguished if (H_{π}^{−∞})^{H} ≠ {0}. Note that if π is not H-distinguished then p_{H}(π) = 1.

4 Lattice point counting: setup

Let G/H be a real algebraic homogeneous space. We further assume that we are given a lattice (a discrete subgroup with finite covolume) Γ ⊂ G, such that Γ_{H} := Γ ∩ H is a lattice in H. We normalize Haar measures on G and H such that:

• vol(G/Γ) = 1.

• vol(H/Γ_{H}) = 1.

Our concern is with the double fibration

\xymatrix{ & G/Γ_{H} \ar[dl] \ar[dr] \\ Z := G/H &  & Y := G/Γ}

Fibre-wise integration yields transfer maps from functions on Z to functions on Y and vice versa. In more precision,

L^{∞}(Y) → L^{∞}(Z), ϕ ↦ ϕ^{H}; ϕ^{H}(gH) := ∫_{H/Γ_{H}} ϕ(ghΓ) d(hΓ_{H})  (4.1)

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

∥ϕ^{H}∥_{∞} ≤ ∥ϕ∥_{∞}  (ϕ ∈ L^{∞}(Y)) .  (4.2)

Likewise we have

L^{1}(Z) → L^{1}(Y), f ↦ f^{Γ}; f^{Γ}(gΓ) := ∑_{γ∈Γ/Γ_{H}} f(gγH) ,  (4.3)

which is contractive, i.e

∥f^{Γ}∥_{1} ≤ ∥f∥_{1}  (f ∈ L^{1}(Z)) .  (4.4)

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

⟨f^{Γ}, ϕ⟩_{L^{2}(Y)} = ⟨f, ϕ^{H}⟩_{L^{2}(Z)}  (4.5)

for all ϕ ∈ L^{∞}(Y) and f ∈ L^{1}(Z). Let us note that (4.5) applied to |f| and ϕ = 1_{Y} readily yields (4.4). We write 1_{R} ∈ L^{1}(Z) for the characteristic function of B_{R} and deduce from the definitions and (4.5):

• 1_{R}^{Γ}(eΓ) = N_{R}(Γ, Z) := #{γ ∈ Γ/Γ_{H} | γ · z_{0} ∈ B_{R}}.

• ∥1_{R}^{Γ}∥_{L^{1}(G/Γ)} = |B_{R}|.

4.1 Weak asymptotics

In the above setup, G/H need not be of reductive type, but we shall assume this again from now on. For spaces with property (I) and Y compact we prove analytically in the following section that

N_{R}(Γ, Z) ∼ |B_{R}| (R → ∞) .  (MT)

For that we will use the following result of [21]:

Theorem 4.1. Let Z = G/H be of reductive type. The smooth vectors for the regular representation of G on L^{p}(Z) vanish at infinity, for all 1 ≤ p < ∞.

With notation from (4.3) we set

F_{R}^{Γ} := \frac{1}{|B_{R}|}1_{R}^{Γ}.

We shall concentrate on verifying the following limit of weak type:

⟨F_{R}^{Γ}, ϕ⟩_{L^{2}(Y)} → ∫_{Y} ϕ̄ dμ_{Y} (R → ∞),  (∀ϕ ∈ C_{0}(Y)) .  (wMT)

Here C_{0} indicates functions vanishing at infinity.

Lemma 4.2. (wMT) ⇒ (MT).

Proof. As in [10] Lemma 2.3 this is deduced from Lemma 2.1 and Lemma 2.2.

5 Main term counting

In this section we will establish main term counting under the mandate of property (I) and Y being compact. Let us call a family of balls (B_{R})_{R>0} well factorizable if it factorizes well to all proper factorizations of type Z → Z_{η}.

5.1 Main theorem on counting

Theorem 5.1. Let G be semi-simple and H a closed reductive subgroup. Suppose that Y is compact and Z admits (I). If (B_{R})_{R>0} is well factorizable, then (wMT) and (MT) hold.

Remark 5.2. In case Z = G/H is real spherical and wavefront, then Z has (I) by Proposition 3.4. If we assume in addition that G has no compact factors and that all proper factorizations are basic, then the family of geometric balls is well factorizable by Corollary 3.3. In particular, Theorem A of the introduction then follows from the above.

The proof is based on the following proposition. For a function space F(Y) consisting of integrable functions on Y we denote by F(Y)_{van} the subspace of functions with vanishing integral over Y.

Proposition 5.3. Let Z = G/H be of reductive type. Assume that there exists a dense subspace A(Y) ⊂ C_{b}(Y)_{van}^{K} such that

ϕ^{H} ∈ C_{0}(Z) for all ϕ ∈ A(Y) .  (5.1)

Then (wMT) holds true.

Proof. We will establish (wMT) for ϕ ∈ C_{b}(Y). As

C_{b}(Y) = C_{b}(Y)_{van} ⊕ ℂ1_{Y} ,

and (wMT) is trivial for ϕ a constant, it suffices to establish

⟨F_{R}^{Γ}, ϕ⟩_{L^{2}(Y)} → 0  (ϕ ∈ C_{b}(Y)_{van}) .  (5.2)

We will show (5.2) is valid for ϕ ∈ A(Y). By density, as F_{R}^{Γ} is K-invariant and belongs to L^{1}(Y), this will finish the proof. Let ϕ ∈ A(Y) and let ϵ > 0. By the unfolding identity (4.5) we have

⟨F_{R}^{Γ}, ϕ⟩_{L^{2}(Y)} = \frac{1}{|B_{R}|}⟨1_{R}, ϕ^{H}⟩_{L^{2}(Z)}.  (5.3)

Using (5.1) we choose K_{ϵ} ⊂ Z compact such that |ϕ^{H}(z)| < ϵ outside of K_{ϵ}. Then

\frac{1}{|B_{R}|}⟨1_{R}, ϕ^{H}⟩_{L^{2}(Z)} = ∫_{K_{ϵ}} + ∫_{Z−K_{ϵ}} \frac{1_{R}(z)}{|B_{R}|}ϕ^{H}(z) dμ_{Z}(z) .

By (4.2), the first term is bounded by \frac{|K_{ϵ}|||ϕ||_{∞}}{|B_{R}|}, which is ≤ ϵ for R sufficiently large. As the second term is bounded by ϵ for all R, we obtain (5.2). Hence (wMT) holds.

Remark 5.4. It is possible to replace (5.1) by a weaker requirement: Suppose that an algebraic sum

A(Y) = ∑_{j∈J} A(Y)_{j}  (5.4)

is given together with a factorization Z_{j}^{⋆} = G/H_{j}^{⋆} for each j ∈ J. Suppose that the balls B_{R} all factorize well to Z_{j}^{⋆}, j ∈ J. Suppose further that ϕ^{H} factorizes to a function

ϕ^{H_{j}^{⋆}} ∈ C_{0}(Z_{j}^{⋆})  (5.5)

for all ϕ ∈ A(Y)_{j} and all j ∈ J. Then the conclusion in Proposition 5.3 is still valid. In fact, using (2.5) the last part of the proof modifies to:

\frac{1}{|B_{R}|}⟨1_{R}, ϕ^{H}⟩_{L^{2}(Z)} = \frac{1}{|B_{R}|}⟨1_{R}^{F}, ϕ^{H_{j}^{⋆}}⟩_{L^{2}(Z_{j}^{⋆})} =

= ∫_{K_{ϵ}^{⋆}} + ∫_{Z_{j}^{⋆}−K_{ϵ}^{⋆}} \frac{1_{R}^{F}(z)}{|B_{R}|}ϕ^{H_{j}^{⋆}}(z) dμ_{Z_{j}^{⋆}}(z)

for ϕ ∈ A(Y)_{j}. As ∥1_{R}^{F}∥_{L^{1}(Z_{j}^{⋆})} = |B_{R}|, the second term is bounded by ϵ for all R. As the balls factorize well to Z_{j}^{⋆} we get the first term as small as we wish with (2.6).

5.2 The space A(Y)

We now construct a specific subspace A(Y) ⊂ C_{b}(Y)_{van}^{K} and verify condition (5.5). Denote by Ĝ_{s} ⊂ Ĝ the K-spherical unitary dual. As Y is compact, the abstract Plancherel-theorem implies:

L^{2}(G/Γ)^{K} ≃ ⨁_{π∈Ĝ_{s}}(H_{π}^{−∞})^{Γ}.

If we denote the Fourier transform by f ↦ f^{∧} then the corresponding inversion formula is given by

f = ∑_{π} a_{v_{π},f^{∧}(π)}.  (5.6)

Here a_{v_{π},f^{∧}(π)} denotes a matrix coefficient for Y with v_{π} ∈ H_{π} normalized K-fixed and f^{∧}(π) ∈ (H_{π}^{−∞})^{Γ}, and the sum in (5.6) is required to include multiplicities. The matrix coefficients for Y are defined as in (1.2), that is

a_{v,ν}(y) = ν(g^{−1} · v)  (y = gH ∈ Y) .  (5.7)

for v ∈ H_{π} and ν ∈ (H_{π}^{−∞})^{Γ}. Note that L^{2}(Y) = L^{2}(Y)_{van} ⊕ ℂ · 1_{Y}. We define A(Y) ⊂ L^{2}(Y)_{van}^{K} to be the dense subspace of functions with finite Fourier support, that is,

A(Y) = span{a_{v,ν} | π ∈ Ĝ_{s} non-trivial, v ∈ H_{π}^{K}, ν ∈ (H_{π}^{−∞})^{Γ}}.

Then A(Y) ⊂ L^{2}(Y)_{van}^{K,∞} is dense and since C^{∞}(Y) and L^{2}(Y)^{∞} are topologically isomorphic, it follows that A(Y) is dense in C(Y)_{van}^{K} as required. The following lemma together with Remark 5.4 immediately implies Theorem 5.1.

Lemma 5.5. Assume that Y is compact and Z has (I), and define A(Y) as above. Then there exists a decomposition of A(Y) satisfying (5.4)-(5.5).

Proof. The map ϕ ↦ ϕ^{H} from (4.1) corresponds on the spectral side to a map (H_{π}^{−∞})^{Γ} → (H_{π}^{−∞})^{H}, which can be constructed as follows. As H/Γ_{H} is compact, we can define for each π ∈ Ĝ_{s}

Λ_{π} : (H_{π}^{−∞})^{Γ} → (H_{π}^{−∞})^{H}, Λ_{π}(ν) = ∫_{H/Γ_{H}} ν ∘ π(h^{−1}) d(hΓ_{H})  (5.8)

by H_{π}^{−∞}-valued integration: the defining integral is understood as integration over a compact fundamental domain F ⊂ H with respect to the Haar measure on H; as the integrand is continuous and H_{π}^{−∞} is a complete locally convex space, the integral converges in H_{π}^{−∞}. It follows from (5.8) that (a_{v,ν})^{H} = m_{v,Λ_{π}(ν)} for all v ∈ H_{π}^{∞} and ν ∈ (H_{π}^{−∞})^{Γ}. Let J denote the set of all factorizations Z^{⋆} → Z, including also Z^{⋆} = Z which we give the index j_{0} ∈ J. For j ∈ J we define A(Y)_{j} ⊂ A(Y) accordingly to be spanned by the matrix coefficients a_{v,ν} for which H_{Λ_{π}(ν)} = H_{j}^{⋆}. Then (5.4) holds. Let ϕ ∈ A(Y)_{j_{0}}, then it follows from (5.6) that

ϕ^{H} = ∑_{π≠1} m_{v_{π},Λ_{π}(ϕ^{∧}(π))} .  (5.9)

Note that H_{η} = H for each distribution vector η = Λ_{π}(ϕ^{∧}(π)) in this sum, by the definition of A(Y)_{j_{0}}. As Z has property (I) the summand m_{v_{π},Λ_{π}(ϕ^{∧}(π))} is contained in L^{p}(G/H) for p > p_{H}(π), and by [19], Lemma 7.2, this containment is then valid for all K-finite generalized matrix coefficients m_{v,Λ_{π}(ϕ^{∧}(π))} of π. Thus m_{v_{π},Λ_{π}(ϕ^{∧}(π))} generates a Harish-Chandra module inside L^{p}(G/H). As m_{v_{π},Λ_{π}(ϕ^{∧}(π))} is K-finite, we conclude that it is a smooth vector. Hence ϕ^{H} ∈ L^{p}(G/H)^{∞}, and in view of Theorem 4.1 we obtain (5.1). The proof of (5.5) for ϕ ∈ A(Y)_{j} for general j ∈ J is obtained by the same reasoning, where one replaces H by H_{j}^{⋆} in (5.8) and (5.9).

This concludes the proof of Theorem 5.1.

6 L^{p}-bounds for generalized matrix coefficients

¿From here on we assume that Z = G/H is wavefront and real spherical. Recall that we assumed that G is semi-simple and that we wrote 𝔤 = 𝔤_{1} ⊕ … ⊕ 𝔤_{m} for the decomposition of 𝔤 into simple factors. It is no big loss of generality to assume that G = G_{1} × … × G_{m} splits accordingly. We will assume that from now on. Further we request that the lattice Γ < G is irreducible, that is, the projection of Γ to any normal subgroup J ⊊ G is dense in J. Let π be an irreducible unitary representation of G. Then π = π_{1} ⊗ … ⊗ π_{m} with π_{j} and irreducible unitary representation of G_{j}. We start with a simple observation.

Lemma 6.1. Let (π, H) be an irreducible unitary representation of G and 0 ≠ ν ∈ (H^{−∞})^{Γ}. If one constituent π_{j} of π is trivial, then π is trivial.

Proof. The element ν gives rise to a G-equivariant injection

H^{∞} ↪ C^{∞}(Y), v ↦ (gΓ ↦ ν(π(g^{−1})v)) .  (6.1)

Say π_{j} is trivial and let J := ∏m_{{i=1 \atop i≠j}} G_{i}. Let Γ_{J} be the projection of Γ to J. Then (6.1) gives rise to a J-equivariant injection H^{∞} ↪ C^{∞}(J/Γ_{J}). As Γ_{J} is dense in J, the assertion follows.

We assume from now on that the cycle H/Γ_{H} ⊂ Y is compact. This technical condition ensures that the vector valued average map (5.8) converges.

Lemma 6.2. Let (π, H) be a non-trivial irreducible unitary representation of G. Let ν ∈ (H_{π}^{−∞})^{Γ} such that η := Λ_{π}(ν) ∈ (H_{π}^{−∞})^{H} is non-zero. Then H_{η}/H is compact.

Proof. Recall from Proposition 3.4 that Z → Z_{η} is weakly basic, and from Lemma 2.8 that then there exists H ⊂ H_{b} ⊂ H_{η} such that H_{η}/H_{b} is compact and Z → Z_{b} is basic. Hence 𝔥_{b} = 𝔥_{I} for some I. As π is irreducible it infinitesimally embeds into C^{∞}(Z_{η}) and hence also to C^{∞}(Z_{b}) on which G_{i} acts trivially for i ∈ I. It follows that π_{i} is trivial for i ∈ I. Hence Lemma 6.1 implies I = ∅ and thus 𝔥_{b} = 𝔥.

In the sequel we use the Plancherel theorem (see [15])

L^{2}(G/Γ)^{K} ≃ ∫ ⊕ ; Ĝ_{s} V_{π,Γ} dμ(π) ,

where V_{π,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 measure μ has support

Ĝ_{Γ,s} := supp(μ) ⊂ Ĝ_{s} .

Given an irreducible lattice Γ ⊂ G we define (cf. Definition 3.5)

p_{H}(Γ) := sup{p_{H}(π) : π ∈ Ĝ_{Γ,s}}  (6.2)

and record the following.

Lemma 6.3. Assume that G = G_{1} × … × G_{m} with all 𝔤_{i} simple and noncompact. Then p_{H}(Γ) < ∞.

Proof. For a unitary representation (π, H) and vectors v, w ∈ H we form the matrix coefficient π_{v,w}(g) := ⟨π(g)v, w⟩. We first claim that there exists a p < ∞ (in general depending on Γ) such that for all non-trivial π ∈ Ĝ_{Γ,s} one has π_{v,w} ∈ L^{p}(G) for all K-finite vectors v, w. In case G has property (T) this follows (independently of Γ) from [7]. The remaining cases contain at least one factor G_{i} of SO_{e}(n, 1) or SU(n, 1) (up to covering) and have no compact factors by assumption. They are treated in [6]. The claim can be interpreted geometrically via the leading exponent Λ_{V} ∈ 𝔞^{∗} which is attached to the Harish-Chandra module of H (see [19], Section 6). The lemma now follows from Prop. 4.2 and Thm. 6.3 in [19] (see the proof of Thm. 7.6 in [19] how these two facts combine to result in integrability).

Let 1 ≤ p < ∞. Let us say that a subset Λ ⊂ Ĝ_{s} is L^{p}-bounded provided that m_{v,η} ∈ L^{p}(Z_{η}) for all π ∈ Λ and v ∈ H_{π}^{∞}, η ∈ (H_{π}^{−∞})^{H}. By definition we thus have that Ĝ_{Γ,s} is L^{p}-bounded for p > p_{H}(Γ). In this section we work under the following:

Hypothesis A: For every 1 ≤ p < ∞ and every L^{p}-bounded subset Λ ⊂ Ĝ_{s} there exists a compact subset Ω ⊂ G and constants c, C > 0 such that the following assertions hold for all π ∈ Λ, η ∈ (H_{π}^{−∞})^{H} and v ∈ H_{π}^{K}:

∥m_{v,η}∥_{L^{p}(Z_{η})} ≤ C∥m_{v,η}∥_{∞} ,  (A1)

∥m_{v,η}∥_{∞} ≤ c∥m_{v,η}∥_{∞,Ω_{η}}  (A2)

where Ω_{η} = ΩH_{η}/H_{η}. Here ∥ · ∥_{∞,ω} denotes the supremum norm taken on the subset ω.

In the sequel we are only interested in the following choice of subset Λ ⊂ Ĝ_{s}, namely

Λ := {π ∈ Ĝ_{Γ,s} | Λ_{π}(ν) ≠ 0 for some ν ∈ V_{π,Γ}} .  (6.3)

An immediate consequence of Hypothesis A is:

Lemma 6.4. Assume that p > p_{H}(Γ). Then there is a C > 0 such that for all π ∈ Ĝ_{Γ,s}, v ∈ H_{π}^{K}, ν ∈ (H_{π}^{−∞})^{Γ} and η := Λ_{π}(ν) ∈ (H_{π}^{−∞})^{H} one has

∥ϕ_{π}^{H}∥_{L^{p}(Z_{η})} ≤ C∥ϕ_{π}∥_{∞}

where ϕ_{π}(gΓ) := ν(π(g^{−1})v).

Proof. Recall from (4.2), that integration is a bounded operator from L^{∞}(Y) → L^{∞}(Z). Hence the assertion follows from (A1).

Recall the Cartan-Killing form κ on 𝔤 = 𝔨 + 𝔰 and choose a basis X_{1}, …, X_{l} of 𝔨 and X′_{1}, …, X′_{s} of 𝔰 such that κ(X_{i}, X_{j}) = −δ_{ij} and κ(X′_{i}, X′_{j}) = δ_{ij}. With that data we form the standard Casimir element

C := − ∑_{j=1}^{l} X_{j}^{2} + ∑_{j=1}^{s}(X′_{j})^{2} ∈ U(𝔤) .

Set Δ_{K} := ∑_{j=1}^{l} X_{j}^{2} ∈ U(𝔨) and obtain the commonly used Laplace element

Δ = C + 2Δ_{K} ∈ U(𝔤)  (6.4)

which acts on Y = G/Γ from the left. Let d ∈ ℕ. For 1 ≤ p ≤ ∞, it follows from [2], Section 3, that Sobolev norms on L^{p}(Y)^{∞} ⊂ C^{∞}(Y) can be defined by

||f||_{p,2d}^{2} = ∑_{j=0}^{d} ||Δ^{j}f||_{p}^{2} .

Basic spectral theory allows one to define ∥ · ∥_{p,d} more generally for any d ≥ 0. Let us define

s := dim 𝔰 = dim G/K = dim Γ\G/K

and

r := dim 𝔞 = rank_{ℝ}(G/K) ,

where 𝔞 ⊂ 𝔰 is maximal abelian. We denote by C_{b}(Y) the space of continuous bounded functions on Y and by C_{b}(Y)_{van} the subspace with vanishing integral.

Proposition 6.5. Assume that

1. Z is a wavefront real spherical space,

2. G = G_{1} × … × G_{m} with all 𝔤_{i} simple and non-compact.

3. Γ < G is irreducible and Y_{H} is compact,

4. Hypothesis A is valid.

Let p > p_{H}(Γ). Then the map

Av_{H} : C_{b}^{∞}(Y)_{van}^{K} → L^{p}(Z)^{K}; Av_{H}(ϕ) = ϕ^{H}

is continuous. More precisely, for all

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

2. k > \frac{r+1}{2}s + 1 if Y is non-compact and Γ is arithmetic there exists a constant C = C(p, k) > 0 such that

∥ϕ^{H}∥_{L^{p}(Z)} ≤ C∥ϕ∥_{∞,k} (ϕ ∈ C_{b}^{∞}(Y)_{van}^{K})

Proof. For all π ∈ Ĝ the operator dπ(C) acts as a scalar λ_{π} and we set

|π| := |λ_{π}| ≥ 0 .

Let ϕ ∈ C_{b}^{∞}(Y)_{van}^{K} and write ϕ = ϕ_{d} + ϕ_{c} for its decomposition in discrete and continuous Plancherel parts. We assume first that ϕ = ϕ_{d}. In case Y is compact we have Weyl’s law: There is a constant c_{Y} > 0 such that

∑_{|π|≤R} m(π) ∼ c_{Y} R^{s/2} (R → ∞) .

Here m(π) = dim V_{π,Γ}. We conclude that

∑_{π} m(π)(1 + |π|)^{−k} < ∞  (6.5)

for all k > s/2 + 1. In case Y is non-compact, we let Ĝ_{μ,d} be the the discrete support of the Plancherel measure. Then assuming Γ is arithmetic, the upper bound in [16] reads:

∑_{{π∈Ĝ_{μ,d} \atop |π|≤R}} m(π) ≤ c_{Y} R^{rs/2} (R > 0) .

For k > rs/2 + 1 we obtain (6.5) as before. Let p > p_{H}(Γ). As ϕ is in the discrete spectrum we decompose it as ϕ = ∑_{π} ϕ_{π} and obtain by Lemmas 6.2 and 6.4

∥ϕ^{H}∥_{p} ≤ ∑_{π} ∥ϕ_{π}^{H}∥_{p} ≤ C ∑_{π} ∥ϕ_{π}∥_{∞} .

The last sum we estimate as follows:

∑_{π} ∥ϕ_{π}∥_{∞} = ∑_{π}(1 + |π|)^{−k/2}(1 + |π|)^{k/2}∥ϕ_{π}∥_{∞}
≤ C ∑_{π}(1 + |π|)^{−k/2}∥ϕ_{π}∥_{∞,k}

with C > 0 a constant depending only on k (we allow universal positive constants to change from line to line). Applying the Cauchy-Schwartz inequality combined with (6.5) we obtain

∥ϕ^{H}∥_{p} ≤ C( ∑_{π} ∥ϕ_{π}∥_{∞,k}^{2})^{\frac{1}{2}}

with C > 0. With Hypothesis (A2) we get the further improvement:

∥ϕ^{H}∥_{p} ≤ C( ∑_{π} ∥ϕ_{π}∥_{Ω,∞,k}^{2})^{\frac{1}{2}}

where the Sobolev norm is taken only over the compact set Ω. To finish the proof we apply the Sobolev lemma on K\G. Here Sobolev norms are defined by the central operator C, whose action agrees with the left action of Δ. It follows that ∥f∥_{∞,Ω} ≤ C∥f∥_{2,k_{1},Ω} with k_{1} > \frac{s}{2} for K-invariant functions f on G. This gives

∥ϕ^{H}∥_{p} ≤ C(∑_{π} ||ϕ_{π}||_{Ω,2,k+k_{1}}^{2})^{\frac{1}{2}} = C||ϕ||_{Ω,2,k+k_{1}} ≤ C||ϕ||_{∞,k+k_{1}}

which proves the proposition for the discrete spectrum. If ϕ = ϕ_{c} belongs to the continuous spectrum, where multiplicities are bounded (see [15]), the proof is simpler. Let μ_{c} be the restriction of the Plancherel measure to the continuous spectrum. As this is just Euclidean measure on r-dimensional space we have

∫Ĝ_{s}(1 + |π|)^{−k} dμ_{c}(π) < ∞  (6.6)

if k > r/2. We assume for simplicity in what follows that m(π) = 1 for almost all π ∈ supp μ_{c}. As sup_{π∈supp μ_{c}} m(π) < ∞ the proof is easily adapted to the general case. Let

ϕ = ∫Ĝ_{s} ϕ_{π} dμ_{c}(π).

As ∥ϕ^{H}∥_{∞} ≤ ∥ϕ∥_{∞} we conclude with Lemma 6.4, (6.6) and Fubini’s theorem that

ϕ^{H} = ∫Ĝ_{s} ϕ_{π}^{H} dμ_{c}(π)

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

∥ϕ^{H}∥_{p} ≤ C∥ϕ∥_{∞,k+k_{1}}

with k > \frac{r}{2} and k_{1} > \frac{s}{2}. This concludes the proof.

7 Error term estimates

Recall 1_{R}, the characteristic function of B_{R}. The first error term for the lattice counting problem can be expressed by

err(R, Γ) := sup_{{ϕ∈C_{b}(Y ) \atop ∥ϕ∥∞≤1}} | ⟨\frac{1_{R}^{Γ}}{|B_{R}|} − 1_{Y} , ϕ⟩ | (R > 0),

and our goal is to give an upper bound for err(R, Γ) as a function of R. According to the decomposition C_{b}(Y) = C_{b}(Y)_{van} ⊕ ℂ1_{Y} we decompose functions as ϕ = ϕ_{o} + ϕ_{1} and obtain

err(R, Γ) = sup_{{ϕ∈C_{b}(Y ) \atop ∥ϕ∥∞≤1}} \frac{|⟨1_{R}^{Γ}, ϕ_{o}⟩|}{|B_{R}|} = sup_{{ϕ∈C_{b}(Y ) \atop ∥ϕ∥∞≤1}} \frac{|⟨1_{R}, ϕ_{o}^{H}⟩|}{|B_{R}|} .

Further, from ∥ϕ_{o}∥_{∞} ≤ 2∥ϕ∥_{∞} we obtain that err(R, Γ) ≤ 2 err_{1}(R, Γ) with

err_{1}(R, Γ) := sup_{{ϕ∈C_{b}(Y )van \atop ∥ϕ∥∞≤1}} \frac{|⟨1_{R}^{Γ}, ϕ⟩|}{|B_{R}|} = sup_{{ϕ∈C_{b}(Y )van \atop ∥ϕ∥∞≤1}} \frac{|⟨1_{R}, ϕ^{H}⟩|}{|B_{R}|} .

7.1 Smooth versus non-smooth counting

Like in the classical Gauss circle problem one obtains much better estimates for the remainder term if one uses a smooth cutoff. Let α ∈ C_{c}^{∞}(G) be a non-negative test function with normalized integral. Set 1_{R,α} := α ∗ 1_{R} and define

err_{α}(R, Γ) := sup_{{ϕ∈C_{b}(Y )^{K}_{o} \atop ∥ϕ∥∞≤1}} \frac{|⟨1_{R,α}^{Γ}, ϕ⟩|}{|B_{R}|} = sup_{{ϕ∈C_{b}(Y )^{K}_{o} \atop ∥ϕ∥∞≤1}} \frac{|⟨1_{R,α}, ϕ^{H}⟩|}{|B_{R}|} .

Lemma 7.1. Let k > s + 1 if Y is compact and k > \frac{r+1}{2}s + 1 otherwise. Let p > p_{H}(Γ) and q be such that \frac{1}{p} + \frac{1}{q} = 1. Then there exists C > 0 such that

err_{α}(R, Γ) ≤ C∥α∥_{1,k}|B_{R}|^{− \frac{1}{p}}  (7.1)

for all R ≥ 1 and all α ∈ C_{c}^{∞}(G).

Proof. First note that

⟨1_{R,α}, ϕ^{H}⟩ = ⟨1_{R,α}, (−1 + Δ)^{k/2}(−1 + Δ)^{−k/2}ϕ^{H}⟩ .

With ψ = (−1 + Δ)^{−k/2}ϕ we have ∥ψ∥_{∞,k} ≤ C∥ϕ∥_{∞} for some C > 0. We thus obtain

err_{α}(R, Γ) ≤ C sup_{{ψ∈C_{b}(Y )^{K}_{o} \atop ∥ψ∥_{∞,k}≤1}} \frac{|⟨1_{R,α}, (−1 + Δ)^{k/2}ψ^{H}⟩|}{|B_{R}|}

≤ \frac{C}{|B_{R}|} sup_{{ψ∈C_{b}(Y )^{K}_{o} \atop ∥ψ∥_{∞,k}≤1}} |⟨1_{R,α}, (−1 + Δ)^{k/2}ψ^{H}⟩|

Moving (−1 + Δ)^{k/2} to the other side we get with Hölder’s inequality and Proposition 6.5 that

err_{α}(R, Γ) ≤ \frac{C}{|B_{R}|}∥(−1 + Δ)^{k/2}α ∗ 1_{R}||_{q} .

Documenta Mathematica 21 (2016) 627–660

Finally,

∥(−1 + Δ)^{k/2}α ∗ 1_{R}∥_{q} ≤ C∥α∥_{1,k}∥1_{R}∥_{q}

and with ∥1_{R}∥_{q} = |B_{R}|^{\frac{1}{q}}, the lemma follows.

Remark 7.2. In the literature results are sometimes stated not with respect to err(R, Γ) but the pointwise error term err_{pt}(R, Γ) = |1_{R}^{Γ}(1) − |B_{R}||. Likewise we define err_{pt,α}(R, Γ). Let B_{Y} be a compact neighborhood of 1Γ ∈ Y and note that

err_{pt,α}(R, Γ) ≤ |B_{R}| sup_{{ϕ∈L^{1}(B_{Y} ) \atop ∥ϕ∥_{1}≤1}} |⟨\frac{1_{R,α}^{Γ}}{|B_{R}|} − 1_{Y} , ϕ⟩| (R > 0).

The Sobolev estimate ∥ϕ∥_{∞} ≤ C∥ϕ∥_{1,k}, for K-invariant functions ϕ on B_{Y} and with k = dim Y/K the Sobolev shift, then relates these error terms:

err_{pt,α}(R, Γ) ≤ |B_{R}| sup_{{ϕ∈C^{∞}_{b}(Y ) \atop ∥ϕ∥_{∞,−k}≤1}} |⟨\frac{1_{R}^{Γ}}{|B_{R}|} − 1_{Y} , ϕ⟩| .

We then obtain

err_{pt,α}(R, Γ) ≤ C|B_{R}|^{1− \frac{1}{p}} (R > 0)

in view of (7.1).

We return to the error bound in Lemma 7.1 and would like to compare err_{1}(R, Γ) with err_{α}(R, Γ). For that we note (by the triangle inequality) that

| err_{1}(R, Γ) − err_{α}(R, Γ)| ≤ sup_{{ϕ∈C_{b}(Y )^{K}_{o} \atop ∥ϕ∥∞≤1}} \frac{|⟨1_{R,α}^{Γ} − 1_{R}^{Γ}, ϕ⟩|}{|B_{R}|} .

Suppose that supp α ⊂ B_{ϵ}^{G} for some ϵ > 0. Then Lemma 2.2 implies that 1_{R,α} is supported in B_{R+ϵ}, and hence

|⟨1_{R,α}^{Γ} − 1_{R}^{Γ}, ϕ⟩| ≤ ∥1_{R,α}^{Γ} − 1_{R}^{Γ}∥_{1}
≤ ∥1_{R,α} − 1_{R}∥_{1}
≤ |B_{R+ϵ}|^{\frac{1}{2}} ∥1_{R,α} − 1_{R}∥_{2}
≤ |B_{R+ϵ}|^{\frac{1}{2}} |B_{R+ϵ}\B_{R}|^{\frac{1}{2}} .

With Lemma 2.1 we get

|B_{R+ϵ}\B_{R}| ≤ Cϵ|B_{R}| (R ≥ 1, ϵ < 1) .

Thus we obtain that

| err_{1}(R, Γ) − err_{α}(R, Γ)| ≤ Cϵ^{\frac{1}{2}} .

Combining this with the estimate in Lemma 7.1 we arrive at the existence of C > 0 such that

err_{1}(R, Γ) ≤ C(ϵ^{−k}|B_{R}|^{− \frac{1}{p}} + ϵ^{\frac{1}{2}} )

for all R ≥ 1 and all 0 < ϵ < 1. The minimum of the function ϵ ↦ ϵ^{−k}c + ϵ^{1/2} is attained at ϵ = (2kc)^{\frac{2}{2k+1}} and thus we get:

Theorem 7.3. Under the assumptions of Proposition 6.5 the first error term err(R, Γ) for the lattice counting problem on Z = G/H can be estimated as follows: for all p > p_{H}(Γ) and k > s + 1 for Y compact, resp. k > \frac{r+1}{2}s + 1 otherwise, there exists a constant C = C(p, k) > 0 such that

err(R, Γ) ≤ C|B_{R}|^{− \frac{1}{(2k+1)p}}

for all R ≥ 1.

Remark 7.4. The point where we lose essential information is in the estimate (6.5) where we used Weyl’s law. In the moment pointwise multiplicity bounds are available the estimate would improve. To compare the results with Selberg on the hyperbolic disc, let us assume that p_{H}(Γ) = 2. Then with r = 1 and s = 2 our bound is err(R, Γ) ≤ C_{ϵ}|B_{R}|^{− \frac{1}{14} +ϵ} while Selberg showed err(R, Γ) ≤ C_{ϵ}|B_{R}|^{− \frac{1}{3} +ϵ}.

8 Triple spaces

In this section we verify our Hypothesis A for triple space Z = G/H where G = G′ × G′ × G′, H = diag(G′) and G′ = SO_{e}(1, n) for some n ≥ 2. Observe that SO_{e}(1, 2) ≅ PSl(2, ℝ). We take K′ := SO(n, ℝ) < G′ as a maximal compact subgroup and set K := K′ × K′ × K′. Further we set 𝔰 := 𝔰′ × 𝔰′ × 𝔰′. A maximal abelian subspace 𝔞 ⊂ 𝔰 is then of the form

𝔞 = 𝔞′_{1} × 𝔞′_{2} × 𝔞′_{3}

with 𝔞′_{i} ⊂ 𝔰′ one dimensional subspaces. We recall the following result from [8].

Proposition 8.1. For the triple space the following assertion hold true:

1. G = KAH if and only if dim(𝔞′_{1} + 𝔞′_{2} + 𝔞′_{3}) = 2.

2. Suppose that all 𝔞′_{i} are pairwise distinct. Then one has P H is open for all minimal parabolics P with Langlands-decomposition P = M_{P} A_{P} N_{P} and A_{P} = A.

We say that the choice of A is generic if all 𝔞′_{i} are distinct and dim(𝔞′_{1}+𝔞′_{2}+𝔞′_{3}) = 2. The invariant measure dz on Z can then be estimated as

∫_{Z} f(z) dz ≤ ∫_{K} ∫_{A} f(ka · z_{0})J(a) da dk (f ∈ C_{c}(Z), f ≥ 0)

with

J(a) = sup_{w∈W} a^{2wρ}  (8.1)

by Lemma 3.2. Note that in this case the Weyl group W is just {±1}^{3}.

8.1 Proof of the Hypothesis A

We first note that for all π ∈ Ĝ_{s} the space of H-invariants

(H_{π}^{−∞})^{H} = ℂI .

is one-dimensional, see [5], Thm. 3.1. Write π = π_{1} ⊗ π_{2} ⊗ π_{3} with each factor a K′-spherical unitary irreducible representation of G′. If we assume that π ≠ 1 has non-trivial H-fixed distribution vectors, then at least two of the factors π_{i} are non-trivial. Let v_{i} be normalized K′-fixed vectors of π_{i} and set v = v_{1}⊗v_{2}⊗v_{3}. Since Z is a multiplicity one space, the functional I ∈ (H_{π}^{−∞})^{H} is unique up to scalars. Our concern is to obtain uniform L^{p}-bounds for the generalized matrix coefficients f_{π} := m_{v,I}:

f_{π}(g_{1}, g_{2}, g_{3}) := I(π_{1}(g_{1})^{−1}v_{1} ⊗ π_{2}(g_{2})^{−1}v_{2} ⊗ π_{3}(g_{3})^{−1}v_{3}) ,

when π belongs to the set Λ of (6.3). We decompose Λ = Λ_{0} ∪ Λ_{1} ∪ {1} with Λ_{0} ⊂ Λ the set of π ∈ Λ with all π_{i} non-trivial, and Λ_{1} the set of π’s with exactly one π_{i} to be trivial. Consider first the case where π ∈ Λ_{1}, i.e. one π_{i} is trivial, say π_{3}. Then π_{2} = π_{1}^{∗}. We identify Z ≃ G′ × G′ via (g, h) ↦ (1, g, h)H and obtain

f_{π}(g, h) = ⟨π_{1}(g)v_{1}, v_{1}⟩ ,

a spherical function. Note that Z_{η} ≃ G′ and Hypothesis A follows from standard properties about K′-spherical functions on G′. To be more specific let G′ = N′A′K′ be an Iwasawa-decomposition with middle-projection a : G′ → A′, then

f_{π}(g, h) = φ_{λ_{1}}(g) := ∫_{K′} a(k′g)^{λ_{1}−ρ′} dk′ .

We use Harish-Chandra’s estimates |φ_{ν}(a)| ≤ a^{ν}φ_{0}(a) and φ_{0}(a) ≤ Ca^{−ρ}(1 + | log a|)^{d} for a ∈ A′ in positive chamber. The condition of π ∈ Λ_{1} implies that ρ−Re λ_{1} > 0 is bounded away from zero and Hypothesis A follows in this case. Suppose now that π ∈ Λ_{0}, i.e. all π_{i} are non-trivial. For a simplified exposition we assume that n = 2, i.e. G′ = PSl(2, ℝ), and comment at the end for the general case. Then π_{i} = π_{λ_{i}} are principal series for some λ_{i} ∈ iℝ^{+} ∪ [0, 1) with H_{π_{i}}^{∞} = C^{∞}(𝕊^{1}) in the compact realization. Set λ = (λ_{1}, λ_{2}, λ_{3}) and set π = π_{λ}.

In order to analyze f_{π} we use G = KAH and thus assume that g = a = (a_{1}, a_{2}, a_{3}) ∈ A. We work in the compact model of H_{π_{i}} = L^{2}(𝕊^{1}) and use the explicit model for I in [3]: for h_{1}, h_{2}, h_{3} smooth functions on the circle one has

I(h_{1} ⊗ h_{2} ⊗ h_{3}) =\frac{1}{(2π)^{3}} ∫_{0}^{2π} ∫_{0}^{2π} ∫_{0}^{2π} h_{1}(θ_{1})h_{2}(θ_{2})h_{3}(θ_{3})·
· K(θ_{1}, θ_{2}, θ_{3}) dθ_{1}dθ_{2}dθ_{3} ,

where

K(θ_{1}, θ_{2}, θ_{3}) = | sin(θ_{2} − θ_{3})|^{(α−1)/2}| sin(θ_{1} − θ_{3})|^{(β−1)/2}| sin(θ_{1} − θ_{2})|^{(γ−1)/2} .

In this formula one has α = λ_{1}−λ_{2}−λ_{3}, β = −λ_{1}+λ_{2}−λ_{3} and γ = −λ_{1}−λ_{2}+λ_{3} where λ_{i} ∈ iℝ ∪ (−1, 1) are the standard representation parameters of π_{i}. According to to [5], Cor. 2.1, the kernel K is absolutely integrable. Set

A′ := \{a_{t} := (\begin{matrix}t & 0 \\ 0 & \frac{1}{t} \end{matrix}) | t > 0\} < G′

Then A′_{i} = k_{ϕ_{i}}A′k_{ϕ_{i}}^{−1} with ϕ_{i} ∈ [0, 2π] and

k_{ϕ} = (\begin{matrix}cos ϕ & − sin ϕ \\ sin ϕ & cos ϕ \end{matrix}) .

Set a_{t,i} = k_{ϕ_{i}}a_{t}k_{ϕ_{i}}^{−1}. Returning to our analysis of f_{π} we now take h_{i}(t_{i}, θ_{i}) = [π_{1}(a_{t_{i},i})v_{i}](θ_{i}) and remark that

h_{i}(t_{i}, θ_{i}) = \frac{1}{(t_{i}^{2} + sin^{2}(θ_{i} − ϕ_{i})(\frac{1}{t_{i}^{2}} − t_{i}^{2}))^{\frac{1}{2} (1+λ_{i})}} .

Let us set |π| := π_{Re λ_{1}} ⊗ π_{Re λ_{2}} ⊗ π_{Re λ_{3}}. Our formulas then show

|f_{π}(a)| ≤ f_{|π|}(a)  (a ∈ A) .  (8.2)

Let c_{i} := 1−| Re λ_{i}| for i = 1, 2, 3. The fundamental estimate in [22], Thm. 3.2, then yields a constant d, independent of π, and a constant C = C(π) > 0 such that for a = (a_{t_{1},1}, a_{t_{2},2}, a_{t_{3},3}) one has

|f_{π}(a)| ≤ C \frac{(1 + | log t_{1}| + | log t_{2}| + | log t_{3}|)^{d}}{[cosh log t_{1}]^{c_{1}} · [cosh log t_{2}]^{c_{2}} · [cosh log t_{3}]^{c_{3}}} .  (8.3)

In view of (8.2) the constant C(π) can be assumed to depend only on the distance of Re λ_{i} to the trivial representation. Looking at the integral representation of f_{π} with the kernel K we deduce a lower bound without the logarithmic factor, i.e. the bound is essentially sharp. Hence (8.1) together with the fact that all f_{π} for π ∈ Λ_{0} are in L^{p}(Z) for some p < ∞ implies that

inf_{π∈Λ_{0}} c_{i}(π) > 0 .  (8.4)

We now claim

sup_{π∈Λ_{0}} ∥f_{π}∥_{p} < ∞ ,  (8.5)

and

sup_{π∈Λ_{0}} ∥f_{π}∥_{∞} < ∞ .  (8.6)

For 0 < ϵ < 1 set Λ_{ϵ,ℝ} = [0, 1 − ϵ] × [0, 1 − ϵ] × [0, 1 − ϵ] and Λ_{ϵ} := i𝔞^{∗} + Λ_{ϵ}. It follows from (8.4) that there exists an ϵ > 0 such that Λ_{0} ⊂ Λ_{ϵ}. We prove the stronger inequalities with Λ_{0} replaced by Λ_{ϵ}. In view of (8.2) and (8.3) we may replace by Λ_{ϵ} by Λ_{ϵ,ℝ}. Let E_{ϵ} be the eight element set of extreme points of Λ_{ϵ,ℝ}. For fixed a = a_{t} and θ = (θ_{1}, θ_{2}, θ_{3}) we let F_{λ}(a, θ) = K(θ)h_{1}(t_{1}, θ_{1})h_{2}(t_{2}, θ_{2})h_{3}(t_{3}, θ_{3}) and note that the assignment Λ_{ϵ,ℝ} → ℝ_{+}, λ ↦ F_{λ}(a, θ) is convex. Therefore we get for all λ ∈ Λ_{ϵ} that

f_{λ}(a) ≤ ∑_{μ∈E_{ϵ}} f_{μ}(a) .

In view of (8.3) the inequalities (8.5) and (8.6) then follow. On the other hand for g = 1 = (1, 1, 1), the value f_{π}(1) is obtained by applying I to the constant function 1 = 1⊗1⊗1. This value has been computed explicitly by Bernstein and Reznikov in [3] as

\frac{Γ((α + 1)/4)Γ((β + 1)/4)Γ((γ + 1)/4)Γ((δ + 1)/4)}{Γ((1 − λ_{1})/2)Γ((1 − λ_{2})/2)Γ((1 − λ_{3})/2)}

where α, β, γ are as before and δ = −λ_{1} − λ_{2} − λ_{3}. Stirling approximation,

|Γ(σ + it)| = const.e^{− \frac{π}{2} |t|}|t|^{σ− \frac{1}{2}} (1 + O(|t|^{−1}))

as |t| → ∞ and σ is bounded, yields a lower bound for f_{π}(1):

inf_{π∈Λ_{0}} |f_{π}(1)| > 0 .  (8.7)

As ∥f_{π}∥_{∞} ≥ |f_{π}(1)| the assertion (A1) of Hypothesis A is readily obtained from (8.5) and (8.7). Likewise (A2) with Ω = {1} follows from (8.6) and (8.7). In general for G′ = SO_{e}(1, n) one needs to compute the Bernstein-Reznikov integral. This was accomplished in [9].

Theorem 8.2. Let Z = G′ × G′ × G′/ diag(G′) for G′ = SO_{e}(1, n) and assume that H/Γ_{H} is compact. Then the first error term err(R, Γ) for the lattice counting problem on Z = G/H can be estimated as follows: for all p > p_{H}(Γ) there exists a C = C(p) > 0 such that

err(R, Γ) ≤ C|B_{R}|^{− \frac{1}{(6n+3)p}}

for all R ≥ 1.

8.2 Cubic lattices

Here we let G_{0} = SO_{e}(1, 2) with the quadratic Q form defining G_{0} having integer coefficients and anisotropic over ℚ, for example

Q(x_{0}, x_{1}, x_{2}) = 2x_{0}^{2} − 3x_{1}^{2} − x_{2}^{2} .

Then, according to Borel, Γ_{0} = G_{0}(ℤ) is a uniform lattice in G_{0}. Next let k be a cubic Galois extension of ℚ. Note that k is totally real. An example of k is the splitting field of the polynomial f(x) = x^{3} +x^{2} −2x−1. Let σ be a generator of the Galois group of k|ℚ. Let O_{k} be the ring of algebraic integers of k. We define Γ < G = G_{0}^{3} to be the image of G_{0}(O_{k}) under the embedding

G_{0}(O_{k}) ∋ γ ↦ (γ, γ^{σ}, γ^{σ^{2}}) ∈ G .

Then Γ < G is a uniform irreducible lattice with trace H ∩ Γ ≃ Γ_{0} a uniform lattice in H ≃ G_{0}.

9 Outlook

We discuss some topics of harmonic analysis on reductive homogeneous spaces which are currently open and would have immediate applications to lattice counting.

9.1 A conjecture which implies Hypothesis A

Hypothesis A falls in the context of a more general conjecture about the growth behavior of families of Harish-Chandra modules. We let Z = G/H be a real spherical space. Denote by A_{Z}^{−} ⊂ A_{Z} the compression cone of Z (see Section 3) and recall that wavefront means that A^{−}A_{H}/A_{H} = A_{Z}^{−} which, however, we do not assume for the moment. We use V to denote Harish-Chandra modules for the pair (𝔤, K) and V^{∞} for their unique moderate growth smooth Fréchet globalizations. These V^{∞} are global objects in the sense that they are G-modules whereas V is defined in algebraic terms. We write V^{−∞} for the strong dual of V^{∞}. We say that V is H-distinguished provided that the space of H-invariants (V^{−∞})^{H} is non-trivial. It is no big loss of generality to assume that A_{Z}^{−} is a sharp cone, as the edge of this cone is in the normalizer of H and in particular acts on the finite dimensional space of H-invariants. As A_{Z}^{−} is pointed it is a fundamental domain for the little Weyl group and as such a simplicial cone (see [17], Section 9). If 𝔞_{Z}^{−} = log A_{Z}^{−}, then we write ω_{1}, …, ω_{r} for a set of generators (spherical co-roots) of 𝔞_{Z}^{−}. Set \overline{Q} := θ(Q) where θ is the Cartan involution determined by the choice of K. Note that V/\overline{𝔮}V is a finite dimensional \overline{Q} module, in particular a finite dimensional A_{Z}-module. Let Λ_{1}, …, Λ_{N} ∈ 𝔞_{Z}^{∗} be the 𝔞_{Z,ℂ}-weight spectrum. Then we define the H-spherical exponent Λ_{V} ∈ 𝔞_{Z}^{∗} of V by

Λ_{V} (ω_{i}) := max_{1≤j≤N} Re Λ_{j}(ω_{i}) .

Further attached to V is a “logarithmic” exponent d ∈ ℕ. Having this data we recall the main bound from [22]

|m_{v,η}(a · z_{0})| ⪅ a^{Λ_{V}} (1 + ∥ log a∥)^{d_{V}} (a ∈ A_{Z}^{−}) .

Conjecture 9.1. Fix a K-type τ, a constant C > 0, and a compact subset Ω ⊂ G. Then there exists a compact set Ω_{A} ⊂ A_{Z}^{−} such that for all Harish-Chandra modules V with ∥Λ_{V} ∥ ≤ C, all v ∈ V [τ] and all η ∈ (V^{−∞})^{H} one has

max_{{a∈A{− \atop Z} \atop g∈Ω}} |m_{v,η}(ga · z_{0})|a^{−Λ_{V}} (1 + ∥ log a∥)^{−d_{V}} =

max_{{a∈Ω_{A} \atop g∈Ω}} |m_{v,η}(ga · z_{0})|a^{−Λ_{V}} (1 + ∥ log a∥)^{−d_{V}} .

It is easily seen that this conjecture implies Hypothesis A if all the generalized matrix coefficients m_{v,η} are bounded, as for example it is the case when Z is wavefront (see Proposition 3.4(1)).

Remark 9.2. It might well be that a slightly stronger conjecture is true. For that we recall that a Harish-Chandra module V has a unique minimal globalization, the analytic model V^{ω}. The space V^{ω} is an increasing union of subspaces V_{ϵ} for ϵ → 0. The parameter ϵ parametrizes left G-invariant neigborhoods Ξ_{ϵ} ⊂ G_{ℂ} of 1 which decrease with ϵ → 0. Further V_{ϵ} consists of those vectors v ∈ V^{ω} for which the orbit map G → V^{ω}, g ↦ g · v extends to a holomorphic map on Ξ_{ϵ}. For fixed ϵ, C > 0 the strengthened conjecture would be that there exists a compact subset Ω_{A} such that for all Harish-Chandra modules V with ∥Λ_{V} ∥ ≤ C and all v ∈ V_{ϵ} one has

max_{a∈A_{Z}^{−}} |m_{v,η}(a · z_{0})|a^{−Λ_{V}} (1 + ∥ log a∥)^{−d_{V}} =

max_{a∈Ω_{A}} |m_{v,η}(a · z_{0})|a^{−Λ_{V}} (1 + ∥ log a∥)^{−d_{V}} .

Note that the compact set Ω is no longer needed, as Ω · V_{ϵ} ⊂ V_{ϵ′}.

9.2 Spectral geometry of Z_{η}

In the general context of a reductive real spherical space it may be possible to establish both main term counting and the error term bound, with the arguments presented here for wavefront spaces, provided the following two key questions allow affirmative answers.

In what follows Z = G/H is a real reductive spherical space and V denotes an irreducible Harish-Chandra module and η ∈ (V^{−∞})^{H}.

Question A: Is H_{η} reductive?

Question B: If for v ∈ V the generalized matrix coefficient m_{v,η} is bounded, then there exists a 1 ≤ p < ∞ such that m_{v,η} ∈ L^{p}(Z_{η}).

In this context we note that issues related to the well-factorization of the intrinsic balls in affine spherical spaces can possibly be resolved with similar methods to those applied here, using volume estimates as described in Theorem 7.17 of [13]. Acknowledgement. We are grateful to an anonymous referee for valuable comments that improved the presentation in the present version of the paper.

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Universität Paderborn Institut für Mathematik Warburger Straße 100 33098 Paderborn Deutschland bkroetz@gmx.de Department of Mathematics Ben Gurion University of the Negev P.O.B. 653 Be’er Sheva 84105, Israel eitan.sayag@gmail.com

University of Copenhagen Department of Mathematics Universitetsparken 5 DK-2100 Copenhagen Ø Denmark schlicht@math.ku.dk