Geometric counting on wavefront real spherical spaces

TL;DR

This paper establishes bounds for eigenfunctions on wavefront real spherical spaces, linking these to lattice counting errors, and extends previous results to higher rank spaces, providing foundational insights into geometric counting.

Contribution

It introduces $L^p$-vs-$L^$ bounds for eigenfunctions on wavefront real spherical spaces, including new higher rank results, and connects these bounds to lattice counting error estimates.

Findings

Eigenfunction bounds on wavefront real spherical spaces

Error term estimates for lattice counting

Extension of results to higher rank spaces

Abstract

We provide $L^p$-versus $L^\infty$-bounds for eigenfunctions on a real spherical space $Z$ of wavefront type. It is shown that these bounds imply a non-trivial error term estimate for lattice counting on $Z$. The paper also serves as an introduction to geometric counting on spaces of the mentioned type. Section 7 on higher rank is new and extends the result from v1 to higher rank. Final version. To appear in Acta Math. Sinica.

Tables

Equations

$$v(z)={\mathrm{vol}}_Z(Bz)(z\in Z)$$

$$|f(z)|\le Cv(z{)}^{-\frac{1}{p}}\Vert f{\Vert }_{p;k}(z\in Z)$$

$$\Vert f{\Vert }_{\infty }\le C\Vert f{\Vert }_{p;k}(f\in C^{\infty }(Z)).$$

$$\Vert f{\Vert }_p\le C\Vert f{\Vert }_{\infty ;l}$$

$$N_R(D,X):=\#\{d\in D\mid d\in B_R\}.$$

$$\underset{R\to \infty }{\lim }\frac{N_R(D,X)}{|B_R|}=1.$$

$$\#\{\gamma \in \Gamma \mid \gamma \cdot F\subset B_R\},$$

$${\int }_{X}f(x)d{\mu }_X(x)={\int }_Z{\int }_{H}f(h\tilde{z})d{\mu }_H(h)d{\mu }_Z(z)(f\in C_c(X))$$

$$\mathrm{vol}(B_R^G)=\mathrm{vol}(B_R^Z)\cdot \mathrm{vol}(B_R^H).$$

$$N_R(\Gamma ,G)=N_R({\Gamma }_H,H)\cdot N_R(\Gamma /{\Gamma }_H,Z)(R\ge R_0).$$

$${\tilde{N}}_R(\Gamma ,G)\to {\tilde{N}}_R(\Gamma /{\Gamma }_H,B_R^Z),\gamma \mapsto \overline{\gamma }$$

$$Z^{\prime }:=U_1\cup U_2\backslash {\overline{U}}_1\cup U_3\backslash ({\overline{U}}_1\cup {\overline{U}}_2)\cup \dots .$$

$$K{\times }_{K_H}V\to G/H,[k,X]\mapsto k\exp (X)H$$

$$B_R:=\{[k,X]\in G/H\mid \Vert X\Vert <R\}.$$

$$\mathfrak{g}=\overline{\mathfrak{u}}\oplus \mathfrak{l}\oplus \mathfrak{u}=\underset{\alpha \in {\Sigma }_{\mathfrak{u}}}{⨁}{\mathfrak{g}}^{-\alpha }\oplus \mathfrak{l}\oplus \underset{\beta \in {\Sigma }_{\mathfrak{u}}}{⨁}{\mathfrak{g}}^{\beta }$$

$$T:\overline{\mathfrak{u}}\to {\mathfrak{l}}_H^{\perp }\oplus \mathfrak{u}\subset \underset{\beta \in \{0\}\cup {\Sigma }_{\mathfrak{u}}}{⨁}{\mathfrak{g}}^{\beta }$$

$$\mathfrak{h}=\mathfrak{l}\cap \mathfrak{h}\oplus \{\overline{X}+T(\overline{X})\mid \overline{X}\in \overline{\mathfrak{u}}\}.$$

$$T=\sum _{\alpha ,\beta }{T}_{\alpha ,\beta }.$$

$$\mathfrak{h}\ni [Y,\overline{X}+T_{\alpha ,\beta }(\overline{X})]=-\alpha (Y)\overline{X}+\beta (Y)T_{\alpha ,\beta }(\overline{X}).$$

$$\mathcal{M}:=\{\alpha +\beta \mid T_{\alpha ,\beta }\ne 0\}$$

$${\mathfrak{a}}_Z^{-}:=\{Y\in \mathfrak{a}\mid (\forall \sigma \in S)\sigma (Y)\le 0\}.$$

$${\mathfrak{a}}^{-}+{\mathfrak{a}}_H\subset {\mathfrak{a}}_Z^{-}.$$

$${\mathfrak{a}}^{-}+{\mathfrak{a}}_H={\mathfrak{a}}_Z^{-}.$$

$$(2),\; (6),\; (9)\; -\; (11),\; (12)\; and\; (13)\; forq=1,\; (20)\; -\; (22).$$

$$\mathfrak{g}=\mathfrak{so}(1,n)\oplus \mathfrak{so}(1,n)\oplus \mathfrak{so}(1,n)and\mathfrak{h}=\mathrm{diag}\mathfrak{so}(1,n)(n\ge 2).$$

$$Q(x_0,x_1,x_2)=2x_0^2-3x_1^2-x_2^2.$$

$$G_0({\mathcal{O}}_k)\ni \gamma \mapsto (\gamma ,{\gamma }^{\sigma },{\gamma }^{{\sigma }^2})\in G.$$

$$Z\hookrightarrow Z_{\mathbb{C}},gH\mapsto g{H}_{\mathbb{C}}$$

$$Z=\Omega A_Z^{-}\mathsf{W}\cdot z_0$$

$$C_1{a}^{-2\rho }\le v(\omega a\mathsf{w}\cdot z_0)\le C_2{a}^{-2\rho }(\omega \in \Omega ,a\in A_Z^{-},\mathsf{w}\in \mathsf{W}).$$

Full text

The second author was partially supported by ISF 1138/10 and ERC 291612

1. Introduction

Given a space $Z$ with a discrete subset $D$ and an increasing and exhausting family $(B_{R})_{R>0}$ of compact subsets, it can be of considerable interest to know the expected number of points from $D$ inside $B_{R}$ for large $R$. In general terms this is called lattice counting on $Z$. This paper is about lattice counting on a homogeneous space $Z=G/H$, and methods from harmonic analysis to approach it. Here $G$ is a real reductive group and $H$ is a closed subgroup with finitely many components and real algebraic Lie algebra.

At this point we do not go into the specifics of the lattice count on $G/H$ and refer right away to Sections 2 and 4 where we give a self-contained exposition aimed at a wide audience. The terminology there is essentially taken from [8] but we emphasize more the underlying fiber bundle structure $H\to G\to G/H$ and relate counting on $Z$ to counting on the total space $G$ and fiber $H$.

The lattice counting problem including non-trivial error terms has a positive solution for all symmetric spaces $Z=G/H$. A central tool there is the so-called wavefront lemma, see [9]. The wavefront lemma however holds more generally for all real spherical spaces of wavefront type. In Section 3 we give an introduction to real spherical space of wavefront type and recall the proof of the wavefront lemma from [24].

The strive for solving the lattice counting problem triggered many interesting developments, perhaps more interesting than the problem itself. Specifically we mention here Selberg’s trace formula for the upper half plane. Here we wish to point out another connection to harmonic analysis. In [30] we have shown that non-trivial error terms for the lattice count are tied to $L^{p}$-versus $L^{\infty}$-bounds of eigenfunctions on the non-compact space $Z$.

We assume now that $Z=G/H$ is unimodular, i.e. carries a $G$-invariant positive Radon measure. Following [3] we measure volume growth on $Z$ via a volume weight

[TABLE]

where $B$ is some fixed neighborhood of ${\bf 1}$ in $G$. Let $1\leq p<\infty$. Then Bernstein’s invariant Sobolev lemma ([3] key lemma p. 686, or [33], Lemma 4.2) implies for all $f\in C^{\infty}(Z)$ that

[TABLE]

where $\|\cdot\|_{p;k}$ is a $k$-th Sobolev norm of the $L^{p}$-norm $\|\cdot\|_{p}$ and $k>{\dim G\over p}$.

We recall from [31] that $\operatorname{\bf v}$ is uniformly bounded from below if and only if $H$ is reductive in $G$. Let us assume this in the sequel. Then we obtain in particular:

[TABLE]

What is relevant for the lattice count are estimates in the other direction. We call $f\in C^{\infty}(Z)$ an eigenfunction if it is an eigenfunction for $\mathcal{Z}(\mathfrak{g})$, the center of the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ of $\mathfrak{g}=\operatorname{Lie}(G)$. Now, given $1\leq p^{\prime}<p<\infty$ we ask whether there exist a number $l=l(Z)>0$ and a constant $C>0$ such that

[TABLE]

holds for all $L^{p^{\prime}}$-eigenfunctions $f$ on $Z$. It is of independent interest to classify all homogeneous spaces $Z=G/H$ which feature (1.3).

The main result of this paper is the verification of (1.3) for all wave-front real spherical spaces (see Section 7). Combined with the harmonic analysis approach of [30] this then leads to a quantitative error bound for the lattice count on all real spherical spaces of wave-front type in Theorem 5.6.

Our approach to (1.3) relies on a generalization of Harish-Chandra’s constant-term approximation of eigenfunctions on reductive groups [14] to real spherical spaces of wave-front type [7]. In Section 6 we first treat the less technical and instructive case of $\operatorname{rank}_{\mathbb{R}}Z=1$: we cut down the techniques from [7] to the absolute necessary and provide a proof of a certain constant term approximation. Let us mention that the constant term approximation obtained in this paper differs from the one in [7], see Remark 7.2. Finally, in Section 7 we adapt the proof from the rank one case to higher rank and obtain (1.3) for all wave-front real spherical spaces.

Acknowledgement: We wish to thank the referee for his insistence on technical detail and clarity of arguments. It eventually made this article much more readable.

2. Geometric counting

A setup for geometric counting needs:

• •

A locally compact space $X$.

• •

A notion of volume on $X$ given by a Radon measure $\mu$.

• •

A discrete set $D\subset X$.

• •

An increasing and exhausting family $\mathcal{B}=(B_{R})_{R>0}$ of relatively compact sets $B_{R}\subset X$.

For $R>0$ we then set

[TABLE]

For a measurable subset $B\subset X$ we use the notation $\operatorname{vol}(B)=|B|=\mu(B)$. We then ask to what extent $N_{R}(D,X)$ approximates $\operatorname{vol}(B_{R})$ for $R\to\infty$. We say that the quadruple $(X,\mu,D,\mathcal{B})$ satisfies main term counting (MTC) provided that

[TABLE]

The mother of all counting problems is the Gauß circle problem (GCP), that is $(\mathbb{R}^{2},dx\wedge dy,\mathbb{Z}^{2},\mathcal{B})$ with $B_{R}=B_{R}^{\rm Eucl}$ the round Euclidean ball of radius $R$. It is almost immediate that the GCP satisfies MTC.

In order to expect MTC in a general setup explained above one needs additional assumptions. In some sense the discrete set needs to be equidistributed at infinity. This might be satisfied if $D$ is freely homogeneous, that is:

• •

There is an infinite discrete group $\Gamma$ acting on $X$ properly, freely and volume-preserving.

• •

There is $x_{0}\in X$ such that $\Gamma\simeq\Gamma\cdot x_{0}=D$.

• •

There is a (locally closed) fundamental domain $F\subset X$ for the $\Gamma$-action with $\operatorname{vol}(X/\Gamma):=\operatorname{vol}(F)=1$ and $x_{0}\in F$.

At least for $F$ relatively compact and the family $(B_{R})_{R>0}$ exhausting the space in a homogeneous manner, we can imagine that $\operatorname{vol}(B_{R})$ is asymptotically approximated by

[TABLE]

i.e. the number of tiles $\gamma\cdot F$ which lie in $B_{R}$. In fact, it is easy to construct a family $\mathcal{B}$ which satisfies MTC. For that let $(F_{R})_{R>0}$ be a relatively compact exhaustion of the fundamental domain $F$ and $(\Gamma_{R})_{R>0}$ an exhaustion of $\Gamma$ by finite subsets. Then $B_{R}:=\Gamma_{R}\cdot F_{R}$ defines an exhaustion of $X$ which satisfies MTC. In practice we certainly wish to take more geometric exhaustions $\mathcal{B}$ than the one constructed above. Typically has a variety of interesting metrics $d$ on $X$ and one would like to take for $\mathcal{B}$ the metric balls $B_{R}:=\{x\in X\mid d(x,x_{0})<R\}$. We return to this issue later on.

Here is a large class of freely homogeneous examples where MTC holds. We let $X=G$ be connected Lie group which admits a lattice $\Gamma<G$. We recall that a lattice in a Lie group $G$ is a discrete subgroup $\Gamma<G$ with finite co-volume with respect to a Haar measure $\mu$ on $G$. We take $D=\Gamma$ and normalize $\mu$ such that $\operatorname{vol}(G/\Gamma)=1$. For the exhausting family $\mathcal{B}$ almost anything will do; a particular nice family would be balls of radius $R$ with respect to a left invariant metric on $X=G$. The quickest way to establish MTC for $G$ is via the wavefront lemma applied to $G$ viewed as a homogeneous space for the two-sided action of $G\times G$, see [9] and Remark 3.9 after Lemma 3.7 below. A further study yields also an error estimate for this case, see Theorem 1.5 of [13].

Starting with a freely homogeneous quadruple $(X,\mu,\Gamma,\mathcal{B})$ which satisfies MTC we let $p:X\to Z:=H\backslash X$ be a principal fiber bundle with fibre $H$. We assume that $H$ is a Lie group. Let $z_{0}=p(x_{0})$ and identify $H$ with $H\cdot x_{0}=p^{-1}(z_{0})$. Let $\mu_{H}$ be a right Haar-measure on $H$ and $\mu_{X}=\mu$. We request that there is a Radon measure $\mu_{Z}$ on $Z$ such that

[TABLE]

where $Z\to X,\ z\mapsto\tilde{z}$ is some measurable cross section.

The following assumptions on the fibre are then natural:

• •

There is a discrete subgroup $\Gamma_{H}<\Gamma$ such that $H\cap(\Gamma\cdot x_{0})=\Gamma_{H}\cdot x_{0}$.

• •

$\Gamma_{H}$ is a co-volume $1$ lattice in $H$.

• •

$(H,\mu_{H},\Gamma_{H},\mathcal{B}^{H})$ satisfies MTC where $\mathcal{B}^{H}:=(B_{R}\cap H)_{R>0}$.

Let now $B_{R}^{Z}:=p(B_{R})$ and $\mathcal{B}^{Z}$ the corresponding family of balls. One might then ask whether $p(\Gamma)\subset Z$ is discrete and $(Z,\mu_{Z},p(\Gamma),\mathcal{B}^{Z})$ satisfies MTC?

In case the principal bundle is homogeneous we have the following:

Conjecture 2.1**.**

Let $G$ be a connected Lie group and $H<G$ a closed subgroup such that $Z:=G/H$ carries a $G$-invariant positive Radon measure $\mu_{Z}$ which satisfies (2.2) with respect to some Haar measures $\mu_{G}$ and $\mu_{H}$ of $G$ and $H$. Assume that $H$ has finitely many connected components. Further let $\Gamma<G$ be a lattice such that

• •

$\operatorname{vol}(G/\Gamma)=1$.

• •

$\Gamma_{H}:=\Gamma\cap H$* is a lattice in $H$ such that $\operatorname{vol}(H/\Gamma_{H})=1$.*

Then there exists an exhausting compact family $\mathcal{B}^{Z}=(B_{R}^{Z})_{R>0}$ of $Z$ such that the quadruple $(Z,\mu_{Z},\Gamma/\Gamma_{H},\mathcal{B}^{Z})$ satisfies MTC.

We point out here that the setup in the above open problem was taken from [8]. Notice that the GCP falls in this setup as well as the Selberg circle problem (SCP) on the upper half plane $Z=\operatorname{SL}(2,\mathbb{R})/\operatorname{SO}(2,\mathbb{R})$ on which we will comment in more detail later on.

Remark 2.2**.**

(Heuristics for Conjecture 2.1)* We now give some heuristics for Conjecture 2.1 in case $Y_{H}:=H/\Gamma_{H}$ is compact. Let us assume first that the fibration $G\to Z=G/H$ is trivial, that is $G\simeq Z\times H$ as a right $H$-space. This is already an interesting class and typical examples arise in the following manner: We let $G$ be semisimple, $G=KAN$ be an Iwasawa decomposition and $N=N_{1}\times N_{2}$. Then for $H=N_{2}$, the principal bundle $G\to Z$ is trivial, as $Z\simeq K\times A\times N_{1}$ as manifolds.*

To continue, we assume that $B_{R}^{G}=B_{R}^{Z}\times B_{R}^{H}$ are product balls. This yields in particular

[TABLE]

As we assume that $Y_{H}:=H/\Gamma_{H}$ is compact we have $H=B_{R}^{H}\Gamma_{H}$ for sufficiently large $R$, say $R\geq R_{0}$. We need to be a bit more specific on the balls $B_{R}^{H}$ and request that they are of the form $B_{R}^{H}=F_{H}\Gamma_{H,R}$ with ${\bf 1}_{H}\in F_{H}\subset H$ a fundamental domain for the right action of $\Gamma_{H}$ on $H$ and ${\bf 1}_{H}\in\Gamma_{H,R}\subset\Gamma_{H}$ an exhausting family of finite subsets. We claim that

[TABLE]

In fact, let $\tilde{N}_{R}(\Gamma,G)=\{\gamma\in\Gamma\mid\gamma\in B_{R}^{G}\}$ so that $\#\tilde{N}_{R}(\Gamma,G)=N_{R}(\Gamma,G)$. Let $\gamma\in\tilde{N}_{R}(\Gamma,G)$ and $\overline{\gamma}=\gamma H$ be its image in $Z$. Clearly $\overline{\gamma}\in B_{R}^{Z}$. Conversely, let $\overline{\gamma}=\gamma H\in B_{R}^{Z}$ with $\gamma\in\Gamma$. Then $\gamma\in B_{R}^{Z}H=B_{R}^{Z}F_{H}\Gamma_{H}$. This allows us to assume that $\gamma\in B_{R}^{Z}F_{H}\subset B_{R}^{Z}B_{R}^{H}$, i.e. $\gamma\in\tilde{N}_{R}(\Gamma,G)$. This shows in particular that the map

[TABLE]

is surjective with fibers isomorphic to $\Gamma_{H,R}=\tilde{N}_{R}(\Gamma_{H},B_{R}^{H})$. We know that MTC holds for groups and all reasonable balls. If we assume now that MTC holds for the above class of balls $B_{R}^{G}$ in $G$ (which in the example of $H=N_{2}$ abelian unipotent can be easily verified), then we deduce from (2.3) and (2.4) that MTC holds in case $G\simeq Z\times H$ is trivial.

To adapt the argument to the case where $G\to Z$ is not trivial we exhibit now an open and measure dense subset $Z^{\prime}\subset Z$ for which the bundle trivializes and which avoids all lattice points. To obtain $Z^{\prime}$ we first find a countable cover of open sets $U_{1},U_{2},\ldots$ of $Z=G/H$ such that $G\to Z$ trivializes over each $\overline{U}_{i}$. We may assume that each $\overline{U}_{i}$ is diffeomorphic to a closed euclidean ball. Now consider the disjoint union

[TABLE]

Clearly $Z^{\prime}$ is open, measure dense in $Z$ and $G\to Z$ trivializes over $Z^{\prime}$. Moreover as $D:=\Gamma/\Gamma_{H}\subset Z$ is countable we can arrange that it avoids $\bigcup_{j=1}^{\infty}\partial U_{j}$, i.e. $D\subset Z^{\prime}$. Now proceed as above.

We keep the setup of Conjecture 2.1. A particular nice class of balls can be defined by the Mostow-decomposition of $G/H$ which we now recall.

Let $K<G$ be a maximal compact subgroup of $G$ such that $K_{H}:=K\cap H$ is a maximal compact subgroup of $H$. Let $N_{K}(H)$ be the normalizer of $H$ in $K$. The Mostow decomposition (see Theorem 8.1 in the appendix) of $Z=G/H$ then asserts the existence of a finite dimensional vector space and $N_{K}(H)$-module $V\subset\mathfrak{g}$ such that

[TABLE]

is a diffeomorphism.

The coordinates given by (2.5) allow us to define a natural family of balls. Let $\|\cdot\|$ be a $N_{K}(H)$-invariant Euclidean norm on $V$ and define $K$-invariant balls as follows

[TABLE]

Following [30] we call these balls intrinsic. Having this terminology we ask:

Problem 2.3**.**

Does Conjecture 2.1 hold true with $\mathcal{B}^{Z}$ a family of intrinsic balls ?

From now on we let $G$ be a real reductive group and $H<G$ be a closed subgroup with finitely many connected components and real algebraic Lie algebra. Further we let $\Gamma<G$ be a lattice in the setup of Conjecture 2.1. Further we set $Y:=G/\Gamma$.

Here is a short history of MTC in the context of Conjecture 2.1. MTC was established via harmonic analysis in [8] for symmetric spaces $G/H$ and certain families of balls, for lattices with $Y_{H}$ compact. In subsequent work [9] the obstruction that $Y_{H}$ is compact was removed and an ergodic approach was presented. The ergodic techniques were refined in [10] and it was discovered that MTC holds for a wider class of reductive spaces: For reductive algebraic groups $G,H$ defined over $\mathbb{Q}$ and arithmetic lattices $\Gamma<G(\mathbb{Q})$ it is sufficient to request that the identity component of $H$ is not contained in a proper parabolic subgroup of $G$ which is defined over $\mathbb{Q}$ 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.

The core of the approach of [9] was a geometric lemma satisfied by symmetric spaces which the authors termed wavefront lemma. In their work on p-adic spherical spaces Sakellaridis and Venkatesh [41] coined the notion of a wavefront p-adic spherical space and showed that these spaces satisfy a p-adic version of the wavefront lemma. Wavefront real spherical spaces were introduced in [24] and it was shown in [24] Lemma 6.3 that they satisfy the wavefront lemma of [9]. In particular real spherical spaces of wavefront type satisfy MTC for all reasonable families of balls (see Theorem 3.8 below).

At this point we remark that all symmetric spaces are real spherical of wavefront type. The latter type of spaces is going to be the main player of this article. Before we continue with error term bounds for the lattice counting problem we insert a section on wavefront spaces and provide a proof of the wavefront lemma.

3. Real spherical spaces

The notational convention for this paper is that we denote Lie groups by upper case Latin letters, e.g. $A,B,C$, and their corresponding Lie algebras with lower case German letters, e.g. $\mathfrak{a},\mathfrak{b},\mathfrak{c}$.

We assume that $Z=G/H$ is real spherical, that is, a minimal parabolic subgroup $P$ of $G$ admits an open orbit on it. By choosing $P$ suitably we can then arrange that its orbit through the origin $z_{0}=H\in Z$ is open, or equivalently that $\mathfrak{g}=\mathfrak{h}+\mathfrak{p}$. All symmetric spaces are known to be real spherical.

According to [26] there is a unique parabolic subgroup $Q\supset P$ with the following two properties:

• •

$QH=PH$.

• •

There is a Levi decomposition $Q=LU$ with $L_{\rm n}\subset Q\cap H\subset L$.

Here $L_{\rm n}$ denotes the analytic subgroup of $L$ for which the Lie algebra $\mathfrak{l}_{\rm n}$ is the sum of all non-compact simple ideals of $\mathfrak{l}$.

We observe that $L\cap P$ is a minimal parabolic subgroup in $L$. It follows that we can choose an Iwasawa decomposition $L=K_{L}A_{L}N_{L}$ such that $A_{L}N_{L}\subset P$. Having fixed that we choose a compatible Iwasawa decomposition $G=KAN$, i.e. $K_{L}<K,A_{L}=A$ and $N_{L}<N$. Then $N$ is the unipotent radical of $P$ and with $M=Z_{K}(A)$ we have the Langlands decomposition $P=MAN$ of $P$.

Set $A_{H}:=A\cap H$ 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 [26]) and denoted by $\operatorname{rank}_{\mathbb{R}}Z$.

Attached to $\mathfrak{a}$ and $P$ are the root system $\Sigma=\Sigma(\mathfrak{a},\mathfrak{g})\subset\mathfrak{a}^{*}\backslash\{0\}$ and its set $\Sigma^{+}$ of positive roots. For the associated root space decomposition $\mathfrak{g}=\oplus_{\alpha\in\{0\}\cup\Sigma}\mathfrak{g}^{\alpha}$ we have $\mathfrak{g}^{0}=\mathfrak{a}\oplus\mathfrak{m}$ and $\mathfrak{n}=\oplus_{\alpha\in\Sigma^{+}}\mathfrak{g}^{\alpha}$. We write $\Sigma_{\mathfrak{u}}\subset\Sigma^{+}$ for the subset of $\mathfrak{a}$-weights of $\mathfrak{u}$, and let $\overline{\mathfrak{n}}$ and $\overline{\mathfrak{u}}$ denote the corresponding sum of negative root spaces. Then

[TABLE]

Attached to $Z$ is a geometric invariant, the so-called compression cone. It is a closed and convex subcone $\mathfrak{a}_{Z}^{-}$ of $\mathfrak{a}_{Z}$, defined as follows. According to [26] there exists a linear map

[TABLE]

such that

[TABLE]

Here $\mathfrak{l}_{H}^{\perp}\subset\mathfrak{a}\oplus\mathfrak{m}$ denotes the orthocomplement of $\mathfrak{l}\cap\mathfrak{h}$ in $\mathfrak{l}$. For $\alpha\in\Sigma_{\mathfrak{u}}$ and $\beta\in\{0\}\cup\Sigma_{\mathfrak{u}}$ we denote by $T_{\alpha,\beta}:\mathfrak{g}^{-\alpha}\to\mathfrak{g}^{\beta}$ the map obtained from (3.1) by restriction of $T$ to $\mathfrak{g}^{-\alpha}$ and projection to $\mathfrak{g}^{\beta}$. Then

[TABLE]

For each $Y\in\mathfrak{a}_{H}$ and all $\overline{X}\in\overline{\mathfrak{u}}$ we find

[TABLE]

Hence if $T_{\alpha,\beta}\neq 0$ we see by comparing with (3.2) that $-\alpha(Y)=\beta(Y)$. Hence

[TABLE]

can be viewed as a subset of $\mathfrak{a}_{H}^{\perp}=\mathfrak{a}_{Z}^{*}$. According to [21], Cor. 12.5, the cone $C_{\mathcal{M}}\subset\mathfrak{a}_{Z}^{*}$ generated by ${\mathcal{M}}$ is simplicial and contains a linearly independent set $S$ of generators such that ${\mathcal{M}}\subset\mathbb{N}_{0}[S]$. Such a set $S$ will be refered to as a set of spherical roots for $Z$. We fix any such set and define

[TABLE]

3.1. Wavefront spaces

In [24], Section 6, we defined the notion of wavefront for real spherical spaces, which will now be recalled.

If we denote by $\mathfrak{a}^{-}\subset\mathfrak{a}$ the closure of the negative Weyl chamber, then it is clear that $\mathfrak{a}^{-}\subset\mathfrak{a}_{Z}^{-}$. Hence

[TABLE]

Since $T_{\alpha,\beta}$ can be zero for many pairs of roots $(\alpha,\beta)$, the above inclusion can be proper in general. By definition $Z$ is called a wavefront space if in fact

[TABLE]

We shall now give some examples of wavefront spaces. We first note that if $Z$ is symmetric, say with corresponding involution $\sigma$ of $\mathfrak{g}$, then the special Iwasawa decomposition which we requested above can be obtained by choosing a Cartan involution $\theta$ that commutes with $\sigma$. Then $T(\overline{X})=\sigma(\overline{X})$ for all $\overline{X}\in\overline{u}$, and $T_{\alpha,\beta}$ is non-zero if and only if $\beta=-\sigma\alpha$ in this case. From this it easily follows that we have the equality in (3.3), that is, all symmetric spaces are wavefront.

Next we give a classification of all non-symmetric wavefront real spherical pairs $(\mathfrak{g},\mathfrak{h})$ with $\mathfrak{g}$ simple and $\mathfrak{h}$ reductive. These are the pairs of the following Table 1. The table is deduced from the classification in [22] together with the following result from [25] Thm. 6.3.

Lemma 3.1**.**

Let $Z=G/H$ be a wavefront real spherical space with $\mathfrak{g}$ simple and $\mathfrak{h}$ reductive. Let $H<H^{\star}<G$ be a closed subgroup such that $Z^{\star}:=G/H^{\star}$ is unimodular. Then $H^{\star}/H$ is compact.

Remark 3.2**.**

We say that $Z$ has real rank one if $\dim\mathfrak{a}_{Z}=1$. In Table 3.1 the following are of real rank one:

[TABLE]

There are many more examples in case $\mathfrak{g}$ is semi-simple and not simple (see the classification in [23]). However, with the exception of two cases, they are not interesting for the lattice count for the following reason: If $G=G_{1}\times\ldots\times G_{n}$ is a product of simple groups, then an irreducible lattice can only exist if the $\mathfrak{g}_{i}\otimes_{\mathbb{R}}\mathbb{C}$ are all isomorphic (see [20]). In view of the classification in [23] one is then left with the group case $(\mathfrak{g},\mathfrak{h})=(\mathfrak{g}_{0}\oplus\mathfrak{g}_{0},\operatorname{diag}(\mathfrak{g}_{0}))$ and the triple spaces

[TABLE]

The triple spaces feature a lot of interesting irreducible lattices. Here we review an example given in [30].

We let $n=2$ and $G_{0}=\operatorname{SO}_{e}(1,2)$ and consider space $Z=G/H$ where $G=G_{0}\times G_{0}\times G_{0}$ and $H=\operatorname{diag}(G_{0})$.

We assume that $G_{0}=\operatorname{SO}_{e}(1,2)$ is defined by a quadratic form $Q$ which has integer coefficients and is anisotropic over $\mathbb{Q}$, for example

[TABLE]

Then, according to Borel, $\Gamma_{0}=G_{0}(\mathbb{Z})$ is a uniform lattice in $G_{0}$.

Next let $k$ be a cubic Galois extension of $\mathbb{Q}$. 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 $\sigma$ be a generator of the Galois group of $k|\mathbb{Q}$. Let ${\mathcal{O}}_{k}$ be the ring of algebraic integers of $k$. We define $\Gamma<G=G_{0}^{3}$ to be the image of $G_{0}({\mathcal{O}}_{k})$ under the embedding

[TABLE]

Then $\Gamma<G$ is a uniform irreducible lattice with $H\cap\Gamma\simeq\Gamma_{0}$ a uniform lattice in $H\simeq G_{0}$.

3.2. The polar decomposition and integral inequalities

In this section $Z=G/H$ is a real spherical space. It is not assumed to be wavefront unless requested otherwise.

Let us denote by $z_{0}=H\in Z$ the standard base point. It will be convenient to assume that there are complex groups $H_{\mathbb{C}}\subset G_{\mathbb{C}}$ such that $G\subset G_{\mathbb{C}}$ is a real form and $H=G\cap H_{\mathbb{C}}$. Set $Z_{\mathbb{C}}=G_{\mathbb{C}}/H_{\mathbb{C}}$ and observe that

[TABLE]

constitutes a $G$-equivariant embedding.

We now recall from [24] (see also [21], Sect. 13) the polar decomposition

[TABLE]

where

• •

$\Omega$ is a compact set of the form $\Omega=FK$ with $F\subset G$ a finite set.

• •

$\mathsf{W}\subset G$ is a finite set with the property that $P\mathsf{w}\cdot z_{0}\subset Z$ is open for all $\mathsf{w}\in\mathsf{W}$. Moreover, when (3.4) is assumed it can be arranged that $\mathsf{W}\cdot z_{0}\subset T\cdot z_{0}\cap Z$ where $T=\exp(i\mathfrak{a})$ and the intersection is taken in $Z_{\mathbb{C}}$.

We recall the definition of the volume weight $\operatorname{\bf v}$ from Section 1 and record the basic estimate from [25]: Let $\rho=\frac{1}{2}\operatorname{tr}\operatorname{ad}_{\mathfrak{u}}\in\mathfrak{a}^{*}$. Assume that $Z$ is unimodular. Then $\rho|_{\mathfrak{a}_{H}}=0$ (cf. Lemma 4.2 in [25]) and hence $\rho$ descends to a functional on $\mathfrak{a}_{Z}$. Moreover, there exists constants $C_{1},C_{2}>0$ such that

[TABLE]

In the sequel it will be convenient to realize $\mathfrak{a}_{Z}$ via $\mathfrak{a}_{H}^{\perp}\subset\mathfrak{a}$. We denote by $\mathfrak{a}_{Z}^{--}$ the interior of $\mathfrak{a}_{Z}^{-}$.

Lemma 3.3**.**

(Integral inequalities)* Let $Z$ be a unimodular real spherical space. Then the following integral inequalities hold:*

1. (1)

There exist an element $a_{1}\in A_{Z}^{--}$ such that for all $a_{0}\in a_{1}A_{Z}^{-}$ the following assertions hold: After replacing $K$ by $\operatorname{Ad}(a_{0})^{-1}K$ there exists constants $c,C>0$ only depending on the normalization of measures, such that for all measurable function $f:Z\to\mathbb{R}_{\geq 0}$ with $\operatorname{supp}f\subset KA_{Z}^{-}\mathsf{W}\cdot z_{0}$ one has

[TABLE]

where $dz,dk,da$ are Haar measures on $Z,K,A_{Z}^{-}$. 2. (2)

Assume that $Z$ is wavefront type. Then there exists a compact neighborhood $B$ of ${\bf 1}$ in $G$ such that $Z=BA_{Z}^{-}\mathsf{W}\cdot z_{0}$ and a constant $C>0$, only depending on $B$ and the normalization of measures, such that for all measurable functions $f:Z\to\mathbb{R}_{\geq 0}$ one has

[TABLE]

with $dg$ a Haar measure on $G$.

The proof of Lemma 3.3 (2) rests on the following geometric lemma which is of independent interest.

Lemma 3.4**.**

(Volumes of balls under projection)* Assume that $Z=G/H$ is a unimodular real spherical space of wavefront type. Fix a compact neighborhood $B_{1}$ of ${\bf 1}$ in $G$. For $E\subset B_{1}$ a measurable subset and $a\in A_{Z}^{-}$ we let $E_{a}\subset B_{1}^{2}=B_{1}\cdot B_{1}$ be the inverse image of $Ea\cdot z_{0}$ under the quotient morphism*

[TABLE]

Then there exists a constant $C=C(B_{1})>0$ such that

[TABLE]

Proof.

Since $EB_{1}\subset B_{1}^{2}$ we first record that

[TABLE]

If $f$ is a compactly supported measurable function on $G$ we denote by $f^{H}$ its $H$-average: $f^{H}(gH)=\int_{H}f(gh)\ dh$. The claim is then a consequence of

[TABLE]

as functions on $Z$, uniformly for all $E\subset B_{1}$ and all $a\in A_{Z}^{-}$. In fact, integrating (3.9) over $Z$ yields (3.7). Now, for $b\in E$ we record

[TABLE]

where we used (3.8) for the first inequality. Thus the lemma will follow from

[TABLE]

for a constant $C$ only depending on $B_{1}$. To see that let $B_{U}$, $B_{\overline{U}}$, $B_{L}$ be convex compact neighborhoods of ${\bf 1}$ in $U,\overline{U},L$ such that $B_{1}\supset B_{U}B_{L}B_{\overline{U}}$. Because $Z$ is assumed to be wavefront, we note that $\operatorname{Ad}(a)^{-1}B_{U}\supset B_{U}$, $\operatorname{Ad}(a)^{-1}B_{\overline{U}}\subset B_{\overline{U}}$ and $\operatorname{Ad}(a)^{-1}B_{L}=B_{L}$.

Thus, if we replace $B_{1}$ by $B_{U}B_{L}B_{\overline{U}}$ we get

[TABLE]

Recall now the vector decomposition $\mathfrak{h}=\mathfrak{l}\cap\mathfrak{h}+{\mathcal{G}}(T)$ with ${\mathcal{G}}(T)\subset\overline{\mathfrak{u}}+(\mathfrak{u}+\mathfrak{l}_{H}^{\perp})$ the graph of the linear map $T$, see (3.1). Let now $Y_{1},\ldots,Y_{d}$ be basis of $\mathfrak{l}\cap\mathfrak{h}$ and $U_{1},\ldots,U_{l}$ be a basis of ${\mathcal{G}}(T)$ with each $U_{i}$ of the form $U_{i}=X_{-\alpha_{i}}+T(X_{-\alpha_{i}})$ for some $X_{-\alpha_{i}}\in\mathfrak{g}^{-\alpha_{i}}\subset\overline{\mathfrak{u}}$. Then it follows from (3.11) that there exists an $\epsilon>0$ such that

[TABLE]

This establishes (3.10) and hence the lemma. ∎

Proof of Lemma 3.3.

We first prove (2). The arguments which follow will show that it is no restriction to assume that $\mathsf{W}=\{{\bf 1}\}$. So let us assume that in order to save notation.

We start with a compact neighborhood $B_{1}$ of ${\bf 1}$ in $G$ such that $B_{1}A_{Z}^{-}\cdot z_{0}=Z$. Let $E\subset B_{1}$ be a measurable subset. For $a\in A_{Z}^{-}$ we let $E_{a}\subset B_{1}^{2}=B_{1}\cdot B_{1}$ be as in Lemma 3.4 above.

Next let $B_{A_{Z}}$ be a fixed compact neighborhood of ${\bf 1}$ in $A_{Z}$. We assume that $B_{A_{Z}}=B_{A_{Z}}^{-1}\subset B_{1}$. We wish to estimate the integral

[TABLE]

First note that there exists a constant $C$ depending on the size of $B_{A_{Z}}$ such that

[TABLE]

Fubini then gives

[TABLE]

Now observe for $a^{\prime}\in B_{A_{Z}}$

[TABLE]

Hence we get from (3.12) and (3.7) that

[TABLE]

In particular we have shown that

[TABLE]

Since $E\subset B_{1}$ was arbitary and $B_{1}A_{Z}^{-}\cdot z_{0}=Z$ we thus get for all measurable functions $f\geq 0$ on $Z$ that

[TABLE]

This implies with $B:=B_{1}^{4}$ that

[TABLE]

and completes the proof for (2) – notice that the flow of arguments showed we can squeeze elements $\mathsf{w}\in\mathsf{W}$ between $a\cdot z_{0}$ without doing any harm.

We move to the proof of (1) which is implicitly contained in the proof of [27], Th. 8.5. We summarize the argument, assuming for simplicity the existence of (3.4).

Fix $\mathsf{w}\in\mathsf{W}$, let $M_{H}:=M\cap H$ and consider the smooth map

[TABLE]

If we assume that $K$ is chosen such that $\mathfrak{k}+\mathfrak{a}+\operatorname{Ad}(a\mathsf{w})\mathfrak{h}=\mathfrak{g}$ for all $a\in A_{Z}^{-}$ (which can be achieved by replacing $K$ with $\tilde{a}_{0}^{-1}K\tilde{a}_{0}$ for an appropriate $\tilde{a}_{0}\in A_{Z}^{--}\subset A$ (see (6.4) in [27]), then $\Psi_{\mathsf{w}}$ has everywhere invertible differential, i.e. $\Psi_{\mathsf{w}}$ is étale. Moreover, as $\Psi_{\mathsf{w}}$ arises from an algebraic map defined over $\mathbb{R}$, it is finite to one. Hence if we pull the invariant differential form on $Z$ back to $K/M_{H}\times A_{Z}^{-}$ via $\Psi_{\mathsf{w}}$ we obtain that

[TABLE]

for a positive function $J_{\mathsf{w}}$ on $A_{Z}^{-}$. Moreover, if $\operatorname{supp}f\subset KA_{Z}^{-}\mathsf{w}\cdot z_{0}$ we obtain a constant $c>0$ such that

[TABLE]

holds as well.

Up to a constant factor $J_{\mathsf{w}}(a)$ equals $|\det d\Psi_{\mathsf{w}}({\bf 1}M_{H},a)|$ for which a formula was provided in the proof of Prop. 4.5 of [25]:

[TABLE]

where ${\bf X}=X_{1}\wedge\ldots\wedge X_{d}$ for a basis $X_{1},\ldots,X_{d}$ of $\mathfrak{h}$, ${\bf Y}=Y_{1}\wedge\dots\wedge Y_{r}$ associated to a basis of $\mathfrak{a}_{Z}$ (viewed as a subspace of $\mathfrak{a}$) and finally ${\bf U}=U_{1}\wedge\ldots\wedge U_{k}$ with $U_{1},\ldots,U_{k}$ a basis of $\mathfrak{m}_{H}^{\perp_{\mathfrak{k}}}\subset\mathfrak{k}$. Now for any $X\in\mathfrak{a}_{Z}^{--}$ we recall from Remark 2.2 in [25] that

[TABLE]

with ${\bf X}_{\lim{}}$ corresponding to a basis of the maximal deformation $\mathfrak{h}_{\lim{}}=\mathfrak{l}\cap\mathfrak{h}+\overline{\mathfrak{u}}$ of $\mathfrak{h}$. In more quantitative terms we have (see (4.4) in [25]) up to normalization

[TABLE]

with $|R(a)|\leq C\max_{\alpha\in S}a^{\alpha}$ with $S$ a set of spherical roots. In particular, we obtain

[TABLE]

provided that $a_{1}\in A_{Z}^{--}$ lies deep enough in the Weyl chamber. Finally notice that the $a_{1}$-shift in the argument of $J_{\mathsf{w}}$ above corresponds to replacing $K$ by $\operatorname{Ad}(a_{1})^{-1}K$. This proves (1).

∎

In the sequel we request that $K$ is chosen such that the conclusion of Lemma 3.3 (1) holds.

For the rest of this article we assume that $Z$ is a real spherical space of wavefront type.

3.3. Property I

We briefly recall some results and notions from [25] and [30].

Let $(\pi,\mathcal{H}_{\pi})$ be a unitary irreducible representation of $G$. We denote by $\mathcal{H}_{\pi}^{\infty}$ the $G$-Fréchet module of smooth vectors and by $\mathcal{H}_{\pi}^{-\infty}$ its dual. Elements in $\mathcal{H}_{\pi}^{-\infty}$ are called distribution vectors. It is known (see [32]) that for a real spherical space the space $(\mathcal{H}_{\pi}^{-\infty})^{H}$ of $H$-fixed distribution vectors is finite dimensional. The representation $\pi$ is said to be $H$-distinguished if $(\mathcal{H}_{\pi}^{-\infty})^{H}\neq\{0\}$.

Let $\eta\in(\mathcal{H}_{\pi}^{-\infty})^{H}$ and $H_{\eta}<G$ the stabilizer of $\eta$. Note that $H<H_{\eta}$ and set $Z_{\eta}:=G/H_{\eta}$. With regard to $\eta$ and $v\in\mathcal{H}^{\infty}$ we form the generalized matrix-coefficient

[TABLE]

which is a smooth function on $Z_{\eta}$.

We recall the following facts from [25] Thm. 7.6 and Prop. 7.7:

Proposition 3.5**.**

Let $Z$ be a wavefront real spherical space with $H<G$ reductive. Then the following assertions hold:

1. (1)

Every generalized matrix coefficient $m_{v,\eta}$ as above is bounded. 2. (2)

Let $(\pi,\mathcal{H})$ be a unitary irreducible representation of $G$ and let $\eta\in(\mathcal{H}_{\pi}^{-\infty})^{H}$. Then $Z_{\eta}$ is unimodular, i.e. carries a positive $G$-invariant Radon measure, and there exists $1\leq p<\infty$ such that $m_{v,\eta}\in L^{p}(Z_{\eta})$ for all $v\in\mathcal{H}_{\pi}^{\infty}$.

The property of $Z=G/H$ that (2) is valid for all $\pi$ and $\eta$ as above is denoted Property (I) in [25]. Note that (1) and (2) together imply $m_{v,\eta}\in L^{q}(Z_{\eta})$ for $q>p$. Assuming Property (I) we can then make the following notation.

Definition 3.6**.**

Given $\pi$ as above, we define $p_{H}(\pi)$ as the smallest index $\geq 1$ such that all $K$-finite generalized matrix coefficients $m_{v,\eta}$ with $\eta\in(\mathcal{H}_{\pi}^{-\infty})^{H}$ belong to $L^{p}(Z_{\eta})$ for any $p>p_{H}(\pi)$.

Notice that $m_{v,\eta}$ belongs to $L^{p}(Z_{\eta})$ for all $K$-finite vectors $v$ once that this is the case for some non-trivial such vector $v$, see [25] Lemma 7.2. For example, this could be the trivial $K$-type, if it exists in $\pi$.

It follows from Proposition 3.5 (2) and finite dimensionality of $(\mathcal{H}_{\pi}^{-\infty})^{H}$ that $p_{H}(\pi)<\infty$. We say that $\pi$ is $H$-tempered if $p_{H}(\pi)=2$. Note that if $\pi$ is not $H$-distinguished (that is, if $(\mathcal{H}_{\pi}^{-\infty})^{H}=0$) then $p_{H}(\pi)=1$.

3.4. The wavefront lemma

Denote $A^{-}=\exp(\mathfrak{a}^{-})$ and notice that the wavefront property (3.3) implies that

[TABLE]

With that we obtain a generalization of the “wavefront lemma” of Eskin-McMullen ([9] Theorem 3.1), see Lemma 6.3 in [24].

Lemma 3.7**.**

(Wavefront Lemma)* Suppose that $Z=G/H$ is a wavefront real spherical space. Then there exists a closed subset $E\subset G$ with the following properties.*

1. (1)

$E\to G/H$* is surjective.* 2. (2)

The family of left translations of $Z$ by elements $g\in E$ is equicontinuous at $z_{0}$, that is, for every neighborhood ${\mathcal{V}}$ of ${\bf 1}$ in $G$, there exists a neighborhood ${\mathcal{U}}$ of ${\bf 1}$ in $G$ such that

[TABLE]

for all $g\in E$.

Proof.

Put

[TABLE]

Note that $E\subset G$ is closed as $A^{-}\subset G$ is closed and both $\Omega$ and $\mathsf{W}$ are compact subsets of $G$. Then (1) follows from the wavefront assumption, by (3.5) and (3.13).

Let ${\mathcal{V}}$ be a neighborhood of ${\bf 1}$ in $G$. By compactness there exists a neighborhood ${\mathcal{V}}_{1}\subset{\mathcal{V}}$ such that $\operatorname{Ad}(x)\mathcal{V}_{1}\subset{\mathcal{V}}$ for all $x\in\Omega$. Then it suffices to establish the implication in (2) for $g\in A^{-}\mathsf{W}$ and with ${\mathcal{V}}_{1}$ instead of ${\mathcal{V}}$.

Since conjugation by $A^{-}$ contracts $\mathfrak{n}$ there exists an open neighborhood ${\mathcal{U}}_{1}\subset\mathcal{V}_{1}\cap P$ of ${\bf 1}$ in $P$ such that $\operatorname{Ad}(a)\mathcal{U}_{1}\subset{\mathcal{U}}_{1}$ for all $a\in A^{-}$. It follows from the openness of $P\mathsf{w}\cdot z_{0}$ for each $\mathsf{w}$ that

[TABLE]

is open. It hence contains ${\mathcal{U}}\cdot z_{0}$ for some neighborhood $\mathcal{U}$ of ${\bf 1}$ in $G$. With that we obtain for $z\in{\mathcal{U}}\cdot z_{0}$ that $\mathsf{w}\cdot z\in{\mathcal{U}}_{1}\mathsf{w}\cdot z_{0}$ for all $\mathsf{w}\in\mathsf{W}$, and hence for $g=a\mathsf{w}\in A^{-}\mathsf{W}$ that

[TABLE]

as claimed. ∎

We refer to [8] or [9] for the notion of well-rounded balls.

Theorem 3.8**.**

Let $Z=G/H$ be a wave-front real spherical space. Then MTC holds for any family $\mathcal{B}$ of well-rounded balls. If in addition $H$ is reductive, then MTC holds for the intrinsic balls.

Proof.

The first part follows from [9]. To be precise: Th. 1.2 (Equidistribution) in [9] only requires the wave-front lemma. MTC, that is [9] Th. 1.4, then follows from Th. 1.2 for any family of well-rounded balls. The last statement follows from [30], Section 2, where it was shown that the intrinsic balls are well rounded for $H$ reductive. ∎

Remark 3.9**.**

The wavefront lemma holds for an arbitrary Lie group $G$ when considered as a homogeneous space $Z=G\times G/\Delta(G)\simeq G$ with isomorphism provided by the map

[TABLE]

We take $E=G\times\{1\}\subset G\times G.$ Now for $(g,{\bf 1})\in E$ and ${\mathcal{U}}=U_{1}\times U_{2}$, a neighborhood of $({\bf 1},{\bf 1})$ in $G\times G$, we have

[TABLE]

On the other hand for the given neighborhood ${\mathcal{V}}=V_{1}\times V_{2}$ we have similarly

[TABLE]

and this contains $gV_{2}^{-1}.$ Thus we just have to require of ${\mathcal{U}}$ that $U_{1}U_{2}^{-1}\subset V_{2}^{-1}.$

4. Generalities on counting with error terms

After the interlude on wavefront real spherical spaces, we pick up the discussion from Section 2 and continue with error terms for the main term count. Assume that we have a quadruple $(X,\mu,D,(B_{R})_{R>0})$ which satisfies main term counting MTC. We then define the pointwise error term

[TABLE]

and one might ask for the optimal $\alpha\leq 1$ such that

[TABLE]

For the GCP one knows that $\alpha\leq\frac{131}{416}=0,3149\ldots$ and some believe that $\alpha=\frac{1}{4}$, the lower threshold of Hardy and Landau, is possible.

In this regard we mention that $\alpha=\frac{1}{2}$ can be achieved for every family of balls which are given by $B_{R}=R\cdot B$ for some absolutely convex bounded set $B\subset\mathbb{R}^{2}$ with $C^{2}$-boundary, see [17], p.8. Also observe that the geometry of $B$ matters for the error count as $\alpha=\frac{1}{2}$ is obviously optimal for the square $B=[-1,1]^{2}$.

Thus if it comes to error term bounds we should use balls which are as round as possible, for instance the intrinsic balls which we just introduced.

In obtaining the reasonably good bound of $\alpha=\frac{1}{3}$ for the GCP elementary techniques from Fourier analysis suffice. As an outsider one might ask at what threshold of $\alpha$ analysis converts to number theory. Before we come to the issue of $\alpha=\frac{1}{3}$ we pin down a more specific general setup.

For the remainder we only consider quadruples $(G/H,\mu,\Gamma/\Gamma_{H},(B_{R})_{R>0})$ which satisfy the MTC. Specifically $Z=G/H$ is a unimodular homogeneous space such that there is a lattice $\Gamma\subset G$ with $\Gamma_{H}=\Gamma\cap H$ a lattice in $H$. We assume that the Haar measures on $G$ and $H$ are normalized such that $G/\Gamma$ and $H/\Gamma_{H}$ both have volume 1.

With this set-up there is then a double fibration

[TABLE]

By fibre-wise integration we obtain maps between functions on $Z$ and $Y$. From $L^{\infty}(Y)$ to $L^{\infty}(Z)$ we thus have $\phi\mapsto\phi^{H}$ defined by

[TABLE]

with

[TABLE]

In the opposite direction, from $L^{1}(Z)$ to $L^{1}(Y)$ we have $f\mapsto f^{\Gamma}$ defined by

[TABLE]

and likewise contractive

[TABLE]

From Fubini’s theorem we obtain the following adjointness relation:

[TABLE]

for all $\phi\in L^{\infty}(Y)$ and $f\in L^{1}(Z)$. In particular, applying (4.5) to $|f|$ and $\phi={\bf 1}_{Y}$ implies (4.4).

We write ${\bf 1}_{R}\in L^{1}(Z)$ for the characteristic function of $B_{R}$ and deduce

• •

${\bf 1}_{R}^{\Gamma}(e\Gamma)=N_{R}(\Gamma,Z):=\#\{\gamma\in\Gamma/\Gamma_{H}\mid\gamma\cdot z_{0}\in B_{R}\}$.

• •

$\|{\bf 1}_{R}^{\Gamma}\|_{L^{1}(G/\Gamma)}=|B_{R}|$.

where the second equality follows from (4.5) with $\phi={\bf 1}_{Y}$.

Example 4.1**.**

Once again we return to the GCP with $Z=G=\mathbb{R}^{2}$ and $\Gamma=\mathbb{Z}^{2}$. Then

[TABLE]

Let us recall how to obtain the bound $\alpha={1\over 3}$ by means of harmonic analysis on $Y=\mathbb{R}^{2}/\mathbb{Z}^{2}$, more precisely the Poisson summation formula. Informally we would like to apply this and deduce

[TABLE]

where $\widehat{{\bf 1}_{R}}$ is the Fourier-transform of the the characteristic function ${\bf 1}_{R}$ and $\Gamma^{\wedge}$ is the dual lattice. However, as the cutoff ${\bf 1}_{R}$ is not smooth, the sum in (4.6) is not absolutely convergent. The remedy is to smoothen ${\bf 1}_{R}$ by convolution with some radial $\varphi\in C_{c}^{\infty}(G)$ of integral $1$, i.e. set ${\bf 1}_{R,\varphi}:=\varphi*{\bf 1}_{R}$ and consider

[TABLE]

Now observe that

[TABLE]

and $\sum_{\gamma\in\mathbb{Z}^{2}\atop\gamma\neq 0}\widehat{{\bf 1}_{R,\varphi}}(\gamma)$ converges. By using for $\varphi$ an approximation of the identity, one derives the estimate as follows.

Since the theory applies equally well in $n$ dimensions we will work in this generality, and fix a non-negative function $\varphi\in C^{\infty}(\mathbb{R}^{n})$, supported in $B_{1}$ and with integral $1$. Let $\varphi_{\epsilon}(x)=\epsilon^{-n}\varphi(\epsilon^{-1}x)$, and observe that

[TABLE]

Hence if we denote $N_{R,\epsilon}=\sum_{\gamma\in\Gamma}{\bf 1}_{R}*\varphi_{\epsilon}(\gamma)\,$ we find

[TABLE]

By explicit calculation (see [43], Ch. IV.3) one finds

[TABLE]

where $J_{\frac{n}{2}}$ is a Bessel function and $C_{1},C_{2}>0$ some constants, which depend only on $n$. It is well-known ([43], Lemma IV.3.11) that the Bessel functions $J_{m}(x)$ for all $m\geq 0$ behave like $x^{-\frac{1}{2}}$ as $x\to\infty$. Hence

[TABLE]

as $R|\lambda|\to\infty$.

Since ${\bf 1}_{R}*\varphi_{\epsilon}\in C^{\infty}_{c}(\mathbb{R}^{n})$ we can apply Poisson summation. By using $\widehat{{\bf 1}_{R}*\varphi_{\epsilon}}=\widehat{{\bf 1}_{R}}\widehat{\varphi_{\epsilon}}$ we obtain from (4.7)

[TABLE]

where $\Gamma^{\wedge}$ denotes the dual lattice. Next we claim that there exists a constant $C>0$ such that

[TABLE]

holds for all $\epsilon>0$. In fact, by the rapid decay of $\widehat{\varphi}$ we have $|\widehat{\varphi}(\epsilon x)|\leq C(1+|\epsilon x|)^{-n-1}$ for all $x\in\mathbb{R}^{n}$ and a fixed constant $C>0$. Hence, by the integral criterion for sums of decaying functions we obtain

[TABLE]

for a constant $c=c(\Gamma^{\wedge})>0$. Finally

[TABLE]

and the claim follows from $n\geq 2$. Combining (4.9) with (4.10) yields

[TABLE]

Now \big{|}|B_{R\pm\epsilon}|-|B_{R}|\big{|}\sim R^{n-1}\epsilon, and by comparing with (4.8) we then conclude that

[TABLE]

for some $C>0$ and all $\epsilon>0$. The best possible value of $\epsilon$ is $R^{\frac{1-n}{n+1}}$ and yields

[TABLE]

For $n=2$ this gives the mentioned bound by $O(R^{\frac{2}{3}})$. For $n=3$ it gives $O(R^{\frac{3}{2}})$, compared to the best known upper bound $O(R^{\frac{4}{3}}\log R)$. For $n\geq 4$ the error is known to behave essentially like $R^{n-2}$ (see [11]).

This method resembles to some extend the approach in [30] where error term bounds were obtained for wavefront spaces: Poisson summation was replaced by spectral analysis and Weyl’s law.

Example 4.2**.**

As a second instance of a classical problem of counting lattice points we review the case of the hyperbolic space. This study was initiated by Delsarte, see [6]. Let the upper half plane $\mathbf{H}=\{z=x+iy\in\mathbb{C}:y>0\}$ be equipped with the invariant metric $ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}$ by which it has constant negative curvature. The Riemanian metric induces a distance function $d$ on $\mathbf{H}$ which we use to define a family of balls

[TABLE]

Their volume is exponentially growing with the radius:

[TABLE]

To set up the counting problem, recall that the hyperbolic plane admits a group of symmetries $G:=\operatorname{Iso}(\mathbf{H})=\operatorname{PSL}_{2}(\mathbb{R})$ and the role of the lattice is played by a discrete subgroup $\Gamma<\operatorname{PSL}_{2}(\mathbb{R})$ of finite co-volume. More precisely, we consider the set $D_{\Gamma}(z)=\Gamma\cdot z,$ with $z\in\mathbf{H}.$ The hyperbolic lattice counting problem then consists of estimating

[TABLE]

as $R\to\infty$.

An interesting feature of this problem is that, unlike the Euclidean lattice point counting, a direct packing argument is not possible since most of the volume of a hyperbolic ball is located near its boundary. This so-called mass concentration phenomenon is a reflection of the exponential volume growth of the Haar measure on the semi-simple Lie group $G$. Thus, different techniques are required to approach this lattice counting problem and even obtaining the main term was a non-trivial achievement. In [6] the problem is studied for co-compact lattices and MTC is proved, that is, with the measure normalized as above,

[TABLE]

for all $z$, as $R\to\infty$. In [38] Selberg developed the trace formula as a tool of obtaining geometric information from spectral information and vice-versa.

In case $\Gamma=\operatorname{PSL}(2,\mathbb{Z})$, Selberg showed [38] that

[TABLE]

and obtained an error term estimate,

[TABLE]

Maybe the most significant result is the connection between the lattice counting problem for a general lattice of finite co-volume $\Gamma\subset\operatorname{PSl}(2,\mathbb{Z})$ and the so-called automorphic spectrum. More precsiely, Selberg provided the following asympotic formula expressing the number of lattice points as a sum over the eigenvalues of the Laplace operator on the Riemann surface $\Gamma\backslash\mathbf{H}=\Gamma\backslash G/K.$ To formulate the exact formula we denote by $\Delta$ the Laplace-Beltrami operator on the Riemannian manifold $\Gamma\backslash\mathbf{H}.$ This is sometimes called the hyperbolic Laplacian. In case $\Gamma\backslash\mathbf{H}$ is compact the spectrum of $\Delta$ is discrete. In the non-compact case, in addition to the discrete spectrum there is a contribution from the continuous spectum. This spectrum consists of Eisenstein series which are parametrized by the interval $[\frac{1}{2},\infty)$ and by cusps. In both cases, we denote by $\{u_{j}\}_{j\geq 0}$ a complete system of orthonormal eigenfunctions for the discrete spectrum of the hyperbolic Laplacian corresponding to $0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\cdots.$ In particular $u_{0}=\operatorname{vol}(\Gamma\backslash\mathbf{H})^{-\frac{1}{2}}$.

Fix $z,w\in\mathbf{H}$ and let

[TABLE]

thus $N_{R}(z)=N(R^{\prime},z,i)$ with $R^{\prime}=2\cosh(R)\sim e^{R}$. By using the spectral expansion of the automorphic kernel (see Theorem 7.1 [18]), Selberg showed in [39] and [40] (see also Theorem 2 of [36] and Theorem 12.1 in [18]) the following formula:

[TABLE]

The relation between $s_{j}$ and the eigenvalue $\lambda_{j}$ is given by the usual translation $\lambda_{j}:=s_{j}(1-s_{j}).$ Notice that in the sum, the value $s_{0}=1$ corresponds to the constant function yielding the main term $\frac{\pi}{\operatorname{vol}(\Gamma\backslash\mathbf{H})}R.$ The rest of the sum is over exceptional eigenvalues, that is those eigenvalues of the Laplacian that satisfy $\lambda_{j}:=s_{j}(1-s_{j})<\frac{1}{4}.$

The moral of the last example is that even if one invokes the full force of the spectral decomposition of $L^{2}(\Gamma\backslash\mathbf{H})$ and the trace formula, understanding the error term in the lattice counting problem requires refined information about the automorphic spectrum. For example, when considering congruence lattices $\Gamma(N)$, the Selberg-$\frac{1}{4}$-conjecture, that there are no exceptional eigenvalues, that is $\lambda_{1}(\Gamma(N)\backslash\mathbf{H})\geq\frac{1}{4}$, is relevant to determining the main term in the corresponding lattice point counting formula.

For the rest of this article we assume now that $Z=G/H$ is a wavefront real spherical space, $H$ is reductive and $\mathcal{B}$ is constituted by the intrinsic balls. Further we restrict ourselves to the cases where the cycle $H/\Gamma_{H}$ is compact.111After a theory for regularization of $H$-periods of Eisenstein series is developed, it is expected that one can drop this assumption. To simplify the exposition here we assume in addition that $\Gamma<G$ is irreducible, i.e. there do not exist non-trivial normal subgroups $G_{1},G_{2}$ of $G$ and lattices $\Gamma_{i}<G_{i}$ such that $\Gamma_{1}\Gamma_{2}$ has finite index in $\Gamma$.

The error we study is measure theoretic in nature, and will be denoted here as $\operatorname{err}(R,\Gamma)$. Thus, $\operatorname{err}(R,\Gamma)$ measures the deviation of two measures on $Y=\Gamma\backslash G$, the counting measure arising from lattice points in a ball of radius $R$, and the invariant measure $d\mu_{Y}$ on $Y$. More precisely, with ${\bf 1}_{R}$ denoting the characteristic function of $B_{R}$ we consider the densities

[TABLE]

Then,

[TABLE]

where $||\,\cdot\,||_{1}$ denotes the total variation of the signed measure. Notice that $|F^{\Gamma}_{R}(e\Gamma)-1|=\frac{|N_{R}(\Gamma,Z)-|B_{R}||}{|B_{R}|}$ is essentially the error term for the point wise count.

Our results on the error term $\operatorname{err}(R,\Gamma)$ allows us to deduce results toward the error term in the smooth counting problem, a classical problem that studies the quantity

[TABLE]

where $\alpha\in C_{c}^{\infty}(G)$ is a positive smooth function of compact support (with integral one) and $F^{\Gamma}_{\alpha,R}=\alpha*F^{\Gamma}_{R}.$ We refer to [30], Remark 5.2 for the comparison of $\operatorname{err}(R,\Gamma)$ with $\operatorname{err}_{pt,\alpha}(R,\Gamma)$.

Below we will introduce an exponent $p_{H}(\Gamma)$ (see (5.1)), which measures the worst $L^{p}$-behavior of any generalized matrix coefficient associated with a spherical unitary representation $\pi$, which is $H$-distinguished and occurs in the automorphic spectrum of $L^{2}(\Gamma\backslash G)$. In [30] we obtained the following error bound for triple spherical spaces.

Theorem 4.3**.**

Let $Z=G_{0}^{3}/\operatorname{diag}(G_{0})$ for $G_{0}=\operatorname{SO}_{e}(1,n)$ and assume that $H/\Gamma_{H}$ is compact. For all $p>p_{H}(\Gamma)$ there exists a $C=C(p)>0$ such that

[TABLE]

for all $R\geq 1$.

5. Counting with error terms on a wavefront real spherical space

We assume from now on that $G$ is semisimple with no compact factors.

First we need some notation. We denote by $\widehat{G}$ the unitary dual of $G$, i.e. the set of equivalence classes of irreducible unitary representations of $G$. We recall the maximal compact subgroup $K<G$. By $\widehat{G}_{s}\subset\widehat{G}$ we understand the subset which corresponds to $K$-spherical representations, i.e. representations which have a non-zero $K$-fixed vector.

With that we define in continuation of Definition 3.6

[TABLE]

and recall that $p_{H}(\Gamma)<\infty$ by [30], Lemma 6.3.

A subset $\Lambda\subset\widehat{G}_{s}$ is called $L^{p}$-bounded provided that $p_{H}(\pi)<p$ for all $\pi\in\Lambda$. Set $\Lambda_{p}:=\{\pi\in\widehat{G}_{s}\mid p_{H}(\pi)<p\}$. We note that

[TABLE]

and

[TABLE]

In [30] we formulated the following hypothesis.

Hypothesis A: For $1\leq p<\infty$ there exists a compact subset $D\subset G$ and constants $c,C>0$ such that the following assertions hold for all $\pi\in\Lambda_{p}$, $\eta\in(\mathcal{H}_{\pi}^{-\infty})^{H}$ and $v\in\mathcal{H}_{\pi}^{K}$:

[TABLE]

[TABLE]

*Here $\|\,\cdot\,\|_{\infty,D}$ denotes the supremum norm taken on the subset $D.$ *

Under this hypothesis we have then shown in [30] Thm. 7.3 that:

Theorem 5.1**.**

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

• •

$G$* is semisimple with no compact factors*

• •

$H$* is reductive and the balls $B_{R}$ are intrinsic*

• •

$\Gamma$* is an arithmetic and irreducible lattice*

• •

$\Gamma_{H}=H\cap\Gamma$* is co-compact in $H$*

• •

$p>p_{H}(\Gamma)$**

• •

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

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

[TABLE]

for all $R\geq 1$. Moreover, if $Y=\Gamma\backslash G$ is compact one can replace the last condition by $k>\dim(G/K)+1$.

In [30] the Hypothesis A was verified for the triple spaces, and thus we could derive Theorem 4.3 from Theorem 5.1.

When writing [30] the harmonic analysis on real spherical spaces was not sufficiently developed to obtain Hypothesis A (or suitable variants thereof) in general. In particular at that time we were not able to derive an error bound for all wavefront real spherical spaces. For symmetric spaces the existence of a non-quantitative error term was established in [2] and improved in [12].

In case of the hyperbolic plane our error term (5.2) is still far from the quality of the bound of A. Selberg. This is because the approach in [30] only uses a weak version of the trace formula, namely Weyl’s law, and uses simple soft Sobolev bounds between eigenfunctions on $Y$.

Here we shall prove a slightly weaker version of Hypothesis A for spaces $Z$ of real rank one. In order to prepare for it we draw from some notions which we used in the lecture notes [33]. Together with every G-continuous norm $p$ on a Harish-Chandra module $V$ there comes a family of Laplace-Sobolev norms $(p_{k})_{k\in\mathbb{N}}$ on $V^{\infty}$. We briefly recall the definition of the $p_{k}$ for $k$ even (one can define Sobolev norms $p_{k}$ for any value of $k\in\mathbb{R}_{\geq 0}$). For that we let $\Delta\in\mathcal{U}(\mathfrak{g})$ be a Laplace element such that $\Delta=\mathcal{C}+2\Delta_{\mathfrak{k}}$ where $\mathcal{C}$ is the Casimir element of $\mathfrak{g}$ and $\Delta_{\mathfrak{k}}\in\mathcal{U}(\mathfrak{k})$ is a Laplace element. Then

[TABLE]

For an irreducible Harish-Chandra module $V$ we let $\chi_{V}\in\mathbb{C}$ be the multiple by which the Casimir element $\mathcal{C}\in\mathcal{Z}(\mathfrak{g})$ acts on $V$. In particular, if $V$ is the Harish-Chandra module of a unitary irreducible representation $(\pi,\mathcal{H})$ we write $|\pi|=|\chi_{V}|$. The following lemma will be used frequently in the sequel to relate between $p_{k}$ and $p$.

Lemma 5.2**.**

Let $V$ be an irreducible Harish-Chandra module. Let $k\in\mathbb{N}_{0}$. Then there exists constants $C_{1},C_{2}>0$, independent of $V$, such that

[TABLE]

Proof.

(For $k$ even) As $v\in V^{K}$ is $K$-fixed we have $\Delta_{\mathfrak{k}}v=0$ and thus for every $j\in\mathbb{N}$

[TABLE]

Hence the assertion follows from (5.3). ∎

In the weaker version of Hypothesis A we replace (A1) by a Sobolev estimate, namely

Hypothesis B: Let $1\leq p^{\prime}<p<\infty$. Then there exist $C>0$ and $l=l(p,p^{\prime})\in\mathbb{N}$ such that

[TABLE]

for all $\pi\in\Lambda_{p^{\prime}}$, $\eta\in(\mathcal{H}_{\pi}^{-\infty})^{H}$ and $v\in\mathcal{H}_{\pi}^{K}$.

Remark 5.3**.**

Assume $\pi$ is non-trivial. Under the assumptions in Thm. 5.1 it follows from [30] Lemma 6.2 that $H_{\eta}/H$ is compact, and hence $Z_{\eta}$ can be replaced by $Z$ both in (A1) and in (B1).

Remark 5.4**.**

Let $1\leq p^{\prime}<p_{1}<p_{2}<\infty$. If (B1) holds for the pair $(p_{1},p^{\prime})$, for some constant $C=C_{1}>0$, then it holds for the pair $(p_{2},p^{\prime})$ as well, for some $C=C_{2}>0$. This follows from the fact that

[TABLE]

for all bounded functions $f$, and thus

[TABLE]

In the hypothesis above it is unnecessary to include an analogue of (A2), as such an analogue can be derived from (B1). This is the content of the next lemma. For $R>0$ we set

[TABLE]

where $\rho_{\mathfrak{n}}=\frac{1}{2}\operatorname{tr}\operatorname{ad}_{\mathfrak{n}}\in\mathfrak{a}^{*}$. Further we set

[TABLE]

and if $D\subset G$ is a compact set

[TABLE]

Lemma 5.5**.**

Let $Z$ be a wavefront real spherical space for which Hypothesis B is valid. With $p^{\prime}$, $p$, and $l=l(p,p^{\prime})$ as above let

[TABLE]

Then there exists a compact subset $D\subset G$ such that

[TABLE]

for all $\pi\in\Lambda_{p^{\prime}}$, $\eta\in(\mathcal{H}_{\pi}^{-\infty})^{H}$ and $v\in\mathcal{H}_{\pi}^{K}$.

Proof.

222 We allow the same symbol $C$ for universal positive constants, independently of their actual values.

We recall the polar decomposition $Z=\Omega A_{Z}^{-}\mathsf{W}\cdot z_{0}$ in (3.5) and the volume bound (3.6) for $\operatorname{\bf v}$. Since $Z$ is wavefront we note that $\operatorname{\bf v}$ is increasing to infinity.

For $f=m_{v,\eta}$ and $g\in\Omega$ we set $f_{g}(z)=f(g\cdot z)$ for $z\in Z$. We normalize $\|f\|_{\infty}=1$. Let now $g_{0}\in\Omega$, $\mathsf{w}\in\mathsf{W}$ and $X_{0}\in\mathfrak{a}_{Z}^{-}$ be such that $|f_{g_{0}}(\exp(X_{0})\mathsf{w}\cdot z_{0})|=\|f\|_{\infty}$ (these elements exist as $f$ decays at infinity by (3.6) and (5.6) below). We recall the invariant Sobolev Lemma from (1.1):

[TABLE]

for $s>{\dim G\over p}$ and all $\phi\in L^{p}(Z)^{\infty}$.

Thus (5.6) applied specifically to $\phi=L_{Y}f_{g}$ for $g\in\Omega$, $Y\in\mathfrak{g}$ with $\|Y\|=1$ and $z=\exp(X)\mathsf{w}\cdot z_{0}$ with $X\in\mathfrak{a}_{Z}^{-}$ yields

[TABLE]

for some $C>0$. For the second inequality we used $\rho(X)\leq 0$ (as $Z$ as wavefront) and $\sup_{g\in\Omega}\|L_{g}f\|_{p,s+1}\leq C\|f\|_{p,s+1}$ for all $f\in L^{p}(Z)^{\infty}$ and a constant $C$ which only depends on $\Omega$ and $s$. The latter follows as the Laplace-Sobolev norms from (5.3) compare to the standard Sobolev norms (see [4], Prop. 3.5).

Now define a function on $K\times A_{Z}^{-}$ by

[TABLE]

and observe that (5.7) implies

[TABLE]

With (B1) and (5.4) we thus obtain

[TABLE]

for some positive constant $C^{\prime}$. Set $\delta:={1\over 4C^{\prime}}(1+|\pi|)^{-{l+s+1\over 2}}$. Then the mean value theorem implies that there exists a neighborhood $U$ of ${\bf 1}$ in $K$ such that

[TABLE]

From Lemma 3.3 (1) we obtain

[TABLE]

We obtain from (5.8) and by applying (5.9) to $\phi=f_{g_{0}}$ a constant $C>0$ such that

[TABLE]

or equivalently (by our choice of $\delta$)

[TABLE]

Now (B1) together with Lemma 5.2 (with the $G$-continuous norm $p$ given by $p(v):=\sup_{z\in Z}|m_{v,\eta}(z)|$ for $v\in V$) yields

[TABLE]

Inserting this inequality into (5.10) gives

[TABLE]

which shows

[TABLE]

for a constant $C$ independent of $\pi$ and

[TABLE]

This shows that the maximum of $|f|$ is attained in a region as asserted (with $D=\Omega A^{-}_{C}$). ∎

Theorem 5.6**.**

Let $Z$ be a wavefront real spherical space for which Hypothesis B is valid, and assume all bulleted items in Thm. 5.1. Let $l=l(p,p^{\prime})$ for $p^{\prime}:=p_{H}(\Gamma)$ and $p>p^{\prime}$ and let $d$ be as in (5.5). Then there exists a constant $C=C(p)>0$ such that

[TABLE]

for all $R\geq 1$, where $s^{\prime}:=\dim G/K+l+2d$.

The bound (5.2) is obtained in [30] Thm. 7.3 from the estimate of Prop. 6.5 via standard techniques quite in resemblance to the Euclidean case, see (4.11) - (4.13). With Hypothesis A replaced by B, the analogue of Prop. 6.5 becomes:

Proposition 5.7**.**

Assume that $Z$ is wavefront real spherical and all the bulleted assumptions from Thm. 5.1. Assume moreover, that Hypothesis B is valid. Let

[TABLE]

Let $p>p_{H}(\Gamma)$. Then the map

[TABLE]

is continuous. More precisely, let $s:=\dim(G/K)$ and $r:=\operatorname{rank}(G/K)$. Then for all

1. (1)

$k>s+1$* if $Y$ is compact.* 2. (2)

$k>\frac{r+1}{2}s+1$* if $Y$ is non-compact and $\Gamma$ is arithmetic*

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

[TABLE]

Here $l=l(p,p_{H}(\Gamma))$ is the constant in Hypothesis B and the constant $d$ is from Lemma 5.5.

Proof.

For the sake of completeness we give the slightly modified proof. In addition we take the opportunity to correct a few mistakes in the proof of Prop. 6.5 of [30].

To begin with we recall part of the Langlands Plancherel-Theorem (see [34] or [1], p. 256, Main Theorem (adelic setup)) for $L^{2}(Y)$ which is of relevance to us, i.e. the $G$-invariant subspace $\mathcal{H}\subset L^{2}(Y)$ which is generated by the $K$-spherical spectrum. For the formulation of the Plancherel Theorem we restrict ourselves to the arithmetic setup, i.e. $G$ is defined over $\mathbb{Q}$ and $\Gamma$ is commensurable to $G(\mathbb{Z})$. In the sequel we use the language from [5] and let $P_{1},\ldots,P_{\ell}$ be a complete set of representatives of the $\Gamma$-conjugacy classes of minimal $\mathbb{Q}$-parabolic subgroups (cusps). Further for each $1\leq j\leq\ell$ we let $[P_{j}]$ be the finite set of parabolic subgroups which contain $P_{j}$. One calls $P,Q\in[P_{j}]$ associated provided that their Levi-subgroups are conjugate. We write $[P_{j}]_{\rm Ass}$ for the corresponding association classes and set ${\mathcal{P}}=\bigcup_{j=1}^{\ell}[P_{j}]_{\rm Ass}$. Tacitly we pick in the sequel for each association class a representative and by slight abuse of notation write $P\in{\mathcal{P}}$ understanding that $P$ is parabolic subgroup of $G$.

The rough Plancherel decomposition is $\mathcal{H}=\bigoplus_{P\in{\mathcal{P}}}\mathcal{H}_{P}$ with $\mathcal{H}_{d}:=\mathcal{H}_{G}$ yielding the discrete spectrum. We let $P=M_{P}A_{P}N_{P}$ be the associated Langlands decomposition and recall that the projection of $\Gamma\cap P$ to $M_{P}\simeq P/A_{P}N_{P}$ yields a lattice in $M_{P}$ (which is anisotropic if $P=P_{j}$ is minimal). Further we denote by $\widehat{M}_{P,d}\subset\widehat{M}$ the $K\cap M_{P}$-spherical discrete spectrum. For each $\sigma\in\widehat{M}_{P,d}$ we denote by $m(\sigma)\in\mathbb{N}$ the multiplicity of $\sigma$ in $L^{2}(M/\Gamma_{M_{P}})$. For $\sigma\in\widehat{M}_{P,d}$ and $\lambda\in i\mathfrak{a}_{P}^{*}$ we denote by $\pi_{\sigma,\lambda}=\operatorname{Ind}_{P}^{G}(\sigma\otimes\lambda)$ the associated unitary principal series (normalized induction). Then one has a unitary equivalence of $G$-modules:

[TABLE]

Let now $\phi\in C_{b}^{\infty}(Y)^{K}_{\rm van}$ and write $\phi=\phi_{d}+\phi_{c}$ for its decomposition in discrete and continuous Plancherel parts. Observe that in general we do not have $\phi_{d},\phi_{c}\in C_{b}^{\infty}(Y)^{K}_{\rm van}$ as $\phi_{d}$ might not be bounded in presence of non-trivial residual spectrum. However, as the arguments below will show, this will not cause us technical problems. We consider first the discrete part $\phi_{d}$.

If $Y$ is compact it follows from Weyl’s law (see [42], Th. 15.2) that the multiplicities in $L^{2}(Y)$ satisfy

[TABLE]

for some constant $c_{Y}>0$. It follows that

[TABLE]

for all $m>s/2+1$. In case $Y$ is non-compact, we let $\widehat{G}_{\Gamma,d}\subset\widehat{G}$ be the the discrete support of the Plancherel measure of $L^{2}(Y)$ and $m(\pi)$ the corresponding multiplicity of $\pi$. Assuming $\Gamma$ is arithmetic it is shown in [19] that

[TABLE]

so that for $m>rs/2+1$ we again obtain (5.13).

Let $p>p_{H}(\Gamma)$. As $\phi_{d}$ is in the discrete spectrum we decompose it as $\phi_{d}=\sum_{\pi}\phi_{\pi}$ according to $\mathcal{H}_{d}\simeq\bigoplus_{\pi}m(\pi)\mathcal{H}_{\pi}$, i.e. $\phi_{\pi}$ is the $\pi$-isotypical part of $\phi_{d}$. Since $\dim\mathcal{H}_{\pi}^{K}\leq 1$ and $\phi_{\pi}$ is $K$-fixed, we observe that $\phi_{\pi}\in m(\pi)\mathcal{H}_{\pi}$ generates an irreducible $G$-submodule of $m(\pi)\mathcal{H}_{\pi}$ which is isomorphic to $\mathcal{H}_{\pi}$ or $\{0\}$ according to whether $\phi_{\pi}$ is zero or not. In particular, we obtain with (B1) that

[TABLE]

Note that $\pi$ is non-trivial in these sums since $\phi_{d}$ has vanishing integral (this is the only place where the vanishing hypothesis on $\phi$ enters the proof). With (B2) we obtain further

[TABLE]

Let $B_{|\pi|}=DA_{d\log(1+|\pi|)}^{-}\mathsf{W}H_{c}\subset G$ where $H_{c}\subset H$ is a compact subset such that $H_{c}\Gamma_{H}=H$. Note that $\|\phi_{\pi}^{H}\|_{\infty,\Omega_{d\log(1+|\pi|)}}\leq\|\phi_{\pi}\|_{\infty,B_{|\pi|}}$ which together with (5.4) allows us to estimate the last sum as follows:

[TABLE]

with $C>0$ a constant depending only on $l$, where we used Lemma 5.2 for the last inequality.

Applying the Cauchy-Schwartz inequality combined with (5.13) we obtain

[TABLE]

with $C>0$.

In the sequel we view functions on $Y$ as right $\Gamma$-invariant functions on $G$. Recall from [33], Sect. 4, the notion of volume weights on a homogeneous space for $G$ and let $\operatorname{\bf v}_{Y}$ be a volume weight for $Y=G/\Gamma$. From the definition of $B_{|\pi|}$ and Lemma 5.8 below applied to $\Omega=\mathsf{W}H_{c}$ we now infer

[TABLE]

The invariant Sobolev lemma (see [33], Lemma 4.2) for $K$-invariant functions on $G/\Gamma$ then gives us with $n>{s\over 2}$ the bound

[TABLE]

again by use of (5.4). Thus we obtain from (5.15) that

[TABLE]

With $k=m+n$ this shows the asserted bound for the discrete spectrum.

We move on to the continuous part $\phi_{c}$ of $\phi$. We recall the Plancherel formula for $\mathcal{H}_{P}$ from (5.12) and note that $\phi_{c}=\sum_{P\in{\mathcal{P}}\atop P\neq G}\phi_{P}$. Accordingly we have $\phi_{P}=\sum_{\sigma\in\widehat{M}_{P,d}}\phi_{P,\sigma}$ where $\phi_{P,\sigma}$ refers to the $\sigma$-isotypical part. Fix now $\sigma\in\widehat{M}_{P,d}$ and unitary model $V_{\sigma}$ of it. We choose $\mathcal{H}_{\sigma,\lambda}:=L^{2}(K\times_{M_{P}\cap K}V_{\sigma})$ as unitary model for the principal series $\pi_{\sigma,\lambda}$. Note that this Hilbert model is uniform in $\lambda\in i\mathfrak{a}_{P}^{*}$.

Let $r_{P}:=\dim\mathfrak{a}_{P}$. Note that $|\pi_{\sigma,\lambda}|\asymp|\sigma|+|\lambda|^{2}$. Hence for $k>{r_{P}\over 2}$ there exists a constant $C>0$ such that

[TABLE]

Let

[TABLE]

be the spectral decomposition of $\phi_{P,\sigma}$. Here $\phi_{\sigma,\lambda}$ are smooth $K$-invariant functions on $Y$ obtained from $G$-equivariant embeddings

[TABLE]

which satisfy

[TABLE]

We recall from Theorem 3.2 of [3] specialized to $Y=G/\Gamma$ (see [3], Ex. 4.3.4) that there exists a constant $C>0$ such that

[TABLE]

where $\|v_{\sigma,\lambda}\|_{n}$ refers to the $n$-th Sobolev norm of $\mathcal{H}_{\sigma,\lambda}$ with $n>{s\over 2}$ as before. Here we are allowed to use the improved Sobolev shift (compared to $n>{\dim G\over 2}$ used in [3]) as $\phi_{\sigma,\lambda}$ is $K$-fixed. For a short proof of (5.20) in the very similar context with $Y$ replaced by $Z=G/H$ we refer to Lemma 9.4 in [33].

It follows from (5.20) that the integral (5.19) is uniformly convergent on compact subsets of $Y$. Hence we obtain from the compactness of $Y_{H}$ and Fubini’s theorem that

[TABLE]

From that we deduce that

[TABLE]

Using (B1) and (B2) we obtain as in the discrete that

[TABLE]

Now (5.22) combined with (5.20) and (5.16) yields that

[TABLE]

Inserting this bound into (5.21) we arrive at

[TABLE]

From this inequality we obtain that

[TABLE]

Finally for ${k\over 2}=m$ as above for the discrete spectrum, the sum $\sum_{\sigma\in\widehat{M}_{P,d}}{1\over(1+|\sigma|)^{{k\over 2}-\frac{r_{P}}{2}}}$ is finite. This completes the proof. ∎

Let $y_{0}=\Gamma\in Y$ be the standard base point and $\operatorname{\bf v}_{Y}$ be an associated volume weight.

Lemma 5.8**.**

Let $\Omega\subset G$ be a compact subset. Then there exists a constant $C>0$ such that

[TABLE]

Proof.

Let $B_{N}$ be a convex compact neighborhood of ${\bf 1}$ in $N$, and likewise $B_{A}\subset A$. Let $B_{\overline{N}}=\theta(B_{N})\subset\overline{N}$. Then $B:=B_{\overline{N}}B_{A}MB_{N}$ is a compact neighborhood of ${\bf 1}$ in $G$, so that we may assume the volume weight is given by $\operatorname{\bf v}_{Y}(y)=\operatorname{vol}_{Y}(By)$ for $y\in Y.$

Note that

[TABLE]

for $a\in A^{-}$. Set $B_{\overline{N}}^{a}:=\operatorname{Ad}(a)^{-1}B_{\overline{N}}$. Now for $y=ag\cdot y_{0}$ with $a\in A^{-}$ and $g\in\Omega$ we note that

[TABLE]

Set $B^{a}:=B_{\overline{N}}^{a}MB_{A}B_{N}$ and observe $B^{a}\subset B$. As $B\Omega$ is compact we thus obtain for all $a\in A^{-}$ and $g\in\Omega$

[TABLE]

It follows that

[TABLE]

Finally, $\operatorname{vol}_{G}(B^{a})\geq Ca^{2\rho_{\mathfrak{n}}}$ by the integral formula for the Bruhat decomposition. ∎

6. Proof of hypothesis B in case $\operatorname{rank}_{\mathbb{R}}Z=1$

In this section we assume that $Z$ is wavefront with $\operatorname{rank}_{\mathbb{R}}Z=1$, i.e. $\dim\mathfrak{a}_{Z}=1$ (see Remark 3.2 for examples, and note in particular the cases (20) - (21), which are far from being symmetric).

6.1. Basic geometry of rank one wavefront spaces

We begin by showing that $\mathfrak{a}_{Z}^{-}$ is a half-line.

Lemma 6.1**.**

Let $Z$ be a wavefront real spherical space of real rank one. Then $\mathfrak{a}_{Z}^{-}\neq\mathfrak{a}_{Z}$.

Proof.

We argue by contradiction and assume that $\mathfrak{a}_{Z}^{-}=\mathfrak{a}_{Z}$. This implies $\mathfrak{a}_{Z}$ equals the edge of the compression cone and thus normalizes $H$. The overgroup $\widehat{H}:=HA_{Z}$ is real spherical and the associated real spherical space $\widehat{Z}=G/\widehat{H}$ has real rank zero. But then [21], Cor. 8.5, implies that $\widehat{H}$ contains a conjugate of $AN$ and this excludes $H$ from being wavefront. ∎

Set $\mathfrak{a}_{Z}^{+}:=-\mathfrak{a}_{Z}^{-}$ and note that $\mathfrak{a}_{Z}=\mathfrak{a}_{Z}^{-}\cup\mathfrak{a}_{Z}^{+}$ with $\mathfrak{a}_{Z}^{-}\cap\mathfrak{a}_{Z}^{+}=\{0\}$.

The first important feature of rank one spaces is a simplified polar decomposition, namely: there exists a compact subset $\Omega_{Z}\subset Z$ such that

[TABLE]

The reason for that is that the standard compactification of $Z$ has only one $G$-orbit in the boundary see [21], Th. 13.7.

In particular we obtain for any non-negative measurable function $f$ on $Z$ that

[TABLE]

We obtain from Lemma 3.3 (1) that

[TABLE]

where $\asymp$ signifies that the ratio of the left hand side by the right hand side is bounded from above and below by positive constants independent of $f$.

In particular, if in addition $f$ is $K$-invariant then we obtain that

[TABLE]

As for proving Hypothesis B it is then not serious to assume that $Z=KA_{Z}^{-}\mathsf{W}\cdot z_{0}$.

We now come to the main technical tool for the verification of Hypothesis B.

6.2. The constant term approximation

Our concern then is with matrix coefficients $f=m_{v,\eta}$ for a unitarizable irreducible Harish-Chandra module $V$ and $v\in V^{K}$ a $K$-fixed vector.

In general, if $V$ is Harish-Chandra module with smooth moderate growth completion $V^{\infty}$, then we refer to $(V,\eta)$ as a spherical pair provided that $\eta:V^{\infty}\to\mathbb{C}$ is a continuous $H$-invariant functional.

Theorem 6.2** (Constant term approximation).**

Let $Z=G/H$ be a wave-front real spherical space of real rank one. There exist constants $C,c_{0}>0$ with the following properties.

Let $(V,\eta)$ be a spherical pair with $V$ irreducible, which satisfies an priori-bound

[TABLE]

for some $r>0$ and a $G$-continuous norm $p$.

Then there exists a number $\mu\in\mathbb{C}$ such that the following holds. Let

[TABLE]

Then for all $v\in(V^{\infty})^{M}$ and $\mathsf{w}\in\mathsf{W}$ there exist $c_{1}(v,\mathsf{w}),c_{2}(v,\mathsf{w})\in\mathbb{C}$ such that

[TABLE]

where

[TABLE]

Moreover, let $I_{0}:=\{i=1,2\mid c_{i}(v,\mathsf{w})\neq 0\text{ for some }v,\mathsf{w}\}$. Then

[TABLE]

and

[TABLE]

for every $r\leq r^{\prime}\leq r+c_{0}$ such that $\operatorname{Re}\lambda_{i}(X)\geq r^{\prime}\rho(X)$ for all $X\in\mathfrak{a}_{Z}^{+}$, $i\in I_{0}$.

Finally, if $V$ is unitarizable, then $\mu\in\mathbb{R}\cup i\mathbb{R}$.

Proof.

In what follows, the elements of $\mathsf{W}$ can be dealt with on an equal footing and for our simplified discussion here we shall assume $\mathsf{W}=\{{\bf 1}\}$. Furthermore, since all ideas are contained in the example of $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$ and $\mathfrak{h}$ one-dimensional, we restrict ourselves here to that case. However, in Remark 6.3 below we indicate the necessary changes for the general rank one case.

Let

[TABLE]

We assume that $\mathfrak{a}_{Z}=\mathbb{R}X$ and $\mathfrak{p}=\operatorname{span}\{X,E\}$. Then $X\in\mathfrak{a}_{Z}^{+}$. Let

[TABLE]

for $t\in\mathbb{R}$.

The Casimir element of $\mathfrak{g}$ is given by

[TABLE]

In general for $Y\in\mathfrak{g}$ and $\psi\in C^{\infty}(Z)$ we set (L_{Y}\psi)(z)={d\over ds}\Big{|}_{s=0}\psi(\exp(-sY)z) for $z\in Z$.

Let now $c\in\mathbb{C}$ be such that $2\mathcal{C}$ acts as $c\operatorname{id}_{V}$ on $V$. Set

[TABLE]

Now note with $f=m_{v,\eta}$ that

[TABLE]

with

[TABLE]

For any smooth function $\psi$ on $Z$, viewed as a right $H$-invariant function on $G$ we have for $a\in A_{Z}^{-}$ that

[TABLE]

with $\alpha$ the positive root and $(R_{F}\psi)(a)={d\over dt}_{t=0}\psi(a\exp(tF))$. Let now $\mathfrak{h}=\mathbb{R}Y$ and note that $\mathfrak{g}=\mathfrak{h}+\mathfrak{p}$ implies that $Y$ can be normalized such that $F=Y+d_{1}X+d_{2}E$ with $d_{1},d_{2}\in\mathbb{R}$. Thus we get

[TABLE]

We let now $\psi=L_{E}f$ and thus obtain from (6.3) that

[TABLE]

for a universal constant $C\geq 0$ and $c_{0}>0$ (specifically for $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$ we have $c_{0}=2$ if $d_{1}\neq 0$ and otherwise $c_{0}=4$, but in general it can be different).

Set

[TABLE]

Then our discussion shows that $\Phi$ satisfies the first order differential equation

[TABLE]

where

[TABLE]

The characteristic polynomial of $\mathsf{A}$ is given by $\lambda^{2}+2\lambda-c$ and thus has solutions

[TABLE]

where

[TABLE]

The general solution formula for (6.13) then is

[TABLE]

Note that $\mathsf{A}$ is semi-simple if and only if $\mu\neq 0$, i.e. $c\neq-1$. For $\lambda\in\operatorname{spec}\mathsf{A}=\{\lambda_{-},\lambda_{+}\}$ we write $E(\lambda)\subset\mathbb{C}^{2}$ for the corresponding generalized eigenspace. Let $0<\delta<c_{0}+r$ and write

[TABLE]

for the projection along the complementary generalized eigenspaces. Next we note that

[TABLE]

in which convergence is ensured by the definition of $\mathsf{P}$, as $\delta>0$.

Set

[TABLE]

This gives us

[TABLE]

for $u\in\mathbb{C}^{2}$ given by

[TABLE]

Next we estimate $I_{1}$ and $I_{2}$ for $t\geq 0$. For that we first recall the Gelfand-Shilov estimate for $\|e^{t\mathsf{A}}\|$ for a complex $N\times N$-matrix with $\sigma:=\max\{\operatorname{Re}\lambda\mid\lambda\in\operatorname{spec}\mathsf{A}\}$:

[TABLE]

and in particular for $N=2$:

[TABLE]

Then,

[TABLE]

For the last line we used the fact, obtained from (5.4), that

[TABLE]

Further note that $C$ does not depend on $\delta$ as $\delta$ was bounded from above by $c_{0}+r$.

Similarly

[TABLE]

With the Lemma 6.4 from below we can find a value $\frac{c_{0}}{4}\leq\delta\leq{c_{0}\over 2}$ and a constant $C>0$ such that $\|\mathsf{P}\|\leq C(1+\|\mathsf{A}\|)^{2}$. This yields then

[TABLE]

We now define ${\bf const}_{\mathsf{w}}(v)(\exp(-tX))$ to be the first coordinate of $e^{tA}u$. By expanding $u=c_{+}u_{+}+c_{-}u_{-}$ into generalized eigenvectors of $\mathsf{A}$ to eigenvalues $\lambda_{\pm}$ we obtain (6.5) for $\lambda_{i}=-\lambda_{\pm}\rho$. Furthermore (6.4) follows from (6.16) and (6.18).

From (6.3) and (6.4) it follows that

[TABLE]

Hence a coefficient $c_{\pm}$ vanishes provided that $\operatorname{Re}\lambda_{\pm}>-r$, as stated in (6.6).

From (6.17) we have $u=\Phi(0)+I_{1}(0)$ and hence $\|u\|\leq Cp_{8}(v)$. If $\operatorname{Re}\lambda_{\pm}\leq-r^{\prime}$ this implies

[TABLE]

and hence (6.7).

Finally, we remark that if $V$ is unitary then the Casimir eigenvalue $c$ is real, and thus the final assertion is a consequence of (6.15). ∎

Remark 6.3**.**

The alert reader might ask where the assumption $v\in V^{M}$ entered the proof. For a general rank one space with parabolic $\mathfrak{q}=\mathfrak{l}+\mathfrak{u}$ we recall the shape of the Casimir operator $\mathcal{C}$. For every root space $\mathfrak{g}^{\alpha}\subset\mathfrak{u}$ we let select a basis $(Y_{\alpha}^{j})_{1\leq j\leq m_{\alpha}}$ with $m_{\alpha}=\dim\mathfrak{g}^{\alpha}$. Then we choose $Y_{-\alpha}^{j}\in\mathfrak{g}^{-\alpha}$ such that $\kappa(Y_{\alpha}^{j},Y_{-\alpha}^{k})=\delta_{jk}$ with $\kappa$ the Cartan-Killing form. Set $\mathcal{C}_{\mathfrak{u}}:=\sum_{\alpha\in\Sigma(\mathfrak{a},\mathfrak{u}+\overline{\mathfrak{u}})}Y_{\alpha}^{j}Y_{-\alpha}^{j}$ and note that

[TABLE]

We note that $2\rho=\kappa(X_{2\rho},\cdot)$ as a functional on $\mathfrak{a}$. The fact that $Z$ is unimodular then implies that $X_{2\rho}\in\mathfrak{a}_{H}^{\perp}$ by [25], Lemma 4.2. Then

[TABLE]

with $\mathcal{C}_{\mathfrak{l}}$ a multiple of the Casimir of $\mathfrak{l}$. Now $\mathfrak{l}=\mathfrak{l}_{\rm n}\oplus\mathfrak{m}_{\mathfrak{l}}\oplus(\mathfrak{z}(\mathfrak{l})\cap\mathfrak{a})$ with $\mathfrak{m}_{\mathfrak{l}}<\mathfrak{m}$ and $\mathfrak{l}_{\rm n}<\mathfrak{l}\cap\mathfrak{h}$ by the local structure theorem. In particular if $f$ is a left $M$-invariant function on $Z=G/H$ and $a\in A_{Z}$, then we have

[TABLE]

and the analogy to the $\mathfrak{sl}(2)$-case becomes apparent: we define $\phi(t)$ as in (6.8) with $X=X_{2\rho}$ and obtain $\Phi$ as in (6.9). The function $\Phi$ satisfies the ODE (6.13) with $r(t)=-2(L_{\mathcal{C}_{\mathfrak{u}}^{-}}f)(\exp(-tX_{2\rho})$. Further, given the shape of $\mathcal{C}_{\mathfrak{u}}^{-}$ we can proceed as in (6.10)-(6.11) to arrive at the remainder term estimate (6.12). The rest of the proof translates verbatim.

Lemma 6.4**.**

Let $0<\nu\leq 1$, $N\in\mathbb{N}$ and $\mathsf{A}\in\operatorname{Mat}_{N}(\mathbb{C})$ with $\operatorname{spec}(\mathsf{A})=\{\lambda_{1},\ldots,\lambda_{r}\}$ such that $\operatorname{Re}\lambda_{1}\leq\ldots\leq\operatorname{Re}\lambda_{r}$. For every $1\leq j\leq r$ let $V_{j}\subset\mathbb{C}^{n}$ be the generalized eigenspace of $\mathsf{A}$ associated to the eigenvalue $\lambda_{j}$. For every $1\leq k\leq r$ we let $E_{k}=\bigoplus_{j=1}^{k}V_{j}$ and $\mathsf{P}_{k}:\mathbb{C}^{N}\to E_{k}$ be the projection along $\bigoplus_{j=k+1}^{r}V_{j}$. Suppose for some $1\leq k\leq r-1$ that $\operatorname{Re}\lambda_{k+1}-\operatorname{Re}\lambda_{k}\geq\nu$. Then there exists a constant $C=C(\nu,N)>0$ such that

[TABLE]

Proof.

Let $R\subset\mathbb{C}$ be the positively oriented and axes-parallel rectangle which intersects the imaginary axis in $\pm i(\|A\|+1)$, and the real axis in $-(\|A\|+1)$, respectively half way between $\operatorname{Re}\lambda_{k}$ and $\operatorname{Re}\lambda_{k+1}$.

Observe that:

• •

$R$ surrounds $\{\lambda_{1},\ldots,\lambda_{k}\}$ but not $\{\lambda_{k+1},\ldots,\lambda_{r}\}$,

• •

$\operatorname{dist}(\operatorname{spec}\mathsf{A},R)\geq\nu/2$,

• •

$|R|\leq 8\|\mathsf{A}\|+8$.

Next we recall that

[TABLE]

and thus

[TABLE]

Cramer’s rule gives

[TABLE]

Using the observations above we get

• •

$|\det(\mathsf{A}-z)|\geq C_{1}$ for all $z\in R$ with $C_{1}=({\nu\over 2})^{n}$.

• •

$|\det(\mathsf{A}-z)_{ij}|\leq C_{2}(1+\|\mathsf{A}\|)^{N-1}$ for all $z\in R$, $1\leq i,j\leq N$, and a constant $C_{2}=C_{2}(N)$ (apply Hadamard’s inequality for the determinant).

The assertion follows. ∎

6.3. Function spaces on $Z$

The results in this subsection are generally valid for any unimodular real spherical space.

We first recall the two standard weight functions for $Z$, the volume weight $\operatorname{\bf v}$ (see Section 1) and the radial weight $\operatorname{\bf r}$ (see [33], Sections 4 and 9), with the bounds:

[TABLE]

for all $\omega\in\Omega,a\in A_{Z}^{-},\mathsf{w}\in\mathsf{W}$. The two weight functions are related to the invariant measure as follows (see [33], Lemma 9.4)

[TABLE]

where $\operatorname{rank}_{\mathbb{R}}Z=\dim\mathfrak{a}_{Z}$.

Given $1\leq p<\infty$, $m>0$ we consider the following norms on $C_{c}^{\infty}(Z)$:

[TABLE]

and

[TABLE]

If $k\in\mathbb{N}_{0}$ we denote by $\|\cdot\|_{p,m;k}$, resp. $q_{p,m;k}$, the $k$-the Sobolev norm of $\|\cdot\|_{p,m}$, resp. $q_{p,m}$ (see (5.3)).

Lemma 6.5**.**

The two families of norms $\|\cdot\|_{p,m;k}$ and $q_{p,m;k}$ are related as follows:

1. (1)

For $pm^{\prime}-m>\operatorname{rank}_{\mathbb{R}}Z$, there exists a constant $C=C(p,m,m^{\prime},k)>0$ such that

[TABLE] 2. (2)

For $k^{\prime}>{\dim G\over p}$ there exists a constant $C=C(p,m,k,k^{\prime})>0$ such that

[TABLE]

Proof.

(1) For $k=0$ (no derivatives) this is an immediate consequence of (6.21). The general case is obtained by applying derivatives to $f$. The inequality in (2) is established in the proof of [33], Lemma 9.5, for $p=2$. The proof for general $p$ is entirely parallel. ∎

In particular it follows from (6.19) and (6.23) for a matrix coefficient $f=m_{v,\eta}$ which is $L^{p}$-integrable that

[TABLE]

with $\|\cdot\|_{p;k}:=\|\cdot\|_{p,0;k}$.

Phrased differently all $L^{p}$-integrable $f=m_{v,\eta}$ satisfy the a-priori bound

[TABLE]

for $r={2\over p}>0$ and the $G$-continuous norm $\|\cdot\|_{p;k}$ as required in (6.3).

6.4. Verification of Hypothesis B

In this subsection it is assumed that $Z$ is wavefront and $\operatorname{rank}_{\mathbb{R}}Z=1$. We also assume that $\mathsf{W}={\bf 1}$ for simplification of the exposition. Furthermore, under the specific assumptions in Theorem 5.1 we have for all non-trivial $\pi$ that $H_{\eta}/H$ is compact (see Remark 5.3) and thus we may as well assume that $H=H_{\eta}$.

Let $1\leq p^{\prime}<p<\infty$. By Remark 5.4 it suffices to establish the hypothesis for $p$ and $p^{\prime}$ close to each other. We shall assume that $\frac{1}{p^{\prime}}-\frac{1}{p}\leq\frac{c_{0}}{2}$ with $c_{0}$ the constant of Theorem 6.2.

Let $f=m_{v,\eta}$ be associated to a representation $\pi\in\Lambda_{p^{\prime}}$. We assume that $f$ is $K$-fixed. With the a priori bound (6.24) for $r=\frac{2}{p}$ we can apply Theorem 6.2 and let $\operatorname{\bf c}(f)(a)={\bf const}_{v}(a,{\bf 1})$ be a constant term for $f$ (see (6.5)). Then the approximation bound (6.4) reads

[TABLE]

As $f$ is $K$-fixed we may assume with (6.2) and the neglect of $\Omega_{Z}$ (on the fixed compact set $\Omega_{Z}$ the $L^{p}$-norm is dominated by the $L^{\infty}$-norm) that

[TABLE]

We now identify $A_{Z}^{-}$ with $[0,\infty[$ via an element $X\in\mathfrak{a}_{Z}^{-}$ with $\rho(X)=-1$, i.e.

[TABLE]

Then (6.26) translates into

[TABLE]

We let $R>0$, to be specified later, and begin with the estimate:

[TABLE]

By assumption $f$ is $L^{p^{\prime}}$-bounded (in particular satisfies the bound (6.24) for $r={2\over p^{\prime}}$), the leading coefficients $\lambda_{i}$ of $\operatorname{\bf c}(f)$ satisfy $\operatorname{Re}\lambda_{i}(X)\leq-{2\over p^{\prime}}$. Hence we obtain from ${2\over p^{\prime}}-{2\over p}\leq c_{0}$ and (6.7) that

[TABLE]

Note that $C$ does not depend on $\pi$.

Further, from (6.25) we obtain

[TABLE]

Set now $\delta:={2\over p^{\prime}}-{2\over p}$ and note that $\delta\leq c_{0}$. Inserting the bounds from (6.29) and (6.30) into (6.28) we thus obtain that

[TABLE]

which, in view of (5.4), yields

[TABLE]

So far, $R>0$ was arbitrary, but we now choose it such that

[TABLE]

holds for the constant $C$ of (6.31), i.e.

[TABLE]

will do. Then we obtain from (6.31)

[TABLE]

and hence

[TABLE]

with $C>0$ independent of $\pi$. It follows with $l:={k+8\over 2\delta p}$

[TABLE]

which is Hypothesis B.

7. $L^{p}$ versus $L^{\infty}$ in higher rank

The purpose of this section is to give a proof of (1.3) for general unimodular wavefront spaces. Observe that (1.3) is more general than Hypothesis B in two respects:

• •

The eigenfunction $f$ is not $K$-fixed.

• •

The $G$-representation generated by $f$ is not necessarily unitarizable.

However, our verification of Hypothesis B for the rank one cases does not really need the assumption that the eigenfunction is $K$-fixed (the argument given allows to sum up $K$-types). Further, the main ingredient is the constant term approximation, which does not require that the underlying Harish-Chandra module is unitarizable.

Regarding the lattice counting problem we note that (1.3) also allows counting for balls which are not $K$-invariant.

We now turn to the heart of the matter:

7.1. The constant term approximation in higher rank

Let $\mathfrak{t}\subset\mathfrak{m}$ be a maximal torus and set $\mathfrak{c}:=\mathfrak{a}+\mathfrak{t}$. Then $\mathfrak{c}<\mathfrak{g}$ is a Cartan algebra and we let $\mathcal{W}_{\mathfrak{c}}$ be the Weyl group associated to the root system $\Sigma(\mathfrak{c}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})$. Further we choose a positive system $\Sigma^{+}(\mathfrak{c}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})$ of $\Sigma(\mathfrak{c}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})$ such that $\Sigma^{+}(\mathfrak{c}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}})|_{\mathfrak{a}}\backslash\{0\}=\Sigma^{+}(\mathfrak{a},\mathfrak{g})$.

For a subset of spherical roots $I\subset S$ we set

[TABLE]

[TABLE]

Finally for an irreducible Harish-Chandra module $V$ we let $\Lambda_{V}\in\mathfrak{c}_{\mathbb{C}}^{*}$ be the associated infinitesimal character. Then with the techniques provided in [7], but essentially the same proof, we obtain the following analogue of Theorem 6.2.

Theorem 7.1** (Constant term approximation - higher rank).**

Let $Z=G/H$ be a wave-front real spherical space. Let $I\subset S$ and $\mathcal{C}_{I}\subset\mathfrak{a}_{I}^{--}$ be a compact subset. Then there exist constants $C,c_{0}>0$, $k,d\in\mathbb{N}$ (depending only on $Z$ and $\mathcal{C}_{I}$) such that the following assertion holds true:

Let $(V,\eta)$ be a spherical pair with $V$ irreducible and which satisfies an a priori-bound

[TABLE]

for some $r>0$ and a $G$-continuous norm $p$. Then there exists a subset $\mathcal{E}_{V}\subset\rho+\mathcal{W}_{\mathfrak{c}}\cdot\Lambda_{V}$ with

[TABLE]

such that for all $v\in V^{\infty}$, $\mathsf{w}\in\mathsf{W}$ and $\lambda\in\mathcal{E}_{V}$ there exist polynomials $c^{I}_{v,\lambda,\mathsf{w}}\in\operatorname{Pol}(\mathfrak{a}_{I})$ of degree bounded by $d$ such that the sum

[TABLE]

satisfies for all $X\in\mathcal{C}_{I}$, $a\in A_{Z}^{-}$, $t\geq 0$, ${v\in V^{\infty}}$ and $\mathsf{w}\in\mathsf{W}$ that

[TABLE]

If in addition there exists an $r\leq r^{\prime}\leq r+c_{0}$ such that the improved bound

[TABLE]

holds, then

[TABLE]

Remark 7.2**.**

The constant term approximation in Theorem 7.1 deviates from the one in [7] in several aspects. The a priori-bound (7.1) used in [7] is

[TABLE]

for some $G$-continuous norm $p$ and $N\in\mathbb{N}$. If we ignore logarithmic terms, then this is in essence the case $r=1$ above, and the presence of logarithmic terms does not pose additional technical difficulties.

The constant term in [7] is unique and obtained from the one above by replacing $\mathcal{E}_{V}$ with $\mathcal{E}_{V}^{\prime}:=\{\lambda\in\mathcal{E}_{V}\mid\operatorname{Re}\lambda=0\}$, i.e. by discarding all exponents $\lambda$ for which $\operatorname{Re}\lambda\neq 0$. As such the constant term approximation in [7] is only uniform if one fixes $\operatorname{Re}\Lambda_{V}$. The constant term approximation above however is uniform (i.e. holds for all $\Lambda_{V}$), but is non-unique and also allows exponents $\lambda$ with $\operatorname{Re}\lambda\neq 0$.

We now provide a proof of Theorem 7.1, that is we will guide the reader through parts of [7] until the analogy to the rank one case becomes fully apparent.

Proof.

We first recall the boundary degenerations $\mathfrak{h}_{I}$ of $\mathfrak{h}$ (see [26], Sect. 3): Let $d:=\dim\mathfrak{h}$ and $X_{I}\in\mathfrak{a}_{I}^{--}$. Then

[TABLE]

exists in the Grassmannian of $d$-dimensional subspaces of $\mathfrak{g}$ and is independent of the choice of $X_{I}\in\mathfrak{a}^{--}$. The geometric meaning of $\mathfrak{h}_{I}$ is approximately as follows: the homogeneous space $Z=G/H$ admits a $G$-equivariant compactification $\widehat{Z}$ with $G$-boundary orbits isomorphic to $\widehat{Z}_{I}=G/\widehat{H}_{I}$ where $\operatorname{Lie}(\widehat{H}_{I})=\mathfrak{h}_{I}+\mathfrak{a}_{I}$. The normal bundles to each $\widehat{Z}_{I}$ are given by $Z_{I}=G/H_{I}$ where $\operatorname{Lie}(H_{I})=\mathfrak{h}_{I}$.

Now choose $Y_{I}\in\mathfrak{a}\cap\mathfrak{z}(\mathfrak{l})$ such that $X_{I}=Y_{I}+\mathfrak{a}_{H}$ and define a parabolic subalgebra

[TABLE]

We write $\overline{\mathfrak{p}}_{I}=\mathfrak{l}_{I}\ltimes\overline{\mathfrak{u}}_{I}$ for the standard Levi-decomposition. A special feature of wavefront spaces then is (see [26], Prop. 5.5):

[TABLE]

Next we recall material from [7], Sect. 3.3. Let $\mathcal{Z}(\mathfrak{g})$ and $\mathcal{Z}(\mathfrak{l}_{I})$ denote the centers of the universal enveloping algebras $\mathcal{U}(\mathfrak{g})$ and $\mathcal{U}(\mathfrak{l}_{I})$. Denote by $\gamma_{I}:\mathcal{Z}(\mathfrak{g})\to\mathcal{Z}(\mathfrak{l}_{I})$ the partial Harish-Chandra homomorphism. We fix a finite-dimensional vector subspace $W\subset\mathcal{Z}(\mathfrak{l}_{I})$ such that $\mathcal{Z}(\mathfrak{l}_{I})\simeq\gamma_{I}(\mathcal{Z}(\mathfrak{g}))\otimes W$. Recall that we assumed the Harish-Chandra module $V$ to be irreducible and we let $\chi_{V}:\mathcal{Z}(\mathfrak{g})\to\mathbb{C}$ the corresponding infinitesimal character. Set ${\mathcal{J}}_{V,I}:=\gamma_{I}(\ker\chi_{V})$ and record

[TABLE]

For a differentiable function $f$ on $Z$ and $X\in\mathfrak{g}$ we write L_{X}f(z)={d\over dt}\big{|}_{t=0}F(\exp(-tX)z) for the left derivative. In case $f=m_{v,\eta}$ for a smooth vector $v\in V^{\infty}$ we record that $L_{X}(m_{v,\eta})=m_{Xv,\eta}$. Now given $f=m_{v,\eta}$ as above we construct a $W^{*}$-valued smooth function $F$ on $Z$ as follows:

[TABLE]

Let $a\in A_{Z}^{-}$ and define a function $F_{I,a}$ on $A_{I}$ by $F_{I,a}(a_{I})=F(aa_{I})$ for $a_{I}\in A_{I}$.

Then there exists a Lie algebra homomorphism

[TABLE]

such that $F_{I,a}$ satisfies the differential equation

[TABLE]

with

[TABLE]

where $(X_{i})_{i=1}^{n}$ is a fixed basis of $\overline{\mathfrak{u}}_{I}$ and $v_{i}=v_{i}(w)\in\mathcal{U}(\mathfrak{g})$ are elements which depend linearly on $w\in W$ and polynomially on $\chi_{V}$ (this is variant of formula (3.23) in [7]). Moreover $\Gamma_{V}$ depends polynomially on $\chi_{V}$ and

[TABLE]

Recall the norm $p$ for which the matrix coefficients $m_{v,\eta}$ satisfy the a priori bound (7.1). Then we obtain from a slight variant of [7], Lemma 3.5 (ii) constants $C>0$ and $k,N\in\mathbb{N}$ which are all independent of $V$ such that

[TABLE]

where $\beta(X):=\max_{\alpha\in\Sigma(\mathfrak{a},\mathfrak{u}_{I})}\alpha(X)<0$.

Also notice the general solution formula for (7.8)

[TABLE]

Now we are ready to make the comparison to the proof in the rank one case. The differential equation (6.13) from the rank one case is replaced by (7.8), and (7.10) will replace the remainder estimate from (6.12). Further the equation (6.15) for the exponents is replaced by (7.9).

As for the spectral gap we let $c_{1}$ be the largest possible value $c>0$ such that $c|\rho(X)|\leq|\beta(X)|$ for all $X\in\mathcal{C}_{I}$. For $\delta>0$ we consider the spectral projection

[TABLE]

with $E(\lambda)$ denoting the generalized eigenspace of $\Gamma_{V}(X)$ on $W^{*}$. Because of (7.9), the pigeonhole principle combined with Lemma 6.4 yields a $\delta=\delta(V,\mathcal{C}_{I})<{c_{1}\over 2|\mathcal{W}_{\mathfrak{c}}|}$ such that

[TABLE]

for constants $C,N>0$ not depending on $V$ (recall that $\Gamma_{V}$ depends polynomially on $\Lambda_{V}$). This finally gives us

[TABLE]

Having said all that we can now proceed as in the rank one case with integrals $I_{1}(t)$ and $I_{2}(t)$ now depending (uniformly) on $X\in\mathcal{C}_{I}$ and the additional variable $a\in A_{Z}^{-}$. This then leads to the asserted bound (7.4). ∎

7.2. Proof of (1.3)

We now prove (1.3) using the constant term approximation from Theorem 7.1. The proof is parallel to the rank one case and we confine ourselves with an explanation of the essential changes which are needed for the adaption of the proof.

Let $\mathfrak{a}_{Z,E}:=\mathfrak{a}_{S}$ and note that $\mathfrak{a}_{Z,E}$ normalizes $\mathfrak{h}$ (it is the Lie algebra of the ”center” of $Z$) and set $A_{Z,E}:=\exp(\mathfrak{a}_{Z,E})$. Since the space of $H$-invariant continuous functionals of $V^{\infty}$ is finite dimensional [32], we obtain an $A_{Z,E}$-finite right action on the eigenfunctions $f=m_{v,\eta}$ attached to $V$. In particular there are no non-zero $L^{p}$-integrable eigenfunctions if $\mathfrak{a}_{Z,E}\neq\{0\}$. We can thus request that $\mathfrak{a}_{Z,E}=0$, and obtain in particular

[TABLE]

We let $B$ be a compact neighborhood of ${\bf 1}$ in $G$ such that the conclusion of Lemma 3.3 (2) holds. In particular,

[TABLE]

where $dz,dg,da$ are invariant measures on $Z,G,A_{Z}^{-}$.

Next we need to decompose the compression cone $\mathfrak{a}_{Z}^{-}$ suitably along its faces so that we can apply Theorem 7.1. For $I\subset S$ and $X\in\mathfrak{a}_{Z}$ we write $X=X_{I}+X^{I}$ with $X_{I}\in\mathfrak{a}_{I}$ and $X^{I}\in\mathfrak{a}_{S\backslash I}$. For $\delta>0$ and $\#S\backslash I=1$ we define

[TABLE]

and for $\#S\backslash I>1$ by recursion

[TABLE]

Then

[TABLE]

and for every $I\subsetneq S$ there exists a compact set $\mathcal{C}_{I}\subset\mathfrak{a}_{I}^{--}$ such that

[TABLE]

Both assertions (7.12) and (7.13) are geometrically obvious for $\dim\mathfrak{a}_{Z}={1,2,3}$ and the general case is an elementary verification based on induction of the dimension. For $X=X_{I}+X^{I}\in D_{I}$ and $a=\exp(X)\in A_{Z}^{-}$ we set $a_{I}=\exp(X_{I})$ and $a^{I}=\exp(X^{I})$ so that $a=a_{I}a^{I}$.

Next we recall the notation on function spaces from Subsection 6.3 and the norm-inequality in (6.22).

Let $f=m_{v,\eta}$ be $L^{p}$-integrable with $v\in V^{\infty}$ a smooth vector. For $g\in B,a\in A_{Z}^{-},\mathsf{w}\in\mathsf{W}$ now set

[TABLE]

Recall the compact set $\mathcal{C}_{I}\subset\mathfrak{a}_{I}^{--}$ from (7.13). Then Theorem 7.1 applied to $r={2\over p}$ implies a constant $c_{0}=c_{0}(V,\mathcal{C}_{I})>0$ such that

[TABLE]

where $k\geq 0$ is the Sobolev-constant from Theorem 7.1 and $k^{\prime}>{\dim G\over p}$. The constant $C$ is universal in the sense that it only depends on $p$ and $Z$. Now, (7.14) combined with (7.13) yields an $\epsilon>0$, only depending on the fixed choice $\delta>0$ and $c_{0}$, such that

[TABLE]

For $R\geq 0$ we now set $D_{I,\geq R}:=\{X\in D_{I}\mid-\rho(X)\geq R\}$ and set

[TABLE]

Further we set $\Omega_{\leq R}:=BA_{R}^{-}\mathsf{W}\cdot z_{0}$ and $\Omega_{\geq R}:=B(A^{-}\backslash A_{R}^{-})\mathsf{W}\cdot z_{0}$ and note that $\operatorname{vol}_{Z}(\Omega_{\leq R})\leq Ce^{2R}$ as a consequence of (7.11).

The starting estimate (6.28) from the rank one case now becomes:

[TABLE]

The next two steps (6.29) and (6.30) from the rank one case are now completely parallel: Instead of (6.26) we use

[TABLE]

which we obtain from (7.11) and the a-priori estimate (6.25) is replaced by the inequalities (7.15). We then arrive at the inequality

[TABLE]

where $\epsilon_{0}>0$ only depends on $\epsilon>0$ and ${1\over p^{\prime}}-{1\over p}>0$.

So far we have not specified $R$ nor requested that $f$ is $K$-finite. We now take the $K$-Fourier series of $f$:

[TABLE]

where

[TABLE]

Then we obviously have $\|f\|_{p}\leq\sum_{\tau\in\widehat{K}}\|f_{\tau}\|_{p}$. The converse is true up to a Sobolev-shift $l=l(K)\in\mathbb{N}$, i.e.

[TABLE]

This is quite standard but we briefly sketch the proof for part of the audience which has not seen that before: Let $\chi_{\tau}\leq 0$ be the eigenvalue of $\Delta_{\mathfrak{k}}$ on the representation $\tau$. Then, first it follows from (7.18) that $\|f_{\tau}\|_{p}\leq(\dim\tau)^{2}\|f\|_{p}$. Thus for $l\in\mathbb{N}$ we get

[TABLE]

Finally the Weyl-dimension formula gives $\sum_{\tau\in\widehat{K}}{(\dim\tau)^{2}\over|\chi_{\tau}|^{l}}<\infty$ provided $l=l(K)$ is large enough and (7.19) is established.

Our reasoning implies in particular that we may assume in the sequel that $f=f_{\tau}$. Next we note that $\Delta f=(\chi_{V}+2\chi_{\tau})f$ and therefore we get from (7.17) that

[TABLE]

which is the analogue of (6.31) in the rank one case. From here the arguments run parallel to the rank one case.

8. Appendix: Mostow decomposition

In this appendix we establish a variant of the Mostow-decomposition which we use in Section 2.

Theorem 8.1**.**

(Mostow, [35])Let $G$ be a real algebraic group and $H<G$ a real algebraic subgroup. Then there exists a maximal compact subgroup $K<G$ such that $K\cap H$ is maximal compact in $H$ and a vector subspace $V\subset\mathfrak{g}$ which is invariant under $N_{K}(H)$ such that the Mostow map

[TABLE]

is a diffeomorphism.

Remark 8.2**.**

The original statement of Mostow is slightly weaker and deviates from the one above in the following sense: Mostow shows that there is a subspace $V\subset\mathfrak{g}$ with $V=V_{1}\times\ldots\times V_{n}$ a product of vector spaces and for $X=(X_{1},\ldots,X_{n})\in V$ he uses instead of $\exp(X)$ the expression $\operatorname{Exp}(X):=\exp(X_{1})\cdot\ldots\cdot\exp(X_{n})$.

Proof.

In the first step we assume that both $G$ and $H$ are reductive and let $\Theta$ be a Cartan-involution of $G$ which fixes $H$. The Cartan involution gives us a maximal compact subgroup $K:=G^{\Theta}$ which has the property that $K\cap H$ is maximal compact in $H$. Further let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ the decomposition into $\pm 1$ -eigenspaces of $\theta:=d\Theta({\bf 1})$. Then we obtain with $V:=\mathfrak{p}\cap\mathfrak{h}^{\perp}$ (where $\perp$ is taken w.r.t. to a fixed non-degenerate invariant form on $\mathfrak{g}$) that $\Phi$ is a diffeomorphism (see [28], Lemma 2.7, for a short proof or [15], Th. 9.3, for a proof using gradient maps). Note that $V$ is also $N_{K}(H)$-invariant.

In the second step we consider the case where $G=L\ltimes U$ is a semi-direct product of a reductive group $L$ and a unipotent group $U$. Further we assume that $H=L_{H}\ltimes U_{H}$ is a Levi-decomposition with $U_{H}<U$ and $L_{H}<L$. We find a maximal compact subgroup $K<G$ which is contained in $L$ and such that $K\cap H$ is maximal compact subgroup of $H$. From the first step applied to $L/L_{H}$ we find a $N_{K}(L_{H})$-invariant subspace $V_{1}\subset\mathfrak{l}$ such that $\Phi_{1}:K\times_{K\cap H}V_{1}\to L/L_{H}$ is a diffeomorphism. By Lemma 8.3 below we find a complementary subspace $V_{2}\subset\mathfrak{u}$ to $\mathfrak{u}_{H}$ which is $N_{K}(U_{H})$-invariant such that $V_{2}\to U/U_{H},\ X\mapsto\exp(X)U_{H}$ is a diffeomorphism. With $V:=V_{1}\times V_{2}$ we finally claim that $\Phi$ is a diffeomorphism. To prepare the proof of the claim let us assume first that $\mathfrak{u}$ is abelian. Then it follows (cf. [37] exercise 7, p. 75, but note the sign error) for $X_{1}\in\mathfrak{l}$ and $X_{2}\in\mathfrak{u}$ that

[TABLE]

If we assume $\mathfrak{u}_{H}\triangleleft\mathfrak{u}$ is a coabelian ideal, then we can apply the above to the semi-direct product $L\ltimes U/U_{H}$ and get for $X=(X_{1},X_{2})\in V=V_{1}\times V_{2}$

[TABLE]

Now $\operatorname{ad}X_{1}$ has real spectrum and therefore ${e^{\operatorname{ad}X_{1}}-{\bf 1}\over\operatorname{ad}X_{1}}$ is invertible on the abelian Lie algebra $\mathfrak{u}/\mathfrak{u}_{H}$. Hence from the fact that $\Phi_{1}$ is a diffeomorphism we obtain that $\Phi$ is a diffeomorphism as well, in case $\mathfrak{u}_{H}$ is a co-abelian ideal. The general case is then obtained via induction on the filtration (see the proof of Lemma 8.3.) Finally $V$ is $N_{K}(H)$-invariant since $N_{K}(H)=N_{K}(L_{H})\cap N_{K}(U_{H})$.

In the third step we consider the case where $G$ is reductive and $H$ is an arbitrary algebraic subgroup of $G$. A well-known theorem of Borel and Tits (see [16], Sect. 30.3, Cor. A) implies the existence of a parabolic subgroup $P=L\ltimes U$ of $G$ with $P\supset H$ and such that $H=L_{H}\ltimes U_{H}$ is a Levi-decomposition of $H$ with $L_{H}=H\cap L$ and $U_{H}=U\cap H$. By step 2 we obtain a maximal compact subgroup $K_{P}\subset L$, a $N_{K_{P}}(H)$-invariant vector subspace $V\subset\mathfrak{p}$, and a diffeomorphism $\Phi_{2}:K_{P}\times_{K_{P}\cap H}V\to P/H$ of the desired form, where $K_{P}$ is an appropriate maximal compact subgroup of $P$. Let $K$ be a maximal compact subgroup of $G$ for which $K_{P}=K\cap P$. In view of

[TABLE]

we deduce the existence of $\Phi$ in this case as well. The Borel-Tits construction of $P\supset H$ (see [16], Sect. 30.3) entails that $P$ can be chosen such that it contains $N_{G}(H)$. It follows that $N_{K}(H)\subset K_{P}$, and hence $V$ is $N_{K}(H)$-invariant.

Finally for most general case where $G=L\ltimes U$ is not necessarily reductive and $H<G$ real algebraic we first consider the intermediate situation with $\tilde{H}=HU$ for which we obtain a $N_{K}(\tilde{H})$-invariant subspace $\tilde{V}\subset\mathfrak{l}$ and a diffeomorphism $\tilde{\Phi}:K\times_{K_{H}}\tilde{V}\to G/\tilde{H}$ by step 3. Further, $\tilde{H}/H\simeq U/U_{H}$ for $U_{H}=U\cap H$. As in step 2 we choose a $K_{H}$-invariant complement $V_{U}$ to $\mathfrak{u}_{H}$ in $\mathfrak{u}$ and obtain a Mostow-map $\Phi$ with $V=\tilde{V}\times V_{U}$. ∎

Lemma 8.3**.**

Let $\mathfrak{u}$ be a nilpotent Lie algebra and $U$ the associated simply connected Lie group. Let $\mathfrak{u}_{H}<\mathfrak{u}$ be a subalgebra and $U_{H}:=\exp(\mathfrak{u}_{H})$. Let $n\in\mathbb{N}$ be the nilpotency degree of $\mathfrak{u}$, i.e. $\mathfrak{u}^{n}=\{0\}$ and $n$ is minimal with respect to this property. For $0\leq j\leq n-1$ we consider the subalgebras $\mathfrak{w}_{j}:=\mathfrak{u}_{H}+\mathfrak{u}^{j}$. Then $\mathfrak{w}_{j+1}\triangleleft\mathfrak{w}_{j}$ is a co-abelian ideal. Further let $V_{j}\subset\mathfrak{w}_{j}$ be any complementary subspace to $\mathfrak{w}_{j+1}$ and set $V:=V_{0}\oplus\ldots\oplus V_{n-1}$. Then $\mathfrak{u}_{H}+V=\mathfrak{u}$ is a direct vector sum and the map

[TABLE]

is a diffeomorphism. If in addition $K_{H}<\operatorname{Aut}(\mathfrak{u})\cap\operatorname{Aut}(\mathfrak{u}_{H})$ is a reductive group of automorphisms, then we can request each $V_{j}$ and hence $V$ to be $K_{H}$-invariant as well.

Proof.

(cf. Lemma 7.5 in [31]) Since all $\mathfrak{u}_{H}^{j+1}\triangleleft\mathfrak{u}_{H}^{j}$ are co-abelian, we obtain by induction on $n$ that the assignment

[TABLE]

is diffeomorphic. Here we expanded $X\in V=V_{0}\oplus\ldots\oplus V_{n-1}$ according to the indicated factors. It remains to show that we can replace $\operatorname{Exp}(X)$ by $\exp(X)$. In case $n=1$ this is clear as $\exp(X)=\operatorname{Exp}(X)$. Moving by induction we obtain with $V^{\prime}:=V_{1}\oplus\ldots\oplus V_{n-1}$ that the map

[TABLE]

is diffeomorphic. The induction step then follows from the observation that $\exp(X_{0}+X^{\prime})W_{1}=\exp(X_{0})W_{1}$ as $\exp:\mathfrak{u}/\mathfrak{w}_{1}\to U/W_{1}$ is a homomorphism.

For the final assertion about $K_{H}$-invariance we note that $K_{H}$ preserves each $\mathfrak{w}_{j}$ and $\mathfrak{u}_{H}$. Hence $K_{H}$ preserves each $\mathfrak{w}_{j}$ and thus, by complete reducibility, we find for each $j$ a $K_{H}$-invariant complement $V_{j}$ to $\mathfrak{w}_{j}$ in $\mathfrak{w}_{j-1}$.

∎