QUANTUM ERGODICITY FOR COMPACT QUOTIENTS OF $\operatorname {SL}_d({\mathbb R})/\textrm {SO}(d)$ IN THE BENJAMINI–SCHRAMM LIMIT

Abstract

We study the limiting behavior of Maass forms on sequences of large-volume compact quotients of $\operatorname {SL}_d({\mathbb R})/\textrm {SO}(d)$, $d\ge 3$, whose spectral parameter stays in a fixed window. We prove a form of quantum ergodicity in this level aspect which extends results of Le Masson and Sahlsten to the higher rank case.

1 Introduction

Let Y be a closed Riemann manifold, and let $\mathscr {B}=\{\psi _i\}$ be an orthonormal basis of $L^2(Y)$ consisting of Laplacian eigenfunctions $\Delta \psi _i=\lambda _i\psi _i$ . A subsequence $\left \{\psi _{i_j}\right \}$ of $\mathscr {B}$ is called quantum ergodic if, for every degree 0 pseudo-differential operator A on Y with principal symbol $a\in C(S^*Y)$ , we have

Here $\mu _L$ is the Liouville measure on the cosphere bundle $S^*Y$ and

indicates normalization by the volume. The quantum ergodicity theorem of Šnirelman [32], Zelditch [35], and Colin de Verdière [10] states that if the geodesic flow on $S^*Y$ is ergodic, one may extract from $\mathscr {B}$ a density $1$ quantum ergodic subsequence. More precisely, they show

(1)

as $\lambda \rightarrow \infty $ , where $N(\lambda )=\lvert \{i: \lambda _i\leq \lambda \}\rvert $ .

In this paper we are concerned with a version of quantum ergodicity where, in contrast to the foregoing semiclassical statement, in which the manifold is fixed and the eigenvalue goes to infinity (the large-frequency regime), we allow the manifold to vary while keeping the eigenvalue constrained to a fixed spectral window (the large-spacial regime).

The most natural setting in which one can formulate such variation is in the Gromov–Hausdorff space of pointed locally compact spaces – or, rather, its space of probability measures – where one has a notion of convergence due to Benjamini and Schramm [5]. Here one may consider a sequence of manifolds which converge, almost everywhere, to their common universal cover. Under certain auxiliary conditions, the question of quantum ergodicity for Benjamini–Schramm convergent sequences was recently settled for a large class of rank $1$ symmetric spaces [2, 24]. The aim of the present paper is to address this question for higher-rank locally symmetric spaces; for the most part we restrict ourselves to compact quotients associated with $\operatorname {SL}_d({\mathbb R})$ .

1.1 The setting of locally symmetric spaces

As our setting will henceforth be that of locally symmetric spaces, we begin by commenting on some of their particular features in higher rank, and reviewing what is known for their quantum ergodic properties in the large-frequency regime.

We let S be a Riemannian globally symmetric space of noncompact type: nonpositively curved and having no Euclidean de Rham local factor. We may write $S=G/K$ , where G is a connected semisimple Lie group with finite center and K is a maximal compact subgroup. The rank of S is defined to be the dimension of a maximal flat subspace in S; equivalently, it is the dimension of a maximal split torus A in G.

In rank $1$ , S is of strictly negative curvature, and A gives rise to the geodesic flow on the cosphere bundle of S via the identification of the latter with $G/M$ , where M is the centralizer of A in K. It is important to note, however, that when the rank is strictly greater than $1$ , such as for $\operatorname {SL}_d({\mathbb R})/\textrm {SO}(d)$ when $d\geq 3$ , the curvature of S, while nonpositive, can vanish, as indeed any geodesic triangle in a maximal flat will be Euclidean.

Now let $\Gamma \subset G$ be a uniform lattice in G and form $Y=\Gamma \backslash S$ , a compact locally symmetric space with geometry S. In higher rank, due to the presence of maximal flats, the geodesic flow is not ergodic on the cosphere bundle of Y. For this reason, the quantum ergodic statement (1) cannot hold, as it is known to be equivalent with the ergodicity of the geodesic flow [36, Theorem 1].

Nevertheless, the higher-rank split torus A does act ergodically – in fact it is mixing – on $\Gamma \backslash G$ , and this is enough for one to expect similar quantum ergodic phenomena to that of Šnirelman’s theorem, at least if one refines the notion of eigenfunction as follows.

Recall that a Maass form on Y is a function $\psi \in L^2(Y)$ which is a joint eigenfunction of the algebra $\mathscr {D}(S)$ of left-G-invariant differential operators on S. Since $\Delta \in \mathscr {D}(S)$ a Maass form is again a Laplacian eigenfunction, but in higher rank, $L^2(Y)$ can be further diagonalized. A Maass form $\psi $ gives rise to an algebra homomorphism $\chi \in \operatorname {Hom}_{{\mathbb C}-{\textrm {alg}}}(\mathscr {D}(S),{\mathbb C})$ verifying $D\psi =\chi (D)\psi $ for all $D\in \mathscr {D}(S)$ . The Harish-Chandra homomorphism allows one to realize $\chi $ as the Weyl group orbit of an element $\nu $ in $\mathfrak {a}^*_{\mathbb C}$ , the complexification of the dual of the Lie algebra $\mathfrak {a}$ of A. We call $\nu $ the spectral parameter of $\psi $ .

One can then formulate a natural extension of formula (1) for Maass forms on Y with growing spectral parameter. To the best of our knowledge, this form of quantum ergodicity has not yet been established, although recent work by Nelson and Venkatesh [28] on microlocal analysis and representation theory should shed light on this problem.

Remark 1.1. There are of course harder conjectures that one can formulate here, such as quantum unique ergodicity [25, Conjecture 1.2]. In the arithmetic setting, certain higher-rank congruence manifolds have been shown to satisfy the arithmetic quantum unique ergodicity property [30, 31], a generalization of the work (and techniques) of Lindenstrauss [26] (see also [33]). We will not make further comments on this active line of research, and refer the reader instead to [24, §1.3] for a discussion of a conjectural strengthening of quantum ergodicity for varying manifolds.

1.2 Our main result

We now pass to the large-spacial regime for higher-rank locally symmetric spaces $Y=\Gamma \backslash S$ , and allow $\Gamma $ to vary along a nonconjugate sequence of torsion-free uniform lattices with growing covolume.

We furthermore require that the sequence $\Gamma $ be uniformly discrete, a property which we now recall. Let d be the Riemannian distance on S. Then the local injectivity radius about a point $x\in Y$ is the quantity

$$ \begin{align*} \operatorname{InjRad}_\Gamma(x)=\frac12\min\{ d(x,\gamma.x): 1\neq \gamma\in \Gamma\}. \end{align*} $$

The global injectivity radius $\operatorname {InjRad}(Y)$ is the infimum of $\operatorname {InjRad}_\Gamma (x)$ over all $x\in Y$ ; this is strictly positive, since Y is compact. We say that that a sequence of uniform torsion-free lattices $\Gamma _n$ is uniformly discrete if $\operatorname {InjRad}(Y_n)$ is bounded away from zero. A conjecture of Margulis [27] states that this is automatic in higher rank; this is a weak form of the Lehmer conjecture on monic integral polynomials.

Our precise result is as follows. We refer to §3.1 for any unexplained notation.

Theorem 1.1. Set $d\geq 3$ . Let $\Gamma _n\subset \operatorname {SL}_d({\mathbb R})$ be a uniformly discrete sequence of torsion-free cocompact lattices such that $\operatorname {vol}(Y_n)\rightarrow \infty $ as $n\rightarrow \infty $ , where $Y_n=\Gamma _n\backslash \operatorname {SL}_d({\mathbb R})/{\mathrm{SO}}(d)$ . Let $a_n$ be a sequence of uniformly bounded, measurable functions on $Y_n$ .

There is $\varrho>1$ such that for sufficiently regular $\nu \in i\mathfrak {a}^*$ , we have

as $n\rightarrow \infty $ , where

denotes the normalization of the integral by the volume of $Y_n$ ,

$$ \begin{align*} B_0(\nu,\varrho)=\{\lambda \in i\mathfrak{a}^*: \lVert\lambda-\nu\rVert_2\leq\varrho\} \end{align*} $$

is the ball of radius $\varrho $ in the unramified tempered spectrum, and

(2) $$ \begin{align} N(B_0(\nu,\varrho),\Gamma_n)=\left\lvert\left\{j: \nu_j^{(n)}\in B_0(\nu,\varrho)\right\}\right\rvert. \end{align} $$

This extends to higher rank the rank $1$ version of the same result in [24] (for hyperbolic surfaces) and [2] (for higher-dimensional hyperbolic manifolds), which themselves built upon the breakthrough results of Anantharaman and Le Masson [3] for large regular graphs. All of these works imposed the following additional hypotheses:

1. (i) The manifolds (or graphs) $Y_n$ converge to S in the sense of Benjamini and Schramm: for every $R>0$ , we have

   $$ \begin{align*} \frac{\operatorname{vol}\left((Y_n)_{\leq R}\right)}{\operatorname{vol}(Y_n)}\rightarrow 0 \end{align*} $$
   as $n\rightarrow \infty $ , where the R-thin part of $Y_n$ is defined as
   (3) $$ \begin{align} (Y_n)_{\leq R}=\left\{x\in Y_n: \operatorname{InjRad}_{\Gamma_n}(x)\leq R\right\}. \end{align} $$

2. (ii) The $Y_n$ have a uniform spectral gap.

The proof of Theorem 1.1 also requires those properties, but they are automatic in higher rank (see [1, §4] and [20]).

We note that in [2] the large-frequency regime for Laplacian eigenfunctions was interpreted as the Benjamini–Schramm convergence to Euclidean space via a rescaling of the Riemannian metric.

Remark 1.2. We point out two differences between the quantum ergodicity theorem of [2] for rank $1$ spaces and the higher-rank version we present in Theorem 1.1. In contrast to our result, the authors of [2] allow for two more general features:

1. (a) They allow the spectral window to shrink with n, thereby isolating in the large-n limit a fixed tempered eigenvalue for the universal cover. This added flexibility was also present in [3, Theorem 1.3].

2. (b) They take more general operators than scalar multiplication by functions $a_n$ . This more advanced formulation was first put forward in [3, Theorem 1.7], using the pseudo-differential calculus for trees developed in [23].

Our Theorem 1.1 therefore more closely resembles the main results of [8, 24], in which neither of these two features is present. By taking more elaborate test functions, we believe we can incorporate (a) into our setup. By contrast, we have so far been unable to extend (b) to this higher-rank setting.

1.3 Comments on the proof

To prove the theorem we first reduce the assertion to several intermediate statements, as explained in §2. This reduction roughly follows along the lines of [24] and the subsequent [2]. To establish these intermediate results, however, we need several new ideas to deal with a number of issues that arise only in rank $2$ or higher:

• • The first main reduction step involves the use of a normalized averaging operator on $S=G/K$ (a kind of wave propagation) with expanding support $C_t$ , $t\rightarrow \infty $ . At a later critical point we need to estimate from above the volume of intersections $C_t\cap g C_t$ with $g\in G$ . In [2, 24] they work with $C_t$ being the metric ball $B_t$ in S of radius t, and exploit the fact that S is a $\text {CAT}(-1)$ space in rank $1$ . In higher rank, S is only a $\text {CAT}(0)$ space, and working with intersection of metric balls becomes problematic. We therefore need to define new types of $C_t$ that look more ‘polytopal’ and are easier to work with in higher rank (see equation (15) and §5.5).

• • We need to establish a suitable lower bound for certain averages of spherical functions. In rank $1$ , this can be dealt with in a relatively straightforward manner, as the elementary spherical functions are basically linear combinations of trigonometric functions in one variable. In higher rank, we need to deal with linear combinations of exponential functions in several variables, which makes the analysis much more delicate (see §4). The techniques of §4 might also be of interest in other contexts.

Additionally, we need to establish a type of local Weyl law/limit multiplicity for the $Y_n$ that gives a lower count for the number of eigenvalues locally around sufficiently regular points in the spectrum of $Y_n$ . We only need the sharp lower bound in the level aspect stated in formula (7), but along the way we prove a stronger version that also yields the right order in the spectral parameter; for the exact statement, see Proposition 2.1. Such a result is also necessary for the rank $1$ situation, but involves a much more careful analysis of the nontempered spectrum in higher rank.

2 Outline of proof and reduction steps

In this section, we shall reduce the proof of Theorem 1.1 to that of two auxiliary estimates: one spectral, one geometric. Let $G=\operatorname {SL}_d({\mathbb R})$ and $K=\textrm {SO}(d)$ . As a preliminary step, we note that by replacing $a_n$ with , we may suppose that the measurable, right-K-invariant functions $a_n$ on $\Gamma _n\backslash G$ satisfy

(4) $$ \begin{align} \int_{\Gamma_n\backslash G} a_n=0 \quad \text{and}\quad \lVert a_n\rVert_\infty\leq 1. \end{align} $$

Under that assumption it will then be enough to prove

(5) $$ \begin{align} \frac{1}{N(B_0(\nu,\varrho),\Gamma_n)}\sum_{j:\nu_j^{(n)}\in B_0(\nu,\varrho)} \left\lvert\left\langle a_n\psi_j^{(n)}, \psi_j^{(n)}\right\rangle_{L^2(Y_n)}\right\rvert^2\rightarrow 0 \end{align} $$

as $n\rightarrow \infty $ . The rest of the paper is devoted to establishing formula (5).

2.1 Spectral estimate

We shall first need to control the spectral counting function defined in equation (2). This is provided in the following result, to be proved in §3 (see [2] for a more abstract approach, in rank $1$ , using the limit-multiplicity result of [1]):

Proposition 2.1. Let G be a connected noncompact simple Lie group with finite center, and K a maximal compact subgroup. Let $S=G/K$ be the associated irreducible Riemannian globally symmetric space of noncompact type. Let $\Gamma _n$ be a sequence of uniformly discrete, torsion-free, cocompact lattices in G such that $Y_n=\Gamma _n\backslash S$ converges, in the sense of Benjamini and Schramm, to S. There exist $0<\varrho _1<1<\varrho _2$ such that the following holds: Fix $\epsilon>0$ , and let $\nu \in i\mathfrak {a}^*$ be $\varepsilon $ -regular. Then

(6) $$ \begin{align} N(B(\nu,\varrho_1),\Gamma_n)\ll \operatorname{vol}(\Gamma_n\backslash G)\tilde\beta(\nu)\ll_\varepsilon N(B(\nu,\varrho_2),\Gamma_n), \end{align} $$

where $B(\nu ,\varrho )=\left \{\lambda \in \mathfrak {a}_{\mathbb C}^*\mid \lVert \lambda - \nu \rVert _2<\varrho \right \}$ , and $N(B(\nu ,\varrho ),\Gamma _n)=\left \lvert \left \{j\mid \nu _j^{(n)}\in B(\nu ,\varrho )\right \}\right \rvert $ is the counting function.

We shall only make use of Proposition 2.1 in the form

(7) $$ \begin{align} \operatorname{vol}(\Gamma_n\backslash G)\ll N(B(\nu,\varrho),\Gamma_n), \end{align} $$

for a sufficiently regular spectral parameter $\nu $ . Indeed formula (7) follows from the second estimate in formula (6) using the fact that $\tilde \beta (\nu )\geq 1$ for all $\nu \in i\mathfrak {a}^*$ . See §3.1 for our notational conventions on the dependence on the implied constants on the assumption of sufficient regularity.

We note that Proposition 2.1 is stated for general symmetric spaces S. By contrast, we have restricted the setting of the next two results, Theorems 2.2 and 2.3, to the symmetric space $S=\operatorname {SL}_d({\mathbb R})/\textrm {SO}(d)$ .

2.2 The averaging subset

Following [8, 24], the main idea behind formula (5) is the strategic use of a self-adjoint normalized averaging operator which on one hand acts nondecreasingly on the spectral average (Theorem 2.2) and on the other hand has small Hilbert–Schmidt norm (Theorem 2.3). This idea can be traced back to the work of Brooks, Le Masson, and Lindenstrauss [8], which presents an alternative proof of [3, Theorem 1.3], using discrete normalized averaging operators.

In the following we give a definition of an exhaustive sequence of sets $E_t\subset \operatorname {SL}_d({\mathbb R})$ , which we shall use in our averaging operators. The difficulty in higher rank is finding a set $E_t$ which satisfies simultaneously the desired spectral and geometric properties (in rank $1$ , the obvious choice of a Riemannian metric ball is shown to work in [[2], 24]).

We begin by introducing some notation. Let $\mathfrak {g}$ be the trace $0$ matrices in $M_d({\mathbb R})$ . Let

$$ \begin{align*} \mathfrak{a}=\left\{X=\textrm{ diag}(X_1,\dotsc ,X_d)\in\mathfrak{g}\right\} \end{align*} $$

be the standard Cartan subalgebra of diagonal matrices. Let $W=N_K(\mathfrak {a})/Z_K(\mathfrak {a})$ be the Weyl group of $(\mathfrak {a},\mathfrak {g})$ ; the action of $W\simeq S_d$ on $\mathfrak {a}$ is by permutation of the coordinates. Let $\mathfrak {a}^+$ be the standard positive Weyl chamber in $\mathfrak {a}$ given by $X_1>X_2>\dotsb >X_d$ . For a vector $X\in \mathfrak {a}$ we will write

$$ \begin{align*} \lVert X\rVert_\infty=\max\{\lvert X_1\rvert,\dotsc, \lvert X_d\rvert\}. \end{align*} $$

The W-invariant norm $\lVert \cdot \rVert _\infty $ on $\mathfrak {a}$ induces a subadditive bi-K-invariant norm on $G=\operatorname {SL}_d({\mathbb R})$ via the Cartan decomposition $G=K\exp \left (\overline {\mathfrak {a}^+}\right )K$ . Namely, if $g\in G$ with $g\in K e^X K$ , then we write $\lvert g\rvert = \lVert X\rVert _\infty $ . We have $\lvert g\rvert \geq 0$ , $\left \lvert g^{-1}\right \rvert =\lvert g\rvert $ , and $\lvert g_1g_2\rvert \leq \lvert g_1\rvert +\lvert g_2\rvert $ .

With this notation, we put

(8) $$ \begin{align} E_t=\{g\in G: \lvert g\rvert \leq t\}. \end{align} $$

This is the averaging set we shall use for our wave propagator. One may view $E_t$ as a radially invariant subset of the symmetric space $G/K$ . See §2.5 for more commentary on the nature of $E_t$ .

2.3 The two main results

We now define

$$ \begin{align*} k_t=\frac{1}{\sqrt{m_G(E_t)}}\textbf{ 1}_{E_t}. \end{align*} $$

Let $\rho _{\Gamma \backslash G}$ denote the right-regular representation of G on $L^2(\Gamma \backslash G)$ , and consider the wave-propagation operator on $L^2(\Gamma \backslash G)$ given by

(9) $$ \begin{align} U_t=\rho_{\Gamma\backslash G}(k_t). \end{align} $$

From $E_t^{-1}=E_t$ it follows that $k_t$ is a self-adjoint operator. For a measurable right-K-invariant function a on $\Gamma \backslash G$ satisfying equation (4), and for $\tau>0$ , we consider the time average

(10) $$ \begin{align} \textbf{ A}(\tau)=\frac{1}{\tau}\int_0^\tau U_taU_t dt. \end{align} $$

In §4 we shall prove the following spectral estimate:

Theorem 2.2 spectral estimate

Let $S=\operatorname {SL}_d({\mathbb R})/{\mathrm{SO}}(d)$ . Let $\Gamma $ a cocompact lattice in $\operatorname {SL}_d({\mathbb R})$ , and put $Y=\Gamma \backslash S$ . Let $\Omega \subseteq i\mathfrak {a}^*$ be compact. There exist constants $c,\tau _0>0$ , depending on $\Omega $ , such that for $\tau \ge \tau _0$ , we have

$$ \begin{align*} \sum_{j: \nu_j\in\Omega} \left\lvert\left\langle a\psi_j, \psi_j\right\rangle_{L^2(Y)}\right\rvert^2\leq c \sum_{j: \nu_j\in\Omega} \left\lvert\left\langle \textbf{ A}(\tau)\psi_j , \psi_j\right\rangle_{L^2(Y)}\right\rvert^2. \end{align*} $$

In §5 we shall use the Nevo mean ergodic theorem [29] (see also [15, Theorem 4.1]) to prove the following geometric estimate. Recall that the Hilbert–Schmidt norm of a bounded operator $\mathbf {A}$ on a separable Hilbert space H is given by

$$ \begin{align*} \lVert\textbf{ A}\rVert_{\textrm{HS}}^2=\sum_i \left\lvert\langle \textbf{ A} e_i,e_i\rangle\right\rvert^2, \end{align*} $$

where $\{e_i\}$ is any orthonormal basis of H.

Theorem 2.3 geometric estimate

Let $S=\operatorname {SL}_d({\mathbb R})/{\mathrm{SO}}(d)$ . Let $\Gamma $ be a cocompact torsion-free lattice in $\operatorname {SL}_d({\mathbb R})$ , and put $Y=\Gamma \backslash S$ . Let a be a measurable function on $L^2(\Gamma \backslash S)$ . There are constants $b, c_1,c_2>0$ , depending only on d, such that for $\tau>0$ ,

$$ \begin{align*} \lVert\textbf{ A}(\tau)\rVert_{\textrm{ HS}}^2\ll_b \frac{\lVert a\rVert_2^2}{\tau}+ \frac{e^{c_1\tau}}{\operatorname{InjRad}(Y)^{\dim S}}\operatorname{vol}\left((\Gamma\backslash G)_{\leq c_2(2\tau+b)}\right)\lVert a\rVert_\infty^2. \end{align*} $$

2.4 Reduction to two main results

We now deduce the main estimate (5) from the foregoing results.

We first note that if $\nu \in i\mathfrak {a}^*$ is sufficiently regular in the sense that $\left \lvert \left \langle \nu ,\alpha ^\vee \right \rangle \right \rvert \ge C_\varrho $ for all simple roots $\alpha $ , with $C_\varrho>0$ a sufficiently large constant depending on $\varrho $ , then $\left \{j\mid \nu _j^{(n)}\in B_0(\nu ,\varrho )\right \}=\left \{j\mid \nu _j^{(n)}\in B(\nu ,\varrho )\right \}$ because of formula (23). By the lower bound (7), which follows from Proposition 2.1, it suffices to prove an upper bound for

$$ \begin{align*} \frac{1}{\operatorname{vol} (\Gamma_n\backslash G)}\sum_{j: \nu_j^{(n)}\in B(\nu,\varrho)} \left\lvert\left\langle a_n\psi_j^{(n)}, \psi_j^{(n)}\right\rangle_{L^2\left(Y_n\right)}\right\rvert^2. \end{align*} $$

From Theorem 2.2, it follows that for $\tau \geq \tau _0$ ,

$$ \begin{align*} \frac{1}{\operatorname{vol} (\Gamma_n\backslash G)}\sum_{j: \nu_j^{(n)}\in B\left(\nu,\varrho\right)} \left\lvert\left\langle a_n\psi_j^{(n)}, \psi_j^{(n)}\right\rangle_{L^2(Y_n)}\right\rvert^2 \ll \frac{1}{\operatorname{vol} (\Gamma_n\backslash G)}\left\lVert \textbf{ A}^{(n)}(\tau)\right\rVert_{\textrm{ HS}}^2. \end{align*} $$

We now apply Theorem 2.3 to the right-hand side. To bound the first term, we use the estimate $\lVert a_n\rVert _2^2\leq \lVert a_n\rVert _\infty ^2 \operatorname {vol}(\Gamma _n\backslash G)$ along with the uniform boundedness of $a_n$ . For the second term we insert the bound $\operatorname {InjRad}(Y_n)\gg 1$ coming from the uniformly discrete hypothesis in the statement of Theorem 1.1. We obtain

(11) $$ \begin{align} \frac{1}{\operatorname{vol} (\Gamma_n\backslash G)}\left\lVert\textbf{ A}^{(n)}(\tau_n)\right\rVert_{\textrm{ HS}}^2\ll \frac{1}{\tau_n}+ e^{c_1\tau_n}\frac{\operatorname{vol}\left((\Gamma_n\backslash G)_{\leq c_2\left(2\tau_n+b\right)}\right)}{\operatorname{vol} (\Gamma_n\backslash G)}. \end{align} $$

We know from [1, §4] that for any sequence $\Gamma _n$ for which $\operatorname {vol}(\Gamma _n\backslash G)\rightarrow \infty $ , the space $\Gamma _n\backslash G/K$ converges in the sense of Benjamini and Schramm to $G/K=\operatorname {SL}_d({\mathbb R})/\textrm {SO}(d)$ (recall our assumption that $d\geq 3$ ). We deduce that there is a sequence $R_n\rightarrow \infty $ such that

(12) $$ \begin{align} \frac{\operatorname{vol}\left((\Gamma_n\backslash G)_{\leq R_n}\right)}{\operatorname{vol}(\Gamma_n\backslash G)}\rightarrow 0 \end{align} $$

as $n\rightarrow \infty $ . Given this sequence $R_n$ , we let $r_n>0$ be a sequence such that

1. (i) $r_n\rightarrow \infty $ as $n\rightarrow \infty $ and

2. (ii) $e^{c_1 r_n}\frac {\operatorname {vol}\left ((\Gamma _n\backslash G)_{\leq c_2\left (2r_n+b\right )}\right )}{\operatorname {vol}(\Gamma _n\backslash G)}\rightarrow 0$ as $n\rightarrow \infty $ .

Taking $\tau _n=r_n$ , both terms in formula (11) go to $0$ with n, establishing formula (5).

Remark 2.1. One could deduce an effective rate of convergence in Theorem 1.1 for congruence subgroups $\Gamma _n$ of $\operatorname {SL}_d({\mathbb Z})$ by inserting, in place of the $o(1)$ result in formula (12), the explicit bounds on the R-thin part for such sequences proved in [1, Theorem 5.2].

2.5 Remarks on the averaging subset

We now make several remarks about $E_t$ :

(i) In rank $1$ , the set $E_t$ recovers the metric ball centered at $i\in \mathbb {H}$ of radius $2t$ for the hyperbolic metric $\left (dx^2+dy^2\right )/y^2$ on the upper half-plane $\mathbb {H}=\operatorname {SL}_2({\mathbb R})/\textrm {SO}(2)$ .

(ii) In higher rank, the set $E_t$ differs substantially from a metric ball centered at the identity for the metric induced by the trace form $\operatorname {Tr}\left (X^\top Y\right )$ (a constant multiple of the Killing form) on $\mathfrak {g}$ .

To see this, let us first compare a geodesic ball and the set $E_t$ by means of the Cartan decomposition of $G=\operatorname {SL}_d({\mathbb R})$ . Let $\mathscr {B}=\left \{X\in \mathfrak {a}: \lVert X\rVert _2\leq 1\right \}$ , where $\lVert X\rVert _2^2=\operatorname {tr}\left (X^2\right )=X_1^2+\dotsb +X_d^2$ . The geodesic ball $B_t$ of radius $t>0$ centered at the origin is given by

(13) $$ \begin{align} B_t=K\exp \left(\mathscr{B}_t^+\right) K, \end{align} $$

where $\mathscr {B}_t^+=t\mathscr {B}\cap \overline {\mathfrak {a}^+}$ (see [4, Proposition 4.2]). On the other hand, let

(14) $$ \begin{align} {\mathscr P}=\left\{X\in\mathfrak{a}\mid \lVert X\rVert_\infty\le 1\right\}. \end{align} $$

Note that ${\mathscr P}$ is a bounded convex polytope in $\mathfrak {a}$ , being the intersection of the half-planes $\pm X_i\leq 1$ . It is clear that

(15) $$ \begin{align} E_t=K\exp\left({\mathscr P}_t^+\right)K, \end{align} $$

where ${\mathscr P}^+_t=t{\mathscr P}\cap \overline {\mathfrak {a}^+}$ . One sees from equations (13) and (15) that an element in $B_t$ or $E_t$ has Cartan radial component contained to, respectively, an expanding ball or a dilated W-invariant polytope in $\mathfrak {a}$ .

(iii) Related to the foregoing description is the difference in the volume asymptotics between $B_t$ and $E_t$ . Both are expressed, naturally, with respect to the half-sum of the positive roots $\rho \in \mathfrak {a}^*$ , as that quantity governs the Jacobian factor in the Cartan decomposition of the Haar measure on G (see §4.1).

For the geodesic ball, a result of Knieper [21], valid more generally for irreducible Riemannian symmetric spaces $S=G/K$ of noncompact type, states that

(16) $$ \begin{align} m_G(B_t)\asymp t^{(r-1)/2}e^{2t\lVert\rho\rVert_2}, \end{align} $$

where r is the rank of S. On the other hand, we show in Corollary A.4 that, for $G=\operatorname {SL}_d({\mathbb R})$ ,

$$ \begin{align*} m_G(E_t)\sim c_2 e^{2t\left\langle \rho, X^0\right\rangle},\quad t\rightarrow\infty, \end{align*} $$

where $X^0\in {\mathscr P}^+$ is the unique point satisfying

(17) $$ \begin{align} \left\langle \rho, X^0\right\rangle=\max_{X\in{\mathscr P}^+}\langle\rho,X\rangle. \end{align} $$

One can compute that

(18) $$ \begin{align} \lVert\rho\rVert_2=\left\langle\rho,\frac{\rho}{\lVert\rho\rVert_2}\right\rangle=\sqrt{d\left(d^2-1\right)/12}, \end{align} $$

whereas using Lemma A.1, we have

$$ \begin{align*} \left\langle\rho,\frac{X^0}{\left\lVert X^0\right\rVert_2}\right\rangle=\begin{cases} d^{3/2}/4, & d \text{ even},\\ (d+1)(d-1)^{1/2}/4, & d \text{ odd}. \end{cases} \end{align*} $$

We have normalized by $\left \lVert X^0\right \rVert _2$ so that $B_t$ is the smallest geodesic ball containing $E_{t/\left \lVert X^0\right \rVert _2}$ . We see that the exponential volume growth of $B_t$ in $\operatorname {SL}_d({\mathbb R})/\textrm {SO}(d)$ is of order $2t \sqrt {d^3/12}$ , whereas that of $E_{t/\left \lVert X^0\right \rVert _2}$ is of order $2t\sqrt {d^3/16}$ , which is smaller by a factor of $\sqrt {3}/2$ .

(iv) Another norm that one often considers in the context of $G=\operatorname {SL}_d({\mathbb R})$ is the restriction to $\operatorname {SL}_d({\mathbb R})\subset M_d({\mathbb R})$ of the Frobenius norm, defined on $M_d({\mathbb R})$ as $\lVert g\rVert ^2=\operatorname {tr}\left (g^\top g\right )$ , where $g^\top $ is the transpose of g. Note that $\left \lVert g^{-1}\right \rVert \neq \lVert g\rVert $ in general. Since invariance under inversion is important for the self-adjointness of the propagator $U_t$ from equation (9), we define the Frobenius ball as

(19) $$ \begin{align} \boldsymbol{E}_t=\left\{g\in G: \max\left\{\lVert g\rVert,\left\lVert g^{-1}\right\rVert\right\}\leq e^t\right\}. \end{align} $$

From the point of view of their large-scale geometry, the Frobenius balls are similar to the sets $E_t$ from equation (8). Indeed, in the proof of Proposition 5.8 we show that $m_G(E_t)\asymp m_G(\boldsymbol {E}_t)$ . Moreover, the very statement of Proposition 5.8, which is a key component of Theorem 2.3, holds equally well for $\boldsymbol {E}_t$ . On the other hand, because they are not defined by their radial Cartan component, Frobenius balls are not amenable to our proof of Theorem 2.2.

3 Weyl-type law

Our aim in this section is to prove Proposition 2.1. Throughout this section only, we let G denote a connected noncompact simple Lie group with finite center and K a maximal compact subgroup.

3.1 Notation

The notation we introduce here will be consistent with that already introduced for the particular case of $G=\operatorname {SL}_d({\mathbb R})$ and $K=\textrm {SO}(d)$ .

Let $\theta $ be the Cartan involution on G for which $K=G^\theta $ . Let $\Theta $ be its differential and let $\mathfrak {g}={\mathfrak p}\oplus {\mathfrak k}$ be the Cartan decomposition of the Lie algebra $\mathfrak {g}$ of G into the $\pm 1$ eigenspaces of $\Theta $ . We may identify ${\mathfrak k}$ with the Lie algebra of K.

Let $\kappa (X,Y)=\operatorname {tr}(\operatorname {ad} X\operatorname {ad} Y)$ denote the Killing form on $\mathfrak {g}$ . Then $-\kappa (X,\Theta Y)$ defines an $\operatorname {Ad}_K$ -invariant inner product on $\mathfrak {g}$ , which induces a left-invariant Haar measure on G, denoted $dg$ or $m_G$ .

The Killing form defines an $\operatorname {Ad}_K$ -invariant inner product $\langle \cdot ,\cdot \rangle $ on ${\mathfrak p}$ , and in particular on $\mathfrak {a}$ , the maximal abelian subalgebra of ${\mathfrak p}$ . Let $W=N_K(\mathfrak {a})/Z_K(\mathfrak {a})$ be the Weyl group. Let $\lVert \cdot \rVert _2$ denote the W-invariant norm ¹ on $\mathfrak {a}$ induced by $\langle \cdot ,\cdot \rangle $ .

Let $\Phi \subset \mathfrak {a}^*$ denote the system of roots for the adjoint action of $\mathfrak {a}$ on $\mathfrak {g}$ . For $\alpha \in \Phi $ , let $\mathfrak {g}_\alpha $ denote the corresponding root space. Let ${\mathfrak m}\subset \mathfrak {g}$ be the centralizer of $\mathfrak {a}$ in $\mathfrak {g}$ . Then we have the root-space decomposition

$$ \begin{align*} \mathfrak{g}={\mathfrak m}\oplus\mathfrak{a}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_\alpha. \end{align*} $$

Choose a positive system of roots $\Phi ^+$ in $\Phi $ . Let $\Delta \subseteq \Phi ^+$ be the set of simple roots.

Let $\mathfrak {a}^*=\mathrm {Hom}(\mathfrak {a},{\mathbb R})$ be the dual vector space of $\mathfrak {a}$ . We may identify $\mathfrak {a}$ with $\mathfrak {a}^*$ via the Killing form. We again denote by $\lVert \cdot \rVert _2$ the induced norm on $\mathfrak {a}^*$ and extend it to a complex bilinear form on the complexification $\mathfrak {a}^*_{\mathbb C}=\mathfrak {a}^*\otimes _{\mathbb R}{\mathbb C}$ . We call $\nu \in i\mathfrak {a}^*$ regular if $\langle \alpha ,\nu \rangle \neq 0$ for all $\alpha \in \Delta $ , and for $c>0$ we call $\nu \in i\mathfrak {a}^* c$ -regular if $\lvert \langle \alpha ,\nu \rangle \rvert \ge c$ for all $\alpha \in \Phi ^+$ . An element $\nu \in i\mathfrak {a}^*$ is sufficiently regular if there exists a sufficiently regular $c>0$ (which may vary at each occurrence) such that $\nu $ is c-regular. Unless otherwise noted, implied constants for statements valid for sufficiently regular $\nu $ can depend on the value of c.

Let ${\mathfrak n}=\sum _{\alpha \in \Phi ^+} \mathfrak {g}_\alpha $ and write $N=\exp {\mathfrak n}$ . Then we have the Iwasawa decomposition $G=NAK$ , where $A=\exp \mathfrak {a}$ , which gives rise to the Iwasawa projection

$$ \begin{align*} H_0: G\rightarrow \mathfrak{a}, \qquad g=nak\mapsto \log a, \end{align*} $$

along the A component.

We may decompose the Riemannian Haar measure $dg$ on G according to the Iwasawa decomposition. We let $da$ denote the Haar measure on A obtained by pushing forward the Lebesgue measure on $\mathfrak {a}$ by means of the exponential. We let $dk$ denote the probability Haar measure $dk$ on K. We normalize the left-invariant Haar measure $du$ on N as in [12, §3.1]. As usual, we put

$$ \begin{align*} \rho=\frac12\operatorname{tr}\left(\operatorname{ad}(\mathfrak{a})\rvert_{\mathfrak n}\right)=\frac12\sum_{\alpha\in\Phi^+}(\dim\mathfrak{g}_\alpha) \alpha. \end{align*} $$

Then we have [14, Proposition 2.4.10]

$$ \begin{align*} dg= c_Ie^{-2\rho \left(H_0\left(g\right)\right)}du da dk, \quad c_I= 2^{-(1/2)\dim N}. \end{align*} $$

Since A normalizes N and $\det \operatorname {Ad}(a)\rvert _{\mathfrak n}=e^{2\rho \left (H_0(a)\right )}$ ,

(20) $$ \begin{align} dg=c_Ida du dk \end{align} $$

in the decomposition $G=ANK$ .

Recall the Cartan decomposition $G=K\exp \left (\overline {\mathfrak {a}^+}\right )K$ , where $\overline {\mathfrak {a}^+}$ denotes the closure of the (open) positive Weyl chamber $\mathfrak {a}^+=\left \{X\in \mathfrak {a}: \alpha (X)>0\;\forall \,\alpha \in \Phi ^+\right \}$ . We have the following decomposition of the Riemannian Haar measure $dg$ into the Cartan decomposition:

(21) $$ \begin{align} dg=c_C J(X) dk_1 da dk_2, \quad c_C=2^{-\dim N}\operatorname{vol} (K)\operatorname{vol}(K/M). \end{align} $$

Here, $\operatorname {vol}(K)$ is the Riemannian volume on K for the measure induced by the inner product $-\kappa (X,Y)$ on $\mathfrak {k}$ , and similarly with $\operatorname {vol}(K/M)$ . Finally, $J(X)$ is the Jacobian factor, given by

(22) $$ \begin{align} J(X)=\prod_{\alpha\in\Phi^+} \sinh(\alpha(X))^{\dim\mathfrak{g}_\alpha},\quad X\in\overline{\mathfrak{a}^+} \end{align} $$

(see [14, Proposition 2.4.11]).

Let $P_0$ be the normalizer of ${\mathfrak m}\oplus \mathfrak {a}\oplus {\mathfrak n}$ in G. Then $P_0$ has Langlands decomposition $P_0=MAN$ , where M is the centralizer of A in G. For $\lambda \in \mathfrak {a}^*_{\mathbb C}$ , let $\pi _\lambda $ denote the unique unramified irreducible subquotient of $\mathrm {Ind}_{P_0}^G\left (1\otimes e^\lambda \otimes 1\right )$ . Let

$$ \begin{align*} \mathfrak{a}^*_{\textrm{ un}}=\left\{\lambda\in\mathfrak{a}^*_{\mathbb C}: \pi_\lambda \text{ unitarizable}\right\} \end{align*} $$

be the spherical unitary dual of G. Furthermore, let

$$ \begin{align*} \mathfrak{a}^*_{\textrm{ hm}}=\bigcup_{w\in W}\left\{\lambda\in\mathfrak{a}^*_{\mathbb C}: w\lambda=-\bar\lambda\right\} \end{align*} $$

denote the spherical Hermitian dual of G [22, §3.3]. Let $i\mathfrak {a}$ be the subspace of $\mathfrak {a}_{\mathbb C}^*$ consisting of $\lambda $ taking on purely imaginary values. We may write $\lambda \in \mathfrak {a}^*_{\mathbb C}$ uniquely as $\lambda =\operatorname {Re}\lambda +\operatorname {Im}\lambda \in \mathfrak {a}^*\oplus i\mathfrak {a}^*$ . We have

(23) $$ \begin{align} i\mathfrak{a}^*\subset \mathfrak{a}^*_{\textrm{ un}}\subset \mathfrak{a}^*_{\textrm{ hm}}\cap\left\{\lambda\in\mathfrak{a}^*_{\mathbb C}: \lVert\operatorname{Re}\lambda\rVert_2\leq \lVert\rho\rVert_2\right\}. \end{align} $$

For $\mu \in i\mathfrak {a}^*$ and $r>0$ , we let

$$ \begin{align*} B_0(\mu,r)=\{\lambda\in i\mathfrak{a}^*: \lVert\lambda-\mu\rVert_2\leq r\} \end{align*} $$

denote the ball of radius r about $\mu $ in the tempered spectrum $i\mathfrak {a}^*$ . When $r=1$ we write $B_0(\mu )$ for $B_0(\mu ,1)$ . We also write

$$ \begin{align*} B(\mu,r)=\left\{\lambda\in \mathfrak{a}^*_{\mathbb C}: \lVert\lambda-\mu\rVert_2\leq r\right\} \end{align*} $$

for the ball of radius r around $\mu $ in $\mathfrak {a}^*_{\mathbb C}$ .

3.2 Plancherel density and the $\mathbf {c}$ -function

We denote by $\beta $ the density function for the Plancherel measure on the spherical unitary dual, which can be identified with $\mathfrak {a}^*_{\mathrm {un}}\big /W$ . Then $\beta $ is a W-invariant function supported on $i\mathfrak {a}^*$ that can be described as a product of $\Gamma $ -functions [17, Ch. IV, §6]. Following [22], we put

$$ \begin{align*} \tilde\beta(t,\lambda)=\prod_{\alpha\in\Phi^+}\left(t+\left\lvert\left\langle \lambda,\alpha^\vee\right\rangle \right\rvert\right)^{\dim\mathfrak{g}_\alpha}, \quad \lambda\in i\mathfrak{a}^*, \ t\geq 1, \end{align*} $$

and $\tilde \beta (\lambda )=\tilde \beta (1,\lambda )$ . From [17, Ch. IV, Theorem 6.14] and standard estimates for the $\Gamma $ -function, it follows that $\beta (\lambda )\ll \tilde \beta (\lambda )$ for all $\lambda \in i\mathfrak {a}^*$ . Since

$$ \begin{align*} 1+\left\lvert\left\langle t\lambda +\mu,\alpha^\vee\right\rangle \right\rvert &\le 1+ t \left\lvert\left\langle \lambda ,\alpha^\vee\right\rangle \right\rvert + \left\lvert\left\langle\mu,\alpha^\vee\right\rangle \right\rvert \\ &\le t (1+ \lVert\lambda\rVert_2) +\left\lvert\left\langle\mu,\alpha^\vee\right\rangle \right\rvert \le (1+ \lVert\lambda\rVert_2)\left(t +\left\lvert\left\langle\mu,\alpha^\vee\right\rangle \right\rvert\right) \end{align*} $$

for all $\lambda ,\mu \in i\mathfrak {a}^*$ and $t\geq 1$ , we get

(24) $$ \begin{align} \tilde\beta(t\lambda+\mu)\ll (1+\lVert\lambda\rVert_2)^{\dim\mathfrak{n}} \tilde\beta(t,\mu) \end{align} $$

for all such $\lambda $ , $\mu $ , and t.

The Harish-Chandra $\mathbf {c}$ -function $\mathbf {c}:\mathfrak {a}^*_{\mathbb C}\longrightarrow {\mathbb C}$ for G asymptotically describes the behavior of the elementary spherical functions $\phi _\lambda \left (e^H\right )$ of G as the group parameter H grows. The quantity $\mathbf {c}(\lambda )$ depends only on the root system of G, and can be explicitly computed as described in [17, Ch. IV, Theorem 6.14]. Up to normalization, the Plancherel density $\beta (\lambda )$ equals $\lvert \mathbf {c}(\lambda )\rvert ^{-2}$ for $\lambda \in i\mathfrak {a}^*$ .

3.3 Test functions

For $\lambda \in \mathfrak {a}^*_{\mathbb C}$ , let

$$ \begin{align*} \varphi_\lambda(g)=\int_K e^{\left\langle \lambda + \rho, H_0\left(kg\right)\right\rangle} dk \end{align*} $$

denote the spherical function on G with spectral parameter $\lambda $ . Here $H_0:G\longrightarrow \mathfrak {a}$ is the Iwasawa projection. Let $C^\infty _c(G // K)$ be the space of complex-valued bi-K-invariant functions on G. The Harish-Chandra transform of a function $k\in C^\infty _c(G // K)$ is defined to be

$$ \begin{align*} \hat{k}(\lambda)=\int_G \varphi_\lambda(g) k(g)dg. \end{align*} $$

For $t>0$ , let $\mathscr {B}_t$ denote the ball of radius t centered at $0$ in $\mathfrak {a}$ with respect to the usual Euclidean norm $\lVert \cdot \rVert _2$ . Let $G_{\leq t}=K\exp \mathscr {B}_t K$ . Let $C^\infty _c(G // K)_{\leq t}$ denote the space of smooth compactly supported bi-K-invariant functions on G, supported on $G_{\leq t}$ . Let $\mathscr {PW}\left (\mathfrak {a}^*_{\mathbb C}\right )_t$ denote the class of Paley–Wiener functions of exponential type t. The Paley–Wiener theorem with supports (see [13, Theorem 3.5]) states that the Harish-Chandra transform is a topological isomorphism $C^\infty _c(G // K)_{\leq t}\xrightarrow {\sim } \mathscr {PW}\left (\mathfrak {a}^*_{\mathbb C}\right )_t^W$ of Fréchet spaces. The inverse map sends $h\in \mathscr {PW}\left (\mathfrak {a}^*_{\mathbb C}\right )_t^W$ to

$$ \begin{align*} \frac{1}{\lvert W\rvert}\int_{i\mathfrak{a}^*}h(\mu)\varphi_\mu(g)\beta(\mu)d\mu. \end{align*} $$

In particular, for $k\in C_c^\infty (G // K)$ we have

(25) $$ \begin{align} k(e)=\frac{1}{\lvert W\rvert}\int_{i\mathfrak{a}^*}\hat{k}(\mu)\beta(\mu)d\mu. \end{align} $$

We shall need a Paley–Wiener function concentrating about $\nu $ and verifying certain positivity properties.

Lemma 3.1 [9, §4]

For $t\geq 1$ and $\nu \in i\mathfrak {a}^*$ , there is whose Harish-Chandra transform $h_{\nu ,t}\in \mathscr {PW}\left (\mathfrak {a}^*_{\mathbb C}\right )_t^W$ satisfies the following:

1. (1) $h_{\nu ,t}$ is real and nonnegative on $\mathfrak {a}^*_{\mathrm {hm}}$ .

2. (2) For all $\lambda \in \mathfrak {a}^*_{\mathrm {un}}$ we have

   $$ \begin{align*} h_{\nu,t}(\lambda)\ll_A (1+\lVert(\operatorname{Im}\lambda-\nu)/t\rVert_2)^{-A}. \end{align*} $$

3. (3) There are constants $0<c_1,c_2<1$ such that for $\lambda \in \mathfrak {a}_{\mathrm {un}}^*$ with $\lVert \operatorname {Im}\lambda -\nu \rVert _2\leq c_1t$ , we have $c_2\leq h_{\nu ,t}(\lambda )\leq 2$ .

4. (4) $k_{\nu ,t}(g)\ll \tilde \beta (\nu )t^r(1+\lVert \nu \rVert _2 d(g,K))^{-1/2}$ , where $r=\dim \mathfrak {a}$ .

3.4 Proof of Proposition 2.1.

Recall that for $\nu \in i\mathfrak {a}^*$ and $c>0$ , we denote by $B(\nu , c)$ the ball $\left \{\lambda \in \mathfrak {a}_{\mathbb C}^*\mid \lVert \lambda -\nu \rVert _2\le c\right \}$ in $\mathfrak {a}_{\mathbb C}^*$ . Moreover, for any set $\Omega \subseteq \mathfrak {a}^*_{\mathbb C}$ , we write $N(\Omega ,\Gamma _n)=\left \lvert \left \{j\mid \nu _j^{(n)}\in \Omega \right \}\right \rvert $ .

In the notation of Lemma 3.1, we write $h_\nu =h_{\nu ,1}$ and $k_\nu =k_{\nu ,1}$ . Then

$$ \begin{align*} \sum_{\lambda\in \Lambda_n}h_\nu(\lambda) = \operatorname{vol}(\Gamma_n\backslash G) k_\nu(1) + \int_{\Gamma_n\backslash G} \sum_{\substack{\gamma\in \Gamma_n\\\gamma \neq 1}} k_\nu\left(g^{-1}\gamma g\right) dg, \end{align*} $$

where $\Lambda _n$ denotes the spectrum of $\Gamma _n\backslash G$ (each $\lambda $ appearing with its respective multiplicity).

To deal with the second term, we use Lemma 3.1(4), as well as the support condition on $k_\nu $ , to deduce

$$ \begin{align*} \int_{\Gamma_n\backslash G} \sum_{\substack{\gamma\in \Gamma_n\\\gamma \neq 1}} k_\nu\left(g^{-1}\gamma g\right) dg\ll \tilde\beta(\nu)\int_{\Gamma_n\backslash G} \lvert\{\gamma\in \Gamma_n\smallsetminus\{1\}: d(g,\gamma g)\leq 1\}\rvert dg. \end{align*} $$

As $\Gamma _n$ is torsion free, the inner sum is empty for all $g\in (\Gamma _n\backslash G)_{>1}$ , so that it suffices to bound

$$ \begin{align*} \int_{(\Gamma_n\backslash G)_{\leq 1}} N_R(g)dg, \end{align*} $$

where

(26) $$ \begin{align} N_R(g)=\lvert\{\gamma\in \Gamma_n: d(g,\gamma g)\leq 1\}\rvert. \end{align} $$

For $g\in (\Gamma _n\backslash G)_{\leq 1}$ we apply [1, Lemma 6.18], which provides a constant $C>0$ , depending only on G, such that $N_R(g)\leq C \operatorname {InjRad}_{\Gamma _n}(g)^{-\dim S}$ . The uniform discreteness of the (torsion-free) $\Gamma _n$ implies that $\operatorname {InjRad}_{\Gamma _n}(g)^{-1} \ll 1$ , uniformly in n and g. Taking these estimates together, we get

(27) $$ \begin{align} \sum_{\lambda\in \Lambda_n}h_\nu(\lambda) =\operatorname{vol}(\Gamma_n\backslash G)k_\nu(1) +O\left(\tilde\beta(\nu)\operatorname{vol}(\Gamma_n\backslash G)_{\leq 1}\right). \end{align} $$

Furthermore, we have

(28) $$ \begin{align} k_\nu(1)\asymp_\epsilon \tilde\beta(\nu), \end{align} $$

where only the lower bound depends on $\varepsilon $ . Indeed, the upper bound results from Lemma 3.1(4). To obtain the lower bound, one applies the Plancherel inversion formula (25) and Lemma 3.1(1) and (3) to get

$$ \begin{align*} k_\nu(1)\geq \int_{\substack{\lambda\in i\mathfrak{a}^*\\ \lVert\lambda-\nu\rVert_2\leq\delta}} h_\nu(\lambda)\beta(\lambda)d\lambda\geq c_2\int_{\substack{\lambda\in i\mathfrak{a}^*\\ \lVert\lambda-\nu\rVert_2\leq\delta}}\beta(\lambda)d\lambda, \end{align*} $$

for any $0<\delta <c_1$ . Recall that $\nu \in i\mathfrak {a}^*$ is $\varepsilon $ -regular. Taking $\delta $ small enough, we may assume that the $\lambda \in i\mathfrak {a}^*$ such that $\lVert \lambda -\nu \rVert _2\leq \delta $ are $\varepsilon /2$ -regular. Then the lower bound follows from the inequality $\beta (\lambda )\gg _\varepsilon \tilde \beta (\lambda )$ for such $\lambda $ (see [12, (3.44a)]).

From formulas (27) and (28), it follows that

$$ \begin{align*} \sum_{\lambda\in \Lambda_n}h_\nu(\lambda) \asymp_\varepsilon \operatorname{vol}(\Gamma_n\backslash G)\tilde\beta(\nu)\left(1+O\left(\frac{\operatorname{vol}(\Gamma_n\backslash G)_{\leq 1}}{\operatorname{vol}(\Gamma_n\backslash G)}\right)\right), \end{align*} $$

where only the lower bound depends on $\varepsilon $ . By the Benjamini–Schramm assumption, we have $\operatorname {vol}(\Gamma _n\backslash G)_{\leq 1}=o(\operatorname {vol}(\Gamma _n\backslash G))$ as $n\rightarrow \infty $ . Thus for n large enough we have

(29) $$ \begin{align} \sum_{\lambda\in \Lambda_n}h_\nu(\lambda) \asymp_\varepsilon \operatorname{vol}(\Gamma_n\backslash G)\tilde\beta(\nu). \end{align} $$

For the first bound in formula (6), it suffices at this point to take $\varrho _1=c_1$ , apply Lemma 3.1(1) and (3), and then quote the upper bound in formula (29) to get

$$ \begin{align*} N(B(\nu,c_1),\Gamma_n)\leq c_2 \sum_{\substack{\lambda\in\Lambda_n\\ \lVert\lambda-\nu\rVert_2\leq c_1}}h_\nu(\lambda)\leq c_2\sum_{\lambda\in \Lambda_n}h_\nu(\lambda) \ll \operatorname{vol}(\Gamma_n\backslash G)\tilde\beta(\nu). \end{align*} $$

For the second bound in formula (6), we must show that the left-hand side in formula (29) approximates $N(B(\nu ,\varrho ),\Gamma _n)$ , for some $\varrho>0$ .

A crucial ingredient for passing from a smooth count as before to a sharp count will be a good upper bound on $N(B(\mu ,t),\Gamma _n)$ , for any center $\mu \in i\mathfrak {a}^*$ and any $t\geq 1$ . This is proved similarly to the preceding analysis. Indeed, from Lemma 3.1 and the preceding geometric argument, we obtain

$$ \begin{align*} \sum_{\substack{\lambda\in\Lambda_n\\\lVert\operatorname{Im}\lambda-\mu\rVert_2\leq t}} 1\ll \sum_{\lambda\in\Lambda_n}h_{\mu,c_1^{-1}t}(\lambda)=\operatorname{vol}(\Gamma_n\backslash G) k_{\mu,c_1^{-1}t}(1) + o\left(\operatorname{vol}(\Gamma_n\backslash G)\tilde\beta(\mu) t^r \right). \end{align*} $$

Now $k_{\mu ,t}(1)\ll \tilde \beta (\mu )t^r$ from Lemma 3.1(4). We conclude that (for n large enough)

(30) $$ \begin{align} N(B'(\mu,t),\Gamma_n)\ll \operatorname{vol}(\Gamma_n\backslash G)\tilde\beta(\mu)t^r, \end{align} $$

where $B'(\mu ,t)=\left \{\lambda \in \mathfrak {a}^*_{\mathbb C}\mid \lVert \operatorname {Im} \lambda - \mu \rVert _2\le t\right \}$ . We now return to the left-hand side of formula (29). We first claim the following:

Lemma 3.2. For every $\varrho>1$ ,

(31) $$ \begin{align} \sum_{\substack{\lambda\in\Lambda_n\\ \lVert\operatorname{Im} \lambda-\nu\rVert_2>\varrho}}h_\nu(\lambda) \ll_A \operatorname{vol}(\Gamma_n\backslash G)\tilde\beta(\nu)\varrho^{-A}. \end{align} $$

Proof. To prove formula (31) we cover $\{\lambda \in i\mathfrak {a}^*: \lVert \lambda -\nu \rVert _2>\varrho \}$ by balls of unit radius with centers based at an affine lattice $\Lambda \subset i\mathfrak {a}^*$ containing $\nu $ . Let $A(\nu ,\varrho )$ denote the set of all $\lambda \in \Lambda $ such that the ball of radius $1$ in $i\mathfrak {a}^*$ around $\lambda $ is entirely contained in the ball of radius $\varrho $ in $i\mathfrak {a}^*$ around $\nu $ .

For any fixed $\mu \in \Lambda -A(\nu ,\varrho )$ , we bound the contribution of $\lambda \in \Lambda _n$ for which $\operatorname {Im}\lambda \in B(\mu )$ by using Lemma 3.1(2) to get

$$ \begin{align*} \sum_{\substack{\lambda\in\Lambda_n\\ \lVert\operatorname{Im}\lambda-\mu\rVert_2\leq 1}}h_\nu(\lambda) \ll_A (1+\lVert\mu-\nu\rVert_2)^{-A}N(B'(\mu,1),\Gamma_n). \end{align*} $$

Using formula (30) with $t=1$ and summing over $\mu \in \Lambda - A(\nu , \varrho )$ we get

$$ \begin{align*} \sum_{\substack{\lambda\in\Lambda_n\\ \lVert\operatorname{Im}\lambda-\nu\rVert_2>\varrho}}h_\nu(\lambda)\ll_A \operatorname{vol}(\Gamma_n\backslash G)\sum_{\mu\in\Lambda- A\left(\nu, \varrho\right)} (1+\lVert\mu-\nu\rVert_2)^{-A}\tilde\beta(\mu). \end{align*} $$

Now by formula (24) we have $\tilde \beta (\mu )\ll (1+\lVert \mu -\nu \rVert _2)^{\dim {\mathfrak n}}\tilde \beta (\nu )$ , so that

$$ \begin{align*} \sum_{\lVert\operatorname{Im} \lambda-\nu\rVert_2>\varrho}h_\nu(\lambda)\ll_A \operatorname{vol}(\Gamma_n\backslash G)\tilde\beta(\nu)\sum_{\mu\in\Lambda-A\left(\nu,\varrho\right)}(1+\lVert\mu-\nu\rVert_2)^{-A + \dim {\mathfrak n}}. \end{align*} $$

For $\varrho $ sufficiently large, this last sum is at most $O_A\left (\varrho ^{-A + \dim {\mathfrak n}}\right )$ , as desired.

From this lemma, we want to deduce the following estimate:

(32) $$ \begin{align} \sum_{\substack{\lambda\in\Lambda_n\\ \lVert\lambda-\nu\rVert_2>\varrho}}h_\nu(\lambda) \ll_A \operatorname{vol}(\Gamma_n\backslash G)\tilde\beta(\nu)\varrho^{-A}. \end{align} $$

For this we split the sum on the left-hand side into three parts:

$$ \begin{align*} \sum_{\substack{\lambda\in\Lambda_n\\ \lVert\operatorname{Im} \lambda-\nu\rVert_2>\varrho}}h_\nu(\lambda) + \sum_{\substack{\lambda\in\Lambda_n\\ \varrho^2-\lVert\rho\rVert_2^2< \lVert\operatorname{Im}\lambda-\nu\rVert_2^2\le \varrho^2 \\ \lVert\operatorname{Re} \lambda\rVert_2^2> \varrho^2-\lVert\operatorname{Im}\lambda-\nu\rVert_2^2 }}h_\nu(\lambda) + \sum_{\substack{\lambda\in\Lambda_n\\ \lVert\operatorname{Im} \lambda-\nu\rVert_2^2\le \varrho^2-\lVert\rho\rVert_2^2 \\ \lVert\operatorname{Re} \lambda\rVert_2^2> \varrho^2-\lVert\operatorname{Im}\lambda-\nu\rVert_2^2 }}h_\nu(\lambda). \end{align*} $$

Using Lemma 3.2, the first sum can be bounded by $\ll _A \operatorname {vol}(\Gamma _n\backslash G) \tilde \beta (\nu ) \varrho ^{-A}$ . The third sum is in fact empty: The conditions on the real and complex parts of $\lambda $ imply that $\lVert \operatorname {Re}\lambda \rVert _2> \lVert \rho \rVert _2$ , which is not possible for $\lambda $ in the spectrum of $L^2(\Gamma _n\backslash G)$ , by formula (23). Finally, for the second term, we extend the sum to all $\lambda \in \Lambda _n$ with $\lVert \operatorname {Im} \lambda -\nu \rVert _2^2\ge \varrho ^2-\lVert \rho \rVert _2^2$ . This is possible because $h_\nu (\lambda )$ is nonzero on the unitary spectrum, by Lemma 3.1. Hence we can use Lemma 3.2 to bound this term by $\ll _A \left (\varrho ^2-\lVert \rho \rVert _2^2\right )^{-A/2}$ .

We deduce from the lower bound in formula (29), as well as from formula (32) that with $A>1$ fixed and $\varrho $ large enough with respect to the implied constant,

$$ \begin{align*} \sum_{\substack{\lambda\in \Lambda_n\\\lVert\lambda-\nu\rVert_2\leq \varrho}}h_\nu(\lambda)\gg_\varepsilon \operatorname{vol}(\Gamma_n\backslash G) \tilde\beta(\nu). \end{align*} $$

On the other hand, by applying Lemma 3.1(3) we have

$$ \begin{align*} N(B(\nu,\varrho),\Gamma_n)\gg \sum_{\substack{\lambda\in\Lambda_n\\ \lVert\lambda-\nu\rVert_2\leq \varrho}}h_\nu(\lambda). \end{align*} $$

Combing these yields the second bound in formula (6), with $\varrho _2=\varrho $ . □

4 Spectral side

We now return to the setting of $G=\operatorname {SL}_d({\mathbb R})$ and prove Theorem 2.2. In the course of the proof, we will make use of both Appendices A and B.

We first examine how $k_t$ acts on eigenfunctions $\psi _\lambda $ . The Harish-Chandra transform of $k_t$ is given by

(33) $$ \begin{align} h_t(\lambda)=\frac{1}{\sqrt{m_G(E_t)}}\int_{E_t} \varphi_\lambda(g)\, dg. \end{align} $$

Then, recalling the definition (9), we have

$$ \begin{align*} U_t\psi_\lambda=h_t(\lambda)\psi_\lambda, \end{align*} $$

which follows from the doubling formula of elementary spherical functions.

Let $\mathbf {A}(\tau )$ be the time-averaging operator of equation (10). Since $k_t$ is self-adjoint, we have

$$ \begin{align*} \left|\langle \textbf{ A}(\tau)\psi_\lambda , \psi_\lambda\rangle_{L^2(Y)}\right|{}^2=\left(\frac{1}{\tau}\int_0^\tau \lvert h_t(\lambda)\rvert^2 dt\right)^2 \left\lvert\langle a\psi_\lambda, \psi_\lambda\rangle_{L^2(Y)}\right\rvert^2. \end{align*} $$

To prove Theorem 2.2, it will therefore be enough to show that $\lvert h_t(\lambda )\rvert $ is bounded away from $0$ on average over t. More precisely, we prove the following higher-rank generalization of [2, §8.1]:

Proposition 4.1. Let $G=\operatorname {SL}_d({\mathbb R})$ . Given a compact nonempty set $\Omega \subseteq i\mathfrak {a}^*$ , there exist constants $C,\tau _0>0$ , depending on $\Omega $ , such that for every $\tau \ge \tau _0$ and $\lambda \in \Omega $ ,

$$ \begin{align*} \frac{1}{\tau}\int_0^\tau \lvert h_t(\lambda)\rvert^2 dt \ge C. \end{align*} $$

4.1 Main idea of proof

We begin by writing the integral defining $h_t$ in polar coordinates, according to the Cartan decomposition. Recall the averaging set $E_t$ from §2.2 and its polar decomposition (15). Letting $f_\lambda (X)=e^{\left \langle \rho ,X\right \rangle }\varphi _\lambda \left (e^X\right )$ , and using the Cartan measure decomposition (21), we find that equation (33) becomes

(34) $$ \begin{align} h_t(\lambda)=\frac{c_C}{\sqrt{m_G(E_t)}}\int_{\mathscr{P}_t^+} f_\lambda(X)J(X)e^{-\left\langle\rho,X\right\rangle}dX. \end{align} $$

The idea of the proof of Proposition 4.1 is to replace $f_\lambda $ by the main term $\Phi _\lambda $ , discussed later, of its Harish-Chandra asymptotic expansion relative to the Levi determined by $X^0$ , defined in equation (17). We do this by showing in Appendix A (see Lemma A.2) that $\mathscr {P}^+$ is the intersection with $\overline {\mathfrak {a}^+}$ of a translated cone $\mathscr {C}=X^0+\mathscr {C}^0$ in $\mathfrak {a}$ . We then show that when $\lambda $ is taken to be rational, $\Phi _\lambda $ behaves like a sum of characters in the direction of $X^0$ . An argument using linear independence of characters then yields the result. We carry out this argument in this section, using some combinatorial results which we establish in Appendix B.

4.2 The main term and periodicity

We wish to replace $f_\lambda $ by the main term in its Harish-Chandra expansion. This is possible thanks to Proposition 4.2. To state it, we must introduce some notation beyond that of §3.1. For simplicity, we continue to take $G=\operatorname {SL}_d({\mathbb R})$ , although Proposition 4.2 applies more generally.

A Levi subgroup of G will be called semistandard if it contains A. Let $\Delta _0\subseteq \Delta $ be a (possibly empty) subset of the set of simple roots, and let $\Phi ^+_0$ be the subset of $\Phi ^+$ generated by $\Delta _0$ . The sets $\Delta _0$ correspond to semistandard Levi subgroups $M\subseteq G$ such that $\Phi ^+_0$ is the set of positive roots of A on M. With this identification, $\Delta _0=\emptyset $ corresponds to $M=A$ and $\Delta _0=\Delta $ to $M=G$ .

For fixed $\Delta _0$ and corresponding M, we further introduce the following notation:

• • $\rho ^M$ is the half-sum of all $\alpha \in \Phi _0^+$ ;

• • $W^M$ is the Weyl group of A in M;

• • $\mathbf {c}^M$ denotes the Harish-Chandra $\mathbf {c}$ -function for $\Phi _0^+$ ;

• • $\varphi ^M_\lambda $ is the spherical function on M for $\lambda \in \mathfrak {a}_{\mathbb C}^*$ – that is,

  $$ \begin{align*} \varphi^M_\lambda(m)= \int_{K\cap M} e^{\left\langle \lambda + \rho^M, H_0(km)\right\rangle} dk; \end{align*} $$

• • for $X\in \mathfrak {a}$ , $\lambda \in \mathfrak {a}_{\mathbb C}^*$ , and $t>0$ , let $f_\lambda ^M(t,X)= e^{\left \langle \rho ^M,tX\right \rangle } \varphi _\lambda ^M\left (e^{tX}\right )$ and (setting $t=1$ ) let

  $$ \begin{align*} f_\lambda^M(X)=e^{\left\langle \rho^M,X\right\rangle} \varphi_\lambda^M\left(e^X\right); \end{align*} $$

• • $\mathfrak {a}_M$ is the common null space of all $\alpha \in \Phi _0^+$ , and $\mathfrak {a}^M$ is its orthocomplement in $\mathfrak {a}$ relative to the inner product $\langle X, Y\rangle =\operatorname {tr} XY$ .

We follow the standard practice of omitting the superscript M from the notation whenever $M=G$ . In this way we recover the $f_\lambda (X)$ introduced earlier. Other than G, the only Levi which will be of importance to us throughout this section is the centralizer of $X^0$ in G. Henceforth, we shall take M to be the centralizer of $X^0$ in G, and $\Delta _0$ such that it corresponds to M. In this case, $X^0\in \mathfrak {a}_M$ .

For $\lambda \in i\mathfrak {a}^*$ regular, we define

(35) $$ \begin{align} \Phi_\lambda(X) = \sum_{w\in W^M\backslash W} \frac{\textbf{ c}(w\lambda)}{\textbf{ c}^M(w\lambda)} f^M_{w\lambda}(X). \end{align} $$

In fact, we can define $\Phi _\lambda (X)$ for any regular $\lambda \in \mathfrak {a}^*_{\mathbb C}$ for which the $\mathbf {c}(w\lambda )$ do not have a pole. We can make this more precise, as follows. For a real number $\eta>0$ , let $A_\eta \subseteq \mathfrak {a}^*$ denote the convex hull of the Weyl group orbit of $\eta \rho $ . Define ${\mathscr T}_\eta \subseteq \mathfrak {a}^*_{\mathbb C}$ to be the set of all $\lambda \in \mathfrak {a}^*_{\mathbb C}$ with $\operatorname {Re}\lambda \in A_\eta $ . Let $\eta $ be so small that for every positive root $\alpha $ and all $\lambda \in {\mathscr T}_\eta $ , we have $\left \langle \operatorname {Re}\lambda , \alpha ^\vee \right \rangle \not \in {\mathbb Z}\smallsetminus \{0\}$ . We fix such a $0<\eta <1$ from now on (we will make our choice more precise later). Note that ${\mathscr T}_\eta $ is W-invariant by construction. We may extend the definition of $\Phi _\lambda (X)$ in equation (35) to $\lambda \in {\mathscr T}_\eta $ regular and $X\in \mathfrak {a}$ . Indeed, the $\mathbf {c}$ -function is nonzero on $\mathfrak {a}^*_{\mathbb C}$ and does not have any poles for regular $\lambda \in {\mathscr T}_\eta $ , by our choice of $\eta $ .

The relevance of $\Phi _\lambda (X)$ is apparent from the next result, due to Trombi and Varadarajan [34, Theorem 2.11.2], providing the full asymptotic expansion of the spherical function. We have taken the exact statement from [14, Theorem 5.9.4]. This extends to all $X\in \mathfrak {a}$ the asymptotic expansion proved by Harish-Chandra and Gangolli for conical subsets of regular elements.

Proposition 4.2. Let $\Omega \subseteq i\mathfrak {a}^*$ be a compact set and $c>0$ . Then there exist $C>0$ and $m\ge 0$ such that for all $H\in \overline {\mathfrak {a}^+}$ with $\beta _{\Delta _0}(H)\ge c$ , where

$$ \begin{align*} \beta_{\Delta_0}(H)=\min_{\alpha\in \Delta\smallsetminus \Delta_0} \alpha(H), \end{align*} $$

and for every regular $\lambda \in \Omega $ , we have

$$ \begin{align*} \left\lvert f_\lambda(H) - \Phi_\lambda(H) \right\rvert \le C (1+\lVert H\rVert_2)^m e^{-2\beta_{\Delta_0}(H)}. \end{align*} $$

We now describe the key periodicity property of $\Phi _\lambda (X)$ . For this we need the following definition:

Definition 4.3. We say that $\lambda \in i\mathfrak {a}^*$ is rational if $i\left \langle w\lambda , X^0\right \rangle \in {\mathbb Q}$ for all $w\in W$ .

If $\lambda $ is rational, we can choose $\tau _1\in {\mathbb R}_{>0}$ such that $\tau _1 \left \langle w\lambda , X^0\right \rangle \in 2i\pi {\mathbb Z}$ for all $w\in W$ . Recall the definition of ${\mathscr C}$ from Lemma A.2.

Lemma 4.4. Let $\lambda \in i\mathfrak {a}^*$ be rational and let $\tau _1$ be defined as before. Set $\tau _n = n\tau _1$ for $n\in {\mathbb N}$ . Then for every $Y\in {\mathscr C}$ , $t\ge 0$ , and $n\in {\mathbb N}$ , we have

$$ \begin{align*} \Phi_\lambda\left(Y + (t+\tau_n)X^0\right) = \Phi_\lambda\left(Y+ tX^0\right). \end{align*} $$

Proof. By the definition of $\tau _1$ , it follows that $e^{\left \langle \lambda , \left (t+\tau _n\right )X^0\right \rangle } =e^{\left \langle \lambda , tX^0\right \rangle }$ for every $n\in {\mathbb N}$ . Further, note that since $\left \langle \rho ^M,X^0\right \rangle =0$ and $H_0\left (ke^{Y+tX^0}\right )=tX^0+H\left (ke^Y\right )$ for $k\in K\cap M$ , we have

$$ \begin{align*} \varphi_\lambda^M\left(e^{Y+tX^0}\right)=e^{\left\langle \rho^M+\lambda,tX^0\right\rangle}\varphi_\lambda^M\left(e^Y\right)=e^{\left\langle \lambda,tX^0\right\rangle}\varphi_\lambda^M\left(e^Y\right). \end{align*} $$

Thus

(36) $$ \begin{align} f^M_\lambda\left(Y+ tX^0\right) = e^{\left\langle \lambda, tX^0\right\rangle} f^M_\lambda(Y). \end{align} $$

Applying these two formulas for every $w\lambda $ , $w\in W^M\backslash W$ , yields the assertion.

4.3 Convergence of an integral

To take advantage of Lemma 4.4, we shall need to replace the integration domain $\mathscr {P}_t^+$ in equation (34) by the translated cone $\mathscr {C}$ . Note that the Jacobian factor (22) can be alternatively written as

(37) $$ \begin{align} J(X)=2^{-\dim\mathfrak{n}}\sum_{w\in W} \sigma(w) e^{2\left\langle w\rho, X\right\rangle}, \quad X\in\overline{\mathfrak{a}^+}, \end{align} $$

where $\sigma (w)\in \{\pm 1\}$ denotes the signature of w. With this in mind, for $t>0$ and $\lambda \in {\mathscr T}_\eta $ , we put

$$ \begin{align*} I(t,\lambda)=\int_{{\mathscr C}^0} \Phi_\lambda\left(Y + tX^0\right) e^{\left\langle \rho, Y\right\rangle} dY \end{align*} $$

whenever it converges.

We now address this question of convergence. From the defining expression for $\Phi _\lambda $ in equation (35), it will be enough to understand the convergence of the integral

$$ \begin{align*} J(t,\lambda) =\int_{{\mathscr C}^0} f^M_{\lambda}\left(Y+ tX^0\right) e^{\left\langle \rho, Y\right\rangle} dY. \end{align*} $$

Lemma 4.5. There is $0<\eta <1$ such that for every $\lambda \in {\mathscr T}_\eta $ and $t\ge 0$ , the integral defining $J(t,\lambda )$ converges absolutely.

Proof. We first observe that it suffices to consider $t=0$ . Indeed, from equation (36) it follows that

(38) $$ \begin{align} J(t,\lambda) = e^{ t \left\langle \lambda, X^0\right\rangle} J(\lambda), \end{align} $$

where we have put $J(\lambda )=J(0,\lambda )$ . There should be no confusion between the integral $J(\lambda )$ and the Jacobian factor $J(X)$ from equation (22), due to the different nature of their arguments.

Set $Y\in {\mathscr C}^0$ and write $Y= Y_M+Y^M$ , with $Y_M\in \mathfrak {a}_M$ and $Y^M\in \mathfrak {a}^M$ . Then

(39) $$ \begin{align} f^M_{\lambda}(Y) = e^{\left\langle \rho^M, Y\right\rangle} \varphi_\lambda^M\left(e^Y\right) = e^{\left\langle\lambda, Y_M\right\rangle} e^{\left\langle \rho^M, Y^M\right\rangle} \varphi_\lambda^M\left(e^{Y^M}\right). \end{align} $$

Proposition B.1 then immediately yields the bound $\left \lvert e^{\left \langle \lambda , Y_M\right \rangle }\right \rvert \leq e^{-\eta C\left \langle \rho ,Y\right \rangle }$ on the first factor on the right-hand side.

To bound the remaining two factors, we first let $Y^M_+$ be a point in the $W^M$ -orbit of $Y^M$ lying in $\overline {\mathfrak {a}^{M,+}}$ . Then by [19, Ch. X, §2, Proposition 2.2, Theorem 2.8], we have

$$ \begin{align*} \left\lvert\varphi_\lambda^M\left(e^{Y^M}\right)\right\rvert \le \varphi_{\operatorname{Re} \lambda}^M\left(e^{Y^M}\right) &= \varphi_{\operatorname{Re} \lambda}^M\left(e^{Y^M_+}\right)\\ & \le \max_{w\in W^M} e^{\left\langle w{\textrm{Re}} \lambda, Y^M_+\right\rangle} \varphi_0^M\left(e^{Y^M_+}\right) \\ & \ll \max_{w\in W^M} e^{\left\langle w{\textrm{Re}} \lambda, Y^M_+\right\rangle} e^{-\left\langle \rho^M, Y^M_+\right\rangle} \left(1+\left\lVert Y^M_+\right\rVert_2\right)^{\dim\mathfrak{n}}. \end{align*} $$

Now $\left \lVert Y^M_+\right \rVert _2 = \left \lVert Y^M\right \rVert _2$ and $\left \langle \rho ^M, Y^M\right \rangle \le \left \langle \rho ^M, Y^M_+\right \rangle $ . Thus there is a $c>0$ such that

$$ \begin{align*} \left\lvert e^{\left\langle \rho^M, Y^M\right\rangle} \varphi_\lambda^M\left(e^{Y^M}\right)\right\rvert \leq c \max_{w\in W^M} e^{\left\langle w\operatorname{Re} \lambda, Y^M_+\right\rangle} \left(1+\left\lVert Y^M\right\rVert_2\right)^{\dim\mathfrak{n}}. \end{align*} $$

Using the W-invariance of $\langle \cdot ,\cdot \rangle $ and $Y^M= Y-Y_M$ , we obtain

$$ \begin{align*} \max_{w\in W^M} \left\langle w\operatorname{Re} \lambda, Y^M_+\right\rangle=\max_{w\in W^M} \left\langle w\operatorname{Re} \lambda, Y^M\right\rangle \leq \max_{w\in W} \lvert\langle w \operatorname{Re} \lambda, Y\rangle\rvert+\max_{w\in W} \lvert\langle w \operatorname{Re} \lambda, Y_M\rangle\rvert. \end{align*} $$

Note that $w\operatorname {Re}\lambda =\operatorname {Re} w\lambda $ and $w\lambda \in \mathcal {T}_\eta $ . Using Remark A.1, we have $ \lvert \langle \operatorname {Re} \lambda , Y\rangle \rvert \ll -\eta \langle \rho , Y\rangle $ for all $\lambda \in {\mathscr T}_\eta $ and $Y\in \mathscr {C}^0$ . From this and Proposition B.1, we deduce that there is a constant $C'>0$ , depending only on d, such that

$$ \begin{align*} \max_{w\in W^M} \left\langle w\operatorname{Re} \lambda, Y^M_+\right\rangle \le - \eta C'\langle\rho, Y\rangle. \end{align*} $$

Similarly, we can find $c>0$ such that $\left \lVert Y^M\right \rVert _2 \le c \lVert Y\rVert _2$ . Hence if $\eta>0$ is sufficiently small, then it follows from equation (39) and our estimates that for some $c'>0$ ,

$$ \begin{align*} \left\lvert f^M_{\lambda}(Y)\right\rvert \le c' e^{-\frac{1}{2}\left\langle \rho, Y\right\rangle}(1+\lVert Y\rVert_2)^{\dim\mathfrak{n}}. \end{align*} $$

We therefore obtain

$$ \begin{align*} \int_{{\mathscr C}^0} \left\lvert f^M_{\lambda}(Y) e^{\left\langle \rho, Y\right\rangle}\right\rvert dY \le \int_{{\mathscr C}^0} e^{\left\langle \frac{1}{2} \rho, Y\right\rangle} (1+\lVert Y\rVert_2)^{\dim\mathfrak{n}} dY. \end{align*} $$

Using Remark A.1 we see that this last integral is finite. Thus the integral defining $J(\lambda )$ , and hence $J(t,\lambda )$ , converges absolutely.

It follows that for $\lambda \in {\mathscr T}_\eta $ we have

(40) $$ \begin{align} I(t,\lambda) = \sum_{w\in W^M\backslash W} \frac{\textbf{ c}(w\lambda)}{\textbf{ c}^M(w\lambda)} J(w\lambda) e^{t \left\langle w\lambda , X^0\right\rangle}, \end{align} $$

where, we recall, $J(\lambda )=J(0,\lambda )$ .

4.4 Main-term replacement

Having established that $I(t,\lambda )$ is well defined, we now show, using Proposition 4.2, that $h_t(\lambda )$ can be approximated by $I(t,\lambda )$ .

Proposition 4.6. There are $c, C>0$ such that for regular $\lambda \in i\mathfrak {a}^*$ ,

$$ \begin{align*} h_t(\lambda)=C I(t,\lambda)+O_\lambda\left(e^{-ct}\right). \end{align*} $$

Before we prove this proposition, we establish the following lemma (see Lemma A.2 for the definition of the $\beta _i^{\vee }$ ):

Lemma 4.7. Set $0<\eta <1$ and define ${\mathscr E}= X^0 -\sum _{i=1}^r [0,\eta ]\beta _i^\vee = \sum _{i=1}^r [1-\eta , 1]\beta _i^\vee $ and ${\mathscr E}^+ = {\mathscr E}\cap {\mathscr P}^+$ . Then if $\eta $ is sufficiently small, we have the following:

1. (i) For every V in the closure $\overline {{\mathscr P}^+\smallsetminus {\mathscr E}^+}$ of ${\mathscr P}^+\smallsetminus {\mathscr E}^+$ , we have

   (41) $$ \begin{align} \langle \rho, V\rangle \le \left\langle\rho, X^0\right\rangle -\delta \end{align} $$
   for some $\delta =\delta (\eta )>0$ uniform in all V.

2. (ii) Let $\Delta ^M\subseteq \Delta $ denote the subset of simple roots in M. Then there exists $c=c(\eta )>0$ such that for every $X\in {\mathscr E}$ and $\alpha \in \Delta \smallsetminus \Delta ^M$ , we have $\alpha (X)\ge c$ .

Proof. Since $X^0$ is the unique point in ${\mathscr P}^+$ at which the linear map $X\mapsto \langle \rho , X\rangle $ attains its maximum, and $\overline {{\mathscr P}^+\smallsetminus {\mathscr E}^+}$ is closed, it will suffice to show that every point in $\overline {{\mathscr P}^+\smallsetminus {\mathscr E}^+}$ is bounded away from $X^0$ . But this is clear, since ${\mathscr P}^+ =\left (X^0 -\sum _{i=1}^r [0,\infty )\beta _i^\vee \right )\cap \mathfrak {a}^+$ by Lemma A.2. Note that this argument holds for any $0<\eta <1$ , with the resulting constant $\delta $ of course depending on $\eta $ .

For the second part, we note that for every $\alpha \in \Delta \smallsetminus \Delta ^M$ , we have $\alpha \left (X^0\right ) = 2$ . Hence if $\eta $ is chosen sufficiently small, each such $\alpha $ stays bounded away from $0$ on ${\mathscr E}$ so that (ii) follows.

Proof of Proposition 4.6. Let

$$ \begin{align*} G(t,\lambda)=e^{-t\left\langle\rho, X^0\right\rangle} \int_{{\mathscr P}^+_t} f_\lambda(X) e^{\left\langle \rho, X\right\rangle} dX. \end{align*} $$

From equations (34) and (37), as well as the volume computation in Corollary A.4, we have

$$ \begin{align*} h_t(\lambda)=C G(t,\lambda)+O\left(e^{-c_1t}\right) \end{align*} $$

for some $c_1, C>0$ . Next, let ${\mathscr E}^+$ be defined as in Lemma 4.7 with $\eta $ sufficiently small. Put

$$ \begin{align*} H(t,\lambda)= \int_{t{\mathscr E}^+} f_\lambda(X) e^{\left\langle\rho, X- tX^0\right\rangle} dX. \end{align*} $$

Then $G(t, \lambda )=H(t,\lambda ) + O\left (e^{-c_2t}\right )$ for some absolute constant $c_2>0$ , by Lemma 4.7(i).

Note that by Lemma 4.7, for any $X\in t{\mathscr E}$ we have $\alpha (X)\ge ct$ for $\alpha \in \Delta \smallsetminus \Delta ^M$ . Using Lemma 4.7(ii) and Proposition 4.2, there exists $c_3>0$ such that

$$ \begin{align*} f_\lambda(X) - \Phi_\lambda(X) \ll_\lambda e^{-c_3t} \end{align*} $$

for every $X\in t{\mathscr E}^+$ and every regular $\lambda \in i\mathfrak {a}^*$ . Hence, for regular $\lambda \in i\mathfrak {a}^*$ , we have

$$ \begin{align*} H(t,\lambda)&= \int_{t{\mathscr E}^+} \Phi_\lambda(X) e^{\left\langle\rho, X- tX^0\right\rangle} dX + O_\lambda\left(e^{-c_3t}\right)\\ &= \int_{t{\mathscr E}^+- tX^0} \Phi_\lambda\left(Y + tX^0\right) e^{\left\langle \rho, Y\right\rangle} dY+O_\lambda\left(e^{-c_3t}\right) \end{align*} $$

after a change of variables. Noting that $t{\mathscr E}^+- tX^0\subseteq {\mathscr C}^0$ , we can then extend the integral to all of ${\mathscr C}^0$ , incurring an additional error of $O_\lambda \left (e^{-c_4t}\right )$ , for some $c_4>0$ .□

4.5 A nonvanishing result

We now show that $I(t,\lambda )$ is generically nonvanishing on $i\mathfrak {a}^*$ . This will follow from the absolute convergence of $I(t,\lambda )$ , the periodic behavior of $\Phi _\lambda $ along $X^0$ for rational $\lambda $ , and a linear-independence-of-characters argument.

Proposition 4.8. There exists a finite number of hyperplanes ${\mathscr H}_1,\dotsc , {\mathscr H}_N\subseteq i\mathfrak {a}^*$ and a locally finite collection ${\mathscr D}$ of $(d-2)$ -dimensional manifolds in $i\mathfrak {a}^*$ such that for every

$$ \begin{align*} \lambda\in i\mathfrak{a}^*\smallsetminus ({\mathscr H}_1\cup\dotsb\cup{\mathscr H}_N\cup {\mathscr D}), \end{align*} $$

there exists $t=t_\lambda \ge 0$ with $I(t,\lambda )\neq 0$ .

Proof. Let $\eta>0$ be sufficiently small and at most as large as in Lemma 4.5. We first claim that there exists a locally finite collection of $(d-2)$ -dimensional complex manifolds $\mathcal {Z}\subseteq {\mathscr T}_\eta $ such that $J(t,w\lambda )\neq 0$ for all $\lambda \in {\mathscr T}_\eta \smallsetminus {\mathscr D}$ , $w\in W$ , and $t\ge 0$ . Indeed, for fixed $m\in M$ , the map $\lambda \mapsto \varphi _\lambda ^M(m)$ is holomorphic in $\lambda \in \mathfrak {a}^*_{\mathbb C}$ . Since the integral defining $J(t,\lambda )$ converges absolutely, $J(t,\lambda )$ is in fact a holomorphic function for $\lambda \in {\mathscr T}_\eta $ . From the nonvanishing at $\lambda =0$ ,

$$ \begin{align*} J(t,0)=\int_{{\mathscr C}^0} e^{\left\langle \rho+\rho^M, Y\right\rangle} \varphi_0^M\left(e^Y\right) dY \neq0, \end{align*} $$

it follows that $J(t,\lambda )$ does not vanish identically in $\lambda $ , hence its zero set $\mathcal {Z}= \{\lambda \in {\mathscr T}_\eta \mid J(t, \lambda )=0\}$ is a locally finite union of $(d-2)$ -dimensional complex manifolds. Put ${\mathscr D} = \mathcal {Z}\cap i\mathfrak {a}^*$ .

By our choice of $\eta $ , $\mathbf {c}(w\lambda )/\mathbf {c}^M(w\lambda )$ has no pole for regular $\lambda $ in ${\mathscr T}_\eta $ . Let ${\mathscr T}_\eta '$ denote the set of all regular $\lambda \in {\mathscr T}_\eta \smallsetminus \mathcal {D}$ such that $\left \langle \lambda , w^{-1}X^0\right \rangle \neq \left \langle \lambda , v^{-1}X^0\right \rangle $ for all $w, v\in W^M\backslash W$ , $w\neq v$ . Then ${\mathscr T}_\eta '$ is in fact dense in ${\mathscr T}_\eta $ , since $w^{-1}X^0\neq v^{-1} X^0$ for all $w\neq v$ , $w, v\in W^M\backslash W$ , so that we can obtain ${\mathscr T}_\eta '$ by removing the hyperplanes $\left \{\lambda \mid \left \langle \lambda , w^{-1}X^0\right \rangle = \left \langle \lambda , v^{-1}X^0\right \rangle \right \}$ for each $v\neq w$ in $W^M\backslash W$ .

Recall the expansion (40). By the definition of ${\mathscr T}_\eta '$ , the phases $\left \langle w\lambda , X^0\right \rangle $ , $w\in W^M\backslash W$ , are pairwise distinct for $\lambda \in {\mathscr T}_\eta '$ . We can therefore apply [16, Lemma 56] (see also [19, Ch. VIII, Lemma 0.1]) to the sum over $w\in W^M\backslash W$ to obtain, for each $\lambda \in {\mathscr T}_\eta '$ ,

$$ \begin{align*} \limsup_{t\rightarrow \infty} \left\lvert I(t,\lambda)\right\rvert^2 \ge \sum_{w\in W^M\backslash W} \left\lvert\frac{\textbf{ c}(w\lambda)}{\textbf{ c}^M(w\lambda)} J(w\lambda)\right\rvert^2. \end{align*} $$

Now for each $w\in W$ and $\lambda \in {\mathscr T}_\eta '$ , the terms $\frac {\mathbf {c}(w\lambda )}{\mathbf {c}^M(w\lambda )} J(w\lambda )$ are nonzero, and hence for each $\lambda \in {\mathscr T}_\eta '$ there exists $t=t_\lambda $ such that $I(t, \lambda )\neq 0$ .

4.6 Final arguments

We now finish the proof of Proposition 4.1.

By enlarging $\Omega $ if necessary, we can assume that $\overline {\Omega ^\circ } = \Omega $ . Let ${\mathscr H}_1,\dotsc , {\mathscr H}_N$ and ${\mathscr D}$ be as in Proposition 4.8. If necessary, we add in more hyperplanes such that the complement of their union in $i\mathfrak {a}^*$ consists of regular points only. Then the set of rational $\lambda $ in

$$ \begin{align*} \Omega\cap \left(i\mathfrak{a}^*\smallsetminus ({\mathscr H}_1\cup\dotsb\cup{\mathscr H}_N\cup{\mathscr D})\right) \end{align*} $$

is dense in $\Omega $ . Since $\lambda \mapsto \frac {1}{T}\int _0^T \lvert h_t(\lambda )\rvert ^2$ is a continuous function, and $\Omega $ is compact, Proposition 4.1 follows from the next result:

Lemma 4.9. For every rational

$$ \begin{align*} \lambda\in i\mathfrak{a}^*\smallsetminus ({\mathscr H}_1\cup\dotsb\cup{\mathscr H}_N\cup{\mathscr D}), \end{align*} $$

there exists $T_0>0$ depending on $\lambda $ such that for all $T>T_0$ , we have

$$ \begin{align*} \frac{1}{T}\int_0^T \lvert h_t(\lambda)\rvert^2 dt>0. \end{align*} $$

Proof. Recall the definition of ‘rational’ from Definition 4.3. Let $\tau _1\in {\mathbb R}_{>0}$ (depending on $\lambda $ ) and $\tau _n=n\tau _1$ for $n\in {\mathbb N}$ be as in the statement of Lemma 4.4. By Proposition 4.8 we can choose $t_0\ge 0$ such that $I(t_0,\lambda )=:a\neq 0$ . Write $t_n=t_0+\tau _n$ for $n\in {\mathbb N}$ . Note that by Lemma 4.4, we have $I(t+\tau _n,\lambda )=I(t,\lambda )$ for all $t\ge 0$ and all $n\in {\mathbb N}$ . Thus $I(t_n,\lambda )=a$ , and we can find a small $\epsilon>0$ such that for every $n\in {\mathbb N}$ and every $s\in (t_n-\epsilon , t_n+\epsilon )$ , we have

$$ \begin{align*} \lvert I(s,\lambda)\rvert\ge \lvert a\rvert/2>0. \end{align*} $$

From Lemma 4.6 we may choose $T_0>0$ so large that $\lvert h_t(\lambda ) - I(t,\lambda )\rvert <\lvert a\rvert /4$ for all $t\ge T_0$ . Let $N\in {\mathbb N}$ be such that $t_n-\epsilon \ge T_0$ for all $n\ge N$ . Then for every $n\ge N$ , we have

$$ \begin{align*} \lvert h_s(\lambda)\rvert\ge \lvert a\rvert/4 \end{align*} $$

for all $s\in (t_n-\epsilon , t_n+\epsilon )$ . We therefore get, for $T>2T_0 + \tau _1$ , that

$$ \begin{align*} \frac{1}{T}\int_0^T \lvert h_t(\lambda)\rvert^2 dt \gg \frac{1}{T}\sum_{n=N}^{\left(T-1-t_0\right)/\tau_1} \int_{-\epsilon}^\epsilon \left\lvert h_{t_n+s}(\lambda)\right\rvert^2 ds \gg_{\lambda} \frac{T-T_0}{T} \lvert a\rvert^2>0, \end{align*} $$

as we wanted to show.

5 Geometric side

The aim of this section is to prove Theorem 2.3. The essential feature of the geometric side is the ergodic properties of the sets $gE_t\cap E_t$ in the quotient $\Gamma \backslash G$ , as $t>0$ and $g\in G$ vary.

5.1 A general bound

In this section, we return to the general setting of §3, with $S=G/K$ and $\Gamma $ a given torsion-free cocompact lattice in G.

For $g,h\in G$ , let $N_R(g,h)$ denote the number of $\gamma \in \Gamma $ for which $d(g, \gamma h)\le R$ . This generalizes the notation $N_R(g)$ from equation (26) when $g=h$ . Let $N_\Gamma (R)=\sup _{\left (g,h\right )}N_R(g,h)$ .

Lemma 5.1. There exists $c>0$ , independent of R and $\Gamma $ , such that

$$ \begin{align*} N_\Gamma(R)\ll \operatorname{InjRad}(\Gamma\backslash S)^{-\dim S} e^{c R}. \end{align*} $$

Proof. Note that $N_R(g,h) = N_R(g,\gamma h)$ for all $\gamma \in \Gamma $ , so that we can assume that h is such that $d(g,h)=\min _{\gamma \in \Gamma } d(g, \gamma h)$ . If $d(g,h)>R$ , then $N_R(g,h)=0$ , so that we can assume $d(g, h)\le R$ .

Now if $\gamma \in \Gamma $ is such that $d(g,\gamma h)\le R$ , then

$$ \begin{align*} d(h,\gamma h) \le d(g,h)+ d(g,\gamma h) \le 2R, \end{align*} $$

so that $N_R(g,h)\le N_{2R}(h,h)$ . Applying [1, Lemma 6.18] to $N_{2R}(h,h)$ gives the assertion of the lemma.

The following lemma is an adaptation of [24, Lemma 5.1] to our setting. Recall the geodesic balls $B_t$ , first introduced in equation (13) in the setting of $G=\operatorname {SL}_d({\mathbb R})$ , but more generally defined for all G using the same notation.

Lemma 5.2. There is $c>0$ , depending only on G, such that the following holds. Let $R>0$ , and let $K\in C(G\times G)$ be invariant under the diagonal action of $\Gamma $ and satisfy $\left \{g^{-1}h: (g,h)\in \mathrm {supp}(K)\right \} \subset B_R$ . Let $\bf {K}$ be an integral operator on $L^2(\Gamma \backslash G)$ with kernel K. Then

$$ \begin{align*} \lVert\textbf{ K}\rVert_{\textrm{ HS}}^2\leq \int_{\Gamma\backslash G} \int_{G}\lvert K(g,h)\rvert^2 dg dh+O\left(\frac{e^{cR}}{\operatorname{InjRad}(\Gamma\backslash S)^{\dim S}}\operatorname{vol}\left((\Gamma\backslash G)_{\leq 2R}\right)\lVert K\rVert_\infty^2\right). \end{align*} $$

Proof. By definition, we have

$$ \begin{align*} \lVert \textbf{ K}\rVert_{\textrm{ HS}}^2=\int_{\Gamma\backslash G}\int_{\Gamma\backslash G} \left\lvert\sum_{\gamma\in \Gamma}K(g,\gamma h)\right\rvert^2 dg dh. \end{align*} $$

Let $D\subset G$ be a fundamental domain for $\Gamma $ . We decompose D into its $2R$ -thin and -thick parts $D=D_{\leq 2R}\cup D_{>2R}$ , where, similar to equation (3), we have put

$$ \begin{align*} D_{\leq 2R}=\{x\in D: \operatorname{InjRad}_\Gamma (x)\leq 2R\}. \end{align*} $$

From the support condition on K, it follows that if $K(g,\gamma h)\neq 0$ then $d(g,\gamma h)\leq R$ .

Now suppose that $h\in D_{>2R}$ and $\gamma ,\delta \in \Gamma $ , $\gamma \neq \delta $ , are such that $d(g,\gamma h)\le R$ and $d(g, \delta h)\le R$ . Then $d(\gamma h, \delta h)\le d(g, \gamma h) + d(g, \delta h)\le 2R$ , but $d(\gamma h, \delta h)>2R$ , since $h\in D_{>2R}$ , leading to a contradiction. Hence there exists at most one $\gamma \in \Gamma $ with $K(g,\gamma h)\neq 0$ . In this way,

$$ \begin{align*} \left\lvert\sum_{\gamma\in\Gamma} K(g,\gamma h)\right\rvert^2=\sum_{\gamma\in\Gamma} \lvert K(g,\gamma h)\rvert^2, \end{align*} $$

so that

$$ \begin{align*} \int_{D_{> 2R}}\int_D \left\lvert\sum_{\gamma\in\Gamma} K(g,\gamma h)\right\rvert^2 dg dh&=\int_{D_{> 2R}}\int_D \sum_{\gamma\in\Gamma} \lvert K(g,\gamma h)\rvert^2 dgdh\\ &\leq \int_D\int_D \sum_{\gamma\in\Gamma} \lvert K(g,\gamma h)\rvert^2 dg dh\\ &= \int_G\int_D \lvert K(g,h)\rvert^2 dg dh\\ & =\int_{\Gamma\backslash G}\int_G \lvert K(g, h)\rvert^2dhdg. \end{align*} $$

To deal with the $2R$ -thin part, we use Cauchy–Schwarz and the support condition on K to obtain

$$ \begin{align*} \left\lvert\sum_{\gamma\in\Gamma} K(g,\gamma h)\right\rvert^2 \leq N_\Gamma(R)\sum_{\gamma\in\Gamma} \lvert K(g,\gamma h)\rvert^2. \end{align*} $$

The factor $N_\Gamma (R)$ is bounded by Lemma 5.1. Furthermore,

$$ \begin{align*} & \int_{(\Gamma\backslash G)_{\leq 2R}}\int_{\Gamma\backslash G} \sum_{\gamma\in\Gamma} \lvert K(g,\gamma h)\rvert^2 dg dh \\ &\leq \sup_{g\in \Gamma \backslash G, h\in G}\lvert K(g, h)\rvert^2\int_{(\Gamma\backslash G)_{\leq 2R}}\int_{\Gamma\backslash G} \sum_{\gamma\in\Gamma} \textbf{ 1}_{B_R}\left(g^{-1}\gamma h\right) dg dh. \end{align*} $$

By unfolding and changing variables, we have

$$ \begin{align*} \int_{\Gamma\backslash G} \sum_{\gamma\in\Gamma} \textbf{ 1}_{B_R}\left(g^{-1}\gamma h\right) dg= \int_G \textbf{ 1}_{B_R}\left(g^{-1} h\right) dg=m_G(B_R)=c_C\int_{\mathscr{B}_R^+}J(X)dX, \end{align*} $$

the last equality coming from the Cartan measure decomposition (21). The Jacobian factor satisfies $J(X)\ll e^{2\left \langle \rho ,X\right \rangle }\ll e^{2R\lVert \rho \rVert _2}$ for $X\in \mathscr {B}^+$ , using equation (37). From this one deduces ² that $m_G(B_R)\ll R^r e^{2R\lVert \rho \rVert _2}$ , which is enough to complete the proof.

5.2 Description of the kernel of $\mathbf {A}(\tau )$

We return to the setting of $G=\operatorname {SL}_d({\mathbb R})$ . We would like to apply Lemma 5.2 to our operator $\mathbf {A}(\tau )$ on $L^2(\Gamma \backslash G)$ from equation (10). For this we shall need a description of its kernel, which we shall denote by $A(\tau )$ ; it is a continuous function on $\Gamma \backslash (G\times G)$ .

Lemma 5.3. The kernel of $\mathbf {A}(\tau )$ is

(42) $$ \begin{align} A(\tau)(g,h)=\frac{1}{\tau}\int_0^\tau \frac{1}{m_G(E_t)}\int_{gE_t\cap hE_t} a(x) dxdt. \end{align} $$

Proof. By definition, the action of $U_taU_t$ on $f\in L^2(\Gamma \backslash G)$ is given by

$$ \begin{align*} (U_taU_t)f(g)=\frac{1}{m_G(E_t)}\int_{E_t}a(gh_1)\int_{E_t} f(gh_1 h_2)dh_2 dh_1. \end{align*} $$

We set $x=gh_1$ and $h=xh_2$ , apply Fubini, and use $E_t^{-1}=E_t$ to obtain

$$ \begin{align*} (U_taU_t)f(g)&=\frac{1}{m_G(E_t)}\int_{gE_t} a(x)\int_{xE_t} f(h)dh dx\\ &=\int_G\left( \frac{1}{m_G(E_t)}\int_{gE_t\cap hE_t} a(x) dx\right)f(h)dh. \end{align*} $$

Averaging over t yields the claim.

5.3 Support and $\sup $ of $\mathbf {A}(\tau )$

In order to bound the second term in Lemma 5.2, we shall need to estimate the support and the supremum of the kernel function $A(\tau )$ . This is accomplished in the next result, which is the higher-rank generalization of [2, Lemma 26]. We recall the norm $\lvert \cdot \rvert $ on $G=\operatorname {SL}_d({\mathbb R})$ introduced in §2.2 and defining the sets $E_t$ .

Proposition 5.4. There exists a constant $b>0$ depending only on d such that $gE_t\cap hE_t$ is empty unless $\left \lvert h^{-1}g\right \rvert \le 2t+b$ . In particular, $A(\tau )(g, h)=0$ unless $\left \lvert h^{-1}g\right \rvert \le 2\tau +b$ . Further,

$$ \begin{align*} \sup_{g, h\in G}\lvert A(\tau)(g, h)\rvert^2\le \lVert a\rVert_\infty^2. \end{align*} $$

Proof. We begin by establishing the following:

Claim: For $x\in \operatorname {SL}_d({\mathbb R})$ , we have $\left \lvert x_{ij}\right \rvert \leq e^{\lvert x\rvert }$ . Moreover, there exists a constant $c>0$ depending only on d, such that for every $x=\left (x_{ij}\right )\in \operatorname {SL}_d({\mathbb R})$ and every $i\in \{1,\dotsc ,d\}$ there exists $j\in \{1,\dotsc , d\}$ with $\left \lvert x_{ij}\right \rvert \ge c e^{-\lvert x\rvert }$ .

Proof of claim: Write $x= k e^Z l$ , with $k=\left (k_{ij}\right )$ , $l = \left (l_{ij}\right ) \in K=\textrm {SO}(d)$ , and $Z\in \overline {\mathfrak {a}^+}$ . Let $\kappa _i=(k_{i1},\dotsc , k_{id})$ and $\lambda _j= \left (l_{1j},\dotsc , l_{dj}\right )\in {\mathbb R}^d$ , for $i,j=1,\dotsc ,d$ . Let $\langle \cdot , \cdot \rangle $ denote the standard inner product on ${\mathbb R}^d$ . Then $\kappa _1,\dotsc ,\kappa _d$ and $\lambda _1,\dotsc , \lambda _d$ are both orthonormal bases of ${\mathbb R}^d$ , and $x_{ij}=\left \langle \kappa _i,\lambda _j e^Z\right \rangle $ . In particular, by Cauchy and Schwarz we have

$$ \begin{align*} \left\lvert x_{ij}\right\rvert=\left\lvert\left\langle\kappa_i,\lambda_j e^Z\right\rangle\right\rvert\leq \left\lVert\lambda_j e^Z\right\rVert^{1/2}\leq e^{\lVert Z\rVert_\infty}=e^{\lvert x\rvert}, \end{align*} $$

proving the first statement.

For the second statement, we begin by observing that for every $j=1,\dotsc ,d$ we have

$$ \begin{align*} \left\langle\kappa_i, \kappa_i e^Z\right\rangle=\sum_{j=1}^de^{Z_j}k_{ij}^2\geq \left(\min_j e^{Z_j}\right)\lVert\kappa_i\rVert^2=\min_j e^{Z_j}\ge e^{-\lVert Z\rVert_\infty}=e^{-\lvert x\rvert}. \end{align*} $$

Writing $\kappa _i$ as a linear combination $\kappa _i= a_1\lambda _1+\dotsb + a_d \lambda _d$ , we obtain

$$ \begin{align*} e^{-\lvert x\rvert} \le \left\langle\kappa_i, \kappa_i e^Z\right\rangle \le \sum_{j=1}^d \left\lvert a_j\right\rvert \left\lvert\left\langle\kappa_i, \lambda_j e^Z\right\rangle\right\rvert. \end{align*} $$

Note that the $a_i$ are uniformly bounded, since k and l are contained in a compact set. Hence there exist $c>0$ and $j\in \{1,\dotsc , d\}$ such that

$$ \begin{align*} \left\lvert x_{ij}\right\rvert=\left\lvert\left\langle\kappa_i, \lambda_j e^Z\right\rangle \right\rvert \ge c e^{-\lvert x\rvert}, \end{align*} $$

as claimed.

Continuing with the proof of the proposition, suppose that there exists $y\in h^{-1}gE_t\cap E_t$ . We can assume without loss that $h^{-1} g= e^Y$ for some suitable $Y=(Y_1,\dotsc , Y_n)\in \overline {\mathfrak {a}^+}$ .

Since $y\in E_t$ , the first part of the claim shows that $\left \lvert y_{ij}\right \rvert \le e^t$ for all $i,j=1,\dotsc , d$ . Moreover, since $y\in e^YE_t$ , we can write $y=e^Y x $ for some $x\in E_t$ . Then $y_{ij} = e^{Y_i} x_{ij}$ , and the second part of the claim shows that for every i there is a $j_i$ such that $\left \lvert x_{i j_i}\right \rvert \ge ce^{-t}$ . Putting this together, we obtain $e^t \ge c e^{Y_i} e^{-t}$ , so that $e^{Y_i}\le c^{-1} e^{2t}$ .

By similar arguments, we also know that $\left \lvert x_{ij}\right \rvert \le e^t$ for all $i,j$ , and that for every i there exists $j_i'$ with $e^{Y_i}\left \lvert x_{i j_i'}\right \rvert =\left \lvert y_{i j_i'}\right \rvert \ge ce^{-t}$ – that is, $e^Y_i \ge c e^{-2t}$ .

These two inequalities together imply the first assertion of the corollary, and hence also the assertion of the support of $A(\tau )$ . The last assertion, on the $\sup $ of $A(\tau )$ , is a direct consequence of the definition (42).

Recall from §2.5(iii) that $E_t\subset B_{t\left \lVert X^0\right \rVert _2}$ . It then follows from Proposition 5.4 that we can take $R=\left \lVert X^0\right \rVert (2\tau +b)$ in Lemma 5.2. Inserting this and the $\sup $ norm bound of Proposition 5.4 into Lemma 5.2, we obtain the second term in Theorem 2.3.

5.4 First term

To establish Theorem 2.3, it remains to estimate the first term in Lemma 5.2, for the operator $\mathbf {A}(\tau )$ with kernel $A(\tau )$ .

For a measurable set $E\subseteq G$ , we write $\rho _{\Gamma \backslash G}(E)$ for the action by convolution of the normalized characteristic function of E on $L^2(\Gamma \backslash G)$ . Thus

$$ \begin{align*} \rho_{\Gamma\backslash G}(E)f(x)=\frac{1}{m_G(E)}\int_E f(xg)dg. \end{align*} $$

We shall in particular be interested in this operator for sets of the form $gE_t\cap E_t$ .

We begin with the following elementary upper bound on the quantity

$$ \begin{align*} \mathscr{A}(\tau)=\int_{\Gamma\backslash G} \int_G\lvert A(\tau)(g,h)\rvert^2 dg dh. \end{align*} $$

Lemma 5.5. For all $\tau> 0$ , we have

$$ \begin{align*} \mathscr{A}(\tau)\leq \frac{1}{\tau^2}\int_{E_{2\tau+b}}\left(\int_{\max\left\{0,\left(\lvert g\rvert-b\right)/2\right\}}^\tau \frac{m_G(gE_t\cap E_t)}{m_G(E_t)}\left\lVert \rho_{\Gamma\backslash G}(gE_t\cap E_t)a\right\rVert_{L^2(\Gamma\backslash G)}dt\right)^2dg, \end{align*} $$

where $b>0$ is as in Proposition 5.4.

Proof. From the definition (42) of $A(\tau )$ , we see that

$$ \begin{align*} \mathscr{A}(\tau)=\frac{1}{\tau^2}\int_{\Gamma\backslash G}\int_G\left\lvert\int_0^\tau \frac{1}{m_G(E_t)}\int_{gE_t\cap hE_t}a(x)dx dt\right\rvert^2 dgdh. \end{align*} $$

Changing variables $x\mapsto hx$ and $g\mapsto h^{-1}g$ , this is

$$ \begin{align*} \frac{1}{\tau^2}\int_{\Gamma\backslash G}\int_G\left\lvert\int_0^\tau \frac{1}{m_G(E_t)}\int_{gE_t\cap E_t}a(hx)dx dt\right\rvert^2 dg dh. \end{align*} $$

In view of Proposition 5.4, the integral over $g\in G$ may be truncated at $\lvert g\rvert \leq 2\tau +b$ , and the lower range of the t integral may be truncated at $\max \{0,\lvert g\rvert -b)/2\}$ . Furthermore,

$$ \begin{align*} \frac{1}{m_G(E_t)}\int_{gE_t\cap E_t}a(hx)dx&=\frac{m_G(gE_t\cap E_t)}{m_G(E_t)}\frac{1}{m_G(gE_t\cap E_t)}\int_{gE_t\cap E_t}a(hx)dx\\ &=\frac{m_G(gE_t\cap E_t)}{m_G(E_t)}\left(\rho_{\Gamma\backslash G}(gE_t\cap E_t)a\right)(h). \end{align*} $$

We conclude the proof by an application of the Minkowski integral inequality.

We briefly return to the level of generality of §3, and let G denote a connected noncompact simple Lie group with finite center. Let $(\pi ,V_\pi )$ be a unitary representation of G. We recall that the integrability exponent $q(\pi )$ of $\pi $ is defined as

$$ \begin{align*} q(\pi)=\inf\{ q>0: \langle \pi(g)v_1,v_2\rangle\in L^q(G) \text{ for}\ v_1,v_2\ \text{in a dense subspace of } V_\pi\}. \end{align*} $$

A classical result of Borel and Wallach [6], Cowling [11], and Howe and Moore [18] states that $\pi $ has a spectral gap precisely when $q(\pi )<\infty $ ; we can effectively take this as our definition of the spectral gap in our setting.

Nevo [29] (see also [15, Theorem 4.1]) has proved a mean ergodic theorem for measure-preserving actions G on a probability space $(X,\mu )$ whose associated unitary representation $L^2_0(X,\mu )$ has a spectral gap. We shall be interested in the action of G on $X=\Gamma \backslash G$ by right translation, where $\Gamma <G$ is a lattice. The associated unitary representation is then $\rho _{\Gamma \backslash G}^0$ , the restriction of the right-regular representation $\rho _{\Gamma \backslash G}$ to the subspace $L^2_0(\Gamma \backslash G)$ of $L^2(\Gamma \backslash G)$ consisting of functions f with $\int _{\Gamma \backslash G} f=0$ . In this setting, they show the following result:

Proposition 5.6 Nevo

Let G be a connected noncompact simple Lie group with finite center. Let $\Gamma <G$ be a lattice. Then there are constants $\theta ,C>0$ , depending on the integrability exponents of $\rho _{\Gamma \backslash G}^0$ , such that for any measurable $E\subseteq G$ of finite measure, we have

$$ \begin{align*} \left\lVert\rho_{\Gamma\backslash G}^0(E)\right\rVert_2 \leq C m_G(E)^{-\theta}. \end{align*} $$

A famous result of Kazhdan [20] states that when G furthermore has rank at least $2$ ,

$$ \begin{align*} \sup_{\pi: \pi^G=0} q(\pi)<\infty, \end{align*} $$

the supremum running over all unitary representations of G having no G-invariant vectors. In this case, the constants of Proposition 5.6 can be taken independently of $\Gamma $ . In particular, this is true of $G=\operatorname {SL}_d({\mathbb R})$ for $d\geq 3$ .

We thus return to $G=\operatorname {SL}_d({\mathbb R})$ , for $d\geq 3$ , and again require $\Gamma <\operatorname {SL}_d({\mathbb R})$ to be a uniform lattice. Recall that $a\in L^2_0(\Gamma \backslash G)$ . We can therefore apply Proposition 5.6 to estimate the quantity

$$ \begin{align*} \left\lVert\rho_{\Gamma\backslash G}(gE_t\cap E_t)a\right\rVert_{L^2(\Gamma\backslash G)}= \left\lVert\rho_{\Gamma\backslash G}^0(gE_t\cap E_t)a\right\rVert_{L^2_0(\Gamma\backslash G)} \end{align*} $$

appearing in Lemma 5.5. Hence there is $\theta>0$ , depending only on d, such that

$$ \begin{align*} \mathscr{A}(\tau) \ll \frac{\lVert a\rVert_{L^2(\Gamma\backslash G)}^2}{\tau^2}\int_{E_{2\tau+b}}\left(\int_{\max\left\{0,\left(\lvert g\rvert-b\right)/2\right\}}^\tau \frac{m_G(gE_t\cap E_t)^{1-\theta}}{m_G(E_t)} dt\right)^2dg. \end{align*} $$

To conclude the proof of Theorem 2.3, it suffices to show the following estimate:

Proposition 5.7. Let $g\in G$ . Then for $\tau \gg 1$ we have

$$ \begin{align*} \int_{E_{2\tau+b}}\left(\int_{\max\left\{0,\left(\lvert g\rvert-b\right)/2\right\}}^\tau \frac{m_G(gE_t\cap E_t)^{1-\theta}}{m_G(E_t)} dt\right)^2dg\ll \tau, \end{align*} $$

the implied constant depending on d.

The main point in the proof of this result is an estimate on the intersection volumes $m_G(gE_t\cap E_t)$ , proved in the next subsection.

5.5 Intersection volumes

We seek to prove the following estimate on the intersection volumes $m_G\left (e^YE_t\cap E_t\right )$ . See §2.5 for a discussion of some geometric aspects.

Proposition 5.8. Set $Y\in \mathfrak {a}^+$ and $t>0$ . Then

(43) $$ \begin{align} m_G\left(e^YE_t\cap E_t \right)\ll e^{-\left\langle \rho, Y\right\rangle}m_G(E_t). \end{align} $$

Proof. Recall the definition of the Frobenius norm $\lVert \cdot \rVert $ from §2.5(iv) and the Frobenius ball $\boldsymbol {E}_t$ from equation (19). We claim that

(44) $$ \begin{align} E_t\subseteq \boldsymbol{E}_{t+\log \sqrt{d}}\subseteq E_{t+\log \sqrt{d}}. \end{align} $$

Note that $\lVert \cdot \rVert $ is bi-K-invariant, so that if $g=k_1 e^X k_2$ , where $X=(X_1,\dotsc ,X_d)\in \mathfrak {a}$ , then $\lVert g\rVert ^2=e^{2X_1}+\dotsb +e^{2X_d}$ . Thus we may write $\boldsymbol {E}_t=K\exp (\boldsymbol {P}_t) K$ , where

$$ \begin{align*} \boldsymbol{P}_t=\left\{X=(X_1,\dotsc ,X_d)\in\mathfrak{a}: \max\left\{e^{2X_1}+\dotsb +e^{2X_d}, e^{-2X_1}+\dotsb +e^{-2X_d}\right\}\leq e^{2t}\right\}. \end{align*} $$

We therefore wish to establish formula (44) for the sets $\boldsymbol {P}_t$ and ${\mathscr P}_t=t{\mathscr P}$ , where ${\mathscr P}$ is defined in equation (14):

1. ⊳ If $\lvert X_i\rvert \leq t$ for all i, we have $\max \left \{e^{2X_1}+\dotsb +e^{2X_d}, e^{-2X_1}+\dotsb +e^{-2X_d}\right \}\leq de^{2t}$ , so that ${\mathscr P}_t\subseteq \boldsymbol {P}_{t+\log \sqrt {d}}$ .

2. ⊲ In the other direction, if $\max \left \{e^{2X_1}+\dotsb +e^{2X_d}, e^{-2X_1}+\dotsb +e^{-2X_d}\right \}\leq e^{2t}$ , then $\lvert X_i\rvert \leq t$ for all i, so that $\boldsymbol {P}_t\subseteq {\mathscr P}_t$ .

It is now enough to prove formula (43) with $E_t$ replaced by $\boldsymbol {E}_t$ . Indeed, suppose that

(45) $$ \begin{align} m_G\left(e^Y\boldsymbol{E}_t\cap \boldsymbol{E}_t\right)\ll e^{-\left\langle\rho,Y\right\rangle}m_G(\boldsymbol{E}_t) \end{align} $$

for any $Y\in \mathfrak {a}$ . We use formula (44) to bound $m_G(e^YE_t\cap E_t)$ by $m_G(e^Y\boldsymbol {E}_{t+\log \sqrt {d}}\cap \boldsymbol {E}_{t+\log \sqrt {d}})$ . Inserting the estimate (45) and the inclusions (44) again, we find

$$ \begin{align*} m_G\left(e^Y\boldsymbol{E}_{t+\log\sqrt{d}}\cap \boldsymbol{E}_{t+\log\sqrt{d}}\right) \ll e^{-\left\langle\rho,Y\right\rangle}m_G\left(\boldsymbol{E}_{t+\log\sqrt{d}}\right)\leq e^{-\left\langle\rho,Y\right\rangle}m_G\left(E_{t+\log\sqrt{d}}\right). \end{align*} $$

Since $m_G\left (E_{t+\log \sqrt {d}}\right )\asymp m_G(E_t)$ , by Corollary A.4, we have reduced formula (43) to formula (45).

We prove formula (45) using the Iwasawa decomposition $G=ANK$ from §3.1. Let $g=auk$ so that $\lVert g\rVert =\lVert au\rVert $ and $\left \lVert g^{-1}\right \rVert =\left \lVert u^{-1}a^{-1}\right \rVert $ . Now $\lVert au\rVert ^2$ just equals the sum of all the squares of the matrix entries of $au$ , so that $\lVert au\rVert =\lVert g\rVert \le e^t$ immediately implies that $\lVert a\rVert \le e^t$ and $\lVert au-a\rVert \le e^t$ . Similarly, $\left \lVert g^{-1}\right \rVert \le e^t$ implies $\left \lVert a^{-1}\right \rVert \le e^t$ and $\left \lVert u^{-1} a^{-1}- a^{-1}\right \rVert \le e^t$ . Thus we obtain the inclusion

$$ \begin{align*} \boldsymbol{E}_t\subseteq \left\{g=auk\in G: \lVert a\rVert, \left\lVert a^{-1}\right\rVert, \lVert au-a\rVert, \left\lVert u^{-1}a^{-1}-a^{-1}\right\rVert \leq e^t\right\}. \end{align*} $$

Applying the same argument to $e^Yg$ , for $Y\in \mathfrak {a}$ , and dropping the last two conditions,

$$ \begin{align*} e^Y\boldsymbol{E}_t\subseteq \left\{g=auk\in G:\left\lVert e^{-Y}a\right\rVert, \left\lVert e^Ya^{-1}\right\rVert \leq e^t\right\}. \end{align*} $$

We conclude that $m_G\left (e^Y\boldsymbol {E}_t\cap \boldsymbol {E}_t\right )\leq I(Y,t)$ , where

$$ \begin{align*} I(Y,t)= \int\limits_{\substack{\left\lVert e^{\pm A}\right\rVert\leq e^t\\ \left\lVert e^{\pm (A-Y)}\right\rVert\leq e^t}} \int\limits_{\substack{\left\lVert e^Au-e^A\right\rVert\leq e^t\\ \left\lVert u^{-1}e^{-A}-e^{-A}\right\rVert\leq e^t}}du dA. \end{align*} $$

Changing variables $u\mapsto e^Au-e^A+I$ in the U-integration, we obtain

$$ \begin{align*} I(Y,t)= I_U(t) \int\limits_{\substack{\left\lVert e^{\pm A}\right\rVert\leq e^t\\ \left\lVert e^{\pm (A-Y)}\right\rVert\leq e^t}} e^{-\left\langle\rho,A\right\rangle}dA, \end{align*} $$

where $I_U(t)=m_U\left (u\in U: \lVert u-I\rVert , \left \lVert u^{-1}-I\right \rVert \leq e^t\right )$ . Changing variables $A\mapsto A-Y$ in the A-integration and dropping a condition, we obtain

$$ \begin{align*} \int\limits_{\substack{\left\lVert e^{\pm A}\right\rVert\leq e^t\\ \left\lVert e^{\pm (A-Y)}\right\rVert\leq e^t}} e^{-\left\langle\rho,A\right\rangle}dA&= e^{-\left\langle \rho,Y\right\rangle} \int\limits_{\substack{\left\lVert e^{\pm (A+Y)}\right\rVert\leq e^t\\ \left\lVert e^{\pm A}\right\rVert\leq e^t}} e^{-\left\langle\rho,A\right\rangle}dA\\ &\leq e^{-\left\langle \rho,Y\right\rangle}\int\limits_{\left\lVert e^{\pm A}\right\rVert\leq e^t} e^{-\left\langle\rho,A\right\rangle}dA. \end{align*} $$

We write this last integral as $I_A(t)$ . Using equation (20), this establishes

(46) $$ \begin{align} m_G\left(e^Y\boldsymbol{E}_t\cap \boldsymbol{E}_t\right)\leq c_I e^{-\left\langle \rho,Y\right\rangle}I_A(t)I_U(t). \end{align} $$

We now go in reverse. For $A\in \mathfrak {a}$ , we change variables $u\mapsto e^{-A}u-e^{-A}+I$ to obtain

$$ \begin{align*} e^{-\left\langle \rho,A\right\rangle}I_U(t)=m_U\left(u\in U: \left\lVert e^Au-e^A\right\rVert, \left\lVert u^{-1}e^{-A}-e^{-A}\right\rVert\leq e^t\right), \end{align*} $$

so that

$$ \begin{align*} I_A(t)I_U(t)=\int\limits_{\left\lVert e^{\pm A}\right\rVert\leq e^t} \int\limits_{\left\lVert\left(e^Au\right)^{\pm 1}-e^{\pm A}\right\rVert\leq e^t} du dA. \end{align*} $$

Again by the triangle inequality and equation (20), this gives

$$ \begin{align*} I_A(t)I_U(t)\leq \iint\limits_{\left\lVert(au)^{\pm 1}\right\rVert\leq 2e^t} da du=c_I^{-1}m_G\left(\boldsymbol{E}_{t+\log 2}\right)\asymp m_G(\boldsymbol{E}_t). \end{align*} $$

Combining this last inequality with formula (46) yields formula (45).

5.6 Proof of Proposition 5.7

From Proposition 5.8 and Lemma A.4 we have, for $Y\in \mathfrak {a}^+$ ,

$$ \begin{align*} \frac{m_G\left(e^YE_t\cap E_t\right)^{1-\theta}}{m_G(E_t)}\ll e^{-(1-\theta)\left\langle \rho, Y\right\rangle}m_G(E_t)^{-\theta} \ll e^{-(1-\theta)\left\langle \rho, Y\right\rangle}e^{-t\theta \left\langle 2\rho, X^0\right\rangle}, \end{align*} $$

so that, using equation (15),

$$ \begin{align*} &\int_{E_{2\tau+b}} \left(\int_{\max\left\{0,(\lvert g\rvert-b)/2\right\}}^\tau \frac{m_G(gE_t\cap E_t)^{1-\theta}}{m_G(E_t)} dt\right)^2dg\\ &\qquad\ll\int_{\mathscr{P}_{2\tau+b}^+}\left(\int_{\max\left\{0,\left(\lVert Y\rVert_\infty-b\right)/2\right\}}^\tau e^{-t\theta\left\langle 2\rho,X^0\right\rangle} dt\right)^2e^{-(1-\theta)\left\langle 2\rho, Y\right\rangle}J(Y)dY. \end{align*} $$

Inserting $J(X)\ll e^{2\left \langle \rho ,X\right \rangle }$ for $X\in \mathfrak {a}^+$ from equation (37) and evaluating the t-integral, we majorize the foregoing expression by

$$ \begin{align*} \int_{\mathscr{P}_{2\tau+b}^+} e^{-\max\left\{0,\lVert Y\rVert_\infty-b\right\}\theta\left\langle 2\rho,X^0\right\rangle} e^{\theta\left\langle 2\rho, Y\right\rangle}dY. \end{align*} $$

We break up the last integral as $I_1+I_2$ , according to $\mathfrak {a}_1^+=\mathscr {P}_{b}^+$ and

$$ \begin{align*} \mathfrak{a}_2^+(\tau)&=\left\{Y\in\mathfrak{a}^+: b\leq \lVert Y\rVert_\infty\leq 2\tau+b\right\}. \end{align*} $$

Since $\mathfrak {a}^+_1$ is independent of $\tau $ (as is the integrand), we have $I_1\ll 1$ . Next, we have

$$ \begin{align*} I_2\ll \int_{\mathfrak{a}_2^+(\tau)} e^{2\theta\left(\left\langle \rho,Y\right\rangle -\lVert Y\rVert_\infty\left\langle \rho,X^0\right\rangle\right)}dY. \end{align*} $$

We will drop several of the conditions on Y in the course of the proof to simplify the integral.

Write $\tau ' = 2\tau +b$ . We distinguish the cases of d being even and odd. Define

(47) $$ \begin{align} s =\begin{cases} d/2 &\text{if}\ d\ \text{is even},\\ (d+1)/2 &\text{if}\ d\ \text{is odd}. \end{cases} \end{align} $$

We first assume that $d\geq 3$ is odd. We can write Y in the integral as

$$ \begin{align*} Y= (a_1,\dotsc, a_s, -a_1-\dotsb -a_s + b_1+\dotsb+ b_s ,-b_s,\dotsc, -b_1), \end{align*} $$

with $\tau '\ge a_1\ge \dotsb \ge a_s\ge 0$ and $\tau '\ge b_1\ge \dotsb \ge b_s\ge 0$ . Then $\langle \rho , Y\rangle = \sum _{i=1}^s (s-i) (a_i+b_i)$ , and using Lemma A.1, $\left \langle \rho , X^0 \right \rangle = 2 A$ with $A=\sum _{i=1}^s(s-i)$ . Moreover, $\lVert Y\rVert _\infty = \max \{a_1, b_1\}$ . Hence

$$ \begin{align*} I_2(\tau)\ll \int\dotsi\int e^{2\theta\left( \sum_{i=1}^s (s-i) \left(a_i+b_i\right)- 2A\max\left\{a_1,b_1\right\}\right)}, \end{align*} $$

where the integral runs over

$$ \begin{align*} \left\{ a_1,\dotsc, a_s, b_1,\dotsc, b_s\bigg\vert \begin{aligned} \tau'\ge a_1\ge0,\; a_1\ge a_2\ge0,\; & \dotsc,\; a_{s-1}\ge a_s\ge0,\\ \tau'\ge b_1\ge0,\; b_1\ge b_2\ge0,\; & \dotsc,\; b_{s-1}\ge b_s\ge0. \end{aligned} \right\} \end{align*} $$

We now integrate first $a_s$ and $b_s$ , and continue successively with $a_{s-1},\dotsc , a_2$ and $b_{s-1},\dotsc , b_2$ , until only $a_1$ and $b_1$ remain to be integrated. In this way we obtain

$$ \begin{align*} I_2(\tau)\ll\iint_0^{\tau'} e^{2\theta A\left(a_1+b_1 - 2\max\left\{a_1, b_1\right\}\right)} da_1 db_1 &= \iint_0^{\tau'} e^{- 2\theta A\left\lvert a_1-b_1\right\rvert} da_1 db_1 \\ & = 2\tau' \int_0^{\tau'} e^{- 2\theta Ax}\, dx \ll \tau. \end{align*} $$

If d is even, then $d\ge 4$ and $s\ge 2$ . We can write $Y= (a_1,\dotsc , a_s, -b_s, \dotsc , -b_1)$ , with $\tau '\ge a_1\ge \dotsb \ge a_s\ge 0$ , $\tau '\ge b_1\ge \dotsb \ge b_s\ge 0$ , and

(48) $$ \begin{align} a_1+\dotsb + a_s = b_1+\dotsb + b_s. \end{align} $$

Then $\lVert Y\rVert _\infty = \max \{a_1, b_1\}$ and

$$ \begin{align*} \langle \rho, Y\rangle = \sum_{i=1}^s (s-i+1/2)(a_i+b_i)= \sum_{i=1}^s (s-i+1)a_i + \sum_{i=1}^{s-1} (s-i)b_i, \end{align*} $$

where we have used equation (48) to replace $b_s$ . Furthermore, using Lemma A.1, we have $\left \langle \rho , X^0\right \rangle =A+ B$ , where we have put

$$ \begin{align*} A=\sum_{i=1}^s (s-i+1)\quad\text{and}\quad B= \sum_{i=1}^s (s-i). \end{align*} $$

Extending the domain of integration if necessary, we obtain

$$ \begin{align*} I_2(\tau)\leq \int\dotsi\int e^{2\theta\left(\sum_{i=1}^s (s-i+1)a_i + \sum_{i=1}^{s-1} (s-i)b_i - (A+B)\max\left\{a_1,b_1\right\}\right)}, \end{align*} $$

where the integral runs over

$$ \begin{align*} \left\{ a_1,\dotsc, a_s, b_1,\dotsc, b_{s-1}\bigg\vert \begin{aligned} \tau'\ge a_1\ge0,\; a_1\ge a_2\ge0,\; & \dotsc,\; a_{s-1} \ge a_s\ge0,\\ \tau'\ge b_1\ge0,\; b_1\ge b_2\ge0,\; & \dotsc,\; b_{s-2}\ge b_{s-1}\ge0 \end{aligned} \right\}. \end{align*} $$

As before, we successively integrate over $a_s,a_{s-1},\dotsc , a_2$ and $b_{s-1}, b_{s-2},\dotsc , b_2$ , obtaining

$$ \begin{align*} I_2(\tau)\ll\iint_{0}^{\tau'} e^{2\theta \left( Aa_1 + Bb_1 - (A+B)\max\left\{a_1, b_1\right\}\right)} da_1 db_1 \le \iint_{0}^{\tau'} e^{-2\theta B \left\lvert a_1-b_1\right\rvert} da_1 db_1 \ll \tau, \end{align*} $$

as desired.□