Quantum Ergodicity for compact quotients of SL d (R)/SO(d) in the Benjamini-Schramm limit
QQUANTUM ERGODICITY FOR COMPACT QUOTIENTS OF SL d ( R ) / SO ( d ) IN THE BENJAMINI–SCHRAMM LIMIT
FARRELL BRUMLEY AND JASMIN MATZ
Abstract.
We study the limiting behavior of Maass forms on sequences of large volumecompact quotients of SL d ( R ) / SO( d ), d ≥
3, whose spectral parameter stays in a fixedwindow. We prove a form of Quantum Ergodicity in this level aspect which extends resultsof Le Masson and Sahlsten to the higher rank case.
Contents
1. Introduction 12. Outline of proof and reduction steps 53. Weyl type law 94. Spectral side 155. Geometric side 21Appendix A. Cones and volumes 29Appendix B. Angles and inner products 32References 351.
Introduction
Let Y be a closed Riemann manifold. Let B = { ψ i } be an orthonormal basis of L ( Y )consisting of Laplacian eigenfunctions ∆ ψ i = λ i ψ i . A subsequence { ψ i j } of B is called quantum ergodic if, for every degree 0 pseudo-differential operator A on Y with principalsymbol a ∈ C ( S ∗ Y ), we have (cid:104) Aψ i j , ψ i j (cid:105) L ( Y ) → − (cid:90) S ∗ Y adµ L , as j → ∞ . Here µ L is the Liouville measure on the cosphere bundle S ∗ Y and − (cid:82) indicates normalizationby the volume. The quantum ergodicity theorem of ˇSnirel (cid:48) man [30], Zelditch [34], and Colinde Verdi`ere [10] states that, if the geodesic flow on S ∗ Y is ergodic, one may extract from B a density one quantum ergodic subsequence. More precisely, they show(1) 1 N ( λ ) (cid:88) λ i ≤ λ (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) Aψ i , ψ i (cid:105) L ( Y ) − − (cid:90) S ∗ Y adµ L (cid:12)(cid:12)(cid:12)(cid:12) → λ → ∞ , where N ( λ ) = |{ i : λ i ≤ λ }| .In this paper we are concerned with a version of quantum ergodicity where, in contrastto the above semi-classical statement in which the manifold is fixed and the eigenvalue goes The first author was supported by ANR grant 14-CE25. a r X i v : . [ m a t h . SP ] O c t o infinity (the large frequency regime), we allow the manifold to vary while keeping theeigenvalue 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 probabilitymeasures – 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, totheir common universal cover. Under certain auxiliary conditions, the question of QuantumErgodicity for Benjamini–Schramm convergent sequences was recently settled for a largeclass of rank one symmetric spaces [24, 2]. The aim of the present paper is to address thisquestion for higher rank locally symmetric spaces; for the most part we restrict ourselves tocompact quotients associated with SL d ( R ).1.1. The setting of locally symmetric spaces.
As our setting will henceforth be thatof locally symmetric spaces, we begin by commenting on some of their particular featuresin higher rank, and review what is known for their quantum ergodic properties in the largefrequency regime.We let S be a Riemannian globally symmetric space of non-compact type: non-positivelycurved and having no Euclidean de Rham local factor. We may write S = G/K where G isa 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 one, S is of constant negative curvature, and A gives rise to the geodesic flowon the cosphere bundle of S , via the identification of the latter with G/M , where M is thecentralizer of A in K . It is important to note, however, that when the rank is strictly greaterthan one, such as for SL d ( R ) / SO( d ) when d ≥
3, the curvature of S is non-constant, asindeed any geodesic triangle in a maximal flat will be Euclidean.Now let Γ ⊂ G be a uniform lattice in G and form Y = Γ \ S , a compact locally symmetricspace with geometry S . In higher rank, due to the presence of maximal flats, the geodesicflow is not ergodic on the cosphere bundle Γ \ G/M of Y . For this reason, the quantumergodic statement (1) cannot hold, as it is known to be equivalent with the ergodicity of thegeodesic flow [35, Theorem 1].Nevertheless, the higher rank split torus A does act ergodically – in fact it is mixing – onΓ \ G , and this is enough for one to expect similar quantum ergodic phenomena to that ofˇSnirel (cid:48) man’s theorem, at least if one refines the notion of eigenfunction as follows.Recall that a Maass form on Y is a function ψ ∈ L ( Y ) which is a joint eigenfunction of thealgebra D ( S ) of left- G -invariant differential operators on S . Since ∆ ∈ D ( S ) a Maass formis again a Laplacian eigenfunction, but in higher rank L ( Y ) can be further diagonalized.A Maass form ψ gives rise to an algebra homomorphism χ ∈ Hom C − alg ( D ( S ) , C ) verifying Dψ = χ ( D ) ψ for all D ∈ D ( S ). The Harish-Chandra homomorphism allows one to realize χ as the Weyl group orbit of an element ν in a ∗ C , the complexification of the dual of the Liealgebra a of A . We call ν the spectral parameter of ψ .One can then formulate a natural extension of (1) for Maass forms on Y with growingspectral parameter. To the best of our knowledge, this form of Quantum Ergodicity hasnot yet been established, although recent work by Nelson and Venkatesh [28] on microlocalanalysis and representation theory should shed light on this problem. emark . There are of course harder conjectures that one can formulate here, such asQuantum
Unique
Ergodicity [25, Conjecture 1.2]. In the arithmetic setting, certain higherrank congruence manifolds have been shown [31, 32] to satisfy the Arithmetic QuantumUnique Ergodicity property, a generalization of the work (and techniques) of Lindenstrauss[26]. We will not make further comments on this active line of research, and refer the readerto [24, § Our main result.
We now pass to the large spacial regime for higher rank locallysymmetric spaces Y = Γ \ S , and allow Γ to vary along a non-conjugate sequence of torsionfree uniform lattices with growing covolume.We furthermore require that the sequence Γ be uniformly discrete, a property which wenow recall. Let d be the Riemannian distance on S . Then the local injectivity radius abouta point x ∈ Y is the quantityInjRad Γ ( x ) = 12 min { d ( x, γ.x ) : 1 (cid:54) = γ ∈ Γ } . The global injectivity radius InjRad( Y ) is the infimum of InjRad Γ ( x ) over all x ∈ Y ; thisis strictly positive since Y is compact. We say that that a sequence of uniform torsion freelattices Γ n is uniformly discrete if InjRad( Y n ) is bounded away from zero. A conjecture ofMargulis [27] states that this is automatic in higher rank; this is a weak form of the Lehmerconjecture on monic integral polynomials.Our precise result is as follows. We refer to § Theorem 1.1.
Let d ≥ . Let Γ n ⊂ SL d ( R ) be a uniformly discrete sequence of torsion freecocompact lattices such that vol( Y n ) → ∞ as n → ∞ , where Y n = Γ n \ SL d ( R ) / SO( d ) . Let a n be a sequence of uniformly bounded, measurable functions on Y n .There is (cid:37) > such that for sufficiently regular ν ∈ i a ∗ we have N ( B ( ν, (cid:37) ) , Γ n ) (cid:88) j : ν ( n ) j ∈ B ( ν,(cid:37) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) a n ψ ( n ) j , ψ ( n ) j (cid:105) L ( Y n ) − − (cid:90) Y n a n d vol Y n (cid:12)(cid:12)(cid:12)(cid:12) → as n → ∞ , where − (cid:82) denotes the normalization of the integral by the volume of Y n , B ( ν, (cid:37) ) = { λ ∈ i a ∗ : (cid:107) λ − ν (cid:107) ≤ (cid:37) } is the ball of radius (cid:37) in the unramified tempered spectrum, and (2) N ( B ( ν, (cid:37) ) , Γ n ) = |{ j : ν ( n ) j ∈ B ( ν, (cid:37) ) }| . This extends to higher rank the rank one version of the same result in the papers [24] (forhyperbolic surfaces) and [2] (for higher dimensional hyperbolic manifolds), which themselvesbuilt upon the breakthrough results of Anantharaman and Le Masson [3] for large regulargraphs. All of these works imposed the following additional hypotheses:(i) that the manifolds (or graphs) Y n converge to S in the sense of Benjamini–Schramm :for every R > Y n ) ≤ R )vol( Y n ) → n → ∞ , where the R -thin part of Y n is defined as(3) ( Y n ) ≤ R = { x ∈ Y n : InjRad Γ n ( x ) ≤ R } ; ii) that the Y n have a uniform spectral gap.The proof of Theorem 1.1 also requires those properties, but they are automatic in higherrank; see [20] and [1, § Remark . We point out two differences between the Quantum Ergodicity theorem of therecent preprint [2] for rank one spaces and the higher rank version we present in Theorem1.1. In contrast to our result, the authors of [2] allow for two more general features, namely(a) they allow the spectral window to shrink with n , thereby isolating in the large n limita fixed tempered eigenvalue for the universal cover. This added flexibility was alsopresent in [3, Theorem 1.3];(b) they take more general operators than scalar multiplication by functions a n . Thismore advanced formulation was first put forward in [3, Theorem 1.7], using thepseudo-differential calculus for trees developed in [23].Our Theorem 1.1 therefore more closely resembles the main results of [8] and [24], in whichneither of these two features is present. By taking more elaborate test functions, we believewe can incorporate (a) into our set-up. 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 toseveral intermediate statements as explained in Section 2. This reduction roughly followsalong the lines of [24] and the subsequent paper [2]. To establish these intermediate results,however, we need several new ideas to deal with a number of issues that arise only in rank2 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 → ∞ . At alater critical point we need to estimate from above the volume of intersections C t ∩ gC t with g ∈ G . In [24, 2] they work with C t being the metric ball B t in S of radius t ,and exploit the fact that S is a CAT( −
1) space in rank 1. In higher rank, S is onlya 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 easierto work with in higher rank; see (13) and Section 5.5. • We need to establish a suitable lower bound for certain averages of spherical func-tions. In rank 1, this can be dealt with in a relatively straightforward manner, asthe elementary spherical functions are basically linear combinations of trigonometricfunctions in one variable. In higher rank, we need to deal with linear combinationsof exponential functions in several variables which makes the analysis much moredelicate; see Section 4. The techniques of that section might also be of interest inother contexts.Additionally, we need to establish a type of local Weyl law/limit multiplicity for the Y n that gives a lower count the number of eigenvalues locally around sufficiently regular pointsin the spectrum of Y n . We only need that sharp lower bound in the level aspect as stated inProposition 2.1, but along the way prove a stronger version that also yields the right order n the spectral parameter, see (24). Such a result is also necessary for the rank 1 situation,but involves a much more careful analysis of the non-tempered 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 = SL d ( R ) and K = SO( d ). As a preliminary step, we notethat by replacing a n by a n − − (cid:82) Y n a n we may suppose that the measurable, right- K -invariantfunctions a n on Γ n \ G satisfy(4) (cid:90) Γ n \ G a n = 0 and (cid:107) a n (cid:107) ∞ ≤ . Under that assumption it will then be enough to prove(5) 1 N ( B ( ν, (cid:37) ) , Γ n ) (cid:88) j : ν ( n ) j ∈ B ( ν,(cid:37) ) |(cid:104) a n ψ ( n ) j , ψ ( n ) j (cid:105) L ( Y n ) | → n → ∞ . The rest of the paper is devoted to establishing (5).2.1. Spectral estimate.
We shall first need to control the spectral counting function de-fined in (2). This is provided in the following result, a sharp spectral lower bound, to beproved in Section 3. See § Proposition 2.1.
Let G be a connected non-compact simple Lie group with finite center,and K a maximal compact subgroup. Let S = G/K be the associated irreducible Riemann-ian globally symmetric space of non-compact type. Let Γ n be a sequence of uniformly dis-crete, torsion free, cocompact lattices in G such that Y n = Γ n \ S converges, in the sense ofBenjamini–Schramm, to S . There is (cid:37) > such that, for any sufficiently regular ν ∈ i a ∗ , vol(Γ n \ G ) (cid:28) N ( B ( ν, (cid:37) ) , Γ n ) , where B ( ν, (cid:37) ) = { λ ∈ a ∗ C | (cid:107) λ − ν (cid:107) < (cid:37) } , and N ( B ( ν, (cid:37) ) , Γ n ) = |{ j | ν ( n ) j ∈ B ( ν, (cid:37) ) }| is thecounting function. The proof of Proposition 2.1 in fact yields more information than what we have recordedhere. See (24) for the precise estimate. In particular, for sufficiently regular ν ∈ i a ∗ we showthe stronger uniform lower bound vol(Γ n \ G ) ˜ β ( ν ) (cid:28) N ( B ( ν, (cid:37) ) , Γ n ).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 below, to thesymmetric space S = SL d ( R ) / SO( d ).2.2. The averaging subset.
Following [8] and [24], the main idea behind (5) is the strategicuse of a self-adjoint normalized averaging operator which, on one hand, acts non-decreasinglyon the spectral average (Theorem 2.2) and on the other hand has small Hilbert–Schmidtnorm (Theorem 2.3). This idea can be traced back to the work of Brooks, Le Masson, andLindenstrauss [8], which presents an alternative proof of [3, Theorem 1.3], using discretenormalized averaging operators.Below we give a definition of an exhaustive sequence of sets E t ⊂ SL d ( R ) which we shalluse in our averaging operators. The difficulty in higher rank is finding a set E t which satisfies imultaneously the desired spectral and geometric properties (in rank 1, the obvious choiceof a Riemannian metric ball is shown to work in [24] and [2]).We begin by introducing some notation. Let g be the trace zero matrices in M d ( R ). Let a = (cid:8) X = diag( X , . . . , X d ) ∈ g (cid:9) be the standard Cartan subalgebra of diagonal matrices. Let W = N K ( a ) /Z K ( a ) be theWeyl group of ( a , g ); the action of W (cid:39) S d on a is by permutation of the coordinates. Let a + be the standard positive Weyl chamber in a given by X > X > · · · > X d . For a vector X ∈ a we will write (cid:107) X (cid:107) ∞ = max {| X | , . . . , | X d |} . The W -invariant norm (cid:107)·(cid:107) ∞ on a induces a sub-additive bi- K -invariant norm on G = SL d ( R )via the Cartan decomposition G = K exp( a + ) K . Namely, if g ∈ G with g ∈ Ke X K , thenwe write | g | = (cid:107) X (cid:107) ∞ . We have | g | ≥ | g − | = | g | , and | g g | ≤ | g | + | g | .With this notation, we put(6) E t = { g ∈ G : | g | ≤ t } . This is the averaging set we shall use for our wave propagator. One may view E t as a radiallyinvariant subset of the symmetric space G/K . See § E t .2.3. The two main results.
We now define k t = 1 (cid:112) m G ( E t ) E t . Let ρ Γ \ G denote the right-regular representation of G on L (Γ \ G ), and consider the wave-propagation operator on L (Γ \ G ) given by(7) U t = ρ Γ \ G ( k t ) . From E − t = E t it follows that k t is a self-adjoint operator. For a measurable right- K -invariant function a on Γ \ G satisfying (4), and for τ >
0, we consider the time average(8) A ( τ ) = 1 τ (cid:90) τ U t aU t dt. In Section 4 we shall prove the following spectral estimate.
Theorem 2.2 (Spectral estimate) . Let S = SL d ( R ) / SO( d ) . Let Γ a cocompact lattice in SL d ( R ) , and put Y = Γ \ S . Let Ω ⊆ i a ∗ be compact. There exist constants c, τ > ,depending on Ω , such that, for τ ≥ τ , we have (cid:88) j : ν j ∈ Ω |(cid:104) aψ j , ψ j (cid:105) L ( Y ) | ≤ c (cid:88) j : ν j ∈ Ω (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) A ( τ ) ψ j , ψ j (cid:105) L ( Y ) (cid:12)(cid:12)(cid:12)(cid:12) . In Section 5, we shall use the Nevo mean ergodic theorem [29] (see also [15, Theorem4.1]) to prove the following geometric estimate. Recall that the Hilbert–Schmidt norm of abounded operator A on a separable Hilbert space H is given by (cid:107) A (cid:107) = (cid:88) i |(cid:104) A e i , e i (cid:105)| , where { e i } is any orthonormal basis of H . heorem 2.3 (Geometric estimate) . Let S = SL d ( R ) / SO( d ) . Let Γ be a cocompact torsionfree lattice in SL d ( R ) , and put Y = Γ \ S . Let a be a measurable function on L (Γ \ S ) . Thereare constants b, c , c > , depending only on d , such that, for τ > , (cid:13)(cid:13) A ( τ ) (cid:13)(cid:13) (cid:28) b (cid:107) a (cid:107) τ + e c τ InjRad( Y ) dim S vol((Γ \ G ) ≤ c (2 τ + b ) ) (cid:107) a (cid:107) ∞ . Reduction to two main results.
We now deduce the main estimate (5) from theabove results.We first note that if ν ∈ i a ∗ is sufficiently regular in the sense that |(cid:104) ν, α ∨ (cid:105)| ≥ C (cid:37) for allsimple roots α with C (cid:37) > (cid:37) , then B ( ν, (cid:37) ) = B ( ν, (cid:37) ) because of (21). By Proposition 2.1 it suffices to prove an upper bound for1vol(Γ n \ G ) (cid:88) j : ν ( n ) j ∈ B ( ν,(cid:37) ) |(cid:104) a n ψ ( n ) j , ψ ( n ) j (cid:105) L ( Y n ) | . From Theorem 2.2 it follows that, for τ ≥ τ ,1vol(Γ n \ G ) (cid:88) j : ν ( n ) j ∈ B ( ν,(cid:37) ) |(cid:104) a n ψ ( n ) j , ψ ( n ) j (cid:105) L ( Y n ) | (cid:28) n \ G ) (cid:13)(cid:13) A ( n ) ( τ ) (cid:13)(cid:13) . We now apply Theorem 2.3 to the right-hand side. To bound the first term, we use theestimate (cid:107) a n (cid:107) ≤ (cid:107) a n (cid:107) ∞ vol(Γ n \ G ) along with the uniform boundedness of a n . For the secondterm we insert the bound InjRad( Y n ) (cid:29) n \ G ) (cid:13)(cid:13) A ( n ) ( τ n ) (cid:13)(cid:13) (cid:28) τ n + e c τ n vol (cid:0) (Γ n \ G ) ≤ c (2 τ n + b ) (cid:1) vol(Γ n \ G ) . We know from [1, §
4] that for any sequence Γ n for which vol(Γ n \ G ) → ∞ the space Γ n \ G/K converges in the sense of Benjamini–Schramm to
G/K = SL d ( R ) / SO( d ) (recall our assump-tion that d ≥ R n → ∞ such that(10) vol ((Γ n \ G ) ≤ R n )vol(Γ n \ G ) → n → ∞ . Given this sequence R n we let r n > r n → ∞ as n → ∞ ,(ii) e c r n vol ( (Γ n \ G ) ≤ c rn + b ) ) vol(Γ n \ G ) → n → ∞ .Taking τ n = r n , both terms in (9) go to 0 with n , establishing (5). Remark . One could deduce an effective rate of convergence in Theorem 1.1 for congruencesubgroups Γ n of SL d ( Z ) by inserting, in place of the o (1) result in (10), the explicit boundson the R -thin part for such sequences proved in [1, Theorem 5.2]. .5. Remarks on the averaging subset.
We now make several remarks about E t :(i) In rank one, the set E t recovers the metric ball centered at i ∈ H of radius 2 t for thehyperbolic metric ( dx + dy ) /y on the upper half-plane H = SL ( R ) / SO(2).(ii) In higher rank, the set E t differs substantially from a metric ball centered at the identityfor the metric induced by the trace form Tr( X (cid:62) Y ) (a constant multiple of the Killing form)on g .To see this, let us first compare a geodesic ball and the set E t by means of the Cartandecomposition of G = SL d ( R ). Let B = { X ∈ a : (cid:107) X (cid:107) ≤ } , where (cid:107) X (cid:107) = tr( X ) = X + · · · + X d . The geodesic ball B t of radius t > B t = K exp( B + t ) K, where B + t = t B ∩ a + ; see [4, Proposition 4.2]. On the other hand, let(12) P = { X ∈ a | (cid:107) X (cid:107) ∞ ≤ } . Note that P is a bounded convex polytope in a , being the intersection of the half planes ± X i ≤
1. It is clear that(13) E t = K exp( P + t ) K, where P + t = t P ∩ a + . One sees from (11) and (13) that an element in B t or E t has Cartanradial component contained, respectively, to an expanding ball or a dilated W -invariantpolytope in a .(iii) Related to the above 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 ρ ∈ a ∗ , as that quantity governs the Jacobian factor in the Cartan decomposition of theHaar measure on G ; see § S = G/K of non-compact type, states that(14) m G ( B t ) (cid:16) t ( r − / e t (cid:107) ρ (cid:107) , where r is the rank of S . On the other hand, we show in Corollary A.4 that (for G = SL d ( R )) m G ( E t ) ∼ c e t (cid:104) ρ,X (cid:105) ( t → ∞ ) , where X ∈ P + is the unique point satisfying(15) (cid:104) ρ, X (cid:105) = max X ∈ P + (cid:104) ρ, X (cid:105) . One can compute that(16) (cid:107) ρ (cid:107) = (cid:10) ρ, ρ (cid:107) ρ (cid:107) (cid:11) = (cid:112) d ( d − / , whereas, using Lemma A.1, we have (cid:10) ρ, X (cid:107) X (cid:107) (cid:11) = (cid:40) d / / , d even( d + 1)( d − / / , d odd . We have normalized by (cid:107) X (cid:107) so that B t is the smallest geodesic ball containing E t/ (cid:107) X (cid:107) .We see that the exponential volume growth of B t in SL d ( R ) / SO( d ) is of order 2 t (cid:112) d / E t/ (cid:107) X (cid:107) is of order 2 t (cid:112) d /
16, which is a factor of √ / iv) Another norm that one often considers in the context of G = SL d ( R ) is the restrictionto SL d ( R ) ⊂ M d ( R ) of the Frobenius norm , defined on M d ( R ) as (cid:107) g (cid:107) = tr( g (cid:62) g ), where g (cid:62) isthe transpose of g . Note that when (cid:107) g − (cid:107) (cid:54) = (cid:107) g (cid:107) in general. Since invariance under inversionis important for the self-adjointness of the propagator U t from (7), we define the Frobeniusball as(17) E t = (cid:8) g ∈ G : max {(cid:107) g (cid:107) , (cid:107) g − (cid:107)} ≤ e t (cid:9) . From the point of view of their large scale geometry, the Frobenius balls are similar to thesets E t from (6). Indeed in the proof of Proposition 5.8 we show that m G ( E t ) (cid:16) m G ( E t ).Moreover, the very statement Proposition 5.8, which is a key component Theorem 2.3, holdsequally well for E t . On the other hand, because they are not defined by their radial Cartancomponent, the 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 non-compact simple Lie group with finite center, K a maximal compactsubgroup.3.1. Notation.
The notation we introduce here will be consistent with that already intro-duced for the particular case of G = SL d ( R ) and K = SO( d ).Let θ be the Cartan involution on G for which K = G θ . Let Θ be its differential and let g = p ⊕ k be the Cartan decomposition of the Lie algebra g of G into the ± k with the Lie algebra of K .Let κ ( X, Y ) = tr(ad X ad Y ) denote the Killing form on g . Then − κ ( X, Θ Y ) defines anAd-invariant inner product on g , which induces a left-invariant Haar measure on G , denoted dg or m G .The Killing form defines an Ad K -invariant inner product (cid:104)· , ·(cid:105) on p , and in particular on a , the maximal abelian subalgebra of p . Let W = N K ( a ) /Z K ( a ) be the Weyl group. Let (cid:107) · (cid:107) denote the W -invariant norm on a induced by (cid:104)· , ·(cid:105) .Let Φ ⊂ a ∗ denote the system of roots for the adjoint action of a on g . For α ∈ Φ let g α denote the corresponding root space. Let m ⊂ g be the centralizer of a in g . Then we havethe root space decomposition g = m ⊕ a ⊕ (cid:77) α ∈ Φ g α . Choose a positive system of roots Φ + in Φ. Let ∆ ⊆ Φ + be the set of simple roots.Let a ∗ = Hom( a , R ) be the dual vector space of a . We may identify a with a ∗ via theKilling form. We again denote by (cid:107) · (cid:107) the induced norm on a ∗ and extend it to a complexbilinear form on the complexification a ∗ C = a ∗ ⊗ R C . We call ν ∈ i a ∗ regular if (cid:104) α, ν (cid:105) (cid:54) = 0for all α ∈ ∆, and for c > ν ∈ i a ∗ c -regular if |(cid:104) α, ν (cid:105)| ≥ c for all α ∈ Φ + . Anelement ν ∈ i a ∗ is sufficiently regular if there exists a sufficiently regular c > ν is c -regular. Unless otherwise noted, implied constantsfor statements valid for sufficiently regular ν can depend on the value of c . When G = SL d ( R ), we introduced in § g , denoted thereby the same symbol (cid:107) · (cid:107) . The Killing form for SL d ( R ), in fact, differs from the trace form by a constantfactor of 2 d . When we return to the specific situation of SL d ( R ) in later sections, we shall always take thenorm (cid:107) · (cid:107) on a to mean (cid:107) X (cid:107) = tr( X ). et n = (cid:80) α ∈ Φ + g α and write N = exp n . Then we have the Iwasawa decomposition G = N AK , where A = exp a , which gives rise to the Iwasawa projection H : G → a , g = nak (cid:55)→ log a, along the A component.We may decompose the Riemannian Haar measure dg on G according to the Iwasawadecomposition. We let da denote the Haar measure on A obtained by pushing forward theLebesgue measure on a my means of the exponential. We let dk denote the probability Haarmeasure dk on K . We normalize the left-invariant Haar measure du on N as in [12, § ρ = 12 tr(ad( a ) | n ) = 12 (cid:88) α ∈ Φ + (dim g α ) α. Then one has [14, Proposition 2.4.10] dg = c I e ρ ( H ( g )) dudadk ( c I = 2 − (1 /
2) dim N ) . Since A normalizes N and det Ad( a ) | n = e ρ ( H ( a )) ,(18) dg = c I dadudk in the decomposition G = AN K .Recall the Cartan decomposition G = K exp( a + ) K , where a + denotes the closure of the(open) positive Weyl chamber a + = { X ∈ a : α ( X ) > ∀ α ∈ Φ + } . We have the followingdecomposition of the Riemannian Haar measure dg into the Cartan decomposition(19) dg = c C J ( X ) dk dadk ( c C = 2 − dim N vol( K ) vol( K/M )) . Here, vol( K ) is the Riemannian volume on K , for the measure induced by the inner product − κ ( X, Y ) on k , and similarly with vol( K/M ). Finally, J ( X ) is the Jacobian factor, given by(20) J ( X ) = (cid:89) α ∈ Φ + sinh( α ( X )) dim g α ( X ∈ a + );see [14, Proposition 2.4.11].Let P be the normalizer of m ⊕ a ⊕ n in G . Then P has Langlands decomposition P = M AN , where M is the centralizer of A in G . For λ ∈ a ∗ C let π λ denote the uniqueunramified irreducible subquotient of Ind GP (1 ⊗ e λ ⊗ a ∗ un = { λ ∈ a ∗ C : π λ unitarizable } be the spherical unitary dual of G . Furthermore, let a ∗ hm = (cid:91) w ∈ W { λ ∈ a ∗ C : wλ = − ¯ λ } denote the spherical hermitian dual of G (see [22, § i a be the subspace of a ∗ C consisting of λ taking on purely imaginary values. We may write λ ∈ a ∗ C uniquely as λ =Re λ + Im λ ∈ a ∗ ⊕ i a ∗ . We have(21) i a ∗ ⊂ a ∗ un ⊂ a ∗ hm ∩ { λ ∈ a ∗ C : (cid:107) Re λ (cid:107) ≤ (cid:107) ρ (cid:107) } . For µ ∈ i a ∗ and r > B ( µ, r ) = { λ ∈ i a ∗ : (cid:107) λ − µ (cid:107) ≤ r } enote the ball of radius r about µ in the tempered spectrum i a ∗ . When r = 1 we write B ( µ ) for B ( µ, B ( µ, r ) = { λ ∈ a ∗ C : (cid:107) λ − µ (cid:107) ≤ r } for the ball of radius r around µ in a ∗ C .3.2. Plancherel density and the c-function.
We denote by β the density function forthe Plancherel measure on the spherical unitary dual, which can be identified with a ∗ un /W .Then β is a W -invariant function supported on i a ∗ that can be described as a product ofΓ-functions [17, Ch. IV, § β ( t, λ ) = (cid:89) α ∈ Φ + ( t + |(cid:104) λ, α ∨ (cid:105)| ) dim g α , λ ∈ i a ∗ , t ≥ , and ˜ β ( λ ) = ˜ β (1 , λ ). From [17, Ch. IV, Theorem 6.14] and standard estimates for theΓ-function it follows that β ( λ ) (cid:28) ˜ β ( λ ) for all λ ∈ i a ∗ . Since1 + |(cid:104) tλ + µ, α ∨ (cid:105)| ≤ t |(cid:104) λ, α ∨ (cid:105)| + |(cid:104) µ, α ∨ (cid:105)| ≤ t (1 + (cid:107) λ (cid:107) ) + |(cid:104) µ, α ∨ (cid:105)| ≤ (1 + (cid:107) λ (cid:107) )( t + |(cid:104) µ, α ∨ (cid:105)| )for all λ, µ ∈ i a ∗ and t ≥
1, we get(22) ˜ β ( tλ + µ ) (cid:28) (1 + (cid:107) λ (cid:107) ) | Φ + | ˜ β ( t, µ )for all such λ, µ , and t .The Harish-Chandra c -function c : a ∗ C −→ C for G asymptotically describes the behaviorof the elementary spherical functions φ λ ( e H ) of G as the group parameter H grows. Thequantity c ( λ ) depends only on the root system of G , and can be explicitly computed asdescribed in [17, Chapter IV, Theorem 6.14]. Up to normalization, the Plancherel density β ( λ ) equals | c ( λ ) | − for λ ∈ i a ∗ .3.3. Test functions.
For λ ∈ a ∗ C let ϕ λ ( g ) = (cid:90) K e (cid:104) λ + ρ,H ( kg ) (cid:105) dk denote the spherical function on G with spectral parameter λ . Here H : G −→ a is theIwasawa projection. Let C ∞ c ( G (cid:12) K ) be the space of complex-valued bi- K -invariant functionson G . The Harish-Chandra transform of a function k ∈ C ∞ c ( G (cid:12) K ) is defined to beˆ k ( ν ) = (cid:90) G ϕ λ ( g ) k ( g ) dg, For t > B t denote the ball of radius t centered at 0 in a with respect to the usualEuclidean norm (cid:107) · (cid:107) . Let G ≤ t = K exp B t K . Let C ∞ c ( G (cid:12) K ) ≤ t denote the space of smoothcompactly supported bi- K -invariant functions on G , supported on G ≤ t . Let PW ( a ∗ C ) t denotethe class of Paley–Wiener functions of exponential type t . The Paley–Wiener theorem withsupports (see [13, Theorem 3.5]) states that the Harish-Chandra transform is a topologicalisomorphism C ∞ c ( G (cid:12) K ) ≤ t ∼ −→ PW ( a ∗ C ) Wt of Fr´echet spaces. The inverse map sends h ∈ PW ( a ∗ C ) Wt to 1 | W | (cid:90) i a ∗ h ( µ ) ϕ µ ( g ) β ( µ ) dµ. n particular, for k ∈ C ∞ c ( G (cid:12) K ) we have(23) k ( e ) = 1 | W | (cid:90) i a ∗ ˆ k ( µ ) β ( µ ) dµ. We shall need a Paley–Wiener function concentrating about ν and verifying certain posi-tivity properties. Lemma 3.1. [9, § For t ≥ and ν ∈ i a ∗ there is k ν,t ∈ C ∞ c ( G (cid:12) K ) ≤ /t whose Harish-Chandra transform h ν,t ∈ PW ( a ∗ C ) Wt satisfies(1) h ν,t is real and non-negative on a ∗ hm ;(2) for all λ ∈ a ∗ un we have h ν,t ( λ ) (cid:28) A (1 + (cid:107) (Im λ − ν ) /t (cid:107) ) − A ; (3) there are constants < c , c < such that for λ ∈ a ∗ un with (cid:107) Im λ − ν (cid:107) ≤ c t wehave c ≤ h ν,t ( λ ) ≤ ;(4) k ν,t ( g ) (cid:28) ˜ β ( ν ) t r (1 + (cid:107) ν (cid:107) d ( g, K )) − / , where r = dim a . Proof of Proposition 2.1.
Recall that for ν ∈ i a ∗ and c >
0, we denote by B ( ν, c )the ball { λ ∈ a ∗ C | (cid:107) λ − ν (cid:107) ≤ c } in a ∗ C . Moreover, for any set Ω ⊆ a ∗ C , we write N (Ω , Γ n ) = |{ j | ν ( n ) j ∈ Ω }| .We shall in fact prove a stronger statement than Proposition 2.1. We shall show that thereexist 0 < (cid:37) < < (cid:37) such that the following holds. Fix (cid:15) >
0, and let ν ∈ i a ∗ be ε -regular.Then(24) N ( B ( ν, (cid:37) ) , Γ n ) (cid:28) vol(Γ n \ G ) ˜ β ( ν ) (cid:28) ε N ( B ( ν, (cid:37) ) , Γ n ) . Proposition 2.1 then follows from the second estimate above and the fact that ˜ β ( ν ) ≥ ν ∈ i a ∗ .In the notation of Lemma 3.1, we write h ν = h ν, and k ν = k ν, . Then (cid:88) λ ∈ Λ n h ν ( λ ) = vol(Γ n \ G ) k ν (1) + (cid:90) Γ n \ G (cid:88) γ ∈ Γ n γ (cid:54) =1 k ν ( g − γg ) dg, where Λ n denotes the spectrum of Γ n \ G (each λ appearing with its respective multiplicity).To deal with the second term, we use part (4) of Lemma 3.1, as well as the supportcondition on k ν , to deduce (cid:90) Γ n \ G (cid:88) γ ∈ Γ n γ (cid:54) =1 k ν ( g − γg ) dg (cid:28) ˜ β ( ν ) (cid:90) Γ n \ G |{ γ ∈ Γ n \ { } : d ( g, γg ) ≤ }| dg. As Γ n is torsion free, the inner sum is empty for all g ∈ (Γ n \ G ) > , so that it suffices tobound (cid:90) (Γ n \ G ) ≤ N R ( g ) dg, where(25) N R ( g ) = |{ γ ∈ Γ n : d ( g, γg ) ≤ }| . For g ∈ (Γ n \ G ) ≤ we apply [1, Lemma 6.18], which provides a constant C >
0, dependingonly on G , such that N R ( g ) ≤ C InjRad Γ n ( g ) − dim S . The uniform discreteness of the (torsion ree) Γ n implies that InjRad Γ n ( g ) − (cid:28)
1, uniformly in n and g . Taking these estimatestogether we get(26) (cid:88) λ ∈ Λ n h ν ( λ ) = vol(Γ n \ G ) k ν (1) + O ( ˜ β ( ν ) vol(Γ n \ G ) ≤ ) . Furthermore, we have(27) k ν (1) (cid:16) (cid:15) ˜ β ( ν ) , where only the lower bound depends on ε . Indeed, the upper bound results from part (4) ofLemma 3.1. To obtain the lower bound, one applies the Plancherel inversion formula (23),and parts (1) and (3) of Lemma 3.1, to get k ν (1) ≥ (cid:90) λ ∈ i a ∗ (cid:107) λ − ν (cid:107) ≤ δ h ν ( λ ) β ( λ ) dλ ≥ c (cid:90) λ ∈ i a ∗ (cid:107) λ − ν (cid:107) ≤ δ β ( λ ) dλ, for any 0 < δ < c . Recall that ν ∈ i a ∗ is ε -regular. Taking δ small enough, we may assumethat the λ ∈ i a ∗ such that (cid:107) λ − ν (cid:107) ≤ δ are ε/ β ( λ ) (cid:29) ε ˜ β ( λ ) for such λ ; see [12, (3.44a)].From (26) and (27) it follows that (cid:88) λ ∈ Λ n h ν ( λ ) (cid:16) ε vol(Γ n \ G ) ˜ β ( ν ) (cid:18) O (cid:18) vol(Γ n \ G ) ≤ vol(Γ n \ G ) (cid:19)(cid:19) , where only the lower bound depends on ε . By the Benjamini–Schramm assumption, we havevol(Γ n \ G ) ≤ = o (vol(Γ n \ G )) as n → ∞ . Thus, for n large enough we have(28) (cid:88) λ ∈ Λ n h ν ( λ ) (cid:16) ε vol(Γ n \ G ) ˜ β ( ν )For the first bound in (24), it suffices at this point to take (cid:37) = c , apply parts (1) and (3)of Lemma 3.1, and then quote the upper bound in (28) to get N ( B ( ν, c ) , Γ n ) ≤ c (cid:88) λ ∈ Λ n (cid:107) λ − ν (cid:107) ≤ c h ν ( λ ) ≤ c (cid:88) λ ∈ Λ n h ν ( λ ) (cid:28) vol(Γ n \ G ) ˜ β ( ν ) . For the second bound in (24), we must show that the left-hand side in (28) approximates N ( B ( ν, (cid:37) ) , Γ n ), for some (cid:37) > N ( B ( µ, t ) , Γ n ), for any center µ ∈ i a ∗ and any t ≥
1. This is provedsimilarly to the preceding analysis. Indeed, from Lemma 3.1, together with the precedinggeometric argument, we obtain (cid:88) λ ∈ Λ n (cid:107) Im λ − µ (cid:107) ≤ t (cid:28) (cid:88) λ ∈ Λ n h µ,c − t ( λ ) = vol(Γ n \ G ) k µ,c − t (1) + o (cid:0) vol(Γ n \ G ) ˜ β ( µ ) (cid:1) . Now k µ,t (1) (cid:28) ˜ β ( µ ) t r from part (4) of Lemma 3.1. We conclude that (for n large enough)(29) N ( B (cid:48) ( µ, t ) , Γ n ) (cid:28) vol(Γ n \ G ) ˜ β ( µ ) t r , where B (cid:48) ( µ, t ) = { λ ∈ a ∗ C | (cid:107) Im λ − µ (cid:107) ≤ t } . We now return to the left-hand side of (28).We first claim: emma 3.2. For every (cid:37) > , (30) (cid:88) λ ∈ Λ n (cid:107) Im λ − ν (cid:107) >(cid:37) h ν ( λ ) (cid:28) A vol(Γ n \ G ) ˜ β ( ν ) (cid:37) − A . Proof.
To prove (30) we cover { λ ∈ i a ∗ : (cid:107) λ − ν (cid:107) > (cid:37) } by balls of unit radius with centersbased at an affine lattice Λ ⊂ i a ∗ containing ν . Let A ( ν, (cid:37) ) denote the set of all λ ∈ Λ suchthat the ball of radius 1 in i a ∗ around λ is entirely contained in the ball of radius (cid:37) in i a ∗ around ν .For any fixed µ ∈ Λ − A ( ν, (cid:37) ), we bound the contribution of λ ∈ Λ n for which Im λ ∈ B ( µ )by using part (2) of Lemma 3.1 to get (cid:88) λ ∈ Λ n (cid:107) Im λ − µ (cid:107) ≤ h ν ( λ ) (cid:28) A (1 + (cid:107) µ − ν (cid:107) ) − A N ( B (cid:48) ( µ, , Γ n ) . Using (29) with t = 1 and summing over µ ∈ Λ − A ( ν, (cid:37) ) we get (cid:88) λ ∈ Λ n (cid:107) Im λ − ν (cid:107) >(cid:37) h ν ( λ ) (cid:28) A vol(Γ n \ G ) (cid:88) µ ∈ Λ − A ( ν,(cid:37) ) (1 + (cid:107) µ − ν (cid:107) ) − A ˜ β ( µ ) . Now by (22) we have ˜ β ( µ ) (cid:28) (1 + (cid:107) µ − ν (cid:107) ) dim n ˜ β ( ν ), so that (cid:88) (cid:107) Im λ − ν (cid:107) >(cid:37) h ν ( λ ) (cid:28) A vol(Γ n \ G ) ˜ β ( ν ) (cid:88) µ ∈ Λ − A ( ν,(cid:37) ) (1 + (cid:107) µ − ν (cid:107) ) − A . For (cid:37) sufficiently large, this last sum is at most O A ( (cid:37) − A ), as desired. (cid:3) From this lemma, we want to deduce the following estimate:(31) (cid:88) λ ∈ Λ n (cid:107) λ − ν (cid:107) >(cid:37) h ν ( λ ) (cid:28) A vol(Γ n \ G ) ˜ β ( ν ) (cid:37) − A . For this we split the sum on the left hand side into three parts, (cid:88) λ ∈ Λ n (cid:107) Im λ − ν (cid:107) >(cid:37) h ν ( λ ) + (cid:88) λ ∈ Λ n (cid:37) −(cid:107) ρ (cid:107) < (cid:107) Im λ − ν (cid:107) ≤ (cid:37) , (cid:107) Re λ (cid:107) >(cid:37) −(cid:107) Im λ − ν (cid:107) h ν ( λ ) + (cid:88) λ ∈ Λ n (cid:107) Im λ − ν (cid:107) ≤ (cid:37) −(cid:107) ρ (cid:107) , (cid:107) Re λ (cid:107) >(cid:37) −(cid:107) Im λ − ν (cid:107) h ν ( λ ) . Using Lemma 3.2 the first sum can be bounded by (cid:28) A vol(Γ n \ G ) ˜ β ( ν ) (cid:37) − A . The third sum isin fact empty: the conditions on the real and complex part of λ imply that (cid:107) Re λ (cid:107) > (cid:107) ρ (cid:107) ,which is not possible for λ in the spectrum of L (Γ n \ G ) by (21). Finally, for the secondterm, we extend the sum to all λ ∈ Λ n with (cid:107) Im λ − ν (cid:107) ≥ (cid:37) − (cid:107) ρ (cid:107) . This is possible as h ν ( λ ) is non-zero on the unitary spectrum by Lemma 3.1. Hence we can use Lemma 3.2 tobound this term by (cid:28) A ( (cid:37) − (cid:107) ρ (cid:107) ) − A/ .We deduce from the lower bound in (28), as well as (31) that, with A > (cid:37) large enough with respect to the implied constant, (cid:88) λ ∈ Λ n (cid:107) λ − ν (cid:107) ≤ (cid:37) h ν ( λ ) (cid:29) ε vol(Γ n \ G ) ˜ β ( ν ) . n the other hand, by applying part (3) of Lemma 3.1 we have N ( B ( ν, (cid:37) ) , Γ n ) (cid:29) (cid:88) λ ∈ Λ n (cid:107) λ − ν (cid:107) ≤ (cid:37) h ν ( λ ) . Combing these yields the second bound in (24), with (cid:37) = (cid:37) . (cid:3) Spectral side
We now return to the setting of G = SL d ( R ) and prove Theorem 2.2. In the course of theproof, we will make use of both Appendices A and B.We first examine how k t acts on eigenfunctions ψ λ . The Harish-Chandra transform of k t is given by(32) h t ( λ ) = 1 (cid:112) m G ( E t ) (cid:90) E t ϕ λ ( g ) dg. Then, recalling the definition (7), we have U t ψ λ = h t ( λ ) ψ λ , which follows from the doubling formula of elementary spherical functions.Let A ( τ ) be the time averaging operator of (8). Since k t is self-adjoint we have (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) A ( τ ) ψ λ , ψ λ (cid:105) L ( Y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) τ (cid:90) τ | h t ( λ ) | dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) aψ λ , ψ λ (cid:105) L ( Y ) (cid:12)(cid:12)(cid:12)(cid:12) . To prove Theorem 2.2, it will therefore be enough to show that | h t ( λ ) | is bounded away from0 on average over t . More precisely, we prove the following higher rank generalization of[2, § Proposition 4.1.
Let G = SL d ( R ) . Given a compact set Ω ⊆ i a ∗ , there exist constants C, τ > , depending on Ω , such that, for every τ ≥ τ and λ ∈ Ω , τ (cid:90) τ | h t ( λ ) | dt ≥ C. Main idea of proof.
We begin by writing the integral defining h t in polar coordi-nates, according to the Cartan decomposition. Recall the averaging set E t from § f λ ( X ) = e (cid:104) ρ,X (cid:105) ϕ λ ( e X ), and using the Cartan measuredecomposition in (19), we find that (32) becomes(33) h t ( λ ) = c C (cid:112) m G ( E t ) (cid:90) P + t f λ ( e X ) J ( X ) e −(cid:104) ρ,X (cid:105) dX. The idea of the proof of Proposition 4.1 is to replace f λ by the main term Φ λ , discussedbelow, of its Harish-Chandra asymptotic expansion relative to the Levi determined by X ,defined in (15). We do this by showing, in Appendix A (see Lemma A.2), that P + is theintersection with a + of a translated cone C = X + C in a . We then show that, when λ istaken to be rational, Φ λ behaves like a sum of characters in the direction of X . An argumentusing linear independence of characters then yields the result. We carry out this argumentin this section, using some combinatorial results which we establish in Appendix B. .2. The main term and periodicity.
We wish to replace f λ by the main term in itsHarish-Chandra expansion. This is possible thanks to Proposition 4.2 below. To state it,we must introduce some notation, beyond that of § G = SL d ( R ), although Proposition 4.2 applies more generally.A Levi subgroup of G will be called semistandard if it contains A . Let ∆ ⊆ ∆ be a(possibly empty) subset of the set of simple roots, and let Φ +0 be the subset of Φ + generatedby ∆ . The sets ∆ correspond to semistandard Levi subgroups M ⊆ G such that Φ +0 is theset of positive roots of A on M . With this identification ∆ = ∅ corresponds to M = A and∆ = ∆ to M = G .For fixed ∆ and corresponding M we further introduce the following notation: • ρ M is the half-sum of all α ∈ Φ +0 , • W M is the Weyl group of A in M , • c M denotes the Harish-Chandra c -function for Φ +0 , • ϕ Mλ is the spherical function on M for λ ∈ a ∗ C , that is, ϕ Mλ ( m ) = (cid:90) K ∩ M e (cid:104) λ + ρ M ,H ( km ) (cid:105) dm. • for X ∈ a , λ ∈ a ∗ C , and t > f Mλ ( t, X ) = e (cid:104) ρ M ,tX (cid:105) ϕ Mλ ( e tX ) and (setting t = 1) let f Mλ ( X ) = e (cid:104) ρ M ,X (cid:105) ϕ Mλ ( e X ) . • a M is the common null space of all α ∈ Φ +0 , and a M is its orthocomplement in a relative to the inner product (cid:104) X, Y (cid:105) = XY .We follow the standard practice of omitting the superscript M from the notation whenever M = G . In this way we recover the f λ ( X ) introduced earlier. Other than G , the only otherLevi which will be of importance to us throughout this section is the centralizer of X in G . Henceforth shall take M to be the centralizer of X in G , and ∆ such that it corresponds to M . In this case X ∈ a M .For λ ∈ i a ∗ regular we define(34) Φ λ ( X ) = (cid:88) w ∈ W M \ W c ( wλ ) c M ( wλ ) f Mwλ ( X ) . In fact, we can define Φ λ ( X ) for any regular λ ∈ a ∗ C for which the c ( wλ ) do not have a pole.We can make this more precise, as follows. For a real number η > A η ⊆ a ∗ denotethe convex hull of the Weyl group orbit of ηρ . Define T η ⊆ a ∗ C to be the set of all λ ∈ a ∗ C with Re λ ∈ A η . Let η be so small that for every positive root α and all λ ∈ T η we have (cid:104) Re λ, α ∨ (cid:105) (cid:54)∈ Z (cid:114) { } . We fix such a 0 < η < T η is W -invariant by construction. We may extend the definitionof Φ λ ( X ) in (34) to λ ∈ T η regular and X ∈ a . Indeed, the c -function is non-zero on a ∗ C anddoes not have any poles for regular λ ∈ T η by our choice of η .The relevance of Φ λ ( X ) is apparent from the next result, due to Trombi and Varadarajan[33, Theorem 2.11.2], providing the full asymptotic expansion of the spherical function. Wehave taken the exact statement from [14, Theorem 5.9.4]. This extends to all X ∈ a theasymptotic expansion proved by Harish-Chandra and Gangolli for conical subsets of regularelements. roposition 4.2. Let Ω ⊆ i a ∗ be a compact set and c > . Then there exists C > and m ≥ such that for all H ∈ a + with β ∆ ( H ) ≥ c , where β ∆ ( H ) = min α ∈ ∆ (cid:114) ∆ α ( H ) , and for every regular λ ∈ Ω , we have | f λ ( H ) − Φ λ ( H ) | ≤ C (1 + (cid:107) H (cid:107) ) m e − β ∆0 ( H ) . We now describe the key periodicity property of Φ λ ( X ). For this we need the followingdefinition. Definition 4.3.
We say that λ ∈ i a ∗ is rational if i (cid:104) wλ, X (cid:105) ∈ Q for all w ∈ W . If λ is rational, we can choose τ ∈ R > such that τ (cid:104) wλ, X (cid:105) ∈ iπ Z for all w ∈ W . Recallthe definition of C from Lemma A.2. Lemma 4.4.
Let λ ∈ i a ∗ be rational and let τ be defined as above. Set τ n = nτ for n ∈ N .Then for every Y ∈ C , t ≥ and n ∈ N we have Φ λ ( Y + ( t + τ n ) X ) = Φ λ ( Y + tX ) . Proof.
By definition of τ it follows that e (cid:104) λ, ( t + τ n ) X (cid:105) = e (cid:104) λ,tX (cid:105) for every n ∈ N . Further notethat, since (cid:104) ρ M , X (cid:105) = 0 and H ( ke Y + tX ) = tX + H ( ke Y ) for k ∈ K ∩ M , we have ϕ Mλ ( e Y + tX ) = e (cid:104) ρ M + λ,tX (cid:105) ϕ Mλ ( e Y ) = e (cid:104) λ,tX (cid:105) ϕ Mλ ( e Y ) . Thus(35) f Mλ ( Y + tX ) = e (cid:104) λ,tX (cid:105) f Mλ ( Y ) . Applying these two formulas for every wλ , w ∈ W M \ W , yields the assertion. (cid:3) Convergence of an integral.
To take advantage of Lemma 4.4, we shall need toreplace the integration domain P + t in (33) by the translated cone C . Note that the Jacobianfactor (20) can be alternatively written as(36) J ( X ) = 2 −| Φ + | (cid:88) w ∈ W σ ( w ) e (cid:104) wρ,X (cid:105) ( X ∈ a + ) , where σ ( w ) ∈ {± } denotes the signature of w . With this in mind, for t > λ ∈ T η , weput I ( t, λ ) = (cid:90) C Φ λ ( Y + tX ) e (cid:104) ρ,Y (cid:105) dY, whenever it converges.In this paragraph we address this question of convergence. From the defining expressionfor Φ λ in (34), it will be enough to understand the convergence of the integral J ( t, λ ) = (cid:90) C f Mλ ( Y + tX ) e (cid:104) ρ,Y (cid:105) dY. Lemma 4.5.
There is < η < such that, for every λ ∈ T η and t ≥ , the integral defining J ( t, λ ) converges absolutely. roof. We first observe that it suffices to consider t = 0. Indeed, from (35) it follows that(37) J ( t, λ ) = e t (cid:104) λ,X (cid:105) J ( λ ) . where we have put J ( λ ) = J (0 , λ ).Let Y ∈ C and write Y = Y M + Y M with Y M ∈ a M and Y M ∈ a M . Then(38) f Mλ ( Y ) = e (cid:104) ρ M ,Y (cid:105) ϕ Mλ ( e Y ) = e (cid:104) λ,Y M (cid:105) e (cid:104) ρ M ,Y M (cid:105) ϕ Mλ ( e Y M ) . Proposition B.1 then immediately yields the bound e (cid:104) λ,Y M (cid:105) ≤ e − ηC (cid:104) ρ,Y (cid:105) on the first factor onthe right-hand side of (38).To bound the remaining two factors, we first let Y M + be a point in the W M -orbit of Y M lying in a M, + . Then by Proposition 2.2 and Theorem 2.8 of [19, Ch. X, §
2] we have (cid:12)(cid:12) ϕ Mλ ( e Y M ) (cid:12)(cid:12) ≤ ϕ M Re λ ( e Y M ) = ϕ M Re λ ( e Y M + ) ≤ max w ∈ W M e (cid:104) w Re λ,Y M + (cid:105) ϕ M ( e Y M + ) (cid:28) max w ∈ W M e (cid:104) w Re λ,Y M + (cid:105) e −(cid:104) ρ M ,Y M + (cid:105) (1 + (cid:107) Y M + (cid:107) ) | Φ + | . Now (cid:107) Y M + (cid:107) = (cid:107) Y M (cid:107) and (cid:104) ρ M , Y M (cid:105) ≤ (cid:104) ρ M , Y M + (cid:105) . Thus there is a c > (cid:12)(cid:12) e (cid:104) ρ M ,Y M (cid:105) ϕ Mλ ( e Y M ) (cid:12)(cid:12) ≤ c max w ∈ W M e (cid:104) w Re λ,Y M + (cid:105) (1 + (cid:107) Y M (cid:107) ) | Φ + | . Using the W -invariance of (cid:104)· , ·(cid:105) and Y M = Y − Y M , we obtainmax w ∈ W M (cid:104) w Re λ, Y M + (cid:105) = max w ∈ W M (cid:104) w Re λ, Y M (cid:105) ≤ max w ∈ W |(cid:104) w Re λ, Y (cid:105)| + max w ∈ W |(cid:104) w Re λ, Y M (cid:105)| . Note that w Re λ = Re wλ and wλ ∈ T η . Using Remark A.1, we have |(cid:104) Re λ, Y (cid:105)| (cid:28) − η (cid:104) ρ, Y (cid:105) for all λ ∈ T η and Y ∈ C . From this, and Proposition B.1, we deduce that there is a constant C (cid:48) >
0, depending only on d , such thatmax w ∈ W M (cid:104) w Re λ, Y M + (cid:105) ≤ − ηC (cid:48) (cid:104) ρ, Y (cid:105) . Similarly, we can find c > (cid:107) Y M (cid:107) ≤ c (cid:107) Y (cid:107) . Hence if η > c (cid:48) > (cid:12)(cid:12) f Mλ ( Y ) (cid:12)(cid:12) ≤ c (cid:48) e − (cid:104) ρ,Y (cid:105) (1 + (cid:107) Y (cid:107) ) | Φ + | . We therefore obtain, (cid:90) C (cid:12)(cid:12) f Mλ ( Y ) e (cid:104) ρ,Y (cid:105) (cid:12)(cid:12) dY ≤ (cid:90) C e (cid:104) ρ,Y (cid:105) (1 + (cid:107) Y (cid:107) ) | Φ + | dY. Using Remark A.1 we see that this last integral is finite. Thus the integral defining J ( λ ),and hence J ( t, λ ), converges absolutely. (cid:3) It follows that for λ ∈ T η we have(39) I ( t, λ ) = (cid:88) w ∈ W M \ W c ( wλ ) c M ( wλ ) J ( wλ ) e t (cid:104) wλ,X (cid:105) , where, we recall, J ( λ ) = J (0 , λ ). .4. Main term replacement.
Having established that I ( t, λ ) is well-defined, we nowshow, using Proposition 4.2, that h t ( λ ) can be approximated by I ( t, λ ). Proposition 4.6.
There are c, C > such that, for regular λ ∈ i a ∗ , h t ( λ ) = CI ( t, λ ) + O λ ( e − ct ) . Before we prove this proposition, we establish the following lemma:
Lemma 4.7.
Let < η < and define E = X − (cid:80) ri =1 [0 , η ] β ∨ i = (cid:80) ri =1 [1 − η, β ∨ i and E + = E ∩ P + . Then if η is sufficiently small, we have:(i) For every V in the closure P + (cid:114) E + of P + (cid:114) E + we have (40) (cid:104) ρ, V (cid:105) ≤ (cid:104) ρ, X (cid:105) − δ for some δ = δ ( η ) > uniform in all V .(ii) Let ∆ M ⊆ ∆ denote the subset of simple roots in M . Then there exists c = c ( η ) > such that for every X ∈ E and α ∈ ∆ (cid:114) ∆ M we have α ( X ) ≥ c .Proof. Since X is the unique point in P + at which the linear map X (cid:55)→ (cid:104) ρ, X (cid:105) attainsits maximum, and P + (cid:114) E + is closed, it will suffice to show that every point in P + (cid:114) E + isbounded away from X . But this is clear, since P + = ( X − (cid:80) ri =1 [0 , ∞ ) β ∨ i ) ∩ a + by LemmaA.2. Note that this argument holds for any 0 < η < δ of coursedepending on η .For the second part, we note that for every α ∈ ∆ (cid:114) ∆ M we have α ( X ) = 2. Henceif η is chosen sufficiently small, each such α stays bounded away from 0 on E so that ( ii )follows. (cid:3) Proof of Proposition 4.6.
Let G ( t, λ ) = e − t (cid:104) ρ,X (cid:105) (cid:90) P + t f λ ( X ) e (cid:104) ρ,X (cid:105) dX. From (33) and (36), as well as the volume computation in Corollary A.4, we have h t ( λ ) = CG ( t, λ ) + O ( e − c t ) , for some c , C >
0. Next, let E + be defined as in Lemma 4.7 with η sufficiently small. Put H ( t, λ ) = (cid:90) t E + f λ ( X ) e (cid:104) ρ,X − tX (cid:105) dX. Then G ( t, λ ) = H ( t, λ ) + O ( e − c t ) for some absolute constant c > i ).Note that for any X ∈ t E we have α ( X ) ≥ t/ α ∈ ∆ (cid:114) ∆ M , where ∆ M ⊆ ∆ denotesthe subset of simple roots in M .Using Lemma 4.7( ii ) and Proposition 4.2, there exists c > f λ ( X ) − Φ λ ( X ) (cid:28) λ e − c t for every X ∈ t E + and every regular λ ∈ i a ∗ . Hence, for regular λ ∈ i a ∗ , we have H ( t, λ ) = (cid:90) t E + Φ λ ( X ) e (cid:104) ρ,X − tX (cid:105) dX + O λ ( e − c t )= (cid:90) t E + − tX Φ λ ( Y + tX ) e (cid:104) ρ,Y (cid:105) dY + O λ ( e − c t ) , fter a change of variables. Noting that t E + − tX ⊆ C , we can then extend the integral toall of C , incurring an additional error of O λ ( e − c t ), for some c > (cid:3) A non-vanishing result.
We now show that I ( t, λ ) is generically non-vanishing on i a ∗ . This will follow from the absolute convergence of I ( t, λ ), the periodic behavior of Φ λ along X for rational λ , and a linear independence of characters argument. Proposition 4.8.
There exists a finite number of hyperplanes H , . . . , H N ⊆ i a ∗ and adiscrete set D ⊂ i a ∗ such that for every λ ∈ i a ∗ (cid:114) ( H ∪ · · · ∪ H N ∪ D ) , there exists t = t λ ≥ with I ( t, λ ) (cid:54) = 0 .Proof. We first claim that there exists a discrete subset A ⊆ T η such that J ( t, wλ ) (cid:54) = 0 forall λ ∈ T η (cid:114) A , w ∈ W , and t ≥
0. Indeed, for fixed m ∈ M , the map λ (cid:55)→ φ Mλ ( m ) isholomorphic in λ ∈ a ∗ C . Let η > J ( t, λ ) converges absolutely, J ( t, λ ) is in fact a holomorphicfunction for λ ∈ T η . From the non-vanishing at λ = 0 J ( t,
0) = (cid:90) C e (cid:104) ρ + ρ M ,Y (cid:105) dY (cid:54) = 0 , it follows that J ( t, λ ) does not vanish identically in λ , hence there exists a discrete subset A ⊆ T η such that J ( t, λ ) (cid:54) = 0 for all λ ∈ T η (cid:114) A .By our choice of η , c ( wλ ) / c M ( wλ ) has no pole for regular λ in T η . Let T (cid:48) η denote the setof all regular λ ∈ T η (cid:114) A such that (cid:104) λ, w − X (cid:105) (cid:54) = (cid:104) λ, v − X (cid:105) for all w, v ∈ W M \ W , w (cid:54) = v .Then T (cid:48) η is in fact dense in T η since w − X (cid:54) = v − X for all w (cid:54) = v , w, v ∈ W M \ W , sothat we can obtain T (cid:48) η by removing the hyperplanes { λ | (cid:104) λ, w − X (cid:105) = (cid:104) λ, v − X (cid:105)} for each v (cid:54) = w in W M \ W .Recall the expansion (39). By definition of T (cid:48) η , the phases (cid:104) wλ, X (cid:105) , w ∈ W M \ W , arepairwise distinct for λ ∈ T (cid:48) η . We can therefore apply [16, Lemma 56] (see also [19, Ch.VIII,Lemma 0.1]) to the sum over w ∈ W M \ W to obtain for each λ ∈ T (cid:48) η ,lim sup t →∞ | I ( t, λ ) | ≥ (cid:88) w ∈ W M \ W (cid:12)(cid:12)(cid:12)(cid:12) c ( wλ ) c M ( wλ ) J ( wλ ) (cid:12)(cid:12)(cid:12)(cid:12) . Now for each w ∈ W and λ ∈ T (cid:48) η , the terms c ( wλ ) c M ( wλ ) J ( wλ ) are non-zero, hence for each λ ∈ T (cid:48) η there exists t = t λ such that I ( t, λ ) (cid:54) = 0. (cid:3) Final arguments.
We now finish the proof of Proposition 4.1.By enlarging Ω if necessary, we can assume that Ω ◦ = Ω. Let H , . . . , H N and D be as inProposition 4.8. If necessary, we add in more hyperplanes such that the complement of theunion of them in i a ∗ consists of regular points only. Then the set of rational λ inΩ ∩ ( i a ∗ (cid:114) ( H ∪ . . . ∪ H N ∪ D ))is dense in Ω. Since λ (cid:55)→ T (cid:82) T | h t ( λ ) | is a continuous function, and Ω is compact, Proposition4.1 follows from the next result. emma 4.9. For every rational λ ∈ i a ∗ (cid:114) ( H ∪ · · · ∪ H N ∪ D ) there exists T > depending on λ such that for all T > T we have T (cid:90) T | h t ( λ ) | dt > . Proof.
Recall the definition of rational from Definition 4.3. Let τ ∈ R > (depending on λ )and τ n = nτ for n ∈ N be as in the statement of Lemma 4.4. By Proposition 4.8 we canchoose t ≥ I ( t , λ ) =: a (cid:54) = 0. Write t n = t + τ n for n ∈ N . Note that, byLemma 4.4, we have I ( t + τ n , λ ) = I ( t, λ ) for all t ≥ n ∈ N . Thus I ( t n , λ ) = a , andwe can find a small (cid:15) > n ∈ N and every s ∈ ( t n − (cid:15), t n + (cid:15) ) we have | I ( s, λ ) | ≥ | a | / > . From Lemma 4.6 we may choose T > | h t ( λ ) − I ( t, λ ) | < | a | / t ≥ T . Let N ∈ N be such that t n − (cid:15) ≥ T for all n ≥ N . Then for every n ≥ N we have | h s ( λ ) | ≥ | a | / s ∈ ( t n − (cid:15), t n + (cid:15) ). We therefore get for T > T + τ that1 T (cid:90) T | h t ( λ ) | dt (cid:29) T ( T − − t ) /τ (cid:88) n = N (cid:90) (cid:15) − (cid:15) | h t n + s ( λ ) | ds (cid:29) λ T − T T | a | > , as we wanted to show. (cid:3) Geometric side
The aim of this section is to prove Theorem 2.3. The essential feature of the geometricside is the ergodic properties of the sets gE t ∩ E t in the quotient Γ \ G , as t > g ∈ G vary.5.1. A general bound.
In this section, we return to the general setting of Section 3, with S = G/K and Γ a given torsion free cocompact lattice in G .For g, h ∈ G let N R ( g, h ) denote the number of γ ∈ Γ for which d ( g, γh ) ≤ R . Thisgeneralizes the notation N R ( g ) from (25) when g = h . Let N Γ ( R ) = sup ( g,h ) N R ( g, h ). Lemma 5.1.
There exists c > , independent of R and Γ , such that N Γ ( R ) (cid:28) InjRad(Γ \ S ) − dim S e cR Proof.
Note that N R ( g, h ) = N R ( g, γh ) for all γ ∈ Γ so that we can assume that h is suchthat d ( g, h ) = min γ ∈ Γ d ( g, γh ). If d ( g, h ) > R , then N R ( g, h ) = 0 so that we can assume d ( g, h ) ≤ R .Now if γ ∈ Γ is such that d ( g, γh ) ≤ R , then d ( h, γh ) ≤ d ( g, h ) + d ( g, γh ) ≤ R so that N R ( g, h ) ≤ N R ( h, h ). Applying [1, Lemma 6.18] to N R ( h, h ) gives the assertion ofthe lemma. (cid:3) he following lemma is an adaptation of [24, Lemma 5.1] to our setting. Recall thegeodesic balls B t , first introduced in (11) in the setting of G = SL d ( R ), but more generallydefined for all G using the same notation. Lemma 5.2.
There is c > , depending only on G , such that the following holds. Let R > .Let K ∈ C ( G × G ) be invariant under the diagonal action of Γ and satisfy { g − h : ( g, h ) ∈ supp( K ) } ⊂ B R . Let K be an integral operator on L (Γ \ G ) with kernel K . Then (cid:107) K (cid:107) ≤ (cid:90) Γ \ G (cid:90) G | K ( g, h ) | dgdh + O (cid:18) e cR InjRad(Γ \ S ) dim S vol ((Γ \ G ) ≤ R ) (cid:107) K (cid:107) ∞ (cid:19) . Proof.
By definition, we have (cid:13)(cid:13) K (cid:13)(cid:13) = (cid:90) Γ \ G (cid:90) Γ \ G | (cid:88) γ ∈ Γ K ( g, γh ) | dgdh. Let D ⊂ G be a fundamental domain for Γ. We decompose D into its 2 R -thin and -thickparts D = D ≤ R ∪ D > R , where, similarly to (3), we have put D ≤ R = { x ∈ D : InjRad Γ ( x ) ≤ R } . From the support condition on K , it follows that if K ( g, γh ) (cid:54) = 0 then d ( g, γh ) ≤ R .On the other hand, if h ∈ D > R , and γ, δ ∈ Γ, γ (cid:54) = δ , then d ( γh, δh ) > R . Hence if d ( g, γh ) ≤ R and d ( g, δh ) ≤ R we get2 R < d ( γh, δh ) ≤ d ( g, γh ) + d ( g, δh ) ≤ R which is a contradiction, proving that in the contribution of the 2 R -thick part. Further notethat for fixed g , such γ = γ g ∈ Γ with d ( g, γ g h ) ≤ R is independent of h as otherwise thiswould similarly lead to a contradiction since D is connected. Hence using the left-invarianceof the Haar measure on G , and the diagonal Γ-invariance of K , we can compute (cid:90) D > R (cid:90) D | (cid:88) δ ∈ Γ K ( g, δh ) | dgdh = (cid:90) D > R (cid:90) D | K ( g, γ g h ) | dgdh ≤ (cid:90) G (cid:90) D | K ( g, γ g h ) | dgdh = (cid:90) D (cid:90) G | K ( g, gh ) | dhdg = (cid:90) Γ \ G (cid:90) G | K ( g, gh ) | dhdg = (cid:90) Γ \ G (cid:90) G | K ( g, h ) | dhdg. To deal with the 2 R -thin part, we use Cauchy–Schwarz and the support condition on K toobtain (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) γ ∈ Γ K ( g, γh ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ N Γ ( R ) (cid:88) γ ∈ Γ | K ( g, γh ) | . he factor N Γ ( R ) is bounded by Lemma 5.1. Furthermore, (cid:90) (Γ \ G ) ≤ R (cid:90) Γ \ G (cid:88) γ ∈ Γ | K ( g, γh ) | dgdh ≤ sup g ∈ Γ \ G,h ∈ G | K ( g, h ) | (cid:90) (Γ \ G ) ≤ R (cid:90) Γ \ G (cid:88) γ ∈ Γ B R ( g − γh ) dgdh. By unfolding and changing variables, we have (cid:90) Γ \ G (cid:88) γ ∈ Γ B R ( g − γh ) dg = (cid:90) G B R ( g − h ) dg = m G ( B R ) = c C (cid:90) B + R J ( X ) dX, the last equality coming from the Cartan measure decomposition (19). The Jacobian factorsatisfies J ( X ) (cid:28) e (cid:104) ρ,X (cid:105) (cid:28) e R (cid:107) ρ (cid:107) for X ∈ B + , using (36). From this one deduces that m G ( B R ) (cid:28) R r e R (cid:107) ρ (cid:107) , which is enough to complete the proof. (cid:3) Description of the kernel of A ( τ ) . We return to the setting of G = SL d ( R ). Wewould like to apply Lemma 5.2 to our operator A ( τ ) on L (Γ \ G ) from (8). For this we shallneed a description of its kernel, which we shall denote by A ( τ ); it is a continuous functionon Γ \ ( G × G ). Lemma 5.3.
The kernel of A ( τ ) is (41) A ( τ )( g, h ) = 1 τ (cid:90) τ m G ( E t ) (cid:90) gE t ∩ hE t a ( x ) dxdt. Proof.
By definition, the action of U t aU t on f ∈ L (Γ \ G ) is given by( U t aU t ) f ( g ) = 1 m G ( E t ) (cid:90) E t a ( gh ) (cid:90) E t f ( gh h ) dh dh . We set x = gh and h = xh , apply Fubini and use E − t = E t , to obtain( U t aU t ) f ( g ) = 1 m G ( E t ) (cid:90) gE t a ( x ) (cid:90) xE t f ( h ) dhdx = (cid:90) G (cid:18) m G ( E t ) (cid:90) gE t ∩ hE t a ( x ) dx (cid:19) f ( h ) dh. Averaging over t yields the claim. (cid:3) Support and sup of A ( τ ) . In order to bound the second term in Lemma 5.2, weshall need to estimate the support and the supremum of the kernel function A ( τ ). This isaccomplished in the next result, which is the higher rank generalization of [2, Lemma 26].We recall the norm | · | on G = SL d ( R ) introduced in § E t . Proposition 5.4.
There exists a constant b > depending only on d such that gE t ∩ hE t isempty unless | h − g | ≤ t + b . In particular, A ( τ )( g, h ) = 0 unless | h − g | ≤ τ + b . Further, sup g,h ∈ G | A ( τ )( g, h ) | ≤ (cid:107) a (cid:107) ∞ . One could also quote the more precise result of Knieper, recalled in (14), but an exponential bound ofany quality is all that is required in our application. roof. We begin by establishing the following
Claim:
For x ∈ SL d ( R ) we have | x ij | ≤ | x | . Moreover, there exists a constant c > d , such that for every x = ( x ij ) ∈ SL d ( R ) and every i ∈ { , . . . , d } thereexists j ∈ { , . . . , d } with | x ij | ≥ ce −| x | . Proof of claim:
Write x = ke Z l with k = ( k ij ) , l = ( l ij ) ∈ K = SO( d ) and Z ∈ a + .Let κ i = ( k i , . . . , k id ) and λ j = ( l j , . . . , l dj ) ∈ R d , for i, j = 1 , . . . , d . Let (cid:104)· , ·(cid:105) denote thestandard inner product on R d . Then κ , . . . , κ d and λ , . . . , λ d are both orthonormal basesof R d , and x ij = (cid:104) κ i , λ j e Z (cid:105) . In particular, by Cauchy–Schwarz we have | x ij | = |(cid:104) κ i , λ j e Z (cid:105)| ≤ (cid:107) λ j e Z (cid:107) / ≤ e (cid:107) Z (cid:107) ∞ = | x | , proving the first statement.For the second statement, we begin by observing that for every j = 1 , . . . , d we have (cid:104) κ i , κ i e Z (cid:105) = d (cid:88) j =1 e Z j k ij ≥ (min j e Z j ) (cid:107) κ i (cid:107) = min j e Z j ≥ e −(cid:107) Z (cid:107) ∞ = e −| x | . Writing κ i as a linear combination κ i = a λ + . . . + a d λ d , we obtain e −| x | ≤ (cid:104) κ i , κ i e Z (cid:105) ≤ d (cid:88) j =1 | a j | |(cid:104) κ i , λ j e Z (cid:105)| . Note that the a i are uniformly bounded since k and l are contained in a compact set. Hencethere exists c > j ∈ { , . . . , d } such that | x ij | = |(cid:104) κ i , λ j e Z (cid:105)| ≥ ce −| x | , as claimed.Continuing with the proof of the proposition, suppose that there exists y ∈ h − gE t ∩ E t .We can assume without loss that h − g = e Y for some suitable Y = ( Y , . . . , Y n ) ∈ a + .Since y ∈ E t , the first part of the claim shows that | y ij | ≤ e t for all i, j = 1 , . . . , d .Moreover, since y ∈ e Y E t , we can write y = e Y x for some x ∈ E t . Then y ij = e Y i x ij and thesecond part of the claim shows that for every i there is j i such that | x ij i | ≥ ce − t . Puttingthis together, we obtain e t ≥ ce Y i e − t , so that e Y i ≤ c − e t .By similar arguments, we also know that | x ij | ≤ e t for all i, j , and that for every i thereexists j (cid:48) i with e Y i | x ij (cid:48) i | = | y ij (cid:48) i | ≥ ce − t , that is, e Yi ≥ ce − t .These two inequalities together imply the first assertion of the corollary, and hence also theassertion the support of A ( τ ). The last assertion, on the sup of A ( τ ), is a direct consequenceof the definition (41). (cid:3) Recall from § E t ⊂ B t (cid:107) X (cid:107) . It then follows from Proposition 5.4 that we cantake R = (cid:107) X (cid:107) (2 τ + b ) in Lemma 5.2. Inserting this and the sup norm bound of Proposition5.4 into Lemma 5.2 we obtain the second term in Theorem 2.3. .4. First term.
To establish Theorem 2.3 it remains to estimate the first term in Lemma5.2, for the operator A ( τ ) with kernel A ( τ ).For a measurable set E ⊆ G we write ρ Γ \ G ( E ) for the action by convolution of thenormalized characteristic function of E on L (Γ \ G ). Thus ρ Γ \ G ( E ) f ( x ) = 1 m G ( E ) (cid:90) E f ( xg ) dg. We shall in particular be interested in this operator for sets of the form gE t ∩ E t .We begin with the following elementary upper bound on the quantity A ( τ ) = (cid:90) Γ \ G (cid:90) G | A ( τ )( g, h ) | dgdh. Lemma 5.5.
For all τ > we have A ( τ ) ≤ τ (cid:90) E τ + b (cid:18)(cid:90) τ max { , ( | g |− b ) / } m G ( gE t ∩ E t ) m G ( E t ) (cid:107) ρ Γ \ G ( gE t ∩ E t ) a (cid:107) L (Γ \ G ) dt (cid:19) dg, where b > is as in Proposition 5.4.Proof. From the definition (41) of A ( τ ), we see that A ( τ ) = 1 τ (cid:90) Γ \ G (cid:90) G (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) τ m G ( E t ) (cid:90) gE t ∩ hE t a ( x ) dxdt (cid:12)(cid:12)(cid:12)(cid:12) dgdh. Changing variables x (cid:55)→ hx and g (cid:55)→ h − g , this is1 τ (cid:90) Γ \ G (cid:90) G (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) τ m G ( E t ) (cid:90) gE t ∩ E t A ( hx, hx ) dxdt (cid:12)(cid:12)(cid:12)(cid:12) dgdh. In view of Proposition 5.4, the integral over g ∈ G may be truncated at | g | ≤ τ + b and thelower range of the t integral may be truncated at max { , | g | − b ) / } . Furthermore,1 m G ( E t ) (cid:90) gE t ∩ E t a ( hx ) dx = m G ( gE t ∩ E t ) m G ( E t ) 1 m G ( gE t ∩ E t ) (cid:90) gE t ∩ E t a ( hx ) dx = m G ( gE t ∩ E t ) m G ( E t ) (cid:0) ρ Γ \ G ( gE t ∩ E t ) a (cid:1) ( h ) . We conclude the proof by an application of the Minkowski integral inequality. (cid:3)
We briefly return to the level of generality of Section 3, and let G denote a connectednon-compact simple Lie group with finite center. Let ( π, V π ) be a unitary representation of G . We recall that the integrability exponent q ( π ) of π is defined as q ( π ) = inf { q > (cid:104) π ( g ) v , v (cid:105) ∈ L q ( G ) for v , v in a dense subspace of V π } . A classical result of Borel–Wallach [6], Cowling [11], and Howe–Moore [18] states that π hasa spectral gap precisely when q ( π ) < ∞ ; we can effectively take this as our definition ofspectral gap in our setting.Nevo [29] (see also [15, Theorem 4.1]) has proved a mean ergodic theorem for measurepreserving actions G on a probability space ( X, µ ) whose associated unitary representation L ( X, µ ) has a spectral gap. We shall be interested in the action of G on X = Γ \ G by right-translation, where Γ < G is a lattice. The associated unitary representation is then ρ \ G , he restriction of the right-regular representation ρ Γ \ G to the subspace L (Γ \ G ) of L (Γ \ G )consisting of functions f with (cid:82) Γ \ G f = 0. In this setting, they show the following result. Proposition 5.6 (Nevo) . Let G be a connected non-compact simple Lie group with finitecenter. Let Γ < G be a lattice. Then exist constants θ, C > , depending on the integrabilityexponents of ρ \ G such that for any measurable E ⊆ G of finite measure, we have (cid:107) ρ \ G ( E ) (cid:107) ≤ Cm G ( E ) − θ . A famous result of Kazhdan [20] states that, when G furthermore has rank at least 2,sup π : π G =0 q ( π ) < , 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 Γ. In particular,this is true of G = SL d ( R ) for d ≥ G = SL d ( R ), for d ≥
3, and again require Γ < SL d ( R ) to be a uniformlattice. Recall that a ∈ L (Γ \ G ). We can therefore apply Proposition 5.6 to estimate thequantity (cid:107) ρ Γ \ G ( gE t ∩ E t ) a (cid:107) L (Γ \ G ) = (cid:107) ρ \ G ( gE t ∩ E t ) a (cid:107) L (Γ \ G ) appearing in Lemma 5.5. Hence there is θ >
0, depending only on d , such that A ( τ ) (cid:28) (cid:107) a (cid:107) L (Γ \ G ) τ (cid:90) E τ + b (cid:18)(cid:90) τ max { , ( | g |− b ) / } m G ( gE t ∩ E t ) − θ m G ( E t ) dt (cid:19) dg. To conclude the proof of Theorem 2.3, it suffices to show the following estimate.
Proposition 5.7.
Let g ∈ G . Then for τ (cid:29) we have (cid:90) E τ + b (cid:18)(cid:90) τ max { , ( | g |− b ) / } m G ( gE t ∩ E t ) − θ m G ( E t ) dt (cid:19) dg (cid:28) τ, the implied constant depending on d . The main point in the proof of the above result is an estimate on the intersections volumes m G ( gE t ∩ E t ), proved in the next paragraph.5.5. Intersection volumes.
We seek to prove the following estimate on the intersectionvolumes m G ( e Y E t ∩ E t ). See § Proposition 5.8.
Let Y ∈ a + and t > . Then (42) m G ( e Y E t ∩ E t ) (cid:28) e −(cid:104) ρ,Y (cid:105) m G ( E t ) . Proof.
Recall the definition of the Frobenius norm (cid:107) · (cid:107) from § E t from (17). We claim that(43) E t ⊆ E t +log √ d ⊆ E t +log √ d . Note that (cid:107) · (cid:107) is bi- K -invariant, so that if g = k e X k , where X = ( X , . . . , X d ) ∈ a , then (cid:107) g (cid:107) = e X + · · · + e X d . Thus we may write E t = K exp( P t ) K , where P t = { X = ( X , . . . , X d ) ∈ a : max { e X + · · · + e X d , e − X + · · · + e − X d } ≤ e t } . We therefore wish to establish (43) for the sets P t and P t = t P , where P is defined in (12): If | X i | ≤ t for all i we have max { e X + · · · + e X d , e − X + · · · + e − X d } ≤ de t , sothat P t ⊆ P t +log √ d . (cid:47) In the other direction, if max { e X + · · · + e X d , e − X + · · · + e − X d } ≤ e t then | X i | ≤ t for all i , so that P t ⊆ P t .It is now enough to prove (42) with E t replaced by E t . Indeed, suppose that(44) m G ( e Y E t ∩ E t ) (cid:28) e −(cid:104) ρ,Y (cid:105) m G ( E t )for any Y ∈ a . We use (43) to bound m G ( e Y E t ∩ E t ) by m G ( e Y E t +log √ d ∩ E t +log √ d ). Insertingthe estimate (44) and the inclusions (43) again we find m G ( e Y E t +log √ d ∩ E t +log √ d ) (cid:28) e −(cid:104) ρ,Y (cid:105) m G ( E t +log √ d ) ≤ e −(cid:104) ρ,Y (cid:105) m G ( E t +log √ d ) . Since m G ( E t +log √ d ) (cid:16) m G ( E t ), by Corollary A.4, we have reduced (42) to (44).We prove (44) using the Iwasawa decomposition G = AN K from § g = auk sothat (cid:107) g (cid:107) = (cid:107) au (cid:107) and (cid:107) g − (cid:107) = (cid:107) u − a − (cid:107) . From the triangle inequality of (cid:107) · (cid:107) on M d ( R ),we see that the conditions (cid:107) a (cid:107) ≤ e t , (cid:107) au − a (cid:107) ≤ e t together imply (cid:107) g (cid:107) ≤ e t . Similarly, theconditions (cid:107) a − (cid:107) ≤ e t , (cid:107) u − a − − a − (cid:107) ≤ e t together imply (cid:107) g − (cid:107) ≤ e t . Thus we obtain theinclusion E t ⊆ { g = auk ∈ G : (cid:107) a (cid:107) , (cid:107) a − (cid:107) , (cid:107) au − a (cid:107) , (cid:107) u − a − − a − (cid:107) ≤ e t } . Applying the same argument to e Y g , for Y ∈ a , and dropping the last two conditions, e Y E t ⊆ { g = auk ∈ G : (cid:107) e − Y a (cid:107) , (cid:107) e Y a − (cid:107) ≤ e t } . We conclude that m G ( e Y E t ∩ E t ) ≤ I ( Y, t ), where I ( Y, t ) = (cid:90) (cid:107) e ± A (cid:107)≤ e t (cid:107) e ± ( A − Y ) (cid:107)≤ e t (cid:90) (cid:107) e A u − e A (cid:107)≤ e t (cid:107) u − e − A − e − A (cid:107)≤ e t dudA. Changing variables u (cid:55)→ e A u − e A + I in the U -integration, we obtain I ( Y, t ) = I U ( t ) (cid:90) (cid:107) e ± A (cid:107)≤ e t (cid:107) e ± ( A − Y ) (cid:107)≤ e t e −(cid:104) ρ,A (cid:105) dA, where I U ( t ) = m U ( u ∈ U : (cid:107) u − I (cid:107) , (cid:107) u − − I (cid:107) ≤ e t ). Changing variables A (cid:55)→ A − Y in the A -integration and dropping a condition, we obtain (cid:90) (cid:107) e ± A (cid:107)≤ e t (cid:107) e ± ( A − Y ) (cid:107)≤ e t e −(cid:104) ρ,A (cid:105) dA = e −(cid:104) ρ,Y (cid:105) (cid:90) (cid:107) e ± ( A + Y ) (cid:107)≤ e t (cid:107) e ± A (cid:107)≤ e t e −(cid:104) ρ,A (cid:105) dA ≤ e −(cid:104) ρ,Y (cid:105) (cid:90) (cid:107) e ± A (cid:107)≤ e t e −(cid:104) ρ,A (cid:105) dA. We write this last integral as I A ( t ). Using (18), this establishes(45) m G ( e Y E t ∩ E t ) ≤ c I e −(cid:104) ρ,Y (cid:105) I A ( t ) I U ( t ) . e now go in reverse. For A ∈ a we change variables u (cid:55)→ e − A u − e − A + I to obtain e −(cid:104) ρ,A (cid:105) I U ( t ) = m U ( u ∈ U : (cid:107) e A u − e A (cid:107) , (cid:107) u − e − A − e − A (cid:107) ≤ e t ) , so that I A ( t ) I U ( t ) = (cid:90) (cid:107) e ± A (cid:107)≤ e t (cid:90) (cid:107) ( e A u ) ± − e ± A (cid:107)≤ e t dudA. Again by the triangle inequality and (18) this gives I A ( t ) I U ( t ) ≤ (cid:90) (cid:90) (cid:107) ( au ) ± (cid:107)≤ e t dadu = c − I m G ( E t +log 2 ) (cid:16) m G ( E t ) . Combining this last inequality with (45) yields (44). (cid:3)
Proof of Proposition 5.7.
From Proposition 5.8 and Lemma A.4 we have, for Y ∈ a + m G ( e Y E t ∩ E t ) − θ m G ( E t ) (cid:28) e − (1 − θ ) (cid:104) ρ,Y (cid:105) m G ( E t ) − θ (cid:28) e − (1 − θ ) (cid:104) ρ,Y (cid:105) e − tθ (cid:104) ρ,X (cid:105) , so that, using (13), (cid:90) E τ + b (cid:18)(cid:90) τ max { , | g |− b ) / } m G ( gE t ∩ E t ) − θ m G ( E t ) dt (cid:19) dg (cid:28) (cid:90) P +2 τ + b (cid:18)(cid:90) τ max { , ( (cid:107) Y (cid:107) ∞ − b ) / } e − tθ (cid:104) ρ,X (cid:105) dt (cid:19) e − (1 − θ ) (cid:104) ρ,Y (cid:105) J ( Y ) dY. Inserting J ( X ) (cid:28) e (cid:104) ρ,X (cid:105) for X ∈ a + from (36), and evaluating the t -integral, we majorizethe above expression by (cid:90) P +2 τ + b e − max { , (cid:107) Y (cid:107) ∞ − b } θ (cid:104) ρ,X (cid:105) e θ (cid:104) ρ,Y (cid:105) dY. We break up the last integral as I + I , according to a +1 = P + b and a +2 ( τ ) = { Y ∈ a + : b ≤ (cid:107) Y (cid:107) ∞ ≤ τ + b } . Since a +1 is independent of τ (as is the integrand), we have I (cid:28)
1. Next, we have I (cid:28) (cid:90) a +2 ( τ ) e θ ( (cid:104) ρ,Y (cid:105)−(cid:107) Y (cid:107) ∞ (cid:104) ρ,X (cid:105) ) dY. We will drop several of the conditions on Y in the course of the proof to simplify the integral.Write τ (cid:48) = 2 τ + b . We distinguish the case of d being even and odd. Define(46) s = (cid:40) d/ d even , ( d + 1) / d odd . We first assume that d ≥ Y in the integral as Y = ( a , . . . , a s , − a − · · · − a s + b + · · · + b s , − b s , . . . , − b ) ith τ (cid:48) ≥ a ≥ · · · ≥ a s ≥ τ (cid:48) ≥ b ≥ · · · ≥ b s ≥
0. Then (cid:104) ρ, Y (cid:105) = (cid:80) si =1 ( s − i )( a i + b i ),and, using Lemma A.1, (cid:104) ρ, X (cid:105) = 2 A with A = (cid:80) si =1 ( s − i ). Moreover, (cid:107) Y (cid:107) ∞ = max { a , b } .Hence I ( τ ) (cid:28) (cid:90) . . . (cid:90) e θ ( (cid:80) si =1 ( s − i )( a i + b i ) − A max { a ,b } ) , where the integral runs over (cid:40) a , . . . , a s , b , . . . , b s (cid:12)(cid:12)(cid:12)(cid:12) τ (cid:48) ≥ a ≥ , a ≥ a ≥ , . . . , a s − ≥ a s ≥ ,τ (cid:48) ≥ b ≥ , b ≥ b ≥ , . . . , b s − ≥ b s ≥ . (cid:41) We now integrate first a s and b s and continue successively with a s − , . . . , a and b s − , . . . , b until only a and b remain to be integrated. In this way we obtain I ( τ ) (cid:28) (cid:90) (cid:90) τ (cid:48) e θA ( a + b − { a ,b } ) da db = (cid:90) (cid:90) τ (cid:48) e − θA | a − b | da db = 2 τ (cid:48) (cid:90) τ (cid:48) e − θAx dx (cid:28) τ. If d is even, then d ≥ s ≥
2. We can write Y = ( a , . . . , a s , − b s , . . . , − b ) with τ (cid:48) ≥ a ≥ · · · ≥ a s ≥ τ (cid:48) ≥ b ≥ · · · ≥ b s ≥
0, and(47) a + · · · + a s = b + · · · + b s . Then (cid:107) Y (cid:107) ∞ = max { a , b } and (cid:104) ρ, Y (cid:105) = s (cid:88) i =1 ( s − i + 1 / a i + b i ) = s (cid:88) i =1 ( s − i + 1) a i + s − (cid:88) i =1 ( s − i ) b i , where we have used (47) to replace b s . Furthermore, using Lemma A.1, we have (cid:104) ρ, X (cid:105) = A + B , where we have put A = s (cid:88) i =1 ( s − i + 1) and B = s (cid:88) i =1 ( s − i ) . Extending the domain of integration if necessary, we obtain I ( τ ) ≤ (cid:90) . . . (cid:90) e θ ( (cid:80) si =1 ( s − i +1) a i + (cid:80) s − i =1 ( s − i ) b i − ( A + B ) max { a ,b } ) , where the integral runs over (cid:40) a , . . . , a s , b , . . . , b s − (cid:12)(cid:12)(cid:12)(cid:12) τ (cid:48) ≥ a ≥ , a ≥ a ≥ , . . . , a s − ≥ a s ≥ ,τ (cid:48) ≥ b ≥ , b ≥ b ≥ , . . . , b s − ≥ b s − ≥ (cid:41) . As before, we successively integrate over a s , a s − , . . . , a and b s − , b s − , . . . , b , obtaining I ( τ ) (cid:28) (cid:90) (cid:90) τ (cid:48) e θ ( Aa + Bb − ( A + B ) max { a ,b } ) da db ≤ (cid:90) (cid:90) τ (cid:48) e − θB | a − b | da db (cid:28) τ as desired. (cid:3) Appendix A. Cones and volumes
Recall the definition of P + and E t in § P + as the intersection of a + with a suitable cone in a and to calculate the asymptotic volume of the set E t . .1. Cones.
We will identify a as before with the subspace of R d consisting of all vectors X = ( X , . . . , X d ) with X + · · · + X d = 0. We also identify a with the set of all trace-zerodiagonal matrices whenever convenient. Lemma A.1.
Let X ∈ P be any point in P satisfying (cid:104) X , ρ (cid:105) = max X ∈ P + (cid:104) X, ρ (cid:105) . Let { e , . . . , e d } denote the usual standard basis of R d . Then (48) X = (cid:40) e + · · · + e s − e s +1 − · · · − e d if d even ; e + · · · + e s − − e s +1 − · · · − e d if d odd , where s is defined in (46) . In particular, X is unique.Proof. We write the half-sum of positive roots ρ ∈ a ∗ in coordinates as ρ = ( ρ , . . . , ρ d ) ∈ R d (cid:39) ( R d ) ∗ , where ρ + · · · + ρ d = 0. We have ρ ≥ · · · ≥ ρ d and ρ i = − ρ d +1 − i for every i ≤ d/ d is odd, then ρ ( d +1) / = 0. Let X = ( X , . . . , X d ) ∈ P . Then (cid:104) X, ρ (cid:105) = (cid:88) i ≤ d/ ( X i − X d +1 − i ) ρ i . Since X ∈ P , we have − ≤ X i − X d +1 − i ≤
2. Furthermore, ρ i > i ≤ d/
2, so this sumis maximized for X i = 1 and X d +1 − i = −
1. This proves our assertion for d even. For d odd,we finally note that X + · · · + X d = 0 forces X ( d +1) / = 0. (cid:3) Lemma A.2.
There is a basis µ , . . . , µ d − ∈ a ∗ such that µ i ( X ) = 1 , i = 1 , . . . , d − , and P + = C ∩ a + , where C = X + C and C = { X ∈ a : µ i ( X ) ≤ } . In particular, if { β ∨ i } is the basis in a which is dual to { µ i } then C = (cid:40) d − (cid:88) i =1 x i β ∨ i : ∀ i x i ≤ (cid:41) and X = β ∨ + · · · + β ∨ d − .Proof. Note that(49) P + = { X ∈ a : X ≤ , − X d ≤ } ∩ a + . We rewrite (49) using the system of fundamental weights (cid:36) i ∈ a ∗ :(50) P + = { X ∈ a : (cid:36) ( X ) ≤ , (cid:36) d − ( X ) ≤ } ∩ a + . Since (cid:36) ( X ) = (cid:36) d − ( X ) = 1, we can put µ = (cid:36) and µ d − = (cid:36) d − . The strategy is thento complete { µ , µ d − } to a basis of linear forms { µ i } in such a way that(1) µ i ( X ) = 1 for all i = 1 , . . . d − X ∈ a + and µ ( X ) , µ d − ( X ) ≤ µ i ( X ) ≤ i = 2 , . . . , d − n this case, property (1) shows that C = X + C = { X ∈ a | ∀ i : µ i ( X − X ) ≤ } = { X ∈ a | ∀ i : µ i ( X ) ≤ } , and then property (2) combines with (50) to establish P + = C ∩ a + .Note that, relative to the standard basis { e i } of R d , µ is the (restriction to a of the) firstcoordinate functional and µ d − is minus the (restriction to a of the) last coordinate functional.Thus if the ± µ i , for i = 2 , . . . , d −
2, are a linearly independent subset of the remaining d − ≥ X ≥ · · · ≥ X d ≥ − ± X i ≤
1. To assure property (1) we simply need to choose the sign suitably and avoid the( d + 1) / d . Thus for d odd we put µ i = (cid:40) (cid:36) i − (cid:36) i − if 2 ≤ i ≤ s − (cid:36) i − (cid:36) i +1 if s ≤ i ≤ d − , and for d even we put µ i = (cid:40) (cid:36) i − (cid:36) i − if 2 ≤ i ≤ s ; (cid:36) i − (cid:36) i +1 if s + 1 ≤ i ≤ d − , where s is defined in (46). Using the definition (48) of X we quickly verify property (1). (cid:3) Remark
A.1 . Using the explicit description of the basis { µ i } ≤ i ≤ d − , one can easily see thatwriting ρ as a linear combination of the µ i , all the coefficients are positive integers. Hence if Y ∈ C , then (cid:104) ρ, Y (cid:105) < Volumes.Proposition A.3.
Suppose P ⊆ a + is a convex polytope. Let V P denote the vertices of P ,and suppose that V P (cid:51) X (cid:55)→ (cid:104) ρ, X (cid:105) takes its maximum at exactly one vertex v ∈ V P . Thenthere exist c, δ > such that (51) (cid:90) P t (cid:88) w ∈ W σ ( w ) e (cid:104) wρ,X (cid:105) dX = ce t (cid:104) ρ,v (cid:105) (cid:0) O ( e − tδ ) (cid:1) as t → ∞ . Here P t = { tX | X ∈ P } and σ ( w ) denotes the signature of w ∈ W . Before starting the proof of this proposition, we note that (cid:88) w ∈ W σ ( w ) e (cid:104) wρ,X (cid:105) ≥ X ∈ a + because of the convexity of the exponential function. Proof.
First note that (cid:104) wρ, X (cid:105) ≤ (cid:104) ρ, X (cid:105) for all X ∈ a + , and in fact (cid:104) wρ, X (cid:105) < (cid:104) ρ, X (cid:105) if X ∈ a + . If P is not simple, we divide it into disjoint simple polytopes P , . . . , P r whoserespective sets of vertices we denote by V , . . . , V r . The vertices V P are contained in theunion (cid:83) j V j . By Brion’s formula [7] there exist non-zero coefficients a jv ∈ R , j = 1 , . . . , r , v ∈ V j , such that (cid:90) P jt e (cid:104) wρ,X (cid:105) dX = (cid:88) v ∈ V j a jv e t (cid:104) wρ,v (cid:105) . e can find δ > j and v ∈ V j , (cid:104) wρ, v (cid:105) ≤ (cid:104) ρ, v (cid:105) − δ unless w − v = v .Hence,(52) (cid:90) P t (cid:88) w ∈ W σ ( w ) e (cid:104) wρ,X (cid:105) dX = (cid:88) j,v,w : w − v = v a jv σ ( w ) e t (cid:104) ρ,v (cid:105) . It remains to argue that this last sum in brackets is a positive number. If v is such that (cid:104) wρ, v (cid:105) = (cid:104) ρ, v (cid:105) if and only if w = 1 and v = v , then the left hand side of (51) tends to + ∞ as t → ∞ . Moreover, the sum on the right hand side of (52) has exactly one non-zero termin that situation which consequently must be positive. If v does not satisfy this condition,we choose another polytope P (cid:48) ⊆ P with regular vertices (i.e., V P (cid:48) ⊆ a + ) that are sufficientlyclose to the original vertices of P . Then (cid:90) P t (cid:88) w ∈ W σ ( w ) e (cid:104) wρ,X (cid:105) dX ≤ (cid:90) P (cid:48) t (cid:88) w ∈ W σ ( w ) e (cid:104) wρ,X (cid:105) dX, and the right hand side grows like ce tc (cid:48) for suitable c > (cid:104) ρ, v (cid:105) ≥ c (cid:48) > (cid:104) ρ, v (cid:105) − δ provided P (cid:48) is chosen sufficiently close to P . This proves that the sum in brackets on theright hand side of (52) is a positive number. (cid:3) We apply the above result to estimate the volume of E t . Corollary A.4.
There exists c, δ > such that we have m G ( E t ) = ce t (cid:104) ρ,X (cid:105) (cid:0) O ( e − tδ ) (cid:1) as t → ∞ .Proof. By Cartan decomposition we have m G ( E t ) = (cid:90) P t J ( X ) dX = 2 −| Φ + | (cid:90) P t (cid:88) w ∈ W σ ( w ) e (cid:104) wρ,X (cid:105) dX where σ ( w ) denotes the signature of w . The assertion of the lemma then follows fromProposition A.3. (cid:3) Appendix B. Angles and inner products
The purpose of this appendix is to prove the following result, used in the course of theproof of Lemma 4.5.Recall the notation of that lemma. Let X be as in (15) and let M denote its centralizerin G . For Y ∈ a we write Y = Y M + Y M for unique Y M ∈ a M and Y M ∈ a M . As before, for η > T η for the set of all λ ∈ a ∗ C such that Re λ lies in the convex hull of theWeyl group orbit of ηρ . The cone C is defined in Lemma A.2. Proposition B.1.
There are constants C , C > , depending only on d , such that for every Y ∈ C , λ ∈ T η , and w ∈ W we have(i) |(cid:104) wρ, Y M (cid:105)| ≤ − C (cid:104) ρ, Y (cid:105) ;(ii) |(cid:104) Re λ, Y M (cid:105)| ≤ − C η (cid:104) ρ, Y M (cid:105) . n particular, there is a constant C > , depending only on d , such that for all Y ∈ C and λ ∈ T η we have |(cid:104) Re λ, Y M (cid:105)| ≤ − ηC (cid:104) ρ, Y (cid:105) . To prove the proposition we shall work abstractly with linear forms on euclidean space R d , equipped with its standard inner product. Namely, we make the usual identification ofthe standard Cartan subalgebra a of gl d ( R ) with R d , so that a = a G identifies with thetrace-zero hyperplane H = (cid:8) X = ( X , . . . , X d ) ∈ R d : X + · · · + X d = 0 (cid:9) . We are interested in the linear form L ( Y ) = (cid:104) ρ, Y (cid:105) on R d or H . It will be convenient todefine a cone C (cid:48) in R d such that C (cid:48) ∩ H coincides with the cone − C of Proposition A.2.Property (i) will then follow from a similar maximizing property of L on C (cid:48) . Property (ii)requires bounds on the angles the vectors in − C can form with X , which we deduce fromthe relative position of X with H .B.1. Positive linear forms.
Let { e , . . . , e d } denote the standard basis of R d . Definition B.2. If d is even, let C (cid:48) ⊂ R d be the closed orthant C (cid:48) = cone R ≥ { e , . . . , e d/ , − e d/ , . . . , − e d } . If d is odd, let C (cid:48) = O + ∪ O − ⊂ R d be the union of the closed orthants O + = cone R ≥ { e , . . . , e ( d +1) / − , e ( d +1) / , − e ( d +1) / , . . . , − e d } O − = cone R ≥ { e , . . . , e ( d +1) / − , − e ( d +1) / , − e ( d +1) / , . . . , − e d } . Remark
B.1 . Recall the explicit description of X ∈ R d as given in (48) of Appendix A.If V d denotes the set of vertices of the unit cube [ − , d , then ± X ∈ V d for d even and ± X ± e ( d +1) / ∈ V d for d odd. In particular, when d is odd − X lies on the edge of[ − , d connecting the vertices X ± e ( d +1) / . We deduce that the origin of R d is a vertex of − X + [ − , d if d is even, and it is the midpoint of an edge of − X + [ − , d if d is odd.It is easy to see that when d is even, − C (cid:48) is the unique orthant in R d containing − X +[ − , d , and when d is odd, − X + [ − , d ⊂ − C (cid:48) . Remark
B.2 . If d is even, then C (cid:48) does not contain any non-trivial linear subspace, and if d is odd, the only non-trivial linear subspace of C (cid:48) is the one-dimensional space R e ( d +1) / . Definition B.3.
We call a linear form L : R d −→ R positive if it is non-negative on C (cid:48) andpositive on C (cid:48) (cid:114) { } if d is even, and positive on C (cid:48) (cid:114) R e ( d +1) / if d is odd. Let L ⊆ R d denote the ray R ≥ X . Since X ∈ C (cid:48) , and C (cid:48) is a cone, we have L ⊂ C (cid:48) .We identify the space of linear forms on R d with ( R d ) ∗ (cid:39) R d via the dual basis to { e , . . . , e d } . If L is a linear form, written as L = ( L , . . . , L d ) with respect to this iden-tification, and σ ∈ S d is a permutation, we define the linear form σL by the rule σL :=( L σ (1) , . . . , L σ ( d ) ). We identify Lemma B.4.
Let L ∈ ( R d ) ∗ be a positive linear form. Let L (cid:48) be a form in the closed convexhull of { σL : σ ∈ S d } ⊆ ( R d ) ∗ . Then for all X ∈ L we have L (cid:48) ( X ) ≤ L ( X ) . roof. A linear form L = ( L , . . . , L d ) is positive if and only if (cid:40) L , . . . , L d/ > , L d/ , . . . , L d < , if d is even; L , . . . , L ( d +1) / − > , L ( d +1) / = 0 , L ( d +1) / , . . . , L d < , if d is odd . It follows that for such L we have L ( X ) = | L | + · · · + | L d | , hence σL ( X ) ≤ L ( X ) forevery σ ∈ S d . Hence if L (cid:48) is in the closed convex hull of { σL : σ ∈ S d } in ( R d ) ∗ , we also have L (cid:48) ( X ) ≤ L ( X ). (cid:3) B.2.
Hyperplane intersections.
We now put C (cid:48)(cid:48) = C (cid:48) ∩ H . Lemma B.5. C (cid:48)(cid:48) is a convex cone in R d of dimension d − that satisfies C (cid:48)(cid:48) ∩ ( − C (cid:48)(cid:48) ) = { } .Moreover, there exists γ > such that the angle between any non-zero vector in C (cid:48)(cid:48) and thevector X is at most π/ − γ .Proof. The first part of the lemma follows directly from the description of C (cid:48) in DefinitionB.2, coupled with Remark B.2.For the assertion on the bound of the angles, we proceed as follows: if d is even, then C (cid:48) is just a single orthant in R d , hence the angle between any two vectors is bounded by π/ X does not lie on the boundary of C (cid:48) , there exists γ > X and any vector in C (cid:48) is bounded from above by π/ − γ .For d odd, recall that C (cid:48) = O + ∪ O − and X ∈ O + ∩ O − , hence the angle between X andany vector in C (cid:48) is at most π/
2. Moreover, it is readily seen from the explicit description of X and O ± that the only vectors in C (cid:48) that are orthogonal to X are the vectors on the line R e ( d +1) / . Since e ( d +1) / (cid:54)∈ H and C (cid:48)(cid:48) = C (cid:48) ∩ H is closed, the angle between X and non-zerovectors in C (cid:48)(cid:48) must be bounded from above by π/ − γ for some γ > (cid:3) Note that L ⊂ C (cid:48)(cid:48) since X ∈ H . Proposition B.6.
Let M ⊆ H be a subvectorspace that is orthogonal to L . Then for every X ∈ L the section ( X + M ) ∩ C (cid:48)(cid:48) is compact. Moreover, if M is of dimension d − , then C (cid:48)(cid:48) = (cid:91) X ∈ L (( X + M ) ∩ C (cid:48)(cid:48) ) , and writing X = rX for a suitable r ≥ we have ( X + M ) ∩ C (cid:48)(cid:48) = r (( X + M ) ∩ C (cid:48)(cid:48) ) .Proof. Follows from Lemma B.5. (cid:3)
B.3.
Application to linear forms.
In the following lemma, we shall take L to be a positivelinear form in the sense of Definition B.3, and M ⊆ H a vector space of dimension d − L . Lemma B.7.
There exists a constant c > such that L ( X + Z ) ≤ cL ( X ) for all X ∈ L and Z ∈ M with X + Z ∈ C (cid:48)(cid:48) .Proof. For fixed X ∈ L , the set of all Z ∈ M such that X + Z ∈ C (cid:48)(cid:48) is compact by PropositionB.6. In particular there is D ⊆ M compact such that ( X + M ) ∩ C (cid:48)(cid:48) = X + D . Hence any + Z ∈ C (cid:48)(cid:48) with X ∈ L and Z ∈ M can be written as X + Z = rX + rZ (cid:48) for some suitable r ≥ Z (cid:48) ∈ D . Putting c = 1 + max Z (cid:48) ∈ D L ( Z (cid:48) ) L ( X )we then get L ( X + Z ) = rL ( X ) + rL ( Z (cid:48) ) ≤ cL ( X ). (cid:3) B.4.
Proof of Proposition B.1.
We now apply the previous results to our situation.Namely, we identify a with R d as usual so that a = a G is the hyperplane H . The cone C (cid:48)(cid:48) = C (cid:48) ∩ H then coincides with the cone − C that we considered in Appendix A, and X coincides with the vector of the same name from Appendix A.We consider the linear form L ( Y ) = (cid:104) ρ, Y (cid:105) , Y ∈ C . Let a M ⊆ a be as before. Then R X ⊆ a M . In fact, if d is even, then a M = R X and a M is orthogonal to a M of dimension d −
2. If d is odd, then a M has dimension 2. In that case, let V ⊆ a M be the orthogonalcomplement to the line R X so that V ⊕ a M is orthogonal to R X of dimension d − Y ∈ C (cid:48)(cid:48) , we write Y = Y M + Y M for unique Y M ∈ a M and Y M ∈ a M . If d iseven, then Y M ∈ R ≥ X , and if d is odd, we can uniquely write Y M = Y (cid:48) M + Y (cid:48)(cid:48) M with unique Y (cid:48) M ∈ R ≥ X and Y (cid:48)(cid:48) M ∈ V .Having set up the notation, we now note that (cid:104) ρ, ·(cid:105) vanishes on V , hence part (i) ofProposition B.1 is an immediate consequence of Lemma B.4.For (ii) we first note that by definition of T η , Re λ is contained in the convex hull of theWeyl group orbit of ηρ so that (cid:104) Re λ, wX (cid:105) ≤ η (cid:104) ρ, X (cid:105) by Lemma B.7 for every w ∈ W .This establishes (ii) for d is even. For d odd, we have η (cid:104) ρ, Y M (cid:105) = η (cid:104) ρ, Y (cid:48) M (cid:105) ≥ (cid:104) Re λ, Y (cid:48) M (cid:105) since (cid:104) ρ, ·(cid:105) vanishes on V . Taking Y (cid:48) M = X , the possible Y (cid:48)(cid:48) M lie in a compact set so that (cid:104) Re λ, Y (cid:48)(cid:48) M (cid:105) is bounded from above by ηc for c some absolute constant. Scaling Y (cid:48) M = X bya scalar, the compact set of Y (cid:48)(cid:48) M gets scaled by the same scalar. Hence there is C > Y ∈ C (cid:48)(cid:48) we have (cid:104) Re λ, Y M (cid:105) ≤ ηC (cid:104) ρ, Y M (cid:105) (cid:3) References [1] M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet. On the growthof L -invariants for sequences of lattices in Lie groups. Ann. of Math. (2), 185(3): 711–790, 2017.[2] M. Abert, N. Bergeron, and E. Le Masson. Eigenfunctions and random waves in the Benjamini–Schramm limit. arXiv preprint https://arxiv.org/abs/1810.05601 [3] N. Arantharaman, E. Le Masson. Quantum ergodicity on large regular graphs.
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