Bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces

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Bottom of the L spectrum of the Laplacian on locally symmetric spaces

Jean-Philippe Anker, Hong-Wei Zhang

To cite this version:

Jean-Philippe Anker, Hong-Wei Zhang. Bottom of the L² spectrum of the Laplacian on locally symmetric spaces. 2020. �hal-02865274v2�

BOTTOM OF THE L² SPECTRUM OF THE LAPLACIAN ON LOCALLY SYMMETRIC SPACES

JEAN-PHILIPPE ANKER AND HONG-WEI ZHANG

In memory of Michel Marias (1953-2020), who introduced us to locally symmetric spaces

Abstract. We estimate the bottom of theL²spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincaré series. Our main result is the higher rank analog of a characterization due to Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves upon previous results obtained by Leuzinger and Weber in higher rank.

1. Introduction

We adopt the standard notation and refer to [Hel00] for more details. LetGbe a semi-simple Lie group, connected, noncompact, with finite center, and K be a maximal compact subgroup of G. The homogeneous space X = G/K is a Riemannian symmetric space of noncompact type. Let g = k⊕p be the Cartan decomposition of the Lie algebra of G. The Killing form of g induces a K-invariant inner product h. , .i on p, hence a G-invariant Riemannian metric on G/K. Fix a maximal abelian subspace a inp. We identify a with its dual a^∗ by means of the inner product inherited from p. Let Γ be a discrete torsion-free subgroup of G that acts freely and properly discontinuously on X. Then Y= Γ\X is a locally symmetric space, whose Riemannian structure is inherited fromX. We denote byd(. , .)the joint Riemannian distance on Xand Y, byntheir joint dimension, and by`their joint rank, which is the dimension ofa.

Let Σ⊂ a be the root system of (g,a) and let W be the associated Weyl group. Choose a positive Weyl chambera⁺⊂aand letΣ⁺⊂Σbe the corresponding subsystem of positive roots.

Denote by ρ = ¹₂P

α∈Σ⁺m_αα the half sum of positive roots counted with their multiplicities.

Occasionally we shall need the reduced root system Σ_red={α∈Σ|^α₂ ∈/Σ}.

Consider the classical Poincaré series P_s(xK, yK) =X

γ∈Γe−sd(xK,γyK) ∀s >0,∀x, y∈G (1.1)

and denote byδ(Γ) = inf{s >0|P_s(xK, yK)<+∞}its critical exponent, which is independent of xK andyK. Recall thatδ(Γ)∈[0,2kρk]may be also defined by

δ(Γ) = lim sup

R→+∞

logN_R(xK, yK)

R ∀x, y∈G,

where N_R(xK, yK) = |{γ ∈ Γ|d(xK, γyK) ≤ R}| denotes the orbital counting function. Fi- nally, let ∆Y be the Laplace-Beltrami operator on Y and let λ0(Y) be the bottom of the L² spectrum of−∆_Y. The following celebrated result, due to Elstrodt ([Els73a], [Els73b], [Els74]), Patterson [Pat76], Sullivan [Sul87] and Corlette [Cor90], expressesλ₀(Y)in terms ofρandδ(Γ) in rank one.

Theorem 1.1. In the rank one case (`= 1), we have λ0(Y) =

(kρk² if 0≤δ(Γ)≤ kρk,

kρk²−(δ(Γ)− kρk)² if kρk ≤δ(Γ)≤2kρk. (1.2)

2020Mathematics Subject Classification. 22E40, 11N45, 35K08, 47A10, 58J50.

Key words and phrases. Locally symmetric space, L² spectrum, Poincaré series , critical exponent, Green function, heat kernel.

1

This result was extended in higher rank as follows by Leuzinger [Leu04] and Weber [Web08].

Let ρmin = min_H∈a+,kHk=1hρ, Hi ∈(0,kρk]. Notice that ρmin = kρk in rank one and thus the following theorem reduces to Theorem 1.1.

Theorem 1.2. In the general case (`≥1), the following estimates hold:

• Upper bound:

λ0(Y)≤

(kρk² if 0≤δ(Γ)≤ kρk, kρk²−(δ(Γ)− kρk)² if kρk ≤δ(Γ)≤2kρk.

• Lower bound:

λ0(Y)≥

(kρk² if 0≤δ(Γ)≤ρmin, max{0,kρk²−(δ(Γ)−ρ_min)²} if ρ_min ≤δ(Γ)≤2kρk.

In other terms,



















λ0(Y) =kρk² if δ(Γ)∈

0, ρmin

, λ₀(Y)∈

kρk²−(δ(Γ)−ρ_min)²,kρk²

if δ(Γ)∈

ρ_min,kρk , λ₀(Y)∈

kρk²−(δ(Γ)−ρ_min)²,kρk²−(δ(Γ)−kρk)²

if δ(Γ)∈

kρk,kρk+ρ_min , λ₀(Y)∈

0,kρk²−(δ(Γ)−kρk)²

if δ(Γ)∈

kρk+ρ_min,2kρk .

In this paper, we first improve the lower bound ofλ₀(Y)in Theorem1.2by a slight modifica- tion of the classical Poincaré series (1.1). Letδ⁰(Γ)denote the critical exponent of the modified Poincaré series (2.7) associated to the polyhedral distance (2.1).

Theorem 1.3. The following lower bound holds for the bottom λ₀(Y) of the L² spectrum of

−∆on Y= Γ\G/K:

λ₀(Y)≥

(kρk² if 0≤δ⁰(Γ)≤ kρk, kρk²−(δ⁰(Γ)− kρk)² if kρk ≤δ⁰(Γ)≤2kρk.

We obtain next a plain analog of Theorem 1.1 by considering a more involved family of Poincaré series. We denote by δ⁰⁰(Γ) the critical exponent ofP_s⁰⁰(xK, yK), see (3.3).

Theorem 1.4. The following characterization holds for the bottom λ₀(Y) of the L² spectrum of −∆on Y= Γ\G/K:

λ₀(Y) =

(kρk² if 0≤δ⁰⁰(Γ)≤ kρk,

kρk²−(δ⁰⁰(Γ)− kρk)² if kρk ≤δ⁰⁰(Γ)≤2kρk. (1.3) Remark 1.5. If Γ is a lattice, i.e., Y = Γ\G/K has finite volume, then λ₀(Y) = 0 and δ⁰⁰(Γ) = 2kρk, hence δ⁰(Γ) = 2kρk. Furthermoreδ(Γ) = 2kρk[Alb99, Theorem 7.4]. As pointed out by Corlette [Cor90] in rank one and by Leuzinger [Leu03] in higher rank, ifGhas Kazhdan’s property (T), then the following conditions are actually equivalent:

(a)Γ is a lattice, (b)λ₀(Y) = 0, (c) δ(Γ) = 2kρk, (d)δ⁰⁰(Γ) = 2kρk.

Remark 1.6. As for the Green function, the heat kernel h^Y_t (ΓxK,ΓyK) =X

γ∈Γh_t(Ky⁻¹γ⁻¹xK)

on a locally symmetric spaceY= Γ\G/K can be expressed and estimated by using the heat kernel ht on the symmetric space X = G/K, whose behavior is well understood [AnJi99; AnOs03].

By adapting straightforwardly the methods carried out in [DaMa88; Web08], and by applying Theorem 1.4 instead of Theorem 1.1 and Theorem 1.2, we refine the Gaussian bounds of h^Y_t

BOTTOM OF THEL² SPECTRUM OF THE LAPLACIAN ON LOCALLY SYMMETRIC SPACES 3

and get rid in particular of ρ_min. The following estimates hold for all t >0 and all x, y∈G:

(i) Assume that δ⁰⁰(Γ)<kρk and letδ⁰⁰(Γ)< s <kρk. Then h^Y_t (ΓxK,ΓyK).t⁻ⁿ² (1 +t)^n−D2 e^−kρk2te⁻d(ΓxK,ΓyK)2

4t P_s⁰⁰(xK, yK).

(ii) Assume that kρk ≤δ⁰⁰(Γ)<2kρk and letδ⁰⁰(Γ)− kρk< s₁ < s₂<kρk. Then h^Y_t (ΓxK,ΓyK).t⁻ⁿ² e⁻(^kρk2−s22)^tP_kρk+s⁰⁰

1(xK, yK).

(iii) Assume that δ⁰⁰(Γ)<2kρk. Let s > δ⁰⁰(Γ) and ε >0. Then h^Y_t(ΓxK,ΓyK).t⁻ⁿ² e⁻(^{kρk2−(δ00(Γ)−kρk)2−2ε})^te⁻

d(ΓxK,ΓyK)2

4(1+ε)t P_s⁰⁰(xK, xK)¹²P_s⁰⁰(yK, yK)¹².

2. First improvement

In this section, we replace the Riemannian distancedon X by apolyhedral distance

d⁰(xK, yK) =h_kρk^ρ ,(y⁻¹x)⁺i ∀x, y∈G, (2.1) where(y⁻¹x)⁺denotes thea⁺-component ofy⁻¹xin the Cartan decompositionG=K(expa⁺)K.

The corresponding balls, which reflect the volume growth ofX at infinity, played an important role in [Ank90] and [Ank91]. More general polyhedral sets were considered in [Ank92] and [AAS10].

Proposition 2.1. d⁰ is aG-invariant distance on X.

Proof. Notice first that (2.1) descends fromG×GtoX×X, as the mapz7→z⁺isK-bi-invariant on G. TheG-invariance of d⁰ is straightforward from the definition (2.1). The symmetry

d⁰(xK, yK) =d⁰(yK, xK) follows from

(y⁻¹)⁺=−w₀.y⁺ and −w0.ρ=ρ, (2.2) wherew₀ denotes the longest element in the Weyl group. Let us check the triangular inequality

d⁰(xK, yK)≤d⁰(xK, zK) +d⁰(zK, yK).

By G-invariance, we may reduce to the case where zK=eK. According to Lemma 2.2below, x⁺+ (y⁻¹)⁺−(y⁻¹x)⁺

belongs to the cone generated by the positive roots.

Lemma 2.2. For every x, y∈G, we have the following inclusion

co[W.(xy)⁺]⊂co[W.(x⁺+y⁺)] (2.3) between convex hulls.

Proof. The inclusion (2.3) amounts to the fact that x⁺+y⁺−(xy)⁺

belongs to the cone generated by the positive roots or, equivalently, to the inequality

hλ,(xy)⁺i ≤ hλ, x⁺i+hλ, y⁺i ∀λ∈a⁺. (2.4) It is enough to prove (2.4) for all highest weights λ of irreducible finite-dimensional complex representations π :G−→GL(V) withK-fixed vectors. According to Weyl’s unitary trick (see for instance [Kna02, Proposition 7.15]), there exists an inner product on V such that

(π(k) is unitary ∀k∈K, π(a) is self-adjoint ∀a∈expa.

Asλis the highest weight of π, then

e^hλ,(xy)+i=kπ(xy)k ≤ kπ(x)kkπ(y)k=e^hλ,x+ie^hλ,y+i =e^hλ,x++y+i.

Remark 2.3. The distance d⁰ is comparable to the Riemannian distance d. Specifically,

ρmin

kρk d(xK, yK)≤d⁰(xK, yK)≤d(xK, yK) ∀x, y∈G. (2.5) This follows indeed from

ρmin

kρk kHk ≤ h_kρk^ρ , Hi ≤ kHk ∀H ∈a⁺. The volume of balls

B_r⁰(xK) ={yK∈X|d⁰(yK, xK)≤r}

was determined in [Ank90, Lemma 6]. For the reader’s convenience, we recall the statement and its proof.

Lemma 2.4. For every x∈Gand r >0, we have ¹

|B_r⁰(xK)|

(rⁿ if 0< r <1, r^`−1e^2kρkr if r≥1.

Remark 2.5. Notice the different large scale behavior, in comparison with the classical ball volume

|B_r(xK)|

(rⁿ if 0< r <1, r^`−12 e^2kρkr if r≥1, see for instance [Str81] or [Kni97].

Proof. By translation invariance, we may assume thatx=e. Recall the integration formula Z

X

dx f(x) = const.

Z

K

dk Z

a⁺

dH ω(H)f(k(expH)K), (2.6) in the Cartan decomposition, with density

ω(H) = Y

α∈Σ⁺

(sinhhα, Hi)^mα Y

α∈Σ⁺

_hα,Hi

1+hα,Hi

mα

e^2hρ,Hi,

since sinht∼ _1+t^t e^t for allt≥0. Thus

|B_r⁰(eK)| = const.

Z

{H∈a⁺|hρ,Hi≤kρkr}

dH ω(H) Z

{H∈a⁺|kHk≤r}

dH Y

α∈Σ⁺

hα, Hi^mα rⁿ if r is small. Let us turn to r large. On the one hand, we estimate from above

|B_r⁰(eK)|. Z

{H∈a⁺|hρ,Hi≤kρkr}

dH e^2hρ,Hi

Z 2kρkr 0

ds s^{`−1</sup>e<sup>s</sup>r<sup>`−1}e^2kρkr. On the other hand, let H₀ ∈a⁺. As

ω(H)e^2hρ,Hi ∀H ∈H0+a⁺, we estimate from below

|B_r⁰(eK)|&

Z

{H∈H0+a⁺|hρ,Hi≤kρkr}

dH e^2hρ,Hi&

Z 2kρkr C0

ds s^{`−1</sup>e<sup>s</sup>r<sup>`−1}e^2kρkr,

whereC₀ >0is a constant depending on H₀.

1The symbolfgbetween two non-negative expressions means that there exist constants0< A≤B <+∞

such thatAg≤f≤Bg.

BOTTOM OF THEL² SPECTRUM OF THE LAPLACIAN ON LOCALLY SYMMETRIC SPACES 5

Consider now the modified Poincaré series P_s⁰(xK, yK) =X

γ∈Γe^{−sd0(xK,γyK)} ∀s >0,∀x, y∈G (2.7) associated with d⁰, its critical exponent

δ⁰(Γ) = inf{s >0|P_s⁰(xK, yK)<+∞} (2.8) and the modified orbital counting function

N_R⁰(xK, yK) =|{γ ∈Γ|d⁰(xK, γyK)≤R}| ∀R≥0,∀x, y∈G. (2.9) The following proposition shows that (2.7), (2.8) and (2.9) share the properties of their classical counterparts.

Proposition 2.6. The following assertions hold:

(i) δ⁰(Γ) does not depend on the choice ofx andy.

(ii) 0≤δ⁰(Γ)≤2kρk.

(iii) For every x, y∈G,

δ⁰(Γ) = lim sup

R→+∞

logN_R⁰(xK, yK)

R . (2.10)

Remark 2.7. It follows from (2.5) that

Ps(xK, yK)≤P_s⁰(xK, yK)≤P^ρmin

kρk s(xK, yK) and

N_R(xK, yK)≤N_R⁰(xK, yK)≤N kρk

ρminR(xK, yK).

Hence

0≤δ(Γ)≤δ⁰(Γ)≤ kρk ρmin

δ(Γ).

Proof. (i) follows from the triangular inequality. More precisely, letx1, y1, x2, y2 ∈Gands >0.

Then

d⁰(x2K, γy2K)≤d⁰(x2K, x1K) +d⁰(x1K, γy1K) +d⁰(γy1K, γy2K)

| {z }

d⁰(y1K, y2K)

∀γ ∈Γ,

hence

X

γ∈Γe^{−s d0(x1K,γy1K)}

| {z }

P_s⁰(x1K,y1K)

≤e^{s{d0(x1K,x2K)+d0(y1K,y2K)}}X

γ∈Γe^{−s d0(x2K,γy2K)}

| {z }

P_s⁰(x2K,y2K)

.

(ii) According to (i), let us show, without loss of generality, that P_s⁰(eK, eK)<+∞ for every s >2kρk. According to Lemma2.9below, there existsr >0 such that the ballsB_r⁰(γK), with γ ∈Γ, are pairwise disjoint inG/K. Let us apply the integration formula (2.6) to the function

f_s(xK) =X

γ∈Γe^{−sd0(γK,eK)}1_B⁰

r(γK)(xK).

On the one hand, as

d⁰(xK, eK)−d⁰(γK, eK)

≤r ∀xK ∈B_r⁰(γK), we have

Z

X

d(xK)fs(xK)X

γ∈Γe^{−sd0(γK,eK)}|B_r⁰(γK)|

| {z }

|B_r⁰(eK)|

P_s⁰(eK, eK).

On the other hand, Z

X

d(xK)f_s(xK). Z

X

d(xK)e^{−sd0(xK,eK)}

Z

a⁺

dH ω(H)e^−sh

ρ kρk,Hi

. Z

a⁺

dH e⁻⁽

s

kρk−2)hρ,Hi

is finite if s >2kρk. ThusP_s⁰(eK, eK)<+∞ in that case, and consequentlyδ⁰(Γ)≤2kρk.

(iii) Denote the right hand side of (2.10) by L(xK, yK) and let us first show that L(xK, yK) is finite. By applying Lemma 2.9below to y⁻¹Γy, we deduce that there existsr >0 such that the ballsB_r⁰(γyK), withγ ∈Γ, are pairwise disjoint. Set

Γ_R⁰(xK, yK) ={γ ∈Γ|d⁰(xK, γyK)≤R} ∀R≥0,∀x, y∈G.

Then the ball B_R+r⁰ (xK) contains the disjoint balls B_r⁰(γyK), with γ ∈ Γ_R⁰(xK, yK). By computing volumes, we estimate

N_R⁰(xK, yK) =|Γ_R⁰(xK, yK)| ≤ |B_R+r⁰ (xK)|

|B_r⁰(eK)| (1 +R)^`−1e^2kρkR.

HenceL(xK, yK)≤2kρk. Let us next show thatL(xK, yK)is actually independent ofx, y∈G.

Givenx₁, y₁, x₂, y₂ ∈G andR₁>0, let

R₂ =R₁+d⁰(x₁K, x₂K) +d⁰(y₁K, y₂K).

Then the triangular inequality

d⁰(x₂K, γy₂K)≤d⁰(x₂K, x₁K) +d⁰(x₁K, γy₁K) +d⁰(γy₁K, γy₂K)

| {z }

d⁰(y1K, y2K)

implies successively

Γ_R⁰₁(x1K, y1K)⊂Γ_R⁰₂(x2K, y2K), N_R⁰₁(x1K, y1K)≤N_R⁰₂(x2K, y2K),

L(x₁K, y₁K)≤L(x₂K, y₂K).

Let us finally prove the equality between δ⁰(Γ)and L=L(eK, eK). For this purpose, observe that

P_s⁰= 1 +X

R∈N^∗

X

γ∈Γ_R⁰rΓ_R−1⁰ e^{−s d0(eK,γK)} 1 +X

R∈N^∗ N_R⁰ −N_R−1⁰ e^−sR

X

R∈NN_R⁰ e^−sR, (2.11)

where we have written for simplicity

P_s⁰ =P_s⁰(eK, eK), Γ_R⁰ = Γ_R⁰(eK, eK) and N_R⁰ =N_R⁰(eK, eK).

One the one hand, let s > Land setε= ^s−L₂ . By definition ofL, N_R⁰ .e^(L+ε)R ∀R ≥0.

Hence

P_s⁰ .X

R∈Ne^−εR<+∞.

One the other hand, lets < L. By definition ofL, there exists a sequence of integers1< R₁ <

R2<· · · →+∞ such that

N_R⁰_j ≥e^sRj ∀j∈N^∗.

Hence the series (2.11) diverges.

BOTTOM OF THEL² SPECTRUM OF THE LAPLACIAN ON LOCALLY SYMMETRIC SPACES 7

Remark 2.8. Here is an example where δ < δ⁰. Consider the product Γ\G/K= (Γ1\G1/K1)×(Γ2\G2/K2)

of two locally symmetric spaces of rank one, with parameters ρ₁, δ₁ and ρ₂, δ₂. Then

δ≤ q

δ₁²+δ₂² (2.12)

and

δ⁰ ≥ q

ρ₁²+ρ₂² max_δ1

ρ1,_ρ^δ2

2 . (2.13)

Hence δ < δ⁰ if ^δ_ρ¹

1 6=_ρ^δ2

2. Notice that there are plenty of such products, starting with the case where δ1= 2ρ1 and δ2= 0. Let us first prove (2.12) and begin with some notation. Write for simplicity

N1,R= (N1)R(eK1, eK1), N2,R= (N2)R(eK2, eK2) and NR=NR(eK, eK),

for every R ≥0. Moreover, for every D⊂R²+, denote by N(D) the number of γ = (γ₁, γ₂) in Γ = Γ1×Γ2 such that d1(γ1K1, eK1), d2(γ2K2, eK2)

belongs to D. In R²₊ we consider the covering of D_R={(R₁, R₂)∈R²₊|R₁²+R²₂≤R²} by the two segments [0, R]× {0}, {0} ×[0, R] and by the squares Qj1, j2= (j1, j1+1]×(j2, j2+1], with j₁²+j₂²< R². Then

NR=N(DR)≤N1,R+N2,R+X

j₁²+j²₂<R²N(Qj1, j2), (2.14) with

N(Q_{j1, j2}) = (N_{1, j1+1}−N_{1, j1})(N_{2, j2+1}−N_{2, j2})≤N_{1, j1+1}N_{2, j2+1}. (2.15) Given s₁> δ₁ and s₂> δ₂, there exist C≥1 such that

N_1,R≤C e^s1R and N_2,R≤C e^s2R (2.16) for every R≥0. By combining (2.14), (2.15) and (2.16), we get

N_R≤C e^s1R+C e^s2R+C²X

j²₁+j₂²<R²e^{s1(j1+1)+s2(j2+1)}. (2.17) Up to a multiplicative constant, the right hand side of (2.17) is bounded above by the integral

Z

R²+∩B(0,R+2)

dR₁dR₂e^s1R1+s2R2 = Z R+2

0

dr r Z ^π

2

0

dθ e^{r(s1cosθ+s2sinθ)}. (2.18)

As the function θ7−→s₁cosθ+s₂sinθ reaches its maximum p

s²₁+s²₂ at θ₀= arctan^s_s²

1, the latter integral is itself bounded above by

π 2

Z R+2 0

dr r e^r

√

s²₁+s²₂ ≤ ^π₂ √^R+2

s²₁+s²₂ e^(R+2)

√

s²₁+s²₂ (2.19) In conclusion, we obtain

logNR

R ≤ ^{2 logC+ logπ−log 2−}

1

2log(s²₁+s²₂)

R +^log(R+2)_R +^R+2_R p

s²₁+s²₂ by combining (2.17), (2.18) and (2.19), hence δ ≤p

s²₁+s²₂ by letting R→+∞, and finally δ ≤p

δ₁²+δ₂² by letting s₁&δ₁ and s₂&δ₂. Let us turn to the proof of (2.13). Set kρk= pρ₁²+ρ₂² and assume that _ρ^δ1

1≥_ρ^δ2

2. As

d⁰(γ K, eK) =_kρk^ρ1 d₁(γ₁K₁, eK₁) +_kρk^ρ2 d₂(γ₂K₂, eK₂), the set {γ∈Γ|d⁰(γ K, eK)≤R} contains the product

{γ1∈Γ1|d1(γ1K1, eK1)≤^kρk_ρ

1 R} × {e}, for every R≥0. Hence N_R⁰ ≥N_1,^kρk

ρ1 R, where N_R⁰ =N_R⁰ (eK, eK), and conseqently δ⁰≥^kρk_ρ

1 δ₁.

Lemma 2.9. ² There existsr >0 such that the balls B_r⁰(γK), withγ ∈Γ, are pairwise disjoint in G/K.

Proof. Let r >0. As Γ is discrete inG, its intersection with the compact subset G_r⁰ ={y∈G|d⁰(yK, eK)≤r}=K exp{H∈a⁺| hρ, Hi ≤ kρkr}

K is finite. Moreover, asΓ is torsion-free,

γ⁺ 6= 0 ∀γ ∈Γ\{e}.

Hence there existsr >0such that Γ∩G_2r⁰ ={e}, which implies that the sets γG_r⁰ are pairwise disjoint inG. In other words, the balls B_r⁰(γK) are pairwise disjoint inG/K.

By usingδ⁰(Γ), we prove Theorem 1.3, which improves the lower bound in Theorem 1.2.

Proof of Theorem 1.3. Let us resume the approach in [Cor90, Section 4] and [Leu04, Section 3].

It consists in studying the convergence of the positive series g^Γ_ζ(ΓxK,ΓyK) =X

γ∈Γg_ζ(Ky⁻¹γ⁻¹xK), (2.20) which expresses the kernelg_ζ^Γof(−∆−kρk²+ζ²)⁻¹ on the locally symmetric spaceY= Γ\G/K in terms of the corresponding Green functiong_ζ on the symmetric spaceX =G/K. Hereζ >0 and ΓxK 6= ΓyK. Recall [AnJi99, Theorem 4.2.2] that

gζ(expH)nY

α∈Σ⁺_red 1 +hα, Hio

kHk^{−`−12 −|Σ+red|}e−hρ,Hi−ζkHk

(2.21) for H ∈a⁺ large, let say kHk ≥ ¹₂, while

g_ζ(expH)

(kHk⁻⁽ⁿ⁻²⁾ if n >2 log_kH¹_k if n= 2 for H small, lets say 0<kHk ≤ ¹₂. Thus (2.20) converges if and only if

X

γ∈Γ

nY

α∈Σ⁺_red 1+hα,(y⁻¹γ⁻¹x)⁺io

×

× d(xK, γyK)^{−`−12 −|Σ+red|}e^−kρkd0(xK,γyK)−ζd(xK,γyK)

(2.22)

converges. Let us compare the series (2.22) with the Poincaré series (1.1) and (2.7). On the one hand, as k(y⁻¹γ⁻¹x)⁺k=d(xK, γyK), (2.22) is bounded from above byP_kρk+ζ⁰ (xK, yK).

On the other hand, as

d(xK, γyK)^{−`−12 −|Σ+red|}&e−εd(xK,γyK)

for every ε >0, (2.22) is bounded from below byPkρk+ζ+ε(xK, yK). Hence (2.22) converges if kρk+ζ > δ⁰(Γ), i.e., ζ > δ⁰(Γ)− kρk, while (2.22) diverges if ζ < δ(Γ)− kρk. We conclude by using the fact [Cor90, Section 4] that λ₀(Y) is the supremum ofkρk²−ζ² over all ζ >0 such

that (2.20) converges.

Next statement is obtained by combining this lower bound with the upper bound in Theorem 1.2.

Corollary 2.10. The following estimates hold for λ₀(Y):







λ0(Y) =kρk² if δ⁰(Γ)≤ kρk,

kρk²−(δ⁰(Γ)− kρk)² ≤λ₀(Y)≤ kρk² if δ(Γ)≤ kρk ≤δ⁰(Γ), kρk²−(δ⁰(Γ)− kρk)² ≤λ₀(Y)≤ kρk²−(δ(Γ)− kρk)² if kρk ≤δ(Γ).

2As observed by the referee, Lemma 3 still holds without the torsion-free assumption, provided thatγruns throughΓ\(Γ∩K).

REFERENCES 9

3. Second improvement

In this section, we obtain the actual higher rank analog of Theorem 1.1 by considering a further family of distances on X, which reflects the large scale behavior (2.21) of the Green function. Specifically, for everys >0and x, y∈G, let

d_s⁰⁰(xK, yK) = min{s,kρk}d⁰(xK, yK) + max{s− kρk,0}d(xK, yK)

= (

s d⁰(xK, yK) if 0< s≤ kρk, kρkd⁰(xK, yK) + (s− kρk)d(xK, yK) if s≥ kρk.

(3.1)

Then (3.1) defines a G-invariant distance on X such that

s d⁰(xK, yK)≤d_s⁰⁰(xK, yK)≤s d(xK, yK) ∀s >0,∀x, y∈G. (3.2) Consider the associated Poincaré series

P_s⁰⁰(xK, yK) =X

γ∈Γe^{−ds00(xK,γyK)} ∀s >0,∀x, y∈G (3.3) and its critical exponent

δ⁰⁰(Γ) = inf{s >0|P_s⁰⁰(xK, yK)<+∞}.

It follows from (3.2) that

0≤δ(Γ)≤δ⁰⁰(Γ)≤δ⁰(Γ)≤2kρk. (3.4) Proof of Theorem 1.4. In the proof of Theorem1.3and Corollary2.10, we compared the series (2.20), or equivalently (2.22), with the Poincaré series (1.1) and (2.7). If we consider instead the Poincaré series (3.3), we obtain in the same way that (2.22) is bounded from above by P_kρk+ζ⁰⁰ (xK, yK)and from below byP_kρk+ζ+ε⁰⁰ (xK, yK), for every ε >0. Hence (2.22) converges if ζ > δ⁰⁰(Γ)− kρk, while (2.22) diverges if ζ < δ⁰⁰(Γ)− kρk. We conclude as in the above-

mentioned proof.

Acknowledgements. The authors are grateful to the referee for checking carefully the manu- script and making several helpful suggestions of improvement. The second author acknowledges financial support from the University of Orléans during his Ph.D. and from the Methusalem Programme Analysis and Partial Differential Equations during his postdoc stay at Ghent Uni- versity.

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Jean-Philippe Anker: anker@univ-orleans.fr

Institut Denis Poisson, Université d’Orléans, Université de Tours & CNRS, Orléans, France Hong-Wei Zhang: hongwei.zhang@ugent.be

Department of Mathematics, Ghent University, Ghent, Belgium

Institut Denis Poisson, Université d’Orléans, Université de Tours & CNRS, Orléans, France