ORBITS OF DISCRETE SUBGROUPS ON A SYMMETRIC SPACE AND THE FURSTENBERG BOUNDARY

ALEXANDER GORODNIK AND HEE OH

Dedicated to Hillel Furstenberg with respect and admiration

Abstract. Let X be a symmetric space of noncompact type and Γ a lattice in the isometry group of X. We study the distribution of orbits of Γ acting on the symmetric space X and its geometric boundary X(∞) generalizing the main equidistribution result of Margulis’ thesis to higher rank symmetric spaces. More precisely, for any y ∈ X and b ∈ X(∞), we investigate the distribution of the set {(yγ, bγ^{−1}) : γ ∈ Γ} in X × X(∞). It is proved, in particular, that the orbits of Γ in the Furstenberg boundary are equidistributed, and that the orbits of Γ in X are equidistributed in “sectors” defined with respect to a Cartan decomposition. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces, which we obtain using Shah’s result based on Ratner’s measure classification theorem.

Contents

1. Introduction 1 2. Main ingredients of the proofs 7 3. Cartan decomposition and the strong wavefront lemma 10 4. Contractions on G/B 15 5. Volume estimates 18 6. Equidistribution of solvable flows 21 7. Distribution of lattice points in sectors and the boundary 28 8. Distribution of lattice points in bisectors 32 9. Measure-preserving lattice actions 35 References 36

1. Introduction

Let 𝔻 denote the hyperbolic unit disc and Γ a torsion free discrete subgroup of the isometry group of 𝔻 such that 𝔻/Γ has finite area. The geometric boundary of 𝔻 is the space of the equivalence classes of geodesic rays in 𝔻. It can be

The first author is partially supported by NSF grant 0400631. The second author partially supported by NSF grant 0333397.

identified with the unit circle 𝕊. Note that the action of Γ on 𝔻 extends to the geometric boundary of 𝔻.

Let x ∈ 𝔻. We denote by B_{T}(x) the ball of radius T centered at x. For an arc Ω ⊂ 𝕊, the sector S_{x}(Ω) in 𝔻 is defined to be the set of points z ∈ 𝔻 such that the end point of the geodesic ray from x to z lies in Ω. Denote by m_{x} the unique probability measure on 𝕊 invariant under the isometries that fix the point x. Then:

A. For any x, y ∈ 𝔻, b ∈ 𝕊 and an arc Ω ⊂ 𝕊,

#\{γ ∈ Γ : bγ^{−1} ∈ Ω, yγ ∈ B_{T}(x)\} ∼_{T →∞} m_{y}(Ω) · \frac{Area(B_{T}(x))}{Area(𝔻/Γ)}.

B. For any x, y ∈ 𝔻 and an arc Ω ⊂ 𝕊,

#\{γ ∈ Γ : yγ ∈ S_{x}(Ω) ∩ B_{T}(x)\} ∼_{T →∞} m_{x}(Ω) · \frac{Area(B_{T}(x))}{Area(𝔻/Γ)}.

C. For every x, y ∈ 𝔻, b ∈ 𝕊 and arcs Ω_{1}, Ω_{2} ⊂ 𝕊,

#{γ ∈ Γ : yγ ∈ S_{x}(Ω_{1}) ∩ B_{T}(x), bγ^{−1} ∈ Ω_{2}}

∼_{T →∞} m_{x}(Ω_{1})m_{y}(Ω_{2}) · \frac{Area(B_{T}(x))}{Area(𝔻/Γ)}.

Statement (A) may be deduced from the work of A. Good [G]. Statement (B) was shown by P. Nicholls [N] (see also [Sh]). Statement (C), which was proved by G. Margulis [M] in his thesis for cocompact Γ, shows that the equidistribution phenomena in (A) and (B) are indeed independent.

Figure 1

The main purpose of this paper is to obtain an analog of statement (C) (note that (C) implies both (A) and (B)) for an arbitrary Riemannian symmetric space of noncompact type (see Theorems 1.1 and 1.2 below). We also generalize statement (B) to the equidistribution of lattice points in a connected noncompact semisimple Lie group G with finite center with respect to both K-components in a Cartan decomposition G = KA^{+}K (see Theorem 1.6 below).

Let X be a Riemannian symmetric space of noncompact type and X(∞) the geometric boundary of X (that is, the space of equivalence classes of geodesic rays in X). Denote by G the identity component of the isometry group of X acting on X from the right. Let Γ be a lattice in G, i.e., a discrete subgroup with finite covolume. The action of G on X extends to X(∞).

For x ∈ X, we denote by B_{T}(x) the Riemannian ball of radius T centered at x, by K_{x} the stabilizer of x in G, and by ν_{x} the probability Haar measure on K_{x}. For x ∈ X and b ∈ X(∞), we denote by m_{b,x} the unique probability K_{x}-invariant measure supported on the orbit bG ⊂ X(∞) (note that G acts transitively on X(∞) only when the rank of X is one, and that K_{x} acts transitively on each G-orbit in X(∞)). Fix a closed Weyl chamber W_{x} ⊂ X at x. According to the Cartan decomposition, we have X = W_{x}K_{x}. Let M_{x} denote the stabilizer of W_{x} in K_{x}.

The following is one of our main theorems:

Theorem 1.1. For x, y ∈ X, b ∈ X(∞), and any Borel subsets Ω_{1} ⊂ K_{x} and Ω_{2} ⊂ bG with boundaries of measure zero,

#\{γ ∈ Γ : yγ ∈ W_{x}Ω_{1}∩B_{T}(x), bγ^{−1} ∈ Ω_{2}\} ∼_{T →∞} ν_{x}(M_{x}Ω_{1})m_{b,y}(Ω_{2})· \frac{Vol(B_{T})}{Vol(G/Γ)},

where Vol(B_{T}) denotes the volume of a ball of radius T in X.

We deduce Theorem 1.1 from a stronger result on the level of Lie groups. Fix the following data:

• G – a connected noncompact semisimple Lie group with finite center,

• G_{n} – the product of all noncompact simple factors of G,

• G = K_{1}A^{+}K_{1} – a Cartan decomposition of G,

• d – an invariant metric on K_{1}\G,

• K_{2} – a maximal compact subgroup of G,

• Q – a closed subgroup of G that contains a maximal connected split

solvable subgroup. Recall that a solvable subgroup S is called split if the eigenvalues of any element of Ad(S) are real for the adjoint representation Ad : G → GL(Lie(G)). It is well-known that a maximal connected split solvable subgroup is a subgroup of the form AN for an Iwasawa decomposition G = KAN. Thus, G = K_{2}Q. Denote by ν_{1} and ν_{2} the probability Haar measure on K_{1} and K_{2} respectively.

Let M_{1} be the centralizer of A in K_{1} and M_{2} = K_{2} ∩ Q. Since any two maximal compact subgroups of G are conjugate to each other, there exists g ∈ G such that K_{2} = g^{−1}K_{1}g. Let Γ be a lattice in G such that \overline{ΓG_{n}Q} = G.

Theorem 1.2. For any Borel subsets Ω_{1} ⊂ K_{1} and Ω_{2} ⊂ K_{2} with boundaries of measure zero,

#{γ ∈ Γ ∩ g^{−1}K_{1}A^{+}Ω_{1} ∩ Ω_{2}Q : d(K_{1}, K_{1}gγ) < T }

∼_{T →∞} ν_{1}(M_{1}Ω_{1})ν_{2}(Ω_{2}M_{2}) · \frac{Vol(G_{T})}{Vol(G/Γ)},

where Vol(G_{T})denotes the volume of a Riemannian ball of radius T in G.

To understand the presence of M_{1} and M_{2} in the above asymptotics, observe that K_{1}A^{+}Ω_{1} = K_{1}A^{+}M_{1}Ω_{1} and Ω_{2}M_{2}Q = Ω_{2}Q.

Remark 1.3. We mention that the continuous version of Theorem 1.2 does not seem obvious either. The method of the proof of Theorem 1.2 also yields the following volume asymptotics:

Vol({h ∈ g^{−1}K_{1}A^{+}Ω_{1} ∩ Ω_{2}Q : d(K_{1}, K_{1}gh) < T })

∼_{T →∞} ν_{1}(M_{1}Ω_{1})ν_{2}(Ω_{2}M_{2}) · \frac{Vol(G_{T})}{Vol(G/Γ)}.

Remark 1.4. If in Theorem 1.2 we replace K_{1}A^{+}Ω_{1} by Ω_{1}A^{+}K_{1}, then the statement of the theorem is false. In fact, we can show that there exist nonempty open subsets Ω_{1} ⊂ K_{1} and Ω_{2} ⊂ K_{2} such that

lim_{T →∞} \frac{1}{Vol(G_{T})} Vol(\{h ∈ g^{−1}Ω_{1}A^{+}K_{1} ∩ Ω_{2}Q : d(K_{1}, K_{1}gh) < T \}) = 0.

To state yet another generalization of statement (B), we fix a Cartan decomposition G = KA^{+}K and an invariant Riemannian metric d on K\G. Let Γ be any lattice in G. Recall that it was shown in [DRS] and [EM] that for a lattice Γ in G,

(1.5) #\{γ ∈ Γ : d(K, Kgγ) < T \} = #(Γ ∩ g^{−1}KA_{T}^{+}K) ∼_{T →∞} \frac{Vol(G_{T})}{Vol(G/Γ)},

where A_{T}^{+} = {a ∈ A^{+} : d(K, Ka) < T }. The following theorem is a generalization of this result:

Theorem 1.6. For g ∈ G and any Borel subsets Ω_{1} ⊂ K and Ω_{2} ⊂ K with boundaries of measure zero,

#(Γ ∩ g^{−1}Ω_{1}A_{T}^{+}MΩ_{2}) ∼_{T →∞} \frac{Vol(g^{−1}Ω_{1}A_{T}^{+}MΩ_{2})}{Vol(G/Γ)} = ν(Ω_{1}M)ν(MΩ_{2}) · \frac{Vol(G_{T})}{Vol(G/Γ)},

where M is the centralizer of A^{+} in K, and ν is the probability Haar measure on K.

We now present several corollaries of (the methods of) the above theorems.

1.1. Lattice action on the Furstenberg boundary. For a connected semisimple Lie group G with finite center, the Furstenberg boundary of G is identified with the quotient space G/P where P is a minimal parabolic subgroup of G (see [GJT, Ch. IV]). In the rank one case, the Furstenberg boundary G/P coincides with the geometric boundary X(∞) of the symmetric space X of G. In the higher rank case, G/P is isomorphic to the G-orbit in X(∞) of any regular geodesic class and can be identified with the space of asymptotic classes of Weyl chambers in X.

It is well-known that the action of a lattice Γ on G/P is minimal, i.e., every Γ-orbit is dense ([Mo, Lemma 8.5]). A natural question is whether each Γ-orbit in G/P is equidistributed. Corollary 1.7, which is a special case of Theorem 1.2, implies an affirmative answer in a much more general setting.

Let d denote an invariant Riemannian metric on the symmetric space X ≃ K\G, where K is a maximal compact subgroup of G.

Corollary 1.7. Let Q be a closed subgroup of G containing a maximal connected split solvable subgroup of G and g ∈ G. Denote by ν_{g} the unique g^{−1}Kg-invariant probability measure on G/Q. Let b ∈ G/Q and Γ be a lattice in G such that \overline{ΓG_{n}b} = G/Q. Then for any Borel subsetΩ ⊂ G/Q such that ν_{g}(∂Ω) = 0,

#\{γ ∈ Γ : γb ∈ Ω, d(K, Kgγ) < T \} ∼_{T →∞} ν_{g}(Ω) · \frac{Vol(G_{T})}{Vol(G/Γ)}.

It follows from the result of N. Shah [S, Theorem 1.1] and Ratner’s topological rigidity [R2] that the condition \overline{ΓG_{n}b} = G/Q is equivalent to the density of the orbit Γb in G/Q.

In the case when Q is a parabolic subgroup of G, a different proof of this result, which is based on ideas developed in [Ma], is recently given in [GM].

In the last decade or so there have been intensive studies on the equidistribution properties of lattice points on homogeneous spaces of G using various methods from analytic number theory, harmonic analysis and ergodic theory ([DRS], [EM],[EMM], [EMS], [GO], [EO], [Go], [L], [Ma], [No], etc.). Of particular interest is the case when the homogeneous space is a real algebraic variety. While most of the attention in this direction is focused on the case of affine homogeneous varieties, there is not so much work done for the projective homogeneous varieties, except for recent works [Go] and [Ma]. In [Go], one studies the distribution of lattice orbits on the real projective homogeneous varieties of G = SL_{n}(ℝ) with respect to the norm given by ∥g∥ = \sqrt{∑ g_{ij}^{2}}, g ∈ SL_{n}(ℝ). In [Ma], one investigates the distribution of lattice orbits on the boundary of a real hyperbolic space. Corollary 1.7 extends both results by proving that an orbit of a lattice in a connected noncompact semisimple real algebraic group G is equidistributed on any projective homogeneous variety of G (with respect to a Riemannian metric).

More generally, we state the following conjecture:

Conjecture 1.8. Let G be a connected semisimple Lie group with finite center, Γ a lattice in G, and Y a compact homogeneous space of G. Then every dense orbit of Γ in Y is equidistributed, i.e., there exists a smooth measure ν on Y such that for any y ∈ Y with \overline{Γy} = Y and for any Borel set Ω ⊂ Y with boundary of measure zero,

#\{γ ∈ Γ : γy ∈ Ω, d(K, Kγ) < T \} ∼_{T →∞} ν(Ω) · \frac{Vol(G_{T})}{Vol(G/Γ)}.

The structure of compact homogeneous spaces of G was studied in [Mor]. We note that the case of the conjecture when Y = G/Γ for a cocompact lattice is also known (see Theorem 1.9).

1.2. Measure-preserving lattice actions. Let G be a connected semisimple noncompact Lie group with finite center and Γ_{1}, Γ_{2} lattices in G. We consider the action of Γ_{1} on G/Γ_{2}. Let d be an invariant Riemannian metric on the symmetric space K\G.

Theorem 1.9. Suppose that for y ∈ G/Γ_{2}, the orbit Γ_{1}y is dense in G/Γ_{2} Then for any g ∈ G and any Borel subset Ω ⊂ G/Γ_{2} with boundary of measure zero,

\{γ ∈ Γ_{1} : γy ∈ Ω, d(K, Kgγ) < T \} ∼_{T →∞} \frac{Vol(Ω) · Vol(G_{T})}{Vol(G/Γ_{1}) Vol(G/Γ_{2})},

where all volumes are computed with respect to one fixed Haar measure on G.

For example, Theorem 1.9 applies to the case when G is a simple connected noncompact Lie group, and Γ_{1} and Γ_{2} are noncommensurable lattices in G. (Recall that the lattices are called commensurable if Γ_{1} ∩ Γ_{2} has finite index in both Γ_{1} and Γ_{2}.) It was first observed by Vatsal (see [V]) that Γ_{1}Γ_{2} is dense in G/Γ_{2}. This is a (simple) consequence of Ratner’s topological rigidity [R2].

Theorem 1.9 was proved in [O] for G = SL_{n}(ℝ) equipped with the norm ∥g∥ = \sqrt{∑ g_{ij}^{2}} and in [GW] for general semisimple Lie groups without compact factors.

1.3. Acknowledgments. The first author wishes to thank for the hospitality of the Department of Mathematics in California Institute of Technology during August 2003 when most of this work has been done.

2. Main ingredients of the proofs

2.1. The strong wavefront lemma. The following theorem is a basic tool which enables us to reduce the counting problems for Γ as in Theorems 1.2 and 1.6 to the study of continuous flows on the homogeneous space Γ\G.

Let G be a connected noncompact semisimple Lie group with finite center, G = KA^{+}K a Cartan decomposition, and M the centralizer of A in K.

Theorem 2.1 (The strong wavefront lemma). Let C be any closed subset of A^{+} with a positive distance from the walls of A^{+}. Then for any neighborhoods U_{1}, U_{2} of e in K and V of e in A, there exists a neighborhood O of e in G such that for any g = k_{1}ak_{2} ∈ KCK,

(1) gO ⊂ (k_{1}U_{1})(aV M)(k_{2}U_{2});

(2) Og ⊂ (k_{1}U_{1})(aV M)(k_{2}U_{2}).

The strong wavefront lemma has been recently generalized to affine symmetric spaces in [GOS].

Remark 2.2. One can check that Theorem 2.1 fails if the set C contains a sequence that converges to a point in a wall of the Weyl chamber A^{+}.

Theorem 2.1 has several geometric implications for the symmetric space K\G:

• Strengthening of the wavefront lemma. Recall that the wavefront lemma introduced by Eskin and McMullen in [EM] says that for any neighborhood O′ of e in G, there exists a neighborhood O of e in G such that

(2.3)  aO ⊂ O′aK for all a ∈ A^{+}.

To see that our strong wavefront lemma (1) implies the wavefront lemma for a ∈ A^{+} with at least a fixed positive distance from the walls of A^{+}, note that O′ contains U_{1}V for some neighborhood U_{1} of e in K and some neighborhood V of e in A, and hence

U_{1}aV MK = U_{1}V aK ⊂ O′aK

By Theorem 2.1 (1), there exists a neighborhood O of e such that

aO ⊂ U_{1}aV MK.

Thus, aO ⊂ O′aK.

To illustrate geometric meaning of the strong wavefront lemma (1), we consider the unit disc 𝔻 equipped with the standard hyperbolic metric and the geodesic flow g_{t} on the unit tangent bundle T^{1}(𝔻) that transports a vector distance t along the geodesic to which it is tangent. Note that with the identification T^{1}(𝔻) ≃ PSL_{2}(ℝ), the geodesic flow g_{t} corresponds to

the left multiplication by ( \begin{matrix}e^{t/2} & 0 \\ 0 & e^{−t/2} \end{matrix} ). Let p ∈ 𝔻 and let K ⊂ T^{1}(𝔻)

be the preimage of p under the projection map π : T^{1}(𝔻) → 𝔻. Note that

Figure 2

K consists of vectors lying over p and pointing in all possible directions, and g_{t}(K) consists of the unit vectors normal to the sphere S_{t}(p) ⊂ 𝔻 of radius t. The wavefront lemma (see (2.3)) implies that one can find a neighborhood O ⊂ T^{1}(𝔻) of a vector v based at p such that g_{t}(O) remains close to g_{t}(K) uniformly for every t ≥ 0.

However, this does not compare g_{t}(O) with the vector g_{t}(v), but rather with the set g_{t}(K). Theorem 2.1 (1) says that we may choose a neighborhood O of v in T^{1}(𝔻) so that g_{t}(O) is close to the vector g_{t}(v) uniformly on t in both angular and radial components (see Fig. 2).

• Uniform openness of the map K × A^{+} × K → G. The product map

K × (interior of A^{+}) × K → G

is a diffeomorphism onto a dense open subset in G, and in particular, it is an open map. Theorem 2.1 (2) shows that this map is uniformly open with respect to the base of neighborhoods Og, where O is a neighborhood of e in G and g ∈ G, on any subset contained in K × A^{+} × K with a positive distance from the walls of A^{+}.

We illustrate the geometric meaning of this property for the case of the hyperbolic unit disc 𝔻. It follows from Theorem 2.1 (2) that for every neighborhood O′ of e in G, there exists a neighborhood O of e in G such that for every a ∈ A^{+} with at least a fixed positive distance from the walls of A^{+},

Oa ⊂ KaO′.

This implies that for every open subset O′ ⊂ T^{1}(𝔻), p ∈ O′, and any t > 0, the set π(g_{t}(O′)) contains a ball centered at π(g_{t}(p)) with a radius independent of t.

• Well-roundness of bisectors. Theorem 2.1 (2) also implies the following corollary, which is a generalization of the well-known property that the Riemannian balls are well-rounded (a terminology used in [EM]). Note that the Riemannian ball {g ∈ G : d(K, Kg) < T } is of the form KA_{T}^{+}K, where A_{T}^{+} = {a ∈ A^{+} : d(K, Ka) < T }.

Corollary 2.4. For Borel subsets Ω_{1}, Ω_{2} ⊂ K whose boundary has measure zero, the family {Ω_{1}A_{T}^{+}Ω_{2} : T > 0} of bisectors is well-rounded, i.e., for every ε > 0, there exists a neighborhood O of e in G such that

Vol(O · ∂(Ω_{1}A_{T}^{+}Ω_{2})) ≤ ε · Vol(Ω_{1}A_{T}^{+}Ω_{2})

for all T > 0.

2.2. Uniform distribution of solvable flows. Using the strong wavefront lemma, Theorem 1.2 is deduced from Theorem 2.5 below, which is also of independent interest from the viewpoint of ergodic theory.

Let K be a maximal compact subgroup of G with the probability Haar measure ν. Let Qbe a closed subgroup of G containing a maximal connected split solvable subgroup of G, and let ρ be a right invariant Haar measure on Q. Fix a Cartan decomposition G = KA^{+}K and g ∈ G. For T > 0 and a subset Ω ⊂ K, we define

Q_{T} (g, Ω) = {q ∈ Q : q ∈ g^{−1}KA^{+}Ω, d(K, Kgq) < T }.

If Ω = K and g = e, the set Q_{T} (g, Ω) is simply {q ∈ Q : d(K, Kq) < T }. Recall that G_{n} denotes the product of all noncompact simple factors of G.

Theorem 2.5. Let G be realized as a closed subgroup of a Lie group L. Let Λ be a lattice in L. Suppose that for y ∈ Λ\L, the orbit yG_{n} is dense in Λ\L. Then for any Borel subset Ω ⊂ K with boundary of measure zero and f ∈ C_{c}(Λ\L),

lim_{T →∞} \frac{1}{ρ(Q_{T} (g, K))} ∫_{Q_{T} (g,Ω)} f(yq^{−1}) dρ(q) = \frac{ν(MΩ)}{μ(Λ\L)} ∫_{Λ\L} f dμ,

where M is the centralizer of A in K, and μ is an L-invariant measure on Λ\L.

A main ingredient of the proof of Theorem 2.5 is the work of N. Shah [S] (see Theorem 6.2 below) on the distribution in Λ\L of translates yUg as g → ∞ for a subset U ⊂ K. Shah’s result is based on Ratner’s classification of measures invariant under unipotent flows [R1] and the work of Dani and Margulis on behavior of unipotent flows [DM]. Implementation of Shah’s theorem in our setting is based on the fundamental property of the Furstenberg boundary B of G: every regular element in a positive Weyl chamber acts on an open subset of full measure in B as a contraction.

Remark 2.6 (on the rate of convergence). The method of the proof of Theorem 6.2 in [S] does not give any estimate on the rate of convergence. In the case when L = G and U = K, Theorem 6.2 was proved by Eskin and McMullen in [EM]. The latter proof is based on the decay of the matrix coefficients of the quasi-regular representation of G on L^{2}(Γ\G) and provides an estimate on the rate of convergence. Combining the strong wavefront lemma (Theorem 2.1) with the method from [EM], we can derive an estimate for the rate of convergence in Theorem 2.5 when L = G provided that one knows the rate of decay of matrix coefficients of L^{2}(Γ\G). In this case, it is also possible to obtain rates of convergence for the theorems stated in the introduction. We hope to address this problem in a sequel paper.

2.3. Equidistribution of lattice points in bisectors. For Ω_{1}, Ω_{2} ⊂ K, and g ∈ G, we define

G_{T} (g, Ω_{1}, Ω_{2}) = {h ∈ G : h ∈ g^{−1}Ω_{1}A^{+}Ω_{2}, d(K, Kgh) < T }.

Using the strong wavefront lemma (Theorem 2.1 (2)), Theorem 1.6 is reduced to showing that the sets G_{T} (g, Ω_{1}, Ω_{2}) are equidistributed in Γ\G in the sense of Theorem 2.5 for any Borel subsets Ω_{1}, Ω_{2} ⊂ K with boundaries of measure zero.

3. Cartan decomposition and the strong wavefront lemma

Let G be a connected noncompact semisimple Lie group with finite center, K a maximal compact subgroup of G, and G = K exp(𝔭) the Cartan decomposition determined by K. A split Cartan subgroup A with respect to K is a maximal connected abelian subgroup of G contained in exp(𝔭). It is well-known that two split Cartan subgroups with respect to K are conjugate to each other by an element of K. Fix a split Cartan subgroup A of G (with respect to K) with the set of positive roots Φ^{+} and the positive Weyl chamber

A^{+} = {a ∈ A : α(log a) ≥ 0 for all α ∈ Φ^{+}}.

Set 𝔞 = log(A) and 𝔞^{+} = log(A^{+}). Let M be the centralizer of A in K. Note that M is finite if and only if G is real split.

The following lemma is well-known (see, for example, [K, Ch. V]).

Lemma 3.1. (Cartan decomposition) For every g ∈ G, there exist a unique element μ(g) ∈ log(A^{+}) such that g ∈ K exp(μ(g))K. Moreover, if k_{1}ak_{2} = k′_{1}ak′_{2} for some a in the interior of A^{+}, then there exists m ∈ M such that k_{1} = k′_{1}m, k_{2} = m^{−1}k′_{2}.

Denote by d an invariant Riemannian metric on the symmetric space K\G.

Lemma 3.2. For every a_{1} and a_{2} in the interior of A^{+} and k ∈ K,

d(Ka_{1}, Ka_{2}) ≤ d(Ka_{1}k, Ka_{2}).

Proof. Let a_{i} = exp(H_{i}) for H_{i} ∈ 𝔞^{+}, i = 1, 2. Then Ka_{1}k = K exp(Ad(k^{−1})H_{1}). Applying the cosine inequality (see [H, Corollary I.13.2]) to the geodesic triangle with vertices Ke, Ka_{1}k, and Ka_{2}, we obtain

d(Ka_{1}k, Ka_{2})^{2} ≥ d(K, Ka_{1}k)^{2} + d(K, Ka_{2})^{2} − 2d(K, Ka_{1}k)d(K, Ka_{2}) cos α
= ∥H_{1}∥^{2} + ∥H_{2}∥^{2} − 2∥H_{1}∥∥H_{2}∥ cos α,

where α is the angle at the vertex Ke. Since

cos α = \frac{⟨Ad(k^{−1})H_{1}, H_{2}⟩}{∥H_{1}∥ · ∥H_{2}∥}

and by Lemma 3.3 below,

⟨Ad(k^{−1})H_{1}, H_{2}⟩ ≤ ⟨H_{1}, H_{2}⟩,

it follows that

d(Ka_{1}k, Ka_{2})^{2} ≥ ∥H_{1}∥^{2} + ∥H_{2}∥^{2} − 2⟨H_{1}, H_{2}⟩ = ∥H_{1} − H_{2}∥^{2} = d(Ka_{1}, Ka_{2})^{2}.

The lemma is proved. □

Lemma 3.3. For any H_{1} and H_{2} in the interior of 𝔞^{+} and for any k ∈ K,

⟨H_{1}, H_{2}⟩ ≥ ⟨Ad(k)H_{1}, H_{2}⟩.

Proof. By [H, Proposition VIII.5.2] and its proof, every G-invariant positive definite form on K\G is of the form ∑_{i} α_{i}B_{i}, where B_{i}’s are the Killing forms of the simple factors of G and α_{i} > 0. Thus, it is sufficient to consider the case when G is simple, and the Riemannian metric is given by the Killing form B of G.

Define the function f(k) = ⟨Ad(k)H_{1}, H_{2}⟩ on K. Let k_{0} ∈ K be a point where f attains its maximum. For every Z ∈ Lie(K),

0 = \frac{d}{dt}|_{t=0}f(k_{0}e^{tZ}) = \frac{d}{dt}|_{t=0}B(Ad(k_{0}) Ad(e^{tZ})H_{1}, H_{2})

= B (Ad(k_{0})\frac{d}{dt}|_{t=0}(Ad(e^{tZ})H_{1}), H_{2}) = B(Ad(k_{0})(ad(Z)H_{1}), H_{2})

= B(Ad(k_{0})[Z, H_{1}], H_{2}) = B(Ad(k_{0})Z, [Ad(k_{0})H_{1}, H_{2}]).

This shows that [Ad(k_{0})H_{1}, H_{2}] ⊥ Lie(K). Since [Ad(k_{0})H_{1}, H_{2}] ∈ Lie(K) and the restriction of B to Lie(K) is negative definite, it follows that [Ad(k_{0})H_{1}, H_{2}] = 0. Therefore, Ad(k_{0})H_{1} ∈ 𝔞. Since the Weyl group W acts transitively on the set of Weyl chambers in A and K contains all representatives of the Weyl group, there exists an element w ∈ K that normalizes A such that Ad(w^{−1}k_{0})H_{1} ∈ 𝔞^{+}. Since H_{1} is in the interior of 𝔞^{+}, it follows from the uniqueness of the Cartan decomposition (Lemma 3.1) that Ad(k_{0})H_{1} = Ad(w)H_{1}. It is easy to see [H, p. 288] that ∥ Ad(w)H_{1} − H_{2}∥, w ∈ W, achieves its minimum at w = e. This implies that ⟨Ad(w)H_{1}, H_{2}⟩ is maximal for w = e and finishes the proof. □

Proposition 3.4. Let C be a closed subset contained in A^{+} with a positive distance from the walls of A^{+}. Then for any neighborhood U_{0} of e in K, there exists ε > 0 such that for any a ∈ C,

{k ∈ K : d(Kak, Ka) < ε} ⊂ MU_{0}.

Proof. Denote by Π the set of simple roots corresponding to the Weyl chamber A^{+}. Without loss of generality, we may assume that

C = {a ∈ A^{+} : α(log a) ≥ C for all α ∈ Π}

for some C > 0. Suppose that in contrary there exist sequences {a_{i}} ⊂ C and {k_{i}} ⊂ K such that d(Ka_{i}k_{i}, Ka_{i}) → 0 as i → ∞, and no limit points of {k_{i}} are contained in M. Passing to a subsequence, we may assume that k_{i} → k_{0} as i → ∞ for somek_{0} ∈ K − M, and for every α ∈ Φ, the sequence {α(log a_{i})} is either bounded or divergent. Set

Φ_{±} = {α ∈ Φ : α(log a_{i}) → ±∞ as i → ∞},
Φ_{0} = {α ∈ Φ : {α(log a_{i})} is bounded}.

Let P^{+} be the standard parabolic subgroup associated to Π − Φ_{+} and P^{−} the standard opposite parabolic subgroup for P^{+}. Note that

P^{−} = {g ∈ G : {a_{i}ga_{i}^{−1}} is bounded}.

Denote by U^{+} and U^{−} the unipotent radicals of P^{+} and P^{−} respectively. Set Z = P^{+} ∩ P^{−}, so that P^{±} = ZU^{±}. It is easy to see that P^{±} ∩ K ⊂ Z. Denote by 𝔤, 𝔲^{+}, 𝔲^{−}, 𝔷 the corresponding Lie algebras and by 𝔤_{α}, α ∈ Φ, the root subspaces in 𝔤. We have

(3.5)  𝔲^{±} = ⊕_{α∈Φ_{±}} 𝔤_{α} and 𝔷 = ⊕_{α∈Φ_{0}} 𝔤_{α}.

\underline{Step 1}: We claim that k_{0} ∈ Z.

There is an embedding π : G/C_{G} → SL_{d}(ℝ), where C_{G} is the center of G, such that π(AC_{G}) is contained in the group of diagonal matrices. We have

π(a_{i}k_{i}a_{i}^{−1})_{st} = π(a_{i})_{ss}π(a_{i})_{tt}^{−1} · π(k_{i})_{st}, s, t = 1, …, d.

Passing to a subsequence, we may assume that for each (s, t), the sequence {π(a_{i})_{ss}π(a_{i})_{tt}^{−1}} is either bounded or divergent. If the sequence {π(a_{i})_{ss}π(a_{i})_{tt}^{−1}} is divergent, then π(k_{i})_{st} → 0 as i → ∞. Thus, π(k_{0})_{st} = 0 for every pair (s, t) such that {π(a_{i})_{ss}π(a_{i})_{tt}^{−1}} is divergent. It follows that π(a_{i}k_{0}a_{i}^{−1}) is bounded. Since the center of G is finite, this proves that the sequence {a_{i}k_{0}a_{i}^{−1}} is bounded. Thus, k_{0} ∈ P^{−}. Since P^{−} ∩ K ⊂ Z, the claim follows.

\underline{Step 2}: We claim that d(Ka_{i}k_{0}a_{i}^{−1}, K) → 0 as i → ∞.

Write k_{i} = k_{0}l_{i} where l_{i} ∈ K and l_{i} → e as i → ∞. Then

(3.6)  d(Ka_{i}k_{i}, Ka_{i}) = d(Ka_{i}k_{0}a_{i}^{−1}, Ka_{i}l_{i}^{−1}a_{i}^{−1}) → 0 as i → ∞.

Thus, the sequence {a_{i}l_{i}^{−1}a_{i}^{−1}} is bounded, and hence, we may assume that it converges. Since l_{i}^{−1} → e as i → ∞ and 𝔤 = 𝔲^{−} ⊕ 𝔷 ⊕ 𝔲^{+}, we obtain that l_{i}^{−1} = u_{i}^{−}z_{i}u_{i}^{+} for some u_{i}^{−} ∈ U^{−}, z_{i} ∈ Z, and u_{i}^{+} ∈ U^{+} such that u_{i}^{−} → e, z_{i} → e, and u_{i}^{+} → e as i → ∞. It follows from (3.5) that a_{i}u_{i}^{−}a_{i}^{−1} → e and a_{i}z_{i}a_{i}^{−1} → e as i → ∞. Hence, a_{i}l_{i}^{−1}a_{i}^{−1} → u^{+} as i → ∞ for some u^{+} ∈ U^{+}. On the other hand, by Step 1, passing to a subsequence, we get a_{i}k_{0}a_{i}^{−1} → z as i → ∞ for some z ∈ Z. Thus, by (3.6), z^{−1}u^{+} ∈ K as i → ∞. Since P^{+} ∩ K ⊂ Z, we deduce that u^{+} = e. Hence, a_{i}l_{i}^{−1}a_{i} → e as i → ∞, and the claim follows from (3.6).

\underline{Step 3}: We claim that a_{i}k_{0}a_{i}^{−1} = d_{i}k_{0}d_{i}^{−1} for some bounded sequence \{d_{i}\} ⊂ C.

Recall that the system of simple roots Π is a basis of the dual space of the Lie algebra of A. Hence, we may write a_{i} = b_{i}c_{i} where b_{i}, c_{i} ∈ A^{+} such that α(log b_{i}) = 0 for every α ∈ Π ∩ Φ_{+} and α(log c_{i}) = 0 for every α ∈ Π − Φ_{+}. Then c_{i} commutes with Z, and since α(b_{i}) = α(a_{i}) for every α ∈ Π − Φ_{+}, the sequence {b_{i}} is bounded. Let c_{0} ∈ A^{+} be such that α(log c_{0}) = C for every α ∈ Π ∩ Φ_{+} and α(log c_{0}) = 0 for every α ∈ Π − Φ_{+}. Then d_{i} = b_{i}c_{0} ∈ C and a_{i}k_{0}a_{i}^{−1} = d_{i}k_{0}d_{i}^{−1} as required.

Taking a subsequence, we obtain that d_{i} → d_{0} as i → ∞ for some d_{0} ∈ C and d_{0}k_{0}d_{0}^{−1} ∈ K. This implies that d_{0}k_{0} = kd_{0} for some k ∈ K. Since d_{0} lies in the interior of A^{+}, it follows from the uniqueness properties of the Cartan decomposition (Lemma 3.1) that k_{0} = k ∈ M. This is a contradiction. □

Theorem 3.7 (The strong wavefront lemma). Let C be a closed subset contained in A^{+} with a positive distance from the walls of A^{+}.Then for any neighborhoods U of e in K and V of e in A, there exists a neighborhood O of e in G such that

Og ∪ gO ⊂ (k_{1}U)(aV M)(k_{2}U)

for any g = k_{1}ak_{2} ∈ KCK.

Proof. Without loss of generality, we may assume that C = A^{+}(C) for some C > 0, where

A^{+}(C) = {a ∈ A^{+} : α(log a) ≥ C for all α ∈ Φ^{+}}.

Replacing V by a smaller subset if necessary, we may assume that A^{+}(C)V is contained in the interior of A^{+}(C/2).

\underline{Step 1}: We claim that there exists an ε-neighborhood O of e in G such that

Og ⊂ KaV K for all g = k_{1}ak_{2} ∈ KA^{+}(C)K.

We take ε > 0 to be sufficiently small such that the ε-neighborhood of e in A is contained in V. Let O be the ε-neighborhood of e in G. For h ∈ Oa, write h = k′_{1}bk′_{2} ∈ KA^{+}K. Then d(Kbk′_{2}a^{−1}, K) = d(Kbk′_{2}, Ka) < ε. Since d(Kb, Ka) ≤ d(Kbk′_{2}, Ka) by Lemma 3.2, we have

(3.8)  d(Kb, Ka) < ε

and hence, h = k′_{1}bk′_{2} ∈ KV aK. This shows that Oa ⊂ KaV K. Since O is K-invariant,

Ok_{1}ak_{2} ⊂ Oak_{2} ⊂ KaV K

This proves the claim.

\underline{Step 2}: We claim that there exists an ε-neighborhood O of e in G such that

(3.9)  Og ⊂ KA^{+}(k_{2}U) for all g = k_{1}ak_{2} ∈ KA^{+}(C)K.

Let U_{0} be a neighborhood of e in K such that kU_{0}k^{−1} ⊂ U for all k ∈ K. Choose ε > 0 that satisfies Step 1 above, and Proposition 3.4 holds with respect to 2ε, U_{0}, and A^{+}(C/2). Take O to be the ε-neighborhood of e in G. Then for any h = k′_{1}bk′_{2} ∈ Oa, we have by (3.8),

d(Kbk′_{2}, Kb) ≤ d(Kbk′_{2}, Ka) + d(Kb, Ka) < 2ε.

Since Oa ⊂ KA^{+}(C/2)K, by Proposition 3.4, this implies that k′_{2} ∈ MU_{0}, and hence Oa ⊂ KA^{+}MU_{0}. Thus,

O(k_{1}ak_{2}) ⊂ KA^{+}MU_{0}k_{2} ⊂ KA^{+}M(k_{2}U).

\underline{Step 3}: We claim that there exists a neighborhood O of e in G such that

Og ⊂ (k_{1}U)A^{+}K for all g = k_{1}ak_{2} ∈ KA^{+}(C)K.

Recall that by the Iwasawa decomposition, G = KAN, where N is the subgroup of G corresponding to the sum of the positive root spaces. Since the Weyl group acts transitively on the sets of simple roots, there exists an element w ∈ K that normalizes A such that wA^{+}(C)w^{−1} = (A^{+}(C))^{−1}.

Let U_{0} be a neighborhood of e in K such that U_{0}(wU_{0}^{−1}w^{−1}) ⊂ U. Let O be the neighborhood from Step 2 with respect to U_{0} and O_{1} a neighborhood of e in AN such that c^{−1}O_{1}c ⊂ O for all c ∈ A^{+}. Since w ∈ K, wOw^{−1} = O. Conjugating (3.9) (with respect to U_{0}) by w, we have for any b ∈ wA^{+}(C)w^{−1},

Ob ⊂ K(A^{+})^{−1}(wU_{0}w^{−1}).

For a ∈ A^{+}(C), a^{−1} ∈ wA^{+}(C)w^{−1} and hence

(3.10)  a^{−1}O_{1} ⊂ Oa^{−1} ⊂ K(A^{+})^{−1}(wU_{0}w^{−1}).

By taking the inverse of (3.10),

(3.11)  O_{1}^{−1}a ⊂ (wU_{0}^{−1}w^{−1})A^{+}K.

Since the product map K × AN → G is a diffeomorphism, U_{0}O_{1}^{−1} is a neighborhood of e in G. Therefore, there exists a neighborhood O_{2} of e in G such that k^{−1}O_{2}k ⊂ U_{0}O_{1}^{−1} for all k ∈ K. Then by (3.11),

O_{2}k_{1}ak_{2} ⊂ k_{1}U_{0}O_{1}^{−1}ak_{2} ⊂ k_{1}U_{0}(wU_{0}^{−1}w^{−1})A^{+}K ⊂ k_{1}UA^{+}K.

This finishes the proof of Step 3.

By the above three steps, we obtain a neighborhood O_{1} of e in G such that for all g = k_{1}ak_{2} ∈ KA^{+}(C)K,

O_{1}g ⊂ KaV K ∩ KA^{+}(k_{2}U) ∩ (k_{1}U)A^{+}K.

By the uniqueness the Cartan decomposition (Lemma 3.1),

KaV K ∩ KA^{+}(k_{2}U) ∩ (k_{1}U)A^{+}K = (k_{1}U)(aV M)(k_{2}U).

Hence, we have shown that for every neighborhood U of e in K and every neighborhood V of e in A, there exists a neighborhood O_{1} of e in G such that

(3.12)  O_{1}g ⊂ (k_{1}U)(aV M)(k_{2}U) for all g = k_{1}ak_{2} ∈ KA^{+}(C)K.

Let U_{0} be a neighborhood of e in K such that kU_{0}^{−1}k^{−1} ⊂ U for every k ∈ K and V_{0} a neighborhood of e in A such that wV_{0}^{−1}w^{−1} ⊂ V, where w is the element of K defined in Step 3. Let O_{2} be a neighborhood of e in G such that

O_{2}g ⊂ (k_{1}U_{0})(aV_{0}M)(k_{2}U_{0}) for all g = k_{1}ak_{2} ∈ KA^{+}(C)K.

By taking the inverse, we have

gO_{2}^{−1} ⊂ (U_{0}^{−1}k_{1})(aV_{0}^{−1}M)(U_{0}^{−1}k_{2}) for all g = k_{1}ak_{2} ∈ KA^{+}(C)^{−1}K.

Conjugating by w, we get

g(wO_{2}^{−1}w^{−1}) ⊂ (k_{1}U)(aV M)(k_{2}U) for all g = k_{1}ak_{2} ∈ KA^{+}(C)K.

Set O = O_{1} ∩ (wO_{2}^{−1}w^{−1}) to finish the proof. □

4. Contractions on G/B

Let G be a connected noncompact semisimple Lie group with finite center, K a maximal compact subgroup of G, and B a maximal connected split solvable subgroup of G. Let N be the unipotent radical of B. The normalizer P of N in G is the unique minimal parabolic subgroup of G containing B.

Lemma 4.1. There exists a split Cartan subgroup A ofG with respect to K and an ordering on the root system Φ such that

• G = KA^{+}K.

• B = AN.

• N is the subgroup generated by all positive root subgroups of G with respect to A.

• P = MAN, where M is the centralizer of A in K and M = K ∩ P.

Proof. Take any split Cartan subgroup A_{0} and a Weyl chamber A_{0}^{+} such that the Cartan decomposition G = KA_{0}^{+}K holds. Set P_{0} = M_{0}A_{0}N_{0}, where M_{0} is the centralizer of A_{0} in K, and N_{0} is the subgroup generated by all positive root subgroups of G with respect to A_{0}. Note that P_{0} is a minimal parabolic subgroup of G (see [W, Sec. 1.2.3]). By the Iwasawa decomposition, G = KP_{0}. Since all minimal parabolic subgroups are conjugate to each other ([loc.cit.]), there exists k ∈ K such that P = kP_{0}k^{−1}. Set A = kA_{0}k^{−1}, A^{+} = kA_{0}^{+}k^{−1}, and M = kM_{0}k^{−1}. It is clear that N = kN_{0}k^{−1}. Since AN is normal in P, it is the unique maximal connected split solvable subgroup in P. Thus, B = AN. □

Let A be a split Cartan subgroup as in Lemma 4.1. Denote by M be the centralizer of A in K and by N^{−} the subgroup generated by all negative root subgroups of G. One can check that the map

N^{−} × M → G/B : (n, m) ↦ nmB

is a diffeomorphism onto its image. Moreover, it follows from the properties of Bruhat decomposition that G/B −π(N^{−} ×M) is a finite union of closed submanifolds of smaller dimensions. On the other hand, by the Iwasawa decomposition, the map

K → G/B : k ↦ kB.

is a diffeomorphism. Thus, we have a map N^{−} × M → K. Since M normalizes B, this map is right M-equivariant. Denote by S ⊂ K the submanifold which is the image of N^{−} × {e} under this map.

Let ν be the probability Haar measure on K and τ the probability Haar measure on M. Then ν = σ ⊗ τ for some finite smooth measure σ on S (see [W, p. 73]), i.e.,

(4.2)  ∫_{K} f dν = ∫_{M} ∫_{S} f(sm) dσ(s)dτ(m), f ∈ C(K).

For C > 0, put

A^{+}(C) = {a ∈ A : α(log a) ≥ C for all α ∈ Φ^{+}}.

Lemma 4.3. Let Ω be a Borel subset of K, s ∈ S, m ∈ M, and k ∈ K. For a ∈ A, r ∈ K, and U ⊂ K, define

Ω_{r}(k, a, U) = {l ∈ Ω : lkB ∈ a^{−1}r^{−1}UB}.

(1) Let V W ⊂ K be an open neighborhood of s where V ⊂ S and W ⊂ M are open subsets. Then for every ε > 0, there exists C > 0 such that

ν((Ω ∩ m^{−1}SW k^{−1}) − Ω_{sm}(k, a, V W)) < ε for all a ∈ A^{+}(C).

(2) Let U ⊂ K be a Borel subset such that s ∉ \overline{UM}. Then for every ε > 0, there exists C > 0 such that

ν(Ω_{sm}(k, a, U)) < ε for all a ∈ A^{+}(C).

Proof. Note that Ω_{sm}(k, a, U) = m^{−1} ·(mΩ)_{s}(k, a, U). Hence, replacing Ω by mΩ, we may assume that m = e. Also, Ω_{s}(k, a, U) = (Ωk)_{s}(e, a, U) · k^{−1}. Thus, we may assume that k = e.

The proof is based on the observation that elements in the interior of A^{+} act on N^{−}B as contractions (cf. [Z, Proposition 8.2.5]). Denote by π the map π : N^{−} → G/B : n ↦ nB. Note that for a ∈ A and V_{0} ⊂ S,

(4.4)  π^{−1}(a^{−1}V_{0}B) = a^{−1}π^{−1}(V_{0}B)a.

Let D be a compact set in S such that σ(S − D) < ε/2.

Suppose that s ∈ V W. Let W_{0} ⊂ W be a compact set such that τ(W − W_{0}) < ε/2. Then s^{−1}V W is a neighborhood of W_{0}. By uniform continuity, there exists a neighborhood V_{0} of e in S such that V_{0}W_{0} ⊂ s^{−1}V W. For each a ∈ A^{+}, the map N^{−} → N^{−} : x → a^{−1}xa expands any neighborhood of e at least by the factor of min_{α∈Φ^{+}} e^{α(log a)}. Hence, there exists C > 0 such that π^{−1}(DB) ⊂ a^{−1}π^{−1}(V_{0}B)a for all a ∈ A^{+}(C). Then by (4.4), DB ⊂ a^{−1}V_{0}B. It follows that DW_{0}B ⊂ a^{−1}V_{0}W_{0}B and

ν(Ω_{s}(e, a, V W)) ≥ ν(Ω ∩ DW_{0}) ≥ ν(Ω ∩ SW) − ν((S − D)M) − ν(S(W − W_{0}))
≥ ν(Ω ∩ SW) − ε.

This proves the first part of the lemma.

To prove the second part, we observe that there exists an open subset V ⊂ S such that s ∈ V ⊂ K − \overline{UM}. Since ν(m^{−1}SMk^{−1}) = 1, it suffices to apply the first part with W = M. □

Lemma 4.5. Let W ⊂ M and Ω ⊂ K be Borel subsets. Then every k ∈ K,

∫_{M} ν(Ω ∩ m^{−1}SW k^{−1}) dτ(m) = ν(Ω)τ(W).

Proof. Without loss of generality, k = e. Since M normalizes N^{−}, we have m^{−1}Sm ⊂ S. Since ν(K −SM) = 0, we may assume that Ω ⊂ SM. Moreover, it suffices to prove the lemma for Ω = Ω_{S}Ω_{M} with Borel sets Ω_{S} ⊂ S and Ω_{M} ⊂ M. By (4.2),

ν(Ω ∩ m^{−1}SW) = σ(Ω_{S})τ(Ω_{M} ∩ m^{−1}W).

For Y ⊂ M, denote by χ_{Y} the characteristic function of Y. We have

∫_{M} τ(Ω_{M} ∩ m^{−1}W) dτ(m) = ∫_{M} ∫_{M} χ_{Ω_{M}} (l)χ_{m^{−1}W} (l) dτ(l)dτ(m)
= ∫_{M} χ_{Ω_{M}} (l) (∫_{M} χ_{W l^{−1}}(m) dτ(m)) dτ(l) = τ(Ω_{M})τ(W).

Since ν(Ω) = σ(Ω_{S})τ(Ω_{M}), this proves the lemma. □

5. Volume estimates

Let G be a connected noncompact semisimple Lie group with finite center and G = KA^{+}K a Cartan decomposition. Let dk denote the probability Haar measure on K, dt the Lebesgue measure on the Lie algebra 𝔞 of A, and da the Haar measure on A derived from dt via the exponential map. We denote by m the Haar measure on G which is normalized so that for any f ∈ C_{c}(G),

(5.1)  ∫_{G} f dm = ∫_{K} ∫_{A^{+}} ∫_{K} f(k_{1}ak_{2})ξ(log a)dk_{1}dadk_{2},

where

(5.2)  ξ(t) = ∏_{α∈Φ^{+}} sinh(α(t))^{m_{α}}, t ∈ 𝔞,

and m_{α} is the dimension of the root subspace corresponding to α. In particular, for any measurable subset D ⊂ A^{+}, we have

(5.3)  Vol(KDK) = ∫_{D} ξ(log a) da.

Let ∥ · ∥ be a Euclidean norm on 𝔞, that is, for some basis v_{1}, …, v_{n} of 𝔞, ∥ ∑_{i} c_{i}v_{i}∥ = \sqrt{∑_{i} c_{i}^{2}}. We assume that ∥ · ∥ is invariant under the Weyl group action. For instance, ∥ · ∥ can be taken to be the norm induced from an invariant Riemannian metric d on the symmetric space K\G, i.e.,

∥t∥ = d(K, K exp(t)) for t ∈ 𝔞.

For T > 0 and 𝔱 ⊂ 𝔞, set

𝔱_{T} = {t ∈ 𝔱 : ∥t∥ < T }.

Let ρ = \frac{1}{2} ∑_{α∈Φ^{+}} m_{α}α. One can check that the maximum of ρ on 𝔞̄_{1} is achieved at a unique point contained in the interior of 𝔞^{+}, which we call the barycenter of 𝔞^{+}.

We present a simple derivation of the asymptotics of the volume of Riemannian balls in a symmetric space of noncompact type (see also [Kn, Theorem 6.2]).

Lemma 5.4. Let 𝔮 be a convex cone in 𝔞^{+} centered at the origin that contains the barycenter of 𝔞^{+} in its interior. Then for some C > 0 (independent of 𝔮),

∫_{𝔮_{T}} ξ(t) dt ∼_{T →∞} C · T^{(r−1)/2}e^{δT} ,

where r = ℝ-rank(G) = dim A and δ = max{2ρ(t) : t ∈ 𝔞̄_{1}}. In particular,

∫_{𝔮_{T}} ξ(t) dt ∼_{T →∞} ∫_{𝔞_{T}^{+}} ξ(t) dt.

Proof. We have

∫_{𝔮_{T}} ξ(t) dt = \frac{1}{2^{|Φ^{+}|}} ∫_{𝔮_{T}} e^{2ρ(t)} dt + “other terms”,

where the “other terms” are linear combinations of integrals of the form ∫_{𝔮_{T}} e^{λ(t)} dt such that 2ρ − λ = ∑_{α∈Φ^{+}} n_{α}α for some n_{α} ≥ 0. In particular, 2ρ > λ in the interior of the Weyl chamber 𝔞^{+}. Since the maximum of2ρ in 𝔞̄_{1} is achieved in the interior of 𝔮,

max{2ρ(t) : t ∈ 𝔮̄_{1}} > max{λ(t) : t ∈ 𝔮̄_{1}} =^{def} δ′.

Thus,

∫_{𝔮_{T}} e^{λ(t)} dt ≤ e^{δ′T} Vol(𝔮_{T}) ≪ e^{δ′T} T^{r} = o(e^{δT})

as T → ∞.It remains to show that for some C > 0 independent of 𝔮,

(5.5)  ∫_{𝔮_{T}} e^{2ρ(t)} dt ∼_{T →∞} C · T^{(r−1)/2}e^{δT} .

Making a change variables and decomposing 𝔮_{1} into slices parallel to the hyperplane {2ρ = 0}, we have

∫_{𝔮_{T}} e^{2ρ(t)} dt = T^{r} ∫_{𝔮_{1}} e^{2T ρ(t)} dt = T^{r} ∫_{0}^{δ} e^{T x}ϕ(x) dx,

where ϕ(x) = Vol_{r−1}(𝔮_{1} ∩ {2ρ = x}).

First, we show that for some c_{1} > 0 independent of 𝔮,

(5.6)  ϕ(x) ∼_{x→δ^{−}} c_{1} · (δ − x)^{(r−1)/2}.

We identify 𝔞 with the set {(t_{1}, ⋯, t_{r}) ∈ ℝ^{r}} and denote by Q the positive quadratic form on 𝔞defined by the norm. After a linear change of variables, we may assume that 2ρ(t) = t_{r} and

Q(t_{1}, …, t_{r}) = ∑_{i=1}^{r−1} α_{i}(t_{i} − β_{i}t_{r})^{2} + α_{r}t_{r}^{2}

for some α_{i} > 0 and β_{i} ∈ ℝ. It is clear that the maximum of 2ρ(t) on the set𝔞̄_{1} = {t : Q(t) ≤ 1} is achieved when t_{i} = β_{i}t_{r} for i = 1, …, r−1 and α_{r}t_{r}^{2} = 1. This implies that δ = α_{r}^{−1/2}. Since for x close to δ, the set 𝔮_{1}^{+}∩{2ρ = x} is defined by the condition Q(t_{1}, …, t_{r−1}, x) ≤ 1, we have

ϕ(x) = Vol_{r−1} ({(t_{1}, …, t_{r−1}) : ∑_{i=1}^{r−1} α_{i}(t_{i} − β_{i}x)^{2} ≤ 1 − α_{r}x^{2}})
= c_{1} · (1 − α_{r}x^{2})^{(r−1)/2} = c_{1} · (1 − x^{2}/δ^{2})^{(r−1)/2}

for a constant c_{1} > 0. This proves (5.6). Now by (5.6) and the L’Hopital’s rule,

β(x) =^{def} ∫_{0}^{x} ϕ(δ − u) du ∼_{x→0^{+}} c_{2} · x^{(r+1)/2}

for some c_{2} > 0. Thus, by the Abelian theorem (see [Wi, Corollary 1a, p. 182]),

∫_{0}^{δ} e^{T x}ϕ(x) dx = e^{δT} ∫_{0}^{δ} e^{−T x}ϕ(δ − x)dx = e^{δT} ∫_{0}^{∞} e^{−T x}dβ(x)
∼_{T →∞} c_{3} · T^{−(r+1)/2}e^{δT}

for some c_{3} > 0. It is clear that c_{3} is independent of 𝔮. This proves (5.5) and the proposition. □

For T > 0 and R ⊂ G, define

(5.7)  R_{T} = {r ∈ R : d(K, Kr) < T }.

Since G_{T} = KA_{T}^{+}K, combining the previous lemma and (5.3), we deduce

Corollary 5.8. For some C > 0,

Vol(G_{T}) ∼_{T →∞} C · T^{(r−1)/2}e^{δT} .

In particular, we have

Lemma 5.9. For some functions a(ε) and b(ε) such that a(ε) → 1 and b(ε) → 1 as ε → 0^{+}, we have

a(ε) ≤ lim inf_{T →∞} \frac{Vol(G_{T −ε})}{Vol(G_{T})} ≤ lim sup_{T →∞} \frac{Vol(G_{T +ε})}{Vol(G_{T})} ≤ b(ε).

For T, C > 0 and R ⊂ A, define

(5.10)  R(C) = {r ∈ R : α(log r) ≥ C for all α ∈ Φ^{+}},
R_{T} (C) = R_{T} ∩ R(C).

Lemma 5.11. Let Q ⊂ A^{+} be a convex cone centered at the origin that contains the barycenter in its interior. Then for any fixed C > 0,

∫_{Q_{T} (C)} ξ(log a) da ∼_{T →∞} ∫_{A_{T}^{+}} ξ(log a) da.

In particular,

Vol(KA_{T}^{+}K) ∼_{T →∞} Vol(KA_{T}^{+}(C)K).

Proof. It suffices to prove the lemma when Q is contained in the interior of A^{+}. Then there exists T_{0} > 0 such that Q_{T} (C) ⊃ Q_{T} − Q_{T_{0}} for all sufficiently large T > 0. Thus, the lemma follows from Lemma 5.4. □

6. Equidistribution of solvable flows

Let G be a connected noncompact semisimple Lie group with finite center which is realized as a closed subgroup of a Lie group L, Λ a lattice in L, and Q a closed subgroup of G that contains a maximal connected split solvable subgroup of G. In this section we investigate the distribution of orbits of Q in the homogeneous space Λ\L.

Let K_{0} be a maximal compact subgroup of G and A_{0} a split Cartan subgroup of G with respect to K_{0}. Then G = K_{0}A_{0}^{+}K_{0} for any positive Weyl chamber A_{0}^{+} in A_{0}. Denote by d an invariant Riemannian metric on K_{0}\G. For g ∈ G, R ⊂ G, Ω ⊂ K_{0}, and T > 0, define

R_{T} (g, Ω) = {r ∈ R : d(K_{0}, K_{0}gr) < T, r ∈ g^{−1}K_{0}A_{0}^{+}Ω}

and R_{T} (g) = R_{T} (g, K_{0}). Note that

R_{T} (k_{0}g, Ω) = R_{T} (g, Ω) = g^{−1}(gR)_{T} (e, Ω)

for any k_{0} ∈ K_{0}.

The main result in this section is Theorem 6.1 on equidistribution of the sets Q_{T} (g, Ω) as T → ∞. Let μ be the Haar measure on L such that μ(Λ\L) = 1, ν_{0} the probability Haar measure on K_{0}, and ρ a right invariant Haar measure on Q. Denote by G_{n} the product of all noncompact simple factors of G.

Theorem 6.1. Suppose that for y ∈ Λ\L, the orbit yG_{n} is dense in Λ\L. Then for g ∈ G, any Borel subset Ω ⊂ K_{0} with boundary of measure zero and f ∈ C_{c}(Λ\L),

lim_{T →∞} \frac{1}{ρ(Q_{T} (g))} ∫_{Q_{T} (g,Ω)} f(yq^{−1}) dρ(q) = ν_{0}(M_{0}Ω) ∫_{Λ\L} f dμ.

The rest of this section is devoted to a proof of Theorem 6.1. We start by stating a theorem of Shah ([S, Corollary 1.2]). Recall that a sequence {g_{i}} ⊂ G is called strongly divergent if for every projection π from G to its noncompact factor, π(g_{i}) → ∞ as i → ∞.

Theorem 6.2 (Shah). Suppose that for y ∈ Λ\L, the orbit yG_{n} is dense in Λ\L. Let {g_{i}} ⊂ G be a strongly divergent sequence. Then for any f ∈ C_{c}(Λ\L) and any Borel subset U in K_{0} with boundary of measure zero,

lim_{i→∞} ∫_{U} f(ykg_{i}) dν_{0}(k) = ν_{0}(U) ∫_{Λ\L} f dμ.

Remark 6.3. Although this theorem was stated in [S] only for the case U = K_{0}, its proof works equally well when U is an open subset of K_{0} with boundary of measure zero, and approximating Borel sets by open sets, one can check that Theorem 6.2 holds in the above generality.

Let K = g^{−1}K_{0}g and B be a maximal connected split solvable subgroup of G. By Lemma 4.1, there exists a a split Cartan subgroup A with an ordering on the root system Φ such that G = KA^{+}K and B = AN, where N is the subgroup generated by all positive root subgroups of A in G. Let M be the centralizer of A in K.

Let m be the Haar measure on G such that (5.1) holds with respect toK_{0} and A_{0}^{+}. It follows from the uniqueness of the Haar measure that for every g ∈ G, there exists c_{g} > 0 such that

(6.4)  ∫_{G} f dm = c_{g} ∫_{K_{0}} ∫_{B} f(g^{−1}kgb) dρ(b)dν_{0}(k), f ∈ C_{c}(G).

In particular,

(6.5)  c_{g}ρ(B_{T} (g)) = Vol(G_{T} (g)) = Vol(G_{T} (e)).

We normalize ρ so that c_{e} = 1. Let ν be the probability Haar measure on K. We use notations from Section 4. In particular, (4.2) holds.

Proposition 6.6. Suppose that for y ∈ Λ\L, the orbit yG_{n} is dense in Λ\L. Let U = V W be an open neighborhood of e in K where V is an open neighborhood of e in S, and W is an open neighborhood of e in M such that σ(∂V) = τ(∂W) = 0and Ω a Borel subset of K_{0} such that ν_{0}(∂Ω) = 0. Then for any f ∈ C_{c}(Λ\L),

\frac{1}{ρ(B_{T} (g))} ∫_{U} ∫_{B_{T} (g,Ω)} f(yb^{−1}k^{−1}) dρ(b)dk → ν(U)ν_{0}(M_{0}Ω) ∫_{Λ\L} f dμ

as T → ∞.

Proof. Since both A_{0} and gAg^{−1} are split Cartan subgroups with respect to K_{0}, there exists k_{0} ∈ K_{0} such that A_{0} = k_{0}gAg^{−1}k_{0}^{−1}. Hence, replacing g by k_{0}g, we may assume that A = g^{−1}A_{0}g.

Let d be the Riemannian metric on K\G induced from the metric on K_{0}\G by the map Kx ↦ Kgx, and notations A_{T}^{+}, A_{T}^{+}(C), and A^{+}(C) be defined as in Section 5. For instance,

A_{T}^{+} = {a ∈ A^{+} : d(K_{0}g, K_{0}ga) < T }.

Since KA_{T}^{+}K = g^{−1}G_{T} (e)g, it follows from (5.3) and (6.5) that for every T > 0,

(6.7)  ρ(B_{T} (e)) = ∫_{A_{T}^{+}} ξ(log a) da

and by Lemma 5.11,

(6.8)  ∫_{A_{T}^{+}−A_{T}^{+}(C)} ξ(log a) da = o(ρ(B_{T} (e))) as T → ∞.

Since B_{T} (g, Ω) = B_{T} (g, M_{0}Ω), we may assume without loss of generality that M_{0}Ω = Ω.

We have

UB_{T} (g, Ω) = g^{−1}K_{0}A_{0 T}^{+} (e)Ω ∩ UB = KA_{T}^{+}(g^{−1}Ωg)g^{−1} ∩ UB
= {k_{1}ak_{2}g^{−1} : k_{1} ∈ K, a ∈ A_{T}^{+}, k_{2} ∈ Ω_{k_{1}}(a)},

where

Ω_{k_{1}}(a) = {k_{2} ∈ g^{−1}Ωg : k_{2}g^{−1}B ∈ a^{−1}k_{1}^{−1}UB}.

By (5.1) and (6.4),

c_{g} ∫_{U} ∫_{B_{T} (g,Ω)} f(yb^{−1}k^{−1}) dρ(b)dν(k) = ∫_{UB_{T} (g,Ω)} f(yx^{−1}) dm(x) (6.9)

= ∫_{k_{1}ak_{2}g^{−1}∈UB_{T} (g,Ω)} f(ygk_{2}^{−1}a^{−1}k_{1}^{−1}) ξ(log a)dν(k_{2})dadν(k_{1})
= ∫_{k_{1}∈K} ∫_{a∈A_{T}^{+}} ∫_{k_{2}∈Ω_{k_{1}}(a)} f(ygk_{2}^{−1}a^{−1}k_{1}^{−1}) ξ(log a)dν(k_{2})dadν(k_{1}).

\underline{Step 1}: We claim that for every m ∈ M,

(6.10) lim_{T →∞} \frac{1}{ρ(B_{T} (e))} ∫_{V} ∫_{A_{T}^{+}} ∫_{Ω_{sm}(a)} f(ygk^{−1}a^{−1}(sm)^{−1}) ξ(log a)dν(k)dadσ(s)

= σ(V) · ν(g^{−1}Ωg ∩ m^{−1}SW k_{g}^{−1}) · ∫_{Λ\L} f dμ,

wherek_{g} ∈ K such that k_{g}B = g^{−1}B.

Let Ω_{m} = g^{−1}Ωg ∩ m^{−1}SW k_{g}^{−1}. One can check that ν(∂Ω_{m}) = 0. To show the claim, we first fix s ∈ V and ε > 0. Note that

Ω_{sm}(a) = {k ∈ g^{−1}Ωg : kk_{g}B ∈ a^{−1}m^{−1}(s^{−1}V)W B}.

By Lemma 4.3(1), there exists C > 0 such that

(6.11)  ∫_{Ω_{m}−Ω_{sm}(a)} |f(ygk^{−1}a^{−1}(sm)^{−1})| dν(k) ≤ ε for all a ∈ A_{T}^{+}(C).

and by Theorem 6.2 for any sufficiently large C > 0 and a ∈ A_{T}^{+}(C), we have

|∫_{Ω_{m}} f(ygk^{−1}a^{−1}(sm)^{−1}) dν(k) − ν(Ω_{m}) ∫_{Λ\L} f dμ| < ε.

Hence,

|∫_{A_{T}^{+}(C)} ∫_{Ω_{sm}(a)} f(ygk^{−1}a^{−1}(sm)^{−1}) ξ(log a)dν(k)da
− ∫_{A_{T}^{+}(C)} ξ(log a) da · ν(Ω_{m}) · ∫_{Λ\L} f dμ|
≤ ∫_{A_{T}^{+}(C)} ∫_{Ω_{m}−Ω_{sm}(a)} |f(ygk^{−1}a^{−1}(sm)^{−1})| ξ(log a)dν(k)da

+ ∫_{A_{T}^{+}(C)} |∫_{Ω_{m}} f(ygk^{−1}a^{−1}(sm)^{−1}) dν(k) − ν(Ω_{m}) ∫_{Λ\L} f dμ| ξ(log a)da

≤ 2ε ∫_{A_{T}^{+}(C)} ξ(log a) da ≤ 2ε ∫_{A_{T}^{+}} ξ(log a) da.

Thus, by (6.7) and (6.8),

|∫_{A_{T}^{+}} ∫_{Ω_{sm}(a)} f(ygk^{−1}a^{−1}(sm)^{−1}) ξ(log a)dν(k)da
−ρ(B_{T} (e))ν(Ω_{m}) ∫_{Λ\L} f dμ| ≤ 2ερ(B_{T} (e)) + o(ρ(B_{T} (e)).

This shows that for every s ∈ V,

lim_{T →∞} \frac{1}{ρ(B_{T} (e))} ∫_{A_{T}^{+}} ∫_{Ω_{sm}(a)} f(ygk^{−1}a^{−1}(sm)^{−1}) ξ(log a)dν(k)da

= ν(Ω_{m}) ∫_{Λ\L} f dμ.

Therefore, (6.10) follows from the Lebesgue dominated convergence theorem. \underline{Step 2}: We claim that for every m ∈ M, (6.12)

lim_{T →∞} \frac{1}{ρ(B_{T} (e))} ∫_{S−\overline{V}} ∫_{A_{T}^{+}} ∫_{Ω_{sm}(a)} f(ygk^{−1}a^{−1}(sm)^{−1}) ξ(log a)dν(k)dadσ(s) = 0.

Let s ∈ S − \overline{V} and ε > 0. By Lemma 4.3(2), there exists C > 0 such that ν(Ω_{sm}(a)) < ε for all a ∈ A_{T}^{+}(C).

Hence, by (6.7),

\frac{1}{ρ(B_{T} (e))} ∫_{A_{T}^{+}(C)} ∫_{Ω_{sm}(a)} |f(ygk^{−1}a^{−1}(sm)^{−1})| ξ(log a)dν(k)da

≤ \frac{ε · sup |f|}{ρ(B_{T} (e))} ∫_{A_{T}^{+}(C)} ξ(log a)da ≤ ε · sup |f|.

Since ε > 0 is arbitrary, it follows from (6.8) that for any s ∈ S − \overline{V},

lim_{T →∞} \frac{1}{ρ(B_{T} (e))} ∫_{A_{T}^{+}} ∫_{Ω_{sm}(a)} f(ygk^{−1}a^{−1}(sm)^{−1}) ξ(log a)dν(k)da = 0.

Hence, (6.12) follows from the Lebesgue dominated convergence theorem.

Since σ(∂V) = 0, combining (6.10) and (6.12), we deduce that for every m ∈ M,

lim_{T →∞} \frac{1}{ρ(B_{T} (e))} ∫_{S} ∫_{A_{T}^{+}} ∫_{Ω_{sm}(a)} f(ygk^{−1}a^{−1}(sm)^{−1}) ξ(log a)dν(k)dadσ(s)

= σ(V) · ν(g^{−1}Ωg ∩ m^{−1}SW k_{g}^{−1}) · ∫_{Λ\L} f dμ.

Thus, by (4.2), the Lebesgue dominated convergence theorem, and Lemma 4.5,

lim_{T →∞} \frac{1}{ρ(B_{T} (e))} ∫_{K} ∫_{A_{T}^{+}} ∫_{Ω_{k_{1}}(a)} f(ygk_{2}^{−1}a^{−1}(k_{1})^{−1}) ξ(log a)dν(k_{2})dadν(k_{1})

= σ(V) · ∫_{M} ν(g^{−1}Ωg ∩ m^{−1}SW k_{g}^{−1}) dτ(m) · ∫_{Λ\L} f dμ

= σ(V)ν(g^{−1}Ωg)τ(W) ∫_{Λ\L} f dμ = ν(U)ν_{0}(Ω) ∫_{Λ\L} f dμ.

Finally, the proposition follows from (6.5) and (6.9). □

Proof of Theorem 6.1. Let B be a maximal connected split solvable subgroup of G contained in Q. Since G = KB, we have Q = DB for D = K ∩ Q. Then

(6.13)  Q_{T} (g, Ω) = DB_{T} (g, Ω)

and for suitable Haar measures ρ_{B} and ν_{D} on B and D respectively.

(6.14)  ∫_{Q} f(q) dρ(q) = ∫_{D} ∫_{B} f(db) dρ_{B}(b)dν_{D}(d), f ∈ C_{c}(Q).

Hence, Theorem 6.1 for Q follows from Theorem 6.1 for B and the Lebesgue dominated convergence theorem. Thus, we may assume that Q = B.

26 ALEXANDER GORODNIK AND HEE OH

Let ε > 0. One can find a neighborhood U of e in K as in Proposition 6.6 and functions f^{+}, f^{−} ∈ C_{c}(Λ\L) such that

f^{−}(xk^{−1}) ≤ f(x) ≤ f^{+}(xk^{−1}) for all x ∈ Λ\L and k ∈ U

and

∫_{Λ\L} |f^{+} − f^{−}| dμ ≤ ε.

Put

F_{T} (k) = \frac{1}{ρ(B_{T} (g))} ∫_{B_{T} (g,Ω)} f(yb^{−1}k^{−1}) dρ(b), k ∈ K.

For any k ∈ U,

(6.15) \frac{1}{ρ(B_{T} (g))} ∫_{B_{T} (g,Ω)} f^{−}(yb^{−1}k^{−1}) dρ(b) ≤ F_{T} (e)

≤ \frac{1}{ρ(B_{T} (g))} ∫_{B_{T} (g,Ω)} f^{+}(yb^{−1}k^{−1}) dρ(b).

Integrating over U ⊂ K, we obtain

(6.16) \frac{1}{ρ(B_{T} (g))} ∫_{U} ∫_{B_{T} (g,Ω)} f^{−}(yb^{−1}k^{−1}) dρ(b)dν(k) ≤ ν(U) · F_{T} (e)

≤ \frac{1}{ρ(B_{T} (g))} ∫_{U} ∫_{B_{T} (g,Ω)} f^{+}(yb^{−1}k^{−1}) dρ(b)dν(k).

Hence, by Proposition 6.6,

ν(U)ν_{0}(M_{0}Ω) ∫_{Λ\L} f^{−} dμ ≤ ν(U) · lim inf_{T →∞} F_{T} (e)

≤ ν(U) · lim sup_{T →∞} F_{T} (e) ≤ ν(U)ν_{0}(M_{0}Ω) ∫_{Λ\L} f^{+} dμ.

This shows that

ν_{0}(M_{0}Ω) · (∫_{Λ\L} f dμ − ε) ≤ lim inf_{T →∞} F_{T} (e)

≤ lim sup_{T →∞} F_{T} (e) ≤ ν_{0}(M_{0}Ω) · (∫_{Λ\L} f dμ + ε) ,

and the theorem follows. □

Since it follows from the result of N. Shah (see [S, Theorem 1.1]) and Ratner’s topological rigidity [R2] that \overline{yQ} = \overline{yG_{n}Q} for every y ∈ Λ\L, one may expect that Theorem 6.1 holds under the condition \overline{yQ} = Λ\L as well.

Lemma 6.17. Let L be a connected semisimple Lie group with finite center and L = L_{c}L_{n} the decomposition of L into the product of compact and noncompact factors. Suppose that for y ∈ Λ\L, we have \overline{yG_{n}} ⊃ yL_{n} and \overline{yG_{n}Q} = Λ\L. Then the conclusion of Theorem 6.1 holds.

Proof. Let B ⊂ Q be a maximal connected split solvable subgroup of G. Then Q = DB for D = K ∩Q. By Ratner’s theorem on orbit closures [R2], the set \overline{yG_{n}} is a homogeneous space yG_{0} with a probability G_{0}-invariant measure μ_{0}, where G_{0} is a closed connected subgroup of L that contains L_{n}. Applying Theorem 6.1 to the space yG_{0} and the subgroup B, we deduce that

lim_{T →∞} \frac{1}{ρ_{B}(B_{T} (g))} ∫_{B_{T} (g,Ω)} f(yb^{−1}) dρ_{B}(b) = ν_{0}(M_{0}Ω) ∫_{yG_{0}} f dμ_{0}

for every f ∈ C_{c}(Λ\L), where ρ_{B} is a right invariant Haar measure on B. Since \overline{yG_{n}Q} = Λ\L, L_{n}Q = L_{n}D, and L_{n} ⊂ G_{0}, we have

(6.18)  Λ\L = \overline{yG_{n}D} = yG_{0}D.

(Here we used that D is compact.) By (6.13), (6.14), and the Lebesgue dominated convergence theorem,

lim_{T →∞} \frac{1}{ρ(Q_{T} (g))} ∫_{Q_{T} (g,Ω)} f(yq^{−1}) dρ(q) = ν_{0}(M_{0}Ω) ∫_{Λ\G} f dμ̃

for every f ∈ C_{c}(Λ\L), where the measure μ̃ is defined by

∫_{Λ\L} f dμ̃ = ∫_{D} ∫_{yG_{0}} f(zd^{−1}) dμ_{0}(z)dν_{D}(d), f ∈ C_{c}(Λ\L).

Since L_{n} is a normal subgroup of L, it is clear that the measure μ̃ is invariant under L_{n}D. This measure corresponds to a Radon measure μ̃ on L which is right (L_{n}D)-invariant and left Stab_{L}(y)-invariant. Namely,

μ̃(f) = ∫_{Λ\L} ( ∑_{λ∈Stab_{L}(y)} f(λg)) dμ̃(g), f ∈ C_{c}(L).

We have a decomposition μ̃ = ∫_{L_{c}} μ̃_{x} dω(x) for a Radon measure ω on L_{c}, where μ̃_{x} is a right L_{n}-invariant measure on the leaf xL_{n} for ω-a.e. x ∈ L_{c}. Since L_{n} commutes with x ∈ L_{c}, μ̃_{x} is left L_{n}-invariant too. It follows that the measure μ̃ is left L_{n}-invariant. Thus, it is left invariant under \overline{Stab_{L}(y)L_{n}} ⊃ G_{0}. Setting E = G_{0} ∩L_{c} and F = DL_{n} ∩L_{c}, we deduce that the measure ω is left E-invariant and right F -invariant. Note that EF is a closed subset of L_{c}, and it follows from (6.18) that yL_{n}EF = Λ\L. Thus, by the Baire category theorem, the set EF contains an open subset of L_{c}. Since the group E × F acts transitively on EF, this implies that EF is open in L_{c}. Thus, L_{c} = EF because L is connected. By [K, Theorem 8.32], ω is a Haar measure on L_{c}. This implies that μ̃ is L-invariant, and the lemma follows. □

7. Distribution of lattice points in sectors and the boundary

In this section, we apply Theorem 6.1 in the case when Λ\L = Γ\G in order to deduce Theorems 1.1 and 1.2. We use notations from Theorem 1.2. In particular, A is a split Cartan subgroup with respect to K_{1}, g^{−1}K_{1}g = K_{2}, M_{1} is the centralizer of A in K_{1}, and M_{2} = K_{2} ∩ Q.

To simplify notations, we make the following conventions in this section: • For R ⊂ G and T > 0,

R_{T} = {r ∈ R : d(K_{1}, K_{1}gr) < T }.

• For T, C > 0, Ω_{1} ⊂ K_{1}, and Ω_{2} ⊂ K_{2},

N_{T} (Ω_{1}, Ω_{2}) = #(Γ_{T} ∩ g^{−1}K_{1}A^{+}Ω_{1} ∩ Ω_{2}Q),
N_{T}^{C}(Ω_{1}, Ω_{2}) = #(Γ_{T} ∩ g^{−1}K_{1}A^{+}(C)Ω_{1} ∩ Ω_{2}Q),

where A^{+}(C) is defined as in (5.10).

• For T, C > 0 and Ω ⊂ K_{1},

Q_{T} (Ω) = Q_{T} ∩ g^{−1}K_{1}A^{+}Ω,
Q_{T}^{C}(Ω) = Q_{T} ∩ g^{−1}K_{1}A^{+}(C)Ω.

Let m denote the Haar measure on G such that (5.1) holds for K_{1} and ρ the right invariant Haar measure on Q such that

(7.1)  ∫_{G} f dm = ∫_{K_{2}} ∫_{Q} f(kq) dρ(q)dν_{2}(k), f ∈ C_{c}(G).

Lemma 7.2. For any C > 0, Ω_{1} ⊂ K_{1}, and Ω_{2} ⊂ K_{2},

lim_{T →∞} \frac{1}{m(G_{T})}(N_{T} (Ω_{1}, Ω_{2}) − N_{T}^{C}(Ω_{1}, Ω_{2})) = 0.

Proof. Fix D > C > 0 and ε > 0. By Theorem 3.7, there exists a neighborhood O of e in G such that

(7.3)  O^{−1}g^{−1}K_{1}A^{+}(D)K_{1} ⊂ g^{−1}K_{1}A^{+}(C)K_{1}.

In addition, we may choose O so that

Γ ∩ O^{−1}O = {e} and OG_{T} ⊂ G_{T +ε} for all T > 0.

It follows from (7.3) that

O · (Γ − g^{−1}K_{1}A^{+}(C)K_{1}) ⊂ G − g^{−1}K_{1}A^{+}(D)K_{1}.

Thus,

N_{T} (Ω_{1}, Ω_{2}) − N_{T}^{C}(Ω_{1}, Ω_{2}) ≤ #{γ ∈ Γ_{T} − g^{−1}K_{1}A^{+}(C)K_{1}}

= \frac{1}{m(O)}m ( ⋃_{γ∈Γ_{T} −g^{−1}K_{1}A^{+}(C)K_{1}} Oγ) ≤ \frac{1}{m(O)}m(G_{T +ε} − g^{−1}K_{1}A^{+}(D)K_{1})

= \frac{1}{m(O)}m(K_{1}(A_{T +ε}^{+} − A_{T +ε}^{+}(D))K_{1}) = o(m(G_{T +ε}))

by (5.3) and Lemma 5.11. Now the lemma follows from Lemma 5.9. □

The proof of the following lemma is similar and is left to the readers.

Lemma 7.4. For any C > 0 and Ω ⊂ K_{1},

lim_{T →∞} \frac{1}{m(G_{T})}(ρ(Q_{T} (Ω)) − ρ(Q_{T}^{C}(Ω))) = 0.

Proof of Theorem 1.2. Without loss of generality, m(Γ\G) = 1. It is easy to check that ν_{1}(∂(M_{1}Ω_{1})) = ν_{2}(∂(Ω_{2}M_{2})) = 0. Thus, we may assume that Ω_{1} = M_{1}Ω_{1} and Ω_{2} = Ω_{2}M_{2}.

We need to show that

N_{T} (Ω_{1}, Ω_{2}) ∼_{T →∞} ν_{1}(Ω_{1})ν_{2}(Ω_{2})m(G_{T}).

Fix any ε > 0 and C > 0. Let U be a neighborhood of e in K_{1} with boundary of measure zero such that ν_{1}(Ω_{1}^{+} − Ω_{1}^{−}) < ε, where

Ω_{1}^{+} = ⋃_{u∈U} uΩ_{1} and Ω_{1}^{−} = ⋂_{u∈U} u^{−1}Ω_{1}.

One can check that ν_{1}(∂Ω_{1}^{±}) = 0.

By the strong wavefront lemma (Theorem 3.7), there exists a symmetric neighborhood O′ of e in G such that

(7.5)  O′g^{−1}K_{1}A^{+}(C) ⊂ g^{−1}K_{1}A^{+}U and O′G_{T} ⊂ G_{T +ε} for all T > 0.

Set O = M_{2}(O′ ∩ Q)M_{2}. Note that O is a symmetric neighborhood of e in Q. Using the fact that M_{2} ⊂ g^{−1}K_{1}g, it is easy to check that (7.5) holds for O as well. We may also assume that ρ(∂O) = 0.

Let f be the characteristic function of Ω_{2}O ⊂ G. Since the decomposition h = h_{K_{2}}h_{Q} for h_{K_{2}} ∈ K_{2} and h_{Q} ∈ Q is uniquely determined modulo M_{2} and M_{2}O = O, we have

f(h) = χ_{Ω_{2}}(h_{K_{2}})χ_{O}(h_{Q}).

We also define a function on Γ\G by

F (h) = ∑_{γ∈Γ} f(γh).

30 ALEXANDER GORODNIK AND HEE OH

\underline{Step 1}: We claim that for any T > 0,

(7.6)  N_{T}^{C}(Ω_{1}, Ω_{2}) ≤ \frac{1}{ρ(O)} ∫_{Q_{T +ε}(Ω_{1}^{+})} F (q^{−1}) dρ(q),

We have

(7.7) ∫_{Q_{T +ε}(Ω_{1}^{+})} F (q^{−1}) dρ(q) = ∫_{Q_{T +ε}(Ω_{1}^{+})} (∑_{γ∈Γ} χ_{Ω_{2}}(γ_{K_{2}})χ_{O}(γ_{Q}q^{−1})) dρ(q)

= ∑_{γ∈Γ: γ_{K_{2}}∈Ω_{2}} ρ(Q_{T +ε}(Ω_{1}^{+}) ∩ Oγ_{Q}).

It follows from (7.5) that

OQ_{T}^{C}(Ω_{1}) ⊂ Q_{T +ε}(Ω_{1}^{+}).

This implies that for every γ ∈ Γ such that γ_{Q} ∈ Q_{T}^{C}(Ω_{1}),

ρ(Q_{T +ε}(Ω_{1}^{+}) ∩ Oγ_{Q}) = ρ(O).

Thus,

∫_{Q_{T +ε}(Ω_{1}^{+})} F (q^{−1}) dρ(q) ≥ #{γ ∈ Γ : γ_{K_{2}} ∈ Ω_{2}, γ_{Q} ∈ Q_{T}^{C}(Ω_{1})} · ρ(O)
= N_{T}^{C}(Ω_{1}, Ω_{2})ρ(O).

This proves (7.6). \underline{Step 2}: We claim that for any T > 0,

(7.8)  N_{T} (Ω_{1}, Ω_{2}) ≥ \frac{1}{ρ(O)} ∫_{Q_{T −ε}^{C}(Ω_{1}^{−})} F (q^{−1}) dρ(q).

As in (7.7),

(7.9)  ∫_{Q_{T −ε}^{C}(Ω_{1}^{−})} F (q^{−1}) dρ(q) = ∑_{γ∈Γ:γ_{K_{2}}∈Ω_{2}} ρ(Q_{T −ε}^{C}(Ω_{1}^{−}) ∩ Oγ_{Q}).

Since UΩ_{1}^{−} ⊂ Ω_{1}, we have by (7.5),

O^{−1}Q_{T −ε}^{C}(Ω_{1}^{−}) ⊂ Q_{T} (Ω_{1}).

Therefore, for γ ∈ Γ such that γ_{Q} ∉ Q_{T} (Ω_{1}),

ρ(Q_{T −ε}^{C}(Ω_{1}^{−}) ∩ Oγ_{Q}) = 0.

By (7.9),

∫_{Q_{T −ε}^{C}(Ω_{1}^{−})} F (q^{−1}) dρ(q) ≤ #{γ ∈ Γ : γ_{K_{2}} ∈ Ω_{2}, γ_{Q} ∈ Q_{T} (Ω_{1})} · ρ(O)
= N_{T} (Ω_{1}, Ω_{2})ρ(O).

This proves (7.8).

Since the boundary of the set Ω_{2}O has measure 0 (this can be checked using (7.1)), the function f can be approximated by continuous functions with compact support and Theorem 6.1 can be applied to the function F (see Lemma 6.17):

lim_{T →∞} \frac{1}{ρ(Q_{T})} ∫_{Q_{T} (Ω_{1}^{±})} F (q^{−1}) dρ(q) = ν_{1}(Ω_{1}^{±}) ∫_{G/Γ} F dm (7.10)

= ν_{1}(Ω_{1}^{±}) ∫_{G} f dm = ν_{1}(Ω_{1}^{±})ν_{2}(Ω_{2})ρ(O).

Note that by (7.1), m(G_{T}) = ρ(Q_{T}). Combining (7.8), Lemma 7.4, and (7.10), we deduce that

lim inf_{T →∞} \frac{N_{T} (Ω_{1}, Ω_{2})}{m(G_{T})} ≥ lim inf_{T →∞} \frac{1}{m(G_{T})ρ(O)} ∫_{Q_{T −ε}^{C}(Ω_{1}^{−})} F (q^{−1}) dρ(q)

≥ (lim inf_{T →∞} \frac{m(G_{T −ε})}{m(G_{T})}) · lim inf_{T →∞} \frac{1}{ρ(O)ρ(Q_{T −ε})} ∫_{Q_{T −ε}(Ω_{1}^{−})} F (q^{−1}) dρ(q)

≥ a(ε)ν_{1}(Ω_{1}^{−})ν_{2}(Ω_{2}) ≥ a(ε)(ν_{1}(Ω_{1}) − ε)ν_{2}(Ω_{2}),

where a(ε) is defined in Lemma 5.9. Since ε > 0 is arbitrary, it follows from Lemma 5.9 that

lim inf_{T →∞} \frac{N_{T} (Ω_{1}, Ω_{2})}{m(G_{T})} ≥ ν_{1}(Ω_{1})ν_{2}(Ω_{2}).

By Lemma 7.2, (7.6), and (7.10),

lim sup_{T →∞} \frac{N_{T} (Ω_{1}, Ω_{2})}{m(G_{T})} = lim sup_{T →∞} \frac{N_{T}^{C}(Ω_{1}, Ω_{2})}{m(G_{T})}

≤ lim sup_{T →∞} \frac{1}{m(G_{T})ρ(O)} ∫_{Q_{T +ε}(Ω_{1}^{+})} F (q^{−1}) dρ(q)

≤ (lim sup_{T →∞} \frac{m(G_{T +ε})}{m(G_{T})}) · lim sup_{T →∞} \frac{1}{ρ(O)ρ(Q_{T +ε})} ∫_{Q_{T +ε}(Ω_{1}^{+})} F (q^{−1}) dρ(q)

≤ b(ε)ν_{1}(Ω_{1}^{+})ν_{2}(Ω_{2}) ≤ b(ε)(ν_{1}(Ω_{1}) + ε)ν_{2}(Ω_{2}),

where b(ε) is defined in Lemma 5.9. Thus, by Lemma 5.9,

lim sup_{T →∞} \frac{N_{T} (Ω_{1}, Ω_{2})}{m(G_{T})} ≤ ν_{1}(Ω_{1})ν_{2}(Ω_{2}),

and the theorem is proved. □

Proof of Theorem 1.1. Note that G is a connected semisimple center-free Lie group with no compact factors, and K_{x} and K_{y} are maximal compact subgroups of G. Since G acts transitively on X (see, for example, [H, Theorem IV.3.3]), there exists g ∈ G such that y = xg. Then K_{y} = g^{−1}K_{x}g. The closed positive Weyl chamber W_{x} at x is of the form xA^{+}, where A^{+} is a positive Weyl chamber in a split Cartan subgroup A of G with respect to K_{x}. The stabilizer of b ∈ X(∞) in G is a parabolic subgroup Q of G (see [GJT, Proposition III.3.8]). In particular, Q contains a maximal connected split solvable subgroup of G. Let π : K_{y} → bG be the map given by k ↦ bk^{−1}. Then for Ω_{1} ⊂ K_{x} and Ω_{2} ⊂ bG,

#{γ ∈ Γ : yγ ∈ W_{x}Ω_{1} ∩ B_{T}(x), bγ^{−1} ∈ Ω_{2}}
= #{γ ∈ Γ ∩ g^{−1}K_{x}A^{+}Ω_{1} ∩ π^{−1}(Ω_{2})Q : d(K_{x}, K_{x}gγ) < T }.

Note that π maps the probability Haar measure on K_{y} to the measure m_{b,y}. Using the fact that the map π is open, one can check that the set π^{−1}(Ω_{2}) has boundary of measure zero if m_{b,y}(∂Ω_{2}) = 0. Hence, Theorem 1.1 follows from Theorem 1.2. □

8. Distribution of lattice points in bisectors

Let G be a connected semisimple Lie group with finite center and G = KA^{+}K a Cartan decomposition of G. To simplify notations, we fix g ∈ G and for Ω_{1}, Ω_{2} ⊂ K and T, C > 0, define

G_{T} = {h ∈ G : d(K, Kgh) < T },
G_{T} (Ω_{1}, Ω_{2}) = g^{−1}Ω_{1}A^{+}Ω_{2} ∩ G_{T} ,
N_{T} (Ω_{1}, Ω_{2}) = #(Γ ∩ G_{T} (Ω_{1}, Ω_{2})).

If we set A_{T}^{+} = {a ∈ A^{+} : d(K, Ka) < T }, then G_{T} (Ω_{1}, Ω_{2}) = g^{−1}Ω_{1}A_{T}^{+}Ω_{2}.

Let G be a closed subgroup of a Lie group L and Λ a lattice in L. Let m be a Haar measure on G such that (5.1) holds and μ the Haar measure on L such that μ(Λ\L) = 1.

Theorem 8.1. Suppose that for y ∈ Λ\L, the orbit yG_{n} is dense in Λ\L, where G_{n} is the product of all noncompact simple factors of G. For any Borel subsets Ω_{1}, Ω_{2} ⊂ K with boundaries of measure zero and f ∈ C_{c}(Λ\L),

lim_{T →∞} \frac{1}{m(G_{T})} ∫_{G_{T} (Ω_{1},Ω_{2})} f(yh^{−1}) dm(h) = ν(Ω_{1})ν(Ω_{2}) ∫_{Λ\L} f dμ.

Proof. By (5.1),

∫_{G_{T} (Ω_{1},Ω_{2})} f(yh^{−1}) dm(h) = ∫_{Ω_{1}} ∫_{A_{T}^{+}} ∫_{Ω_{2}} f(yk_{2}^{−1}a^{−1}k_{1}^{−1}g) ξ(log a)dk_{2}dadk_{1} (8.2)

= ∫_{Ω_{1}} ∫_{A_{T}^{+}} ∫_{Ω_{2}^{−1}} f(yk_{2}a^{−1}k_{1}^{−1}g) ξ(log a)dk_{2}dadk_{1}.

Since ν(Ω_{2}^{−1}) = ν(Ω_{2}), by Theorem 6.2, for every ε > 0, there exists C > 0 such that

(8.3)  |∫_{Ω_{2}^{−1}} f(yk_{2}a^{−1}k_{1}^{−1}g) dk_{2} − ν(Ω_{2}) ∫_{Λ\L} f dμ| < ε

for all a ∈ A_{T}^{+}(C). Since m(G_{T}) = ∫_{A_{T}^{+}} ξ(log a) da, it follows from Lemma 5.11 that

(8.4)  ∫_{A_{T}^{+}−A_{T}^{+}(C)} ξ(log a) da = o(m(G_{T})) as T → ∞.

Combining (8.3) and (8.4), we get

|∫_{A_{T}^{+}} ∫_{Ω_{2}^{−1}} f(yk_{2}a^{−1}k_{1}^{−1}g) dk_{2}ξ(log a)da − ν(Ω_{2}) · ∫_{A_{T}^{+}} ξ(log a) da · ∫_{Λ\L} f dμ|

≤ ∫_{A_{T}^{+}(C)} |∫_{Ω_{2}^{−1}} f(yk_{2}a^{−1}k_{1}^{−1}g) dk_{2} − ν(Ω_{2}) ∫_{Λ\L} f dμ| ξ(log a)da

+ 2 sup |f| ∫_{A_{T}^{+}−A_{T}^{+}(C)} ξ(log a) da ≤ ε ∫_{A_{T}^{+}(C)} ξ(log a) da + o(m(G_{T}))

≤ ε · m(G_{T}) + o(m(G_{T})) as T → ∞.

This proves that for every k_{1} ∈ K,

lim_{T →∞} \frac{1}{m(G_{T})} ∫_{A_{T}^{+}} ∫_{Ω_{2}^{−1}} f(yk_{2}a^{−1}k_{1}^{−1}g) ξ(log a)dk_{2}da = ν(Ω_{2}) ∫_{Λ\L} f dμ.

Now the statement follows from (8.2) and the Lebesgue dominated convergence theorem. □

Proof of Theorem 1.6. We need to show that

N_{T} (Ω_{1}M, MΩ_{2}) ∼_{T →∞} ν(Ω_{1}M)ν(MΩ_{2}) · \frac{m(G_{T})}{m(Γ\G)}.

We may assume without loss of generality that Ω_{1}M = Ω_{1}, MΩ_{2} = Ω_{2}, and m(Γ\G) = 1. Since M contains all compact factors of G, we may assume that G contains no compact factors.

Fix ε > 0 and C > 0. There exists a neighborhood U of e in K with boundary of measure zero such that ν(Ω_{i}^{+} − Ω_{i}^{−}) < ε, i = 1, 2, where

Ω_{i}^{+} = ⋃_{u∈U} Ω_{i}u  and  Ω_{i}^{−} = ⋂_{u∈U} Ω_{i}u^{−1}.

Note that ν(∂Ω_{i}^{±}) = 0, Ω_{i}U ⊂ Ω_{i}^{+}, and Ω_{i}^{−}U ⊂ Ω_{i}. By the strong wavefront lemma (Theorem 3.7), there exists a neighborhood O of e in G such that

O^{−1}Ω_{1}A^{+}(C)Ω_{2} ⊂ Ω_{1}^{+}A^{+}Ω_{2}^{+},
(8.5)  OΩ_{1}^{−}A^{+}(C)Ω_{2}^{−} ⊂ Ω_{1}A^{+}Ω_{2},
O^{±1}G_{T} ⊂ G_{T +ε} for all T > 0.

Let

G_{T}^{C}(Ω_{1}, Ω_{2}) = g^{−1}KA^{+}(C)K ∩ G_{T} (Ω_{1}, Ω_{2}),
N_{T}^{C}(Ω_{1}, Ω_{2}) = #(Γ ∩ G_{T}^{C}(Ω_{1}, Ω_{2})).

It is not hard to show (see Lemmas 7.2 and 7.4 for a similar argument) that

(8.6)  m(G_{T} (Ω_{1}, Ω_{2}) − G_{T}^{C}(Ω_{1}, Ω_{2})) = o(m(G_{T})),
(8.7)  N_{T} (Ω_{1}, Ω_{2}) − N_{T}^{C}(Ω_{1}, Ω_{2}) = o(m(G_{T}))

as T → ∞.

Let f ∈ C_{c}(G) be such that f ≥ 0, supp(f) ⊂ O and ∫_{G} f dm = 1. Define a function on Γ\G by

F (h) = ∑_{γ∈Γ} f(γh).

Clearly, ∫_{Γ\G} F dm = 1. We claim that

N_{T}^{C}(Ω_{1}, Ω_{2}) ≤ ∫_{G_{T +ε}(Ω_{1}^{+},Ω_{2}^{+})} F (h^{−1}) dm(h) (8.8)

and

N_{T} (Ω_{1}, Ω_{2}) ≥ ∫_{G_{T −ε}^{C}(Ω_{1}^{−},Ω_{2}^{−})} F (h^{−1}) dm(h). (8.9)

First, we observe that

∫_{G_{T +ε}(Ω_{1}^{+},Ω_{2}^{+})} F (h^{−1}) dm(h) = ∑_{γ∈Γ} ∫_{G_{T +ε}(Ω_{1}^{+},Ω_{2}^{+})γ^{−1}} f(h^{−1}) dm(h).

By (8.5),

O^{−1}G_{T}^{C}(Ω_{1}, Ω_{2}) ⊂ G_{T +ε}(Ω_{1}^{+}, Ω_{2}^{+}).

Thus, for every γ ∈ Γ ∩ G_{T}^{C}(Ω_{1}, Ω_{2}), we have O^{−1} ⊂ G_{T +ε}(Ω_{1}^{+}, Ω_{2}^{+})γ^{−1} and

∫_{G_{T +ε}(Ω_{1}^{+},Ω_{2}^{+})γ^{−1}} f(h^{−1}) dm(h) = 1.

This implies (8.8).

To prove (8.9), we use that

∫_{G_{T −ε}^{C}(Ω_{1}^{−},Ω_{2}^{−})} F (h^{−1}) dm(h) = ∑_{γ∈Γ} ∫_{G_{T −ε}^{C}(Ω_{1}^{−},Ω_{2}^{−})γ^{−1}} f(h^{−1}) dm(h).

By (8.5),

OG_{T −ε}^{C}(Ω_{1}^{−}, Ω_{2}^{−}) ⊂ G_{T} (Ω_{1}, Ω_{2}).

Therefore, for γ ∈ Γ − G_{T} (Ω_{1}, Ω_{2}), we have O^{−1} ∩ G_{T −ε}^{C}(Ω_{1}^{−}, Ω_{2}^{−})γ^{−1} = ∅ and

∫_{G_{T −ε}^{C}(Ω_{1}^{−},Ω_{2})γ^{−1}} f(h^{−1}) dm(h) = 0.

This proves (8.9).

By (8.9), (8.6), and Theorem 8.1,

lim inf_{T →∞} \frac{N_{T} (Ω_{1}, Ω_{2})}{m(G_{T})} ≥ lim inf_{T →∞} \frac{1}{m(G_{T})} ∫_{G_{T −ε}^{C}(Ω_{1}^{−},Ω_{2}^{−})} F (h^{−1}) dm(h)

≥ (lim inf_{T →∞} \frac{m(G_{T −ε})}{m(G_{T})}) · lim inf_{T →∞} \frac{1}{m(G_{T −ε})} ∫_{G_{T −ε}^{C}(Ω_{1}^{−},Ω_{2}^{−})} F (h^{−1}) dm(h)

≥ a(ε)ν(Ω_{1}^{−})ν(Ω_{2}^{−}) ≥ a(ε)(ν(Ω_{1}) − ε)(ν(Ω_{2}) − ε),

where a(ε) is defined in Lemma 5.9. Since ε > 0 is arbitrary, it follows from Lemma 5.9 that

lim inf_{T →∞} \frac{N_{T} (Ω_{1}, Ω_{2})}{m(G_{T})} ≥ ν(Ω_{1})ν(Ω_{2}).

The opposite inequality for lim sup is proved similarly using (8.7), (8.8), and Theorem 8.1. □

9. Measure-preserving lattice actions

Let G be a connected semisimple Lie group with finite center and Γ_{1}, Γ_{2} lattices in G. Let L = G × G, Λ = Γ_{1} × Γ_{2}, H = {(g, g) : g ∈ G}, and Q a closed subgroup of H containing a maximal connected split solvable subgroup of H. Fix an invariant Riemannian metric d on the symmetric space K\G, and for T > 0 and g ∈ G, define

Q_{T} (g) = {(q, q) ∈ Q : d(K, Kgq) < T }.

Corollary 9.1. Suppose that for y ∈ Λ\L, the orbit yQ is dense inΛ\L. Then for any f ∈ C_{c}(Λ\L),

lim_{T →∞} \frac{1}{ρ(Q_{T} (g))} ∫_{Q_{T} (g)} f(yq^{−1}) dρ(q) = ∫_{Λ\L} f dμ,

where ρ is a right Haar measure on Q, and μ is the L-invariant probability measure on Λ\L.

Proof. It suffices to check the conditions of Lemma 6.17. Namely, we need to show that

(9.2)  \overline{yH_{n}} ⊃ yL_{n},

where H_{n} and L_{n} denote the product of all noncompact simple factors of H and L respectively. We also denote by H_{c} and L_{c} the product of all compact simple factors of H and L. By Ratner’s theorem on orbit closures [R2], \overline{yH_{n}} = yH_{0} for some closed subgroup H_{0} of L containing H_{n}. Then yH_{0}H_{c} = \overline{yH} = Λ\L, and it follows from the Baire category theorem (cf. proof of Lemma 6.17) that H_{0}H_{c} is an open subset of L. Since H_{0}H_{c} is also closed, we conclude that L = H_{0}H_{c}. Now (9.2) follows from the Lemma 9.3 below. □

Lemma 9.3. Let S be a connected subgroup of L that contains H_{n}. Then S = (S ∩ L_{n})(S ∩ L_{c}).

Proof. Let 𝔥_{n} ⊂ 𝔰 ⊂ 𝔩 = 𝔩_{n} ⊕ 𝔩_{c} be the corresponding Lie algebras. We have decompositions

𝔥_{n} = ⨁_{i} 𝔥_{n}^{i} and 𝔩_{n} = ⨁_{i} 𝔩_{n}^{i},

where 𝔥_{n}^{i} and 𝔩_{n}^{i} are the simple ideals of 𝔥_{n} and 𝔩_{n} respectively such that 𝔥_{n}^{i} ⊂ 𝔩_{n}^{i}. Note that 𝔥_{n}^{i} is a maximal subalgebra of 𝔩_{n}^{i}. In particular, 𝔥_{n}^{i} is its own normalizer in 𝔩_{n}^{i}.

It suffices to show that for every s = (∑_{i} s_{i}) + s_{c} ∈ 𝔰 with s_{i} ∈ 𝔩_{n}^{i} and s_{c} ∈ 𝔩_{c}, we have s_{i}, s_{c} ∈ 𝔰. Clearly,

[𝔥_{n}^{i}, s] = [𝔥_{n}^{i}, s_{i}] ⊂ 𝔰.

If [𝔥_{n}^{i}, s_{i}] + 𝔥_{n}^{i} ≠ 𝔥_{n}^{i}, then [𝔥_{n}^{i}, s_{i}] + 𝔥_{n}^{i} generates 𝔩_{n}^{i}, and hence, 𝔩_{n}^{i} ⊂ 𝔰. Otherwise, s_{i} normalizes 𝔥_{n}^{i}, and it follows that s_{i} ∈ 𝔥_{n}^{i}. In both cases, s_{i} ∈ 𝔰. Then s_{c} = s − ∑_{i} s_{i} ∈ 𝔰 too. This proves the lemma. □

Theorem 1.9 can be deduced from Corollary 9.1, as it was explained in [GW] or [O].

References

[DM] S. G. Dani and G. A. Margulis. Asymptotic behavior of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci. 101 (1991), no. 1, 1–17.

[DRS] W. Duke, Z. Rudnick and P. Sarnak. Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), no. 1, 143–179.

[EMM] A. Eskin, G. Margulis and S. Mozes. Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2) 147 (1998), no. 1, 93–141.

[EM] A. Eskin and C. McMullen. Mixing, counting and equidistribution on Lie groups, Duke Math. J. 71 (1993), no. 1, 181–209.

[EMS] A. Eskin, S. Mozes and N. Shah. Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math. (2) 143 (1996), no. 2, 253–299.

[EO] A. Eskin and H. Oh. Integer points on a family of homogeneous varieties and unipotent flows, To appear in Duke Math. J.

[H] S. Helgason. Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, 80. Academic Press, Inc., New York-London, 1978.

[GO] W. T. Gan and H. Oh. Equidistribution of integer points on a family of homogeneous varieties: a problem of Linnik, Compositio Math. 136 (2003), no. 3, 323–352.

[G] A. Good. Local analysis of Selberg’s trace formula, Lecture Notes in Mathematics, 1040, Springer-Verlag, Berlin, 1983.

[Go] A. Gorodnik. Lattice action on the boundary of SL_{n}(ℝ), Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1817–1837.

[GM] A. Gorodnik and F. Maucourant. Proximality and equidistribution on the Furstenberg boundary, Geom. Dedicata 113 (2005), 197–213.

[GOS] A. Gorodnik, H. Oh and N. Shah. In preparation.

[GW] A. Gorodnik and B. Weiss, Distribution of lattice orbits on homogeneous varieties, To appear in GAFA.

[GJT] Y. Guivarc’h, L. Ji and J. C. Taylor. Compactifications of symmetric spaces, Progress in Mathematics, 156. Birkhäuser Boston, Inc., Boston, MA, 1998.

[K] A. Knapp. Representation theory of semisimple Lie groups, An overview based on examples, Princeton university press, 1986.

[Kn] G. Knieper. On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal. 7 (1997), no. 4, 755–782.

[L] F. Ledrappier. Ergodic properties of some linear actions, J. Math. Sci. (New York) 105 (2001), no. 2, 1861–1875.

[M] G. Margulis. On some aspects of the theory of Anosov systems, Springer-Verlag, Berlin, 2004.

[Ma] F. Maucourant. Approximation diophantienne, dynamique des chambres de Weyls et répartitions d’orbites de réseaux, PhD Thesis, Université de Lille, 2002.

[Mor] D. Morris Witte. Cocompact subgroups of semisimple Lie groups, Lie algebra and related topics (Madison, WI, 1988), 309–313, Contemp. Math., 110, Amer. Math. Soc., Providence, RI, 1990.

[Mo] G. D. Mostow. Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78. Princeton University Press, 1973.

[N] P. Nicholls. A lattice point problem in hyperbolic space, Michigan Math. J. 30 (1983), no. 3, 273–287.

[No] A. Nogueira. Orbit distribution on ℝ^{2} under the natural action of SL(2, ℤ), Preprint, 2000.

[O] H. Oh. Lattice action on finite volume homogeneous spaces, J. Korean Math. Soc. 42 (2005), no. 4, 635–653.

[R1] M. Ratner. On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545–607.

[R2] M. Ratner. On Raghunathan’s topological conjecture and distribution of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280.

[S] N. Shah. Limit distribution of expanding translates of certain orbits on homogeneous spaces on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 2, 105–125.

[Sh] R. Sharp. Sector estimates for Kleinian groups, Port. Math. (N.S.) 58 (2001), no. 4, 461– 471.

[V] N. Vatsal. Uniform distribution of Heegner points, Invent. Math. 148 (2002), no. 1, 1–46.

[W] G. Warner. Harmonic Analysis on Semi-simple Lie groups I, Springer-Verlag, New York-Heidelberg, 1972.

[Wi] D. Widder. The Laplace Transform, Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941.

[Z] R. Zimmer. Ergodic theory and semisimple groups, Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984.

Mathematics 253-37, Caltech, Pasadena, CA 91106

E-mail address: gorodnik@caltech.edu

Mathematics 253-37, Caltech, Pasadena, CA 91106

E-mail address: heeoh@caltech.edu

Current address: Math Department, 151 Thayer St., Brown University, Providence, RI 02912

E-mail address: heeoh@math.brown.edu