LATTICE ACTION ON THE BOUNDARY OF SL(n, ℝ)

ALEXANDER GORODNIK

Abstract. Let Γ be a lattice in G = SL(n, ℝ) and X = G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of Γ in X analogous to the classical equidistribution on torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of Borel subgroup of G.

1. Introduction

Let G = SL(n, ℝ) and Γ a lattice in G; that is, Γ is a discrete subgroup of G with finite covolume. We study the action of Γ on a compact homogeneous space X of algebraic origin. Namely, X = G/S where S is a closed subgroup of G which contains the connected component of a real algebraic subgroup H of G such that G/H is compact. An important example is provided by the Furstenberg boundary of G [F63]. In this case, X = G/B where B is the subgroup of upper triangular matrices in G.

It is possible to deduce from a result of Dani [St, Theorem 13.1] that every orbit of Γ in X is dense. We will prove a quantitative estimate for the distribution of orbits.

Introduce a norm on G:

∥g∥ = (Tr(^{t}gg))^{1/2} = (∑_{i,j} g_{ij}^{2})^{1/2} for g = (g_{ij}) ∈ G.  (1)

For T > 0, Ω ⊆ X, and x_{0} ∈ X, define a counting function

N_{T} (Ω, x_{0}) = |{γ ∈ Γ : ∥γ∥ < T, γ · x_{0} ∈ Ω}|.  (2)

Let m be a normalized SO(n)-invariant measure on X. It follows from the Iwasawa decomposition (see (8) below) that X is a homogeneous space of SO(n). Therefore, the measure m is unique. The following theorem shows that orbits of Γ in X are uniformly distributed with respect to the measure m. Theorem 1. For a Borel set Ω ⊆ X such that m(∂Ω) = 0 and x_{0} ∈ X,

N_{T} (Ω, x_{0}) ∼ \frac{γ_{n}}{μ̄(Γ\G)}m(Ω)T^{n(n−1)} as T → ∞,  (3)

where γ_{n} is a constant (defined in (14) below), and μ̄ is a finite G-invariant measure on Γ\G (defined in (10)).

It would be interesting to obtain an estimate for the error term in (3). This, however, would demand introducing different techniques than those employed here.

This article is a part of author’s PhD thesis at Ohio State University done under supervision of Prof. Bergelson.

Theorem 1 is analogous to the result of Ledrappier [L99] (see also [N00]) who investigated the distribution of dense orbits of a lattice Γ ⊂ SL(2, ℝ) acting on ℝ^{2}. Ledrappier used the equidistribution property of the horocycle flow. Similarly, we deduce Theorem 1 from an equidistribution property of orbits of Borel subgroup.

Denote by B^{o} the connected component of the upper triangular subgroup of G. To prove Theorem 1, we use the following ergodic theorem for the right action of B^{o} on Γ\G.

Theorem 2. Let ϱ be a right Haar measure on B^{o}, and ν the normalized G-invariant measure on Γ\G. Then for any f̃ ∈ C_{c}(Γ\G) and y ∈ Γ\G,

\frac{1}{ϱ(B_{T}^{o})} ∫_{B_{T}^{o}} f̃(yb^{−1})dϱ(b) → ∫_{Γ\G} f̃dν as T → ∞,

where B_{T}^{o} = {b ∈ B^{o} : ∥b∥ < T }.

Remarks.

1. One can consider the analogous limit for a left Haar measure on B^{o}. In this case, it may happen that the limit is 0 for some y ∈ Γ\G and all f̃ ∈ C_{c}(Γ\G) (see Proposition 8).

2. Since B^{o} is solvable (hence, amenable), one might expect that convergence for a.e. y ∈ Γ\G follows from known ergodic theorems for amenable group actions. Moreover, since ν is the only normalized B^{o}-invariant measure on Γ\G, one could expect that convergence holds for every y. However, this approach does not work because the sets B_{T}^{o} do not form a Følner sequence, and the space Γ\G is not compact in general.

3. To prove Theorem 2, we use Ratner’s classification of ergodic measures for unipotent flows [R91a]. In fact, we don’t need the full strength of this result. Since the subgroup U (defined in (37) below) is horospherical, it is enough to know classification of ergodic measures for horospherical subgroups. The situation is much easier in this special case (see [St, §13]).

4. We expect that analogs of Theorems 1 and 2 hold for a noncompact semisimple Lie group and its irreducible lattice with balls B_{T}^{o} defined by the Riemann metric. The main difficulty here is to show that the measure of B_{T}^{o} is “concentrated” on the “cone” B_{T}^{C} (cf. Lemma 7).

5. It was pointed out by P. Sarnak that it might be possible to prove the results of this article using harmonic analysis on Γ\G. In particular, Corollary 3 below can be deduced from the result of Good (Corollary on page 119 of [G]). Note that his method gives an estimate on the error term.

The paper is arranged as follows. In the next section, we give examples of applications of Theorem 1. In Section 3 we set up notations and prove some basic lemmas. Theorem 1 is deduced from Theorem 2 in Section 4. In Sections 5 and 6, we review results on the structure of unipotent flows and prove auxiliary facts about finite-dimensional representations of SL(n, ℝ). Finally, Theorem 2 is proved in Section 7.

2. Examples

1. Let X = ℝ ∪ {∞}, which is considered as the boundary of the hyperbolic upper half plane. The group G = SL(2, ℝ) acts on X by fractional linear transformations:

g · x = \frac{ax + b}{cx + d} for x ∈ X, g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G.  (4)

Let Γ be a lattice in SL(2, ℝ). For Ω ⊆ X and x_{0} ∈ X, define the counting function N_{T} (Ω, x_{0}) as in (2). Its asymptotics can be derived from Theorem 1. Note that the asymptotics of N_{T} (X, x_{0}) as T → ∞ provides a solution of the so-called hyperbolic circle problem (cf. [T, p. 266] and references therein).

Corollary 3. (of Theorem 1) For x_{0} ∈ X and −∞ ≤ a < b ≤ +∞,

N_{T} ((a, b), x_{0}) ∼ c_{Γ} (∫_{a}^{b} \frac{dt}{1 + t^{2}} ) T^{2} as T → ∞,

where c_{Γ} = \frac{1}{2μ̄(Γ\G)} (μ̄ is the G-invariant measure on Γ\G defined in (10)).

Proof. It is easy to see from (4) that G acts transitively on X, and the stabilizer of ∞ in G is the group of upper triangular matrices B. Thus, Theorem 1 is applicable to the space X.

Note that

K = SO(2) = \{k_{θ} = ( \begin{matrix}cos 2πθ & sin 2πθ \\ − sin 2πθ & cos 2πθ \end{matrix} ) : θ ∈ [0, 1)\} ,

and the normalized Haar measure on K is given by dk = dθ. The measure m on X can be defined as the image of dk under the map K → X : k → k · ∞. By (4), k_{θ} · ∞ = −ctan 2πθ. Then

m((a, b)) = ∫_{k·∞∈(a,b)} dk = \begin{matrix}∫ & −ctan 2πθ∈(a,b) \\ θ∈[0,1) \end{matrix} dθ = \frac{1}{π} ∫_{a}^{b} \frac{dt}{1 + t^{2}} .

We have used the substitution t = −ctan 2πθ.

Finally, by Theorem 1,

N_{T} ((a, b), x_{0}) ∼_{T →∞} \frac{γ_{2}m((a, b))}{μ̄(Γ\G)}T^{2} = c_{Γ} (∫_{a}^{b} \frac{dt}{1 + t^{2}} ) T^{2}.

Note that γ_{2} = \frac{π}{2} by (14) below. □

2. Let X = ℙ^{n−1} be the projective space (more generally X = G_{n,k}, Grassmann variety, or X = F_{n}, flag variety), and m be the rotation invariant normalized measure on X. Then the asymptotic estimate (3) holds for the standard action of G = SL(n, ℝ) on X. This is a special case of Theorem 1 because X can be identified with G/S where S is a closed subgroup of G that contains B, the group of upper triangular matrices.

3. Basic facts

For s = (s_{1}, …, s_{n}) ∈ ℝ^{n}, define

a(s) = diag(e^{s_{1}}, …, e^{s_{n}}).

For t = (t_{ij} : 1 ≤ i < j ≤ n), t_{ij} ∈ ℝ, denote by n(t) the unipotent upper triangular matrix with entries t_{ij} above the main diagonal.

We use the following notations:

G = SL(n, ℝ),
K = SO(n),
A^{o} = {a(s)| s ∈ ℝ^{n}, ∑_{i} s_{i} = 0},
N = {n(t)| t_{ij} ∈ ℝ, 1 ≤ i < j ≤ n},
B^{o} = A^{o}N = NA^{o}.

For s ∈ ℝ^{n}, ∑_{i} s_{i} = 0, denote α_{ij}(s) = s_{i} − s_{j}, where i, j = 1, …, n, i < j. These are the positive roots of the Lie algebra of G. Note that

Ad_{a(s)}n ({t_{ij}}) = a(s)n ({t_{ij}}) a(s)^{−1} = n({e^{α_{ij}(s)}t_{ij}}).  (5)

Let

δ(s) = \frac{1}{2} ∑_{i<j} α_{ij}(s) = ∑_{1≤k≤n}(n − k)s_{k}.  (6)

For C ∈ ℝ, define

A^{C} = {a(s) ∈ A^{o}| s_{i} > C, i = 1, …, n − 1}.

Also put B^{C} = A^{C}N.

Let dk be the normalized Haar measure on K. A Haar measure on N is given by dn = dt = ∏_{i<j} dt_{ij}. A Haar measure on A^{o} is da = ds = ds_{1} … ds_{n−1}.

The product map A^{o} × N → B^{o} is a diffeomorphism. The image of the product measure under this map is a left Haar measure on B^{o}. Denote this measure by λ. Then a right Haar measure ϱ on B^{o} can be defined by

ϱ(f) = ∫_{B^{o}} f(b^{−1})λ(b) = ∫_{A^{o}×N} f(a(s)n(t))e^{2δ(s)}dsdt, f ∈ C_{c}(B^{o}).  (7)

The map corresponding to the Iwasawa decomposition

(k, a, n) ↦ kan : K × A^{o} × N → G  (8)

is a diffeomorphism. One can define a Haar measure μ on G in terms of this decomposition:

∫_{G} fdμ = ∫_{K×A^{o}×N} f(ka(s)n(t))e^{2δ(s)}dkdsdt = ∫_{K×B^{o}} f(kb)dkdϱ(b)  (9)

for f ∈ C_{c}(G). For a lattice Γ in G, there exists a finite measure μ̄_{Γ} on Γ\G such that

∫_{G} fdμ = ∫_{Γ\G} ∑_{γ∈Γ} f(γg)dμ̄_{Γ}(g), f ∈ C_{c}(G).  (10)

Let β be an automorphism of G. Then β(Γ) is a lattice too. Moreover, the following lemma holds.

Lemma 4. Define a map

β̄ : Γ\G → β(Γ)\G : gΓ ↦ β(g)β(Γ)

Then β̄(μ̄_{Γ}) = μ̄_{β(Γ)}.

Proof. Since the automorphism group of G is a finite extension of the group of inner automorphisms, and G is unimodular, it follows that that the measure μ is β-invariant.

Every f̃ ∈ C_{c}(β(Γ)\G) can be represented as ∑_{γ∈β(Γ)} f(γg) for some f ∈ C_{c}(G) (see [R, Ch. 1]). Then

β̄(μ̄_{Γ})(f̃) = ∫_{Γ\G} ∑_{γ∈β(Γ)} f(γβ(g))dμ̄_{Γ}(g) = ∫_{G} f(β(g))dμ(g)

= ∫_{G} f(g)dμ(g) = μ̄_{Γ}(f̃).
□

For a subset D ⊆ G and T > 0, put

D_{T} = {d ∈ D : ∥d∥ < T }.

Note that

B_{T}^{o} = {a(s)n(t) : ∑_{1≤i≤n} e^{2s_{i}} + ∑_{1≤i<j≤n} e^{2s_{i}}t_{ij}^{2} < T^{2}} .  (11)

For s ∈ ℝ^{n}, define

N(s) = ∑_{i} e^{2s_{i}}.  (12)

Lemma 5. For C ∈ ℝ,

ϱ(B_{T}^{C}) = c_{n} ∫_{A_{T}^{C}} (T^{2} − N(s))^{\frac{n(n−1)}{4}} exp (∑_{k}(n − k)s_{k}) ds,

where c_{n} = π^{n(n−1)/4}/Γ(1 + n(n − 1)/4).

Proof. Use formulas (7), (6), (11), and make change of variables t_{ij} → e^{−s_{i}}t_{ij}. Then the above formula follows from the fact that the volume of the unit ball in ℝ^{m} is π^{m/2}/Γ(1 + m/2). □

It follows from Lemma 5 that ϱ(B_{T}^{o}) = O (T^{(n^{2}−n)}) as T → ∞. In fact, more precise statement is true: Lemma 6.

ϱ(B_{T}^{o}) ∼ γ_{n}T^{(n^{2}−n)} as T → ∞,  (13)

where

γ_{n} = \frac{π^{n(n−1)/4}}{2^{n−1}Γ (\frac{n^{2}−n+2}{2})} ∏_{k=1}^{n−1} Γ (\frac{n − k}{2} ) .  (14)

Proof. Since the norm (1) is K-invariant, G_{T} = KB_{T}^{o}. By (9), μ(G_{T}) = ϱ(B_{T}^{o}). The asymptotics of μ(G_{T}) as T → ∞ was computed in [DRS93, Appendix 1]. □

Lemma 7. For any C ∈ ℝ, ϱ(B_{T}^{C}) ∼ ϱ(B_{T}^{o}) as T → ∞.

Proof. For i_{0} = 1, …, n − 1, put

A_{T}^{i_{0}} = {a(s) ∈ A_{T}^{o} : s_{i_{0}} ≤ C} and B_{T}^{i_{0}} = {a(s)n(t) ∈ B_{T}^{o} : s_{i_{0}} ≤ C}.

We claim that ϱ(B_{T}^{i_{0}}) = o(ϱ(B_{T}^{o})) as T → ∞. It follows from (11) that if a(s) ∈ A_{T}^{o}, then s_{i} < log T for every i = 1, …, n. Then by Lemma 5,

ϱ(B_{T}^{i_{0}}) ≤ c_{n}T^{\frac{n(n−1)}{2}} ∫_{A_{T}^{i_{0}}} exp (∑_{k}(n − k)s_{k}) ds

≪ T^{\frac{n(n−1)}{2}} ∏_{k<n,k≠i_{0}} ∫_{−∞}^{log T} e^{(n−k)s_{k}}ds_{k} ≪ T^{n(n−1)−(n−i_{0})}.

(Here and later on A ≪ B means A < c · B for some absolute constant c > 0.) Now the claim follows from (13). Since B_{T}^{o} − B_{T}^{C} = ∪_{i_{0}<n}B_{T}^{i_{0}}, ϱ(B_{T}^{o} − B_{T}^{C}) = o(ϱ(B_{T}^{o})) as T → ∞. Therefore, ϱ(B_{T}^{C}) ∼ ϱ(B_{T}^{o}) as T → ∞. □

Next, we show that Theorem 2 fails for a left Haar measure on B^{o}. Proposition 8. Let Γ be a lattice in G = SL(2, ℝ), and y ∈ Γ\G be such that the orbit yN is periodic. Then for every f̃ ∈ C_{c}(Γ\G),

\frac{1}{λ(B_{T}^{o})} ∫_{B_{T}^{o}} f̃(yb^{−1})dλ(b) → 0 as T → ∞.

Proof. For C ∈ ℝ, put Â^{C} = {a(s) ∈ A^{o} : s_{1} < C} and B̂^{C} = Â^{C}N. As in Lemma 7, one can show that λ(B_{T}^{o} − B̂_{T}^{C}) = o(λ(B_{T}^{o})) as T → ∞. Therefore, for every C ∈ ℝ,

\frac{1}{λ(B_{T}^{o})} ∫_{B_{T}^{o} −B̂_{T}^{C}} f̃(yb^{−1})dλ(b) → 0 as T → ∞.

On the other hand, according to [St, Lemma 14.2], yN^{−1}a(s)^{−1} → ∞ as s_{1} → −∞. Thus, there exists C ∈ ℝ such that y(B̂_{T}^{C})^{−1} ∩ supp(f̃) = ∅. Then

∫B̂_{T}^{C} f̃(yb^{−1})dλ(b) = 0.

This proves the proposition. □

4. Proof of Theorem 1

Claim. It is enough to prove Theorem 1 for X = G/B^{o}.

Proof. Suppose that the theorem is proved for X = G/B^{o}.

At first, we consider a special case:

X = G/(B^{o})^{g_{0}} for some g_{0} ∈ G,  (15)

where (B^{o})^{g_{0}} = g_{0}^{−1}B^{o}g_{0}. By the Iwasawa decomposition (8), (B^{o})^{g_{0}} = (B^{o})^{k_{0}} for some k_{0} ∈ K.

The normalized K-invariant measure m on G/(B^{o})^{k_{0}} is defined as

m(C) = ∫_{k:k(B^{o})^{k_{0}}∈C} dk for Borel set C ⊆ G/(B^{o})^{k_{0}}.

Similarly, one defines the normalized K-invariant measure m^{∗} on G/B^{o}. Consider a map

β : G/B^{o} → G/(B^{o})^{k_{0}} : gB^{o} ↦ g^{k_{0}}(B^{o})^{k_{0}}.

Clearly, β is a diffeomorphism. Using that K is unimodular, one proves that

m^{∗}(β^{−1}(C)) = m(C) for Borel set C ⊆ G/(B^{o})^{k_{0}}.  (16)

Take

Ω ⊆ G/(B^{o})^{k_{0}} and x_{0} = h_{0}(B^{o})^{k_{0}} ∈ G/(B^{o})^{k_{0}}

such that m(∂Ω) = 0. Set

Ω^{∗} = β^{−1}(Ω) ⊆ G/B^{o} and x_{0}^{∗} = h_{0}^{k_{0}^{−1}}B^{o} ∈ G/B^{o}.

By (16), m^{∗}(∂Ω^{∗}) = 0 too. For γ ∈ Γ, γ · x_{0} ∈ Ω iff γ^{k_{0}^{−1}} · x_{0}^{∗} ∈ Ω^{∗}. Note also that ∥γ^{k_{0}^{−1}}∥ = ∥γ∥. Therefore,

N_{T} (Ω, x_{0}) = |{γ ∈ Γ^{k_{0}^{−1}} : ∥γ∥ < T, γ · x_{0}^{∗} ∈ Ω^{∗}}|

Applying the assumption to the lattice Γ^{∗} = Γ^{k_{0}^{−1}}, one gets

N_{T} (Ω, x_{0}) ∼ \frac{γ_{n}}{μ̄^{∗}(Γ^{∗}\G)}m^{∗}(Ω^{∗})T^{n(n−1)} as T → ∞,

where μ̄^{∗} is the measure on Γ^{∗}\G defined in (10). Now the special case (15) follows from Lemma 4 and (16).

Let us consider the general case. Let S be a closed subgroup of G such that S ⊇ H^{o}, where H is a real algebraic subgroup of G, and G/H is compact. Since H has finitely many connected components, G/H^{o} is compact too. Recall that the homogeneous space G/H^{o} is compact iff H_{ℂ} contains a maximal connected ℝ-split solvable ℝ-subgroup of G_{ℂ} (see, for example, [PR, Theorem 3.1]). Since maximal connected ℝ-split solvable ℝ-subgroups of G_{ℂ} are conjugate over G_{ℝ} (see [BT65], or Theorem 15.2.5 and Exercise 15.4.8 in [Sp]), for some g_{0} ∈ G, B^{g_{0}} ⊆ H. Hence, (B^{o})^{g_{0}} ⊆ H^{o} ⊆ S.

Denote by π the projection map G/(B^{o})^{g_{0}} → G/S. Take

Ω ⊆ G/S and x_{0} ∈ G/S

such that m(∂Ω) = 0. Set

Ω^{∗} = π^{−1}(Ω) ⊆ G/(B^{o})^{g_{0}} and x_{0}^{∗} ∈ π^{−1}(x_{0}).

Let m^{∗} be the K-invariant normalized measure on G/(B^{o})^{g_{0}}. Then m = π(m^{∗}) is the K-invariant normalized measure on G/S. It is easy to check that π(∂Ω^{∗}) ⊆ ∂Ω. Hence, m^{∗}(∂Ω^{∗}) ≤ m^{∗}(π^{−1}(∂Ω)) = m(∂Ω) = 0. Finally,

N_{T} (Ω, x_{0}) = N_{T} (Ω^{∗}, x_{0}^{∗}) ∼_{T →∞} \frac{γ_{n}}{μ̄(Γ\G)}m^{∗}(Ω^{∗})T^{n(n−1)} = \frac{γ_{n}}{μ̄(Γ\G)}m(Ω)T^{n(n−1)}.

□

We need the following proposition that follows easily from Theorem 2.

Proposition 9. Let f be the characteristic function of a relatively compact Borel subset Z ⊆ G such that μ(∂Z) = 0. Then for any y ∈ Γ\G,

\frac{1}{ϱ(B_{T}^{o})} ∫_{B_{T}^{o}} f̃(yb^{−1})dϱ(b) ⟶ \frac{1}{μ̄(Γ\G)} ∫_{G} fdμ as T → ∞,  (17)

where f̃(Γg) = ∑_{γ∈Γ} f(γg) ∈ C_{c}(Γ\G).

Proof. The argument is quite standard. One chooses functions ϕ_{n}, ψ_{n} ∈ C_{c}(G), n ≥ 1, such that ϕ_{n} ≤ f ≤ ψ_{n} and ∫_{G}(ψ_{n} − ϕ_{n})dμ < \frac{1}{n}. By Theorem 2 and (10),

lim_{T →∞} \frac{1}{ϱ(B_{T}^{o})} ∫_{B_{T}^{o}} ϕ̃_{n}(yb^{−1})dϱ(b) = \frac{1}{μ̄(Γ\G)} ∫_{Γ\G} ϕ̃_{n}dμ̄ = \frac{1}{μ̄(Γ\G)} ∫_{G} ϕ_{n}dμ,

and

lim_{T →∞} \frac{1}{ϱ(B_{T}^{o})} ∫_{B_{T}^{o}} ψ̃_{n}(yb^{−1})dϱ(b) = \frac{1}{μ̄(Γ\G)} ∫_{G} ψ_{n}dμ

for every n ≥ 1. This implies (17). □

The proof of Theorem 1 should be compared with similar arguments in [DRS93], [EM93], [EMS96], [EMM98] where other counting problems were also reduced to asymptotics of certain integrals.

Proof of Theorem 1. Let α : K → G/B^{o} be a map defined by α(k) = kB^{o}. By (8), α is a diffeomorphism. The measure m is given by

m(C) = ∫_{α^{−1}(C)} dk for Borel set C ⊆ G/B^{o}.  (18)

Since α is surjective, x_{0} = k_{0}^{−1}B^{o} for some k_{0} ∈ K. It follows from the Iwasawa decomposition (8) that the product map

K × B^{o} : (k, b) ↦ kk_{0}^{−1}bk_{0} ∈ G

is a diffeomorphism. For g ∈ G, define k_{g} ∈ K and b_{g} ∈ B^{o} such that

g = k_{g}k_{0}^{−1}b_{g}k_{0}.

Since G and K are unimodular, it follows from (9) that for f ∈ C_{c}(G),

∫_{G} fdμ = ∫_{K×B^{o}} f(kk_{0}^{−1}bk_{0})dkdϱ(b).

Let ϕ be the characteristic function of Ω̃ =^{def} α^{−1}(Ω)k_{0} ⊆ K, and ψ_{O} the characteristic function of an open bounded symmetric neighborhood O of 1 in B^{o} with boundary of measure 0 normalized so that ∫_{B^{o}} ψ_{O}dϱ = 1. Then ∫_{B^{o}} ψ_{O}dλ = 1 too. Note that for g ∈ G, gx_{0} ∈ Ω iff k_{g} ∈ Ω̃. Put f_{O}(g) = ϕ(k_{g})ψ_{O}(b_{g}). Let f̃_{O}(Γg) = ∑_{γ∈Γ} f_{O}(γg). Now Proposition 9 can be applied to f̃_{O}:

\frac{1}{ϱ(B_{T}^{o})} ∫_{B_{T}^{o}} f̃_{O}(Γk_{0}^{−1}b^{−1}k_{0})dϱ(b) ⟶_{T →∞} \frac{1}{μ̄(Γ\G)} ∫_{G} f_{O}(gk_{0})dμ(g)

= \frac{1}{μ̄(Γ\G)} ∫_{K} ϕdk · ∫_{B^{o}} ψ_{O}dϱ = \frac{1}{μ̄(Γ\G)} ∫Ω̃ dk = \frac{m(Ω)}{μ̄(Γ\G)}.  (19)

The last equality follows from (18).

Take r > 1. There exists a bounded open symmetric neighborhood O of identity in B^{o} (with boundary of measure 0) such that for any b ∈ O = O^{−1} and x ∈ M(n, ℝ),

r^{−1}∥x∥ ≤ ∥b^{−1}x∥ ≤ r∥x∥.  (20)

Then for O as above,

∑_{γ∈Γ} ∫_{B_{T}^{o}} f_{O}(γk_{0}^{−1}b^{−1}k_{0})dϱ(b) = ∑_{γ∈Γ} ∫_{(B_{T}^{o})^{−1}} f_{O}(k_{γ}k_{0}^{−1}b_{γ}bk_{0})dλ(b)

= ∑_{γ∈Γ} ∫_{b_{γ}(B_{T}^{o})^{−1}} f_{O}(k_{γ}k_{0}^{−1}bk_{0})dλ(b) = ∑_{γ∈Γ} ∫_{∥b^{−1}b_{γ}∥<T} ϕ(k_{γ})ψ_{O}(b)dλ(b)

= ∑_{γ:k_{γ}∈Ω̃} ∫_{∥b^{−1}b_{γ}∥<T} ψ_{O}(b)dλ(b) = ∑_{γ:γ·x_{0}∈Ω} ∫_{∥b^{−1}b_{γ}∥<T} ψ_{O}(b)dλ(b).

The integral

I_{γ} =^{def} ∫_{∥b^{−1}b_{γ}∥<T} ψ_{O}(b)dλ(b)

is not greater than 1. By (20), I_{γ} = 0 for γ ∈ Γ such that ∥γ∥ = ∥b_{γ}∥ ≥ rT. Therefore,

N_{rT} (Ω, x_{0}) ≥ ∑_{γ∈Γ} ∫_{B_{T}^{o}} f_{O}(γk_{0}^{−1}b^{−1}k_{0})dϱ(b).  (21)

By (20), I_{γ} = 1 for γ ∈ Γ such that ∥γ∥ = ∥b_{γ}∥ < r^{−1}T. Thus,

N_{r^{−1}T} (Ω, x_{0}) ≤ ∑_{γ∈Γ} ∫_{B_{T}^{o}} f_{O}(γk_{0}^{−1}b^{−1}k_{0})dϱ(b).  (22)

It follows from (13) that ϱ(B_{r^{−1}T}^{o}) ∼ γ_{n}r^{n−n^{2}}T^{n^{2}−n} as T → ∞. Then using (21) and (19), we get

lim inf_{T →∞} \frac{N_{T} (Ω, x_{0})}{T^{n^{2}−n}} ≥ lim inf_{T →∞} \frac{γ_{n}}{r^{n^{2}−n}ϱ(B_{r^{−1}T}^{o})} ∫_{B_{r^{−1}T}^{o}} f̃_{O}(γk_{0}^{−1}b^{−1}k_{0})dϱ(b)

= \frac{γ_{n}m(Ω)}{r^{n^{2}−n}μ̄(Γ\G)}.

This inequality holds for any r > 1. Hence,

lim inf_{T →∞} \frac{N_{T} (Ω, x_{0})}{T^{n^{2}−n}} ≥ \frac{γ_{n}m(Ω)}{μ̄(Γ\G)}.

Similarly, using (22) and (19), one can prove that

lim sup_{T →∞} \frac{N_{T} (Ω, x_{0})}{T^{n^{2}−n}} ≤ \frac{γ_{n}m(Ω)}{μ̄(Γ\G)}.

This finishes the proof of Theorem 1. □

5. Behavior of unipotent flows

In this section we review some deep results on equidistribution of unipotent flows, which are crucial for the proof of Theorem 2. Note that there are two different approaches available: Ratner [R91b] and Dani, Margulis [DM93]. Both of these methods rely on Ratner’s measure rigidity [R91a]. We follow the method of Dani and Margulis. The results below were proved in [DM91, DM93] for the case of one-dimensional flows and extended to higher dimensional flows and even polynomial trajectories in [Sh94, EMS97, Sh96]. See [St, §19] and [KSS02] for more detailed exposition.

Appropriate adjustments are made for the right G-action on Γ\G instead of left G-action on G/Γ.

Notations: Let G be a connected semisimple Lie group without compact factors, and Γ a lattice in G. Let 𝔤 be the Lie algebra of G. For positive integers d and n, denote by P_{d,n}(G) the set of functions q : ℝ^{n} → G such that for any a, b ∈ ℝ^{n}, the map

t ∈ ℝ ↦ Ad(q(at + b)) ∈ 𝔤

is a polynomial of degree at most d with respect to some basis of 𝔤.

Let V_{G} = ⊕_{i=1}^{dim 𝔤} ∧^{i} 𝔤. There is a natural action of G on V_{G} induced from the adjoint representation. Fix a norm on V_{G}. For a Lie subgroup H of G with Lie algebra 𝔥, take a unit vector p_{H} ∈ ∧^{dim 𝔥}𝔥.

The following theorem allows us to estimate divergence of polynomial trajectories. For its proof, see [Sh96, Theorems 2.1–2.2]. Theorem 10. There exist closed subgroups U_{i} (i = 1, …, r) such that each U_{i} is the unipotent radical of a maximal parabolic subgroup, ΓU_{i} is compact in Γ\G, and for any d, n ∈ ℕ, ε, δ > 0, there exists a compact set C ⊆ Γ\G such that for any q ∈ P_{d,n}(G) and a bounded open convex set D ⊆ ℝ^{n}, one of the following holds:

1. There exist γ ∈ Γ and i = 1, …, r such that sup_{t∈D} ∥q(t)^{−1}γ · p_{U_{i}}∥ ≤ δ.

2. ℓ(t ∈ D : Γq(t) ∉ C) < εℓ(D), where ℓ is the Lebesgue measure on ℝ^{n}. Denote by H_{Γ} the family of all proper closed connected subgroups H of G such that Γ∩H

is a lattice in H, and Ad(H ∩ Γ) is Zariski-dense in Ad(H). Theorem 11. The set H_{Γ} is countable. For any H ∈ H_{Γ}, Γ · p_{H} is discrete.

For proofs, see [R91a, Theorem 1.1] and [DM93, Theorems 2.1 and 3.4].

Fix a subgroup U generated by 1-parameter unipotent subgroups. For a closed subgroup H of G, denote

X(H, U) = {g ∈ G : gU ⊆ Hg}.

Define a singular set

Y = ⋃_{H∈H_{Γ}} ΓX(H, U) ⊆ Γ\G.  (23)

It follows from Dani’s generalization of Borel density theorem and Ratner’s topological rigidity that y ∈ Y iff yU is not dense in Γ\G (see [St, Lemma 19.4]).

One needs to estimate behavior of polynomial trajectories near the singular set Y. The following result can be deduced from [Sh94, Proposition 5.4]. It is formulated in [Sh96] and [KSS02]. Note that it is analogous to Theorem 10 with the point at infinity being replaced by the singular set.

Theorem 12. Let d, n ∈ ℕ, ε > 0, H ∈ H_{Γ}. For any compact set C ⊆ ΓX(H, U), there exists a compact set F ⊆ V_{G} such that for any neighborhood Φ of F in V_{G}, there exists a neighborhood Ψ of C in Γ\G such that for any q ∈ P_{d,n}(G) and a bounded open convex set D ⊆ ℝ^{n}, one of the following holds:

1. There exists γ ∈ Γ such that q(D)^{−1}γ · p_{H} ⊆ Φ.

2. ℓ(t ∈ D : Γq(t) ∈ Ψ) < εℓ(D), where ℓ is the Lebesgue measure on ℝ^{n}.

6. Representations of SL(n, ℝ)

In order to be able to use the results from the previous section, we collect here some information about representations of SL(n, ℝ).

The next lemma is essentially Lemma 5.1 from [Sh96]. We present its proof because more precise information about dependence on β in the inequality (24) is needed. Lemma 13. Let V be a finite dimensional vector space with a norm ∥ · ∥, 𝔫 be a nilpotent Lie subalgebra of End(V) with a basis {b_{i} : i = 1, …, m}, and N = exp(𝔫) ⊆ GL(V) be the Lie group of 𝔫. Define a map Θ : ℝ^{m} → N:

Θ(t_{1}, …, t_{m}) = exp (∑_{i=1}^{m} t_{i}b_{i}) , (t_{1}, …, t_{m}) ∈ ℝ^{m}.

For β > 0, put D(β) = Θ ([0, β] × ⋯ × [0, β]). Let

W = {v ∈ V : 𝔫 · v = 0}.

Denote by pr_{W} the orthogonal projection on W with respect to some scalar product on V. Then there exists a constant c_{0} > 0 such that for any β ∈ (0, 1) and v ∈ V,

sup_{n∈D(β)} ∥pr_{W} (n · v)∥ ≥ c_{0}β^{d}∥v∥,  (24)

where d is the degree of the polynomial map Θ.

Proof. Let

I = {(i_{1}, …, i_{m}) ∈ ℤ^{m} : i_{k} ≥ 0, ∑_{k} i_{k} ≤ d} .

For t ∈ ℝ^{m} and I = (i_{1}, …, i_{m}) ∈ I, denote t^{I} = ∏_{k} t_{k}^{i_{k}}, and |I| = ∑_{k} i_{k}.

One can write Θ(t) = ∑_{I∈I} t^{I}B_{I} for some B_{I} ∈ End(V). Then

pr_{W} (Θ(t)v) = ∑_{I∈I} t^{I}pr_{W} (B_{I}v).

Consider a map T : V → ⊕_{I∈I}W defined by

T v = ∑_{I∈I} pr_{W} (B_{I}v),

and a map A_{t} : ⊕_{I∈I}W → W for t ∈ ℝ^{m} defined by

A_{t} (⨁_{I∈I} w_{I}) = ∑_{I∈I} t^{I}w_{I}.

For I ∈ I, take fixed s_{I} ∈ ℝ^{m} such that 0 < s_{I,k} < 1 and s_{I_{1},k} ≠ s_{I_{2},k} for I_{1} ≠ I_{2}, and put t_{I} = βs_{I}. Let

A = ⨁_{I∈I} A_{t_{I}} : ⨁_{I∈I} W → ⨁_{I∈I} W.

The map A has a matrix form (t_{I}^{J})_{I,J∈I}. This matrix is a Kronecker product of Vandermonde matrices which implies that A is invertible. Using elementary row and column operations, one can write

(t_{I}^{J})_{I,J∈I} = B · diag (β^{|I|} : I ∈ I) · C  (25)

for some B, C ∈ GL (⊕_{I∈I}W), which are independent of β. It is convenient to use a norm on ⊕_{I∈I}W defined by

∥⨁_{I∈I} w_{I} ∥ = max_{I∈I} ∥w_{I}∥, w_{I} ∈ W.

Then by (25), for w ∈ ⊕_{I∈I}W,

∥Aw∥ ≥ ∥B^{−1}∥^{−1} · β^{d} · ∥C^{−1}∥^{−1} · ∥w∥.  (26)

It follows from Lie-Kolchin theorem that T is injective (see [Sh96, Lemma 5.1]). Therefore, there exists a constant c_{1} > 0 such that ∥T v∥ ≥ c_{1}∥v∥ for v ∈ V. Then using (26), we get

sup_{n∈D(β)} ∥pr_{W} (n · v)∥ = sup_{t:0<t_{i}<β} ∥A_{t}T v∥ ≥ ∥AT v∥ ≥ c_{0}β^{d}∥v∥,

where c_{0} = ∥B^{−1}∥^{−1} · ∥C^{−1}∥^{−1} · c_{1}. □

We will need the following elementary observation: Lemma 14. Let F : ℝ^{m} → ℝ^{m} be a C^{1} bijection such that F (0) = 0, and F^{−1} is C^{1} too. Fix a norm on ℝ^{m} and denote by D(r) a ball of radius r centered at the origin. Then there are c_{1}, c_{2} > 0 such that

D(c_{1}r) ⊆ F (D(r)) ⊆ D(c_{2}r)

for every r ∈ (0, 1).

Let 𝔤 be the Lie algebra of G = SL(n, ℝ), and 𝔤_{ℂ} = 𝔤 ⊗ ℂ. Recall the root space decomposition of 𝔤_{ℂ}:

𝔤_{ℂ} = 𝔤_{0} ⊕ ∑_{i≠j} 𝔤_{ij},

where 𝔤_{0} is the subalgebra of diagonal matrices of 𝔤_{ℂ}, and 𝔤_{ij} = ℂE_{ij} (E_{ij} is the matrix with 1 in position (i, j) and 0’s elsewhere).

Introduce fundamental weights of 𝔤_{ℂ}:

ω_{i}(s) = s_{1} + ⋯ + s_{i}, 1 ≤ i ≤ n − 1,  (27)

where s ∈ ℂ^{n} and ∑_{i} s_{i} = 0. Dominant weights are defined as linear combinations with non-negative integer coefficients of the fundamental weights ω_{i}, 1 ≤ i ≤ n − 1.

A highest weight of a representation of 𝔤_{ℂ} is a weight that is maximal with respect to the ordering on the dual space of 𝔤_{0}. Recall that irreducible representations of 𝔤_{ℂ} are classified by their highest weights (see, for example, [GW, Ch. 5]). The highest weights are precisely the dominant weights defined above.

Lemma 15. Let π be a representation of G on a finite dimensional complex vector space V. Let x ∈ V − {0} be such that π(N)x = x. Then x is a sum of weight vectors with dominant weights. Moreover, if V does not contain non-zero G-fixed vectors, every weight in this sum is not zero.

Proof. Consider a representation π̃ of 𝔤_{ℂ} on V induced by the representation π. Since this representation is completely reducible, it is enough to consider the case when it is irreducible.

We claim that in this case, x is a vector with the highest weight. Write x = ∑_{k} x_{k} where each x_{k} ∈ V is a weight vector with a weight λ_{k}. We may assume that λ_{k} ≠ λ_{l} for k ≠ l. Since π(N)x = x, π̃(E_{ij})x = 0 for i < j. Thus, ∑_{k} π̃(E_{ij})x_{k} = 0. Since π̃(E_{ij})x_{k} is either 0 or a weight vector with the weight λ_{k} + α_{ij}, the non-zero terms in the sum are linearly independent. Hence, π̃(E_{ij})x_{k} = 0 for i < j. Suppose that λ_{k} is not the highest weight. Note that π̃(𝔤_{0})x_{k} = ℂx_{k}, and π̃(E_{ji})x_{k} has weight λ_{k} − α_{ij} < λ_{k} for i < j. By Poincaré-Birkhoff-Witt theorem, the universal enveloping algebra U(𝔤_{ℂ}) = U(𝔟^{−}) ⊕ U(𝔤_{ℂ})𝔫, where 𝔟^{−} is the space of lower triangular matrices, and 𝔫 is the Lie algebra of N. Therefore, the space π̃(U(𝔤_{ℂ}))x_{k} = π̃(U(𝔟^{−}))x_{k} does not contain a vector with the highest weight. This contradicts irreducibility of π̃. Thus, each x_{k} is of the highest weight, and x is a highest weight vector. Since every highest weight is dominant, the lemma is proved. □

For a fixed g_{0} ∈ G, define q_{s}(t) = g_{0}n(t)^{−1}a(s)^{−1}. We are going to study q_{s} using techniques from Section 5. The next lemma guarantees that the first possibility in Theorems 10 and 12 does not occur.

For β > 0, define

D(β) = {n(t) ∈ N : ∑_{i<j} t_{ij}^{2} < β^{2}} .  (28)

Lemma 16. Let π be a nontrivial representation of G on a real vector space V equipped with a norm ∥ · ∥. Let V_{0} = {v ∈ V : π(G)v = v}, and V_{1} be a G-invariant complement for V_{0}. (Note that V_{1} exists because π is completely reducible.) Denote by Π the projection on V_{1}.

Then for any relatively compact set K ⊆ V and r > 0, there exist α ∈ (0, 1) and C_{0} > 0 such that for any s with a(s) ∈ A_{T}^{C_{0}} and x ∈ V with ∥Π(x)∥ > r,

π (q_{s} (D(e^{−αs_{1}})))^{−1} · x ⊈ K.  (29)

Proof. It is convenient to extend π to V_{ℂ} = V ⊗ ℂ. (V_{0})_{ℂ} is the space of G-fixed vectors in V_{ℂ}. Thus, we may assume V to be complex. Also dealing with projections on V_{1}, we may assume that V has no G-fixed vectors.

Since {g_{0}^{−1} · x : ∥x∥ > r} ⊆ {x : ∥x∥ > r_{1}} for some r_{1} > 0, we may assume that g_{0} = 1. For some R > 0, K ⊆ {x ∈ V : ∥x∥ < R}. If (29) fails for some s ∈ ℝ^{n} and x ∈ V, then

sup_{n∈D(e^{−αs_{1}})} ∥π(a(s)n)x∥ < R.  (30)

Let W = {x ∈ V : π(N)x = x}. Clearly, the statement of the lemma is independent of the norm used. It is convenient to use the max-norm with respect to a basis {v_{i}} of V consisting of weight vectors, i.e.

∥∑_{i} u_{i}v_{i}∥ = max_{i} |u_{i}|, u_{i} ∈ ℂ,

and each v_{i} is an eigenvector of A. Moreover we can choose the basis {v_{i}} so that it contains a basis of W. Let pr_{W} be the projection onto W with respect to this basis. Then pr_{W} commutes with a(s). Note that there exists C′ > 0 such that for any v ∈ V,

∥v∥ ≥ C′∥pr_{W} (v)∥.  (31)

Let K ⊆ ℕ be such that k ∈ K iff v_{k} has a non-zero dominant weight. Denote this weight by λ_{k}. By Lemma 15, W ⊆ ⟨v_{k} : k ∈ K⟩. In particular, for n ∈ N,

pr_{W} (π(n)x) = ∑_{k∈K} u_{k}(n)v_{k} for some u_{k}(n) ∈ ℂ.  (32)

Therefore,

∥pr_{W} (π(n)x)∥ = max_{k∈K} |u_{k}(n)|.  (33)

Using the fact that pr_{W} and π(a(s)) commute, (31), (32), and (33), we have that for any n ∈ N,

∥π(a(s)n)x∥ ≥ C′∥pr_{W} (π(a(s))π(n)x)∥ = C′ ∥π(a(s)) (∑_{k∈K} u_{k}(n)v_{k})∥

= C′ max_{k∈K} (|u_{k}(n)|e^{λ_{k}(s)}) ≥ C′ exp (min_{k∈K} λ_{k}(s)) ∥pr_{W} (π(n)x)∥.  (34)

Let 𝔫 be the Lie algebra of N. Denote by π̃ the representation of 𝔤 induced by π. Since 𝔤 is simple, π̃ is faithful. Thus, π̃ defines an isomorphism between 𝔫 and π̃(𝔫). Since the exponential map 𝔫 → N is a polynomial isomorphism, the coordinates on N used in Lemma 13 and the coordinates {t_{ij}} are connected by a polynomial isomorphism too. By Lemma 14, (24) holds for the set D(β) defined in (28). Therefore,

sup_{n∈D(e^{−αs_{1}})} ∥pr_{W} (π(n)x)∥ ≥ c_{0}(e^{−αs_{1}})^{d}∥x∥ ≥ c_{0}re^{−αds_{1}}  (35)

for some positive integer d. It follows from (34) and (35) that if (30) holds, then

exp (min_{k∈K} λ_{k}(s) − αds_{1}) ≤ \frac{R}{c_{0}C′r}.

Take α < d^{−1}. Since each λ_{k} is a non-zero dominant weight, it follows from (27) that that λ_{k}(s) − αds_{1} → ∞ as C → ∞ for s such that a(s) ∈ A^{C}. Hence, there exists C_{0} > 0 such that (30) does not hold for s with a(s) ∈ A_{T}^{C_{0}}. Since (30) fails, (29) holds. □

Lemma 17. Use notations from Lemma 16. Let H be a subgroup of G, and x ∈ V such that π(H)x is discrete in V. Then Π(π(H)x) is discrete in V_{1}.

Proof. Denote by x_{0} ∈ V_{0} and x_{1} ∈ V_{1} the components of x with respect to the decomposition V = V_{0} ⊕ V_{1}. Then Π(π(H)x) = π(H)x_{1}. Suppose that for some {h_{n}} ⊆ H, π(h_{n})x_{1} → y for some y ∈ V_{1}. Then π(h_{n})x converges to x_{0} + y. It follows that π(h_{n})x is constant for large n. Therefore, π(h_{n})x_{1} = π(h_{n})x − x_{0} is constant for large n too. □

7. Proof of Theorem 2

Let Z = (Γ\G) ∪ {∞} be the 1-point compactification of Γ\G. For T > 0, define a normalized measure on Z by

ν_{T} (f̃) = \frac{1}{ϱ(B_{T}^{o})} ∫_{B_{T}^{o}} f̃(yb^{−1})dϱ(b), f̃ ∈ C_{c}(Γ\G).

To prove Theorem 1, we need to show that ν_{T} → ν in weak-∗ topology as T → ∞. Since the space of normalized measures on Z is compact, it is enough to prove that if ν_{T_{i}} → η as T_{i} → ∞ for some normalized measure η on Z, then η is G-invariant, and η({∞}) = 0.

It follows from Lemma 7 that for any C ∈ ℝ,

η(f̃) = lim_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫_{B_{T_{i}}^{C}} f̃(yb^{−1})dϱ(b), f̃ ∈ C_{c}(Γ\G).  (36)

Let

U = {n(t) ∈ N : t_{ij} = 0 for i < j < n}.  (37)

Lemma 18. The measure η is U-invariant.

Proof. For f̃ ∈ C_{c}(Γ\G), and g_{0} ∈ G, define f̃^{g_{0}}(Γg) = f̃(Γgg_{0}) ∈ C_{c}(Γ\G).

Let f̃ ∈ C_{c}(Γ\G). Take M > 0 such that |f̃| < M.

For T > 0 and s ∈ ℝ^{n−1}, define a set

D_{s,T} = {n ∈ N : ∥a(s)n∥ < T }.  (38)

Denote by χ_{s,T} (n) the characteristic function of the set D_{s,T}. Then we can rewrite (36) as

η(f̃) = lim_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫_{A_{T_{i}}^{C}} ∫_{N} f̃(yn^{−1}a(−s))χ_{s,T_{i}}(n)e^{2δ(s)}dnds.  (39)

Let

A_{T}^{′C} = {a(s) ∈ A_{T}^{C} : T^{2} − N(s) > T },  (40)

where N(s) is defined in (12), and B_{T}^{′C} = A_{T}^{′C}N ∩B_{T}^{o}. We claim that the equality (39) holds when A_{T_{i}}^{C} is replaced by A_{T_{i}}^{′C}. By Lemma 5,

∫_{A_{T_{i}}^{C}−A_{T_{i}}^{′C}} ∫_{N} f̃(yn^{−1}a(−s))χ_{s,T_{i}}(n)e^{2δ(s)}dnds

≪ ∫_{A_{T_{i}}^{C}−A_{T_{i}}^{′C}} (T_{i}^{2} − N(s))^{\frac{n(n−1)}{4}} exp (∑_{k}(n − k)s_{k}) ds

= O (Ti^{\frac{3n(n−1)}{4}} ) as T_{i} → ∞.

Now the claim follows from (13).

Take u ∈ U. Let u(s) = Ad_{a(−s)}(u). Then

|η(f̃^{u}) − η(f̃)| ≤ lim sup_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫_{A_{T_{i}}^{′C}} ∫_{N} |f̃(yn^{−1}u(s)a(−s))χ_{s,T_{i}}(n)

−f̃(yn^{−1}a(−s))χ_{s,T_{i}}(n)|dne^{2δ(s)}ds.  (41)

We estimate the last integral:

∫_{N} |f̃(yn^{−1}u(s)a(−s))χ_{s,T_{i}}(n) − f̃(yn^{−1}a(−s))χ_{s,T_{i}}(n)| dn
= ∫_{N} |f̃(yn^{−1}a(−s))| · |χ_{s,T_{i}}(u(s)n) − χ_{s,T_{i}}(n)| dn
≤ M ∫_{N} |χ_{s,T_{i}}(u(s)n) − χ_{s,T_{i}}(n)| dn.  (42)

Recall that α_{i,n}(−s) = −s_{i}+s_{n}. Therefore, by (5), u(s)_{in} = e^{−s_{i}+s_{n}}u_{in} for i = 1, …, n−1. It follows from the triangle inequality that

∥a(s)u(s)n∥ ≤ ∥a(s)n∥ + \sqrt{∑_{i=1}^{n−1} e^{2s_{i}}u(s)_{in}^{2}} ≤ ∥a(s)n∥ + e^{s_{n}}∥u∥,

and similarly,

∥a(s)n∥ ≤ ∥a(s)u(s)n∥ + e^{s_{n}}∥u^{−1}∥ = ∥a(s)u(s)n∥ + e^{s_{n}}∥u∥.

Hence,

χ_{s,T_{i}−e^{sn}∥u∥}(n) ≤ χ_{s,T_{i}}(u(s)n) ≤ χ_{s,T_{i}+e^{sn}∥u∥}(n)

for n ∈ N. Therefore,

∫_{N} |χ_{s,T_{i}}(u(s)n) − χ_{s,T_{i}}(n)| dn ≤ ∫_{N} (χ_{s,T_{i}+e^{sn}∥u∥} − χ_{s,T_{i}−e^{sn}∥u∥}) dn.  (43)

Let ε > 0. We claim that there exists C_{0} > 0 such that for C > C_{0} and a(s) ∈ A_{T_{i}}^{′C},

∫_{N} (χ_{s,T_{i}+e^{sn}∥u∥} − χ_{s,T_{i}−e^{sn}∥u∥}) dn ≤ ε ∫_{N} χ_{s,T_{i}}dn.  (44)

Similarly to Lemma 5,

∫_{N} χ_{s,T_{i}}dn = c_{n} (T_{i}^{2} − N(s))^{\frac{n(n−1)}{4}} exp (∑_{k}(n − k)s_{k}) .

Also e^{s_{n}} → 0 for a(s) ∈ A_{T_{i}}^{′C} as C → ∞. Therefore, the equation (44) will follow from the following.

Claim. There exists d_{0} = d_{0}(ε) > 0 such that for any d ∈ (0, d_{0}) and a(s) ∈ A_{T_{i}}^{′C},

((T_{i} + d)^{2} − N(s))^{\frac{n(n−1)}{4}} − ((T_{i} − d)^{2} − N(s))^{\frac{n(n−1)}{4}} < ε((T_{i} − d)^{2} − N(s))^{\frac{n(n−1)}{4}} . (45)

Note that by (40), (T_{i} − d)^{2} − N(s) > 0 for a(s) ∈ A_{T_{i}}^{′C} and d < 1/2. The inequality (45) is equivalent to

(T_{i} + d)^{2} − N(s) < (1 + ε̄)((T_{i} − d)^{2} − N(s)),

where ε̄ = (1 + ε)^{\frac{4}{n(n−1)}} − 1. By (40), the last inequality follows from

(T_{i} + d)^{2} + ε̄(T_{i}^{2} − T_{i}) < (1 + ε̄)(T_{i} − d)^{2},

or equivalently,

T_{i}(4d − ε̄+ 2dε̄) < ε̄d^{2}.

If one takes d < d_{0} = ε̄/(2ε̄+ 4), the left hand side is negative. This proves the claim (45). Thus, (44) holds. Then by (41), (42), (43), and (44),

|η(f̃^{u}) − η(f̃)| ≤ (Mε) lim sup_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫_{A_{T_{i}}^{′C}} ∫_{N} χ_{s,T_{i}}(n)e^{2δ(s)}dnds
= (Mε) lim sup_{T_{i}→∞} \frac{ϱ(B_{T_{i}}^{′C})}{ϱ(B_{T_{i}}^{o})} ≤ Mε.

This shows that η(f̃^{u}) = η(f̃). □

Lemma 19. Let α ∈ (0, 1). Let

Ã_{T}^{C} = {a(s) ∈ A_{T}^{C}| (T^{2} − N(s))^{1/2} > exp ( max_{1≤j≤n−1}{s_{j}} − αs_{1})} ,  (46)

where N(s) is as in (12), and B̃_{T}^{C} = Ã_{T}^{C}N ∩ B_{T}^{o}. Then for C > 0,

η(f̃) = lim_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫B̃_{T_{i}}^{C} f̃(yb^{−1})dϱ(b), f̃ ∈ C_{c}(Γ\G).

Proof. By (36), it is enough to show that

\frac{ϱ (B_{T_{i}}^{C} − B̃_{T_{i}}^{C})}{ϱ (B_{T_{i}}^{o})} → 0 as T_{i} → ∞.  (47)

As in Lemma 5,

ϱ (B_{T_{i}}^{C} − B̃_{T_{i}}^{C}) = c_{n} ∫_{A_{T_{i}}^{C}−Ã_{T_{i}}^{C}} (T_{i}^{2} − N(s))^{\frac{n(n−1)}{4}} exp (∑_{k}(n − k)s_{k}) ds.

Therefore,

ϱ (B_{T_{i}}^{C} − B̃_{T_{i}}^{C}) ≤ c_{n} ∫_{A_{T_{i}}^{C}} exp (\frac{n(n − 1)}{2} max_{1≤j≤n−1}\{s_{j}\}

−\frac{αn(n − 1)}{2}s_{1} + ∑_{k}(n − k)s_{k})ds ≤ c_{n} ∑_{1≤j≤n−1} ∫_{A_{T_{i}}^{C}} exp (\frac{n(n − 1)}{2}s_{j}

−\frac{αn(n − 1)}{2}s_{1} + ∑_{k}(n − k)s_{k})ds

Then as in the proof of Lemma 7, for j ≠ 1

∫_{A_{T_{i}}^{C}} exp (\frac{n(n − 1)}{2}s_{j} − \frac{αn(n − 1)}{2}s_{1} + ∑_{k}(n − k)s_{k}) ds

≤ ∫_{−∞}^{log T_{i}} exp ((−\frac{αn(n − 1)}{2} + n − 1) s_{1}) ds_{1}

· ∫_{−∞}^{log T_{i}} exp ((\frac{n(n − 1)}{2} + n − j) s_{j}) ds_{j} · ∏_{k<n,k≠1,j} ∫_{−∞}^{log T_{i}} e^{(n−k)s_{k}}ds_{k}

≪ T_{i}^{n(n−1)−αn(n−1)/2}

as T_{i} → ∞. For j = 1, the same estimate can be obtained by a similar calculation. Now (47) follows from (13). □

Let y = Γg_{0} for g_{0} ∈ G. Define q_{s}(t) = g_{0}n(t)^{−1}a(s)^{−1}. We apply the results of Section 5 to the map q_{s}. Lemma 20. η({∞}) = 0.

Proof. Let ε, δ > 0. Apply Theorem 10 to the map q_{s}(t). By Theorem 11, the set Γ·p_{U_{i}} ⊆ V_{G} is discrete. Write V_{G} = V_{0} ⊕ V_{1}, where V_{0} is the space of vectors fixed by G, and V_{1} is its G-invariant complement. Denote by Π the projection on V_{1}. By Lemma 17, Π(Γ · p_{U_{i}}) is discrete. Also 0 ∉ Π(Γ · p_{U_{i}}). Otherwise p_{U_{i}} is fixed by G, and it would follow that U_{i} is normal in G which is a contradiction. Therefore, there exists r > 0 such that

∥Π(x)∥ > r for x ∈ ⋃_{i} Γ · p_{U_{i}}.

Now we can apply Lemma 16. Let

K = {x ∈ V_{G} : ∥x∥ ≤ δ}.

By Lemma 16, there exist α ∈ (0, 1) and C_{0} > 0 such that the first case of Theorem 10 fails for q_{s} when D is a bounded open convex set which contains D(e^{−αs_{1}}) (it is defined in (28)), and s is such that a(s) ∈ A_{T_{i}}^{C_{0}}. Therefore, for some compact set C ⊆ Γ\G,

ω ({n(t) ∈ D : Γq_{s}(t) ∉ C}) < εω(D)  (48)

when D ⊇ D(e^{−αs_{1}}) and a(s) ∈ A_{T_{i}}^{C_{0}}, where ω = dt denotes the Lebesgue measure on N. Let D_{s,T_{i}} be as in (38). By Lemma 19,

η(f̃) = lim_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫Ã_{T_{i}}^{C} ∫_{D_{s,T_{i}}} f̃(Γq_{s}(t))e^{2δ(s)}dtds, f̃ ∈ C_{c}(Γ\G).  (49)

Note that

D_{s,T_{i}} = {n(t) ∈ N : ∑_{i<j} e^{2s_{i}}t_{ij}^{2} < T_{i}^{2} − N(s)} ,

where N(s) is defined in (12) (cf. (11)). It follows that D_{s,T_{i}} contains D(β) for

β < (T_{i}^{2} − N(s))^{1/2} exp (− max_{1≤i≤n−1}{s_{i}}) .

When a(s) ∈ Ã_{T_{i}}^{C_{0}}, the right hand side is greater then e^{−αs_{1}} (see (46)). Therefore, D_{s,T_{i}} ⊇ D(e^{−αs_{1}}) when a(s) ∈ Ã_{T_{i}}^{C_{0}}. By (48),

ω ({n(t) ∈ D_{s,T_{i}} : Γq_{s}(t) ∉ C}) < εω(D_{s,T_{i}}) for a(s) ∈ Ã_{T_{i}}^{C_{0}}.  (50)

Let χ_{C} be the characteristic function of the set C. Take f̃ ∈ C_{c}(Γ\G) such that χ_{C} ≤ f̃ ≤ 1. Then using (49) and (50), we get

η(supp(f̃)) ≥ lim_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫Ã_{T_{i}}^{C_{0}} ∫_{D_{s,T_{i}}} χ_{C}(Γq_{s}(t))e^{2δ(s)}dtds

≥ lim_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫Ã_{T_{i}}^{C_{0}}(1 − ε)ω(D_{s,T_{i}})e^{2δ(s)}ds

= (1 − ε) lim_{T_{i}→∞} \frac{ϱ(B̃_{T_{i}}^{C_{0}})}{ϱ(B_{T_{i}}^{o})} = 1 − ε.

Hence, η({∞}) ≤ η(supp(f̃)^{c}) ≤ ε for every ε > 0. □

Recall that the singular set Y of U was defined in (23). Lemma 21. η(Y) = 0.

Proof. By (23) and Theorem 11, it is enough to show that η(ΓX(H, U)) = 0 for any H ∈ H_{Γ}. Moreover, since ΓX(H, U) is σ-compact, we just need to show that η(C) = 0 for any compact set C ⊆ ΓX(H, U).

Take ε > 0. Apply Theorem 12 to the map q_{s}(t). Let F ⊆ V_{G} be as in Theorem 12. Fix a relatively compact neighborhood Φ of F in V_{G}. Take Ψ ⊇ C as in Theorem 12. By Theorem 11, the set Γ · p_{H} is discrete. Let Π be as in the proof of Lemma 20. By Lemma 17, Π(Γ·p_{H}) is discrete. If 0 ∈ Π(Γ·p_{H}), the vector p_{H} is fixed by G, and H is normal in G, which is impossible. Therefore, for some r > 0, ∥Π(x)∥ > r for every x ∈ Γ · p_{H}. Applying Lemma 16 with K = Φ, one gets that there exist α ∈ (0, 1) and C_{0} > 0 such that the first case of Theorem 10 fails for q_{s} when D is a bounded open convex set containing D(e^{−αs_{1}}), and s is such that a(s) ∈ A_{T_{i}}^{C_{0}}. Therefore, the second case should hold:

ω ({n(t) ∈ D : Γq_{s}(t) ∈ Ψ}) < εω(D)  (51)

when D ⊇ D(e^{−αs_{1}}) and a(s) ∈ A_{T_{i}}^{C_{0}}.

Let D_{s,T_{i}} be as in (38). It was shown in the proof of Lemma 20 that D_{s,T_{i}} ⊇ D(e^{−αs_{1}}) when a(s) ∈ Ã_{T_{i}}^{C_{0}}. It follows from (51) that

ω ({n(t) ∈ D_{s,T_{i}} : Γq_{s}(t) ∈ Ψ}) < εω(D_{s,T_{i}}) for a(s) ∈ Ã_{T_{i}}^{C_{0}}.  (52)

Take a function f̃ ∈ C_{c}(Γ\G) such that f̃ = 1 on C, supp(f̃) ⊆ Ψ, and 0 ≤ f̃ ≤ 1. Let χ_{Ψ} be the characteristic function of Ψ. Then using (49) and (52), we get

η(C) ≤ lim_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫Ã_{T_{i}}^{C_{0}} ∫_{D_{s,T_{i}}} χ_{Ψ}(Γq_{s}(t))e^{2δ(s)}dtds

≤ lim_{T_{i}→∞} \frac{1}{ϱ(B_{T_{i}}^{o})} ∫Ã_{T_{i}}^{C_{0}} εω(D_{s,T_{i}})e^{2δ(s)}ds = ε lim_{T_{i}→∞} \frac{ϱ(B̃_{T_{i}}^{C_{0}})}{ϱ(B_{T_{i}}^{o})} = ε.

This shows that η(C) = 0. Hence, η(Y) = 0. □

Recall that by Lemma 18, η is U-invariant. By Ratner’s measure classification [R91a], an ergodic component of η is either G-invariant or supported on Y ∪ {∞}. By Lemmas 20 and 21, the set of ergodic components of the second type has measure 0. Therefore, η is G-invariant, and η = ν. This finishes the proof of Theorem 2.

Acknowledgments

The author wishes to thank his advisor V. Bergelson for constant support, encouragement, and help with this paper, N. Shah and Barak Weiss for insightful comments about the paper, and H. Furstenberg and G. Margulis for interesting discussions.

References

[BT65] A. Borel and J. Tits. Groupes réductifs. Inst. Hautes tudes Sci. Publ. Math. 27 (1965), 55–150.

[DM91] 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), 1–17.

[DM93] S. G. Dani and G. A. Margulis. Limit distributions of orbits of unipotent flows and values of quadratic forms. I. M. Gelfand Seminar, 91–137, Adv. Soviet Math. 16, Part 1, AMS, Providence, RI, 1993.

[DRS93] W. Duke, Z. Rudnick and P. Sarnak. Density of integer points on affine homogeneous varieties. Duke Math. J. 71 (1993), 143–179.

[EM93] A. Eskin and C. McMullen. Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71 (1993), 181–209.

[EMM98] A. Eskin, G. Margulis and S. Mozes. Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Ann. Math. 147 (1998), 93–141.

[EMS96] A. Eskin, S. Mozes and N. Shah. Nimish Unipotent flows and counting lattice points on homogeneous varieties. Ann. Math. 143 (1996), 253–299.

[EMS97] A. Eskin, S. Mozes and N. Shah. Non-divergence of translates of certain algebraic measures.Geom. Funct. Anal. 7 (1997), 48–80.

[F63] H. Furstenberg. A Poisson formula for semi-simple Lie groups. Ann. Math. 77 (1963), 335–386.

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

[GW] R. Goodman and N. R. Wallach. Representations and invariants of the classical groups. Cambridge University Press, Cambridge, 1998.

[KSS02] D. Kleinbock, N. Shah and A. Starkov. Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. in Handbook of dynamical systems, Vol. 1A, 813–930, North-Holland, Amsterdam, 2002.

[L99] F. Ledrappier, Distribution des orbites des réseaux sur le plan réel. C. R. Acad. Sci. Paris Sr. I Math. 329 (1999), 61–64.

[N00] A. Nogueira, Orbit distribution on ℝ^{2} under the natural action of SL(2, ℤ), preprint, 2000.

[PR] V. Platonov and A. Rapinchuk. Algebraic groups and number theory. Academic Press, Inc., Boston, MA, 1994.

[R] M. S. Raghunathan. Discrete Subgroups of Lie Groups. Springer-Verlag, New York, 1972.

[R91a] M. Ratner. On Raghunathan’s measure conjecture. Ann. Math. 134 (1991), 545–607.

[R91b] M. Ratner. Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63 (1991), 235–280.

[Sh94] N. A. Shah. Limit distributions of polynomial trajectories on homogeneous spaces. Duke Math. J. 75 (1994), 711–732.

[Sh96] N. Shah. Limit distributions of expanding translates of certain orbits on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 105–125.

[Sp] T. A. Springer. Linear algebraic groups. Birkhäuser, Boston, 1998.

[St] A. N. Starkov. Dynamical systems on homogeneous spaces. Translations of Mathematical Monographs 190, AMS, Providence, RI, 2000.

[T] A. Terras. Harmonic analysis on symmetric spaces and applications I. Springer-Verlag, New York, 1985.

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109

  E-mail address: gorodnik@umich.edu