Geometric And Functional Analysis 1016-443X/97/0x001-xx$1.50+0.20/0 Vol. 7, No. x (1997) \begin{matrix}c \\ ◯ \end{matrix}1997 Birkhäuser Verlag, Basel Geometric And Functional Analysis 1016-443X/97/0x001-xx$1.50+0.20/0 Vol. 7, No. x (1997) \begin{matrix}c \\ ◯ \end{matrix}1997 Birkhäuser Verlag, Basel

NON-DIVERGENCE OF TRANSLATES OF CERTAIN ALGEBRAIC MEASURES

A. Eskin, S. Mozes and N. Shah

Abstract

Let G and H ⊂ G be connected reductive real algebraic groups defined over ℚ, and admitting no nontrivial ℚ-characters. Let Γ ⊂ G(ℚ) be an arithmetic lattice in G, and π : G → Γ\G be the natural quotient map. Let μ_{H} denote the H-invariant probability measure on the closed orbit π(H). Suppose that π(Z(H)) is compact, where Z(H) denotes the centralizer of H in G. We prove that the set {μ_{H} · g : g ∈ G} of translated measures is relatively compact in the space of all Borel probability measures on Γ\G, where μ_{H} · g(E) = μ_{H}(Eg^{−1}) for all Borel sets E ⊂ Γ\G.

1 Introduction

Let G = G(ℝ)^{0} and H = H(ℝ)^{0} for reductive algebraic groups G and H ⊂ G defined over ℚ. Suppose that G and H do not admit nontrivial characters defined over ℚ. Let Γ be an arithmetic subgroup of G with respect to the ℚ-structure, i.e. Γ ⊂ G(ℚ) and Γ ∩ G(ℤ) is of finite index in G(ℤ) as well as in Γ. By a theorem of Borel and Harish-Chandra [BH], Γ and H ∩ Γ are a lattices in G and H, respectively. Let π : G → Γ\G be the natural quotient map. Then the orbit π(H) ≅ (H ∩ Γ)\H admits a (unique) H-invariant probability measure, say μ_{H}. Such a measure and its ‘translates’ are referred to as algebraic measures.

For a measure μ on Γ\G, its translate by g ∈ G is defined as

μ · g(E) = μ(Eg^{−1}) , ∀ measurable E ⊂ Γ\G .

This also defines a continuous G-action on the space of Borel probability measures on Γ\G endowed with the weak-∗ convergence. The measure μ_{H}g is supported on π(H)g and it is invariant under g^{−1}Hg.

The second author is sponsored in part by the Edmund Landau Center for research in Mathematical Analysis supported by the Minerva Foundation (Germany), and by the Israel Academy of Sciences

In [DuRuS] and [EMc], it was shown that if H is the connected component of the identity of an affine symmetric subgroup (i.e. the set of fixed points of an involutive automorphism of G) and G is ℚ-simple, then μ_{H}g_{i} converges to the Haar measure on Γ\G, provided the image of {g_{i}} on H\G has no convergent subsequence. Using this result they obtained asymptotic estimates for the number of integral points with norm less than T (as T → ∞) on rational affine symmetric varieties (i.e. isomorphic to H\G over ℚ) contained in ℝ^{n}. Their method of proof suggests that, in the general case, if there is an appropriate ‘algebraic’ criterion for the sequences {g_{i}} ⊂ G such that μ_{H} · g_{i} → μ_{G} as i → ∞, then one can provide solutions to the following general counting problem: Suppose that G is realized as a ℚ subgroup of GL_{n}(ℝ) and there exists p ∈ ℤ^{n} such that the orbit V = p · G is closed. The problem is to estimate the cardinality of V ∩ ℤ^{n} ∩ B_{T}, as T → ∞, where B_{T} is the ball of radius T around 0 in ℝ^{n}. (Note that V is a rational affine variety, H is the stabilizer of p in G, and V ≅ H\G over ℚ).

The above mentioned approach to the counting problem motivated us to study the limiting distributions for the collection of measures {μ_{H} ·g}_{g∈G}. We study this question in two steps. First we would like to describe the sequences {g_{i}} ⊂ G for which the measures μ_{H} · g_{i} have convergent subsequences in the space of all probability measures on Γ\G. Then we can describe the probability measures on Γ\G which arise as the limits of such sequences, and obtain some algebraic condition on {g_{i}} to ensure that the limit measure is G-invariant. In this paper we address the first part under some additional assumptions on H. The second part is answered in a more general setup in [EMozSh], using the classification of ergodic invariant measures for unipotent flows due to Ratner [R].

Consider the following: Since H is a reductive ℚ-subgroup, the same holds for Z(H) (the centralizer of H in G). Therefore there exists a linear representation of G defined over ℚ and a ℚ-vector v_{0} in the representation space such that the stabilizer of v_{0} in G is Z(H) (see [B, Prop. 7.7]). Hence the orbit v_{0}·Γ is discrete (see [B, Prop. 7.13]). In particular, the set Z(H)·Γ is closed in G, and hence the orbit π(Z(H)) is closed in G/Γ. Now if π(Z(H)) is not compact, then there exists a sequence {z_{i}} ⊂ Z(H) such that {π(z_{i})} has no convergent subsequence. Thus for any compact sets K ⊂ Γ\G and C ⊂ π(H), we have Cz_{i} ∩ K = ∅ for all but finitely many i. Hence μ_{H}z_{i}(K) → 0 as i → ∞. In other words, {μ_{H}z_{i}} converges to the unit mass on the point at infinity in the one-point compactification of Γ\G. Thus if π(Z_{G}(H)) is not compact, then certain sequences of translated measures ‘escape to infinity’.

The main goal of this article is to show that if π(Z_{G}(H)) is compact, then any sequence {μ_{H} · g_{i}} has a convergent subsequence. In other words:

Vol. 7, 1997 NON-DIVERGENCE OF TRANSLATES OF CERTAIN ALGEBRAIC MEASURES 3

Theorem 1.1. Let the notation be as above. Suppose that π(Z(H)) is compact. Then given ϵ > 0 there exists a compact set K ⊂ Γ\G such that

μ_{H}g(K) ≥ 1 − ϵ , ∀ g ∈ G .

Remark 1.2. The condition that π(Z(H)) is compact, is equivalent to saying that H is not contained in any proper ℚ-parabolic subgroup of G, or that any ℚ-subgroup of G containing H is reductive (see Lemma 5.1 for a proof).

In particular, if a reductive subgroup H is a proper maximal connected ℚ-subgroup of G, then π(Z(H)) is compact, and hence the conclusion of theorem 1.1 holds. Note that the symmetric subgroups considered in [DuRuS] are proper maximal over ℚ.

The above theorem is related to the results due to Dani and Margulis [DM2] on recurrence properties of individual unipotent trajectories on finite volume homogeneous spaces (see also [M],[D1,2]). We first generalize their result to higher-dimensional trajectories of a more general type. The generalized result provides an algebraic condition to determine when a given compact subset of Γ\G has positive relative measure with respect to a given trajectory in Γ\G. The major component of the proof of Theorem 1.1 is to show that the algebraic condition, which is given in terms of certain linear representation of G, is satisfied for the sequence of trajectories under consideration; namely for {π(Ω)g_{i}}, where Ω is a relatively compact neighbourhood of identity in H.

Thanks to Theorem 1.1, given any sequence {g_{i}} ⊂ G, there exists a subsequence {g_{i_{k}}} ⊂ {g_{i}} such that the sequence {μ_{H} · g_{i_{k}}} converges to a probability measure μ on Γ\G as i_{k} → ∞. In [EMozSh] we show that any such limit measure is ‘algebraic’; i.e. for any c ∈ π^{−1}(supp(μ)) there exists a connected real algebraic ℚ-subgroup L of G containing H such that μ is supported on π(L)c and it is c^{−1}Lc-invariant. These results are then applied to obtain some new results on estimating the number of integral points on certain closed rational orbits of linear actions of G. For instance, one may determine the asymptotics of the number of integral matrices A in a large ball in M_{n}(ℝ), whose characteristic polynomial det(I − λA) is a fixed monic irreducible polynomial p(λ).

Theorem 1.1 can be reformulated as follows:

Theorem 1.3. Let the notation be as before. Suppose that π(Z(H)) is compact. Then given a Borel set H ⊂ H of positive Haar measure, there exists a compact set C ⊂ G such that G = HΓC.

As an application of the above formulation we have the following.

Corollary 1.4. Let H be a connected reductive real algebraic subgroup of SL_{n}(ℝ) defined over ℚ and acting irreducibly over ℚ on ℝ^{n}. Let W be a linear subspace of ℝ^{n} defined over ℚ. Then for any linear subspace V of ℝ^{n}, there exists h ∈ H such that

dim(W · h ∩ V) ≤ \frac{1}{n} dim(W) dim(V) .

Note that the set of h satisfying the above condition is Zariski open in H.

We are not aware of any direct proof of this result. We note that John Stalker obtained a nice direct proof for the analogous assertion over ℝ, in place of ℚ.

The paper is organized as follows: In section 2 we introduce certain spaces of functions (denoted by E(n, m, Λ)) and describe some of their basic properties. These functions arise as matrix coefficients of finite dimensional linear representations. Section 3 provides a generalization of the theorem of Dani and Margulis [DM2] for E(m, n, Λ) type of trajectories. The main part of the paper, which is section 5, is to show that the conditions of the generalized Dani-Margulis theorem hold under the hypothesis of Theorem 1.1. Some results about linear representations of G which are required in section 5 are proved in section 4. The Corollary 1.4 is deduced at the end of section 5.

For the proof of Theorem 1.1 in Section 5, we shall only need the statement of Theorem 3.9 out of Sections 2 and 3. The reader may find it convenient to first assume the statement of Theorem 3.9, and then start from Section 4.

We would like to thank S.G. Dani and John Stalker for helpful discussions and suggestions. We would like to thank the referee for helpful suggestions.

2 Certain Growth Properties for a Class of Functions

Certain growth properties of polynomials of bounded degrees observed in [M] have played an important role in the study of limit distributions of orbits of unipotent flows on homogeneous spaces of Lie groups (cf. [DM3]). We show that the same properties, interpreted appropriately, hold for a much larger class of functions. This generalization is useful in studying the limit distribution of translates of algebraic measures in homogeneous spaces.

Functions of one variable

Definition 2.1. Let n ∈ ℕ and Λ ≥ 0 be given. Let E(n, Λ) be the set of functions f : ℝ → ℂ of the form

f(t) = ∑_{i=1}^{n} ∑_{l=0}^{n−1} a_{i,l}t^{l}e^{λ_{i}t} (∀ t ∈ ℝ) , (1)

where a_{i,l} ∈ ℂ, and λ_{i} ∈ ℂ with |λ_{i}| ≤ Λ.

The spaces E(n, Λ) arise naturally as follows.

Lemma 2.2. Let n ∈ ℕ and X ∈ M_{n}(ℂ). Then for any v, w ∈ ℂ^{m}, the map f : ℝ → ℂ defined by

f(t) = ⟨w · exp(tX), v⟩ , ∀ t ∈ ℝ ,

is in E(n, Λ) for some Λ ≥ 0; in fact the λ_{i}’s appearing in equation (1) for f are precisely the eigenvalues of X.

Proof. Use Jordan decomposition. □ Observe the following.

Lemma 2.3. If f ∈ E(n, Λ), t_{0} ∈ ℝ, and η > 0, the map f_{1} : ℝ → ℝ defined as f_{1}(t) = f(ηt + t_{0}), ∀ t ∈ ℝ, is in E(n, ηΛ). □

The main tool for studying these spaces is provided by the following lemma.

Lemma 2.4. Let n ∈ ℕ, λ⃗ = (λ_{1}, …, λ_{n}) ∈ ℂ^{n}, and t⃗ = (t_{1}, …, t_{n^{2}}) ∈ ℝ^{n^{2}}. Let D(t⃗, λ⃗) be the determinant of the n^{2} ×n^{2} matrix whose entries are t_{k}^{l}e^{λ_{i}t_{k}}, where 0 ≤ l ≤ n − 1, 1 ≤ i ≤ n, and 1 ≤ k ≤ n^{2}. Then

D(t⃗, λ⃗) = ∏_{j<k}(t_{k} − t_{j}) ∏_{i<p}(λ_{i} − λ_{p})^{m_{i,p}}F (t⃗, λ⃗) ,

where m_{i,p} ∈ ℕ and F is an entire function on ℂ^{n^{2}} ×ℂ^{n} such that F (0, λ⃗) ≠ 0.

Proof. See [Ha]. □

Note 2.5. In view of Lemma 2.4, given n ∈ ℕ and Λ ≥ 0, there exist δ_{0}(n, Λ) > 0 and c_{0} > 0 such that |F (t⃗, λ⃗)| ≥ c_{0} for all t⃗ = (t_{1}, …, t_{n^{2}}) with |t_{k}| ≤ δ_{0}(n, Λ) for each k = 1, …, n, and for all λ⃗ = (λ_{1}, …, λ_{n}) with |λ_{i}| ≤ Λ for each i = 1, …, n.

We fix n, Λ, and δ_{0} = δ_{0}(n, Λ) for this section.

Lemma 2.6. Let 0 ≤ t_{1} < … < t_{n^{2}} ≤ δ_{0} be given. Then there exist entire functions G_{j} : ℂ × ℂ^{n} → ℂ for j = 1, …, n^{2} such that for any f ∈ E(n, Λ) expressed as in equation (1), and s ∈ ℝ, we have that

f(s) = \frac{1}{F (t⃗,λ⃗)} ∑_{j=1}^{n^{2}} G_{j}(s, λ⃗) · ( ∏_{k>j} \frac{t_{k}−s}{t_{k}−t_{j}} )f(t_{j}) , (2)

where t⃗ = (t_{1}, …, t_{n^{2}}) and λ⃗ = (λ_{1}, …, λ_{n}).

Proof. For s ∈ ℂ and j = 1, …, n^{2}, put

v⃗_{j}(s) = (t_{1}, …, t_{j−1}, s, t_{j+1}, …, t_{n^{2}}) .

Then by Lagrange’s interpolation formula,

f(s) = ∑_{j=1}^{n^{2}} \frac{D(v⃗_{j}(s),λ⃗)}{D(t⃗,λ⃗)}f(t_{j}) .

Using Lemma 2.4 we obtain that

\frac{D(v⃗_{j}(s),λ⃗)}{D(t⃗,λ⃗)} = \frac{F (v⃗_{j}(s),λ⃗)}{F (t⃗,λ⃗)} ∏_{k>j} \frac{t_{k}−s}{t_{k}−t_{j}} .

Put G_{j}(s, λ⃗) = F (v⃗_{j}(s), λ⃗) for all s ∈ ℂ and λ⃗ ∈ ℂ^{n}. Then G_{j} : ℂ × ℂ^{n} → ℂ is an entire function and equation (2) holds. □

Corollary 2.7. Given δ ∈ (0, δ_{0}], there exists an M_{δ} > 0 such that for any f ∈ E(n, Λ), we have

sup_{t∈[0,δ_{0}]} |\frac{df}{dt}(t)| ≤ M_{δ} · sup_{t∈[0,δ]} |f(t)| . (3)

Further,

sup_{t∈[0,δ]} |\frac{df}{dt}(t)| ≤ (δ_{0}/δ)M_{δ_{0}} · sup_{t∈[0,δ]} |f(t)| . (4)

Proof. Fix 0 ≤ t_{1} < … < t_{n^{2}} ≤ δ. Express f as in equation (2). Since G_{j}(s, λ⃗) and \frac{∂G_{j}}{∂s} (s, λ⃗) are entire functions on ℂ × ℂ^{n}, and since s and λ⃗ vary over compact sets, the absolute values of these functions are bounded above. Also by our choice of δ_{0}, we have that 1/|F (t⃗, λ⃗)| is bounded above by 1/c_{0}. Therefore there exists M_{δ} > 0 such that

|\frac{df}{ds}(s)| ≤ M_{δ} · sup_{1≤k≤n^{2}} |f(t_{k})| , ∀ s ∈ [0, δ_{0}] .

This proves equation (3).

To obtain equation (4) define g(t) = f((δ/δ_{0})t). Then g ∈ E(n, (δ/δ_{0})Λ) ⊂ E(n, Λ). Now apply equation (3) for g, δ_{0}, and δ_{0}^{2}/δ in place of f, δ, and δ_{0}, respectively.

Corollary 2.8. Given η > 0 there exists M_{η} ≥ 1 such that for any δ ∈ (0, δ_{0}] and any subinterval I of length at least ηδ in [0, δ], and any f ∈ E(n, Λ),

sup_{t∈[0,δ]} |f(t)| ≤ M_{η} · sup_{t∈I} |f(t)| .

Proof. In view of Lemma 2.3, it is enough to prove the result for δ = δ_{0} and I = [0, ηδ_{0}]. Fix 0 ≤ t_{1} < … < t_{n^{2}} ≤ ηδ_{0} and express f as in equation (2).

Since each G_{j}(s, λ⃗) is entire, and s and λ⃗ vary over compact sets, the value of G_{j}(s, λ⃗) is bounded. Also 1/|F (t⃗, λ⃗)| is bounded above by 1/c_{0}. Therefore there exists M_{η} ≥ 1 such that

|f(s)| ≤ M_{η} · sup_{k=1,...,n^{2}} |f(t_{k})| , ∀ s ∈ [0, δ_{0}] .

This completes the proof. □

Corollary 2.9. Any nonzero function E(n, Λ) admits at most n^{2} − 1 distinct zeroes in [0, δ_{0}].

Proof. Let f ∈ E(n, Λ). If there are n^{2} distinct zeroes of f in [0, δ_{0}], we choose 0 ≤ t_{1} < … < t_{n^{2}} ≤ δ_{0} to be zeroes of f. But then from the expression of f as in Lemma 2.6, we obtain that f is identically zero. □

We are now ready to prove the main growth properties of functions in E(n, λ).

Corollary 2.10. (cf. [M, DM2]) Given ϵ > 0, there exists M > 0 such that for any f ∈ E(n, Λ) and any interval I of length at most δ_{0},

|{t ∈ I : |f(t)| < (1/M) sup_{t∈I} |f(t)|}| ≤ ϵ · |I| .

Proof. Without loss of generality we may assume that f ≠ 0 and I ⊂ [0, δ_{0}]. For any a > 0 put

I(a) = {t ∈ I : −a < f(t) < a} .

Since f(t) − a and f(t) + a are in E(n + 1, Λ), due to Corollary 2.9, f(t) ± a has at most (n + 1)^{2} − 1 distinct zeros in [0, δ_{0}]. Therefore I(a) has at most (n + 1)^{2} connected components. Let η = ϵ/(n + 1)^{2}. By Corollary 2.8, there exists M ≥ 1 depending only on n and Λ, such that for any subinterval J of I with |J| ≥ η · |I|,

sup_{t∈I} |f(t)| ≤ (M/2) sup_{t∈J} |f(t)| .

Putting a = (1/M) sup_{t∈I} |f(t)|, we obtain that |J| ≤ η · |I| for every connected component J of I(a). Hence

|I(a)| ≤ (n + 1)^{2}η · |I| = ϵ · |I| .

This completes the proof. □

Corollary 2.11. (cf. [M],[DM2]) Let δ ∈ [0, δ_{0}] and σ > 1 be given and let δ′ = σ^{−2n^{2}}δ. Then there exist constants 0 < β_{1} ≤ β_{2} such that the following holds: For any f ∈ E(n, Λ) such that sup_{t∈[0,δ′]} |f(t)| = 1, there exists k ∈ {1, …, n^{2}} such that

β_{1} ≤ |f(t)| ≤ β_{2} , ∀ t ∈ [σ^{2k−1}δ′, σ^{2k}δ′] . (5)

Proof. By Lemma 2.6 there exists β_{2} ≥ 1 such that for all f ∈ E(n, Λ),

sup_{t∈[0,δ]} |f(t)| ≤ β_{2} · sup_{t∈[0,δ′]} |f(t)| .

For any choice of t_{k} ∈ [σ^{2k−1}δ′, σ^{2k}δ′], k = 1, …, n^{2}, one has

t_{k+1} − t_{k} ≥ (σ^{2k+1} − σ^{2k})δ′ .

Hence applying Lemma 2.6 as in the proof of Corollary 2.8, there exists a constant M ≥ 1 such that for any choice of t_{k} as above we have

sup_{t∈[0,δ]} |f(t)| ≤ M · sup_{k=1,...,n^{2}} |f(t_{k})| .

Now if we choose β_{1} = 1/(2M), then for some k ∈ {1, …, n^{2}},

inf_{t∈[σ^{2k−1}δ′,σ^{2k}δ′]} |f(t)| ≥ β_{1} .

This completes the proof. □

Functions of several variables. Now we generalize some of the above results for functions of several variables.

Definition 2.12. For m ∈ ℕ, n ∈ ℕ, and Λ ≥ 0 let E(m, n, Λ) be the set of functions f : ℝ^{m} → ℂ of the form

f(t_{1}, …, t_{m}) = ∑_{i_{1},...,i_{m}=1}^{n} ∑_{l_{1},...,l_{m}=0}^{n−1} a_{i⃗,l⃗} · ∏_{k=1}^{m} t_{k}^{l_{k}}e^{λ_{k,i_{k}} t_{k}} , (6)

where i⃗ = (i_{1}, …, i_{m}) and l⃗ = (l_{1}, …, l_{m}), a_{i⃗,l⃗} ∈ ℂ, and λ_{k,j} ∈ ℂ with |λ_{k,j}| ≤ Λ.

Our interest in considering this space comes from the following observations.

Lemma 2.13. Let m ∈ ℕ, n ∈ ℕ, and X_{1}, …, X_{m} ∈ M_{n}(ℂ). For any v, w ∈ ℂ^{n}, define f : ℝ^{m} → ℂ as

f(t) = ⟨v · exp(t_{1}X_{1}) ⋯ exp(t_{m}X_{m}), w⟩ , ∀ t = (t_{1}, …, t_{m}) ∈ ℝ^{m} .

Then f ∈ E(m, n, Λ), where Λ is the maximum of the absolute values of all the eigenvalues of all the X_{k}’s.

Proof. It is straightforward to verify this using Lemma 2.2. □ From the definition we obtain the following two consequences.

Lemma 2.14. For m ∈ ℕ, n ∈ ℕ, Λ ≥ 0, f ∈ E(m, n, Λ), v ∈ ℝ^{m} with ∥v∥ = 1, and t_{0} ∈ ℝ^{m}, define a function ϕ : ℝ → ℂ as

ϕ(t) = f(tv + t_{0}) , ∀ t ∈ ℝ .

Then ϕ ∈ E(mn, \sqrt{m}Λ). □

Lemma 2.15. Let P be a complex polynomial in k variables and with total degree d. For f_{1}, …, f_{k} ∈ E(m, n, Λ), define

g(t) = P (f_{1}(t), …, f_{k}(t)) , ∀ t ∈ ℝ^{m} .

Then g ∈ E(m, n^{d}, d · Λ). □

Note 2.16. Fix m ∈ ℕ, n ∈ ℕ, and Λ ≥ 0. We define δ_{0} = δ_{0}(mn, \sqrt{m}Λ) as in Note 2.5.

Corollary 2.17. Given η > 0, there exists M ≥ 1 such that for any δ ∈ (0, δ_{0}], any ball B of diameter δ in ℝ^{m}, any ball D of diameter at least ηδ contained in B, and any f ∈ E(m, n, Λ),

(7)  sup_{t∈B} |f(t)| ≤ M · sup_{t∈D} |f(t)| .

Proof. Let t_{0} be the center of D. Let S be the unit sphere in ℝ^{m} centered at 0. Take any v ∈ S, and consider the function ϕ_{v} : ℝ → ℂ defined as ϕ_{v}(t) = f(tv + t_{0}), ∀ t ∈ ℝ. Then ϕ_{v} ∈ E(mn, \sqrt{m}Λ).

By Corollary 2.8, there exists M_{η} > 0 such that for any ψ ∈ E(mn, \sqrt{m}Λ), any interval I of length at most δ, and any interval J ⊂ I of length at least ηδ,

sup_{t∈I} |ψ(t)| ≤ M_{η} · sup_{t∈J} |ψ(t)| . (8)

Now for each v ∈ S, by putting ψ = ϕ_{v}, I = {t ∈ ℝ : tv + t_{0} ∈ B}, and J = {t ∈ ℝ : tv + t_{0} ∈ D} in equation (8), we conclude that equation (7) holds. □

Corollary 2.18. Given M > 1 there exists η > 0 such that for any f ∈ E(m, n, Λ), any δ ∈ (0, δ_{0}], and a closed ball B in ℝ^{m} of diameter δ there exists a closed ball D ⊂ B of diameter ηδ such that

(9)  sup_{t∈B} |f(t)| ≤ M · inf_{t∈D} |f(t)| .

Proof. Let t_{0} ∈ \overline{B} be such that sup_{t∈B} |f(t)| = |f(t_{0})|. For every v ∈ S, define ϕ_{v}(t) = f(tv + t_{0}), ∀ t ∈ ℝ. Then ϕ_{v} ∈ E(mn, \sqrt{m}Λ). Let I_{v} = \{t ≥ 0 : tv + t_{0} ∈ B}. Since sup_{t∈I_{v}} |ϕ_{v}(t)| ≤ |f(t_{0})|, by Corollary 2.7 there exists M_{0} > 0 such that

(10)  sup_{t∈I_{v}} |\frac{dϕ_{v}}{dt} (t)| ≤ (M_{0}/|I_{v}|)|f(t_{0})| , ∀ v ∈ S .

Put η = (1/M_{0})(1 − 1/M). Then due to equation (10), for any 0 ≤ t ≤ η|I_{v}|, we get

|ϕ_{v}(t)| = |ϕ_{v}(0) + t · \frac{dϕ_{v}}{dt} (t_{1})| , for some t_{1} ∈ [0, t]
≥ (1 − M_{0}η)|f(t_{0})|
≥ (1/M)|f(t_{0})| .

Clearly

D = ⋃_{v∈S}[0, η|I_{v}|] · v + t_{0}

is a ball of diameter ηδ contained in B, and equation (9) holds. □

3 An Algebraic Condition for Returning to Compact Sets

In this section we extend an important result due to Dani and Margulis [DM2] about large compact sets in finite volume homogeneous spaces having relative measures close to 1 on trajectories of unipotent flows. The generalization is made for the following class of trajectories in a Lie group G in place of unipotent one-parameter subgroups.

Definition 3.1. Let G be a Lie group and 𝔤 be its Lie algebra equipped with a positive definite inner product. For m ∈ ℕ, n ∈ ℕ, and Λ ≥ 0, let E_{G}(m, n, Λ) be the set of all functions f : ℝ^{m} → G such that for any v, w ∈ 𝔤, the map

t ↦ ⟨v · Ad(f(t)), w⟩ , ∀ t ∈ ℝ^{m}

is in E(m, n, Λ). Denote by E_{G}(n, Λ) the set E_{G}(1, n, Λ).

Corollary 3.2. Let G be a Lie group and 𝔤 its Lie algebra. For X_{1}, …, X_{m} ∈ 𝔤, define a map ϕ : ℝ^{m} → G as

ϕ(t) = exp(t_{1}X_{1}) ⋯ exp(t_{m}X_{m}) , ∀ t = (t_{1}, …, t_{m}) ∈ ℝ^{m} .

Then ϕ ∈ E_{G}(m, n, Λ), where n = dim G and Λ is the maximum of the absolute values of the eigenvalues of all the ad X_{i}’s.

Proof. It is a direct consequence of Lemma 2.13. □

Geometric condition for return in measure. In order to derive some of the results for higher-dimensional trajectories from the one-dimensional case, we make an elementary observation.

Lemma 3.3. Let B be a bounded open convex subset of ℝ^{m}. Let S be the unit sphere in ℝ^{m} centered at the origin. Fix x_{0} ∈ \overline{B}. For every v ∈ S, and a Borel measurable subset E of B, define

E_{v} = {t ≥ 0 : x_{0} + tv ∈ E} .

Then

|E|/|B| ≤ m · sup_{v∈S} |E_{v}|/|B_{v}| ,

where |A| denotes the standard Lebesgue measure of a measurable subset A ⊂ ℝ^{m}.

Proof. Let σ denote the rotation invariant probability measure on S such that the volume of a unit ball in ℝ^{m} is σ(S)/m. Using polar decomposition of B at the pole t_{0}, we have

|E| = ∫_{v∈S} dσ(v) · ∫_{0}^{|B_{v}|} χ_{E_{v}}(t)t^{m−1} dt

≤ ∫_{v∈S} dσ(v) · |E_{v}| · |B_{v}|^{m−1}

≤ m( sup_{v∈S} |E_{v}|/|B_{v}|) ∫_{v∈S} |B_{v}|^{m}/m dσ(v)

= m( sup_{v∈S} |E_{v}|/|B_{v}|)|B| .  □

Theorem 3.4. Let G be a Lie group, Γ be a lattice in G, and π : G → Γ\G be the natural quotient map. Given m, n ∈ ℕ, Λ > 0, a compact set C ⊂ Γ\G, T > 0 and an ϵ > 0, there exists a larger compact set C′ ⊂ Γ\G such that for any ϕ ∈ E_{G}(m, n, Λ) and a ball B of diameter at most T in ℝ^{m}, one of the following holds:

1. π(ϕ(B)) ∩ C = ∅.

2. |{t ∈ B : π(ϕ(t)) ∈ C′}| ≥ (1 − ϵ)|B|.

Proof. In [DM3, Theorem 6.1] the result is stated for an Ad-unipotent oneparameter subgroup u : ℝ → G, in place of ϕ as above.

We observe that the only property of the map u : ℝ → G used in that proof is that u ∈ E_{G}(n_{1}, 0) for some n_{1} ∈ ℕ depending only on G. Specifically the relevant properties of E(n, 0) are Corollary 2.8, Corollary 2.9, and Corollary 2.11.

Hence using the above mentioned corollaries, the same proof works to give the result for all ϕ ∈ E_{G}(n, Λ).

Now choose a compact set C′ ⊂ Γ\G such that the conclusion of the theorem is valid for ϵ/m in place of ϵ, and all ψ ∈ E_{G}(mn, \sqrt{m}Λ) in place of ϕ, and δ_{0} > 0 as in the Note 2.5 depending only on G, m, n, and Λ.

First assume that T ≤ δ_{0}. Let ϕ ∈ E_{G}(m, n, Λ). Suppose that condition 1 above does not hold. Then there exists t_{0} ∈ B such that π(ϕ(t_{0})) ∈ C. Define

E = {t ∈ B : π(ϕ(t)) ∉ C′} .

Let the notation be same as in Lemma 3.3. Take v ∈ S. Define a map ψ(t) = ϕ(tv + t_{0}), ∀ t ∈ ℝ. By Lemma 2.14 ψ ∈ E_{G}(mn, \sqrt{m}Λ). Therefore by the above hypothesis,

|E_{v}| = |{t ∈ B_{v} : π(ψ(t)) ∉ C′}| ≤ (ϵ/m)|B_{v}| .

Therefore by Lemma 3.3,

|E| ≤ ϵ · |B| .

For the general case, observe that there exists a natural number N depending only on m, T, and δ_{0} such that the following holds: In ℝ^{m}, any ball B of diameter at most T can be covered by a sequence {B_{i}}_{i=1}^{N} of balls of diameters at most δ_{0} such that

|B_{i} ∩ B_{i+1}| > (1/4) sup{|B_{i}|, |B_{i+1}|} , i = 1, …, N − 1 .

Applying the theorem for T = δ_{0}, ϵ = 1/4, and the given C, we obtain a sequence {C_{j}}_{j=1}^{N} of compact sets in Γ\G such that C_{1} = C, C_{j} ⊂ C_{j+1}, and the conclusion of the theorem holds for C = C_{j} and C′ = C_{j+1}, where j = 1, …, N − 1.

Again apply the theorem for δ_{0}, ϵ/N, and ∪_{j=1}^{N}C_{j} in place of T, ϵ, and C, respectively, and obtain a compact set C′ such that the conclusion of the theorem holds.

Now let B be any ball of diameter at most T in ℝ^{m} and suppose that ϕ(B) ∩ C ≠ ∅. Cover B by a sequence of balls {B_{i}}_{i=1}^{N} as above. Then by the choice of {C_{j}}, we have ϕ(B_{i}) ∪ ⋃_{j=1}^{N} C_{j} ≠ ∅. Hence by the choice of C′,

|{t ∈ B_{i} : ϕ(t) ∉ C′}| < (ϵ/N)|B_{i}| , i = 1, …, N .

Hence |{t ∈ B : ϕ(t) ∈ C′}| > (1 − ϵ)|B|. □

Algebraic formulation of the geometric condition for returning to compact sets. The usefulness of the previous theorem is enhanced by the next result which provides an algebraic condition, in terms of certain representations of G, equivalent to the geometric condition π(ϕ(B)) ∩ C = ∅. For simplicity we consider only the case of arithmetic lattices. In order to formulate the result, we need some notation.

Let G a connected semisimple real algebraic group defined over ℚ. Let Γ ⊂ G be an arithmetic lattice with respect to the Q-structure on G and π : G → Γ\G be the natural quotient map. Let r be the ℚ-rank of G. We can assume that r ≥ 1; if r = 0, then by Godement’s compactness criterion (see [B, Theorem 8.4]), Γ\G is cocompact and the results of this section are trivial in this case.

Let P be a minimal ℚ-parabolic subgroup of G. Then by [B, Theorem 15.6], there exists a finite set F ⊂ G(ℚ) such that

G(ℚ) = Γ · F · P (ℚ) . (11)

Let S be a maximal ℚ-split torus of G contained in P. Let Π be the set of ℚ-roots with respect to S. The roots whose corresponding root spaces are contained in the Lie subalgebra of 𝔤 associated to P form a system of positive roots. Let Δ = {α_{1}, …, α_{r}} denote the corresponding system of simple roots.

Take i ∈ {1, …, r}. Let P_{i} denote the standard maximal parabolic subgroup associated to the set of simple roots Δ \ {α_{i}}. Let U_{i} be the unipotent radical of P_{i}. For each i construct a representation of G on a finite dimensional vector space V_{i} as follows: Let 𝔲_{i} be the Lie subalgebra of 𝔤 associated to U_{i}. Put l_{i} = dim U_{i}. Let V_{i} = ∧^{l_{i}}𝔤, the l_{i}-th exterior power, and consider the ∧^{l_{i}} Ad-action of G on V_{i}. Fix any Euclidean norm on 𝔤 and let {e_{1}, …, e_{n}} be an orthonormal basis of 𝔤. Define a Euclidean norm on V_{i} such that the basis

{e_{i_{1}} ∧ ⋯ ∧ e_{i_{l_{i}}} ∈ V_{i} : 1 ≤ i_{1} < ⋯ < i_{l_{i}} ≤ n}

is orthonormal in V_{i}.

Note that since G is defined over ℚ, its Lie algebra 𝔤 inherits a ℚstructure. Now V_{i} also has a natural ℚ-structure and ∧^{l_{i}}𝔲_{i} is a rational one-dimensional subspace of V_{i}. Now take p_{i} ∈ ∧^{l_{i}}𝔲_{i}(ℚ) \ {0}. Then for any g ∈ P_{i}(ℝ), we have

p_{i} · g = det(Ad g|_{𝔲_{i}}) · p_{i} . (12)

Define a function d_{i} : G → ℝ^{∗} as

d_{i}(g) = ∥p_{i} · g∥^{2} ∀ g ∈ G .

Theorem 3.5. Let m ∈ ℕ, n ∈ ℕ, Λ > 0, T > 0, and α > 0 be given. Then there exists a compact set C ⊂ Γ\G such that for any ball B ⊂ ℝ^{m} of diameter at most T and any ϕ ∈ E_{G}(m, n, Λ), one of the following conditions is satisfied:

1. There exists i ∈ {1, …, r} and λ ∈ ΓF such that

d_{i}(λ^{−1}ϕ(t)) = ∥p_{i} · λ^{−1}ϕ(t)∥^{2} < α , ∀ t ∈ B .

2. π(ϕ(B)) ∩ C ≠ ∅.

This theorem is proved in [DM2] for unipotent one parameter subgroups; i.e. for ϕ ∈ E_{G}(1, n, 0). The organization of that proof crucially uses the one-dimensionality of the trajectory. Here we modify their proof to take care of the higher dimensional cases.

Algebraic description of compact sets in Γ\G Γ\G Γ\G. First we recall a result in [DM2] which gives an algebraic description of relatively compact open connected subsets of Γ\G using the reduction theory.

Let I ⊂ {1, …, r}, and put J = {1, …, r} \ I. Define

P_{I} = ⋂_{i∈I} P_{i}
Q_{I} = {g ∈ P_{I} : d_{i}(g) = 1 , ∀ i ∈ I} .

Note that each Q_{I} is a ℚ-algebraic group with no nontrivial characters defined over ℚ, and P ∩Q_{I} is a minimal ℚ-parabolic subgroup of Q_{I}. Therefore by [B, Theorem 15.6], there exists a finite set F_{I} ⊂ Q_{I}(ℚ) such that

Q_{I}(ℚ) = (Γ ∩ Q_{I})F_{I}(P ∩ Q_{I})(ℚ) . (13)

We define Λ(I) = F_{I}^{−1}(Γ ∩ Q_{I}). Note that P_{∅} = Q_{∅} = G and Λ(∅) = F^{−1}Γ.

Lemma 3.6. Let j ∈ {1, …, r}, I ⊂ {1, …, r} \ {j}, and I′ = I ∪ {j}. Then there exists a finite set E ⊂ P (ℚ) such that

Λ(I′)Λ(I) ⊂ E · Λ(I) .

Proof. By definition

Λ(I′)Λ(I) = F_{I′}^{−1}(Q_{I′} ∩ Γ) · F_{I}^{−1}(Q_{I} ∩ Γ) .

Note that any g ∈ G(ℚ), gΓg^{−1} ∩ Γ is a subgroup of finite index in Γ (see [B, Corollary 7.13]). Therefore there exists a finite set L ⊂ Q_{I}(ℚ) such that

(Q_{I′} ∩ Γ) · F_{I}^{−1} ⊂ (Q_{I} ∩ Γ)F_{I}^{−1} ⊂ L(Q_{I} ∩ Γ) .

Now by equation (13), there exists a finite set E_{1} ⊂ (P ∩ Q_{I})(ℚ) such that

(F_{I′}^{−1}L)^{−1} ⊂ (Γ ∩ Q_{I})F_{I}E_{1} .

Hence

Λ(I′)Λ(I) ⊂ (F_{I′}^{−1}L)(Q_{I} ∩ Γ)
⊂ E_{1}^{−1}F_{I}^{−1}(Q_{I} ∩ Γ)
= E · Λ(I) ,

where E = E_{1}^{−1}F_{I}^{−1} ⊂ P (ℚ). □

Notation. Let I be the collection of all ordered p-tuples, where 0 ≤ p ≤ r; by a 0-tuple we mean the empty set. Let I = (i_{1}, …, i_{p}) ∈ I. Then there exists a finite set L(I) ⊂ G(ℚ) such that

Λ({i_{1}, …, i_{p−1}}) ⋯ Λ({i_{1}})Λ(∅) = L(I)Γ .

We define L(∅) = {e}.

For positive reals 0 < a < b and α > 0, and any λ ∈ L(I)Γ, define

W_{α,a,b}(I, λ) = {g ∈ G : d_{j}(θλg) > α, ∀ j ∈ {1, …, r} \ I and ∀ θ ∈ Λ(I) ,

and a ≤ d_{i}(λg) ≤ b, ∀ i ∈ I} .

Note that for any γ ∈ Γ,

W_{α,a,b}(I, λγ) = γ^{−1}W_{α,a,b}(I, λ) .

Define the following subsets of Γ\G:

W_{α,a,b}(I) = ⋃_{λ∈L(I)} \overline{π(W (I, λ))} = ⋃_{λ∈L(I)Γ} \overline{π(W (I, λ))} .

The following result is obtained in [DM2, Prop. 1.8].

Proposition 3.7. The set W_{α,a,b}(I) is compact.

In order to prove Theorem 3.5, we need to find constants 0 < a < b and β > 0 (which are independent of ϕ and B) such that if condition 1 fails to hold, then π(ϕ(t)) ∈ W_{β,a,b}(I) for some t ∈ B and I ∈ I.

Basic proposition. The following proposition is a main technical tool for finding a, b, and β. It is based on some ideas from [DM1, Appendix].

By Lemma 2.15, there exists n_{1} ∈ ℕ and Λ_{1} > 0 such that for any ϕ ∈ E_{G}(m, n, Λ), any g ∈ G, and any i ∈ {1, …, r}, the map f : ℝ^{m} → ℝ defined by f(t) = d_{i}(gϕ(t)) is in E(m, n_{1}, Λ_{1}). Let δ_{0} = δ_{0}(m, n_{1}, Λ_{1}) be as in Note 2.16.

Proposition 3.8. Let α > 0 and D ⊂ ℝ^{m} be a ball of diameter at most δ_{0}. Suppose a family F ⊂ E(m, n_{1}, Λ_{1}) satisfies the following conditions:

1. For any t ∈ D and any β > 0,

#{f ∈ F : |f(t)| < β} < ∞ .

2. For every f ∈ F,

sup_{t∈D} |f(t)| > α .

Then at least one of the following conditions is satisfied:

(a) There exists a t_{0} ∈ D such that |f(t_{0})| > α for all f ∈ F.

(b) There exist a ball D_{1} ⊂ D and f_{0} ∈ F such that: (i) f_{0}(D_{1}) ⊂ (α/2, α); (ii) there exists M ≥ 1 (depending only on m, n_{1}, and Λ_{1}) such that

for all f ∈ F,

sup_{t∈D_{1}} |f(t)| > α/M .

Proof. Let t_{0} be the center of D. If (a) does not hold then due to condition 1, there exists a finite set F_{1} ⊂ F such that |f(t_{0})| ≥ α for all f ∈ F \ F_{1}. Now let E be the ball centered at t_{0} of some diameter 0 < δ < δ_{0} such that

1) sup_{t∈E} |f(t)| ≥ α, ∀ f ∈ F_{1}, and

2) there exists f_{0} ∈ F_{1} such that sup_{t∈E} |f_{0}(t)| = α.

By Corollary 2.18, there exist a constant η > 0 (depending only on m, n_{1}, and Λ_{1}) and a ball D_{1} of diameter ηδ contained in E such that

inf_{t∈D_{1}} |f_{0}(t)| ≥ α/2 .

Now by Corollary 2.17, there exists a constant M ≥ 1 (depending only on η, m, n_{1}, and Λ_{1}) such that for any f ∈ F we have

sup_{t∈D_{1}} |f(t)| ≥ α/M .

This completes the proof. □

Proof of Theorem 3.5. Let α > 0 be given. Let B ⊂ ℝ^{m} be a ball of diameter at most δ_{0}, and let ϕ ∈ E_{G}(m, n, Λ). Suppose that condition 1 of the theorem fails to hold.

By a stepwise construction we shall obtain I ∈ I, λ ∈ L_{I}Γ, and constants 0 < a_{I} < b_{I} and α_{I} > 0 depending only on I and α such that

π(ϕ(B)) ∩ π(W_{α_{I},a_{I},b_{I}}(I, λ)) ≠ ∅ .

In view of Proposition 3.7 this will imply that the second condition of the theorem holds.

Consider the following procedure: Suppose I ∈ I, λ ∈ L_{I}Γ, a ball D ⊂ ℝ^{m} of diameter at most δ_{0}, and constants 0 < a_{I} < b_{I} are such that

(A)  d_{i}(λϕ(B)) ⊂ (a_{I}, b_{I}) , ∀ i ∈ I .

Let F(I, λ) denote the family of all functions f : ℝ^{m} → ℝ_{>0} of the form f(t) = d_{j}(θλϕ(t)) for all t ∈ ℝ^{m}, where θ ∈ Λ(I) and j ∈ J = {1, …, r} \ I. Suppose further that for some α_{I} > 0, we have

(B)  sup_{t∈D} |f(t)| ≥ α_{I} , ∀ f ∈ F(I, λ) .

Observe that condition 1 of Proposition 3.8 is satisfied for the family F(I, λ), because the set p_{j} · Λ(I)L_{I} · Γ is discrete in V_{j} for every j ∈ J. Condition 2 of Proposition 3.8 follows from condition (B) as above. Therefore due to the proposition, one of the following holds:

(a) There exists t_{0} ∈ D such that d_{j}(θλϕ(t_{0})) ≥ α_{I} for all θ ∈ Λ(I) and all j ∈ J. In this case by condition (A) we have Γϕ(t_{0}) ∈ W_{α_{I},a_{I},b_{I}}(I). We fix this I ∈ I, t_{0} ∈ B, and constants 0 < a_{I} < b_{I} and α_{I} > 0 and stop the procedure.

(b) There exist j_{0} ∈ J, θ_{0} ∈ Λ(I), and a ball D_{1} ⊂ D such that the following holds:

(i) d_{j_{0}}(θ_{0}λϕ(D_{1})) ⊂ (α_{I}/2, α_{I}).

(ii) For all θ ∈ Λ(I) and j ∈ J,

sup_{t∈D_{1}} d_{j}(θλϕ(t)) ≥ α_{I}/M .

In this case, let I_{1} = I ∪ j_{0}, and λ_{1} = θ_{0}λ. We will now show that conditions (A) and (B) are satisfied for D_{1}, I_{1} and λ_{1}, with suitable constants a_{I_{1}}, b_{I_{1}}, and α_{I_{1}}. Since d_{i}(θ_{0}g) = d_{i}(g), ∀ i ∈ I and ∀ g ∈ G, condition (A) is satisfied with a_{I_{1}} = α_{I}/2 and b_{I_{1}} = α_{I}. By Lemma 3.6, there exists a finite set E ⊂ P(ℚ) (depending only on I and j_{0}) such that for any θ ∈ Λ(I ∪ {j_{0}}), there exists θ′ ∈ Λ(I) and x ∈ E such that θθ_{0} = xθ′. Hence for every j ∈ J \ {j_{0}},

sup_{t∈D_{1}} d_{j}(θλ_{1}ϕ(t)) = sup d_{j}(θθ_{0}λϕ(t))
= sup d_{j}(xθ′λϕ(t))
= sup d_{j}(x) · d_{j}(θ′λϕ(t))
≥ β · α_{I}/M ,

where β = min_{x∈E} d_{j}(x) > 0 depends only on I and j_{0}. Therefore condition (B) is also satisfied for the family F(I_{1}, λ_{1}) and α_{I_{1}} = βα_{I}/M > 0. This completes the description of our procedure.

To prove the theorem, we start with I = ∅, λ = e, and D = B. Then condition (A) is vacuously satisfied. We can assume that condition 1 in the statement of the theorem does not hold. Then condition (B) is satisfied for F(∅, e).

We can repeatedly apply the above procedure till we get I, λ ∈ L_{I}Γ, and constants 0 < a_{I} < b_{I} and α_{I} > 0 such that d_{i}(θλϕ(B)) > α_{I} for all θ ∈ Λ(I); at which step we are through. Since the cardinality of I increases each time we apply the procedure, it must stop after at most r steps. This completes the proof for T ≤ δ_{0}.

For the general case, cover the ball B with finitely many balls of radius δ_{0} such that for any two intersecting balls the measure of their intersection is at least 1/4 of the measure of each. The number of balls required for this purpose depends only on m, T, and δ_{0}. Suppose condition 2 fails to hold. Then condition 1 holds for any one of the smaller balls. Now apply Corollary 2.17 successively for η = 1/4. Condition 1 holds for the full ball B, with α replaced by a bounded multiple of it. Since the multiple depends only on m, n_{1}, Λ_{1} and the number of smaller balls, the theorem follows. □

Combining Theorem 3.4 and Theorem 3.5 we obtain the following stronger version.

(For the notation used the the statement of the following theorem, we refer the reader to Corollary 3.2 and the definitions and notation stated just before the statement of Theorem 3.5.)

Theorem 3.9. Given m ∈ ℕ, n ∈ ℕ, Λ ≥ 0, α > 0, ϵ > 0, and T > 0 there exists a compact set C ⊂ Γ\G such that for any ball B ⊂ ℝ^{m} of diameter at most T and any ϕ ∈ E_{G}(m, n, Λ), one of the following conditions is satisfied:

1. There exists i ∈ {1, …, r} and λ ∈ ΓF such that

d_{i}(λ^{−1}ϕ(B)) = ∥p_{i} · λ^{−1}ϕ(B)∥^{2} < α .

2. |{t ∈ B : π(ϕ(t)) ∈ C}| ≥ (1 − ϵ)|B|.

Remark 3.10. Condition 1 in Theorem 3.9 can be reformulated as:

1′. There exist i ∈ {1, …, r}, λ ∈ F (which is a finite set), and γ ∈ Γ such that for p = p_{i} · λ^{−1} we have

∥p · γ^{−1}ϕ(B)∥^{2} < α .

Note that p is contained in the one-dimensional subspace of V_{i} = ∧^{l_{i}}𝔤 associated to the Lie algebra 𝔲 of the unipotent radical of P = γ^{−1}P_{i}γ, which is a maximal proper ℚ-parabolic subgroup of G. Also p ∈ V_{i}(ℚ) \ {0} and P = {g ∈ G : g · p ∈ ℝ^{×} · p} (see equation (12)).

Remark 3.11. Note that Theorem 1 of [DM2] for unipotent one-parameter subgroups can be recovered from Theorem 3.9 by putting m = 1, n = dim 𝔤, and Λ = 0. Since Λ = 0, by changing the parameterization, here we can avoid the dependence of C on the choice of T.

Remark 3.12. See [Sh2, Theorem 2.2] for a result similar to theorem 3.9 for any Lie group G and any lattice Γ in G.

See also a recent preprint “Flows on homogeneous spaces and Diophantine approximation on manifolds” by D. Y. Kleinbock and G. A. Margulis for related results.

4 A Result About Linear Representations of G G G

In this section we obtain a result in linear algebra with allows us to apply the results of the previous section in order to prove Theorem 1.1 when H is an algebraic torus. The proof uses a certain decomposition of the group G (Proposition 4.4) and the following properties of finite dimensional spaces of functions on a set.

Lemma 4.1. Let X be any set and {β_{1}, …, β_{n}} be a set of linearly independent complex valued functions on X. Then there exists a finite set {t_{1}, …, t_{n}} ⊂ X such that the matrix A = (β_{i}(t_{j}))_{i,j=1,...,n} is nonsingular. In particular, for any a = (a_{1}, …, a_{n}) ∈ ℂ^{n}, we have

sup_{j=1,...,n} | ∑_{i=1}^{n} a_{i}β_{i}(t_{j})| ≥ ∥A^{−1}∥^{−1} · ∥a∥ , (14)

where ∥T ∥ denotes the operator norm of a linear transformation T.

Proof. By induction we choose t_{1}, …, t_{n−1} such that (β_{i}(t_{j}))_{i,j=1,...,n−1} is a nonsingular matrix. Suppose there does not exist t_{n} such that A is nonsingular, then

det((β_{i}(t_{j}))_{i,j=1,...,n}) = 0 , ∀ t_{n} ∈ X .

This is a linear dependence relation between the β_{i}, i = 1, …, n as functions in the variable t_{n}. It is a nontrivial relation because the coefficient of β_{n}(t_{n}) is ± det((β_{i}(t_{j}))_{i,j=1,...,n−1}) ≠ 0. This contradicts the linear independence of the β_{i}’s, and the proof is complete. □

Corollary 4.2. Let X be a set and S be a subset of a finite dimensional subspace of the space of complex valued functions on X. Then there exists a finite set X_{0} ⊂ X such that for any f ∈ S, if f(X_{0}) = 0 then f(X) = 0. □

Using the fact that an analytic function on a compact neighborhood Ω of 0 in ℂ^{m} is determined by its restriction to ℝ^{m} ∩ Ω one has:

Corollary 4.3. Let S be a subset of a finite dimensional subspace of the space of analytic functions on a compact neighborhood Ω of 0 in ℂ^{m}. Then there exists M > 0 such that for any f ∈ S,

(15)  sup_{t∈Ω} |f(t)| ≤ M · sup_{t∈Ω∩ℝ^{m}} |f(t)| .  □

The next proposition is the main result of this section. In the proof of Theorem 1.1, it takes care of the central torus of H and reduces the problem to the case of semisimple H.

Proposition 4.4. Let G be a connected semisimple real algebraic group and T a real algebraic torus in G. Then there exists a closed subset Y of G such that the following holds.

1. G = Z_{G}(T) · Y.

2. For any linear representation of G on ℝ^{n}, and a given neighborhood Ω of e in T, there exists c > 0 such that

sup_{t∈Ω} ∥v · ty∥ ≥ c · ∥v∥ , ∀ v ∈ ℝ^{n} and ∀ y ∈ Y .

For instance, if G = SL_{n}(ℝ) and T a diagonal subgroup containing a regular semisimple element, we get Y = NK in the theorem, where N is the lower triangular unipotent subgroup and K = SO_{n}(ℝ). Proof. Let S be a maximal complex algebraic torus in G(ℂ) containing T (ℂ). Fix a complex Borel subgroup B of G(ℂ) containing S. Consider the root system of G(ℂ) with respect to S. Consider the system of positive roots associated to the Borel B. Let Δ denote the corresponding system of simple roots. Let Ψ ⊂ Δ be the set of roots which are trivial on T. Let P denote the standard complex parabolic subgroup of G(ℂ) associated to Ψ. Let U be the unipotent radical of P. Then P = Z_{G(ℂ)}(T (ℂ))U. Put M = Z_{G}(T). Then M(ℂ) = Z_{G(ℂ)}(T (ℂ)).

Let θ be a Cartan involution of G which stabilizes T. Extend the differential of θ on the Lie algebra of G to a conjugate linear automorphism on the complexification of the Lie algebra. This lifts to a Cartan involution of G_{ℂ}, also denoted by θ, where G(ℂ) is treated as a real algebraic semisimple group. The complex conjugation on G_{ℂ}, denoted by σ, is an involution on G(ℂ). Note that the involutions θ and σ commute with each other.

Let K be the maximal compact subgroup of G(ℂ) corresponding to θ. Then

G(ℂ) = P K = M(ℂ)UK .

Note that σ and θ stabilize M(ℂ). Therefore M = {m ∈ M(ℂ) : σ(m) = m} is a symmetric subgroup of M(ℂ). Treating M(ℂ) as a real algebraic group, there exists a real algebraic ℝ-split torus A in M(ℂ) such that

θ(a) = a^{−1} = σ(a) , ∀ a ∈ A ,

and

M(ℂ) = M · A · (K ∩ M(ℂ))

(see [Sc, Prop. 7.1.2-3]). Since M(ℂ) normalizes U, we have

G(ℂ) = MAUK .

Put Y = (AUK) ∩ G. Since M ⊂ G, we have

G = M · Y .

We need to show that Y satisfies condition 2. The linear G-action on ℝ^{n} extends to a complex linear G(ℂ)-action on ℂ^{n}. Let ρ : G(ℂ) → SL_{n}(ℂ) denote the associated representation. Claim 1. Let Ω be a neighborhood of e in T. Then there exists a constant c_{1} > 0 such that for any q ∈ ℂ^{n} and u ∈ U,

sup_{t∈Ω} ∥q · (tut^{−1})∥ ≥ c_{1} · ∥q∥ .

To prove the claim, let S′ be a maximal complex algebraic torus of SL_{n}(ℂ) and B′ a Borel subgroup in SL_{n}(ℂ) containing S′ such that ρ(S) ⊂ S′ and ρ(B) ⊂ B′. The set of characters on S′ with respect to the adjoint action on 𝔰𝔩_{n}(ℂ) forms a root system. Consider the system of positive roots associated to the Borel B′. Let Δ′ be the corresponding system of simple roots. Let Ψ′ ⊂ Δ′ be the set of roots which are trivial on ρ(T). Let P′ be the standard parabolic subgroup of SL_{n}(ℂ) associated to Ψ′. Let U′ be the unipotent radical of P′. Then it is straightforward to verify that

ρ(U) ⊂ U′ .

Hence to prove the claim we can assume that u ∈ U′.

We can choose an orthonormal basis {e_{1}, …, e_{n}} of ℂ^{n} (used only for the proof of Claim 1) such that for each 1 ≤ k ≤ n,

e_{k} · S′ ⊂ ℂ-span{e_{k}} and e_{k} · B′ ⊂ ℂ-span{e_{1}, …, e_{k}} .

Note that the elements of ℂ^{n} are row vectors and SL_{n}(ℂ) acts from the right.

Let α_{ij} (1 ≤ i ≤ n, 1 ≤ j ≤ n) be the characters on T such that for any matrix X = (x_{ij}) ∈ M_{n}(ℂ) and t ∈ T, the (i, j)-th coordinate of ρ(t)Xρ(t)^{−1} is α_{ij}(t)x_{ij}. When for some i_{0} < j_{0}, the character α_{i_{0}j_{0}} = 1, then by the definition of U′, we have u_{i_{0}j_{0}} = 0 for every (u_{ij}) ∈ U′.

Let q = ∑_{i=1}^{n} q_{i}e_{i} ∈ ℂ^{n}, u = (u_{ij}) ∈ U′, and t ∈ T. Then

q · (tut^{−1}) = ∑_{i=1}^{n} y_{i}(t)e_{i} ,

where

y_{i}(t) = q_{i} + ∑_{j=i+1}^{n} α_{ij}(t)u_{ij}q_{j} .

Fix i ∈ {1, …, n}. Since distinct characters on T are linearly independent on Ω, we can choose characters β_{0} = 1, β_{1}, …, β_{k} on T, for some k, which are linearly independent on Ω and

{β_{0}, …, β_{k}} = {1} ∪ {α_{ij} : i < j} .

Moreover since u_{ij} = 0 if α_{ij} = 1, we have

y_{i}(t) = q_{i} + ∑_{l=1}^{k} β_{l}(t)z_{l} ,

where each z_{l} is obtained by summing the coefficients u_{ij}q_{j} corresponding to those α_{ij} which are same as β_{l}. Now for every i = 1, …, n, by Lemma 4.1 there exists a constant C_{i} > 0 such that

sup_{t∈Ω} |y_{i}(t)| ≥ C_{i} · |q_{i}| .

22  A. ESKIN, S. MOZES AND N. SHAH  GAFA

Therefore there exists a constant c_{1} > 0 such that

c_{1} · ∥q∥ ≤ sup_{t∈Ω} ∥q · (tut^{−1})∥ . (16)

This completes the proof of Claim 1.

Since Ω is relatively compact, there exists a constant c′ > 0 such that for any v ∈ ℂ^{n} and t ∈ Ω, ∥v∥ ≥ c′ · ∥v · t^{−1}∥ . (17)

Therefore by equations (16) and (17), for any q ∈ ℂ^{n} and u ∈ U, we have

sup_{t∈Ω} ∥q · tu∥ ≥ c_{2} · ∥q∥ , (18)

where c_{2} = c′c_{1} > 0. Claim 2. There exists a constant c_{3} > 0 such that for any v ∈ ℝ^{n} and a ∈ A, ∥v · a∥ ≥ c_{3} · ∥v∥. (19)

(Note that when T is an ℝ-split torus, A is compact and the claim is trivial.)

To prove the claim, let Λ be the finite set of characters on A such that if we define

V^{λ} = {v ∈ ℂ^{n} : v · a = v · λ(a) , ∀ a ∈ A} , ∀ λ ∈ Λ ,

then

ℂ^{n} = ⊕_{λ∈Λ}V^{λ} .

For every v ∈ ℂ^{n}, write v = ∑_{λ∈Λ} v_{λ}, where v_{λ} ∈ V^{λ} for all λ ∈ Λ. Then there exists c_{3} > 0 such that for all v ∈ ℂ^{n},

c_{3} · ∥v∥ ≤ sup_{λ∈Λ} ∥v_{λ}∥ ≤ c_{3}^{−1} · ∥v∥ . (20)

Then for every λ ∈ Λ and a ∈ A, ∥v · a∥ ≥ c_{3} · |λ(a)| · ∥v_{λ}∥ . (21)

Since σ preserves norm ∥ · ∥ on ℂ^{n} and σ(a) = a^{−1} for all a ∈ A, for any v ∈ ℝ^{n} and a ∈ A,

∥v · a^{−1}∥ = ∥v · σ(a)∥ = ∥σ(v · a)∥ = ∥v · a∥ .

Therefore due to equation (21),

∥v · a∥ ≥ c_{3} · ∥v_{λ}∥ , ∀ λ ∈ Λ .

Now by equation (20),

∥v · a∥ ≥ c_{3}^{2} · ∥v∥ .

This completes the proof of Claim 2.

Now since K is compact, there exists c_{4} > 0 such that for any v ∈ ℂ^{n} and k ∈ K,

∥v · k∥ ≥ c_{4} · ∥v∥ . (22)

Let v ∈ ℝ^{n}. Given y ∈ Y there exist a ∈ A, u ∈ U and k ∈ K, such that y = auk. Then by equations (18), (19), and (22), we get

sup_{t∈Ω} ∥v · ty∥ ≥ c_{4} · sup_{t∈Ω} ∥v · tau∥
= c_{4} · sup_{t∈Ω} ∥(v · a)tu∥
≥ c_{4}c_{2} · ∥v · a∥
≥ c_{4}c_{2}c_{3} · ∥v∥.

This completes the proof of Proposition 4.4. □

5 Returning to Compact Sets

In this section we complete the proof of Theorem 1.1.

Lemma 5.1. Let G and H ⊂ G be connected reductive real algebraic ℚgroups admitting no nontrivial ℚ-characters. Then the following conditions are equivalent:

1. (H ∩ Γ)\H is compact for any arithmetic lattice Γ ⊂ G(ℚ) in G with respect to the ℚ-structure on G.

2. Z_{G}(H) is ℚ-anisotropic; that is, it contains no nontrivial ℚ-split torus.

3. H is not contained in any proper ℚ-parabolic subgroup of G.

4. Every ℚ-subgroup of G containing H is reductive.

5. H does not normalize any nontrivial unipotent subgroup of G defined

over ℚ. Proof. 1 ⇔ 2: This equivalence is the Godement’s compactness criterion (see [B]).

2 ⇒ 3: Suppose H is contained in a proper ℚ-parabolic subgroup P. The centralizer of any Levi subgroup of P defined over ℚ contains a nontrivial ℚ-split torus. Now H being reductive and defined over ℚ, it is contained in a Levi subgroup of P defined over ℚ. Therefore Z_{G}(H) contains a nontrivial ℚ-split torus.

3 ⇒ 2: Since G admits no nontrivial ℚ-characters, the same holds for G/[G, G], where [G, G] is the commutator of G and it is semisimple. Now since G/[G, G] is a torus defined over ℚ, it does not contain a nontrivial ℚ-split torus. Suppose that Z_{G}(H) contains a nontrivial a ℚ-split torus, say T. Then T ⊂ [G, G]. Hence Z_{G}(T) is the Levi part of a proper ℚ-parabolic subgroup of G, necessarily containing H.

3 ⇒ 5: Follows from the fact that a subgroup normalizing a nontrivial unipotent subgroup defined over ℚ is contained in a proper ℚ-parabolic subgroup (see [Mo]).

Clearly 5 ⇒ 3 and 5 ⇒ 4.

4 ⇒ 5: If H normalizes a unipotent ℚ-subgroup U, then N_{G}(U) is a non-reductive ℚ-subgroup containing H. □

Proof of Theorem 1.1. We shall prove the theorem by induction on dim G.

Reduction to the case of semisimple G: Let Z be the maximal central torus in G. Then G_{1} = Z\G is a semisimple real algebraic group defined over ℚ, and the quotient homomorphism q : G → G_{1} is defined over ℚ. Now H_{1} := q(H) is a reductive ℚ-subgroup with ℚ-anisotropic centralizer in G_{1}. Also Γ_{1} := q(Γ) is an arithmetic lattice in G_{1}. Let ϕ : Γ\G → Γ_{1}\G_{1} be the natural quotient map, whose fibers are orbits of Z. By Lemma 5.1, applied to G itself in place of H there, we have that π(Z) is compact. Therefore every Z-orbit in Γ\G is compact. Hence, if the conclusion of the theorem for G_{1}, H_{1}, and Γ_{1} holds, it will also hold for G, H, and Γ. Thus if dim Z > 0, viz. if G is not semisimple, the result follows by induction on dim G. Therefore we can assume that G is semisimple.

Applying Theorem 3.9: Let 𝔥 be the Lie algebra of H, m = dim H, and {X_{1}, …, X_{m}} be a basis of 𝔥 (over ℝ). Put H_{ℂ} = H(ℂ). Define a map Θ : ℂ^{m} → H_{ℂ} as

Θ(t_{1}, …, t_{m}) = exp(t_{1}X_{1}) ⋯ exp(t_{m}X_{m}) , ∀ (t_{1}, …, t_{m}) ∈ ℂ^{m} .

By Corollary 3.2, there exist n ∈ ℕ and Λ > 0 be such that for any g ∈ G, if we define ϕ(t) = Θ(t)g (∀ t ∈ ℝ^{m}), then ϕ ∈ E_{G}(m, n, Λ).

Now to prove Theorem 1.1, it is enough to show that given any ϵ > 0 and any bounded ball J in ℝ^{m} the following holds: there exists a compact set K ⊂ Γ\G such that

|{t ∈ J : π(Θ(t))g ∈ K}| > (1 − ϵ)|J| , ∀ g ∈ G . (23)

Suppose that there does not exist such a compact set K. Take any sequence α_{i} → 0 and obtain a sequence of compact sets C_{i} ⊂ Γ\G as in the statement of Theorem 3.9. By our assumption, there exists a sequence {g_{i}} ⊂ G such that the equation (23) fails to hold for K = C_{i} and g = g_{i} for each i. Hence for each i and ϕ(t) = Θ(t)g_{i}, condition 2 in the conclusion of Theorem 3.9 fails to hold. Therefore condition 1′ in Remark 3.10 must be satisfied for each i. Therefore after passing to a subsequence, the following holds: There exist a representation of G over ℚ on a vector space V with a ℚ-structure, a point p ∈ V (ℚ) \ {0} and a sequence {γ_{i}} ⊂ Γ such that

(24)  lim_{i→∞} sup_{t∈J} ∥p · γ_{i}Θ(t)g_{i}∥ = 0 ,

where ∥ · ∥ is a fixed Euclidean norm on V (ℂ).

Reduction to the case of semisimple H: Let T be the center of H. There exists a neighborhood Ω of e in T and a nonempty ball J_{1} in ℝ^{m} centered at 0 such that

Θ(J_{1})Ω ⊂ Θ(J) . (25)

By Proposition 4.4, there exists a closed subset Y ⊂ G such that

G = Z_{G}(T) · Y (26)

and there exists c > 0 such that for any v ∈ V (ℝ) and y ∈ Y,

sup_{ω∈Ω} ∥v · ωy∥ ≥ c∥v∥. (27)

From equations (24), (25), (26), and (27) one can deduce that there exists a sequence {z_{i}} ⊂ Z_{G}(T) such that

(28)  lim_{i→∞} sup_{t∈J_{1}} ∥p · γ_{i}Θ(t)z_{i}∥ = 0 .

Since the orbit p · Γ is discrete in V (ℝ), the set p · ΓC is bounded away from 0 for any given compact set C ⊂ G. Hence there exists an i_{0} ∈ ℕ, depending on C, such that

π(Θ(J_{1})) · z_{i} ∩ π(C) = ∅ , ∀ i ≥ i_{0} . (29)

Note that G_{2} = Z_{G}(T) is a reductive real algebraic group defined over ℚ and Z_{G_{2}}(H) ⊂ Z_{G}(H) is ℚ-anisotropic. Since Z_{G_{2}}(G_{2}) ⊂ Z_{G_{2}}(H) is ℚanisotropic, G_{2} admits no nontrivial characters defined over ℚ. Hence Γ_{2} = Γ ∩ G_{2} is an arithmetic lattice in G_{2}. Let π′ : G_{2} → Γ_{2}\G_{2} be the natural quotient map. In view of the natural injection Γ_{2}\G_{2} ↪ Γ\G, we have that π′(H) = π(H). Therefore μ_{H} is the H-invariant probability measure on π′(H).

Let ϵ = (1/2) · μ_{H}(Θ(J_{1})) > 0. Then by equation (29), there exists a sequence {z_{i}} ⊂ G_{2} such that for any compact set K ⊂ Γ_{2}\G_{2},

μ_{H}(Kz_{i}^{−1}) < 1 − ϵ ∀ i ≥ i_{0} , (30)

for some i_{0} ∈ ℕ depending on K.

If dim G_{2} < dim G, this contradicts the induction hypothesis. Therefore we may assume that G_{2} = G. Since T is central in G_{2} and G is semisimple, we have that dim T = 0; in other words, H is semisimple.

Taking care of compact factors of H via complexification of Γ\G: It is straightforward to verify that the collection of entire functions of the form ℂ^{m} ∋ t ↦ ∥v · Θ(t)g∥^{2}, where v ∈ V (ℂ) and g ∈ G, is contained in a finite dimensional subspace of the space of complex valued functions on ℂ^{m}. Therefore by Corollary 4.3 and equation (24), there exists a ball D of positive radius in ℂ^{m} such that

(31)  lim_{i→∞} sup_{t∈D} ∥p · γ_{i}Θ(t)g_{i}∥ = 0 .

By restriction of scalars from ℚ[i] to ℚ, we can treat G(ℂ), H(ℂ), and V (ℂ) as real algebraic groups defined over ℚ.

Let Γ_{ℂ} be an arithmetic lattice in G(ℂ) with respect to the ℚ-structure of R_{ℚ[i]/ℚ}(G) such that Γ ∩ Γ_{ℂ} is of finite index in Γ. Hence without loss of generality we may assume that Γ ⊂ Γ_{ℂ}. Let π_{ℂ} : G(ℂ) → Γ_{ℂ}\G(ℂ) be the natural quotient map. Since p ∈ V (ℚ) and Γ_{ℂ} is arithmetic, the orbit p · Γ_{ℂ} is discrete in V (ℂ). From this and equation (31) it follows that for any compact set K ⊂ Γ_{ℂ}\G(ℂ), there exists i_{0} ∈ ℕ such that

π_{ℂ}(Θ(D))g_{i} ∩ K = ∅ , ∀ i ≥ i_{0} . (32)

Since H(ℂ) is a complex semisimple group, it is generated by unipotent one-parameter subgroups. Therefore by the Moore’s ergodicity theorem and Birkhoff ergodic theorem, there exists a unipotent one-parameter subgroup {u(t) : t ∈ ℝ} ⊂ H(ℂ) such that the orbit {π_{ℂ}(u(t)) : t > 0} is uniformly distributed with respect to μ_{H(ℂ)}, which is the H(ℂ)-invariant probability measure supported on the closed orbit π_{ℂ}(H(ℂ)). Hence there exists T_{0} ≥ 0 such that for any T ≥ T_{0},

(33)(1/T)|{t ∈ [0, T] : π_{ℂ}(u(t)) ∈ π_{ℂ}(Θ(D))}| > (1/2)μ_{H(ℂ)}(π_{ℂ}(Θ(D))) .

Put ϵ = (1/2)μ_{H(ℂ)}(π_{ℂ}(Θ(D)). For every i ∈ ℕ, put u_{i}(t) = g_{i}^{−1}u(t)g_{i} for all t ∈ ℝ. Now given any compact set K ⊂ Γ_{ℂ}\G(ℂ), by equations (32) and (33), there exists i_{0} ∈ ℕ such that

(1/T)|{t ∈ [0, T] : π_{ℂ}(g_{i})u_{i}(t) ∈ K}| < (1 − ϵ) , ∀ i ≥ i_{0} and ∀ T ≥ T_{0} . (34)

Take any sequence α_{i} → 0. For each i, we apply Theorem 1.1 of [DM2] (see Theorem 3.9 and Remark 3.11) to the ℚ-group G(ℂ), the arithmetic lattice Γ_{ℂ}, α_{i} in place of α, and the unipotent one-parameter subgroup {u_{i}(t) : t ∈ ℝ} (or ϕ(t) = g_{i}u_{i}(t)). Hence by inequality (34), condition 2 of Theorem 3.9 fails to hold. Therefore condition 1′ of Remark 3.10 must be satisfied.

Now since the collection {p_{i}·λ^{−1} : 1 ≤ i ≤ r, λ ∈ F } as in this condition 1′ is finite, after passing to a subsequence of {g_{i}}, there exists a proper ℚparabolic subgroup P of G(ℂ) and a sequence {γ_{i}} ⊂ Γ_{ℂ} such that the following holds: Let 𝔤_{ℂ} be the Lie algebra of G(ℂ) with the inherited ℚstructure. Let 𝔫 be the Lie subalgebra of 𝔤_{ℂ} associated to the unipotent radical of P and defined over ℚ. Put d = ℝ- dim 𝔫. Consider the d-th exterior of the Adjoint representation of G(ℂ) on W = ∧^{d}𝔤_{ℂ} (here 𝔤_{ℂ} is treated as a vector space defined over ℝ). Then ∧^{d}𝔫 is a one-dimensional rational subspace of W. Let q ∈ ∧^{d}𝔫 ∩ W (ℚ) \ {0}. Then

∥q · γ_{i}g_{i}u_{i}(t)∥ < α_{i} , ∀ t > 0 . (35)

Hence

q · γ_{i}g_{i}u_{i}(t) = q · γ_{i}g_{i} , ∀ i ∈ ℕ and ∀ t ∈ ℝ .

Hence

q · γ_{i} · u(t) = q · γ_{i} , ∀ i ∈ ℕ and ∀ t ∈ ℝ . (36)

Let G(ℂ)_{v} denote the stabilizer of an element v of W in G(ℂ). Note that since the orbit q · Γ_{ℂ} is discrete, the orbit π_{ℂ}(G(ℂ)_{qγ_{i}}) is closed in Γ_{ℂ}\G(ℂ). Since π({u(t) : t ∈ ℝ}) is dense in π(H(ℂ)) and u(t) ∈ G(ℂ)_{q·γ_{i}}, we have H(ℂ) ⊂ G(ℂ)_{qγ_{i}}. Now since P = N_{G(ℂ)}(𝔫),

H(ℂ) ⊂ γ_{i}^{−1}P γ_{i} , ∀ i ∈ ℕ . (37)

Reduction to the case of γ_{i} = γ_{j}, ∀ i, j: Let S be a maximal ℚ-split torus in G. From the standard construction of a Siegel domain (see [B]), we deduce that there exists a finite set F ⊂ G(ℚ) and a compact set C ⊂ G such that

G = ΓF SC . (38)

In view of equations (35) and (38), we obtain sequences {λ_{i}} ⊂ ΓF and {a_{i}} ⊂ S such that

lim_{i→∞} ∥q · γ_{i}λ_{i} · a_{i}∥ = 0 . (39)

Since S is a ℚ-split torus, there exists a finite set Φ consisting of ℚcharacters on S such that if we define

W^{ν} = {v ∈ W : v · a = v · ν(a) ∀ a ∈ S} , ∀ ν ∈ Φ ,

then W^{ν} is the set of ℝ-points of a ℚ-subspace of W and

W = ⊕_{ν∈Φ}W^{ν} .

Let

Φ^{−} = {ν ∈ Φ : lim_{i→∞} ν(a_{i}) = 0} and Φ^{0+} = Φ \ Φ^{−} .

28  A. ESKIN, S. MOZES AND N. SHAH  GAFA

Then

lim sup_{i→∞} |ν(a_{i})| > 0 , ∀ ν ∈ Φ^{0+} . (40)

Put

W^{−} = ⊕_{ν∈Φ^{−}}W^{ν} and W^{0+} = ⊕_{ν∈Φ^{0+}}W^{ν} .

Then W = W^{−} ⊕ W^{0+}. Since W^{−} and W^{0+} are ℝ-points of ℚ-subspaces, the projection π_{0+} : W → W^{0+} with kernel W^{−} is defined over ℚ.

Since F ⊂ G(ℚ) is finite and q ∈ W (ℚ), there exists k ∈ ℕ such that q · Γ_{ℂ}F ⊂ W (\frac{1}{k}ℤ) . (41) (see [B, Prop. 7.12]). Hence inf \{∥π_{0+}(v)∥ : v ∈ W (\frac{1}{k}ℤ)\W^{−}\} > 0 . (42) By equations (39), (40), (41), and (42), there exists i_{0} ∈ ℕ such that q · γ_{i_{0}}λ_{i_{0}} ⊂ W^{−} . (43)

Put q_{0} = qγ_{i_{0}} and P_{0} = γ_{i_{0}}^{−1}P γ_{i_{0}}. (Note that as a subgroup of G(ℂ), P_{0} is defined over ℚ.) Then by equation (37), H(ℂ) ⊂ P_{0} . (44)

Put b_{i} = λ_{i_{0}}a_{i}λ_{i_{0}}^{−1}, ∀ i ∈ ℕ. Since λ_{i_{0}} ∈ ΓF ⊂ G, we have {b_{i}}_{i∈ℕ} ⊂ G . (45) By equation (43) and the definition of W^{−}, we get

lim_{i→∞} ∥q_{0} · b_{i}∥ = 0 . (46)

The group P_{0} ∩ G: Since Z(H) is ℚ-anisotropic, by Lemma 5.1, H does not normalize a nontrivial unipotent ℚ-subgroup of G. By equation (44) we have H ⊂ P_{0} ∩ G. In view of Lemma 5.1, this will lead to a contradiction if we prove the following: Claim. The group P_{0} ∩G normalizes a nontrivial unipotent subgroup of G defined over ℚ.

To prove the claim let N_{0} be the unipotent radical of P_{0}, 𝔭_{0} be the Lie algebra of P_{0} and 𝔫_{0} be the Lie algebra of N_{0}. For any g ∈ P_{0},

(47) | det(Ad g|_{𝔭_{0}})| = | det(Ad(g)|_{𝔫_{0}})| .

Fix a Euclidean norm on 𝔤_{ℂ}. Now for every g ∈ G(ℂ) and a subspace E ⊂ 𝔤_{ℂ}, let J(Ad g|_{E}) denote the Jacobian of Ad g on the subspace E. Let K be a maximal compact subgroup of G(ℂ) such that Ad(K) preserves the norm on 𝔤_{ℂ}. Also we have G(ℂ) = P_{0}K. Therefore by equation (47), J(Ad g|_{𝔭_{0}}) = J(Ad g|_{𝔫_{0}}) , ∀ g ∈ G . (48)

Note that ∧^{d}𝔫_{0} is a one-dimensional subspace of W = ∧^{d}𝔤_{ℂ} containing q_{0}. Therefore (see equation (12))

J(Ad g|_{𝔫_{0}}) = ∥q_{0} · g∥/∥q_{0}∥ , ∀ g ∈ G . (49)

Let σ be the complex conjugation. Then σ commutes with Ad g for all g ∈ G. Now since the Jacobian of σ restricted to any subspace of 𝔤_{ℂ} is 1, we get

J(Ad g|_{σ(𝔫_{0})}) = J(Ad g|_{𝔫_{0}}) , ∀ g ∈ G . (50)

Now by equations (45), (46), (48), (49), and (50), we obtain that

lim_{i→∞} J(Ad b_{i}|_{𝔭_{0}}) = 0 = lim_{i→∞} J(Ad b_{i}|_{σ(𝔫_{0})}) .

Therefore

lim_{i→∞} J(Ad b_{i}|_{(𝔭_{0}+σ(𝔫_{0}))}) = 0 . (51)

Now since G is reductive, J(Ad g|_{𝔤_{ℂ}}) = 1 for all g ∈ G(ℂ). Hence we have that

𝔭_{0} ⊕ σ(𝔫_{0}) ≠ 𝔤_{ℂ} .

Therefore, since

dim 𝔭_{0} + dim σ(𝔫_{0}) = dim 𝔭_{0} + dim 𝔫_{0} = dim 𝔤_{ℂ} ,

we get

𝔭_{0} ∩ σ(𝔫_{0}) ≠ {0} . (52)

Since P_{0} ∩ σ(N_{0}) is a unipotent group normalizing N_{0}, the group

U = (P_{0} ∩ σ(N_{0})) · N_{0}

is unipotent. Note that U is normalized by P_{0} ∩ σ(P_{0}). Hence U ∩ G is a normal subgroup of P_{0} ∩ G.

Take any X_{1} ∈ 𝔭_{0} ∩ σ(𝔫_{0}). Put X = X_{1} + σ(X_{1}). Then X is in the Lie algebra of U ∩ G. If X ≠ 0, the unipotent radical of P_{0} ∩ G is nontrivial and the claim holds. Otherwise

σ(X_{1}) = −X_{1} , ∀ X_{1} ∈ 𝔭_{0} ∩ σ(𝔫_{0}) . (53)

Then 𝔭_{0} ∩ σ(𝔫_{0}) ⊂ 𝔫_{0}. Hence by equations (52) and (53), the Lie algebra σ(𝔫_{0}) ∩ 𝔫_{0} ≠ 0, and it consists of purely imaginary vectors in 𝔤_{ℂ} = 𝔤 ⊗ ℂ. Also since σ(𝔫_{0})∩𝔫_{0} is the Lie algebra of a unipotent group, it consists of adnilpotent elements. Therefore i(σ(𝔫_{0}) ∩ 𝔫_{0}) is a nonzero Lie subalgebra of 𝔤 consisting of nontrivial ad-nilpotent elements. Observe that it is Ad(P_{0} ∩ G)- invariant, and it is the set of ℝ-points of a ℚ-subspace of 𝔤. Thus the Lie group associated to i(σ(𝔫_{0}) ∩ 𝔫_{0}) is a nontrivial unipotent subgroup of G which is defined over ℚ and normalized by P_{0} ∩ G. This completes the proof of the claim, and the proof of the theorem. □

Proof of Corollary 1.4. Let V be a real subspace of ℝ^{n}. Let V^{⊥} be a complementary subspace for V. Let r = dim(V). For t > 0, let a(t) ∈ SL_{n}(ℝ) be such that

v · a(t) = \{ \begin{matrix}t^{−(n−r)}v & if v ∈ V \\ t^{r}v & if v ∈ V^{⊥} . \end{matrix} .

Let W be a real subspace of ℝ^{n}. Let d = dim(W). Then ⋀^{d} W is a one-dimensional subspace of ⋀^{d} ℝ^{n}. Let p_{W} ∈ ⋀^{d} W \ {0}.

We claim the following:

(54) p_{W} · a(t) → 0 as t → ∞ ⇔ dim(V ∩ W) > \frac{1}{n} dim(V) dim(W) .

To prove the claim, let k = dim(V ∩ W). We extend a basis {w_{1}, …, w_{k}} of V ∩ W to a basis {w_{1}, …, w_{d}} of W such that p_{W} = w_{1} ∧ ⋯ ∧ w_{d}. For k + 1 ≤ j ≤ d, we may write w_{j} = w_{j}^{−} + w_{j}^{+}, where w_{j}^{−} ∈ V, w_{j}^{+} ∈ V^{⊥}. Note that p′′_{W} := w_{1} ∧ ⋯ ∧ w_{k} ∧ w_{k+1}^{+} ∧ ⋯ ∧ w_{d}^{+} ≠ 0. Expressing p_{W} in terms of the eigenvectors of a(t) of the form w_{1} ∧ ⋯ ∧ w_{k} ∧ w_{k+1}^{±} ∧ ⋯ ∧ w_{d}^{±}, we see that as t → ∞,

p_{W} · a(t) → 0 ⇔ p′′_{W} · a(t) = t^{−(n−r)k+r(d−k)}p′′_{W} → 0 .

Thus p_{W} · a(t) → 0 as t → ∞ if and only if nk > rd; i.e. n dim(V ∩ W) > dim(V) dim(W). This completes the proof of the claim.

Now assume that W is a ℚ-subspace of ℝ^{n}. Then we can choose p_{W} to be a rational vector in ⋀^{d} ℝ^{n}. Our hypothesis that the action of H is irreducible over ℚ is equivalent to the condition that H is not contained in any proper ℚ-parabolic subgroup of G = SL_{n}(ℝ). Let Γ = SL_{n}(ℤ) and π : G → G/Γ the natural quotient map. Then by Lemma 5.1, π(Z(H)) is compact, so that Theorem 1.1 applies.

Suppose the proposition is false; then by equation (54) and its proof, p_{W} · ha(t) → 0, ∀ h ∈ H and the convergence is uniform over compact subsets of H. Let H ⊂ H be a compact set with positive Haar measure. By Theorem 1.3 there exists a compact set C ⊂ G such that G = HΓC. For each t > 0, let h_{t} ∈ H such that a(t) ⊂ h_{t}ΓC. Then p_{W} · h_{t}^{−1}a(t) ⊂ p_{W} · ΓC. Since p_{W} is a rational vector, p_{W} · Γ is discrete. Thus p_{W} · ΓC is bounded away from 0, which contradicts p_{W} · h_{t}^{−1}a(t) → 0 as t → ∞. □

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Alex Eskin  Shahar Mozes  Nimish Shah
Dept. of Mathematics Inst. of Mathematics School of Mathematics
University of Chicago Hebrew University  Tata Institute of
Chicago, IL 60637  Jerusalem 91904  Fundamental Res.
USA  Israel  Mumbai 400005, India

Emails: eskin@math.uchicago.edu, mozes@math.huji.ac.il,
nimish@math.tifr.res.in

Submitted: December 1995 Revised version: December 1996