REPRESENTATIONS OF INTEGERS BY AN INVARIANT POLYNOMIAL AND UNIPOTENT FLOWS

ALEX ESKIN AND HEE OH

1. Introduction

Let f be an integral homogeneous polynomial of degree d in n variables. A basic problem in Diophantine analytic number theory is to understand the behavior of the integral representations of integers m by f as m tends to infinity.

For each m ∈ ℕ, consider the level variety V_{m} := {x ∈ ℝ^{n} : f(x) = m}. For instance, if f(x_{1}, ⋯, x_{n}) = x_{1}^{2} + ⋯ + x_{n}^{2} (n ≥ 3), then V_{m} is the sphere of radius \sqrt{m} centered at the origin and the set V_{m}(ℤ) = V_{m} ∩ ℤ^{n} consists of integral vectors the sum of whose squares is equal to m. In this case, the asymptotic of #V_{m}(ℤ) is well known. For n ≥ 5 the classical Hardy-Littlewood circle method applies and for n = 4 the Kloosterman sum method works. For n = 3, Linnik gave a conditional answer and later Iwaniec gave a complete answer (see [Sa], [Iw]).

In the case when V_{m} is non-compact, the number #V_{m}(ℤ) may be infinite. In this case one asks if there exists an asymptotic density for V_{m}(ℤ) as m → ∞ . To be more precise, for a compact subset Ω of V_{1}, set

(1.1)  N_{m}(f, Ω) := #V_{m}(ℤ) ∩ m^{1/d}Ω.

Or equivalently,

N_{m}(f, Ω) = #V_{m}(ℤ) ∩ ℝ^{+}Ω

where ℝ^{+}Ω is the radial cone {x ∈ V : tx ∈ Ω for some t ∈ ℝ^{+}}. The question we study in this paper is if there exists a sequence of numbers ω_{m}, independent of the compact subset Ω, such that

N_{m}(f, Ω) ∼_{m→∞} ω_{m} · vol(Ω).

This formulation is basically due to Linnik ([Li2], see also [Sa]).

The only known general method is the Hardy-Littlewood circle method. However for this method to work, the number of variables needs to be much larger than the degree of the polynomial in general.

In this paper, we focus on the polynomials that are invariant under an action of a semisimple real algebraic group. In such cases, the level varieties admit actions of a semisimple algebraic group and the dynamics of such groups plays a crucial role in understanding this question. In particular, when V_{1} is a homogeneous space of a semisimple real algebraic group with the stabilizer being generated by unipotent flows, we are able to use a well developed theory of unipotent flows on a homogeneous space of a Lie group.

The first author is partially supported by Packard foundation.

The second author partially supported by NSF grants DMS 0070544 and DMS 0333397.

We formulate our main results for a family of varieties V_{m} which are more general than level sets of a given polynomial. Let V be a finite dimensional real vector space with a ℤstructure. Let G be a semisimple real algebraic group defined over ℚ and let ρ : G → GL(V) be a ℚ-rational representation of G. Fix a non-zero vector v_{0} ∈ V (ℤ) such that v_{0}G is Zariski closed and a sequence {λ_{m} ∈ ℝ^{+} : m ∈ ℕ} of strictly increasing numbers. We set

V_{m} := λ_{m}v_{0}G for each m ∈ ℕ.

Denote by A_{V} be the collection of all arithmetic subgroups Γ ⊂ G(ℚ) of G such that V (ℤ)Γ ⊂ V (ℤ). Since ρ is defined over ℚ, A_{V} is non-empty. Consider a sequence {O_{m} ⊂ V_{m}(ℤ) : m ∈ ℕ} of Γ-invariant subsets of V_{m}(ℤ) for some Γ ∈ A_{V}. For a compact subset Ω of V_{1}, we define

N(O_{m}, Ω) := # pr(O_{m}) ∩ Ω

where pr : V_{m} → V_{1} denotes the radial projection given by pr(x) = λ_{1}λ_{m}^{−1}x.

Let H denote the stabilizer of v_{0} in G. The notation H^{0} and G^{0} denote the identity components of H and G respectively. In the rest of the introduction, we assume that H is semisimple without compact factors.

Theorem 1.2. Suppose that H^{0} is a maximal connected closed subgroup of G^{0}. Let {O_{m} ⊂ V_{m}(ℤ)} be a sequence of Γ-invariant subsets for some Γ ∈ A_{V}, for example, O_{m} = V_{m}(ℤ). Suppose that for each m_{0} ∈ ℕ,

(1.3)  #{m ∈ ℕ : pr(O_{m_{0}}) = pr(O_{m})} < ∞.

Fix a G-invariant Borel measure μ on V_{1}. Then for any compact subset Ω of V_{1} with boundary of measure 0,

(1.4)  N(O_{m}, Ω) ∼_{m→∞} ω(O_{m}) · μ(Ω)

where ω(O_{m}) is given below (1.6).

An immediate consequence of the above theorem is that the sequence {pr(O_{m})} of projections is equidistributed on V_{1} as m → ∞, since it follows from (1.4) that for any two compact subsets Ω_{1} and Ω_{2} of V_{1} with boundary measure 0,

\frac{N_{m}(O_{m}, Ω_{1})}{N_{m}(O_{m}, Ω_{2})} ∼_{m→∞} \frac{μ(Ω_{1})}{μ(Ω_{2})}.

The condition (1.3) says that there is no infinite sequence of varieties V_{m_{i}} where all the integer points in O_{m_{i}} are simply the radial dilations of a fixed O_{m_{1}}. Clearly this is a necessary condition for the conclusion of the above theorem. We remark that the condition (1.3) is also equivalent to saying lim_{m→∞} ω(O_{m}) = ∞ (see Lemma 4.3).

In order to give the formula of the number ω(O_{m}) in Theorem 1.2, let μ_{G} and μ_{H} denote Haar measures on G and H respectively so that the triple (μ, μ_{G}, μ_{H}) are topologically compatible in the sense of Weil [We], namely, for any continuous function f on G with compact support,

(1.5)  ∫_{G} f dμ_{G}(g) = ∫_{H\G} dμ(ḡ) ∫_{H} f(hg)dμ_{H}(h)

where μ, by slight abuse of notation, denotes the G-invariant measure on H\G pulled back from μ on V_{1} via the identification of H\G with V_{1} by Hg ↦ v_{0}g.

The number ω(O_{m}) is defined as follows: for any Γ ∈ A_{V} preserving each O_{m},

(1.6)  ω(O_{m}) := \frac{∑_{ξΓ⊂O_{m}} μ_{H} ((H ∩ g_{ξ}^{−1}Γg_{ξ})\H)}{μ_{G}(Γ\G)},

where the sum is taken over the set of disjoint Γ-orbits in O_{m} and g_{ξ} ∈ G is an element such that v_{0} = pr(ξ)g_{ξ}. Under our assumption, ω(O_{m}) < ∞, and moreover ω(O_{m}) does not depend on the choice of Γ in A_{V} preserving O_{m}, justifying our notation. This is shown in [Oh] for O_{m} = V_{m}(ℤ) but the same proof works for any O_{m}. For the case of O_{m} = V_{m}(ℤ), we simply write

(1.7)  ω_{m} = ω(V_{m}(ℤ)).

Note also that the product ω(O_{m})·μ(Ω) is independent of the choice of compatible measures (μ, μ_{H}, μ_{G}). For this reason, we sometimes write ω(O_{m}) · vol(Ω) in what follows.

We remark that Theorem 1.2 applies to the cases where V_{1} is an affine symmetric space, i.e., H is the set of fixed points of an involution of G, with additional assumptions that G is ℚ-simple and that H is semisimple without compact factors. It is so since H^{0} is then a maximal connected closed subgroup of G^{0} (cf. [Bo, Lemma 8.0]).

We present some examples which follow from Theorem 1.2 for the case O_{m} = V_{m}(ℤ). See (1.1) for the notation N_{m}(f, Ω). The following three theorems are proven in section 6 where we realize each f as an invariant polynomial of a certain representation of a semisimple algebraic group. Once we do that, the number ω_{m} is defined as in (1.6) and (1.7).

Theorem 1.8. Let n ≥ 2 and Det denote the determinant polynomial on the vector space M_{n}(ℝ) of n × n matrices. For any compact subset Ω of V_{1} = {X ∈ M_{n}(ℝ) : Det(X) = 1} of boundary of measure 0, we have

N_{m}(Det, Ω) ∼_{m→∞} ω_{m} · vol (Ω).

Theorem 1.8 was proven first by Linnik [Li1] (for n = 2) and by Linnik and Skubenko (for n ≥ 3) [LS] in the early sixties using methods in analytic number theory. A different proof was also given using methods based on Hecke operators (see [Sa], [COU] and [GO]).

Let V be the subspace of M_{2n}(ℝ) consisting of the skew-symmetric matrices, i.e., V = {X ∈ M_{2n}(ℝ) : X^{t} = −X} (n ≥ 2) and consider the Pfaffian as a polynomial on V defined by Pff(X)^{2} = Det(X) and

Pff (\begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix}) = 1

where I_{n} denotes the identity matrix of order n.

Theorem 1.9. Let n ≥ 2. For any compact subset Ω of V_{1} = {X ∈ V : Pff(X) = 1} of boundary of measure 0, we have

N_{m}(Pff, Ω) ∼_{m→∞} ω_{m} · vol (Ω).

An integer m is called a fundamental discriminant if and only if m is either a square free integer congruent to 1 mod 4, or 4 times of a square free integer which is congruent to 2 or 3 mod 4.

Theorem 1.10. Let Q be an integral quadratic form of signature (r, s) where r + s ≥ 4, r ≥ 2 and s ≥ 1. For any compact subset Ω of V_{1} = {X ∈ ℝ^{r+s} : Q(X) = 1} of boundary of measure 0,

N_{m}(Q, Ω) ∼ ω_{m} · vol (Ω)

as m → ∞ along the fundamental discriminants.

In fact, the Hardy-Littlewood circle method together with the Kloosterman sum method (needed for r +s = 4) (cf. [Va], [Es]) also gives an asymptotic density in Theorem 1.10, in the form of product of local densities. By comparing the two different forms of the asymptotic for N_{m}(Q, Ω), one obtains a new proof of Siegel mass formula for quadratic forms (see [Oh]) in the same spirit of the work of Eskin, Rudnick and Sarnak [ERS]. The above theorem is still true for the case of r + s = 3 and rs > 0 though our method does not apply. This follows from a theorem of Duke [Du].

More generally, a natural question is whether the asymptotic ω_{m} · vol(Ω) in Theorem 1.2 coincides with the heuristics predicted by the Hardy-Littlewood circle method. It is shown in [Oh] that this is true in many cases but not always.

To state our main result in a more general setting without the maximality assumption on H, we define the following: the notation Z(H^{0}) denotes the centralizer of H^{0} in G.

Definition 1.11. A sequence {O_{m} ⊂ V_{m}(ℤ)} is called focused if, for every Γ ∈ A_{V} preserving {O_{m}}, there exist a proper connected closed subgroup L of G^{0} in which L ∩ Γ is a Zariski dense lattice, g ∈ G with g^{−1}H^{0}g ⊂ L and a compact subset C ⊂ v_{0}Z(H^{0})gL such that for every compact subset Ω ⊂ V_{1}, there exists γ_{Ω} ∈ Γ with

(1.12)  lim sup_{m→∞} \frac{∑_{ξΓ⊂O_{m}}\{ω(ξΓ) : pr(ξΓ) ∩ Ω ⊂ C(L ∩ Γ)γ_{Ω}\}}{ω(O_{m})} > 0.

In the above and also in the rest of the paper, the notation ∑_{ξΓ⊂O_{m}} means that the sum is taken over the set of disjoint Γ-orbits contained in O_{m}.

Theorem 1.13. Suppose that {O_{m} : m ∈ ℕ} is not focused. Then for any compact subset Ω of V_{1} with boundary of measure 0,

N(O_{m}, Ω) ∼_{m→∞} ω(O_{m}) · vol(Ω).

Note that in the case when O_{m} consists of one Γ-orbit, the focusing of {O_{m}} implies that for every given compact subset Ω of V_{1}, there exists an infinite sequence m_{i} such that pr(O_{m_{i}}) ∩ Ω lies completely inside a proper subvariety of V_{1}. In general, if {O_{m}} is focused, one expects in view of Theorem 1.13 that a positive proportion of points in pr(O_{m_{i}}) ∩ Ω lies inside a proper subvariety of V_{1} of the form v_{0}Z(H)gLγ for some γ ∈ Γ and for some proper subgroup L of G^{0}.

Assuming H connected for simplicity, we explain some of schemes in the proofs. For each ξΓ ⊂ O_{m}, consider the H-invariant measure ν_{ξΓ} on Γ\G supported on the closed orbit

Γ\Γg_{ξ}H with the total measure ω(ξΓ). Let σ_{m} denote the averaging measure ^{\frac{∑_{ξΓ⊂Om} ν_{ξΓ}}{ω(O_{m})}} on Γ\G. We first observe that the equidistribution of pr(O_{m}) (Theorem 1.13) follows if σ_{m} converges to the G-invariant probability measure on Γ\G as m → ∞ (Proposition 2.2). We then show that under the non-focusing assumption of {O_{m}}, the H-orbits Γ\Γg_{ξ}H which stay outside a given compact subset C as well as the H-orbits Γ\Γg_{ξ}H which stay inside Γ\ΓLg_{ξ} for some proper subgroup L of G are ignorable, in considering the weak limits of σ_{m}. Then by applying main ergodic results of Dani and Margulis in [DM1-2], we show that each H-orbit Γ\Γg_{ξ}H is getting longer and longer and moreover uniformly distributed on Γ\G as m → ∞; hence the average measure σ_{m} tends to the Haar measure on Γ\G as m → ∞.

In particular in the case where the subgroup H is a maximal closed subgroup of G^{0}, the equidistribution of {pr(O_{m})} is a consequence of the phenomenon that for any sequence of non-repeated individual Γ-orbits {pr(ξ)Γ ⊂ pr(O_{m})}, the corresponding sequence {Γ\Γg_{ξ}H} of H-orbits is uniformly distributed on Γ\G and hence pr(ξ)Γ is equidistributed on V_{1} as m → ∞. Remark Gan and Oh [GO] showed that for any invariant polynomial f in the above setting but with a more general H (not necessarily semisimple), if V_{d_{0}}(ℤ) contains at least one integer point for some d_{0} ∈ ℕ, then there exist explicitly computable constants c and r depending only on G, ρ and deg (f) such that pr(V_{cd_{0}m^{r}}(ℤ)) becomes dense on V_{1} in a strong sense as m → ∞. In particular it follows that the limit supreme in (1.12) is strictly less than 1. Acknowledgment The second named author would like to thank Peter Sarnak for suggesting this problem as well as for helpful discussions.

2. Measure theoretic formulation of a counting problem

Let G be a real semisimple algebraic group defined over ℚ. This means that there exists a connected semisimple algebraic group \underline{G} defined over ℚ such that G is a closed subgroup of \underline{G}(ℝ) containing the identity component \underline{G}(ℝ)^{∘}. Let ρ : G → GL(V) be a ℚ-rational representation for a finite dimensional real vector space V defined over ℤ. Fix a non-zero vector v_{0} ∈ V such that v_{0}G is Zariski closed and a sequence {λ_{m} : m ∈ ℕ} of strictly increasing positive numbers. For each m ∈ ℕ, we set

V_{m} = λ_{m}v_{0}G.

Let H denote the stabilizer of v_{0} in G. Since ρ is rational, H is a real algebraic subgroup of G. In particular, the identity component H^{0} is a finite index normal subgroup of H. Assume that H^{0} has no non-trivial ℝ-character. Let μ_{G} denote a Haar measure on G. Since both G and H are unimodular, we may choose a G-invariant Borel measure μ on V_{1} and an H-invariant measure μ_{H} on H so that the triple (μ, μ_{G}, μ_{H}) is compatible in the sense of (1.5).

Let A_{V} be defined as in the introduction, and let Γ ∈ A_{V}. Since G acts transitively on V_{1}, for any ξ ∈ V_{m}, there exists g_{ξ} ∈ G such that v_{0} = pr(ξ)g_{ξ}. The choice of g_{ξ} is unique only up to modulo H. If H_{ξ} denotes the stabilizer of ξ in G for ξ ∈ V_{m}(ℤ), then H_{ξ} is a ℚ-subgroup of G and H_{ξ} = g_{ξ}Hg_{ξ}^{−1}. Therefore H_{ξ}^{0} has no non-trivial ℚ-character and hence by a theorem of Borel and Harish-Chandra, H_{ξ} ∩ Γ is a lattice in H_{ξ}. Hence

ω(ξΓ) := \frac{μ_{H}(H ∩ g_{ξ}^{−1}Γg_{ξ}\H)}{μ_{G}(Γ\G)} < ∞.

Observe that the definition of ω(ξΓ) depends only on the Γ-orbit not on its representative.

Let {O_{m} ⊂ V_{m}(ℤ)} be a sequence of non-empty Γ-invariant subsets of V_{m}(ℤ) for some Γ ∈ A_{V}. Since each V_{m} is Zariski closed, by a theorem of Borel and Harish-Chandra, the number of Γ-orbits in V_{m}(ℤ) and hence in O_{m} is finite. Hence

ω(O_{m}) := ∑_{ξΓ⊂O_{m}} ω(ξΓ) < ∞.

Lemma 2.1 (Oh). The number ω(O_{m}) is independent of the choice of Γ ∈ A_{V} preserving O_{m}.

The space P(Γ\G) of the probability measures on Γ\G is equipped with the weak^{∗}- topology. Now fix any Γ ∈ A_{V} which preserves each O_{m}. For each Γ-orbit ξΓ ⊂ O_{m}, let ν_{ξΓ} denote the unique H-invariant measure on Γ\G supported on the closed orbit Γ\Γg_{ξ}H and with the total measure given by ω(ξΓ). Hence

\frac{1}{ω(O_{m})} ∑_{ξΓ⊂O_{m}} ν_{ξΓ} ∈ P(Γ\G).

Here is a main proposition suggested by Sarnak which translates the counting problem to the question of whether the weak-limits of the above measures are G-invariant.

Proposition 2.2. If

lim_{m→∞} \frac{1}{ω(O_{m})} ∑_{ξΓ⊂O_{m}} ν_{ξΓ} = \frac{1}{μ_{G}(Γ\G)}μ_{G} in P(Γ\G)

then for any compact subset Ω of V_{1} with boundary of measure 0,

N(O_{m}, Ω) ∼_{m→∞} ω(O_{m}) · μ(Ω).

Proof. Without loss of generality, we assume μ_{G}(Γ\G) = 1. Let ϕ be any continuous function with compact support on H\G = V_{1}. Define a function F_{ϕ}^{m} as follows: for each g ∈ G

F_{ϕ}^{m}(g) := \frac{1}{ω(O_{m})} ∑_{ξΓ⊂O_{m}} ∑_{γ∈(H_{ξ}∩Γ)\Γ} ϕ(pr(ξ)γg).

Since F_{ϕ}^{m} is left Γ-invariant, it may be considered as a function on Γ\G. Let ψ be a continuous function on Γ\G with compact support. Note that

ω(O_{m}) · ⟨F_{ϕ}^{m}, ψ⟩ = ∑_{ξΓ⊂O_{m}} ∫_{Γ\G} ( ∑_{γ∈(H_{ξ}∩Γ)\Γ} ϕ(pr(ξ)γg)ψ(g)) dμ_{G}(g)

= ∑_{ξΓ⊂O_{m}} ∫_{g∈(H_{ξ}∩Γ)\G} ϕ(pr(ξ)g)ψ(g) dμ_{G}(g)

= ∑_{ξΓ⊂O_{m}} ∫_{t∈(H∩g_{ξ}^{−1}Γg_{ξ})\G} ϕ(pr(ξ)g_{ξ}t) ψ(g_{ξ}t) dμ_{G}(t)

= ∑_{ξΓ⊂O_{m}} ∫_{g∈H\G} ϕ(v_{0}g) (∫_{h∈(H∩g_{ξ}^{−1}Γg_{ξ})\H} ψ(g_{ξ}hg)dμ_{H}(h)) dμ(g)

= ∑_{ξΓ⊂O_{m}} ∫_{g∈H\G} ϕ(v_{0}g) (∫_{s∈Γ\G} ψ(sg) dν_{ξ}(s)) dμ(g)

Consider a function ψ_{g} on Γ\G defined by ψ_{g}(s) := ψ(sg). Then

∫_{Γ\G} ψ_{g} dμ_{G} = ∫_{Γ\G} ψ dμ_{G}.

Hence by the assumption,

lim_{m→∞} \frac{1}{ω(O_{m})} ∑_{ξΓ⊂O_{m}} ∫_{s∈Γ\G} ψ(sg) dν_{ξ}(s) = ∫_{Γ\G} ψ dμ_{G}.

Now by the Lebesgue dominated convergence theorem,

lim_{m→∞}⟨F_{ϕ}^{m}, ψ⟩ = ∫_{H\G} ϕ(v_{0}g)dμ(g) · ∫_{Γ\G} ψ dμ_{G}.

If follows that

(2.3)  lim_{m→∞}⟨F_{χ_{Ω}}^{m}, ψ⟩ = μ(Ω) · ∫_{Γ\G} ψ dμ_{G}

where χ_{Ω} denotes the characteristic function of Ω.

Fix ϵ > 0. Let U_{ϵ} be a symmetric neighborhood of e in G such that

μ(Ω_{ϵ}^{+} − Ω_{ϵ}^{−}) ≤ ϵ

where Ω_{ϵ}^{+} = ∪_{u∈U_{ϵ}}Ωu and Ω_{ϵ}^{−} = ∩_{u∈U_{ϵ}}Ωu. Then for all g ∈ U_{ϵ},

(2.4)  F_{χ_{Ω_{ϵ−}}}^{m} (g) ≤ F_{χ_{Ω}}^{m}(e) ≤ F_{χ_{Ω_{ϵ+}}}^{m} (g).

Let ψ_{ϵ} be a non-negative continuous function on Γ\G with support in U_{ϵ} and ∫_{Γ\G} ψ_{ϵ} dμ_{G} = 1. Integrating (2.4) against ψ_{ϵ} now gives

⟨F_{χ_{Ω_{ϵ−}}}^{m} , ψ_{ϵ}⟩ ≤ F_{χ_{Ω}}^{m}(e) ≤ ⟨F_{χ_{Ω_{ϵ+}}}^{m} , ψ_{ϵ}⟩.

Since both sides tend to μ(Ω_{ϵ±}) respectively as m → ∞ by (2.3) and ϵ > 0 is arbitrary, we have

F_{χ_{Ω}}^{m}(e) → μ(Ω) as m → ∞.

Since

F_{χ_{Ω}}^{m}(e) = \frac{N(O_{m}, Ω)}{ω(O_{m})},

this proves the claim. □

3. Asymptotic behavior of unipotent flows

We recall the following fundamental result of Dani and Margulis.

Theorem 3.1 (DM2, Theorem 6.1). Let G be a connected Lie group and Γ a lattice in G. Given a compact subset C ⊂ Γ\G and an ϵ > 0, there exists a compact set K ⊂ Γ\G such that the following holds: for any x ∈ C, any unipotent one-parameter subgroup {u(t)} of G, and any T > 0,

|{t ∈ [0, T] : xu(t) ∈ K}| > (1 − ϵ)T

where | · | denotes the Lebesgue measure on ℝ.

Let G be a connected semisimple real algebraic group defined over ℚ, and H a connected semisimple real algebraic subgroup of G. Let Γ ⊂ G(ℚ) be an arithmetic subgroup of G. The results in this section have meanings only when Γ\G is non-compact, which we assume. Consider the one point compactification Γ\G ∪ {∞} of Γ\G. The space P(Γ\G ∪ {∞}) of the probability measures on Γ\G ∪ {∞} equipped with the weak^{∗}-topology is weak^{∗} compact.

Let {g_{m} ∈ G} be a sequence such that g_{m}Hg_{m}^{−1} is a ℚ-subgroup of G for each m. By a theorem of Borel and Harish-Chandra [BH], it follows that g_{m}^{−1}Γg_{m} ∩ H is a lattice in H. Hence each Γ\Γg_{m}H is closed in Γ\G (cf. [Rag]) and there exists the unique H-invariant probability measure μ_{m} in Γ\G supported on Γ\Γg_{m}H.

Proposition 3.2. Assume either that H has no compact factors or that g_{m}Hg_{m}^{−1} is ℚ-simple for each m. Then the following are equivalent:

(1) There exists a compact subset C of Γ\G such that

Γ\Γg_{m}H ∩ C ≠ ∅ for all sufficiently large m ∈ ℕ.

(2) Every weak limit of {μ_{m}} in P(Γ\G ∪ {∞}) is supported on Γ\G.

Proof. Assume that (1) is true. Without loss of generality, we may assume that Γ\Γg_{m} ∈ C for all m ∈ ℕ. Let H_{N} denote the unique maximal connected normal closed subgroup of H without compact factors. Let U = {u(t)} be a unipotent one parameter subgroup in H_{N} not contained in any proper normal subgroup of H_{N}. Such a subgroup exists (see for example, [MS, Lemma 2.3]). Under our assumption, either H = H_{N} or g_{m}^{−1}Γg_{m} ∩ H is an irreducible lattice in H. Hence it follows from Moore’s ergodicity theorem (cf. Theorem 2.1 in [BM]) that U acts ergodically with respect to each μ_{m}. Moreover by the Birkhoff ergodic theorem (cf. [BM]), for almost all h ∈ H, Γ\Γg_{m}hu(t) is uniformly distributed on Γ\G with respect to μ_{m}, i.e., for any f ∈ C_{c}(Γ\G),

lim_{T →∞} \frac{1}{T} ∫_{0}^{T} f(Γ\Γg_{m}hu(t)) dt = ∫ f dμ_{m}.

Therefore we may assume that for each m ∈ ℕ, there exists h_{m} ∈ H such that Γ\Γg_{m}h_{m} ∈ C and Γg_{m}h_{m}u(t) is uniformly distributed on Γ\G with respect to μ_{m}. For any given ϵ > 0, let K be a compact subset of Γ\G as in Theorem 3.1 with respect to C. Then for each m ∈ ℕ,

μ_{m}(K) ≥ 1 − ϵ.

Therefore μ(K) ≥ 1 − ϵ for any weak limit μ of {μ_{m}}. Since ϵ > 0 is arbitrary, we have μ(Γ\G) = 1, proving that (2) holds.

Now suppose that (1) fails. First write Γ\G as ∪_{i=1}^{∞}C_{i} where C_{i} are compact subsets such that C_{i} ⊂ C_{i+1} for all i. Then for each i, there exists m_{i} such that

Γ\Γg_{m_{i}}H ∩ C_{i} = ∅.

Since C_{i} is increasing, we have

(3.3)  Γ\Γg_{m_{j}}H ∩ C_{i} = ∅ for all j ≥ i.

This implies that any weak limit of {μ_{m_{i}}} cannot be supported on Γ\G, for if so, then for some i_{0}, μ_{m_{i}}(C_{i_{0}}) > 1/2 for infinitely many i. This is contradiction to (3.3). Hence (2) implies (1). □

Let {𝔒_{m} ⊂ H\G} be a sequence of a finitely many union of Γ-orbits. For each H\Hg ∈ 𝔒_{m}, we assume that gHg^{−1} is a ℚ-subgroup of G. For each Γ-orbit ηΓ ⊂ 𝔒_{m}, set

ω(ηΓ) := \frac{μ_{H}((H ∩ g_{η}^{−1}Γg_{η})\H)}{μ_{G}(Γ\G)} and ω(𝔒_{m}) := ∑_{ηΓ⊂𝔒_{m}} ω(ηΓ).

Here g_{η} ∈ G is such that η = H\Hg_{η}.

Let ν_{ηΓ} denote the H-invariant measure on Γ\G supported on Γ\Γg_{η}H with the total measure given by ω(ηΓ). Define an H-invariant probability measure σ_{m} on Γ\G:

σ_{m} = \frac{1}{ω(𝔒_{m})} ∑_{ηΓ⊂𝔒_{m}} ν_{ηΓ}.

The notation H_{N} denotes the unique maximal connected normal closed subgroup of H without compact factors.

Proposition 3.4. Assume either that H has no compact factors or that g_{η}Hg_{η}^{−1} is ℚ-simple for any η ∈ 𝔒_{m}. Suppose that g_{η}H_{N}g_{η}^{−1} is not contained in any proper ℚ-parabolic subgroup of G for any η ∈ ∪_{m}𝔒_{m}. Then any weak limit of {σ_{m} : m ∈ ℕ} in P(Γ\G∪{∞}) is supported in Γ\G.

Proof. Without loss of generality we may assume that {σ_{m}} converges in P(Γ\G ∪ {∞}). It suffices to show that, for any ϵ > 0, there exists a compact subset K ⊂ Γ\G such that

σ_{m}(K) > 1 − ϵ for all sufficiently large m.

Assume not; then for any compact subset K ⊂ Γ\G, (after going to a subsequence) there exists η_{m} ∈ 𝔒_{m} such that

ν_{η_{m}Γ}(K) < (1 − ϵ)ω(η_{m}Γ) for each m ∈ ℕ.

Let U = {u(t)} be a unipotent one-parameter subgroup of H_{N} as in the proof of previous proposition. Let R be the set of h ∈ H such that Γ\Γg_{η_{m}}hu(t) is uniformly distributed in Γ\Γg_{η_{m}}H with respect to the probability measure \frac{1}{ω(η_{m}Γ)}ν_{η_{m}Γ}. Then R has the full measure in H (see the proof of the previous proposition). Fix any h ∈ R. Then for each m ∈ ℕ, there exists T_{m} > 0 (depending on h) such that

\frac{1}{T} |\{t ∈ [0, T] : Γ\Γg_{η_{m}}hu(t) ∈ K\}| < 1 − ϵ/2

for all T > T_{m} where | · | denotes the Lebesgue measure on ℝ. Applying a theorem of Dani and Margulis [DM1, Theorem 2] (see also [EMS1]), we obtain that for any given α_{m} > 0 with lim_{m→∞} α_{m} = 0, after passing to a subsequence, there exist a proper parabolic ℚsubgroup P of G, a non-zero vector q ∈ ∧^{k} Lie(W)(ℚ) (W being the unipotent radical of P and k = dim(W)) and a sequence {γ_{m}(h) ∈ Γ} such that for all m ∈ ℕ and t > 0,

∥q.γ_{m}(h)g_{η_{m}}hu(t)∥ < α_{m}

where the action is through the k-th exterior of the adjoint representation of G on ∧^{k} Lie(G). Since u(t) acts as a unipotent one-parameter subgroup on ∧^{n} Lie(G) and any orbit of a unipotent one-parameter subgroup is unbounded except for a fixed point, it follows that for all 0 ≤ t < ∞ and for all m ∈ ℕ,

q.γ_{m}^{h}g_{η_{m}}hu(t) = q.γ_{m}^{h}g_{η_{m}}h.

Hence

U ⊂ (γ_{m}^{h}g_{η_{m}}h)^{−1}P (γ_{m}^{h}g_{η_{m}}h),

since the latter group contains the stabilizer of the vector q.γ_{m}^{h}g_{η_{m}}h. Hence we have shown that for almost all h ∈ H and for any m ∈ ℕ,

hUh^{−1} ⊂ (γ_{m}^{h}g_{η_{m}})^{−1}P (γ_{m}^{h}g_{η_{m}})

for some γ_{m}^{h} ∈ Γ. Since Γ is countable, it follows that for each m ∈ ℕ, there exist an element γ_{m} ∈ Γ and a subset S_{m} ⊂ H of positive measure such that

hUh^{−1} ⊂ (γ_{m}g_{η_{m}})^{−1}P (γ_{m}g_{η_{m}}) for all h ∈ S_{m}.

Since the set

{h ∈ H : hUh^{−1} ⊂ (γ_{m}g_{η_{m}})^{−1}P (γ_{m}g_{η_{m}})}

is a real analytic submanifold of H with a positive measure, it is indeed equal to H. Hence

hUh^{−1} ⊂ (γ_{m}g_{η_{m}})^{−1}P (γ_{m}g_{η_{m}}) for all h ∈ H.

Since U is not contained in any proper normal subgroup of H_{N}, it follows that

g_{η_{m}}H_{N}g_{η_{m}}^{−1} ⊂ γ_{m}^{−1}P γ_{m}.

This contradicts the assumption since γ_{m}^{−1}P γ_{m} is a proper parabolic ℚ-subgroup of G. □

4. Projections of O_{m} and stabilizer subgroups

We recall the following theorem of Dani and Margulis: let G be any connected Lie group and Γ a discrete subgroup of G. We fix a left invariant Riemannian metric on G. Let M be any closed subgroup of G such that M ∩ Γ is a lattice in M. Then Γ\ΓM is a closed Riemannian submanifold of Γ\G and hence it has a right M-invariant Riemannian volume form, denoted by V, induced by the Riemannian metric.

Theorem 4.1 (DM2, Theorem 5.1). For any c > 0, let W_{c} be the collection of all closed connected subgroups of G such that Γ\ΓM is closed in Γ\G and V((M ∩ Γ)\M) ≤ c. Then there are only finitely many subgroups of the form M ∩ Γ with M ∈ W_{c}.

We also need the following simple consequence of a theorem of Kazhdan and Margulis ([KM], [Ra, Theorem 11.8]):

Lemma 4.2. Let G be a connected linear semisimple Lie group without compact factors. There exists a constant c > 0 such that for any discrete subgroup Γ of G, the co-volume of Γ in G (with respect to a fixed Haar measure on G) is at least c.

We keep the same notation from section 2 for G, ρ : G → GL(V), Γ, H, H_{ξ}, v_{0}, etc. Let O_{m} be a Γ-invariant subset of V_{m}(ℤ) for each m. We assume that H is semisimple without compact factors. Denote by N(H) the normalizer of H in G.

Lemma 4.3. Assume that [N(H) : H] < ∞ and that for each ξ ∈ O_{m}, H_{ξ}^{0} is not contained in any proper ℚ-parabolic subgroup of G. Then the following are equivalent:

(1) Suppose that for each m_{0} ∈ ℕ,

#{m ∈ ℕ : pr(O_{m_{0}}) = pr(O_{m})} < ∞.

(2) lim_{m→∞} ω(O_{m}) = ∞.

Proof. It is easy to see that pr(O_{m}) = pr(O_{k}) implies ω(O_{m}) = ω(O_{k}). Hence (2) implies (1). Assume now that (2) fails. Then by passing to a subsequence we may assume that ω(O_{m}) is uniformly bounded. By Lemma 4.2 which we may apply since H^{0} has finite index in H, there exists some c > 0 such that

ω(ξΓ) = \frac{μ_{H}((H ∩ g_{ξ}^{−1}Γg_{ξ})\H)}{μ_{G}(Γ\G)} > c for all ξ ∈ ∪_{m}O_{m}.

Since ω(O_{m}) ≥ h_{m} ·c where h_{m} is the number of disjoint Γ-orbits in O_{m}, we may also assume that h_{m} is constant, say r, for all m, by passing to a subsequence. Now write O_{m} = ∪_{i=1}^{r}ξ_{m_{i}}Γ. It suffices to show that for each 1 ≤ i ≤ r, pr(ξ_{m_{i}}Γ) is the same set for infinitely many m. For, this implies that pr(O_{m}) is the same set for infinitely many m, which contradicts (1). Fix 1 ≤ i ≤ r and set ξ_{m_{i}} = ξ_{m} for simplicity.

It follows from Propositions 3.2 and 3.4 that there exists a compact subset C of Γ\G such that Γ\Γg_{ξ_{m}}H ∩ C ≠ ∅ for all m ∈ ℕ. Hence we may choose g_{ξ_{m}} so that {g_{ξ_{m}} : m ∈ ℕ} is relatively compact.

On the other hand, if δ_{g} denotes the factor by which the volumes of subsets gets multiplied under the transformation h → ghg^{−1}, h ∈ H, then

V((H_{ξ_{m}} ∩ Γ)\H_{ξ_{m}}) = δ_{g_{ξm}} · ω(ξ_{m}Γ)

up a uniform constant multiple depending only on the choice of Haar measure μ_{H}. Since {g_{ξ_{m}} : m ∈ ℕ} is relatively compact, sup_{m} δ_{g_{ξm}} < ∞.

Therefore

sup_{m} V((H_{ξ_{m}} ∩ Γ)\H_{ξ_{m}}) ≤ sup_{m} δ_{g_{ξm}} · sup_{m} ω(O_{m}) < ∞.

By Theorem 4.1, this implies that H_{ξ_{m}} ∩ Γ are all equal to each other by passing to a subsequence. Since H_{ξ_{m}} ∩ Γ is Zariski dense in H_{ξ_{m}} by Borel density theorem, it follows that H_{ξ_{m}} are all equal to each other, that is, g_{ξ_{m}}^{−1}g_{ξ_{k}} ∈ N(H) for all m, k. Since [N(H) : H] < ∞, by passing to a subsequence, we have

g_{ξ_{m}}^{−1}g_{ξ_{k}} ∈ H, and hence pr(ξ_{m}Γ) = pr(ξ_{k}Γ)

for all m, k. This finishes the proof. □

Observe that for compact subsets Ω ⊂ V_{1}, C_{0} ⊂ G such that H\HC_{0} = Ω and ξ ∈ V_{m}(ℤ),

(4.4)  N(ξΓ, Ω) = 0 if and only if Γ\Γg_{ξ}H ∩ Γ\ΓC_{0}^{−1} = ∅.

Proposition 4.5. Consider a sequence {ξ_{m} ∈ V_{m}(ℤ)} such that {pr(ξ_{m})} is relatively compact in V_{1}. Suppose that L is a closed subgroup of G containing H_{ξ_{m}}^{0} for all m. such that L ∩ Γ is a lattice in L. Then for any compact subset Ω of V_{1}, there exists a finite subset Λ_{Ω} ⊂ Γ such that for all m,

pr(ξ_{m})Γ ∩ Ω ⊂ pr(ξ_{m})(L ∩ Γ)Λ_{Ω}.

Proof. Let Ω_{0} be a compact subset of G such that H\HΩ_{0} = Ω. Write H as a disjoint union ∪_{i=1}^{k}h_{i}H^{0}. By the assumption, there exists a choice of {g_{ξ_{m}}} so that {g_{ξ_{m}} ∈ G} is relatively compact. Let Ω_{1} ⊂ G be a compact subset which contains {g_{ξ_{m}}h_{i} : 1 ≤ i ≤ k, m ∈ ℕ}Ω_{0}.

Since L ∩ Γ is a lattice in L, Γ\ΓL is closed in Γ\G [Rag], and this implies easily that L\LΓ is closed in L\G. Since L\LΓ is a closed countable subset of L\G, it follows from Baire category theorem that there exists at least one isolated point. Since Γ acts transitively on L\LΓ, every point of L\LΓ is an isolated point. Therefore L\LΓ is discrete in L\G. Hence there exists a finite subset Λ_{Ω} of Γ such that

LΩ_{1} ∩ LΓ ⊂ LΛ_{Ω}.

Note that

g_{ξ_{m}}HΩ_{0} ∩ g_{ξ_{m}}H^{0}g_{ξ_{m}}^{−1}Γ ⊂ Lg_{ξ_{m}}(∪_{i=1}^{k}h_{i})Ω_{0} ∩ LΓ ⊂ LΩ_{1} ∩ LΓ ⊂ LΛ_{Ω}.

Hence

(4.6)  HΩ_{0} ∩ H^{0}g_{ξ_{m}}^{−1}Γ ⊂ g_{ξ_{m}}^{−1}LΛ_{Ω}.

If v_{0}x ∈ pr(ξ_{m}Γ) ∩ Ω for x ∈ G, then

x = hg_{ξ_{m}}^{−1}γ = h′w

for some h, h′ ∈ H, γ ∈ Γ and w ∈ Ω_{0}.

Then

g_{ξ_{m}}^{−1}γ ∈ HΩ_{0} ∩ H^{0}g_{ξ_{m}}^{−1}Γ,

and hence by (4.6),

g_{ξ_{m}}^{−1}γ = g_{ξ_{m}}^{−1}gγ_{1} for some g ∈ L and γ_{1} ∈ Λ_{Ω}.

In particular, g = γγ_{1}^{−1} ∈ L ∩ Γ. Therefore

x = hg_{ξ_{m}}^{−1}gγ_{1} ∈ Hg_{ξ_{m}}^{−1}(L ∩ Γ)Λ_{Ω}

proving

pr(ξ_{m}Γ) ∩ Ω ⊂ pr(ξ_{m})(L ∩ Γ)Λ_{Ω}.
□

5. Asymptotic behavior of O_{m}

We start by recalling the following theorem of Dani and Margulis. For any two closed subgroups U and L of a connected Lie group G, set

X(U, L) := {g ∈ G : gU ⊂ Lg}.

Theorem 5.1 (DM2, Theorem 3). Let G be a connected Lie group and Γ a lattice in G. Let U = {u(t)} be a unipotent one-parameter subgroup of G and let ψ be a bounded continuous function of Γ\G. Let K be a compact subset of Γ\G and let ϵ > 0 be given. Then there exist finitely many proper closed subgroups L_{1}, ⋯, L_{k} such that L_{i} ∩ Γ is a lattice in L_{i} for each 1 ≤ i ≤ k, and compact subsets C_{1}, ⋯, C_{k} of X(U, L_{1}), ⋯, X(U, L_{k}) respectively, for which the following holds: for any compact subset F ⊂ K − ∪_{i=1}^{k}Γ\ΓC_{i}, there exists T_{0} ≥ 0 such that for all x ∈ F and T > T_{0},

|\frac{1}{T} ∫_{0}^{T} ψ(xu(t)) dt − ∫_{Γ\G} ψ dμ_{G}| ≤ ϵ.

In fact, it is shown in the proof of the above theorem [DM2] that the subgroups L_{i} can be taken so that Ad(L_{i} ∩ Γ) is Zariski dense in Ad(L_{i}) as well where Ad denotes the adjoint representation of G.

We keep the same notation from section 2 for G, ρ : G → GL(V), Γ, v_{0}, etc. Let O_{m} be a Γ-invariant subset of V_{m}(ℤ). As before, we assume that the subgroup H, which is the stabilizer of v_{0}, is a semisimple real algebraic subgroup of G without compact factors. Set X = Γ\G. We assume without loss of generality μ_{G}(X) = 1. For each ξ ∈ V_{m}(ℤ), we denote by ν_{ξΓ} the unique H-invariant measure on X supported on Γ\Γg_{ξ}H with the total measure given by ω(ξΓ).

Denote by π the canonical projection from H^{0}\G to H\G. By the identification of V_{1} = v_{0}G with H\G, we consider pr(O_{m}) as a subset of H\G. Set 𝔒_{m} = π^{−1}(pr(O_{m})). Note that 𝔒_{m} is Γ-invariant and has finitely many Γ-orbits. For each ηΓ ⊂ 𝔒_{m}, the notation ν_{ηΓ} denotes the H^{0}-invariant measure on Γ\G supported on Γ\Γg_{η}H^{0} with the total measure given by

ω(ηΓ) := \frac{μ_{H^{0}}((H^{0} ∩ g_{η}^{−1}Γg_{η})\H^{0})}{μ_{G}(Γ\G)}

where g_{η} is any element in G such that H^{0}\H^{0}g_{η} = η. Here μ_{H^{0}} is simply the restriction of μ_{H} to H^{0}. Note that if π(η_{1}Γ) = π(η_{2}Γ), then ω(η_{1}Γ) = ω(η_{2}Γ) and ω(ηΓ) ≤ ω(ξΓ) if π(η) = pr(ξ).

For ξΓ ⊂ O_{m}, it is not hard to check that

ν_{ξΓ} = ∑_{ηΓ⊂π^{−1}(pr(ξΓ))} ν_{ηΓ}

where the sum is taken over the disjoint Γ-orbits ηΓ in 𝔒_{m} such that π(ηΓ) = pr(ξΓ). The number such Γ-orbits is clearly bounded by [H : H^{0}].

Therefore

(5.2)  ∑_{ξΓ⊂O_{m}} ν_{ξΓ} = ∑_{ηΓ⊂𝔒_{m}} ν_{ηΓ} and ω(O_{m}) = ∑_{ηΓ⊂𝔒_{m}} ω(ηΓ).

Let U = {u(t)} be a unipotent one-parameter subgroup of H^{0} not contained in any proper closed normal subgroup of H^{0}. For given compact subset K ⊂ X, ϵ > 0 and a bounded continuous function ψ on X, let L_{i} and C_{i}, 1 ≤ i ≤ k be as in the above theorem, with respect to the given triples (K, ϵ, ψ). Set

S(K, ψ, ϵ) := K ∩ (∪_{i=1}^{k}Γ\ΓC_{i})

and let G(K, ψ, ϵ) denote the complement of S(K, ψ, ϵ) inside K.

Proposition 5.3. Fix a compact subset K of X with a non-empty interior. Suppose that for any ϵ > 0 and for any continuous function ψ on X with compact support,

lim_{m→∞} \frac{∑_{ηΓ⊂𝔒_{m}}\{ω(ηΓ) : Γ\Γg_{η}H^{0} ∩ G(K, ψ, ϵ) ≠ ∅\}}{ω(O_{m})} = 1.

Then

\frac{1}{ω(O_{m})} ∑_{ξΓ⊂O_{m}} ν_{ξΓ} → μ_{G} as m → ∞.

Proof. We set

A_{m}(ϵ) := {Γ\Γg_{η}H^{0} : ηΓ ⊂ 𝔒_{m}, Γ\Γg_{η}H^{0} ∩ K ⊂ S(K, ψ, ϵ)}

and

B_{m}(ϵ) := {Γ\Γg_{η}H^{0} : Γ\Γg_{η}H^{0} ∩ G(K, ψ, ϵ) ≠ ∅}.

The assumption implies that (5.4)

lim_{m→∞} \frac{∑\{ω(ηΓ) : Γ\Γg_{η}H^{0} ∈ A_{m}(ϵ)\}}{ω(O_{m})} = 0; lim_{m→∞} \frac{∑\{ω(ηΓ) : Γ\Γg_{η}H^{0} ∈ B_{m}(ϵ)\}}{ω(O_{m})} = 1.

By the Birkhoff ergodic theorem and Moore’s ergodicity theorem, U acts ergodically with respect to each \frac{1}{ω(ηΓ)}ν_{ηΓ} and the following subset R has the zero co-measure in H^{0}:

R = \{h ∈ H^{0} : Γ\Γg_{η}hu(t) is uniformly distributed in Γ\Γg_{η}H^{0} w. r. t. \frac{1}{ω(ηΓ)}ν_{ηΓ}\}.

Let Γ\Γg_{η}H^{0} ∈ B_{m}(ϵ). Since G(K, ψ, ϵ) is open in K and R has co-measure 0 in H^{0}, we may assume by a suitable choice for g_{η} that Γ\Γg_{η} ∈ G(K, ψ, ϵ) and

lim_{T →∞} \frac{1}{T} ∫_{0}^{T} ψ(Γ\Γg_{η}u(t))dt = \frac{1}{ω(ηΓ)} ∫_{X} ψ dν_{ηΓ}.

Therefore by applying Theorem 5.1 to each singleton F = {Γ\Γg_{η}}, we obtain that for any Γ\Γg_{η}H^{0} ∈ B_{m}(ϵ),

(5.5)  |\frac{1}{ω(ηΓ)} ∫_{X} ψ dν_{ηΓ} − ∫_{X} ψ dμ_{G}| ≤ ϵ.

Now

| ∑_{ηΓ⊂𝔒_{m}} ∫_{X} ψ dν_{ηΓ} − ∫_{X} ψ dμ_{G}| ≤ ∑_{Γ\Γg_{η}H^{0}∈A_{m}(ϵ)} |∫_{X} ψ dν_{ηΓ} − ω(ηΓ) ∫_{X} ψ dμ_{G}|

+ ∑_{Γ\Γg_{η}H^{0}∈B_{m}(ϵ)} |∫_{X} ψ dν_{ηΓ} − ω(ηΓ) ∫_{X} ψ dμ_{G}|

By (5.5), the above is again less than or equal to

(∥ψ∥_{∞} + ∥ψ∥_{1})(∑{ω(ηΓ) : Γ\Γg_{η}H^{0} ∈ A_{m}(ϵ)}) + ϵ(∑{ω(ηΓ) : Γ\Γg_{η}H^{0} ∈ B_{m}(ϵ)}).

By applying Lebesgue dominated convergence theorem, we deduce from (5.4)

lim sup_{m→∞} |\frac{1}{ω(O_{m})} ∑_{ηΓ⊂𝔒_{m}} (∫_{X} ψ dν_{ηΓ}) − ∫ ψdμ_{G}| ≤ ϵ.

Since ϵ > 0 is arbitrary, we have for any bounded continuous function ψ on X,

lim_{m→∞} \frac{1}{ω(O_{m})} ∑_{ηΓ⊂𝔒_{m}} (∫_{X} ψ dν_{ηΓ}) = ∫ ψ dμ_{G}.

This proves our claim by (5.2). □

Lemma 5.6 (EMS2, Lemma 5.1). Let G, H and L be connected real algebraic groups such that H ⊂ L ⊂ G. If at least one of G, H, and L is reductive, then X(H, L) is a union of finitely many closed double cosets of the form L · g · Z(H) where g ∈ X(H, L).

Proof of Theorem 1.13 Since G^{0} has a finite index in G and V_{1} consists of finitely many open G^{0} orbits, it suffices to prove the theorem for each G^{0}-orbit. Hence we may assume that G is connected without loss of generality. Since {O_{m}} is not focused, there exists an arithmetic subgroup Γ ⊂ G(ℚ) which preserves each O_{m} and {O_{m}} is not focused with respect to Γ.

First, for some compact subset C of V_{1},

lim sup_{m→∞} \frac{∑_{ξΓ⊂O_{m}}\{ω(ξΓ) : pr(ξ)Γ ∩ C = ∅\}}{ω(O_{m})} = 0.

Note that the same holds for any compact subset of V_{1} containing C.

Hence it follows from the observation (4.4) that for some relatively compact open subset C_{0} of G, we have

(5.7)  lim sup_{m→∞} \frac{∑_{ηΓ⊂𝔒_{m}}\{ω(ηΓ) : Γ\Γg_{η}H^{0} ∩ Γ\ΓC_{0} = ∅\}}{ω(O_{m})} = 0.

Set K = Γ\(Γ\overline{C_{0}}) and K′ = Γ\(ΓC_{0}). By Propositions 2.2, and 5.3, it suffices to show that for any ϵ > 0 and for any bounded continuous function ψ on X,

lim_{m→∞} \frac{∑_{ηΓ⊂𝔒_{m}}\{ω(ηΓ) : Γ\Γg_{η}H^{0} ∩ G(K, ψ, ϵ) ≠ ∅\}}{ω(O_{m})} = 1.

Suppose not. Since the orbits Γ\Γg_{η}H^{0} disjoint from K′ can be ignored by (5.7), it follows that there exist a bounded continuous function ψ on X and an ϵ > 0 such that

(5.8)  lim sup_{m→∞} \frac{∑_{ηΓ⊂𝔒_{m}}\{ω(ηΓ) : ∅ ≠ K′ ∩ Γ\Γg_{η}H^{0} ⊂ S(K, ψ, ϵ)\}}{ω(O_{m})} > 0

Let L_{i} and C_{i}, 1 ≤ i ≤ k be the subgroups and compact subsets in X(U, L_{i}) respectively, used in the definition of S(K, ψ, ϵ). Since S(K, ψ, ϵ) is contained in the finite union ∪_{i=1}^{k}Γ\ΓC_{i}, there exists 1 ≤ i ≤ k such that (5.8) holds with Γ\ΓC_{i} in place of S(K, ψ, ϵ). Without loss of generality, we assume i = 1. By Lemma 5.6, there exists g ∈ G such that gUg^{−1} ⊂ L_{1} and

(5.9)  lim sup_{m→∞} \frac{∑_{ηΓ⊂𝔒_{m}}\{ω(ηΓ) : ∅ ≠ Γg_{η}H^{0} ∩ K′ ⊂ Γ(C_{1} ∩ L_{1}gZ(U))\}}{ω(O_{m})} > 0.

Whenever

∅ ≠ Γg_{η}H^{0} ∩ K′ ⊂ Γ(C_{1} ∩ L_{1}gZ(U))

we may assume that g_{η} ∈ C_{1} ∩ L_{1}gZ(U) by replacing η and g_{η} by suitable elements in π^{−1}(pr(ξΓ)) and g_{η}H^{0} respectively. We may also assume that the set {h ∈ H^{0} : g_{η}h ∈ C_{1} ∩ L_{1}gZ(U)} has a positive measure, since K′ is open. Note that g_{η}h ∈ C_{1} ∩ L_{1}gZ(U) implies that hUh^{−1} ⊂ g_{η}^{−1}L_{1}g_{η}. By a similar argument as in the proof of Proposition 3.4, it follows that H_{ξ}^{0} ⊂ L_{1}, i.e., g_{ξ} ∈ X(H^{0}, L_{1}), whenever ∅ ≠ Γg_{η}H^{0} ∩ K′ ⊂ Γ(C_{1} ∩ L_{1}gZ(U)) and π(η) ∈ pr(ξΓ). Applying Lemma 5.6 again, we deduce from (5.9) that for some g_{0} ∈ G such that g_{0}H^{0}g_{0}^{−1} ⊂ L_{1},

lim sup_{m→∞} \frac{∑_{ηΓ⊂𝔒_{m}}\{ω(ηΓ) : g_{η} ∈ C_{1} ∩ L_{1}g_{0}Z(H^{0})\}}{ω(O_{m})} > 0.

Since g_{η} ∈ C_{1} ∩ L_{1}g_{0}Z(H) and π(η) = pr(ξ) implies that g_{η}H^{0}g_{η}^{−1} = H_{ξ}^{0} ⊂ L_{1}, it follows from Proposition 4.5 that for any compact subset Ω of V_{1}, there exists a finite subset Λ_{Ω} ⊂ Γ such that for all g_{η} ∈ C_{1} ∩ L_{1}g_{0}Z(H),

π(η)Γ ∩ Ω ⊂ (Hg_{η}^{−1})(L_{1} ∩ Γ)Λ_{Ω} ⊂ H(L_{1}g_{0}Z(H) ∩ C_{0})^{−1}(L_{1} ∩ Γ)Λ_{Ω}.

Hence we have shown that for any compact subset Ω ⊂ V_{1},

lim sup_{m→∞} \frac{∑_{ηΓ⊂𝔒_{m}}\{ω(ηΓ) : π(η)Γ ∩ Ω ⊂ H(Z(H)g_{0}^{−1}L_{1} ∩ C_{0}^{−1})(L_{1} ∩ Γ)Λ_{Ω}\}}{ω(O_{m})} > 0,

for some finite subset Λ_{Ω} ⊂ Γ. Hence for some γ_{Ω} ∈ Γ,

lim sup_{m→∞} \frac{∑_{ξΓ⊂O_{m}}\{ω(ξΓ) : pr(ξ)Γ ∩ Ω ⊂ v_{0}(Z(H)g_{0}^{−1}L_{1} ∩ C_{0}^{−1})(L_{1} ∩ Γ)γ_{Ω}\}}{ω(O_{m})} > 0,

By the remark following Theorem 5.1, L_{1} ∩ Γ is a Zariski dense lattice in L_{i}.

Hence the sequence {O_{m}} is focused, yielding contradiction. This finishes the proof. Proof of Theorem 1.2 Without loss of generality we may assume that G is connected. Fix any Γ ∈ A_{V} preserving each O_{m}. If (1.4) does not hold, then {O_{m}} is focused by Theorem 1.13. Since H^{0} is a maximal connected closed subgroup of G, it follows that there exists g ∈ G such that for any compact subset Ω ⊂ V_{1}

lim sup_{m→∞} \frac{∑_{ξΓ⊂O_{m}}\{ω(ξΓ) : pr(ξ)Γ ∩ Ω ⊂ v_{0}Z(H^{0})H^{0}gΓ\}}{ω(O_{m})} > 0.

We claim that

(5.10)  lim sup_{m→∞} \frac{∑_{ξΓ⊂O_{m}}\{ω(ξΓ) : pr(ξ)Γ ∩ v_{0}Z(H^{0})H^{0}gΓ ≠ ∅\}}{ω(O_{m})} > 0

Suppose not. Then it follows that for any compact subset Ω of V_{1},

lim sup_{m→∞} \frac{∑_{ξΓ⊂O_{m}}\{ω(ξΓ) : pr(ξ)Γ ∩ Ω = ∅\}}{ω(O_{m})} > 0.

This is equivalent to saying that for any compact subset C of Γ\G,

lim sup_{m→∞} \frac{∑_{ηΓ⊂𝔒_{m}}\{ω(ηΓ) : Γ\Γg_{η}H^{0} ∩ C = ∅\}}{ω(O_{m})} > 0.

Hence there exists a sequence η_{m} ∈ 𝔒_{m} such that for any compact subset C ⊂ X, there exists m such that Γ\Γg_{η_{m}}H^{0} ∩ C = ∅. On the other hand, since H^{0} is a proper maximal closed subgroup of G^{0}, by Proposition 3.4, any weak limit of {ν_{η_{m}Γ}} in P(X ∪ {∞}) is supported on X. This is a contradiction by Proposition 3.2. Hence (5.10) is proved.

It is easy to check that pr(ξ)Γ ∩ v_{0}Z(H^{0})H^{0}gΓ ≠ ∅ implies γH_{ξ}γ^{−1} = gHg^{−1} for some γ ∈ Γ. Hence (5.10) implies:

lim sup_{m→∞} \frac{∑_{ξΓ⊂O_{m}}\{ω(ξΓ) : γH_{ξ}γ^{−1} = gHg^{−1} for some γ ∈ Γ\}}{ω(O_{m})} > 0

It follows that there exists ξ_{0} ∈ O_{m_{0}} for some m_{0} such that

lim sup_{m→∞} \frac{∑_{ξΓ⊂O_{m}}\{ω(ξΓ) : γH_{ξ}γ^{−1} = H_{ξ_{0}} for some γ ∈ Γ\}}{ω(O_{m})} > 0

Observe that the condition γH_{ξ}γ^{−1} = H_{ξ_{0}} implies that g_{ξ_{0}}^{−1}γg_{ξ} ∈ N(H), and the condition g_{ξ_{0}}^{−1}γg_{ξ} ∈ H implies that pr(ξ)Γ = pr(ξ_{0})Γ.

Since H has a finite index in the normalizer N(H), it follows that

(5.11)  lim sup_{m→∞} \frac{∑_{ξΓ⊂O_{m}}\{ω(ξΓ) : pr(ξ)Γ = pr(ξ_{0})Γ)\}}{ω(O_{m})} > 0.

Note that if pr(ξ)Γ = pr(ξ_{0})Γ, then ω(ξΓ) = ω(ξ_{0}Γ). Since there can be at most one Γ-orbit ξΓ in O_{m} such that pr(ξ)Γ = pr(ξ_{0})Γ, (5.11) implies that

ω(ξ_{0}Γ) · lim sup_{m→∞} \frac{1}{ω(O_{m})} > 0 or equivalently lim inf_{m→∞} ω(O_{m}) < ∞.

By Lemma 4.3, this contradicts the assumption on {O_{m}}. Hence the proof is now complete.

6. Examples

Theorem 6.1. Let Q be an integral quadratic form of signature (r, s) where r + s ≥ 4, r ≥ 2 and s ≥ 1. For any compact subset Ω of V_{1} with boundary of measure 0,

N_{m}(Q, Ω) ∼ ω_{m} · vol (Ω)

as m → ∞ along the fundamental discriminants.

Proof. Consider the standard representation of the orthogonal group O(Q) on V := ℝ^{r+s}. Let V_{m} be the level set {x ∈ V : Q(x) = m}. By Witt’s theorem the orthogonal group O(Q) acts transitively on each V_{1}. The stabilizer of a vector v_{0} in V_{1} is isomorphic to O(r − 1, s). Note that the assumptions on the size of the parameters r and s guarantee that O(r − 1, s) is non-compact and simple. It is well known that O(r − 1, s)^{0} is a maximal connected subgroup of O(r, s)^{0}. If we set Γ := O(Q) ∩ SL_{r+s}(ℤ), then Γ ∈ A_{V}. Under the assumption that r + s ≥ 4 and rs = 1, V_{m}(ℤ) ≠ ∅ for all fundamental discriminants m (see [Oh]). Hence we may take O_{m} = V_{m}(ℤ) to apply theorem 1.2. To check the condition (1.3), note that if m ≠ k are fundamental discriminants, then \sqrt{m}^{−1}V_{m}(ℤ) ∩ \sqrt{k}^{−1}V_{k}(ℤ) = ∅; otherwise this would imply that \sqrt{(m/k)} ∈ ℚ, which can be seen to be false by an easy computation. Therefore Theorem 1.2 implies the claim. □

Let V := {X ∈ M_{2n}(ℝ) : X^{t} = −X} be the space of skew-symmetric matrices, so that the Pfaffian on V is defined by

Pff^{2}(X) = Det(X) and Pff(v_{0}) = 1

where

v_{0} = (\begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix}) .

E.g., for n = 2, we have Pff(x_{1}, ⋯, x_{6}) = x_{1}x_{2} − x_{3}x_{4} + x_{5}x_{6}.

Theorem 6.2. Let n ≥ 2. For any compact subset Ω of V_{1} with boundary of measure 0,

N_{m}(Pff, Ω) ∼_{m→∞} ω_{m} · vol (Ω).

Proof. Consider the representation ρ : SL_{2n}(ℝ) → GL(V) defined by

ρ(A)(X) = A^{t}XA

where A ∈ SL_{2n}(ℝ) and X ∈ V. It is well known that SL_{2n}(ℝ) acts transitively on V_{1} = {X ∈ V : Pff(X) = 1}, so that V_{m} = m^{1/n}v_{0} SL_{2n}(ℝ). The stabilizer H of v_{0} is the symplectic group Sp_{2n}(ℝ) corresponding to v_{0}, which is a maximal connected closed subgroup of SL_{2n}(ℝ). Clearly V_{m}(ℤ) ≠ ∅ and SL_{2n}(ℤ) preserves V_{m}(ℤ) for each m.

To check the condition 1.3 of Theorem 1.2, suppose for a given m_{0} ∈ ℕ that pr(V_{m_{0}}(ℤ)) = pr(V_{m}(ℤ)). Then

m^{−1/n} (\begin{matrix}& & m & 0 \\ & & 0 & I_{n−1} \\ −m & 0 \\ 0 & −I_{n−1} \end{matrix}) ∈ m_{0}^{−1/n}V (ℤ).

Hence m^{−1/n} = m_{0}^{−1/n}k for some k ∈ ℕ. This leads to m = m_{0}k^{−n} ≤ m_{0}. Hence

|{m ∈ ℕ : pr(V_{m_{0}}(ℤ)) = pr(V_{m}(ℤ))}| ≤ m_{0} < ∞.

Hence the claim follows from Theorem 1.2. □

Since the explicit representatives of SL_{2n}(ℤ)-orbits on V_{m}(ℤ) can easily be written down in the Pff case, we can compute ω_{m}, using the local density formula given in [GY]. For instance, for a prime p,

ω_{p} = C · ∑_{i=0}^{2n−2} p^{i}

for some constant C independent of p.

Theorem 6.3. Let n ≥ 2. For any compact subset Ω of V_{1} = SL_{n}(ℝ) with boundary of measure 0,

N_{m}(Det, Ω) ∼_{m→∞} ω_{m} · vol (Ω).

Proof. Consider the representation SL_{n}(ℝ) × SL_{n}(ℝ) on the space V = M_{n}(ℝ) given by X(A, B) = AXB^{−1} where X ∈ V and A, B ∈ SL_{n}(ℝ). The stabilizer of I_{n} is given by the diagonal embedding of SL_{n}(ℝ) in the product SL_{n}(ℝ) × SL_{n}(ℝ), which is a maximal connected closed subgroup. Clearly V_{m}(ℤ) ≠ ∅ and SL_{n}(ℤ) × SL_{n}(ℤ) preserves V (ℤ). The condition 1.3 can be checked similarly as in the case of Pfaffian. Hence our claim Theorem 1.2. □

For the determinant case, the constant ω_{m} is well known from the theory of Hecke operators. (cf. [Sa], or [GO]). For instance, for any fixed k ∈ ℕ and a prime p,

ω_{p^{k}} ∼_{p→∞} C · p^{k(n−1)}.

References

[Bo] A. Borel Values of indefinite quadratic forms at integral points and flows on spaces of lattices, Bull. AMS, Vol 32, 1995, pp. 184–204.

[BP] A. Borel and G. Prasad Values of isotropic quadratic forms at S-integral points, Compositio Math. Vol 83, 1992, pp. 347–372.

[BH] A. Borel and Harish-Chandra Arithmetic subgroups of algebraic groups, Ann. Math., Vol 75, 1962, pp. 485–535.

[BM] M. B. Bekka and M. Mayer Ergodic theory and Topological Dynamics of group actions on homogeneous spaces, LMS Lecture Note series 269, Cambridge

[COU] L. Clozel, H. Oh and E. Ullmo Hecke operators and equidistribution of Hecke points, Invent. Math., Vol 144, 2001, pp. 327–351.

[DM1] S. G. Dani and G. Margulis Asymptotic behavior of trajectories of unipotent flows on homogeneous spaces, Proc. Indian. Acad. Sci. (Math. Sci), Vol 101, No 1, 1991, pp. 1–17.

[DM2] S. G. Dani and G. Margulis Limit distribution of orbits of unipotent flows and values of quadratic forms, Advances in Soviet Math., Vol 16, 1993, pp. 91–137.

[Du] W. Duke Hyperbolic distribution problems and half integral weight Maass forms Inven. Math. Vol 92 (1988) 73–90

[EM] A. Eskin and C. McMullen Mixing, counting and equidistribution in Lie groups, Duke Math. J. Vol 71, No. 1, 1993, 181–209.

[EMS1] A. Eskin, S. Mozes and N. Shah Non-divergence of translates of certain algebraic measures, GAFA, Vol 7, 1997, pp. 48–80.

[EMS2] A. Eskin, S. Mozes and N. Shah Unipotent flows and counting lattice points on homogeneous varieties, Ann. Math, Vol 143, 1996, pp. 149–159.

[ERS] A. Eskin, Z. Rudnick and P. Sarnak A proof of Siegel’s weight formula, IMRN Vol 5, (1991) 65–69 [Es] T. Estermann A new applications of the Hardy-Littlewood-Kloosterman method, Proc. London. Math.

Soc, Vol 12, 1962, pp. 425–444

[GO] W. T. Gan and H. Oh Equi-distribution of integer points on a family of homogeneous varieties: a problem of Linnik, Compositio Math. Vol 136, 323-352 (2003)

[GY] W. T. Gan and J.-K. Yu Group schemes and local densities, Duke. M. J., Vol 105, 2000, pp. 497–524 [Iw] H. Iwaniec Fourier coefficients of modular forms of half integral weight, Inv. Math (1987), Vol 87, pp.

385–401.

[KM] D. Kazhdan and G. Margulis A proof of Selberg’s conjecture, Math. Sbornik, 1968, Vol 4, pp. 147–151.

[Li1] Y. V. Linnik Ergodic properties of algebraic fields, Ergebnisse der Mathematik und Grenzgebiete, Band 45, Springer-Verlag New York, 1968

[Li2] Y. V. Linnik Additive problems and eigenvalues of the modular operators, Proc. Int. Cong. Math. Stockholm, 1962, pp. 270–284.

[LS] Y. V. Linnik and B. F. Skubenko Asymptotic distribution of integral matrices of third order, Vest. Leniner. Univ. Ser. Math. Vol. 13, 1964, pp. 25–36.

[Ma] G. Margulis Discrete subgroups of semisimple Lie groups, Springer-Verlag

[MS] S. Mozes and N. Shah On the space of ergodic invariant measures of unipotent flows, Ergod. Th. & Dynam. Sys., Vol 15, 1995, pp. 149–159.

[Oh] H. Oh Hardy-Littlewood system and representations of integers by an invariant polynomial, To appear in Geom. Anal. Funct. Anal.

[Ra] M. Ratner On Raghunathan’s measure conjecture, Ann. of Math., Vol 134, 1991, pp. 545–607.

[Rag] M. S. Raghunathan Discrete subgroups of Lie groups, Ergebnisse Der Mathematik, Springer-Verlag [Sa] P. Sarnak Diophantine problems and linear groups, Proc. Int. Cong. Math. 1990, Vol 1, pp. 459–471.

[Va] R. C. Vaughan The Hardy-Littlewood method, 2nd ed. Cambridge Tracts in Math. 1997, Vol 125, Cambridge-New York

[We] A. Weil Adeles and algebraic groups, Boston, Birkhäuser, 1982

Mathematics Department, University of Chicago, Chicago, IL 60637

   E-mail address: eskin@math.uchicago.edu

Mathematics Department, Princeton University, Princeton, NJ 08544, Current address: Math 253-37, Caltech, Pasadena, CA 91125

   E-mail address: heeoh@its.caltech.edu