UNIFORM DISTRIBUTION OF ORBITS OF LATTICES ON SPACES OF FRAMES

ALEXANDER GORODNIK

Abstract. We study distribution of orbits of a lattice Γ ⊆ SL(n, ℝ) in the the space V_{n,l} of l-frames in ℝ^{n} (1 ≤ l ≤ n − 1). Examples of dense Γ-orbits are known from the work of Dani, Raghavan, and Veech. We show that dense orbits of Γ are uniformly distributed in V_{n,l} with respect to an explicitly described measure. We also establish analogous result for lattices in Sp(n, ℝ) that act on the space of isotropic n-frames.

1. Introduction

Let G = SL(n, ℝ) and V_{n,l} be the space of l-frames in ℝ^{n} (i.e. the space of l-tuples of linearly independent vectors in ℝ^{n}), 1 ≤ l ≤ n. The group G acts on this space as follows:

g · (v_{1}, …, v_{l}) = (gv_{1}, …, gv_{l}), g ∈ SL(n, ℝ), (v_{1}, …, v_{l}) ∈ V_{n,l}.

The action is transitive for l < n. Let Γ be a lattice in G; that is, a discrete subgroup in G such that the factor space Γ\G has finite volume (e.g. Γ = SL(n, ℤ)). The main result of this paper concerns distribution of Γ-orbits in V_{n,l}.

When l = n, every orbit of Γ is discrete. The situation becomes much more interesting for l < n. Let us recall known results: Theorem 1. (Dani, Raghavan [DR80]) Let Γ = SL(n, ℤ), and v = (v_{1}, …, v_{l}) be an l-frame in ℝ^{n}, 1 ≤ l ≤ n − 1. Then the orbit Γ · v is dense in V_{n,l} iff the space spanned by {v_{i} : i = 1, …, l} contains no nonzero rational vectors. Theorem 2. (Veech [Ve77]) If Γ is a cocompact lattice in G, then every orbit of Γ in V_{n,l}, 1 ≤ l ≤ n − 1, is dense.

Theorems 1 and 2 provide examples of dense Γ-orbits in V_{n,l}. Here we show that dense Γ-orbits are uniformly distributed with respect to an explicitly described measure on V_{n,l}. This measure is \frac{dv}{Vol(v)}, where dv is the Lebesgue measure on ∏_{i=1}^{l} ℝ^{n}, and Vol(v) is the l-dimensional volume of the frame v.

Note that the measure dv is G-invariant, and it is unique up to a constant. However, orbits of Γ are equidistributed with respect to the measure \frac{dv}{Vol(v)}, which is not G-invariant. This phenomenon was already observed by Ledrappier [Le99].

Define a norm on M(n, ℝ) by

∥x∥ = (Tr(^{t}xx))^{1/2} = (∑_{i,j} x_{ij}^{2})^{1/2} for x = (x_{ij}) ∈ M(n, ℝ).  (1)

2000 Mathematics Subject Classification. Primary 37A15, 22E40, 37A17.

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

For T > 0, Ω ⊆ V_{n,l}, v^{0} ∈ V_{n,l}, put

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

We determine asymptotic behavior of N_{T} (Ω, v^{0}) as T → ∞. This result gives a quantitative strengthening of Theorems 1 and 2, and it can be interpreted as uniform distribution of dense orbits of Γ in V_{n,l}. Theorem 3. Let Γ be a lattice in SL(n, ℝ). Let v^{0} ∈ V_{n,l} be an l-frame in ℝ^{n} such that Γ·v^{0} is dense in V_{n,l}. Let Ω be a relatively compact Borel subset of V_{n,l} such that ∫_{∂Ω} dv = 0. Then

N_{T} (Ω, v^{0}) ∼ a_{n,l} \frac{Vol(v^{0})^{1−n}}{μ̄(Γ\G)} (∫_{Ω} \frac{dv}{Vol(v)}) T^{(n−1)(n−l)} as T → ∞,  (3)

where a_{n,l} is a constant (which is computed in (45) below), and μ̄ is a G-invariant measure on Γ\G (which is defined in (29) below). Remark. The term T^{(n−1)(n−l)} in (3) comes from the asymptotics of the volume of the set {h ∈ H : ∥h∥ < T } in the stabilizer H of v^{0} with respect to the measure on H which is determined by the choice of the Haar measures on G and V_{n,l} = G · v^{0} (see Section 2).

For n = 2 and l = 1, this theorem was proved by Ledrappier [Le99] for general Γ and by Nogueira [No00] for Γ = SL(2, ℤ) and max-norm using different methods.

Combining Theorems 1 and 3, we get: Corollary 4. Let Γ = SL(n, ℤ). Let v^{0} = (v_{1}^{0}, …, v_{l}^{0}) ∈ V_{n,l} be an l-frame in ℝ^{n} such that the space ⟨v_{1}^{0}, …, v_{l}^{0}⟩ contains no nonzero rational vectors. Let Ω be a relatively compact Borel subset of V_{n,l} such that ∫_{∂Ω} dv = 0. Then

N_{T} (Ω, v^{0}) ∼ b_{n,l}Vol(v^{0})^{1−n} (∫_{Ω} \frac{dv}{Vol(v)}) T^{(n−1)(n−l)} as T → ∞,  (4)

where b_{n,l} is a constant (which is computed in (83) below).

Example Figure 1 shows a part of the the orbit SL(2, ℤ)v^{0} for v^{0} = ^{t}(\sqrt{2}, \sqrt{3}). By the result of Ledrappier, this orbit is uniformly distributed in ℝ^{2} with respect to the measure \frac{dxdy}{\sqrt{x^{2}+y^{2}}} .

Dani and Raghavan also considered orbits of frames under Sp(n, ℤ). Denote

J = ( \begin{matrix}0 & E \\ −E & 0 \end{matrix} ) ,

where E is the identity n × n matrix. The symplectic form (x, y) ↦ ^{t}xJy will be denoted by J too. Let

G = Sp(n, ℝ) = {g ∈ SL(2n, ℝ) : ^{t}gJg = J}

and Γ = Sp(n, ℤ). A frame (v_{1}, …, v_{s}) is called isotropic if the symplectic form J is 0 on the space spanned by {v_{i} : i = 1, …, s}. Theorem 5. (Dani, Raghavan [DR80]) Let v = (v_{1}, …, v_{n}) be an isotropic frame in ℝ^{2n}. Then Γ · v is dense in the space of isotropic n-frames iff the space spanned by {v_{i} : i = 1, …, n} contains no nonzero rational vectors.

Figure 1

A result similar to Theorem 3 holds in this case too. Denote by W_{n} the space of 2ndimensional n-frames that are isotropic with respect to the standard symplectic form J. Note that W_{n} is an open subset of an algebraic set in ∏_{i=1}^{n} ℝ^{2n}. Since by Witt’s theorem Sp(n, ℝ) acts transitively on W_{n}, W_{n} is a submanifold of ∏_{i=1}^{n} ℝ^{2n}.

We improve Theorem 5 by showing that dense orbits of Γ are uniformly distributed:

Theorem 6. Let Γ be a lattice in Sp(n, ℝ), and v^{0} ∈ W_{n} be such that Γ · v^{0} is dense in W_{n}. Let Ω be a relatively compact Borel subset of W_{n} such that the boundary of Ω has measure 0 in the Lebesgue measure class. Then

N_{T} (Ω, v^{0}) ∼ λ_{v^{0}}(Ω)T^{n(n+1)/2} as T → ∞

for some measure λ_{v^{0}} on W_{n} in the Lebesgue measure class, which can be explicitly computed.

Note that the measure λ_{v^{0}} is not Sp(n, ℝ)-invariant.

In the next section we show how to derive asymptotic distribution for counting functions similar to N_{T} (Ω, v^{0}) from uniform distribution of orbits of subgroups of G in the space Γ\G. In section 3, we consider the case G = SL(n, ℝ). First, for convenience of the reader, we sketch an easy proof of Theorems 1 and 2 based on topological rigidity of unipotent flows, which was established by Ratner [Rat91b]. Then we introduce a decomposition of G based on the Iwasawa decomposition, and obtain results on volume of balls in the subgroup B_{l}^{o}, which is defined below. This allows us to use results from Section 2 to prove Theorem 3 and Corollary 4 modulo ergodic theorem along balls in the group B_{l}^{o} (Theorem 20). In Section 4, we prove the ergodic theorem for B_{l}^{o}. Note that for l = n − 1 it is a special case of the result of Shah [Sh94]. The proof of the ergodic theorem is similar to the proof of Theorem 2 from [Go02]. Finally, in Section 5 we consider the case G = Sp(n, ℝ), and prove Theorem 6. The method of the proof is similar to one used for Theorem 3: we use Iwasawa decomposition for Sp(n, ℝ) and uniform distribution of large unipotent subgroups due to Shah [Sh94]. In the Appendix, we prove some technical volume estimates and Corollary 4.

Remark. In the definition of N_{T} (Ω, v^{0}), we used the norm (1). The fact that this norm is invariant under the orthogonal group made our calculations easier. However, one can prove similar results for every norm on M(n, ℝ) with possibly different limit measure in the Lebesgue measure class. Acknowledgment: I would like to thank H. Furstenberg for fruitful discussions and Barak Weiss for some comments and for pointing out an error in the preliminary version of this paper. I am also very grateful to V. Bergelson for his encouragement and for many interesting discussions.

2. Some limit theorems

In this section we establish asymptotics of some counting functions.

Let G ⊆ SL(n, ℝ) be a Lie group, Γ a lattice in G, and H a Lie subgroup of G. Denote by ϱ a right Haar measure on H. Let μ be a Haar measure on G, and μ̄ be the measure on Γ\G such that

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

Throughout this section, we assume that for some M > 0 and every c > 0, m ∈ ℝ,

lim_{T →∞} \frac{ϱ(H_{cT +m})}{ϱ(H_{T})} = c^{M}.  (5)

where H_{T} = {h ∈ H : ∥h∥ < T }, and for every x ∈ Γ\G such that xH is dense in Γ\G and f̃ ∈ C_{c}(Γ\G),

\frac{1}{ϱ(H_{T})} ∫_{H_{T}} f̃(xh^{−1})dϱ(h) → \frac{1}{μ̄(Γ\G)} ∫_{Γ\G} f̃dμ as T → ∞.  (6)

First, we prove an elementary lemma: Lemma 7. Let (V, ∥ · ∥) be a normed vector space, G be a topological group, and ρ : G → B(V)^{∗} (= the space of bounded invertible linear operators on V) be a continuous map (w.r.t. norm topology). Then for any g_{0} ∈ G and k > 1, there exists a neighborhood O_{g_{0}} of g_{0} in G such that for any g ∈ O_{g_{0}} and v ∈ V,

k^{−1}∥ρ(g_{0})v∥ ≤ ∥ρ(g)v∥ ≤ k∥ρ(g_{0})v∥.

Proof. Take

O_{g_{0}} = {g : ∥ρ(g_{0})ρ(g)^{−1}∥ < k, ∥ρ(g)ρ(g_{0})^{−1}∥ < k}.

Then for g ∈ O_{g_{0}},

∥ρ(g_{0})v∥ ≤ ∥ρ(g_{0})ρ(g)^{−1}∥ · ∥ρ(g)v∥ < k∥ρ(g)v∥.

Similarly, for g ∈ O_{g_{0}},

∥ρ(g)v∥ ≤ ∥ρ(g)ρ(g_{0})^{−1}∥ · ∥ρ(g_{0})v∥ < k∥ρ(g_{0})v∥.
□

For g ∈ SL(n, ℝ), denote by ĝ : x ↦ gxg^{−1} the inner automorphism corresponding to g. For a subgroup L of G, denote by N_{G}(L) the normalizer of L in G. For g ∈ N_{G}(H), define

Δ_{H}(g) = |det (Ad(g)|_{Lie(H)})|  (7)

where Ad(g) is the adjoint transformation of the Lie algebra of G. Proposition 8. Let x_{0} ∈ G be such that Γx_{0}H is dense in Γ\G. Let

g ↦ c_{g} : G → N_{G}(H),
g ↦ d_{g} : G → H,
g ↦ e_{g} : G → ℝ,
g ↦ m_{g} : G → ℝ_{+},

be continuous maps that factor through G/x̂_{0}(H). Then for every f ∈ C_{c}(G),

lim_{T →∞} \frac{1}{ϱ(H_{T})} ∑_{γ∈Γ} ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} f(γx_{0}h^{−1}x_{0}^{−1})dϱ(h)  (8)

= \frac{1}{μ̄(Γ\G)} ∫_{G} \frac{f(g)}{m_{g}^{M} · Δ_{H}(c_{g})}dμ(g).

Proof. We shall assume without loss of generality that f ≥ 0.

There exist real M_{1} and M_{2} such that M_{1} ≤ e_{g} ≤ M_{2} for g ∈ supp(f)x̂_{0}(H). Then for g ∈ supp(f)x̂_{0}(H),

{h : m_{g}^{2}∥ĉ_{g}(hd_{g})∥^{2} + e_{g} < T^{2}} ⊆ {h : m_{g}∥ĉ_{g}(hd_{g})∥ < (T^{2} − M_{1})^{1/2}}.  (9)

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

Let r > 1 and ε > 0. By Lemma 7, for any g_{0} ∈ G there exists a neighborhood O_{g_{0}} of g_{0} such that

r^{−1}∥ĉ_{g_{0}}(v)∥ < ∥ĉ_{g}(v)∥ < r∥ĉ_{g_{0}}(v)∥,  (10)
r^{−1}∥vĉ_{g_{0}}(d_{g_{0}})∥ < ∥vĉ_{g_{0}}(d_{g})∥ < r∥vĉ_{g_{0}}(d_{g_{0}})∥  (11)

for all g ∈ O_{g_{0}} x̂_{0}(H) and v ∈ M(n, ℝ). Moreover, O_{g_{0}} can be taken such that

|\frac{1}{m_{g}^{M} · Δ_{H}(c_{g})} − \frac{1}{m_{g_{0}}^{M} · Δ_{H}(c_{g_{0}})}| < ε,

and

r^{−1}m_{g_{0}} ≤ m_{g} ≤ rm_{g_{0}}

for all g ∈ O_{g_{0}} x̂_{0}(H).

Note that for every v ∈ N_{G}(H), Γx_{0}vH is dense in Γ\G. Therefore, by (6), for every u ∈ G and v ∈ N_{G}(H),

lim_{T →∞} \frac{1}{ϱ(H_{T})} ∫_{H_{T}} f̃(Γx_{0}vh^{−1}u)dϱ(h) = \frac{1}{μ̄(Γ\G)} ∫_{Γ\G} f̃(yu)dμ̄(y)
= \frac{1}{μ̄(Γ\G)} ∫_{Γ\G} f̃dμ̄.  (12)

To prove (8), we first suppose that supp(f) ⊆ O_{g_{0}} for some g_{0} ∈ G. Put c_{0} = c_{g_{0}}, d_{0} = d_{g_{0}}, m_{0} = m_{g_{0}}. Then using (9), (10), and (11), we get

∑_{γ∈Γ} ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} f(γx_{0}h^{−1}x_{0}^{−1})dϱ(h)

≤ ∑_{γ∈Γ} ∫_{m_{γ}∥ĉ_{γ}(hd_{γ})∥<(T^{2}−M_{1})^{1/2}} f(γx_{0}h^{−1}x_{0}^{−1})dϱ(h)

≤ ∑_{γ∈Γ} ∫_{r^{−2}m_{0}∥ĉ_{0}(h)ĉ_{0}(d_{γ})∥<(T^{2}−M_{1})^{1/2}} f(γx_{0}h^{−1}x_{0}^{−1})dϱ(h)

≤ ∑_{γ∈Γ} ∫_{r^{−3}m_{0}∥ĉ_{0}(h)ĉ_{0}(d_{0})∥<(T^{2}−M_{1})^{1/2}} f(γx_{0}h^{−1}x_{0}^{−1})dϱ(h)

= ∫_{∥h∥<r^{3}m_{0}^{−1}(T^{2}−M_{1})^{1/2}} f̃(Γx_{0}c_{0}^{−1}ĉ_{0}(d_{0})h^{−1}c_{0}x_{0}^{−1})\frac{dϱ(h)}{Δ_{H}(c_{0})}.

Thus, by (12) and (5),

lim sup_{T →∞} \frac{1}{ϱ(H_{T})} ∑_{γ∈Γ} ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} f(γx_{0}h^{−1}x_{0}^{−1})dϱ(h)

≤ lim sup_{T →∞} \frac{ϱ(H_{r^{3}m_{0}^{−1}(T^{2}−M_{1})^{1/2}})}{ϱ(H_{T})} · \frac{1}{μ̄(Γ\G)Δ_{H}(c_{0})} ∫_{Γ\G} f̃dμ̄

= \frac{r^{3M}}{m_{0}^{M} · μ̄(Γ\G) · Δ_{H}(c_{0})} ∫_{Γ\G} f̃dμ̄

= \frac{r^{3M}}{m_{0}^{M} · μ̄(Γ\G) · Δ_{H}(c_{0})} ∫_{G} fdμ.

Now let f be arbitrary. There exists a finite cover supp(f) ⊆ ∪_{i}O_{g_{i}}. Let c_{i} = c_{g_{i}} and m_{i} = m_{g_{i}}. Let α_{i} ∈ C_{c}(G) be a partition of unity for {O_{g_{i}}} such that ∑_{i} α_{i} = 1 on supp(f). Then

lim sup_{T →∞} \frac{1}{ϱ(H_{T})} ∑_{γ∈Γ} ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} f(γx_{0}h^{−1}x_{0}^{−1})dϱ(h)

= lim sup_{T →∞} \frac{1}{ϱ(H_{T})} ∑_{γ∈Γ} ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} \{ ∑_{i} f(γx_{0}h^{−1}x_{0}^{−1})

·α_{i}(γx_{0}h^{−1}x_{0}^{−1})\}dϱ(h) ≤ \frac{r^{3M}}{μ̄(Γ\G)} ∑_{i} ∫_{G} \frac{f(g)}{m_{i}^{M}Δ_{H}(c_{i})}α_{i}(g)dμ(g)

≤ \frac{r^{3M}}{μ̄(Γ\G)} ∑_{i} ∫_{G} (\frac{f(g)}{m_{g}^{M} · Δ_{H}(c_{g})} + εf(g)) α_{i}(g)dμ(g)

= \frac{r^{3M}}{μ̄(Γ\G)} ∫_{G} \frac{f(g)}{m_{g}^{M} · Δ_{H}(c_{g})}dμ(g) + \frac{r^{3M}ε}{μ̄(Γ\G)} ∫_{G} fdμ

for every r > 1 and ε > 0. Therefore,

lim sup_{T →∞} \frac{1}{ϱ(H_{T})} ∑_{γ∈Γ} ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} f(γx_{0}hx_{0}^{−1})dϱ(h)

≤ \frac{1}{μ̄(Γ\G)} ∫_{G} \frac{f(g)}{m_{g}^{M} · Δ_{H}(c_{g})}dμ(g).  (13)

Similarly, one can prove the lower bound for (8). □

Proposition 9. Let f be the characteristic function of a relatively compact Borel subset Z ⊆ G such that μ(∂Z) = 0. Let x_{0} ∈ G be such that Γx_{0}H is dense in Γ\G. Then (8) holds for f.

Proof. Denote by int(Z) and \overline{Z} the interior and the closure of Z respectively.

Let W_{0} be an open relatively compact subset such that \overline{Z} ⊆ W_{0}. There exists C > 0 such that (m_{g}^{M} · Δ_{H}(c_{g}))^{−1} ≤ C for g ∈ W_{0}.

Let ε > 0. There exist a compact subset V ⊆ int(Z) and an open subset W such that \overline{Z} ⊆ W ⊆ W_{0} and μ(W − V) < ε. Take functions f_{1}, f_{2} ∈ C_{c}(G) such that 0 ≤ f_{i} ≤ 1, f_{1} = 1 on V, f_{1} = 0 outside int(Z), f_{2} = 1 on \overline{Z}, and f_{2} = 0 outside W. Then f_{1} ≤ f ≤ f_{2}. By Proposition 8 applied to f_{2},

lim sup_{T →∞} \frac{1}{ϱ(H_{T})} ∑_{γ∈Γ} ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} f(γx_{0}h^{−1}x_{0}^{−1})dϱ(h)

≤ \frac{1}{μ̄(Γ\G)} ∫_{G} \frac{f_{2}(g)}{m_{g}^{M} · Δ_{H}(c_{g})}dμ(g)
≤ \frac{1}{μ̄(Γ\G)} (∫_{G} \frac{f(g)}{m_{g}^{M} · Δ_{H}(c_{g})}dμ(g) + ∫_{G} \frac{f_{2}(g) − f_{1}(g)}{m_{g}^{M} · Δ_{H}(c_{g})}dμ(g))
≤ \frac{1}{μ̄(Γ\G)} ∫_{G} \frac{f(g)}{m_{g}^{M} · Δ_{H}(c_{g})}dμ(g) + \frac{Cμ(W − V)}{μ̄(Γ\G)}
≤ \frac{1}{μ̄(Γ\G)} ∫_{G} \frac{f(g)}{m_{g}^{M} · Δ_{H}(c_{g})}dμ(g) + \frac{Cε}{μ̄(Γ\G)}

for every ε > 0. This shows (13). The dual inequality for lim inf can be proved similarly. □

Suppose that for a closed subset Y of G, the product map Y × x̂_{0}(H) → G be a homeomorphism. For g ∈ G, define y_{g} ∈ Y and h_{g} ∈ H such that g = y_{g}x̂_{0}(h_{g}). The map

α : y ↦ y · x̂_{0}(H) : Y → G/x̂_{0}(H).

is a homeomorphism too. Let ν_{1} be a measure on Y such that

∫_{G} fdμ = ∫_{Y} ∫_{H} f(yx̂_{0}(h))dϱ(h)dν_{1}(y), f ∈ C_{c}(G).  (14)

Note that such a measure exists because μ and ϱ are right invariant. Let ν be the measure on G/x̂_{0}(H) which is the image of ν_{1} under α, i.e.

∫_{Y} f(α(y))dν_{1}(y) = ∫_{G/x̂_{0}(H)} fdν, f ∈ C_{c}(G/x̂_{0}(H)).  (15)

Note that the measure ν depends on the choice of the section Y. Proposition 10. Use notations as in Proposition 8. Let Ω be relatively compact Borel subset of G/x̂_{0}(H) such that ν(∂Ω) = 0. Let

N_{T} (Ω) = |{γ ∈ Γ : m_{γ}^{2}∥ĉ_{γ}(h_{γ}d_{γ})∥^{2} + e_{γ} < T^{2}, γ · x̂_{0}(H) ∈ Ω)}|.

Then

\frac{N_{T} (Ω)}{ϱ(H_{T})} ∼ \frac{1}{μ̄(Γ\G)} ∫_{α^{−1}(Ω)} \frac{1}{m_{y}^{M} · Δ_{H}(c_{y})}dν_{1}(y)

as T → ∞.

Proof. Let

O_{ε} = {h ∈ H : ∥h − 1∥ < ε, ∥h^{−1} − 1∥ < ε}

for ε > 0.

Let ϕ be the characteristic function of α^{−1}(Ω) ⊆ Y. Take ψ_{ε} to be the characteristic function of O_{ε} normalized so that ∫_{H} ψ_{ε}dϱ = 1. Let

f_{ε}(g) = ϕ(y_{g})ψ_{ε}(h_{g}) for g ∈ G.  (16)

Note that f satisfies conditions of Proposition 9, but before applying this proposition, we need a lemma.

Lemma 11. For every r > 1, there exists ε > 0 such that

N_{r^{−1}T} (Ω) ≤ ∑_{γ∈Γ} ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} f_{ε}(γx_{0}h^{−1}x_{0}^{−1})dϱ(h) ≤ N_{rT} (Ω).  (17)

Proof. Note that f_{ε}(γx_{0}h^{−1}x_{0}^{−1}) = 0 for all h ∈ H unless

y_{γ} ∈ α^{−1}(Ω),  (18)

and if the above condition holds,

∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} f_{ε}(γx_{0}h^{−1}x_{0}^{−1})dϱ(h)
= ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<T^{2}} ψ_{ε}(h_{γ}h^{−1})dϱ(h)
= ∫_{m_{γ}^{2}∥ĉ_{γ}(hh_{γ}d_{γ})∥^{2}+e_{γ}<T^{2}} ψ_{ε}(h^{−1})dϱ(h)
= ∫_{m_{γ}^{2}∥ĉ_{γ}(h)ĉ_{γ}(h_{γ}d_{γ})∥^{2}+e_{γ}<T^{2}} ψ_{ε}(h)dϱ(h).  (19)

Let

I_{γ} =^{def} ∫_{m_{γ}^{2}∥ĉ_{γ}(h)ĉ_{γ}(h_{γ}d_{γ})∥^{2}+e_{γ}<T^{2}} ψ_{ε}(h)dϱ(h).

For γ as in (18), there exists C > 0 such that

∥ĉ_{γ}(v)∥ ≤ C∥v∥ and ∥ĉ_{γ}^{−1}(v)∥ ≤ C∥v∥ for all v ∈ M(n, ℝ).

It follows that for every ε > 0,

O_{ε/C} ⊆ ĉ_{γ} (O_{ε}) ⊆ O_{Cε}.

Therefore, by Lemma 7, there exists ε > 0 such that

r^{−1}∥v∥ ≤ ∥ĉ_{γ}(h)v∥ ≤ r∥v∥  (20)

for every v ∈ M(n, ℝ), h ∈ O_{ε}, and γ ∈ Γ such that (18) holds.

Let γ ∈ Γ be such that

m_{γ}^{2}∥ĉ_{γ}(h_{γ}d_{γ})∥^{2} + e_{γ} < r^{−2}T^{2},

and (18) holds. Then by (20),

m_{γ}^{2}∥ĉ_{γ}(h)ĉ_{γ}(h_{γ}d_{γ})∥^{2} + e_{γ} ≤ r^{2}m_{γ}^{2}∥ĉ_{γ}(h_{γ}d_{γ})∥^{2} + e_{γ} < T^{2}

for h ∈ O_{ε}. It follows that the I_{γ} = 1. This proves the first inequality in (17). Note that I_{γ} ≤ 1. Let γ ∈ Γ be such that I_{γ} ≠ 0. Then (18) holds, and for some h ∈ O_{ε},

m_{γ}^{2}∥ĉ_{γ}(h)ĉ_{γ}(h_{γ}d_{γ})∥^{2} + e_{γ} < T^{2}.

Using (20), we deduce that

m_{γ}^{2}∥ĉ_{γ}(h_{γ}d_{γ})∥^{2} + e_{γ} < r^{2}T^{2}.

This proves the second inequality in (17). □

Now we can use Proposition 9 to find asymptotics for N_{T} (Ω). By Lemma 11, for every r > 1 there exists ε > 0 such that

lim sup_{T →∞} \frac{N_{T} (Ω)}{ϱ(H_{T})}
≤ lim sup_{T →∞} \frac{1}{ϱ(H_{T})} ∑_{γ∈Γ} ∫_{m_{γ}^{2}∥ĉ_{γ}(hd_{γ})∥^{2}+e_{γ}<r^{2}T^{2}} f_{ε}(γx_{0}h^{−1}x_{0}^{−1})dϱ(h).

Therefore, by Proposition 9, (5), (14), and (15),

lim sup_{T →∞} \frac{N_{T} (Ω)}{ϱ(H_{T})}
≤ (lim sup_{T →∞} \frac{ϱ(H_{rT})}{ϱ(H_{T})}) \frac{1}{μ̄(Γ\G)} ∫_{G} \frac{f_{ε}(g)}{m_{g}^{M} · Δ_{H}(c_{g})}dμ(g)

= \frac{r^{M}}{μ̄(Γ\G)} ∫_{Y ×H} \frac{f_{ε}(yx̂_{0}(h))}{m_{y}^{M} · Δ_{H}(c_{y})}dν_{1}(y)dϱ(h)

= \frac{r^{M}}{μ̄(Γ\G)} ∫_{Y} \frac{ϕ(y)}{m_{y}^{M} · Δ_{H}(c_{y})}dν_{1}(y) ∫_{H} ψ_{ε}(h)dϱ(h)

= \frac{r^{M}}{μ̄(Γ\G)} ∫_{α^{−1}(Ω)} \frac{dν_{1}(y)}{m_{y}^{M} · Δ_{H}(c_{y})}.

Taking r → 1+, we get

lim sup_{T →∞} \frac{N_{T} (Ω)}{ϱ(H_{T})} ≤ \frac{1}{μ̄(Γ\G)} ∫_{α^{−1}(Ω)} \frac{dν_{1}(y)}{m_{y}^{M} · Δ_{H}(c_{y})}.

Similarly, one can prove that

lim inf_{T →∞} \frac{N_{T} (Ω)}{ϱ(H_{T})} ≥ \frac{1}{μ̄(Γ\G)} ∫_{α^{−1}(Ω)} \frac{dν_{1}(y)}{m_{y}^{M} · Δ_{H}(c_{y})}.

This proves the Proposition. □

3. Uniform distribution for a lattice in SL(n, ℝ)

3.1. Density of orbits.

In this section we derive Theorems 1 and 2 from the following result on topological rigidity of unipotent flow, which was proved by M. Ratner: Theorem 12. (Ratner [Rat91b]) Let G be a connected Lie group, Γ be a lattice in G, and U be a subgroup of G generated by Ad-unipotent 1-parameter subgroups. Then for every x ∈ Γ\G, \overline{xU} = xH, where H is a closed connected subgroup of G such that U ⊆ H, and xH supports finite H-invariant Borel measure.

Note that the proofs of Dani, Raghavan, Veech are different from the proofs that are presented here. In fact, their proofs can be considered as the first important steps towards the general result on topological rigidity – Theorem 12.

We start the proof of Theorem 1 with a simple lemma: Lemma 13. Let {v_{i} : i = 1, …, s} ⊆ ℝ^{n}, 1 ≤ s ≤ n − 2, be linearly independent vectors such that ⟨v_{i} : i = 1, …, s⟩ contains no nonzero rational vectors. Then there exists v_{s+1} such that v_{i}, i = 1, …, s + 1, are linearly independent, and ⟨v_{i} : i = 1, …, s + 1⟩ contains no nonzero rational vectors.

Proof. Let V = ⟨v_{i} : i = 1, …, s⟩. Since s ≤ n − 2, for any v ∈ ℝ^{n}, ⟨V, v⟩ is a proper subspace of ℝ^{n}. Therefore one can take a vector v_{s+1} outside ∪_{v∈ℚ^{n}} ⟨V, v⟩. If v = ∑_{i=1}^{s+1} α_{i}v_{i} is rational and nonzero for some α_{i} ∈ ℝ, then α_{s+1} ≠ 0, and v_{s+1} ∈ ⟨V, v⟩. This is a contradiction. Thus, v_{s+1} is as required. □

Proof of Theorem 1. It is easy to see that if the condition of the theorem is not satisfied, the orbit cannot be dense. The hard part is to show that the above condition implies density. By Lemma 13, we may assume that l = n − 1.

Denote G = SL(n, ℝ), Γ = SL(n, ℤ), and

U_{0} = ( \begin{matrix}E & ∗ \\ 0 & 1 \end{matrix} ) ,

where E is the identity (n − 1) × (n − 1) matrix. Let g_{0} ∈ G be such that g_{0}v_{i} = e_{i} for i = 1, …, n − 1. Here {e_{i} : i = 1, …, n} is the standard basis of ℝ^{n}. Then the stabilizer of v in G is U = g_{0}^{−1}U_{0}g_{0}. Note that any nontrivial U-invariant subspace of ℝ^{n} is contained in ⟨v_{i} : i = 1, …, n − 1⟩. Consider U-orbit ΓU ⊂ Γ\G. By Ratner’s theorem (Theorem 12), \overline{ΓU} = ΓH where H is a closed connected subgroup of G containing U, and H ∩Γ is a lattice in H. Moreover by [Sh91, Proposition 3.2], H is the connected component of the smallest real algebraic ℚ-subgroup containing U, and the radical of H is unipotent. Let R be the radical of H. Since R is defined over ℚ and unipotent, the space V^{R} of R-fixed vectors is nonzero and defined over ℚ. Also V^{R} is H-invariant because R is normal in H. Thus, if V^{R} ≠ ℝ^{n}, V^{R} ⊆ ⟨v_{i} : i = 1, …, n − 1⟩. However, this contradicts our hypothesis on v. Therefore, V^{R} = ℝ^{n} and R = 1, i.e. H is semisimple. We claim that H = G. To simplify notations, we work with the group H_{0} =^{def} g_{0}Hg_{0}^{−1}. Let 𝔥 and 𝔲 be the Lie algebras of H_{0} and U_{0} respectively. Since U_{0} ⊆ H_{0},

𝔲 = ⟨E_{in} : i = 1, …, n − 1⟩ ⊆ 𝔥.

Here E_{ij} denotes a matrix with 1 at the place (i, j) and 0 elsewhere. Using that the Killing form k(x, y) = Tr(xy) for x, y ∈ 𝔰𝔩(n, ℝ) is nondegenerate on 𝔥, one shows that the projection map from 𝔥 to the space ⟨E_{ni} : i = 1, …, n − 1⟩ with respect to the basis {E_{ij}} is surjective. Thus for i = 1, …, n−1, there exists h_{i} = E_{ni} +h̃_{i} ∈ 𝔥 with h̃_{i} in the normalizer of 𝔲. Then

𝔥 ⊇ [h_{i}, 𝔲] + 𝔲 ⊇ [E_{ni}, 𝔲], i = 1, …, n − 1.

It follows that 𝔥 = 𝔰𝔩(n, ℝ) and H = G. Thus, \overline{ΓU} = G. Finally,

\overline{Γv} = \overline{ΓUv} ⊇ \overline{ΓU}v = Gv = V_{n,l}.  (21)
□

Proof of Theorem 2. It is sufficient to prove the claim for l = n − 1.

Let U be as in the proof of Theorem 1. By (21), we just need to show that ΓU is dense in G. By Ratner’s theorem (Theorem 12), \overline{ΓU} = ΓH where H is a closed connected subgroup of G containing U, and H ∩ Γ is a lattice in H. By Lemma 3.8 and Proposition 3.10 from [Sh91], one of the following two possibilities holds: H is reductive, or W ∩ Γ is a lattice in W where W is the unipotent radical of a proper parabolic subgroup of G. Since Γ is cocompact, it follows from Godement’s criterion that Γ has no nontrivial unipotent elements. This contradicts the second possibility. Thus, H is reductive, and the Killing form is nondegenerate on the Lie algebra of H. Now one can show by the same argument as in the proof of Theorem 1 that H = G. Hence, ΓU is dense in Γ\G. This implies Theorem 2. □

3.2. Iwasawa decomposition for SL(n, ℝ).

Fix l = 1, …, n − 1.

For s = (s_{1}, …, s_{n}) ∈ ℝ^{n}, ∑_{i=1}^{n} s_{i} = 0, define

a(s) = diag(e^{s_{1}}, …, e^{s_{n}}) ∈ SL(n, ℝ).

For a vector s as above, define decomposition

s = s^{−} + s^{+}

with s^{−} = (s_{1}, …, s_{l}, r, …, r), r = \frac{1}{n−l}(−s_{1} − ⋯ − s_{l}), s^{+} = s − s^{−}. Note that r is chosen so that a(s^{+}), a(s^{−}) ∈ SL(n, ℝ).

For t = (t_{ij} : 1 ≤ i < j ≤ l), t_{ij} ∈ ℝ, denote by n^{−}(t) the unipotent upper triangular matrix which entries above diagonal are equal t_{ij} for i < j ≤ l and 0 otherwise. Similarly, for t = (t_{ij} : 1 ≤ i < j ≤ n, j > l), t_{ij} ∈ ℝ, denote by n^{+}(t) the unipotent upper triangular matrix which entries above diagonal are equal t_{ij} for 1 ≤ i < j ≤ n, j > l and 0 otherwise.

We use the following notations:

G = SL(n, ℝ),
K = SO(n, ℝ),
A_{l−}^{o} = {a(s^{−}) : s ∈ ℝ^{n}, ∑_{i=1}^{n} s_{i} = 0}

A_{l+}^{o} = {a(s^{+}) : s ∈ ℝ^{n}, ∑_{i=1}^{n} s_{i} = 0} ,

A^{o} = A_{l−}^{o}A_{l+}^{o} = {a(s) : s ∈ ℝ^{n}, ∑_{i=1}^{n} s_{i} = 0} ,
N_{l−} = {n^{−}(t) : t_{ij} ∈ ℝ, 1 ≤ i < j ≤ l},
N_{l+} = {n^{+}(t) : t_{ij} ∈ ℝ, 1 ≤ i < j ≤ n, j > l},
N = N_{l−}N_{l+} = “unipotent upper triangular group”,
B_{l}^{o} = A_{l+}^{o}N_{l+} = N_{l+}A_{l+}^{o}.

Denote by dk the normalized Haar measure on K.

Let

dn^{+} = dt^{+} = ∏_{i<j≤l} dt_{ij} and dn^{−} = dt^{−} = ∏_{max(i,l)<j} dt_{ij}.

These measures are Haar measures for N_{l+} and N_{l−} respectively. The subgroup N_{l+} is normal in N, and the product map

N_{l−} × N_{l+} → N

is a diffeomorphism. Also the image of product of dn^{−} and dn^{+} under this map is a Haar measure on N. Let us denote it by dn:

∫_{N} f(n)dn = ∫_{N_{l−}×N_{l+}} f(n^{−}n^{+})dn^{−}dn^{+}, f ∈ C_{c}(N).  (22)

Haar measures on A_{l−}^{o} and A_{l+}^{o} are defined by

da^{−} = ds^{−} = ∏_{i=1}^{l} dsi^{−} and da^{+} = ds^{+} = ∏_{i=l+1}^{n−1} dsi^{+}

respectively. Then a Haar measure da on A^{o} = A_{l−}^{o}A_{l+}^{o} is the product measure:

∫_{A^{o}} f(a)da = ∫_{A_{l−}^{o}×A_{l+}^{o}} f(a^{−}a^{+})da^{−}da^{+}, f ∈ C_{c}(A^{o}).  (23)

The product map A_{l+}^{o} × N_{l+} → B_{l}^{o} is a diffeomorphism, and the image of the product measure under this map is a left Haar measure on B_{l}^{o}. Denote this measure by λ_{l}. Then a right Haar measure ϱ_{l} on B_{l}^{o} can be defined by

ϱ_{l}(f) = ∫_{B_{l}^{o}} f(b^{−1})λ_{l}(b) = ∫_{A_{l+}^{o}×N_{l+}} f(a(s^{+})n^{+})δ_{l}^{+}(s^{+})ds^{+}dn^{+}  (24)

for f ∈ C_{c}(B_{l}^{o}), where

δ_{l}^{+}(a) = δ_{l}^{+}(s) = exp {2 ∑_{i=l+1}^{n}(n − i)s_{i}}  (25)

for a = diag(e^{s_{1}}, …, e^{s_{n}}) ∈ SL(n, ℝ).

By Iwasawa decomposition, the map

(k, n, a) ↦ kna : K × N × A^{o} → G  (26)

is a diffeomorphism. Lemma 14. Let e^{0} = (e_{1}, …, e_{l}) be the standard orthonormal frame in ℝ^{n}. Then for k ∈ K, n ∈ N, and a = diag(e^{s_{1}}, …, e^{s_{n}}) ∈ A^{o},

Vol(knae^{0}) = exp {∑_{i=1}^{l} s_{i}} .

Proof. Let g = kna. Recall that Iwasawa decomposition is proved using Gramm-Schmidt orthogonalization for basis v_{i} = ge_{i}. Let {w_{i}} be an orthonormal basis such that ⟨v_{k} : k ≤ i⟩ = ⟨w_{k} : k ≤ i⟩ for 1 ≤ i ≤ n. Then e^{s_{i}} = v_{i} ·w_{i}, i.e. e^{s_{i}} is the length of projection of v_{i} onto the orthogonal complement of ⟨v_{k} : k ≤ i − 1⟩ in ⟨v_{k} : k ≤ i⟩. Now the statement is obvious. □

Define

δ_{l}^{−}(a) = δ_{l}^{−}(s) = exp {∑_{i=1}^{l} s_{i}} ,  (27)

where a = diag(e^{s_{1}}, …, e^{s_{n}}) ∈ SL(n, ℝ).

The image of the product measure under the map (26) is a Haar measure on G [He, Proposition X.1.12]. Let us denote this image by μ:

∫_{G} fdμ = ∫_{K×N×A^{o}} f(kna)dkdnda, f ∈ C_{c}(G).  (28)

For a lattice Γ in G, there exists a measure μ̄ on Γ\G such that

∫_{G} fdμ = ∫_{Γ\G} ∑_{γ∈Γ} f(γg)dμ̄(g), f ∈ C_{c}(G).  (29)

For our purposes, we modify the Iwasawa decomposition as follows:

(k, n^{−}, a^{−}, b) ↦ kn^{−}a^{−}b : K × N_{l−} × A_{l−}^{o} × B_{l}^{o} → G.  (30)

Since A_{l−}^{o} normalizes B_{l}^{o}, this map is a diffeomorphism too.

Fix g_{0} ∈ G. By (30) the map

(k, n^{−}, a^{−}, b) ↦ kn^{−}a^{−}g_{0}b : K × N_{l−} × A_{l−}^{o} × (B_{l}^{o})^{g_{0}} → G  (31)

is a diffeomorphism. Here we use notation: X^{g_{0}} = g_{0}^{−1}Xg_{0}.

For g ∈ G define k_{g} ∈ K, ag^{±} ∈ A_{l±}^{o}, and ng^{±} ∈ N_{l±} such that

g = k_{g}ng^{−}ag^{−}g_{0}(ag^{+}ng^{+})^{g_{0}}.  (32)

Also define

b_{g}^{−} = ng^{−}ag^{−}, b_{g} = ag^{+}ng^{+},

UNIFORM DISTRIBUTION OF ORBITS OF LATTICES ON SPACES OF FRAMES 14

and b′_{g} ∈ GL(l, ℝ) such that

b_{g}^{−} = ( \begin{matrix}b′_{g} & 0 \\ 0 & ∗ \end{matrix} ) .  (33)

Lemma 15. For f ∈ C_{c}(G) and g_{0} ∈ G,

∫_{G} fdμ = ∫_{K×N_{l−}×A_{l−}^{o}×B_{l}^{o}} f(kn^{−}a(s^{−})g_{0}b^{g_{0}})δ_{l}^{−}(s^{−})^{n}dkdn^{−}ds^{−}dϱ_{l}(b),

where δ_{l}^{−} is defined in (27). Proof. By (28), (22), and (23),

∫_{G} fdμ = ∫_{K×N_{l−}×N_{l+}×A_{l−}^{o}×A_{l+}^{o}} f(kn^{−}n^{+}a^{−}a^{+})dkdn^{−}dn^{+}da^{−}da^{+}.  (34)

The Jacobian of the map

N_{l+} → N_{l+} : n ↦ a^{−1}na

for a = a(s^{−})a(s^{+}) is equal to δ_{l}^{−}(s^{−})^{−n}δ_{l}^{+}(s^{+})^{−1}. Thus, it follows from (24) and (34) that

∫_{G} fdμ = ∫_{K×N_{l−}×A_{l−}^{o}×B_{l}^{o}} f(kn^{−}a(s^{−})b)δ_{l}^{−}(s^{−})^{n}dkdn^{−}ds^{−}dϱ_{l}(b).

Then since G is unimodular,

∫_{G} fdμ = ∫_{G} f(gg_{0})dμ(g)
= ∫_{K×N_{l−}×A_{l−}^{o}×B_{l}^{o}} f(kn^{−}a(s^{−})bg_{0})δ_{l}^{−}(s^{−})^{n}dkdn^{−}ds^{−}dϱ_{l}(b)
= ∫_{K×N_{l−}×A_{l−}^{o}×B_{l}^{o}} f(kn^{−}a(s^{−})g_{0}b^{g_{0}})δ_{l}^{−}(s^{−})^{n}dkdn^{−}ds^{−}dϱ_{l}(b).
□

Lemma 16. Let e^{0} = (e_{1}, …, e_{l}) be the standard frame in ℝ^{n}. For f ∈ C_{c}(V_{n,l}),

∫_{V_{n,l}} f(v)dv = d_{n,l} ∫_{K×N_{l−}×A_{l−}^{o}} f(kn^{−}a(s^{−})e^{0})δ_{l}^{−}(s^{−})^{n}dkdn^{−}ds^{−},  (35)

where d_{n,l} is a constant computed in (73). Proof. The measure on the left side of (35) is G-invariant. We claim that the measure on the right side is G-invariant too. It is easy to see that the map

gB_{l}^{o} ↦ ge^{0} : G/B_{l}^{o} → V_{n,l}

is proper. Thus, every function f ∈ C_{c}(V_{n,l}) can be lifted to a function f_{1} ∈ C_{c}(G/B_{l}^{o}). Then the function f can be represented as

f(ge^{0}) = f_{1}(gB_{l}^{o}) = ∫_{B_{l}^{o}} f_{2}(gb)dϱ_{l}(b) for some f_{2} ∈ C_{c}(G) (see [Rag, Ch. 1]). Then by Lemma 15,

∫_{K×N_{l−}×A_{l−}^{o}} f(kn^{−}a(s^{−})e^{0})δ_{l}^{−}(s^{−})^{n}dkdn^{−}ds^{−} = ∫_{G} f_{2}dμ.

It follows that the measure on the right side of (35) is G-invariant. By uniqueness of Haar measure, the integrals are equal up to a scalar multiple d_{n,l}. This constant is computed in the Appendix. □

3.3. Volume estimates.

For a set S ⊆ G and T > 0, define

S_{T} = {s ∈ S : ∥s∥ < T }.

We compute the asymptotics of ϱ_{l}(B_{l,T}^{o}) as T → ∞: Lemma 17.

ϱ_{l}(B_{l,T}^{o}) ∼ γ_{n,l}T^{(n−1)(n−l)} as T → ∞,  (36)

where the constant γ_{n,l} is given in (80).

The proof, which is given in the Appendix, follows the method of Duke, Rudnick, Sarnak [DRS93].

For C ∈ ℝ, define

A_{l+}^{C} = {a(s^{+}) : si^{+} > C, i = l + 1, …, n − 1},
B_{l}^{C} = A_{l+}^{C}N_{l+}.

The following “measure concentration” result plays a crucial role in our proof. Lemma 18. For C ∈ ℝ,

ϱ_{l}(B_{l,T}^{C}) ∼ ϱ_{l}(B_{l,T}^{o}) as T → ∞.

This lemma is proved in the Appendix.

3.4. Uniform distribution.

Theorem 19. Let Γ be a lattice in G = SL(n, ℝ). Fix g_{0} ∈ (KN_{l−}A_{l−}^{o})^{−1} such that Γ(B_{l}^{o})^{g_{0}} is dense in G. Let Y = KN_{l−}A_{l−}^{o}g_{0}, and ν_{1} be a measure on Y such that

∫_{G} fdμ = ∫_{Y} ∫_{B_{l}^{o}} f(yb^{g_{0}})dϱ_{l}(b)dν_{1}(y), f ∈ C_{c}(G).  (37)

Let ν be a measure on G/(B_{l}^{o})^{g_{0}} induced by ν_{1}. For T > 0 an Ω ⊆ G/(B_{l}^{o})^{g_{0}}, denote

N_{T} (Ω, g_{0}) = |{γ ∈ Γ : ∥γ∥ < T, γ(B_{l}^{o})^{g_{0}} ∈ Ω}|.  (38)

Then for relatively compact Borel subset Ω of G/(B_{l}^{o})^{g_{0}} such that ν(∂Ω) = 0,

N_{T} (Ω, g_{0}) ∼ ϱ(B_{l,T}^{o})\frac{δ_{l}^{−}(a_{0})^{1−n}}{μ̄(Γ\G)} ∫_{Ω} \frac{dν(x)}{δ_{l}^{−}(ax^{−})} as T → ∞,

where a_{0} and ax^{−} are the A_{l−}^{o}-components of g_{0}^{−1} and x with respect to decomposition (30) respectively.

Note that a similar result holds for every g_{0} ∈ G because of the decomposition (30).

Proof. Write g_{0}^{−1} = k_{0}b_{0}^{−} for k_{0} ∈ K, b_{0}^{−} = n_{0}a_{0} ∈ N_{l−}A_{l−}^{o}.

It is convenient to use decomposition (32). The product map Y × (B_{l}^{o})^{g_{0}} → G is a diffeomorphism. For g ∈ G, denote y_{g} = k_{g}ng^{−}ag^{−}g_{0}, the Y -component of g. The map

α : Y → G/(B_{l}^{o})^{g_{0}} : y ↦ y(B_{l}^{o})^{g_{0}}

is a diffeomorphism. Clearly, γ(B_{l}^{o})^{g_{0}} ∈ Ω iff y_{γ} ∈ α^{−1}(Ω).

For g ∈ G,

∥g∥ = ∥k_{g}b_{g}^{−}b_{g}g_{0}∥ = ∥b_{g}^{−}b_{g}(b_{0}^{−})^{−1}∥.  (39)

Note that for ( \begin{matrix}∗ & X \\ 0 & Y \end{matrix} ) ∈ SL(n, ℝ),

b_{g}^{−} ( \begin{matrix}∗ & X \\ 0 & Y \end{matrix} ) (b_{0}^{−})^{−1} = ( \begin{matrix}∗ & β_{0}^{−1}(b′_{g} · X) \\ 0 & β_{0}^{−1}β_{g}Y \end{matrix} ) ,  (40)

where

β_{0} = det(b′_{0})^{− \frac{1}{n−l}} and β_{g} = det(b′_{g})^{− \frac{1}{n−l}}  (41)

(here b′_{0} and b′_{g} are defined as in (33)). Put

c_{g} = b_{g}^{−} and m_{g} = |β_{0}^{−1}β_{g}|.  (42)

It follows from (39) and (40) that for g ∈ G,

∥g∥^{2} = m_{g}^{2}∥ĉ_{g}(b_{g})∥^{2} + e_{g},

where e_{g} is a continuous function depending only on the b^{−}-components of g with respect to decomposition (32).

Using previous notations, we have

N_{T} (Ω, g_{0}) = |{γ ∈ Γ : m_{γ}^{2}∥ĉ_{γ}(b_{γ})∥^{2} + e_{γ} < T^{2}, y_{γ} ∈ α^{−1}(Ω)}|.  (43)

To derive asymptotics of N_{T} (Ω, g_{0}), we can use Proposition 10 with H = B_{l}^{o}, h_{γ} = b_{γ}, and d_{γ} = e. Note that by Lemma 17 the condition (5) for H = B_{l}^{o} is satisfied with M = (n − 1)(n − l), and by Theorem 20 below, the condition (6) holds. Therefore, applying Proposition 10, we get

N_{T} (Ω, g_{0}) ∼ \frac{ϱ(B_{l,T}^{o})}{μ̄(Γ\G)} ∫_{α^{−1}(Ω)} \frac{1}{m_{y}^{(n−1)(n−l)} · Δ_{H}(c_{y})}dν_{1}(y)

as T → ∞, where Δ_{H} is defined in (7). By (41) and (42),

m_{g} = (δ_{l}^{−}(a_{0})^{−1}δ_{l}^{−}(ag^{−}))^{− \frac{1}{n−l}} .

Also

Δ_{B_{l}^{o}}(c_{g}) = \frac{det(b′_{g})^{n−l}}{β_{g}^{l(n−l)}} = δ_{l}^{−}(ag^{−})^{n}.

Thus,

∫_{α^{−1}(Ω)} \frac{1}{m_{y}^{(n−1)(n−l)} · Δ_{H}(c_{y})}dν_{1}(y) = ∫_{α^{−1}(Ω)} \frac{δ_{l}^{−}(a_{0})^{1−n}}{δ_{l}^{−}(ay^{−})} dν_{1}(y)

= δ_{l}^{−}(a_{0})^{1−n} ∫_{Ω} \frac{dν(x)}{δ_{l}^{−}(ax^{−})}.

This proves the theorem. □

Proof of Theorem 3. For some g_{0} ∈ (KN_{l−}A_{l−}^{o})^{−1}, v^{0} = g_{0}^{−1}e^{0} where e^{0} = (e_{1}, …, e_{l}) is the standard frame. The condition that Γv^{o} is dense in V_{n,l} is equivalent to ΓG_{l}^{g_{0}} is dense in G where

G_{l} = ( \begin{matrix}E & ∗ \\ 0 & SL(n − l, ℝ) \end{matrix} ) .

Since B_{l}^{o} is epimorphic in G_{l}, it follows from [SW00, Corollary 1.3] that Γ(B_{l}^{o})^{g_{0}} is dense in G.

Consider a map

α : G/(B_{l}^{o})^{g_{0}} → V_{n,l} ≃ G/(G_{l})^{g_{0}} : g(B_{l}^{o})^{g_{0}} ↦ gv^{0}.

Note that this map is proper and G-equivariant. Put Ω^{∗} = α^{−1}(Ω). Then Ω^{∗} is relatively compact, and N_{T} (Ω, v^{0}) = N_{T} (Ω^{∗}, g_{0}), where N_{T} (Ω^{∗}, g_{0}) is defined in (38).

Let ν be the measure on G/(B_{l}^{o})^{g_{0}} defined in (37). It follows from Lemmas 15 and 16 that α(ν) = d_{n,l}^{−1}dv, where d_{n,l} is defined in (73) and dv is the Lebesgue measure on V_{n,l}.

One can check that α(∂Ω^{∗}) ⊆ ∂Ω. Therefore,

ν(∂Ω^{∗}) ≤ ν(α^{−1}(∂Ω)) = d_{n,l}^{−1} ∫_{∂Ω} dv = 0.

By Theorem 19,

N_{T} (Ω, v^{0}) ∼ ϱ(B_{l,T}^{o})\frac{δ_{l}^{−}(a_{0})^{1−n}}{μ̄(Γ\G)} ∫_{Ω^{∗}} \frac{dν(x)}{δ_{l}^{−}(ax^{−})} as T → ∞.

By Lemma 14, δ_{l}^{−}(a_{0}) = Vol(v^{0}). Using decomposition (32), we have gv^{0} = k_{g}ng^{−}ag^{−}e^{0}. Thus, by Lemma 14, δ_{l}^{−}(ag^{−}) = Vol(gv^{0}). Hence,

N_{T} (Ω, v^{0}) ∼ ϱ(B_{l,T}^{o})\frac{Vol(v^{0})^{1−n}}{d_{n,l}μ̄(Γ\G)} (∫_{Ω} \frac{dv}{Vol(v)}) as T → ∞.

Finally, using Lemma 17 and (73), we have

N_{T} (Ω, v^{0}) ∼ a_{n,l} \frac{Vol(v^{0})^{1−n}}{μ̄(Γ\G)} (∫_{Ω} \frac{dv}{Vol(v)}) T^{(n−1)(n−l)} as T → ∞,  (44)

where

a_{n,l} = \frac{γ_{n,l}}{d_{n,l}} .  (45)

The constants γ_{n,l} and d_{n,l} are computed in the Appendix. □

The proof of Corollary 4 is presented in the Appendix.

4. Ergodic Theorem

The main result of this section is the following ergodic theorem along balls in B_{l}^{o}. Theorem 20. Let 1 ≤ l ≤ n − 1. Let Γ be a lattice in G, and y ∈ Γ\G be such that yB_{l}^{o} is dense in Γ\G. Denote by ν the probability G-invariant measure on Γ\G. Then for any f̃ ∈ C_{c}(Γ\G),

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

If l = n − 1, the group B_{l}^{o} is unipotent, and Theorem 20 is a special case of the result of Shah [Sh94]. Thus, we may assume that l < n − 1.

4.1. Representations of SL(n, ℝ).

Before starting the proof, we prepare some auxiliary results on representations of G = SL(n, ℝ).

Denote by 𝔤 the Lie algebra of G. We have the root space decomposition of 𝔤_{ℂ} = 𝔤 ⊗ ℂ:

𝔤_{ℂ} = 𝔤_{0} ⊕ ∑_{i≠j} ℂE_{ij},

where 𝔤_{0} is the diagonal subalgebra of 𝔤_{ℂ}, and E_{ij} is the matrix with 1 in position (i, j) and 0’s elsewhere. It is convenient to identify 𝔤_{0} with the space of vectors s = (s_{1}, …, s_{n}) ∈ ℂ^{n}, ∑_{i} s_{i} = 0. Introduce the roots of 𝔤_{ℂ}:

α_{ij}(s) = s_{i} − s_{j}, i ≠ j,

and the fundamental weights of 𝔤_{ℂ}:

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

The simple roots of 𝔤_{ℂ} are α_{i} = α_{i,i+1}, i = 1, …, n − 1. For i < j,

α_{ij} = α_{i} + ⋯ + α_{j}.

The dominant weights are linear combinations with nonnegative integer coefficients of the fundamental weights. A highest weight of a finite-dimensional representation of 𝔤_{ℂ} is a weight that is maximal with respect to the ordering on the dual space of 𝔤_{0}. This weight is unique. Irreducible representations of 𝔤_{ℂ} are in one-to-one correspondence with the dominant weights. The corresponding dominant weight is the highest weight of the representation.

Let 𝔤_{0}^{+} be the Lie subalgebra of 𝔤_{ℂ} that corresponds to A_{l+}^{o}. That is, 𝔤_{0}^{+} consists of diagonal matrices with entries

(0, …, 0, s_{l+1}, …, s_{n}), s_{l+1} + ⋯ + s_{n} = 0,

on the diagonal. Lemma 21. Let π be a representation of G on a finite-dimensional complex vector space V. Let

V_{0} = {v ∈ V : π(B_{l}^{o})v = v}.  (46)

Then every vector v̄ ∈ V/V_{0} − {0} such that π(N_{l+})v̄ = v̄ is a sum of weight vectors of 𝔤_{0}^{+} with nonzero dominant weights.

Proof. First, we show that there are no nonzero vectors in V/V_{0} fixed by B_{l}^{o}. Let W be the maximal B_{l}^{o}-invariant subspace on which B_{l}^{o} acts unipotently. The space W can be constructed inductively as follows. Let W_{0} be the space of vectors fixed by B_{l}^{o}, W_{1} ⊇ W_{0} be the space such that W_{1}/W_{0} is the space of vectors in V/W_{0} fixed by B_{l}^{o}, and so on. After finitely many steps, we get W. We claim that W = V_{0}. Note A_{l+}^{o} acts trivially on W. Take w ∈ W. Suppose that π̃(E_{ij})w ≠ 0 for some E_{ij} ∈ 𝔫^{+}, where 𝔫^{+} is the Lie algebra of N_{l+}. Then it is a weight vector with weight α_{ij}|_{𝔤_{0}^{+}} ≠ 0 with respect to 𝔤_{0}^{+}. This is a contradiction. Thus, π̃(𝔫^{+})w = 0 for every w ∈ W, and W = V_{0}. Clearly, the space V/W does not contain any vectors that are fixed by B_{l+}^{o}. This proves the claim.

Let

G̃ = ( \begin{matrix}E & 0 \\ 0 & SL(n − l, ℝ) \end{matrix} ) and Ñ = G̃ ∩ N_{l+}.

Since A_{l+}^{o}Ñ is an epimorphic subgroup of G̃, the space V_{0} is G̃-invariant. Take a vector v̄ ∈ V/V_{0} − {0} such that π(N_{l+})v̄ = v̄. By [Go02, Lemma 15] applied to G̃, v̄ = ∑_{k=1}^{m} v̄_{k} where v̄_{k}, k = 1, …, m, are weight vectors with dominant weights λ_{k} with respect to 𝔤_{0}^{+}. Without loss of generality, λ_{k} ≠ λ_{j} for k ≠ j. For E_{ij} ∈ 𝔫^{+}, we have ∑_{k=1}^{m} π̃(E_{ij})v̄_{k} = 0. Using that π̃(E_{ij})v̄_{k} is either 0 or has weight λ_{k} + α_{ij}, we conclude that π̃(E_{ij})v̄_{k} = 0 for every k = 1, …, m. Thus, π(N_{l+})v̄_{k} = v̄_{k}. Since V_{0} = W, the vector v̄_{k} cannot be fixed by π(A_{l+}^{o}). Hence, λ_{k} ≠ 0. □

We now modify slightly our notations. For s = (s_{l+1}, …, s_{n}) ∈ ℝ^{n−l}, ∑_{i=l+1}^{n} s_{i} = 0, denote

a^{+}(s) = diag(1, …, 1, e^{s_{l+1}}, …, e^{s_{n}}).

For β > 0, define

D(β) = {t = (t_{ij} : max(i, l) < j) : ∑_{max(i,l)<j} t_{ij}^{2} < β^{2}} .  (47)

Lemma 22. Let π be a representation of G on a finite-dimensional real vector space V, V_{0} be defined as in (46), and V̄ = V/V_{0}. Introduce a norm on V̄. Then for every relatively compact subset K ⊆ V̄ and r > 0, there exists α ∈ (0, 1) and C_{0} > 0 such that for every s such that a^{+}(s) ∈ A_{l+}^{C_{0}} and x ∈ V̄ such that ∥x∥ > r,

π(a^{+}(s)n^{+}(D(e^{−αs_{l+1}})))x ⊈ K.  (48)

Proof. The proof is the same as the proof of Lemma 16 in [Go02]. We will just sketch the main idea.

We need to get a lower estimate for

sup{∥π(a^{+}(s))π(n^{+}(t))x∥ : t ∈ D(e^{−αs_{l+1}})}.

Let W̄ = {v̄ ∈ V̄ : π(N_{l+})v̄ = v̄}, and prW̄ : V̄ → W̄ is a projection on W̄ that commutes with π(a^{+}(s)). By Lemma 21, W̄ is spanned by weight vectors of A_{l+}^{o} with nonzero dominant weights. Using that the character s ↦ e^{s_{l+1}} is the smallest nontrivial dominant weight of A_{l+}^{o}, one concludes that for every y ∈ V̄ and a^{+}(s) ∈ A_{l+}^{C} with C > 0,^{1}

∥π(a^{+}(s))y∥ ≫ ∥π(a^{+}(s))prW̄ (y)∥ ≫ e^{s_{l+1}}∥prW̄ (y)∥.  (49)

By a lemma due to Shah [Sh96] (see [Go02, Lemma 13]),

sup{∥prW̄ (π(n^{+}(t))x)∥ : t ∈ D(e^{−αs_{l+1}})} ≫ (e^{−αs_{l+1}})^{d}∥x∥.  (50)

for some positive integer d. Combining (49) and (50), we get

sup{∥π(a^{+}(s))π(n^{+}(t))x∥ : t ∈ D(e^{−αs_{l+1}})} ≫ e^{cs_{l+1}}∥x∥,  (51)

where c = 1 − αd. Choose α ∈ (0, 1) such that c > 0. Then for a^{+}(s) ∈ A_{l+}^{C} the right hand side of (51) gets arbitrarily large as C → ∞. This proves (48). □

1A ≪ B means A < c · B for some absolute constant c > 0.

4.2. Proof of Theorem 20.

Now we are ready to start the proof of Theorem 20. Let X = (Γ\G) ∪ {∞} be the one-point compactification of Γ\G. For T > 0, define a normalized measure on X by

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

We need to show that ν_{T} → ν as T → ∞ in weak^{∗} topology. Since the space of normalized measures on Z is compact in weak^{∗} topology, it is enough to show that every limit point of ν_{T}, T → ∞, is equal to ν. Let ν_{T_{i}} → η as T_{i} → ∞ for some normalized measure η on X. By Lemma 18, for every C ∈ ℝ,

η(f̃) = lim_{T_{i}→∞} \frac{1}{ϱ_{l}(B_{l,T_{i}}^{o})} ∫_{B_{l,T_{i}}^{C}} f̃(yb^{−1})dϱ_{l}(b)  (52)

Let U = {n^{+}(t) ∈ N : t_{ij} = 0 for i < j < n}. Lemma 23. The measure η is U-invariant.

Up to minor modifications, the proof is the same as the proof of Lemma 18 in [Go02]. Lemma 24. For α ∈ (0, 1), define

Ã_{l+,T}^{C} = {a^{+}(s) ∈ A_{l+,T}^{C} : (T^{2} − N(s) − l)^{1/2} > exp (max_{l<i<n}{s_{i}} − αs_{l+1})} ,  (53)

where N(s) is defined in (82), and

B̃_{l,T}^{C} = (Ã_{l+,T}^{C} N_{l+}) ⋂ B_{l,T}^{o} .

Then for every C > 0,

η(f̃) = lim_{T_{i}→∞} \frac{1}{ϱ_{l}(B_{l,T_{i}}^{o})} ∫B̃_{l,T_{i}}^{C} f̃(yb^{−1})dϱ_{l}(b).

The proof is routine computation based on Lemma 28 in the Appendix. See Lemma 19 in [Go02] or the proof of Lemma 18 above for a similar argument.

Write y = Γg_{0} for some g_{0} ∈ G. Let q_{s}(t) = n^{+}(t)^{−1}a^{+}(s)^{−1}.

Next, we review some deep results on distribution of polynomial trajectories due to Dani, Margulis, Shah, and Ratner. See [KSS02] and [St, §19] for more comprehensive exposition. These results will be applied to the polynomial map

t ↦ g_{0}q_{s}(t) : ℝ^{m} → SL(n, ℝ),

where m = \frac{1}{2}(n − l)(n + l − 1).

Let 𝔤 be the Lie algebra of G, and V_{G} = ⊕_{i=1}^{dim g} ∧^{i} 𝔤. Fix a norm in V_{G}. For every Lie subgroup H of G with Lie algebra 𝔥, take a unit vector p_{H} ∈ ∧^{dim 𝔥}𝔥 ⊆ V_{G}. Also define

X(H, U) = {g ∈ G : gU ⊆ Hg}.

Denote by H_{Γ} the family of all proper closed connected subgroups H of G such that Γ ∩ H is a lattice in H, and Ad(Γ ∩ H) is Zariski dense in Ad(H).

The singular set of U is

Y = ⋃_{H∈H_{Γ}} ΓX(H, U) ⊆ Γ\G.

The set Y is precisely the set of y ∈ Γ\G such that yU is not dense in Γ\G.

The following facts and results will be used in the sequel:

(I) The set H_{Γ} is countable.

(II) For every H ∈ H_{Γ}, Γ · p_{H} is discrete in V_{G}. Thus, ΓN_{G}^{1}(H) is closed in Γ\G, where

N_{G}^{1}(H) = {g ∈ G : g · p_{H} = p_{H}}. (III) Assume that Γ is not cocompact. Then there exist closed subgroups U_{i}, i = 1, …, r,

such that each U_{i} is the unipotent radical of a parabolic subgroup, ΓU_{i} is closed in Γ\G, and for every ε, δ > 0, there exists a compact set C ⊆ Γ\G such that for every bounded open convex subset D ⊆ ℝ^{m}, one of the following holds:

1. There exist γ ∈ Γ and i = 1, …, r such that

sup_{t∈D} ∥q_{s}(t)^{−1}g_{0}^{−1}γ · p_{U_{i}}∥ ≤ δ.

2. ω({t ∈ D : Γg_{0}q_{s}(t) ∉ C}) < εω(D), where ω is the Lebesgue measure on ℝ^{m}. (IV) Let ε > 0 and H ∈ H_{Γ}. For every compact set C ⊆ ΓX(H, U), there exists a

compact set F ⊆ V_{G} such that for every neighborhood Φ of F in V_{G} there exists a neighborhood Ψ of C in Γ\G such that for every bounded open convex set D ⊆ ℝ^{m}, one of the following holds:

1. There exists γ ∈ Γ such that q_{s}(D)^{−1}g_{0}^{−1}γ · p_{H} ⊆ Φ.

2. ω({t ∈ D : Γg_{0}q_{s}(t) ∈ Ψ}) < εω(D). (I) is proved in [Rat91a, Theorem 1.1] and [DM93, Theorem 2.1]. For the proof of (II),

see [DM93, Theorem 3.4]. (III) is a special case of [Sh96, Theorems 2.1–2.2]. (IV) is based on [Sh94, Proposition 5.4]. It is formulated in [Sh96].

To simplify our notations, we put V = V_{G}. Let V_{0} be defined as in (46). Lemma 25. For H ∈ H_{Γ}, g_{0}^{−1}Γ · p_{H} ⊆ V − V_{0}.

Proof. Suppose that g_{0}^{−1}γ · p_{H} ∈ V_{0} for some γ ∈ Γ. Then (γ^{−1}g_{0}B_{l}^{o}g_{0}^{−1}γ) · p_{H} = p_{H}. Thus, γ^{−1}g_{0}B_{l}^{o}g_{0}^{−1}γ ⊆ N_{G}^{1}(H). By (II), ΓN_{G}^{1}(H) is not dense in Γ\G. It follows that Γg_{0}B_{l}^{o} is not dense in Γ\G too. This is a contradiction. □

In the case when Γ is not cocompact, we prove the following lemma: Lemma 26. η({∞}) = 0.

Proof. Write

V = V_{0} ⊕ V_{1},  (54)

where V_{1} is A_{l+}^{o}-invariant complement. For a vector v ∈ V, denote by v_{0} ∈ V_{0} and v_{1} ∈ V_{1} its components with respect to the decomposition (54). Fix norms on V_{0} and V_{1}. Define a norm on V by

∥v∥ = max{∥v_{0}∥, ∥v_{1}∥}, v_{0} ∈ V_{0}, v_{1} ∈ V_{1}.

The space V_{1} is naturally isomorphic with V/V_{0}. The norm on V_{1} induces a norm on V/V_{0} through this isomorphism.

We use (III). Let ε, δ > 0. Let

P = ⋃_{i=1}^{r} g_{0}^{−1}Γ · p_{U_{i}},

and P_{1} = {p ∈ P : ∥p_{0}∥ > δ}. For p ∈ P_{1},

∥q_{s}(0)^{−1}p∥ ≥ ∥p_{0}∥ > δ.  (55)

By Lemma 25, P ⊆ V −V_{0}, and by (II), P is discrete. Therefore, there exists r > 0 such that ∥p_{1}∥ > r for p ∈ P − P_{1}. Since the factor-map V → V/V_{0} is continuous and B_{l}^{o}-equivariant, for some M > 0,

∥q_{s}(t)^{−1} · v∥ ≥ M∥q_{s}(t)^{−1} · v̄∥, v ∈ V.  (56)

Now we apply Lemma 22 with K = \{v̄ ∈ V/V_{0} : ∥v̄∥ ≤ \frac{δ}{M} \}. There exist α ∈ (0, 1) and C_{0} > 0 such that for every s such that a^{+}(s) ∈ A_{l+}^{C_{0}} and every v̄ ∈ V/V_{0} such that ∥v̄∥ > r,

q_{s}(D(e^{−αs_{l+1}}))^{−1} · v̄ ⊈ K.

In particular, this holds for v̄ = p̄ with p ∈ P − P_{1}. Thus, by (56),

sup_{t∈D(e^{−αs_{l+1}})} ∥q_{s}(t)^{−1} · p∥ > δ  (57)

for p ∈ P − P_{1}. In fact, (57) holds for p ∈ P_{1} because of (55). Thus, the case (a) of (III) does not occur when a^{+}(s) ∈ A_{l+,T_{i}}^{C_{0}} and D is a bounded open convex subset such that D ⊇ D(e^{−αs_{l+1}}). It follows that for some compact set C ⊆ Γ\G,

ω({t ∈ D : Γg_{0}q_{s}(t) ∉ C}) < εω(D).  (58)

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

We have

η(f̃) = lim_{T_{i}→∞} \frac{1}{ϱ_{l}(B_{l,T_{i}}^{o})} ∫Ã_{l+,T_{i}}^{C} ∫_{D_{s,T_{i}}} f̃(Γg_{0}q_{s}(t))δ_{l}^{+}(s)dt^{+}ds^{+}, f̃ ∈ C_{c}(Γ\G), (59)

where

D_{s,T_{i}} = {n(t) ∈ N : ∑_{i≤l;l<j} t_{ij}^{2} + ∑_{l<i<j} e^{2s_{i}}t_{ij}^{2} < T_{i}^{2} − N(s) − l} ,  (60)

and N(s) is defined in (82). Note that D_{s,T_{i}} contains D(β), which is defined in (47), for

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

When a^{+}(s) ∈ Ã_{l+,T_{i}}^{C_{0}}, the right hand side is greater then e^{−αs_{l+1}} (see (53)). Therefore, D_{s,T_{i}} ⊇ D(e^{−αs_{l+1}}) when a^{+}(s) ∈ Ã_{l+,T_{i}}^{C_{0}}. By (58),

ω ({t ∈ D_{s,T_{i}} : Γg_{0}q_{s}(t) ∉ C}) < εω(D_{s,T_{i}}) for a^{+}(s) ∈ Ã_{l+,T_{i}}^{C_{0}}.  (61)

Let χ_{C} be the characteristic function of the set C. Take f̃ ∈ C_{c}(Γ\G) such that χ_{C} ≤ f̃ ≤ 1. Then using (59) and (61), we get

η(supp(f̃)) ≥ lim_{T_{i}→∞} \frac{1}{ϱ_{l}(B_{l,T_{i}}^{o})} ∫Ã_{l+,T_{i}}^{C_{0}} ∫_{D_{s,T_{i}}} χ_{C}(Γg_{0}q_{s}(t))δ_{l}^{+}(s)dt^{+}ds^{+}

≥ lim_{T_{i}→∞} \frac{1}{ϱ_{l}(B_{l,T_{i}}^{o})} ∫Ã_{l+,T_{i}}^{C_{0}}(1 − ε)ω(D_{s,T_{i}})δ_{l}^{+}(s)ds^{+}

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

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

Lemma 27. η(Y) = 0.

Proof. Since H_{Γ} is countable, it is enough to show that η(ΓX(H, U)) = 0 for every H ∈ H_{Γ}. Moreover, it is enough to show that η(C) = 0 for every compact set C ⊆ ΓX(H, U).

We use the notations from the proof of Lemma 26, in particular, decomposition (54). We apply (IV). Take ε > 0. Let F be a compact subset of V as in (IV). Take a relatively

compact neighborhood Φ of F. Let Ψ ⊃ C be as in (IV). Denote P = g_{0}^{−1}Γ · p_{H} and P_{1} = {p ∈ P : ∥p_{0}∥ > δ} with δ = sup{∥v_{0}∥ : v ∈ Φ}. Then (55) holds, so that

q_{s}(0) · p ∉ Φ.  (62)

As in the proof of the previous lemma, there exists r > 0 such that ∥p_{1}∥ > r for p ∈ P − P_{1}, and applying Lemma 22, we deduce that there exist α ∈ (0, 1) and C_{0} > 0 such that for every s such that a^{+}(s) ∈ A_{l+}^{C_{0}} and every p ∈ P − P_{1},

q_{s}(D(e^{−αs_{l+1}}))^{−1} · p ⊈ Φ.  (63)

By (62), (63) holds for every p ∈ P. Thus, case (a) of (IV) fails. Therefore, case (b) holds:

ω({t ∈ D : Γg_{0}q_{s}(t) ∈ Ψ}) < εω(D),  (64)

when a^{+}(s) ∈ A_{l+,T_{i}}^{C_{0}} and D is an open convex set such that D ⊇ D(e^{−αs_{l+1}}). Recall that D_{s,T_{i}} was defined in (60). It is easy to see from (60) that D_{s,T_{i}} ⊇ D(e^{−αs_{l+1}}) when a^{+}(s) ∈ Ã_{l+,T_{i}}^{C_{0}}. Thus, (64) holds for D = D_{s,T_{i}} with a^{+}(s) ∈ Ã_{l+,T_{i}}^{C_{0}}.

Take a function f̃ ∈ C_{c}(Γ\G) such that f̃ = 1 on C, supp(f̃) ⊆ Ψ, and 0 ≤ f̃ ≤ 1. Let χ_{Ψ} be the characteristic function of Ψ. Then using (59) and (64) with D = D_{s,T_{i}}, we get

η(C) ≤ lim_{T_{i}→∞} \frac{1}{ϱ_{l}(B_{l,T_{i}}^{o})} ∫Ã_{l+,T_{i}}^{C_{0}} ∫_{D_{s,T_{i}}} χ_{Ψ}(Γg_{0}q_{s}(t))δ_{l}^{+}(s)dt^{+}ds^{+}

≤ lim_{T_{i}→∞} \frac{1}{ϱ_{l}(B_{T_{i}}^{o})} ∫Ã_{l+,T_{i}}^{C_{0}} εω(D_{s,T_{i}})δ_{l}^{+}(s)ds^{+} = ε lim_{T_{i}→∞} \frac{ϱ_{l}(B̃_{l,T_{i}}^{C_{0}})}{ϱ_{l}(B_{T_{i}}^{o})} = ε.

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

By Lemma 23, the measure η is U-invariant. Consider the ergodic decomposition of η into U-ergodic measures. By Ratner’s measure classification [Rat91a], an ergodic component of η is either G-invariant or supported on Y ∪ {∞}. By Lemmas 26 and 27, the set of ergodic components of the second type has measure 0. Therefore, η is G-invariant, and η = ν. This proves Theorem 20.

5. Uniform distribution for a lattice in Sp(n, ℝ)

5.1. Density of orbits.

Proof of Theorem 5. Clearly, the condition is neccesary for denseness. Suppose that the condition is satisfied. Let {e_{i} : i = 1, …, 2n} be the standard basis of ℝ^{2n}, and e = (e_{1}, …, e_{n}). Then e is an isotropic frame, and by Witt’s Theorem, the space of isotropic n-frames is Ge. The stabilizer of e in G is

U_{0} = \{( \begin{matrix}E & X \\ 0 & E \end{matrix} ) : ^{t}X = X\} .

Let g_{0} ∈ G be such that g_{0}v = e. Then the stabilizer of v in G is U = g_{0}^{−1}U_{0}g_{0}. It is not hard to check that any U_{0}-invariant subspace is either contained in ⟨e_{1}, …, e_{n}⟩ or contains it. It follows that any U-invariant subspace is either contained in V_{0} =^{def} ⟨v_{1}, …, v_{n}⟩ or contains it.

As in the proof of Theorem 1, \overline{ΓU} = ΓH where H is the connected component of the smallest real ℚ-algebraic subgroup containing U, and the radical of H is unipotent. Let R be the radical of H. The space of R-fixed vectors V^{R} is defined over ℚ and H-invariant. Since R is unipotent, V^{R} ≠ 0. Thus, by the condition on v, V^{R} ⊈ V_{0}. Then V_{0} ⊆ V^{R}. Suppose that V^{R} ≠ ℝ^{2n}. Since V_{0}^{⊥} = V_{0}, J|_{V^{R}} is degenerate. Then 0 ≠ Rad(J|_{V^{R}}) ⊆ V_{0}. This is a contradiction because the space Rad(J|_{V^{R}}) is defined over ℚ. Hence, V^{R} = ℝ^{2n}, R = 1, and H is semisimple.

We claim that H = G. Let H_{0} = g_{0}Hg_{0}^{−1}. Denote by 𝔤, 𝔥, and 𝔲 the Lie algebras of G, H_{0}, and U_{0} respectively. The Killing form on 𝔤 is defined by k(x, y) = Tr(xy) for x, y ∈ 𝔤. Since H_{0} is semisimple, k is nondegenerate on 𝔥.

Recall the root decomposition for 𝔤. A Cartan subalgebra of 𝔤 is

𝔞 = {diag(h_{1}, …, h_{n}, −h_{1}, …, −h_{n}) : h_{i} ∈ ℝ}.

Let α_{i}(h) = h_{i} for h ∈ 𝔞. The root system of 𝔤 is

Δ = {α_{i} − α_{j}, ±(α_{p} + α_{q}) : 1 ≤ i ≠ j ≤ n, 1 ≤ p ≤ q ≤ n},

and the following root space decomposition holds:

𝔤_{α_{i}−α_{j}} = ⟨E_{i,j} − E_{j+n,i+n}⟩ ,
𝔤_{α_{p}+α_{q}} = ⟨E_{p,q+n} + E_{q,p+n}⟩ ,
𝔤_{−α_{p}−α_{q}} = ⟨E_{p+n,q} + E_{q+n,p}⟩ ,
𝔤 = 𝔞 ⊕ ∑_{α∈Δ} 𝔤_{α}.  (65)

Since U_{0} ⊆ H_{0},

𝔲 = ∑_{1≤p≤q≤n} 𝔤_{α_{p}+α_{q}} ⊆ 𝔥.

Note that k(𝔤_{α}, 𝔤_{β}) = 0 if α + β ≠ 0. Since the Killing form k is nondegenerate on 𝔥, the projection map from 𝔥 to the space ∑_{1≤p≤q≤n} 𝔤_{−α_{p}−α_{q}} with respect to the decomposition (65) is surjective. Thus for 1 ≤ p ≤ q ≤ n, there exists h_{pq} = x_{pq} + h̃_{pq} ∈ 𝔥 where x_{pq} is a generator of the space 𝔤_{−α_{p}−α_{q}}, and h̃_{pq} is in the normalizer of 𝔲. Then

𝔥 ⊇ [h_{pq}, 𝔲] + 𝔲 ⊇ [x_{pq}, u], 1 ≤ p ≤ q ≤ n.

Using that [𝔤_{α}, 𝔤_{β}] = 𝔤_{α+β} for α, β, α + β ∈ Δ, we conclude that

∑_{α∈Δ} 𝔤_{α} ⊆ 𝔥.

It follows that 𝔥 = 𝔤, and H = G. Finally,

\overline{Γv} = \overline{ΓUv} ⊇ \overline{ΓU}v = Gv = W_{n}.
□

5.2. Iwasawa decomposition for Sp(n, ℝ).

Let G = Sp(n, ℝ). We use the following notations:

K = G ∩ SO(2n, ℝ),
N_{+} = \{( \begin{matrix}E & N \\ 0 & E \end{matrix} ) : ^{t}N = N\} ,
N_{−} = \{( \begin{matrix}M & 0 \\ 0 & ^{t}M^{−1} \end{matrix} ) : M is upper triangular, unipotent\} ,
A = \{( \begin{matrix}A & 0 \\ 0 & A^{−1} \end{matrix} ) : A is positive, diagonal\} ,
B = N_{−}A,
N = N_{−}N_{+}.

We have Iwasawa decomposition:

(k, n, a) ↦ kna : K × N × A → G

(see, for example, [Te, p. 286]). This map is a diffeomorphism. It is easy to check that the product map N_{−} ×N_{+} → N is a diffeomorphism, N_{−} normalizes N_{+}, and A normalizes N_{+}. Thus, modified Iwasawa decomposition holds:

(k, b, n) ↦ kbn : K × B × N_{+} → G.  (66)

Fix g_{0} ∈ G. We also have decomposition:

(k, b, n) ↦ kbg_{0}n^{g_{0}} : K × B × N_{+} → G.  (67)

For g ∈ G, define k_{g} ∈ K, b_{g} ∈ B, and n_{g} ∈ N_{+} such that

g = k_{g}b_{g}g_{0}n_{g}^{g_{0}}.

Also define b′_{g} ∈ GL(n, ℝ) such that

b_{g} = ( \begin{matrix}b′_{g} & 0 \\ 0 & ^{t}(b′_{g})^{−1} \end{matrix} ) .

Let μ be a Haar measure on G, and μ̄ be the measure on Γ\G such that

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

Let ϱ be the Lebesgue measure on N_{+} ≃ ℝ^{\frac{n(n+1)}{2}} , and ν be the measure on G/N_{+}^{g_{0}} such that

∫_{G} fdμ = ∫_{G/N_{+}^{g_{0}}} ∫_{N_{+}} f(gn^{g_{0}})dϱ(n)dν(g), f ∈ C_{c}(G).

Note that

ϱ(N_{+,T}) ∼ CT^{\frac{n(n+1)}{2}} as T → ∞  (68)

for some C > 0.

5.3. Uniform distribution.

Proof of Theorem 6. We can write v^{0} = g_{0}^{−1}e^{0} for some g_{0} ∈ G, where e^{0} = (e_{1}, …, e_{n}) is the standard frame. Without loss of generality, g_{0}^{−1} = k_{0}b_{0} for some k_{0} ∈ K, and b_{0} ∈ B (see (66)).

Let Y = KBg_{0}. By (67), the product map Y × N_{+}^{g_{0}} → G is a diffeomorphism. For g ∈ G, denote y_{g} = k_{g}b_{g}g_{0}, the Y -component of g. The map

α : Y → W_{n} ≃ G/N_{+}^{g_{0}} : y ↦ yv^{0}

is a diffeomorphism. Denote by ν_{1} the measure on Y which is the pull-back of the measure ν by the map α.

For g ∈ G, gv^{0} = y_{g}v^{0}. This shows that gv^{0} ∈ Ω iff y_{g} ∈ α^{−1}(Ω). Write

n_{g} = ( \begin{matrix}E & l_{g} \\ 0 & E \end{matrix} ) .

Then

∥g∥ = ∥k_{g}b_{g}n_{g}g_{0}∥ = ∥b_{g}n_{g}b_{0}^{−1}∥ = ∥( \begin{matrix}∗ & b′_{g}l_{g}(^{t}b′_{0}) \\ 0 & ∗ \end{matrix} )∥ .

Thus,

∥g∥^{2} = ∥ĉ_{g}(n_{g}d_{g})∥^{2} + e_{g},

where

c_{g} = b_{g}, d_{g} = ( \begin{matrix}(b′_{0})^{−1}b′_{g} & 0 \\ 0 & (^{t}b′_{0})(^{t}b′_{g})^{−1} \end{matrix} ) ,

and e_{g} is a continuous function depending only on the B-components of g. We can use Proposition 10 with H = N_{+}, h_{g} = n_{g}, and m_{g} = 1. Since Γ · v^{0} is dense in W_{n}, ΓN_{+}^{g_{0}} is dense in G. By (68), the condition (5) holds for H = N_{+}. Since N_{+} is unipotent, the condition (6) for H = N_{+} holds too [Sh94].

Applying Proposition 10, we get

N_{T} (Ω, v^{0}) ∼ (\frac{1}{μ̄(Γ\G)} ∫_{α^{−1}(Ω)} \frac{dν_{1}(y)}{Δ_{H}(c_{y})}) ϱ(N_{+,T})

as T → ∞, where Δ_{H} is defined in (7). Thus, by (68),

N_{T} (Ω, v^{0}) ∼ λ_{v^{0}}(Ω)T^{\frac{n(n+1)}{2}} as T → ∞,

UNIFORM DISTRIBUTION OF ORBITS OF LATTICES ON SPACES OF FRAMES 27

where

λ_{v^{0}}(Ω) = \frac{C}{μ̄(Γ\G)} ∫_{Ω} \frac{dν(x)}{Δ_{N_{+}}(c_{x})}.

□

Appendix

Proof of Lemma 16 (part 2). To find the constant d_{n,l}, we calculate measures of the set

D = {v = (v_{1}, …, v_{l}) ∈ V_{n,l} : ∥v_{i}∥ < 1, 1 ≤ i ≤ l}.

Denote by V_{k} the Lebesgue measure of a k-dimensional unit ball. Recall that

V_{k} = \frac{π^{k/2}}{Γ(1 + k/2)}.  (69)

Clearly,

∫_{D} dv = V_{n}^{l} = \frac{π^{nl/2}}{Γ(1 + n/2)^{l}} .  (70)

For k ∈ K, n ∈ N_{l−}, and a ∈ A_{l−}^{o}, knae^{0} ∈ D iff ∥knae_{i}∥ < 1 for i = 1, …, l. We have

∥kn^{−}(t)a(s^{−})e_{i}∥^{2} = exp(2si^{−}) (1 + t_{1i}^{2} + ⋯ + t_{i−1i}^{2}) .  (71)

Let as introduce new coordinates on A_{l−}^{o}: a_{i} = exp(si^{−}), 1 ≤ i ≤ l. The Haar measure on A_{l−}^{o} (23) is given by da = ∏_{i=1}^{l} \frac{da_{i}}{a_{i}}. By (71), the set of (k, n^{−}(t), a) ∈ K × N_{l−} × A_{l−}^{o} such that kn^{−}(t)ae^{0} ∈ D is described by conditions:

0 < a_{i} < 1  i = 1, …, l,
∥t_{∗i}∥ < (\frac{1 − a_{i}^{2}}{a_{i}^{2}} )^{1/2}  i = 2, …, l.

Thus,

∫_{kn^{−}a^{−}e^{0}∈D} δ_{l}^{−}(a^{−})^{n}dkdn^{−}da^{−}

= ∫_{0}^{1} ⋯ ∫_{0}^{1} ∏_{i=2}^{l} V_{i−1} (\frac{1 − a_{i}^{2}}{a_{i}^{2}} )^{\frac{i−1}{2}} δ_{l}^{−}(a^{−})^{n}da^{−}
= ∏_{i=1}^{l} V_{i−1} ∫_{0}^{1}(1 − a_{i}^{2})^{\frac{i−1}{2}} a_{i}^{n−i}da_{i}
= ∏_{i=1}^{l} \frac{V_{i−1}}{2} ∫_{0}^{1}(1 − b_{i})^{\frac{i−1}{2}} bi^{\frac{n−i−1}{2}} db_{i}
= 2^{−l} ∏_{i=1}^{l} V_{i−1}B (\frac{i + 1}{2}, \frac{n − i + 1}{2})
= \frac{π^{l(l−1)/4}}{2^{l}Γ(1 + n/2)^{l}} ∏_{i=1}^{l} Γ (\frac{n − i + 1}{2}) .  (72)

In the last step, we have used (69) and the well-known identity for Γ-function and Bfunction. Finally, by (70) and (72),

d_{n,l} = \frac{2^{l}π^{l(2n−l+1)/4}}{Γ(n/2)Γ((n − 1)/2) ⋯ Γ((n − l + 1)/2)}.  (73)

□

Proof of Lemma 17. Let

d(ā) = diag (1, …, 1, a_{l+1}, …, a_{n})

for ā = (a_{l+1}, …, a_{n}) ∈ ℝ^{n−l}, and

α(λ) = diag (1, …, 1, λ^{\frac{1}{n−l}} , …, λ^{\frac{1}{n−l}} ) .

For b = n^{+}(t)d(ā), define

Λ(b) = ∑_{l<i≤n} a_{i}^{2} + ∑_{max(i,l)<j} a_{j}^{2}t_{ij}^{2}.  (74)

Note that Λ(b) = ∥b∥^{2} − l. Thus, it is enough to compute asymptotics of the function

ϕ(x) =^{def} ∫_{Λ(b)<x} dϱ_{l}(b).

as x → ∞. By Tauberian theorem [Wid, Ch. V, Theorem 4.3], it can be deduced from asymptotics of the function

ψ(λ) =^{def} ∫_{0}^{∞} e^{−λx}dϕ(x) = ∫_{B_{l}^{o}} exp{−λΛ(b)}dϱ_{l}(b).  (75)

as λ → 0+. It is more convenient to work with the function

ψ̃(λ) = ψ (λ^{\frac{2}{n−l}} ) .  (76)

Let B̄_{l}^{o} = N_{l+}d(ℝ_{+}^{d}) = B_{l}^{o}α(ℝ_{+}). One can check that

∫_{N_{l+}×ℝ_{+}^{n−l}} f(nd(ā))dn^{+} \frac{da_{l+1}}{a_{l+1}} ⋯ \frac{da_{n}}{a_{n}} = ∫_{B_{l}^{o}×ℝ_{+}} f(bα(λ))dϱ_{l}(b)\frac{dλ}{λ}  (77)

for f ∈ L^{1}(B̄_{l}^{o}). (In fact, each of the integral defines a right Haar measure on B̄_{l}^{o}.)

Consider Mellin transform of the function ψ̃:

F (z) = ∫_{0}^{∞} ψ̃(λ)λ^{z} \frac{dλ}{λ} =^{(75)} ∫_{B_{l}^{o}×ℝ_{+}} exp \{−Λ(bα(λ))\} λ^{z}dϱ_{l}(b)\frac{dλ}{λ}

=^{(77)} ∫_{N_{l+}×ℝ_{+}^{n−l}} exp \{−Λ(n^{+}d(ā))\} ( ∏_{i=l+1}^{n} a_{i})^{z} dn^{+} \frac{da_{l+1}}{a_{l+1}} ⋯ \frac{da_{n}}{a_{n}}

=^{(74)} ∫_{N_{l+}×ℝ_{+}^{n−l}} exp {− ∑_{max(i,l)<j}(a_{j}t_{ij})^{2} − ∑_{i=l+1}^{n} a_{i}^{2}}

× ( ∏_{i=l+1}^{n} a_{i})^{z−1} ∏_{max(i,l)<j} dt_{ij} ∏_{i=l+1}^{n} da_{i}.

Using that ∫_{ℝ} e^{−x^{2}}dx = \sqrt{π}, we get

F (z) = π^{\frac{(n+l−1)(n−l)}{4}} ∫_{ℝ_{+}^{n−l}} exp \{− ∑_{i=l+1}^{n} a_{i}^{2}\} ( ∏_{i=l+1}^{n} a_{i}^{z−i}) da_{l+1} … da_{n}.

Making substitution u_{i} = a_{i}^{2}, we get

F (z) = \frac{π^{\frac{(n+l−1)(n−l)}{4}}}{2^{n−l}} ∏_{j=l+1}^{n} Γ (\frac{z − j + 1}{2}) .  (78)

By Mellin inversion formula, for sufficiently large u,

ψ̃(λ) = \frac{1}{2πi} ∫_{Re(z)=u} F (z)λ^{−z}dz.  (79)

Since Γ-function decays fast on vertical strips, we can shift the line of integration to the left. By (78), the first pole of F (z) occurs at z = n − 1. Therefore, it follows from (79) that

ψ̃(λ) ∼ \frac{π^{\frac{(n+l−1)(n−l)}{4}}}{2^{n−l−1}} ∏_{j=l+1}^{n−1} Γ (\frac{n − j}{2} ) λ^{−(n−1)} as λ → 0 + .

By (76),

ψ(λ) ∼ \frac{π^{\frac{(n+l−1)(n−l)}{4}}}{2^{n−l−1}} ∏_{j=l+1}^{n−1} Γ (\frac{n − j}{2} ) λ^{− \frac{(n−1)(n−l)}{2}} as λ → 0 + .

Finally, the asymptotic estimate for ϕ(x) as x → ∞ follows from Tauberian Theorem [Wid, Ch. V, Theorem 4.3]. We have

γ_{n,l} = \frac{π^{(n+l−1)(n−l)/4}}{2^{n−l−1}Γ (\frac{(n−1)(n−l)}{2} + 1)} ∏_{j=l+1}^{n−1} Γ (\frac{n − j}{2} ) .  (80)

This proves the lemma. □

The following lemma is used in the proof of Lemma 18.

Lemma 28. For C ∈ ℝ and T > 0,

ϱ_{l}(B_{l,T}^{C}) = c_{n,l} ∫_{A_{T}^{C}} (T^{2} − N(s^{+}) − l)^{\frac{(n−l)(n+l−1)}{4}} exp \{ ∑_{k=l+1}^{n}(n − k)sk^{+}\} ds^{+},  (81)

where

c_{n,l} = \frac{π^{(n−l)(n+l−1)/4}}{Γ(1 + (n − l)(n + l − 1)/4)},
N(s^{+}) = ∑_{i=l+1}^{n} exp{2si^{+}}.  (82)

Proof. Note that

B_{l,T}^{C} = {a(s^{+})n(t) : l + N(s^{+}) + ∑_{l<i<j≤n} exp{2si^{+}}t_{ij}^{2} + ∑_{1≤i≤l;l<j≤n} t_{ij}^{2} < T^{2}} .

We use the formula (24) for ϱ_{l} and make the change of variables

t_{ij} → exp{−si^{+}}t_{ij}

for l < i < j ≤ n. The formula (81) follows from the fact that the volume of a unit ball in ℝ^{m} is π^{m/2}/Γ(1 + m/2). □

Proof of Lemma 18. For i_{0} = l + 1, …, n − 1, put

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

We claim that ϱ_{l}(B_{l,T}^{i_{0}} ) = o(ϱ(B_{l,T}^{o})) as T → ∞. If a(s^{+}) ∈ A_{l+,T}^{o}, then si^{+} < log T for every i = l + 1, …, n. Then as in Lemma 28,

ϱ_{l}(B_{l,T}^{i_{0}} ) ≤ c_{n,l}T^{\frac{(n−l)(n+l−1)}{2}} ∫_{A_{l+,T}^{i_{0}}} exp \{ ∑_{k=l+1}^{n}(n − k)sk^{+}\} ds^{+}

≪ T^{\frac{(n−l)(n+l−1)}{2}} ∏_{l<k<n,k≠i_{0}} ∫_{−∞}^{log T} exp\{(n − k)sk^{+}\}dsk^{+}
≪ T^{(n−1)(n−l)−(n−i_{0})}.

Now the claim follows from (36). Since

B_{l,T}^{o} − B_{l,T}^{C} = ⋃_{l<i_{0}<n} B_{l,T}^{i_{0}} ,

we have

ϱ_{l}(B_{l,T}^{o} − B_{l,T}^{C}) = o(ϱ_{l}(B_{l,T}^{o})) as T → ∞.

Therefore, ϱ_{l}(B_{l,T}^{C}) ∼ ϱ_{l}(B_{l,T}^{o}) as T → ∞. □

Proof of Corollary 4. By Theorem 1, Γv^{0} is dense in V_{n,l}. By Theorem 3, (44) holds. The volume of Γ\G was computed by Minkowski. For the measure μ̄, we have

μ̄(Γ\G) = 2^{−(n−1)} ∏_{i=2}^{n} π^{−i/2}Γ(i/2)ζ(i)

(see [Sh00, Theorem 5.6]).

Therefore,

N_{T} (Ω, v^{0}) ∼ b_{n,l}Vol(v^{0})^{1−n} (∫_{Ω} \frac{dv}{Vol(v)}) T^{(n−1)(n−l)} as T → ∞,

where

b_{n,l} = \frac{a_{n,l}}{μ̄(Γ\G)} = \frac{π^{n(n−l)/2}}{Γ (\frac{(n−1)(n−l)}{2} + 1) Γ (\frac{n−l}{2})} ∏_{i=2}^{n} ζ(i)^{−1}.  (83)

Here we used that Γ(\frac{1}{2}) = \sqrt{π}. □

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

  E-mail address: gorodnik@umich.edu