DISTRIBUTION OF LATTICE ORBITS ON HOMOGENEOUS VARIETIES

ALEX GORODNIK AND BARAK WEISS

Abstract. Given a lattice Γ in a locally compact group G and a closed subgroup H of G, one has a natural action of Γ on the homogeneous space V = H\G. For an increasing family of finite subsets {Γ_{T} : T > 0}, a dense orbit v · Γ, v ∈ V and compactly supported function φ on V, we consider the sums S_{φ,v}(T) = ∑_{γ∈Γ_{T}} φ(vγ). Understanding the asymptotic behavior of S_{φ,v}(T) is a delicate problem which has only been considered for certain very special choices of H, G and {Γ_{T} }. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have S_{φ,v}(T) ∼ ∫_{G_{T}} φ(vg) dg, where G_{T} = {g ∈ G : ∥g∥ < T } and Γ_{T} = G_{T} ∩Γ. We also show that the asymptotics of S_{φ,v}(T) is governed by ∫_{V} φ dν, where ν is an explicit limiting density depending on the choice of v and ∥ · ∥.

Contents

1. Introduction 2 2. Notation and Statement of Results 8 3. Induction 15 4. Duality 18 5. The limiting density 22 6. Properties of distance functions 25 7. Matrix norms and volume computations 28 8. Balanced semisimple groups 36 9. Riemannian skew balls and volume computations 38 10. Ratner theory and linearization 46 11. Applications 49 12. Examples 52 References 60

Date: October 16, 2006.

1. Introduction

Let V be a manifold equipped with a transitive right action of a Lie group G, so that V is identified with H\G for some subgroup H of G stabilizing a point of V. Let Γ be a lattice in G, that is, a discrete subgroup of finite co-volume. In this paper we study the asymptotic distribution of Γ-orbits. More precisely, we fix a proper function D : G → [0, ∞), set

G_{T} = {g ∈ G : D(g) < T },

and study, for an arbitrary φ ∈ C_{c}(V) and v ∈ V, the asymptotic behavior of the sum

S_{φ,v}(T) = ∑_{γ∈Γ∩G_{T}} φ(v · γ)

as T → ∞.

We have not assumed that there is an invariant measure on V, and certainly not a finite one. Thus this problem does not belong to the classical framework of ergodic theory, and has many surprising features.

1.1. A simple case. Perhaps the simplest nontrivial case is when

V = ℝ^{2} ∖ {0}, G = SL(2, ℝ), D(g) = ∥g∥,

where

∥(\begin{matrix}a & b \\ c & d \end{matrix})∥ = (a^{p} + b^{p} + c^{p} + d^{p})^{1/p} , 1 ≤ p < ∞.

Then H is (up to conjugation) the one-parameter group of upper-triangular unipotent matrices. The distribution of SL(2, ℤ)-orbits in V was considered by Nogueira [No], and the case of a general lattice by Ledrappier [Le]. Ledrappier showed that for φ ∈ C_{c}(V) and v ∈ V such that v · Γ is dense in V, one has:

S_{φ,v}(T) ∼ (c_{Γ} ∫ \frac{φ(Y)}{∥v∥ · ∥Y ∥}dY ) T,

where dY denotes Lebesgue measure on ℝ^{2}, ∥ · ∥ is the p-norm on ℝ^{2}, and c_{Γ} is an explicit constant depending on the lattice Γ (here and throughout the paper, the notation A(T) ∼ B(T) means that \frac{A(T)}{B(T)} →_{T →∞} 1).

In this example, one already notices that the limiting measure \frac{1}{∥v∥} \frac{dY}{∥Y ∥} is not Γ-invariant and depends essentially on the choice of v and the choice of norm used on G. A computation with the Haar measure of G shows that in this case the asymptotics of S_{φ,v}(T) is the same as that of the corresponding average along the (unique) G-orbit, namely

S_{φ,v}(T) ∼ S̃_{φ,v}(T), where S̃_{φ,v}(T) = ∫_{G_{T}} φ(v · g) dm(g),

and m is a suitably normalized Haar measure on G.

We reconsider and generalize this example in §12.4.

1.2. The general problem. It is natural to try to extend this result to a general case. Namely, given an arbitrary locally compact second countable topological group G, a lattice Γ in G, a transitive G-space V and a proper function D : G → [0, ∞), one would like to:

• Obtain an explicit expression for the asymptotics of S_{φ,v}(T).

• Show that S_{φ,v}(T) ∼ S̃_{φ,v}(T). A number of special cases of this program were carried out in work of

the first-named author, all involving G = SL(n, ℝ), D(g) = \sqrt{tr (^{t}g g)} and a general lattice Γ: in [Go1], the case in which V is the Furstenberg boundary of G, and in [Go2], the case in which V is the space of k-frames in ℝ^{n}, 1 ≤ k < n. Additional cases were studied in [LePo, Mau, Oh, GoMau, GoOh].

Our goal in this paper is twofold. First we abstract some ideas of the abovementioned papers and develop a general axiomatic framework for studying the problem. We specify certain explicit conditions on G, H, D, Γ under which the above problems can be solved. Our axiomatic framework can be applied in principle to investigate previously known equidistribution results (as we illustrate by the example from Section 1.1), but verification of the conditions on G, H, D, Γ may be quite involved. In the second part of the paper, we check these conditions in two important new cases: when H is semisimple, or when G is semisimple and the Γ-action admits a finite invariant measure.

1.3. Our two main theorems. For convenience of the reader, in order to introduce our results we begin with two special cases and some concrete examples. We postpone the formulation of our results in their most general form until Section 2. Let G be a connected Lie group, Γ a lattice in G, and H a connected simple noncompact subgroup of G. We take the function D to be one of the following:

• Let Ψ : G → GL(d, ℝ) be a representation with compact kernel, ∥ · ∥ any norm on Mat_{d}(ℝ), and

D(g) = max{1, ∥Ψ(g)∥}.  (1)

• Let G be closed subgroup of a semisimple group L, d the Cartan-Killing metric on the symmetric space Ω of L, P : L → Ω the natural projection, and

D(g) = d(P (g), P (e)).  (2)

In the celebrated paper [EsMoSh], the asymptotics of discrete Γ-orbits in H\G were determined for a specific choice of D as in (1). Here we give the complete answer for the case of dense Γ-orbits:

Theorem 1.1. For every φ ∈ C_{c}(H\G) and x ∈ H\G such that \overline{xΓ} = H\G,

S_{φ,x}(T) ∼ S̃_{φ,x}(T) as T → ∞.

Moreover,

lim_{T →∞} \frac{S_{φ,x}(T)}{λ(H ∩ G_{T})} = ∫_{H\G} φ dν_{x}

where λ is a Haar measure on H and ν_{x} is an explicit smooth measure on H\G.

Now we consider an action of a lattice Γ in a simple noncompact Lie group G, acting by measure preserving transformations on a probability measure space (X, μ), which we assume is algebraic. Namely, we suppose that X = L/Λ where L is a Lie group, Λ is a lattice in L, μ is the normalized Haar measure, and one of the following holds:

• The group G is a closed subgroup of L and Γ acts on X = L/Λ by left multiplication.

• The group G is a closed subgroup of Aut(L) and Γ acts on X = L/Λ

by automorphisms (stabilizing Λ). For the sets G_{T} defined by either (1) or (2), we have the following equidistribution result:

Theorem 1.2. For every φ ∈ C_{c}(X) and x_{0} ∈ X such that \overline{Γx_{0}} = X,

\frac{1}{|Γ ∩ G_{T} |} ∑_{γ∈Γ∩G_{T}} φ(γ^{−1}x_{0}) → ∫_{X} φ dμ

as T → ∞.

1.4. Methods. We use two strategies to prove our results. First, in the presence of a Γ-invariant probability measure as in Theorem 1.2, the standard construction of an induced action gives a G-action, and one may understand the asymptotic behavior of the Γ-orbits by studying the corresponding Gorbits. Secondly, when this assumption is not satisfied (as in Theorem 1.1) one may use ‘duality’ to replace the question about Γ-orbits on H\G with a problem about H-orbits on G/Γ (a similar approach was introduced in [DRS] and employed in [EsMoSh]). Although more general, the second strategy is more difficult to implement because the question about H-orbits involves averaging along certain ‘skew-balls’ with respect to the function D.

Following either strategy, one is led to a question about the distribution of an orbit of a connected Lie group on a finite measure homogeneous space. Such questions have been thoroughly studied in recent years, building on Ratner’s classification of measures invariant under unipotent flows and subsequent work of many authors. The precise result we require follows from work of Nimish Shah [Sh]. We refer the reader to [KlShSt] for a recent and detailed account of this theory.

Our strategy also requires a precise understanding of the asymptotics of volumes of certain ‘skew balls’ in H, with respect to a norm in a linear representation, or with respect to the natural G-invariant metric on the symmetric space of G. For example we determine the asymptotics of the volume of the sets {g ∈ G : ∥g∥ < T }, where G is a semisimple Lie group realized as a matrix group in Mat_{d}(ℝ) and ∥ · ∥ is an arbitrary linear norm on Mat_{d}(ℝ). Even this very natural question has only been settled in the literature in certain special cases.

1.5. Applications. In the next section we establish notation and state our results in more detail, and at the end of the section, describe the organization of the paper. To conclude this introduction, we list some applications which illustrate the scope of our methods. The results of this section are proved in Section 11.

1.5.1. Values of indefinite irrational quadratic forms on frames. Let Q be a nondegenerate indefinite quadratic form in d ≥ 3 variables, which is not a scalar multiple of a rational form. Recall that the Oppenheim conjecture, proved by Margulis [Ma], states that the set Q(ℤ^{d}) of its values at integer points is dense in ℝ. A strengthening of this result was obtained by Dani and Margulis and by Borel and Prasad. Denote by F_{d} the space of unimodular frames in ℝ^{d}, i.e.

F_{d} = {(f_{1}, …, f_{d}) : f_{i} ∈ ℝ^{d}, Vol (ℝ^{d}/(ℤf_{1} ⊕ ⋯ ⊕ ℤf_{d})) = 1} .

Let F_{d}(ℤ) be the space of integer unimodular frames. For a frame f ∈ F_{d}, denote by Q̄(f) the corresponding Gram matrix:

Q̄(f) = (Q(f_{i}, f_{j}))_{i,j=1,...,d} ∈ Mat_{d}(ℝ)

(note that we denote by Q(·, ·) the bilinear form corresponding to Q(·)). Then it was observed in [DaMa] and [BoPr] that the set Q̄(F_{d}(ℤ)) is dense in Q̄(F_{d}). Our results imply a quantitative strengthening of this fact. Let

G = {g ∈ GL(d, ℝ) : det g = ±1}.

Recall that the map

g ∈ G ↦ g e ∈ F_{d},  (3)

where e is the frame consisting of the standard basis vectors, is a diffeomorphism. This endows F_{d} with a natural measure μ coming from Haar measure m on G, which we normalize by requiring that m (G/G(ℤ)) = 1. Note that Q̄(F_{d}) is a submanifold of Mat_{d}(ℝ), consisting of symmetric matrices of a given signature. We have:

Corollary 1.3. Let Q be as above. Fix any norm ∥·∥ on ℝ^{d} and let A ⊂ Q̄(F_{d}) be a bounded set whose boundary has measure zero. Then

# {f ∈ F_{d}(ℤ) : ∥f∥ < T, Q̄(f) ∈ A} ∼ μ ({f ∈ F_{d} : ∥f∥ < T, Q̄(f) ∈ A})

(here ∥(f_{1}, …, f_{d})∥ = max_{i=1,...,d} ∥f_{i}∥). Explicitly, these quantities are asymptotic to

\begin{matrix}CT^{p(q−1)} & if p < q \\ C log T T^{p(p−1)} & if p = q \end{matrix}

where (p, q), 1 ≤ p ≤ q is the signature of Q and C is an explicitly computable constant depending on A, Q and ∥ · ∥.

Note that in contrast with the quantitative version of the Oppenheim conjecture, proved by Eskin, Margulis and Mozes [EsMaMo], in this result there are no difficulties arising in signatures (1, 2) and (2, 2).

1.5.2. Dense projections of lattices. Suppose G is a semisimple non-simple group, and Γ is an irreducible lattice, i.e. the projection of Γ onto each noncocompact factor of G is dense. Our results imply a quantitative version of this fact. Here we formulate two special cases.

Let G be a Lie group which is a direct product G = SH of its closed subgroups S and H. Suppose Γ is a lattice in G such that HΓ is dense in G. Denote by P : G → S the projection map. Suppose H is simple, and Ψ_{H} : H → GL(V_{H}), Ψ_{S} : S → GL(V_{S}) are two irreducible representations with compact kernels.

Let m be a Haar measure on G, normalized by requiring that the covolume of Γ in G is equal to 1. Let λ and ν Haar measures on H and S respectively, with m = λ ⊗ ν (note that since G is unimodular, H and S are unimodular).

Corollary 1.4. Let the notation be as above.

(i) Let Ψ : G → GL(V_{H} ⊕ V_{S}) be the direct sum representation of G, and for a linear norm ∥ · ∥ on V_{H} ⊕ V_{S} let

H_{T} = {h ∈ H : ∥Ψ(h)∥ < T }, Γ_{T} = {γ ∈ Γ : ∥Ψ(γ)∥ < T }.

For every φ ∈ C_{c}(S) and every s_{0} ∈ S,

\frac{1}{λ(H_{T})} ∑_{γ∈Γ_{T}} φ(s_{0}P (γ)) ⟶_{T →∞} ∫_{S} φ dν.

(ii) Let Ψ : G → GL(V_{H} ⊗ V_{S}) be the tensor product representation of G. Choose bases {v_{1}, …, v_{k}}, {u_{1}, …, u_{ℓ}} of V_{H} and V_{S}, and for 1 ≤ p < ∞ let ∥ · ∥ denote both the p-norm on V_{S} associated with the basis {u_{1}, …, u_{ℓ}} and the p-norm on V_{H} ⊗V_{S} associated with the basis

{v_{i} ⊗ u_{j}}. Then for every φ ∈ C_{c}(S) and every s_{0} ∈ S,

\frac{1}{λ(H_{T})} ∑_{γ∈Γ_{T}} φ(s_{0}P (γ)) ⟶_{T →∞} c ∫_{S} φ(s) \frac{dν(s)}{∥Ψ_{S}(s_{0}s)∥^{m}} ,

where m and c are explicitly given positive constants.

1.5.3. Lattice actions by translations. Let L be a noncompact simple connected Lie group, and Λ and Δ lattices in L. The group Λ acts on L/Δ by left translation: if π : L → L/Δ is the quotient map, then the action is given by

λ π(g) = π(λg).

It is a simple application of Ratner’s theorems that for any x = π(g) ∈ L/Δ, Λx is either finite (in case Λ and gΔg^{−1} are commensurable) or dense. Note that a special case of this observation was used recently by Vatsal [Va] in his study of Heegner points. A direct application of Theorem 1.2 yields an equidistribution result for the dense Λ-orbits in L/Δ, generalizing recent results of [Oh, GoOh].

Let m be the Haar measure on L, normalized so that m (L/Δ) = 1. We denote the induced finite measures on L/Δ and L/Λ also by m. We assume that L is a closed subgroup of GL(d, ℝ), fix a norm ∥ · ∥ on Mat_{d}(ℝ), and set

L_{T} = {ℓ ∈ L : ∥ℓ∥ < T }.

Corollary 1.5. Let π(g_{0}) ∈ L/Δ be such that \overline{Λπ(g_{0})} = L/Δ. Then for φ ∈ C_{c}(L/Δ),

\frac{1}{m(L_{T})} ∑_{λ∈Λ∩L_{T}} φ(λ^{−1}π(g_{0})) ⟶_{T →∞} \frac{1}{m(L/Λ)} ∫_{L/Δ} φ dm.

1.5.4. Toral automorphisms. Consider Γ = SL(d, ℤ) acting by linear isomorphisms on the torus 𝕋 = ℝ^{d}/ℤ^{d}. It is well-known that an orbit Γx for this action is either finite (if x ∈ 𝕋(ℚ)) or dense. Theorem 1.2 implies a quantitative strengthening of this fact.

Corollary 1.6. Fix a norm ∥ · ∥ on Mat_{d}(ℝ). Let φ ∈ C_{c}(𝕋). Then for any x ∈ 𝕋 ∖ 𝕋(ℚ),

\frac{∑_{∥γ∥<T} φ(γ^{−1}x_{0})}{# \{γ ∈ Γ : ∥γ∥ < T \}} ⟶_{T →∞} ∫_{𝕋} φ dμ

where μ is the Lebesgue probability measure on 𝕋.

Corollary 1.6 was independently proved by N. Shah (private communication).

1.6. Acknowledgements. We thank Amos Nevo and Ralf Spatzier for many useful and insightful conversations. Thanks also to Hee Oh, Dave Morris and Michael Levin. We thank the Center for Advanced Studies in Mathematics at Ben-Gurion University for funding Gorodnik’s visit to Be’er Sheva, when this work was conceived, and the Technion’s hospitality, which made additional progress possible. The first author is partially supported by NSF grant 0400631, and the second by the Israel Science Foundation.

2. Notation and Statement of Results

2.1. The general setup. Let G be a second countable locally compact noncompact topological group, let Γ be a lattice in G, and let π : G → G/Γ be the natural quotient map.

There is a natural left-action of G and any of its subgroups on G/Γ defined by g_{1} π(g_{2}) = π(g_{1}g_{2}).

Since it admits a lattice, G is unimodular, i.e. the Haar measure m is invariant under both left and right multiplication. Let m′ denote the Ginvariant measure on G/Γ induced by m, that is,

m′(X) =^{def} m (Ω ∩ π^{−1}(X)) ,

for some Borel fundamental domain Ω for the right-action of Γ on G. It follows from the invariance of m that m′ is independent of the choice of Ω. Normalize m so that m′(G/Γ) = 1.

By a distance function on G we mean a function G → [0, ∞) which is continuous and proper. Let D be a distance function on G. The words the general setup holds mean that G, Γ, D, m, m′ are as above.

Fixing D, for a subset L ⊂ G we write

L_{T} =^{def} {g ∈ L : D(g) < T }.

We list some hypotheses about this setup.

UC Right uniform continuity of log D. For any ε > 0 there is a neighborhood U of identity in G such that for all g ∈ G, u ∈ U,

D(gu) < (1 + ε)D(g).  (4)

I1 Moderate volume growth for balls in G. For any ε > 0 there are δ > 0 and T_{0} such that for all T > T_{0}:

m (G_{(1+δ)T} ) ≤ (1 + ε)m (G_{T}) .  (5)

I2 Γ-points equidistributed in G w.r.t. D.

#Γ_{T} ∼ m(G_{T}).

2.2. Inducing the action. Suppose X is a space on which Γ acts on the right, preserving a finite invariant measure. There is a standard construction (see §3) of a left G-space Y with a G-map π_{G/Γ} : Y → G/Γ, such that the fiber over π(e) = [Γ] ∈ G/Γ is isomorphic to X. In case X = H\G, it is simply the action of G on the product H\G × G/Γ, given by

g · (τ(g_{2}), π(g_{1})) = (τ(g_{2}g^{−1}), π(gg_{1}))  (6)

where τ : G → H\G denotes the natural factor map.

The following reduces the study of asymptotic behavior of Γ-orbits on X to that of G-orbits on Y.

Proposition 2.1. Suppose hypotheses I1, I2 and UC hold. Let x_{0} ∈ X, and let y_{0} ∈ Y be the corresponding point in the fiber π_{G/Γ}^{−1} ([Γ]). Suppose that there is a measure μ on X such that:

(∗) For any F ∈ C_{c}(Y),

\frac{1}{m(G_{T})} ∫_{G_{T}} F (g^{−1} · y_{0}) dm(g) ⟶_{T →∞} ∫_{Y} F dν,

where dν = dμ dm′.

Then for any φ ∈ C_{c}(X),

\frac{1}{#Γ_{T}} ∑_{γ∈Γ_{T}} φ(x_{0} γ) ⟶_{T →∞} ∫_{X} φ dμ.

2.3. The general setup, continued. Suppose the general setup holds, let H be a closed subgroup of G, and let τ : G → H\G be the natural quotient map. Let λ be a left Haar measure on H. There is a natural right-action of G on H\G defined by τ(g_{1}) g_{2} = τ(g_{1}g_{2}). We have the following diagram.

\xymatrix{ & G \ar[dl]_{τ} \ar[dr]^{π} \\ H\G &  & G/Γ}

If H has been defined, the words the general setup holds mean in addition that H, λ and τ are as above.

If g_{1}, g_{2}, g ∈ G and L ⊂ G, we write

L_{T} [g_{1}, g_{2}] =^{def} {ℓ ∈ L : D(g_{1}ℓg_{2}) < T }, L_{T} [g] =^{def} L_{T} [e, g].  (7)

We sometimes call these ‘skew balls’. We list some additional hypotheses.

S Locally continuous section. For any x ∈ H\G there is a Borel map σ : H\G → G which is continuous in a neighborhood of x and satisfies τ∘σ = Id_{H\G} .

D1 Uniform volume growth for skew-balls in H. For any bounded B ⊂ G and any ε > 0 there are T_{0} and δ > 0 such that for all T > T_{0}

and all g_{1}, g_{2} ∈ B we have:
λ (H_{(1+δ)T} [g_{1}, g_{2}]) ≤ (1 + ε)λ (H_{T} [g_{1}, g_{2}]) .  (8)

D2 Limit volume ratios. For any g_{1}, g_{2} ∈ G, the limit

α(g_{1}, g_{2}) =^{def} lim_{T →∞} \frac{λ(H_{T} [g_{1}^{−1}, g_{2}])}{λ(H_{T})}  (9)

exists and is positive and finite.

2.4. Duality. The following result establishes a link between the asymptotic behavior of a Γ-orbit on H\G and the behavior of a corresponding H-orbit on G/Γ.

Theorem 2.2. Suppose the general setup holds, and conditions S, D1, UC are satisfied. Let g_{0} ∈ G and assume the following:

(∗∗) For any b ∈ G and any F ∈ C_{c}(G/Γ),

\frac{1}{λ (H_{T} [g_{0}, b])} ∫_{H_{T} [g_{0},b]} F (h^{−1}π(g_{0})) dλ(h) ⟶_{T →∞} ∫_{G/Γ} F (x) dm′(x).

Then for every non-negative compactly supported φ on H\G,

∑_{γ∈Γ_{T}} φ(τ(g_{0})γ) ∼ ∫_{G_{T}} φ(τ(g_{0})g) dm(g).  (10)

2.5. The limiting density. The conclusion of Theorem 2.2 describes the asymptotics of a Γ-orbit in terms of the asymptotics of a G-orbit. In order to calculate the latter we need some more terminology, and an additional assumption. Note that we are calculating the asymptotics of a G-orbit in a space on which G acts transitively. Nevertheless the computation is not trivial.

We have not assumed that H is unimodular, and therefore a G-invariant measure on H\G need not exist. To remedy this, we need to discuss measures on H\G. We suppose that condition D2 is satisfied, and define α(g_{1}, g_{2}) by (9).

Let Y be a lift of H\G to G. That is, Y is a Borel subset of G such that the product map

H × Y → G : (h, y) ↦ hy

is a Borel isomorphism. Since the measures m and λ are left H-invariant,

dm(hy) = dλ(h) dν_{Y} (y)  (11)

for some Borel measure ν_{Y} on Y . Moreover (11) determines ν_{Y} uniquely. When H is unimodular, ν_{Y} is identified with a measure on H\G which we denote by ν_{H\G}. It is independent of Y and is the unique (up to scaling) G-invariant measure on H\G.

For x_{0} = τ(g_{0}) ∈ H\G, define a measure ν_{x_{0}} on Y by

dν_{x_{0}}(y) =^{def} α(g_{0}, y) dν_{Y} (y).  (12)

This is easily seen to be well-defined (independent of the choice of g_{0}). Pushing forward the measures ν_{x_{0}} and ν_{Y} via the map τ|_{Y} defines measures on H\G, which we denote by the same letters. Although ν_{Y} (as a measure on H\G) depends on the choice of Y, the measure ν_{x_{0}} depends only on x_{0}. See Proposition 5.1 below for more details.

Theorem 2.3. Assume the general setup holds, and conditions UC, D1, D2 are satisfied. For x_{0} ∈ H\G and for every compactly supported continuous function φ on H\G,

\frac{1}{λ(H_{T})} ∫_{G_{T}} φ(x_{0} g) dm(g) ⟶_{T →∞} ∫_{H\G} φ dν_{x_{0}}.  (13)

Note that the conclusion of Theorem 2.3 may fail if condition D2 does not hold, see §12.2.

As an immediate corollary we obtain an equidistribution result for certain Γ-orbits. To state it, for A ⊂ H\G and x_{0} ∈ H\G, let

N_{T} (A, x_{0}) =^{def} #{γ ∈ Γ_{T} : x_{0} γ ∈ A}.

Corollary 2.4. Assume the general setup holds, and conditions S, UC, D1, D2 are satisfied. Let g_{0} ∈ G, x_{0} = τ(g_{0}), and assume (∗∗) holds. Then for any φ ∈ C_{c}(H\G) we have:

\frac{1}{λ(H_{T})} ∑_{γ∈Γ_{T}} φ(x_{0} γ) ⟶_{T →∞} ∫_{H\G} φ dν_{x_{0}},  (14)

and for any bounded A ⊂ H\G with ν_{x_{0}}(∂A) = 0 we have:

\frac{N_{T} (A, x_{0})}{λ(H_{T})} ⟶_{T →∞} ν_{x_{0}}(A)  (15)

2.6. Distance functions. In order to apply our general results we must show that the conditions listed above hold in some cases of interest. Showing this can be quite complicated, and requires specific methods for specific cases. We verify the conditions under several hypotheses on D, G and H. In all cases we consider in this paper, G will be a Lie group and H a Lie subgroup of G. In this setup condition S holds by a standard application of the implicit function theorem, see e.g. [W, Thm. 3.58]. Note that if G is not assumed to be a Lie group, condition S might not hold (see §12.1). In this paper we will always assume furthermore that at least one of H and G is semisimple and connected.

2.6.1. Linear norms. Suppose G is realized, via a linear representation Ψ : G → GL(d, ℝ), as a closed subgroup of Mat_{d}(ℝ), and ∥ · ∥ is a norm on Mat_{d}(ℝ) (considered as a vector space). Assume that ker Ψ is compact. In this setting we call

D(g) =^{def} max {1, ∥Ψ(g)∥}

a matrix norm distance function. Although D depends on both Ψ and ∥ · ∥, to simplify notation we omit this dependence from the notation.

2.6.2. Symmetric space distance functions. Suppose G is a connected, semisimple Lie group, K a maximal compact subgroup of G, and H a semisimple subgroup. Thus K\G (resp., G/K) is the right (resp. left) symmetric space of G. It is equipped with a natural G-invariant Riemannian metric induced by the Cartan–Killing form on the Lie algebra 𝔤 of G (see [He1, Ch. IV] or [Eb, Ch. 2]). Let P : G → K\G (resp., P′ : G → G/K) be the natural projection. Let d (resp. d′) be the corresponding metric on K\G (resp., G/K). In this setting we call

D(g) =^{def} exp (d (P (e), P (g)))

(respectively exp(d′(P′(e), P′(g)))) a symmetric space distance function. Note that D is proper since K is compact and x ↦ d(x_{0}, x) is proper on K\G.

Definition 2.5. Suppose the general setup holds. We say that G, D are standard if G is semisimple and connected, and D is either a matrix norm distance function or a symmetric space distance function. We say that G, H, D are standard in either of the following two cases:

• H is semisimple and connected and D is a matrix norm distance function.

• Both G and H are semisimple and connected and D is a symmetric space distance function.

Proposition 2.6. If G, D are standard and G is balanced then UC, I1 and I2 are satisfied. If G, H, D are standard then conditions UC, D1, and D2 are satisfied.

We deduce the proposition from the following two results, which are of independent interest, regarding the asymptotics of volumes of ‘balls’ with respect to a standard distance function.

Theorem 2.7. Let H be connected noncompact semisimple Lie group and let Ψ : H → GL(V) be a representation with compact kernel. Then for any matrix norm distance function we have

λ(H_{T}) ∼ C(log T)^{ℓ}T^{m},

where ℓ ∈ ℤ_{+} and m > 0 are explicitly computable constants depending only on Ψ, and C > 0 is an explicitly computable constant depending continuously on the choice of norm on Mat_{d}(ℝ).

In §7 we introduce a technical condition, called condition G, about Ψ (see Definition 7.1). The condition holds for ‘most’ choices of Ψ (cf. Remark 7.2). We prove Theorem 7.4, which is a precise version of Theorem 2.7, under the assumption that condition G holds, and assuming that Ψ is irreducible. The proof of Theorem 2.7 without making these assumptions is a rather lengthy computation which will appear in a separate paper.

In §9 we treat the case of a symmetric space distance function, and prove:

Theorem 2.8. Let G be a connected semisimple Lie group, H a connected semisimple subgroup, and let D : G → [0, ∞) be a symmetric space distance function. There are non-negative constants ℓ and m, and for any g_{1}, g_{2} ∈ G, a positive constant C = C(g_{1}, g_{2}) such that

λ (H_{T} [g_{1}, g_{2}]) ∼ C(log T)^{ℓ}T^{m}.  (16)

The constants ℓ ≥ 0, m > 0, C(g_{1}, g_{2}) > 0 are given explicitly, C(g_{1}, g_{2}) is continuous in g_{1}, g_{2}, and the convergence in (16) is uniform for g_{1}, g_{2} in compact subsets of G.

2.7. Balanced semisimple groups. We will see below that when H is simple and G, H, D are standard, condition (∗∗) holds whenever Hπ(g_{0}) is dense in G/Γ. However, in the case of a general semisimple group, we need to make an additional assumption.

Suppose that H is a semisimple non-simple Lie group with finite center, and thus a nontrivial almost direct product of its simple factors. It may happen that some of these factors do not contribute to the volume growth of balls in H, and therefore should be ignored when computing the asymptotics of H-orbits. To make this precise, we make the following definition.

Let H = H_{1} ⋯ H_{t} be the decomposition of H into an almost direct product of its simple factors. Fix measurable sections σ_{i} : H → H_{i}, i = 1, …, t; that is, σ_{i}(h) ∈ H_{i} and h = σ_{1}(h) ⋯ σ_{t}(h) for each h ∈ H.

Definition 2.9. We say that H is balanced if for every j ∈ {1, …, t}, every g_{1}, g_{2} ∈ G and every compact L ⊂ H_{j},

lim_{T →∞} \frac{λ (\{h ∈ H_{T} [g_{1}, g_{2}] : σ_{j}(h) ∈ L\})}{λ(H_{T} [g_{1}, g_{2}])} = 0.  (17)

Note that this definition does not depend on the choice of the sections because for any two sections σ_{i}, σ′_{i} we have σ′_{i}(h) ∈ Z σ_{i}(h) for every h ∈ H, where Z is the (finite) center of H. Note also that if H has compact factors then H is not balanced by our definition.

We remark that if D is a matrix norm distance function, corresponding to a representation Ψ : G → GL_{d}(ℝ) ⊂ Mat_{d}(ℝ) with compact kernel, and a norm ∥ · ∥ on Mat_{d}(ℝ), then the condition that H is balanced depends on Ψ but not on ∥ · ∥ (see Proposition 8.1).

The importance of the assumption that H is balanced lies in the following:

Theorem 2.10. Suppose the general setup holds, G, H, D are standard and H is balanced. Suppose g_{0} ∈ G satisfies \overline{Hπ(g_{0})} = G/Γ. Then (∗∗) holds.

Note that the conclusion of theorem 2.10 may fail if H is not balanced, see §12.3.

2.8. Equidistribution results. Collecting the results stated previously, we obtain the following results concerning the distribution of lattice orbits:

Corollary 2.11. Suppose the general setup holds, G, D are standard, and G is balanced. Suppose there is a Γ-invariant probability measure μ on H\G, and x_{0} ∈ H\G satisfies H\G = \overline{x_{0} Γ}.

Then for any φ ∈ C_{c}(H\G),

\frac{1}{#Γ_{T}} ∑_{γ∈Γ_{T}} φ(x_{0} γ) ⟶_{T →∞} ∫_{H\G} φ dμ.

Corollary 2.12. Suppose the general setup holds, G, H, D are standard, and H is balanced. Suppose x ∈ H\G with H\G = \overline{x Γ}. Then for any φ ∈ C_{c}(V),

S_{φ,x}(T) ∼ S̃_{φ,x}(T)  (18)

and

\frac{S_{φ,x}(T)}{λ(H_{T})} ⟶_{T →∞} ∫_{H\G} φ dν_{x}.  (19)

Also, for any bounded A ⊂ H\G with ν_{H\G} (∂A) = 0,

lim_{T →∞} \frac{N_{T} (A, x)}{λ(H_{T})} = ν_{x}(A).

2.9. Organization of the paper. In Section 3, we develop a framework for analyzing action preserving finite measure and prove a general version of Proposition 2.1. In Section 4, we prove Theorem 2.2, which relates distributions of Γ orbits in H\G and H-orbits in G/Γ. In Section 5, we discuss the limiting density and prove Theorem 2.3. In Section 6, we verify some of the basic properties of distance functions and prove Theorem 2.6. In Section 7, we compute volume asymptotics for sets defined by matrix norms proving Theorem 2.7, and in Section 9, we compute asymptotics for the Riemannian metric case proving Theorem 2.8. Section 8 contains discussion of ‘balancedness’. In Section 10, we prove Theorem 2.10. Finally, the results stated in Section 1 are proved in Section 11. In Section 12, we present examples to demonstrate necessity of conditions S, D2, and ‘balanced’, and discuss the example from Section 1.1 in more detail.

3. Induction

Let Γ act continuously on a locally compact Hausdorff space X from the right. In this section we recall the definition of the induced action of G, and relate the asymptotics of a Γ-orbit with that of a corresponding G-orbit.

On Ỹ =^{def} X × G, there is a left action of G given by

g_{1} · (x, g) = (x, g_{1}g),

and a right action of Γ given by

(x, g) · γ = (x · γ, gγ).

It is clear that these actions commute, so that G acts on the quotient Y = Ỹ/Γ, and the quotient map π_{G/Γ} : Y → G/Γ is a continuous G-map. Since Γ is discrete in G, the projection is a covering map, so that Y inherits (locally) the topological properties of Ỹ. As a G-space, Y is a fiber bundle with base G/Γ and fiber isomorphic to X. In particular, if μ is a Γ-invariant probability measure on X then dμ dm′ defines a finite G-invariant measure on Y.

Assume that σ : G/Γ → G is a Borel section. Then

γ(g) = g^{−1} σ(π(g)) ∈ Γ

and the map

π_{X} = π_{X}^{σ} : Y → X, π_{X} : y = [(x, g)] ↦ x · γ(g)

is well-defined (does not depend on the representative (x, g) ∈ Ỹ of the Γ-orbit [(x, g)] ∈ Y).

Note that when X = H\G is as in §2.2, the space Y is isomorphic to H\G × G/Γ, with the action given by (6) where the isomorphism is defined by the map

(x, g) ↦ (xg^{−1}, g), x ∈ X, g ∈ G.

The following reduces the study of asymptotic behavior of Γ-orbits on X to that of G-orbits on Y. It immediately implies Proposition 2.1.

Proposition 3.1. Suppose hypotheses I1 and UC hold. Let x_{0} ∈ X, let ỹ_{0} = (x_{0}, e) ∈ Ỹ and let y_{0} = [ỹ_{0}] ∈ Y. Suppose that there is a measure μ on X such that for any F ∈ C_{c}(Y),

\frac{1}{m(G_{T})} ∫_{G_{T}} F (g^{−1} y_{0}) dm(g) ⟶_{T →∞} ∫_{Y} F dν,  (20)

where ν = dμ dm′.

Then for any φ ∈ C_{c}(X),

\frac{1}{m(G_{T})} ∑_{γ∈Γ_{T}} φ(x_{0} · γ) ⟶_{T →∞} ∫_{X} φ dμ.  (21)

If in addition I2 holds then

\frac{1}{#Γ_{T}} ∑_{γ∈Γ_{T}} φ(x_{0} · γ) ⟶_{T →∞} ∫_{X} φ dμ.  (22)

Proof of Proposition 3.1. There is no loss of generality in assuming that φ ≥ 0.

Let ε > 0. By condition I1, there are positive δ, T_{0} such that for all T > T_{0},

\frac{m(G_{(1+δ)T})}{m(G_{T})} < 1 + ε.  (23)

By condition UC, there exists a symmetric neighborhood O of the identity in G such that for every T > 0,

G_{T} O ⊂ G_{(1+δ)T} .  (24)

Let χ ∈ C_{c}(G) be a non-negative function such that supp χ ⊂ O and ∫_{G} χ dm = 1. Define functions F̃ : Ỹ → ℝ, F : Y → ℝ by

F̃(ỹ) = F̃(x, g) = χ(g) φ(x), F (y) = ∑_{γ∈Γ} F̃(ỹγ),

where ỹ ∈ Ỹ, y = [ỹ] ∈ Y. Since F̃ is compactly supported, the sum in the definition of F is actually finite, F ∈ C_{c}(Y), and

∫_{Y} F dν = ∫_{G/Γ} ∫_{X} F (y) dμ(π_{X}(y)) dm′(π_{G/Γ}(y))

= ∑_{γ∈Γ} ∫_{σ(G/Γ)} ∫_{X} χ(gγ)φ(x · γ) dμ(x) dm(g)

= ∫_{X} φ dμ (∑_{γ∈Γ} ∫_{σ(G/Γ)} χ(gγ) dm(g))

= ∫_{X} φ dμ ∫_{G} χ dm = ∫_{X} φ dμ.

Applying (20) we obtain:

\frac{1}{m(G_{T})} ∫_{G_{T}} F (g^{−1} y_{0}) dm(g) ⟶_{T →∞} ∫_{X} φ dμ.  (25)

Now let

\begin{aligned}I_{T} (x_{0}) & =^{def} ∑_{γ∈Γ} (∫_{G_{T}} χ(g^{−1}γ) dm(g)) φ(x_{0} · γ) \\ & = ∑_{γ∈Γ} ∫_{G_{T}} F̃ ((x_{0}, g^{−1}) · γ) dm(g) \\ & = ∫_{G_{T}} F (g^{−1} y_{0}) dm(g). \end{aligned}  (26)

We claim that

I_{T/(1+δ)}(x_{0}) ≤ ∑_{γ∈Γ_{T}} φ(x_{0} · γ) ≤ I_{(1+δ)T} (x_{0}).  (27)

Let g ∈ G ∖ G_{(1+δ)T}. Then by (24), g ∉ G_{T} O, and hence g^{−1}γ ∉ O for all γ ∈ Γ_{T}. This implies that χ(g^{−1}γ) = 0, that is,

γ ∈ Γ_{T} ⟹ ∫_{G_{(1+δ)T}} χ(g^{−1}γ) dm(g) = ∫_{G} χ(g^{−1}γ) dm(g) = 1.

The right hand inequality in (27) follows.

Now if γ ∈ Γ ∖ Γ_{T} then for all g ∈ G_{T/(1+δ)} we have g^{−1}γ ∉ O, hence χ(g^{−1}γ) = 0 and so

γ ∉ Γ_{T} ⟹ ∫_{G_{T/(1+δ)}} χ(g^{−1}γ) dm(g) = 0.

This implies the second inequality in (27).

We obtain, for all large enough T:

∑_{γ∈Γ_{T}} φ(x_{0} · γ) ≤^{(27)} I_{T (1+δ)}(x_{0})

≤^{(25)} (1 + ε)m(G_{(1+δ)T}) ∫_{X} φ dν

≤^{(23)} (1 + ε)^{2}m(G_{T}) ∫_{X} φ dν.

Since ε > 0 was arbitrary,

lim sup_{T →∞} \frac{1}{m(G_{T})} ∑_{γ∈Γ_{T}} φ(x_{0} · γ) ≤ ∫_{X} φ dν.

The opposite inequality for lim inf is similarly established. This proves (21) and (22) immediately follows in light of condition I2. □

4. Duality

In this section we prove Theorem 2.2.

We first record the following useful consequence of condition UC.

Proposition 4.1. Suppose the general setup holds and condition UC is satisfied. Then for any bounded B ⊂ G and any η > 0 there is a neighborhood O of identity in G such that for all g ∈ G, h ∈ O, b ∈ B, we have

|\frac{D(ghb)}{D(gb)} − 1| < η.  (28)

Proof. Let ε = η and let U be as in the formulation of condition UC. Since B is bounded, there is a small enough neighborhood O of identity in G such that for all h ∈ O, b ∈ B we have b^{−1}hb ∈ U ∩ U^{−1}. This implies that for g ∈ G, h ∈ O, b ∈ B,

D(ghb) = D(gb b^{−1}hb) < (1 + η)D(gb)

and

D(gb) = D(ghb b^{−1}h^{−1}b) ≤ (1 + η)D(ghb).

Putting these together we obtain

\frac{D(gb)}{1 + η} < D(ghb) < (1 + η)D(gb),

and (28) follows. □

We now make some reductions:

Claim 1. There is no loss of generality in assuming:

(a) g_{0} = e.

(b) There is an open set U ⊂ H\G and a Borel section σ : H\G → G such that σ|_{U} is continuous and supp φ ⊂ U.

In order to reduce to (a), let H′ =^{def} g_{0}^{−1}Hg_{0}, λ′ a left Haar measure on H′, τ′ : G → H′\G the natural quotient map and φ′(τ′(g)) =^{def} φ(τ(g_{0}g)) and note that the hypotheses and conclusion of the theorem for H′, λ′, τ′, φ′, e are equivalent to those for H, λ, τ, φ, g_{0}.

Using condition S, for each x ∈ supp φ there is a section σ_{x} : H\G → G which is continuous in a neighborhood U_{x} of x. Using the compactness of supp φ we may choose a finite subset of {U_{x} : x ∈ supp φ} covering supp φ, and using a standard partition of unity argument for this cover we may assume that for some x ∈ H\G, supp φ ⊂ U_{x} and take σ = σ_{x}. This reduces our problem to the case (b). The claim is proved.

Let

B =^{def} σ(supp φ),

a compact subset of G, in view of Claim 1. Also let

h_{g} =^{def} σ̃(g)g^{−1} ∈ H, where σ̃ =^{def} σ ∘ τ : G → G.

Given c > 1 we will find T_{0} such that for all T > T_{0},

∑_{γ∈Γ_{T}} φ(τ(γ)) < c ∫_{G_{T}} φ(g) dm(g).  (29)

Let

0 < ε < \sqrt{c} − 1,  (30)

and let δ > 0 be as in hypothesis D1. Let η > 0 be small enough so that

(1 + η)^{4} < 1 + δ.  (31)

Using hypothesis UC and Proposition 4.1, let O be a small enough neighborhood of the identity in H so that for all g ∈ G, h ∈ O, b ∈ B,

D(ghb) < (1 + η)D(gb).

Let ψ : H → ℝ be a non-negative continuous function such that

supp ψ ⊂ O and ∫_{H} ψ(h) dλ(h) = 1.

Define a function f : G → ℝ by

f(g) = ψ(h_{g})φ(τ(g)).

Note that f is non-negative, continuous, of compact support, and

τ(g) ∉ supp φ ⟹ ∀h ∈ H, f(h^{−1}g) = 0.  (32)

We now show:

Claim 2. For any T > 0,

∑_{γ∈Γ_{T}} φ(τ(γ)) ≤ ∑_{γ∈Γ} ∫_{H_{(1+η)T} [σ̃(γ)]} f(h^{−1}γ) dλ(h)  (33)

and

∫_{G_{T}} φ(τ(g)) dm(g) ≥ ∫_{G} ∫_{H_{T/(1+η)}[σ̃(g)]} f(h^{−1}g)dλ(h) dm(g).  (34)

Let g ∈ G_{T}, that is, D(h_{g}^{−1}σ̃(g)) = D(g) < T. Suppose τ(g) ∈ supp φ. Then for h ∈ O one has

D(h_{g}^{−1}hσ̃(g)) < (1 + η)T,

that is, h_{g}^{−1}h ∈ H_{(1+η)T} [σ̃(g)]. Since supp ψ ⊂ O, this shows that

supp ψ ⊂ h_{g}H_{(1+η)T} [σ̃(g)].

20 ALEX GORODNIK AND BARAK WEISS

So, using left-invariance of λ and the fact that h_{g}h_{0} = h_{h_{0}^{−1}g}:

φ(τ(g)) = φ(τ(g)) ∫_{h_{g}H_{(1+η)T} [σ̃(g)]} ψ(h) dλ(h)
= φ(τ(g)) ∫_{H_{(1+η)T} [σ̃(g)]} ψ(h_{g}h) dλ(h)
= ∫_{H_{(1+η)T} [σ̃(g)]} ψ(h_{h^{−1}g})φ(τ(h^{−1}g)) dλ(h)
= ∫_{H_{(1+η)T} [σ̃(g)]} f(h^{−1}g) dλ(h).

Specializing to g = γ ∈ Γ_{T} one obtains (33). Let G̃ =^{def} τ^{−1}(supp φ). Similarly, it is easy to check that for g ∈ G̃ ∖ G_{T} ,

supp ψ ∩ h_{g}H_{T/(1+η)}[σ̃(g)] = ∅.

Therefore, arguing as in the previous computation:

∫_{G} ∫_{H_{T/(1+η)}[σ̃(g)]} f(h^{−1}g)dλ(h) dm(g)

=^{(32)} ∫G̃ ∫_{H_{T/(1+η)}[σ̃(g)]} f(h^{−1}g)dλ(h) dm(g)

≤ [∫_{G_{T}} + ∫_{G̃∖G_{T}}] (φ(τ(g)) ∫_{h_{g}H_{T/(1+η)}[σ̃(g)]} ψ(h)dλ(h)) dm(g)

= ∫_{G_{T}} φ(τ(g)) ∫_{h_{g}H_{T/(1+η)}[σ̃(g)]} ψ(h)dλ(h) dm(g)

≤ ∫_{G_{T}} φ(τ(g)) dm(g). This proves (34). We now claim:

Claim 3. There are b_{1}, …, b_{N} ∈ B and a symmetric neighborhood O of the identity in G, such that for x ∈ Ob_{i}, i = 1, …, N,

H_{(1+η)T} [x] ⊂ H_{(1+η)^{2}T} [b_{i}]

and such that

B ⊂ ⋃_{i=1}^{N} Ob_{i}.  (35)

To see this, using hypothesis UC and Proposition 4.1 we let O be a small enough symmetric neighborhood of identity in G such that for all g ∈ H, h ∈ O and b ∈ B, we have (1−η/2)D(gb) < D(ghb). This implies that H_{(1+η)T} [x] ⊂ H_{(1+η)^{2}T} [b] for all x ∈ Ob. Taking a finite subcover {Ob_{i} : i = 1, …, N} of the cover {Ob : b ∈ B} proves the claim.

Using a partition of unity subordinate to the cover (35), there is no loss of generality in assuming that for some b ∈ B,

supp φ ⊂ τ(Ob).  (36)

Thus for some b ∈ B, and for all γ ∈ Γ,

∫_{H_{(1+η)T} [σ̃(γ)]} f(h^{−1}γ) dλ(h) ≤ ∫_{H_{(1+η)^{2}T} [b]} f(h^{−1}γ) dλ(h).  (37)

Now defining

F (π(g)) =^{def} ∑_{γ∈Γ} f(gγ)

(actually a finite sum for each g) one obtains via the monotone convergence theorem:

\begin{matrix}∑_{γ∈Γ} ∫_{H_{(1+η)T} [σ̃(γ)]} f(h^{−1}γ) dλ(h) \\ ≤^{(37)} ∑_{γ∈Γ} ∫_{H_{(1+η)^{2}T} [b]} f(h^{−1}γ) dλ(h) \\ = ∫_{H_{(1+η)^{2}T} [b]} F (h^{−1}π(e)) dλ(h). \end{matrix}  (38)

Using (∗∗), there is T_{1} such that for all T > T_{1}:

|\frac{∫_{H_{(1+η)^{2}T} [b]} F (h^{−1}π(e)) dλ(h)}{λ (H_{(1+η)^{2}T} [b])} − ∫_{G/Γ} F (x) dm′(x)| < ε,

hence

∫_{H_{(1+η)^{2}T} [b]} F (h^{−1}π(e)) dλ(h) ≤ (1 + ε)λ (H_{(1+η)^{2}T} [b]) ∫_{G/Γ} F (x) dm′(x). (39)

Reversing the argument with G in place of Γ yields:

\begin{matrix}λ (H_{(1+η)^{2}T} [b]) ∫_{G/Γ} F (x) dm′(x) \\ ≤^{(8), (31)} (1 + ε)λ (H_{T/(1+η)^{2}}[b]) ∫_{G} f(g) dm(g) \\ = (1 + ε) ∫_{H_{T/(1+η)^{2}}[b]} ∫_{G} f(h^{−1}g) dm(g) dλ(h) \\ = (1 + ε) ∫_{G} ∫_{H_{T/(1+η)^{2}}[b]} f(h^{−1}g) dλ(h) dm(g) \\ ≤^{(32), (36)} (1 + ε) ∫_{G} ∫_{H_{T/(1+η)}[σ̃(g)]} f(h^{−1}g) dλ(h) dm(g) \\ ≤^{(34)} (1 + ε) ∫_{G_{T}} φ(τ(g)) dm(g). \end{matrix}  (40)

Now putting together (33), (38), (39), (40) and (30) proves (29).

The proof of the second inequality

∑_{γ∈Γ_{T}} φ(τ(γ)) > \frac{1}{c} ∫_{G_{T}} φ(τ(g))dm(g)  (41)

is similar and is left to the reader. The two inequalities (29) and (41), along with the assumption that g_{0} = e, imply (10). □

5. The limiting density

Let the notation be as in §2.5. In particular assume condition D2. In this section we discuss the measures {ν_{x} : x ∈ H\G} and prove Theorem 2.3. We first introduce some additional notation.

Let N_{G}(H) denote the normalizer of H in G. There is a homomorphism Δ : N_{G}(H) → ℝ_{+} such that for any Borel subset A of H and any n ∈ N_{G}(H),

λ(nAn^{−1}) = Δ(n) λ(A).  (42)

Note that Δ|_{H} is the modular function of H.

Proposition 5.1. Let g_{0} ∈ G, x_{0} = τ(g_{0}). Suppose that condition D2 is satisfied. Then α(g_{1}, g_{2}) and the measure ν_{x_{0}} have the following properties:

(i) For h ∈ H, n ∈ N_{G}(H) and g_{1}, g_{2} ∈ G,

α(nhg_{1}, ng_{2}) = Δ(n) α(g_{1}, g_{2}).
In particular, for h_{1}, h_{2} ∈ H, g_{1}, g_{2} ∈ G,
α(h_{1}g_{1}, h_{2}g_{2}) = Δ(h_{2}) α(g_{1}, g_{2}).

(ii) ν_{x_{0}} is well-defined (does not depend on g_{0}) and does not depend on the

section Y. (iii) If G, H are Lie groups then ν_{x_{0}} is absolutely continuous with respect

to the smooth measure class on H\G. (iv) If H is unimodular then

α_{x_{0}} : H\G → ℝ_{+}, α_{x_{0}}(τ(g)) = α(g_{0}, g)

is well-defined and dν_{x_{0}}(x) = α_{x_{0}}(x) dν_{H\G}(x), where ν_{H\G} is the Ginvariant measure on H\G.

Proof. For h ∈ H, n ∈ N_{G}(H), g_{1}, g_{2} ∈ G we have

H_{T} [(nhg_{1})^{−1}, ng_{2}] = {h′ ∈ H : D(g_{1}^{−1}(nh)^{−1}h′ng_{2}) < T }
= {h′ ∈ H : D(g_{1}^{−1}(nh)^{−1}n n^{−1}h′n g_{2}) < T }
= n {h′′ ∈ H : D(g_{1}^{−1}h^{−1}h′′g_{2}) < T } n^{−1}
= nh {h′′′ ∈ H : D(g_{1}^{−1}h′′′g_{2}) < T } n^{−1}
= nh H_{T} [g_{1}^{−1}, g_{2}] n^{−1},

hence, using (42) and the left-invariance of λ,

λ(H_{T} [(nhg_{1})^{−1}, ng_{2}]) = Δ(n) λ(H_{T} [g_{1}, g_{2}])

and (i) follows.

It follows from (i) that α(hg_{0}, y) = α(g_{0}, y) for all h ∈ H, so ν_{x_{0}} is welldefined. Suppose Y, Z are two different sections for τ. There are maps h̄ : Z → H, ȳ : Z → Y defined by the formula z = h̄(z)ȳ(z) for all z ∈ Z. Let ȳ_{∗}ν_{Z} denote the measure on Y obtained by pushing forward ν_{Z} via ȳ. Using (11) we have for each h ∈ H,

\begin{aligned}dm(hz) & = dm(hh̄(z)ȳ(z)) = dλ(hh̄(z)) dν_{Y} (ȳ(z)) \\ & = Δ(h̄(z)) dλ(h) dȳ_{∗}ν_{Z}(y). \end{aligned}

By the uniqueness of the decomposition (11) we have Δ(h̄(z)) dȳ_{∗}ν_{Z}(y) = dν_{Y} (y). In particular, setting ȳ(z) = y, h̄(z) = h = zy^{−1} we have

α(g_{0}, y)dν_{Y} (y) = α(g_{0}, yz^{−1} z)Δ(zy^{−1}) dȳ_{∗}ν_{Z}(y)
= Δ(yz^{−1}) α(g_{0}, z)Δ(zy^{−1})dν_{Z}(z)
= α(g_{0}, z)dν_{Z}(z).

proving (ii).

If H is a Lie subgroup of a Lie group G then there is a manifold structure on H\G which is obtained locally by finding, for each x_{0} ∈ H\G, a submanifold U ⊂ G such that x_{0} ∈ τ(U) and τ|_{U} is one to one. Now, using (ii), we can assume that the lift Y contains U. It is clear from (11) and G-invariance of m that ν_{Y} |_{U} is smooth and (iii) follows.

The last assertion follows immediately from the preceding ones. □

Proof of Theorem 2.3. By separating φ into its negative and positive parts, we may assume without loss of generality that φ ≥ 0. Assume also that φ ≠ 0 (otherwise there is nothing to prove). Let Y = σ(H\G) where σ : H\G → G is a Borel section, and let B = σ(supp φ). We may assume that B is bounded, e.g. by applying condition S and a partition of unity argument. Let ν_{Y} be the measure on Y satisfying (11). In view of Proposition 5.1, there is no loss of generality in defining ν_{x_{0}} by (12). We have:

\begin{aligned}∫_{G_{T}} φ(τ(g_{0})g) dm(g) & = ∫_{\{g:D(g_{0}^{−1}g)<T \}} φ(τ(g)) dm(g) \\ & = ∫_{Y} ∫_{\{h:D(g_{0}^{−1}hy)<T \}} φ(τ(y)) dλ(h) dν_{Y} (y) \\ & = ∫_{Y} φ(τ(y))λ(H_{T} [g_{0}^{−1}, y]) dν_{Y} (y). \end{aligned}  (43)

Let T_{0} be as in condition D1. We claim that there is C such that for all T > T_{0} and all y ∈ B,

\frac{λ(H_{T} [g_{0}^{−1}, y])}{λ(H_{T})} < C.

Let C_{1} > 1. By Proposition 4.1, there exists a neighborhood O of identity in G such that for all h ∈ O and b ∈ B, D(g_{0}^{−1}hb) < C_{1} D(g_{0}^{−1}b). Choose a finite cover B ⊂ ∪_{i=1}^{N}Ob_{i} for some b_{i} ∈ B. Then for every y ∈ B,

H_{T} [g_{0}^{−1}, y] ⊂ ⋃_{i=1}^{N} H_{C_{1}T} [g_{0}^{−1}, b_{i}]

and

λ(H_{T} [g_{0}^{−1}, y]) ≤ N max_{i} λ(H_{C_{1}T} [g_{0}^{−1}, b_{i}]).

Using condition D1 a finite number of times (depending on C_{1}) we find a constant C_{2} such that λ(H_{C_{1}T} [g_{0}^{−1}, b_{i}]) ≤ C_{2}λ(H_{T} [g_{0}^{−1}, b_{i}]). Since the limit defining α(g_{0}, b_{i}) exists, there is a constant C_{3} > 0 such that

\frac{λ(H_{T} [g_{0}^{−1}, b_{i}])}{λ(H_{T})} < C_{3}, for all T > T_{0} and i = 1, …, N.

Therefore

\frac{λ(H_{T} [g_{0}^{−1}, y])}{λ(H_{T})} < N C_{2} C_{3} =^{def} C.

Hence, by (9), (43), and Lebesgue’s dominated convergence theorem,

lim_{T →∞} \frac{1}{λ(H_{T})} ∫_{G_{T}} φ(τ(g_{0})g) dm(g) = ∫_{B} φ(τ(y)) lim_{T →∞} \frac{λ(H_{T} [g_{0}^{−1}, y])}{λ(H_{T})} dν_{Y} (y)
= ∫_{B} φ(τ(y))α(g_{0}, y) dν_{Y} (y)
= ∫_{Y} φ dν_{x_{0}}.

□

Proof of Corollary 2.4. Statement (14) is immediate from (10) and (13), and (15) follows by a standard argument for approximating 1_{A} from above and below by continuous functions, which we omit. □

6. Properties of distance functions

Our next goal will be to verify the conditions we have listed. Most of the verification is reduced to the computation of asymptotics of volumes of certain sets. We make the reduction in this section, and also list some simple relations between our conditions.

We first show that I2 follows from I1, UC, and a very special case of (∗∗). Namely, we have:

Proposition 6.1. Suppose that I1 and UC hold, and for any F ∈ C_{c}(G/Γ),

\frac{1}{m(G_{T})} ∫_{G_{T}} F (g^{−1}π(e)) dm(g) ⟶_{T →∞} ∫_{G/Γ} F dm′.  (44)

Then I2 holds, that is, #Γ_{T} ∼ m(G_{T}) as T → ∞.

Proof. Let ε > 0. By UC, there exists a symmetric neighborhood O of e in G such that

G_{T} O ⊂ G_{(1+ε)T}  (45)

for every T > 0. Let f ∈ C_{c}(G) be such that

f ≥ 0, supp(f) ⊂ O, ∫_{G} fdm = 1,  (46)

and let F ∈ C_{c}(G/Γ) be defined by F (π(g)) = ∑_{γ∈Γ} f(gγ). We have

∫_{G_{T}} F (π(g^{−1})) dm(g) = ∑_{γ∈Γ} ∫_{G_{T}} f(g^{−1}γ) dm(g)

=^{(45),(46)} ∑_{γ∈Γ_{(1+ε)T}} ∫_{G_{T}} f(g^{−1}γ) dm(g)

= ∑_{γ∈Γ_{(1+ε)T}} ∫_{G_{T}^{−1}γ} f dm ≤^{(46)} #Γ_{(1+ε)T} ,

and

∫_{G_{T}} F (π(g^{−1})) dm(g) = ∑_{γ∈Γ} ∫_{G_{T}} f(g^{−1}γ) dm(g)

≥^{(45),(46)} ∑_{γ∈Γ_{T/(1+ε)}} ∫_{G_{T}} f dm = #Γ_{T/(1+ε)}.

This implies that

lim sup_{T →∞} \frac{#Γ_{T}}{m(G_{T})} ≤ lim sup_{T →∞} \frac{1}{m(G_{T})} ∫_{G_{(1+ε)T}} F (π(g^{−1})) dm(g)

≤^{(44)} lim sup_{T →∞} \frac{m(G_{(1+ε)T})}{m(G_{T})} ∫_{G/Γ} F dm′ = lim sup_{T →∞} \frac{m(G_{(1+ε)T})}{m(G_{T})}

for every ε > 0. Thus, by I1,

lim sup_{T →∞} \frac{#Γ_{T}}{m(G_{T})} ≤ 1.

The lower estimate for lim inf is proved similarly. □

Proposition 6.2. If D is a matrix norm distance function then D satisfies condition UC.

Proof. The set

Mat_{d}^{1}(ℝ) =^{def} {z ∈ Mat_{d}(ℝ) : ∥z∥ = 1}

is compact. Hence, given ε > 0, there is a small enough neighborhood V of identity in Mat_{d}(ℝ) such that for all x ∈ V and all z ∈ Mat_{d}^{1}(ℝ),

∥zx∥ < (1 + ε).

Let U = Ψ^{−1}(V). There is a bounded set B ⊂ G such that for all g ∈ G ∖ B and all g′ ∈ U we have ∥Ψ(g)∥ > 1 and ∥Ψ(gg′)∥ > 1 so that

D(g) = ∥Ψ(g)∥ and D(gg′) = ∥Ψ(gg′)∥.

For any g ∈ G ∖ B, z = \frac{Ψ(g)}{∥Ψ(g)∥} ∈ Mat_{d}^{1}(ℝ), hence for any g′ ∈ U we have

\frac{D(gg′)}{D(g)} = \frac{∥Ψ(gg′)∥}{∥Ψ(g)∥} = ∥zΨ(g′)∥ < 1 + ε.

By making U smaller if necessary we can also ensure that

\frac{D(gg′)}{D(g)} < 1 + ε

for all g ∈ B and g′ ∈ U. This implies UC. □

Proposition 6.3. Suppose (X, d) is a metric space equipped with a right (respectively, left) action of G which is continuous and isometric. Then for any x_{0} ∈ X, the function

D(g) = exp(d(x_{0}, x_{0}g)) (resp., exp(d(x_{0}, gx_{0})))

satisfies UC.

Proof. Suppose the action is on the right. Given ε > 0, let

U =^{def} {g ∈ G : d(x_{0}, x_{0}g) < log(1 + ε)}.

Then for every g ∈ G and u ∈ U,

d(x_{0}, x_{0}gu) − d(x_{0}, x_{0}g) = d(x_{0}u^{−1}, x_{0}g) − d(x_{0}, x_{0}g)
≤ d(x_{0}u^{−1}, x_{0}) < log(1 + ε).

This implies UC. The proof for a left action is similar. □

Proposition 6.4. Suppose F : [0, ∞) → [0, ∞) and C : G × G → (0, ∞) are continuous functions such that for all δ > 0,

\frac{F ((1 + δ)T)}{F (T)} ⟶_{T →∞} 1.

If the general setup holds and m(G_{T}) ∼ F (T) then I1 holds.

If the general setup holds and

λ (H_{T} [g_{1}, g_{2}]) ∼ C(g_{1}, g_{2}) F (T)

as T → ∞, uniformly for g_{1}, g_{2} in compact subsets of G, then conditions D1 and D2 are satisfied.

Proof. Given a bounded B ⊂ G and ε > 0, there is T_{1} such that for all T > T_{1} and all g_{1}, g_{2} ∈ B,

\frac{C(g_{1}, g_{2}) F (T)}{(1 + ε)^{1/3}} < λ (H_{T} [g_{1}, g_{2}]) < (1 + ε)^{1/3} C(g_{1}, g_{2}) F (T).

Also there is T_{2} > 0 such that for all T > T_{2},

\frac{F ((1 + δ)T)}{F (T)} < (1 + ε)^{1/3}.

Then setting T_{0} = max{T_{1}, T_{2}} we obtain for T > T_{0},

λ (H_{(1+δ)T} [g_{1}, g_{2}]) < (1 + ε)^{1/3} C(g_{1}, g_{2}) F ((1 + δ)T)
< (1 + ε)^{2/3}C(g_{1}, g_{2}) F (T)
< (1 + ε)λ (H_{T} [g_{1}, g_{2}]) .

This proves D1.

Now for any g_{1}, g_{2} ∈ G we have

lim_{T →∞} \frac{λ(H_{T} [g_{1}^{−1}, g_{2}])}{λ(H_{T})} = lim_{T →∞} \frac{λ(H_{T} [g_{1}^{−1}, g_{2}])}{F (T)} \frac{F (T)}{λ(H_{T})} = \frac{C(g_{1}^{−1}, g_{2})}{C(e, e)},

and we obtain D2. The proof of I1 is similar and is omitted. □

Proof of Proposition 2.6. Condition UC follows from Propositions 6.2, 6.3. If G, D are standard and G is balanced, then condition I1 follows from Theorems 2.7 and 2.8 and Proposition 6.4, and condition I2 follows from Proposition 6.1 and Theorem 2.10.

If G, H, D are standard, then conditions D1, D2 follow from Theorems 2.7 and 2.8 and Proposition 6.4. □

7. Matrix norms and volume computations

In this section we consider the asymptotics of volumes of ‘balls’ H_{T} with respect to a matrix norm distance function. We first recall some standard details on Cartan (or polar) decomposition and Haar measure.

Let H be a connected semisimple Lie group. Let K be a maximal compact subgroup of H and let A be the associated split Cartan subgroup of H; by this we mean that A is a maximal connected group which is invariant under a Cartan involution associated to K, such that Ad(A) is diagonalizable over ℝ. Denote by 𝔥 and 𝔞 the Lie algebras of H and A respectively. One can write

𝔥 = 𝔥_{0} ⊕ ⨁_{α∈Φ} 𝔥_{α}

for Φ ⊂ 𝔞^{∗}, the dual space of 𝔞, where

𝔥_{α} = {X ∈ 𝔥 : ∀Y ∈ 𝔞, Ad(exp(Y))X = e^{α(Y)}X},

and Φ is the restricted root system of H relative to A. Let Φ^{+} be the set of positive roots with respect to some ordering and

𝔞^{+} = {Y ∈ 𝔞 : ∀α ∈ Φ^{+}, α(Y) ≥ 0}

the corresponding (closed) Weyl chamber. Let ρ ∈ 𝔞^{∗} be defined by

ρ = \frac{1}{2} ∑_{α∈Φ} m_{α}α.  (47)

Let Δ = {α_{1}, …, α_{r}} ⊂ Φ^{+} be the set of simple roots that corresponds to 𝔞^{+} and {β̃_{1}, …, β̃_{r}} the dual basis of 𝔞, that is,

α_{i}(β̃_{j}) = δ_{ij}.

It will be convenient to rescale each β̃_{i} according to ρ. Namely, let

β_{i} = \frac{β̃_{i}}{2ρ(β̃_{i})},

so that

2ρ (∑_{i=1}^{r} t_{i}β_{i}) = ∑_{i=1}^{r} t_{i} and i ≠ j ⟹ α_{i}(β_{j}) = 0.  (48)

Every h ∈ H can be written as h = k_{1}ak_{2}, where k_{1}, k_{2} ∈ K and a ∈ A^{+} =^{def} exp(𝔞^{+}). Note that this decomposition is not unique. Nevertheless, λ can be expressed in terms of this decomposition of G (see [He2, p. 186]). Namely, letting dk denote the probability Haar measure on K we have for f ∈ C_{c}(H):

∫_{H} f(h) dλ(h) = ∫_{K} ∫_{𝔞^{+}} ∫_{K} f(k_{1} exp(Y)k_{2})ξ(Y) dk_{1} dY dk_{2},  (49)

where dY is a scalar multiple of the Lebesgue measure on 𝔞, and

ξ(Y) = ∏_{α∈Φ^{+}} sinh^{m_{α}}(α(Y)), m_{α} = dim 𝔥_{α}.  (50)

Now suppose Ψ : H → GL(V) is an irreducible representation. Let d = dim V and V = V_{1} ⨁ ⋯ ⨁ V_{s} be a direct sum decomposition into the weightspaces of 𝔞 with weights λ_{1}, …, λ_{s} ∈ 𝔞^{∗}, that is, for all Y ∈ 𝔞, v ∈ V_{i},

Ψ(exp(Y))v = e^{λ_{i}(Y)}v.  (51)

One of the λ_{i}’s is a highest weight – indeed, this is well-known over ℂ and holds also over ℝ, see e.g. [GJT, Chap. IV]. By re-ordering the basis of V assume with no loss of generality that the highest weight is λ_{1}, i.e.

∀Y ∈ 𝔞^{+}, ∀j ∈ {2, …, s}, λ_{1}(Y) ≥ λ_{j}(Y).  (52)

Now let

m_{1} = min_{i=1,...,r} λ_{1}(β_{i}).  (53)

By re-ordering β_{1}, …, β_{r} assume with no loss of generality that

m_{1} = λ_{1}(β_{1}).

We now make the following assumption.

Definition 7.1. Say that Ψ satisfies condition G if

m_{2} = min_{j=2,...,r} λ_{1}(β_{j}) > m_{1} = λ_{1}(β_{1}).  (54)

Remark 7.2. Irreducible representations of H are usually described in terms of the corresponding dominant weights. Note that if condition G fails, then for some i ≠ j, the dominant weight λ of Ψ satisfies

ρ(β̃_{j})λ(β̃_{i}) = ρ(β̃_{i})λ(β̃_{j}).

Thus the condition fails only for a subset of Ψ whose dominant weight is contained in a finite union of proper linear subspaces of 𝔞^{∗}.

Let

J = {j ∈ {1, …, s} : λ_{j}(β_{1}) = m_{1}}.

Note that 1 ∈ J, and by (52), we have for j ∈ J,

λ_{j}(β_{1}) = max_{t=1,...,s} λ_{t}(β_{1}).

We fix a basis V such that Ψ(A) is a diagonal subgroup. For j = 1, …, d let E_{j} ∈ Mat_{d}(ℝ) denote the matrix whose j, j-th entry is 1 and all other entries are zero, and for τ = (t_{2}, …, t_{r}) ∈ ℝ^{r−1}, define

τ̄ = ∑_{i=2}^{r} t_{i}β_{i},  E_{τ} = ∑_{(j,k): k∈J ,E_{j}∈V_{k}} e^{λ_{k}(τ̄)}E_{j}.  (55)

Also, let

Φ̂ = {α ∈ Φ : α(β_{1}) = 0},

that is, Φ̂ contains those roots whose expression as a linear combination of simple roots does not involve α_{1}, and let

ξ̂(τ) = (\frac{1}{2})^{∑_{α∉Φ̂} m_{α}} ∏_{α∈Φ̂} (\frac{1}{2} − \frac{1}{2e^{2α(τ̄)}} )^{m_{α}} e^{∑_{i=2}^{r} t_{i}}.

We say that a collection N of norms on Mat_{d}(ℝ) is bounded if there is c > 1 such that for any two norms ∥ · ∥_{1}, ∥ · ∥_{2} in N, and any nonzero A ∈ Mat_{d}(ℝ), \frac{1}{c} < \frac{∥A∥_{1}}{∥A∥_{2}} < c.

Proposition 7.3. For any norm ∥ · ∥ on Mat_{d}(ℝ), the indefinite integral

D = ∫_{τ∈[0,∞)^{r−1}} \frac{ξ̂(τ)}{∥E_{τ}∥^{1/m_{1}}} dτ  (56)

converges, and the convergence is uniform for ∥ · ∥ in a bounded collection of norms.

Theorem 7.4. Let H be connected and semisimple, and let Ψ : H → GL(V) be an irreducible representation satisfying condition G. Choosing a basis of V, identify Ψ(H) with a subset of Mat_{d}(ℝ), where d = dim V. Then for any linear norm ∥ · ∥ on Mat_{d}(ℝ) we have

λ(H_{T}) ∼ CT^{m},

where m = \frac{1}{m_{1}} , m_{1} is given by (53), and

C = ∫_{K} ∫_{K} ∫_{τ∈[0,∞)^{r−1}} \frac{ξ̂(τ)}{∥Ψ(k_{1})E_{τ}Ψ(k_{2})∥^{m}} dτ dk_{1} dk_{2}.

Remark 7.5. (1) In the above expression, m depends only on Ψ, and C depends continuously on ∥ · ∥; this means that for any ε > 0 there is δ > 0 such that for a norm | · |,

∀v ≠ 0, |1 − \frac{∥v∥}{|v|}| < δ ⟹ |C(∥ · ∥) − C(| · |)| < ε.

(2) Since for any k_{1}, k_{2} ∈ G, the map A ↦ ∥Ψ(g_{1})AΨ(g_{2})∥ is a linear norm on Mat_{d}(ℝ), the integral for C converges in light of Proposition 7.3.

(3) In the general case, that is when Ψ is reducible or does not satisfy condition G, there exist k ∈ ℤ_{+}, m > 0 and C > 0, where k and m depend only on Ψ and C depends continuously on ∥ · ∥, such that

λ(H_{T}) ∼ C(log T)^{k}T^{m}.

Details will appear elsewhere.

Proposition 7.6. Suppose N is a bounded collection of norms on Mat_{d}(ℝ). Then for any δ > 0 there is T_{0} such that for all T ≥ T_{0} and all ∥ · ∥ ∈ N we have:

|∫_{𝔞^{+}(T,∥·∥)} ξ(Y)dY − DT^{m}| < δ T^{m},  (57)

where

𝔞^{+}(T, ∥ · ∥) = {Y ∈ 𝔞^{+} : ∥Ψ(exp(Y))∥ < T },

D is as in (56), and m = 1/m_{1}.

Proof of Theorem 7.4 assuming Proposition 7.6. Note that the collection of norms

{x ↦ ∥Ψ(k_{1})xΨ(k_{2})∥ : k_{1}, k_{2} ∈ K}

is bounded. Thus the result follows using (49). □

Proof of Proposition 7.3. Let N be a bounded collection of norms on Mat_{d}(ℝ). For the purpose of this proof, the notation X ≪ Y will mean that X and Y are quantities depending on various parameters, and there is C, depending only on N and independent of the other parameters, such that X ≤ CY.

First note that for a fixed basis B of Mat_{d}(ℝ), for every ∥ · ∥ ∈ N, and every A = ∑_{E∈B} a_{E}E ∈ Mat_{d}(ℝ),

∥A∥ ≪ max_{E∈B} |a_{E}| ≪ ∥A∥.

Since E_{1}, …, E_{d} can be completed to a basis of Mat_{d}(ℝ), this implies that ∥E_{τ}∥ is bounded below by a positive constant independent of τ ∈ [0, ∞)^{r−1} and of ∥ · ∥ ∈ N. Thus we need only consider the behavior of the integrand as τ → ∞. For E_{j} ∈ V_{1}, we have

e^{m_{2} ∑_{i=2}^{r} t_{i}} ≪^{(54)} e^{∑_{i=2}^{r} t_{i}λ_{1}(β_{i})}∥E_{j}∥ = e^{λ_{1}(τ̄)}∥E_{j}∥ ≪^{(55)} ∥E_{τ}∥.  (58)

On the other hand it is clear that

ξ̂(τ) ≪ e^{∑_{i=2}^{r} t_{i}}.  (59)

Putting together (58) and (59), and using condition G we obtain

D = ∫_{τ∈[0,∞)^{r−1}} \frac{ξ̂(τ)}{∥E_{τ}∥^{1/m_{1}}} dτ

≪^{(58)} ∫_{τ∈[0,∞)^{r−1}} \frac{e^{∑_{i=2}^{r} t_{i}}}{(e^{m_{2} ∑_{i=2}^{r} t_{i}})^{1/m_{1}}} dτ

= ∫_{τ∈[0,∞)^{r−1}} e^{(1−m_{2}/m_{1}) ∑_{i=2}^{r} t_{i}}dτ ≪^{(54)} ∞.

□

Proof of Proposition 7.6. We first show that for all sufficiently large M, for all T > 0, we have

∫_{𝔟^{+}(M,T,∥·∥)} ξ(Y)dY < (\frac{δ}{3}) T^{m},  (60)

where

𝔟^{+}(M, T, ∥ · ∥) = {∑_{i=1}^{r} t_{i}β_{i} ∈ 𝔞^{+}(T, ∥ · ∥) : ∃j ∈ {2, …, r}, t_{j} ≥ M} . (61)

It suffices to show that for any j ∈ {2, …, r} there is M_{0} such that for M > M_{0} and any T > 0

∫_{𝔟} ξ(Y)dY < (\frac{δ}{3r}) T^{m},  (62)

where

𝔟 = {∑_{i=1}^{r} t_{i}β_{i} ∈ 𝔞^{+}(T, ∥ · ∥) : t_{j} ≥ M} .

To prove (62) we will need the following easy lemma, which is proved by induction on k:

Lemma 7.7. Given k ∈ ℕ and 0 < m_{1} < ⋯ < m_{k}, there is c > 0 such that for all S > 0,

∫_{Δ} e^{∑_{i=1}^{k} t_{i}}dt_{1} ⋯ dt_{k} ≤ ce^{S/m_{1}},

where

Δ = {(t_{1}, …, t_{k}) : ∀i, t_{i} ≥ 0, ∑_{1}^{k} m_{i}t_{i} ≤ S} .

By (52), and by comparing with the max-norm, we find that there is C̃ such that for all ∥ · ∥ ∈ N and all T > 0,

𝔞^{+}(T, ∥ · ∥) ⊂ {∑_{i=1}^{r} s_{i}β_{i} : ∀i, s_{i} ≥ 0, ∑_{i=1}^{r} s_{i}λ_{1}(β_{i}) ≤ log T + C̃} . (63)

Suppose j ∈ {2, …, r} and t_{j} ≥ M. To simplify notation suppose j = r. Then we obtain that

𝔟 ⊂ {∑_{i=1}^{r} s_{i}β_{i} : ∀i, s_{i} ≥ 0, s_{r} ≥ M, ∑_{i=1}^{r} s_{i}λ_{1}(β_{i}) < log T + C̃}

Now using condition G, choose m̃_{1} < m̃_{2} < ⋯ < m̃_{r} such that

m̃_{1} = m_{1} and m̃_{j} ≤ λ_{1}(β_{j}), j ∈ {2, …, r}.

Then it follows that 𝔟 ⊂ Mβ_{r} + 𝔠 (that is, the translation of 𝔠 by the vector Mβ_{r}) where

𝔠 = {∑_{i=1}^{r} s_{i}β_{i} : ∀i, s_{i} ≥ 0, ∑_{i=1}^{r} s_{i}m̃_{i} < log T + C̃ − m_{2}M} .

Applying Lemma 7.7 we find that

∫_{𝔟} ξ(Y)dY ≤ ∫_{𝔟} e^{2ρ(∑_{1}^{r} t_{i}β_{i})}dt_{1} ⋯ dt_{r} ≤ ∫_{Mβ_{r}+𝔠} e^{∑_{1}^{r} t_{i}}dt_{1} ⋯ dt_{r}
= e^{M} ∫_{𝔠} e^{∑_{1}^{r} t_{i}}dt_{1} ⋯ dt_{r} ≤ ce^{M+(log T +C̃−m_{2}M)/m_{1}}
≤ ceC̃e^{(1−m_{2}/m_{1})M} T^{1/m_{1}}.

Since m_{2} > m_{1}, this implies (62), and we have proved (60).

Now set

C_{τ} = \frac{ξ̂(τ)}{∥E_{τ}∥^{m}} ,  (64)

so that

D = ∫_{[0,∞)^{r−1}} C_{τ}dτ.  (65)

Using Proposition 7.3, we may assume by enlarging M that

|∫_{ℝ^{r−1}∖[0,M]^{r−1}} C_{τ} dτ| < δ/3.  (66)

We will show below that

Claim 4. There is T_{0} such that for all ∥ · ∥ ∈ N, all T > T_{0} and all τ = (t_{2}, …, t_{r}) ∈ [0, M]^{r−1},

|\frac{∫_{𝔡(τ,T)} ξ (∑_{i=1}^{r} t_{i}β_{i}) dt_{1}}{C_{τ}T^{m}} − 1| < \frac{δ}{3M^{r−1} max_{τ∈[0,M]^{r−1}} C_{τ}} ,  (67)

where

𝔡(τ, T) = {t_{1} : (t_{1}, t_{2}, …, t_{r}) ∈ 𝔞^{+}(T, ∥ · ∥)} .

Assuming the validity of Claim 4, and writing

𝔞 = 𝔞^{+}(T, ∥ · ∥), 𝔟 = 𝔟^{+}(M, T, ∥ · ∥),

we have for all T > T_{0}:

|∫_{𝔞} ξ(Y)dY − DT^{m}|

<^{(65)} |∫_{𝔞∖𝔟} ξ(Y)dY − T^{m} ∫_{[0,M]^{r−1}} C_{τ}dτ|

+ |∫_{𝔟} ξ(Y)dY | + T^{m} |∫_{ℝ^{r−1}∖[0,M]^{r−1}} C_{τ}dτ|

<^{(60),(66)} |∫_{[0,M]^{r−1}} [∫_{𝔡(τ,T)} ξ(t_{1}β_{1} + τ̄) dt_{1} − C_{τ}T^{m}] dτ| + \frac{2δ}{3} T^{m}

<^{(67)} δT^{m}.

In order to prove the claim, we will first show that for any η > 0 there is T_{0} such that for all ∥ · ∥ ∈ N and all T > T_{0},

[0, \frac{log T − log ∥E_{τ}∥ − η}{m_{1}} ] ⊂ 𝔡(τ, T) ⊂ [0, \frac{log T − log ∥E_{τ}∥ + η}{m_{1}} ] . (68)

Indeed, suppose that t_{1} ∈ 𝔡(τ, T), that is,

∥Ψ (exp(∑_{i=1}^{r} t_{i}β_{i}))∥ < T.

Write u = max_{s∉J} λ_{s}(β_{1}) < m_{1}. Then we have:

∥Ψ (exp(∑_{i=1}^{r} t_{i}β_{i}))∥ = ∥ ∑_{(j,k): E_{j}∈V_{k}} e^{λ_{k}(∑_{i=1}^{r} t_{i}β_{i})}E_{j}∥

= ∥ ∑_{(j,k): E_{j}∈V_{k}} e^{t_{1}λ_{k}(β_{1})} e^{∑_{i=2}^{r} t_{i}λ_{k}(β_{i})}E_{j}∥

= ∥e^{m_{1}t_{1}}E_{τ} + ∑_{(j,k):k∉J ,E_{j}∈V_{k}} e^{t_{1}λ_{k}(β_{1})} e^{∑_{i=2}^{r} t_{i}λ_{k}(β_{i})}E_{j}∥

≥ e^{m_{1}t_{1}}∥E_{τ}∥ − Ce^{ut_{1}}

(here C is a constant depending only on N and M). Adjusting C if necessary we obtain:

e^{m_{1}t_{1}}∥E_{τ}∥(1 − Ce^{(u−m_{1})t_{1}}) < T.  (69)

Taking t_{1} and T large enough we can ensure that

− log (1 − Ce^{(u−m_{1})t_{1}}) < η,

and by plugging this in (69) and taking logs we find:

m_{1}t_{1} < log T − log ∥E_{τ}∥ + η.

This proves the right hand inclusion in (68). The proof of the left hand inclusion is similar.

For any α ∈ Φ ∖ Φ̂, α(∑_{i=1}^{r} t_{i}β_{i}) →_{t_{1}→∞} ∞ and hence

\frac{sinh (α(∑_{i=1}^{r} t_{i}β_{i}))}{e^{α(∑_{i=1}^{r} t_{i}β_{i})}} ⟶_{t_{1}→∞} \frac{1}{2}.

Hence for fixed τ,

\frac{ξ (∑_{i=1}^{r} t_{i}β_{i})}{e^{t_{1}}} = \frac{ξ(∑_{i=1}^{r} t_{i}β_{i})}{e^{∑_{i=1}^{r} t_{i}}} e^{∑_{i=2}^{r} t_{i}}

= \frac{∏_{α∈Φ} sinh (α(∑_{i=1}^{r} t_{i}β_{i}))^{m_{α}}}{e^{2ρ(∑_{i=1}^{r} t_{i}β_{i})}} e^{∑_{i=2}^{r} t_{i}}

= ∏_{α∈Φ} (\frac{sinh (α(∑_{i=1}^{r} t_{i}β_{i}))}{e^{α(∑_{i=1}^{r} t_{i}β_{i})}})^{m_{α}} e^{∑_{i=2}^{r} t_{i}}

⟶_{t_{1}→∞} (\frac{1}{2})^{∑_{α∉Φ̂} m_{α}} ∏_{α∈Φ̂} (\frac{sinh (α(∑_{i=1}^{r} t_{i}β_{i}))}{e^{α(∑_{i=1}^{r} t_{i}β_{i})}})^{m_{α}} e^{∑_{i=2}^{r} t_{i}}

= ξ̂(τ) .

Note also that the convergence is uniform for τ in compact sets. It follows that

\frac{ξ̂(τ) ∫_{𝔡(τ,T)} e^{t_{1}}dt_{1}}{∫_{𝔡(τ,T)} ξ (∑_{i=1}^{r} t_{i}β_{i}) dt_{1}} ⟶_{T →∞} 1,  (70)

and the convergence is uniform for τ ∈ [0, M]^{r−1}.

We now have:

lim_{T →∞} \frac{∫_{𝔡(τ,T)} ξ (∑_{i=1}^{r} t_{1}β_{i}) dt_{1}}{C_{τ}T^{m}} =^{(70)} lim_{T →∞} \frac{ξ̂(τ) ∫_{𝔡(τ,T)} e^{t_{1}}dt_{1}}{C_{τ}T^{m}}

=^{(68)} lim_{T →∞} \frac{ξ̂(τ) ∫_{0}^{(log T −log ∥E_{τ} ∥)/m_{1}} e^{t_{1}}dt_{1}}{C_{τ}T^{m}}

= lim_{T →∞} \frac{ξ̂(τ)}{C_{τ}T^{m}} (\frac{T}{∥E_{τ}∥})^{1/m_{1}} =^{(64)} 1.

The convergence in the above expression is uniform in τ, therefore Claim 4 is valid. This completes the proof of the Proposition. □

8. Balanced semisimple groups

As a corollary of the computations in the previous sections, we obtain some information about balanced semisimple groups (cf. Definition 2.9). First we have:

Proposition 8.1. If D is a matrix norm distance function, corresponding to a representation Ψ : G → GL_{d}(ℝ) ⊂ Mat_{d}(ℝ) with compact kernel and a norm ∥ · ∥ on Mat_{d}(ℝ), then the condition that H is balanced depends on Ψ but not on ∥ · ∥.

Proof. Let ∥·∥, ∥·∥′ be two norms on Mat_{d}(ℝ), and suppose that H is balanced with respect to ∥ · ∥. Then for some C > 1 and every x ∈ Mat_{d}(ℝ),

\frac{∥x∥}{C} ≤ ∥x∥′ ≤ C∥x∥.  (71)

Let H_{i}, i = 1, …, t be the simple factors of H and let σ_{i} : H → H_{i} be measurable sections. Given j ∈ {1, …, t}, g_{1}, g_{2} ∈ G and a compact L ⊂ H_{j} let

\begin{aligned}S_{T} & = S_{T} [g_{1}, g_{2}] = \{h ∈ H_{T} [g_{1}, g_{2}] : σ_{j}(h) ∈ L\}, \\ H′_{T} & = H′_{T} [g_{1}, g_{2}] = \{h ∈ H : ∥g_{1}hg_{2}∥′ < T \}, \\ S′_{T} & = S′_{T} [g_{1}, g_{2}] = \{h ∈ H′_{T} [g_{1}, g_{2}] : σ_{j}(h) ∈ L\}. \end{aligned}

It follows from (71) that

S′_{T} ⊂ S_{CT} and H_{T/C} ⊂ H′_{T} .

It is a consequence of Theorem 2.7 that

lim sup_{T →∞} \frac{λ(H_{CT})}{λ(H_{T/C})} < ∞.

Therefore

\frac{λ(S′_{T})}{λ(H′_{T})} ≤ \frac{λ(S_{CT})}{λ(H_{T/C})} = \frac{λ(S_{CT})}{λ(H_{CT})} \frac{λ(H_{CT})}{λ(H_{T/C})} ⟶_{T →∞} 0.

This shows that H is balanced with respect to the norm ∥ · ∥′. □

We also have the following result, showing that balanced representations of semisimple nonsimple groups are rather atypical.

Proposition 8.2. Let H be a semisimple nonsimple Lie group, realized as a matrix group via an irreducible representation Ψ : H → GL(V). Suppose condition G is satisfied. Then H is not balanced.

Proof. Let H = H_{1} ⋯ H_{t}, t ≥ 2 be a representation of H as an almost direct product, let K be a maximal compact subgroups of H, let A be an associated split Cartan subgroup of H, and for each i ∈ {1, …, t} write A_{i} = H_{i} ∩ A, so that A_{i} is a split Cartan subgroup of H_{i}. We write Φ = Φ_{1} ∪ ⋯ ∪ Φ_{t} where Φ_{i} is the root system corresponding to (H_{i}, A_{i}). Also fix measurable sections σ_{i} : H → H_{i}.

Let the notations be as in §7. There is a partition

Δ = Δ_{1} ∪ ⋯ ∪ Δ_{t}

of the simple roots of 𝔞 such that for each j ∈ {1, …, t}, Δ_{j} is a set of simple roots for 𝔞_{j}. Assume by reordering that α_{1} ∈ Δ_{1}, and let j ∈ {2, …, t}. Since Δ is indexed by {1, …, r}, we think of Δ_{j} as a subset of {1, …, r}. For each M > 0 and T > 0, and each norm ∥ · ∥ on Mat_{d}(ℝ), define

𝔞^{+}(M, T, ∥ · ∥) = {Y = ∑_{i=1}^{r} s_{i}β_{i} ∈ 𝔞^{+}(T, ∥ · ∥) : ∀i ∈ Δ_{j}, s_{i} ≤ M} .

Then 𝔞_{j} ∩ 𝔞^{+}(M, T, ∥ · ∥) is compact for each M > 0 and hence

L = L(M) = ⋃_{T} σ_{j} (K exp(𝔞^{+}(M, T, ∥ · ∥))K) ⊂ H_{j}

is precompact.

Now let g_{1} = g_{2} = e, let C be as in Theorem 7.4, let 0 < δ < C, and let 𝔟(M, T, ∥·∥) be as in (61). Since the complement of 𝔟(M, T, ∥·∥) is contained in 𝔞^{+}(M, T, ∥ · ∥), we see that for large M (60) contradicts (17). □

9. Riemannian skew balls and volume computations

Suppose G is a connected semisimple Lie group and H is its connected semisimple Lie subgroup. The main result of this section is a computation of the asymptotics of the volume growth for certain ‘skew balls’ in H, with respect to a symmetric space distance function.

Before formulating the precise result we introduce some notation. We are given a semisimple Lie group G, a semisimple Lie subgroup H and a maximal compact subgroup K. It follows from a theorem of Mostow [Mos] that for a conjugate H′ of H, we can choose a maximal compact subgroup L of H′ respectively so that L ⊂ K and split Cartan subgroups D and A of G and H′ (associated to K and L) respectively so that A ⊂ D. Let 𝔤, 𝔥, 𝔡, 𝔞 denote the corresponding Lie algebras.

Let X = K\G be the right symmetric space of G, and let P : G → X, d(·, ·) be as in §2.6.2. For the case of the left symmetric space G/K, see Remark 9.5 below. We will write ḡ = P (g) to simplify notation. Our goal will be to determine the asymptotics of the volumes of all skew balls H_{T} [g_{1}, g_{2}], as in (7). Since applying a conjugation in G permutes the set of skew balls, with no loss of generality we will replace H with H′. The Riemannian metric on X induces a scalar product (·, ·) on 𝔡. We denote the corresponding norm by ∥ · ∥, and write

𝔡^{0} = {Y ∈ 𝔡 : ∥Y ∥ = 1}, 𝔞^{0} = 𝔞 ∩ 𝔡^{0}.

We recall the following facts about the geometry of X, see e.g. [Ba, Eb] for more details.

Proposition 9.1. (1) X is a complete Riemannian manifold of nonpositive curvature.

(2) The map 𝔡 → X, Y ↦ P (exp(Y)) is an isometry. In particular, the submanifold P (D) is a totally geodesic subset and for any Y ∈ 𝔡^{0}, the path

t ↦ γ_{Y} (t) = ē exp(tY)

is a unit speed geodesic. There is a continuous Busemann function β : X × 𝔡^{0} → ℝ such that

lim_{t→∞} d(γ_{Y} (t), x) − t = β(x, Y)

and the convergence is uniform over compact subsets of X × 𝔡^{0}.

(3) For any g ∈ G, x ∈ X, and Y ∈ 𝔡^{0},

lim_{t→∞} d(ḡ exp(tY), x) − d(ḡ exp(tY), ē) = β(x, Y)

and the convergence is uniform over compact subsets of G × X × 𝔡^{0}.

Proof. Parts (1) and (2) are well-known. To prove (3), note that

d (ē exp(tY), ḡ exp(tY)) = d(ē, ḡ),

that is, {ḡ exp(tY) : t ≥ 0} stays within bounded distance of the geodesic {ē exp(tY) : t ≥ 0}. Now the result can be proved using [Ba, Proposition 2.5]. □

Let 𝔞^{+} be a positive Weyl chamber in 𝔞 that corresponds to a system of positive roots Φ^{+} ⊂ Φ. For T > 0, put 𝔞_{T}^{+} = {a ∈ 𝔞^{+} : ∥a∥ ≤ T }. We will require the following standard fact concerning the functional ρ defined in (47).

Lemma 9.2. There is a unique Y_{max} ∈ 𝔞_{1}^{+} ∩ int 𝔞^{+} such that ρ(Y_{max}) = max_{Y ∈𝔞_{1}^{+}} ρ(Y).

With this notation we have:

Theorem 9.3. Let G be a semisimple Lie group and H a connected semisimple subgroup. Then for any g_{1}, g_{2} ∈ G,

λ (H_{T} [g_{1}, g_{2}]) ∼ C(g_{1}, g_{2})T^{(d−1)/2}e^{δT} ,  (72)

where d = dim A, δ = 2ρ(Y_{max}), and

C(g_{1}, g_{2}) = (∫_{L} exp (−δβ (ḡ_{1}ℓ, −Y_{max})) dℓ)
× (∫_{L} exp (−δβ (\overline{g_{2}^{−1}}ℓ^{−1}, Y_{max})) dℓ) .

The convergence in (72) is uniform for g_{1}, g_{2} in compact subsets of G.

We will need another standard result about integration of exponential functions on balls. Lebesgue measure on ℝ^{r} is denoted by dY. We call S ⊂ ℝ^{r} a convex cone S is convex and for any s ∈ S, the ray {ts : t > 0} is contained in S.

Lemma 9.4. Let λ be a linear functional on ℝ^{r}, and let δ = max_{Y ∈𝔞_{1}^{+}} λ(Y). Assume that S ⊂ 𝔞^{+} is an open convex cone such that δ = max_{Y ∈𝔞_{1}^{+}∩S} λ(Y). Then there is a constant C such that

∫_{𝔞_{T}^{+}∩S} e^{λ(Y)} dY ∼ ∫_{𝔞_{T}^{+}} e^{λ(Y)} dY ∼ CT^{(d−1)/2}e^{δT} .

Proof of Theorem 9.3 assuming Lemmas 9.2 and 9.4. Define ξ by (50). Let

𝔞_{T}^{+}[g_{1}, g_{2}] = {Y ∈ 𝔞^{+} : d(ḡ_{1} exp(Y)g_{2}, ē) ≤ T }
𝔞_{T}^{+} = 𝔞_{T}^{+}[e, e] = {Y ∈ 𝔞^{+} : ∥Y ∥ ≤ T }.

First, we determine the asymptotics of

ψ_{g_{1},g_{2}}(T) =^{def} ∫_{𝔞_{T}^{+}[g_{1},g_{2}]} ξ(Y)dY

as T → ∞ with uniform convergence for g_{1} and g_{2} in a fixed compact set E ⊂ G. To do this we will replace ξ by an exponential function.

By the triangle inequality, there exists C = C(E) > 0 such that for g_{1}, g_{2} ∈ E,

𝔞_{T −C}^{+}[e, e] ⊂ 𝔞_{T}^{+}[g_{1}, g_{2}] ⊂ 𝔞_{T +C}^{+}[e, e].  (73)

Expanding (50) we obtain that there are λ_{1}, …, λ_{k} ∈ 𝔞^{∗}, a_{1}, …, a_{k} ∈ ℝ such that

ξ(Y) = \frac{e^{2ρ(Y)}}{2^{m}} + ∑_{i=1}^{k} a_{i}e^{λ_{i}(Y)},

where m = ∑_{α∈Φ} m_{α} and for all Y ∈ int 𝔞^{+},

2ρ(Y) > max_{i} λ_{i}(Y).

Therefore

ψ_{g_{1},g_{2}}(T) = \frac{1}{2^{m}} ∫_{𝔞_{T}^{+}[g_{1},g_{2}]} e^{2ρ(Y)} dY + ∑_{i=1}^{k} a_{i} ∫_{𝔞_{T}^{+}[g_{1},g_{2}]} e^{λ_{i}(Y)} dY.

Let C′ be a constant such that for all T > 0 and all g_{1}, g_{2} ∈ E, the Lebesgue measure of 𝔞_{T}^{+}[g_{1}, g_{2}] is at most C′ T^{r}. Let Y_{max} be as in Lemma 9.2. Using the compactness of 𝔞_{1}^{+}, we find that there is η > 0 such that for each i ∈ {1, …, k},

max_{Y ∈𝔞_{1}^{+}} λ_{i}(Y) ≤ δ − η.

This implies that

\begin{aligned}\frac{ψ_{g_{1},g_{2}}(T) − 2^{−m} ∫_{𝔞_{T}^{+}[g_{1},g_{2}]} e^{2ρ(Y)} dY}{e^{δT}} & = \frac{∑_{i=1}^{k} a_{i} ∫_{𝔞_{T}^{+}[g_{1},g_{2}]} e^{λ_{i}(Y)} dY}{e^{δT}} \\ & ≤ \frac{∑_{i=1}^{k} C′T^{r} |a_{i}| max_{Y ∈𝔞_{T +C}^{+}} e^{λ_{i}(Y)}}{e^{δT}} \\ & ≤ (e^{max_{i} λ_{i}(C)} C′T^{r} ∑_{i=1}^{k} |a_{i}|) e^{−ηT} \\ & ⟶_{T →∞} 0. \end{aligned} (74)

Hence, in order to derive the asymptotics of ψ_{g_{1},g_{2}}(T), it suffices to find D_{g_{1},g_{2}} > 0 such that

∫_{𝔞_{T}^{+}[g_{1},g_{2}]} e^{2ρ(Y)} dY ∼ D_{g_{1},g_{2}} T^{(d−1)/2}e^{δT}  (75)

as T → ∞ uniformly on g_{1}, g_{2} ∈ E.

We will use polar coordinates on 𝔞^{+}. Thus we will represent each Y ∈ 𝔞^{+} as tω, where t = ∥Y ∥ and ω ∈ 𝔞^{1} = 𝔞^{0} ∩ 𝔞^{+}.

Let

r_{T} = r_{T} (ω, g_{1}, g_{2}) = inf {r > 0 : rω ∉ 𝔞_{T}^{+}[g_{1}, g_{2}]}
= inf \{r > 0 : d (\overline{g_{1}} exp(rω), \overline{g_{2}^{−1}}) > T \} ,
R_{T} = R_{T} (ω, g_{1}, g_{2}) = sup {r > 0 : rω ∈ 𝔞_{T}^{+}[g_{1}, g_{2}]}
= sup \{r > 0 : d (\overline{g_{1}} exp(rω), \overline{g_{2}^{−1}}) ≤ T \} .

Then

{rω : ω ∈ 𝔞^{1}, 0 ≤ r < r_{T} } ⊂ 𝔞_{T}^{+}[g_{1}, g_{2}] ⊂ {rω : ω ∈ 𝔞^{1}, 0 ≤ r ≤ R_{T} }. (76)

By (73), r_{T} (ω, g_{1}, g_{2}) → ∞ as T → ∞ uniformly for g_{1}, g_{2} ∈ E and ω ∈ 𝔞^{1}. Let s_{T} > 0 be such that d (\overline{g_{1}} exp(s_{T} ω), \overline{g_{2}^{−1}}) = T and s_{T} → ∞. It follows

from continuity that this condition holds for s_{T} = r_{T} as well as for s_{T} = R_{T}. By Proposition 9.1(3),

lim_{T →∞}(T − s_{T})

= lim_{T →∞} [d (\overline{g_{1}} exp(s_{T} ω), \overline{g_{2}^{−1}}) − d (\overline{g_{1}} exp(s_{T} ω), \overline{e}) + d (\overline{g_{1}} exp(s_{T} ω), \overline{e}) − s_{T} ]

= β (\overline{g_{2}^{−1}}, ω) + lim_{T →∞} (d (ēexp(−s_{T} ω), \overline{g_{1}}) − s_{T})

= β (\overline{g_{2}^{−1}}, ω) + β (\overline{g_{1}}, −ω) .

(77)

In particular, this shows that

R_{T} (ω, g_{1}, g_{2}) − r_{T} (ω, g_{1}, g_{2}) → 0 as T → ∞  (78)

with uniform convergence for ω ∈ 𝔞^{1} and g_{1}, g_{2} ∈ E. By (76),

∫_{𝔞^{1}} Λ(r_{T} (ω, g_{1}, g_{2})) dω ≤ ∫_{𝔞_{T}^{+}[g_{1},g_{2}]} e^{2ρ(Y)} dY
≤ ∫_{𝔞^{1}} Λ(R_{T} (ω, g_{1}, g_{2})) dω,

where

Λ(r) = ∫_{0}^{r} e^{2sρ(ω)}s^{d−1}ds,

and dω is a volume form on 𝔞^{1} such that dY = t^{d−1} dtdω. One can check that for every ε > 0, there exists δ > 0 such that Λ(r + δ) ≤ (1 + ε)Λ(r) for all sufficiently large r. Thus, it follows from (78) that

∫_{𝔞^{1}} Λ(R_{T} (ω, g_{1}, g_{2})) dω ∼ ∫_{𝔞^{1}} Λ(r_{T} (ω, g_{1}, g_{2})) dω

as T → ∞, uniformly on g_{1}, g_{2} ∈ E, and to prove (75), it suffices to derive asymptotics of one of these integrals.

Let

c_{g_{1},g_{2}} = β (\overline{g_{2}^{−1}}, Y_{max}) + β (\overline{g_{1}}, −Y_{max}) .

Let ε > 0. Since the Busemann function is continuous, we may choose an open cone S ⊂ 𝔞^{+} sufficiently close to the ray through Y_{max}, so that for ω ∈ S ∩ 𝔞^{1} and g_{1}, g_{2} ∈ E,

|β (\overline{g_{2}^{−1}}, ω) + β (\overline{g_{1}}, −ω) − c_{g_{1},g_{2}} | < ε.

Then by (77) for all sufficiently large T, g_{1}, g_{2} ∈ E, and ω ∈ S ∩ 𝔞^{1},

|T − c_{g_{1},g_{2}} − r_{T} (ω, g_{1}, g_{2})| < 2ε.

This implies that

S ∩ 𝔞_{T −c_{g_{1},g_{2}}−2ε}^{+} ⊂ {rω : ω ∈ 𝔞^{0} ∩ S, 0 ≤ r < r_{T} } ⊂ S ∩ 𝔞_{T −c_{g_{1},g_{2}}+2ε}^{+},

hence using Lemma 9.4

lim sup_{T →∞} \frac{∫_{𝔞^{1}} Λ(r_{T} (ω, g_{1}, g_{2})) dω}{T^{(d−1)/2} e^{δT}} = lim sup_{T →∞} \frac{∫_{𝔞_{T −cg_{1},g_{2} +2ε}^{+}} e^{2ρ(Y)} dY}{T^{(d−1)/2} e^{δT}}

≤ lim sup_{T →∞} \frac{(T − c_{g_{1},g_{2}} + 2ε)^{(d−1)/2}e^{δ(T −c_{g_{1},g_{2}}+2ε)}}{T^{(d−1)/2}e^{δT}} = e^{δ(−c_{g_{1},g_{2}}+2ε)}.

Since c_{g_{1},g_{2}} is bounded, convergence is uniform for g_{1}, g_{2} ∈ E.

Similarly, one has

lim inf_{T →∞} \frac{∫_{𝔞^{1}} Λ(r_{T} (ω, g_{1}, g_{2})) dω}{T^{(d−1)/2}e^{δT}} ≥ e^{δ(−c_{g_{1},g_{2}}−2ε)},

and since ε was arbitrary, we have

∫_{𝔞^{1}} Λ(r_{T} (ω, g_{1}, g_{2})) dω ∼ e^{−δc_{g_{1},g_{2}}} · T^{(d−1)/2}e^{δT}

as T → ∞. This proves (75) with

D_{g_{1},g_{2}} = e^{−δc_{g_{1},g_{2}}} = exp (−δ (β (\overline{g_{2}^{−1}}, Y_{max}) + β (\overline{g_{1}}, −Y_{max}))) .

Thus we have proved that

ψ_{g_{1},g_{2}}(T) ∼ D_{g_{1},g_{2}}T^{(d−1)/2}e^{δT} ,

as T → ∞, with uniform convergence for g_{1}, g_{2} ∈ E.

We now obtain (72) via (49). □

Remark 9.5. We use notations from Section 2.6.2. To treat the case of the left symmetric space G/K, we observe that the transformation g ↦ g^{−1} maps Riemannian balls {g : d(P (e), P (g)) < T } to the balls {g : d′(P′(e), P′(g)) < T }. Hence, denoting

H′_{T} [g_{1}, g_{2}] = {h ∈ H : d′(P′(e), P′(g_{1}hg_{2})) < T },

we have

H′_{T} [g_{1}, g_{2}] = H_{T} [g_{2}^{−1}, g_{1}^{−1}]^{−1}.

This implies Theorem 9.3 for the balls H′_{T} [g_{1}, g_{2}].

Proof of Lemma 9.2. For any χ ∈ 𝔞^{∗}, let v_{χ} ∈ 𝔞 so that for all Y ∈ 𝔞,

(v_{χ}, Y) = χ(Y).

Since 𝔞_{1} is strictly convex, max_{Y ∈𝔞_{1}} ρ(Y) is attained at a unique point which we denote by Y_{max}. It is a standard application of Lagrange multipliers that Y_{max} = \frac{v_{ρ}}{∥v_{ρ}∥}, so it remains to show that v_{ρ} ∈ int 𝔞^{+}, that is, for all α ∈ Φ,

(v_{α}, v_{ρ}) > 0.  (79)

The inner product (·, ·) is invariant under the action of the Weyl group, that is, for any α, β, γ ∈ Φ,

(v_{r_{α}(β)}, v_{r_{α}(γ)}) = (v_{β}, v_{γ}) ,

where r_{α} : 𝔞^{∗} → 𝔞^{∗} denotes the reflection in the root α. Let

Ψ = Φ^{+} ∖ span(α), ρ′ = \frac{1}{2} ∑_{β∈Ψ} m_{β}β,

so that

ρ = ρ′ + \frac{1}{2} ∑_{β∈Φ^{+}∖Ψ}m_{β}β.

It is a standard fact about root systems (see e.g. [Wa, Prop. 1.1.2.5]) that r_{α}(ρ′) = ρ′. This implies that

(v_{α}, v_{ρ′}) = (v_{r_{α}(α)}, v_{r_{α}(ρ′)}) = (v_{−α}, v_{ρ′}) = −(v_{α}, v_{ρ′}),

so that (v_{α}, v_{ρ′}) = 0. Moreover any β ∈ Φ^{+} ∖ Ψ is a positive multiple of α and hence satisfies (v_{α}, v_{β}) > 0. Therefore

(v_{α}, v_{ρ}) = (v_{α}, v_{ρ′}) + \frac{1}{2} ∑_{β∈Φ^{+}∖Ψ}m_{β} (v_{α}, v_{β}) > 0,

and we have (79). □

Proof of Lemma 9.4. Let

g_{T} (z) = \{ \begin{matrix}e^{−z}z^{(r−1)/2} (2 − \frac{z}{T} )^{(r−1)/2} & when 0 ≤ z ≤ 2T \\ 0 & when z > 2T \end{matrix} ,

and let

g(z) = 2^{(r−1)/2} e^{−z}z^{(r−1)/2}.

By Lebesgue’s dominated convergence theorem,

∫_{0}^{∞} g_{T} (z) dz ⟶_{T →∞} ∫_{0}^{∞} g(z) dz = C.  (80)

By rescaling, we may assume that δ = 1. Let Vol_{r−1} denote Lebesgue measure on the subspace ker λ. By translation it induces a measure on each affine subspace parallel to ker λ. For each x ∈ [−T, T],

B_{T,x} = {Y ∈ B(0, T) : λ(Y) = x}

is a ball of radius \sqrt{T^{2} − x^{2}} in a translate of ker λ and hence

Vol_{r−1}(B_{T,x}) = C′ (T^{2} − x^{2})^{(r−1)/2} ,

where C′ is a constant depending only on r.

Decomposing the integration into slices parallel to ker λ, we have

\frac{∫_{B(0,T)} e^{λ(Y)} dY}{e^{T} T^{(r−1)/2}} = ∫_{−T}^{T} \frac{e^{x} Vol_{r−1}(B_{T,x})}{e^{T} T^{(r−1)/2}} dx

= C′ ∫_{−T}^{T} \frac{e^{x} (T^{2} − x^{2})^{(r−1)/2}}{e^{T} T^{(r−1)/2}} dx

= C′ ∫_{0}^{2T} e^{−z} (\frac{z(2T − z)}{T})^{(r−1)/2} dz

= C′ ∫_{0}^{∞} g_{T} (z) dz.

Applying (80) we obtain

∫_{B(0,T)} e^{λ(Y)} dY ∼ C T^{(r−1)/2} e^{T} .  (81)

Since B(0, 1) is strictly convex, the maximum max_{Y ∈B(0,1)} λ(Y) is attained at a unique point Y_{0}. By the hypothesis Y_{0} ∈ S ⊂ int 𝔞^{+}. There is ε > 0 such that for all Y ∈ B(0, 1)∖S, λ(Y) < δ −ε. Furthermore there is a cube C ⊂ ℝ^{r} of side length 2, such that

B(0, 1) ∖ S ⊂ C

and

max_{Y ∈C} λ(Y) ≤ δ − ε.

For T > 0 let

T C = {tc : c ∈ C, t ∈ [0, T]} .

We have

∫_{B(0,T)∖S} e^{λ(Y)} dY ≤ ∫_{T C} e^{λ(Y)} dY

≤ (2T)^{r} max_{Y ∈T C} e^{λ(Y)}

≤ C′′T^{r}e^{(δ−ε)T} .

Comparing with (81) we obtain

\frac{∫_{B(0,T)∖S} e^{λ(Y)} dY}{∫_{B(0,T)} e^{λ(Y)} dY} ⟶_{T →∞} 0,

and so ∫_{B(0,T)∩S} e^{λ(Y)} dY ∼ ∫_{B(0,T)} e^{λ(Y)} dY. The assertion follows. □

10. Ratner theory and linearization

Our goal in this section is to prove Theorem 2.10. We will derive it from the following result:

Theorem 10.1. Suppose the general setup holds, where H is connected, semisimple, and balanced, and assume also that UC and D1 hold. Suppose g_{0} ∈ G satisfies \overline{Hπ(g_{0})} = G/Γ. Then for every F ∈ C_{c}(G/Γ) and any g_{1}, g_{2} ∈ G we have

\frac{1}{λ (H_{T} [g_{1}, g_{2}])} ∫_{H_{T} [g_{1},g_{2}]} F (hπ(g_{0})) dλ(h) ⟶_{T →∞} ∫_{G/Γ} F dm′.  (82)

The proof of Theorem 10.1 relies on the fundamental results of Ratner on the dynamics of unipotent flows on homogeneous spaces, and subsequent work of Dani, Margulis, Mozes, Shah, and others. Specifically, we use a result of Nimish Shah.

We say that a sequence {h_{n}} ⊂ H is strongly divergent if its projection on every simple factor is divergent. In the case of a matrix group, and using the notation of §2.7, {h_{n}} is strongly divergent if and only if {σ_{i}(h_{n})} ⊂ H_{i} has no convergent subsequence for every simple factor H_{i} of H.

Theorem 10.2 (Shah). Let H be a connected semisimple Lie subgroup of a Lie group G. Let A be a split Cartan subgroup of H, A^{+} a closed Weyl chamber and K a maximal compact subgroup of H. Let Γ be a lattice in G. Let μ be a finite Borel measure on K that is absolutely continuous with respect to Haar measure on K. Suppose that for x ∈ G/Γ, Hx is dense in G/Γ. Then for every f ∈ C_{c}(G/Γ) and every strongly divergent sequence {a_{n}} ⊂ A^{+},

∫_{K} f(a_{n}kx) dμ(k) → μ(K) ∫_{G/Γ} f dm′ as n → ∞.

Remark 10.3. Theorem 10.2 is proved, but not formulated explicitly, in [Sh] (see the proof of Corollary 1.2).

Proof of Theorem 10.1. Let F ∈ C_{c}(G/Γ), and assume without loss of generality that F ≥ 0. Let g_{0}, g_{1}, g_{2} be as in the statement of the theorem. In proving (82), to conserve ink and computer memory, we will omit g_{1} and g_{2} from the notation. Thus H_{T} stands for H_{T} [g_{1}, g_{2}], D(x) stands for D(g_{1}xg_{2}) and so on.

Fix ε > 0, let ε_{0} > 0 such that (1 + ε_{0})^{2} < 1 + ε, and by D1, let δ > 0 such that for all large enough T,

λ (H_{(1+δ)^{2}T} ) ≤ (1 + ε_{0})λ (H_{T}) .

By UC, there exists a symmetric neighborhood O of identity in K such that for all g ∈ G and k ∈ O,

D(gk) ≤ (1 + δ)D(g).  (83)

Since K is compact, there exists a finite cover

K = ⋃_{i=1}^{N} ℓ_{i}O, ℓ_{1}, …, ℓ_{N} ∈ K.

Denote by κ the probability Haar measure on K. Using a partition of unity, there are measures μ_{i}, i = 1, …, N on K, absolutely continuous with respect to κ, such that

supp μ_{i} ⊂ ℓ_{i}O, i = 1, …, N  and ∑_{i=1}^{N} μ_{i} = κ.  (84)

Let ν denote the measure on 𝔞^{+} which is equal to ξ(Y) dY, where dY is Lebesgue measure. We decompose λ as in (49), so that

dλ(h) = dκ(k_{1}) dν(Y) dκ(k_{2})

for k_{i} ∈ K, Y ∈ 𝔞^{+}, h = k_{1} exp(Y)k_{2}.

Let H = H_{1} ⋯ H_{t} be the decomposition of H into an almost direct product of simple factors and let σ_{j} : H → H_{j} be measurable sections. Taking x = π(g_{0}) in Theorem 10.2, we find that there are compact subsets C_{j} ⊂ H_{j}, j = 1, …, t such that for any h ∈ H̃

|∫_{K} F (hkπ(g_{0}))dμ_{i}(k) − μ_{i}(K) ∫_{G/Γ} F dm′| < ε_{0},  (85)

where

H̃ = {h ∈ H : σ_{j}(h) ∉ C_{j}, j = 1, …, t}.

For each k_{1}, k_{2} ∈ K and T > 0 write

𝔟(k_{1}, k_{2}, T) = {Y ∈ 𝔞^{+} : ∀j, σ_{j}(exp(Y)) ∉ C_{j}, D(k_{1} exp(Y)k_{2}) < T }.

By enlarging each C_{j} we may assume that (85) holds for any h = k_{1} exp(Y), with k_{1} ∈ K, Y ∈ 𝔞 such that σ_{j}(exp(Y)) ∉ C_{j} for j = 1, …, t. Further, we may assume that

H̃_{T} = ⋃_{k_{1},k_{2}∈K} k_{1} exp (𝔟(k_{1}, k_{2}, T)) k_{2}.

It follows from (83) and (84) that if k_{2} ∈ supp μ_{i} then for all T > 0,

𝔟(k_{1}, k_{2}, T) ⊂ 𝔟(k_{1}, ℓ_{i}, (1 + δ)T) ⊂ 𝔟(k_{1}, k_{2}, (1 + δ)^{2}T).  (86)

Writing x = π(g_{0}), we have

∫H̃_{T} F (hx) dλ(h)
= ∫_{K} dκ(k_{1}) ∫_{K} dκ(k_{2}) ∫_{𝔟(k_{1},k_{2},T)} F (k_{1} exp(Y)k_{2}x) dν(Y)

=^{(84)} ∑_{i=1}^{N} ∫_{K} dκ(k_{1}) ∫_{ℓ_{i}O} dμ_{i}(k_{2}) ∫_{𝔟(k_{1},k_{2},T)} F (k_{1} exp(Y)k_{2}x) dν(Y)

≤^{(86)} ∑_{i=1}^{N} ∫_{K} dκ(k_{1}) ∫_{𝔟(k_{1},ℓ_{i},(1+δ)T)} dν(Y) ∫_{K} F (k_{1} exp(Y)k_{2}x) dμ_{i}(k_{2})

≤^{(85)}(1 + ε_{0}) ∫_{G/Γ} F dm′ ∑_{i=1}^{N} ∫_{K} dκ(k_{1}) ∫_{𝔟(k_{1},ℓ_{i},(1+δ)T)} μ_{i}(K) dν(Y)

≤^{(86)}(1 + ε_{0}) ∫_{G/Γ} F dm′ ∑_{i=1}^{N} ∫_{K} dκ(k_{1}) ∫_{K} dμ_{i}(k_{2}) ∫_{𝔟(k_{1},k_{2},(1+δ)^{2}T)} dν(Y)

≤(1 + ε_{0}) λ \binom{}{H_{(1+δ)^{2}T}} ∫_{G/Γ} F dm′
≤(1 + ε_{0})^{2} λ (H_{T}) ∫_{G/Γ} F dm′.

Thus

\frac{1}{λ (H_{T})} ∫H̃_{T} F (hx) dλ(h) < (1 + ε) ∫_{G/Γ} F dm′.

Since H is balanced and ε was arbitrary we have

lim sup_{T →∞} \frac{1}{λ (H_{T})} ∫_{H_{T}} F (hx) dλ(h) ≤ ∫_{G/Γ} F dm′.

The proof of the opposite inequality for lim inf is similar, and we have proved (82). □

Proof of Theorem 2.10. Clearly (∗∗) follows from (82) if we replace h by h^{−1}. To justify this, let

D̃(g) = D(g^{−1})

and apply Theorem 10.1 to G, H, D̃ instead of G, H, D. Note that D1 for D̃ holds since H is unimodular, and UC for D̃ can be proved as in Propositions 6.2 and 6.3. □

11. Applications

In this section we prove the results stated in Section 1.5.1. Note that Theorem 1.1 follows from Corollary 2.12.

Proof of Theorem 1.2. Suppose that the general setup holds for a balanced semisimple Lie group G and a standard distance function D. Let Γ be a lattice in G.

First, we suppose that G is a closed subgroup of a Lie group L, Λ is a lattice in L, and Γ acts on X = L/Λ by right multiplication. Let H = G × L and G̃ be the diagonal embedding of G in H, and let D((g, g)) = D(g), g ∈ H. Our choices of G̃, D show that the general setup holds, and that G̃, D are standard. Since Γπ(g_{0}) is dense in L/Λ, G̃π′(y_{0}) is dense in H/(Γ × Λ), where y_{0} = (g_{0}, e) and π′ : H → H/(Γ × Λ) is the quotient map. By Theorem 2.10, we find that (∗∗) holds for G̃. Since the G̃-action on H/(Γ × Λ) is isomorphic with the G̃-action on Y = Γ\G × L/Λ, it follows from Theorem 2.10 that (∗) holds for the G̃-action on Y. Using Proposition 2.6 we find that I1, I2 and UC hold, and using Proposition 3.1 we obtain the desired result.

Now we suppose that G acts smoothly on a Lie group L by automorphisms and Γ · Λ ⊂ Λ. Thus, Γ acts on X = L/Λ. By Proposition 2.6, I1, I2 and UC are satisfied. Define Ỹ = G × X and Y = Y/Γ as in §3. The map G ⋉ L → G × X induces a G-equivariant homeomorphism of (G ⋉ L)/(Γ ⋉ Λ) and Y. Let π′ : Ỹ → Y be the quotient map. Since \overline{Gπ′(e, x_{0})} = Ỹ, the statement follows from Theorem 2.10 and Proposition 3.1. □

Remark 11.1. Arguing as in the proof of Theorem 2.10, it is possible in Theorem 1.2 to replace γ^{−1} with γ.

Proof of Corollary 1.3. In terms of the identification (3) of F_{d} with G, the map

F_{d} → Mat_{d}(ℝ), f ↦ Q̄(f)

is given by

G → Mat_{d}(ℝ), g ↦ ^{t}gA_{Q}g,  (87)

where A_{Q} is the matrix of the quadratic form Q with respect to the standard basis e. The set

Q̄(F_{d}) = {^{t}gA_{Q}g : g ∈ G}

consists of symmetric matrices that have the same determinant and the same signature as A_{Q}, and we have an identification

ρ : H\G → Q̄(F_{d}), ρ(τ(g)) = ^{t}gA_{Q}g = Q̄(g e),

where H is the orthogonal group of Q. It can be shown that Q̄(F_{d}) is an algebraic variety and that ρ is an isomorphism of algebraic varieties.

Given a norm ∥ · ∥ on ℝ^{d}, we define a norm on Mat_{d}(ℝ), which we also denote by ∥ · ∥, by

∥g∥ = max_{j=1,...d} ∥g_{∗j}∥, g = (g_{ij}) ∈ Mat_{d}(ℝ).

Then for g ∈ G, ∥g e∥ < T if and only if ∥g∥ < T. Let Γ = G(ℤ) and A ⊂ Q̄(F_{d}) as in the statement of the corollary. We have

# {f ∈ F_{d}(ℤ) : ∥f∥ < T, Q̄(f) ∈ A}
=# {γ ∈ Γ : ∥γ∥ < T, Q̄(γ e) ∈ A}
=# {γ ∈ Γ : ∥γ∥ < T, τ(γ) ∈ ρ^{−1}(A)} .

Let H^{o} be the connected component of the identity in H. Since d ≥ 3, the group H^{o} is a connected semisimple Lie group, and it is simple unless Q is of signature (2, 2). If H is simple, then it is balanced. If H is of signature (2, 2), then it can be shown by a direct computation that it is balanced.

Let P : H^{o}\G → H\G be the projection map. Since P has finite fibers, the set Ã = P^{−1}(ρ^{−1}(A)) is a relatively compact subset with boundary of measure zero with respect to the smooth measure class on H^{o}\G. It is a well-known consequence of Ratner’s orbit-closure theorem (see e.g. [BoPr]) that H^{o}π(e) is dense in G/Γ. Thus we may apply Corollary 2.12, and obtain that

# {γ ∈ Γ : ∥γ∥ < T, γH^{o} ∈ Ã}
∼ m ({g ∈ G : ∥g∥ < T, gH^{o} ∈ Ã})
= m ({g ∈ G : ∥g∥ < T, Q̄(g e) ∈ A})
= m ({f ∈ F_{d} : ∥f∥ < T, Q̄(f) ∈ A}) .

This proves the first assertion, and shows that these quantities are asymptotic to ν(A)λ(H_{T}), where ν is the measure ν_{x_{0}} for x_{0} = τ(e) defined by (12).

For the second assertion, we need to compute the asymptotics of λ(H_{T}). We use the notations of §7. For the volume computation, we may assume that Q is given by

Q(x_{1}, …, x_{p+q}) = x_{1}x_{p+q} + ⋯ + x_{p}x_{q+1} + x_{p+1}^{2} + … + x_{q}^{2}.

Then the split Cartan subgroup of SO(Q) can be chosen to be

A = diag(e^{s_{1}}, …, e^{s_{p}}, 1, …, 1, e^{−s_{p}}, …, e^{−s_{1}}),

and the weights of the representation map Y = (s_{1}, …, s_{p}, 0, −s_{p}, …, −s_{1}) to its coordinates s_{1}, …, s_{p}, 0, −s_{1}, …, −s_{p}. Choosing an order on roots so that the positive root spaces are upper triangular, we find that the positive roots are

s_{i} − s_{j}, 1 ≤ i < j ≤ p, multiplicity = 1,
s_{i}, 1 ≤ i ≤ p, multiplicity = q − p,
s_{i} + s_{j}, 1 ≤ i < j ≤ p, multiplicity = 1.

The dominant weight is λ_{1}(s) = s_{1} and

2ρ(s) = ∑_{i=1}^{p}(p + q − 2i)s_{i}.

First, we consider the case when p < q. Then the system of simple roots is

{s_{1} − s_{2}, …, s_{p−1} − s_{p}, s_{p}}.

The dual basis of 𝔞 is β̃_{k}, k = 1, …, p, where

β̃_{k} = (1, …, 1, 0, …, 0) (k ones).

and

β_{k} = \frac{β̃_{k}}{2ρ(β̃_{k})} = (\frac{1}{k(p + q − k − 1)}, …, \frac{1}{k(p + q − k − 1)}, 0, …, 0) .

We have λ_{1}(β_{k}) = \frac{1}{k(p+q−k−1)}. Note that λ_{1}(β_{k}) is strictly decreasing for 0 ≤ k ≤ p ≤ \frac{p+q−1}{2}. Thus, m_{1} = \frac{1}{p(q−1)} and condition G is satisfied, so by Theorem 7.4

λ(H_{T}) ∼ CT^{p(q−1)}.

Now if p = q, it may be shown that condition G is not satisfied. Recall that in Remark 7.5(3) we mentioned a generalization of Theorem 7.4 in the case when condition G does not hold. Using this generalization we are able to show that in this case the asymptotics is C(log T)T^{p(p−1)}. We omit the details. □

Proof of Corollary 1.4. For both assertions we apply Corollary 2.12 to the matrix norm distance function corresponding to Ψ and ∥ · ∥, with H\G = S and x = P (s_{0}). Note that all conditions are satisfied, and ν_{x} in our case is equal to α(s_{0}, s) dν(s). So it remains to calculate the density α.

We begin with the first case. Since V = V_{H} ⊕ V_{S}, for any s ∈ S, h ∈ H, Ψ(sh) = Ψ(s) + Ψ(h) − Id, where Id is the identity matrix on V. Therefore, given a bounded S_{0} ⊂ S there is M > 0 such that for h ∈ H and s ∈ S_{0},

D(h) − M ≤ D(sh) ≤ D(h) + M.

This implies that for s_{1}, s_{2} ∈ S,

λ(H_{T −M}[e, e]) ≤ λ(H_{T} [s_{1}, s_{2}]) ≤ λ(H_{T +M}[e, e]).

Via Theorem 2.7 we obtain that α ≡ 1, and the first assertion follows.

For the second assertion, we proceed as before to calculate α. Note that for s ∈ S and h ∈ H and using the p-norms on V_{H} and V_{S},

∥Ψ(sh)∥ = (dim V_{S} · dim V_{H})^{−1/p} · ∥Ψ_{S}(s)∥ · ∥Ψ_{H}(h)∥.

Therefore

H_{T} [s_{0}, s] = {h ∈ H : ∥Ψ(s_{0}hs)∥ < T }
= {h ∈ H : ∥Ψ(hs_{0}s)∥ < T }
= {h ∈ H : ∥Ψ_{H}(h)∥ < c_{1}T },

where

c_{1} = \frac{(dim V_{S} · dim V_{H})^{1/p}}{∥Ψ_{S}(s_{0}s)∥} .

It follows using Theorem 2.7 that

α(s_{0}, s) = lim_{T →∞} \frac{λ (H_{T} [s_{0}, s])}{λ (H_{T})}

= lim_{T →∞} \frac{C (log c_{1}T)^{ℓ}(c_{1}T)^{m}}{C (log T)^{ℓ} T^{m}}

= c∥Ψ_{S}(s_{0}s)∥^{−m}

with

c = (dim V_{S} · dim V_{H})^{m/p}.

This proves the second statement. □

12. Examples

In this section we collect some examples showing that our hypotheses are not automatically satisfied, and that they are important for the validity of our results.

12.1. Condition S. Let

S^{1} =^{def} {z ∈ ℂ : |z| = 1}, G =^{def} ∏_{1}^{∞} S^{1}, H =^{def} ∏_{1}^{∞}{±1},

equipped with the Tychonov topology, and let τ : G → H\G be the natural map. Then G is compact and H is a compact subgroup. Any section σ : H\G → G induces a section {±1}\S^{1} → S^{1} in each factor. Let U be an open subset of H\G. By the definition of the product topology on G, τ^{−1}(U) contains a subset of the form V_{1} × ⋯ × V_{r} × ∏_{r+1}^{∞} S^{1}, where V_{1}, …, V_{r} are open, and in particular contains a copy of S^{1} in one factor. Since there is no continuous section {±1}\S^{1} → S^{1}, there is no continuous section defined on U.

12.2. Condition D2. Let G =^{def} SL(3, ℝ) and let

H = \{(\begin{matrix}e^{t} cos t & −e^{t} sin t & x \\ e^{t} sin t & e^{t} cos t & y \\ 0 & 0 & e^{−2t} \end{matrix}) : x, y, t ∈ ℝ\} .

In coordinates t, x, y, the left Haar measure on H is given by dλ = e^{2t} dx dy dt.

For a constant c > 1 to be specified below, let D be a matrix norm distance function given by D(g) = ∥g∥, where

∥(a_{ij})∥ =^{def} max \{\sqrt{ca_{11}^{2} + a_{12}^{2}}, \sqrt{ca_{22}^{2} + a_{21}^{2}}, \sqrt{a_{13}^{2} + a_{23}^{2}}, |a_{31}|, |a_{32}|, |a_{33}|\} .

We first compute the volume λ(H_{T}) along two subsequences. In (t, x, y) coordinates, H_{T} is given by the inequalities

\sqrt{x^{2} + y^{2}} < T, e^{−2t} < T,

and

f_{c}(t) < T, where f_{c}(t) = e^{t}\sqrt{c^{2} cos^{2} t + sin^{2} t}.  (88)

For c > 1 small enough, f_{c}(t) is a monotonically increasing function of t. This means that (88) is satisfied along a ray of the form (−∞, τ), where T = f_{c}(τ).

\begin{aligned}λ(H_{T}) & = ∫_{H_{T}} e^{2t} dt dx dy \\ & = ∫_{− log T/2}^{τ} ∫_{x^{2}+y^{2}<T^{2}} e^{2t} dt dx dy \\ & = πT^{2} ∫_{− log T/2}^{τ} e^{2t}dt \\ & = \frac{π}{2} T^{2} (e^{2τ} − (2/T)^{2}) . \end{aligned}  (89)

Now let τ = τ_{n} = 2nπ, for n ∈ ℕ. We have for T_{n} = f_{c}(τ_{n}) = ce^{2nπ} :

λ(H_{T_{n}}) = \frac{π}{2} T_{n}^{2} ((T_{n}/c)^{2} − (2/T_{n})^{2}) ∼ \frac{π}{2c^{2}} T_{n}^{4} as n → ∞.  (90)

Similarly, for τ_{n} =^{def} (2n + 1/2)π, S_{n} = f_{c}(τ_{n}) we have

λ(H_{S_{n}}) = \frac{π}{2} S_{n}^{2} (S_{n}^{2} − (2/S_{n})^{2}) ∼ \frac{π}{2} S_{n}^{4} as n → ∞.  (91)

In particular the quantity λ(H_{T}) is not asymptotic to a function of the form kT^{4} as T → ∞ for any constant k.

Now taking

g_{1} = g_{2} = (\begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) ,

and computing the shape of H_{T} [g_{1}^{−1}, g_{2}] in (t, x, y) coordinates we find the role of sin and cos switched in (88). This implies that the roles of T_{n} and S_{n} are reversed, that is,

λ(H_{T_{n}}[g_{1}^{−1}, g_{2}]) ∼ \frac{π}{2} T_{n}^{4}, λ(H_{S_{n}}[g_{1}^{−1}, g_{2}]) ∼ \frac{π}{2c^{2}} S_{n}^{4}.

In particular,

\frac{λ(H_{T_{n}}[g_{1}^{−1}, g_{2}])}{λ(H_{T_{n}})} →_{n→∞} c^{2}, \frac{λ(H_{S_{n}}[g_{1}^{−1}, g_{2}])}{λ(H_{S_{n}})} →_{n→∞} \frac{1}{c^{2}} ,

and D2 is not satisfied.

To show that condition D2 is indeed necessary for the validity of our results, we have the following:

Theorem 12.1. Let H, G, ∥·∥, {S_{n}}, {T_{n}} be as in the above example. Then there are two equivalent but different measures ν_{1}, ν_{2} on H\G such that for any lattice Γ in G, any x_{0} ∈ H\G such that H\G = \overline{x_{0} Γ}, and any bounded A ⊂ H\G with ν_{i}(∂A) = 0 we have

\frac{N_{S_{n}}(A, x_{0})}{λ(H_{S_{n}})} ⟶_{n→∞} ν_{1}(A)  (92)

and

\frac{N_{T_{n}}(A, x_{0})}{λ(H_{T_{n}})} ⟶_{n→∞} ν_{2}(A).  (93)

Sketch of proof. Suppose Γ is given. The subgroup of H given by the requirement t = 0 is the unipotent radical of a parabolic subgroup of G. Using the methods of [Sh] one can show that condition (∗∗) holds for any g_{0} ∈ G for which \overline{x_{0} Γ} = H\G (where x_{0} = τ(g_{0})) .

We now claim that D1 is satisfied. Let N be any bounded collection of norms on Mat_{3}(ℝ). For | · | ∈ N, let

H_{T}^{|·|} = {h ∈ H : |h| < T }.

We will show that there is a function f(| · |, T), which depends continuously on both its arguments, such that

λ (H_{T}^{|·|}) ∼ f(| · |, T) T^{4}  (94)

and such that for all | · | ∈ N and all T > 0,

f(| · |, aT) = f(| · |, T), where a = e^{2π}.  (95)

It is easily seen that this implies D1.

For

ω = (x_{0}, y_{0}, s_{0}) ∈ S^{2} = {(x_{0}, y_{0}, s_{0}) : x_{0}^{2} + y_{0}^{2} + s_{0}^{2} = 1}

and for T, r > 0, write

h(ω, r) = (\begin{matrix}rs_{0} cos log(rs_{0}) & −rs_{0} sin log(rs_{0}) & rx_{0} \\ rs_{0} sin log(rs_{0}) & rs_{0} cos log(rs_{0}) & ry_{0} \\ 0 & 0 & (rs_{0})^{−2} \end{matrix})

and

τ(ω, T, | · |) = {r ≥ 0 : |h(ω, r)| < T }.

Then, letting dω denote the standard volume form on S^{2}, we have

λ (H_{T}^{|·|}) = ∫_{S^{2}} ∫_{τ(ω,T,|·|)} s_{0}r^{3} dr dω.

Also write

E_{θ} = (\begin{matrix}cos θ & − sin θ & 0 \\ sin θ & cos θ & 0 \\ 0 & 0 & 1 \end{matrix}) , E_{1} = (\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) ,

E_{2} = (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) , E_{3} = (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) .

Then we have, for ω = (x_{0}, y_{0}, s_{0}) ∈ S^{2},

τ(ω, T, | · |) = \{r ≥ 0 : |rs_{0}E_{log(rs_{0})} + rx_{0}E_{1} + ry_{0}E_{2} + \frac{1}{(rs_{0})^{2}} E_{3}| < T \} .

Define

\begin{aligned}τ̃(ω, T, | · |) & = \{r ≥ 0 : |rs_{0}E_{log(rs_{0})} + rx_{0}E_{1} + ry_{0}E_{2}| < T \} \\ & = \{r ≥ 0 : r < T/|E(r, ω)|\} , \end{aligned}

where

E(r, ω) = s_{0}E_{log(rs_{0})} + x_{0}E_{1} + y_{0}E_{2}.

By the computation (89), and since the collection N is bounded, there are constants c_{1}, c_{2} such that for all | · | ∈ N:

0 < c_{1} ≤ lim inf_{T →∞} \frac{λ(H_{T}^{|·|})}{T^{4}} ≤ lim sup_{T →∞} \frac{λ(H_{T}^{|·|})}{T^{4}} ≤ c_{2} < ∞.  (96)

Given ε > 0, there is a constant C such that for each ω with s_{0} ≥ ε, and each | · | ∈ N, the symmetric difference of the sets τ̃(ω, T, | · |) and τ(ω, T, | · |) is contained in an interval of length C/T. Using this and (96) it is not hard to show that

λ (H_{T}^{|·|}) ∼ ∫_{S^{2}} ∫_{τ̃(ω,T,|·|)} r^{3} dr dω.  (97)

Since E(r, ω) = E(ar, ω) we have, for all T > 0 and ω ∈ S^{2},

τ̃(ω, aT) = aτ̃(ω, T),

therefore by a change of variables, for any ω ∈ S^{2}:

∫_{τ̃(ω,aT)} r^{3} dr dω = a^{4} ∫_{τ̃(ω,T)} r^{3} dr dω,  (98)

hence

f(| · |, T) = \frac{∫_{S^{2}} ∫_{τ̃(ω,T,|·|)} r^{3} dr dω}{T^{4}}

depends continuously on both its parameters and satisfies (95). Now (94) follows from (97) and (98).

Now we may apply Theorem 2.2 to obtain, for any φ ∈ C_{c}(H\G),

S_{φ,x_{0}}(T) ∼ S̃_{φ,x_{0}}(T).

Fix a sequence t_{n} = e^{2πn}t_{0}, for any t_{0} > 0. The ratio

α_{t_{0}}(g_{1}, g_{2}) = lim_{n→∞} \frac{λ (H_{t_{n}}[g_{1}^{−1}, g_{2}])}{λ (H_{t_{n}})}

exists, is positive, and depends continuously on g_{1}, g_{2} by (94). Let ν_{t_{0}} be defined by (12), but using α_{t_{0}} in place of α. Now repeating the arguments of §5, but taking limits as n → ∞, we obtain the conclusion of Corollary 15. In particular, taking in turn t_{n} = S_{n}, t_{n} = T_{n} we obtain (92) and (93). Note that the two limiting measures in this case are different by (90) and (91). □

12.3. Non-balanced semisimple groups. We now construct a non-balan-ced semisimple group and show that Theorem 2.10 fails for this group, i.e. find an action of this group on a homogeneous space which does not satisfy (∗∗).

Let

H_{1} = H_{2} = SL(2, ℝ), H = H_{1} × H_{2}.

Abusing notation, we consider H_{1} and H_{2} as subgroups of H.

Denote by A_{i} a Cartan subgroup of H_{i}, 𝔞_{i} its (one-dimensional) Lie algebra, Y_{i} a generator of 𝔞_{𝔦}. Then A = A_{1}A_{2} is a Cartan subgroup of H. We will write (s_{1}, s_{2}) = s_{1}Y_{1} + s_{2}Y_{2} ∈ 𝔞. With respect to these coordinates, a root system of (H, A) is {±α_{1}, ±α_{2}}, where

α_{i}(s_{1}, s_{2}) = 2s_{i}

and

𝔞^{+} = {(s_{1}, s_{2}) : s_{1} ≥ 0, s_{2} ≥ 0}.

In the notation of §7 we have

ρ = \frac{1}{2} (α_{1} + α_{2}) , β_{i} = \frac{Y_{i}}{2} , i = 1, 2.

We consider H as a subgroup of G = SL(2ℓ, ℝ) where H is embedded in G via the tensor product of irreducible representations of H_{1} and H_{2} of dimensions 2 and ℓ > 2 respectively. Note that the set of weights of 𝔞 is

{(s_{1}, s_{2}) ↦ is_{1} + js_{2} : i ∈ {±1}, j ∈ {1 − ℓ, 3 − ℓ, …, ℓ − 1}} ,

and a highest weight corresponding to the choice of 𝔞^{+} is

λ_{1}(s_{1}, s_{2}) = s_{1} + (ℓ − 1)s_{2}.

We have

λ_{1}(β_{1}) = \frac{1}{2} < \frac{ℓ − 1}{2} = λ_{1}(β_{2}),

so condition G is satisfied, and by Proposition 8.2, H is not balanced, that is, there is a bounded open L ⊂ H_{2} such that

c = lim sup_{T →∞} \frac{λ(H_{T}^{L})}{λ(H_{T})} > 0, where H_{T}^{L} = \{h = (h_{1}, h_{2}) : ∥h∥ < T, h_{2} ∈ L\}

for some (any, see Proposition 8.1) norm ∥ · ∥ on Mat_{2ℓ}(ℝ). We have:

Claim 5. There is a ℚ-subgroup M of G containing H_{1} and an element m ∈ M such that m^{−1}Hm is not contained in any proper ℚ-subgroup of G. In particular, setting x = π(m) and Γ = SL(2ℓ, ℤ) we have that H_{1}x ⊂ Mπ(e) = \overline{Mπ(e)} and \overline{Hx} = G/Γ.

Assuming the claim is true, let X_{0} = LMπ(e) ⊂ G/Γ. This is the closure of a locally closed submanifold of G/Γ, being the image of M/Δ × L ⊂ H under the proper map (gΔ, ℓ) ↦ ℓgx, where Δ = M ∩ mΓm^{−1}. Since dim M < dim G, we have m′(X_{0}) = 0. Therefore, for any compact K_{0} ⊂ G/Γ we can find φ ∈ C_{c}(G/Γ) such that ∫_{G/Γ} φ dm′ < c/2 and φ|_{X_{0}∩K_{0}} ≡ 1. It follows from the non-divergence results used in [Sh] that we can make K_{0} large enough so that

lim sup_{T →∞} \frac{λ \{h ∈ H_{T} : hx ∉ K_{0}\}}{λ(H_{T})} < c/2.

This yields

lim inf_{T →∞} \frac{1}{λ(H_{T})} ∫_{H_{T}} φ(hx) dλ(h)

≥ lim inf_{T →∞} \frac{λ \{h ∈ H_{T} : hx ∈ K_{0} ∩ X_{0}\}}{λ(H_{T})}

≥ lim inf_{T →∞} \frac{λ(H_{T}^{L})}{λ(H_{T})} − lim sup_{T →∞} \frac{λ \{h ∈ H_{T} : hx ∉ K_{0}\}}{λ(H_{T})}

≥ c − c/2 > ∫_{G/Γ} φ dm′,

and (∗∗) fails.

It remains to prove Claim 5. Let {u_{1}, u_{2}} and {v_{1}, …, v_{ℓ}} be the standard bases of ℝ^{2} and ℝ^{ℓ} respectively, and let B = {u_{i}⊗v_{j}}, a basis of ℝ^{2}⊗ℝ^{ℓ} = ℝ^{2ℓ}. The ℚ-structure on G is defined via B, and the H_{1}-action (respectively, the H_{2}) action on ℝ^{2ℓ} is induced by its action on the u_{i}’s (respectively, v_{j}’s). Now let M be the subgroup of G leaving invariant each of the subspaces V_{j} = span{u_{1}⊗v_{j}, u_{2}⊗v_{j}}. Clearly each V_{j} is H_{1}-invariant and hence H_{1} ⊂ M. It is also clear that M is defined over ℚ and hence Mπ(e) is closed. It remains to show that there exists m ∈ M so that m^{−1}Hm is not contained in any proper ℚ-subgroup of G. Suppose otherwise; since the number of ℚ-subgroups of G is countable, this would imply that there is a fixed proper ℚ-subgroup T ⊂ G such that for all m ∈ M, m^{−1}Hm ⊂ T. However it is not difficult to show (we omit the computation) that the set {m^{−1}hm : m ∈ M, h ∈ H} generates G, and this is a contradiction.

12.4. A simple case revisited. Our results also enable us to generalize Ledrappier’s result, discussed in §1.1, to general norms. Namely we have:

Theorem 12.2. Let Γ be a lattice in SL(2, ℝ) and let ∥ · ∥ be a norm on Mat_{2}(ℝ). Suppose that v ∈ V = ℝ^{2} satisfies \overline{v · Γ} = V, and let dw denote Lebesgue measure on ℝ^{2}. Then for every φ ∈ C_{c}(V),

S_{φ,v}(T) ∼ (c_{Γ} ∫_{V} φ(w)α_{v}(w) dw) T,  (99)

where c_{Γ} > 0 is a constant depending on Γ and

α_{v}(w) = ∥( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )∥ · ∥( \begin{matrix}−v_{2}w_{1} & −v_{2}w_{2} \\ v_{1}w_{1} & v_{1}w_{2} \end{matrix} )∥^{−1} .

Proof. Via the map

τ : G → V, τ : g ↦ (1, 0) · g,

the space V ∖ {0} is identified with H\G, where G = SL(2, ℝ) and

H = \{u_{t} = ( \begin{matrix}1 & 0 \\ t & 0 \end{matrix} ) : t ∈ ℝ\} .

Note that Haar measures on H and G are only defined up to a constant multiple; we equip H with the Haar measure dt, and choose Haar measure μ on G so that ν_{H\G} is Lebesgue measure. This induces a choice μ′ of Ginvariant measure on G/Γ. Since our results were formulated for the choice making μ′ a probability measure, we set c_{Γ} = \frac{1}{μ′(G/Γ)}, and m′ = c_{Γ}μ′.

We check that the hypotheses of Theorems 2.2 and 2.3 hold in this case. For g_{1}, g_{2} ∈ G,

H_{T} [g_{1}, g_{2}] = {u_{t} : ∥a + tb∥ < T }

for a = g_{1}g_{2} and b = g_{1} ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) g_{2}. This implies

\{u_{t} : |t| < \frac{T − ∥a∥}{∥b∥}\} ⊂ H_{T} [g_{1}, g_{2}] ⊂ \{u_{t} : |t| < \frac{T + ∥a∥}{∥b∥}\} .

Hence

λ(H_{T} [g_{1}, g_{2}]) ∼ \frac{2T}{∥b∥},  (100)

with uniform convergence for g_{1}, g_{2} in a compact subset of G. In particular, hypotheses D1 and D2 are satisfied. Hypothesis (∗∗) follows from the equidistribution of the horocycle flow [DaSm]. Thus, Corollary 2.4 applies, and (99) holds for the function α_{v}(w) as in Proposition 5.1(iv).

To calculate α_{v}(w), define V_{0} = {(x_{1}, x_{2}) ∈ V : x_{1} ≠ 0} and assume that v ∈ V_{0} and supp φ ⊂ V_{0}. Consider a measurable section σ : V → G whose restriction to V_{0} is continuous and defined by

σ(x_{1}, x_{2}) = ( \begin{matrix}x_{1} & x_{2} \\ 1 & \frac{x_{2}+1}{x_{1}} \end{matrix} ) .

By (9) and (100),

α_{v}(w) = lim_{T →∞} \frac{λ(H_{T} [σ(v)^{−1}, σ(w)])}{λ(H_{T})}

= ∥( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )∥ · ∥σ(v)^{−1} ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) σ(w)∥^{−1}

for v, w ∈ V_{0}. This implies the corollary. If v ∉ V_{0} or supp φ ⊄ V_{0} we complete the proof by taking a different section in the obvious way. □

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California Institute of Technology, Pasadena, CA 91125 gorodnik@caltech.edu

Ben Gurion University, Be’er Sheva, Israel 84105 barakw@math.bgu.ac.il