Nonexistence of invariant distributions supported on

the limit set

Ulrich Bunke and Martin Olbrich

June 27, 2001

Contents

1 Statement of the result 0

2 Geometry of geometrically finite discrete subgroups 5

3 Proof of Theorem 1.8 7

1 Statement of the result

Let X be a symmetric space of negative curvature. Then X either belongs to one of the three families of real, complex, or quaternionic hyperbolic spaces, or it is the Cayley hyperbolic plane.

Let G be a connected linear Lie group which finitely covers the isometry group of X. Furthermore, let Γ ⊂ G be a discrete subgroup. We assume that Γ is geometrically finite. We refer to Definition 2.1 for a precise explanation of this notion. If X is a real hyperbolic space, then Γ is geometrically finite iff it admits a fundamental domain with finitely many totally geodesic faces. In the other cases the definition is more complicated. Essentially, Γ is geometrically finite if the corresponding locally symmetric space Γ\X has finitely many cusps and can be compactified by adding a geodesic boundary and closing the cusps. In particular, if Γ is cocompact, or convex-cocompact, or the locally symmetric space Γ\X has finite volume, then Γ is geometrically finite.

We adjoin the geodesic boundary ∂X to X and obtain a compact manifold with boundary X̄ := X ∪ ∂X. A point of ∂X is an equivalence class of geodesic rays where two rays are in the same class if they run in bounded distance to each other. The action of G extends naturally to X̄. Let Λ_{Γ} ⊂ ∂X denote the limit set of Γ. It is defined as the set of accumulation points of any orbit Γo in X̄ for o ⊂ X.

We consider a G-equivariant irreducible complex vector bundle V → ∂X and a finitedimensional representation (φ, V_{φ}) of Γ. Furthermore, by Λ → ∂X we denote the Gequivariant bundle of densities on ∂X. To V we associate the G-equivariant bundle

Ṽ := Hom(V, Λ) .

The space C^{−∞}(∂X, V) of distribution sections of V is then, by definition, the topological dual of C^{∞}(∂X, Ṽ). We define the space of invariant distribution sections of V with twist φ by

Definition 1.1

I(Γ, V, φ) := (C^{∞}(∂X, V) ⊗ V_{φ})^{Γ} .

Next we introduce some real quantities which represent growth properties of the geometric objects introduced so far. We first define the number ρ ∈ ℝ which is a measure of the volume growth of the symmetric space X. We use this number in order to normalize the critical exponents below. Let o be any point of X, and let B(r, o) denote the ball of radius r centered at o.

Definition 1.2

ρ := \frac{1}{2} lim_{r→∞} \frac{log vol B(r, o)}{r} .

The growth of the action of G on the bundle V is measured by the quantity s(V) ∈ ℝ. Note that Λ is the complexification of a real orientable line bundle. It is therefore trivial if considered merely as a vector bundle, but it is not trivial as a G-equivariant bundle. The bundle Λ can be represented by a cocycle of positive transition functions. If α ∈ ℂ, then raising the transition functions to the power α, we obtain a new cocycle which represents the G-equivariant bundle Λ^{α}.

Definition 1.3 s(V)is defined as the unique number such that V ⊗ Λ^{s(V)} ≅ V^{♯} as Gequivariant bundles, where V^{♯} denotes the complex conjugate bundle of Ṽ.

For example, if V = ∂X × ℂ is the trivial bundle, then s(V) = 1. More generally, s(Λ^{α}) = 1 − 2Re(α).

The normalized growth of Γ is expressed by the critical exponent

Definition 1.4

d_{Γ} := \frac{1}{ρ} inf\{ν | ∑_{g∈Γ} dist(go, o)^{−ρ−ν} < ∞\} .

This definition is independent of the choice of o ∈ X. Since Γ is discrete and infinite we have d_{Γ} ∈ (−1, 1].

The exponent d_{φ} is a measure for the growth of φ. It is defined by

Definition 1.5

d_{φ} := \frac{1}{ρ} inf\{ν | sup_{g∈Γ} ∥φ(g)∥dist(go, o)^{−ν} < ∞\} ,

where we have fixed any norm ∥.∥ on End(V_{φ}) and any point o ∈ X. Since Γ is finitely generated, we have d_{φ} < ∞.

A cusp of Γ is, by definition, a Γ-conjugacy class [P]_{Γ} of proper parabolic subgroups P ⊂ G such that Γ ∩ P is infinite and π(Γ ∩ P) ⊂ L is precompact, where π is the projection onto the semisimple quotient L given by the sequence

0 → N → P →^{π} L → 0

with N ⊂ P denoting the unipotent radical of P. Note that if [P]_{Γ} is a cusp of Γ, then Γ_{P} := Γ ∩ P again satisfies our assumptions. The limit set of Γ_{P} consists of the unique fixed point ∞_{P} ⊂ ∂X of P. Since Γ_{P} acts properly on Ω_{Γ_{P}} := ∂X \ {∞_{P} } and Γ_{P} \(Λ_{Γ} \{∞_{P} }) ⊂ Γ_{P} \Ω_{Γ_{P}} is compact (see Lemma 2.2) we can choose a smooth function χ^{Γ_{P}} on Ω_{Γ_{P}} such that supp(χ^{Γ_{P}}) ∩ Λ_{Γ} is a compact subset of Ω_{Γ_{P}}, {supp(g^{∗}χ^{Γ_{P}})}_{g∈Γ_{P}} is a locally finite covering of Ω_{Γ_{P}}, and ∑_{g∈Γ_{P}} g^{∗}χ^{Γ_{P}} ≡ 1. Assume that s(V) > d_{Γ} + d_{φ}.

Definition 1.6 We say that f ∈ I(Γ, V, φ) is strongly supported on the limit set if

1. f is supported on the limit set as a distribution.

2. For any h ∈ V^{∞}(∂X, Ṽ) ⊗ Ṽ_{φ} and cusp [P]_{Γ} of Γ we have

⟨f, h⟩ = ∑_{g∈Γ_{P}}⟨χ^{Γ_{P}} f_{|Ω_{Γ_{P}}} , φ̃(g)^{−1}g^{∗}h⟩ .

In order to see that the second condition is well-defined note that supp(χ^{Γ_{P}} f_{|Ω_{Γ_{P}}}) ⊂ supp(χ^{Γ_{P}}) ∩ Λ_{Γ} is a compact subset of Ω_{Γ_{P}}. Therefore the pairing ⟨χ^{Γ_{P}} f_{|Ω_{Γ_{P}}} , φ̃(g)^{−1}g^{∗}h⟩ is defined. The sum converges because of our assumption s(V) > d_{Γ} + d_{φ} ≥ d_{Γ_{P}} + d_{φ}, which implies that ∑_{g∈Γ_{P}} φ̃(g)^{−1}g^{∗}h_{|Ω_{Γ_{P}}} converges in the space of smooth functions. In fact, the argument proving [1], Lemma 4.2, applies in the more general case when Γ is merely geometrically finite. In Lemma 2.4 we will verify that this definition is independent of the choice of χ^{Γ_{P}}.

In [3] and [2] we have expressed the condition ”strongly supported on the limit set” in the form res^{Γ}(f) = 0. While this definition works for all values of s(V) there we must assume that f is ”deformable”. Because in the present paper we are in the ”domain of convergence” we can use the simpler and more general definition above.

Definition 1.7 By I_{Λ_{Γ}}(Γ, V, φ) we denote the subspace of all f ∈ I(Γ, V, φ) which are strongly supported on the limit set.

The main result of the present paper can now be formulated as follows.

Theorem 1.8 If s(V) > d_{Γ} + d_{φ} + max_{[P]_{Γ}}(0, d_{Γ_{P}} − d_{Γ} + 1) (where the maximum is taken over all cusps of Γ), then I_{Λ_{Γ}}(Γ, V, φ) = 0.

Let us note the following special case which was already shown in [1], Thm 4.7. The group Γ is called convex cocompact if it acts freely and cocompactly on X̄ \ Λ_{Γ}. In this case Γ has no cusps and I_{Λ_{Γ}}(Γ, V, φ) is just the space invariant distribution sections of V with twist φ which are supported on Λ_{Γ}.

Corollary 1.9 If Γ is convex cocompact and s(V) > d_{Γ} + d_{φ}, then I_{Λ_{Γ}}(Γ, V, φ) = 0.

Back to the general case of a geometrically finite discrete group let 1 be the trivial representation of Γ. Then we have d_{1} = 0. In the place of V we consider Λ^{\frac{1−d_{Γ}}{2}} . Note that s(Λ^{\frac{1−d_{Γ}}{2}} ) = d_{Γ} = d_{Γ} + d_{φ}. The space I_{Λ_{Γ}}(Γ, Λ^{\frac{1−d_{Γ}}{2}} , 1) is spanned by the Patterson-Sullivan measure [6], [8], [4], [5], hence dim I_{Λ_{Γ}}(Γ, Λ^{\frac{1−d_{Γ}}{2}} , 1) = 1. Here we must use the definition of the condition ”strongly supported on the limit set” in terms of res^{Γ} given in [3].

In order to construct some twisted examples we consider a finite-dimensional M-spherical representation (π, V_{π}) of G. Here (π, V_{π}) is called M-spherical, if for any parabolic subgroup P ⊂ G there exists a vector 0 ≠ v ∈ V_{π} and a character χ : P → ℝ such that π(p)v = χ(p)v for all p ∈ P. There is a natural inclusion

I_{Λ_{Γ}}(Γ, Λ^{\frac{1−d_{Γ}}{2}} , 1) ↪ I_{Λ_{Γ}}(Γ, Λ^{\frac{1−d_{Γ}−dπ}{2}} , π)

showing that I_{Λ_{Γ}}(Γ, Λ^{\frac{1−d_{Γ}−dπ}{2}} , π) ≠ 0. On the other hand, s(Λ^{\frac{1−d_{Γ}−dπ}{2}} ) = d_{Γ} + d_{π}.

These examples show that our estimate can not be improved in general for convex cocompact Γ. On the other hand, even for geometrically finite Γ we do not know any counterexample to the assertion that already s(V) > d_{Γ}+d_{φ} implies that I_{Λ_{Γ}}(Γ, V, φ) = 0.

2 Geometry of geometrically finite discrete subgroups

If Γ ⊂ G is a discrete subgroup and Λ_{Γ} is its limit set, then Γ acts on X̄ \ Λ_{Γ} properly discontinuously. Let Ȳ_{Γ} denote the manifold with boundary Ȳ_{Γ} := Γ\(X̄ \ Λ_{Γ}). If [P]_{Γ} is a cusp of Γ, then we form the manifold with boundary Ȳ_{Γ_{P}} := Γ_{P} \(X̄ \ {∞_{P} }).

Definition 2.1 The group Γ is called geometrically finite if the following conditions hold:

1. Γ has finitely many cusps.

2. There is a bijection between the set of ends of Ȳ_{Γ} and and the set of cusps of Γ.

3. If [P]_{Γ} is a cusp of Γ, then there exists a representative Ȳ_{P} of the corresponding end of Ȳ_{Γ} and embedding e_{P} : Ȳ_{P} → Ȳ_{Γ_{P}} which is isometric in the interior such that its image e_{P} (Ȳ_{P}) represents the end of Ȳ_{Γ_{P}}.

Lemma 2.2 If [P]_{Γ} is a cusp of Γ, then Γ_{P} \(Λ_{Γ} \{∞_{P} }) is a compact subset of Γ_{P} \Ω_{Γ_{P}}.

Proof. It suffices to show that Γ_{P} \(Λ_{Γ} \{∞_{P} }) is compact in Ȳ_{Γ_{P}}. Note that (Λ_{Γ} \{∞_{P} }) is closed in X̄ \ {∞_{P} }. Therefore, Γ_{P} \(Λ_{Γ} \ {∞_{P} }) is closed in Ȳ_{Γ_{P}}. Furthermore, it is contained in the compact set Ȳ_{Γ_{P}} \ e_{P} (Ȳ_{P}) (note that Ȳ_{P} is open). The assertion now follows. □

Let o ∈ X be any point. We consider the Dirichlet domain F ⊂ X of Γ with respect to o. It is a fundamental domain given by

F := {x ∈ X|dist(x, o) ≤ dist(hx, o) ∀h ∈ Γ} .

If [P]_{Γ} is a cusp of Γ, then let χ^{Γ_{P}} be the cut-off function introduced before Definition 1.6.

Lemma 2.3 We can decompose F as F_{0} ∪ F_{1} ∪ … F_{r}, where r is the number of cusps [P_{i}]_{Γ}, i = 1, … r, of Γ, and the subsets F_{i} satisfy

1. The closure of F_{0} in X̄ \ Λ_{Γ} is compact.

2. \overline{Γ_{P_{i}}F_{i}} ∩ (Λ_{Γ} \ \{∞_{P_{i}}\}) ∩ supp(χ^{Γ_{P_{i}}}) = ∅ for i = 1, …, r.

Proof. By Ȳ_{0} we denote the compact subset Ȳ_{Γ} \ ⋃_{i=1,...r} Ȳ_{P_{i}} of Ȳ_{Γ}. Then we define F_{0} := ΓȲ_{0} ∩ F, where ΓȲ_{0} denotes the preimage of Ȳ_{0} under the projection X̄ \ Λ_{Γ} → Ȳ_{Γ}. By definition, F̄_{0} ⊂ (X̄ \ Λ_{Γ}).

For i = 1, … r we define F_{i} := ΓȲ_{P_{i}}∩F. We then have \overline{Γ_{P_{i}}F_{i}}∩(Λ_{Γ}\\{∞_{P_{i}}\})∩supp(χ^{Γ_{P_{i}}}) = ∅ since the contrary this would imply e(Ȳ_{P_{i}}) ∩ Γ_{P_{i}}\(Λ_{Γ} \ {∞_{P_{i}}}) ≠ ∅. □

Lemma 2.4 Let χ_{1}, χ_{2} be two choices of the cut-off function χ^{Γ_{P}} in Definition 1.6. Then

∑_{g∈Γ_{P}}⟨χ_{1}f_{|Ω_{Γ_{P}}} , φ̃(g)^{−1}g^{∗}h⟩ = ∑_{g∈Γ_{P}}⟨χ_{2}f_{|Ω_{Γ_{P}}} , φ̃(g)^{−1}g^{∗}h⟩ ,

where [P]_{Γ}, f, and h are as 1.6.

Proof. The estimates given in the proof of [1], Lemma 4.2, show that all sums below converge absolutely. This justifies the resummations in the following computation. In the first and the last equality we use the Γ-invariance of f.

∑_{g∈Γ_{P}}⟨χ_{1}f_{|Ω_{Γ_{P}}} , φ̃(g)^{−1}g^{∗}h⟩ = ∑_{g∈Γ_{P}}⟨g^{∗}χ_{1}f_{|Ω_{Γ_{P}}} , h⟩
= ∑_{g∈Γ_{P}} ∑_{l∈Γ_{P}}⟨g^{∗}χ_{1}l^{∗}χ_{2}f_{|Ω_{Γ_{P}}} , h⟩
= ∑_{l∈Γ_{P}} ∑_{g∈Γ_{P}}⟨g^{∗}χ_{1}l^{∗}χ_{2}f_{|Ω_{Γ_{P}}} , h⟩
= ∑_{l∈Γ_{P}}⟨l^{∗}χ_{2}f_{|Ω_{Γ_{P}}} , h⟩
= ∑_{g∈Γ_{P}}⟨χ_{2}f_{|Ω_{Γ_{P}}} , φ̃(g)^{−1}g^{∗}h⟩ .

□

3 Proof of Theorem 1.8

We adapt the argument of the proof of [1], Thm.4.7 given there in the special case of a convex-cocompact group Γ to the present situation where Γ is geometrically finite.

Definition 3.1 We call the bundle V spherical, if V = Λ^{\frac{1−t(V )}{2}} for some t(V) ∈ ℂ.

Note that Re t(V) = s(V). We first show the following special case.

Proposition 3.2 Theorem 1.8 is true if V is spherical.

Proof. Let f ∈ I_{Λ_{Γ}}(Γ, V, φ). Then we must show that ⟨f, h⟩ = 0 for any h ∈ C^{∞}(∂X, Ṽ)⊗ Ṽ_{φ}.

Lemma 3.3 If Γ does not contain any hyperbolic element, then ⟨f, h⟩ = 0.

Proof. If Γ does not contain any hyperbolic element, then Γ = Γ_{P} for the unique cusp [P]_{Γ} of Γ. Since f is supported on Λ_{Γ} = {∞_{P} } as a distribution we have we have f_{|Ω_{Γ_{P}}} = 0. This implies ⟨f, h⟩ = ∑_{g∈Γ_{P}} ⟨χ^{Γ_{P}} f_{|Ω_{Γ_{P}}} , φ̃(g)^{−1}g^{∗}h⟩ = 0. □

It remains to consider the case that Γ contains a hyperbolic element which we will denote by g_{0}.

Lemma 3.4 If Γ does contain a hyperbolic element, say g_{0}, then ⟨f, h⟩ = 0.

Proof. Let b_{±} ∈ ∂X denote the attracting and repelling fixed points of g_{0}. We can write h = h_{+} + h_{−} such that h_{±} vanishes in a neighbourhood of b_{∓}. It suffices to show that ⟨f, h⟩ = 0 for any h which vanishes in a neighbourhood of say b_{+}.

We fix the origin o ∈ X such that o is on the unique geodesic connecting b_{−} with b_{+}. Let F̃ ⊂ X be the Dirichlet domain of Γ with respect to this choice of the origin. Furthermore, let ¯ F̃ be the closure of F̃ in X \ Λ_{Γ}. The Dirichlet domain D_{<g_{0}>} with respect to o of the group < g_{0} > generated by g_{0} separates X \ D_{<g_{0}>} into two connected components X_{+} and X_{−}. Let ∂X_{±} := X̄_{±} ∩ ∂X. We can assume that b_{±} ∈ ∂X_{±}.

Replacing o, if necessary, by g_{0}^{j}o, j ∈ ℕ_{0} sufficiently large, we can assume that supp(h) ⊂ ∂X_{−}. Then we define F := g_{0}^{i}F̃, F̄ := g_{0}^{i} ¯F̃, where we choose i ∈ ℕ_{0} sufficiently large such that F ⊂ X_{+}.

We use the polar coordinates (a, k) ∈ ℝ_{+} × ∂X in order to parametrize points x ∈ X \ {o} such that a(x) = exp(dist(o, x)) and k(x) ∈ ∂X is represented by the geodesic ray through x starting in o. Using these coordinates we extend h to the interior of X setting h̃(x) = χ(a(x))h(k(x)), where χ ∈ C^{∞}(R_{+}) is some cut-off function which is equal to one near infinity and vanishes for a < 1. Note that supp(h̃) ⊂ X \ X_{+}.

Note that by our assumption V is spherical and s(V) > 0. Therefore, the Poisson transformation

P : C^{−∞}(∂X, V) → C^{∞}(X)

is injective (we refer to [1] and the literature cited therein (e.g. [7]) for a definition of the Poisson transformation and its properties). We use the same symbol P in order to denote the extension of the Poisson transform to the tensor product by V_{φ}.

Using the polar coordinates we pull-back the volume form of the unit sphere in T_{o}X to ∂X and thus obtain a volume form dk on ∂X. Then the inverse of the Poisson transformation is given by the following limit formula

⟨f, h⟩ = c_{1} lim_{a→∞} a^{ρ(1−t(V))} ∫_{∂X}⟨P f(a(x), k), h(k)⟩dk

for some constant c_{1}. Using the fact that for large a the volume form dx can be written as dx = c_{2}a^{2ρ}dadk + O(a^{2ρ−1}) we deduce

⟨f, h⟩ = c lim_{a→∞} a^{−ρ(1+t(V))} ∫_{{x∈G|a≤a(x)≤a_{0}a}}⟨P f(x), h̃(x)⟩dx .

where c depends on c_{1}, a_{0} > 1, and c_{2}. We now employ the covering of X by translates of the fundamental domain gF, g ∈ Γ, and the Γ-invariance of P f

P f(gx) = φ(g)P f(x) .

We get

⟨f, h⟩ = c lim_{a→∞} a^{−ρ(1+t(V))} ∑_{g∈Γ} ∫_{{x∈F |a≤a(gx)≤a_{0}a}}⟨φ(g)P f(x), h̃(gx)⟩dx .

Since supp(h̃) ⊂ X \ X_{+} we have gF ∩ X \ X_{+} ≠ ∅ if g ∈ Γ contributes to the sum above. The triangle inequality for X gives a(x)a(g) ≥ a(gx), where we write a(g) for a(go). We will also need the following converse version of the triangle inequality.

Lemma 3.5 There exists a_{1} ∈ ℝ_{+} such that for all g ∈ Γ with gF ∩ X \ X_{+} ≠ ∅ and x ∈ F we have a(g)a(x) ≤ a_{1}a(gx).

We postpone the proof of this lemma and continue the argument for Lemma 3.4. Using 3.5 we obtain

{x ∈ F | a ≤ a(gx) ≤ a_{0}a} ⊂ {x ∈ F | a ≤ a(x)a(g) ≤ a_{1}a}
= {x ∈ F | aa(g)^{−1} ≤ a(x) ≤ a_{1}aa(g)^{−1}}

for all g ∈ Γ with gF ∩ X \ X_{+} ≠ ∅. Taking into account that h̃ is bounded and that for given ϵ > 0 there exists a constant C_{0} such that for all g ∈ Γ we have ∥φ(g)∥ ≤ C_{0}a(g)^{ρ(d_{φ}+ϵ)} we obtain

| ∫_{{x∈F |a≤a(gx)≤a_{0}a}}⟨φ(g)P f(x), h̃(gx)⟩dx|
≤ C_{1}a(g)^{ρ(d_{φ}+ϵ)} ∫_{{x∈F |aa(g)^{−1}≤a(x)≤a_{1}aa(g)^{−1}}} |P f(x)|dx ,  (1)

where C is independent of g ∈ Γ and a ∈ ℝ_{+}. In order to proceed further we employ the following crucial estimate.

Lemma 3.6 There is a constant C_{1} such that

∫_{{x∈F |b≤a(x)≤a_{1}b}} |P f(x)|dx ≤ C_{1}b^{ρ(s(V)+1−μ)}

for all sufficiently small μ > 0 and all b ≥ 1.

We again postpone the proof of Lemma 3.6 and continue with the proof of Lemma 3.4. If we insert the estimate claimed in Lemma 3.6 into (1) and sum over Γ, then we obtain

∑_{g∈Γ} | ∫_{{x∈F |a≤a(gx)≤a_{0}a}}⟨φ(g)P f(x), h̃(gx)⟩dx|

≤ C_{2} ∑_{g∈Γ} a(g)^{ρ(d_{φ}+ϵ+μ−s(V)−1)}a^{ρ(s(V)+1−μ)} ,

where C_{3} is independent of a ≥ 1. If we choose μ, ϵ > 0 so small such that d_{φ} + ϵ + μ − s(V) < −d_{Γ}, then the sum converges and the right-hand side can be estimated by C_{3}a^{ρ(s(V)+1−μ)} with C_{3} independent of a ≥ 1. We conclude that

lim_{a→∞} a^{−ρ(t(V)+1)} ∑_{g∈Γ} ∫_{{x∈F |a≤a(gx)≤a_{0}a}}⟨φ(g)P f(x), h̃(gx)⟩dx = 0 ,

and thus ⟨f, h⟩ = 0.

It remains to prove Lemma 3.5 and Lemma 3.6.

Proof. [of Lemma 3.5] Note that for all g ∈ Γ one of the following two conditions fails:

gg_{0}^{i}o ∈ X_{+}
gF ∩ (X \ X_{+}) ≠ ∅ .

Indeed, if the first condition holds, then gF ∩ X_{+} ≠ ∅. We conclude that gF ⊂ X_{+} and hence gF ∩ (X \ X_{+}) = ∅. Further note that

\overline{\{gg_{0}^{i}o|g ∈ Γ and gF ∩ (X \ X_{+}) ≠ ∅\}} ∩ ∂X = \overline{\{go|g ∈ Γ and gF ∩ (X \ X_{+}) ≠ ∅\}} ∩ ∂X .

We see that F̄ ∩ ∂X ⊂ int∂X_{+} and \overline{\{go|g ∈ ΓandgF ∩ (X \ X_{+}) ≠ ∅\}} ∩ ∂X ⊂ ∂X \ ∂X_{+} are disjoint. We now obtain the desired inequality from Corollary 2.5 of [1]. □

Proof. [of Lemma 3.6]

It is at this point where we use that Γ is geometrically finite. Namely, let F_{0} ∪F_{1} ∪… F_{r} be the decomposition of F given in Lemma 2.3.

Since Λ_{Γ} and F̄_{0} are separated we can use [1], Lemma 6.2 (2), in order to get the estimate

∫_{{x∈X|b≤a(x)≤a_{1}b}∩F_{0}} |P f(x)|dx ≤ Cb^{ρ(1−s(V))} ,

where C is independent of b ≥ 1. This is the required estimate for the contribution of F_{0}.

It remains to consider the contributions of the cusps, i.e of F_{i}, i > 0. Let now [P]_{Γ}, P = P_{i} for one i > 0, be a cusp of Γ. Then for v ∈ Vφ̃ and x ∈ X we have

⟨P f(x), v⟩ = ∑_{g∈Γ_{P}}⟨φ(g)P (χ^{Γ_{P}} f_{|Ω_{Γ_{P}}})(g^{−1}x), v⟩ .

Indeed, let p_{x,v} ∈ C^{∞}(∂X, Ṽ)⊗Vφ̃ denote the integral kernel of the map f ↦ ⟨P (f)(x), v⟩. Then using the invariance properties of the kernel p_{x,v} and that f is strongly supported on the limit set we get

⟨P f(x), v⟩ = ⟨p_{x,v}, f⟩
= ∑_{g∈Γ_{P}}⟨χ^{Γ_{P}} f_{|Ω_{Γ_{P}}} , φ̃(g)^{−1}g^{∗}p_{x,v}⟩
= ∑_{g∈Γ_{P}}⟨χ^{Γ_{P}} f_{|Ω_{Γ_{P}}} , p_{g^{−1}x,φ̃(g)v}⟩
= ∑_{g∈Γ_{P}}⟨φ(g)^{−1}P (χ^{Γ_{P}} f_{|Ω_{Γ_{P}}})(gx), v⟩ .

Since \overline{Γ_{P} F_{i}} ∩ supp(χ^{Γ_{P_{i}}}f_{|Ω_{Γ_{P}}}) = ∅ we again apply [1], Lemma 6.2 (2), in order to get the estimate

|P (χ^{Γ_{P}} f_{|Ω_{Γ_{P}}})(gx)| ≤ Ca(gx)^{−ρ(s(V)+1)} ,  (2)

where C is independent of x ∈ F_{i} and g ∈ Γ_{P}. In order to estimate the sum over g ∈ Γ_{P} we need the following geometric lemma.

Lemma 3.7 There is a constant a_{3} ∈ ℝ_{+} such that for all x ∈ F and g ∈ Γ_{P} we have a(gx) ≥ a_{3} max(a(g)a(x)^{−1}, a(x)).

Let us postpone the proof of the lemma und continue with the estimates. We choose ν > 0 sufficiently small such that d_{Γ_{P}} + d_{φ} − s(V) + ν < 0. For those ν using Lemma 3.7 and (2) we obtain for all x ∈ F_{i}

|P (f)(x)| ≤ C ∑_{g∈Γ_{P}} ∥φ(g)^{−1}∥a(gx)^{−ρ(1+s(V))}
≤ C_{1} ∑_{g∈Γ_{P}} a(g)^{ρd_{φ}}a(gx)^{−ρ(1+s(V))}
≤ C_{2} ∑_{g∈Γ_{P}} a(g)^{−ρ(1+d_{Γ_{P}} +ν)}a(x)^{ρ(−s(V)+1+2d_{Γ_{P}} +2d_{φ}+2ν)}
≤ C_{3}a(x)^{ρ(−s(V)+1+2d_{Γ_{P}} +2d_{φ}+2ν)}

Since s(V) > d_{φ} + d_{Γ_{P}} + 1 we can choose κ, ν, μ > 0 such that −s(V) + 1 + 2d_{Γ_{P}} + 2d_{φ} + 2ν + 2 + κ < s(V) + 1 − μ. Then

|P (f)(x)| ≤ C_{3}a(x)^{ρ(1+s(V)−μ)}a(x)^{−ρ(2+κ)} .

Lemma 3.6 now follows from the fact that the function X ∋ x ↦ a(x)^{−ρ(2+κ)} ∈ ℝ is integrable. In fact,

∫_{{x∈F_{i}|b≤a(x)≤a_{1}b}} |P f(x)|dx ≤ C_{3}b^{ρ(1+s(V)−μ)} ∫_{X} a(x)^{−ρ(2+κ)}dx ≤ C_{4}b^{ρ(1+s(V)−μ)} ,

where C_{4} is independent of b. □

Proof. [of Lemma 3.7] We consider the triangle inequality for the triangle (o, gx, go) in X and obtain dist(o, gx) + dist(go, gx) ≥ dist(o, go). Since dist(go, gx) = dist(o, x) we conclude that a(x)a(gx) ≥ a(g).

Recall that F is a Dirichlet domain of Γ with respect to g_{0}^{i}o. We conclude that dist(g_{0}^{i}o, x) ≤ dist(g_{0}^{i}o, gx) for all g ∈ Γ_{P} and x ∈ F. Using this and the triangle inequality for the triangle (o, x, g_{0}^{i}o) we obtain

dist(o, x) ≤ dist(g_{0}^{i}o, x) + dist(o, g_{0}^{i}o)
≤ dist(g_{0}^{i}o, gx) + dist(o, g_{0}^{i}o)
≤ 2dist(g_{0}^{i}o, o) + dist(o, gx)

and therefore a(gx) ≥ a(g_{0}^{i})^{−2}a(x) for all g ∈ Γ_{P} and x ∈ F. The assertion of the lemma holds true if we set a_{3} := min(1, a(g_{0}^{i})^{−2}) □

We now have finished the proof of Lemma 3.4 and therefore of Proposition 3.2. □

Proof. [end of the proof of Theorem 1.8] By Proposition 3.2 we know that Theorem 1.8 holds true under the additional asumption that V is spherical. We twist by finitedimensional representations of G in order to conclude the general case. Fix any parabolic subgroup P ⊂ G. Then we can write ∂X = G/P. If (π, V_{π}) is a finite-dimensional representation of G, then we can form the G-equivariant bundle V (π) = G ×_{P} V_{π} on ∂X. The idea of twisting is based on the fact that there is an G-equivariant isomorphism

C^{∞}(∂X, V ⊗ V (π)) ⊗ V_{φ} →^{≅} C^{∞}(∂X, V) ⊗ V_{π} ⊗ V_{φ} .

In particular, there is an isomorphism

j : I_{Λ_{Γ}}(Γ, V ⊗ V (π), φ) →^{≅} I_{Λ_{Γ}}(Γ, V, φ ⊗ π) .

If V is an irreducible G-equivariant bundle on ∂X, then there exists an irreducible representation (π, V_{π}) of G and a G-equivariant embedding

i : V ↪ Λ^{\frac{1−t(V )−dπ}{2}} ⊗ V (π) ,

where t(V) ∈ ℂ is defined such that V ⊗ Λ^{t(V)} = Ṽ. In particular, Re(t(V)) = s(V). For these facts we refer to [1], p. 108 ( in particular, to the formulas (33), (34)). The embedding i composed with the isomorphism j gives an embedding

j ∘ i : I_{Λ_{Γ}}(Γ, V, φ) ↪ I_{Λ_{Γ}}(Γ, Λ^{\frac{1−t(V )−dπ}{2}} , φ ⊗ π) .

We can apply Prop. 3.2 to the right-hand side. Indeed,

s(Λ^{\frac{1−t(V )−dπ}{2}} ) = s(V)+d_{π} > d_{Γ}+d_{φ}+d_{π}+max_{[P]_{Γ}}(0, d_{Γ_{P}} −d_{Γ}+1) = d_{Γ}+d_{φ⊗π}+max_{[P]_{Γ}}(0, d_{Γ_{P}} −d_{Γ}+1) ,

and hence I_{Λ_{Γ}}(Γ, Λ^{\frac{1−t(V )−dπ}{2}} , φ ⊗ π) = 0. This implies I_{Λ_{Γ}}(Γ, V, φ) = 0. □

14 REFERENCES

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[2] U. Bunke and M. Olbrich.  Regularity of invariant distributions.  Preprint

arXiv:math.DG/0103144.

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[5] K. Corlette and A. Iozzi. Limit sets of discrete groups of isometries of exotic hyperbolic spaces. Trans. AMS, 351 (1999), 1507–1530.

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[8] D. Sullivan. The density at infinity of a discrete group of hyperbolic motions. I.H.E.S. Publ. Math., 50(1979), 171–209.

Ulrich Bunke and Martin Olbrich Universität Göttingen Mathematisches Institut Bunsenstr. 3-5 37073 Göttingen GERMANY bunke@uni-math.gwdg.de, olbrich@uni-math.gwdg.de