Small discrete subgroups and the Plancherel theorem

Ulrich Bunke^{∗}and Martin Olbrich^{†} December 27, 1996

Contents

1 Small discrete subgroups Γ 1 2 The Plancherel theorem for L^{2}(G) 3 3 Γ-averages 7 4 Fourier transform for L^{2}(Γ\G)(δ) 11 5 Schottky-construction 19

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, Email:bunke@cfgauss.uni-math.gwdg.de

†Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, Email:olbrich@cfgauss.uni-math.gwdg.de

6 Examples 29

1 Small discrete subgroups Γ

Throughout this paper G is a real semisimple Lie group in the Harish-Chandra class. Let Γ ⊂ G be a torsion-free discrete subgroup. Then a natural question is to describe the decomposition of L^{2}(Γ\G) in terms of irreducible unitary representations of G.

If Γ is trivial, then the decomposition of L^{2}(G) is the content of the Plancherel theorem which was proved by Harish-Chandra [3]. We will describe this theorem in detail in Section 2. Let P denote the set of association classes of cuspidal parabolic subgroups of G. Any class [P] ∈ P contributes a direct summand L^{2}(G)_{[P]} to L^{2}(G) which is a sum of direct integrals of unitary principal series representations induced from P.

Let K ⊂ G be a maximal compact subgroup which we fix once and for all. If we take a K-type δ ∈ K̂, then we can consider the subspace L^{2}(G)(δ) of functions which transform according to δ with respect to the action of K by right translations.

Given δ there is a minimal subset P(δ) ⊂ P of association classes of parabolic subgroups such that L^{2}(H)_{[P]}(δ) = {0} for [P] ∉ P(δ). For example, if δ is the trivial K-type, then P(δ) consists of the class of minimal parabolic subgroups of G.

In this paper we consider discrete subgroups Γ ⊂ G which are small in the sense that with respect to the analysis on Γ\G/K they behave more like the trivial supgroup than, say, cocompact subgroups or subgroups of finite covolume.

We first define smallness with respect to [P] ⊂ P, P ≠ G. Let P ∈ [P] and let Z_{P} := G/P be the corresponding generalized flag variety. On Z_{P} we have a unique Kinvariant volume form vol such vol(Z_{P}) = 1. The following conditions only depend on the class [P].

Axiom 1.1 A([P]): There is an Γ-invariant open subset Ω_{P} ⊂ Z_{P} of full measure on which Γ acts freely and properly. There exists a fundamental domain F ⊂ Ω_{P} for Γ such that

∑_{g∈Γ} vol(gF)^{1/2} < ∞ .

This condition allows us to perform a certain kind of Γ-averages of smooth vectors of unitary principal series representations induced from P. We want to apply this to the decomposition of L^{2}(Γ\G)(δ), δ ∈ K̂. Therefore we need the following condition.

Axiom 1.2 A(δ) : We say that Γ satisfies A(δ) if A([P]) is satisfied for all [P] ∈ P(δ).

It is a natural question at this point wether there exist subgroups Γ satisfying these axioms. Besides the trivial group there are cyclic groups Γ =< γ > which satisfy A(δ) for all δ ∈ K̂. We consider these as the trivial examples.

On the one hand if rank_{R}(G) = 1, then there are Γ ⊂ G which are convex cocompact. If the critical exponent δ_{Γ} (see [2] for definitions) is negative, then then Γ satisfies A(δ) for all δ ∈ K̂. It is easy to construct Schottky groups satisfying this condition (see Section 6). One the other hand we do not know any example of a non-trivial subgroup of a simple group G of rank ≥ 2 satisfying A([P]) for all proper parabolic subgroups. But we will construct examples satisfying A([P]) for some classes [Q]. By the following Lemma these examples always satisfy A([P_{0}]), where P_{0} is a minimal parabolic subgroup, and thus A(1), where 1 ∈ K̂ is the trivial K-type.

Lemma 1.3 If Γ ⊂ G satisfies A([P]), then for any Q ⊂ P also A([Q]) holds.

Proof. The natural projection π : Z_{Q} → Z_{P} has compact fibres. Let Ω_{Q} := π^{−1}(Ω_{P}). Then Ω_{Q} has full measure and Γ acts freely and properly on Ω_{Q}. If F_{P} is a fundamental domain of Γ on Ω_{P}, then we obtain a fundamental domain of Γ on Ω_{Q} by π^{−1}(F_{P}). Let h := vol(π^{−1}(z)) for any z ∈ Z_{P}. Then vol(γF_{Q}) = hvol(γF_{P}). This implies A([Q]). □

2 THE PLANCHEREL THEOREM FOR L^{2}(G) 3

We now describe the content of the following sections. In Section 2 we describe in detail Harish-Chandra’s Plancherel theorem for L^{2}(G).

In Section 3 we study Γ averages in unitary principal series representations induced from P assuming that A([P]) is satsified.

In Section 4 we introduce Fourier transforms for the locally symmetric space and consider the Plancherel theorem for L^{2}(Γ\G)(δ) assuming A(δ).

In Section 5 we consider a Schottky construction of discrete subgroups satisfying A([P]).

In the final Section 6 we discuss examples for particular groups G.

2 The Plancherel theorem for L^{2}(G)

The aim of the present section is to describe the Plancherel theorem for L^{2}(G). Let θ be the Cartan involution of G such that K is the set of fixed points θ.. We fix a minimal parabolic subgroup P_{0} ⊂ G with Langlands decomposition P_{0} = M_{0}A_{0}N_{0} such that A_{0} is θ-stable. Let P denote the set of all cuspidal parabolic subgroups P ⊂ G containing M_{0}A_{0}. For P ∈ P let P = M_{P} A_{P} N_{P} be the Langlands decomposition such that M_{P} and A_{P} are θ-stable. For any P ∈ P let M̂_{P}^{2} denote the set of equivalence classes of discrete series representations (σ, V_{σ}). In each equivalence class we choose a representative.

Let a_{P} denote the Lie algebra of A_{P} and let (a_{P})_{C}^{∗} be its complexified dual. For any λ ∈ (a_{P})_{C}^{∗} we consider the principal series representation

H_{P}^{σ,λ} := Ind_{P}^{G}(σ_{λ}) := {f : G → V_{σ} | f(gman) = a^{λ−ρ_{P}} σ(m)^{−1}f(g), f_{|K} ∈ L^{2}(K, V_{σ})} .

Here ρ_{P} ∈ a_{P}^{∗} is given by ρ_{P} (H) := \frac{1}{2}tr ad(H)_{n_{P}}, ∀H ∈ a_{P}, where n_{P} denotes the Lie algebra of N_{P}. If λ ∈ ıa_{P}^{∗}, then H^{σ,λ} is an unitary representation of G with scalar product given by

⟨f, f′⟩ := ∫_{K}⟨f(k), f′(k)⟩dk.

By H_{P,♯}^{σ,λ}, ♯ ∈ {K, ∞, −∞}, we denote the space of K-finite, smooth, or distribution vectors of H_{P}^{σ,λ} (see [7], Ch.11).

Let W denote the Weyl group N_{K}(A_{0})/M_{0}. For each w ∈ W we choose a representative in N_{K}(A_{0}) which we also denote by w. For P, Q ∈ P we put

W (P, Q) = {s : A_{P} →^{∼} A_{Q} | ∃w ∈ W : A_{P}^{w} = A_{Q}, s = w_{|A_{P}} } ,

where we employ the notation a^{w} := waw^{−1}. For any P, Q ∈ P and s ∈ W (P, Q) there is a meromorphic family of intertwining operators

Ĵ_{λ}(s) : H_{P,♯}^{σ,λ} → H_{Q,♯}^{σ^{w},λ^{w}}, λ ∈ (a_{P})_{C}^{∗} , ♯ ∈ {K, ∞, −∞}

where σ^{w} is the representation of M_{Q} = M_{P}^{w} on V_{σ} given by σ(m) = σ(w^{−1}mw), λ^{w} = λ ∘ Ad(w^{−1}), and w ∈ W represents s. The equivalence class of σ^{w} is independent of the choice of w representing s and thus can also be denoted by σ^{s}. Similarly we can write λ^{s} := λ^{w}. Note that the definition of Ĵ_{λ}(s) involves the choice of an identification of σ^{w} with the previously chosen representative of σ^{w}. The intertwining operator Ĵ_{λ}(s) is given in its domain of convergence by

(Ĵ_{λ}(s)f)(g) = ∫_{N_{Q}/N_{Q}∩N_{P}} f(gnw)dn .

There exist normalizations J_{λ}(s) of Ĵ_{λ}(s), ∀Q, P ∈ P, λ ∈ (a_{P})_{C}^{∗}, s ∈ W (P, Q), such that if P, Q, R ∈ P with s ∈ W (P, Q), t ∈ W (Q, R), then J_{λ}(ts) = J_{λ^{s}}(t) ∘ J_{λ}(s) and J_{λ}(s) : H_{P}^{σ,λ} → H_{Q}^{σ^{w},λ^{w}} is unitary for λ ∈ ıa_{P}^{∗} (see [4], Ch. XIV.6).

We consider {(P, σ) | P ∈ P, σ ∈ M̂_{P}^{2} } as the set of objects of a category S with

Hom_{S}((P, σ), (Q, τ)) := {s ∈ W (P, Q), σ^{s} = τ} .

This category is in fact a groupoid. The intertwining operators J_{λ}(s) implement a unitary representation H of the groupoid S, i.e a functor from S to the category of Hilbert spaces such that all morphisms are mapped to unitary isomorphisms. For (P, σ) ∈ S, λ ∈ ıa_{P}^{∗} let L_{2}(H_{P}^{σ,λ}) denote the Hilbert space of Hilbert-Schmid operators on H_{P}^{σ,λ}. Then H associates to any object (P, σ) ∈ S the direct integral of Hilbert spaces and G-bimodules

H(P, σ) := ∫_{ıa_{P}^{∗}} L_{2}(H_{P}^{σ,λ}) p_{P,σ}(dλ) .

Here p_{P,σ}(dλ) denotes the Plancherel measure which is absolutely continuous to the Lebesgue measure on ıa_{P}^{∗}. In order to make the direct integral explicit we use the identification of Hilbert spaces H_{P}^{σ,λ} = L^{2}(K, V_{σ})^{K_{P}}, K_{P} := M_{P} ∩ K. Then we can write H(P, σ) = L^{2}(ıa_{P}^{∗} , L_{2}(H_{P}^{σ,λ}), p_{P,σ}(dλ)). The functor H associates to any morphism s ∈ Hom_{S}((P, σ), (Q, τ)) the unitary operator H(s) : H(P, σ) → H(Q, τ) given by

H(s)(A)(λ) := J_{λ}(s)A(λ)J_{λ}(s)^{∗} , A ∈ L^{2}(ıa_{P}^{∗} , L_{2}(H_{P}^{σ,λ}), p_{P,σ}(dλ)) .

Note that if G ∈ P, then a_{G} = {0}. In this case for σ ∈ M̂_{G}^{2} the positive number p_{G,σ}(0) is the formal dimension of the square integrable representation σ.

A section of the representation H is a family {A(P, σ) ∈ H(P, σ) | (P, σ) ∈ S} which satisfies H(s)A(P, σ) = A(Q, τ) for all s ∈ Hom_{S}((P, σ), (Q, τ)). We define the support of a section A of H to be the set supp(A) := {(P, σ) ∈ S | A(P, σ) ≠ 0}. We define a scalar product between sections A, B of H by

⟨A, B⟩ := ∑_{(P,σ)∈S} ∫_{ıa_{P}^{∗}} TrA(P, σ)(λ)^{∗}B(P, σ)(λ) p_{P,σ}(dλ) .

Let Γ(H) be the corresponding Hilbert space of sections of H. It is a unitary Gbimodule by

((g, h)A)(P, σ)(λ) := π^{σ,λ}(h)A(P, σ)(λ)π^{σ,λ}(g)^{−1} , g, h ∈ G , A ∈ Γ(H) .

The Plancherel theorem states that L^{2}(G) and Γ(H) are unitary equivalent G-bimodules. In order to write down the equivalences explicitly we introduce suitable subspaces. We consider H̄_{P,∞}^{σ,λ} ⊗ H_{P,K}^{σ,λ} as a subspace of L_{2}(H_{P}^{σ,λ}) by (ϕ ⊗ ψ)(κ) = ⟨ϕ, κ⟩ψ, ϕ ∈ H̄_{P,∞}^{σ,λ}, ψ ∈ H_{P,K}^{σ,λ}, κ ∈ H_{P}^{σ,λ}. Here H̄_{P,∞}^{σ,λ} is the Hermitean conjugate space of H_{P,∞}^{σ,λ} and ⊗ is the algebraic tensor product. Let Γ_{0}(H) ⊂ Γ(H) denote the subspace of left smooth and right-K-finite vectors A ∈ Γ(H) such that for all (P, σ) ∈ S the function

ıa_{P}^{∗} ∋ λ ↦ A(P, σ)(λ) ∈ H̄_{P,∞}^{σ,λ} ⊗ H_{P,K}^{σ,λ}

is smooth and for any X ∈ U(g), n ∈ N, and constant coefficient differential operator D on a_{P}^{∗} we have

max_{(P,σ)∈S} sup_{λ∈ıa_{P}^{∗}} ∥((X, 1)A)(P, σ)(λD)∥_{1}(1 + |λ|)^{n} < ∞ ,

where ∥.∥_{1} is the trace norm. Note that a right K-finite section of H has finite support.

We can define maps U : Γ_{0}(H) → C^{∞}(G)_{K} and U^{∗} : C_{c}^{∞}(G)_{K} → Γ_{0}(H) by

U(A)(g) := ∑_{(P,σ)∈S} ∫_{ıa_{P}^{∗}} Tr π^{σ,λ}(g)A(P, σ)(λ) p_{P,σ}(dλ), A ∈ Γ_{0}(H), g ∈ G,
U^{∗}(f)(P, σ)(λ) := ∫_{G} f(g)π^{σ,λ}(g)^{−1} dg , f ∈ C_{c}^{∞}(G)_{K} .

By [1] we have indeed U^{∗}(f) ∈ Γ_{0}(H), ∀f ∈ C^{∞}(G)_{K}. Moreover we have U(A) ∈ L^{2}(G)_{K}, ∀A ∈ Γ_{0}(H), by the Plancherel Theorem.

Theorem 2.1 ([3], [7]) The map U extends by continuity to an unitary isomorphism of G-bimodules U : Γ(H) ≅ L^{2}(G). Moreover U^{∗} has a continuous extension to all of L^{2}(G) such that U^{∗}U = id, UU^{∗} = id.

Let P ∈ P and let [P] ∈ P be its association class. Then we set Γ(H)_{[P]} := {A ∈ Γ(H)|(Q, τ) ∈ supp(A) ⇒ [Q] = [P]}. Then L^{2}(G)_{[P]} := U(Γ(H)_{[P]}) is an invariant subspace with respect to the right action of G.

Let δ ∈ K̂. Then U and U^{∗} restrict to unitary isomorphisms between the δ-isotypic components Γ_{0}(H)(δ), L^{2}(G)(δ). Let S(δ) ⊂ S be the subgroupoid of those (P, σ) such that H^{σ,0}(δ) ≠ 0. By P(δ) we denote the image of S(δ) by the natural projection S → P. Then P(δ) is the set of association classes of elements of P(δ). We have

L^{2}(G)(δ) = ⊕_{[P]∈P(δ)}L^{2}(G)_{[P]}(δ) .

3 Γ-averages

In the present section we fix a proper cuspidal parabolic subgroup P ∈ P. Let Γ ⊂ G be a discrete subgroup satisfying A([P]). By Ω_{P} ⊂ Z_{P} we denote the Γ-invariant subset of full measure on which Γ acts freely and properly. We choose a fundamental domain F_{P} ⊂ Ω. The same notation F_{P} will be used for the set F_{P} := {k ∈ K | kK_{P} ∈ F_{P} } ⊂ K, where K_{P} := M_{P} ∩ K.

Lemma 3.1 Let (P, σ) ∈ S, λ ∈ ıa_{P}^{∗}, and ϕ ∈ C^{∞}(K, V_{σ})^{K_{P}}. Then

p_{∗}(λ)(ϕ) := ∑_{g∈Γ}(π^{σ,λ}(g)ϕ)_{|F_{P}} ∈ L^{2}(F_{P} , V_{σ})^{K_{P}}

converges in norm and satisfies

∥p_{∗}(λ)(ϕ)∥_{L^{2}(F_{P})} ≤ sup_{k∈K} ∥ϕ(k)∥ ∑_{γ∈Γ} vol(γF_{P})^{1/2} .

Proof. We can decompose g ∈ G into a product κ_{P} (g)m_{P} (g)a_{P} (g)n_{P} (g), where κ_{P} (g) ∈ K, m_{P} (g) ∈ M_{P}, a_{P} (g) ∈ A_{P}, n_{P} (g) ∈ N_{P}, and κ_{P} (g) (m_{P} (g)) are well defined up to right (left) multiplication by elements of K_{P}. Then

(π^{σ,λ}(g)ϕ)(k) = a_{P} (g^{−1}k)^{λ−ρ_{P}} σ(m_{P} (g^{−1}k))^{−1}ϕ(κ_{P} (g^{−1}k)) .

and we can estimate :

∥(π^{σ,λ}(g)ϕ)(k)∥ ≤ sup_{k∈K} ∥ϕ(k)∥a_{P} (g^{−1}k)^{−ρ_{P}} .

The assertion of the Lemma now follows from :

∥(π^{σ,λ}(g)ϕ)∥_{L_{2}(F_{P})}^{2} ≤ sup_{k∈K} ∥ϕ(k)∥^{2} ∫_{F_{P}} a(g^{−1}k)^{−2ρ_{P}} dk
= sup_{k∈K} ∥ϕ(k)∥^{2} ∫_{gF_{P}} 1dk
= sup_{k∈K} ∥ϕ(k)∥^{2}vol(gF_{P}) .
□

In the following we will often write p_{∗} for p_{∗}(λ). Let B_{P} := Γ\Ω_{P} and for λ ∈ (a_{P})_{C}^{∗} we let σ_{λ} be the representation of P on V_{σ} defined by P = M_{P} A_{P} N_{P} ∋ man ↦ a^{ρ_{P} −λ}σ(m). On B_{P} we consider the bundle of Hilbert spaces

V_{B_{P}} (σ_{λ}) := Γ\(G ×_{P,σ_{λ}} V_{σ})_{|Ω_{P}} .

If λ ∈ ıa_{P}^{∗}, then we can form the Hilbert space of sections L^{2}(B_{P} , V_{B_{P}} (σ_{λ})), where the scalar product is given by

⟨ψ, ϕ⟩ = ∫_{F_{P}} ⟨ψ(k), ϕ(k)⟩dk .

By Lemma 3.1 we have a continuous map p_{∗}(λ) : C^{∞}(K, V_{σ})^{K_{P}} → L^{2}(B_{P} , V_{B_{P}} (σ_{λ})).

Definition 3.2 We define ext(λ) : L^{2}(B_{P} , V_{B_{P}} (σ_{λ})) → (C^{∞}(K, V_{σ})^{K_{P}})^{∗} to be the adjoint of p_{∗}(λ).

It is easy to check that ext(λ) has values in ^{Γ}H_{P,−∞}^{σ,λ}, where we have identified the space (C^{∞}(K, V_{σ})^{K_{P}})^{∗} with a subspace of the distribution vectors of the principal series representation H_{P,−∞}^{σ,λ}. Again we often write ext for ext(λ).

The extension map ext was introduced in our previous work on convex-cocompact subgroups of semisimple groups of rank one. We refer to [2] for proofs of the basic properties of ext.

For later reference we collect two useful identities. Let χ ∈ C^{∞}(Ω_{P}) be a cut-off function such that ∑_{γ∈Γ} γ^{∗}χ ≡ 1 and ♯{γ ∈ Γ | (γ^{∗}χ)χ ≢ 0} < ∞. Such a function exists since Γ acts properly on Ω_{P}. If ϕ ∈ L^{2}(B_{P} , V_{B_{P}} (σ_{λ})), then we can consider ϕ as a Γ-equivariant function ϕ : Ω_{P} → V_{σ}. The product χϕ belongs to L^{2}(K, V_{σ}). If ψ ∈ C^{∞}(K, V_{σ})^{K_{P}}, then

⟨ext ϕ, ψ⟩ = ⟨ϕ, p_{∗} ψ⟩  (1)
= ∑_{g∈Γ}⟨χϕ, π^{σ,λ}(g)ψ⟩ ,

where the sum is absolutely convergent. If ψ, ϕ ∈ H_{P,∞}^{σ,λ} = C^{∞}(K, V_{σ})^{K_{P}}, then we have

⟨ext ∘ p_{∗}ϕ, ψ⟩ = ∑_{g∈Γ}⟨χp_{∗}ϕ, π^{σ,λ}(g)ψ⟩  (2)
= ∑_{g,h∈Γ}⟨χπ^{σ,λ}(h)ϕ, π^{σ,λ}(g)ψ⟩
= ∑_{g,h∈Γ}⟨π^{σ,λ}(g)χπ^{σ,λ}(h)ϕ, ψ⟩
= ∑_{g∈Γ}⟨π^{σ,λ}(g)ϕ, ψ⟩ ,

where the sums are again absolutely convergent.

Lemma 3.3 For (P, σ), (Q, τ) ∈ S, s ∈ Hom_{S}((P, σ), (Q, τ)), λ ∈ ıa_{P}^{∗}, we have

J_{λ}(s) ∘ ext ∘ p_{∗} = ext ∘ p_{∗} ∘ J_{λ}(s) .

Proof. Let ϕ ∈ H_{P,∞}^{σ,λ}, ψ ∈ H_{Q,∞}^{σ^{s},λ^{s}}. Then by (2)

⟨J_{λ}(s) ∘ ext ∘ p_{∗} ϕ, ψ⟩ = ⟨ext ∘ p_{∗} ϕ, J_{λ^{s}}(s^{−1})ψ⟩
= ∑_{g∈Γ}⟨ϕ, π^{σ,λ}(g)J_{λ^{s}}(s^{−1})ψ⟩
= ∑_{g∈Γ}⟨ϕ, J_{λ^{s}}(s^{−1})π^{σ^{s},λ^{s}}(g)ψ⟩
= ∑_{g∈Γ}⟨J_{λ}(s)ϕ, π^{σ^{s},λ^{s}}(g)ψ⟩
= ⟨ext ∘ p_{∗} ∘ J_{λ}(s)ϕ, ψ⟩ .
□

Let V_{σ,±∞} be the space of smooth/distribution vectors of V_{σ}. Let V (σ_{λ}, ∞) := G×_{P,σ_{λ}}V_{σ,∞} and V_{B_{P}} (σ_{λ}, ∞) := Γ\V (σ_{λ}, ∞)_{|B}. Let C^{−∞}(B_{P} , V_{B_{P}} (σ_{λ}, −∞)) be the dual space of C_{c}^{∞}(B_{P} , V_{B_{P}} (σ_{λ}, ∞)).

Definition 3.4 We define the restriction

res(λ) : ^{Γ}H_{P,−∞}^{σ,λ} → C^{−∞}(B_{P} , V_{B_{P}} (σ_{λ}, ∞))

to be the composition of the restriction to Ω_{P} and the identification of invariant distributions over Ω_{P} with distributions over B_{P}.

We will often write res := res(λ). There is a natural embedding L^{2}(B_{P} , V_{B_{P}} (σ_{λ})) ↪ C^{−∞}(B_{P} , V_{B_{P}} (σ, −∞)), and we have res ∘ ext = id.

If (P, σ), (Q, τ) ∈ S and Hom_{S}((P, σ), (Q, τ)) ≠ ∅, then [P] = [Q]. In particular, Γ satisfies A([Q]).

Definition 3.5 For (P, σ), (Q, τ) ∈ S, s ∈ Hom_{S}((P, σ), (Q, τ)), λ ∈ ıa_{P}^{∗}, we define the scattering matrix S_{λ}(s) : L^{2}(B_{P} , V_{B_{P}} (σ_{λ})) → C^{−∞}(B_{Q}, V_{B_{Q}}(σ_{λ^{s}}^{s}, −∞)) to be the composition

S_{λ}(s) := res ∘ J_{λ}(s) ∘ ext .

Lemma 3.6 For (P, σ), (Q, τ) ∈ S, s ∈ Hom_{S}((P, σ), (Q, τ)), λ ∈ ıa_{P}^{∗}, ψ ∈ C_{c}^{∞}(B_{Q}, V_{B_{Q}}(σ_{λ^{s}}^{s}, ∞)) and ϕ ∈ C_{c}^{∞}(B_{P} , V_{B_{P}} (σ_{λ}, ∞)) we have

⟨S_{λ}(s)ϕ, ψ⟩ = ⟨ϕ, S_{λ^{s}}(s^{−1}) ψ⟩ .

Proof. Let χ ∈ C^{∞}(Ω_{P}) be a cut-off function as in (1). Applying this identity twice and using that J_{λ}(s) is unitary we obtain

⟨S_{λ}(s)ϕ, ψ⟩ = ⟨res ∘ J_{λ}(s) ∘ ext ϕ, ψ⟩
= ⟨J_{λ}(s) ∘ ext ϕ, χψ⟩
= ⟨ext ϕ, J_{λ}(s)^{∗}(χψ)⟩
= ∑_{g∈Γ}⟨χϕ, π^{σ,λ}(g)J_{λ}(s)^{∗}(χψ)⟩
= ∑_{g∈Γ}⟨π^{σ^{s},λ^{s}}(g)J_{λ}(s)(χϕ), χψ⟩
= ⟨χϕ, J_{λ^{s}}(s^{−1}) ∘ ext ψ⟩
= ⟨ϕ, res ∘ J_{λ^{s}}(s^{−1}) ∘ ext ψ⟩
= ⟨ϕ, S_{λ^{s}}(s^{−1}) ψ⟩ .
□

4 FOURIER TRANSFORM FOR L^{2}(Γ\G)(δ) 11

We would like to know that S_{λ}(s), λ ∈ ıa_{P}^{∗}, is unitary, and that it satisfies the relations S_{λ^{s}}(t) ∘ S_{λ}(s) = S_{λ}(ts), where t ∈ Hom_{S}((Q, τ), (R, κ)). If Γ is a convex-cocompact subgroup of a semisimple group of rank one, then these relations were proved in [2].

4 Fourier transform for L^{2}(Γ\G)(δ)

In the present section we fix a K-type δ ∈ K̂ and assume that Γ satisfies A(δ). Let U(g)^{K} ⊂ U(g) the subalgebra of K-invariant elements. We will introduce two representations R_{±∞}^{δ} of the groupoid S(δ) which are associated to Γ. For (P, σ) ∈ S(δ), P ≠ G, λ ∈ ıa_{P}^{∗}, we consider the U(g)^{K}-modules ^{Γ}Hom(H_{P,∞}^{σ,λ}, H_{P}^{σ,λ}(δ)) and Hom(^{Γ}H_{P,−∞}^{σ,λ}, H_{P}^{σ,λ}(δ)). Here the (g, K)-module structure is induced from H_{P,K}^{σ,λ}(δ), while the condition of Γinvariance for ϕ ∈ ^{Γ}Hom(H_{P,∞}^{σ,λ}, H_{P}^{σ,λ}(δ)) is ϕ(π^{σ,λ}(γ)ψ) = ϕ(ψ) for all γ ∈ Γ and ψ ∈ H_{P,∞}^{σ,λ}.

The representation R_{±∞}^{δ} associates to (P, σ) ∈ S(δ), P ≠ G, the U(g)^{K}-modules

R_{∞}^{δ}(P, σ) := {ıa_{P}^{∗} ∋ λ ↦ Φ_{λ} ∈ ^{Γ}Hom(H_{P,∞}^{σ,λ}, H_{P}^{σ,λ}(δ))}
R_{−∞}^{δ}(P, σ) := {ıa_{P}^{∗} ∋ λ ↦ Φ_{λ} ∈ Hom(^{Γ}H_{P,−∞}^{σ,λ}, H_{P}^{σ,λ}(δ))} ,

and the trivial module to (G, σ) ∈ S(δ). If s ∈ Hom_{S}((P, σ), (Q, τ)), then we define the map R_{±∞}^{δ}(s) : R_{±∞}^{δ}(P, σ) → R_{±∞}^{δ}(Q, τ) by (R_{±∞}^{δ}(s)Φ)_{λ} := J_{λ}(s) ∘ Φ_{λ} ∘ J_{λ}(s)^{∗}. Later we will specify regularity conditions on the families Φ_{λ}.

Definition 4.1 We define Fourier transforms F_{±∞} : C_{c}^{∞}(Γ\G)(δ) → Γ(R_{±∞}^{δ}) by

F_{−∞}(f)_{P,σ,λ} := ∫_{Γ\G} f(Γg)π^{σ,λ}(g)^{−1}dg, (P, σ) ∈ S(δ), λ ∈ ıa_{P}^{∗}
F_{∞}(f)_{P,σ,λ} := F_{−∞}(f)_{P,σ,λ} ∘ ext ∘ p_{∗}

Using the intertwining property of J_{λ}(s) it is easy to check that F_{−∞}(f) is a section of R_{−∞}^{δ}. It follows from Lemma 3.3 that F_{∞}(f) is a section of R_{∞}^{δ}. If U(g)^{K} acts on C_{c}^{∞}(Γ\G)(δ) by the right action, then the Fourier transforms F_{±∞} are U(g)^{K}-intertwining operators.

Let S(ıa_{P}^{∗} , L^{2}(F_{P} , V_{σ})^{K_{P}}) be the Schwartz space of L^{2}(F_{P} , V_{σ})^{K_{P}} -valued functions on ıa_{P}. For each λ ∈ ıa_{P}^{∗} we identify L^{2}(F_{P} , V_{σ})^{K_{P}} with L^{2}(B_{P} , V_{B_{P}} (σ_{λ})). Let Γ_{0}(R_{∞}^{δ}) ⊂ Γ(R_{∞}^{δ}) be the subspace spanned by sections Φ ∈ Γ(R_{∞}^{δ}) of finite support in S(δ) which are given by

Φ_{P,σ,λ}(ψ) := ⟨ext ϕ_{P,σ,λ}, ψ⟩ϕ̂_{P,σ,λ} , ∀ψ ∈ H_{P,∞}^{σ,λ}

for some ϕ ∈ S̄(ıa_{P}^{∗} , L^{2}(F_{P} , V_{σ})^{K_{P}}) and ϕ̂_{P,σ,λ} ∈ C^{∞}(K, V_{σ})^{K_{P}} (δ) is independent of λ. Let Γ_{c}(R_{∞}^{δ}) ⊂ Γ_{0}(R_{∞}^{δ}) be the subspace of sections which have compact support with respect to λ. Note that Γ_{0}(R_{∞}^{δ}) and Γ_{c}(R_{∞}^{δ}) are U(g)^{K}-modules.

Using ext⊗id we identify Γ_{0}(R_{∞}^{δ}) with a subspace of ⊕_{(P,σ)∈S(δ)}S̄(ıa_{P}^{∗} , L^{2}(F_{P} , V_{σ})^{K_{P}})⊗ C^{∞}(K, V_{σ})^{K_{P}} (δ). Let Ψ ∈ ⊕_{(P,σ)∈S(δ)}S̄(ıa_{P}^{∗} , L^{2}(F_{P} , V_{σ})^{K_{P}}) ⊗ C^{∞}(K, V_{σ})^{K_{P}} (δ). If (ext ⊗ id)Ψ ∈ Γ_{0}(R_{∞}^{δ}), then Ψ satisfies the compatibility conditions

Ψ_{Q,σ^{s},λ^{s}} = S_{λ}(s) ⊗ J_{λ}(s)(Ψ_{P,σ,λ})

for all (P, σ), (Q, τ) ∈ S(δ), s ∈ Hom_{S}((P, σ), (Q, τ)), λ ∈ ıa_{P}^{∗}. We would like to know that this condition is also sufficient for (ext ⊗ id)Ψ ∈ Γ_{0}(R_{∞}^{δ}).

On Γ_{0}(R_{∞}^{δ}) we can define a scalar product by

⟨(ext ϕ ⊗ ϕ̂), (ext ϕ′ ⊗ ϕ̂′)⟩ := ∫_{ıa_{P}^{∗}} ⟨ϕ′_{λ}, ϕ_{λ}⟩⟨ϕ̂, ϕ̂′⟩p_{P,σ}(dλ) .

where ϕ ⊗ ϕ̂, ϕ′ ⊗ ϕ̂′ ∈ S̄(ıa_{P}^{∗} , L^{2}(F_{P} , V_{σ})^{K_{P}}) ⊗ C^{∞}(K, V_{σ})^{K_{P}} (δ).

Definition 4.2 We define the U(g)^{K}-module intertwining operator V : Γ_{0}(R_{∞}^{δ}) → C^{∞}(Γ\G)(δ) by

V (ext ϕ ⊗ ϕ̂)(g) := ∫_{ıa_{P}} ⟨π^{σ,λ}(g)^{−1}ext ϕ_{λ}, ϕ̂⟩p_{P,σ}(dλ) ,
ϕ ⊗ ϕ̂ ∈ S̄(ıa_{P}^{∗} , L^{2}(F_{P} , V_{σ})^{K_{P}}) ⊗ C^{∞}(K, V_{σ})^{K_{P}} (δ) .

We can write V in the following more intiutive way. Let

Φ_{λ} = ext ϕ_{λ} ⊗ ϕ̂_{λ} ∈ ^{Γ}Hom(H_{P,∞}^{σ,λ}, H_{P}^{σ,λ}(δ)) ⊂ Hom(H_{P,∞}^{σ,λ}, H_{P,∞}^{σ,λ}) ,

then we have

V (ext ϕ ⊗ ϕ̂)(g) = ∫_{ıa_{P}} Tr π^{σ,λ}(g)Φ_{λ}p_{P,σ}(dλ) .

The estimate of Lemma 3.1 shows that (ext∘p_{∗}⊗id)Γ_{0}(H)(δ) ⊂ Γ_{0}(R_{∞}^{δ}). By L^{2}(G)_{d}(δ) we denote the subspace of L^{2}(G)(δ) spanned by the matrix coefficients of the discrete series representations. Let L^{2}(G)_{c}(δ) be its orthogonal complement. We consider another push-down p_{∗} : C^{∞}(G)_{c}(δ) → C_{c}^{∞}(Γ\G)(δ) which is defined by p_{∗}(f)(Γg) := ∑_{γ∈Γ} f(γg).

Lemma 4.3 Let f ∈ C_{c}^{∞}(G) ∩ L^{2}(G)_{c}(δ). Then we have the following identity:

V ∘ (ext ∘ p_{∗} ⊗ id)U^{∗}(f) = p_{∗}(f) .

Proof. The sums and integrals below are absolutely convergent by the estimates of Lemma 3.1. This justifies the interchange of summation and integration.

V ∘ (ext ∘ p_{∗} ⊗ id)U^{∗}(f)(g) = ∑_{(P,σ)∈S(δ),P ≠G} ∫_{ıa_{P}^{∗}} ∑_{γ∈Γ} Tr π^{σ,λ}(γg)U^{∗}(f)(P, σ)(λ)p_{P,σ}(dλ)
= ∑_{γ∈Γ} ∑_{(P,σ)∈S(δ),P ≠G} ∫_{ıa_{P}^{∗}} Tr π^{σ,λ}(γg)U^{∗}(f)(P, σ)(λ)p_{P,σ}(dλ)
= ∑_{γ∈Γ}(UU^{∗}(f))(γg)
= p_{∗}(f) .

In the term V ∘ (ext ∘ p_{∗} ⊗ id)U^{∗}(f)(g) we view U^{∗}(f) as an element of

⊕_{(P,σ)∈S}C^{∞}(ıa_{P}^{∗} , C̄^{∞}(K, V_{σ})^{K_{P}} ⊗ C^{∞}(K, V_{σ})^{K_{P}} (δ)

which satisfies further estimates and Weyl invariance conditions. In the remaining computation we consider U^{∗}(f)(P, σ)(λ) as a trace class operator on H_{P}^{σ,λ}. □

We now turn to the definition of the inverse of V. Since Γ ⊂ G is discrete, we can choose a right-K-invariant function χ ∈ C^{∞}(G) with values in [0, 1] such that ∑_{γ∈Γ} χ(γg) ≡ 1. If f ∈ C_{c}^{∞}(Γ\G)(δ), then we define

(V^{∗}f)_{P,σ,λ} := (ext ∘ p_{∗} ⊗ 1) ∫_{G} χ(g)f(g)π^{σ,λ}(g)dg = (ext ∘ p_{∗} ⊗ 1)U^{∗}(χf) ,  (3)

∀(P, σ) ∈ S(δ), P ≠ G, and we set (V^{∗}f)_{G,σ} := 0. Since χf ∈ C_{c}^{∞}(G)(δ) we have U^{∗}(χf) ∈ Γ_{0}(H)(δ) and hence V^{∗}f ∈ Γ_{0}(R_{∞}^{δ}). Equation (3) is just another way of writing the Fourier transform F_{∞}. Indeed we have

Lemma 4.4 V^{∗} = F_{∞}.

Proof. We consider (V^{∗}f)_{P,σ,λ} ∈ ^{Γ}Hom(H_{P,∞}^{σ,λ}, H_{P}^{σ,λ}(δ). Let ϕ ∈ H_{P,∞}^{σ,λ}, ψ ∈ H_{P}^{σ,λ}(δ)). Using the shorthands (V^{∗}f)_{P,σ,λ} =: V^{∗}(f), (F_{∞}f)_{P,σ,λ} = F_{∞}(f) we have

⟨V^{∗}(f)(ϕ), ψ⟩ = ⟨ϕ, ext ∘ p_{∗} ∫_{G} χ(g)f̄(g)π^{σ,λ}(g)ψdg⟩
= ⟨ext ∘ p_{∗}ϕ, ∫_{G} χ(g)f̄(g)π^{σ,λ}(g)ψdg⟩
= ⟨∫_{G} χ(g)f(g)π^{σ,λ}(g)^{−1}ext ∘ p_{∗}ϕdg, ψ⟩
= ⟨F_{∞}(f)(ϕ), ψ⟩ .
□

The following Lemma justifies the notation V^{∗}.

Lemma 4.5 If f ∈ C_{c}^{∞}(Γ\G)(δ) and Φ ∈ Γ_{0}(R_{∞}^{δ}), then

⟨f, V (Φ)⟩ = ⟨V^{∗}(f), Φ⟩ .

Proof. Let Φ := (ext⊗id)ϕ⊗ϕ̂ for some ϕ⊗ϕ̂ ∈ C̄_{c}^{∞}(ıa_{P}^{∗} , L^{2}(F_{P} , V_{σ})^{K_{P}})⊗C^{∞}(K, V_{σ})^{K_{P}} (δ). We further choose a finite orthonormal base {η_{i}} of C^{∞}(K, V_{σ})^{K_{P}} (δ). Then we have

⟨f, V (Φ)⟩ = ∫_{Γ\G} f̄(g)V (Φ)(g)dg
= ∫_{Γ\G} ∫_{ıa_{P}} f̄(g)⟨π^{σ,λ}(g)^{−1}ext ϕ_{λ}, ϕ̂⟩p_{P,σ}(dλ)dg
= ∫_{ıa_{P}} ∫_{Γ\G} f̄(g)⟨π^{σ,λ}(g)^{−1}ext ϕ_{λ}, ϕ̂⟩dgp_{P,σ}(dλ)
= ∫_{ıa_{P}} ⟨∫_{G} χ(g)f̄(g)π^{σ,λ}(g)^{−1}ext ϕ_{λ}dg, ϕ̂⟩p_{P,σ}(dλ)
= ∫_{ıa_{P}} ∑_{i}⟨∫_{G} χ(g)f̄(g)π^{σ,λ}(g)^{−1}ext ϕ_{λ}dg, η_{i}⟩⟨η_{i}, ϕ̂⟩p_{P,σ}(dλ)

= ∫_{ıa_{P}} ∑_{i}⟨ϕ_{λ}, p_{∗} ∫_{G} χ(g)f(g)π^{σ,λ}(g)η_{i}dg⟩⟨η_{i}, ϕ̂⟩p_{P,σ}(dλ)
= ⟨∑_{i} ext ∘ p_{∗} ∫_{G} χ(g)f(g)π^{σ,λ}(g)η_{i} ⊗ η_{i}, ϕ_{λ} ⊗ ϕ̂⟩
= ⟨V^{∗}(f), Φ⟩ .
□

Lemma 4.6 Let f ∈ C_{c}^{∞}(G) ∩ L^{2}(G)_{c}(δ). Then we have the following identity:

V V^{∗}(p_{∗}f) = p_{∗}(f) .

Proof. Let ϕ, ψ ∈ H_{P,∞}^{σ,λ}. Using the shorthand U^{∗}(f)(P, σ)(λ) = U^{∗}(f) we have for any f ∈ C_{c}^{∞}(G)

⟨ext ∘ p_{∗}V^{∗}(p_{∗}f)(ψ), ϕ⟩ = ∑_{γ∈Γ}⟨π^{σ,λ}(γ)V^{∗}(p_{∗}f)(ψ), ϕ⟩
= ∑_{γ∈Γ}⟨π^{σ,λ}(γ) ∫_{G} χ(g) ∑_{h∈Γ} f(hg)π^{σ,λ}(g)ψdg, ϕ⟩
= ∑_{γ∈Γ} ∑_{h∈Γ} ∫_{G}⟨f(g)π^{σ,λ}(γ)χ(hg)π^{σ,λ}(h)π^{σ,λ}(g)ψ, ϕ⟩dg
= ∑_{γ∈Γ}⟨π^{σ,λ}(γ) ∫_{G} f(g)π^{σ,λ}(g)ψdg, ϕ⟩
= ⟨ext ∘ p_{∗}U^{∗}(f)(ψ), ϕ⟩ .

The assertion of the Lemma now follows from 4.3. □

Let Γ_{2}(R_{∞}^{δ}) be the Hilbert space closure of Γ_{0}(R_{∞}^{δ}). Furthermore let L^{2}(Γ\G)_{c}(δ) be the closure of p_{∗}(C_{c}^{∞}(G) ∩ L^{2}(G)_{c}(δ)) and L^{2}(Γ\G)_{d}(δ) be its orthogonal complement in L^{2}(Γ\G)(δ). By Lemma 4.6 the restriction of V^{∗} to the subspace C_{c}^{∞}(G) ∩ L^{2}(Γ\G)_{c}(δ) extends to an unitary embedding onto a subspace of Γ_{2}(R_{∞}^{δ}).

We now study L^{2}(Γ\G)_{d}(δ). Let M_{G}^{2}(δ) ⊂ M_{G}^{2} be the finite set of equivalence classes of discrete series representations containing the K-type δ. For σ ∈ M_{G}^{2}(δ) let I_{σ} ⊂ U(g)^{K} be the annihilator of V_{σ}(δ). We define I(δ) ⊂ U(g)^{K} by I(δ) := ∩_{σ∈M_{G}^{2}(δ)}I_{σ}. Note that C^{∞}(G) ∩ L^{2}(Γ\G)(δ) carries a representation of U(g)^{K}.

Lemma 4.7 If f ∈ C^{∞}(G) ∩ L^{2}(Γ\G)_{d}(δ), then I(δ)f = 0.

Proof. All σ ∈ M_{G}^{2}(δ) are unitary representations. Thus if A ∈ I(δ), then also A^{∗} ∈ I(δ). If h ∈ C_{c}^{∞}(G)(δ), then for any A ∈ I(δ) we have A^{∗}h ∈ L^{2}(G)_{c}(δ). For any f ∈ C^{∞}(G) ∩ L^{2}(Γ\G)(δ) we conclude

⟨Af, p_{∗}(h)⟩ = ⟨f, A^{∗}p_{∗}(h)⟩
= ⟨f, p_{∗}(A^{∗}h)⟩ .

Now p_{∗}C_{c}^{∞}(G)(δ) is dense in L^{2}(Γ\G)(δ). Thus if f ⊥ L^{2}(Γ\G)_{c}(δ), then Af = 0. □

The following Lemma expresses L^{2}(Γ\G)_{d} in terms of matrix coefficients of discrete series representations of G. In order to make a precise statement we need the following fact. If σ is an irreducible representation of G on V_{σ}, and V_{σ}(δ) ≠ 0, then σ is determined by the irreducible representation χ_{σ} of U(g)^{K} on V_{σ}(δ) (see [6], Ch. II).

Lemma 4.8 There is a finite U(g)^{K}-invariant orthogonal decomposition

L^{2}(Γ\G)_{d}(δ) = ⊕_{σ∈M_{G}^{2}(δ)}L^{2}(Γ\G)_{σ}(δ) ,

where U(g)^{K} acts on L^{2}(Γ\G)_{σ}(δ) by the representation χ_{σ}. For all σ ∈ M_{G}^{2}(δ) we have

dim L^{2}(Γ\G)_{σ}(δ) = ∞ .

Proof. An elliptic regularity argument in combination with Lemma 4.7 shows that L^{2}(Γ\G)_{d}(δ) ∩ C^{∞}(G) = L^{2}(Γ\G)_{d}(δ). We have I(δ)L^{2}(Γ\G)_{d}(δ) = 0. The representations χ_{σ}, σ ∈ M_{G}^{2}(δ), are irreducible and pairwise non-equivalent. Recall the definition I(δ) := ∩_{σ∈M_{G}^{2}(δ)}I_{σ} of the ideal I(δ) ⊂ U(g)^{K}. In particular, I(δ) ⊂ U(g)^{K} is a cofinite. The decomposition of L^{2}(Γ\G)_{d}(δ) according to the irreducible representations χ_{σ}, σ ∈ M_{G}^{2}(δ), now follows. It is easy to check that this decomposition is orthogonal.

It remains to show that dim L^{2}(Γ\G)_{σ_{0}}(δ) = ∞ for any σ_{0} ∈ M̂_{G}^{2}(δ). We fix A_{0} ∈ ∩_{σ∈M̂_{G}^{2}(δ),σ≠σ_{0}}I_{σ} with A_{0}^{∗} = A_{0} and A_{0} ∉ I_{σ_{0}}. There are a spectral decompositions of L^{2}(Γ\G)(δ) with respect to the abelian algebras Z(g) and Z(g)(A_{0}). Let c ⊂ g be a Cartan subalgebra and W_{c} be the Weyl group of (g, c). Then by the Harish-Chandra isomorphism the space of characters of Z(g) can be identified with X := ^{W_{c}}c_{C}^{∗}. There is a measurable field of Hilbert spaces {H_{χ}}_{χ∈X}, a measure p(dχ) on X, and a unitary equivalence

S : L^{2}(Γ\G)(δ) →^{∼} ∫_{X} H_{χ}p(dλ)

such that SAS^{∗} is the multiplication by the function X ∋ χ → χ(A) ∈ C, A ∈ Z(g). If f ∈ L^{2}(Γ\G)_{c}(δ), then f = V V^{∗}(f). It follows that S(f) is supported on the projection X_{prin} to X of a locally finite set of affine subspaces of c_{C}^{∗} with associated vector spaces contained in ıc^{∗} (the set of infinitesimal characters of unitary principal series representations of G which contain the K-type δ).

If f ∈ L^{2}(Γ\G)(δ)_{d}, then S(f) is supported in the real discrete subset X_{dis} of the infinitesimal characters of discrete series representations of G. It is known that X_{dis} ∩ X_{prin} = ∅.

Let χ ∈ X_{dis}. Then H_{χ} is invariant with respect to Z(g)(A_{0}). In order to show that σ_{0} occurs on L^{2}(Γ\G)_{d} with infinite multiplicity it is sufficient to show that p({χ}) ≠ 0 and the range of A_{0} on H_{χ} is infinite dimensional. Note that the spectrum of A_{0} on H_{χ} is finite. We show that p({χ}) ≠ 0 and that there is an eigenvalue μ ≠ 0 of A_{0} on H_{χ} of infinite multiplicity by constructing a Weyl sequence.

The assumption A(δ) implies A[P] for some proper parabolic subgroup P. Let P_{0} be a minimal parabolic subgroup of G contained in P. Since the fibration G/P_{0} → G/P has compact fibres we can assume A[P_{0}]. Let F_{P_{0}} ⊂ Ω_{P_{0}} be a fundamental domain. In the union of positive Weyl chambers of G/K corresponding to F_{P_{0}} we can find an infinite sequence of balls B(x_{n}, n) ⊂ G/K, n ∈ N, of radius n with center x_{n} such that γ ∈ Γ and γB(x_{n}, n) ∩ B(x_{m}, m) ≠ ∅ implies that γ = 1 and n = m. We choose a sequence of cut-off functions χ_{n} ∈ C_{c}^{∞}(B(x_{n}, n)) with χ_{n}(x) = 1 if dist(x, x_{n}) ≤ n/2. Let U(g)^{>} be the subspace of all elements without constant term. We further require that for any A ∈ U(g)^{>}

sup_{g∈G} |χ_{n}(gA)| < C(A)/n .

The multiplicity of σ_{0} in L^{2}(G) is infinite. Thus we can choose an infinite orthonormal sequence ϕ_{i} ∈ L^{2}(G)(δ) such that A_{0}ϕ_{i} = μϕ_{i} for some μ ≠ 0 and L_{g}ϕ_{i} ⊥ ϕ_{j} for all g ∈ G, i ≠ j.

For any prime p and m ∈ N we choose l(p, m) ∈ N such that there exists g_{p,m} ∈ G with ∥L_{g_{p,m}}ϕ_{p^{m}}∥_{L^{2}(B(x_{p^{l(p,m)}},p^{l(p,m)}/2))} ≥ 1−1/p^{m}. We now define ψ_{p,m} := p_{∗}χ_{p^{l(p,m)}}L_{g_{p,m}}ϕ_{p^{m}}. Then for two primes p ≠ q

⟨ψ_{p,m}, ψ_{q,n}⟩ ≤ \frac{2}{max(p, q)}

while ∥ψ_{p,m}∥ ≥ 1 − 1/p^{m}. Elliptic regularity implies that for any B ∈ U(g)

∥BL_{g_{p,m}}ϕ_{p^{m}}∥_{L^{2}(B(x_{p^{l(p,m)}} ,p^{l(p,m)})\B(x_{p^{l(p,m)}} ,p^{l(p,m)}/2))} ≤ C(B)1/p^{m} .

For any A ∈ Z(g) we thus have ∥(A − χ(A))ψ_{p,m}∥ ≤ C(A)/p^{m}. Moreover, ∥(A_{0} − μ)ψ_{p,m}∥ ≤ C/p^{m}. It follows that ψ_{p,m} is a Weyl sequence for the character χ of Z(g) and the eigenvalue μ of A_{0}. This finishes the proof of the infinite multiplicity of σ_{0} in L^{2}(Γ\G)_{d}. □

Theorem 4.9 Let δ ∈ K̂ be a K-type such that A(δ) holds. Then we have an orthogonal decomposition

L^{2}(Γ\G)(δ) = L^{2}(Γ\G)_{c}(δ) ⊕ L^{2}(Γ\G)_{d}(δ) .

The Fourier transform F_{∞} provides an isometry of L^{2}(Γ\G)_{c}(δ) onto a subspace of Γ_{2}(R_{∞}^{δ}). Moreover,

L^{2}(Γ\G)_{d}(δ) = ⊕_{σ∈M_{G}^{2}(δ)}L^{2}(Γ\G)_{σ}(δ)

is a finite orthogonal decomposition into infinite dimensional U(g)^{K}-invariant subspaces, where U(g)^{K} acts on L^{2}(Γ\G)_{σ}(δ) by χ_{σ}.

We would like to know that if f ∈ C_{c}^{∞}(Γ\G)(δ), and f_{c} is its component in L^{2}(Γ\G)_{c}(δ), then F_{∞}(f) = F_{∞}(f_{c}). Furthermore, we would like to know that F_{∞} maps onto Γ_{2}(R_{∞}^{δ}).

5 Schottky-construction

In this section we consider a Schottky construction of free discrete subgroups of G. We start by giving an overview on the contents of the present section.

First we will study the dynamics of cyclic subgroups on the generalized flag varieties. Let P ⊂ G be a proper parabolic subgroup such that P_{0} ⊂ P. Let V ∈ m_{0}, H ∈ a_{0}^{+}, X := V + H ∈ g, and γ := exp(X) ∈ G. Then we study the flow on Z_{P} given by R ∋ t ↦ exp(tX) ∈ G. It turns out that this flow is a Morse flow associated to a Morse function f_{X} and an appropriately choosen K-invariant Riemannian metric on Z_{P}. We will define an attracting set A and an repelling set B such that the cyclic group Γ :=< γ > acts cocompactly on Z_{P} \ (A ∪ B). We will use f_{X} in order to construct a fundamental domain F for this action. In order to verify Axiom 1.1 for Schottky groups we will study the volume and ball contraction properties of Γ.

Next we discuss the Schottky construction of free subgroups generated by two elements. Let γ_{1}, γ_{2} be two elements of G conjugated to elements of M_{0}A_{0} with regular A_{0}-parts (obtained like γ above). Then there are subsets A_{i}, B_{i}, F_{i}, i = 1, 2, associated to γ_{i} as explained above. We consider the following condition.

Axiom 5.1 Z_{P} \ F_{1} ⊂ F_{2} and Z_{P} \ F_{2} ⊂ F_{1}.

If 5.1 is satisfied, then Γ :=< γ_{1}, γ_{2} > is a free discrete subgroup of G acting cocompactly on an open subset Ω_{P} ⊂ Z_{P}. Condition 5.1 keeps being satisfied if we replace γ_{i} by γ_{i}^{m} for any m ∈ N. Let Γ_{m} :=< γ_{1}^{m}, γ_{2}^{m} >, Ω_{P}^{m} be its domain of discontinuity and Λ_{P}^{m} := Z_{P} \Ω_{P}^{m}. We will estimate the Hausdorff-dimension dim_{H}(Λ_{P}^{m}) from above. In particular, if m is sufficiently large, then we will show that dim_{H}(Λ_{P}^{m}) < dim(Z_{P}). This implies that Ω_{P}^{m} is dense and has full measure.

The natural question is now how to construct pairs γ_{1}, γ_{2} ∈ G satisfying 5.1. To this end we consider the following weaker condition.

Axiom 5.2 (A_{1} ∪ B_{1}) ∩ (A_{2} ∪ B_{2}) = ∅.

If γ_{1}, γ_{2} satisfy 5.2, then for n_{1}, n_{2} sufficiently large γ_{1}^{n_{1}}, γ_{2}^{n_{2}} satisfy 5.1. Thus we have to find examples of pairs γ_{1}, γ_{2} satisfying 5.2. In order to construct such examples we consider γ = exp(X) as above. Then we will formulate conditions on γ ∈ P ⊂ G which imply that for a generic g ∈ G in a neighbourhood of 1 the pair γ_{1} := γ, γ_{2} := gγg^{−1} satifies 5.2. We will present examples where these contitions hold.

Using the volume contraction and Hausdorff dimension estimates mentioned above we will show that if γ_{1}, γ_{2} satify 5.1 and an additional genericity hypothesis, then for n_{1}, n_{2} sufficiently large the group Γ :=< γ_{1}^{n_{1}}, γ_{2}^{n_{2}} > satisfies 1.1 for [P], i.e. A([P]). Though it is easy to extend our arguments to free subgroups generated by a finite number of elements γ_{1}, …, γ_{r} we restrict ourselves for simplicity reasons to the case r = 2.

Let P = M_{P} A_{P} N_{P} be the θ-stable Langlands decomposition of a proper parabolic subgroup P ⊂ G and p = m_{P} ⊕ a_{P} ⊕ n_{P} be the corresponding decomposition of the Lie algebra p of P. Furthermore, let c := t_{P} ⊕ a_{P} be a Cartan algebra of g such that t is a Cartan algebra of m. There exists a complex finite-dimensional representation (q, F) of G with a highest weight μ ∈ c^{∗} such that μ_{|t} = 0 and μ_{|a_{P}} ∈ (a_{P}^{∗})^{+}, i.e. μ = ∑_{α∈Σ_{P}} c_{α}α, c_{α} > 0 for all α ∈ Σ_{P}, where Σ_{P} denotes the system of simple roots of (a_{P} , g). Let G^{d} be a compact form of G to which q extends. On F we choose a G^{d}-invariant scalar product.

Let Sym(F) := {A ∈ End(F) | A^{∗} = A}. Then G^{d} acts on Sym(F) by (g, A) ↦ q(g)Aq(g)^{∗}, and we define a G^{d}-invariant Riemannian metric on Sym(F) by ⟨A, B⟩ := Re Tr AB. There is an embedding P(F) ↪ Sym(F) which associates to the line l ⊂ F the orthogonal projection P_{l} ∈ Sym(F). This embedding is G^{d}-equivariant and induces a G^{d}-invariant Riemannian metric on P(F). Let now e_{μ} ∈ F be a highest weight vector and l_{μ} := Ce_{μ}. Then Z_{P} = G/P ∋ gP ↦ P_{q(g)l_{μ}} defines an G-equivariant embedding Z_{P} ↪ P(F) inducing a K-invariant Riemannian metric on Z_{P}.

Let X = V + H, V ∈ m_{0}, and H ∈ a_{0}^{+}. Then we define the function f_{X} : P(F) → R by f_{X}(l) := Re Tr q(X)P_{l}. We denote the restriction of f_{X} to Z_{P} by the same symbol.

The following lemma is based on ideas we have taken from [5].

Lemma 5.3 The flow R ∋ t ↦ Φ_{t} := exp(tX) ∈ G on Z_{P} is the gradient flow of f_{X} with respect to the Riemannian metric defined above.

Proof. It is sufficient to show that the flow R ∋ t ↦ q(exp(tX)) ∈ G on P(F) is the gradient flow of f_{X}. Let l ∈ P(F). Then P_{t} := q(e^{tX})P_{l}q(e^{−tX}) is a projection onto P_{q(e^{tX})l}, but fails to be orthogonal. The corresponding orthogonal projection can be written as L_{t} := P_{t}P_{t}^{∗}(P_{t}P_{t}^{∗} + Q_{t}^{∗}Q_{t}), where Q_{i} := 1 − P_{t}. One can show that

\frac{d}{dt}_{|t=0}L_{t} = \{q(X), P_{l}\} − 2P_{l}q(X)P_{l} ,

where {A, B} := AB + BA. We consider the functions α, β : Sym(F) → R given by α(A) := ReTr q(X)A^{2} and β(A) := ReTr q(X)A^{3}. Then we have

dα(A)(B) = \frac{1}{2}ReTr(\{q(X), A\} + \{q(X), A\}^{∗})B
dβ(A)(B) = \frac{1}{2}ReTr((\{q(X), A^{2}\} + Aq(X)A) + (\{q(X), A^{2}\} + Aq(X)A)^{∗})B .

Using this and the definition of the Riemannian metric on Sym(F) we can write

\frac{d}{dt}_{|t=0}L_{t} = 3gradα(P_{l}) − 2gradβ(P_{l}) = gradα(P_{l}) ,

where we have used the fact that P_{l}^{2} = P_{l} and its consequence α_{|P(F)} = β_{|P(F)}. □

Let Δ_{0} be the system of roots of (a_{0}, g) and Δ_{0}^{P} := {α ∈ Δ_{0} | α_{|a_{P}} = 0}. Let W_{P} ⊂ W be the subgroup generated by reflections at the walls corresponding to the roots in Δ_{0}^{P}. We choose a set of representatives W ⊂ N_{K}(M_{0}) of the classes in W/W_{P}. Then we have a Bruhat decomposition G = ∪_{w∈W}N_{0}wP. Since H ∈ a_{0} is regular the set {wP | w ∈ W} is exactly the set of fixed points of the flow generated by X and thus also the set of critical points of f_{X}. We define W_{A} := {w ∈ W | ρ_{P}^{w}(H) ≥ 0} and set A := ∪_{w∈W_{A}}N_{0}wP. Let N̄_{0} = θ(N_{0}). Then we have another decomposition G = ∪_{w∈W} N̄_{0}wP. We define W_{B} := {w ∈ W | ρ_{P}^{w}(H) < 0} and B := ∪_{w∈W_{B}} N̄_{0}wP. Then W_{A} ∩ W_{B} = ∅ and W_{A} ∪ W_{B} = W. It follows that A ∩ B = ∅ and A ∪ B contains all fixed points of the flow. Let r ∈ R be such that f_{X}(B) < r < f_{X}(A). Then r is a regular value of f_{X} and {f_{X} = r} is a cross section for the flow generated by X. Let γ := exp(X). We define a fundamental domain for the action of Γ :=< γ > on Z_{P} \ (A ∪ B) by

F := {exp(tX)z | z ∈ Z_{P} , f_{X}(z) = r, t ∈ [0, 1)} = γ{f < r} \ {f < r} .

Let r_{1} := max{f_{X}(exp(tX)z) | z ∈ Z_{P} , f_{X}(z) = r, t ∈ [0, 1]}. Then {r ≤ f_{X} ≤ r_{1}} is compact and covers F. Since B = ∩_{n∈Z}γ^{n}{f < r} and A = Z_{P} \ ∪_{n∈Z}γ^{n}{f < r}, the action of Γ on Z_{P} \ (A ∪ B) is cocompact.

We now discuss the volume contraction properties of Γ. We define d := min_{w∈W} |ρ_{P}^{w}(H)|.

Lemma 5.4 Let V′ ⊂ Z_{P} \ (A ∪ B) be any compact subset and V ⊂ V′. Then there is a constant C < ∞ depending on V′ such that

vol(γ^{m}V) ≤ Cvol(V′)e^{−2md} , ∀m ∈ Z .  (4)

Proof. We have

vol(γ^{m}V) = ∫_{V} a(γ^{−m}k)^{−2ρ_{P}} dk ,

where we identify V with the subset {k ∈ K | kP ∈ V } ⊂ K and dk is the normalized Haar measure of K. Since V′ ∩ A = ∅ we have V′ ∩ N_{0}wP = V_{w}wP for all w ∉ W_{A}, where V_{w} ⊂ N_{0} are certain compact subsets. We have for n ∈ V_{w}

a(γ^{−m}κ(nw))^{−2ρ_{P}} = a(exp(−mH)nwn(nw)^{−1}a(nw)^{−1})^{−2ρ_{P}}
= a(nw)^{2ρ_{P}} e^{2mρ_{P}^{w}(H)}a(n^{exp(−mH)}w)^{−2ρ_{P}} .

Now a(nw)^{2ρ_{P}} and a(n^{exp(−mH)}w)^{−2ρ_{P}} are uniformly bounded for all n ∈ V_{w}, m ∈ N. Moreover 2mρ_{P}^{w}(H) ≤ −2md. Thus there exists C < ∞ only depending on V′ such that for all m ∈ N, n ∈ V_{w}

a(γ^{−m}κ(nw))^{−2ρ_{P}} ≤ Ce^{−2md} .  (5)

In a similar way using the decomposition G = ∪_{w∈W} N̄_{0}wP one can show (4) for all m ∈ −N. □

We now fix an arbitrary Riemannian metric on Z_{P} and study how Γ acts on metric balls. Let g = ∑_{α∈Δ_{0}} g^{α} ⊕m_{0} ⊕a_{0} be the root-space decomposition. We define for w ∈ W the nilpotent subgroup N(w) ⊂ N by N(w) := exp(span{X ∈ g^{α} | α ∈ Δ_{0}(w)}), where Δ_{0}(w) := {α ∈ Δ_{0} | α > 0, α_{|a_{P}^{+}}^{w^{−1}} < 0}. Then one can check that N_{0}wP = N(w)wP. Analogously we define Δ̄_{0}(w) := {α ∈ Δ_{0} |α < 0, α_{|a_{P}^{+}}^{w^{−1}} < 0} and N̄_{0}(w) := exp(span{X ∈ g^{α} | α ∈ Δ̄_{0}(w)}) such that N̄_{0}wP = N̄(w)wP. We define e := − max_{α∈Δ_{P}} α(H) and recall that d := max_{w∈W} |ρ_{P}^{w}(H)|. Let B̃ be any open neighbourhood of B.

Lemma 5.5 There are constants C_{4}, C_{5} depending only on B̃ such that for any metric ball D ⊂ Z_{P} \B̃ of radius r the set γ^{m}D can be covered by C_{4}e^{−m(d+dim(Z_{P})e)} balls of radius C_{5}e^{m}r.

Proof. We use the Bruhat decomposition of Z_{P} in order to linearize the action of γ on Z_{P}. Note that ∪_{w∈W} N̄_{0}wP = Z_{P} is a cell decomposition of Z_{P}. We want to cover Z_{P} \ B̃ by a union of open sets of the type U(w)Ū(w)wP, where w ∈ W_{A}, and U(w), Ū(w) are precompact subsets of N(w), N̄(w) which are either open and invariant under contractions and M_{0}, or reduce to a point. We require that the map ϕ_{w} : U(w)^{γ} × Ū(w) → Z_{P} given by ϕ_{w}(u, ū) = uūwP is a diffeomorphism onto its image. We construct these sets by induction on the dimension of N̄(w).

We let s ∈ W be the unique maximal element characterized by dim N̄(s) = 0. Let Ū(s) := {0} and U(s) ⊂ N_{0}(s) be an open, M_{0}- and contraction invariant, precompact subset.

We now describe the induction step. Assume that we already have constructed U(w), Ū(w) for w ∈ W_{<}. Put W_{≥} := W \ W_{<}. Let w ∈ W_{≥} be such that dim N̄(w) is minimal. Since the N̄_{0} orbits in Z_{P} form a cell decomposition, we can find an open, M_{0}- and contraction invariant, precompact subset Ū(w) such that

N̄(w)wP ⊂ ∪_{w∈W_{<}}U(w)Ū(w)wP ∪ Ū(w)wP .

Let n(w) denote the Lie-algebra of N(w). The restriction of fundamental vector fields of n(w) to the orbit N̄(w)wP gives an identification of N̄(w)wP × n(w) with the normal bundle to this orbit. Since Ū(w) is precompact there is an open, M_{0}- and contraction invariant, precompact subset U(w) ⊂ N(w) such that the map ϕ_{w} : U(w)^{γ_{1}} × Ū(w) → Z_{P} is a diffeomorphism onto an open neighbourhood of Ū(w)wP. Since U(w) ⊂ U(w)^{γ} this completes the induction step.

We can assume that Z_{P} \ B̃ ⊂ ∪_{w∈W_{a}}U(w)Ū(w)wP. We choose M_{0}-invariant scalar products on n, n̄. By restriction we obtain scalar products on n(w), n̄(w). Let u(w) := log(U(w)), ū(w) := log(Ū(w)), and let Φ_{w} : Ad(γ)u(w) × ū(w) → Z_{P} be the composition of ϕ_{w} with the corresponding exponential maps. Then Φ_{w} is a quasi isometry for all w ∈ W.

In the following we describe a procedure P roc. We consider a subset D ⊂ Z_{P} \ B̃ that can be covered by a ball of radius r. Then P roc produces a decomposition D = D_{0}∪D\D_{0} and a ball cover of γ^{m_{1}}D_{0}, where 1 < m_{1} ≤ m.

Let 1 < m_{1} ≤ m be the minimum number for which there exists w_{0} ∈ W_{A} such that D ∩ U(w_{0})Ū(w_{0})w_{0}P ≠ ∅ and γ^{m_{1}}D ∩ U(w_{0})^{γ}Ū(w_{0})w_{0}P = ∅. If this condition is never satified, then we set w_{0} = id and m_{1} := m. Since U(w_{0}) is dilatation invariant, γ^{l}D ∩ U(w_{0})Ū(w_{0})w_{0}P = ∅ for all m ≥ l > m_{1}.

Let D_{0} := D ∩ U(w_{0})Ū(w_{0})w_{0}P. Then γ^{m_{1}}D_{0} ⊂ U(w_{0})^{γ}Ū(w)wP. There is a constant C < ∞ independend of m and D such that Φ_{w_{0}}^{−1}(D_{0}) ⊂ ×_{α∈Δ(w_{0})∪Δ̄(w_{0})}B_{w_{0},α,s}, where B_{w_{0},α,s} ⊂ g^{α} is a ball of radius s = Cr. We have γ^{m_{1}}D_{0} ⊂ Φ_{w_{0}}(×_{α∈Δ(w_{0})∪Δ̄(w_{0})}Ad(γ^{m_{1}})B_{w_{0},α,s}). Now Ad(γ^{m_{1}})B_{w_{0},α,s} is a ball of radius e^{m_{1}α(H)}s. We cover each ball Ad(γ^{m_{1}})B_{w_{0},α,s} by balls of radius e^{m_{1}e}s. It follows that there are constants C_{1}, C_{2} which are independend of m and D such that Φ_{w_{0}}(γ^{m_{1}}D_{0}) can be covered by C_{1} \frac{e^{m_{1}f}}{e^{m_{1} dim(Z_{P})e}} balls of radius C_{2}e^{m_{1}e}r, where f := max_{w∈W_{A}} ∑_{α∈Δ(w)∪Δ̄(w)} α(H). Note that f < −d. Finally we can cover γ^{m_{1}}D_{0} by the same number of balls of radius C_{3}e^{m_{1}e}r, where C_{3} is independend of m and D. This completes the description of the procedure P roc applied to D.

Let now D ⊂ Z_{P} \ B̃ be a metric ball of radius r. In the first step we apply P roc to D. In the second we apply P roc to D \ D_{0} and to the balls of the cover of γ^{m_{1}}D_{0} given by P roc. We carry on applying P roc at most ♯W_{a}(♯W_{a} − 1)/2-times. There are constants C_{4}, C_{5} which are independent of m, D such that γ^{m}D is covered by C_{4}e^{−m(d+dim(Z_{P})e)} balls of radius C_{5}e^{me}r. □

This finishes our investigation of cyclic subgroups. Let now γ_{i} ∈ G be conjugated to exp(V_{i} + H_{i}) with V_{i} ∈ m_{0} and H_{i} ∈ a_{0}^{+}. Then we have d_{1} := min_{w∈W} |ρ_{P}^{w}(H_{1})|, d_{2} := min_{w∈W} |ρ_{P}^{w}(H_{2})|. Let A_{i},B_{i}, F_{i}, be the attracting sets, repelling sets, and the fundamental domains for the groups < γ_{i} >, i = 1, 2. We assume that the pair γ_{1}, γ_{2} satisfies 5.1. After a generic small deformation of H we can assume in addition that

Axiom 5.6 d = min(d_{1}, d_{2}) > 0.

Then Γ :=< γ_{1}, γ_{2} > is a free, discrete subgroup of G acting cocompactly on an open subset Ω_{P} of Z_{P}. A fundamental domain for Γ is e.g. F := F_{1} ∩ F_{2}. Let C = min(C_{1}, C_{2}), where C_{i} are the constants given in Lemma 5.4 for γ_{i} and F_{i} in the place of γ and V′.

Lemma 5.7 If 2d > log(C) and α > \frac{log(4)}{2d−log(C)}, then ∑_{γ∈Γ} vol(γF)^{α} < ∞.

Proof. Let ∥.∥ : Γ → N_{0} be the word metric defined by

∥γ_{1}^{n_{1}}γ_{2}^{n_{2}} … γ_{1}^{n_{2r−1}}γ_{2}^{n_{2r}}∥ = ∑_{i=1}^{r} |n_{r}| .

If γ ∈ Γ, then γ_{2}γF ⊂ F_{1} and γ_{1}F ⊂ F_{2}. We repeatedly apply Lemma 5.4 with F_{i} in the place of V′. Let γ = γ_{1}^{n_{1}}γ_{2}^{n_{2}} … γ_{1}^{n_{2r−1}}γ_{2}^{n_{2r}}. Then

vol(γ_{1}^{n_{1}}γ_{2}^{n_{2}} … γ_{1}^{n_{2r−1}}γ_{2}^{n_{2r}}F) ≤ Ce^{−2d|n_{1}|}vol(γ_{2}^{n_{2}} … γ_{1}^{n_{2r−1}}γ_{2}^{n_{2r}}F)
≤ C^{2}e^{−2d(|n_{1}|+|n_{2}|)}vol(γ_{1}^{n_{3}} … γ_{1}^{n_{2r−1}}γ_{2}^{n_{2r}}F)
⋮
≤ C^{∥γ∥}e^{−2d∥γ∥}

Since ♯{γ ∈ Γ | ∥γ∥ = n} = 4^{n}, n ∈ N, we have

∑_{γ∈Γ} vol(γF)^{α} ≤ ∑_{n≥0} 4^{n}C^{αn}e^{−2dαn} .

The assertion of the Lemma now follows. □

If we replace γ_{i} by γ_{i}^{n} for n ∈ N, then 5.1 is still satisfied, but d is replaced by nd. Thus if we choose n large enough, then we can satisfy the condition for the convergence of

∑_{γ∈Γ} vol(γF)^{1/2} ,

i.e. a part of condition A([P]).

Let γ_{1}, γ_{2} be as above and e_{i} := − max_{α∈Δ_{P}} α(H_{i}), We define e := min(e_{1}, e_{2}). For m ∈ N we consider the group Γ_{m} :=< γ_{1}^{m}, γ_{2}^{m} >. Let Ω_{P}^{m} be its domain of discontinuity and Λ_{P}^{m} := Z_{P} \ Ω_{P}^{m}.

Lemma 5.8 As m → ∞ the Hausdorff dimension of Λ_{P}^{m} satisfies

dim_{H}(Λ_{P}^{m}) ≤ dim(Z_{P}) + \frac{d}{e} + o(1/m) .

In particular, for m sufficiently large we have dim_{H}(Λ_{P}^{m}) < dim(Z_{P}). In this case Ω_{P}^{m} is dense and has full measure.

Proof. We must count the number of balls of radius r needed to cover Λ_{P}^{m} as r → 0. Here we can take the balls with respect to any Riemannian metric on Z_{P}. We will employ the action of the group Γ^{m} in order to produce from a given cover G_{n} of Λ_{P}^{m} by balls of radius r_{n} a new cover G_{n+1} by balls of radius r_{n+1} such that r_{n} → 0 as n → ∞.

To be more precise let Ã_{i} and B̃_{i} be open neighbourhoods of A_{i}, B_{i}, i = 1, 2 such that Λ_{P}^{m} ⊂ Ã_{1} ∪ Ã_{2} ∪ B̃_{1} ∪ B̃_{2} for all m ∈ N. We can further assume that γ_{1} maps Ã_{2} ∪ B̃_{2} ∪ Ã_{1} into Ã_{1} and similar conditions for γ_{1}^{−1} and γ_{2}^{±1}.

Assume that we have constructed the cover G_{n} by balls of radius r_{n}. Applying γ_{i}^{±m}, i = 1, 2, to this cover we obtain a new cover G̃_{n+1}. When compared with balls the elements of G̃_{n+1} may, of course, look very deformed. If ♯G_{n}, ♯G̃_{n+1} denote the number of elements of the covers, then it is easy to check that

♯G̃_{n+1} = 3♯G_{n} .

We now cover the elements of G̃_{n+1} by balls using Lemma 5.5. Let C_{4,±i} and C_{5,±i} be the constants obtained in this Lemma for γ = γ_{i}^{±1} and C_{4} := max C_{4,±i} and C_{5} := max C_{5,±i}. Then each element of G̃_{n+1} can be covered by C_{4}e^{−m(d+dim(Z_{P})e)} balls of radius r_{n+1} := C_{5}e^{me}r_{n}. This is the cover G_{n+1}. By induction we have

♯G_{n} ≤ C_{4}^{n}3^{n}e^{−nm(d+dim(Z_{P})e)}♯G_{0}, r_{n} = C_{5}^{n}e^{nme}r_{0} .

If m is sufficiently large, then r_{n} → 0 as n → ∞. We have

−\frac{log ♯G_{n}}{log r_{n}} ≤ \frac{mn(dim(Z_{P})e + d) − log ♯G_{0} − n log C_{4} − n log 3}{log r_{0} + n log C_{5} + nme} .

This shows that

dim_{H}(G_{n}) ≤ \frac{m(dim(Z_{P})e + d) − log C_{4} − log 3}{log C_{5} + me}

and the assertion of the Lemma follows. □

Now the problem of constructing examples of non-trivial Γ satisfying A([P]) is reduced to the construction of pairs γ_{1}, γ_{2} satisfying 5.2. Indeed, if γ_{1}, γ_{2} is such an example and m is sufficiently large, then Γ =< γ_{1}^{m}, γ_{2}^{m} > satisfies A([P]) by Lemmas 5.7 and 5.8.

In order to construct a pair γ_{1}, γ_{2} satisfying 5.2 and 5.6 we start with γ_{1} = γ = exp(V + H) such that V ∈ m_{0} and H ∈ a_{0}^{∗} is generic such that d_{1} > 0. We try to define γ_{2} := gγ_{1}g^{−1} for g ∈ G generic in a neighbourhood of 1. Then the pair γ_{1}, γ_{2} automatically satsifies 5.6 since d_{2} = d_{1} > 0. The difficulty is to fulfill 5.2.

In fact there are examples of P ⊂ G where 5.2 can never be fulfilled. Consider e.g. G = SL(3, R) and let P ⊂ G be a maximal parabolic subgroup such that Z_{P} = PR^{2}. If γ_{1} = exp(H) with H ∈ a_{0}^{+}, then the flow generated by X has three fixed points. In this case A_{1} ∪ B_{1} is the union of a hyperplane and a point. If γ_{2} is another element which is conjugated to exp(H′), H_{2} ∈ a_{0}^{+}, then A_{2} ∪B_{2} is the union of another pair of a hyperplane and a point. Since any two hyperplanes of PR^{2} have a non-trivial intersection, condition 5.2 can not be satisfied. Using the classification of projective surfaces one can show that there is no discrete subgroup of SL(3, R) with the following property: There exists an open subset Ω ⊂ PR^{2} of full measure on which Γ acts freely an cocompactly (We learned this observation in a conversation with M. Karpovich).

We now discuss a sufficient condition for performing the Schottky construction. This condition depends on the triple γ ∈ P ⊂ G. In Section 6 we will verify this condition for γ ∈ P_{0} ⊂ SL(3, R). In the same section we discuss triples γ ∈ P ⊂ SL(4, R) on which one can perform the Schottky group construction but the condition below does not hold.

Let A, B be the attracting and repelling set for γ.

Axiom 5.9 There exists β^{±} ∈ Δ_{0}, ±β^{±} < 0, such that β^{−} ∉ Δ_{0} \ {β^{−}} + N_{0}Δ̄_{0}(w), for all w ∈ W_{B}, β^{+} ∉ Δ_{0} \ {β^{+}} + N_{0}Δ_{0}(w) for all w ∈ W_{A}, and such that (β^{+})^{w^{−1}} < 0, ∀w ∈ W_{A}, and (β^{−})^{w^{−1}} < 0, ∀w ∈ W_{B}.

Lemma 5.10 If γ ⊂ P ⊂ G satisfies 5.9, then for a generic g ∈ G in a neighbourhood of 1 the pair γ_{1} := γ, γ_{2} := gγ_{2}g^{−1} satisfies 5.2.

Proof. If g ∈ G and γ_{2} = gγg^{−1}, then A_{2} = gA, B_{2} = gB. Let U ∈ g and U^{∗} ∈ C^{∞}(Z_{P} , T Z_{P}) be the associated fundamental vector field on Z_{P}. If U^{∗} is transverse to A∪B, then for s ∈ R, |s| sufficiently small and g := exp(sU) we have g(A∪B)∩(A∪B) = ∅.

We decompose U = ∑_{α∈Δ_{0}} U_{α} + U_{0}, where U_{α} ∈ g^{α} and U_{0} ∈ m_{0} ⊕ a_{0}. Let β^{±} be the roots, the existence of which is asserted in 5.9. We claim that if U_{β^{±}} ≠ 0, then U^{∗} is transverse to A ∪ B. We consider U^{∗} at the point nw ∈ N(w)w ⊂ A. Then U^{∗}(nw) = (L_{n})_{∗}(Ad(n^{−1})U)^{∗}(w). Since β^{+} ∉ Δ_{0} \ {β^{+}} + N_{0}Δ_{0}(w) we have a decomposition Ad(n^{−1})U = U_{β^{−}} + ∑_{α∈Δ_{0}\{β^{+}}} Ũ_{α} + Ũ_{0}, where Ũ_{α} ∈ g^{α} and Ũ_{0} ∈ m_{0} ⊕ a_{0}. It follows that (Ad(n^{−1})U)^{∗}(w) is transverse to A. Thus U^{∗}(nw) = (L_{n})_{∗}(Ad(n^{−1})U)^{∗}(w) is transverse to A. Analogously one shows that U^{∗} is transverse to B. □

6 Examples

Let G be a semisimple group of real rank one. Then there is a unique association class of proper parabolic subgroups [P_{0}]. Let P_{0} = M_{0}A_{0}N_{0} be a Langlands decomposition and a_{0} be the Lie algebra of A_{0}. Then dim(a_{0}) = 1. We fix H ∈ a_{0}^{+}. The Weyl group is W = Z_{2} = {1, −1}. We obtain W_{A} = {1} and W_{B} = {−1}. There is an element α ∈ a_{0}^{∗} such that Δ_{0} = {±α} or Δ_{0} = {±α, ±2α}. We set β^{±} := ∓α or β^{±} := ∓2α, respectively. It is easy to check that 5.9 holds and thus Schottky groups exist (this is of course a well known fact). It follows that G admits subgroups satisfying A([P_{0}]) and A(δ) for all δ ∈ K̂.

Let not H be a semisimple group of real rank one. Let P_{0} ⊂ H be a proper parabolic subgroup. We consider h_{1}, h_{2} ∈ H satisfying 5.1. Then γ_{1} := (h_{1}, h_{1}), γ_{2} := (h_{2}, h_{2}) ∈ G satisfy 5.1 for the association classes of maximal proper parabolic subgroups P_{0} × H and H × P_{0} of G. Since 5.1 is invariant under small deformations of γ_{1}, γ_{2} we can construct Zariski dense subgroups of G satisfying A([P_{0} × H]), A([H × P_{0}]). By Lemma 1.3 these groups also satisfy A([P_{0} ×P_{0}]), thus A([Q]) for all association classes of proper parabolic subgroups of G. As a consequence these groups satisfy A(δ) for all δ ∈ K̂. Up to our knowledge the examples above are the only ones satisfying A(δ) for all δ ∈ K̂. Are there simple groups of real rank ≥ 2 admitting discrete subgroups satisfying A(δ) for all δ ∈ K̂?

Now we consider G := SL(3, R). As already indicated our construction of Schottky groups fails for P ⊂ G maximal. Thus consider the minimal parabolic subgroup P_{0} formed by the upper triangular matrices. Let P_{0} = M_{0}A_{0}N_{0} be a Langlands decomposition such that A_{0} = {diag(a_{1}, a_{2}, a_{3}) | a_{1}a_{2}a_{3} = 1}. Then a_{0} = {diag(t_{1}, t_{2}, t_{3}) | t_{1} + t_{2} + t_{3} = 0}. Let e_{i} ∈ a_{0}^{∗} be defined by e_{i}(diag(t_{1}, t_{2}, t_{3})) = t_{i}, i = 1, 2, 3. Then Δ_{0} = {e_{1} − e_{2}, e_{1} − e_{3}, e_{2} − e_{3}, e_{3} − e_{1}, e_{3} − e_{2}, e_{2} − e_{1}}. The first three roots in that list are positive, the remaining roots negative. We have ρ_{0} = e_{1} − e_{3}. The Weyl group W is isomorphic to the permutation group S_{3}. If σ ∈ W, then diag(t_{1}, t_{2}, t_{3})^{σ} = diag(t_{σ(1)}, t_{σ(2)}, t_{σ(3)}). We put H = diag(1, 0, −1) ∈ a_{0}^{+}. Then it is easy to check that W_{A} = {(123), (132), (213)}, and W_{B} = {(231), (321), (312)}. We have e = −2 and d = 1. If we take β^{+} := e_{3} − e_{1} and β^{−} := e_{1} − e_{3}, then 5.9 holds. It follows that Schottky groups Γ exist for P_{0}. Let Γ_{m} ⊂ Γ be the subgroup generated by the m’th powers of the generators of Γ and Λ_{P_{0}}^{m} be its ”limit set”. Then the estimate of the Hausdorff dimension of Λ_{P_{0}}^{m} reads

dim_{H}(Λ_{P_{0}}^{m}) ≤ 3 − 1/2 + o(m^{−1}) .

One would need dim_{H}(Λ_{P_{0}}^{m}) < 2 in order to obtain an open dense domain of discontinuity Ω_{P} for the Schottky group Γ_{m} acting on Z_{P} for P ⊂ G maximal.

We conclude that SL(3, R) admits subgroups satisfying A([P_{0}]) and thus A(1), where 1 ∈ K̂ is the trivial K-type. It seems reasonable to conjecture that SL(3, R) does not admit subgroups satisfying A([P]) for P a maximal proper parabolic subgroup.

We now consider G := SL(4, R). In this case condition 5.9 can not be used in order to construct Schottky groups for certain maximal proper parabolic subgroups of G. Therefore we prefer a more geometric argument. Let {e_{1}, e_{2}, e_{3}, e_{4}} be the standard base of R^{4}. By [e_{1}, e_{2}] we will denote the linear subspace of R^{4} generated by e_{1}, e_{2}, etc. A minimal parabolic subgroup P_{0} ⊂ G is given as the stabilizer of the complete flag [e_{1}] ⊂ [e_{1}, e_{2}] ⊂ [e_{1}, e_{2}, e_{3}]. There are three intermediate parabolic subgroups P_{i}, i = 1, 2, 3 which contain P_{0}. Let P_{1} be the stabilizer of the flag [e_{1}] ⊂ [e_{1}, e_{2}], P_{2} be the stabilizer of [e_{1}] ⊂ [e_{1}, e_{2}, e_{3}] and P_{3} be the stabilizer of [e_{1}, e_{2}] ⊂ [e_{1}, e_{2}, e_{3}]. We have maximal parabolic subgroups P_{12}, P_{13}, P_{23} being the stabilizers of [e_{1}], [e_{1}, e_{2}] and [e_{1}, e_{2}, e_{3}]. Then Z_{P_{12}} ≅ PR^{4}, Z_{P_{23}} ≅ P(R^{4})^{∗}, and Z_{P_{13}} = G_{2,4}(R). Let γ = diag(a_{1}, a_{2}, a_{3}, a_{4}) be such that a_{1} > a_{2} > a_{3} > a_{4} > 0 and a_{1}a_{2}a_{3}a_{4} = 1. We first consider the dynamics of γ on Z_{P_{12}} ≅ PR^{4}. There are four fixed points [e_{1}], [e_{2}], [e_{3}], [e_{4}], where the first is stable, the second and third are hyperbolic, and the last one is instable. We let A := [e_{1}, e_{2}] ⊂ PR^{4}, B := [e_{3}, e_{4}] ⊂ PR^{4}. Let g ∈ G be a generic element. Then g(A ∪ B) ∩ (A ∪ B) = ∅ and 5.2 holds. Thus Schottky groups exist for P_{12} ⊂ G. In a similar manner we can argue for P_{23}. Since P_{0} ⊂ P_{1}, P_{2} ⊂ P_{12} and P_{0} ⊂ P_{2}, P_{3} ⊂ P_{23} by Lemma 1.3 there are discrete subgroups Γ of SL(4, R) satisfying A([Q]) for all proper parabolic subgroups Q ∉ [P_{13}].

Next we show what goes wrong for P_{13}. γ has 6 fixed points

[e_{1}, e_{2}], [e_{1}, e_{3}], [e_{1}, e_{4}], [e_{2}, e_{3}], [e_{2}, e_{4}], [e_{3}, e_{4}]

on G_{2,4}(R), where the first is stable, the next two are hyperbolic of index one, the following two are hyperbolic of index two, and the last one is instable. We are forced to put A := {F ⊂ R^{4} | dim(F) = 2, F ⊂ [e_{1}, e_{2}, e_{3}]} and B := {F ⊂ R^{4} | dim(F) = 2, e_{4} ∈ F }. One can check that for any g ∈ G we have gA ∩ A ≠ 0, gB ∩ B ≠ 0. It seems again reasonable to conjecture that there are no subgroups of SL(4, R) satisfying A([P_{13}]).

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