Journal of Lie Theory Volume 13 (2003) 465–479 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Invariant Control Sets on Flag Manifolds and Ideal Boundaries of Symmetric Spaces ^{∗}

M. Firer and O. G. do Rocio

Communicated by J. D. Lawson

Abstract. Let G be a semisimple real Lie group of non-compact type, K a maximal compact subgroup and S ⊆ G a semigroup with nonempty interior. We consider the ideal boundary ∂_{∞}(G/K) of the associated symmetric space and the flag manifolds G/P_{Θ} . We prove that the asymptotic image ∂_{∞}(Sx_{0}) ⊆ ∂_{∞}(G/K), where x_{0} ∈ G/K is any given point, is the maximal invariant control set of S in ∂_{∞}(G/K). Moreover there is a surjective projection π: ∂_{∞}(Sx_{0}) → ⋃_{Θ⊆Σ} C_{Θ} , where C_{Θ} is the maximal invariant control set for the

action of S in the flag manifold G/P_{Θ} , with P_{Θ} a parabolic subgroup. The points that project over C_{Θ} are exactly the points of type Θ in ∂_{∞} (Sx_{0}) (in the sense of the type of a cell in a Tits Building). Key words: semigroups, semi-simple Lie groups, control sets, ideal boundary. AMS 1991 subject classification: Primary 20M20, 93B29; Secondary 22E46.

1. Introduction

The concept of invariant control sets (i.c.s.) of a semigroup was first introduced by Arnold and Kliemann [1]. We consider the special instance where G is a semisimple real Lie group of non-compact type with finite center and S ⊆ G a semigroup with non-empty interior. If P ⊆ G is a parabolic subgroup, the homogeneous space G/P is a compact manifold (a generalized flag manifolds). The study of invariant control sets for the left action of S on G/P has been systematically used and developed by San Martin [7], [8]. One of the basic results in this context is the existence of a unique i.c.s., whose set of transitivity is given by the points fixed by elements in the interior of S.

From another side, we have the concept of ideal boundary ∂_{∞} (X) of a Hadamard manifold X, that was first introduced by Eberlein and Oneil [3] as a way to compactify Hadamard manifolds. Later, the special case of a symmetric space X = G/K (G is again a semisimple real Lie group of non-compact type with finite center and K a maximal compact subgroup) was exploited by M. Gromov

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag [2] in the study of many important results, such as Margulis’ Lemma and Mostow’s Rigidity Theorem.

∗ This research was partially supported by CAPES, an Institituion of the Brazilian Government, PROCAD no. 0186/00-7.

In this article we determine the i.c.s. of a semigroup S ⊆ G in the ideal boundary ∂_{∞} (G/K): it is just the ideal boundary ∂_{∞} (Sx_{0}), the set of points in the ideal boundary ∂_{∞} (G/K) that belong to the closure on any orbit Sx_{0}, where x_{0} is an arbitrary point of the symmetric space G/K (Theorem 4.4). Moreover, we consider a minimal parabolic subgroup P ⊆ G and the set {P_{Θ}|Θ ⊆ Σ} of parabolic subgroups of G containing P (here Σ is a simple root system determined by P). Then, to each such Θ there is a flag manifold G/P_{Θ} and a (unique) i.c.s. C_{Θ} ⊆ G/P_{Θ}. All those i.c.s. are incorporated in ∂_{∞} (Sx_{0}) (Theorem 4.3) in the sense that there is a surjective projection π : ∂_{∞} (Sx_{0}) → ⋃_{Θ⊆Σ} C_{Θ}.

2. Basic constructions

Let X be a symmetric space of non-compact type. We let G = Isom^{0} (X) be the identity component of the isometry group of X and K the stabilizer (in G) of a point x_{0} ∈ X. Then X = G/K, G is a real semi-simple Lie group and K a maximal compact subgroup of G. The choice of the base point is immaterial, since their stabilizers are conjugate in G. In this section we introduce the main concepts and notations concerning semisimple Lie algebras and groups and associated symmetric spaces. The standard reference for this section is [5].

2.1. Lie algebra structure.

Since G is semi-simple the Cartan-Killing form

B (X, Y) = Tr (adX ∘ adY)

is a non-degenerate bilinear form on 𝔤×𝔤, where 𝔤 is the Lie algebra of G. If we denote by 𝔨 the Lie algebra of K and by 𝔵 its orthogonal complement we get a Cartan decomposition 𝔤 = 𝔨⊕𝔵 (direct sum), with [𝔨,𝔨] ⊆ 𝔨, [𝔵,𝔵] ⊆ 𝔨 and [𝔨,𝔵] ⊆ 𝔵. We get another Cartan decompositions if we consider another maximal compact subgroup K′ ⊂ G. All such subgroups are conjugate and so are their algebras: 𝔨′ = e^{ad(X)} (𝔨) , for some X ∈ 𝔤.

A Cartan involution of 𝔤 is an automorphism ν : 𝔤 ⟶ 𝔤 such that

ν (X_{𝔨} + X_{𝔵}) = X_{𝔨} − X_{𝔵},

where X_{𝔨}+X_{𝔵} is the decomposition of X relative to a given Cartan decomposition of 𝔤. The quadratic form

⟨X, Y ⟩ = −B (X, ν (Y))

is a positive definite quadratic form on 𝔤 invariant under the action of Ad (K). We choose (and fix) a maximal abelian subalgebra 𝔞 ⊂ 𝔵. The real

rank rank^{ℝ} (𝔤) is the dimension of 𝔞. The rank does not depend on the choice neither of 𝔵 nor of 𝔞. The root space decomposition of 𝔤 is given by

𝔤 = 𝔤_{0} ⊕ ∑_{λ∈Λ} 𝔤_{λ}

where λ ∈ Hom (𝔞, ℝ) ,

𝔤_{λ} = {Y ∈ 𝔤| [H, Y] = λ (H) Y, for all H ∈ 𝔞}

and

Λ = {λ ∈ Hom (𝔞, ℝ) |𝔤_{λ} ≠ {0}} .

The λ’s in Λ are called roots of 𝔤 and each 𝔤_{λ} a root subspace. Each root λ ∈ Λ determines a hyperplane H_{λ} = {H ∈ 𝔞|λ (H) = {0}}. Each component of

𝔞 \ ⋃_{λ∈Λ} H_{λ}

is said to be an open Weyl chamber. A Weyl chamber is the closure of an open Weyl Chamber. A Weyl chamber \overline{𝔞}^{+} determines a set of positive roots

Π^{+} = \{λ ∈ Λ|λ (H) ≥ 0 for every H ∈ \overline{𝔞}^{+}\}

and of negative roots Π^{−} = −Π^{+}.

It also determines a set of simple roots, that is, a linearly independent set Σ = {λ_{1}, λ_{2}, ..., λ_{r}} of positive roots such that every root may be written as a linear combination λ = ∑_{i=1}^{r} m_{i}λ_{i} with all the m_{i} having the same sign: m_{i} ≥ 0 if λ ∈ Π^{+} and m_{i} ≤ 0 if λ ∈ Π^{−}. Geometrically, a root λ is simple (relatively to a given chamber \overline{𝔞}^{+}) if and only if H_{λ} ∩ \overline{𝔞}^{+} has dimension r (𝔤) − 1.

Given a subset Θ ⊂ Σ, the subspace

H_{Θ} = {H ∈ 𝔞|λ (H) = 0 for all λ ∈ Θ}

is an abelian subalgebra of 𝔤 and every abelian subalgebra of maximal rank is conjugate to one of those subspaces. The intersection of H_{Θ} with a closed Weyl chamber \overline{𝔞}^{+} is said to be the Θ-wall \overline{𝔞}_{Θ} of \overline{𝔞}^{+}. In fact, this wall is determined by the simple roots (relatively to 𝔞^{+}) contained in Θ, that is, \overline{𝔞}_{Θ} = \overline{𝔞}_{Θ∩Σ}, where Σ is the set of simple roots of the chamber 𝔞^{+}. An open wall 𝔞_{Θ} is the interior of \overline{𝔞}_{Θ} in 𝔞. We denote by \overline{𝔞}_{Θ}^{+} the intersection H_{Θ} ∩ \overline{𝔞}^{+} and call the closed Θ-Weyl wall of \overline{𝔞}^{+}. The open Θ-Weyl wall of \overline{𝔞}^{+} is the interior 𝔞_{Θ}^{+} of \overline{𝔞}_{Θ}^{+} in H_{Θ}.

A Weyl chamber 𝔞^{+} (alternatively, a set of positive roots Π^{+} or a set of associated simple roots Σ) determines maximal nilpotent subalgebras

𝔫^{±} = ∑_{λ∈Π^{±}} 𝔤_{λ}.

We denote by 𝔪 the centralizer of 𝔞 in 𝔨. A minimal parabolic subalgebra is any algebra conjugate in 𝔤 to

𝔭 = 𝔪 ⊕ 𝔞 ⊕ 𝔫^{+}.

The subalgebras 𝔞, 𝔫^{+} and 𝔪 are determined by the choice of a Weyl chamber 𝔞^{+}. Since all such chambers are conjugate, so are the minimal parabolic subalgebras.

More generally, for a subset Θ ⊆ Σ we denote by 𝔭_{Θ} the parabolic subalgebra

𝔭_{Θ} = 𝔫^{−} (Θ) ⊕ 𝔭.

Here 𝔫^{−} (Θ) stands for the subalgebra spanned by the root spaces 𝔤_{−λ}, for λ ∈ ⟨Θ⟩, where ⟨Θ⟩ is the set of (positive) roots generated by Θ. Particularly, 𝔭_{Ø} = 𝔭 and 𝔭_{Σ} = 𝔤.

An Iwasawa decomposition of 𝔤 is given by 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫^{+}. As in the case of parabolic algebras, all Iwasawa decompositions are conjugate one to the other.

2.2. The structure of the Lie group.

For all objects introduced so far in the Lie algebra 𝔤 we find corresponding objects in the Lie group G and the symmetric space X = G/K. We denote by x_{0} the base point π̃ (Id). The subspace 𝔵 ⊂ 𝔤 given by the Cartan decomposition is identified with the tangent space of X at x_{0} by the map dπ̃|_{𝔵} : 𝔵 → T_{x_{0}}X. Moreover, geodesics in X with initial point x_{0} are defined as η (t) = exp (tY) x_{0}, for some vector Y ∈ 𝔵, with ∥Y ∥ = 1.

By defining A = exp 𝔞, K = exp 𝔨 and N^{+} = exp 𝔫^{+}, we get an Iwasawa decomposition G = KAN^{+}.

Here A is a maximal abelian subgroup and N^{+} a maximal nilpotent subgroup. A flat in X is an isometrically embedded Euclidean space. It can be easily proved that flats in X containing the point x_{0} correspond (by the exponential map) to subalgebras of 𝔞 and their conjugate. So, F = Ax_{0} is a maximal flat in X and every maximal flat in X is of the form F′ = gF = gAx_{0}, with g ∈ G. The rank of a symmetric space is the dimension of a maximal flat and, by the preceding argument, it equals the dimension of A.

The structure of Weyl chambers in 𝔞 carries over to the subgroup A = exp 𝔞 and to the flat F = Ax_{0} ⊂ X: if we denote by 𝔞^{+} a Weyl chamber of 𝔞 and by A^{+} = exp 𝔞^{+} its image in G, we shall call gA^{+}x_{0} a Weyl chamber, to any g ∈ G. The point gx_{0} ∈ gA^{+}x_{0} is called the base point of the chamber. A subalgebra H_{Θ} gives rise to Θ-flats gF_{Θ} := g exp (H_{Θ}) x_{0}. In a similar way, we say that g\overline{A}_{Θ}^{+}x_{0} := g exp (\overline{𝔞}_{Θ}^{+}) x_{0} is the Θ-wall of the chamber gA^{+}x_{0}.

A minimal parabolic subalgebra 𝔭 = 𝔪 ⊕ 𝔞 ⊕ 𝔫^{+} determines a minimal parabolic subgroup P = MAN^{+}, where M is the normalizer of A in K. The subgroup P is the normalizer of the algebra 𝔭 via the adjoint action of G:

P = {g ∈ G|Ad (g) 𝔭 = 𝔭}.

Similarly, a parabolic subalgebra 𝔭_{Θ} = 𝔫^{−} (Θ) ⊕ 𝔭 determines a parabolic subgroup

P_{Θ} = {g ∈ G|Ad (g) 𝔭_{Θ} = 𝔭_{Θ}}.

The Weyl group of G is the quotient W = M/M′, where M′ := Z_{K} (A) is the centralizer of A in K. It is a finite group that acts simply transitively on the set of Weyl chambers of 𝔞, respecting the incidence relation of walls.

Each parabolic subgroup determines a (compact) flag manifold 𝔹_{Θ} = G/P_{Θ} that is realize as the set {Ad (g) 𝔭_{Θ}|g ∈ G}. Since Cartan subalgebras and subgroups are all conjugate, the same is true for the root systems determined by the groups A and gAg^{−1}. So, if λ is a root determined by 𝔞, then gλ is the root of Ad (g) 𝔞 defined by the formula

gλ (H) = Ad (g) ∘ λ ∘ Ad (g^{−1}) (H) , for all H ∈ Ad (g) 𝔞.

A parabolic subgroup is said to be of type Θ if it is determined by a set of roots of the form g (Θ). By doing so, the flag manifold 𝔹_{Θ} may be viewed as the set of all type Θ parabolic subgroups. In particular, 𝔹 := 𝔹_{Ø} = G/P is the set of all minimal parabolic subgroups.

Parabolic subgroups are partially ordered by inclusion, with P_{Θ_{1}} ⊂ P_{Θ_{2}} if Θ_{1} ⊂ Θ_{2}. Hence there is a natural fibration

π̂_{Θ_{2}}^{Θ_{1}} : 𝔹_{Θ_{1}} → 𝔹_{Θ_{2}}, gP_{Θ_{1}} ↦ gP_{Θ_{2}}

The set of all Weyl chambers and walls is also partially ordered by inclusion, but with respect to the reverse order: \overline{𝔞}_{Θ_{2}} ⊂ \overline{𝔞}_{Θ_{1}} if Θ_{1} ⊂ Θ_{2}.

3. Ideal boundaries and symmetric spaces

The concept of ideal boundary was first introduced by P. Eberlein and B. O’Neill [3]. The approach adopted here is the one found in [2]. Although this concept may be defined for every metric space, this general definition is highly non geometric and give us not much intuition to work with. So, we will restrict ourselves to a geometric definition that holds for a sufficiently wide family of spaces, the so called Hadamard spaces.

3.1. Ideal boundaries of Hadamard spaces.

We start this section with the definition of Comparision Inequalities of Alexandrov and Topogonov, shortly denoted by CAT inequalities:

Let X be a geodesic metric space and X (ϵ) a surface of constant curvature ϵ, that is, a sphere of radius \frac{1}{ϵ} when ϵ > 0, a Euclidean plane when ϵ = 0 or an hyperbolic plane with curvature ϵ when ϵ < 0. Given a geodesic triangle △(x, y, z), with vertices x, y and z in X, we can construct a comparison triangle ˜△´(x̃, ỹ, z̃) in X (ϵ) having sides of the same length as △ (just taking the care that, when ϵ > 0, △ has sides no longer than \frac{π}{2ϵ}). We say that X satisfies inequality CAT (ϵ) if the following condition holds:

For every triangle △(x, y, z) ⊂ X and every point a in the segment \overline{xy} we have that d (z, a) ≤ d_{ϵ} (z̃, ã), where ã is the corresponding point in the segment \overline{x̃ỹ} of the comparison triangle ˜△ and d_{ϵ} is the metric in X (ϵ). We note that Riemannian manifolds with curvature bounded from above by ϵ satisfy CAT(ϵ).

A Hadamard space (manifold) is a simply connected geodesic metric space (manifold) satisfying CAT (0).

We will now define the ideal boundary of a Hadamard space (X , d (·, ·)). Two geodesic rays γ, β : ℝ^{+} ⟶ X are said to be asymptotic if there is

a constant a ≥ 0 such that d (γ (t) , β (t)) ≤ a, for every t ≥ 0. This defines an equivalence relation on the set of all geodesic rays in X. We call the set of equivalence classes of asymptotic geodesic rays the ideal boundary of X. We denote this space by ∂_{∞}X and the equivalence class determined by γ we denote by γ (∞).

Ther is a natural metric on the ideal boundary:

1. Given η, ξ ∈ ∂_{∞}X and x ∈ X , one can prove that there are representative geodesic rays γ, β with γ (0) = β (0) = x and γ (∞) = η, β (∞) = ξ. The function \frac{1}{t}d (γ (t) , β (t)) is a bounded convex function, hence its limit exists

and we define d_{l} (η, ξ) = lim_{t→∞} \frac{1}{t}d (γ (t) , β (t))

If X is a Hadamard differentiable manifold, we can define two more metrics:

2. Given η, ξ ∈ ∂_{∞}X and x ∈ X , we choose as before representatives γ, β with γ (0) = β (0) = x and γ (∞) = η, β (∞) = ξ and define∠_{x} (η, ξ) to be the angle between the geodesics γ and β at x. Then we define the Tits metric by

d_{T} (η, ξ) = sup_{x∈X} ∠_{x} (η, ξ)

3. Since X is simply connected, given any class γ (∞) and any point x_{0} ∈ X there is one and only one geodesic ray β : ℝ^{+} ⟶ X with β (0) = x_{0} and β (∞) = γ (∞) , so that we can identify ∂_{∞}X with the unit tangent sphere and give it the usual metric of a unit sphere. We shall denote this metric by d_{S} (·, ·).

The metrics d_{l} and d_{T} do not only determines the same topology, but are surprisingly related [2] by the same relation we find between the extrinsic and the intrinsic metric of a unit sphere embedded in ℝ^{n}

d_{l} (η, ξ) = 2 sin (\frac{d_{T} (η, ξ)}{2})

while the metric d_{S} usually defines a different topology. These topologies coincide only when X is a Euclidean space. On the other hand, (∂_{∞}X , d_{l}) is a discrete metric space when the curvature of X is bounded from above by a constant C < 0, whereas (∂_{∞}X , d_{S}) is never discrete. The discreetness of (∂_{∞}X , d_{l}) reflects the fact that we can find a geodesic asymptotic to any given pair of geodesic rays.

Let X be a Hadamard manifold. We fix a point x_{0} ∈ X and for a given η ∈ ∂_{∞}X we choose the unique geodesic ray η (s) such that η (0) = x_{0} and η (∞) = η. Given a sequence (x_{n})_{n∈ℕ} of points in X, consider the sequence of geodesic rays (η_{n} (s))_{n=1}^{∞} such that η_{n} (0) = x_{0} and η_{n} (d (x_{0}, x_{n})) = x_{n}. We say that x_{n} converges to η if lim_{n→∞} d (x_{0}, x_{n}) = ∞ and lim_{n→∞} η′_{n} (0) = η′ (0), or equivalently, if

lim_{n→∞} η_{n} (∞) = η

in (∂_{∞}X , d_{S}). This defines a topology on \overline{X} := X ∪ ∂_{∞}X that coincides with the metric topology on X and the sphere metric d_{S} in ∂_{∞}X, and such that ∂_{∞}X is closed. The convergence from points of X to a point in ∂_{∞}X is defined as above. This turns X into a compact topological space.

The rest of this section will be devoted to the study of the ideal boundary of a symmetric space X = G/K of non-compact type and real rank at least 2.

For a subset C ⊂ X, we define its ideal boundary ∂_{∞}C := ∂C ∩∂_{∞}X, where ∂C stands for the usual boundary in \overline{X}. If C is convex, then

∂_{∞}C = {η (∞) |η (s) is a geodesic ray contained in C}.

Since every Weyl chamber B_{g} = g\overline{A}^{+}x_{0} is convex, we have that

g\overline{A}^{+} (∞) = ∂_{∞} (g\overline{A}^{+}x_{0})
= {η (∞) |η (s) = g (exp sX) x_{0}, X ∈ 𝔞^{+}}.

This is called a Weyl chamber at infinity. The Weyl chambers at infinity are either equal or disjoint.

Similar definitions hold also for walls at infinity and flats at infinity, denoted by gA_{Θ}^{+} (∞) and gA (∞) respectively. We will consider only closed chambers g\overline{A}^{+} (∞) and walls at infinity:

g\overline{A}_{Θ}^{+} (∞) = ∂_{∞} (g\overline{A}_{Θ}^{+}x_{0}) = \{η (∞) |η (s) = g (exp sX) x_{0}, X ∈ 𝔞_{Θ}^{+}\} .

3.2. The structure of ideal chambers.

In order to prove our results, we must characterize the ideal points of a symmetric space according to their parabolic type. To put it explicitly, we define a map

π : ∂_{∞}X → ⋃_{Θ⊂Σ} G/P_{Θ},

as follows: each η ∈ ∂_{∞}X is of the form η (∞) with η (s) = (g exp sX) x_{0} with

|X| = 1 and X ∈ ⋃^{∘}_{Θ⊆Σ}𝔞_{Θ}^{+}, where we are considering the open chambers and open walls, so that the union is disjoint. So, we associate to η the parabolic subgroup π (η) := gP_{Θ}g^{−1} (where X ∈ 𝔞_{Θ}^{+}). This association is independent of the choice of g. Alternatively, we could associate to η the (open) Weyl chamber or wall at infinity gA_{Θ_{0}} (∞) that contains it.

We denote by ∂_{∞}^{Θ}X the inverse image π^{−1} (G/P_{Θ}), the set of all Θ-singular geodesic rays. We note that π^{−1} (G/P_{Ø}) = ∂_{∞}^{Ø}X is an open and dense subset of ∂_{∞}X, whenever we consider either the Tits metric or the spherical metric. Also,

⋃^{∘}_{λ∈Σ}∂_{∞}^{{λ}}X is open and dense in ∂_{∞}X \∂_{∞}^{Ø}X. In the same way, we find that

⋃^{∘}\begin{matrix}Θ⊆Σ \\ |Θ|=k \end{matrix}∂_{∞}^{Θ}X is open and dense in ∂_{∞}X \ (⋃^{∘}\begin{matrix}Φ⊆Σ \\ |Φ|<k \end{matrix}∂_{∞}^{Φ}X ), where |Θ| is just the

cardinality of Θ and k ≤ r (𝔤). Again, this fact is independent of the metric topology we work with. The projection π : ∂_{∞}X → ⋃_{Θ⊆Σ} G/P_{Θ} splits as a set of projections

π^{Θ} : ∂_{∞}^{Θ}X → G/P_{Θ}, Θ ⊆ Σ.

Remark 3.1. This structure is actually a geometric realization of a spherical Tits building. Shortly, the set of apartments is

A = {gA (∞) |g ∈ G} = {All flats at infinity},

and the chambers and walls are given by

Δ = \{g\overline{A}_{Θ}^{+} (∞) |g ∈ G, Θ ⊆ Σ\} = \{All chambers and walls at infinity\}.

Since for every pair of chambers or walls at infinity there is a flat at infinity that contains both of them, the adjacency relation is just the usual one defined in the flats at infinity. More details about this structure may be found in [2, Appendix

4. Semigroups and invariant control sets

Let X = G/L be a homogeneous manifold. We denote respectively by clD and intD the closure and the interior of a subset D (of X or G, to be clearly understood from the context). A set S of diffeomorphisms of M is a semigroup if the composition of elements of S (with possible restrictions of domains) is still in S. An invariant control set for S (an S -i.c.s.) is a subset C ⊆ X with int(C) ≠ Ø satisfying the conditions:

(i) For all x ∈ C, cl Sx = cl C,

(ii) C is maximal with property (i).

For the simplicity of the presentation, we assume that G is a semisimple Lie group of non-compact type and S a subsemigroup of G, even if some of the results do not depend on the semisimplicity of G. Regarding the control sets in a compact homogeneous space X = G/L we have the following:

Proposition 4.1. 7, Proposition 2.1 Let X = G/L be a compact homogeneous space and S a subsemigroup of G with intS ≠ Ø. Let C ⊂ X be an S -i.c.s and let C_{0} = (int S) C. Then:

(i) C_{0} = int (Sx) for all x ∈ C_{0}.

(ii) SC_{0} ⊂ C_{0} = Sy = (int S) y for all y ∈ C_{0}.

(iii) C_{0} = {x ∈ C|∃g ∈ int S with gx = x}.

(iv) C_{0} = {x ∈ C|∃g ∈ int S with g^{−1}x ∈ C}.

(v) cl C_{0} = C.

Because of property (ii) in the proposition above, C_{0} is called the set of transitivity of C.

The product MA is a closed subgroup of G. The homogeneous space G/MA may be seen as the set of Weyl chambers in either 𝔤 or Weyl chambers in G with base point at the identity. Alternatively, it may be seen as the choice of a Weyl chamber decomposition in each of the flats gAx_{0} of X. Each Weyl chamber b = gMA is conjugate to the base chamber A^{+}: b = gA^{+}g^{−1}.

We assume throughout the rest of the paper that S has non-empty interior. Then, it has a unique S invariant control set C [7, Theorem 3.1]. If we put

Δ = {b = gA^{+}g^{−1} ∈ G/MA|b ∩ int S ≠ Ø} ,

we have the following:

Theorem 4.2. [8, Theorem 3.1] Let C be the unique S -i.c.s. in G/P and C_{0} be its set of transitivity. Let

p : G/MA → G/MAN^{+}

be the canonical projection. Then

C_{0} = p (Δ) .

4.1. Ideal Boundaries and invariant control sets.

In this section we consider subsemigroups of a semisimple Lie group of non-compact type G. Without loss of generality, we assume that G has finite center. We also assume that S has non-empty interior and show how to construct invariant control sets in the ideal boundary ∂_{∞} (X) of the associated symmetric space X = G/K. This is done simply by considering the ideal boundary of an orbit of S in X, as stated in our main Theorem:

Theorem 4.3. Let S be a sub-semigroup of a semisimple Lie group G, with non-empty interior. Consider the boundary of an orbit Sx_{0} in G/K and let D be the ideal boundary ∂_{∞} (Sx_{0}). Then D is the invariant control set of S. Moreover, let C_{Θ} denote the unique S -i.c.s. in G/P_{Θ} and D^{Θ} = D ∩ ∂_{∞}^{Θ} (X). Then,

π^{Θ} (D^{Θ}) = C_{Θ}.

We start proving a particular instance of this theorem, the case Θ = Ø.

Theorem 4.4. Let S be a sub-semigroup of a semisimple Lie group G, with non-empty interior. Let C be the unique S -i.c.s. in G/P = G/P_{Ø}. Consider the boundary of an orbit Sx_{0} in G/K and let D = ∂_{∞} (Sx_{0}) and D^{Ø} = D ∩ ∂_{∞}^{Ø} (X). Then,

π^{Ø} (D^{Ø}) = C.

Actually, we could consider the orbit of any point gx_{0} ∈ X instead of the orbit Sx_{0}:

Proposition 4.5. For any two points gx_{0}, hx_{0} ∈ X, the ideal boundaries of their S -orbits coincide, that is, ∂_{∞} (Sgx_{0}) = ∂_{∞} (Shx_{0}).

Proof. Given η ∈ ∂_{∞} (Sgx_{0}), there is a sequence (s_{i})_{i=1}^{∞} , s_{i} ∈ S such that lim_{i→∞} s_{i}gx_{0} = η, the limit considered in \overline{X} = X ∪ ∂_{∞}x. But

d (s_{i}gx_{0}, s_{i}hx_{0}) = d (gx_{0}, hx_{0})

is bounded, so that lim_{i→∞} s_{i}hx_{0} = η.

We note that D := ∂_{∞} (Sgx_{0}) = cl\overline{X} (Sx_{0}) \ X.

The proof of Theorem 4.4 is obtained from the next two lemmas.

Lemma 4.6. With the notation above defined,

C_{0} ⊆ π^{Ø} (D^{Ø}) .

Proof. Let b̃ ∈ C_{0}. By Theorem 4.2 there is b ∈ Δ such that p (b) = b̃ and an element g ∈ b ∩ intS. We consider an open ball B_{r} (g) ⊂ intS. For each h ∈ b∩B_{r} (g), we have some Y_{h} ∈ 𝔤 with ∥Y_{h}∥ = 1 such that hx_{0} = exp (t_{h}Y_{h}) x_{0}. Then, h^{n} = exp (nt_{h}Y_{h}) ∈ b∩intS for every n ≥ 1 so that h^{n}x_{0} ⊂ Sx_{0}. Moreover, if we put η_{h} (t) = exp (tY_{h}) x_{0}, we find that

lim_{n→∞} h^{n}x_{0} = lim_{n→∞} η_{h} (nt_{h}) = η_{h} (∞) .

Since b may be considered as an open Weyl chamber, we get that η_{h} (∞) ∈ D^{Ø} = D∩∂_{∞}^{Ø} (X) and, from the fact that h^{n} ∈ b we deduce that π^{Ø} (η_{h} (∞)) = p (b) = b̃.

Lemma 4.7. With the notation above defined, let int (D^{Ø}) stands for the intersection of D^{Ø} with the interior of D as a subset of ∂_{∞}X. Then,

π^{Ø} (int (D^{Ø})) ⊆ \overline{C_{0}}.

Proof. First of all we notice that D := ∂_{∞} (Sx_{0}) = ∂_{∞} ((intS) x_{0}). Indeed, if h_{n} is a point in the boundary of S, there is an interior point g_{n} of S which distance (in G, with any left invariant metric) from h_{n} at most 1, so that the sequences h_{n}x_{0} and g_{n}x_{0} have bounded distance in X and one of them converges to an ideal point if and only if the other one converges to the same ideal point.

Let η = η (∞) ∈ int (D^{Ø}), and put η (t) = exp (tY_{0}) x_{0}, with ∥Y_{0}∥ = 1, with Y_{0} ∈ 𝔵. Since η ∈ D, there is a sequence (g_{n})_{n=1}^{∞} , g_{n} ∈ S such that lim_{n→∞} g_{n}x_{0} = η. If we put g_{n} = exp t_{n}Y_{n}, with Y_{n} ∈ 𝔵 and ∥Y_{n}∥, it means that t_{n} → ∞ and the angle θ_{n} = ∢ (Y_{0}, Y_{n}) between Y_{0} and Y_{n} goes to 0. Since ∂_{∞} (Sx_{0}) = ∂_{∞} ((intS) x_{0}), we may assume that g_{n} ∈ intS. The same reasoning used in Lemma 4.7 implies that η_{n} (t) = exp (tY_{n}) x_{0} defines an element η_{n} := η_{n} (∞) such that π^{Ø} (η_{n}) ∈ C_{0}.

On the other hand, since η is an interior point of D, considering the spherical metric in ∂_{∞}X, the sets

B_{θ,r} (η) = {exp (tY) x_{0}|t > r; ∢ (Y_{0}, Y) < θ} , r > 0, θ > 0

constitute a base for the neighborhoods of η in \overline{X}. Since lim_{n→∞} g_{n}x_{0} = η, we find that lim_{n→∞} Y_{n} = Y so that

lim_{n→∞} η_{n} = η_{0}.

But the projection π^{Ø} : ∂_{∞}^{Ø} (X) → G/P is continuous and this implies that π^{Ø} (η) ∈ \overline{C_{0}}.

Now we can prove Theorem 4.4.

Proof. [Proof (of Theorem 4.4)] Since \overline{C_{0}} = C, Lemma 4.6 assures that

\overline{C_{0}} = C ⊆ π^{Ø} (D^{Ø}).

Since π^{Ø} is continuous, we find that π^{Ø} (\overline{A}) ⊆ \overline{π^{Ø} (A)} for every subset A ⊆ ∂_{∞} (X), the closures being taken, respectively, in ∂_{∞} (X) and G/P_{Ø}. So, by Lemma 4.7, we find that

π^{Ø} (\overline{int (D^{Ø})}) = π^{Ø} ((D^{Ø})) ⊆ \overline{\overline{C_{0}}} = \overline{C} = C

and therefore

π^{Ø} (D^{Ø}) = C.

Now we can prove our main theorem:

Proof. [Proof of Theorem 4.3] We consider now another parabolic subgroup P_{Θ}, Θ ⊆ Σ. Again, the compactness of the manifold G/P_{Θ} assures the existence of a unique S -invariant control set C_{Θ} ⊆ G/P_{Θ}. It is known that C_{Θ} = ρ_{Θ} (C), where ρ_{Θ} : G/P → G/P_{Θ} is the natural projection.

The closed Weyl chamber \overline{𝔞}^{+} is a cone generated by a family \{H_{α}^{⊥}\}_{α∈Σ} of unit vectors where β (H_{α}^{⊥}) = 0 whenever β ≠ α. It means that H_{α}^{⊥} is contained in the intersection of all the hyperplanes orthogonal to H_{β} for every root β ≠ α. Then, the open Weyl chamber may be described as

𝔞^{+} = {∑_{α∈Σ} c_{α}H_{α}^{⊥}|c_{α} > 0 for every α ∈ Σ}.

Then, we may define a projection p_{Θ} : 𝔞^{+} → 𝔞_{Θ}^{+} by

p_{Θ} (∑_{α∈Σ} c_{α}H_{α}^{⊥}) = ∑_{α∈Σ\Θ} c_{α}H_{α}^{⊥}.

This projection is not orthogonal (with respect to the Cartan-Killing form) but it is clearly surjective. Moreover, the diagram is clearly commutative.

[figure: ∂_{∞}^{Ø}(X); ∂_{∞}^{Θ}(X); G/P_{Ø}; G/P_{Θ}; π^{Ø}; π^{Θ}; p_{Θ}; ρ_{Θ}]

Diagram 1

Since ∂_{∞}^{Ø} (X) is open and dense in ⋃_{Ø≠Θ⊆Σ} ∂_{∞}^{Θ} (X), the same is true for the ideal boundary of an orbit of the semigroup: ∂_{∞}^{Ø} (Sx_{0}) is open and dense in ∂_{∞} (Sx_{0}) and its boundary is contained in ⋃_{Ø≠Θ⊆Σ} ∂_{∞}^{Θ} (Sx_{0}), so that

∂_{∞}^{Θ} (Sx_{0}) ⊆ p_{Θ} (∂_{∞}^{Ø} (Sx_{0}))

From this and the fact that the Diagram 1 commutes, we find that

π^{Θ} (∂_{∞}^{Θ} (Sx_{0})) ⊆ π^{Θ} (p_{Θ} (∂_{∞}^{Ø} (Sx_{0})))
= ρ_{Θ} ∘ π^{Ø} (∂_{∞}^{Ø} (Sx_{0}))
= ρ_{Θ} (C)
= C_{Θ}.

Moreover, given b̃ ∈ (C_{Θ})_{0} (the set of transitivity of C_{Θ}), by definition there is an element g ∈ int (S) such that gb̃ = b̃. As in the proof of Lemma 4.6, we find that g^{n}x_{0} converges (in \overline{X}) to a point η = η (∞) ∈ ∂_{∞}^{Θ} (Sx_{0}). The theorem follows by taking the appropriates closures, as we did in the proof of Theorem 4.4.

Example 4.8. If G is semisimple Lie group of non-compact type and real rank 1, then the Weyl chambers are one-dimensional, so that their asymptotic images in ∂_{∞}X have dimension 0, that is, they are just points. It follows that the projection π : ∂_{∞}X → G/P is actually a bijection so that the ideal boundary and the Furstenberg boundary may be identified.

As a special case, we consider the group G = SL(2, ℝ) and the semigroup S = SL_{+}(2, ℝ) of matrices with non-negative entries. The symmetric space is just the hyperbolic plane. We consider the Lobatchevsky model, the halfplane ℍ^{2} = {z ∈ ℂ|Im(z) > 0}. Its ideal boundary is just the set {z ∈ ℂ|Im(z) = 0} ∪ {∞}. The group G acts on ℍ^{2} by Möbius transformations

A = (\begin{matrix}a & b \\ c & d \end{matrix}) : z → A(z) := \frac{az + b}{cz + d}

If A ∈ SL_{+}(2, ℝ), the image of the point z = i
\frac{ai + b}{ci + d} = \frac{(ac + bd) + i(ad − bc)}{c^{2} + d^{2}}
= \frac{ac + bd + i}{c^{2} + d^{2}}

has non-negative real part, so that ∂^{−}∞(Si) ⊂ {0 ≤ t ∈ ℝ} ∪ ∞. On the other hand, for any λ ≥ 0, t > 0, the matrices

A_{t} = (\begin{matrix}\frac{1}{t} + λt & λt \\ t & t \end{matrix})

belong to SL_{+}(2, ℝ) and lim_{t→∞}A_{t}(i) = (λ+ \frac{1}{2t^{2}} )+ \frac{i}{2t^{2}} = λ, so that \{0 ≤ t ∈ ℝ\} ⊂ Si The ideal point ∞ is attained as the limit of the orbit of the one-parameter subgroup of hyperbolic isometries

B_{t} = (\begin{matrix}t & 0 \\ 0 & \frac{1}{t} \end{matrix}) .

Example 4.9. We consider the group SL(3, ℝ). The associated symmetric space X = Sl(3, ℝ)/SO(3, ℝ) may be identified with the space of all 3 × 3 symmetric matrices, such that vBv^{T} > 0 for every 0 ≠ v ∈ ℝ^{3}. This identification is obtained from the action of Sl(3, ℝ) on X defined by A·x = AxA^{T}. This space is 5-dimensional and its ideal boundary is thus a 4-dimensional sphere. This boundary may be identified with a set of 2-dimensional strips in ℝ^{3}, as in [2]. Since X = Sl(3, ℝ)/SO(3, ℝ) is a Hadamard manifold, its ideal boundary is determined by the asymptotic image of geodesic rays starting at the point x_{0} = Id_{3×3} (section 3.). The tangent vector η′(0) of a unit speed geodesic ray η : [0, ∞] → X is a symmetric matrix A with trace 0 and norm ∥A∥^{2} = Tr(AA^{T}) = 1. Since A is symmetric, it has 3 eigenvalues a, b and c. The norm 1 and trace 0 conditions imply that a^{2} + b^{2} + c^{2} = 1 and a + b + c = 0. We lose no generality by assuming that a ≥ b ≥ c. Since A is symmetric, it has orthogonal eigenvectors v_{a}, v_{b} and v_{c}. We associate to this ray the strip s_{η} := (ℝv_{a} + [−r, r]v_{b}) ⊂ ℝ^{3}, where r := (b − c)/(a − b) ∈ [0, ∞] equals 0 if b = c and ∞ if a = b. The map η ↦ s_{η} is well defined and a bijection onto the space

Str := {ℝv_{1} + [−r, r]v_{2} ⊂ ℝ^{3}| ∥v_{1}∥ = ∥v_{2}∥ = 1, v_{1} orthogonal to v_{2}, r ∈ [0, ∞]}

of all strips in ℝ^{3}. We note that, {v_{a}, v_{b}, v_{c}} being an orthogonal base, at least one of the vectors {±v_{a}, ±v_{b}, ±v_{c}} is contained in the positive orthant

ℝ_{(+,+,+)}^{3} := {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3}| x_{1}, x_{2}, x_{3} ≥ 0}

We consider now the set

𝔰𝔩_{+}(3, ℝ) := {(a_{ij})_{i,j=1}^{3} ∈ 𝔰𝔩(3, ℝ)| a_{ij} ≥ 0 for i ≠ j}

of all matrices with nonnegative entries (outside the diagonal) and trace 0. It is clearly a closed convex cone with non empty interior. So, its image under the exponential map generates a semigroup S with non empty interior [6]. We note that S is contained (but not equal) to the semigroup SL_{+}(3, ℝ) of the matrices with nonnegative entries.

We claim that the invariant control set of S in ∂_{∞}X, is contained in the set of strips

S_{(+,+,+)} := {(ℝv_{a} + [−r, r]v_{b}) ∈ Str| : v_{a} ∈ ℝ_{(+,+,+)}^{3}}.

By definition, given η ∈ ∂_{∞}(Sx_{0}), there is a sequence A_{n} ∈ S such that A_{n}x_{0} converges to η. We start proving there is an element B ∈ S such that η = lim_{n↦∞} B^{n}x_{0}. First of all we note that, if we repeat terms of the sequence (e.g. considering A_{1}, A_{1}, A_{2}, A_{2}, ...instead of A_{1}, A_{2}, ...) or delete terms of the sequence (i.e., consider subsequences of the given one), we get a sequence A′_{n} such that lim_{n↦∞} A_{n}x_{0} = lim_{n↦∞} A′_{n}x_{0}. However, the first substitution (adding repeated terms) may increase the rate of growth of d(x_{0}, A_{n}x_{0}), while the second substitution may decrease it. So, we lose no generality by assuming that lim_{n↦∞} \frac{1}{n}d(x_{0}, A_{n}x_{0}) = k, for any given k ≥ 0. We consider now a vector X ∈ 𝔰𝔩 (3, ℝ) with ∥X∥ = 1 such that lim_{t↦∞} exp(tX)x_{0} = η. If η ∈ int(∂_{∞}(Sx_{0})), we find that B := exp(t_{0}X) ∈ int(S) for some t_{0} > 0. We let k_{0} be the distance d(x_{0}, Bx_{0}). Since B^{n} = exp(n · (t_{0}X)), we find that lim_{n↦∞} B^{n}x_{0} = lim_{n↦∞} A_{n}x_{0} = η and B^{n} ∈ int(S), for every n ∈ ℕ. It follows that we may consider only elements η ∈ ∂_{∞}(Sx_{0}) such that η = lim_{n↦∞} B^{n}x_{0}, for some element B ∈ int(S). Moreover, since the set of matrices in S with different eigenvalues (the split regular elements) is open and dense in S, we loose no generality by restricting ourselves to matrices B ∈ S with different eigenvalues.

We consider a split regular matrix B ∈ S and let η := lim_{n↦∞} B^{n}x_{0}. Let a > b > c be the eigenvalues of B, with corresponding eigenvectors v_{a}, v_{b} and v_{c}. We want to prove that v_{a} or its opposite −v_{a} is in the orthant ℝ_{(+,+,+)}^{3}.

If we look at the projective space ℙ (ℝ^{3}) over ℝ^{3}, the sequence (B^{n})_{n∈ℕ} determines a quasi-projective transformation in the sense of Goldsheid and Margulis [4]. It follows that every line ℝv ∈ (ℙ (ℝ^{3}) − ℙ (ℝv_{b} + ℝv_{c})) is attracted by the line ℝv_{a}, since a is the greatest eigenvalue, that is,

lim_{n→∞} B^{n} (ℝv) = ℝv_{a}

the limit being taken in ℙ (ℝ^{3}) [4, Corollary 2.4]. However, any matrix B with nonnegative entries leaves the positive orthant invariant (actually, S is the compression semigroup of this orthant [7]). It follows that the eigenvector v_{a} (or its opposite −v_{a}) must be contained in this orthant and so ∂_{∞}(Sx_{0}) ⊆ S_{(+,+,+)}.

We note that the inclusion above is a strict one, that is, not every strip in S_{(+,+,+)} is contained in ∂_{∞}(Sx_{0}). Indeed, given an orthogonal base {v_{a}, v_{b}, v_{c}}, and a, b, c ∈ ℝ, the unique symmetric matrix that has eigenvalues a, b, c with corresponding eigenvectors (relative to a fixed orthogonal base of unit length vectors) {v_{a}, v_{b}, v_{c}} is the matrix

B := V (\begin{matrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}) V^{T}

where V has columns {v_{a}, v_{b}, v_{c}}. If the coordinates of the eigenvectors are given by

v_{a} = (x_{1}, y_{1}, z_{1}), v_{b} = (x_{2}, y_{2}, z_{2}), and v_{c} = (x_{3}, y_{3}, z_{3}),

the condition that B has nonnegative entries outside the diagonal is equivalent to the inequalities

∗ \begin{cases}x_{1}y_{1}a + x_{2}y_{2}b + x_{3}y_{3}c & ≥ & 0 \\ x_{1}z_{1}a + x_{2}z_{2}b + x_{3}z_{3}c & ≥ & 0 \\ y_{1}z_{1}a + y_{2}z_{2}b + y_{3}z_{3}c & ≥ & 0. \end{cases}

Since r = \frac{b−c}{a−b} and a + b + c = 0, this system of equations is equivalent to

∗ ∗ \begin{cases}x_{1}y_{1} (r + 2) + x_{2}y_{2} (r − 1) + x_{3}y_{3} (2r + 1) & ≥ & 0 \\ x_{1}z_{1} (r + 2) + x_{2}z_{2} (r − 1) + x_{3}z_{3} (2r + 1) & ≥ & 0 \\ y_{1}z_{1} (r + 2) + y_{2}z_{2} (r − 1) + y_{3}z_{3} (2r + 1) & ≥ & 0 . \end{cases}

If we take for example the pair of orthogonal vectors v_{a} := \frac{1}{\sqrt{3}}(1, 1, 1) and v_{b} := \frac{1}{\sqrt{14}}(3, −1, −2), then, by direct calculations, we can show that those inequalities are satisfied only for r ≤ \frac{1}{10} , that is, the strip (ℝv_{a} + [−r, r]v_{b}) ∈ ∂_{∞}(Sx_{0}) if and only if r ≤ \frac{1}{10} . So, not every strip in S_{(+,+,+)} is in the ideal boundary of Sx_{0}. However, since the projections π^{Θ} are surjective, we find that, for any choice of unit length orthogonal vectors v_{a}, v_{b}, with v_{a} in the first orthant, the system (∗∗) of inequalities has a solution. Actually, this system is the precise description of the strips attained in ∂_{∞}(Sx_{0}).

Remark 4.10. As we saw in remark 3.1, the ideal boundary ∂_{∞} (X) may be considered, in a canonical way as the geometric realization of a spherical Tits Building. With this structure, each connected component of ∂_{∞}^{Θ} (X) is a cell of type Θ in the building structure. This fact encourages the investigation of invariant control sets semigroups of algebraic groups and (B, N) pairs. Acknowledgement We thanks the referees for the careful correction and many important suggestions.

References

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Marcelo Firer Instituto de Matemática Universidade Estadual de Campinas - UNICAMP Cx. Postal 6065 13.081-970 - Campinas - SP Brasil email: mfirer@ime.unicamp.br Osvaldo do Rocio Centro de Ciências Exatas Universidade Estadual de Maringá - UEM Avenida Colombo, 5790 87020-900 - Maringá - PR Brazil e-mail: rocio@uem.br

Received March 19, 2002 and in final form December 11, 2002