Journal of Lie Theory Volume 8 (1998) 111–128 ◯^{C}1998 Heldermann Verlag

Homogeneous spaces admitting transitive semigroups

Luiz A. B. San Martin^{∗}

Communicated by K. H. Hofmann

Abstract.

Let G be a semi-simple Lie group with finite center and S ⊂ G a semigroup with int S ≠ Ø. A closed subgroup L ⊂ G is said to be S -admissible if S is transitive in G/L. In [10] it was proved that a necessary condition for L to be S -admissible is that its action in B (S) is minimal and contractive where B (S) is the flag manifold associated with S, as in [9]. It is proved here, under an additional assumption, that this condition is also sufficient provided S is a compression semigroup. A subgroup with a finite number of connected components is admissible if and only if its component of the identity is admissible, and if L is a connected admissible group then L is reductive and its semi-simple component E is also admissible. Moreover, E is transitive in B (S) which turns out to be a flag manifold of E.

Key words: semigroups, semi-simple groups, flag manifolds, transitive groups, minimal actions. AMS 1991 subject classification: primary 20M20, 54H15; secondary 93B

1. Introduction

Let G be a connected and noncompact semi-simple Lie group. The problem we address here is that of finding the pairs (S, L) formed by a semigroup S ⊂ G with nonvoid interior and a closed subgroup L of G such that S is transitive in the homogeneous space G/L. In such a pair we say that L is S -admissible. We approach this problem by looking at the actions of S and L in the flag manifolds of G. In [9] one of these flag manifolds, say B (S), was intrinsically attached to S in such a way that the invariant control set for S on B (S) contracts to one point through iterations of regular elements inside the interior of S. Thus it becomes natural to search conditions on L which involve its action on B (S). There is indeed the necessary condition for the transitivity of S on G/L, proved in [10], which ensures that the action of L on B (S) must be minimal and contractive. In this article we prove that these conditions are also sufficient as long as S is a

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag compression semigroup and the reflection θ (L) of L under a Cartan involution of G satisfies the same conditions as L. Once we have this necessary and sufficient conditions for transitivity we proceed to analyze those subgroups satisfying them. As we shall see they impose severe restrictions on L so we get, as a rule, that transitivity of proper semigroups is a rare event. We consider mainly connected subgroups and make the basic assumption that G is a simple group. In this case an application of a result by Vinberg [12], namely that a triangular linear group has a fixed point in any compact invariant subset of the projective space, allow us to show that any admissible connected subgroup L is reductive and is such that its semi-simple part E is noncompact and transitive in B (S). Moreover, E is also admissible and B (S) is a flag manifold of E. So that the connected admissible subgroups are essentially those noncompact semi-simple subgroups of G which have a common flag manifold with G. In particular, if B (S) is the maximal flag manifold then there are no S -admissible proper connected subgroups.

∗Research partially supported by CNPq grant n^{o} 301060/94-0

As to the case where L is not connected we observe first that if it has a finite number of connected components then L is admissible for some S if and only if its identity component L_{0} is admissible so that this case is easily reduced to the connected case. On the other hand, we have little to say about the case where L has an infinite number of components: We remark that the algebraic closure zc (L) of L is also admissible and has a finite number of components so that the semi-simple part of the identity component of zc (L) is also admissible. A consequence of this fact is that the S -admissible subgroups are discrete if G is simple and B (S) is the maximal flag manifold of G. Also we reproduce here a proof for the well known fact in control theory which says that any semigroup with nonvoid interior in G is transitive in G/L if this homogeneous space admits a finite invariant measure. This shows in particular that any lattice in G is admissible for any semigroup with interior points.

2. Preliminaries on semigroups and flags

In this section we recall some results about semigroups in semi-simple Lie groups which will be needed afterwards. Throughout the paper we let G be a noncompact semi-simple Lie group with finite center and denote by 𝔤 its Lie algebra. Let 𝔤 = 𝔨 ⊕ 𝔰 be a Cartan decomposition and denote by θ the corresponding Cartan involution either of 𝔤 or of G. Select a maximal abelian subalgebra 𝔞 ⊂ 𝔰 and let Π be the set of roots of the pair (𝔤, 𝔞). We shall say that H ∈ 𝔞 is regular real provided α (H) ≠ 0 for every root α. More generally, H ∈ 𝔤 is regular real if it is conjugate to a regular real element in 𝔞, or equivalently if it is regular real in some abelian subalgebra conjugate to 𝔞. In a similar way, we say that h ∈ G is regular real if h = exp H for some regular real H ∈ 𝔤.

Choose a simple system of roots Σ ⊂ Π and denote by Π^{+} the corresponding set of positive roots. Let 𝔪 be the centralizer of 𝔞 in 𝔨. The standard minimal parabolic subalgebra of 𝔤 is given by 𝔭 = 𝔪 ⊕ 𝔞 ⊕ 𝔫 where

𝔫 = ∑_{α∈Π^{+}} 𝔤_{α}

is the nilpotent constituent of the Iwasawa decomposition. As usual we denote by K and N the connected subgroups whose Lie algebras are 𝔨 and 𝔫 respectively. The normalizer P of 𝔭 in G is a minimal parabolic subgroup and G/P is the maximal flag manifold of G. It is well known that 𝔭 is the Lie algebra of P.

Given Θ ⊂ Σ let ⟨Θ⟩ be the subset of positive roots generated by Θ and denote by 𝔫^{−} (Θ) the subalgebra spanned by the root spaces 𝔤_{−α}, α ∈ ⟨Θ⟩. We denote by 𝔭_{Θ} the parabolic subalgebra

𝔭_{Θ} = 𝔫^{−} (Θ) ⊕ 𝔭 .

Its normalizer P_{Θ} in G is a parabolic subgroup whose Lie algebra is 𝔭_{Θ}. We put B_{Θ} = G/P_{Θ} for the corresponding flag manifold. If M_{Θ} = P_{Θ} ∩ K then B_{Θ} = K/M_{Θ}.

Recall that if h ∈ G is regular real then it has a finite number of fixed points in B_{Θ}. Each fixed point is hyperbolic and there is just one attractor, i.e., whose stable manifold is open and dense. The decomposition of B_{Θ} into stable manifolds of h is the so called Bruhat decomposition. In fact, the stable manifolds are the orbits of the group N^{−} = exp 𝔫^{−} where 𝔫^{−} is the subalgebra opposed to 𝔫. We note furthermore that the open stable manifold is also given by N_{Θ}^{−}b_{0} where b_{0} = P_{Θ} is the origin in B_{Θ} = G/P_{Θ} and N_{Θ}^{−} = exp 𝔫_{Θ}^{−}. Here

𝔫_{Θ}^{−} = ∑_{α} 𝔤_{α} ,

with the sum extended to the negative roots outside −⟨Θ⟩, is the subalgebra spanned by root spaces complementary to 𝔭_{Θ} in 𝔤.

Denote by W the Weyl group of the pair (𝔤, 𝔞), and by W_{Θ} the subgroup of W generated by the reflections with respect to the roots in Θ ⊂ Σ. In [9, Section 4] it was associated with a semigroup S ⊂ G with int S ≠ Ø a subgroup W (S) ⊂ W which accounts for the number of S -control sets on the maximal boundary B. It was shown that W (S) = W_{Θ_{S}} for some subset Θ_{S} of the simple system of roots. We use the notation B (S) = B_{Θ_{S}}. The main property of B (S) which will be used here is that if C ⊂ B (S) stands for the invariant control set for S then C is contained in the stable manifold of the attractor b in B (S) of any h ∈ int S which is regular real. This implies that C is contractive with respect to h in the sense that h^{k}x → b as k → ∞ for all x ∈ C.

In the sequel we shall use often the notion of contractive sequences in G (see Guivarc’h and Raugi [3]): Let g_{k} be a sequence in G and write its polar decomposition as g_{k} = v_{k}a_{k}u_{k} with v_{k}, u_{k} ∈ K and a_{k} ∈ cl A^{+}. Here A^{+} = exp 𝔞^{+}, where 𝔞^{+} ⊂ 𝔞 is an open Weyl chamber. For a root α ∈ Π and a ∈ exp 𝔞, put ϕ_{α} (a) = exp (α (log a)). The sequence g_{k} is said to be contractive if ϕ_{α} (a_{k}) → 0 as k → +∞ for all negative roots α. Moreover, the sequence is said to be contractive with respect to a flag manifold B_{Θ} if ϕ_{α} (a_{k}) → 0 for every negative root α which is not in the subset −⟨Θ⟩ of roots spanned by −Θ.

In general, given a sequence g_{k} = v_{k}a_{k}u_{k} there exists a subsequence g_{k_{n}} = v_{k_{n}}a_{k_{n}}u_{k_{n}} such that v_{k_{n}} → v and u_{k_{n}} → u in K, and there is a map τ : 𝔫_{Θ}^{−} → 𝔫_{Θ}^{−} such that for every Y ∈ 𝔫_{Θ}^{−}

g_{k_{n}}u^{−1} exp (Y) b_{0} → v exp (τY) b_{0}.  (1)

(see [3] and [10, Prop. 2.5]). The linear map τ is diagonal and its eigenvalues are lim ϕ_{α} (a_{k}) with α a negative root outside −⟨Θ⟩. Because of this the limit in (1) is uniform for Y in a compact subset of 𝔫_{Θ}^{−}. The subsequence is contractive with respect to B_{Θ} if and only if τ = 0. In this case (1) implies that g_{k_{n}}x → vb_{0} for all x ∈ u^{−1}N_{Θ}^{−}b_{0}, that is, for all x of the form x = u^{−1} exp (Y) b_{0} with Y ∈ 𝔫_{Θ}^{−}.

From the limit in (1) we get the following criteria: If g_{k}x → x_{0} for all x in an open subset of B_{Θ} then g_{k} admits a contractive subsequence. In fact, it follows from (1) that τY = 0 for Y in an open subset of 𝔫_{Θ}^{−} so that τ = 0 (c.f. Corollary 2.6 in [10]).

Another fact about contractive sequences we shall need below is that if g_{k} is a contractive sequence with respect to B_{Θ} then for every h, l ∈ G, hg_{k}l admits a contractive subsequence. This follows immediately from Corollary 2.6 in [10] (see also [3, Cor. 2.3]). In fact, if g_{k} is contractive then g_{k}l converges to a point for an open and dense subset. So that the same happens to hg_{k}l and hence this sequence admits a contractive subsequence.

In the sequel we shall say that the action of a subgroup L ⊂ G on a flag manifold B is contractive, or simply that L is contractive on B, if there is a contractive sequence g_{k} ∈ L with respect to B.

Recall that the action of a group L on the topological space X is said to be minimal if every orbit Lx, x ∈ X is dense in X. Equivalently, the action is minimal if there is no proper invariant closed subset. In the sequel we say simply that L is minimal on X if its action on X is minimal.

3. Transitive semigroups

Let S ⊂ G be a semigroup with int S ≠ Ø. A closed subgroup L ⊂ G is said to be S -admissible if S is transitive in G/L. From [10] we have the following necessary conditions for a subgroup to be S -admissible.

Theorem 3.1. Let S ⊂ G be a semigroup with int S ≠ Ø and L ⊂ G a closed subgroup. In order that S is transitive on G/L it is necessary that

1. the action of L on B (S) is minimal, and

2. L admits a contractive sequence with respect to B (S).

The objective of this section is to prove a partial converse of this theorem, namely that the conditions on L are sufficient as far as S is a compression semigroup and the subgroup θ (L) satisfies the same conditions as L, where θ is a Cartan involution of G.

We consider here only those compression semigroups of their invariant control sets in a flag manifold. More precisely, let C be the invariant control set for S in B (S). Then S is the compression semigroup of C if

S = {g ∈ G : gC ⊂ C}.  (2)

We note that such semigroups indeed exist. In fact, let S_{0} be a semigroup with int S_{0} ≠ Ø and denote by C its invariant control set on B (S_{0}). If S is defined by (2) then C is the invariant control set for S because S_{0} ⊂ S. Any maximal semigroup with nonempty interior in G is the compression semigroup of its invariant control set in some minimal flag manifold (see [9]).

We start by proving the following lemma. For its statement we fix an Iwasawa decomposition G = KAN and the minimal parabolic subgroup P = MAN. If Q ⊃ P is a parabolic subgroup we denote by b_{0} the origin in the flag manifold B = G/Q.

Lemma 3.2. Let σ stand for the open Bruhat component containing b_{0}. Take k ∈ K and let C ⊂ kσ be a compact subset. Suppose that g_{i} ∈ G is a sequence such that g_{i}b_{0} → kb_{0}. Then for some subsequence g_{i_{j}} we have that C ⊂ θ (g_{i_{j}} ) σ, j ≥ j_{0}.

Proof. Let g_{i} = u_{i}a_{i}n_{i} stand for the Iwasawa decomposition of g_{i} with u_{i} ∈ K, a_{i} ∈ A and n_{i} ∈ N. Then g_{i}b_{0} = u_{i}b_{0} because b_{0} is invariant under a_{i}n_{i}. Also, θ (g_{i}) = u_{i}a_{1}^{−1}θ (n_{i}), and since σ is invariant under AN^{−}, θ (g_{i}) σ = u_{i}σ. By taking a subsequence we can assume that u_{i} → u ∈ K. Since u_{i}b_{0} → kb_{0} and u_{i}b_{0} → ub_{0} we have that ub_{0} = kb_{0}. Therefore if we write B = K/M_{Θ} then u = km with m ∈ M_{Θ}.

Now, M_{Θ} normalizes N_{Θ}^{−} and fixes b_{0} so that mσ = σ. Therefore if we put v_{i} = u_{i}m^{−1} then v_{i} → k, v_{i}σ = u_{i}σ and kv_{i}^{−1} → 1. On the other hand, the action of K on B is continuous with respect to the compact-open topology on the set of continuous maps of B. Therefore the fact that C ⊂ kσ implies that kv_{i}^{−1}C ⊂ kσ for i big enough. This means that v_{i}^{−1}C ⊂ σ, that is, C ⊂ v_{i}σ = u_{i}σ. Since θ (g_{i}) σ = u_{i}σ, the lemma follows.

The proof that a semigroup is transitive on a homogenous space is simplified by the following device.

Lemma 3.3. Let G be a topological group, L ⊂ G a closed subgroup and S ⊂ G a semigroup with int S ≠ Ø. We have,

1. If S is transitive on G/L then int gSg^{−1} ∩ L ≠ Ø for all g ∈ G.

2. Reciprocally, suppose that G/L is connected. Then S is transitive in G/L if int gSg^{−1} ∩ L is not empty for all g ∈ G.

Proof. See Corollary 2.2 in [10].

As mentioned above we shall prove the converse of Theorem 3.1 under the additional assumption that the action of θ (L) is contractive and minimal. At this regard we note that contractivity of the L-action ensures trivially that the action of θ (L) is contractive. In fact, let g_{n} ∈ L be a contractive sequence and write g_{n} = u_{n}a_{n}v_{n} for its polar decomposition. Then θ (g_{n}^{−1}) = v_{n}^{−1}a_{n}u_{n}^{−1} has the same radial component as g_{n}, so that θ (g_{n}^{−1}) is also contractive.

We can prove now the main result of this section.

Theorem 3.4. Let S ⊂ G be a semigroup with int S ≠ Ø and denote by C its invariant control set on B (S). Suppose that S is the compression semigroup of C, i.e.,

S = {g ∈ G : gC ⊂ C}.

Let L be a closed subgroup and assume that its action on B (S) is contractive. Assume also that both L and θ (L) are minimal on B (S). Then S is transitive on G/L.

Proof. We keep fixed an Iwasawa decomposition of G which gives rise to a standard parabolic subgroup which is assumed to be the isotropy at the origin b_{0} of B (S). The corresponding open Bruhat component N^{−}b_{0} will be denoted by σ. We can assume without loss of generality that C ⊂ σ.

Take g ∈ G. We wish to show that L ∩ int gSg^{−1} ≠ Ø. The fact that S is a compression semigroup implies that

gSg^{−1} = {h ∈ G : hgC ⊂ gC}.

Therefore in order to prove that the above intersection is not trivial it is enough to show that there exists l ∈ L such that lgC ⊂ int gC. In fact, the action of G on B (S) is continuous with respect to the compact-open topology so that l belongs to int (gSg^{−1}) if and only if it maps gC into its interior. This l ∈ L will be provided by a contracting sequence in L.

Let g_{i} = x_{i}a_{i}y_{i} ∈ L with a_{i} ∈ cl A^{+} and x_{i}, y_{i} ∈ K be a contracting sequence such that x_{i} → x and y_{i} → y^{−1} with x, y ∈ K. The limit in (1) implies that g_{i}b → xb_{0} for all b ∈ yσ. Since L is minimal on B (S), we can find h ∈ L such that b_{1} = hxb_{0} belongs to int gC. Hence the contracting sequence l_{i} = hg_{i} ∈ L satisfies l_{i}b → b_{1} for all b ∈ yσ.

On the other hand, write g = kan, k ∈ K, a ∈ A, n ∈ N^{−} for the Iwasawa decomposition of g. Then gσ = kσ and gC ⊂ kσ because C ⊂ σ. Since the action of θ (L) is minimal on B (S), there exists a sequence m_{i} ∈ θ (L) such that m_{i}yb_{0} → kb_{0}. By Lemma 3.2 there exists i big enough such that gC ⊂ θ (m_{i}y) σ. Since θ (m_{i}y) = θ (m_{i}) y we get m ∈ L such that gC ⊂ myσ.

Now, take the contracting sequence h_{i} = l_{i}m^{−1} ∈ L. Then h_{i}b → b_{1} for all b ∈ myσ. In particular, this sequence contracts gC to b_{1}, and the convergence is uniform in gC because this set is compact. But b_{1} ∈ int gC, so that there exists i_{0} such that h_{i_{0}}gC ⊂ int gC. Since h_{i_{0}} ∈ L this shows that L intercepts the interior of the compression semigroup of gC, concluding the proof of the theorem.

This theorem raises the question of whether the minimal action of L implies that the action of θ (L) is also minimal. For the time being we do not know the complete answer to this question. However there are some cases where the answer is in the affirmative. In fact, given a flag manifold G/Q there is the dual flag manifold G/θ (Q) which is diffeomorphic to G/Q through the commutative diagram

[figure: G/Q; G/θ(Q); \overline{θ}; G; G; θ]

where \overline{θ} (gQ) = θ (g) θ (Q). The actions of G on G/Q and G/θ (Q) are interchanged by \overline{θ} through the formula \overline{θ} ∘g = θ (g)∘\overline{θ}. This implies that L is minimal on G/Q if and only if θ (L) is minimal on G/θ (Q). In some cases θ (Q) is conjugate to Q and hence G/θ (Q) coincides with G/Q. In these cases the minimal action of L on G/Q is equivalent to the minimal action of θ (L). Conjugacy of Q and θ (Q) holds for instance if Q is a minimal parabolic subgroup or for any parabolic subgroup if θ is an inner automorphism of G. We recall that if 𝔤 is a real form of a simple complex Lie algebra then θ is an inner automorphism if the diagram of the roots of the pair (𝔤, 𝔞) is not of the type A_{l}, D_{l}, l odd or E_{6}. Hence in a good deal of groups the minimal action of L implies that of θ (L). In addition to these conditions on the flag manifolds, we mention that if L is connected then it will be proved in the next section that L is reductive. This implies that L is invariant under some Cartan involution. Two Cartan involutions θ_{1} and θ_{2} are related by θ_{1} = θ_{2} ∘ C_{g} for some g ∈ G where C_{g} (h) = ghg^{−1}. Hence if the action of L is minimal on G/Q, the same happens to the action of θ (L).

Regarding still the minimal action of θ (L) we note that if L is S -admissible then θ (L) is minimal in B (S). In fact, S is transitive in G/L if and only if S^{−1} is transitive. Now, it follows from the theory in [9] that B (S^{−1}) = G/θ (Q) if B (S) = G/Q. Therefore L is minimal on G/θ (Q), which by the above duality implies that θ (L) is minimal on B (S).

As a final comment about Theorem 3.4 we stress that it ensures that S is transitive in G/L only in case S is the compression semigroup of the subset C. This condition is needed because our method of proof consists in showing that some element of L maps C into its interior. We do not know, however, if the conditions of Theorem 3.4 remain sufficient for more general semigroups with nonvoid interior.

4. Connected subgroups

The conditions of Theorem 3.1 open the way to a close analysis of the admissible subgroups. In this section we start this analysis by reducing the possibilities for the connected subgroups. We do not need here the contractive property of an admissible subgroup, but only its minimal action in some flag manifold. Therefore, along this section B stands for a flag manifold of G and L for a connected Lie subgroup whose action on B is minimal. We do not assume in advance that L is closed in G.

Since we only consider semi-simple groups with finite center, we may assume without loss of generality that the center is trivial, so we are actually working in the identity component of the adjoint group Ad (G). Also, in order to simplify most of the arguments we assume from now on that 𝔤 is simple.

In this context we shall prove below that L is reductive, i.e., its Lie algebra 𝔩 is the direct sum of its center 𝔷 and a semi-simple component, say 𝔢. With the further assumption that L is closed it turns out that its action on B is actually transitive.

The proof that L is reductive uses successively the following lemma which is a consequence of a theorem of Vinberg [12]. In the sequel we say that a nilpotent subalgebra 𝔥 ⊂ 𝔤 is triangular in 𝔤 if the weights of the adjoint representation of 𝔥 in 𝔤 are real.

Lemma 4.1. Let Q be a parabolic subgroup and put B = G/Q. Let also L ⊂ G be a connected subgroup with Lie algebra 𝔩, and suppose that its action on B is minimal. Then {0} is the only nilpotent triangular ideal of 𝔩.

Proof. Let 𝔦 be a nilpotent triangular ideal. The connected subgroup I whose Lie algebra is 𝔦 is normal in L. If we check that I has a fixed point in B then we are done. In fact, if x is fixed by I then any y ∈ Lx is also fixed because I is normal. Since Lx is dense in B this implies that Iz = z for all z ∈ B. But the action of G on B is effective because we are assuming G to be simple and centerless. Therefore I has a fixed point in B if and only if I = {1}, or equivalently, 𝔦 = 0.

The existence of a point fixed by I follows from [12]: Since 𝔦 is triangular the adjoint action of 𝔦 and I on 𝔤 can be represented as a group of triangular matrices. This property is preserved when passing to exterior products of the adjoint representation. Since Q is the normalizer of its Lie algebra, B can be realized as a compact projective G-orbit in some exterior product. Now [12] ensures that any triangular group has a fixed point in a compact invariant subset of a projective space. It follows that I has a fixed point in B, showing the lemma.

From this lemma we get at once that the solvable radical 𝔵 of 𝔩, i.e., the maximal solvable ideal, is abelian. In fact, by the Theorem of Lie on representations of solvable Lie algebras, the derived algebra 𝔵′ is triangular in 𝔤. Hence the lemma implies that 𝔵′ = 0, so that 𝔵 is abelian. Taking into account the Levi decomposition of 𝔩, we conclude:

Lemma 4.2. Let the notations and assumptions be as in Lemma 4.1. Then

𝔩 = 𝔢 ⊕ 𝔳

with 𝔢 semi-simple and 𝔳 and abelian ideal of 𝔩.

Once we have this lemma we proceed to show that 𝔩 is a reductive Lie algebra, or equivalently, that X ∈ 𝔳 commutes with 𝔩. In order to show this we must look in detail at the adjoint representation of 𝔳 in 𝔤, and afterwards at the adjoint representation of 𝔢 in 𝔳.

For X ∈ 𝔳 denote by S_{X} and N_{X} respectively the semi-simple and nilpotent parts of the Jordan decomposition of ad_{𝔤} (X). Since 𝔤 is semi-simple, there are unique X_{S}, X_{N} ∈ 𝔤 such that S_{X} = ad (X_{S}) and N_{X} = ad (X_{N}). We have X = X_{S} + X_{N}, and of course these elements of 𝔤 commute between themselves.

Let s_{𝔳} denote the subspace of 𝔤 spanned by X_{S} with X running through 𝔳. Similarly, denote by n_{𝔳} the subspace spanned by X_{N} with X ∈ 𝔳. We have,

Lemma 4.3. The subspaces s_{𝔳}, n_{𝔳} and s_{𝔳} + n_{𝔳} are abelian subalgebras of 𝔤. Moreover, n_{𝔳} is triangular whereas the elements of s_{𝔳} are semi-simple.

Proof. From the general theory of representations of nilpotent Lie algebras (see e.g. [11]), we can decompose the complexification 𝔤_{ℂ} of 𝔤 into weight spaces for the adjoint representation of 𝔳. For each X ∈ 𝔳 the restriction of ad_{𝔤_{ℂ}} (X) to a weight space has just one eigenvalue. Hence the restriction of S_{X} to a weight space is a scalar matrix. This implies that

ad_{𝔤}[X_{S}, Y_{S}] = ad_{𝔤_{ℂ}}[X_{S}, Y_{S}] = [S_{X}, S_{Y}] = 0

for any pair X, Y ∈ 𝔳. Since 𝔤 is centerless, it follows that [X_{S}, Y_{S}] = 0 so that s_{𝔳} is abelian. Using again the fact that S_{X} is a scalar matrix inside a weight space and taking into account that each weight space is N_{X} -invariant, we get for all X, Y ∈ 𝔳, that [X_{S}, Y_{N}] = [S_{X}, N_{Y}] = 0. Since 𝔳 is abelian, it follows that

0 = [X, Y] = [X_{N}, Y_{N}].

Hence n_{𝔳} is an abelian subalgebra commuting with s_{𝔳}, so that s_{𝔳} + n_{𝔳} is also abelian.

The last statement follows from the fact that N_{X}, X ∈ 𝔳, are simultaneously triangular, and the complexifications of S_{X}, X ∈ 𝔳, are simultaneously diagonalizable.

From the construction of the subspaces, it is clear that

𝔳 ⊂ s_{𝔳} + n_{𝔳}, s_{𝔳} ⊂ 𝔳 + n_{𝔳} and n_{𝔳} ⊂ 𝔳 + s_{𝔳}.  (3)

In order to continue we must look at the bracket relations between these subspaces and the semi-simple component 𝔢 of 𝔩. For this note first that since 𝔳 is an abelian ideal of 𝔩, ad_{𝔩} (X)^{2} = 0 for all X ∈ 𝔳. On the other hand, the linear maps S_{X} and N_{X}, X ∈ 𝔳 are polynomials in ad_{𝔤} (X) so that they leave invariant both 𝔳 and 𝔩. Hence the restriction of S_{X} to 𝔩 is zero. This implies that [𝔢, s_{𝔳}] = 0. In particular, s_{𝔳} is invariant under ad_{𝔤} (𝔢). Combining this with the inclusions (3) we conclude that s_{𝔳} + n_{𝔳} is an invariant subspace under the adjoint representation of 𝔢 in 𝔤. We show below that n_{𝔳} is also invariant. Before that we need the following lemma, which will be needed also afterwards in the proof that L is transitive.

Lemma 4.4. Let X ∈ s_{𝔳} then the eigenvalues of ad_{𝔤} (X) are purely imaginary.

Proof. Since ad_{𝔤} (X) is semi-simple, X belongs to some Cartan subalgebra, say 𝔧 (see [13, Prop. 1.3.5.4]). For this subalgebra there exists a Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔰 such that

𝔧 = 𝔧_{𝔨} ⊕ 𝔧_{𝔰}

where 𝔧_{𝔨} = 𝔧 ∩ 𝔨 and 𝔧_{𝔰} = 𝔧 ∩ 𝔰. The eigenvalues of the adjoint in 𝔤 of the elements in 𝔧_{𝔨} are purely imaginary and those of 𝔧_{𝔰} are real. Let θ be the Cartan involution associated with the Cartan decomposition and consider the inner product B_{θ} (Y, Z) = −⟨Y, θZ⟩ where ⟨·, ·⟩ is the Cartan-Killing form of 𝔤. With respect to B_{θ}, ad_{𝔤} (Y), Y ∈ 𝔨 is skew-symmetric while ad_{𝔤} (Z), Z ∈ 𝔰 is symmetric.

Now, put 𝔩_{1} = 𝔩+(s_{𝔳} + n_{𝔳}) and take a basis of 𝔤 which is orthonormal with respect to B_{θ} and such that its first elements are in 𝔩_{1}. Since S_{X} annihilates 𝔩 and s_{𝔳} + n_{𝔳} is abelian it follows that the matrix of ad_{𝔤} (X) in this basis is of the form

ad_{𝔤} (X) = ( \begin{matrix}0 & ∗ \\ 0 & ∗ \end{matrix} ) .

On the other hand if we write X = X_{𝔨} + X_{𝔰} with X_{𝔨} ∈ 𝔧_{𝔨} and X_{𝔰} ∈ 𝔧_{𝔰} then

ad_{𝔤} (X_{𝔨}) = ( \begin{matrix}a & b \\ −b^{t} & c \end{matrix} )

with a and c skew-symmetric because X_{𝔨} ∈ 𝔨. Also, X_{𝔰} ∈ 𝔰 hence

ad_{𝔤} (X_{𝔰}) = ( \begin{matrix}α & β \\ β^{t} & γ \end{matrix} )

with α and γ symmetric. Comparing these matrices we get a = α = 0 and β = b. Moreover, X_{𝔨}, X_{𝔰} ∈ 𝔧 and 𝔧 is abelian. Hence

ad_{𝔤}[X_{𝔨}, X_{𝔰}] = ( \begin{matrix}2bb^{t} & ∗ \\ ∗ & ∗ \end{matrix} ) = 0,

which shows that b = β = 0. Therefore X_{𝔰} centralizes 𝔩_{1}. We can now apply Lemma 4.1 to show that X_{𝔰} = 0: The subspace spanned by X_{𝔰} is an ideal of 𝔩_{1}. Let L_{1} be the connected subgroup whose Lie algebra is 𝔩_{1}. Since L is assumed to be connected, L ⊂ L_{1}, hence L_{1} is minimal on B. Since ad_{𝔤} (X_{𝔰}) has real eigenvalues, the ideal spanned by X_{𝔰} is triangular so that X_{𝔰} = 0 concluding the proof of the lemma.

Now we can prove that n_{𝔳} is invariant under the adjoint representation of 𝔢.

Lemma 4.5. Let s_{𝔳}^{⊥} denote the orthogonal complement of s_{𝔳} with respect to the Cartan-Killing form ⟨·, ·⟩ of 𝔤. Then n_{𝔳} = s_{𝔳}^{⊥} ∩ (s_{𝔳} + n_{𝔳}), and n_{𝔳} is ad_{𝔤} (𝔢)- invariant.

Proof. If X ∈ s_{𝔳} and Y ∈ n_{𝔳} then ad_{𝔤} (X) ad_{𝔤} (Y) is upper triangular. Hence

tr (ad_{𝔤} (X) ad_{𝔤} (Y)) = 0.

This means that n_{𝔳} ⊂ s_{𝔳}^{⊥} ∩ (s_{𝔳} + n_{𝔳}). On the other hand, Lemma 4.4 implies that ⟨X, X⟩ < 0 if 0 ≠ X ∈ s_{𝔳}. It follows that the restriction to s_{𝔳} of ⟨·, ·⟩ is nondegenerate so that s_{𝔳}^{⊥} ∩ s_{𝔳} = 0 and 𝔤 = s_{𝔳}^{⊥} ⊕ s_{𝔳}. Therefore

s_{𝔳} + n_{𝔳} = s_{𝔳} ⊕ (s_{𝔳}^{⊥} ∩ (s_{𝔳} + n_{𝔳})) .

Since n_{𝔳} ⊂ s_{𝔳}^{⊥} ∩ (s_{𝔳} + n_{𝔳}) these subspaces are equal.

Finally, note that the ad_{𝔤} (𝔢)-invariance of s_{𝔳} implies that s_{𝔳}^{⊥} is also invariant. Since s_{𝔳} + n_{𝔳} is invariant we conclude that n_{𝔳} is an invariant subspace under the adjoint action of 𝔢.

From this lemma it is easy to apply Lemma 4.1 to show that the subspace n_{𝔳} reduces to zero: Put 𝔩_{1} = 𝔩+ (s_{𝔳} + n_{𝔳}). Then 𝔩_{1} is a subalgebra because s_{𝔳} + n_{𝔳} is ad_{𝔤} (𝔢)-invariant. Since n_{𝔳} is invariant under ad_{𝔤} (𝔢), it is an ideal in 𝔩_{1}. By construction n_{𝔳} is triangular. Let L_{1} be the connected subgroup whose Lie algebra is 𝔩_{1}. The assumption that L is connected implies that L ⊂ L_{1} so that L_{1} is minimal on B. Therefore by Lemma 4.1, n_{𝔳} = 0. This implies that 𝔳 = s_{𝔳} and since s_{𝔳} commutes with 𝔩, 𝔳 is the center of 𝔩.

Summarizing all the previous discussion in this section we have:

Theorem 4.6. Suppose that a connected subgroup L ⊂ G acts minimally on a flag manifold B of G. Let 𝔩 be the Lie algebra of L. Then 𝔩 is reductive, that is, 𝔩 = 𝔢 ⊕ 𝔷 with 𝔢 semi-simple and 𝔷 the center of 𝔩. Moreover, for all X ∈ 𝔷, ad_{𝔤} (X) is semi-simple and has purely imaginary eigenvalues.

Up to this point the only requirement about L, in addition to its minimal action, was that it is a connected Lie subgroup. We impose now the condition that L is closed and prove that its action on B is actually transitive. Note first that since we are assuming that G is centerless, the fact that the eigenvalues of ad_{𝔤} (X) are imaginary implies that the connected subgroup Z whose Lie algebra is 𝔷 is relatively compact in G. Hence Z is relatively compact in L if L is closed. Its closure \overline{Z} is contained in the center of L. Hence the Lie algebra of \overline{Z} is 𝔷. Since \overline{Z} is connected it follows that \overline{Z} = Z and Z is compact.

Theorem 4.7. Suppose that a closed and connected subgroup L ⊂ G acts minimally on a flag manifold B of G. Then L is transitive on B.

Proof. Write as above 𝔩 = 𝔢 + 𝔷 with 𝔢 semi-simple and 𝔷 the center of 𝔩. Denote by E and Z the connected subgroups whose Lie algebras are 𝔢 and 𝔷 respectively. Every g ∈ L is a product of exponentials of elements of 𝔩. Since Z is a central subgroup, L = ZE with Z is compact. However E has a compact orbit in B because it is a semi-simple subgroup. Let Ex_{0} be this orbit. Then Lx_{0} = ZEx_{0} is compact. But the action of L is minimal. Hence Lx_{0} = B, and L is transitive.

5. Semi-simple subgroups

In the previous section we restricted attention to subgroups acting minimally on some flag manifold. We return here to the admissible groups, which satisfy also a contractive property.

For the admissible groups we have a slightly stronger result than Theorem 4.7, namely that the semi-simple component of L is also admissible. In fact, let S ⊂ G be a semigroup with int S ≠ Ø and suppose that S is transitive in G/L. Then L = ZE with Z a central compact subgroup and E semi-simple. Hence L/E is a compact group so that the canonical fibration

G/E ⟶ G/L

is a connected principal bundle with compact structure group L/E. This implies that S is transitive on G/L if and only if it is transitive on G/E (see [6] or [2]).

Hence E is S -admissible provided L is S -admissible and if this is the case then E is also transitive on the flag B (S), as follows from Theorem 4.7.

After these preliminaries we proceed to the analysis of the admissible semisimple groups.

Throughout this section we let E stand for an admissible, noncompact and connected semi-simple subgroup of G. As before we denote by S a semigroup with nonempty interior which is transitive on G/E, so that the action of E on B (S) is transitive and contractive.

Let 𝔢 be the Lie algebra of E and write 𝔢 = 𝔨_{1} + 𝔰_{1} for a Cartan decomposition of 𝔢. It is well known that there exists a Cartan decomposition 𝔤 = 𝔨 + 𝔰 such that 𝔨_{1} ⊂ 𝔨 and 𝔰_{1} ⊂ 𝔰. Also, we can choose compatible abelian subalgebras 𝔞_{1} ⊂ 𝔞 with 𝔞_{1} maximal in 𝔰_{1} and 𝔞 maximal in 𝔰. With these choices we denote by Π the set of roots of the pair (𝔤, 𝔞) and by Π_{1} the set of roots of (𝔤_{1}, 𝔞_{1}).

Given a root α ∈ Π_{1} and H ∈ 𝔞_{1}, α (H) is an eigenvalue of ad_{𝔢} (H) and hence of ad_{𝔤} (H). However the eigenvalues of ad_{𝔤} (H) are given by the values in H of the roots in Π because 𝔞_{1} ⊂ 𝔞. Therefore every root in Π_{1} is the restriction to 𝔞_{1} of a root in Π. In general a set of positive roots in Π does not restrict to a set of positive roots of Π_{1}. But we can choose a simple system of roots for Π compatible with a simple system for Π_{1}, that is, in such a way that the positive roots in Π contain the positive roots in Π_{1}. In fact, let H ∈ 𝔞_{1} be a regular real element of 𝔢. Then H belongs to the closure of a Weyl chamber in 𝔞. Let 𝔞^{+} be one of these chambers, and denote by Σ the corresponding simple system of roots and by Π^{+} the set of positive roots thus obtained. The roots in Π^{+} assume nonnegative values in H. By the same token, the values assumed in H by the roots in Π^{−} = −Π^{+} are ≤ 0.

On the other hand, H is regular real in 𝔢 so that it belongs to a Weyl chamber 𝔞_{1}^{+} ⊂ 𝔞_{1}. Let

Π_{1}^{+} = {α ∈ Π_{1} : α (H) > 0}

be the set of positive roots and denote by Σ_{1} the simple system associated with 𝔞_{1}^{+}. For α ∈ Π_{1}^{+} we have α (H) > 0, and since the values of the roots in Π^{−} are nonpositive in H, α is the restriction to 𝔞_{1} of some β ∈ Π^{+}. Therefore, we have

Lemma 5.1. Let Π_{1}^{+} be a positive system of roots for (𝔢, 𝔞_{1}) and denote by 𝔞_{1}^{+} the corresponding Weyl chamber. Pick H ∈ 𝔞_{1}^{+}. Then there is a positive system Π^{+} for (𝔤, 𝔞) such that α (H) ≥ 0 for all α ∈ Π^{+}. In this case Π_{1}^{+} and Π^{+} are compatible, i.e., every β ∈ Π_{1}^{+} is the restriction to 𝔞_{1} of some α ∈ Π^{+}.

Compatible positive systems induce compatible Iwasawa decompositions. In fact, with the same notations as in the lemma, put

𝔫_{1} = ∑_{β∈Π_{1}^{+}} 𝔢_{β}.

Then 𝔫_{1} is the sum of eigenspaces of ad_{𝔢} (H) associated to positive eigenvalues. Since these eigenspaces are contained in 𝔫 = ∑_{α∈Π^{+}} it follows that 𝔫_{1} ⊂ 𝔫. We remark that even if the simple systems are compatible it may happen in general that 𝔞_{1}^{+} is not contained in cl 𝔞^{+}. This is due to the fact that some root α ∈ Π^{+} may be negative in some H′ ∈ 𝔞_{1}^{+}. Nevertheless we shall prove below that for admissible subgroups the inclusion between the Weyl chambers actually holds.

Let B be a flag manifold of G. If we take a simple system Σ for Π, then B = B_{Θ} for some subset Θ ⊂ Σ. Let Ψ ⊂ Σ be the subset of simple roots which vanish identically on 𝔞_{1}. The next statement relates Θ and Ψ in case the action of E on B_{Θ} is contractive.

Proposition 5.2. Take compatible simple systems Σ_{1} and Σ, and suppose that Ψ ⊂ Θ. Then the action of E on B_{Θ} is contractive.

Reciprocally, if the action of E on the flag manifold B is contractive then there are Weyl chambers 𝔞_{1}^{+} ⊂ 𝔞_{1} and 𝔞^{+} ⊂ 𝔞, defining compatible simple systems Σ_{1} and Σ such that if B = B_{Θ} with Θ ⊂ Σ, and Ψ ⊂ Σ is as above the annihilator of 𝔞_{1} in Σ then Ψ ⊂ Θ.

Proof. Keeping the notations as above, 𝔞_{1} intercepts the closure of 𝔞^{+} because the simple systems were chosen to be compatible. Set

𝔟 = {H ∈ 𝔞 : α (H) = 0 for all α ∈ Ψ}

if Ψ ≠ Ø and 𝔟 = 𝔞 if Ψ = Ø. Clearly, 𝔞_{1} ⊂ 𝔟. The intersection of 𝔟 with the closure of 𝔞^{+} is given by those elements H ∈ 𝔟 such that α (H) ≥ 0 for all α ∈ Σ. This is a cone in 𝔟 whose interior is the “subchamber”

𝔟^{+} = {H ∈ 𝔟 : α (H) > 0, α ∈ Σ − Ψ}.

By the choice of 𝔞^{+}, 𝔞_{1} intercepts the closure of 𝔟^{+}. Actually, 𝔞_{1} intercepts 𝔟^{+} itself because Ψ is exactly the subset of simple roots which vanish on 𝔞_{1}. Hence there exists H ∈ 𝔞_{1} ∩ 𝔟^{+}. This means that α (H) > 0 for every root α ∈ Σ − Ψ. But by assumption Ψ ⊂ Θ, so that α (H) < 0 for every negative root outside −⟨Θ⟩. Therefore the sequence exp (nH), n ≥ 0 is contractive in B_{Θ}, showing the first statement.

For the converse take at first an arbitrary Weyl chamber 𝔞^{+} ⊂ 𝔞, and use the notations A = exp 𝔞, A^{+} = exp 𝔞^{+} and A_{1} = exp 𝔞_{1}. Then B = B_{Θ} for some Θ ⊂ Σ, where Σ is the simple system associated with 𝔞^{+}. Let g_{n} ∈ E be a sequence which is contractive w.r.t. B. Then there are the two polar decompositions

• g_{n} = u_{n}h_{n}v_{n}, with u_{n}, v_{n} ∈ K and h_{n} ∈ cl A^{+}, and

• g_{n} = s_{n}a_{n}t_{n} with s_{n}, t_{n} ∈ K_{1} and a_{n} ∈ A_{1}.

The second of these decompositions is intrinsic in E. Since A_{1} ⊂ A it follows that a_{n} ∈ A. Hence there exists w_{n} in the Weyl group of 𝔞 such that b_{n} = w_{n}a_{n}w_{n}^{−1} belongs to the closure of A^{+}. By the second decomposition we can write g_{n} = s_{n}w_{n}^{−1}b_{n}w_{n}t_{n}, so that the uniqueness of the radial part in a polar decomposition ensures that h_{n} = b_{n} = w_{n}a_{n}w_{n}^{−1}.

By assumption g_{n} is contractive. Hence ϕ_{α} (h_{n}) → 0 for α outside −⟨Θ⟩. This means that ϕ_{α} (w_{n}a_{n}w_{n}^{−1}) → 0, or, equivalently, ϕ_{w_{n}^{−1}α} (a_{n}) → 0 for all α < 0, α ∉ −⟨Θ⟩. Taking a subsequence we can assume that w_{n} = w is independent of n. Then there exists a ∈ A_{1} such that ϕ_{w^{−1}α} (a) < 1 for every negative root α ∉ −⟨Θ⟩. In other words there exists H ∈ 𝔞_{1} such that α (H) < 0 for every negative root α ∉ −⟨w^{−1}Θ⟩. By continuity we can take H to be regular real in 𝔞_{1}. Now consider the simple system Σ′ = w^{−1}Σ and the subset Θ′ = w^{−1}Θ ⊂ Σ′. Of course B = B_{Θ′} with the construction made from Σ′. Let W_{Θ′} be the subgroup generated by the reflections with respect to the roots in Θ′. By Lemma 5.3 below there exists u ∈ W_{Θ′} such that (u^{−1}α) (H) = α (uH) ≥ 0 for all α ∈ Θ′. We have that u^{−1} leaves ⟨Θ⟩ invariant. Hence (u^{−1}α) (H) > 0 for every positive root α ∉ ⟨Θ⟩. Therefore if we put Σ′′ = u^{−1}Σ′, Θ′′ = u^{−1}Θ′ and let Ψ′′ be the subset of roots in Σ′′ which are identically zero on 𝔞_{1}, then Ψ′′ ⊂ Θ′′ and the data H ∈ 𝔞_{1}, Ψ′′, Θ′′ and Σ′′ accomplish the requirements.

Lemma 5.3. For a subset Θ ⊂ Σ denote by W_{Θ} the subgroup generated by the reflections with respect to the roots in Θ. Take H ∈ 𝔞. Then there exists u ∈ W_{Θ} such that α (uH) ≥ 0 for all α ∈ Θ.

Proof. For α ∈ 𝔞^{∗} let H_{α} ∈ 𝔞 be defined by α (·) = ⟨H_{α}, ·⟩. Denote by 𝔞_{Θ} the subspace generated by H_{α}, α ∈ Θ. The subset {H_{α} : α ∈ ⟨Θ⟩} is a root system in 𝔞_{Θ} containing Σ_{Θ} = {H_{α} : α ∈ Θ} as a simple system and whose Weyl group is W_{Θ}. Write H = H_{1} + H_{2} with H_{1} ∈ 𝔞_{Θ} and H_{2} orthogonal to 𝔞_{Θ}. Then there exists u ∈ W_{Θ} such that wH_{2} belongs to the Weyl chamber defined by Σ_{Θ}. Since uH_{2} = H_{2} for all u ∈ W_{Θ} the lemma follows.

We can prove now that B (S) is a flag manifold for semi-simple admissible subgroups.

Theorem 5.4. Let E ⊂ G be a connected semi-simple subgroup and suppose that its action on the flag manifold B of G is contractive and transitive. Then B is also a flag manifold of E.

Proof. Let P_{1} = M_{1}A_{1}N_{1} be a minimal parabolic subgroup of E. We must show that P_{1} has a fixed point in B. Take an abelian subgroup A ⊂ G such that A_{1} ⊂ A. For a Weyl chamber A^{+} ⊂ A denote by Σ the corresponding simple system of roots. By the above proposition we can choose a chamber A^{+} ⊂ A and a regular real h ∈ A_{1} such that h belongs to the closure of A^{+} and α (log h) < 0 for every negative root α ∉ −⟨Θ⟩, where Θ ⊂ Σ is the subset defining the standard parabolic subgroup P_{Θ} which gives B = G/P_{Θ}. Let x_{0} be the origin in G/P_{Θ}. By the choice of h ∈ A_{1}, we have that x_{0} is its attractor with stable manifold N^{−}x_{0}, that is, h^{n}x → x_{0} as n → ∞ for all x ∈ N^{−}x_{0} where N^{−} is the nilpotent group given by the negative roots defined by Σ.

In the minimal parabolic subgroup P_{1} ⊂ E we can take N_{1} to be exp 𝔫_{1} where 𝔫_{1} is the sum of the root spaces associated with the roots of 𝔞_{1} which are positive on log h. With this choice we have compatible Iwasawa decompositions so that N_{1} ⊂ N. This implies that A_{1}N_{1}x_{0} = x_{0}.

On the other hand, take m ∈ M_{1}. Then mx_{0} is fixed by h because hm = mh. From this we get that x_{0} is fixed under the identity component (M_{1})_{0} of M_{1}. In fact, since the stable manifold N^{−}x_{0} is open, there is a neighborhood of the identity U ⊂ M_{1} such that Ux_{0} ⊂ N^{−}x_{0}. But the only h-fixed point in N^{−}x_{0} is x_{0} itself. Hence Ux_{0} = x_{0} which shows that (M_{1})_{0} x_{0} = x_{0}.

Let Q ⊂ E be the isotropy at x_{0} for the E -action. So far we have shown that (M_{1})_{0} A_{1}N_{1} is contained in Q. This implies that the Lie algebra 𝔪_{1} + 𝔞_{1} + 𝔫_{1} of (M_{1})_{0} A_{1}N_{1} is contained in the isotropy subalgebra 𝔮 at x_{0}. Therefore 𝔮 is a parabolic subalgebra of 𝔢. Let Q_{1} be the corresponding parabolic subgroup. Then Q is a subgroup of finite index in Q_{1} so that the canonical equivariant fibration

π : B = E/Q ⟶ E/Q_{1}

is a covering. The equivariance of π implies that y = π (x_{0}) is an attractor for the action of h in E/Q. Hence the elements in the fiber over y are also attractors. However the stable manifold at x_{0} is dense, so that there is just one attractor and B = E/Q_{1}, showing that B is indeed a flag manifold of E.

Corollary 5.5. Let E be as in the theorem above and suppose that the Weyl chambers 𝔞_{1}^{+} and 𝔞^{+} in 𝔢 and 𝔤 are compatible in the sense of Lemma 5.1. Then 𝔞_{1}^{+} ⊂ cl 𝔞^{+}.

Proof. Since B is a flag manifold of E there is a point x_{0} ∈ B which is a common attractor for h ∈ A_{1}^{+}. The stable manifold of this attractor is N_{Θ}^{+}x_{0} = (exp 𝔫_{Θ}^{+}) x_{0} where

𝔫_{Θ}^{+} = ∑ 𝔤_{α}

with the sum running through the negative roots α ∈ −⟨Θ⟩. On the other hand, let 𝔟^{+} be as in the proof of Proposition 5.2 and suppose that H ∈ 𝔞_{1}^{+} is not in the closure of 𝔟^{+}. Then there exists a negative root α ∉ −⟨Θ⟩ such that α (H) > 0 because Ψ ⊂ Θ and hence x_{0} is not an attractor for exp H. Therefore every H ∈ 𝔞_{1}^{+} is in the closure of 𝔞^{+}.

With the above theorem it is virtually possible to get all the admissible connected subgroups by checking the list of flags for the different noncompact semi-simple real Lie groups. As an example we mention that the projective space ℙ^{n−1} is a flag manifold for Sl (n, ℝ). The list of semi-simple groups transitive on ℙ^{n−1} was provided by Boothby and Wilson [1]. The connected subgroups which are admissible for a semigroup S with B (S) = ℙ^{n−1} are in that list. By checking it we can see that the only one which is contractive and hence admissible is Sp (m, ℝ) with n = 2m (c.f. [10]).

In addition to this specific example we also obtain that G is the only connected subgroup admissible for semigroups associated to the maximal flag manifold:

Proposition 5.6. Suppose that in Theorem 5.4 the group G is simple and B is the maximal flag manifold of G. Then E = G.

Proof. Take compatible Cartan decompositions 𝔢 = 𝔨_{1} ⊕ 𝔰_{1} and 𝔤 = 𝔨 ⊕ 𝔰, and let 𝔞_{1} ⊂ 𝔰_{1} be a maximal abelian subalgebra. By Proposition 5.2 there exists H ∈ 𝔞_{1} which is regular real in 𝔤. Since B is a flag manifold of E, the compact group K_{1} = exp 𝔨_{1} is transitive on it. On the other hand since B is the maximal flag manifold of G it is the Ad (K)-orbit of H and therefore B = Ad (K_{1}) H. This orbit is contained in 𝔰_{1} and generates 𝔰. Hence 𝔰_{1} = 𝔰. Since 𝔤 is simple 𝔰 generates 𝔤 so that 𝔢 = 𝔤.

6. Nonconnected subgroups

In the preceding sections we made the basic assumption that the admissible subgroup L is connected. Here we make some comments about the nonconnected admissible subgroups. First of all, the case where there are a finite number of connected components is easily reduced to the connected case. In fact, if L_{0} stands for the identity component in L then there is the canonical fibration

G/L_{0} ⟶ G/L

which makes G/L_{0} a principal bundle with L/L_{0} as structure group. If the number of connected components of L is finite then L/L_{0} is finite and hence a semigroup S ⊂ G with int S ≠ Ø is transitive in G/L_{0} if and only if it is transitive in G/L (see [6], [2]). Therefore L is admissible if and only if L_{0} is admissible and we are back to the situation of the previous sections.

We say now a few words about subgroups with infinite connected components. Suppose that G is centerless. Then G is the identity component of an algebraic group and we can consider it in the Zariski topology. For a subset C ⊂ G we let zc (C) stand for its Zariski closure. If L ⊂ G is a subgroup then zc (L) is a closed subgroup with a finite number of connected components. Moreover, since L ⊂ zc (L) we have that zc (L) is admissible for the semigroup S if L is S -admissible. Let (zc (L))_{0} be the identity component of zc (L) and put L_{1} = L ∩ (zc (L))_{0}. Then L/L_{1} is isomorphic to a subgroup of (zc (L)) / (zc (L))_{0} so L_{1} is a subgroup of finite index in L which is also admissible. We have thus proved that any admissible subgroup has a subgroup of finite index which is Zariski dense in a connected admissible group.

Recall that if a subspace of 𝔤 is invariant under L then it is also invariant under the algebraic closure (zc (L))_{0}. Therefore the Lie algebra 𝔩 of L is an ideal of the Lie algebra 𝔩̃ of zc (L). Hence if (zc (L))_{0} is simple either 𝔩 = 0 or 𝔩 = 𝔩̃. In the latter case L contains (zc (L))_{0} so it has a finite number of connected components and we are back to the previous situation. On the other hand if 𝔩 = 0 then L is discrete. Combining this fact with Proposition 5.6 we obtain:

Proposition 6.1. Suppose that G is simple with finite center and B (S) is the maximal flag manifold. Suppose also that L ≠ G is a subgroup such that S is transitive on G/L. Then L is discrete.

Concerning the discrete infinite subgroups which are admissible we mention that there is an easy case, namely when L is a lattice in G. Recall that a discrete subgroup H of a topological group is said to be a lattice if there exists a Ginvariant probability measure μ on G/H, that is, μ (g^{−1}A) = μ (A) for all g ∈ G and measurable sets A ∈ G/H. In case L is a lattice, the Recurrence Theorem implies that any semigroup S ⊂ G with int S ≠ Ø is transitive in G/L. This is a well known fact in the context of control systems (see [5], [8]), whose proof extends to more general semigroup actions. For the sake of completeness we provide here a proof which works for the action of a semigroup of homeomorphisms in a topological space. We need first the

Lemma 6.2. Let M be a topological space and T a semigroup of homeomorphisms of M. Denote by H the group of homeomorphisms of M generated by T ∪ T^{−1}. Assume that

(1) There exists a finite measure μ on the Borel subsets of M which is invariant under H and such that μ (A) > 0 for any open set A ⊂ M.

(2) The action of H on M is transitive. Let U ⊂ M be a non empty open subset which is invariant under T. Then U is dense in M.

Proof. Since H is transitive in M, we must show that cl U is H -invariant. This invariance follows at once if we show that cl U is invariant under T^{−1}. In order to check this, suppose to the contrary that there exists x ∈ cl U, an open subset V with V ∩cl U = Ø and g ∈ T such that g^{−1}x ∈ V . Apply the Recurrence Theorem (see e.g. [4]) to g to get a subset F ⊂ gV with μ (F) = μ (gV) such that for any y ∈ F there exists an integer k > 1 such that g^{k}y ∈ gV . Since μ is positive on open sets and μ (F) = μ (gV) it follows that F is dense in gV . Hence U ∩ F ≠ Ø because U ∩ gV is open in gV . If y ∈ U ∩ F and k > 1 is such that g^{k}y ∈ gV then g^{k−1}y ∈ V which contradicts the fact that U is T -invariant.

From this Lemma we obtain sufficient conditions for the transitivity of semigroups.

Proposition 6.3. Suppose that T and M are as in the previous lemma and assume further that for all x ∈ M, int (T x) and int (T^{−1}x) are not empty. Then T is transitive on M.

Proof. Pick x, y ∈ M. By the lemma int (T x) is dense in M. Hence int (T^{−1}y) ∩ int (T x) ≠ Ø which shows that there exists g ∈ T such that gx = y and hence T is transitive.

Finally, we have the following hereditary property for admissible subgroups.

Proposition 6.4. Suppose that L is S -admissible and let H ⊂ L be a closed subgroup such that S ∩ L is transitive in L/H. Then S is transitive in G/H.

Proof. Consider the canonical fibration

π : G/H ⟶ G/L

with fiber L/H. Let x_{0} = L be the origin in G/L. The assumption that S ∩ L is transitive in L/H implies that S is transitive along the fiber π^{−1} (x_{0}). Since S is transitive in G/L it follows that S is transitive in G/H.

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Instituto de Matemática Universidade Estadual de Campinas Cx. Postal 6065 13.081-970 Campinas, SP, Brasil smartin@ime.unicamp.br Received May 22, 1997 and in final form September 24, 1997