Journal of Lie Theory Volume 8 (1998) 335–350 ◯^{C}1998 Heldermann Verlag

Order and Domains of Attraction of Control Sets in Flag Manifolds

Luiz A. B. San Martin^{∗}

Communicated by J. Hilgert

Abstract. Let G be a real semi-simple noncompact Lie group and S ⊂ G a subsemigroup with int S ≠ Ø. This article relates the Bruhat-Chevalley order in the Weyl group W of G to the ordering of the control sets for S in the flag manifolds of G by showing that the one-to-one correspondence between the control sets and the elements of a double coset W (S) \W/W_{Θ} of W reverses the orders. This fact is used to show that the domain of attraction of a control set is a union of Schubert cells.

Key words: semigroups, semi-simple groups, flag manifolds, control sets, Bruhat-Chevalley order. AMS 1991 subject classification: primary 20M20, 54H15; secondary 93B

1. Introduction

For the action of a semigroup of maps in some state space the transitivity structure is described by the control sets, which are lasting regions of the evolution of the semigroup, and by the transience between the control sets. In this article we consider a noncompact semi-simple Lie group G and look at the transitivity properties of the action of a semigroup S ⊂ G in the flag manifolds of G. A basic assumption is that the semigroup has nonvoid interior in G. The control sets for these actions were studied in [9], which provides us with the following picture: Let W be the Weyl group of G. Then there is a parabolic subgroup W (S) of W attached to S such that the control sets for the S -action in the maximal flag manifold B of G are in one-to-one correspondence with the cosets in W (S) \W. This fact extends to any other flag manifold B_{Θ} establishing a bijection between the control sets in B_{Θ} and the double cosets in W (S) \W/W_{Θ} where W_{Θ} is the parabolic subgroup of W associated with B_{Θ}.

These results bring to the study of the semigroups in G, and their control sets in the flag manifolds, the combinatorics of the Weyl group. The purpose of

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag this paper is to relate the Weyl group also to the transience behavior between the control sets. This is made as follows: There is a natural partial order ≤ between the control sets, which is defined by putting D_{1} ≤ D_{2} if it is possible to steer the points of D_{1} into D_{2}. Clearly, this order describes the transience between the control sets. On the other hand, there is the well known Bruhat-Chevalley partial order in the Weyl group W. These two orders are related by the above bijections. In fact, in Theorem 4.1 below we show that the order between the control sets in the maximal flag manifold is obtained by reversing the Bruhat-Chevalley order in W and projecting onto W (S) \W so that the bijection is order reversing. This fact is extended to the other flag manifolds in Proposition 7.1 where it is proved that the control sets in a flag manifold B_{Θ} are ordered according to the reverse of the order induced by the Weyl-Chevalley order on the double coset W (S) \W/W_{Θ}.

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

Intimately related to the order of the control sets is the concept of domain of attraction of a control set D, which is defined to be the subset of points in the state space that can be steered by the semigroup into D. It is not hard to prove that D_{1} ≤ D_{2} if and only if D_{1} is contained in the domain of attraction of D_{2} (see Proposition 2.1 below). In the Weyl group side this fact parallels the classical Chevalley-Borel-Tits Theorem which characterizes the Schubert cells from the Bruhat-Chevalley order and the Bruhat cells. In fact, we use this theorem in Section 6. to show that the domain of attraction of a control set in the maximal flag manifold is a union of Schubert cells. This is achieved in two steps. First we construct a Schubert cell by starting from a point in B and exhausting successively subsets with fibers of projections from B onto smaller flag manifolds. This construction resembles the Bott-Samelson [1] construction of the desingularization of a Schubert cell. Once we have this characterization of a Schubert cell we apply it to show that the same exhausting procedure, starting now from the minimal control set, yields the domain of attraction of a control set. In this procedure the choice of the smaller flag manifolds is determined by the element of the Weyl group associated with the control set.

The results proved in this paper show that the order of the control sets reduces to the order of the Weyl group. The combinatorics of the Bruhat-Chevalley ordering is extensively studied in the literature. We refer to [4], [5], [6], and references therein, for a description of this order in the different Weyl groups. This description provides the order of the control sets for the semigroups.

The author is thankful to J. Hilgert for pointing out to the relation between the order of the control sets and the Weyl-Chevalley order. In particular, Theorem 4.1 below was conjectured by him in a personal talk.

2. Control sets

Let G be a Lie group, S ⊂ G a semigroup with int S ≠ Ø and G/H a compact homogeneous space with H ⊂ G a closed subgroup. Recall that a subset D ⊂ G/H is said to be a control set for the S -action in G/H provided it satisfies

1. D ⊂ cl (Sx) for every x ∈ D,

2. int D ≠ Ø, 3. the subset D_{0} = {x ∈ D : x ∈ (int S) x} is not empty, and

4. D is maximal with these properties.

The subset D_{0} appearing in the third condition is open and dense in D (see [9]). We refer to it as the set of transitivity or the core of D.

The domain of attraction A (D) of a control set D is the subset of those x ∈ G/H such that there exists g ∈ S with gx ∈ D.

The control sets for S on G/H are ordered by putting D_{1} ≤ D_{2} if there are x ∈ D_{1} and g ∈ S such that gx ∈ D_{2}. Equivalently, D_{1} ≤ D_{2} if D_{1} ∩A (D_{2}) ≠ Ø. The following statement clarifies the relation between the domain of attraction and the order of the control sets.

Proposition 2.1. The domain of attraction A (D) of the control set D is open and if x ∈ A (D) then there exists g ∈ int S such that gx ∈ D_{0}. Moreover, for the control sets D_{1} and D_{2} the following statements are equivalent:

1. D_{1} ≤ D_{2}.

2. There exists x ∈ (D_{1})_{0} and g ∈ int S such that gx ∈ (D_{2})_{0}.

3. For any y ∈ (D_{1})_{0} and z ∈ (D_{2})_{0} there exists g ∈ int S such that gy = z.

4. D_{1} ⊂ A (D_{2}).

Proof. Take x ∈ A (D) and h ∈ S such that hx ∈ D. Then S (hx) contains a dense subset of D so that there exists g ∈ S with gx ∈ D_{0}. Since any point in D_{0} is fixed by some element in int S we can choose g ∈ int S as claimed. This implies that the open set g^{−1} (D_{0}) contains x and is contained in A (D). Hence A (D) is open.

As to the equivalent statements, suppose that D_{1} ≤ D_{2}. Then (D_{1})_{0} ∩ A (D_{2}) ≠ Ø because (D_{1})_{0} is dense in D_{1} and A (D_{2}) is open. Hence (2) follows from the statement about A (D). Now, take x and g as in (2). If y ∈ (D_{1})_{0} and z ∈ (D_{2})_{0} then there are h_{1}, h_{2} ∈ S such that h_{1}y = x and h_{2}gx = z (see [9, Prop. 2.2]). So that (3) follows. Item (3) means that (D_{1})_{0} ⊂ A (D_{2}). Since for any x ∈ D_{1} there is g ∈ S such that gx ∈ (D_{1})_{0} we have that D_{1} ⊂ A (D_{2}) as well. Finally, D_{1} ≤ D_{2} if D_{1} ⊂ A (D_{2}) as follows from the definitions.

In the sequel we write y ⇝ z if z ∈ (int S) y. From the third equivalent statement in this proposition we have that y ⇝ z for any y ∈ (D_{1})_{0} and z ∈ (D_{2})_{0} if D_{1} ≤ D_{2}.

The order of the control sets for S^{−1} is given by reversing the order of the control sets for S. In fact, there is a mapping D ↦ D^{−} which associates to a control set D for S the control set D^{−} for S^{−1} which is related to D by D_{0}^{−} = D_{0}. From the third equivalent property in the above lemma we have that D_{1}^{−} ≤ D_{2}^{−} if and only if D_{1} ≥ D_{2}.

Note that, since we are assuming that G/H is compact, a control set D is invariant (i.e., Sx ⊂ D for all x ∈ D) if and only if D is maximal with respect to the order. Also, a control set D is minimal if and only if D^{−} is S^{−1}-invariant. We refer to [3], [9] for further results about control sets.

3. Flag manifolds

In what follows we are interested in semigroup actions on G/H with G a connected and noncompact semi-simple Lie group and H a parabolic subgroup. We assume throughout that G has finite center. For these groups we use the following standard notation and terminology.

Let 𝔤 be the Lie algebra of G. Take a Cartan decomposition 𝔤 = 𝔨⊕𝔰 with 𝔨 the compactly embedded subalgebra and denote by θ the corresponding Cartan involution. Let 𝔞 be a maximal abelian subalgebra contained in 𝔰 and denote by Π the set of roots of the pair (𝔤, 𝔞). Fix a simple system of roots Σ ⊂ Π. Denote by Π^{+} the set of positive roots and by 𝔞^{+} the Weyl chamber

𝔞^{+} = {H ∈ 𝔞 : α (H) > 0 for all α ∈ Σ}.

Let

𝔫 = ∑_{α∈Π^{+}} 𝔤_{α}

be the direct sum of the root spaces corresponding to the positive roots and

𝔫^{−} = θ (𝔫) = ∑_{α∈Π^{+}} 𝔤_{−α}

the opposed subalgebra.

The notations K, N and N^{−} are used to indicate the connected subgroups whose Lie algebras are 𝔨, 𝔫 and 𝔫^{−} respectively.

Let W be the Weyl group of G. It is constructed either as the subgroup of reflections generated by the roots of (𝔤, 𝔞) or as the quotient M^{∗}/M where M^{∗} and M are respectively the normalizer and the centralizer of 𝔞 in K.

A minimal parabolic subalgebra of 𝔤 is given by

𝔭 = 𝔪 ⊕ 𝔞 ⊕ 𝔫

where 𝔪, the Lie algebra of M, is the centralizer of 𝔞 in 𝔨.

Let P be the minimal parabolic subgroup with Lie algebra 𝔭 and put B = G/P for the maximal flag manifold. We denote by b_{0} the base point in G/P. This flag manifold fibers over the other boundaries of G, which are built from subsets of Σ as follows: 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⟩. Then

𝔭_{Θ} = 𝔫_{Θ}^{−} ⊕ 𝔭.

The normalizer P_{Θ} of 𝔭_{Θ} in G is a parabolic subgroup which contains P. The corresponding flag manifold B_{Θ} = G/P_{Θ} is the base space for the natural fibration π_{Θ} : B → B_{Θ} whose fiber is P_{Θ}/P. This fiber is a flag manifold of a semi-simple subgroup M_{Θ} ⊂ G whose rank is the order of Θ (see [11]). In particular, the group M_{Θ} is of rank one if Θ is singleton. The Weyl group of M_{Θ} is the subgroup W_{Θ} generated by the reflections with respect to the simple roots in Θ.

A conjugate Ad (g) H, g ∈ G, H ∈ 𝔞^{+} is said to be split-regular in 𝔤. Similarly, a split-regular element in G is a exponential h = exp (H) with H ∈ 𝔤 split-regular. A split-regular H ∈ 𝔤 belongs to a unique Weyl chamber in 𝔤 (a conjugate of 𝔞^{+}). If h_{0} ∈ A^{+} = exp 𝔞^{+} then h_{0} has a finite number of fixed points in B, namely, the base point b_{0}(= P) and its orbit under the subgroup M^{∗} ⊂ K, the normalizer of 𝔞 in K. The action of M^{∗} in this orbit factors through M so that the fixed points are given by wb_{0}, w ∈ W = M^{∗}/M. The same way, the fixed points in B of a split-regular h = gh_{0}g^{−1} with g ∈ G and h_{0} ∈ A^{+}, are the points gwb_{0}. In what follows we say that gwb_{0} is the fixed point of type w for h and denote it by ϕ (h, w). We note that ϕ (h, 1) is the only attractor for the action of h in B, in the sense that it is a hyperbolic fixed point with an open stable manifold. On the other hand, h has just one repeller, i.e., an attractor for h^{−1}, which is ϕ (h, w_{0}) where w_{0} is the principal involution of W, that is, the only element of W which satisfies w_{0}Σ = −Σ.

Any h ∈ A leaves invariant the fiber π_{Θ}^{−1}π_{Θ} (b_{0}) for every Θ ⊂ Σ, and if h is regular then its fixed points in this fiber are wb_{0} with w ∈ W_{Θ}, that is, the orbit of b_{0} under the subgroup W_{Θ}. In particular, if Θ is singleton there are just two fixed points for the action of h in this fiber, one of them being an attractor and the other one a repeller.

Regarding still the notations, we shall say that a root α ∈ Π is indivisible if α/2 is not a root. The set of indivisible roots is denoted by Δ with a superscript Δ^{±} to indicate the positive or negative ones.

We consider now a semigroup S ⊂ G with int S ≠ Ø. In [9] the control sets for the action of S on the flag manifolds were described by means of the Weyl group W. For this description it is assumed that G has finite center. It is proved that int S contains enough split-regular elements so that we have a mapping

w ⟼ D (w)  (1)

which associates to w ∈ W a control set D (w) in such a way that the core D (w)_{0} is the set of the fixed points of type w for the split-regular elements in int S. There is just one invariant control set D (1) whose core is the set of attractors for the split-regular elements in int S. Similarly the repellers of the split-regular elements in int S form the core of a unique minimal control set D (w_{0}) where w_{0} is the principal involution of W.

The level sets of (1) are described by the subset

W (S) = {w ∈ W : D (w) = D (1)}.

This subset is a parabolic subgroup of W, and for w_{1}, w_{2} ∈ W, D (w_{1}) = D (w_{2}) if and only if w_{1}w_{2}^{−1} ∈ W (S) (see [9, Prop. 4.2]). Hence the control sets on the maximal boundary B are in one-to-one correspondence with the cosets in W (S) \W.

These facts also apply to the inverted semigroup S^{−1} = {g^{−1} : g ∈ S} and we have a mapping

w ⟼ D^{−} (w)

into the control sets of S^{−1}. The control set D^{−} (w) contains the fixed points of type w for the split-regular elements in int S^{−1}. Hence, if h ∈ int S is splitregular then ϕ (h^{−1}, w) ∈ D^{−} (w)_{0}. If w_{0} is the principal involution of W then h^{−1} = w_{0}h_{0}w_{0}^{−1} for some h_{0} in the same chamber as h. Therefore, the fixed point of type w for h^{−1} is ϕ (h, w_{0}w). Since this fixed point belongs to the core of D (w_{0}w) we have the following equality between the cores of these control sets.

Proposition 3.1. D^{−} (w)_{0} = D (w_{0}w)_{0}.

4. Order of the control sets

Recall the Bruhat-Chevalley order of the Weyl group [2, 4]: Keeping fixed a simple system of roots, take for w ∈ W a reduced expression w = s_{1} ⋯ s_{n} as a product of reflections with respect to the simple roots. Then w_{1} ≤ w if and only if there are integers 1 ≤ i_{1} < ⋯ < i_{j} ≤ n such that w_{1} = s_{i_{1}} ⋯ s_{i_{j}} is a reduced expression for w_{1}.

In general the order on W depends on the choice of the simple system of roots Σ, that is, on the set of generators of W. Note however that the order obtained from −Σ coincides with the order coming from Σ because both simple systems of roots define the same set of generators of W.

A useful elementary fact about the order in W is that w_{1} ≤ w if and only if w_{1}^{−1} ≤ w^{−1}. This follows directly from the definition and the remark that w^{−1} = s_{n} ⋯ s_{1} is a reduced expression for w^{−1} if the decomposition w = s_{1} ⋯ s_{n} is reduced.

From [2, Thm. 3.13] we have that w ≤ w_{1} if and only if wb_{0} ∈ cl (Nw_{1}b_{0}) where b_{0} is the base point in G/P.

We prove next that the order of the control sets in G/P for a semigroup S ⊂ G with nonvoid interior is precisely the reverse of the Bruhat-Chevalley ordering of the Weyl group. We have

Theorem 4.1. For u ∈ W let D (u) denote the control set in G/P given by (1). Let w_{1}, w_{2} ∈ W. Then the following statements are equivalent:

1. D (w_{1}) ≤ D (w_{2}).

2. There exists w ∈ W such that w_{1} ≥ w and w ∈ W (S)w_{2}, that is, D (w) = D (w_{2}).

Proof. We choose the simple system of roots such that the corresponding Weyl chamber A^{+} intersects int S. The ordering of the control sets will be provided by the action of a split-regular h ∈ A^{+} ∩ int S. The h-fixed points in G/P are wb_{0}, w ∈ W. Recall that wb_{0} ∈ D (w)_{0} for all w ∈ W.

Suppose that w_{1} ≥ w and w_{2} ∈ W (S)w. Then wb_{0} ∈ cl (Nw_{1}b_{0}). Since wb_{0} ∈ D (w)_{0} we have that Nw_{1}b_{0} ∩ D(w)_{0} ≠ Ø. Hence there exists n ∈ N such that nw_{1}b_{0} ∈ D(w)_{0}. But h^{−k}nw_{1}b_{0} → w_{1}b_{0} as k → ∞, which ensures the existence of an integer k > 0 such that h^{−k}nw_{1}b_{0} ∈ D(w_{1})_{0}. Therefore there exists g ∈ int S^{−1} and x ∈ D (w)_{0} such that gx ∈ D (w_{1})_{0} so that Proposition 2.1 implies that D (w_{1}) ≤ D (w) = D (w_{2}).

Conversely assume that D (w_{1}) ≤ D (w_{2}) and take a reduced expression

w_{1} = s_{1} ⋯ s_{n}

where s_{i} is the reflection with respect to the simple root α_{i}, i = 1, …, n. Let P_{i} = P_{{α_{i}}} be the parabolic subgroup defined by Θ = {α_{i}} and denote by π_{i} : G/P → G/P_{i} the projection from the maximal flag manifold.

We have from Proposition 2.1 that

w_{1}b_{0} w_{2}b_{0}.

If g ∈ int S is such that gw_{1}b_{0} = w_{2}b_{0} then each fibration π_{i} is equivariant under g so that g interchanges fibers. In particular, g maps the fiber of w_{1}b_{0} onto the fiber of w_{2}b_{0}. We shall exploit these fibrations to show that b_{0} w_{2}s_{i_{k}} ⋯ s_{i_{1}}b_{0} for some integers 1 ≤ i_{1} < ⋯ < i_{k} ≤ n.

Consider the reflection s_{n} and the corresponding fibration π_{n} : G/P → G/P_{n}. The fixed points of h in the fiber of w_{1}b_{0} are w_{1}b_{0} itself and w_{1}s_{n}b_{0}. To see this note that w_{1}^{−1} maps the fiber of w_{1}b_{0} onto the fiber of b_{0}. Also, w_{1}^{−1} interchanges the fixed points of h and those of h_{1} = w_{1}^{−1}hw_{1}. Since h_{1} is splitregular its fixed points in the fiber of b_{0} are b_{0} itself and s_{n}b_{0}. Hence w_{1}b_{0} and w_{1}s_{n}b_{0} are the fixed points of h in the fiber of w_{1}b_{0} as claimed.

By the same reason w_{2}b_{0} and w_{2}s_{n}b_{0} are the h-fixed points in the fiber through w_{2}b_{0}.

Now, if g ∈ int S maps w_{1}b_{0} into w_{2}b_{0} then gw_{1}s_{n}b_{0} belongs to the same fiber as w_{2}b_{0}. One of the two fixed points w_{2}b_{0} or w_{2}s_{n}b_{0} is the attractor for the action of h inside this fiber. Therefore as k → ∞ we have that h^{k}gw_{1}s_{n}b_{0} converges either to w_{2}b_{0} or to w_{2}s_{n}b_{0}. This implies that D (w_{1}s_{n}) ≤ D (w_{2}) or D (w_{1}s_{n}) ≤ D (w_{2}s_{n}) and hence we have one of the two possibilities

w_{1}s_{n}b_{0} w_{2}b_{0} or w_{2}s_{n}b_{0}.

Now we can repeat this argument with w_{1}s_{n}b_{0} in place of w_{1}b_{0} and with the fibration π_{n−1} : G/P → G/P_{n−1} instead of π_{n}. The fixed points in the π_{n−1}- fiber of w_{1}s_{n}b_{0} are w_{1}s_{n}b_{0} itself and w_{1}s_{n}s_{n−1}b_{0}. Hence according to the above possibilities we get that w_{1}s_{n}s_{n−1}b_{0} w_{2}b_{0} or w_{2}s_{n−1}b_{0} if w_{1}s_{n}b_{0} w_{2}b_{0}. Otherwise, we have w_{1}s_{n}s_{n−1}b_{0} w_{2}s_{n}b_{0} or w_{2}s_{n}s_{n−1}b_{0} if w_{1}s_{n}b_{0} w_{2}s_{n}b_{0}. Hence at least one of the following cases happens

w_{1}s_{n}s_{n−1}b_{0} w_{2}b_{0} or w_{2}s_{n}b_{0} or w_{2}s_{n−1}b_{0} or w_{2}s_{n}s_{n−1}b_{0}.

Continuing this way we arrive, after n steps, that there are integers 1 ≤ i_{1} < ⋯ < i_{k} ≤ n such that

w_{1}s_{n} ⋯ s_{1}b_{0} = b_{0} w_{2}s_{i_{k}} ⋯ s_{i_{1}}b_{0}.

This implies that w_{2}s_{i_{k}} ⋯ s_{i_{1}}b_{0} belongs to the invariant control set D (1). Therefore

w_{2}s_{i_{k}} ⋯ s_{i_{1}} ∈ W (S)

so that

w_{2} ∈ W (S)s_{i_{1}} ⋯ s_{i_{k}}.

Since w_{1} ≥ s_{i_{1}} ⋯ s_{i_{k}} the second condition in the statement is satisfied with w = s_{i_{1}} ⋯ s_{i_{k}}.

We conclude this section with the following fact about the Bruhat-Chevalley order, which will be needed later. Although this fact is known in the literature (see [8], Example 3, p.119) we give here a proof via semigroups, as a consequence of the above theorem.

Corollary 4.2. Let w_{0} be the principal involution of W, and take w_{1}, w_{2} ∈ W. Then w_{1} ≤ w_{2} if and only if w_{0}w_{1} ≥ w_{0}w_{2}.

Proof. Let S be a semigroup such that W (S) = {1}. The existence of such a semigroup is implicit in the theory of [9]: If S is the compression semigroup of some subset C contained in an open Bruhat cell N^{−}b_{0} then W (S) = 1. By the theorem w_{1} ≤ w_{2} if and only if D (w_{1}) ≥ D (w_{2}). On the other hand, we have from Proposition 3.1 that D^{−} (w_{0}w_{1})_{0} = D (w_{1})_{0} and D^{−} (w_{0}w_{2})_{0} = D (w_{2})_{0}. Hence in the order of the control sets for S^{−1} we have

D^{−} (w_{0}w_{1}) ≤ D^{−} (w_{0}w_{2}) .

Applying the theorem again we get w_{0}w_{1} ≥ w_{0}w_{2}.

5. Schubert cells

A Schubert cell in the maximal flag manifold B is the closure of an orbit of the subgroup N or any of its conjugates. As is well known, the N -orbits on B, the so called Bruhat cells, are Nwb_{0} with w running through the Weyl group W. The purpose of this section is to provide a geometric description of the Schubert cells in B in terms of smaller flag manifolds.

In the discussion to follow we keep fixed a simple system of roots Σ. For a finite sequence (with possible repetitions) α_{1}, …, α_{n} of simple roots we let s_{1}, …, s_{n} be the reflections with respect to these roots. Also, we denote by P_{i} = P_{{α_{i}}} the parabolic subgroup defined by Θ = {α_{i}}. The corresponding flag manifold is denoted by B_{i} = G/P_{i}. Associated with B_{i} there is the canonical fibration π_{i} : B → B_{i}. For i = 1, …, n we let γ_{i} stand for the operation of exhausting a subset of B with the fibers of π_{i}, that is, if X ⊂ B then

γ_{i} (X) = π_{i}^{−1}π_{i} (X)

We recall that π_{i} is equivariant under g ∈ G. This implies that γ_{i} is also equivariant under g, i.e., gγ_{i} (X) = γ_{i} (gX) for any subset X ⊂ B.

In the sequel we shall prove that a Schubert cell is a subset of the type γ_{1} ⋯ γ_{n} (b) for some sequence of simple roots and b ∈ B.

For w ∈ W put N^{w} = wNw^{−1} and let 𝔫^{w} = Ad (w) 𝔫 be its Lie algebra. Every Schubert cell is the image under some g ∈ G of cl (N^{w}b_{0}) for a w ∈ W. So we start by looking at these cells.

Let α ∈ Σ be a simple root such that −α ∈ w (Δ^{+}) ∩ Δ^{−} and consider the parabolic subgroup P_{α} = P_{{α}}. Denote by B_{α} the flag manifold G/P_{α} and let π : B → B_{α} be the canonical fibration. Also, let N_{α}^{−} stand for the nilpotent group generated by exp (𝔤_{−α} + 𝔤_{−2α}). The orbit P_{α}b_{0} is the fiber π^{−1}π (b_{0}) while N_{α}^{−}b_{0} is dense in this fiber. The fact that −α ∈ w (Δ^{+}) ∩ Δ^{−} implies that 𝔤_{−α} + 𝔤_{−2α} is contained in 𝔫^{w}. Hence N_{α}^{−}b_{0} ⊂ N^{w}b_{0}. Therefore we have that

γ (b_{0}) ⊂ cl (N^{w}b_{0}) .  (2)

where for a subset X ⊂ B, γ (X) = π^{−1}π (X). From this inclusion we get the following statement which is the main step in the proof of Theorem 5.3 below.

Proposition 5.1. As above, take α ∈ Σ such that −α ∈ w (Δ^{+}) ∩ Δ^{−}. Then

cl (N^{w}b_{0}) = γ (cl (N^{w}sb_{0}))

where s is the reflection with respect to the simple root α.

Proof. The fiber γ (b_{0}) contains sb_{0} so that π (b_{0}) = π (sb_{0}). This implies that the orbits N^{w}π (b_{0}) and N^{w}π (sb_{0}) in B_{α} coincide. Hence π (N^{w}b_{0}) = π (N^{w}sb_{0}) because π is equivariant. Taking closures and using the fact that π is a continuous and closed map we conclude that

π (cl (N^{w}b_{0})) = π (cl (N^{w}sb_{0})) .  (3)

Therefore cl (N^{w}b_{0}) ⊂ γ (cl (N^{w}sb_{0})).

In order to show the reverse inclusion, take x ∈ γ (cl (N^{w}sb_{0})). Then (3) ensures that π (x) ∈ cl (N^{w}π (b_{0})). Hence there exists a sequence n_{k} ∈ N^{w} such that n_{k}π (b_{0}) → π (x). By trivializing the bundle π : B → B_{α} around π (x) it is easy to see that there exists a sequence z_{k} ∈ B such that π (z_{k}) = n_{k}π (b_{0}) and z_{k} → x. Let y_{k} = n_{k}^{−1}z_{k}. Then π (y_{k}) = π (b_{0}) for all k so that we have from the inclusion in (2) that y_{k} ∈ cl (N^{w}b_{0}) and hence z_{k} = n_{k}y_{k} ∈ cl (N^{w}b_{0}). Since z_{k} → x we have that x ∈ cl (N^{w}b_{0}) concluding the proof.

From this proposition we can proceed by induction to get the desired description of the Schubert cells. Take a reduced expression

w = s_{1} ⋯ s_{n}

with s_{i} the reflection with respect to the simple, and hence indivisible root α_{i} ∈ Σ. It is well known (see e.g. [10, Thm. 4.15.10]) that the indivisible positive roots that are mapped into negative roots by w are

s_{n} ⋯ s_{2}α_{1}, …, s_{n}α_{n−1}, α_{n}.

Applying w to these roots we get that

w (Δ^{+}) ∩ Δ^{−} = {−α_{1}, −s_{1}α_{2}, …, −s_{1} ⋯ s_{n−1}α_{n}}.

For the inductive step it is needed the following lemma.

Lemma 5.2. Let w = s_{1} ⋯ s_{n} be a reduced product of reflections with respect to simple roots in Σ. For k = 1, …, n − 1 put t_{k} = s_{1} ⋯ s_{k} and consider the simple system of roots Σ_{k} = t_{k}Σ. Denote by Δ_{k}^{±} the positive (respectively negative) indivisible roots defined by Σ_{k}.

Then t_{k}s_{k+1}t_{k}^{−1} is the reflection with respect to the simple root t_{k}α_{k+1} ∈ Σ_{k} and −t_{k}α_{k+1} ∈ w (Δ^{+}) ∩ Δ_{k}^{−}.

Proof. For a root β denote by r_{β} the reflection it defines. Then r_{uβ} = ur_{β}u^{−1} for any u ∈ W. Hence t_{k}s_{k+1}t_{k}^{−1} is the reflection with respect to the simple root t_{k}α_{k+1} ∈ Σ_{k}. We have that −t_{k}α_{k+1} ∈ Δ_{k}^{−} and if k = n − 1 then −t_{k}α_{k+1} = wα_{n} so that −t_{k}α_{k+1} ∈ w (Δ^{+}). On the other hand, if k < n − 1 then

w^{−1} (−t_{k}α_{k+1}) = s_{n} ⋯ s_{k+2}α_{k+1}

and, as mentioned above, this is one of the positive roots in Δ^{+} which is mapped into Δ^{−}. Therefore −t_{k}α_{k+1} ∈ w (Δ^{+}) ∩ Δ_{k}^{−} as required.

With this background at hand we can give a geometric description of the Schubert cell cl (N^{w}b_{0}).

Theorem 5.3. Let w = s_{1} ⋯ s_{n} be a reduced expression as a product of reflections with respect to the simple roots in Σ. Then for any k = 1, …, n, we have

cl (N^{w}b_{0}) = γ_{1} ⋯ γ_{k} (cl (N^{w}s_{1} ⋯ s_{k}b_{0})) .  (4)

In particular cl (N^{w}b_{0}) = γ_{1} ⋯ γ_{n} (b_{w}) where b_{w} = wb_{0} is the only N^{w} -fixed point in B.

Proof. By induction on k. For k = 1, (4) reduces to the formula in Proposition 5.1 with s = s_{1} and α = α_{1}.

Suppose then that (4) holds for 1 < k ≤ n − 1. Let t_{k} = s_{1} ⋯ s_{k} and put Σ_{k} = t_{k}Σ, Δ_{k}^{±} = t_{k}Δ^{±}. Then t_{k}α_{k+1} ∈ Σ_{k} and the above lemma ensures that −t_{k}α_{k+1} ∈ w (Δ^{+}) ∩ Δ_{k}^{−}. This implies that if u_{k} = t_{k}^{−1}w then −α_{k+1} ∈ u_{k} (Δ^{+}) ∩ Δ^{−}. Hence we can apply Proposition 5.1 with α = α_{k+1} and u_{k} in place of w. In this case s = s_{k+1} and γ = γ_{k+1} so we have

cl (N^{u_{k}}b_{0}) = γ_{k+1} cl (N^{u_{k}}s_{k+1}b_{0}) .  (5)

Applying t_{k} to the left hand side gives

t_{k} cl (N^{u_{k}}b_{0}) = t_{k} cl (N^{u_{k}}t_{k}^{−1}t_{k}b_{0}) = cl (N^{w}t_{k}b_{0}) .

On the other hand, the image under t_{k} of the right hand side of (5) is

t_{k}γ_{k+1} cl (N^{u_{k}}s_{k+1}b_{0}) = γ_{k+1} cl (N^{w}t_{k}s_{k+1}b_{0}) = γ_{k+1} cl (N^{w}t_{k+1}b_{0}) .

Therefore cl (N^{w}t_{k}b_{0}) = γ_{k+1} cl (N^{w}t_{k+1}b_{0}) and

γ_{1} ⋯ γ_{k} cl (N^{w}t_{k}b_{0}) = γ_{1} ⋯ γ_{k}γ_{k+1} cl (N^{w}t_{k+1}b_{0})

completing the induction step.

Remark: The above description of a Schubert cell is comparable to the classical result known as Bott-Samelson desingularization which states that the cell cl (Nwb_{0}) is the image of the map ϕ : P_{1} × ⋯ × P_{n} → B defined by

ϕ (g_{1}, …, g_{n}) = g_{1} ⋯ g_{n}b_{0}

(see [1, 6, 7]). Here w = s_{1} ⋯ s_{n} and P_{i} is as above the parabolic subgroup associated with s_{i}. In the light of this result Theorem 4.1 provides an indirect proof that successive applications of the parabolic subgroups P_{i} yields successive exhaustions of the corresponding fibers.

The other Schubert cells are easily obtained from cl (N^{w}b_{0}). In the sequel we are particularly interested in the cells cl (N^{−}wb_{0}), w ∈ W, where N^{−} is the group opposed to N. For these cells we have

Corollary 5.4. Let w_{0} be the principal involution and suppose that w_{0}w = s_{n} ⋯ s_{1} is a reduced expression. Then

cl (N^{−}wb_{0}) = γ_{1} ⋯ γ_{n} (w_{0}b_{0})

where γ_{i} = π_{i}^{−1}π_{i} comes from the reflection s_{i}.

Proof. Since N^{−} = w_{0}Nw_{0}^{−1} we have that N^{−}wb_{0} = wN^{w^{−1}w_{0}}b_{0} so that

cl (N^{−}wb_{0}) = wγ_{1} ⋯ γ_{n} (w^{−1}w_{0}b_{0})

and the corollary follows from the equivariance of γ_{i}.

6. Domain of Attraction of a Control Set

In this section we apply the previous results about the Schubert cells and the order of control sets to show that the domain of attraction of a control set is a union of Schubert cells.

For this we need the following consequences of Theorem 4.1: From Borel and Tits [2, Thm. 3.13] we know that a Schubert cell is given by

cl (Nwb_{0}) = ⋃_{w_{1}≤w} Nw_{1}b_{0}.  (6)

In this formula N and b_{0} are linked by the fact that b_{0} is the only N -fixed point. If we take N^{−} instead of N then N^{−} = w_{0}Nw_{0}^{−1} where w_{0} is the principal involution of W. This subgroup is linked to b^{−} = w_{0}b_{0}. For N^{−} and b^{−} formula (6) reads

cl (N^{−}wb^{−}) = ⋃_{w_{1}≤w} N^{−}w_{1}b^{−}  (7)

(recall that the orders in W defined by Σ and −Σ coincide). We need now a formula similar to (7) where in the left hand side b^{−} is replaced by b_{0}.

Proposition 6.1. The Schubert cell cl (N^{−}wb_{0}) is given by

cl (N^{−}wb_{0}) = ⋃_{s≥w} N^{−}sb_{0}.

Proof. We have

cl (N^{−}wb_{0}) = cl (N^{−} (ww_{0}) w_{0}b_{0}) = cl (N^{−} (ww_{0}) b^{−}) .

Applying the Borel-Tits formula (7) to the right hand side of this equality we get

cl (N^{−}wb_{0}) = ⋃_{w_{1}≤ww_{0}} N^{−}w_{1}b^{−} = ⋃_{w_{1}≤ww_{0}} N^{−} (w_{1}w_{0}) b_{0}.

So that by putting s = w_{1}w_{0} we have

cl (N^{−}wb_{0}) = ⋃{N^{−}sb_{0} : sw_{0} ≤ ww_{0}}.

Now sw_{0} ≤ ww_{0} if and only if w_{0}s^{−1} ≤ w_{0}w^{−1} and by Corollary 4.2 this happens if and only if s^{−1} ≥ w^{−1}. So that the above union runs through s such that s ≥ w as claimed.

As a consequence of this result we can prove that certain Schubert cells are contained in the domain of attraction of a control set. This fact will be needed below in the proof of the main result of this section.

Proposition 6.2. If A^{+} ∩ int S ≠ Ø then cl (N^{−}wb_{0}) ⊂ A (D (w)) for every w ∈ W.

Proof. From the above proposition we have

cl (N^{−}wb_{0}) = ⋃_{s≥w} N^{−}sb_{0}.

In this formula each sb_{0} belongs A (D (w)). In fact, sb_{0} ∈ D (s)_{0}. But if s ≥ w then D (s) ≤ D (w) so that D (s) ⊂ A (D (w)). On the other hand, if n ∈ N^{−} and h ∈ A^{+} ∩int S then h^{k}nsb_{0} → sb_{0}, as k → ∞, and since sb_{0} is in the core of a control set this implies that nsb_{0} sb_{0}. Hence nsb_{0} wb_{0} so that nsb_{0} ∈ A (D (w)).

We can now prove our main result about domains of attraction.

Theorem 6.3. Let C^{∗} = D (w_{0}) be the minimal control set. Then for all w ∈ W the domain of attraction A (D (w)) of D (w) is given by

A (D (w)) = γ_{1} ⋯ γ_{n} (C^{∗}) .  (8)

Here the sequence γ_{1}, …, γ_{n} comes from a reduced expression

w_{0}w = s_{n} ⋯ s_{1}

where w_{0} is the principal involution of W.

Proof. The inclusion γ_{1} ⋯ γ_{n} (C^{∗}) ⊂ A (D (w)) is a consequence of the previous characterization of the Schubert cells and the above proposition. In fact, it is easily checked by induction that γ_{1} ⋯ γ_{n} (C^{∗}) = ⋃_{b∈C^{∗}} γ_{1} ⋯ γ_{n} (b) so it is required to show that

γ_{1} ⋯ γ_{n} (b) ⊂ A (D (w))  (9)

for all b ∈ C^{∗}. Actually it is enough to prove this for b ∈ C_{0}^{∗} because if b ∈ C^{∗} then there exists g ∈ S such that gb ∈ C_{0}^{∗}. Since gγ_{1} ⋯ γ_{n} (b) = γ_{1} ⋯ γ_{n} (gb) we have (9) from the inclusion for gb. Now, if b ∈ C_{0}^{∗} then there exists a split-regular h ∈ int S such that b is the repeller of h in B. So that if we take our basic objects such that h ∈ A^{+} then from Corollary 5.4 we have that the left hand side of (9) is the Schubert cell cl (N^{−}wb_{0}), which is contained in A (D (w)) by the above proposition.

The proof that A (D (w)) is contained in the right hand side of (8) is by induction on n, the length ℓ (w_{0}w) of w_{0}w. If n = 0 then w_{0}w = 1, w = w_{0} and D (w) = C^{∗}. So that (8) holds because A (C^{∗}) = C^{∗}.

If ℓ (w_{0}w) = n, let w_{0}w = s_{n} ⋯ s_{2}s_{1} be a reduced expression and define

w_{1} = w_{0}s_{n} ⋯ s_{2}.

Then ℓ (w_{0}w_{1}) = ℓ (w_{0}w) − 1 so we have by the induction hypothesis that

A (D (w_{1})) = γ_{2} ⋯ γ_{n} (C^{∗}) .  (10)

Let π_{1} : B → B_{1} be the projection corresponding to s_{1}. By construction we have that π_{1} (b_{0}) = π_{1} (s_{1}b_{0}). Applying w_{1} to this equality we get that π_{1} (w_{1}b_{0}) = π_{1} (wb_{0}). This implies that there exists a control set, say E, in B_{1} such that

π_{1} (D (w)_{0}) = π_{1} (D (w_{1})_{0}) = E_{0}

(see [9, Prop. 5.1]). Now, take y ∈ A (D (w)) and g ∈ S such that gy ∈ D (w)_{0}. Then gπ_{1} (y) ∈ E_{0} and since π_{1}^{−1}π_{1} (gy) intersects D (w_{1})_{0} and g interchanges fibers, there exists z in the same fiber as y such that gz ∈ D (w_{1})_{0}. By definition z ∈ A (D (w_{1})) so (10) implies that

z ∈ γ_{2} ⋯ γ_{n} (C^{∗}) .

Hence π_{1} (y) = π_{1} (z) ∈ π_{1}γ_{2} ⋯ γ_{n} (C^{∗}), that is,

y ∈ γ_{1}γ_{2} ⋯ γ_{n} (C^{∗})

concluding the proof.

7. The other flag manifolds

As before we denote by B_{Θ} the flag manifold defined by Θ ⊂ Σ and by π_{Θ} : B → B_{Θ} the natural projection from the maximal flag manifold B. If D ⊂ B is a control set then π_{Θ} (D_{0}) is the core of a control set in B_{Θ} and reciprocally if E is a control set in B_{Θ} then there exists w ∈ W such that π_{Θ} (D (w)_{0}) = E_{0} (see [9, Prop. 5.1]). This gives a well defined map

w ⟼ E (w)  (11)

onto the control sets in B_{Θ}. Since we have further that for w_{1}, w_{2} ∈ W, π_{Θ} (D (w_{1})_{0}) = π_{Θ} (D (w_{2})_{0}) if and only if there exists w ∈ W (S) w_{2} such that π_{Θ} (wb_{0}) = π_{Θ} (w_{2}b_{0}) this map factors through W (S) \W/W_{Θ} establishing a oneto-one correspondence between these double cosets and the control sets in B_{Θ}. From this correspondence we can detect the order and domains of attraction of the control sets in B_{Θ}.

The Bruhat-Chevalley order in W induces an order in W/W_{Θ} by putting w_{1}W_{Θ} ≤ w_{2}W_{Θ} if one of the following equivalent conditions hold:

1. There exists a ∈ w_{1}W_{Θ} such that a ≤ w_{2}.

2. There exists b ∈ w_{2}W_{Θ} such that w_{1} ≤ b.

3. There are a ∈ w_{1}W_{Θ} and b ∈ w_{2}W_{Θ} such that a ≤ b.

It is not hard to check, either from the definition of the order or from its relation with the order of the control sets, that these are indeed equivalent conditions. Actually, it can be shown that in each coset wW_{Θ} there is just one element of minimal length, whose order provide the order between the cosets. In [5] the elements of minimal length in the cosets were written down explicitly for the classical diagrams.

Clearly a similar order is defined in the quotient W_{Θ}\W. Since W (S) is a parabolic subgroup of W we have in particular that W (S) \W is ordered. Note that by Theorem 4.1 the control sets in B are ordered according to this order in W (S) \W. We can now factor this order once again and get the order of the control sets in B_{Θ}.

Proposition 7.1. For u ∈ W let E (u) be the control set in B_{Θ} as given by (11). Take w_{1}, w_{2} ∈ W. Then E (w_{1}) ≤ E (w_{2}) if and only if there exists w ∈ w_{2}W_{Θ} such that D (w_{1}) ≤ D (w).

Proof. If w is as in the statement then there are g ∈ int S, x ∈ D (w_{1})_{0} and y ∈ D (w)_{0} such that gx = y. By the definition in (11) it follows that π_{Θ} (x) ∈ E (w_{1})_{0}, π_{Θ} (y) ∈ E (w)_{0} and E (w_{2}) = E (w). Therefore gπ_{Θ} (x) ∈ E (w_{2})_{0} showing that E (w_{1}) ≤ E (w_{2}).

For the converse fix a split-regular h ∈ int S and let b_{0} be its attractor. Then ub_{0} ∈ D (u) for all u ∈ W. Hence for i = 1, 2, π_{Θ} (w_{i}b_{0}) belongs to E (w_{i})_{0}. Therefore there exists g ∈ int S which maps the fiber through w_{1}b_{0} onto the fiber through w_{2}b_{0}. Since h has an attractor in the latter fiber, it follows that there exists w ∈ w_{2}W_{Θ} such that D (w_{1}) ≤ D (w) concluding the proof.

Corollary 7.2. For w_{1}, w_{2} ∈ W the following assertions are equivalent

1. E (w_{1}) ≤ E (w_{2}).

2. There exists w ≤ w_{1} such that w ∈ W (S) w_{2}W_{Θ}, that is, E (w) = E (w_{2}).

Proof. From the above proposition we have that E (w_{1}) ≤ E (w_{2}) if and only if there is w_{3} ∈ w_{2}W_{Θ} such that D (w_{1}) ≤ D (w_{3}). On the other hand, by Theorem 4.1 this inequality holds if and only if there is w ∈ W (S) w_{3} such that w_{1} ≥ w. These two equivalences together imply the corollary.

This corollary implies at once that the order in W factors through the double coset W (S) \W/W_{Θ} in such a way that the bijection with the control sets is order reversing.

We turn now to the domains of attraction of the control sets in B_{Θ}. These are projections of the domains of attraction of the control sets in B. In fact,

Proposition 7.3. As above let E (w) be a control set in B_{Θ}. Then

A (E (w)) = π_{Θ} (A (D (w))) .

Proof. If x ∈ A (D (w)) then there exists g ∈ int S such that gx ∈ D (w)_{0}. Since π_{Θ} (D (w)_{0}) = E (w)_{0}, equivariance of π_{Θ} implies that gπ_{Θ} (x) ∈ E (w)_{0} so that π_{Θ} (x) ∈ A (E (w)) showing that π_{Θ} (A (D (w))) ⊂ A (E (w)).

For the other inclusion take y ∈ A (E (w)). Then there exists g ∈ int S such that gy = π_{Θ} (wb_{0}) because π_{Θ} (wb_{0}) ∈ E (w)_{0}. Hence g maps the fiber π_{Θ}^{−1} (y) onto the fiber through wb_{0}. Therefore there exists x ∈ π_{Θ}^{−1} (y) such that gx = wb_{0}. We have that x ∈ A (D (w)) and π_{Θ} (x) = y showing that A (E (w)) ⊂ π_{Θ} (A (D (w))).

As a consequence of this proposition we have that the domains of attraction of the control sets in B_{Θ} are also unions of Schubert cells. In fact, the Schubert cells in B_{Θ} are projections of the Schubert cells in B. Therefore we have from Theorem 6.3 that

A (E (w)) = π_{Θ}γ_{1} ⋯ γ_{n} (C^{∗})

is the union of the Schubert cells in B_{Θ} determined by the split-regular elements in int S.

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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 August 13, 1997 and in final form September 9, 1997