Journal of Lie Theory Volume 8 (1998) 415–428 ◯^{C}1998 Heldermann Verlag

Projective subspaces in the variety of normal sections and tangent spaces to a symmetric space

Walter Dal Lago, Alicia García and Cristián U. Sánchez^{∗}

Communicated by Karl Strambach

Abstract. In the present article we continue the study of the variety X [M] associated to pointwise planar normal sections of a natural imbedding for a flag manifold M. When M = G/T is the manifold of complete flags of a compact simple Lie group G, we obtain two results about subspaces of the tangent space T_{[T]} (M), invariant by the torus action, which give rise to real projective spaces in X [M]. The first result determines their maximal dimension. While the other one characterizes those of maximal dimension as tangent spaces to the inner symmetric space G/K (the one of largest dimension for the group G) at a fixed point of the natural action of the torus T. The last section contains a nice application of these results.

1. Introduction

In the present article we continue our study of the variety X [M] of directions of pointwise planar normal sections for a manifold of complete flags M. The nature of this variety gives information about the extrinsic geometry of a natural imbedding of a flag manifold. For instance, as we saw in [5], this variety happens to be a projective space if and only if M is an extrinsic symmetric submanifold.

In previous papers [5] and [6] we have gotten some results that, we fell, help to understand the nature of these varieties. In [5] we found a characterization of the set of pointwise planar normal sections in terms of the tensors α and D on a general flag manifold M which indicates that this set is in fact a real algebraic variety. In that paper we also computed its Euler-Poincaré characteristic showing that it depends only on the dimension and hence gives little information about the geometric nature of the flag manifold M.

In [6] we studied the presence of projective subspaces in the variety X [M] , for a manifold of complete flags M = G_{u}/T. The existence of these subspaces indicates that the variety X [M] is rather special. The main result of that paper

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag describes a natural family of maximal projective subspaces contained in X [M] . On the other hand [6, Th. 4.2] (Theorem 2.2 in the present paper) characterizes those subspaces of the tangent space T_{[T]} (M), invariant by the torus action, which give rise to real projective spaces contained in X [M]. This result motivated our interest in a deeper study of these subspaces and gave rise to the present paper.

∗Research partially supported by CONICET, CONICOR and SECYT-UNCba, Argentina.

Here we present the following two theorems on the variety X [M], for M = G_{u}/T manifold of complete flags, and an interesting consequence of them. In both of them G_{u} denotes a compact simply connected simple Lie group.

The meaning of the name presymmetric set is explained at the end of the next section where all our notation is presented. Our first result identifies the largest possible cardinality for these sets of positive roots and the second one shows that each one of them gives the tangent space, at a specific point, of the inner symmetric space G_{u}/K (the one of largest dimension d (G_{u}) for the group G_{u}). These results show an interesting connection of the variety X [M] and the geometry of the natural fibration G_{u}/T → G_{u}/K. The result obtained as application of these facts (Theorem 4.1) is a nice expression of this connection.

Theorem 1.1. For each G_{u}, the number \frac{1}{2}d (G_{u}) is the largest possible cardinality for a presymmetric set.

This means that if 𝔭̃ ⊂ T_{[T]} (M) is invariant by the natural action of the torus T and gives rise to a projective subspace of X [M] then dim 𝔭̃ ≤ d (G_{u}) .

Theorem 1.2. For each G_{u} and each presymmetric set Δ̃ whose cardinality is \frac{1}{2}d (G_{u}) , the subspace 𝔭̃ = ∑_{γ∈_{Δ}} 𝔪_{γ} is tangent to the inner symmetric space G_{u}/K at a fixed point of the action of the torus T.

It is clear that every tangent space to G_{u}/K at a fixed point of the torus action is of this form, i.e. comes from a presymmetric set of roots. So the last Theorem characterizes the tangent spaces to the inner symmetric space G_{u}/K at those points.

Notice that the hypothesis of Theorem 1.2 mean that 𝔭̃ ⊂ T_{[T]} (M) is invariant by the natural action of the torus T, its dimension is d (G_{u}) and it gives rise to a projective subspace of X [M].

This paper is organized as follows. In the next section we include the basic facts obtained in [5] and [6] as well as notations used throughout the paper. In Section 3 we study presymmetric sets of roots and obtain the proofs of Theorems 1.1 and 1.2. In Section 4 we present the mentioned application of these results.

2. Basic facts

In the present section we introduce some of the basic notation to be used throughout the paper. All unexplained notation will have the same meaning as in [5].

Let G be a simply connected, complex, simple Lie group and let 𝔤 be its Lie algebra. Let 𝔥 be a Cartan subalgebra of 𝔤 and Δ = Δ (𝔤, 𝔥) the root system of 𝔤 relative to 𝔥. We may write

𝔤 = 𝔥⊕ ∑_{γ∈Δ^{+}} 𝔤_{γ} ⊕ 𝔤_{−γ}

where Δ^{+} indicates the set of positive roots with respect to some order. Let us consider in 𝔤 the Borel subalgebra

𝔟 = 𝔥⊕ ∑_{γ∈Δ^{+}} 𝔤_{−γ}.

Let B be the analytic subgroup of G corresponding to the subalgebra 𝔟. B is closed and its own normalizer in G. The quotient space M = G/B is a complex homogeneous space called the manifold of complete flags of G.

Let π = {α_{1}, ..., α_{n}} ⊂ Δ^{+} be a system of simple roots. We may take in 𝔤 a Weyl basis [11, III, 5] {X_{γ} : γ ∈ Δ} and {H_{β} : β ∈ π} . The following set of vectors provides a basis of a compact real form 𝔤_{u} of 𝔤.

\{ \begin{matrix}U_{γ} = \frac{1}{\sqrt{2}} (X_{γ} − X_{−γ}) & γ ∈ Δ^{+} \\ U_{−γ} = \frac{i}{\sqrt{2}} (X_{γ} + X_{−γ}) & γ ∈ Δ^{+} \\ iH_{β} & β ∈ π. \end{matrix}  (1)

We shall denote by 𝔥_{u} the real vector space generated by {iH_{β} : β ∈ π} and by 𝔪_{γ} that of {U_{γ}, U_{−γ}} . Then we may write 𝔤_{u} = 𝔥_{u} ⊕ ∑_{γ∈Δ^{+}} 𝔪_{γ} = 𝔥_{u} ⊕ 𝔪.

Let G_{u} be the analytic subgroup of G corresponding to 𝔤_{u}. G_{u} is compact and acts transitively on M which can be written as

M = G_{u}/T

where the subgroup T = G_{u} ∩ B = exp 𝔥_{u} is a maximal torus in G_{u}. The manifold M is then a compact simply connected complex manifold and it is well known that it is the orbit of a regular element E ∈ 𝔤_{u} by the adjoint action of G_{u} on 𝔤_{u}. Then we have a natural embedding of M on 𝔤_{u} which we may assume isometric by taking in 𝔤_{u} the inner product given by the opposite of the Killing form.

For this embedding we consider the algebraic variety X [M] of directions of pointwise planar normal sections of the flag manifold M which was introduced in [5].

Associated to each simple group G_{u} we have its family of symmetric spaces of type I [11, p. 518] and among them, those which are inner (i.e. the spaces in which the symmetry at each point belongs to the group G_{u}). These are, among all symmetric spaces, the only ones that are related with our algebraic variety X [M] , as we saw in [6]. They are those of the form G_{u}/K, where K is a subgroup of maximal rank in G_{u}. The ones which are not inner in the list in [11, p. 518] are AI, AII, BDI (p + q = 2n, p odd, 1 ≤ p ≤ n), EI and EIV.

By conjugating K if necessary, we may assume that K contains T.

Let 𝔨 be the Lie algebra of K and write 𝔤_{u} = 𝔨⊕𝔭 where 𝔭 is the orthogonal complement to 𝔨 with respect to the Killing form. Then 𝔥_{u} ⊂ 𝔨 and 𝔭 ⊂ 𝔪.

Now we recall a few results from [6] which we need in the present paper. We shall denote by RP (𝔮) the real projective space associated to a real vector space 𝔮.

Proposition 2.1. [6, 4.1] The tangent space 𝔭 of the inner symmetric space G_{u}/K at [K] gives rise to a projective space RP (𝔭) contained in X [M] .

The subspaces 𝔭̃ ⊂ 𝔪 invariant by the natural action of the torus T (see [5]) are those of the form 𝔭̃ = ∑_{γ∈_{Δ}} 𝔪_{γ} where Δ̃ is any subset of Δ^{+}(an example of this subspace is 𝔭 of above proposition). For these subspaces we have the following results.

Theorem 2.2. [6, 4.2] If 𝔭̃ = ∑_{γ∈_{Δ}} 𝔪_{γ} with Δ̃ ⊂ Δ^{+} then

RP (𝔭̃) ⊂ X [M] ⟺ (ε, ρ ∈ Δ̃ ⇒ ε + ρ ∉ Δ̃) .

The condition ε, ρ ∈ Δ̃ ⇒ ε + ρ ∉ Δ̃ is weaker than the one usually given in the definition of symmetric space. For this reason, in order to simplify our notation, a subset Δ̃ ⊂ Δ^{+} with this property will be called a presymmetric set.

For a set A, # (A) will denote its cardinality.

Theorem 2.3. [6, 4.3] Let 𝔭 be the tangent space of the inner symmetric space G_{u}/K at [K] . Then RP (𝔭) is maximal among the projective spaces RP \binom{}{𝔭} contained in X [M] , with 𝔭̃ of the form 𝔭̃ = ∑_{γ∈_{Δ}} 𝔪_{γ} for Δ̃ ⊂ Δ^{+}.

3. Presymmetric sets

The list of irreducible symmetric spaces [11, p. 518] indicates that the irreducible inner symmetric spaces of maximal dimension are those included in the following table, with their respective dimensions.

Table I

𝔤 name N = G_{u}/K  dimN
𝔞_{l} AIII SU(l + 1)/S(U(k + 1) × U(k))  l = 2k  \frac{1}{2}l(l + 2)
SU(l + 1)/S (U (k + 1) × U (k + 1)) l = 2k + 1 \frac{1}{2} (l + 1)^{2}
𝔟_{l} BDI SO(2l + 1)/SO(l + 1) × SO(l)  l(l + 1)
𝔠_{l} CI  Sp(l)/U(l)  l(l + 1)
𝔡_{l} BDI SO(2l)/SO(l) × SO(l)  l even  l^{2}
SO(2l)/SO(l + 1) × SO(l − 1)  l odd  l^{2} − 1
𝔢_{6} EII  E_{6}/SU(6)Sp(1)  40
𝔢_{7} EV  E_{7}/(SU(8)/Z_{2})  70
𝔢_{8} EVIII E_{8}/(Spin(16)/Z_{2)}  128
𝔣_{4} FI  F_{4}/Sp(3)Sp(1)  28
𝔤_{2} G  G_{2}/SO(4)  8

We shall denote by d (G_{u}) the dimension dim (G_{u}/K) indicated in the table. From Proposition 2.1 and Theorem 2.2 we conclude that if 𝔭 = ∑_{γ∈_{Δ}} 𝔪_{γ}

is the tangent space at [K] , for each one of the spaces in the table, then Δ̃ is a presymmetric set. It is also clear that the cardinality of Δ̃ is # \binom{}{Δ} = \frac{1}{2}d (G_{u}) .

We denote by Γ the set of positive roots of odd height [13, p. 47] with respect to the system of simple roots π.

In order to give the proof of Theorem 1.1 we need the following four lemmas.

Lemma 3.1. #(Γ) = \frac{1}{2}d (G_{u}) .

Proof. It is clear that the statement is true for the exceptional algebras (see [10, pp. 527-531]). We give a proof for each family of classical simple Lie algebras.

𝔞_{l}. For each h ≥ 1, the roots of height h are of the form ∑_{k≤i≤k+h−1} α_{i} with 1 ≤ k ≤ l − h + 1.

If l = 2k, there are 2 (k − r) positive roots of height 2r + 1 with 0 ≤ r ≤ k − 1. Therefore

#(Γ) = ∑_{0≤r≤k−1} 2 (k − r) = \frac{1}{4}l (l + 2) .

If l = 2k + 1, there are 2 (k − r) + 1 positive roots of height 2r + 1 with 0 ≤ r ≤ k. Therefore

#(Γ) = ∑_{0≤r≤k} 2 (k − r) + k + 1 = \frac{1}{4} (l + 1)^{2} .

𝔟_{l} (and 𝔠_{l}). For each positive root α let j (α) = j be the number of coefficients equal to 2 of α with respect to the system π. Clearly 0 ≤ j ≤ l − 1 and the set of roots in Γ with j = 0 has as many elements as in the algebra 𝔞_{l}, that is \frac{1}{4}l (l + 2) or \frac{1}{4} (l + 1)^{2} according to wether l is even or odd.

Let us assume first that l = 2k. In that set, the amount of odd positive roots with 1 ≤ j ≤ l − 1 is k − s if j = 2s or j = 2s + 1. Then

\begin{aligned}#(Γ) & = & \frac{1}{4}l (l + 2) + ∑_{1≤s≤k−1}(k − s) + ∑_{0≤s≤k−1} (k − s) \\ & = & \frac{1}{2}l (l + 1) . \end{aligned}

Now, if l = 2k + 1 we have k − s + 1 or k − s odd positive roots when j = 2s or j = 2s + 1 respectively. Thus

\begin{aligned}#(Γ) & = & \frac{1}{4} (l + 1)^{2} + ∑_{1≤s≤k}(k − s + 1) + ∑_{0≤s≤k−1} (k − s) \\ & = & \frac{1}{2}l (l + 1) . \end{aligned}

𝔡_{l}. The number of elements of Γ such that the coefficients of α_{l−1} and α_{l} coincide is the same as for the case 𝔟_{l−2} with the addition of \frac{1}{2}l − 1 if l is even and with the addition of \frac{1}{2} (l − 1) if l is odd. On the other hand, the number of elements such that the coefficients of α_{l−1} and α_{l} are different is l or l − 1 according to wether l is even or odd.

\begin{aligned}# (Γ) & = & \frac{1}{2} (l − 1) (l − 2) + \frac{1}{2}l − 1 + l & = & \frac{1}{2}l^{2} & l even \\ # (Γ) & = & \frac{1}{2} (l − 1) (l − 2) + \frac{1}{2} (l − 1) + l − 1 & = & \frac{1}{2} (l^{2} − 1) & l odd \end{aligned}

Lemma 3.2. For each simple Lie algebra 𝔤, if Δ̃ ⊂ Δ^{+} (𝔤, 𝔥) is a presymmetric set and satisfies

i) # \binom{}{Δ} = \frac{1}{2}d (G_{u}) ,

ii) π ⊂ Δ̃;
then Δ̃ = Γ.

Proof. Let Λ be the subset of Δ̃ of those roots that have even height. If Λ = ϕ the result follows from Lemma 3.1. Let us assume now that Λ ≠ ϕ.

Consider first the case in which 𝔤 is of classical type. We make the following claim: Ψ : Λ → Γ defined by Ψ (γ) = γ − α_{j_{γ}} where α_{j_{γ}} is the first simple root such that γ − α_{j_{γ}} is a root of 𝔤, is injective.

If β and γ are two roots in Λ such that Ψ (γ) = Ψ (β) i.e. γ−α_{j_{γ}} = β −α_{j_{β}} and either j_{γ} = j_{β} or the coefficients of γ and β,with respect to π, are all different from 2 then it is easy to see that γ = β. This proves that the claim is true for the algebras of type 𝔞_{l}.

Assume now that the algebra 𝔤 is of type 𝔟_{l} or 𝔡_{l}. If our β ∈ Λ has some of its coefficients, say k_{i} (β) , equal to 2 then β has an even number of coefficients k_{t} (β) equal to 1 for t < i. Then j_{β} is the index corresponding to the first coefficient equal to 1 of β. Under this condition on β the root γ must have also a coefficient equal to 2 and (for analogous reasons to the previous case) j_{γ} corresponds also to the first coefficient equal to 1 in the root γ. Then it is clear that γ = β.

Let us consider now the case 𝔠_{l}. If our β ∈ Λ has some of its coefficients, say k_{i} (β) , equal to 2 then β has an odd number of coefficients k_{t} (β) equal to 1 for t < i. Then j_{β} is the index corresponding to the first coefficient equal to 1 of β. Under this condition on β the root γ must have also a coefficient equal to 2 and (for analogous reasons to the previous case) j_{γ} corresponds also to the first coefficient equal to 1 in the root γ. Then it is clear that γ = β.

For the exceptional Lie algebras one may construct an injective function Ψ by choosing for each γ ∈ Λ a convenient simple root α_{j(γ)}. This is seen by checking the roots of even height of these algebras. (see [10, pp. 528-530]).

This proves our claim.

Now we observe that if the algebra 𝔤 is not of type 𝔞_{l} with l even, then the maximal root μ has odd height. In this case if ν is of maximal height in Λ, then there is a simple root α_{j} such that ν + α_{j} ∈ Γ. Since Δ̃ is a presymmetric set and due to hypothesis (ii) we have that Ψ (Λ) ⊂ (Γ − Δ̃) and ν + α_{j} ∈ Γ − Δ̃.

Therefore

# \binom{}{Δ} = # (Λ) + # (Γ ∩ Δ̃) = # (Ψ(Λ)) + # (Γ ∩ Δ̃) < # (Γ)

and by (i) this is a contradiction.

To finish our proof it remains to reach the same contradiction for the algebras of type 𝔞_{l} with l even. If μ ∉ Δ̃ the proof is the same as above. Now if μ ∈ Δ̃ then μ − α_{l} ∈ Γ − (Ψ (Λ) ∪ Δ̃) and so it also follows that

# \binom{}{Δ} < # (Γ) .

Our proof is now complete.

Lemma 3.3. If Δ̃ ⊂ Δ^{+} (𝔤, 𝔥) is a presymmetric set and the maximal root μ is in Δ̃ then # \binom{}{Δ} ≤ \frac{1}{2}d (G_{u}) .

Proof. We indicate first some notation that will be used in this proof.

If 𝔤 is not of type 𝔞_{l} there is only one root α_{j} ∈ π such that μ − α_{j} ∈ Δ and μ is the unique positive root for which k_{j} (μ) = 2. In this case set Δ_{0} = {γ ∈ Δ^{+} : k_{j} (γ) = 0} , Δ_{1} = {γ ∈ Δ^{+} : k_{j} (γ) = 1} .

When 𝔤 is of type 𝔞_{l}, μ − α_{j} ∈ Δ for j = 1, l (l ≥ 2) and we set Δ_{0} = {γ ∈ Δ^{+} : k_{1} (γ) = k_{l} (γ) = 0} , Δ_{1} = {γ ∈ Δ^{+} : k_{1} (γ) + k_{l} (γ) = 1} .

Then, for any 𝔤, we have Δ^{+} = Δ_{0}∪Δ_{1}∪{μ} and by inspection, we observe that the set Δ_{1} is union of pairs of positive roots γ_{j} and β_{j} such that γ_{j} + β_{j} = μ.

Let now Δ̃ ⊂ Δ^{+} (𝔤, 𝔥) be a presymmetric set containing μ. It is clear that # \binom{}{Δ} = # (Δ̃ ∩ Δ_{0}) + # (Δ̃ ∩ Δ_{1}) + 1 and # (Δ̃ ∩ Δ_{1}) ≤ \frac{1}{2}# (Δ_{1}) .

To study # (Δ̃ ∩ Δ_{0}) and # (Δ_{1}) (and therefore estimate # \binom{}{Δ}) we must separate the proof into different cases according to the type of the simple Lie algebra in question. The proof in each case follows the same pattern but the numbers involved are different. In the first place one does it, by induction, for the Lie algebras of classical type and then one considers each exceptional algebra using the result obtained for the classical ones. For sake of brevity we will do only one case since we think that the reader will have no difficulty in reconstructing the proof in the other cases.

Let 𝔤 be an algebra of type 𝔞_{l}. The lemma is immediate for l = 1, 2. We proceed with the proof by induction on l. Let l ≥ 3. With Δ_{0} and Δ_{1} indicated above for this case we have # (Δ_{1}) = 2 (l − 1) , (Δ̃ ∩ Δ_{0}) is a presymmetric set for an algebra of type 𝔞_{l−2} (α_{2}, ..., α_{l−1} are its simple roots) and by the inductive hypothesis its cardinality is

# (Δ̃ ∩ Δ_{0}) ≤ \{ \begin{matrix}\frac{1}{4}l (l − 2) & l even \\ \frac{1}{4} (l − 1)^{2} & l odd. \end{matrix}

Then

# \binom{}{Δ} ≤ \{ \begin{matrix}\frac{1}{4}l (l − 2) + (l − 1) + 1 = \frac{1}{4}l (l + 2) & l even \\ \frac{1}{4} (l − 1)^{2} + (l − 1) + 1 = \frac{1}{4} (l + 1)^{2} & l odd. \end{matrix}

Lemma 3.4. For each simple Lie algebra 𝔤, if Δ̃ ⊂ Δ^{+} (𝔤, 𝔥) is a presymmetric set such that # \binom{}{Δ} = \frac{1}{2}d (G_{u}) then there exist elements γ in Δ̃ and σ in the Weyl group of Δ (𝔤, 𝔥) such that σ (γ) is the maximal root μ.

Proof. Since in each simple Lie algebra the maximal root μ is long, it suffices to find a long root in Δ̃ (see [13, p. 53]). This is clear when all roots have the same length. It is necessary to consider only the algebras of type 𝔟_{l}, 𝔠_{l}, 𝔣_{4} and 𝔤_{2}.

If the algebra 𝔤 is if type 𝔟_{l} then it has only l positive short roots . Since \frac{1}{2}d (SO (2l + 1)) = \frac{1}{2}l (l + 1) > l there is at least a long root γ in Δ̃.

For the algebras 𝔣_{4} and 𝔤_{2} the conclusion is reached similarly.

If 𝔤 is of type 𝔠_{l} there are l (l − 1) positive short roots while \frac{1}{2}d (G_{u}) = \frac{1}{2}l (l + 1) . In this case we will prove the lemma, by induction on l, showing that if Δ̃ is a set of positive short roots then its cardinality is less than \frac{1}{2}l (l + 1) . For l = 3 this is true because the set of positive short roots is

{α_{1}, α_{2}, α_{1} + α_{2}, α_{2} + α_{3}, α_{1} + α_{2} + α_{3}, α_{1} + 2α_{2} + α_{3}}

and clearly is not a presymmetric set. For l = 4 the set of positive short roots has twelve elements and it is easy to find a partition into four subsets of the form {β, γ, β + γ}. Then any presymmetric set of short root has at most eight elements and since \frac{1}{2}l (l + 1) is 10 in this case, the assertion follows.

Let now l ≥ 5. It is easy to see that ([13, p. 64])

ν = α_{1} + ∑_{1<j<l} 2α_{j} + α_{l}

is a short root.

Let us assume first that ν ∈ Δ̃ and consider the following three subsets

Δ̃_{r} = {γ ∈ Δ̃ : k_{2} (γ) = r} r = 0, 1, 2.

Let us notice that:

i) Since for each short root of the form ∑_{1≤j<l} k_{j}α_{j} + α_{l} the first nonzero coefficient k_{j} (1 ≤ j < l) is equal to 1 it follows that Δ̃_{2} = {v} .

ii) Set Δ_{1} = {γ ∈ Δ^{+} : k_{2} (γ) = 1} . The cardinality of Δ_{1} is 4l −8 and by inspection we can see that the set Δ_{1} is the union of pairs of positive short roots γ_{j}, β_{j} such that γ_{j} + β_{j} = ν. Therefore # \binom{}{Δ_{1}} = # (Δ̃ ∩ Δ_{1}) ≤ 2l − 4.

iii) Δ̃_{0} − {α_{1}} is a presymmetric set of positive short roots for a subalgebra of type 𝔠_{l−2} and then, by the inductive hypothesis, # (Δ̃_{0} − {α_{1}}) < \frac{1}{2} (l − 2) (l − 1) .

Then, since Δ̃_{0}, Δ̃_{1} and Δ̃_{2} conform a partition of Δ̃, we have

# \binom{}{Δ} ≤ \frac{1}{2}(l − 2)(l − 1) + 1 + 2l − 4 + 1 < \frac{1}{2}l (l + 1) .

Assume now that ν ∉ Δ̃. By taking any short root γ in Δ̃ there is a σ in the Weyl group such that σ (γ) = ν. Now σ (Δ̃ ∪ (−Δ̃)) ∩ Δ^{+} is a presymmetric set of short roots which contains ν and therefore its cardinality is less than \frac{1}{2}l (l + 1) . Then so does Δ̃. This completes the proof of the lemma.

Proof of Theorem 1.1. If the maximal root μ ∈ Δ̃ then conclusion follows from Lemma 3.3. Let assume now that μ ∉ Δ̃ and # \binom{}{Δ} > \frac{1}{2}d (G_{u}) and consider Ω ⊂ Δ̃ such that # (Ω) = \frac{1}{2}d (G_{u}) . By Lemma 3.4 there exists a σ in the Weyl group of Δ (𝔤, 𝔥) such that μ ∈ σ (Ω) . Since μ ∈ σ \binom{}{Δ} , σ (Δ̃ ∪ (−Δ̃)) ∩ Δ^{+} is

a presymmetric set and its cardinality is # \binom{}{Δ} , we reach a contradiction with Lemma 3.3. This finishes the proof.

Our next objective is the proof of Theorem 1.2 and to that end we give the following Lemma 3.5. For each simple Lie algebra 𝔤, if Δ̃ ⊂ Δ^{+} (𝔤, 𝔥) is a presymmetric set with # \binom{}{Δ} = \frac{1}{2}d (G_{u}) , then there exists an element σ in the Weyl group of

Δ (𝔤, 𝔥) such that σ (Δ̃ ∪ (−Δ̃)) ∩ Δ^{+} is the set Γ of positive roots of odd height.

Proof. We separate the proof into different cases according to the type of the simple Lie algebra in question. We keep the notation from Lemma 3.3 and in the same manner that in that Lemma, the proof in each case follows the same pattern but the numbers involved are different. For sake of brevity we will do only the proof for types 𝔞_{l} and 𝔟_{l}, because we think that the reader will have no difficulty in reconstructing it in the other cases.

Let 𝔤 be an algebra of type 𝔞_{l}. We proceed with the proof by induction on l. The lemma is immediate for l = 1, 2. Set l ≥ 3. By Lemma 3.4 we may assume that μ ∈ Δ̃. Due to Lemma 3.2, the assertion that we want to prove is equivalent to find an element σ in the Weyl group such that the system of simple roots π is contained in σ (Δ̃ ∪ (−Δ̃)) .With Δ_{0} and Δ_{1} indicated in Lemma 3.3 for the algebra 𝔞_{l}, we have # (Δ_{1}) = 2 (l − 1) and (Δ̃ ∩ Δ_{0}) is a presymmetric set for an algebra of type 𝔞_{l−2} (α_{2}, ..., α_{l−1} are its simple roots). By Theorem 1.1, its cardinality is

# (Δ̃ ∩ Δ_{0}) ≤ \{ \begin{matrix}\frac{1}{4}l (l − 2) & l even \\ \frac{1}{4} (l − 1)^{2} & l odd \end{matrix}  (2)

and due to the fact that μ ∈ Δ̃,

# (Δ̃ ∩ Δ_{1}) ≤ l − 1.  (3)

Now since

# \binom{}{Δ} = # (Δ̃ ∩ Δ_{0}) + # (Δ̃ ∩ Δ_{1}) + 1 = \{ \begin{matrix}\frac{1}{4}l (l + 2) & l even \\ \frac{1}{4} (l + 1)^{2} & l odd \end{matrix}

we get that formulas (2) and (3) are in fact equalities.

Then (Δ̃ ∩ Δ_{0}) is a presymmetric set of maximal cardinality for an algebra of type 𝔞_{l−2}. Thus, by inductive hypothesis, there exists τ_{1} in the Weyl group of 𝔞_{l−2} such that

τ_{1} [(Δ̃ ∩ Δ_{0}) ∪ (−(Δ̃ ∩ Δ_{0}))] ⊃ {α_{2}, ..., α_{l−1}} .

Let τ be the natural extension of τ_{1} to the Weyl group of 𝔞_{l}. Since τ is product of reflections σ_{α_{j}} with 2 ≤ j ≤ l − 1, it is clear that τ (μ) = μ and therefore

{μ, α_{2}, α_{3}, ..., α_{l−1}} ⊂ τ (Δ̃ ∪ (−Δ̃)) .

In order to finish the proof, for the algebras of type 𝔞_{l}, we need to consider the following four possibilities.

1) α_{1} and α_{l} ∈ τ (Δ̃ ∪ (−Δ̃)) , then taking σ = τ we have the Lemma.

2) Neither α_{1} nor α_{l} belong to τ (Δ̃ ∪ (−Δ̃)) then, keeping in mind that in formula (3) equality holds, we have that ± (μ − α_{1}) and

± (μ − α_{1}) ∈ τ (Δ̃ ∪ (−Δ̃)) .

Since the reflection σ_{μ} corresponding to the maximal root μ satisfies

σ_{μ} (α_{j}) = \{ \begin{matrix}α_{j} & j ≠ 1, l \\ α_{j} − μ & j = 1, l \end{matrix}

by taking σ = σ_{μ}τ, we have the Lemma.

3) α_{1} ∈ τ (Δ̃ ∪ (−Δ̃)) and α_{l} ∉ τ (Δ̃ ∪ (−Δ̃)). If we call π′ = {−μ, α_{1}, ..., α_{l−1}}, this is also a system of simple roots for Δ (𝔤, 𝔥) and therefore there exists a θ in the Weyl group of 𝔞_{l} such that θ (π′) = π. Since π′ is contained in τ (Δ̃ ∪ (−Δ̃)) then by taking σ = θτ we have the Lemma.

4) It is clear that the situation α_{l} ∈ τ (Δ̃ ∪ (−Δ̃)) and α_{1} ∉ τ (Δ̃ ∪ (−Δ̃)) is totally analogous to (3) and so we also have the Lemma in this case. This concludes the proof of the Lemma for the algebras of type 𝔞_{l}.

Let 𝔤 be an algebra of type 𝔟_{l}. We proceed with the proof by induction on l. The lemma is immediate for l = 1, 2 (𝔟_{1} ≈ 𝔞_{1}). Set l ≥ 3. By Lemma 3.4 we may assume that μ ∈ Δ̃. Due to Lemma 3.2, the assertion that we want to prove is equivalent to find an element σ in the Weyl group such that the system of simple roots π is contained in σ (Δ̃ ∪ (−Δ̃)) .

With Δ_{0} and Δ_{1} indicated in Lemma 3.3 for the algebra 𝔟_{l}, we have # (Δ_{1}) = 2 (2l − 3) and (Δ̃ ∩ Δ_{0}) − {α_{1}} is a presymmetric set for an algebra of type 𝔟_{l−2} (α_{3}, ..., α_{l} are its simple roots). By Theorem 1.1, its cardinality is

# ((Δ̃ ∩ Δ_{0}) − \{α_{1}\}) ≤ \frac{1}{2} (l − 2) (l − 1)  (4)

and

# (Δ̃ ∩ Δ_{1}) ≤ 2l − 3  (5)

because μ ∈ Δ̃.

Now since

# \binom{}{Δ} = # (Δ̃ ∩ Δ_{0}) + # (Δ̃ ∩ Δ_{1}) + 1 = \frac{1}{2}l (l + 1)

we get that formulas (4) and (5) are in fact equalities and α_{1} ∈ Δ̃. Then (Δ̃ ∩ Δ_{0}) − {α_{1}} is a presymmetric set of maximal cardinality for an algebra of type 𝔟_{l−2}. Thus, by inductive hypothesis, there exists τ_{1} in the Weyl group of 𝔟_{l−2} such that

τ_{1} [((Δ̃ ∩ Δ_{0}) − {α_{1}}) ∪ − ((Δ̃ ∩ Δ_{0}) − {α_{1}})] ⊃ {α_{3}, ..., α_{l}} .

Let τ be the natural extension of τ_{1} to the Weyl group of 𝔟_{l}. Since τ is product of reflections σ_{α_{j}} with 3 ≤ j ≤ l, it is clear that τ (α_{1}) = α_{1} and therefore

{α_{1}, α_{3}, ..., α_{l}} ⊂ τ (Δ̃ ∪ (−Δ̃)) .

If α_{2} ∈ τ (Δ̃ ∪ (−Δ̃)) then taking σ = τ we have the Lemma.

If α_{2} ∉ τ (Δ̃ ∪ (−Δ̃)) then, keeping in mind that in formula (5) equality

holds, we have that ± (μ − α_{2}) ∈ τ (Δ̃ ∪ (−Δ̃)) . Since the reflection σ_{μ} corresponding to the maximal root μ satisfies

σ_{μ} (α_{j}) = \{ \begin{matrix}α_{j} & j ≠ 2 \\ α_{2} − μ & j = 2, \end{matrix}

by taking σ = σ_{μ}τ we conclude the proof for the algebras of type 𝔟_{l}.

Proof of Theorem 1.2. In the paper [6, Remark 4.1] we proved that if 𝔭 is the tangent space of the inner symmetric space G_{u}/K at [K] then there exists a root γ^{∗} ∈ π such that 𝔭 is of the form

𝔭 = ∑_{γ∈Δ^{∗}} 𝔪_{γ}, where Δ^{∗} = {γ ∈ Δ^{+} : k_{γ}(γ^{∗}) = 1} .

Furthermore by Proposition 2.1 we know that RP (𝔭) ⊂ X [M] . Now from Theorem 2.2 and Lemma 3.5 it follows that there exists an element τ in the Weyl group of Δ (𝔤, 𝔥) such that τ (Δ^{∗} ∪ (−Δ^{∗})) ∩ Δ^{+} = Γ (set of positive roots of odd height). Then, if 𝔭_{Γ} = ∑_{γ∈Γ} 𝔪_{γ} there exists h in the normalizer N_{G_{u}} (T) of the maximal torus T in the group G_{u} such that Ad (h) 𝔭_{Γ} = 𝔭.

By Lemma 3.5, for our subspace 𝔭̃ there exits a g in N_{G_{u}} (T) such that Ad (g) 𝔭̃ = 𝔭_{Γ}. Thus Ad (hg) 𝔭̃ = 𝔭 and since Ad (hg) is orthogonal with respect to the Killing form in 𝔤_{u}, it takes 𝔨̃ = 𝔭̃^{⊥} onto 𝔨 = 𝔭^{⊥}. Since the set of fixed points of the torus T acting on G_{u}/K is the orbit of the point [K] by the normalizer N_{G_{u}} (T) the Theorem follows.

4. Some applications.

We present here some consequences and comments concerning the results of the previous sections. First of all we introduce some notation and terminology which is convenient for our purposes.

Let us recall a definition, as given for instance in [12, p. 70]. If Y ⊂ RP^{n} is an algebraic variety and G (k, n) denotes the Grassmannian of real projective subspaces RP^{k} contained in the projective space RP^{n} [12, p. 63], we may consider the set of k-planes contained in Y

F_{k} (Y) = {Λ ∈ G (k, n) : Λ ⊂ Y } .

This is an algebraic variety ([12, p. 70]) which is a subvariety of the Grassmannian. For each k this variety is a very interesting geometric object naturally associated to the original variety Y which it is usually called the k-th Fano variety associated to the variety Y .

We want to take a look here to a particular Fano variety associated to the variety X [M] for our flag manifold M = G_{u}/T and present an application of our results.

Let us take the Fano variety F_{d−1} (X [M]) ⊂ G (d − 1, n − 1) where d = d (G_{u}) and n = dim (M) . The points of F_{d−1} (X [M]) are the real projective subspaces of dimension d − 1 contained in the variety X [M] . It is clear that the projective subspaces described in Theorems 2.2 and 2.3 are points in F_{d−1} (X [M]) and so this variety is not empty. In order to write down our result we need to introduce some notation.

Recall that for each compact connected simple Lie group G_{u} we have the corresponding flag manifold of complete flags M = G_{u}/T and also associated to the group G_{u} we have an inner symmetric space G_{u}/K of maximal dimension d (G_{u})

(see Table I). There is one of them for each group G_{u} and so the flag manifold M determines uniquely the inner symmetric space and hence the subgroup K (without loosing generality we may assume T ⊂ K).

Associated to the subgroup K we have its normalizer N_{G_{u}} (K) in the group G_{u}. It is well known that the Lie algebra of N_{G_{u}} (K) coincides with that of K. This means that the quotient group Σ = N_{G_{u}} (K) /K is discrete and hence finite.

There is a right free action of the group Σ on the symmetric space G_{u}/K (compare [2, p. 24]) given by

(gK).(hK) = ghK, for hK ∈ Σ .

In this situation we may consider the quotient of the symmetric space by the free action, which is then a manifold, and denote it by Σ \ (G_{u}/K) .

As usual we denote by χ(L) the Euler-Poincaré characteristic of the topological space L. Theorems 1.1 and 1.2 in this paper and results from [6] allow us to present the following theorem concerning the Euler-Poincaré characteristic of the variety F_{d−1} (X [M]) .

Theorem 4.1. The Euler-Poincaré characteristic of the (d − 1) − th Fano variety associated to the variety X [M] coincides with that of the locally symmetric manifold Σ \ (G_{u}/K); i.e.

χ(F_{d−1} (X [M])) = χ (Σ \ (G_{u}/K)) .

Proof. Set o = eT ∈ M. The linear action (via Ad_{G_{u}} (T)) of T on 𝔪 = T_{o} (M) induces an action of T on the Grassmannian

G (d, T_{o} (M)) = G (d − 1, RP (T_{o} (M))) = G (d − 1, n − 1) .

Since the variety X [M] is defined in terms of the invariant tensors α an D it is itself invariant by the action of the torus T (see [5, pp. 227 and 231]). This clearly implies that there is a natural action of the torus T on the Fano variety F_{d−1} (X [M]) .

It is a classical fact (see for instance [1, p. 163, 10.9] or [5, p. 234]) that the Euler-Poincaré characteristic of a compact topological space supporting an action of a torus, is equal to that of its fixed point set. For our action of T on F_{d−1} (X [M]) the fixed point set clearly consists of those subspaces of the form RP (𝔮) where 𝔮 is a subspace of T_{o} (M) invariant by the torus action on this tangent space and such that RP (𝔮) ⊂ X [M]. Theorem 2.2 identifies these subspaces precisely, and so we see that the Euler-Poincaré characteristic of F_{d−1} (X [M]) is just the number of these subspaces.

On the other hand Theorem 1.2 indicates that all these subspaces are tangent to the symmetric space G_{u}/K at the fixed points of the natural action of the torus T ⊂ K on G_{u}/K.

Since that action of the torus has only isolated fixed points it is also clear, and well known, (by the same classical result mentioned above) that the Euler-Poincaré characteristic of G_{u}/K is precisely the cardinality of this finite set Θ of fixed points i.e. χ (G_{u}/K) = |Θ| .

Since the action of Σ on G_{u}/K is free the space G_{u}/K is a |Σ|-sheeted covering of the manifold Σ \ (G_{u}/K) and then

χ (Σ \ (G_{u}/K)) = \frac{χ (G_{u}/K)}{|Σ|} = \frac{|Θ|}{|Σ|}.

The group Σ acts simply transitively on the set

Ω = {s ∈ G_{u}/K : (G_{u})_{s} = K} .

In fact, if s = uK ∈ Ω then uKu^{−1} = K which means u ∈ N_{G_{u}} (K) and so the action is transitive (we already mentioned that the action of Σ is free). Of course the same thing is true for each of the sets hΩ = {h(s) ∈ G_{u}/K : (G_{u})_{h(s)} = hKh^{−1}} for each h ∈ N_{G_{u}} (T) .

Now we observe that the union of all these sets is Θ

Θ = ⋃_{h∈N_{Gu}(T)} hΩ

(since the sets hΩ are in fact orbits of Σ, they either coincide or they are disjoint), notice that h ∈ N_{K} (T) implies hΩ = Ω;

For each s ∈ Ω the isotropy subalgebra is precisely 𝔨 and so the tangent spaces T_{s} (G_{u}/K) (the subspace 𝔭 orthogonal to 𝔨 by the Killing form in 𝔤_{u}) coincide as subspaces of 𝔪. Thus they give a single RP (𝔭) contained in X [M] and therefore one point in F_{d−1} (X [M]) fixed by the torus T. Naturally the same thing is true for each of the sets hΩ. This proves that there are \frac{|Θ|}{|Σ|} points in the fixed point set F (F_{d−1} (X [M]) , T) and concludes de proof of the theorem.

The group Σ is a well known object associated to the inner symmetric space G_{u}/K. It ”counts”, in some sense, how many points in that space have the same isotropy subgroup or, as we have preferred to do above, how many points have ”the same” tangent space (as long as we identify the tangent space at a point with the orthogonal complement of the isotropy subalgebra in 𝔤_{u} with respect to the Killing form).

The reader may find a list of the corresponding groups Σ for the classical irreducible inner symmetric spaces for which it is not trivial in [2, p. 24]. The authors indicate there that the group is trivial for G_{2}/SO(4).

We have been able to see that the group Σ is also trivial for the symmetric spaces F_{4}/Sp (3) Sp (1) and E_{6}/SU (6) Sp (1) . This was obtained from the last theorem, by computing specifically all the presymmetric sets of roots for the algebras 𝔣_{4} and 𝔢_{6} and noticing that the number of those sets coincides with the Euler-Poincaré characteristic of the symmetric spaces.

There are many problems that we leave open because their consideration seems beyond the scope of our methods.

References

[1] Bredon, G., “Introduction to compact transformation groups,” Pure and Applied Math. 46, Academic Press, New York and London, 1972.

[2] Burstall F. E., and J. H. Rawnsley, “Twistor theory for Riemannian Symmetric Spaces,” Lecture Notes in Math. 1424, Springer Verlag Berlin etc. 1995.

[3] Chen, B. Y., Differential geometry of submanifolds with planar normal sections, Ann. Mat.Pura Appl. 130 (1982), 59–66.

[4] Chen, B. Y., and P. Verheyen, P., Sous-variétés dont les sectiones normales sont des géodésiques, C. R. Acad. Sci. Paris, Sér. A, 293 (1981), 611–613.

[5] Dal Lago, W., A. García, and C. Sánchez, Planar normal sections on the natural imbedding of a flag manifold, Geom. Dedicata 53 (1994), 223–235.

[6] —, Maximal projective subspaces in the variety of planar normal sections of a flag manifold, Geom. Dedicata, to appear.

[7] Deprez, J., and P. Verheyen, Immersions with circular normal sections and normal sections of product immersions, Geom. Dedicata 20 (1986), 335–344.

[8] Ferus, D., Immersionen mit paralleler zweiter Fundamentalform: Beispiele und Nicht-Beispiele, Manuscripta math. 12 (1974), 153–162.

[9] —, Symmetric submanifolds of Euclidean spaces, Math. Ann. 247 (1980), 81–93.

[10] Freudenthal, H., and H. deVries, “Linear Lie Groups, ” Academic Press, New York and London, 1969.

[11] Helgason, S., “Differential Geometry, Lie Groups and Symmetric Spaces,” Academic Press, New York and London, 1978.

[12] Harris, J., “Algebraic Geometry. A First Course, ” Graduate Texts in Mathematics 133, Springer Verlag Berlin etc., 1992.

[13] Humphreys, J. E., “Introduction to Lie Algebras and Representation Theory,” Springer-Verlag Berlin etc., 1972.

Fa.M.A.F. Universidad Nacional de Córdoba Ciudad Universitaria 5000, Córdoba, Argentina. csanchez@famaf.unc.edu.ar

Received January 20, 1998 and in final form February 19, 1998