LOWER BOUNDS FOR THE CURVATURE INVARIANT p(G/K) ASSOCIATED WITH RIEMANNIAN SYMMETRIC SPACES

YOSHIO AGAOKA AND EIJI KANEDA

Abstract. We propose a systematic method to obtain a lower bound for the curvature invariant p(G/K) of a Riemannian symmetric space G/K, which we have previously defined in [3] to estimate the least dimension of the Euclidean space R R R^{N} into which G/K can be locally isometrically imbedded. It is shown that in many cases this lower bound coincides with p(G/K). We also determine p(G/K) for all compact rank one Riemannian symmetric spaces G/K, by which we can show the non-existence theorem of local isometric imbeddings of these spaces in low-codimensions.

1. Introduction

Let M be a Riemannian manifold. In our paper [3], we defined a Z Z Z-valued function p_{M} on M, which is a curvature invariant of M. As we have shown, p_{M} plays an important role to estimate the least dimension of the Euclidean space into which G/K can be locally isometrically imbedded (see Proposition 1.1 of [3]).

In the case where M is a Riemannian symmetric space, the function p_{G/K} takes a constant value and can be reformulated in terms of Lie algebras. Let G/K be a Riemannian symmetric space and let 𝔤 = 𝔨 + 𝔪 be the canonical decomposition of the Lie algebra 𝔤 of G associated with the Riemannian symmetric pair (G, K). We select and fix a maximal abelian subspace 𝔞 in 𝔪. By 𝔨_{0} we denote the centralizer of 𝔞 in 𝔨, i.e., 𝔨_{0} = {X ∈ 𝔨 | [X, 𝔞] = 0}. We call a subspace V of 𝔪 pseudo-abelian if it satisfies

[V, V ] ⊂ 𝔨_{0}.

Let us denote by p(G/K) the maximum dimension of pseudo-abelian subspaces in 𝔪. As is easily seen, p(G/K) does not depend on the choice of 𝔞, which we call the pseudo-nullity of G/K. We have shown that the constant p_{G/K} coincides with the pseudo-nullity p(G/K) (see Proposition 2.1 of [3]). Applying Proposition 1.1 of [3] to G/K, we have Theorem 1.1 ([3]). Let G/K be a Riemannian symmetric space. Then, any open set of G/K cannot be isometrically imbedded into the Euclidean space R R R^{N} with N ≤ 2dim G/K − p(G/K) − 1.

Date: December 8, 2001.

2000 Mathematics Subject Classification. 17B20, 53B25, 53C35.

Key words and phrases. Curvature invariant, isometric imbedding, Riemannian symmetric space.

In [3] and [4] we have calculated the pseudo-nullity p(G/K) for the following compact irreducible Riemannian symmetric spaces:

(1) The spheres S^{n} (n ≥ 2).

(2) Those spaces G/K satisfying rank(G/K) = rank(G), i.e., AI, CI, EI, EV, EV III, F I, G, BI : SO(2n + 1)/SO(n + 1) × SO(n) (n ≥ 2), DI : SO(2n)/SO(n) × SO(n) (n ≥ 3).

(3) Compact Lie groups:

Sp(n) (n ≥ 1), SU(n) (2 ≤ n ≤ 5), SO(n) (3 ≤ n ≤ 9, n ≠ 4), G_{2}.

(Precisely, see [3] and [4]. For reducible Riemannian symmetric spaces and non-compact irreducible Riemannian symmetric spaces, see Proposition 1.2 of [3]).

For each symmetric space G/K listed above, we obtain an estimate on the least dimension of the Euclidean space into which G/K can be locally isometrically imbedded. Especially, for the Riemannian symmetric spaces CI : Sp(n)/U(n) (n ≥ 1) and Sp(n) (n ≥ 1), our estimates indicate that the canonical isometric imbeddings of G/K defined in Kobayashi [10] are the least dimensional isometric imbedding of G/K into the Euclidean spaces.

Unfortunately, it seems a hard algebraic problem to determine the pseudo-nullities p(G/K) for the other compact irreducible Riemannian symmetric spaces G/K. In this paper, we try to give a lower bound p_{cat}(G/K) of p(G/K) for each compact irreducible Riemannian symmetric space G/K.

Our method of this paper is composed of two steps. As the first step we localize the problem. Let Σ be the set of all non-zero restricted roots associated with G/K. We denote by 𝔪(μ) the root subspace of 𝔪 corresponding to a non-zero restricted root μ ∈ Σ. We denote by n(μ) the maximum dimension of pseudo-abelian subspaces contained in 𝔪(μ) and call it the local pseudo-nullity associated with μ. Our first task is to determine n(μ) for each restricted root μ. We can perform this task by combining n(μ) with the multiplicity m(μ) of μ (see Theorem 3.2). Our results are summarized in Table 3.

Our second step is explained as follows: Let Γ be a strongly orthogonal subset of Σ (for the definition of strongly orthogonal subsets, see § 4). Let V (μ) be a pseudo-abelian subspace contained in 𝔪(μ) with dim V (μ) = n(μ). Then, the sum ∑_{μ∈Γ} V (μ) plus a suitable subspace of 𝔞 forms a pseudo-abelian subspace of 𝔪 (see Proposition 4.1). We denote by p_{cat}(G/K) the maximum dimension of all pseudo-abelian subspaces constructed in the manner stated above. We call p_{cat}(G/K) the categorical pseudo-nullity of G/K. Clearly, it holds p_{cat}(G/K) ≤ p(G/K) and therefore p_{cat}(G/K) gives a lower bound of the pseudo-nullity p(G/K). Our second task is to determine the categorical pseudo-nullity p_{cat}(G/K) for each compact irreducible Riemannian symmetric space G/K with G simple. In Table 4 and Table 5 we exhibit the results of this task.

LOWER BOUNDS FOR THE CURVATURE INVARIANT p(G/K) 3

Our method explained above is a generalization of the one used in the proof of Proposition 6.3 in [3], where we gave an approximate value of the pseudo-nullity for each compact simple Lie group. In this paper, we mainly develop our arguments by assuming that G is simple. However, it is noted that they are also valid in case G/K is a compact simple Lie group (see Remark 3).

Although the categorical pseudo-nullity p_{cat}(G/K) does not directly serve to determine the least dimensional local isometric imbeddings of G/K, it gives a fairly good estimate on p(G/K). It will be shown that the equality p(G/K) = p_{cat}(G/K) holds for Riemannian symmetric spaces G/K listed above (see Table 4 and Table 5).

In § 5, we will determine the pseudo-nullities p(G/K) for compact rank one Riemannian symmetric spaces G/K. As a result, for these spaces the equality p(G/K) = p_{cat}(G/K) holds except the 2-dimensional complex projective space P^{2}(C C C) (see Theorem 5.1). On the basis of this result we obtain an estimate on the least dimension of the Euclidean space into which compact rank one Riemannian symmetric spaces G/K can be locally isometrically imbedded (see Theorem 5.6). In the case where G/K = P^{n}(H H H) (n ≥ 2) or P^{2}(Cay Cay Cay), Theorem 5.6 improves the estimate obtained in [2].

It is expected that the equality p(G/K) = p_{cat}(G/K) holds for more wide class of Riemannian symmetric spaces G/K, whose proof will be investigated as a main subject in our future works.

2. Restricted roots and multiplicities

In the subsequent sections we follow the notations in Introduction. Let G/K be a compact irreducible Riemannian symmetric space with G simple. The aim of this section is to recall the multiplicities of the restricted roots associated with G/K.

Let B be the Killing form of 𝔤. We introduce an inner product ( , ) of 𝔤 by setting

(X, Y) = −B(X, Y), X, Y ∈ 𝔤.

Let 𝔱 be a Cartan subalgebra satisfying 𝔱 ⊃ 𝔞. We set 𝔟 = 𝔱 ∩ 𝔨. Then we have

𝔱 = 𝔞 + 𝔟 (orthogonal direct sum).

Let 𝔤^{c} be the complexification of 𝔤. For each α ∈ 𝔱 we define a subspace 𝔤_{α} of 𝔤^{c} by

𝔤_{α} = \{Z ∈ 𝔤^{c} | [H, Z] = \sqrt{−1}(α, H)Z, ∀H ∈ 𝔱 \}.

An element α ∈ 𝔱 is called a root of 𝔤^{c} if 𝔤_{α} ≠ 0. We denote by Δ the set of non-zero roots of 𝔤^{c}.

Let λ ∈ 𝔱. By λ_{𝔞} we mean the 𝔞-component of λ with respect to the orthogonal decomposition 𝔱 = 𝔞 + 𝔟. An element μ ∈ 𝔞 is called a restricted root if there is a root α such that α_{𝔞} = μ. Let us denote by Σ the set of all non-zero restricted roots. As is well-known, Σ forms an irreducible (possibly non-reduced) root system.

Let μ ∈ Σ. We denote by Δ(μ) the set of all roots α ∈ Δ such that α_{𝔞} = μ. The cardinal number #Δ(μ) is called the multiplicity of μ ∈ Σ and is denoted by m(μ).

Let μ ∈ Σ. We define two subspaces 𝔨(μ) ⊂ 𝔨 and 𝔪(μ) ⊂ 𝔪 by setting

𝔨(μ) = {X ∈ 𝔨 | ad (H)^{2}(X) = −(μ, H)^{2}X, ∀H ∈ 𝔞},
𝔪(μ) = {Y ∈ 𝔪 | ad (H)^{2}(Y) = −(μ, H)^{2}Y, ∀H ∈ 𝔞}.

As is easily seen, we have 𝔨(−μ) = 𝔨(μ), 𝔪(−μ) = 𝔪(μ). It is convenient to set 𝔨(0) = 𝔨_{0}, 𝔪(0) = 𝔞 and 𝔨(μ) = 𝔪(μ) = 0 in case μ ∉ Σ ∪ {0}.

Let θ be the involution of 𝔤 induced from the geodesic symmetry at the origin of G/K. Let us introduce a linear order “<”of 𝔱 satisfying

H > 0, H ∉ 𝔟 ⟹ θH < 0.

Such an order “<”is called a θ-order of 𝔱. It is easy to see that “<” naturally induces a linear order of 𝔞. We denote by Δ^{+} (resp. Σ^{+}) the set of positive roots of Δ (resp. Σ) with respect to this order “<”. Then the following assertion is well-known (see [11]): Proposition 2.1. (1) dim 𝔨(μ) = dim 𝔪(μ) = m(μ).

(2) It holds the following decompositions.

𝔪 = 𝔞 + ∑_{μ∈Σ^{+}}𝔪(μ) (orthogonal direct sum),
𝔨 = 𝔨_{0} + ∑_{μ∈Σ^{+}}𝔨(μ) (orthogonal direct sum).

(3) Let μ_{1}, μ_{2} ∈ Σ ∪ {0}. Then it holds

[𝔨(μ_{1}), 𝔨(μ_{2})] ⊂ 𝔨(μ_{1} + μ_{2}) + 𝔨(μ_{1} − μ_{2}),
[𝔪(μ_{1}), 𝔪(μ_{2})] ⊂ 𝔨(μ_{1} + μ_{2}) + 𝔨(μ_{1} − μ_{2}),
[𝔨(μ_{1}), 𝔪(μ_{2})] ⊂ 𝔪(μ_{1} + μ_{2}) + 𝔪(μ_{1} − μ_{2}).

Since the restricted root system Σ forms an irreducible root system, Σ contains at most three sorts of roots with different lengths. It is also known that restricted roots with the same length have the same multiplicity (see Appendix of [2]). Accordingly, the multiplicity m(μ)(μ ∈ Σ) takes at most three distinct values.

We now divide Σ into a disjoint union of three subsets Σ_{1}, Σ_{2}, Σ_{3} of Σ as follows: In the case where Σ is not reduced, i.e., Σ is of type BC_{n} (n ≥ 1), we set

Σ_{2} = {μ ∈ Σ | 2μ ∈ Σ},  Σ_{3} = {μ ∈ Σ | (1/2)μ ∈ Σ_{2}},  Σ_{1} = Σ \ (Σ_{2} ∪ Σ_{3}).

A restricted root μ contained in Σ_{2} (resp. Σ_{3}) is called a multipliable root (resp. divisible root). In the case where Σ is reduced, Σ contains at most two sorts of roots with different lengths. We say a root α ∈ Σ is short if there is a root β ∈ Σ such that |α| < |β|, where |α| (resp. |β|) denotes the length of α (resp. β). We denote by Σ_{short} the set of all short roots of Σ and set Σ_{long} = Σ \ Σ_{short}. Under these preparations, we set

Σ_{1} = Σ_{long},  Σ_{2} = Σ_{short},  Σ_{3} = ∅.

By the above definition we easily know that Σ = Σ_{1} ∪ Σ_{2} ∪ Σ_{3} (disjoint union) and that each Σ_{i} (i = 1, 2, 3) consists of restricted roots with the same length. Note that Σ_{2} = Σ_{3} = ∅ in case Σ consists of roots with the same length.

Let m_{i} (i = 1, 2, 3) be the multiplicity of the restricted roots in Σ_{i}. We promise that m_{i} = 0 if Σ_{i} = ∅. Then from the classification table in Araki [7], we can get the triplet M(G/K) = {m_{1}, m_{2}, m_{3}} (see Table 3). As we have stated above, by Table 3 we can obtain the multiplicities of all restricted roots μ ∈ Σ.

3. Local pseudo-nullities n(μ)

As in the previous section we assume that G/K is a compact, irreducible Riemannian symmetric space with G simple. As we have stated in Introduction, it is a hard algebraic problem to determine the pseudo-nullity p(G/K). In this section, as the first step to estimate the pseudo-nullity p(G/K), we localize the problem and determine all local pseudo-nullities associated with restricted roots of Σ.

Let P denote the family of pseudo-abelian subspaces of 𝔪. Let U ∈ P. U is called local if U is contained in some root subspace 𝔪(μ)(μ ∈ Σ). Let μ ∈ Σ. We define an integer n(μ) by setting

n(μ) = max {dim U | U ⊂ 𝔪(μ), U ∈ P} .

The number n(μ) is called the local pseudo-nullity associated with μ.

We first note: Proposition 3.1. The local pseudo-nullity is invariant under the action of the Weyl group W (Σ), i.e., n(wμ) = n(μ) holds for w ∈ W (Σ) and μ ∈ Σ. Consequently, if two restricted roots μ and μ′ are in the same length, i.e., |μ| = |μ′|, then it holds n(μ) = n(μ′).

Proof. Let w ∈ W (Σ). As is well-known, there is an element k ∈ K such that Ad(k)𝔞 = 𝔞 and Ad(k)μ = wμ. Then it can be easily observed that Ad(k)𝔪(μ) = 𝔪(wμ) and Ad(k)𝔨_{0} = 𝔨_{0}. Therefore, a subspace U of 𝔪(μ) is pseudo-abelian if and only if Ad(k)U is a pseudo-abelian subspace of 𝔪(wμ). This implies n(wμ) = n(μ). Moreover, in the case where |μ| = |μ′|, we can take an element of w ∈ W (Σ) such that μ′ = wμ. Therefore, we have n(μ′) = n(μ). □

Let n_{i} (i = 1, 2, 3) be the local pseudo-nullity of the restricted roots in Σ_{i}. In the following we concentrate to determine the triplet of the integers N (G/K) = {n_{1}, n_{2}, n_{3}}. Our aim of this section is to prove Theorem 3.2. Let G/K be a compact irreducible Riemannian symmetric space with G simple. Let M(G/K) = {m_{1}, m_{2}, m_{3}} (resp. N (G/K) = {n_{1}, n_{2}, n_{3}}) be the multiplicities (resp. local pseudo-nullities) of G/K. Then it holds

n_{1} = m_{1}, n_{2} = m_{2}/(1 + m_{3}), n_{3} = m_{3}.

Before proceeding to the proof of Theorem 3.2, we first note Lemma 3.3. (1) [𝔪(μ), 𝔪(μ)] ⊂ 𝔨(2μ) + 𝔨_{0}.

(2) [𝔨(2μ), 𝔪(μ)] ⊂ 𝔪(μ).

Proof. From (3) of Proposition 2.1 the assertion (1) follows directly. We also have the assertion (2), because 3μ ∉ Σ. □

We now start the proof of Theorem 3.2. First assume that μ is not multipliable, i.e., 2μ ∉ Σ. Then we have [𝔪(μ), 𝔪(μ)] ⊂ 𝔨_{0} and hence 𝔪(μ) ∈ P. Consequently, if μ ∈ Σ_{1} ∪ Σ_{3} then it follows n(μ) = m(μ). This proves that n_{1} = m_{1} and n_{3} = m_{3}. Similarly, if μ ∈ Σ_{2} and Σ_{3} = ∅ (equivalently m_{3} = 0), then it holds n_{2} = m_{2}.

In the following we assume that μ is multipliable, i.e., μ ∈ Σ_{2} and Σ_{3} ≠ ∅. This case occurs only in the case where Σ is of type BC_{n}. In view of Table 3, we know that such restricted roots are exhausted by the following G/K:

AIII_{p,q} (p > q ≥ 1), CII_{p,q} (p > q ≥ 1), DIII_{n} (n = 2m + 1), EIII, F II.

We also know that the multiplicity m_{3} is equal to 1, 3 or 7 and that restricted roots μ satisfying m(2μ) > 1 can be found only in CII_{p,q} (p > q ≥ 1) : G^{p,q}(H H H) and F II : P^{2}(Cay Cay Cay).

Let X ∈ 𝔨(2μ). Then, by (2) of Lemma 3.3, we know that ad X induces a linear endomorphism of 𝔪(μ), which we denote by X^{†}, i.e.,

X^{†}(Y) = [X, Y ], Y ∈ 𝔪(μ).

It is easy to see that X^{†} is a skew-symmetric endomorphism of 𝔪(μ) with respect to the inner product ( , ). The following statement is useful in the remaining discussions. Lemma 3.4. Let V be a subspace of 𝔪(μ). Then V is pseudo-abelian if and only if it holds (X^{†}(V), V) = 0 for each X ∈ 𝔨(2μ).

Proof. From the well-known equality

(X^{†}(Y), Z) = ([X, Y ], Z) = (X, [Y, Z]), ∀Y, ∀Z ∈ 𝔪(μ),

we have (X^{†}(V), V) = (X, [V, V ]). Hence, if [V, V ] ⊂ 𝔨_{0}, then for each X ∈ 𝔨(2μ) we have (X^{†}(V), V) = 0. Conversely, if (X^{†}(V), V) = 0 holds for all X ∈ 𝔨(2μ), then we have (𝔨(2μ), [V, V ]) = 0. This implies that [V, V ] ⊂ 𝔨_{0} (see (1) of Lemma 3.3). □

The set 𝔨(2μ)^{†} composed of all X^{†} (X ∈ 𝔨(2μ)), which is a subspace of the space of endomorphisms of 𝔪(μ), has the following special feature. In § 6 of this paper, we will give the proof of the following Theorem 3.5. Let μ ∈ Σ. Assume that μ is multipliable, i.e., 2μ ∈ Σ. Then:

(1) If m(2μ) = 1, then dim 𝔪(μ) is even and there is an element I of 𝔨(2μ) such that I^{†} determines a complex structure of 𝔪(μ), i.e., I^{†}^{2} = −111_{𝔪(μ)}.

(2) If m(2μ) = 3, i.e., if G/K is of type CII_{p,q} (p > q ≥ 1), then dim 𝔪(μ) is a multiple of 4 and there are elements I, J, K of 𝔨(2μ) such that the triplet {I^{†}, J^{†}, K^{†}} determines a quaternion structure of 𝔪(μ), i.e.,

I^{†}^{2} = J^{†}^{2} = K^{†}^{2} = −111_{𝔪(μ)},
I^{†}J^{†} = −J^{†}I^{†} = K^{†}, J^{†}K^{†} = −K^{†}J^{†} = I^{†}, K^{†}I^{†} = −I^{†}K^{†} = J^{†}.

In the above (1) and (2), 111_{𝔪(μ)} implies the identity mapping of 𝔪(μ).

By virtue of these structures, in many cases, the determination of n(μ) can be reduced to an easy problem.

We first consider the case where m(2μ) = 3, i.e., G/K is of type CII_{p,q} (p > q ≥ 1). We note that I^{†}, J^{†} and K^{†} stated in the above theorem are orthogonal transformations of 𝔪(μ). Let U be an arbitrary local pseudo-abelian subspace of 𝔪(μ). Let us denote by V the sum of four subspaces U, I^{†}(U), J^{†}(U) and K^{†}(U), i.e.,

V = U + I^{†}(U) + J^{†}(U) + K^{†}(U).

We can prove that the above summation is an orthogonal direct sum. In fact, by Lemma 3.4 we have

(U, I^{†}(U)) = (U, J^{†}(U)) = (U, K^{†}(U)) = 0.

Moreover, since I^{†} is an orthogonal endomorphism of 𝔪(μ), we can prove

(I^{†}(U), J^{†}(U)) = (U, I^{†}J^{†}(U)) = (U, K^{†}(U)) = 0.

Similarly, we have (J^{†}(U), K^{†}(U)) = (K^{†}(U), I^{†}(U)) = 0. Consequently, V is an orthogonal direct sum of four subspaces U, I^{†}(U), J^{†}(U) and K^{†}(U). Therefore, we have dim U ≤ dim 𝔪(μ)/4, which proves that n(μ) ≤ m(μ)/4.

We now show the converse. Utilizing the quaternion structure {I^{†}, J^{†}, K^{†}}, we can get a subspace U_{0} of 𝔪(μ) such that dim U_{0} = m(μ)/4 and

𝔪(μ) = U_{0} + I^{†}(U_{0}) + J^{†}(U_{0}) + K^{†}(U_{0}) (orthogonal direct sum).

Since (U_{0}, I^{†}(U_{0})) = (U_{0}, J^{†}(U_{0})) = (U_{0}, K^{†}(U_{0})) = 0 and since {I, J, K} forms a basis of 𝔨(2μ), we have (U_{0}, X^{†}(U_{0})) = 0 for any X ∈ 𝔨(2μ). This proves that U_{0} is pseudoabelian and hence n(μ) ≥ m(μ)/4. Accordingly, we have the equality n(μ) = m(μ)/4 if m(2μ) = 3.

In a similar manner, we can prove that n(μ) = m(2μ)/2 for those symmetric spaces G/K satisfying m(2μ) = 1.

Finally, we treat the case where G/K is of type F II, i.e., G/K = P^{2}(Cay Cay Cay). As in the case of CII, a certain complicated algebraic structure on 𝔪(μ) may be induced from 𝔨(2μ)^{†}. However, to solve our problem it is sufficient to assert the following simple proposition. Proposition 3.6. Assume that G/K = P^{2}(Cay Cay Cay) and that μ, 2μ ∈ Σ. Let Y be a non-zero element of 𝔪(μ). Then, dim[𝔨(2μ), Y ] = 7.

Proof. Since dim 𝔨(2μ) = m(2μ) = 7, it suffices to prove that there is no non-trivial element X of 𝔨(2μ) satisfying [X, Y ] = 0. Now we suppose [X, Y ] = 0 holds for some X ∈ 𝔨(2μ).

Since ad μ gives an isomorphism between 𝔪(2μ) and 𝔨(2μ), we can find Y′ ∈ 𝔪(2μ) such that X = [μ, Y′]. Consequently, we have

[Y, [μ, Y′]] = 0.  (3.1)

Applying ad μ to (3.1), we have

[[μ, Y ], [μ, Y′]] = 4(μ, μ)^{2}[Y, Y′].  (3.2)

(Note that [μ, [μ, Y′]] = −4(μ, μ)^{2}Y′.) In § 5, we will prove that Y ∈ 𝔪(μ) and Y′ ∈ 𝔪(2μ) must satisfy the following equality (see (1) of Lemma 5.3):

[[μ, Y ], [μ, Y′]] = 2(μ, μ)^{2}[Y, Y′].  (3.3)

Comparing the equalities (3.2) and (3.3), we can easily conclude [Y, Y′] = 0. This implies that two vectors Y and Y′ span an abelian subspace of 𝔪. However, since rank(P^{2}(Cay Cay Cay)) = 1, Y′ is a scalar multiple of Y. Hence we get Y′ = 0, because Y ∈ 𝔪(μ) and Y′ ∈ 𝔪(2μ). Thus, we have X = 0, which proves the proposition. □

We now proceed to the determination of the local pseudo-nullity n(μ) for G/K = P^{2}(Cay Cay Cay). Let V be a local pseudo-abelian subspace in 𝔪(μ). Let Y be a non-zero element of V. By Lemma 3.4 we know that V is necessarily orthogonal to [𝔨(2μ), Y]. Therefore, since dim[𝔨(2μ), Y ] = 7 and m(μ) = 8, we have dim V ≤ 1. This proves n(μ) = 1 = 8/(1 + 7).

By the above discussions, we complete the proof of Theorem 3.2. □ For each compact irreducible Riemannian symmetric space G/K with G simple we will

give the list of local pseudo-nullities N (G/K) = {n_{1}, n_{2}, n_{3}} in Table 3.

4. Categorical pseudo-nullities p_{cat}(G/K)

As the second step to estimate the pseudo-nullities p(G/K), we construct pseudo-abelian subspaces of 𝔪 by using local pseudo-abelian subspaces on the basis of strongly orthogonal subsets of Σ.

Let Γ be a subset of the restricted root system Σ. Γ is called a strongly orthogonal subset (= SOS) of Σ if it satisfies the following:

α, β ∈ Γ, α ≠ β ⟹ α ± β ∉ Σ ∪ {0}.

It is noted that any two distinct elements α, β in a SOS Γ are orthogonal with respect to ( , ), i.e., (α, β) = 0. By SOS we denote the family of all SOS’s of Σ.

We now define the notion of categorical subspace of 𝔪. Let V be a subspace of 𝔪. V is called categorical if V satisfies the following conditions:

(1) V is represented by

V = V (0) + ∑_{μ∈Σ^{+}}V (μ) (direct sum),

where V (μ) = V ∩ 𝔪(μ) (μ ∈ Σ^{+} ∪ {0}).

(2) The support Γ of V, which is defined by Γ = {μ ∈ Σ^{+} | V (μ) ≠ 0}, is a SOS of Σ. Using the notion of categorical subspaces, we can produce many non-local pseudo-abelian subspaces of 𝔪. In fact: Proposition 4.1. Let V = V (0) + ∑_{μ}V (μ) be a categorical subspace of 𝔪 and let Γ the

support of V. Then, V is pseudo-abelian if and only if the following two conditions are satisfied:

(1) V (0) ⊥ Γ, i.e., (V (0), Γ) = 0.

(2) For each μ ∈ Γ, V (μ) is a local pseudo-abelian subspace.

Proof. Let μ_{1}, μ_{2} be two distinct restricted roots in Γ. Then, by Proposition 2.1, we have

[V (μ_{1}), V (μ_{2})] ⊂ 𝔨(μ_{1} + μ_{2}) + 𝔨(μ_{1} − μ_{2}).

Since Γ is a SOS, we have μ_{1} ± μ_{2} ∉ Σ ∪ {0}. Hence, [V (μ_{1}), V (μ_{2})] = 0. Therefore, it is easy to see that V is pseudo-abelian if and only if

[V (0), V (μ)] ⊂ 𝔨(0),  (4.1)
[V (μ), V (μ)] ⊂ 𝔨(0)  (4.2)

hold for each μ ∈ Γ. Obviously, (4.2) implies that V (μ) is a local pseudo-abelian subspace. On the other hand, (4.1) is equivalent to [V (0), V (μ)] = 0, because [V (0), V (μ)] ⊂ [𝔞, 𝔪(μ)] ⊂ 𝔨(μ). It is easy to verify that [V (0), V (μ)] = 0 holds if and only if (V (0), μ) = 0. □

Let P_{cat} denote the family of all categorical pseudo-abelian subspaces of 𝔪. We define a number p_{cat}(G/K) ∈ Z Z Z by setting

p_{cat}(G/K) = max{ dim V | V ∈ P_{cat}}.

We call the number p_{cat}(G/K) the categorical pseudo-nullity of G/K. In the following we will determine the pseudo-nullities p_{cat}(G/K) for all compact irreducible symmetric spaces G/K.

Let Γ be a SOS of Σ. We define a quantity b(Γ) ∈ Z Z Z by setting

b(Γ) = rank(G/K) − #Γ + ∑_{μ∈Γ}n(μ).

We now show the following equality. Proposition 4.2. p_{cat}(G/K) = max{b(Γ) | Γ ∈ SOS}.

Proof. Let V be an element of P_{cat} and Γ be the support of V. By Proposition 4.1, we have dim V (0) ≤ dim 𝔞 − #Γ and dim V (μ) ≤ n(μ) for μ ∈ Γ. Therefore we have dim V ≤ b(Γ) and hence p_{cat}(G/K) ≤ max_{Γ}{b(Γ)}.

Now we show the converse. Let Γ ∈ SOS. Then, there is a categorical pseudo-abelian subspace V whose support coincides with Γ. In fact, define V by V = R R RΓ^{⊥} + ∑_{μ∈Γ}U(μ),

where R R RΓ^{⊥} denotes the orthogonal complement of R R RΓ in 𝔞 and U(μ) a pseudo-abelian subspace of 𝔪(μ) such that dim U(μ) = n(μ). Then, it is easy to see that V ∈ P_{cat} and dim V = b(Γ). This proves p_{cat}(G/K) ≥ max_{Γ}{b(Γ)}. □

The following assertion is fundamental to calculate p_{cat}(G/K). Proposition 4.3. (1) b is invariant under the action of the Weyl group of Σ, i.e., b(wΓ) = b(Γ) holds for any w ∈ W (Σ) and Γ ∈ SOS.

(2) Let Γ, Γ′ ∈ SOS. Suppose that Γ ⊂ Γ′. Then b(Γ) ≤ b(Γ′).

Proof. It is obvious that wΓ ∈ SOS, #(wΓ) = #Γ. Since n(wμ) = n(μ) (see Proposition 3.1), we have b(wΓ) = b(Γ). This proves (1).

By the definition, we easily have

b(Γ′) − b(Γ) = −#(Γ′ \ Γ) + ∑_{μ∈Γ′\Γ}n(μ) = ∑_{μ∈Γ′\Γ}(n(μ) − 1).

Since every one-dimensional subspace of 𝔪(μ) is pseudo-abelian, we have n(μ) ≥ 1, and hence we get b(Γ′) ≥ b(Γ). □

In our previous paper [5], for all irreducible root systems Σ, we classified the equivalence classes of maximal SOS’s in Σ under the action of the Weyl group W (Σ). We listed up the representative maximal SOS’s for these equivalence classes (see §§ 3–5 of [5]). In view of Proposition 4.3 we know that in order to determine p_{cat}(G/K) we have only to evaluate b(Γ) for representative maximal SOS’s of Σ.

We now recall the conclusions of [5] more closely. Let Γ be a maximal SOS of Σ. First we assume that Σ is isomorphic to a reduced, irreducible root system X_{n} (X = A ∼ G), where n implies the rank of G/K. Then, in our terminology in [5], Γ is equivalent to Γ (X_{n})^{s}, where the superscript s indicates the cardinal number of short roots contained in Γ, i.e., s = #(Γ ∩ Σ_{2}) (see § 3 and § 5 of [5]). According to the type of Σ, s takes a value in Table 1.

Table 1. Range of s

Type of Σ  Range
A_{n} (n ≥ 1), D_{n} (n ≥ 4), E_{i} (i = 6, 7, 8)  s = 0
B_{n} (n = 2m + 1, m ≥ 1), G_{2}  s = 1
B_{n} (n = 2m, m ≥ 1), F_{4}  0 ≤ s ≤ 1
C_{n} (n ≥ 3)  0 ≤ s ≤ [n/2]

We next assume that Σ is not reduced, i.e., Σ is isomorphic to BC_{n} (n = rank(G/K)). Then Γ is equivalent to Γ (BC_{n})^{r,s}, where the superscript r implies the number of multipliable roots in Γ and s implies the number of roots in Γ which are not multipliable nor divisible (see § 4 of [5]). In other words, we have r = #(Γ ∩ Σ_{2}) and s = #(Γ ∩ Σ_{1}). Moreover, the pair (r, s) has to satisfy the condition: 0 ≤ r ≤ 1 and 0 ≤ s ≤ [(n − r)/2].

We now prove the following main result of this paper:

Theorem 4.4. Let Σ be the restricted root system of a compact irreducible Riemannian symmetric space G/K with G simple. Let N (G/K) = {n_{1}, n_{2}, n_{3}} be the local pseudonullities of G/K. Then:

(1) Assume that Σ is isomorphic to a reduced, irreducible root system X_{n} (X = A ∼ G), where n = rank(G/K). Then it holds

p_{cat}(G/K) = rank(G/K) + max_{s} {(n_{1} − 1)#Γ (X_{n})^{s} + (n_{2} − n_{1})s},

where s runs through the range listed in Table 1.

(2) Assume that Σ is isomorphic to BC_{n} (n = rank(G/K)). Then it holds

p_{cat}(G/K) = n_{3} rank(G/K) + max_{(r,s)} {(n_{2} − n_{3})r + (n_{1} − 2n_{3} + 1)s},

where the pair (r, s) satisfies 0 ≤ r ≤ 1 and 0 ≤ s ≤ [(n − r)/2].

Proof. Let Γ be a maximal SOS of Σ. We set Γ_{i} = Γ ∩ Σ_{i} (i = 1, 2, 3). Then by the definition of b(Γ) we have

b(Γ) = rank(G/K) − #Γ + n_{1}#Γ_{1} + n_{2}#Γ_{2} + n_{3}#Γ_{3}.

If Σ is isomorphic to X_{n} (X = A ∼ G) and if Γ is equivalent to Γ (X_{n})^{s}, then we have #Γ = #Γ (X_{n})^{s}, #Γ_{3} = 0, #Γ_{2} = s and #Γ_{1} = #Γ − s. Putting these equalities into the above formula of b(Γ), we have easily the assertion (1).

On the other hand, if Σ is isomorphic to BC_{n} and if Γ is equivalent to Γ (BC_{n})^{r,s}, then we have #Γ = #Γ (BC_{n})^{r,s} = n − s (see Theorem 4.1 in [5]), #Γ_{1} = s, #Γ_{2} = r and #Γ_{3} = #Γ −r −s. Putting these equalities into the formula of b(Γ), we have the assertion (2). □

The result of calculations of p_{cat}(G/K) is summarized in Table 4 and Table 5. Remark 1. In view of Table 4, we have p_{cat}(S^{p}) = p − 1 (p ≥ 2) (see the types BII_{p} and DII_{p}). This proves that p(S^{p}) = p_{cat}(S^{p}) holds for p ≥ 2. Remark 2. Assume that G/K satisfies rank(G/K) = rank(G). For such G/K we have proved that p(G/K) = rank(G/K) (see Proposition 2.3 of [3]). By our method of this paper we also obtain: p_{cat}(G/K) = rank(G/K). In fact, under our assumption, 𝔞 becomes a Cartan subalgebra of 𝔤. Thus, Δ = Σ and hence Σ does not contain any multipliable root nor any divisible root and m(μ) = 1 holds for any μ ∈ Σ. Therefore, we have n(μ) = 1. By the definition of b(Γ), for any Γ ∈ SOS, it holds b(Γ) = rank(G/K). This implies that p_{cat}(G/K) = rank(G/K). Consequently, we get the equality p_{cat}(G/K) = p(G/K). Remark 3. Let G be a compact connected simple Lie group and 𝔤 be the Lie algebra of G. As is well-known, G endowed with a bi-invariant metric can be represented by a compact, irreducible Riemannian symmetric space G = G̃/K̃, where G̃ = G × G and K̃ denotes the diagonal subgroup of G × G. In the previous sections we developed our discussions for compact irreducible Riemannian symmetric spaces G/K with G simple. We note that these discussions are also valid for compact simple Lie groups G = G̃/K̃.

For example, the notions of restricted roots, multiplicities, (local) pseudo-nullities and strongly orthogonal subsets, etc. can also be well-defined. In addition, the notion of categorical pseudo-nullities p_{cat}(G̃/K̃) can be defined and Proposition 4.2 is also true in this situation.

In [3] we have proved an inequality concerning p(G̃/K̃), which can be expressed as p(G̃/K̃) ≥ rank(G) + s_{0}(G) in the terminology of this paper (see Proposition 6.3 of [3]), where s_{0}(G) denotes the integer given by

s_{0}(G) = \{ \begin{matrix}[(n + 1)/2] & if G = SU(n + 1), \\ 2[n/2] & if G = SO(2n) or Spin(2n), \\ 4 & if G = E_{6}, \\ rank(G) & otherwise. \end{matrix}

Here, let us reconsider the above estimate p(G̃/K̃) ≥ rank(G) + s_{0}(G) in the line of this paper. We can prove: Proposition 4.5. Let G = G̃/K̃ be a compact simple Lie group. Then

p_{cat}(G̃/K̃) = rank(G) + s_{0}(G).

Proof. It is well-known that the Lie algebra 𝔤̃ (resp. 𝔨̃) of G̃ (resp. K̃) is given by 𝔤̃ = 𝔤⊕𝔤 (resp. 𝔨̃ = {(X, X) ∈ 𝔤 ⊕ 𝔤 | X ∈ 𝔤}). Putting 𝔪̃ = {(X, −X) ∈ 𝔤 ⊕ 𝔤 | X ∈ 𝔤}, we get the canonical decomposition 𝔤̃ = 𝔨̃ + 𝔪̃. Let 𝔱 be a Cartan subalgebra of 𝔤. Then 𝔱̃ = 𝔱 ⊕ 𝔱 defines a Cartan subalgebra of 𝔤̃ and 𝔞̃ = {(H, −H) ∈ 𝔪̃ | H ∈ 𝔱} defines a maximal abelian subspace of 𝔪̃.

Let Δ (resp. Δ̃) be the set of non-zero roots of 𝔤^{c} (resp. 𝔤̃^{c}) with respect to 𝔱 (resp. 𝔱̃). As is known, Δ̃ is composed of roots written in the form α^{+} = (α, 0) or α^{−} = (0, −α), where α ∈ Δ. Since α^{+}𝔞̃ = α^{−}𝔞̃ = 1/2 · (α, −α), the set of non-zero restricted roots associated with G̃/K̃ can be written by Σ̃ = {1/2 · (α, −α) | α ∈ Δ}.

By these facts we can verify the following:

(1) For each restricted root μ ∈ Σ̃, it holds m(μ) = 2.

(2) Σ̃ does not contain a multipliable root. Accordingly, it holds n(μ) = m(μ) = 2 for each μ ∈ Σ̃.

(3) For each SOS Γ̃ of Σ̃, there is a SOS Γ of Δ such that Γ̃ = {1/2 · (γ, −γ) | γ ∈ Γ }. Therefore, by Proposition 4.2 we have p_{cat}(G̃/K̃) = rank(G) + max_{Γ}{#Γ }, where Γ runs over all SOS’s of Δ. In [5], we have determined maximal SOS’s of Δ for all irreducible root systems Δ. In view of Theorems 3.1 and 5.1 of [5], we easily get the equality max_{Γ}{#Γ } = s_{0}(G), which proves the proposition. □

Note that in view of the results of [3] and [4], we have the equality p(G̃/K̃) = p_{cat}(G̃/K̃) for the following compact simple Lie groups G:

SU(n) (2 ≤ n ≤ 5), SO(n) (3 ≤ n ≤ 9, n ≠ 4), Sp(n) (n ≥ 1), G_{2}.

We conjecture that the equality p(G̃/K̃) = p_{cat}(G̃/K̃) holds for all compact simple Lie groups G = G̃/K̃.

5. Rank one symmetric spaces

The purpose of this section is to determine the pseudo-nullities p(G/K) for compact rank one Riemannian symmetric spaces G/K not isomorphic to any sphere S^{n}, i.e.,

(1) The complex projective spaces P^{n}(C C C) (n ≥ 2).

(2) The quaternion projective spaces P^{n}(H H H) (n ≥ 2).

(3) The Cayley projective plane P^{2}(Cay Cay Cay). In the rest of this section G/K denotes one of the Riemannian symmetric spaces listed above. In view of Table 3, we know that the restricted root system Σ of G/K is isomorphic to BC_{1}. Let us select a multipliable root μ ∈ Σ. Then for these G/K we have obtained the following table:

Table 2. Basic data for rank one symmetric spaces

\begin{matrix}Type & G/K & n(μ) & n(2μ)(= m(2μ)) & p_{cat}(G/K) \\ AIII_{n,1} & P^{n}(C C C)(n ≥ 2) & n − 1 & 1 & n − 1 \\ CII_{n,1} & P^{n}(H H H)(n ≥ 2) & n − 1 & 3 & max\{3, n − 1\} \\ F II & P^{2}(Cay Cay Cay) & 1 & 7 & 7 \end{matrix}

Now we prove Theorem 5.1. Let G/K be a compact rank one Riemannian symmetric space not isomorphic to any sphere S^{n}. Then it holds

p(G/K) = \begin{cases}p_{cat}(G/K) & if G/K = P^{n}(C C C) (n ≥ 3), P^{n}(H H H) (n ≥ 2) or P^{2}(Cay Cay Cay), \\ 2 & if G/K = P^{2}(C C C). \end{cases}

In order to prove this theorem we have to estimate the dimensions of non-categorical pseudo-abelian subspaces. It is shown that the dimension of any non-categorical pseudoabelian subspace is fairly small. In fact, we have Proposition 5.2. (1) Let V be a non-categorical pseudo-abelian subspace of 𝔪. Then it holds dim V ≤ 2.

(2) If G/K = P^{n}(C C C) (n ≥ 2), there is a non-categorical pseudo-abelian subspace V satisfying dim V = 2.

As is easily seen, Theorem 5.1 immediately follows from this proposition and Table 2. Before proceeding to the proof of Proposition 5.2, we prepare several lemmas. Let μ be

a multipliable root of Σ. Then we have the following decomposition.

𝔪 = 𝔞 + 𝔪(μ) + 𝔪(2μ) (orthogonal direct sum), 𝔞 = R R Rμ.

Lemma 5.3. Let Y_{1} ∈ 𝔪(μ) and Y_{2} ∈ 𝔪(2μ). Then:

(1) [[μ, Y_{1}], [μ, Y_{2}]] = 2(μ, μ)^{2}[Y_{1}, Y_{2}].

(2) [[μ, Y_{2}], Y_{1}] = 2[[μ, Y_{1}], Y_{2}].

Proof. We first note that [Y_{1}, Y_{2}] ∈ 𝔨(μ) (see Proposition 2.1). Consequently, we have

(ad μ)^{2}[Y_{1}, Y_{2}] = −(μ, μ)^{2}[Y_{1}, Y_{2}].

Since (ad μ)^{2}Y_{1} = −(μ, μ)^{2}Y_{1}, (ad μ)^{2}Y_{2} = −4(μ, μ)^{2}Y_{2}, we have

(ad μ)^{2}[Y_{1}, Y_{2}] = [(ad μ)^{2}Y_{1}, Y_{2}] + 2[[μ, Y_{1}], [μ, Y_{2}]]
+ [Y_{1}, (ad μ)^{2}Y_{2}]
= 2[[μ, Y_{1}], [μ, Y_{2}]] − 5 (μ, μ)^{2}[Y_{1}, Y_{2}].

Hence we immediately get the equality (1).

Similarly, applying ad μ to the equality (1), we can easily get the equality (2). □

Lemma 5.4. Let V be an arbitrary pseudo-abelian subspace of 𝔪.

(1) If V ⊄ 𝔪(μ), then dim V ≤ 1 + m(2μ).

(2) If V ⊄ 𝔪(2μ), then dim V ≤ 1 + n(μ).

Proof. We show the assertion (1). We first prove V ∩ 𝔪(μ) = 0. By the assumption, V contains an element Y = H+Y_{1}+Y_{2} (H ∈ 𝔞, Y_{1} ∈ 𝔪(μ), Y_{2} ∈ 𝔪(2μ)) such that H+Y_{2} ≠ 0. Let Y′ be an arbitrary element of V ∩ 𝔪(μ). Then we have

[Y, Y′] = [H + Y_{2}, Y′] + [Y_{1}, Y′] ∈ 𝔨_{0}.

Since [H + Y_{2}, Y′] ∈ 𝔨(μ) and [Y_{1}, Y′] ∈ 𝔨_{0} + 𝔨(2μ) (see Proposition 2.1), we have [H + Y_{2}, Y′] = 0. This implies that the subspace spanned by H + Y_{2} and Y′ is an abelian subspace of 𝔪. On the other hand, since rank(G/K) = 1, we have Y′ = 0, which proves V ∩ 𝔪(μ) = 0. Consequently, we have dim V ≤ dim(𝔞 + 𝔪(2μ)) = 1 + m(2μ).

Next, we show the assertion (2). In the same manner as above, we can prove that V ∩ 𝔪(2μ) = 0. Set r = dim V. Since dim(V ∩ (𝔪(μ) + 𝔪(2μ))) ≥ r − 1, we get elements Y^{i} = Y_{1}^{i} + Y_{2}^{i} ∈ V (1 ≤ i ≤ r − 1) such that Y_{1}^{i} ∈ 𝔪(μ) and Y_{2}^{i} ∈ 𝔪(2μ). Moreover, since V ∩ 𝔪(2μ) = 0, we may assume that the elements Y_{1}^{i} (1 ≤ i ≤ r − 1) are linearly independent. Now, since [Y^{i}, Y^{j}] ∈ 𝔨_{0}, we have

[Y_{1}^{i} + Y_{2}^{i}, Y_{1}^{j} + Y_{2}^{j}] ∈ 𝔨_{0}.

On the other hand, since [Y_{2}^{i}, Y_{2}^{j}] ∈ 𝔨_{0}, [Y_{1}^{i}, Y_{1}^{j}] ∈ 𝔨_{0}+𝔨(2μ) and [Y_{1}^{i}, Y_{2}^{j}]+[Y_{2}^{i}, Y_{1}^{j}] ∈ 𝔨(μ), we have

[Y_{1}^{i}, Y_{1}^{j}] ∈ 𝔨_{0}, 1 ≤ i ≠ j ≤ r − 1.

This implies that the subspace spanned by the elements {Y_{1}^{i} (1 ≤ i ≤ r − 1)} is a pseudoabelian subspace of 𝔪(μ). This proves that r − 1 ≤ n(μ), completing the proof of the assertion (2). □

Lemma 5.5. Let V be a non-categorical pseudo-abelian subspace of 𝔪. Then it holds

dim V ≤ min{1 + m(2μ), 1 + n(μ)}.

Proof. Since V is not categorical, it follows that V ⊄ 𝔪(μ) and V ⊄ 𝔪(2μ). Therefore, by the above lemma we have dim V ≤ 1 + m(2μ) and dim V ≤ 1 + n(μ). □

Proof of Proposition 5.2. By Lemma 5.5, we can prove (1) of Proposition 5.2 for the spaces P^{n}(C C C) (n ≥ 2), P^{2}(H H H) and P^{2}(Cay Cay Cay). In fact, we have m(2μ) = 1 if G/K = P^{n}(C C C) (n ≥ 2) and n(μ) = 1 if G/K = P^{2}(H H H) or P^{2}(Cay Cay Cay).

For the remaining spaces G/K = P^{n}(H H H) (n ≥ 3) we directly show (1). Suppose that there is a non-categorical pseudo-abelian subspace V with dim V ≥ 3. As in the proof of Lemma 5.4, we may assume that there are two elements Y = Y_{1} + Y_{2}, Y′ = Y′_{1} + Y′_{2} ∈ V (Y_{1}, Y′_{1} ∈ 𝔪(μ), Y_{2}, Y′_{2} ∈ 𝔪(2μ)) such that Y_{1} and Y′_{1} are linearly independent and the subspace {Y_{1}, Y′_{1}} is pseudo-abelian. Further, since [Y_{2}, Y′_{2}] ∈ 𝔨_{0} + 𝔨(2μ) and [Y_{2}, Y′_{1}] + [Y_{1}, Y′_{2}] ∈ 𝔨(μ), we have

[Y_{2}, Y′_{1}] = [Y′_{2}, Y_{1}].  (5.1)

By Lemma 3.4 we know that the condition [Y_{1}, Y′_{1}] ∈ 𝔨_{0} is equivalent to

(Y′_{1}, I^{†}Y_{1}) = (Y′_{1}, J^{†}Y_{1}) = (Y′_{1}, K^{†}Y_{1}) = 0.  (5.2)

(Note that 𝔨(2μ) is spanned by I, J and K.) Applying ad μ to the equality (5.1), we have

[[μ, Y_{2}], Y′_{1}] + [Y_{2}, [μ, Y′_{1}]] = [[μ, Y′_{2}], Y_{1}] + [Y′_{2}, [μ, Y_{1}]].

Using (2) of Lemma 5.3, we have

X_{2}^{†}(Y′_{1}) = X′_{2}^{†}(Y_{1}),  (5.3)

where we set X_{2} = [μ, Y_{2}], X′_{2} = [μ, Y′_{2}]. Applying X_{2}^{†} to the both sides of (5.3), we have

(X_{2}^{†})^{2}(Y′_{1}) = X_{2}^{†}X′_{2}^{†}(Y_{1}).  (5.4)

Since X_{2}^{†} and X′_{2}^{†} are linear combinations of I^{†}, J^{†} and K^{†}, it follows that (X_{2}^{†})^{2} = c111_{𝔪(μ)} (c ∈ R R R, c ≠ 0) and X_{2}^{†}X′_{2}^{†} is written as a linear combination of 111_{𝔪(μ)}, I^{†}, J^{†} and K^{†}. Consequently, by (5.4) we know that Y′_{1} can be written as a linear combination of Y_{1}, I^{†}(Y_{1}), J^{†}(Y_{1}) and K^{†}(Y_{1}). This together with (5.2), we can conclude that Y′_{1} equals a scalar multiple of Y_{1}. This contradicts the assumption that Y_{1} and Y′_{1} are linearly independent. Therefore, we have dim V ≤ 2.

Finally, we prove (2) of Proposition 5.2. Assume that G/K = P^{n}(C C C) (n ≥ 2). Consider a subspace V ⊂ 𝔪 spanned by the vectors μ + Y and [I, μ − 2Y ], where Y ∈ 𝔪(μ) satisfies (Y, Y) = 2(μ, μ)^{2}(I, I). It is easily shown that dim V = 2, because (0 ≠)[I, μ − 2Y ] ∈ 𝔪(μ) + 𝔪(2μ) but μ + Y ∉ 𝔪(μ) + 𝔪(2μ). To show V is pseudo-abelian it suffices to prove

[μ + Y, [I, μ − 2Y ]] ∈ 𝔨_{0}.

By (1) of Lemma 5.3 and (ad μ)^{2}I = −4(μ, μ)^{2}I we have

[I, [μ, Y ]] = [−\frac{1}{4(μ, μ)^{2}} [μ, [μ, I]], [μ, Y ]]

= −\frac{2(μ, μ)^{2}}{4(μ, μ)^{2}} [[μ, I], Y ]

= −\frac{1}{2}[[μ, I], Y ].

Consequently, we have

[μ, [I, Y ]] = [[μ, I], Y ] + [I, [μ, Y ]] = \frac{1}{2}[[μ, I], Y ].

Therefore, by a simple calculation we have

[μ + Y, [I, μ − 2Y ]] = 2 {2(μ, μ)^{2}I − [Y, [I, Y ]]} .

We note that the right hand side of the above equality is contained in 𝔨_{0} + 𝔨(2μ). Since 𝔨(2μ) = R R RI and since

(I, 2(μ, μ)^{2}I − [Y, [I, Y ]]) = 2(μ, μ)^{2}(I, I) − ([I, Y ], [I, Y ])
= 2(μ, μ)^{2}(I, I) − (Y, Y) = 0,

we have [μ + Y, [I, μ − 2Y ]] ∈ 𝔨_{0}. Accordingly, we get (2) of Proposition 5.2. □

By Theorem 5.1 we can prove Theorem 5.6. Let G/K be a compact rank one Riemannian symmetric space not isomorphic to any sphere S^{n}. Define an integer Q(G/K) by

Q(G/K) = \begin{cases}min\{4n − 2, 3n + 1\}, & if G/K = P^{n}(C C C) (n ≥ 2), \\ min\{8n − 3, 7n + 1\}, & if G/K = P^{n}(H H H) (n ≥ 2), \\ 25, & if G/K = P^{2}(Cay Cay Cay). \end{cases}

Then, any open set of G/K cannot be isometrically imbedded into the Euclidean space R R R^{N} with N ≤ Q(G/K) − 1. Remark 4. Concerning the complex projective space P^{n}(C C C), we refer to the result of Agaoka [1]. He investigated directly the solvability of the Gauss equation associated with isometric imbeddings of P^{n}(C C C), and obtained the following Proposition 5.7 ([1]). Any open set of the complex projective space P^{n}(C C C) cannot be isometrically imbedded into the Euclidean space R R R^{N} with N ≤ [16n/5] − 1.

As is easily seen, Agaoka’s result gives a better estimate in case n is large enough (n ≥ 10). It is noted that our estimate obtained by the pseudo-nullity p(P^{n}(C C C)) does not necessarily give the least dimension of Euclidean spaces into which P^{n}(C C C) can be locally isometrically imbedded. This is an interesting phenomenon compared with the spaces Sp(n)/U(n) and Sp(n), where the least dimension are just determined by p(G/K).

6. Proof of Theorem 3.5

In this section we prove Theorem 3.5. Before starting the proof, we prepare some general lemmas. Notations are the same that are used in Introduction and § 2.

Let G/K be a compact irreducible Riemannian symmetric space with G simple. Let τ be the conjugation of 𝔤^{c} with respect to 𝔤. As is known (see [2]), there is a set of vectors {Z_{α} ∈ 𝔤_{α} | α ∈ Δ} of 𝔤^{c} satisfying

(1) θZ_{α} = Z_{θα}, τZ_{α} = Z_{−α},

(2) [Z_{α}, Z_{−α}] = 2\sqrt{−1} α/(α, α).

Let α, β ∈ Δ. We define an integer A_{α,β} by setting A_{α,β} = 2(α, β)/(β, β). Then, by induction we obtain the following formula, which is a well-known fact in the theory of Lie algebras (see [9]): Lemma 6.1. Assume that α + β ∉ Δ ∪ {0}. Then:

ad Z_{β}(ad Z_{−β})^{k}(Z_{α}) = k (−A_{α,β} + k − 1) (ad Z_{−β})^{k−1}(Z_{α}), k ∈ Z Z Z, k ≥ 0.

Let us set Δ_{0} = Δ ∩ 𝔟. For a root α ∈ Δ\Δ_{0}, we define a subspace 𝔤(α) of 𝔤^{c} by

𝔤(α) = 𝔤_{α} + 𝔤_{−α} + 𝔤_{θα} + 𝔤_{−θα}.

As is easily seen, 𝔤(α) satisfies the following properties. Lemma 6.2. Let α ∈ Δ\Δ_{0}. Then:

(1) 𝔤(α) = 𝔤(−α) = 𝔤(θα) = 𝔤(−θα).

(2) dim 𝔤(α) = 4 if θα ≠ −α; dim 𝔤(α) = 2 if θα = −α.

(3) Let β ∈ Δ\Δ_{0} satisfy β ≠ ±α, ±θα. Then 𝔤(β) is orthogonal to 𝔤(α) with respect to the inner product ( , ), i.e., (𝔤(α), 𝔤(β)) = 0.

We also have the following lemma. The proof is left to the readers. Lemma 6.3. Let Σ be the restricted root system of G/K and μ ∈ Σ. Then:

(1) Let α ∈ Δ(μ). Then −θα ∈ Δ(μ).

(2) The following decomposition holds.

𝔨(μ)^{c} + 𝔪(μ)^{c} = ∑_{α∈Δ(μ), −θα≤α} 𝔤(α) (orthogonal direct sum).

(3) Let α ∈ Δ(μ). Define vectors X(α)^{±} and Y (α)^{±} of 𝔤(α) by

X(α)^{+} = Z_{α} + Z_{−α} + Z_{θα} + Z_{−θα},
X(α)^{−} = \sqrt{−1} (Z_{α} − Z_{−α} + Z_{θα} − Z_{−θα}) ,
Y (α)^{+} = \sqrt{−1} (Z_{α} − Z_{−α} − Z_{θα} + Z_{−θα}) ,
Y (α)^{−} = Z_{α} + Z_{−α} − Z_{θα} − Z_{−θα}.

Then, it holds X(α)^{±} ∈ 𝔨(μ), Y (α)^{±} ∈ 𝔪(μ) and X(−θα)^{±} = ±X(α)^{±}, Y (−θα)^{±} = ±Y (α)^{±}.

(4) The set of vectors {X(α)^{±} | α ∈ Δ(μ)} (resp. {Y (α)^{±} | α ∈ Δ(μ)} ) spans 𝔨(μ) ( resp. 𝔪(μ) ).

These being prepared, we start the proof of Theorem 3.5. In the following we assume that Σ is of type BC_{n} and μ ∈ Σ is a multipliable root, i.e., 2μ ∈ Σ. Under this assumption, we have m(μ) = even, m(2μ) = odd and μ ∉ Δ, 2μ ∈ Δ (see [7] or Table 3).

We first prove Proposition 6.4. (1) Let α ∈ Δ(μ). Then A_{α,2μ} = 1 and α−2μ ∈ Δ, but α+2μ ∉ Δ∪{0}.

(2) Set I = Z_{2μ} + Z_{−2μ}. Then I ∈ 𝔨(2μ) and I^{†} determines a complex structure of 𝔪(μ), i.e., I^{†}^{2} = −111_{𝔪(μ)}.

Proof. It is clear that I ∈ 𝔨(2μ). Now let α ∈ Δ(μ). We consider the 2μ-series of roots containing α. Since A_{α,2μ} = 2(α, 2μ)/(2μ, 2μ) = 1, it follows that α − 2μ ∈ Δ. On the contrary, since the 𝔞-component of α + 2μ is equal to 3μ, it follows that α + 2μ ∉ Δ ∪ {0}. Therefore, by Lemma 6.1 we have

(ad I)^{2}(Z_{α}) = [Z_{2μ}, [Z_{−2μ}, Z_{α}]]
= −Z_{α}.

Moreover, since ad I ·θ = θ ·ad I and ad I ·τ = τ ·ad I, we have (ad I)^{2}(Z_{α′}) = −Z_{α′}, where α′ = ±α or ±θα. Therefore we have I^{†}^{2} = −111_{𝔪(μ)}, because the vectors Y (α)^{±} (α ∈ Δ(μ)) generate 𝔪(μ). □

The above lemma shows the assertion (1) of Theorem 3.5. Therefore, in what follows, we may assume that m(2μ) = 3. We first consider the sets Δ(2μ) and Δ(μ). Lemma 6.5. (1) There is a root ν ∈ Δ_{0} such that Δ(2μ) = {2μ, 2μ ± ν} and (ν, ν) = 4(μ, μ).

(2) Let α ∈ Δ(μ). Then (α, α) = 4(μ, μ).

(3) Let α, α′ ∈ Δ(μ). Assume that α′ ≠ α, −θα. Then, one and only one of the following assertions holds.

(a) A_{α′,α} = 1, A_{α′,−θα} = 0.

(b) A_{α′,α} = 0, A_{α′,−θα} = 1.

Proof. In Appendix of [2], we have proved that for a restricted root ψ ∈ Σ satisfying m(ψ) = odd and m(ψ) > 1, there is a root ν ∈ Δ_{0} such that ψ±ν ∈ Δ. Applying this to the case ψ = 2μ, we have the first part of the assertion (1). Since (2μ+ν)±(2μ−ν) ∉ Δ∪{0}, we have (2μ + ν, 2μ − ν) = 0. This shows that (ν, ν) = 4(μ, μ).

We now prove the assertion (2). Consider the α-series of roots containing 2μ. By (1) of Proposition 6.4, we have 2μ + α ∉ Δ ∪ {0} and 2μ − α ∈ Δ. On the other hand, by the fundamental property of symmetric spaces we have 2μ − 2α = −(α + θα) ∉ Δ. It is also seen that 2μ − 2α ≠ 0. Therefore we have A_{2μ,α} = 1. Since A_{α,2μ} = 1 (see (1) of Proposition 6.4) we have (α, α) = (2μ, 2μ).

Finally, we prove (3). Since α − θα = 2μ and since (α, α) = (θα, θα) = (2μ, 2μ), we have

A_{α′,α} + A_{α′,−θα} = \frac{2(α′, α)}{(α, α)} + \frac{2(α′, −θα)}{(θα, θα)} = \frac{2(α′, α − θα)}{(2μ, 2μ)} = A_{α′,2μ} = 1.

We also have |A_{α′,α}| ≤ 1 and |A_{α′,−θα}| ≤ 1, because α′ ≠ ±α, ±θα, (α′, α′) = (α, α). Then the assertion (3) immediately follows from these facts. □

In the following discussion we fix an element ν ∈ Δ_{0} stated in (1) of Lemma 6.5. Lemma 6.6. Let α ∈ Δ(μ). Then :

(1) It holds A_{α,ν} = ±1. Moreover, (a) A_{α,ν} = 1 ⟺ α − ν ∈ Δ.

(b) A_{α,ν} = −1 ⟺ α + ν ∈ Δ.

(2) α ± 2ν ∉ Δ ∪ {0}.

Proof. Since α − θα = 2μ ∈ Δ and 2μ + ν ∈ Δ, it follows that [Z_{ν}, [Z_{α}, Z_{−θα}]] ≠ 0. Hence, we have [Z_{ν}, Z_{α}] ≠ 0 or [Z_{ν}, Z_{−θα}] ≠ 0. Therefore, we have either α + ν ∈ Δ or −θα + ν ∈ Δ.

Now assume that α + ν ∈ Δ. Then we have α + ν ∈ Δ(μ) and hence by Lemma 6.5 we have (α + ν, α + ν) = 4(μ, μ). Since (α, α) = 4(μ, μ) and (ν, ν) = 4(μ, μ), we have (α, ν) = −2(μ, μ). This implies A_{α,ν} = −1. Conversely, if A_{α,ν} = −1, then we have α + ν ∈ Δ. This proves the assertion (a).

Next assume that −θα+ν ∈ Δ. Then, since −θα+ν = −θ(α−ν), we have α−ν ∈ Δ. In this case, by the same method stated above, we have (α, ν) = 2(μ, μ) and hence A_{α,ν} = 1. Conversely, if A_{α,ν} = 1, then we have α − ν ∈ Δ, which proves the assertion (b).

Finally, we show (2). It is clear that the 𝔞-component of α ± 2ν is μ. Hence α ± 2ν ≠ 0. Since (α±2ν, α±2ν) ≠ 4(μ, μ), we have α±2ν ∉ Δ(μ). This proves α±2ν ∉ Δ∪{0}. □

We now define an action of 𝔨_{0} on 𝔪(μ). Since [𝔨_{0}, 𝔪(μ)] ⊂ 𝔪(μ) (see Proposition 2.1), each element ad X (X ∈ 𝔨_{0}) induces a skew-symmetric endomorphism of 𝔪(μ), which is also denoted by X^{†}. This together with the action of 𝔨(2μ) defined in § 3, we get the action of 𝔨_{0} + 𝔨(2μ) on 𝔪(μ). By the definition we directly have

[X, X′]^{†} = X^{†}X^{′†} − X^{′†}X^{†} = [X^{†}, X^{′†}], X, X′ ∈ 𝔨_{0} + 𝔨(2μ).

Set ν̂ = 2ν/(ν, ν), P = Z_{ν} + Z_{−ν} and Q = \sqrt{−1}(Z_{ν} − Z_{−ν}). Then we easily have ν̂, P, Q ∈ 𝔨_{0} and

[ν̂, P ] = 2Q, [ν̂, Q] = −2P, [P, Q] = 2ν̂.

We now prove Proposition 6.7. The triplet {ν^{†}, P^{†}, Q^{†}} determines a quaternion structure of 𝔪(μ), i.e.,

(ν^{†})^{2} = P^{†}^{2} = Q^{†}^{2} = −111_{𝔪(μ)},
ν^{†}P^{†} = −P^{†}ν^{†} = Q^{†}, Q^{†}ν^{†} = −ν^{†}Q^{†} = P^{†}, P^{†}Q^{†} = −Q^{†}P^{†} = ν^{†}.

For the proof, we prepare the following Lemma 6.8. Let α ∈ Δ(μ). Then:

(1) [Z_{ν}, [Z_{−ν}, Z_{α}]] + [Z_{−ν}, [Z_{ν}, Z_{α}]] = −Z_{α}.

(2) [ν̂, [Z_{±ν}, Z_{α}]] + [Z_{±ν}, [ν̂, Z_{α}]] = 0.

Proof. Assume that α − ν ∈ Δ. Then we have A_{α,ν} = 1 and α + ν ∉ Δ ∪ {0} (see Lemma 6.6). By Lemma 6.1 we have

[Z_{ν}, [Z_{−ν}, Z_{α}]] = −A_{α,ν}Z_{α} = −Z_{α},
[ν̂, [Z_{−ν}, Z_{α}]] = \sqrt{−1}A_{α−ν,ν}[Z_{−ν}, Z_{α}] = −\sqrt{−1}[Z_{−ν}, Z_{α}],
[Z_{−ν}, [ν̂, Z_{α}]] = \sqrt{−1}A_{α,ν}[Z_{−ν}, Z_{α}] = \sqrt{−1}[Z_{−ν}, Z_{α}].

By these equalities and [Z_{ν}, Z_{α}] = 0, we get the assertions (1) and (2). Similarly, in the case α + ν ∈ Δ we can prove (1) and (2). □

We now proceed to the proof of Proposition 6.7. We first note that since θν = τν = ν, the endomorphism ad ν̂ commutes with θ and τ. Similarly, since θP = τP = P (resp. θQ = τQ = Q), ad P (resp. ad Q) commutes with θ and τ. Let α ∈ Δ(μ). By Lemmas 6.8, 6.6 and a direct calculation, we have

(ad ν̂)^{2}(Z_{α}) = [ν̂, \sqrt{−1}A_{α,ν}Z_{α}] = −A_{α,ν}^{2}Z_{α} = −Z_{α},
(ad P)^{2}(Z_{α}) = (ad Q)^{2}(Z_{α}) = [Z_{ν}, [Z_{−ν}, Z_{α}]] + [Z_{−ν}, [Z_{ν}, Z_{α}]] = −Z_{α}.

Therefore, by the same reason stated in the proof of (2) of Proposition 6.4 we can conclude (ν^{†})^{2} = P^{†}^{2} = Q^{†}^{2} = −111_{𝔪(μ)}. Further, by Lemmas 6.8, 6.6 and a direct calculation, we can also prove [P, [Q, Z_{α}]] = −[Q, [P, Z_{α}]], [ν̂, [P, Z_{α}]] = −[P, [ν̂, Z_{α}]] and [ν̂, [Q, Z_{α}]] = −[Q, [ν̂, Z_{α}]]. By the same reason as above, we have P^{†}Q^{†} = −Q^{†}P^{†}, ν^{†}P^{†} = −P^{†}ν^{†} and ν^{†}Q^{†} = −Q^{†}ν^{†}. From these equalities, it follows

ν^{†}P^{†} = (1/2)(ν^{†}P^{†} − P^{†}ν^{†}) = (1/2)[ν̂, P ]^{†} = Q^{†},
Q^{†}ν^{†} = (1/2)(Q^{†}ν^{†} − ν^{†}Q^{†}) = (1/2)[Q, ν̂]^{†} = P^{†},
P^{†}Q^{†} = (1/2)(P^{†}Q^{†} − Q^{†}P^{†}) = (1/2)[P, Q]^{†} = ν^{†}.

This completes the proof of the proposition. □

Finally, we prove Proposition 6.9. It holds I^{†} = ε ν^{†}, where ε ∈ R R R and ε^{2} = 1.

If the above proposition is true, we can get Theorem 3.5. In fact, set J = −(1/2)[I, Q], K = (1/2)[I, P ]. Then we have J, K ∈ 𝔨(2μ) and

J^{†} = −(1/2)[I^{†}, Q^{†}] = −(ε/2)[ν̂, Q]^{†} = ε P^{†},
K^{†} = (1/2)[I^{†}, P^{†}] = (ε/2)[ν̂, P ]^{†} = ε Q^{†}.

Consequently, it follows that the triplet {εI^{†}, εJ^{†}, εK^{†}}(⊂ 𝔨(2μ)^{†}) determines a quaternion structure of 𝔪(μ).

Now we start the proof of Proposition 6.9. For each α ∈ Δ(μ) let us define a complex number ρ_{α} by

[Z_{−2μ}, Z_{α}] = \sqrt{−1}ρ_{α}Z_{θα}.  (6.1)

ρ_{α} is well-defined, because −2μ + α = θα ∈ Δ and hence [Z_{−2μ}, Z_{α}] ∈ 𝔤_{θα}. Lemma 6.10. (1) ρ_{α}^{2} = 1, ρ_{−θα} = −ρ_{α}.

(2) For each α ∈ Δ(μ), it holds [I, Y (α)^{±}] = ±ρ_{α}Y (α)^{∓}.

Proof. From (6.1), we easily get

[I, Z_{α}] = \sqrt{−1}ρ_{α}Z_{θα}.  (6.2)

Applying θ to the both sides of (6.2), we have [I, Z_{θα}] = \sqrt{−1}ρ_{α}Z_{α}. Hence we have

[I, [I, Z_{α}]] = [I, \sqrt{−1}ρ_{α}Z_{θα}] = −ρ_{α}^{2}Z_{α}.

Since (ad I)^{2}(Z_{α}) = −Z_{α}, the above equality implies ρ_{α}^{2} = 1.

Applying τ and θτ to the both sides of (6.2), we have [I, Z_{−α}] = −\sqrt{−1}ρ_{α}Z_{−θα} and [I, Z_{−θα}] = −\sqrt{−1}ρ_{α}Z_{−α}. From the latter equality, it follows that ρ_{−θα} = −ρ_{α}. Moreover, by an easy calculation we obtain the assertion (2). □

We need one more lemma concerning the values ρ_{α} (α ∈ Δ(μ)). Lemma 6.11. Let ζ ∈ Δ_{0}, α ∈ Δ(μ). Assume that α + ζ ∈ Δ(μ). Then:

(a) If ζ ≠ ±ν, then ρ_{α+ζ} = ρ_{α}.

(b) If ζ = ν or −ν, then ρ_{α+ζ} = −ρ_{α}.

Proof. First note that [Z_{ζ}, Z_{α}] ∈ 𝔤_{α+ζ}, [Z_{ζ}, Z_{α}] ≠ 0. We also note

[[Z_{−2μ}, Z_{ζ}], Z_{α}] = [Z_{−2μ}, [Z_{ζ}, Z_{α}]] − [Z_{ζ}, [Z_{−2μ}, Z_{α}]]
= \sqrt{−1}(ρ_{α+ζ} − ρ_{α})θ[Z_{ζ}, Z_{α}].

Assume that ζ ≠ ±ν. Then since −2μ + ζ ∉ Δ ∪ {0} (see (1) of Lemma 6.5), we have [Z_{−2μ}, Z_{ζ}] = 0. Hence we have ρ_{α+ζ} = ρ_{α}. On the contrary, if ζ = ν or −ν, then −2μ + ζ ∈ Δ and −2μ + ζ + α = θ(α + ζ) ∈ Δ. Hence we have [[Z_{−2μ}, Z_{ζ}], Z_{α}] ≠ 0. Consequently, we have ρ_{α+ζ} − ρ_{α} ≠ 0. Since ρ_{α+ζ}^{2} = ρ_{α}^{2} = 1, we have ρ_{α+ζ} = −ρ_{α}. □

By a simple calculation, we have

[ν̂, Y (α)^{±}] = ∓A_{α,ν}Y (α)^{∓}.

Compare this equality with (2) of Lemma 6.10. Then, we know that Proposition 6.9 immediately follows from the following Lemma 6.12. Let α, α′ ∈ Δ(μ). Then it holds A_{α′,ν}/ρ_{α′} = A_{α,ν}/ρ_{α}.

Proof. If α′ = α, then there is nothing to prove. Next we consider the case α′ = −θα. By Lemma 6.10 (1), we have ρ_{−θα} = −ρ_{α}. On the other hand, we have

A_{−θα,ν} = \frac{2(−θα, ν)}{(ν, ν)} = −\frac{2(α, θν)}{(ν, ν)} = −\frac{2(α, ν)}{(ν, ν)} = −A_{α,ν}.

This shows that the lemma is true for the case α′ = −θα.

Now we assume that α′ ≠ α, −θα. Replacing α by −θα if necessary, we may assume that A_{α′,α} = 1 (see Lemma 6.5 (3)). Then, setting ζ = α′ − α, we have ζ ∈ Δ_{0}. In view of Lemma 6.6, we have A_{α,ν} = ±1, A_{α′,ν} = ±1. First consider the case A_{α′,ν} = A_{α,ν}. Then, since A_{ζ,ν} = A_{α′,ν} − A_{α,ν} = 0, we have ζ ≠ ±ν. Therefore, by Lemma 6.11, we have ρ_{α′} = ρ_{α+ζ} = ρ_{α}. This implies that A_{α′,ν}/ρ_{α′} = A_{α,ν}/ρ_{α}.

Next assume that A_{α′,ν} = −A_{α,ν}. Then we have A_{ζ,ν} = A_{α′,ν} − A_{α,ν} = −2A_{α,ν} = ±2, which implies (ζ, ν) = ±(ν, ν). Since (α, α) = (α′, α′) = (ν, ν) (see Lemma 6.5), we have

(ζ, ζ) = (α′ − α, α′ − α) = (α, α) (2 − A_{α′,α}) = (ν, ν).

Consequently, we have ζ = ν or −ν. Therefore, by Lemma 6.11 we have ρ_{α′} = ρ_{α+ζ} = −ρ_{α}. Hence, in this case, we also get A_{α′,ν}/ρ_{α′} = A_{α,ν}/ρ_{α}. □

By the above discussions, we have completed the proof of Theorem 3.5.

References

[1] Y. Agaoka, A note on local isometric imbeddings of complex projective spaces, J. Math. Kyoto Univ. 27 (1987), 501–505.

[2] Y. Agaoka and E. Kaneda, On local isometric immersions of Riemannian symmetric spaces, Tôhoku Math. J. 36 (1984), 107–140.

[3] , An estimate on the codimension of local isometric imbeddings of compact Lie groups, Hiroshima Math. J. 24 (1994), 77–110.

[4] , Local isometric imbeddings of symplectic groups, Geometriae Dedicata 71 (1998), 75–82.

[5] , Strongly orthogonal subsets in root systems, (to appear in Hokkaido Math. J.).

[6] , Local isometric imbeddings of Grassmann manifolds, (in preparation).

[7] S. Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34.

[8] N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 et 6, Hermann, Paris (1968).

[9] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York (1978).

[10] S. Kobayashi, Isometric imbeddings of compact symmetric spaces, Tôhoku Math. J. 20 (1968), 21–25.

[11] M. Takeuchi, Modern Spherical Functions, Translations of Math. Monographs vol. 135 (1994), Amer. Math. Soc., Providence, Rhode Island.

(Yoshio AGAOKA) Faculty of Integrated Arts and Sciences, Hiroshima University 1-7-1 Kagamiyama, Higashi-Hiroshima City, Hiroshima, 739-8521, Japan

E-mail address: agaoka@mis.hiroshima-u.ac.jp

(Eiji KANEDA) Department of International Studies, Osaka University of Foreign Studies 8-1-1 Aomadani-Higashi, Minoo City, Osaka, 562-8558, Japan

E-mail address: kaneda@post01.osaka-gaidai.ac.jp

LOWER BOUNDS FOR THE CURVATURE INVARIANT p(G/K) 23

Table 3. Multiplicities and local pseudo-nullities

Type  G/K  Σ  M(G/K)  N (G/K)
AI_{n} SU(n + 1)/SO(n + 1) (n ≥ 1)  A_{n}  {1, 0, 0}  {1, 0, 0}
AII_{n} SU(2(n + 1))/Sp(n + 1) (n ≥ 1) A_{n}  {4, 0, 0}  {4, 0, 0}
AIII_{p,q} SU(p + q)/S(U(p) × U(q)) (p ≥ q ≥ 1, p ≥ 2)
(p > q > 1)  BC_{q}  {2, 2(p − q), 1} {2, p − q, 1}
(p = q > 1)  C_{q}  {1, 2, 0}  {1, 2, 0}
(p > q = 1)  BC_{1}  {0, 2(p − 1), 1} {0, p − 1, 1}
BI_{p,q} SO(p + q)/SO(p) × SO(q) (p + q = odd, p > q ≥ 2)
B_{q}  {1, p − q, 0}  {1, p − q, 0}
BII_{p} SO(p + 1)/SO(p) (p = even ≥ 2) A_{1}  {p − 1, 0, 0}  {p − 1, 0, 0}
CI_{n} Sp(n)/U(n) (n ≥ 2)  C_{n}  {1, 1, 0}  {1, 1, 0}
CII_{p,q} Sp(p + q)/Sp(p) × Sp(q) (p ≥ q ≥ 1)
(p > q > 1)  BC_{q}  {4, 4(p − q), 3} {4, p − q, 3}
(p = q > 1)  C_{q}  {3, 4, 0}  {3, 4, 0}
(p > q = 1)  BC_{1}  {0, 4(p − 1), 3} {0, p − 1, 3}
(p = q = 1)  A_{1}  {3, 0, 0}  {3, 0, 0}
DI_{p,q} SO(p + q)/SO(p) × SO(q) (p + q = even, p ≥ q ≥ 2, (p, q) ≠ (2, 2))
(p ≥ q + 2)  B_{q}  {1, p − q, 0}  {1, p − q, 0}
(p = q)  D_{q}  {1, 0, 0}  {1, 0, 0}
DII_{p} SO(p + 1)/SO(p) (p = odd ≥ 3)  A_{1}  {p − 1, 0, 0}  {p − 1, 0, 0}
DIII_{n} SO(2n)/U(n) (n ≥ 4)
(n = 2m)  C_{m}  {1, 4, 0}  {1, 4, 0}
(n = 2m + 1)  BC_{m}  {4, 4, 1}  {4, 2, 1}
EI  E_{6}/Sp(4)  E_{6}  {1, 0, 0}  {1, 0, 0}
EII E_{6}/SU(6) · SU(2)  F_{4}  {1, 2, 0}  {1, 2, 0}
EIII E_{6}/Spin(10) · SO(2)  BC_{2}  {6, 8, 1}  {6, 4, 1}
EIV E_{6}/F_{4}  A_{2}  {8, 0, 0}  {8, 0, 0}
EV  E_{7}/SU(8)  E_{7}  {1, 0, 0}  {1, 0, 0}
EV I E_{7}/Spin(12) · SU(2)  F_{4}  {1, 4, 0}  {1, 4, 0}
EV II E_{7}/E_{6} · SO(2)  C_{3}  {1, 8, 0}  {1, 8, 0}
EV III E_{8}/Spin(16)  E_{8}  {1, 0, 0}  {1, 0, 0}
EIX E_{8}/E_{7} · SU(2)  F_{4}  {1, 8, 0}  {1, 8, 0}
F I  F_{4}/Sp(3) · SU(2)  F_{4}  {1, 1, 0}  {1, 1, 0}
F II F_{4}/Spin(9)  BC_{1}  {0, 8, 7}  {0, 1, 7}
G  G_{2}/SO(4)  G_{2}  {1, 1, 0}  {1, 1, 0}

24 YOSHIO AGAOKA AND EIJI KANEDA

Table 4. Categorical pseudo-nullities p_{cat}(G/K) (Classical type)

Type  Maximal SOS  #Γ  b(Γ)  p_{cat}(G/K)
AI_{n}  Γ (A_{n})^{0}  [(n + 1)/2]  n  n
AII_{n}  Γ (A_{n})^{0}  [(n + 1)/2]  n + 3[(n + 1)/2]  n + 3[(n + 1)/2]

AIII_{p,q}

(p > q > 1) Γ (BC_{q})^{0,s} q − s q + s

(0 ≤ s ≤ [q/2])

Γ (BC_{q})^{1,t} q − t p + t − 1
(0 ≤ t ≤ [(q − 1)/2])  max{[3q/2],
(p = q > 1) Γ (C_{q})^{s} q − s q + s [(2p + q − 3)/2]}

(0 ≤ s ≤ [q/2])

(p > q = 1) Γ (BC_{1})^{0,0}  1  1
Γ (BC_{1})^{1,0}  1  p − 1
BI_{p,q}  Γ (B_{q})^{0}(q = even)  q  q  p − 1
Γ (B_{q})^{1}  2[(q − 1)/2] + 1  p − 1
BII_{p}  Γ (A_{1})^{0}  1  p − 1  p − 1
CI_{n}  Γ (C_{n})^{s}  n − s  n  n

(0 ≤ s ≤ [n/2])

CII_{p,q}

(p > q > 1) Γ (BC_{q})^{0,s} q − s 3q − s

(0 ≤ s ≤ [q/2])

Γ (BC_{q})^{1,t} q − t p + 2q − t − 3
(0 ≤ t ≤ [(q − 1)/2])  max{3q, p + 2q − 3}
(p = q > 1) Γ (C_{q})^{s} q − s 3q − s

(0 ≤ s ≤ [q/2])

(p > q = 1) Γ (BC_{1})^{0,0}  1  3
Γ (BC_{1})^{1,0}  1  p − 1
(p = q = 1) Γ (A_{1})^{0}  1  3

DI_{p,q}

(p ≥ q + 2) Γ (B_{q})^{0}(q = even)  q  q
Γ (B_{q})^{1}  2[(q − 1)/2] + 1  p − 1  max{p − 1, q}
(p = q)  Γ (D_{q})^{0}  2[q/2]  q
DII_{p}  Γ (A_{1})^{0}  1  p − 1  p − 1

DIII_{n}

(n = 2m) Γ (C_{m})^{s} m − s m + 3s m + 3[m/2]

(0 ≤ s ≤ [m/2])

(n = 2m + 1) Γ (BC_{m})^{0,s} m − s m + 3s [5m/2]

(0 ≤ s ≤ [m/2])

Γ (BC_{m})^{1,t} m − t m + 3t + 1

(0 ≤ t ≤ [(m − 1)/2])

LOWER BOUNDS FOR THE CURVATURE INVARIANT p(G/K) 25

Table 5. Categorical pseudo-nullities p_{cat}(G/K) (Exceptional type)

Type  Maximal SOS  #Γ b(Γ) p_{cat}(G/K)
EI  Γ (E_{6})^{0}  4  6  6
EII  Γ (F_{4})^{0}  4  4  5
Γ (F_{4})^{1}  3  5
EIII  Γ (BC_{2})^{0,s} (s = 0, 1) 2 − s 2 + 5s  7
Γ (BC_{2})^{1,0}  2  5
EIV  Γ (A_{2})^{0}  1  9  9
EV  Γ (E_{7})^{0}  7  7  7
EV I  Γ (F_{4})^{0}  4  4  7
Γ (F_{4})^{1}  3  7
EV II Γ (C_{3})^{s} (s = 0, 1)  3 − s 3 + 7s  10
EV III Γ (E_{8})^{0}  8  8  8
EIX  Γ (F_{4})^{0}  4  4  11
Γ (F_{4})^{1}  3  11
F I  Γ (F_{4})^{s} (s = 0, 1)  4 − s  4  4
F II  Γ (BC_{1})^{0,0}  1  7  7
Γ (BC_{1})^{1,0}  1  1
G  Γ (G_{2})^{1}  2  2  2