LIE-ALGEBRAIC CHARACTERIZATION OF TANGENTIALLY DEGENERATE ORBITS OF s-REPRESENTATIONS

OSAMU IKAWA, TAKASHI SAKAI, AND HIROYUKI TASAKI

Abstract. In this paper we study tangential degeneracy of the orbits of srepresentations in the sphere. We show that the orbit of an s-representation is tangentially degenerate if and only if it is through a long root, or a short root of restricted root system of type G_{2}. Moreover these orbits provide many new examples of tangentially degenerate submanifolds which satisfy the Ferus equality.

1. Introduction

The linear isotropy representation of a Riemannian symmetric pair is called an s-representation, which was coined by Ozeki-Takeuchi [13]. This representation is orthogonal, so we regard its orbit of a point as a submanifold of the hypersphere in the representation space. A submanifold whose Gauss map is degenerate is said to be tangentially degenerate. The purpose of this paper is to give a Liealgebraic characterization of tangentially degenerate orbits of s-representations. For the purpose we describe the kernels of the differentials of the Gauss maps of the orbits by the restricted root systems of the Riemannian symmetric pair which determine s-representations. The description of the kernels of the differentials of the Gauss maps of the orbits lead us to a Lie-algebraic characterization of tangentially degenerate orbits and the classification of them. We can obtain the rank of the Gauss map of them and many examples of orbits satisfying the Ferus equality. We shall explain the Ferus equality in the next paragraph.

Ferus [5] obtained a remarkable result for tangentially degenerate submanifolds in the sphere. He showed that for a submanifold, there exists a number, called the Ferus number, with property that if the rank of the Gauss map is less than the Ferus number, then the submanifold must be a totally geodesic sphere. If the rank of the Gauss map is equal to the Ferus number, we call this equality the Ferus equality. Many examples of submanifolds which satisfy the Ferus equality have not been found. In their papers [10, 11, 12], Ishikawa, Kimura and Miyaoka studied submanifolds with degenerate Gauss mappings in the sphere via a method of isoparametric hypersurfaces. They showed that Cartan hypersurfaces and some focal submanifolds of homogeneous isoparametric hypersurfaces are tangentially degenerate. Moreover, some of them satisfy the Ferus equality.

We want to emphasize the importance of the orbits of s-representations. In fact, every homogeneous hypersurface in a sphere is an orbit of s-representation of Riemannian symmetric spaces of rank 2, by Hsiang-Lawson [8], and they are all isoparametric, by Takagi-Takahashi [15]. We have already studied austere orbits and weakly reflective orbits of s-representations and classified them in our previous paper [9]. Ishikawa, Kimura and Miyaoka showed some relationship between tangentially degeneracy and the property ‘austere’ of isoparametric hypersurfaces and their focal submanifolds. In this paper we study tangentially degenerate orbits of s-representations via methods of symmetric spaces and obtain that the spaces of relative nullity of them. The following theorem is the main result of this paper.

2000 Mathematics Subject Classification. 53C40 (Primary), 53C35 (Secondary).

Key words and phrases. Gauss map, tangentially degenerate, s-representation, symmetric space.

Theorem 1.1. Let (G, K) be an irreducible compact symmetric pair. An orbit of the s-representation is tangentially degenerate in the sphere if and only if either one of the followings is valid:

(1) The orbit is through a longest root of the restricted root system of (G, K).

(2) The restricted root system of (G, K) is of type G_{2} and the orbit is through

a short root. In such cases the space of relative nullity of the orbit is equal to the root space of the root.

It is well known that an s-representation admits a subspace which intersects all orbits orthogonally. An orthogonal representation of a compact Lie group which satisfies such a property is called a polar representation. Dadok [4] showed that with few exceptions s-representations occupy all polar representations of compact Lie groups and that any orbit of a polar representation is an orbit of an s-representation. By this result our main theorem gives the classification of tangentially degenerate orbits of irreducible polar representations in the spheres.

We have showed that the orbit of any root is weakly reflective in [9], so a tangentially degenerate orbit is weakly reflective. After the classification of tangentially degenerate orbits of s-representations we shall observe that these orbits provide many new examples of tangentially degenerate submanifolds in the sphere which satisfy the Ferus equality.

The organization of this paper is as follows:

1. Introduction

2. Preliminaries

3. Proof of Theorem 1.1 (Sufficiency for tangential degeneracy)

4. Spaces of relative nullity

5. Proof of Theorem 1.1 (Necessity for tangential degeneracy)

6. Lemmas on quaternionic symmetric spaces

7. Ferus equalities In Section 2, we review the definition of the Gauss map of a submanifold in

a sphere and its tangential degeneracy, and results concerning them. We also review fundamental facts on the orbits of s-representations and obtain a basic result on tangentially degenerate orbits mentioned in Proposition 2.5, which states that a tangentially degenerate orbit is always through a restricted root. After this proposition it is enough to consider only the orbits of restricted roots.

In Section 3, we show that the orbits satisfying one of the conditions (1) and (2) in Theorem 1.1 are tangentially degenerate.

Section 4 describes the spaces of relative nullity of the orbits of restricted roots. Proposition 4.1 gives a fundamental description of those spaces and leads Theorem 4.5 which determines the spaces of relative nullity of tangentially degenerate orbits. Using these we prove the last assertion of Theorem 1.1.

LIE-ALGEBRAIC CHARACTERIZATION OF TANGENTIALLY DEGENERATE ORBITS 3

In Section 5, we show that the orbits which do not satisfy (1) or (2) in Theorem 1.1 are not tangentially degenerate. At the last of this section we list all irreducible compact symmetric pair such that the orbits of their s-representations have degenerate Gauss maps.

In Section 5, we collect some results on restricted root systems of compact quaternionic symmetric pairs. The last lemma 6.7 is used in Subsection 5.6.

In Section 7, we review the definition of the Ferus number and collect its properties we need. Using these we show new examples of tangentially degenerate submanifolds which satisfy the Ferus equality.

We wish to thank the referee for helpful suggestions on improvement of our manuscript.

2. Preliminaries

Let f : M ⟶ S^{n} be an immersion of an l-dimensional manifold M into an n-dimensional sphere S^{n}. The Gauss map γ of f is defined as a mapping from M to a Grassmannian manifold G_{l+1}(R^{n+1}) of all (l + 1)-dimensional subspaces in R^{n+1} by:

γ : M ⟶ G_{l+1}(R^{n+1})
x ⟼ Rf(x) ⊕ T_{f(x)}(f(M)).

We denote by r the maximal rank of the Gauss map γ of f. If the Gauss map is degenerate, i.e. r < l, then an immersed submanifold f(M) ⊂ S^{n} is said to be tangentially degenerate or developable. We note that γ is constant, i.e. r = 0, if and only if f(M) is a part of a totally geodesic sphere.

We denote by h the second fundamental form of f and by A_{ξ} the shape operator of f with respect to a normal vector ξ. Chern and Kuiper [3] introduced the notion of the index of relative nullity at x ∈ M, that is the dimension of the vector space

N_{x} = {X ∈ T_{x}(M) | h(X, Y) = 0, ^{∀}Y ∈ T_{x}(M)}
= ⋂_{ξ∈T_{x}^{⊥}(M)} ker(A_{ξ}).

It is easy to show ker(dγ)_{x} = N_{x}, therefore the index of relative nullity is equal to the degeneracy of the Gauss map at each point.

A linear isotropy representation of a Riemannian symmetric pair is called an s-representation. In the following section, we will study orbits of s-representations which are tangentially degenerate. For this purpose, we shall provide some fundamental notions of orbits of s-representations in this section.

Let G be a compact, connected Lie group and K a closed subgroup of G. Assume that θ is an involutive automorphism of G and G_{θ}^{0} ⊂ K ⊂ G_{θ}, where

G_{θ} = {g ∈ G | θ(g) = g}

and G_{θ}^{0} is the identity component of G_{θ}. Then (G, K) is a compact symmetric pair with respect to θ. We denote the Lie algebras of G and K by 𝔤 and 𝔨, respectively. The involutive automorphism of 𝔤 induced from θ will be also denoted by θ. Then we have

𝔨 = {X ∈ 𝔤 | θ(X) = X}.

Take an inner product ⟨ , ⟩ on 𝔤 which is invariant under θ and the adjoint representation of G. Set

𝔪 = {X ∈ 𝔤 | θ(X) = −X},

then we have a canonical orthogonal direct sum decomposition

𝔤 = 𝔨 + 𝔪.

Fix a maximal abelian subspace 𝔞 in 𝔪 and a maximal abelian subalgebra 𝔱 in 𝔤 containing 𝔞. For α ∈ 𝔱 we set

(2.1)  𝔤̃_{α} = \{X ∈ 𝔤^{C} | [H, X] = \sqrt{−1}⟨α, H⟩X (H ∈ 𝔱)\}

and define the root system R̃ of 𝔤 by

(2.2)  R̃ = {α ∈ 𝔱 − {0} | 𝔤̃_{α} ≠ {0}}.

For λ ∈ 𝔞 we set

𝔤_{λ} = \{X ∈ 𝔤^{C} | [H, X] = \sqrt{−1}⟨λ, H⟩X (H ∈ 𝔞)\}

and define the restricted root system R of (𝔤, 𝔨) by

R = {λ ∈ 𝔞 − {0} | 𝔤_{λ} ≠ {0}}.

Set

R̃_{0} = R̃ ∩ 𝔨

and denote the orthogonal projection from 𝔱 to 𝔞 by H ↦ H̄. Then we have

R = {ᾱ | α ∈ R̃ − R̃_{0}}.

We take a basis of 𝔱 extended from a basis of 𝔞 and define the lexicographic orderings > on 𝔞 and 𝔱 with respect to these bases. Then for H ∈ 𝔱, H̄ > 0 implies H > 0. We denote by F̃ the set of simple roots of R̃ with respect to the ordering >. Set

F̃_{0} = F̃ ∩ R̃_{0},

then the set of simple roots F of R with respect to the ordering > is given by

F = {ᾱ | α ∈ F̃ − F̃_{0}}.

We set

R̃_{+} = {α ∈ R̃ | α > 0}, R_{+} = {λ ∈ R | λ > 0}.

Then we have

R_{+} = {ᾱ | α ∈ R̃_{+} − R̃_{0}}.

We also set

𝔨_{0} = {X ∈ 𝔨 | [X, H] = 0 (H ∈ 𝔞)},

and define

𝔨_{λ} = 𝔨 ∩ (𝔤_{λ} + 𝔤_{−λ}), 𝔪_{λ} = 𝔪 ∩ (𝔤_{λ} + 𝔤_{−λ})

for λ ∈ R_{+}. Under these notations, we have the following lemma whose statements are simple consequences of results of Chapters VI and VII in Helgason’s book [6].

Lemma 2.1. (1) We have orthogonal direct sum decompositions

𝔨 = 𝔨_{0} + ∑_{λ∈R_{+}} 𝔨_{λ}, 𝔪 = 𝔞 + ∑_{λ∈R_{+}} 𝔪_{λ}.

(2) If H ∈ 𝔞 and ⟨λ, H⟩ ≠ 0, then ad(H) gives a linear isomorphism between 𝔪_{λ} and 𝔨_{λ}.

LIE-ALGEBRAIC CHARACTERIZATION OF TANGENTIALLY DEGENERATE ORBITS 5

We define a subset D of 𝔞 by

D = ⋃_{λ∈R}{H ∈ 𝔞 | ⟨λ, H⟩ = 0}.

A connected component of 𝔞 − D is a Weyl chamber. We set

C = {H ∈ 𝔞 | ⟨λ, H⟩ > 0 (λ ∈ F)}.

Then C is an open convex subset of 𝔞 and the closure of C is given by

C̄ = {H ∈ 𝔞 | ⟨λ, H⟩ ≥ 0 (λ ∈ F)}.

For a subset Δ ⊂ F, we define

C^{Δ} = {H ∈ C̄ | ⟨λ, H⟩ > 0 (λ ∈ Δ), ⟨μ, H⟩ = 0 (μ ∈ F − Δ)}.

Lemma 2.2. (1) For Δ_{1} ⊂ F, the decomposition

\overline{C^{Δ_{1}}} = ⋃_{Δ⊂Δ_{1}} C^{Δ}

is a disjoint union. In particular, C̄ = ⋃_{Δ⊂F} C^{Δ} is a disjoint union.

(2) For Δ_{1}, Δ_{2} ⊂ F, Δ_{1} ⊂ Δ_{2} if and only if C^{Δ_{1}} ⊂ \overline{C^{Δ_{2}}}.

For each λ ∈ F, we take H_{λ} ∈ 𝔞 such that

⟨H_{λ}, μ⟩ = \{ \begin{matrix}1 & (μ = λ), \\ 0 & (μ ≠ λ) \end{matrix} (μ ∈ F).

Then, for Δ ⊂ F, we have

C^{Δ} = { ∑_{λ∈Δ} t_{λ}H_{λ} | t_{λ} > 0} .

We set

R^{Δ} = R ∩ (F − Δ)_{Z}, R_{+}^{Δ} = R^{Δ} ∩ R_{+}.

Under these notations, we have the following lemma. Lemma 2.3 ([7]). Fix a subset Δ ⊂ F. For H ∈ C^{Δ} we have the following:

(1) R^{Δ} = {μ ∈ R | ⟨μ, H⟩ = 0},

(2) R_{+}^{Δ} = {μ ∈ R_{+} | ⟨μ, H⟩ = 0}.

Now we shall study an orbit Ad(K)H of the linear isotropy representation of (G, K) through H ∈ 𝔪. We set

Z_{K}^{H} = {k ∈ K | Ad(k)H = H}.

Then Z_{K}^{H} is a closed subgroup of K and the orbit Ad(K)H is diffeomorphic to the coset manifold K/Z_{K}^{H}. The Lie algebra 𝔷_{K}^{H} of Z_{K}^{H} is given by

𝔷_{K}^{H} = {X ∈ 𝔨 | [H, X] = 0}.

An orbit Ad(K)H is a submanifold of the hypersphere S of radius ∥H∥ in 𝔪. From [7], Ad(K)H is connected. Since

𝔪 = ⋃_{k∈K} Ad(k)C̄,

without loss of generality we may assume H ∈ C̄. Moreover, from Lemma 2.2, there exists Δ ⊂ F such that H ∈ C^{Δ}. From Lemma 2.1 we have the following lemma.

Lemma 2.4 ([9]). For Δ ⊂ F and H ∈ C^{Δ}, the tangent space T_{H}(Ad(K)H) of the orbit Ad(K)H at H and the normal space T_{H}^{⊥}(Ad(K)H) in the hypersphere can be expressed as

(2.3)  T_{H}(Ad(K)H) = ∑_{μ∈R_{+}−R_{+}^{Δ}} 𝔪_{μ},

(2.4)  T_{H}^{⊥}(Ad(K)H) = 𝔞 ∩ H^{⊥} + ∑_{ν∈R_{+}^{Δ}} 𝔪_{ν} = Ad((Z_{K}^{H})_{0})(𝔞 ∩ H^{⊥}),

where (Z_{K}^{H})_{0} is the identity component of the stabilizer Z_{K}^{H} of H in K.

Proposition 2.5. If the orbit Ad(K)H through H ∈ 𝔞 is tangentially degenerate, then H is a constant multiple of a restricted root.

Proof. First we note that

A_{ξ} = Ad(k)^{−1}A_{Ad(k)ξ}Ad(k)

for any ξ ∈ 𝔞 ∩ H^{⊥} and k ∈ (Z_{K}^{H})_{0}. From this we have

⋂_{ξ∈T_{H}^{⊥}(Ad(K)H)} ker A_{ξ} = ⋂_{ξ∈Ad((Z_{K}^{H})_{0})(𝔞∩H^{⊥})} ker A_{ξ}

= ⋂_{{ξ∈𝔞∩H^{⊥} \atop k∈(Z^{H}_{K})_{0}}} ker A_{Ad(k)ξ}

= ⋂_{{ξ∈𝔞∩H^{⊥} \atop k∈(Z^{H}_{K})_{0}}} ker(Ad(k)A_{ξ}Ad(k)^{−1})

= ⋂_{{ξ∈𝔞∩H^{⊥} \atop k∈(Z^{H}_{K})_{0}}} ker(A_{ξ}Ad(k)^{−1})

= ⋂_{{ξ∈𝔞∩H^{⊥} \atop k∈(Z^{H}_{K})_{0}}} Ad(k) ker A_{ξ}

= ⋂_{k∈(Z_{K}^{H})_{0}} Ad(k) ⋂_{ξ∈𝔞∩H^{⊥}} ker A_{ξ}.

For ξ ∈ 𝔞 ∩ H^{⊥} the set of eigenvalues of A_{ξ} is given by

\{−\frac{⟨λ, ξ⟩}{⟨λ, H⟩} | λ ∈ R_{+} − R_{+}^{Δ}\} ,

and the eigenspace associated with eigenvalue −⟨λ, ξ⟩/⟨λ, H⟩ is given by

∑_{− \frac{⟨μ,ξ⟩}{⟨μ,H⟩} =− \frac{⟨λ,ξ⟩}{⟨λ,H⟩}} 𝔪_{μ}.

See [9] for details. The space ker A_{ξ} is nothing but the eigenspace associated with 0-eigenvalue. Thus

ker A_{ξ} = ∑_{⟨μ,ξ⟩=0} 𝔪_{μ}.

Therefore we have

⋂_{ξ∈𝔞∩H^{⊥}} ker A_{ξ} = ⋂_{ξ∈𝔞∩H^{⊥}} ∑_{⟨μ,ξ⟩=0} 𝔪_{μ} = ∑_{μ//H} 𝔪_{μ},

where μ // H means that μ and H are parallel. Hence

(2.5)  ⋂_{ξ∈T_{H}^{⊥}(Ad(K)H)} ker A_{ξ} = ⋂_{k∈(Z_{K}^{H})_{0}} Ad(k) ∑_{μ//H} 𝔪_{μ} ⊂ ∑_{μ//H} 𝔪_{μ}.

Consequently, if Ad(K)H is tangentially degenerate, then H must be a constant multiple of a restricted root. □

3. Proof of Theorem 1.1 (Sufficiency for tangential degeneracy)

We retain the notation in Section 2. Let (G, K) be an irreducible compact symmetric pair. By the conjugacy of a maximal abelian subspace 𝔞 of 𝔪 under the action of K, every Ad(K)-orbit in 𝔪 intersects with 𝔞. The restricted root system R is the root system of 𝔤 with respect to 𝔞. Since we are concerned with the tangential degeneracy of Ad(K)-orbit, by Proposition 2.5 we can restrict our attention to Ad(K)-orbit through roots in R. Since the tangentially degeneracy of the orbit is invariant under scalar multiples on the vector space 𝔪, we do not discriminate the difference of the length of a vector H. When (G, K) is of rank 1, K acts on the sphere in 𝔪 transitively. Therefore we only consider a symmetric pair whose rank is greater than or equal to 2. In this section we prove that Ad(K)λ is tangentially degenerate, if either one of (1) and (2) in the Theorem 1.1 is valid.

We put H = λ ∈ R_{+} and

Δ = {μ ∈ F | ⟨μ, λ⟩ > 0}.

Then we have λ ∈ C^{Δ}. If 2λ ∉ R_{+}, then 𝔨_{0} + 𝔨_{λ} is a Lie subalgebra of 𝔨. We denote by K(λ) the analytic subgroup of K which corresponds to 𝔨_{0} + 𝔨_{λ}.

Lemma 3.1. If λ ∈ R_{+} satisfies (a) 2λ ∉ R_{+},

(b) λ + ν ∉ R and λ − ν ∉ R for all ν ∈ R_{+}^{Δ},

then Ad(K)λ is tangentially degenerate.

Proof. Since the tangent space of the orbit Ad(K)λ at λ is given as in (2.3), the image of λ by the Gauss map γ is

γ(λ) = Rλ + ∑_{μ∈R_{+}−R_{+}^{Δ}} 𝔪_{μ},

and its orthogonal complement in 𝔪 is

γ(λ)^{⊥} = 𝔞 ∩ λ^{⊥} + ∑_{ν∈R_{+}^{Δ}} 𝔪_{ν}.

From a rule of the bracket product of root spaces and the assumption (b), we have

[𝔨_{0}, 𝔞 ∩ λ^{⊥} + ∑_{ν∈R_{+}^{Δ}} 𝔪_{ν} ] ⊂ ∑_{ν∈R_{+}^{Δ}} 𝔪_{ν},  [𝔨_{λ}, 𝔞 ∩ λ^{⊥} + ∑_{ν∈R_{+}^{Δ}} 𝔪_{ν} ] = {0}.

Therefore

[𝔨_{0} + 𝔨_{λ}, 𝔞 ∩ λ^{⊥} + ∑_{ν∈R_{+}^{Δ}} 𝔪_{ν} ] ⊂ 𝔞 ∩ λ^{⊥} + ∑_{ν∈R_{+}^{Δ}} 𝔪_{ν}.

This yields

Ad(K(λ)) (𝔞 ∩ λ^{⊥} + ∑_{ν∈R_{+}^{Δ}} 𝔪_{ν} ) = 𝔞 ∩ λ^{⊥} + ∑_{ν∈R_{+}^{Δ}} 𝔪_{ν}.

Hence

Ad(K(λ)) · γ(λ) = γ(λ).

Since γ is K-equivariant, we have

γ(Ad(k)λ) = Ad(k)γ(λ) = γ(λ)

for any k ∈ K(λ). This means that γ is constant on Ad(K(λ))λ. It is clear that Ad(K(λ))λ is not a point, since T_{λ}(Ad(K(λ))λ) = 𝔪_{λ}. Consequently Ad(K)λ is tangentially degenerate. □

We denote by δ ∈ R_{+} the highest root of R.

Lemma 3.2 ([16]). For λ ∈ R_{+},

\frac{⟨λ, δ⟩}{∥δ∥^{2}} = \{ \begin{matrix}0 & (when λ ⊥ δ), \\ 1 & (when λ = δ), \\ 1/2 & (otherwise). \end{matrix}

When ⟨λ, δ⟩/∥δ∥^{2} = 0, then λ − δ is not a root. When ⟨λ, δ⟩/∥δ∥^{2} = 1/2, then λ − δ is a root.

Wolf [16] showed this lemma in the case where R is the root system of a simple Lie algebra. The proof of Lemma 3.2 is similar, so we omit its proof.

The following corollaries 4.3 and 4.4 show that the orbit is tangentially degenerate ‘if’ either one of (1) and (2) in Theorem 4.1 is valid.

From Lemmas 3.1 and 3.2 we have the following corollaries.

Corollary 3.3. The orbit Ad(K)λ through a longest root λ of R is tangentially degenerate.

Corollary 3.4. The orbit through a short root in a restricted root system of type G_{2} is tangentially degenerate.

Proof. If α and β in a restricted root system R of type G_{2} are orthogonal, then α±β ∉ R. In particular, a short root in R satisfies the condition (b) of Lemma 3.1. It also satisfies (a) of Lemma 3.1 and its orbit is tangentially degenerate. □

4. Spaces of relative nullity

If the orbit Ad(K)H through H ∈ 𝔞 is tangentially degenerate, then H is a constant multiple of a restricted root because of Proposition 2.5. We describe the spaces of relative nullity of the orbits of restricted roots and prove the last assertion of Theorem 1.1 in this section. In order to determine the spaces of relative nullity of these orbits we give the following criterion for an orbit of an s-representation to be tangentially degenerate.

Proposition 4.1. The orbit Ad(K)λ through a restricted root λ ∈ R is tangentially degenerate if and only if there exists a non-zero subspace of ∑_{μ//λ} 𝔪_{μ} which is invariant under ad(𝔷_{K}^{λ}). More precisely, the following equality is valid:

(4.6)  ker(dγ)_{λ} = ⋂_{k∈(Z_{K}^{λ})_{0}} Ad(k) ∑_{μ//λ} 𝔪_{μ}

and ker(dγ)_{λ} is the maximal subspace of ∑_{μ//λ} 𝔪_{μ} which is invariant under ad(𝔷_{K}^{λ}).

Proof. From (2.5) we have (4.6) immediately. Thus the orbit Ad(K)λ is tangentially degenerate if and only if the right-hand side of (4.6) is a non-zero vector space.

If there exists a non-zero subspace V of ∑_{μ//λ} 𝔪_{μ} which is invariant under Ad((Z_{K}^{λ})_{0}), then

⋂_{k∈(Z_{K}^{λ})_{0}} Ad(k) ∑_{μ//λ} 𝔪_{μ} ⊃ ⋂_{k∈(Z_{K}^{λ})_{0}} Ad(k)V = V ≠ {0}.

Hence Ad(K)λ is tangentially degenerate. Conversely, we assume that Ad(K)λ is tangentially degenerate. Then

⋂_{k∈(Z_{K}^{λ})_{0}} Ad(k) ∑_{μ//λ} 𝔪_{μ} ⊂ ∑_{μ//λ} 𝔪_{μ}

is a non-zero subspace, and we denote it by V. Then for any g ∈ (Z_{K}^{λ})_{0} we have

Ad(g)V = Ad(g) ⋂_{k∈(Z_{K}^{λ})_{0}} Ad(k) ∑_{μ//λ} 𝔪_{μ}

= ⋂_{k∈(Z_{K}^{λ})_{0}} Ad(gk) ∑_{μ//λ} 𝔪_{μ} = V.

Thus V is invariant under Ad((Z_{K}^{λ})_{0}). Consequently, the orbit Ad(K)λ is tangentially degenerate if and only if there exists a non-zero subspace of ∑_{μ//λ} 𝔪_{μ} invariant under Ad((Z_{K}^{λ})_{0}). Since 𝔷_{K}^{λ} is the Lie algebra of a connected Lie group (Z_{K}^{λ})_{0}, we obtain the assertion. □

In particular, for an orbit of the adjoint representation of a compact Lie group we have the following corollary.

Corollary 4.2. An adjoint orbit of a compact, connected semisimple Lie group through a root α is tangentially degenerate if and only if there exists a non-zero subspace of

𝔤 ∩ (𝔤_{α} ⊕ 𝔤_{−α})

which is invariant under ad(𝔷_{G}^{α}).

Lemma 4.3. Let λ be a root and V a non-zero subspace of 𝔪_{λ}. Then V is invariant under ad(𝔷_{K}^{λ}) if and only if V is invariant under ad(𝔨_{0}) and satisfies

[ ∑_{ν∈R_{+}^{Δ}} 𝔨_{ν}, V ] = {0}.

In addition, if the action of 𝔨_{0} on 𝔪_{λ} is irreducible then V = 𝔪_{λ}.

Proof. Since

𝔷_{K}^{λ} = {X ∈ 𝔨 | [X, λ] = 0} = 𝔨_{0} ⊕ ∑_{ν∈R_{+}^{Δ}} 𝔨_{ν},

V is invariant under ad(𝔷_{K}^{λ}) if and only if V is invariant under ad(𝔨_{0}) and

[ ∑_{ν∈R_{+}^{Δ}} 𝔨_{ν}, V ] ⊂ V ⊂ 𝔪_{λ}.

On the other hand,

[ ∑_{ν∈R_{+}^{Δ}} 𝔨_{ν}, V ] ⊂ [ ∑_{ν∈R_{+}^{Δ}} 𝔨_{ν}, 𝔪_{λ}] ⊂ ∑_{ν∈R_{+}^{Δ}}(𝔪_{λ+ν} ⊕ 𝔪_{λ−ν}).

Hence we have

[ ∑_{ν∈R_{+}^{Δ}} 𝔨_{ν}, V ] ⊂ (𝔪_{λ} ∩ ∑_{ν∈R_{+}^{Δ}}(𝔪_{λ+ν} ⊕ 𝔪_{λ−ν})) = {0}.

□

Lemma 4.4. The root space 𝔪_{λ} corresponds to a longest root λ is a subspace of ∑_{μ//λ} 𝔪_{μ} invariant under ad(𝔷_{K}^{λ}).

Proof. We can suppose that λ is the highest root δ by the action of the Weyl group. The Lie algebra 𝔷_{K}^{δ} of Z_{K}^{δ} is given by

𝔷_{K}^{δ} = {X ∈ 𝔨 | [X, δ] = 0} = 𝔨_{0} ⊕ ∑_{⟨ν,δ⟩=0} 𝔨_{ν}.

From Lemma 3.2, we have δ ± ν ∉ R for any ν ∈ R_{+} which is perpendicular to δ. Hence from Lemma 4.3, 𝔪_{δ} is invariant under ad(𝔷_{K}^{δ}). □

From this lemma, we have the following theorem. Four cases in the following theorem are equivalent with two cases in Theorem 1.1. Hence the following theorem shows the last assertion of Theorem 1.1.

Theorem 4.5. Let (G, K) be a compact symmetric pair. If λ in R is one of the following list, then ker(dγ)_{λ} = 𝔪_{λ}.

(1) a long root except in the case where R is of type BC,

(2) any root in the case where R is of type G_{2},

(3) a longest root in the case where (G, K) is a Hermitian symmetric pair with restricted root system of type BC_{p} and p ≥ 2,

(4) a long root in the case where (G, K) = (Sp(2p + n), Sp(p) × Sp(p + n)) (p ≥ 2, n ≥ 1).

Proof. We divide the proof into the four cases.

(1) The conclusion follows directly from Proposition 4.1 and Lemma 4.4.

(2) For any root λ ∈ R and ν ∈ R which satisfies ⟨ν, λ⟩ = 0, we have λ ± ν ∉ R, because R is of type G_{2}. Hence ker(dγ)_{λ} = 𝔪_{λ} by Proposition 4.1.

Before treating the cases (3) and (4), we recall the restricted root system of type BC_{p}. In this case we can put

R = {±2e_{i} | 1 ≤ i ≤ p} ∪ {±e_{i} | 1 ≤ i ≤ p} ∪ {±e_{i} ± e_{j} | 1 ≤ i < j ≤ p},

λ = 2e_{1}.

We already know that the space of relative nullity N_{λ} of Ad(K)λ satisfies

𝔪_{2e_{1}} ⊂ N_{λ} ⊂ 𝔪_{2e_{1}} + 𝔪_{e_{1}}

and invariant under ad(𝔷_{K}^{λ}). Since

R_{+}^{Δ} = {μ ∈ R_{+} | ⟨λ, μ⟩ = 0}
= {2e_{i} | 2 ≤ i ≤ p} ∪ {e_{i} | 2 ≤ i ≤ p} ∪ {e_{i} ± e_{j} | 2 ≤ i < j ≤ p},

we have

𝔷_{K}^{λ} = 𝔨_{0} + ∑_{μ∈R_{+}^{Δ}} 𝔨_{μ} = 𝔨_{0} + ∑_{2≤i≤p} 𝔨_{2e_{i}} + ∑_{2≤i≤p} 𝔨_{e_{i}} + ∑_{2≤i<j≤p} 𝔨_{e_{i}±e_{j}}.

(3) In order to prove the assertion in this case, we recall the following two lemmas.

Lemma 4.6 ([14] Lemma 2.3). For a Hermitian symmetric space, the complex structure J translates restricted root spaces as following:

J𝔪_{e_{i}±e_{j}} = 𝔪_{e_{i}∓e_{j}}, J𝔪_{e_{i}} = 𝔪_{e_{i}}, J𝔞 = ∑_{i=1}^{p} 𝔪_{2e_{i}}.

We denote the Hopf fibration by π : S^{2n+1} ⟶ CP^{n}.

Lemma 4.7 ([11] Lemma 2.2). Let M ⊂ CP^{n} be a complex submanifold of complex dimension k. Then π^{−1}(M) is a submanifold of dimension 2k + 1 with degenerate Gauss map of S^{2n+1} . Moreover, if M is compact and not a complex projective subspace, then the rank of Gauss map is equal to 2k.

Without loss of generality we can put λ = 2e_{1}, and we consider the orbit Ad(K)λ through λ. The tangent space of Ad(K)λ at λ is given by

T_{λ}(Ad(K)λ) = ∑_{μ∈R_{+}−R_{+}^{Δ}} 𝔪_{μ} = 𝔪_{2e_{1}} + 𝔪_{e_{1}} + ∑_{2≤i≤p} 𝔪_{e_{1}±e_{i}}.

We denote by π : S ⟶ CP^{n} the Hopf fibration from the hypersphere S in 𝔪 to the complex projective space. Then the image π(Ad(K)λ) of the orbit Ad(K)λ is a submanifold of CP^{n}, and its tangent space at π(λ) is given by

T_{π(λ)}(π(Ad(K)λ)) = 𝔪_{e_{1}} + ∑_{2≤i≤p} 𝔪_{e_{1}±e_{i}}.

Therefore from Lemma 4.6, π(Ad(K)λ) is a complex submanifold of CP^{n}. Obviously π(Ad(K)λ) is not a complex projective subspace when p ≥ 2. Thus from Lemma 4.7 the index of the relative nullity of Ad(K)λ ⊂ S is equal to 1. Hence N_{λ} = 𝔪_{2e_{1}}.

(4) We shall give the restricted root space decomposition of (G, K) = (Sp(2p + n), Sp(p) × Sp(p + n)). We express 𝔤 as

𝔤 = 𝔰𝔭(2p + n) = {X ∈ M_{2p+n}(H) | ^{t}X̄ + X = 0}.

We define an involutive automorphism θ on 𝔤 by

θ : 𝔤 ⟶ 𝔤; X ⟼ [ \begin{matrix}I_{p} \\ & −I_{p+n} \end{matrix} ] X [ \begin{matrix}I_{p} \\ & −I_{p+n} \end{matrix} ] ,

where I_{r} denotes the r × r identity matrix. Then the eigenspaces 𝔨 and 𝔪 of θ associated to eigenvalues ±1 are given by

𝔨 = \{[ \begin{matrix}X \\ & Y \end{matrix} ] | X ∈ 𝔰𝔭(p), Y ∈ 𝔰𝔭(p + n)\} ,
𝔪 = \{[ \begin{matrix}& X \\ −^{t}X̄ \end{matrix} ] | X ∈ M_{p,p+n}(H)\} .

We take a maximal abelian subspace 𝔞 of 𝔪 by

𝔞 = \{[ \begin{matrix}& T \\ −T \end{matrix} ] | T = t_{1}E_{11} + ⋯ + t_{p}E_{pp}, t_{i} ∈ R\} ,

where E_{ij} denotes a matrix whose (i, j) element is 1 and all other elements are 0. We define e_{i} ∈ 𝔞 by

e_{i} = [ \begin{matrix}& E_{ii} \\ −E_{ii} \end{matrix}  ] .

Then the restricted root system of (𝔤, 𝔨) is of type BC_{p}. We note that, when n = 0, the restricted root system is of type C_{p}.

In the case of type BC, the restricted root spaces 𝔨_{e_{i}} and 𝔪_{e_{i}} which correspond to e_{i} are given by

𝔪_{e_{i}} = {∑_{j=1}^{n}(x_{j}E_{i,2p+j} − x̄_{j}E_{2p+j,i}) | x_{j} ∈ H} ,

𝔨_{e_{i}} = {∑_{j=1}^{n}(y_{j}E_{p+i,2p+j} − ȳ_{j}E_{2p+j,p+i}) | y_{j} ∈ H} .

In order to prove the assertion in this case, we will show that N_{λ} does not contain 𝔪_{e_{1}}-component. We take X ∈ 𝔪_{e_{1}} arbitrarily. Then [𝔨_{e_{2}}, X] ⊂ 𝔪_{e_{1}+e_{2}} + 𝔪_{e_{1}−e_{2}}. Since N_{λ} is invariant under ad(𝔷_{K}^{λ}), we have that if X ∈ N_{λ} then [𝔨_{e_{2}}, X] ⊂ N_{λ} ⊂ 𝔪_{2e_{1}} + 𝔪_{e_{1}}. Therefore, if X ∈ N_{λ} then [𝔨_{e_{2}}, X] = {0}. We can express X = ∑_{j=1}^{n}(x_{j}E_{1,2p+j} − x̄_{j}E_{2p+j,1}) ∈ 𝔪_{e_{1}}. Then

[𝔨_{e_{2}}, X] = {(∑_{j=1}^{n} x_{j}ȳ_{j}) E_{1,p+2} − (∑_{j=1}^{n} y_{j}x̄_{j}) E_{p+2,1} | y_{j} ∈ H} .

This yields X = 0. Thus N_{λ} does not contain 𝔪_{e_{1}}-component. Hence N_{λ} = 𝔪_{λ}. □ 5. Proof of Theorem 1.1 (Necessity for tangential degeneracy)

In this section we prove that Ad(K)λ is tangentially degenerate, “only if” either one of (1) and (2) in Theorem 1.1 is valid. We divide the proof into six cases which are treated in the following six subsections. Before beginning the subsections we prepare the following lemma.

Lemma 5.1. If the restricted root system R is not of type G_{2}, then for any short root λ there exists a root μ which is orthogonal to λ and λ ± μ ∈ R.

Proof. We will follow the notations of root systems in [2].

In the case of type B, the restricted root system is given by

R = {±e_{i} | 1 ≤ i ≤ p} ∪ {±e_{i} ± e_{j} | 1 ≤ i < j ≤ p}.

If we add ±e_{j} to a short root ±e_{i} (i ≠ j), then it becomes a root again. In the case of type C, the restricted root system is given by

R = {±2e_{i} | 1 ≤ i ≤ p} ∪ {±e_{i} ± e_{j} | 1 ≤ i < j ≤ p}.

Short roots are ±e_{i} ± e_{j}. By the action of the Weyl group, it suffices to consider a short root e_{1} + e_{2}. The set of roots which are perpendicular to e_{1} + e_{2} is

{±(e_{1} − e_{2})} ∪ {±2e_{i} | 3 ≤ i ≤ p} ∪ {±e_{i} ± e_{j} | 3 ≤ i < j ≤ p}

and

(e_{1} + e_{2}) + (e_{1} − e_{2}) = 2e_{1} ∈ R, (e_{1} + e_{2}) − (e_{1} − e_{2}) = 2e_{2} ∈ R.

In the case of type BC, the restricted root system is given by

R = {±2e_{i}, ±e_{i} | 1 ≤ i ≤ p} ∪ {±e_{i} ± e_{j} | 1 ≤ i < j ≤ p}.

We can see that short roots ±e_{i}, ±e_{i} ± e_{j} satisfy the condition mentioned in the lemma by a similar way in the case of types B and C.

The root system of F_{4} contains a root system of type B_{2} as a sub-system. Then a short root of type F_{4} can be regarded as a short root of type B_{2}. Thus in this case a short root satisfies the condition mentioned in the lemma. □

5.1. The symmetric spaces with R of types G_{2} and A, D, E_{6}, E_{7}, E_{8}. We have already proved that in this case Ad(K)λ is tangentially degenerate, only if (2) in Theorem 1.1 is valid, by Proposition 2.5.

5.2. Group manifolds of type B, C, F_{4}. Group manifolds of the other types has been already treated in the previous subsection. The following proposition shows that in this case Ad(K)λ is tangentially degenerate, only if (1) in Theorem 1.1 is valid.

Proposition 5.2. Let G be a compact connected simple Lie group which is not of type G_{2}. The adjoint orbit of G through a short root is not tangentially degenerate.

Proof. We show that the orbit Ad(G)α through a short root α ∈ R_{+} is not tangentially degenerate. Assume that V is a subspace of 𝔤 ∩ (𝔤_{α} ⊕ 𝔤_{−α}) invariant under ad(𝔷_{G}^{α}). Then the complexification V^{C} ⊂ 𝔤_{α} ⊕ 𝔤_{−α} of V is a complex vector space which is invariant under ad(𝔷_{G}^{α}). We take v ∈ V^{C} and express as v = X_{α} + X_{−α} (X_{±α} ∈ 𝔤_{±α}). In this case, from Lemma 5.1, there exists β ∈ R_{+} which satisfies ⟨β, α⟩ = 0 and α±β ∈ R. We take a non-zero vector X_{β} ∈ 𝔤_{β}. Then

[X_{β}, v] = [X_{β}, X_{α}] + [X_{β}, X_{−α}] ∈ (𝔤_{β+α} ⊕ 𝔤_{β−α}) ∩ V^{C} = {0}.

This shows X_{±α} = 0, since [𝔤_{β}, 𝔤_{±α}] = 𝔤_{β±α}. Thus we obtain V = {0}. Hence from Corollary 4.2, Ad(G)α is not tangentially degenerate. □

5.3. Hermitian symmetric spaces.

Proposition 5.3. Let (G, K) be a Hermitian symmetric pair. (Then the restricted root system of (G, K) is of type C or BC.) The orbit Ad(K)λ through a short root λ is not tangentially degenerate.

Proof. Without loss of generality we can put λ = e_{1} + e_{2}. It is sufficient to prove that if X ∈ 𝔪_{e_{1}+e_{2}} satisfies [𝔨_{e_{1}−e_{2}}, X] = {0}, then X = 0. From the assumption,

0 = J[𝔨_{e_{1}−e_{2}}, X] = [𝔨_{e_{1}−e_{2}}, JX].

Therefore we have

0 = ⟨𝔞, [𝔨_{e_{1}−e_{2}}, JX]⟩ = ⟨[𝔞, 𝔨_{e_{1}−e_{2}}], JX⟩ = ⟨𝔪_{e_{1}−e_{2}}, JX⟩.

From Lemma 4.6 we have JX ∈ 𝔪_{e_{1}−e_{2}}. This implies JX = 0, hence X = 0. □

5.4. Normal real forms of type B, C, F_{4}. We recall some definitions. A real form 𝔤 of a semisimple Lie algebra 𝔩 over C is called normal if in each Cartan decomposition 𝔤 = 𝔨 + 𝔪 the space 𝔪 contains a maximal abelian subalgebra of 𝔤. It is known that there exists a normal real form for each semisimple Lie algebra over C, moreover that is unique up to isomorphism ([6, Ch. IX, Theorem 5.10]).

A compact symmetric pair (G, K) is called a compact symmetric pair corresponding to a normal real form if the dual (𝔤^{∗}, 𝔨) of the orthogonal symmetric Lie algebra (𝔤, 𝔨) of (G, K) is a normal real form of the complexification 𝔤^{C} of 𝔤. Those of type B are (SO(2p + 1), SO(p) × SO(p + 1)), those of type C are (Sp(p), U(p)), and that of type F_{4} is (F_{4}, SU(2) · Sp(3)).

Proposition 5.4. Let (G, K) be a compact symmetric pair which corresponds to a normal real form with a restricted root system of type B, C, or F_{4}. Then the orbit through a short root is not tangentially degenerate.

Proof. Since (G, K) is a compact symmetric pair which corresponds to a normal real form, 𝔨 and 𝔪 can be expressed as

𝔨 = ∑_{α∈R_{+}} RF_{α}, 𝔪 = 𝔱 ⊕ ∑_{α∈R_{+}} RG_{α}, 𝔨_{α} = RF_{α}, 𝔪_{α} = RG_{α},

where F_{α} = (E_{α}−E_{−α})/\sqrt{2} and G_{α} = \sqrt{−1}(E_{α}+E_{−α})/\sqrt{2}. Here E_{α} ∈ 𝔤_{α} satisfies that, for α, β ∈ R, if α + β ∈ R then [E_{α}, E_{β}] = N_{α,β}E_{α+β} and N_{α,β} is non-zero real number which satisfies N_{α,β} = −N_{−α,−β}.

When α is a short root, from Lemma 5.1, there exists β ∈ R_{+} such that α ⊥ β and α ± β ∈ R. Then we have

[𝔨_{β}, 𝔪_{α}] = R(N_{α,β}G_{α+β} − N_{−α,β}G_{α−β}) ≠ {0}.

Thus, from Lemma 4.3, the orbit Ad(K)α through α is not tangentially degenerate. □ 5.5. Real and quaternionic Grassmannians.

Proposition 5.5. In the cases of (G, K) = (SO(2p + n), S(O(p) × O(p + n))) (p ≥ 2, n ≥ 1) and (Sp(2p + n), Sp(p) × Sp(p + n)) (p ≥ 2, n ≥ 0), the orbit through a root which is not longest are not tangentially degenerate.

Proof. We first consider the case of the real Grassmannians. In this case the restricted root system R of (G, K) is of type B_{p}, that is R = {±e_{i} | 1 ≤ i ≤ p} ∪ {±e_{i} ± e_{j} | 1 ≤ i < j ≤ p}. Without loss of generality we can put λ = e_{1}. The action of 𝔨_{0} = 𝔬(n) on 𝔪_{λ} = R^{n} is irreducible, thus 𝔪_{λ} is the only non-zero subspace of 𝔪_{λ} invariant under 𝔨_{0}. Restricted root spaces 𝔪_{e_{i}}, 𝔨_{e_{i}} (1 ≤ i ≤ p) are given by

𝔪_{e_{i}} = \{( \begin{matrix}& X \\ −^{t}X \end{matrix} ) | X = x_{1}E_{i1} + ⋯ + x_{n}E_{in}, x_{j} ∈ R\} ,

𝔨_{e_{i}} = \{( \begin{matrix}& −X \\ ^{t}X \end{matrix} ) | X = x_{1}E_{i1} + ⋯ + x_{n}E_{in}, x_{j} ∈ R\} .

Therefore, when i ≥ 2, we have that e_{i} is perpendicular to e_{1} and

[𝔨_{e_{i}}, 𝔪_{e_{1}}] = R ( \begin{matrix}& −E_{1i} \\ E_{i1} \end{matrix}  ) ⊂ 𝔪_{e_{1}−e_{i}} ⊕ 𝔪_{e_{1}+e_{i}}.

Hence, from Lemma 4.3, the orbit Ad(K)λ is not tangentially degenerate.

We next consider the quaternionic Grassmannians. Without loss of generality we can put λ = e_{1} + e_{2}. When n ≥ 1, the restricted root system of (G, K) is of type BC_{p}. And when n = 0, the restricted root system is of type C_{p}. However, we shall consider both cases uniformly. In order to prove the proposition, it suffices to show that {0} is the only subspace of 𝔪_{e_{1}+e_{2}} invariant under ad(𝔷_{K}^{λ}).

Let V be a subspace of 𝔪_{e_{1}+e_{2}} invariant under ad(𝔷_{K}^{λ}). We take X ∈ V arbitrarily. Then [𝔨_{e_{1}−e_{2}}, X] ⊂ V ⊂ 𝔪_{e_{1}+e_{2}}. On the other hand, [𝔨_{e_{1}−e_{2}}, X] ⊂ 𝔪_{2e_{1}} ⊕ 𝔪_{2e_{2}}. Therefore we have [𝔨_{e_{1}−e_{2}}, X] = {0}.

Under the notation of the proof of Theorem 4.5 (4), restricted root spaces 𝔪_{e_{i}+e_{j}} and 𝔨_{e_{i}−e_{j}} are given by

𝔪_{e_{i}+e_{j}} = {x(E_{i,p+j} + E_{p+i,j}) − x̄(E_{p+j,i} + E_{j,p+i}) | x ∈ H},
𝔨_{e_{i}−e_{j}} = {y(E_{ij} + E_{p+i,p+j}) − ȳ(E_{ji} + E_{p+j,p+i}) | y ∈ H}.

We put X = x(E_{1,p+2} + E_{p+1,2}) − x̄(E_{p+2,1} + E_{2,p+1}) ∈ V. Then

[𝔨_{e_{1}−e_{2}}, X] = {(xȳ − yx̄)(E_{1,p+1} + E_{p+1,1}) + (x̄y − ȳx)(E_{2,p+2} + E_{p+2,2}) | y ∈ H}.

Therefore x must be zero for the right-hand side to be {0}. Hence V = {0}. Consequently we have that {0} is the only subspace of 𝔪_{e_{1}+e_{2}} invariant under ad(𝔷_{K}^{λ}). □

5.6. Quaternionic symmetric spaces EII, EV I, EIX. In this subsection we shall show when

(G, K) = (E_{6}, SU(2) · SU(6)), (E_{7}, SU(2) · SO(12)), (E_{8}, SU(2) · E_{7}),

the orbit through a short root λ is not tangentially degenerate. In these cases, G/K is a compact quaternionic symmetric space whose restricted root system is of type F_{4}. See Section 6 in detail. From Lemmas 4.3 and 6.7, it is sufficient to prove that the condition (B) in Lemma 6.7 holds for the short root λ.

We shall prove the above claim for each of the three cases.

Proposition 5.6. In the case of (G, K) = (E_{6}, SU(2) · SU(6)), the orbit Ad(K)λ through a short root λ is not tangentially degenerate.

Proof. We may put λ = π(Φ(α_{1})). Then λ is a short root, and

(πΦ)^{−1}(λ) = {α ∈ R̃ | π(Φ(α)) = λ} = {α_{1}, α_{6}}.

We set ν = π(Φ(α_{3} + α_{4} + α_{5} + α_{6})). Then ν is a short root perpendicular to λ, and

(πΦ)^{−1}(ν) = {α_{3} + α_{4} + α_{5} + α_{6}, α_{1} + α_{3} + α_{4} + α_{5}}.

Now we assume that

Y = x_{1}E_{α_{1}} + y_{1}E_{−α_{1}} + x_{2}E_{α_{6}} + y_{2}E_{−α_{6}} ∈ ∑_{α∈R̃,π(Φ(α))=λ}(𝔤_{α} ⊕ 𝔤_{−α})

satisfies the condition [Y, Ω_{ν}] = 0. We note that the set of roots of the form α_{3} + α_{4} + α_{5} + α_{6} ± α where α ∈ (πΦ)^{−1}(λ) is

{(α_{3} + α_{4} + α_{5} + α_{6}) + α_{1}, (α_{3} + α_{4} + α_{5} + α_{6}) − α_{6}}.

Therefore we have

[E_{α_{3}+α_{4}+α_{5}+α_{6}}, Y]

= x_{1}N_{α_{3}+α_{4}+α_{5}+α_{6},α_{1}}E_{α_{1}+α_{3}+α_{4}+α_{5}+α_{6}} + y_{2}N_{α_{3}+α_{4}+α_{5}+α_{6},−α_{6}}E_{α_{3}+α_{4}+α_{5}}.

This shows that the condition [E_{α_{3}+α_{4}+α_{5}+α_{6}}, Y] = 0 yields x_{1} = y_{2} = 0. Similarly the condition [E_{−(α_{3}+α_{4}+α_{5}+α_{6})}, Y] = 0 yields y_{1} = x_{2} = 0. Hence we obtain Y = 0. □

The following two propositions can be proved in a similar way to the proof of Proposition 5.6. So we write only the essentials of their proofs.

Proposition 5.7. In the case of (G, K) = (E_{7}, SU(2) · SO(12)), the orbit Ad(K)λ through a short root λ is not tangentially degenerate.

Proof. We may put λ = π(Φ(α_{4})). Then λ is a short root, and

(πΦ)^{−1}(λ) = {α_{4}, α_{4} + α_{5}, α_{2} + α_{4}, α_{2} + α_{4} + α_{5}}.

We set ν = π(Φ(α_{3} + α_{4})). Then ν is a short root perpendicular to λ, and

(πΦ)^{−1}(ν) = {α_{3} + α_{4}, α_{3} + α_{4} + α_{5}, α_{2} + α_{3} + α_{4}, α_{2} + α_{3} + α_{4} + α_{5}}.

We get the assertion from the following: The set of roots of the form α_{3} + α_{4} ± α where α ∈ (πΦ)^{−1}(λ) is

{(α_{3} + α_{4}) − α_{4}, (α_{3} + α_{4}) + (α_{2} + α_{4} + α_{5})}.

The set of roots of the form α_{3}+α_{4}+α_{5}±α where α ∈ (πΦ)^{−1}(λ)−{α_{4}, α_{2}+α_{4}+α_{5}} is

{(α_{3} + α_{4} + α_{5}) − (α_{4} + α_{5}), (α_{3} + α_{4} + α_{5}) + (α_{2} + α_{4})}.
□

Proposition 5.8. In the case of (G, K) = (E_{8}, SU(2) · E_{7}), the orbit Ad(K)λ through a short root λ is not tangentially degenerate.

Proof. We may put λ = π(Φ(α_{1})). Then λ is a short root, and

(πΦ)^{−1}(λ) = \{ \begin{matrix}α_{1}, α_{1} + α_{3}, α_{1} + α_{3} + α_{4}, α_{1} + α_{2} + α_{3} + α_{4}, \\ α_{1} + α_{3} + α_{4} + α_{5}, α_{1} + α_{2} + α_{3} + α_{4} + α_{5}, \\ α_{1} + α_{2} + α_{3} + 2α_{4} + α_{5}, α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5} \end{matrix} \} .

We set ν = π(Φ(α_{1} + α_{2} + α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + α_{7})). Then ν is a short root perpendicular to λ, and

(πΦ)^{−1}(ν) = \{ \begin{matrix}α_{1} + α_{2} + α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + α_{7}, \\ α_{1} + α_{2} + 2α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + α_{7}, \\ α_{1} + α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + 2α_{6} + α_{7}, \\ α_{1} + 2α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + 2α_{6} + α_{7}, \\ α_{1} + α_{2} + 2α_{3} + 3α_{4} + 3α_{5} + 2α_{6} + α_{7}, \\ α_{1} + 2α_{2} + 2α_{3} + 3α_{4} + 3α_{5} + 2α_{6} + α_{7}, \\ α_{1} + 2α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}, \\ α_{1} + 2α_{2} + 3α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7} \end{matrix} \} .

We get the assertion from the following: The set of roots of the form α_{1} + α_{2} + α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + α_{7} ± α where α ∈ (πΦ)^{−1}(λ) is

\{ \begin{matrix}(α_{1} + α_{2} + α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + α_{7}) − α_{1}, \\ (α_{1} + α_{2} + α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + α_{7}) + (α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5}) \end{matrix} \} .

The set of roots of the form α_{1} + α_{2} + 2α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + α_{7} ± α where

α ∈ (πΦ)^{−1}(λ) − {α_{1}, α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5}}

is

\{ \begin{matrix}(α_{1} + α_{2} + 2α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + α_{7}) − (α_{1} + α_{3}), \\ (α_{1} + α_{2} + 2α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + α_{7}) + (α_{1} + α_{2} + α_{3} + 2α_{4} + α_{5}) \end{matrix} \} .

The set of roots of the form α_{1} + α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + 2α_{6} + α_{7} ± α where

α ∈ (πΦ)^{−1}(λ) − \{ \begin{matrix}α_{1}, α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5}, \\ α_{1} + α_{3}, α_{1} + α_{2} + α_{3} + 2α_{4} + α_{5} \end{matrix} \}

is

\{ \begin{matrix}(α_{1} + α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + 2α_{6} + α_{7}) − (α_{1} + α_{3} + α_{4}), \\ (α_{1} + α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + 2α_{6} + α_{7}) + (α_{1} + α_{2} + α_{3} + α_{4} + α_{5}) \end{matrix} \} .

The set of roots of the form α_{1} + 2α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + 2α_{6} + α_{7} ± α where

α ∈ (πΦ)^{−1}(λ) − \{ \begin{matrix}α_{1}, α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5}, \\ α_{1} + α_{3}, α_{1} + α_{2} + α_{3} + 2α_{4} + α_{5}, \\ α_{1} + α_{3} + α_{4}, α_{1} + α_{2} + α_{3} + α_{4} + α_{5} \end{matrix} \}

is

\{ \begin{matrix}(α_{1} + 2α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + 2α_{6} + α_{7}) − (α_{1} + α_{2} + α_{3} + α_{4}), \\ (α_{1} + 2α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + 2α_{6} + α_{7}) + (α_{1} + α_{3} + α_{4} + α_{5}) \end{matrix} \} .

□

According to the classification of irreducible compact symmetric pairs, we have proved the ‘only if’ part of Theorem 1.1.

5.7. List of tangential degeneracy. At the end of this section, we give the list of all irreducible compact symmetric pairs whose ranks are equal or greater than 2 such that the orbits of their s-representations have degenerate Gauss maps. All of them are orbits through long roots except the case of type G_{2}. In the case of type G_{2} both of orbits through a long root and a short root have degenerate Gauss maps, and both of them have the same dimension and the same rank of Gauss map. In Table 1, we denote the dimension of the orbit by l and the rank of Gauss map by r. Then tangentially degeneracy is equal to l − r.

When (G, K) is of rank 2, the results above were studied by Ishikawa, Kimura and Miyaoka [11].

type rank 𝔤  𝔨  l  r  l − r
A  p 𝔰𝔲(p + 1)  𝔰𝔬(p + 1)  2p − 1  2p − 2  1
p 𝔰𝔲(p + 1)^{2} 𝔰𝔲(p + 1)  2(2p − 1)  2(2p − 2)  2
p 𝔰𝔲(2(p + 1)) 𝔰𝔭(p + 1)  4(2p − 1)  4(2p − 2)  4
2 𝔢_{6}  𝔣_{4}  24  16  8
B  p 𝔰𝔬(2p + 1)^{2} 𝔰𝔬(2p + 1)  8p − 10  8p − 12  2
p 𝔰𝔬(2p + n) 𝔰𝔬(p) ⊕ 𝔰𝔬(p + n)  4p + 2n − 7 4p + 2n − 8 1
C  p 𝔰𝔭(p)  𝔲(p)  2p − 1  2p − 2  1
p 𝔰𝔭(p)^{2}  𝔰𝔭(p)  4p − 2  4p − 4  2
p 𝔰𝔭(2p)  𝔰𝔭(p) ⊕ 𝔰𝔭(p)  8p − 5  8p − 8  3
p 𝔰𝔲(2p)  𝔰𝔲(p) ⊕ 𝔰𝔲(p) ⊕ R  4p − 3  4p − 4  1
p 𝔰𝔬(4p)  𝔲(2p)  8p − 7  8p − 8  1
3 𝔢_{7}  𝔢_{6} ⊕ R  33  32  1
D  p 𝔰𝔬(2p)  𝔰𝔬(p) ⊕ 𝔰𝔬(p)  4p − 7  4p − 8  1
p 𝔰𝔬(2p)^{2}  𝔰𝔬(2p)  2(4p − 7)  2(4p − 8)  2
E_{6}  6 𝔢_{6}  𝔰𝔭(4)  21  20  1
6 𝔢_{6} ⊕ 𝔢_{6}  𝔢_{6}  42  40  2
E_{7}  7 𝔢_{7}  𝔰𝔲(8)  33  32  1
7 𝔢_{7} ⊕ 𝔢_{7}  𝔢_{7}  66  64  2
E_{8}  8 𝔢_{8}  𝔰𝔬(16)  57  56  1
8 𝔢_{8} ⊕ 𝔢_{8}  𝔢_{8}  114  112  2
F_{4}  4 𝔣_{4}  𝔰𝔲(2) ⊕ 𝔰𝔭(3)  15  14  1
4 𝔣_{4} ⊕ 𝔣_{4}  𝔣_{4}  30  28  2
4 𝔢_{6}  𝔰𝔲(2) ⊕ 𝔰𝔲(6)  21  20  1
4 𝔢_{7}  𝔰𝔲(2) ⊕ 𝔰𝔬(12)  33  32  1
4 𝔢_{8}  𝔰𝔲(2) ⊕ 𝔢_{7}  57  56  1
G_{2}  2 𝔤_{2}  𝔰𝔬(4)  5  4  1
2 𝔤_{2} ⊕ 𝔤_{2}  𝔤_{2}  10  8  2
BC p 𝔰𝔲(2p + n) 𝔰𝔲(p) ⊕ 𝔰𝔲(p + n) ⊕ R 4p + 2n − 3 4p + 2n − 4 1
p 𝔰𝔬(4p + 2) 𝔲(2p + 1) 8p − 3 8p − 4 1
p 𝔰𝔭(2p + n) 𝔰𝔭(p) ⊕ 𝔰𝔭(p + n)  8p + 4n − 5 8p + 4n − 8 3
2 𝔢_{6} 𝔰𝔬(10) ⊕ R 21 20 1

Table 1

6. Lemmas on quaternionic symmetric spaces

A 4n-dimensional Riemannian manifold is called quaternion-Kähler if its holonomy group is contained in Sp(n) · Sp(1). A quaternion-Kähler manifold is called quaternionic symmetric if it is a Riemannian symmetric space ([1, p. 396]).

We will review a construction of a quaternionic symmetric space from a compact simple Lie algebra 𝔤 whose rank is greater than or equal to 2 (see [16] in detail). Set G = Int(𝔤), which is a compact connected semisimple Lie group. We denote by ⟨ , ⟩ a biinvariant Riemannian metric on G. Take a maximal torus T in G and denote its Lie algebra by 𝔱. For α ∈ 𝔱 we set 𝔤̃_{α} as (2.1), and define root system R̃ by (2.2). We have then

𝔤^{C} = 𝔱^{C} + ∑_{α∈R̃} 𝔤̃_{α}.

For α ∈ R̃ we can take E_{α} ∈ 𝔤̃_{α} such that

E_{α} − E_{−α} ∈ 𝔤, \sqrt{−1}(E_{α} + E_{−α}) ∈ 𝔤, [E_{α}, E_{−α}] = −\sqrt{−1}α,

∥\frac{1}{\sqrt{2}}(E_{α} − E_{−α})∥ = ∥\begin{matrix}√ & \overline{−1} \\ √ & \overline{2} \end{matrix}(E_{α} + E_{−α})∥ = 1,

and that if we define N_{α,β} by [E_{α}, E_{β}] = N_{α,β}E_{α+β}, then N_{α,β} = −N_{−α,−β} where we put N_{α,β} = 0 if α + β ∉ R̃. Let F̃ be a fundamental system of R̃ and denote by R̃_{+} the set of positive roots with respect to F̃. For α ∈ R̃_{+} set

F_{α} = \frac{1}{\sqrt{2}}(E_{α} − E_{−α}), G_{α} = \begin{matrix}√ & \overline{−1} \\ √ & \overline{2} \end{matrix}(E_{α} + E_{−α}),

then we have

(6.1)  𝔤 = 𝔱 + ∑_{α∈R̃_{+}}(RF_{α} + RG_{α}), ∥F_{α}∥ = ∥G_{α}∥ = 1, [F_{α}, G_{α}] = α.

For each α ∈ R̃_{+}, we define a subalgebra 𝔤(α) of 𝔤 by

𝔤(α) = Rα + 𝔤 ∩ (𝔤̃_{α} + 𝔤̃_{−α}) = Rα + RF_{α} + RG_{α},

which is isomorphic to 𝔰𝔲(2). We denote the highest root by δ ∈ R̃_{+}. By Lemma 3.2,

s = exp ad (\frac{2π}{∥δ∥^{2}} δ)

is an involutive automorphism of 𝔤. The fixed points set 𝔨 of s in 𝔤 is given by

𝔨 = 𝔱 + RF_{δ} + RG_{δ} + ∑_{α⊥δ}(RF_{α} + RG_{α})
= 𝔤(δ) + 𝔱 ∩ δ^{⊥} + ∑_{α⊥δ}(RF_{α} + RG_{α}).

The subalgebras 𝔤(δ) and 𝔱 ∩ δ^{⊥} + ∑_{α⊥δ}(RF_{α} + RG_{α}) are ideals of 𝔨. The (−1)- eigenspace 𝔪 of s is given by

𝔪 = ∑_{α∈R̃_{+}^{𝔪}}(RF_{α} + RG_{α}) where R̃_{+}^{𝔪} = \{α ∈ R̃_{+} | \frac{⟨α, δ⟩}{∥δ∥^{2}} = \frac{1}{2} \} .

Since there exists a subset R̃_{+}(δ) in R̃_{+}^{𝔪} such that

(6.2)  𝔪 = ∑_{α∈R̃_{+}(δ)}(RF_{α} + RG_{α} + RF_{δ−α} + RG_{δ−α}),

the dimension of 𝔪 is a multiple of 4.

We also denote by s the involutive automorphism of G induced from s. Since the fixed point set of s in G is closed and G is compact, the identity component K of the fixed points set is also compact. The Lie algebra of K coincides with 𝔨 and (G, K) is a compact symmetric pair. Hence the coset manifold G/K is a compact Riemannian symmetric space. Moreover G/K is a quaternionic symmetric space since (6.2) defines a quaternionic structure. Conversely it is known that every compact quaternionic symmetric space is obtained in this way. We omit its proof. See [16] in detail.

Quaternionic symmetric spaces have a similar property with Hermitian symmetric spaces as we shall mention below: Two roots γ_{1}, γ_{2} ∈ R̃_{+}(δ) are said to be strongly orthogonal if γ_{1} ± γ_{2} ∉ R̃.

Proposition 6.1. Let G/K be a compact quaternionic symmetric space of rank p. Then there exist R̃_{+}(δ) which satisfies (6.2) and a subset {γ_{i}}_{1≤i≤p} of R̃_{+}(δ) consisting of strongly orthogonal roots such that

𝔞 = ∑_{i=1}^{p} RF_{γ_{i}}

is a maximal abelian subspace of 𝔪.

We can prove this proposition in a way similar to the proof of Proposition 7.4 in Helgason’s book [6] (p.385) by using Lemma 6.4.

Lemma 6.2. If α, β ∈ R̃_{+}^{𝔪} and α + β ∈ R̃, then α + β = δ.

Proof. Since α, β ∈ R̃_{+}^{𝔪}, we have

\frac{⟨α + β, δ⟩}{∥δ∥^{2}} = 1.

Using Lemma 3.2, α + β ∈ R̃ implies α + β = δ. □

Corollary 6.3. [𝔤̃_{α}, 𝔤̃_{β}] ⊂ 𝔤̃_{δ} for α, β ∈ R̃_{+}^{𝔪}.

Proof. If α + β ∈ R̃, Lemma 6.2 implies [𝔤̃_{α}, 𝔤̃_{β}] = 𝔤̃_{δ}. If α + β ∉ R̃, then [𝔤̃_{α}, 𝔤̃_{β}] = {0}. □

If Q is any subset of R̃_{+}^{𝔪}, let

𝔪_{Q} = ∑_{α∈Q}(𝔤̃_{α} + 𝔤̃_{−α}).

Remark that 𝔪R̃_{+}^{𝔪} = 𝔪^{C}. For the lowest root γ in Q, put

Q(γ) = {β ∈ Q − {γ} | β ± γ ∉ R̃}.

Then β ± γ ∉ R̃ ∪ {0} for β ∈ Q(γ).

Lemma 6.4. We denote by 𝔷_{𝔪_{Q}}(E_{γ} + E_{−γ}) the centralizer of E_{γ} + E_{−γ} in 𝔪_{Q}. Then

𝔷_{𝔪_{Q}}(E_{γ} + E_{−γ}) = 𝔪_{Q(γ)} + C(E_{γ} + E_{−γ}).

We can prove this lemma in a way similar to the proof of Lemma 7.5 in Helgason’s book [6] (p.385) by using Corollary 6.3, so we omit it.

Hence 𝔪 is given by the following:

𝔪 = 𝔞 + ∑_{i=1}^{p}(RG_{γ_{i}} + RF_{δ−γ_{i}} + RG_{δ−γ_{i}})

+ ∑_{α∈R̃_{+}(δ)−{γ_{1},··· ,γ_{p}}}(RF_{α} + RG_{α} + RF_{δ−α} + RG_{δ−α})

When the root system of G is not of type G_{2}, then ∥γ_{1}∥ = ⋯ = ∥γ_{p}∥. Set

𝔟 = 𝔱 ∩ {γ_{1}, ⋯, γ_{p}}^{⊥}, 𝔱′ = 𝔞 + 𝔟,

then 𝔱′ is a maximal abelian subalgebra of 𝔤 containing 𝔞. We define the Cayley transform Φ by

Φ = exp \frac{π}{2} ad (∑_{j=1}^{p} \frac{G_{γ_{j}}}{∥γ_{j}∥}) ∈ Aut(𝔤),

and set λ_{i} = ∥γ_{i}∥F_{γ_{i}}, then

Φ(γ_{i}) = λ_{i}, Φ(H) = H (H ∈ 𝔟).

Hence the Cayley transform Φ maps 𝔱 onto 𝔱′. We denote by R the restricted root system of (G, K) with respect to 𝔞. Let π : 𝔱′ = 𝔞 + 𝔟 → 𝔞 be the orthogonal projection, then R = π(Φ(R̃)). Since

α ≡ ∑_{i=1}^{p} \frac{⟨α, γ_{i}⟩}{∥γ_{i}∥^{2}}γ_{i} mod 𝔟 for α ∈ R̃,

we have

Φ(α) ≡ ∑_{i=1}^{p} \frac{⟨α, γ_{i}⟩}{∥γ_{i}∥^{2}}λ_{i} mod 𝔟,

which implies that

(6.3)  π(Φ(α)) = ∑_{i=1}^{p} \frac{⟨α, γ_{i}⟩}{∥γ_{i}∥^{2}}λ_{i}.

In particular

\{λ_{1}, ⋯, λ_{p}\} ⊂ R = \{∑_{i=1}^{p} \frac{⟨α, γ_{i}⟩}{∥γ_{i}∥^{2}}λ_{i} | α ∈ R̃\} .

The multiplicity m(λ) of λ = π(Φ(α)) ∈ Σ (α ∈ R̃) is given by

m(λ) = #{β ∈ R̃ | ⟨α, γ_{i}⟩ = ⟨β, γ_{i}⟩}.

By (6.3), we have

∥π(Φ(α))∥^{2} = ∑_{i=1}^{p} ⟨α, \frac{γ_{i}}{∥γ_{i}∥}⟩^{2} ≤ ∥α∥^{2},

22 OSAMU IKAWA, TAKASHI SAKAI, AND HIROYUKI TASAKI

and the equality holds if and only if α ∈ span{γ_{1}, ⋯, γ_{p}}. Hence ∥π(Φ(α))∥^{2} = ∥α∥^{2} for any α ∈ R̃ if and only if p = rank(G). Lemma 6.5. RG_{γ_{i}} ⊂ 𝔪_{λ_{i}}, Rγ_{i} ⊂ 𝔨_{λ_{i}}. Proof. For H = ∑ x_{j}λ_{j} ∈ 𝔞, we have

[H, G_{γ_{i}}] = ∑ x_{j}[∥γ_{j}∥F_{γ_{j}}, G_{γ_{i}}] = x_{i}∥γ_{i}∥[F_{γ_{i}}, G_{γ_{i}}]
= x_{i}∥γ_{i}∥^{2} \frac{γ_{i}}{∥γ_{i}∥} = ⟨H, λ_{i}⟩\frac{γ_{i}}{∥γ_{i}∥},
[H, \frac{γ_{i}}{∥γ_{i}∥}] = ∑ x_{j}∥γ_{j}∥ [F_{γ_{j}}, \frac{γ_{i}}{∥γ_{i}∥}] = −x_{i}∥γ_{j}∥^{2}G_{γ_{j}}
= −⟨H, λ_{i}⟩G_{γ_{j}},

where we used (6.1). □ Lemma 6.6. For any restricted root λ ∈ R

𝔨_{λ} + 𝔪_{λ} = Φ ( ∑_{α∈R̃,π(Φ(α))=λ}(RF_{α} + RG_{α})) .

Proof. Since

𝔨_{λ} + 𝔪_{λ} = {X ∈ 𝔤 | [H, [H, X]] = −⟨λ, H⟩^{2}X (H ∈ 𝔞)},
RF_{α} + RG_{α} = 𝔤 ∩ (𝔤̃_{α} + 𝔤̃_{−α})
= {X ∈ 𝔤 | [H, [H, X]] = −⟨α, H⟩^{2}X (H ∈ 𝔱)},

we have

Φ ( ∑_{α∈R̃,π(Φ(α))=λ}(RF_{α} + RG_{α}))

= Φ (  ∑_{α∈R̃,π(Φ(α))=λ}{X ∈ 𝔤 | [H, [H, X]] = −⟨α, H⟩^{2}X (H ∈ 𝔱)})
= ∑_{α∈R̃,π(Φ(α))=λ}{Y ∈ 𝔤 | [Φ(H), [Φ(H), Y]] = −⟨Φ(α), Φ(H)⟩^{2}Y (H ∈ 𝔱)}
= ∑_{α∈R̃,π(Φ(α))=λ}{Y ∈ 𝔤 | [H, [H, Y]] = −⟨Φ(α), H⟩^{2}Y (H ∈ 𝔱′)}
⊂ ∑_{α∈R̃,π(Φ(α))=λ}{Y ∈ 𝔤 | [H, [H, Y]] = −⟨π(Φ(α)), H⟩^{2}Y (H ∈ 𝔞)}
= 𝔨_{λ} + 𝔪_{λ}.

Here dim(𝔨_{λ} + 𝔪_{λ}) = 2m(λ). Since Φ is a linear isomorphism, we have

dim Φ (  ∑_{α∈R̃,π(Φ(α))=λ}(RF_{α} + RG_{α})) = dim ∑_{α∈R̃,π(Φ(α))=λ}(RF_{α} + RG_{α})
= 2#{α ∈ R̃ | π(Φ(α)) = λ}
= 2m(λ).

Hence we get the assertion. □

Lemma 6.7. Let (𝔤, 𝔨) be a compact quaternionic symmetric pair and let λ, ν ∈ R_{+} with λ ⊥ ν. We denote Ω_{λ} := ∑_{α∈R̃,π(Φ(α))=λ}(𝔤_{α}+𝔤_{−α}). Let us consider the following

two conditions:

(A) Let X ∈ 𝔪_{λ}. Then [𝔨_{ν}, X] = 0 implies X = 0.

(B) Let Y ∈ Ω_{λ}. Then [Y, Ω_{ν}] = 0 implies Y = 0. Then the condition (B) implies (A).

Proof. Let ν be in R_{+} such that ν ⊥ λ. Note that [ν, 𝔪_{λ}] = {0}. We take X ∈ 𝔪_{λ} arbitrarily. We prove the condition (A) holds. Now we assume that [𝔨_{ν}, X] = 0. Then, from the Jacobi identity and (2) of Lemma 2.1, we have

0 = [ν, [𝔨_{ν}, X]] = [[ν, 𝔨_{ν}], X] + [𝔨_{ν}, [ν, X]] = [𝔪_{ν}, X].

Hence [𝔨_{ν} + 𝔪_{ν}, X] = 0. Applying the inverse Φ^{−1} of the Cayley transform to the equality above, we have

[ ∑_{α∈R̃,π(Φ(α))=ν}(RF_{α} + RG_{α}), Φ^{−1}(X)] = 0.

Here we used Lemma 6.6. Since

( ∑_{α∈R̃,π(Φ(α))=ν}(RF_{α} + RG_{α}))^{C} = Ω_{ν},

we have [Ω_{ν}, Φ^{−1}(X)] = 0. Using Lemma 6.6 again, we have

Φ^{−1}(X) ∈ Φ^{−1}(𝔪_{λ}) ⊂ Φ^{−1}(𝔨_{λ} + 𝔪_{λ}) = ∑_{α∈R̃,π(Φ(α))=λ}(RF_{α} + RG_{α}) ⊂ Ω_{λ}.

Hence the condition (B) implies Φ^{−1}(X) = 0 and X = 0. □

7. Ferus equalities

Let f : M ⟶ S^{n} be an immersion of a compact, connected manifold M of dimension l. We denote by r the maximal rank of the Gauss map γ of f. Ferus [5] showed that there exists a number F (l), which only depends on the dimension l of M, such that the inequality r < F (l) implies r = 0. Then f(M) must be an l-dimensional great sphere in S^{n}. Here the number F (l) is called the Ferus number and defined by

F (l) = min{k | A(k) + k ≥ l},

where A(k) is the Adams number, that is the maximal number of linearly independent vector fields at each point on the (k − 1)-dimensional sphere S^{k−1}. Any positive integer k can be written as (2s + 1)2^{t} by some non-negative integers s and t. We write t = c + 4d by some 0 ≤ c ≤ 3 and 0 ≤ d. In this situation the Adams number A(k) can be calculated by

A(k) = 2^{c} + 8d − 1.

Regarding the Ferus inequality, Ishikawa, Kimura and Miyaoka posed the following problem:

Problem 7.1 ([11]). (1) Is the inequality r < F (l) best possible for the implication r = 0? Do there exist tangentially degenerate immersions M^{l} → S^{n} with r = F (l)?

(2) If the above problem is true, classify tangentially degenerate immersions M^{l} → S^{n} with r = F (l).

In the list of Section 5.7, we can find many new orbits which satisfy the equality r = F (l). In order to observe this we state some properties of the Ferus number. The definition of the Ferus number immediately implies F (l) ≤ l.

Lemma 7.2. F (l) ≤ F (l + 1).

Proof. The relation {k | A(k) + k ≥ l + 1} ⊂ {k | A(k) + k ≥ l} implies

F (l + 1) = min{k | A(k) + k ≥ l + 1} ≥ min{k | A(k) + k ≥ l} = F (l).
□

Lemma 7.3. F (2^{q}) = 2^{q}.

Proof. It is sufficient to show A(k)+k < 2^{q} for k < 2^{q}. We write k = 2^{q} −(2s+1)2^{t} by some non-negative integers s and t, and t = c + 4d by some 0 ≤ c ≤ 3 and d ≥ 0. Then t < q and we get

A(k) = A(2^{q} − 2^{t}(2s + 1)) = A(2^{t}(2^{q−t} − (2s + 1))) = 2^{c} + 8d − 1.

Thus

A(k) + k = 2^{q} − {2^{c+4d}(2s + 1) − 2^{c} − 8d + 1}.

Here

2^{c+4d}(2s + 1) − 2^{c} − 8d + 1 ≥ 2^{c+4d} − 2^{c} − 8d + 1
= 2^{c}(2^{4d} − 1) − 8d + 1
≥ 2^{4d} − 8d ≥ 1.

Therefore we obtain A(k) + k < 2^{q}. □

Proposition 7.4. Assume q ≥ 1 and write q = c + 4d (0 ≤ c ≤ 3, d ≥ 0). Then

F (2^{q} + a) = 2^{q}

holds for any 0 ≤ a ≤ 2^{c} + 8d − 1.

Proof. Since q ≥ 1, we have c ≥ 1 or d ≥ 1. Thus A(2^{q}) = 2^{c} + 8d − 1 ≥ 1. This shows A(2^{q}) + 2^{q} = 2^{q} + 2^{c} + 8d − 1. From Lemmas 7.2 and 7.3 we get

2^{q} ≥ F (2^{q} + 2^{c} + 8d − 1) ≥ F (2^{q}) = 2^{q}.
□

The above proposition shows the following equalities:

F (2^{q} + 1) = 2^{q} (q ≥ 1),
F (2^{q} + 2) = 2^{q} (q ≥ 2),
F (2^{q} + 3) = 2^{q} (q ≥ 2),
F (2^{q} + 4) = 2^{q} (q ≥ 3).

By the use of the above equalities, we have many new orbits of the s-representations which satisfy the Ferus equality F (l) = r in Table 1. For example, the orbits of the s-representations of the following symmetric pairs through a long root satisfy F (l) = r:

(𝔰𝔲(2^{q−1} + 2), 𝔰𝔬(2^{q−1} + 2)) (q ≥ 1),

(𝔰𝔲(2^{q−2} + 2)^{2}, 𝔰𝔲(2^{q−2} + 2)) (q ≥ 2),

(𝔰𝔲(2(2^{q−3} + 2)), 𝔰𝔭(2^{q−3} + 2)) (q ≥ 3),

(𝔢_{6}, 𝔣_{4}),

(𝔰𝔬(2p + n), 𝔰𝔬(p) ⊕ 𝔰𝔬(p + n)) (4p + 2n − 7 = 2^{q} + 1, p ≥ 2, n ≥ 1, q ≥ 1), (𝔰𝔭(2^{q−1} + 1), 𝔲(2^{q−1} + 1)) (q ≥ 1), (𝔰𝔭(2^{q−2} + 1)^{2}, 𝔰𝔭(2^{q−2} + 1)) (q ≥ 2), (𝔰𝔭(2(2^{q−3} + 1)), 𝔰𝔭(2^{q−3} + 1) ⊕ 𝔰𝔭(2^{q−3} + 1)) (q ≥ 3), (𝔰𝔲(2(2^{q−2} + 1)), 𝔰𝔲(2^{q−2} + 1) ⊕ 𝔰𝔲(2^{q−2} + 1) ⊕ R) (q ≥ 2), (𝔰𝔬(4(2^{q−3} + 1)), 𝔲(2(2^{q−3} + 1))) (q ≥ 3), (𝔢_{7}, 𝔢_{6} ⊕ R), (𝔰𝔬(2(2^{q−2} + 2)), 𝔰𝔬(2^{q−2} + 2) ⊕ 𝔰𝔬(2^{q−2} + 2)) (q ≥ 2), (𝔰𝔬(2(2^{q−3} + 2))^{2}, 𝔰𝔬(2(2^{q−3} + 2))) (q ≥ 3), (𝔢_{6} ⊕ 𝔢_{6}, 𝔢_{6}), (𝔢_{7}, 𝔰𝔲(8)), (𝔢_{7} ⊕ 𝔢_{7}, 𝔢_{7}), (𝔢_{8}, 𝔰𝔬(16)), (𝔢_{8} ⊕ 𝔢_{8}, 𝔢_{8}), (𝔢_{7}, 𝔰𝔲(2) ⊕ 𝔰𝔬(12)), (𝔢_{8}, 𝔰𝔲(2) ⊕ 𝔢_{7}), (𝔰𝔲(2p + n), 𝔰𝔲(p) ⊕ 𝔰𝔲(p + n) ⊕ R) (4p + 2n − 3 = 2^{q} + 1, p ≥ 2, n ≥ 1, q ≥ 1), (𝔰𝔭(2p + n), 𝔰𝔭(p) ⊕ 𝔰𝔭(p + n)) (8p + 4n − 5 = 2^{q} + 3, p ≥ 2, n ≥ 1, q ≥ 2).

Furthermore the orbits of s-representations of symmetric pairs

(𝔤_{2}, 𝔰𝔬(4)) and (𝔤_{2} ⊕ 𝔤_{2}, 𝔤(2))

through a long root or a short root satisfy the Ferus equality F (5) = 4 or F (10) = 8.

References

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[12] R. Miyaoka, Homogeneity of isoparametric hypersurfaces with six principal curvatures, preprint.

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[15] R. Takagi and T. Takahashi, On the principal curvature of homogeneous hypersurfaces in a sphere, Differential geometry, in honor of K. Yano, Kinokuniya, Tokyo, 1972, 469 – 481.

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Department of General Education, Fukushima National College of Technology, Iwaki, Fukushima, 970-8034 Japan

   E-mail address: ikawa@fukushima-nct.ac.jp

Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka-shi, Osaka, 558-8585 Japan

   E-mail address: tsakai@sci.osaka-cu.ac.jp

Graduate School of Pure and Applied Science, University of Tsukuba, Tsukuba, Ibaraki, 305-8571 Japan

   E-mail address: tasaki@math.tsukuba.ac.jp