WEAKLY REFLECTIVE SUBMANIFOLDS AND AUSTERE SUBMANIFOLDS

OSAMU IKAWA, TAKASHI SAKAI, AND HIROYUKI TASAKI

Abstract. An austere submanifold is a minimal submanifold where for each normal vector, the set of eigenvalues of its shape operator is invariant under the multiplication by −1. In the present paper, we introduce the notion of weakly reflective submanifold, which is an austere submanifold with a reflection for each normal direction, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of linear isotropy representations of Riemannian symmetric spaces.

1. Introduction

Orbits of an s-representation, that is a linear isotropy representation of a Riemannian symmetric pair, are important examples of homogeneous submanifolds in the hypersphere of a Euclidean space. For example, a homogeneous isoparametric hypersurface in the hypersphere, which many mathematicians have investigated, can be obtained as a principal orbit of an s-representation of a Riemannian symmetric pair of rank two. It is known that there exists a unique minimal isoparametric hypersurface in each parallel family of isoparametric hypersurfaces. Furthermore, typical examples of minimal submanifolds in the hypersphere are given as orbits of s-representations. Hirohashi-Song-Takagi-Tasaki [7] showed that there exists a unique minimal orbit in each strata of the stratification of orbit types. However, in general we can not explicitly point out which orbit among each strata is a minimal submanifold.

Harvey-Lawson [4] introduced the notion of austere submanifold, which is a minimal submanifold whose second fundamental form has a certain symmetry. They showed that one can construct a special Lagrangian cone, therefore absolutely areaminimizing, in a complex Euclidean space as the twisted normal bundle of an austere submanifold in a sphere (see [4], [2]). As we mentioned above, the complete list of minimal orbits of s-representations in the hypersphere is unknown at the moment. Therefore we first attempt to determine all austere orbits. We give a necessary and sufficient condition for an orbit to be an austere submanifold in the hypersphere in terms of the restricted root system of a Riemannian symmetric pair. By this criterion, we can determine all orbits which are austere submanifolds in the hypersphere. Since the definition is focused on a symmetry of its second fundamental form, the notion of austere submanifold is an infinitesimal property of a submanifold. However, we observe that some of austere orbits, which we classified, have a certain global symmetry. This symmetry is a globalization of the notion of austere submanifold and a weakened condition of a reflective submanifold. Therefore we shall call them weakly reflective submanifolds, and study some fundamental properties of them. Finally we determine all weakly reflective orbits of s-representations.

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

Key words and phrases. reflective submanifold, austere submanifold, symmetric space, srepresentation, R-space.

The organization of this paper is as follows. In Section 2, we will give the definition of weakly reflective submanifold (Definition 2.1), and recall some related notions. We study their relationship and fundamental properties. In Section 3, we summarize the geometry of orbits of s-representations of Riemannian symmetric pairs. This will be a preliminary for the sections below. In Section 4, we shall give the list of orbits of s-representations which are weakly reflective submanifolds in the hypersphere (Theorem 4.1). We show that these orbits are weakly reflective submanifold in the hypersphere there, however, we will show that the list gives all weakly reflective orbits later. In Section 5, we will give a criterion of austere orbits (Lemma 5.3), and determine all orbits which are austere submanifolds in the hypersphere (Theorem 5.1). Furthermore we show that austere orbits which are not enumerated in the list of weakly reflective orbits are not weakly reflective submanifolds. Then we will complete the proof of the list of weakly reflective orbits. In Section 6, we will study relationships between weakly reflective submanifolds in a sphere and those in Euclidean spaces or complex projective spaces.

The authors are profoundly grateful to Makoto Kimura and Osami Yasukura for their helpful suggestion on Proposition 4.4. Before we wrote this paper, Kimura, Yasukura and the third named author showed a previous version of Proposition 4.4 which is unpublished, that is, the orbit of the highest root of a compact Lie group under the adjoint action is an austere submanifold in the hypersphere. The authors would also like to thank Reiko Miyaoka for her valuable comments. In fact, Proposition 2.9 was essentially suggested by her. Finally the authors are grateful to the referee who gave them many useful comments.

2. Definitions and fundamental results

We begin with recalling the definition of reflective submanifold given by Leung [8]. Let M̃ be a complete Riemannian manifold. A connected component of the fixed point set of an involutive isometry of M̃ is called a reflective submanifold. A reflective submanifold is a complete totally geodesic submanifold. The involutive isometry which defines a reflective submanifold M can be determined uniquely. We call it the reflection of M and denote by σ_{M}. If M is a reflective submanifold in M̃ and σ_{M} is its reflection, then for any normal vector ξ ∈ T_{x}^{⊥}M

σ_{M}(x) = x,  (dσ_{M})_{x}ξ = −ξ,  σ_{M}(M) = M

hold. Taking notice of these properties, we define a weakly reflective submanifold as follows.

Definition 2.1. Let M be a submanifold of a Riemannian manifold M̃. For each normal vector ξ ∈ T_{x}^{⊥}M at each point x ∈ M, if there exists an isometry σ_{ξ} of M̃ which satisfies

σ_{ξ}(x) = x,  (dσ_{ξ})_{x}ξ = −ξ,  σ_{ξ}(M) = M,

then we call M a weakly reflective submanifold and σ_{ξ} a reflection of M with respect to ξ.

In the case where M is a hypersurface, σ_{ξ} is independent of the choice of ξ at each point x. In this paper mainly we deal with orbits of some isometric actions of compact Lie groups. We note that if M is an extrinsic homogeneous submanifold in M̃, that is an orbit of an isometric action of a Lie group on M̃, then it suffices to ascertain that at one point of M it satisfies the condition to be a weakly reflective submanifold.

Remark 2.2. For a reflective submanifold, there exists a reflection which is independent of the choice of a normal vector. So it is clear that a reflective submanifold is always a weakly reflective submanifold.

Example 2.3.

S^{n−1}(1) × S^{n−1}(1) = {(x, y) | x, y ∈ S^{n−1}(1)}

is a weakly reflective submanifold in (2n − 1)-dimensional sphere S^{2n−1}(\sqrt{2}) of radius \sqrt{2}.

Proof. Since S^{n−1}(1) × S^{n−1}(1) is a homogeneous submanifold of S^{2n−1}(\sqrt{2}), it suffices to ascertain that at one point of S^{n−1}(1)×S^{n−1}(1) it satisfies the condition to be a weakly reflective submanifold. The tangent space of S^{n−1}(1) × S^{n−1}(1) at

x = (1, 0, …, 0, 1^{{n+1 \atop ⌣}}, 0, …, 0) ∈ S^{n−1}(1) × S^{n−1}(1)

is given by

T_{x}(S^{n−1}(1) × S^{n−1}(1)) = {(0, x_{2}, …, x_{n}, 0, y_{2}, …, y_{n}) | x_{i}, y_{j} ∈ R},

and the normal space in S^{2n−1}(\sqrt{2}) is

T_{x}^{⊥}(S^{n−1}(1) × S^{n−1}(1)) = R(1, 0, …, 0, −1^{{n+1 \atop ⌣}}, 0, …, 0).

Now we define an isometry σ of S^{2n−1}(\sqrt{2}) by

σ(x_{1}, …, x_{n}, y_{1}, …, y_{n}) = (y_{1}, …, y_{n}, x_{1}, …, x_{n})

for (x_{1}, …, x_{n}, y_{1}, …, y_{n}) ∈ S^{2n−1}(\sqrt{2}). Then

σ(x) = x, σ(S^{n−1}(1) × S^{n−1}(1)) = S^{n−1}(1) × S^{n−1}(1)

and dσ_{x} acts on T_{x}^{⊥}(S^{n−1}(1) × S^{n−1}(1)) as −id. Thus S^{n−1}(1) × S^{n−1}(1) is a weakly reflective submanifold in S^{2n−1}(\sqrt{2}). □

Definition 2.4. Let M be a submanifold of a Riemannian manifold M̃. We denote the shape operator of M by A. M is called an austere submanifold if for each normal vector ξ ∈ T_{x}^{⊥}M, the set of eigenvalues with their multiplicities of A_{ξ} is invariant under the multiplication by −1. It is obvious that an austere submanifold is a minimal submanifold.

The notion of austere submanifold was first given by Harvey-Lawson [4].

Proposition 2.5. A weakly reflective submanifold is an austere submanifold.

Proof. Let M be a weakly reflective submanifold in a Riemannian manifold M̃. Then for each normal vector ξ ∈ T_{x}^{⊥}M, there exists an isometry σ_{ξ} of M̃ which satisfies

σ_{ξ}(x) = x,  (dσ_{ξ})_{x}ξ = −ξ,  σ_{ξ}(M) = M.

For a normal vector ξ ∈ T_{x}^{⊥}M, we denote by A_{ξ} the shape operator of M with respect to ξ and by h the second fundamental form of M. For X, Y ∈ T_{x}M, we take vector fields X̃ and Ỹ defined on a neighborhood of x in M̃ which are tangent to M and X̃_{x} = X and Ỹ_{x} = Y. Since σ_{ξ} satisfies σ_{ξ}(M) = M, vector fields dσ_{ξ}X̃ and dσ_{ξ}Ỹ are tangent to M. Let ¯∇ denote the covariant derivative of M̃. Then we have

h((dσ_{ξ})_{x}X, (dσ_{ξ})_{x}Y) = ( ¯∇_{dσ_{ξ}X̃}dσ_{ξ}Ỹ)_{x}^{⊥} = ((dσ_{ξ})_{x} ¯∇X̃ Ỹ)^{⊥}
= (dσ_{ξ})_{x}( ¯∇X̃ Ỹ)^{⊥} = (dσ_{ξ})_{x}h(X, Y).

From the following calculation

⟨A_{ξ}(dσ_{ξ})_{x}X, (dσ_{ξ})_{x}Y ⟩ = ⟨h((dσ_{ξ})_{x}X, (dσ_{ξ})_{x}Y), ξ⟩
= ⟨(dσ_{ξ})_{x}h(X, Y), ξ⟩ = ⟨h(X, Y), (dσ_{ξ})_{x}^{−1}ξ⟩
= ⟨h(X, Y), −ξ⟩ = −⟨A_{ξ}X, Y ⟩,

we have (dσ_{ξ})_{x}^{−1}A_{ξ}(dσ_{ξ})_{x} = −A_{ξ}. This implies that (dσ_{ξ})_{x} provides an isomorphism between eigenspaces of A_{ξ} for eigenvalues λ and −λ. Thus M is an austere submanifold. □

In the rest of this section, we shall study weakly reflective orbits of isometric actions of Lie groups on Riemannian manifolds. First we shall provide some preliminaries. Let G be a Lie group acting isometrically on a Riemannian manifold M̃ and G_{x} be the isotropy subgroup at x, that is, G_{x} = {g ∈ G | gx = x}. Then the orbit G(x) is diffeomorphic to the coset manifold G/G_{x}. An orbit G(x) is a principal orbit if, for any y ∈ M̃, there exists g ∈ G such that G_{x} ⊂ gG_{y}g^{−1}. It is known that there exists a principal orbit. The codimension of a principal orbit is called the cohomogeneity of the action of G on M̃. An orbit which is not principal is called a singular orbit. The differential of the action of G_{x} defines a linear representation of G_{x} on T_{x}M̃ called the linear isotropy representation. The tangent space T_{x}(G(x)) and the normal space T_{x}^{⊥}(G(x)) of G(x) at x are invariant subspaces of the linear isotropy representation. The restriction of the linear isotropy representation to T_{x}^{⊥}(G(x)) is called the slice representation at x.

Theorem 2.6. (Slice representation theorem [9, Theorem 1.1], [11, Theorem 4.6], [12, Proposition 5.4.7]). The cohomogeneity of a slice representation equals the cohomogeneity of the action of G on M̃. Moreover, G(x) is a principal orbit if and only if the slice representation at x is trivial.

Proposition 2.7. Any singular orbit of a cohomogeneity one action on a Riemannian manifold is a weakly reflective submanifold.

Proof. Suppose that the isometric action of a Lie group G on a Riemannian manifold M̃ is cohomogeneity one. Let G(x) be a singular orbit.

First we consider the case where the codimension of G(x) is greater than or equal to 2. From the slice representation theorem, the isotropy subgroup G_{x} acts transitively on the hypersphere in T_{x}^{⊥}(G(x)). In particular, for any ξ ∈ T_{x}^{⊥}(G(x)) there exists g ∈ G_{x} such that dg_{x}(ξ) = −ξ. Therefore g becomes a reflection of G(x) at x with respect to ξ. Since G(x) is a homogeneous submanifold, G(x) has a reflection with respect to any normal vector at any point. Thus G(x) is a weakly reflective submanifold in M̃.

When the codimension of G(x) is 1, dim T_{x}^{⊥}(G(x)) = 1 and the dimension of a nontrivial orbit of the slice representation is equal to 0 by Theorem 2.6. Moreover the slice representation at x is not trivial, because G(x) is a singular orbit. Therefore the image of the slice representation is not SO(1) but O(1) and for any ξ ∈ T_{x}^{⊥}(G(x)) there exists g ∈ G_{x} such that dg_{x}(ξ) = −ξ. Thus, by the same discussion with above, G(x) is a weakly reflective submanifold in M̃. □

Remark 2.8. Podestá [13] proved that any singular orbit of a cohomogeneity one action is an austere submanifold. However, essentially he showed Proposition 2.7.

Proposition 2.9. Let G be a connected Lie group acting isometrically on a complete, connected Riemannian manifold M̃. Suppose that the action of G on M̃ is cohomogeneity one with two singular orbits. If there exists a principal orbit which is a weakly reflective submanifold in M̃, then it has a same distance from two singular orbits and two singular orbits are isometric.

Proof. Since there exist two singular orbits, the orbit space M̃/G is homeomorphic to a closed interval (Mostert [10], Bergery [1]). Orbits of interior points are principal and those of end points are singular. Moreover principal orbits are hypersurfaces in M̃, because the cohomogeneity of the action of G is one. Suppose that G(x) is a principal orbit which is a weakly reflective submanifold. Then, by the slice representation theorem, there exists a unit normal vector field ξ on G(x), which is invariant under the action of G. We take a geodesic γ(t) of M̃ which satisfies an initial condition

γ(0) = x, γ′(0) = ξ_{x}.

Then γ(t) is a section of the action of G on M̃, namely γ(t) meets all orbits perpendicularly ([12] Proposition 5.6.2). Since dg_{x}(ξ_{x}) = ξ_{gx} for any g ∈ G, gγ(t) is a geodesic of M̃ which satisfies an initial condition

gγ(0) = gx, (gγ)′(0) = ξ_{gx}.

Since G(x) is a weakly reflective submanifold of M̃, there exists an isometry σ of M̃ which satisfies

σ(x) = x, dσ_{x}(ξ_{x}) = −ξ_{x}, σ(G(x)) = G(x),

that is a reflection of G(x) with respect to ξ_{x}. We set

G(x)_{±} = {y ∈ G(x) | dσ_{y}(ξ_{y}) = ±ξ_{σ(y)}}.

The sets G(x)_{+} and G(x)_{−} are closed subsets of G(x), and G(x) is a disjoint union of G(x)_{+} and G(x)_{−} because G(x) is a hypersurface in M̃. Since G(x) is connected and x ∈ G(x)_{−}, we have G(x) = G(x)_{−}. This implies that dσ_{y}(ξ_{y}) = −ξ_{σ(y)} for any y ∈ G(x). For any g ∈ G, σgγ(t) is a geodesic which satisfies an initial condition

σgγ(0) = σ(gx), (σgγ)′(0) = dσ_{gx}(gγ)′(0) = dσ_{gx}(ξ_{gx}) = −ξ_{σgx}.

Now we take g_{1} ∈ G such that g_{1}x = σ(gx). Then σgγ(t) and g_{1}γ(−t) are geodesics of same initial conditions, hence σgγ(t) = g_{1}γ(−t) ∈ G(γ(−t)). Therefore we have σ(G(γ(t))) ⊂ G(γ(−t)) for each t. Since σ^{−1} is also a reflection of G(x) at x, we also have σ^{−1}(G(γ(−t))) ⊂ G(γ(t)) by the same discussion for σ^{−1} and γ(−t). Thus σ(G(γ(t))) = G(γ(−t)) and σ induces a homeomorphism of M̃/G. This implies that σ maps one singular orbit to the other one. Hence two singular orbits can be expressed as G(γ(t_{1})) and G(γ(−t_{1})) for some t_{1}. Consequently we have the conclusion. □

3. Orbits of s-representations

A linear isotropy representation of a Riemannian symmetric pair is called an s-representation as we mentioned in Introduction. In the following sections, we will study orbits of s-representations which are austere submanifolds or weakly reflective submanifolds. 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 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.

Henceforth we assume that the symmetric pair (G, K) is irreducible, namely K acts irreducibly on 𝔪.

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

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

and define the root system R̃ of 𝔤 by

R̃ = {α ∈ 𝔱 − {0} | 𝔤̃_{α} ≠ {0}}.

For α ∈ 𝔞 we set

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

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

R = {α ∈ 𝔞 − {0} | 𝔤_{α} ≠ {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 fundamental system of R̃ with respect to the ordering >. Set

F̃_{0} = F̃ ∩ R̃_{0},

then the fundamental system F of R with respect to the ordering > is given by

F = {ᾱ | α ∈ F̃ − F̃_{0}}.

WEAKLY REFLECTIVE SUBMANIFOLDS AND AUSTERE SUBMANIFOLDS 7

We set

R̃_{+} = {α ∈ R̃ | α > 0}, R_{+} = {α ∈ R | α > 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_{α} = 𝔨 ∩ (𝔤_{α} + 𝔤_{−α}), 𝔪_{α} = 𝔪 ∩ (𝔤_{α} + 𝔤_{−α})

for α ∈ R_{+}. Under these notations, we have the following lemma. Lemma 3.1 ([14]). (1) We have orthogonal direct sum decompositions

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

(2) For each α ∈ R̃_{+} − R̃_{0}, there exist S_{α} ∈ 𝔨 and T_{α} ∈ 𝔪 such that

{S_{α} | α ∈ R̃_{+}, ᾱ = λ}, {T_{α} | α ∈ R̃_{+}, ᾱ = λ} are respectively orthonormal bases of 𝔨_{λ} and 𝔪_{λ} and that for H ∈ 𝔞 [H, S_{α}] = ⟨α, H⟩T_{α}, [H, T_{α}] = −⟨α, H⟩S_{α}, [S_{α}, T_{α}] = ᾱ,

Ad(exp H)S_{α} = cos⟨α, H⟩S_{α} + sin⟨α, H⟩T_{α},
Ad(exp H)T_{α} = − sin⟨α, H⟩S_{α} + cos⟨α, H⟩T_{α}.

We define a subset D of 𝔞 by

D = ⋃_{α∈R}{H ∈ 𝔞 | ⟨α, H⟩ = 0}.

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

C = {H ∈ 𝔞 | ⟨α, H⟩ > 0 (α ∈ F)}.

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

C̄ = {H ∈ 𝔞 | ⟨α, H⟩ ≥ 0 (α ∈ F)}.

For a subset Δ ⊂ F, we define

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

Lemma 3.2. (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 α ∈ F, we take H_{α} ∈ 𝔞 such that

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

Then we have

C̄ = { ∑_{α∈F} t_{α}H_{α} | t_{α} ≥ 0} ,

and for Δ ⊂ F

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

We set

R^{Δ} = R ∩ (F − Δ)_{Z},
R_{+}^{Δ} = R^{Δ} ∩ R_{+},
𝔤^{Δ} = 𝔨_{0} + 𝔞 + ∑_{α∈R_{+}^{Δ}}(𝔨_{α} + 𝔪_{α}).

We also set

𝔨^{Δ} = 𝔤^{Δ} ∩ 𝔨 = 𝔨_{0} + ∑_{α∈R_{+}^{Δ}} 𝔨_{α},

𝔪^{Δ} = 𝔤^{Δ} ∩ 𝔪 = 𝔞 + ∑_{α∈R_{+}^{Δ}} 𝔪_{α}.

Then we have an orthogonal direct sum decomposition

𝔤^{Δ} = 𝔨^{Δ} + 𝔪^{Δ}.

For 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}.

Under these notations, we have the following lemma.

Lemma 3.3 ([6]). Fix a subset Δ ⊂ F. For H ∈ C^{Δ} we have the following: (1) R_{+}^{Δ} = {α ∈ R_{+} | ⟨α, H⟩ = 0},

(2) R^{Δ} = {α ∈ R | ⟨α, H⟩ = 0},

(3) 𝔤^{Δ} = {X ∈ 𝔤 | [H, X] = 0},

(4) (𝔤^{Δ}, 𝔨^{Δ}) is a symmetric pair and its canonical decomposition is given by

𝔤^{Δ} = 𝔨^{Δ} + 𝔪^{Δ},

(5) 𝔨^{Δ} is the Lie algebra of Z_{K}^{H}.

Now we shall study an orbit Ad(K)H of the linear isotropy representation of (G, K) through H ∈ 𝔪. An orbit Ad(K)H is a submanifold of the hypersphere S of radius ∥H∥ in 𝔪. From [6], Ad(K)H is connected. Since

𝔪 = ⋃_{k∈K} Ad(k)C̄,

without loss of generalities we may assume H ∈ C̄. Moreover, from Lemma 3.2, there exists Δ ⊂ F such that H ∈ C^{Δ}. For X ∈ 𝔨 we define a vector field X^{∗} on 𝔪 by

X_{x}^{∗} = \frac{d}{dt}|_{t=0} Ad(exp tX)x = [X, x]

at x ∈ 𝔪. Then X^{∗}|_{Ad(K)H} is a tangent vector field on Ad(K)H. From Lemma 3.1 we have the following lemma.

Lemma 3.4. 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

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

T_{H}^{⊥}(Ad(K)H) = H^{⊥} ∩ 𝔞 + ∑_{α∈R_{+}^{Δ}} 𝔪_{α} = ⋃_{k∈Z_{K}^{H}} Ad(k)(H^{⊥} ∩ 𝔞).

Let h denote the second fundamental form of Ad(K)H at H in the hypersphere S. Then we have

h(X_{H}^{∗}, Y_{H}^{∗}) = [Y, [X, H]]^{N},

where [Y, [X, H]]^{N} is T_{H}^{⊥}(Ad(K)H)-component of [Y, [X, H]].

Proof. From Lemma 3.1 we have

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

T_{H}^{⊥}(Ad(K)H) = H^{⊥} ∩ 𝔞 + ∑_{α∈R_{+}^{Δ}} 𝔪_{α} = H^{⊥} ∩ 𝔪^{Δ}

Moreover, from Lemma 3.3

𝔪^{Δ} = ⋃_{k∈Z_{K}^{H}} Ad(k)𝔞.

Since Ad(k)H = H for k ∈ Z_{K}^{H}, we have

H^{⊥} ∩ 𝔪^{Δ} = ⋃_{k∈Z_{K}^{H}} Ad(k)(H^{⊥} ∩ 𝔞).

The calculation of X_{x}^{∗} mentioned above shows the representation of the second fundamental form. □

For orbits of s-representations which are minimal submanifolds in the hypersphere, the following theorem is known.

Theorem 3.5 ([7]). Fix a hypersphere S in 𝔪 centered at 0. For each subset Δ ⊂ F, there exists a unique H ∈ C^{Δ} ∩ S such that Ad(K)H is a minimal submanifold in S.

However, in general we can not determine H where Ad(K)H is a minimal submanifold in S explicitly. In the following two sections, we will give the complete lists of H where Ad(K)H is an austere submanifold or a weakly submanifold in S.

4. Weakly reflective orbits of s-representations

In this section, we shall study orbits of irreducible s-representations which are weakly reflective submanifolds in the hypersphere. In the next section, we will study austere orbits. Since these two properties of orbits are invariant under scalar multiples on the vector spaces, we do not discriminate the difference of the length of a vector. The following theorem is the main result of this section. We shall follow the notations of root systems in [3].

Theorem 4.1. An orbit of an irreducible s-representation which is a weakly reflective submanifold in the hypersphere is one of the following list:

(1) an orbit through a restricted root vector (Proposition 4.4),

(2) the orbit through the vector 2e_{1} − e_{2} − e_{3} and the orbit through the vector e_{1} + e_{2} − 2e_{3} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i}−e_{j})} of type A_{2} (Proposition 4.5),

(3) the orbit through the vector e_{1}+e_{2}−e_{3}−e_{4} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i}−e_{j})} of type A_{3} (Proposition 4.6),

(4) the orbit through the vector e_{1} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i} ± e_{j})} of type D_{l} (l ≥ 4) (Proposition 4.7),

(5) the orbit through the vector e_{1} +e_{2} +e_{3} +e_{4} and the orbit through the vector e_{1}+e_{2}+e_{3}−e_{4} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i}±e_{j})} of type D_{4} (Proposition 4.8).

Remark 4.2. In the case of the restricted root system of type A_{2}, the orbit through a restricted root is a principal orbit called a Cartan hypersurface, that is a minimal isoparametric hypersurface with three distinct principal curvatures. And the orbits through the vectors 2e_{1} − e_{2} − e_{3} and e_{1} + e_{2} − 2e_{3} are its focal submanifolds, called Veronese surfaces. These two singular orbits are not equal but isometric with each other.

In the case of the restricted root system of type D_{4}, from the proof of Proposition 4.8 we can see that three orbits through e_{1}, e_{1} + e_{2} + e_{3} + e_{4} and e_{1} + e_{2} + e_{3} − e_{4} are isometric with each other by the “triality”.

Here we prove that orbits listed above are weakly reflective submanifolds. In the next section, we will classify all austere orbits of irreducible s-representations and show that all weakly reflective orbits can be obtained in Theorem 4.1. For this purpose, we shall first give the following lemma.

Lemma 4.3. For H ∈ 𝔞, the orbit Ad(K)H is a weakly reflective submanifold in the hypersphere S if and only if, for any ξ ∈ H^{⊥} ∩ 𝔞, there exists a linear isometry σ_{ξ} of 𝔪 which satisfies

(4.1)  σ_{ξ}(H) = H,  σ_{ξ}(ξ) = −ξ,  σ_{ξ}(Ad(K)H) = Ad(K)H.

Proof. From Lemma 3.4, the normal space of the orbit Ad(K)H at H in S is given by

T_{H}^{⊥}(Ad(K)H) = H^{⊥} ∩ 𝔞 + ∑_{α∈R_{+}^{Δ}} 𝔪_{α} = ⋃_{k∈Z_{K}^{H}} Ad(k)(H^{⊥} ∩ 𝔞).

If Ad(K)H is a weakly reflective submanifold in S, then for ξ ∈ H^{⊥} ∩ 𝔞 there exists a linear isometry σ_{ξ} of 𝔪 which satisfies

σ_{ξ}(H) = H,  (dσ_{ξ})_{H}ξ = −ξ,  σ_{ξ}(Ad(K)H) = Ad(K)H.

Here we have (dσ_{ξ})_{H} = σ_{ξ}, since σ_{ξ} is a linear isometry.

Conversely, assume that Ad(K)H satisfies the condition (4.1). We take an arbitrary normal vector ξ ∈ T_{H}^{⊥}(Ad(K)H). From Lemma 3.4, there exists k_{0} ∈ Z_{K}^{H} such that Ad(k_{0})ξ ∈ H^{⊥} ∩ 𝔞. Then, from the assumption, there exists a linear isometry σ which satisfies

σ(H) = H,  σAd(k_{0})ξ = −Ad(k_{0})ξ,  σ(Ad(K)H) = Ad(K)H.

We now define σ_{ξ} = Ad(k_{0})^{−1}σAd(k_{0}). Then σ_{ξ} satisfies

σ_{ξ}(H) = H,  σ_{ξ}(ξ) = −ξ,  σ_{ξ}(Ad(K)H) = Ad(K)H.

Thus σ_{ξ} is a reflection of Ad(K)H with respect to a normal vector ξ at H. Since Ad(K)H is a homogeneous submanifold, we have a reflection with respect to any normal vector at an arbitrary point. Consequently Ad(K)H is a weakly reflective submanifold in S. □

Proposition 4.4. An orbit through a restricted root vector of the linear isotropy representation of an irreducible compact symmetric pair is a weakly reflective submanifold in the hypersphere S.

Proof. Let α_{0} be a restricted root vector and put H = α_{0}. The reflection s_{α_{0}} on 𝔞 with respect to α_{0} is given by

s_{α_{0}}(X) = X − \frac{2⟨α_{0}, X⟩}{⟨α_{0}, α_{0}⟩}α_{0} (X ∈ 𝔞)

and satisfies

s_{α_{0}}(H) = −H, s_{α_{0}}|_{𝔞∩H^{⊥}} = 1_{𝔞∩H^{⊥}}.

The reflection s_{α_{0}} is an element of the Weyl group, hence there exists k_{0} ∈ N_{K} such that Ad(k_{0})|_{𝔞} = s_{α_{0}}, where

N_{K} = {k ∈ K | Ad(k)𝔞 = 𝔞}.

Therefore

−H = Ad(k_{0})H ∈ Ad(K)H,

and we have Ad(K)(−H) = Ad(K)H. We define a linear isometry σ of 𝔪 by

σ = −Ad(k_{0})|_{𝔪}.

Then σ satisfies

σ(H) = H,  σ|_{𝔞∩H^{⊥}} = −1|_{𝔞∩H^{⊥}},  σ(Ad(K)H) = Ad(K)H.

Thus, from Lemma 4.3, Ad(K)H is a weakly reflective submanifold in S. □

Proposition 4.5. The orbit through the vector 2e_{1} − e_{2} − e_{3} and the orbit through the vector e_{1} +e_{2} −2e_{3} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i} − e_{j})} of type A_{2} is a weakly reflective submanifold in the hypersphere S.

Proof. Since the symmetric pair (G, K) is of rank 2, the action of K on S is cohomogeneity one. The vector 2e_{1} − e_{2} − e_{3} (resp. e_{1} + e_{2} − 2e_{3}) is orthogonal to a restricted root e_{2} −e_{3} (resp. e_{1} −e_{2}). Therefore the orbit of K through 2e_{1} −e_{2} −e_{3} (resp. e_{1} + e_{2} − 2e_{3}) is a singular orbit. Hence from Proposition 2.7, this orbit is a weakly reflective submanifold in S. □

Proposition 4.6. The orbit through the vector e_{1} + e_{2} − e_{3} − e_{4} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i} − e_{j})} of type A_{3} is a weakly reflective submanifold in the hypersphere S.

Proof. Put H = e_{1} + e_{2} − e_{3} − e_{4}. The set R_{+}^{Δ} of all positive restricted roots which are orthogonal to H is given by

R_{+}^{Δ} = {e_{1} − e_{2}, e_{3} − e_{4}}.

Let s_{e_{1}−e_{2}} and s_{e_{3}−e_{4}} be the reflections with respect to restricted roots e_{1} − e_{2} and e_{3} −e_{4}, respectively. Then s_{e_{1}−e_{2}} and s_{e_{3}−e_{4}} are elements of the Weyl group, hence there exist k_{0}, k_{1} ∈ N_{K} such that

s_{e_{1}−e_{2}} = Ad(k_{0})|_{𝔞}, s_{e_{3}−e_{4}} = Ad(k_{1})|_{𝔞}.

We now define a linear isometry of 𝔪 by

σ(X) = Ad(k_{0})Ad(k_{1})X (X ∈ 𝔪).

Then σ satisfies

σ(H) = H,  σ|_{𝔞∩H^{⊥}} = −1_{𝔞∩H^{⊥}},  σ(Ad(K)H) = Ad(K)H.

Thus from Lemma 4.3, Ad(K)H is a weakly reflective submanifold in S. □

Proposition 4.7. The orbit through the vector e_{1} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±e_{i} ± e_{j}} of type D_{l} is a weakly reflective submanifold in the hypersphere S.

Proof. An irreducible compact symmetric pair with the restricted root system of type D_{l} is one of (SO(2l) × SO(2l), SO(2l)^{∗}) and (SO(2l), SO(l) × SO(l)).

First we consider the case of (SO(2l) × SO(2l), SO(2l)^{∗}). In this case, 𝔪 can be identified with 𝔬(2l) in a natural manner. We take a maximal abelian subalgebra

𝔞 = \{diag \{[ \begin{matrix}0 & −t_{1} \\ t_{1} & 0 \end{matrix} ] , …, [ \begin{matrix}0 & −t_{l} \\ t_{l} & 0 \end{matrix} ]\} | t_{1}, …, t_{l} ∈ R\}

of 𝔬(2l), and put

H = e_{1} = diag \{[ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] , 0, …, 0\} .

We define a linear isometry σ of 𝔬(2l) by

σ(X) = sXs (X ∈ 𝔬(2l)),

where

s = diag \{[ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] , [ \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ] , …, [ \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ]\} ∈ O(2l).

Then σ is an isometry of S and satisfies

σ(H) = H,  σ|_{𝔞∩H^{⊥}} = −id_{𝔞∩H^{⊥}},  σ(Ad(K)H) = Ad(K)H.

Hence from Lemma 4.3, Ad(K)H is a weakly reflective submanifold in S.

Second, we consider the case of (SO(2l), SO(l) × SO(l)). We take a maximal abelian subspace

𝔞 = \{[ \begin{matrix}0 & X \\ −X & 0 \end{matrix} ] | X = diag(t_{1}, …, t_{l}), t_{i} ∈ R\} ,

and put

H = e_{1} = [ \begin{matrix}0 & X_{0} \\ −X_{0} & 0 \end{matrix} ] ∈ 𝔬(2l),

where

X_{0} = diag{1, 0, …, 0} ∈ M_{l}(R).

We define a linear isometry σ of 𝔪 by

σ(X) = [ \begin{matrix}s & 0 \\ 0 & I_{l} \end{matrix} ] X [ \begin{matrix}s & 0 \\ 0 & I_{l} \end{matrix} ] (X ∈ 𝔪),

where

s = diag{1, −1, …, −1} ∈ O(l).

Then σ is an isometry of S and satisfies

σ(H) = H,  σ|_{𝔞∩H^{⊥}} = −id_{𝔞∩H^{⊥}},  σ(Ad(K)H) = Ad(K)H.

Hence from Lemma 4.3, Ad(K)H is a weakly reflective submanifold in S. □

Proposition 4.8. The orbit through the vector e_{1} + e_{2} + e_{3} + e_{4} and the orbit through the vector e_{1} +e_{2} +e_{3} −e_{4} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±e_{i} ± e_{j}} of type D_{4} is a weakly reflective submanifold in the hypersphere S.

Proof. We take a fundamental system of {±e_{i} ± e_{j}} of type D_{4}:

α_{1} = e_{1} − e_{2}, α_{2} = e_{2} − e_{3}, α_{3} = e_{3} − e_{4}, α_{4} = e_{3} + e_{4}.

The automorphism group of the Dynkin diagram is the permutation group of {α_{1}, α_{3}, α_{4}}. So there exists an automorphism of 𝔪 mapping α_{1} to α_{4} and fixing α_{3}, which gives an equivalence of the orbits through e_{1} and e_{1} + e_{2} + e_{3} + e_{4}. Thus the orbit through e_{1} + e_{2} + e_{3} + e_{4} is also a weakly reflective submanifold in the hypersphere. Similarly the permutation of α_{1} and α_{3} gives an equivalence of the orbits through e_{1} and e_{1} + e_{2} + e_{3} − e_{4}. Thus the orbit through e_{1} + e_{2} + e_{3} − e_{4} is also a weakly reflective submanifold in the hypersphere. □

In the case of (SO(8), SO(4) × SO(4)) we can explicitly represent a reflection of the orbit though e_{1} + e_{2} + e_{3} + e_{4}. The linear isotropy representation is equivalent to

(g_{1}, g_{2}) · X = g_{1}Xg_{2}^{−1} ((g_{1}, g_{2}) ∈ SO(4) × SO(4), X ∈ M_{4}(R)).

Let e_{i} denotes an element of M_{4}(R) whose (i, i) component is 1 and others are 0. Then the orbit through e_{1} + e_{2} + e_{3} + e_{4} is SO(4) in M_{4}(R).

For z_{1} ⊗ z_{2} ∈ H ⊗ H, we define

ϕ_{z_{1}⊗z_{2}} : H → H ; z ↦ z_{1}zz̄_{2}.

Since H ≅ R^{4}, we can regard ϕ_{z_{1}⊗z_{2}} as an element of M_{4}(R) and ϕ induces an isomorphism M_{4}(R) ≅ H ⊗ H of real algebras. We define an involutive isometry σ of H ⊗ H by

σ : H ⊗ H → H ⊗ H ; z_{1} ⊗ z_{2} ↦ z_{1} ⊗ z̄_{2}.

We also denote by σ the linear isometry of M_{4}(R) induced from σ through ϕ. We note that

SO(4) = {ϕ_{z_{1}⊗z_{2}} | z_{1}, z_{2} ∈ Sp(1)}.

Moreover

{z_{1} ⊗ z_{2} | z_{1}, z_{2} ∈ Sp(1)} ⊂ H ⊗ H

is invariant under σ. Therefore SO(4) is invariant under σ. The identity element I is fixed by the action of σ. The normal space of SO(4) at I in S^{15}(2) is given by

T_{I}^{⊥}(SO(4)) = {X ∈ M_{4}(R) | X : symmetric, tr(X) = 0}.

It is easy to see that T_{I}^{⊥}(SO(4)) is contained in the eigenspace of σ for an eigenvalue −1. Thus σ is a reflection of SO(4) with respect to an arbitrary normal vector at I.

5. Austere orbits of s-representations

In this section we classify all orbits of irreducible s-representations which are austere submanifolds in the hypersphere S. In the previous section we showed that all orbits through restricted root vectors (or their scalar multiples) are weakly reflective, hence austere. Therefore, hereafter we shall concern with other orbits. We will also determine austere orbits which are not weakly reflective submanifolds. Then we will complete to prove Theorem 4.1.

The classification of austere orbits is following:

Theorem 5.1. An orbit of an irreducible s-representation which is an austere submanifold in the hypersphere is one of the following list:

(1) an orbit through a restricted root vector,

(2) the orbit through the vector 2e_{1} − e_{2} − e_{3} and the orbit through the vector e_{1} + e_{2} − 2e_{3} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i} −e_{j})} of type A_{2} (Proposition 4.5 or 5.4),

(3) the orbit through the vector e_{1}+e_{2}−e_{3}−e_{4} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i}−e_{j})} of type A_{3} (Proposition 5.4),

(4) the orbit through the vector e_{1} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i} ± e_{j})} of type D_{l} (l ≥ 4) (Proposition 5.5),

(5) the orbit through the vector e_{1} +e_{2} +e_{3} +e_{4} and the orbit through the vector e_{1}+e_{2}+e_{3}−e_{4} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±(e_{i}±e_{j})} of type D_{4} (Proposition 5.5),

(6) the orbit through the vector e_{1} + \frac{e_{1}+e_{2}}{\sqrt{2}} of the linear isotropy representation of a compact symmetric pair with the restricted root system {±e_{i}, ±e_{i} ± e_{j}} of type B_{2} whose multiplicities are constant (Proposition 5.6),

(7) the orbit through the vector α_{1} + \frac{α_{2}}{\sqrt{3}} of the linear isotropy representation of a compact symmetric pair with the restricted root system of type G_{2}, where α_{1} = e_{1} − e_{2} and α_{2} = −2e_{1} + e_{2} + e_{3} (Proposition 5.8).

Remark 5.2. In the case where the rank of the symmetric pair is equal to two, any principal orbit of an s-representation is an isoparametric hypersurface in the hypersphere. It is known that the number g of distinct principal curvatures of an isoparametric hypersurface in the sphere is one of 1, 2, 3, 4 and 6. There exists a unique minimal isoparametric hypersurface in each parallel family of isoparametric hypersurfaces. Theorems 4.1 and 5.1 shows some of minimal isoparametric hypersurfaces and their focal submanifolds are austere, furthermore weakly reflective, and some of them are not. More precisely;

• When g = 1, a minimal isoparametric hypersurface is nothing but a great sphere.

• When g = 2, a minimal isoparametric hypersurface is weakly reflective (austere) if and only if the multiplicities of two distinct principal curvatures are constant as in Example 2.3.

• When g = 3, that corresponds to the case of type A_{2}, a minimal isoparametric hypersurface is weakly reflective. Moreover both of its two focal submanifolds are also weakly reflective.

• When g = 4, that corresponds to the case of types B_{2} = C_{2} or BC_{2}, a minimal homogeneous isoparametric hypersurface is austere, but not weakly reflective, if and only if the multiplicities of two distinct principal curvatures are constant. Both of focal submanifolds are always weakly reflective.

• When g = 6, that corresponds to the case of type G_{2}, a minimal homogeneous isoparametric hypersurface is austere, but not weakly reflective. Both of its two focal submanifolds are weakly reflective.

Before giving a proof of Theorem 5.1 we shall provide some preliminaries. Let (G, K) be an irreducible compact symmetric pair. We shall use the notations in previous sections. From Lemma 3.4, for a normal vector ξ ∈ T_{H}^{⊥}(Ad(K)H), the shape operator A_{ξ} of Ad(K)H in the hypersphere S is given by

(5.2)  ⟨A_{ξ}(X^{∗}), Y^{∗}⟩ = ⟨h(X^{∗}, Y^{∗}), ξ⟩ = ⟨[Y, [X, H]], ξ⟩ = −⟨[X, H], [Y, ξ]⟩.

For simplification, we discuss a normalization of a normal vector ξ. From Lemma 3.4, there exists k ∈ Z_{K}^{H} such that Ad(k)ξ ∈ H^{⊥} ∩ 𝔞. Then

⟨A_{ξ}(X^{∗}), Y^{∗}⟩ = ⟨Ad(k)h(X^{∗}, Y^{∗}), Ad(k)ξ⟩
= ⟨h(Ad(k)X^{∗}, Ad(k)Y^{∗}), Ad(k)ξ⟩
= ⟨A_{Ad(k)ξ}Ad(k)X^{∗}, Ad(k)Y^{∗}⟩
= ⟨Ad(k)^{−1}A_{Ad(k)ξ}(Ad(k)X^{∗}), Y^{∗}⟩.

Thus we have

A_{ξ} = Ad(k)^{−1}A_{Ad(k)ξ}Ad(k).

This implies that eigenvalues of A_{Ad(k)ξ} and their multiplicities coincide with those of A_{ξ}. Hence, in order to show whether an orbit Ad(K)H is austere, it suffices to check eigenvalues of A_{ξ} for ξ ∈ H^{⊥} ∩ 𝔞. Hereafter we assume that ξ ∈ H^{⊥} ∩ 𝔞.

From Lemmas 3.1 and 3.4

{T_{α} | α ∈ R̃_{+} − R̃_{+}^{Δ}}

is an orthonormal basis of T_{H}(Ad(K)H). For α, β ∈ R̃_{+} − R̃_{+}^{Δ} we have

⟨A_{ξ}((S_{α}^{∗})_{H}), (S_{β}^{∗})_{H}⟩ = ⟨α, H⟩⟨β, H⟩⟨A_{ξ}(T_{α}), T_{β}⟩.

On the other hand, from (5.2), we have

⟨A_{ξ}((S_{α}^{∗})_{H}), (S_{β}^{∗})_{H}⟩ = −⟨[S_{α}, H], [S_{β}, ξ]⟩ = −⟨α, H⟩⟨β, ξ⟩δ_{αβ}.

Therefore we have

A_{ξ}(T_{α}) = −\frac{⟨α, ξ⟩}{⟨α, H⟩}T_{α}.

This shows that T_{α} is an eigenvector of A_{ξ} and its eigenvalue is

−\frac{⟨α, ξ⟩}{⟨α, H⟩}.

Hence Ad(K)H is an austere submanifold in S if and only if, for any ξ ∈ H^{⊥} ∩ 𝔞, the set

\{ \frac{⟨α, ξ⟩}{⟨α, H⟩} | α ∈ R̃_{+} − R̃_{+}^{Δ}\}

with multiplicities is symmetric by the multiplication of −1. We shall describe this condition in terms of a finite subset of a Euclidean space.

Let A be a finite subset of a Euclidean space V. We consider a condition that, for any v ∈ V, the set

{⟨a, v⟩ | a ∈ A}

with multiplicities is symmetric by the multiplication of −1. This condition is equivalent to a condition that A is symmetric by the multiplication of −1 on V. Indeed, it is obvious that {⟨a, v⟩ | a ∈ A} is symmetric whenever A is symmetric. Conversely, fix an arbitrary a ∈ A. From the assumption we have

V = ⋃_{b∈A}{v ∈ V | ⟨a, v⟩ = −⟨b, v⟩}.

If −a ∉ A, then the right hand side consists of finite union of hyperplanes of V. This is a contradiction. Therefore −a ∈ A. Consequently A is symmetric by the multiplication of −1 on V.

Let p_{H} : 𝔞 → H^{⊥} ∩ 𝔞 denote the orthogonal projection. An orbit Ad(K)H is austere in S if and only if the set

\{ \frac{p_{H}(α)}{⟨α, H⟩} | α ∈ R_{+} − R_{+}^{Δ}\}

with multiplicities is symmetric by the multiplication of −1 on H^{⊥} ∩ 𝔞. By this criterion, we can easily see that orbits listed in Theorem 5.1 are austere submanifolds in the hypersphere S. Hereafter we shall prove that all austere orbits can be obtained in Theorem 5.1.

We set RR = {xα | x ∈ R, α ∈ R}. We have already showed that the orbit through any element in RR is weakly reflective in the hypersphere, so we consider the orbits through elements in 𝔞 − RR

Lemma 5.3. For H ∈ 𝔞 − RR, the orbit Ad(K)H is an austere submanifold in S if and only if there exist a mapping f : R_{+} − R_{+}^{Δ} → R_{+} − R_{+}^{Δ} without fixed points, and constants n_{α} ≠ 0, ϵ_{α} = ±1 for each α ∈ R_{+} − R_{+}^{Δ} such that

(5.3)  H = n_{α} (\frac{α}{∥α∥} + ϵ_{α} \frac{f(α)}{∥f(α)∥}) ,

and

(5.4)  ∑_{{μ∈R_{+}−R^{Δ}_{+} \atop μ//α}} m(μ) = ∑_{{ν∈R_{+}−R^{Δ}_{+} \atop ν//f(α)}} m(ν).

Here we denote by m(μ) the multiplicity of a restricted root μ, and μ // α means that μ and α are linearly dependent.

Excepting the case where the restricted root system R is of type BC, the equality (5.4) is equivalent to m(α) = m(f(α)), moreover #(R_{+} − R_{+}^{Δ}) is even and f^{2} = 1.

Proof. The orthogonal projection p_{H} is defined by

p_{H}(X) = X − \frac{⟨X, H⟩}{⟨H, H⟩}H (X ∈ 𝔞).

Therefore Ad(K)H is an austere submanifold in S if and only if the set

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

with multiplicities is symmetric by the multiplication of −1 on H^{⊥} ∩ 𝔞. In other words, there exists a mapping f : R_{+} − R_{+}^{Δ} → R_{+} − R_{+}^{Δ} which satisfies

(5.5)  \frac{f(α)}{⟨f(α), H⟩} − \frac{H}{⟨H, H⟩} = −\frac{α}{⟨α, H⟩} + \frac{H}{⟨H, H⟩}

and

∑ \{m(μ) | μ ∈ R_{+} − R_{+}^{Δ}, \frac{μ}{⟨μ, H⟩} − \frac{H}{⟨H, H⟩} = \frac{α}{⟨α, H⟩} − \frac{H}{⟨H, H⟩} \}
= ∑ \{m(ν) | ν ∈ R_{+} − R_{+}^{Δ}, \frac{ν}{⟨ν, H⟩} − \frac{H}{⟨H, H⟩} = \frac{f(α)}{⟨f(α), H⟩} − \frac{H}{⟨H, H⟩} \}

for any α ∈ R_{+} − R_{+}^{Δ}. This condition for the multiplicities is equivalent to (5.4). From (5.5), if f has a fixed point α, then H ∈ RR. Thus f has no fixed points.

If we assume (5.5), then there exist non-zero real numbers x, y so that H = xα + yf(α). Applying this to the equation (5.5), we have a quadratic equation

∥f(α)∥^{2}y^{2} = ∥α∥^{2}x^{2}

with respect to x and y. Thus we have

y = ±\frac{∥α∥}{∥f(α)∥}x,

hence H can be expressed as

H = xα ± \frac{∥α∥}{∥f(α)∥}xf(α) = x∥α∥ (\frac{α}{∥α∥} ± \frac{f(α)}{∥f(α)∥}) .

Since this equality holds for any α ∈ R_{+} − R_{+}^{Δ}, we have the condition (5.3). Replacing α in (5.5) by f(α), we have

(5.6)  \frac{f^{2}(α)}{⟨f^{2}(α), H⟩} + \frac{f(α)}{⟨f(α), H⟩} = \frac{2H}{⟨H, H⟩}.

From equations (5.5) and (5.6), we have

\frac{α}{⟨α, H⟩} = \frac{f^{2}(α)}{⟨f^{2}(α), H⟩}.

The above discussion stands for any restricted root systems R, including of type BC. Henceforth we assume that R is not of type BC. Then α is the only element of R_{+} − R_{+}^{Δ} which is a scalar multiple of α. Thus f^{2}(α) = α. Since f has no fixed points, #(R_{+} − R_{+}^{Δ}) is even. This completes the proof. □

Here we mention some results concerning with the Weyl group needed later. The action of the Weyl group maps the restricted root system R onto itself ([5]). We can see that, for any restricted roots α and β with the same length, there exists an element s in the Weyl group such that β = sα by the classification of the restricted root systems. In this case, m(β) = m(α) holds. In particular, if the restricted root system R is one of the types A_{l}, D_{l}, E_{6}, E_{7} and E_{8}, then the Weyl group acts transitively on R and all restricted roots have constant multiplicities, since all restricted roots have the same length.

Proposition 5.4. In the case where R is of type A_{l}, an austere orbit is one of the following except orbits through a restricted root vector:

(1) when l = 2, the orbit through H = 2e_{1} − e_{2} − e_{3} and the orbit through

e_{1} + e_{2} − 2e_{3},
(2) when l = 3, the orbit through H = e_{1} + e_{2} − e_{3} − e_{4}.

Proof. In the case of R = A_{l}, R_{+} is given by R_{+} = {e_{i} − e_{j} | i < j}. Since all restricted roots have constant multiplicities, the condition (5.4) of Lemma 5.3 is always satisfied. From Lemma 5.3, without loss of generalities, we may assume that H = ±(a positive root) ± (a positive root), since all restricted roots have the same length. Moreover since any root can be translated to e_{1} − e_{2} by the action of the Weyl group, we may assume that H = (e_{1}−e_{2})±(a positive root). The positive root in the second term of H is one of e_{1} − e_{i} (3 ≤ i), e_{2} − e_{j} (3 ≤ i), e_{i} − e_{j} (3 ≤ i < j).

In the case of H = (e_{1} − e_{2}) ± (e_{1} − e_{i}) (3 ≤ i), e_{i} can be translated to e_{3} by the action of an element of the Weyl group which fixes both e_{1} and e_{2}. Therefore we can put

H = (e_{1} − e_{2}) ± (e_{1} − e_{3}) = \{ \begin{matrix}(e_{1} − e_{2}) + (e_{1} − e_{3}) = 2e_{1} − e_{2} − e_{3} \\ (e_{1} − e_{2}) − (e_{1} − e_{3}) = −e_{2} + e_{3} (root) \end{matrix}

Similarly, in the case of H = (e_{1} − e_{2}) ± (e_{2} − e_{i}) (3 ≤ i), we can put

H = (e_{1} − e_{2}) ± (e_{2} − e_{3}) = \{ \begin{matrix}e_{1} − 2e_{2} + e_{3} ∼ e_{1} + e_{2} − 2e_{3} \\ e_{1} − e_{3} (root) \end{matrix}

Here, for H_{1}, H_{2} ∈ 𝔞, we express H_{1} ∼ H_{2} when H_{1} can be translated to H_{2} by some element of K. In other words, H_{1} is equivalent to H_{2} under the action of the Weyl group.

In the case of H = (e_{1} − e_{2}) ± (e_{i} − e_{j}) (3 ≤ i < j), there exists an element of the Weyl group which fixes e_{1}, e_{2} and translates e_{i} to e_{3} and e_{j} to e_{4}. Therefore we can put

H = (e_{1} − e_{2}) ± (e_{3} − e_{4}) = \{ \begin{matrix}e_{1} + e_{3} − e_{2} − e_{4} \\ e_{1} + e_{4} − e_{2} − e_{3} \end{matrix}

By the action of the Weyl group, these vectors are equivalent to e_{1} + e_{2} − e_{3} − e_{4}. Consequently, it suffices to consider orbits through

H = 2e_{1} − e_{2} − e_{3}, e_{1} + e_{2} − 2e_{3}, e_{1} + e_{2} − e_{3} − e_{4},

which have a possibility to be austere.

In the case of H = 2e_{1} − e_{2} − e_{3}, the only possibility to be the form H = (a positive root) ± (a positive root) is H = (e_{1} − e_{2}) + (e_{1} − e_{3}). Thus, from Lemma 5.3, the set R_{+} − R_{+}^{Δ} must be

R_{+} − R_{+}^{Δ} = {e_{1} − e_{2}, e_{1} − e_{3}}.

When l ≥ 3, since ⟨e_{3} − e_{4}, H⟩ ≠ 0, we have e_{3} − e_{4} ∈ R_{+} − R_{+}^{Δ}. This is a contradiction. Hence l = 2 and then Ad(K)H is austere in S. Similarly, the orbit through H = e_{1} + e_{2} − 2e_{3} is also austere.

In the case of H = e_{1}+e_{2}−e_{3}−e_{4}, possibilities to be the form H = (a positive root)± (a positive root) are

H = (e_{1} − e_{3}) + (e_{2} − e_{4}) = (e_{1} − e_{4}) + (e_{2} − e_{3}).

Thus R_{+} − R_{+}^{Δ} must satisfy

R_{+} − R_{+}^{Δ} ⊂ {e_{1} − e_{3}, e_{2} − e_{4}, e_{1} − e_{4}, e_{2} − e_{3}}

When l ≥ 4, since ⟨e_{4} − e_{5}, H⟩ ≠ 0, we have e_{4} − e_{5} ∈ R_{+} − R_{+}^{Δ}. This is a contradiction. Hence l = 3, and then Ad(K)H is austere in S. □

Proposition 5.5. In the case where R is of type D_{l}, an austere orbit is one of the following except orbits through a restricted root vector:

(1) the orbit through H = e_{1},

(2) when l = 4, the orbits through H = e_{1} + e_{2} + e_{3} + e_{4}, e_{1} + e_{2} + e_{3} − e_{4}.

Proof. In the case of R = D_{l}, R_{+} is given by R_{+} = {e_{i} ± e_{j} | i < j}. Since all restricted roots have constant multiplicities, the condition (5.4) of Lemma 5.3 is always satisfied. It is easy to see that the orbit through e_{1} (or its scalar multiple) is austere. Therefore we consider other orbits. From Lemma 5.3, we can assume H = ±(a positive root) ± (a positive root), since all restricted roots have the same length. Since any root can be translated to e_{1} +e_{2} by the action of the Weyl group, we can assume H = (e_{1} + e_{2}) ± (a positive root). Furthermore any root can be translated to one of

e_{1} ± e_{2}, e_{1} + e_{3}, e_{2} + e_{4}, e_{3} + e_{4}, e_{3} − e_{4}

by the action of elements of the Weyl group which fix e_{1}, e_{2}. Therefore H is one of the following:

H = (e_{1} + e_{2}) ± (e_{1} − e_{2}) = 2e_{1}, 2e_{2} ∼ 2e_{1},
H = (e_{1} + e_{2}) ± (e_{1} + e_{3}) = \{ \begin{matrix}2e_{1} + e_{2} + e_{3}, \\ e_{2} − e_{3} (root), \end{matrix}
H = (e_{1} + e_{2}) ± (e_{2} + e_{4}) = \{ \begin{matrix}2e_{2} + e_{1} + e_{4} ∼ 2e_{1} + e_{2} + e_{3}, \\ e_{1} − e_{4} (root), \end{matrix}
H = (e_{1} + e_{2}) ± (e_{3} + e_{4}) ∼ e_{1} + e_{2} + e_{3} + e_{4},
H = (e_{1} + e_{2}) ± (e_{3} − e_{4}) ∼ e_{1} + e_{2} + e_{3} − e_{4}.

Consequently, it suffices to consider orbits through

H = 2e_{1} + e_{2} + e_{3}, e_{1} + e_{2} + e_{3} + e_{4}, e_{1} + e_{2} + e_{3} − e_{4}

which have a possibility to be austere.

In the case of H = 2e_{1} + e_{2} + e_{3}, the only possibility to be the form H = (a positive root) ± (a positive root) is

H = (e_{1} + e_{2}) + (e_{1} + e_{3}).

Thus R_{+} − R_{+}^{Δ} must be

R_{+} − R_{+}^{Δ} = {e_{1} + e_{2}, e_{1} + e_{3}}.

Since ⟨e_{1} − e_{2}, H⟩ ≠ 0, we have e_{1} − e_{2} ∈ R_{+} − R_{+}^{Δ}. This is a contradiction. Hence this orbit is not austere.

In the case of H = e_{1}+e_{2}+e_{3}+e_{4}, possibilities of the form H = (a positive root)± (a positive root) are

H = (e_{1} + e_{2}) + (e_{3} + e_{4}) = (e_{1} + e_{3}) + (e_{2} + e_{4}) = (e_{1} + e_{4}) + (e_{2} + e_{3}).

Thus R_{+} − R_{+}^{Δ} must satisfy

R_{+} − R_{+}^{Δ} ⊂ {e_{1} + e_{2}, e_{3} + e_{4}, e_{1} + e_{3}, e_{2} + e_{4}, e_{1} + e_{4}, e_{2} + e_{3}}.

When l ≥ 5, since ⟨e_{4} + e_{5}, H⟩ ≠ 0, we have e_{4} + e_{5} ∈ R_{+} − R_{+}^{Δ}. This is a contradiction. Hence l = 4, and then the orbit Ad(K)H is austere in S. In the case of H = e_{1} + e_{2} + e_{3} − e_{4}, similarly we have l = 4, and then Ad(K)H is austere. □

Proposition 5.6. In the case where R is one of types B_{l}, C_{l} and BC_{l}, an austere orbit except orbits through restricted root vectors is the following:

When R = B_{2} where the multiplicities of the restricted roots are constant, the orbit through

H = e_{1} + \frac{e_{1} + e_{2}}{\sqrt{2}}

is austere. This orbit is a principal orbit.

Remark 5.7. In the case of R = B_{2}, there exist two singular orbits and these are not isometric. Hence from Proposition 2.9, a principal austere orbit in Proposition 5.6 is not a weakly reflective submanifold.

Proof. First we consider the case of R = B_{l}, where R_{+} = {e_{i}, e_{i} ± e_{j} | i < j}. From Lemma 5.3, we can assume

H = \frac{α}{∥α∥} ± \frac{β}{∥β∥} (α, β ∈ R_{+}).

i) When α and β are both short roots, we can put H = α ± β. Furthermore, since any short root α can be translated to e_{1} by the action of the Weyl group, we can assume H = e_{1} ± β. If β = e_{1}, then H = 2e_{1} and this is equivalent to the orbit through a root vector. If β = e_{j} (j ≥ 2), then H = e_{1} ± e_{j} is a root vector.

ii) When α and β are both long roots, we can put H = α ± β. Since any long root α can be translated to e_{1} + e_{2} by the action of the Weyl group, we can assume H = (e_{1} + e_{2}) ± β. Furthermore β can be translated to one of

β = e_{1} ± e_{2}, e_{1} + e_{3}, e_{2} + e_{3}, e_{3} + e_{4}

by the action of elements of the Weyl group which fix e_{1} and e_{2}.

In the case of β = e_{1} ± e_{2}, H is equivalent to a root or zero vector.

In the case of β = e_{1} + e_{3},

H = (e_{1} + e_{2}) ± (e_{1} + e_{3}) = \{ \begin{matrix}2e_{1} + e_{2} + e_{3}, \\ e_{2} − e_{3} (root). \end{matrix}

When H = 2e_{1} +e_{2} +e_{3}, the only possibility to be the form H = (a positive root)± (a positive root) is H = (e_{1} + e_{2}) + (e_{1} + e_{3}). Thus R_{+} − R_{+}^{Δ} must be R_{+} − R_{+}^{Δ} = {e_{1} + e_{2}, e_{1} + e_{3}}. On the other hand, since ⟨e_{1}, H⟩ ≠ 0, we have e_{1} ∈ R_{+} − R_{+}^{Δ}. This is a contradiction. Hence this orbit is not austere.

In the case of β = e_{2} + e_{3},

H = (e_{1} + e_{2}) ± (e_{2} + e_{3}) = \{ \begin{matrix}e_{1} + 2e_{2} + e_{3} ∼ 2e_{1} + e_{2} + e_{3}, \\ e_{1} − e_{3} (root). \end{matrix}

In the case of β = e_{3} + e_{4},

H = (e_{1} + e_{2}) ± (e_{3} + e_{4}) = \{ \begin{matrix}e_{1} + e_{2} + e_{3} + e_{4}, \\ e_{1} + e_{2} − e_{3} − e_{4} ∼ e_{1} + e_{2} + e_{3} + e_{4}. \end{matrix}

In this case,

H = (e_{1} + e_{2}) + (e_{3} + e_{4}) = (e_{1} + e_{3}) + (e_{2} + e_{4}) = (e_{1} + e_{4}) + (e_{2} + e_{3})

are possibilities to be the form H = α ± β. Thus R_{+} − R_{+}^{Δ} must satisfy

R_{+} − R_{+}^{Δ} ⊂ {e_{1} + e_{2}, e_{3} + e_{4}, e_{1} + e_{3}, e_{2} + e_{4}, e_{1} + e_{4}, e_{2} + e_{3}}.

On the other hand, since ⟨e_{1}, H⟩ ≠ 0, we have e_{1} ∈ R_{+}−R_{+}^{Δ}. This is a contradiction. Hence this orbit is not austere.

iii) When α is a short root and β is a long root, we can assume α = e_{1} and

H = e_{1} ± \frac{β}{\sqrt{2}} where β = e_{1} + e_{2}, e_{2} + e_{3}.

In the case of H = e_{1} + \frac{e_{1}+e_{2}}{\sqrt{2}} , if l ≥ 3, then e_{3} ∈ R_{+} − R_{+}^{Δ}. On the other hand, there is no μ ∈ R_{+} such that

H = n (e_{3} ± \frac{μ}{∥μ∥}) .

Thus we have l = 2. In this case R_{+} − R_{+}^{Δ} = {e_{1}, e_{2}, e_{1} + e_{2}, e_{1} − e_{2}}. If we define f : R_{+} − R_{+}^{Δ} → R_{+} − R_{+}^{Δ} by

f(e_{1}) = e_{1} + e_{2}, f(e_{2}) = e_{1} − e_{2},

then H satisfies the condition (5.3) of Lemma 5.3. Hence this orbit is austere if the multiplicities of the restricted roots are constant.

In the case of H = e_{1} − \frac{e_{1}+e_{2}}{\sqrt{2}} , we can express H as

H = −\frac{1}{\sqrt{2} + 1} \{(1 + \frac{1}{\sqrt{2}}) e_{2} − \frac{1}{\sqrt{2}}e_{1}\} .

Permuting e_{1} and e_{2} by the action of the Weyl group and replacing e_{2} ↦ −e_{2}, we have that this orbit is equivalent to the orbit through

H = e_{1} + \frac{e_{1} + e_{2}}{\sqrt{2}} .

In the case of β = e_{2} + e_{3},

H = e_{1} ± \frac{e_{2} + e_{3}}{\sqrt{2}} ∼ e_{1} + \frac{e_{2} + e_{3}}{\sqrt{2}} .

In this case e_{3} ∈ R_{+} − R_{+}^{Δ}. On the other hand, there is no μ ∈ R_{+} such that

H = n (e_{3} ± \frac{μ}{∥μ∥}) .

Hence this orbit is not austere.

Second we consider the case of R = C_{l}, where R_{+} = {2e_{i}, e_{i} ± e_{j} | i < j}. For this purpose we shall use the dual mapping and transfer the result of the case R = B_{l} by the dual mapping. A mapping

𝔞 − \{0\} → 𝔞 − \{0\}; H ↦ H^{∗} = \frac{2H}{⟨H, H⟩}

is called a dual mapping. This maps a root system to a root system, more precisely, a long root is moved to a short root and a short root is moved to a long root. The root systems of type B_{l} and C_{l} are dual by this mapping, and other irreducible root systems are self-dual. If there exists f which satisfies (5.3) for H ∈ 𝔞, then there exists f^{∗} which satisfies (5.3) for H^{∗}.

In the above discussion, in the case of R = B_{l} (l ≥ 3), we showed that there are no austere orbits except orbits through a restricted root vector. Thus we also have that there are no austere orbits except orbits through a restricted root vector in the case of R = C_{l} (l ≥ 3). When l = 2, C_{2} = B_{2}.

Finally we shall consider the case of R = BC_{l}, where R_{+} = {e_{i}, 2e_{i}, e_{i} ±e_{j} | i < j}. From Lemma 5.3, we can put

H = \frac{α}{∥α∥} ± \frac{β}{∥β∥} (α, β ∈ R_{+}).

If ∥α∥ = ∥β∥, then we can put H = α ± β. When α and β are both short roots or both long roots, H is a scalar multiple of a root vector. When α and β are both middle roots, we can assume α = e_{1} + e_{2} and

H = (e_{1} + e_{2}) ± β (β = e_{1} + e_{3}, e_{2} + e_{3}, e_{3} + e_{4}).

By the action of the Weyl group, these are equivalent to

H = 2e_{1} + e_{2} + e_{3}, H = e_{1} + e_{2} + e_{3} + e_{4}

or a restricted root vector. In the case of H = 2e_{1} + e_{2} + e_{3}, we have l = 3 and

R_{+} − R_{+}^{Δ} = {e_{1}, e_{2}, e_{3}, 2e_{1}, 2e_{2}, 2e_{3}, e_{1} ± e_{2}, e_{1} ± e_{3}, e_{2} + e_{3}}.

Since there is no f which satisfies (5.3), this orbit is not austere. In the case of H = e_{1} + e_{2} + e_{3} + e_{4}, we have l = 4 and

R_{+} − R_{+}^{Δ} = {e_{i}, 2e_{i} | 1 ≤ i ≤ 4}
∪{e_{1} + e_{2}, e_{1} + e_{3}, e_{1} + e_{4}, e_{2} + e_{3}, e_{2} + e_{4}, e_{3} + e_{4}}.

Since there is no f which satisfies (5.3), this orbit is not austere.

It remains the case where ∥α∥ < ∥β∥. When α is a short root and β is a long root, H is a scalar multiple of a root vector. By the dual mapping, we can identify two cases, where α is a short root and β is a middle root, and where α is a middle root and β is a long root. Therefore we shall discuss the former. In this case, we can assume α = e_{1} and

H = e_{1} ± \frac{β}{\sqrt{2}} where β = e_{1} + e_{2}, e_{2} + e_{3}.

Similarly with the case of the restricted root system of type B,

H = e_{1} + \frac{e_{1} + e_{2}}{\sqrt{2}} , H = e_{1} + \frac{e_{2} + e_{3}}{\sqrt{2}}

have a possibility to be austere. When H = e_{1} + \frac{e_{2}+e_{3}}{\sqrt{2}} , we have l = 3 and

R_{+} − R_{+}^{Δ} = {e_{1}, e_{2}, e_{3}, 2e_{1}, 2e_{2}, 2e_{3}, e_{1} ± e_{2}, e_{1} ± e_{3}, e_{2} + e_{3}}.

Since there is no f which satisfies (5.3), this orbit is not austere. When H = e_{1} + \frac{e_{1}+e_{2}}{\sqrt{2}} , the orbit has a possibility to be austere if l = 2. In this case, the orbit is a principal orbit. This orbit is austere if the sum of the multiplicities of long roots and short roots coincides with the multiplicity of middle roots. From the classification of symmetric pairs, there does not exist such a symmetric pair. □

Proposition 5.8. In the case where R is of type G_{2}, the orbit through

H = α_{1} + \frac{α_{2}}{\sqrt{3}}

is the only austere orbit except orbits through a restricted root vector. This orbit is a principal orbit.

Remark 5.9. By the same discussion in Remark 5.7, this principal austere orbit is not a weakly reflective submanifold from Proposition 2.9.

Proof. The fundamental system F of the restricted root system of type G_{2} is given by F = {α_{1} = e_{1} − e_{2}, α_{2} = −2e_{1} + e_{2} + e_{3}} and the set R_{+} of positive roots is

R_{+} = F ∪ \{ \begin{matrix}α_{1} + α_{2} = −e_{1} + e_{3}, 2α_{1} + α_{2} = −e_{2} + e_{3}, \\ 3α_{1} + α_{2} = e_{1} − 2e_{2} + e_{3}, 3α_{1} + 2α_{2} = −e_{1} − e_{2} + 2e_{3} \end{matrix} \} .

In the case of G_{2}, since all restricted roots have constant multiplicities, the condition (5.4) of Lemma 5.3 is always satisfied. From Lemma 5.3, we can put

H = \frac{α}{∥α∥} ± \frac{β}{∥β∥} (α, β ∈ R_{+}).

When α and β are both short roots, we can put H = α±β (α ≠ β). Furthermore, since any short root α can be translated to α_{1} by the action of the Weyl group, we can assume

H = α_{1} ± β where β = α_{1} + α_{2}, 2α_{1} + α_{2}.

In the case of β = α_{1} + α_{2},

H = α_{1} ± (α_{1} + α_{2}) = 2α_{1} + α_{2}, −α_{2}.

Then H is a root vector. In the case of β = 2α_{1} + α_{2},

H = α_{1} ± (2α_{1} + α_{2}) = 3α_{1} + α_{2}, −α_{1} − α_{2}.

Then H is a root vector.

When α and β are both long roots, we can put H = α ± β (α ≠ β). Since any long root α can be translated to α_{2} by the action of the Weyl group, we can assume

H = α_{2} ± β where β = 3α_{1} + α_{2}, 3α_{1} + 2α_{2}.

In the case of β = 3α_{1} + α_{2},

H = α_{2} ± (3α_{1} + α_{2}) = −3α_{1}, 3α_{1} + 2α_{2}.

Then H is a scalar multiple of a root vector. In the case of β = 3α_{1} + 2α_{2},

H = α_{2} ± (3α_{1} + 2α_{2}) = −3α_{1} − α_{2}, 3(α_{1} + α_{2}).

Then H is a scalar multiple of a root vector.

When α is a short root and β is a long root, we can assume α = α_{1} and

H = α_{1} ± \frac{β}{\sqrt{3}} where β = α_{2}, 3α_{1} + α_{2}, 3α_{1} + 2α_{2}.

We note that the orbit though H is a principal orbit. In these cases, H is equivalent to a scalar multiple of α_{1} + \frac{α_{2}}{\sqrt{3}} by the action of the Weyl group. In the case of H = α_{1} + \frac{α_{2}}{\sqrt{3}},

H = \frac{1}{\sqrt{3} + 1} (α_{1} + α_{2} + \frac{3α_{1} + α_{2}}{\sqrt{3}} ) = \frac{1}{\sqrt{3} + 2} (2α_{1} + α_{2} + \frac{3α_{1} + 2α_{2}}{\sqrt{3}} ) .

Thus from Lemma 5.3 this orbit is austere. This completes the proof. □

Proposition 5.10. In the case of R = F_{4}, there are no austere orbits except orbits through a restricted root vector.

Proof. In this case R_{+} is given by

R_{+} = \{e_{i}\}_{1≤i≤4} ∪ \{e_{i} ± e_{j}\}_{1≤i<j≤4} ∪ \{\frac{1}{2}(e_{1} ± e_{2} ± e_{3} ± e_{4})\}

From Lemma 5.3, we can assume

H = \frac{α}{∥α∥} ± \frac{β}{∥β∥} (α, β ∈ R_{+}).

When α and β are both short roots, we can put H = α ± β. In this case, since any short root can be translated to e_{1} by the action of the Weyl group, we can put

H = e_{1} ± β where β = e_{i} (i ≥ 2), \frac{1}{2}(e_{1} ± e_{2} ± e_{3} ± e_{4}).

In the case of β = e_{i}, H is a root vector. In the case of β = \frac{1}{2}(e_{1} ± e_{2} ± e_{3} ± e_{4}).

H = e_{1} ± \frac{1}{2}(e_{1} ± e_{2} ± e_{3} ± e_{4}) = \{ \begin{matrix}e_{1} + \frac{1}{2}(e_{1} ± e_{2} ± e_{3} ± e_{4}), \\ \frac{1}{2}(e_{1} ± e_{2} ± e_{3} ± e_{4}) (a root). \end{matrix}

Thus we consider the case of H = e_{1} + \frac{1}{2}(e_{1} ±e_{2} ±e_{3} ±e_{4}). In this case, ⟨e_{4}, H⟩ ≠ 0, however, there does not exist β ∈ R_{+} such that

H = n (e_{4} ± \frac{β}{∥β∥}) .

Hence from Lemma 5.3 this orbit is not austere.

When α and β are both long root, we can assume H = e_{1} + e_{2} ± e_{i} ± e_{j} (i < j). Moreover, we exclude H which is a scalar multiple of a root vector. Then

H = 2e_{1} + e_{2} ± e_{i}, e_{1} + 2e_{2} ± e_{i} (i = 3, 4).

The reflection s_{e_{3}−e_{4}} permutes e_{3} and e_{4}, and fixes e_{1}, e_{2}. Therefore we can put

H = 2e_{1} + e_{2} ± e_{3}.

In this case, ⟨e_{3}, H⟩ ≠ 0, however, there does not exist β ∈ R_{+} such that

H = n (e_{3} ± \frac{β}{∥β∥}) .

Hence this orbit is not austere.

When α is a short root and β is a long root, we can put

H = e_{1} + \frac{±e_{i} ± e_{j}}{\sqrt{2}} (i < j).

Moreover, by the action of the Weyl group we can assume

H = e_{1} + \frac{±e_{1} + e_{2}}{\sqrt{2}} or H = e_{1} + \frac{e_{2} + e_{3}}{\sqrt{2}} .

In the case of H = e_{1} + \frac{e_{2}+e_{3}}{\sqrt{2}} , we have ⟨e_{3}, H⟩ ≠ 0. However, there does not exist β ∈ R_{+} such that

H = n (e_{3} ± \frac{β}{∥β∥}) .

Hence this orbit is not austere. In the case of H = e_{1} + \frac{±e_{1}+e_{2}}{\sqrt{2}} , we have ⟨e_{2} + e_{3}, H⟩ ≠ 0. However, there does not exist β ∈ R_{+} such that

H = n (\frac{e_{2} + e_{3}}{\sqrt{2}} + \frac{β}{∥β∥}) .

Hence this orbit is not austere. □

Proposition 5.11. In the case of R = E_{8}, there are no austere orbits except the orbits through a restricted root vector.

Proof. In the case of R = E_{8}, R_{+} is given by

R_{+} = \{±e_{i} + e_{j} | 1 ≤ i < j ≤ 8\} ∪ \{ \frac{1}{2}(e_{8} + ∑_{i=1}^{7}(−1)^{ν(i)}e_{i}) | ∑_{i=1}^{7} ν(i) ∈ 2Z\}

Since all restricted roots have constant multiplicities, the condition (5.4) of Lemma 5.3 is always satisfied. From Lemma 5.3, we can put H = e_{1} + e_{2} + β where

β = \{ \begin{matrix}±e_{1} ± e_{2}, \\ ±e_{1} ± e_{i} & (3 ≤ i ≤ 8), \\ ±e_{2} ± e_{i} & (3 ≤ i ≤ 8), \\ ±e_{i} ± e_{j} & (3 ≤ i < j ≤ 8), \\ \frac{1}{2} ∑_{i=1}^{ν(i)}(−1)^{ν(i)}e_{i} & (∑_{i=1}^{8} ν(i) ∈ 2Z). \end{matrix}

i) In the case of β = ±e_{1} ± e_{2},

H = \{ \begin{matrix}e_{1} + e_{2} + e_{1} + e_{2} = 2(e_{1} + e_{2}) (twice of a root), \\ e_{1} + e_{2} + e_{1} − e_{2} = 2e_{1}, \\ e_{1} + e_{2} − e_{1} + e_{2} = 2e_{2}, \\ e_{1} + e_{2} − e_{1} − e_{2} = 0. \end{matrix}

When H = 2e_{1},

R − R^{Δ} = \{±e_{1} ± e_{j} | 2 ≤ j ≤ 8\} ∪ \{±\frac{1}{2}(e_{1} + ∑_{i=2}^{8}(−1)^{ν(i)}e_{i}) | ∑_{i=2}^{8} ν(i) ∈ 2Z\}

Then there does not exist β ∈ R_{+} such that H = n(\frac{1}{2} ∑_{i=1}^{8} e_{i} ± β). Hence this orbit is not austere. When H = 2e_{2}, since the reflection s_{e_{1}−e_{2}} permutes e_{1} and e_{2}, this orbit is equivalent to the orbit through H = 2e_{1}. Hence this orbit is not austere.

ii) In the case of β = ±e_{1} ± e_{i} (3 ≤ i ≤ 8),

H = \{ \begin{matrix}2e_{1} + e_{2} ± e_{i}, \\ e_{2} ± e_{i} (a root). \end{matrix}

The reflection s_{e_{3}−e_{i}} (i ≥ 4) fixes e_{1}, e_{2} and permutes e_{3} and e_{i}. Thus the orbit through H = 2e_{1}+e_{2}±e_{i} is equivalent to the orbit through H = 2e_{1}+e_{2}±e_{3}. Then ⟨e_{1}+e_{4}, H⟩ ≠ 0, however, there does not exist β ∈ R_{+} such that H = n(e_{1}+e_{4}±β). Hence this orbit is not austere.

iii) In the case of β = ±e_{2} ± e_{i} (3 ≤ i ≤ 8),

H = \{ \begin{matrix}e_{1} + 2e_{2} ± e_{i}, \\ e_{1} ± e_{i} (a root). \end{matrix}

By the action of the Weyl group, the orbit through H = e_{1} + 2e_{2} ± e_{i} is equivalent to the orbit through H = 2e_{1} + e_{2} ± e_{i}. Hence this orbit is not austere.

iv) In the case of β = ±e_{i} ± e_{j} (3 ≤ i < j ≤ 8), we can assume

H = e_{1} + e_{2} ± e_{3} ± e_{4}.

Then ⟨e_{1} + e_{5}, H⟩ ≠ 0, however, there does not exist β ∈ R_{+} such that H = n(e_{1} + e_{5} ± β). Thus this orbit is not austere. v) v-1) In the case of

β = \frac{1}{2}(−e_{1} − e_{2} + ∑_{i=3}^{8}(−1)^{ν(i)}e_{i}), ∑_{i=3}^{8} ν(i) ∈ 2Z,

then

H = \frac{1}{2}(e_{1} + e_{2} + ∑_{i=3}^{8}(−1)^{ν(i)}e_{i}) (a root).

v-2) In the case of

β = \frac{1}{2}(−e_{1} + e_{2} + ∑_{i=3}^{8}(−1)^{ν(i)}e_{i}), ∑_{i=3}^{8} ν(i) ∈ 2Z + 1,

then

H = \frac{1}{2}(e_{1} + 3e_{2} + ∑_{i=3}^{8}(−1)^{ν(i)}e_{i}) ∼ \frac{1}{2}(3e_{1} + e_{2} + ∑_{i=3}^{8}(−1)^{ν(i)}e_{i}).

v-3) In the case of

β = \frac{1}{2}(e_{1} − e_{2} + ∑_{i=3}^{8}(−1)^{ν(i)}e_{i}), ∑_{i=3}^{8} ν(i) ∈ 2Z + 1,

then

H = \frac{1}{2}(3e_{1} + e_{2} + ∑_{i=3}^{8}(−1)^{ν(i)}e_{i}).

In this case, there exists an i such that ν(i) = 1. Permuting e_{i} and e_{3} by the action of the Weyl group, we have

H = \frac{1}{2}(3e_{1} + e_{2} − e_{3} + ∑_{i=4}^{8}(−1)^{ν(i)}e_{i}), ∑_{i=4}^{8} ν(i) ∈ 2Z.

Then ⟨e_{2} − e_{3}, H⟩ ≠ 0, however, there does not exist β ∈ R_{+} such that H = n(e_{2} − e_{3} ± β). Thus this orbit is not austere.

vi) In the case of

β = \frac{1}{2}(e_{1} + e_{2} + ∑_{i=3}^{8}(−1)^{ν(i)}e_{i}), ∑_{i=3}^{8} ν(i) ∈ 2Z,

then

H = \frac{1}{2}(3e_{1} + 3e_{2} + ∑_{i=3}^{8}(−1)^{ν(i)}e_{i}).

In this case ⟨e_{1} + e_{3}, H⟩ ≠ 0, however, there does not exist β ∈ R_{+} such that H = n(e_{1} + e_{3} ± β). Thus this orbit is not austere. □

Proposition 5.12. In the case of R = E_{7}, there are no austere orbits except the orbits through a restricted root vector.

Proof. In the case of R = E_{7}, all restricted roots have constant multiplicities. Thus the condition (5.4) of Lemma 5.3 is always satisfied. 𝔞 = {∑_{i=1}^{8} ξ_{i}e_{i} | ξ_{8} = −ξ_{7}} and

R_{+} = {±e_{i} + e_{j} | 1 ≤ i < j ≤ 6} ∪ {e_{8} − e_{7}}

∪ \{ \frac{1}{2}(e_{8} − e_{7} + ∑_{i=1}^{6}(−1)^{ν(i)}e_{i}) | ∑_{i=1}^{6} ν(i) ∈ 2Z + 1\}

From Lemma 5.3, we can assume H = e_{7} − e_{8} + β where

β = \{ \begin{matrix}±e_{i} ± e_{j} (1 ≤ i < j ≤ 6), \\ ±\frac{1}{2}(e_{7} − e_{8} + ∑_{i=1}^{6}(−1)^{ν(i)}e_{i}). \end{matrix}

In the case of β = ±e_{i} ± e_{j} (1 ≤ i < j ≤ 6), we take k such that 1 ≤ k ≤ 6 and k ≠ i, j. Then e_{j} + e_{k} ∈ R_{+} and ⟨H, e_{j} + e_{k}⟩ ≠ 0, however, there does not exist α ∈ R such that H = n(e_{j} + e_{k} + α). Thus this orbit is not austere.

In the case of β = ±\frac{1}{2}(e_{7} − e_{8} + ∑_{i=1}^{6}(−1)^{ν(i)}e_{i}) then

H = \{ \begin{matrix}\frac{1}{2}(3e_{7} − 3e_{8} + ∑_{i=1}^{6}(−1)^{ν(i)}e_{i}), \\ \frac{1}{2}(e_{7} − e_{8} − ∑_{i=1}^{6}(−1)^{ν(i)}e_{i}) (a root). \end{matrix}

Therefore it suffices to consider the case of H = \frac{1}{2}(3e_{7} − 3e_{8} + ∑_{i=1}^{6}(−1)^{ν(i)}e_{i}). In this case, either e_{1} + e_{2} or e_{1} − e_{2} is an element of R_{+} − R_{+}^{Δ}. Denote this element by α. Then there does not exist β ∈ R_{+} such that H = n(α ± β). Hence this orbit is not austere. This completes the proof. □

Proposition 5.13. In the case of R = E_{6}, there are no austere orbits except orbits through a restricted root vector.

Proof. In the case of R = E_{6}, all restricted roots have constant multiplicities. Thus the condition (5.4) of Lemma 5.3 is always satisfied. 𝔞 = {∑_{i=1}^{8} ξ_{i}e_{i} | ξ_{6} = ξ_{7} = −ξ_{8}} and

R_{+} = {±e_{i} + e_{j} | 1 ≤ i < j ≤ 5}

∪ \{ \frac{1}{2}(e_{8} − e_{7} − e_{6} + ∑_{i=1}^{5}(−1)^{ν(i)}e_{i}) | ∑_{i=1}^{5} ν(i) ∈ 2Z\}

From Lemma 5.3, we can assume H = e_{1} + e_{2} + β where

β = \{ \begin{matrix}±(e_{1} − e_{2}), \\ ±e_{2} ± e_{i} (3 ≤ i ≤ 5), \\ ±e_{i} ± e_{j} (3 ≤ i < j ≤ 5), \\ ±\frac{1}{2}(e_{8} − e_{7} − e_{6} + ∑_{i=1}^{5}(−1)^{ν(i)}e_{i}). \end{matrix}

i) In the case of β = ±(e_{1} − e_{2}), then H = 2e_{1}, 2e_{2}. For

α = \frac{1}{2}(e_{8} − e_{7} − e_{6} + e_{1} + e_{2} + e_{3} − e_{4} − e_{5}) ∈ R_{+} − R_{+}^{Δ}

there does not exist β ∈ R_{+} such that H = n(α±β). Thus this orbit is not austere. ii) In the case of β = ±e_{2} ± e_{i} (3 ≤ i ≤ 5), then

H = e_{1} + e_{2} ± e_{2} ± e_{i} = \{ \begin{matrix}e_{1} + 2e_{2} ± e_{i}, \\ e_{1} ± e_{i} (root). \end{matrix}

Therefore it suffices to consider the case of H = e_{1} + 2e_{2} ± e_{i}. In this case, for j with 3 ≤ j ≤ 5 and j ≠ i, we have e_{1} + e_{j} ∈ R_{+} − R_{+}^{Δ}. However, there does not exist β ∈ R_{+} such that H = n(e_{1} + e_{j} ± β). Thus this orbit is not austere.

iii) In the case of β = ±e_{i} ± e_{j} (3 ≤ i < j ≤ 5), then H = e_{1} + e_{2} ± e_{i} ± e_{j}. For k with 3 ≤ k ≤ 5 and k ≠ i, j, we have e_{1} + e_{k} ∈ R_{+} − R_{+}^{Δ}. However, there does not exist β ∈ R_{+} such that H = n(e_{1} + e_{k} ± β). Thus this orbit is not austere.

iv) In the case of

β = ±\frac{1}{2}(e_{8} − e_{7} − e_{6} + ∑_{i=1}^{5}(−1)^{ν(i)}e_{i}) where ∑_{i=1}^{5} ν(i) ∈ 2Z,

then

H = e_{1} + e_{2} ± \frac{1}{2}(e_{8} − e_{7} − e_{6} + ∑_{i=1}^{5}(−1)^{ν(i)}e_{i}).

Here

H = \{ \begin{matrix}e_{1} + e_{2} + \frac{1}{2}(e_{8} − e_{7} − e_{6} − e_{1} − e_{2} + ∑_{i=3}^{5}(−1)^{ν(i)}e_{i}), \\ e_{1} + e_{2} − \frac{1}{2}(e_{8} − e_{7} − e_{6} + e_{1} + e_{2} + ∑_{i=3}^{5}(−1)^{ν(i)}e_{i}) \end{matrix}

are root vectors.

In the case of

H = \{ \begin{matrix}e_{1} + e_{2} + \frac{1}{2}(e_{8} − e_{7} − e_{6} + e_{1} − e_{2} + ∑_{i=3}^{5}(−1)^{ν(i)}e_{i}), \\ e_{1} + e_{2} + \frac{1}{2}(e_{8} − e_{7} − e_{6} − e_{1} + e_{2} + ∑_{i=3}^{5}(−1)^{ν(i)}e_{i}), \\ e_{1} + e_{2} − \frac{1}{2}(e_{8} − e_{7} − e_{6} + e_{1} − e_{2} + ∑_{i=3}^{5}(−1)^{ν(i)}e_{i}), \\ e_{1} + e_{2} − \frac{1}{2}(e_{8} − e_{7} − e_{6} − e_{1} + e_{2} + ∑_{i=3}^{5}(−1)^{ν(i)}e_{i}), \end{matrix}

we have ⟨e_{1} − e_{2}, H⟩ ≠ 0. However there does not exist β ∈ R_{+} such that H = n(e_{1} − e_{2} ± β). Thus this orbit is not austere.

Finally, in the case of

H = \{ \begin{matrix}e_{1} + e_{2} + \frac{1}{2}(e_{8} − e_{7} − e_{6} + e_{1} + e_{2} + ∑_{i=3}^{5}(−1)^{ν(i)}e_{i}), \\ e_{1} + e_{2} − \frac{1}{2}(e_{8} − e_{7} − e_{6} − e_{1} − e_{2} + ∑_{i=3}^{5}(−1)^{ν(i)}e_{i}) \end{matrix}

we have ⟨e_{1} + e_{3}, H⟩ ≠ 0. However there does not exist β ∈ R_{+} such that H = n(e_{1} + e_{3} ± β). Thus this orbit is not austere. This completes the proof. □

By discussions above, we completed the proof of Theorem 5.1 and Theorem 4.1.

6. Miscellaneous results

In this section, we shall concern with some results on weakly reflective submanifolds besides orbits of s-representations.

Proposition 6.1. Let M_{1} and M_{2} be weakly reflective submanifolds in Riemannian manifolds M̃_{1} and M̃_{2}, respectively. Then M_{1} × M_{2} is a weakly reflective submanifold in M̃_{1} × M̃_{2}.

Proof. We take (x_{1}, x_{2}) ∈ M_{1}×M_{2} and a normal vector (ξ_{1}, ξ_{2}) ∈ T_{(x_{1},x_{2})}^{⊥}(M_{1}×M_{2}). Let σ_{ξ_{1}} and σ_{ξ_{2}} be reflections of M_{1} and M_{2} in M̃_{1} and M̃_{2} with respect to ξ_{1} and ξ_{2}, respectively. Then σ_{ξ_{1}} × σ_{ξ_{2}} is an isometry of M̃_{1} × M̃_{2} and satisfies

(σ_{ξ_{1}} × σ_{ξ_{2}})(x_{1}, x_{2}) = (x_{1}, x_{2}),
d(σ_{ξ_{1}} × σ_{ξ_{2}})_{(x_{1},x_{2})}(ξ_{1}, ξ_{2}) = −(ξ_{1}, ξ_{2}),
(σ_{ξ_{1}} × σ_{ξ_{2}})(M_{1} × M_{2}) = M_{1} × M_{2}.

Thus M_{1} × M_{2} is a weakly reflective submanifold in M̃_{1} × M̃_{2}. □

The following proposition states that the cone over a weakly reflective submanifold in a sphere is also a weakly reflective submanifold in a Euclidean space.

Proposition 6.2. Let M be a weakly reflective submanifold in a unit sphere S^{n−1}(1). Then the cone

C(M) = {tx | t ∈ R, t > 0, x ∈ M}

over M is a weakly reflective submanifold in a Euclidean space R^{n}.

Proof. Fix x ∈ M. We note that for arbitrary t ∈ R (t > 0), we have

T_{tx}^{⊥}(C(M)) = T_{x}^{⊥}M ⊂ T_{x}S^{n−1}(1).

For ξ ∈ T_{x}^{⊥}M, a reflection σ_{ξ} of M with respect to ξ satisfies

σ_{ξ}(x) = x,  (dσ_{ξ})_{x}ξ = −ξ,  σ_{ξ}(M) = M.

Since σ_{ξ} is an isometry of S^{n−1}(1), it can be expressed as an orthogonal matrix. Thus σ_{ξ} acts on R^{n} and satisfies

σ_{ξ}(tx) = tσ_{ξ}(x) = tx, (dσ_{ξ})_{x}ξ = −ξ.

In addition, for x′ ∈ M, t′ ∈ R, t′ > 0, we have

σ_{ξ}(t′x′) = t′σ_{ξ}(x′) ∈ C(M).

Therefore σ_{ξ}(C(M)) = C(M). Hence C(M) is a weakly reflective submanifold in R^{n}. □

Next we shall describe the relationship between weakly reflective submanifolds in an odd dimensional sphere and in a complex projective space.

Proposition 6.3. Let M be a weakly reflective submanifold in the hypersphere S ⊂ C^{n+1}. If M is invariant under the U(1)-action, that is the multiplication of e^{iθ} on C^{n+1}, and if a reflection of M with respect to each normal vector is a unitary transformation, then the image P (M) of M is a weakly reflective submanifold in CP^{n}, where P is the natural projection P : S → CP^{n}.

Proof. By the definition of P, T_{x}(U(1)x) = ker dP_{x} at each point x ∈ S. Thus dP_{x} : T_{x}^{⊥}(U(1)x) → T_{x}(CP^{n}) is an isometric linear isomorphism. Since M is invariant under the U(1)-action, P (M) is a submanifold in CP^{n}. Moreover, since T_{x}(U(1)x) ⊂ T_{x}M for x ∈ M, we have T_{x}^{⊥}(U(1)x) ⊃ T_{x}^{⊥}M. Thus dP_{x} : T_{x}^{⊥}M → T_{p(x)}^{⊥}(P (M)) also gives an isometric linear isomorphism. Let σ_{ξ} denote a reflection of M with respect to ξ ∈ T_{x}^{⊥}M. From the assumption, σ_{ξ} is a unitary transformation of C^{n+1}. Hence σ_{ξ} induces an isometry of CP^{n}. Since σ_{ξ}(x) = x and σ_{ξ}(M) = M, we have σ_{ξ}(P (x)) = P (x) and σ_{ξ}(P (M)) = P (M). In addition we have

dσ_{ξ}(dP_{x}(ξ)) = d(σ_{ξ} ∘ P)_{x}(ξ) = d(P ∘ σ_{ξ})_{x}(ξ)
= dP_{x}σ_{ξ}(ξ) = dP_{x}(−ξ) = −dP_{x}(ξ).

Thus σ_{ξ} is a reflection of P (M) with respect to a normal vector dP_{x}(ξ) at P (x). Hence P (M) is a weakly reflective submanifold in CP^{n}. □

Corollary 6.4. An orbit of the s-representation of an irreducible compact Hermitian symmetric pair through a restricted root vector induces a weakly reflective submanifold in a complex projective space.

Proof. The center of the linear isotropy subgroup of an irreducible compact Hermitian symmetric pair is U(1) ([5]). Thus all orbits are invariant under U(1). Furthermore an orbit in the hypersphere S through a restricted root vector is a weakly reflective submanifold. From the proof of Proposition 4.4, a reflection of an orbit through a restricted root vector with respect to each normal vector is a unitary transformation. Hence, from Proposition 6.3, we have the conclusion. □

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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