Journal of Lie Theory Volume 13 (2003) 535–563 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Compactification of Parahermitian Symmetric Spaces

and its Applications, II: Stratifications and Automorphism Groups

Soji Kaneyuki

Communicated by S. Gindikin

Abstract. A simple parahermitian symmetric space is a symplectic symmetric space of a simple Lie group G with two invariant Lagrangian foliations. Such a symmetric space has a nice G-equivariant compactification. In this paper, we obtain the stratification of the compactification, whose strata are G-orbits. By using this, we determine the automorphism group of the double foliation for each simple parahermitian symmetric space.

Introduction

Let M be a smooth manifold. A pair (F^{±}, ω) is called a parakähler structure (or bi-Lagrangian structure) on M if ω is a symplectic form on M and F^{±} are two Lagrangian foliations. A significant property of parakähler structures is that a coadjoint orbit of a semisimple Lie group is hyperbolic if and only if it admits an invariant parakähler structure ([4]). A symmetric space G/H of a Lie group G is called a parahermitian symmetric space (or bi-Lagrangian symmetric space)([10]) if G/H admits a G-invariant parakähler structure (F^{±}, ω). The simplest example of parahermitian symmetric spaces is the symmetric space SL(2, ℝ)/ℝ^{∗}, realized as the one-sheeted hyperboloid x^{2} + y^{2} − z^{2} = 1 in ℝ^{3} = Lie SL(2, ℝ). The Lagrangian foliations F^{±} are given by the two families of rulings of the hyperboloid. Parahermitian symmetric spaces of semisimple Lie groups were classified and characterized group-theoretically in [10,5]. A semisimple symmetric space G/H is parahermitian if and only if H is an open subgroup of the Levi subgroup of a parabolic subgroup with abelian nilradical. Semisimple parahermitian symmetric spaces G/H are in one-to-one correspondence (up to covering) with semisimple graded Lie algebras (shortly, GLAs) of the 1st kind 𝔤 = 𝔤_{−1} + 𝔤_{0} + 𝔤_{1}, in such a way that 𝔤 = Lie G and 𝔤_{0} = Lie H. For the explicit forms of simple parahermitian symmetric pairs, see the tables in 6.4 and 7.2.

Now let 𝔤 = 𝔤_{−1} + 𝔤_{0} + 𝔤_{1} be a simple GLA, and let M = G/G_{0} be the parahermitian symmetric space corresponding to the symmetric pair (𝔤, 𝔤_{0}), where G is the largest possible open subgroup of the automorphism group of 𝔤 such that

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag G/G_{0} is realized as an Ad 𝔤-orbit in 𝔤. The subgroups U^{±} := G_{0} exp 𝔤_{±1} are the parabolic subgroups with Lie U^{±} = 𝔤_{0} +𝔤_{±1}. The flag manifolds M^{±} = G/U^{±} are symmetric R-spaces. Let r be the rank of M^{±}. Then there are exactly r numbers of 𝔰𝔩(2, ℝ)-triplets in 𝔤 which are pairwise commutative and whose direct sum is expressed as a graded subalgebra 𝔞_{−1} + 𝔞_{0} + 𝔞_{1} in 𝔤 (cf. [8]). One has the root system Δ(𝔤, 𝔞_{0}) of 𝔤 with respect to the abelian subspace 𝔞_{0}. Δ(𝔤, 𝔞_{0}) is of BC_{r} -type or C_{r} -type ([8,1]). We say that G/G_{0} and the GLA 𝔤 are of BC_{r} -type or C_{r} -type, if Δ(𝔤, 𝔞_{0}) is.

A fundamental problem of the geometry of parahermitian symmetric spaces (M = G/H, F^{±}, ω) is to determine the automorphism group Aut(M, F^{±})— the group consisting of diffeomorphisms of M leaving the double foliation F^{±} invariant. The aim of this paper is to settle this problem for an arbitrary simple Lie group G. A partial answer was given by Tanaka [15] under the assumption that G is classical simple. Let us describe our procedure to determine the automorphism group. The first step is to obtain the G-orbit structure of M̃ = M^{−} × M^{+}, which is the natural G-equivariant compactification of M (cf. [6] and Sections 2 and 3). The second step is to show that the G-orbit decomposition gives M̃ a stratification whose strata are G-orbits. This is done in Sections 5 and 6.

The third step is concerned with BC_{r} -type. We now assume that M is of BC_{r} -type. In terms of the root system Δ(𝔤, 𝔞_{0}), we construct a grading 𝔤 = ∑_{k=−2}^{2} 𝔤_{k}(r) of the 2nd kind having the property that 𝔤_{±1}(r) are expressed as the direct sum of two equi-dimensional abelian subspaces, 𝔤_{±1}^{+}(r) + 𝔤_{±1}^{−}(r) (cf. (4.10),(4.14)). Such a grading is called a pseudo-product grading of 𝔤 (Tanaka [15]). Let Q_{r} be the isotropy subgroup of G at the base point of the lowest dimensional G-orbit M_{0}. The third step is to show that Q_{r} is the parabolic subgroup with Lie Q_{r} = 𝔤_{−2}(r)+𝔤_{−1}(r)+𝔤_{0}(r) (Proposition 4.8) and that its Levi subgroup coincides with the automorphism group of the pseudo-product grading (cf. Lemma 7.4 and Remark 7.1). The flag manifold M_{0} = G/Q_{r} has the double foliation F_{0}^{±} induced from the product structure of M̃. We denote by Aut(M_{0}, F_{0}^{±}) the group of diffeomorphisms of M_{0} leaving F_{0}^{±} invariant. Then Tanaka’s theory [15] of Cartan connections for pseudo-product manifolds, together with the third step guarantees the validity of the relation Aut(M_{0}, F_{0}^{±}) = G.

For the case where M is of C_{r} -type, the minimal G-orbit M_{0} coincides with G/U^{−} = M^{−}, and the double foliation F_{0}^{±} becomes trivial. But, in turn, M^{−} has the generalized conformal structure K, which is obtained from the cone defined as the union of singular G_{0}-orbits in 𝔤_{1} ([2]). We determined in [2] the conformal automorphism group Aut(M^{−}, K) for each symmetric R-space M^{−}.

The fourth and the last step is to obtain the injective homomorphism of Aut(M, F^{±}) into Aut(M_{0}, F_{0}^{±}) or into Aut(M^{−}, K). For this purpose, the stratification of M̃ is essential. Let f ∈ Aut(M, F^{±}). Then f extends to M̃ as an automorphism f̃ of the product structure of M̃. It follows that f̃ is an automorphism of the stratification of M̃ (Corollary 6.14). In particular f̃ leaves M_{0} stable. It is shown that the assignment f ↦ f̃|_{M_{0}} gives the injective homomorphism as desired. The main results are Theorems 8.1 and 8.4.

We want to supplement some details on the stratification of M̃, since it is a rather independent topic. By a stratification of a real analytic manifold X, we mean a partition X = ∐_{k=0}^{s} A_{k} which satisfies the following conditions: (S1)

Each A_{k} is an analytic submanifold of X, (S2) the closure \overline{A_{k}} of A_{k} is an analytic set of X and coincides with A_{≤k} := ∐_{i=0}^{k} A_{i} (0 ≤ k ≤ s), and (S3) the singular locus Sing(\overline{A_{k}}) is given by A_{≤k−1} (1 ≤ k ≤ s − 1). Let M̃ = ∐_{k=0}^{r} M_{k} be the G-orbit decomposition (G acts on M̃ diagonally), where dim M_{k} > dim M_{k−1} and M_{r} = M is open dense in M̃. For the G-orbit decomposition of M̃, the properties (S1) and (S2) were already proved in [6]. We will verify (S3) in this paper.

Suppose first that the GLA 𝔤 is of BC_{r} -type. We consider the two abelian subspaces of 𝔤: 𝔤_{1} = 𝔤_{2}(r) + 𝔤_{1}^{−}(r), 𝔤′_{1} := 𝔤_{2}(r) + 𝔤_{1}^{+}(r). The direct sum 𝔤_{1} ⊕ 𝔤′_{1} is imbedded in M̃ as an open subset. We identify 𝔤_{1} ⊕ 𝔤′_{1} with its image in M̃. Let M_{k}^{∗} := M_{k} ∩ (𝔤_{1} ⊕ 𝔤′_{1}), which is open dense in M_{k}. The closure \overline{M_{k}^{∗}} of M_{k}^{∗} in 𝔤_{1} ⊕ 𝔤′_{1} coincides with M_{≤k}^{∗} := ∐_{i=0}^{k} M_{i}^{∗} and it is an algebraic variety in 𝔤_{1} ⊕ 𝔤′_{1} (Theorem 5.14). Obviously we have Sing(M_{≤k}) = G(Sing(M_{≤k}^{∗})). Thus, in order to find the singular locus of M_{≤k}, it is enough to find that of M_{≤k}^{∗}. Now we look at 𝔤_{2}(r). The Levi subgroup L of Q_{r} corresponding to 𝔤_{0}(r) acts on 𝔤_{2}(r). We have a partition 𝔤_{2}(r) = ∐_{k=0}^{r} V_{k}, where V_{k} is a union of equi-dimensional L-orbits with dim V_{k} > dim V_{k−1}. As a conclusion, the above partition is a stratification of 𝔤_{2}(r). The validity of (S1) and (S2) was proved in [8]; as was shown in [8], V_{≤k} (0 ≤ k ≤ r − 1) is an algebraic variety in 𝔤_{2}(r). By Proposition 6.3, the problem of finding the singular locus of M_{≤k}^{∗} is reduced to finding that of V_{≤k}. In the case where the GLA 𝔤 is of C_{r} -type, we have 𝔤_{1} = 𝔤′_{1} = 𝔤_{2}(r). Therefore it is enough to look at the determinantal varieties V_{≤k} in 𝔤_{1} for the case where the GLA 𝔤 is of C_{r} -type. In the realization of 𝔤_{1} as a matrix space, V_{≤k} is a complex or real determinantal variety of classical or exceptional type. The determination of Sing(V_{≤k}) will be carried out in Section 6. Levasseur-Stafford [11] is a good reference for the complex classical case. The final result on the stratification of M̃ is given by Theorem 6.13.

The class of simple parahermitian symmetric spaces of C_{r} -type contains an interesting sub-class of symmetric spaces of Cayley type, which are causal symmetric spaces. For a symmetric space M of Cayley type, M̃ is the causal compactification of M (cf. [12]). As an application of the results of the present paper, one can determine the full causal automorphism group of M. In the forthcoming paper, we will treat this topic.

The author would like to thank MSRI, Berkeley, where most of this work was completed during his stay in the fall, 2001. The author happily express his thanks to Simon Gindikin for frequent valuable conversations.

The paper is organized as follows:

1. Preliminaries on parahermitian symmetric spaces.

2. Double foliation of M.

3. Orbit structure of M̃.

4. Isotropy subgroups for boundary orbits.

5. Siegel-type realization of orbits.

6. Stratification of M̃.

7. Double foliation on the minimal boundary orbit.

8. Determination of automorphism groups of M.

Throughout this paper, a diffeomorphism always means a C^{∞}- diffeomorphism. The group of diffeomorphisms of a smooth (i.e., C^{∞}) manifold M is denoted by Diffeo(M).

1. Preliminaries on parahermitian symmetric spaces

Let M be a connected 2n-dimensional smooth manifold, and let F^{±} be two ndimensional completely integrable distributions on M. (F^{±}) is called a paracomplex structure on M ([10]) if the tangent bundle T M of M can be expressed as the Whitney sum F^{+} ⊕ F^{−}. In this case (M, F^{±}) is called a paracomplex manifold. A paracomplex manifold (M, F^{±}) is called a parakähler manifold ([10]) if there exists a symplectic form ω on M with respect to which F^{±} are Lagrangian subbundles. For a parakähler manifold (M, F^{±}, ω), one can consider the two kinds of automorphisms: By a paracomplex automorphism of M we mean a diffeomorphism of M which leaves F^{±} invariant. By a paracomplex isometry of M we mean a paracomplex automorphism leaving ω invariant. We denote by Aut(M, F^{±}) (resp. Aut(M, F^{±}, ω)) the group of paracomplex automorphisms (resp. paracomplex isometries) of M. The group Aut(M, F^{±}, ω) is always a finite-dimensional Lie group, but Aut(M, F^{±}) is not in general.

Definition 1.1. ([10]). Let M = G/H be an almost effective symmetric coset space of a Lie group G, and let (F^{±}, ω) be a parakähler structure on M. If G acts on M as paracomplex isometries with respect to (F^{±}, ω), then (M = G/H, F^{±}, ω) is called a parahermitian symmetric space.

For each parahermitian symmetric space M = G/H, the Lie algebra 𝔤 = Lie G has the structure of a GLA of the first kind 𝔤 = 𝔤_{−1} + 𝔤_{0} + 𝔤_{1} ([10]). Under the assumption of semisimplicity of 𝔤, the assignment M ⇝ 𝔤 = 𝔤_{−1} + 𝔤_{0} + 𝔤_{1} induces a bijection between the set of local isomorphism classes of parahermitian symmetric spaces and the set of isomorphism classes of effective semisimple GLA of the first kind ([5]). In this case the original parakähler structure on M can be recovered by the grading of 𝔤.

Let us start with a real simple GLA of the first kind

𝔤 = 𝔤_{−1} + 𝔤_{0} + 𝔤_{1}.  (1.1)

The automorphism group of the Lie algebra 𝔤 is denoted by Aut 𝔤. Let (Z, τ) be the associated pair of the GLA: Z is the characteristic element of the GLA, that is, Z is a unique element of 𝔤_{0} satisfying the condition ad Z = k1 on 𝔤_{k}, k = 0, ±1, and τ is a Cartan involution of 𝔤 satisfying τ(Z) = −Z. Let σ = Ad exp(πiZ). Then σ is the involutive automorphism of 𝔤 such that σ = 1 on 𝔤_{0} and −1 on 𝔪 := 𝔤_{−1} + 𝔤_{1}. Thus we have a symmetric triple (𝔤, 𝔤_{0}, σ). Let G_{0} be the centralizer of Z in Aut 𝔤, and let G be the open subgroup of Aut 𝔤 generated by G_{0} and Ad 𝔤. Note that Lie G_{0} = 𝔤_{0} and that G_{0} coincides with the group of grade-preserving automorphisms of the GLA (1.1).

Proposition 1.2. The coset space M = G/G_{0} is a parahermitian symmetric space corresponding to the symmetric triple (𝔤, 𝔤_{0}, σ). The group G acts on M effectively by paracomplex isometries.

Proof. If we put σ̃(a) = σaσ, a ∈ G, then σ̃ is an involutive automorphism of Aut 𝔤. σ̃ leaves G stable. Let Gσ̃ be the subgroup of G consisting of all σ̃-fixed elements of G. Then, from the definition of σ, it follows that G_{0} is an open subgroup of Gσ̃. So M = G/G_{0} is a symmetric space. By the definition of G_{0}, M is realized in 𝔤 as the adjoint G-orbit through Z ∈ 𝔤. Therefore the Kirillov-Kostant form

ω̃(X, Y) = (Z, [X, Y]),  X, Y ∈ 𝔤  (1.2)

induces a G-invariant symplectic form ω on M,where ( , ) denotes the Killing form of 𝔤. Let 0 ∈ M be the origin of M = G/G_{0}. We identify 𝔪 with the tangent space T_{0}M of M at 0. Then the two G_{0}-invariant subspaces 𝔤_{±1} extend to G-invariant distributions F^{±} on M, which are Lagrangian with respect to ω by (1.2). The complete integrability of F^{±} has been proved in two ways, one in [10] by a differential geometric argument, the other by an algebraic method using dipolarizations. To prove effectivity of the G-action on M, first note that the natural G_{0}-action on 𝔪 can be identified with the linear isotropy group at 0 ∈ M = G/G_{0}. Let a ∈ G and suppose that a acts on M as the identity. Then a ∈ G_{0} and a acts on 𝔪 as the identity. Since 𝔤 is simple, we have 𝔤_{0} = [𝔤_{1}, 𝔤_{−1}]. So it follows that a acts on 𝔤_{0} as the identity. This implies that a is the unit element of G.

The parahermitian symmetric space (M = G/G_{0}, F^{±}, ω) thus constructed is called the parahermitian symmetric space associated to a simple GLA (1.1). The parahermitian symmetric space M = G/G_{0}, which is a hyperbolic Ad G-orbit, is the bottom space with respect to the covering relation among the parahermitian symmetric spaces corresponding to the symmetric triple (𝔤, 𝔤_{0}, σ).

2. Double foliations of M

Let (M = G/G_{0}, F^{±}, ω) be the parahermitian symmetric space associated to a simple GLA (1.1). We denote by F^{±}(p) the leaves of F^{±} through a point p ∈ M. It is easy to see that the leaves F^{±}(0) through the origin 0 ∈ M are given by the orbits (exp 𝔤_{±1}) · 0. Consider the parabolic subgroups U^{±} := G_{0} exp 𝔤_{±1} of G. Note that 𝔲^{±} := Lie U^{±} = 𝔤_{0} + 𝔤_{±1}. The flag manifolds G/U^{±} are called the symmetric R-spaces associated with M or with the GLA (1.1).

Lemma 2.1. Let M^{±} be the sets of leaves of F^{±} on M. Then M^{±} are identified with the flag manifolds G/U^{±}.

Proof. Since F^{±} are G-invariant, any element of G induces a permutation on the sets of leaves of F^{±}, which means that G acts on M^{±}. The transitivity of G on M^{±} follows from that of G on M. Now let g ∈ G and suppose for example gF^{+}(0) = F^{+}(0). Then the point g · 0 is in F^{+}(0). One can write g · 0 = exp X · 0 for some X ∈ 𝔤_{1}. Therefore g^{−1} exp X ∈ G_{0}, which implies g ∈ U^{+}, and M^{+} is expressed as G/U^{+}.

Lemma 2.2. For any point p ∈ M, we have F^{+}(p) ∩ F^{−}(p) = {p}.

Proof. One can assume p to be the origin 0. Any point q ∈ F^{+}(0)∩F^{−}(0) can be expressed as q = exp X · 0 = exp Y · 0 for X ∈ 𝔤_{1} and Y ∈ 𝔤_{−1}. This implies that exp X ∈ (exp Y)G_{0} ⊂ U^{−}. Consequently exp X ∈ U^{+} ∩ U^{−} = G_{0} and hence exp X ∈ (exp 𝔤_{1}) ∩ G_{0} = (1), which implies X = 0. Thus we have q = p.

We denote the points F^{±}(0) ∈ M^{±} by 0^{±}, and let us consider the product manifolds

M̃ = M^{−} × M^{+}  (2.1)

with the origin (0^{−}, 0^{+}). M̃ has a double foliation arising from the product structure. The leaves through the point (g_{1}0^{−}, g_{2}0^{+}), g_{1}, g_{2} ∈ G, are denoted by M^{±}(g_{1}0^{−}, g_{2}0^{+}). M^{−}(g_{1}0^{−}, g_{2}0^{+}) is called the horizontal leaf and M^{+}(g_{1}0^{−}, g_{2}0^{+}) is called the vertical leaf. They are given by

\begin{aligned}M^{−}(g_{1}0^{−}, g_{2}0^{+}) & = Gg_{1}0^{−} × \{g_{2}0^{+}\} = G/g_{1}U^{−}g_{1}^{−1} × \{g_{2}0^{+}\}, \\ M^{+}(g_{1}0^{−}, g_{2}0^{+}) & = \{g_{1}0^{−}\} × Gg_{2}0^{+} = \{g_{1}0^{−}\} × G/g_{2}U^{+}g_{2}^{−1}. \end{aligned}  (2.2)

Let us define a map φ of M to M̃ by putting

φ(p) = (F^{−}(p), F^{+}(p)),  p ∈ M.  (2.3)

Lemma 2.3. φ is a G-equivariant open imbedding of M into M̃ and preserves the double foliations on M and M̃; actually we have φ(F^{∓}(p)) ⊂ M^{±}(φ(p)), p ∈ M.

Proof. Let g ∈ G. Then φ(g ·0) = (F^{−}(g ·0), F^{+}(g ·0)) = (g ·0^{−}, g ·0^{+}). From this and Lemma 2.2 it follows that φ is G-equivariant imbedding. The openness of φ follows from dimension counting. Now let q ∈ F^{−}(p). Then F^{−}(q) = F^{−}(p) and hence φ(q) = (F^{−}(p), F^{+}(q)), which implies that φ(q) lies on the vertical leaf through φ(p).

Since U^{−} ∩ U^{+} = G_{0}, M has the structure of the double fibration over M^{±}. The projections π^{±}: M → M^{±} are given by

π^{±}(g · 0) = g · 0^{±},  g ∈ G.  (2.4)

Lemma 2.4. For each point p ∈ M, we have

F^{±}(p) = (π^{±})^{−1}(π^{±}(p)).

Proof. It is enough to prove the assertion for the case where p is the origin. Choose a point (exp X)0 ∈ F^{+}(0), X ∈ 𝔤_{1}. Then we have π^{+}((exp X)0) = (exp X)0^{+} = 0^{+}, and hence (exp X)0 ∈ (π^{+})^{−1}(0^{+}) = (π^{+})^{−1}(π^{+}(0)). Conversely, let p ∈ (π^{+})^{−1}(0^{+}). Then π^{+}(p) = 0^{+}. Consequently p can be written as p = u0, where u ∈ U^{+}. If we write u = (exp Y)h, Y ∈ 𝔤_{1}, h ∈ G_{0}, then we have p = (exp Y)0 ∈ F^{+}(0).

If we denote the projections by ϖ^{±}: M̃ → M^{±}, then we have ϖ^{±} · φ = π^{±} (cf. (2.4)). Therefore, under the identification of M with φ(M), the double fibration of M is the restriction of the trivial double fibration of M̃ to M. Later on we always identify M with its φ-image in M̃. As was seen in the proof of Lemma 2.3, M is an orbit through the origin (0^{−}, 0^{+}) ∈ M̃ under the diagonal G-action.

3. Orbit structure of M̃

We wish to consider the orbit structure of M̃ under the diagonal G-action. We start with a simple GLA (1.1). Recall the decomposition 𝔤 = 𝔤_{0} + 𝔪 by σ (cf. §1). Also we have the Cartan involution τ satisfying τ(Z) = −Z. The property τ(Z) = −Z means that τ is grade-reversing, i.e., τ(𝔤_{k}) = 𝔤_{−k}, k = 0, ±1. Let 𝔤 = 𝔨 + 𝔭 be the Cartan decomposition by τ, where τ = 1 on 𝔨 and −1 on 𝔭. Since σ and τ commute, we have the decomposition

𝔤 = 𝔨_{0} + 𝔪_{𝔨} + 𝔭_{0} + 𝔪_{𝔭},  (3.1)

where 𝔨_{0} = 𝔤_{0} ∩ 𝔨, 𝔪_{𝔨} = 𝔪 ∩ 𝔨, 𝔭_{0} = 𝔤_{0} ∩ 𝔭 and 𝔪_{𝔭} = 𝔪 ∩ 𝔭. Note that Z ∈ 𝔭_{0}. Choose a maximal abelian subspace 𝔞 of 𝔭 such that Z ∈ 𝔞. Then 𝔞 is contained in 𝔭_{0}. Let Δ be the root system of 𝔤 with respect to 𝔞. Let ( , ) denote the Killing form of 𝔤. Then we have the partition of Δ corresponding to the grading of 𝔤:

\begin{aligned}Δ & = Δ_{−1} ⨿ Δ_{0} ⨿ Δ_{1}, \\ Δ_{k} & = \{ α ∈ Δ : (α, Z) = k \},  k = 0, ±1. \end{aligned}  (3.2)

Choose a linear order in Δ in such a way that Δ_{1} ⊂ Δ^{+} ⊂ Δ_{0} ∪ Δ_{1}, where Δ^{+} denotes the positive system of Δ with respect to that order. Then choose a maximal system of strongly orthogonal roots, Γ = {β_{1}, …, β_{r}} in Δ_{1}, such that each β_{i} has the same length and that θ = β_{1} > β_{2} > ⋯ > β_{r}, θ being the highest root in Δ. Here the number r is the split rank of the symmetric pair (𝔤, 𝔤_{0}). Note that r is equal to the rank of the symmetric R-space G/U^{−}.

Moreover choose a root vector E_{i} in the root space 𝔤^{β_{i}} ⊂ 𝔤_{1} (1 ≤ i ≤ r) in such a way that

[E_{i}, E_{−i}] = β̌_{i} = \frac{2}{(β_{i}, β_{i})}β_{i}, 1 ≤ i ≤ r,

where E_{−i} = −τ(E_{i}) ∈ 𝔤^{−β_{i}} ⊂ 𝔤_{−1}. Put X_{i} := E_{i}+E_{−i} ∈ 𝔪_{𝔭} and Y_{i} := E_{i}−E_{−i} ∈ 𝔪_{𝔨} (1 ≤ i ≤ r). Then 𝔠 = ∑_{i=1}^{r} ℝX_{i} is a maximal abelian subspace of 𝔪_{𝔭}. 𝔠 is a split Cartan subalgebra of the symmetric pair (𝔤, 𝔤_{0}). Note that 𝔠 is also a Cartan subalgebra of the noncompact dual of the symmetric R-space G/U^{−}. It is well-known that the root system Δ(𝔤, 𝔠) of 𝔤 with respect to 𝔠 is of C_{r} -type or BC_{r} -type. Correspondingly we say that the GLA (1.1) and the parahermitian symmetric space M = G/G_{0} are of C_{r} -type or BC_{r} -type, respectively. Let 𝔞_{0} be the subspace of 𝔞 spanned by β_{1}, …, β_{r} and ϖ be the orthogonal projection of 𝔞 onto 𝔞_{0} with respect to ( , ). Then either one of the following two cases occurs ([13,6]):

\{ \begin{matrix}ϖ(Δ_{1}) & = \{ \frac{1}{2}(β_{i} + β_{j}) : 1 ≤ i ≤ j ≤ r \}, \\ ϖ(Δ_{0}^{+}) − (0) & = \{ \frac{1}{2}(β_{i} − β_{j}) : 1 ≤ i < j ≤ r \}, \end{matrix}  (3.3)

\{ \begin{matrix}ϖ(Δ_{1}) & = \{\frac{1}{2}(β_{i} + β_{j}) (1 ≤ i ≤ j ≤ r), \frac{1}{2}β_{i} (1 ≤ i ≤ r) \}, \\ ϖ(Δ_{0}^{+}) − (0) & = \{\frac{1}{2}(β_{i} − β_{j}) (1 ≤ i < j ≤ r), \frac{1}{2}β_{i} (1 ≤ i ≤ r)\}, \end{matrix}  (3.4)

according as Δ(𝔤, 𝔠) is of C_{r} -type or BC_{r} -type, respectively. Here Δ_{0}^{+} = Δ_{0}∩Δ^{+}. Now let K be the subgroup of G consisting of elements which commute with τ.

Then K is the maximal compact subgroup of G with Lie K = 𝔨. We denote the identity component of K by K^{0}, and define the elements a_{l} (0 ≤ l ≤ r) in the normalizer N_{K^{0}}(𝔞) of 𝔞 in K^{0} by putting

\{ \begin{matrix}a_{l} & = exp(−\frac{π}{2} ∑_{i=1}^{l} Y_{i}),  1 ≤ l ≤ r, \\ a_{0} & = 1. \end{matrix}  (3.5)

The following theorem gives the G-orbit structure of M̃.

Theorem 3.1.

(i) The points (0^{−}, a_{l} 0^{+}) ∈ M̃, 0 ≤ l ≤ r, are a complete set of representatives of G-orbits in M̃.

(ii) Let M_{l} = G(0^{−}, a_{r−l} 0^{+}), 0 ≤ l ≤ r. Then the closure \overline{M_{l}} of M_{l} in M̃ is given by

\overline{M_{l}} = M_{l} ⨿ M_{l−1} ⨿ ⋯ ⨿ M_{0}, 0 ≤ l ≤ r.

(iii) M_{r} = M is a single open G-orbit, and hence M̃ is a G-equivariant compactification of M.

(iv) A single closed G-orbit M_{0} has the property: If M is of C_{r} -type, then M_{0} = M^{−} and a_{r} U^{+}a_{r}^{−1} = U^{−} holds. If M is of BC_{r} -type, then M_{0} is a flag manifold of the second kind. M_{0} has the double fibration:

G/U^{−} = M^{−}(0^{−}, a_{r} 0^{+}) ⟵ M_{0} ⟶ M^{+}(0^{−}, a_{r} 0^{+}) = G/a_{r} U^{+}a_{r}^{−1}.

Proof. Let us denote by G^{0} the identity component of G. In [6] we proved the theorem for the G^{0}-action. But, by Theorem 4.12 in [6], dim M_{l} is strictly increasing, as l increases. Let g ∈ G. Since g normalizes G^{0}, g(M_{l}) is still a G^{0}-orbit which has the same dimension as M_{l}. Therefore g(M_{l}) = M_{l}. In other words, G leaves each G^{0}-orbit stable.

4. Isotropy subgroups for boundary orbits

We go back to a real simple GLA 𝔤 in (1.1). We wish to construct a certain class of gradings of 𝔤 of the second kind in terms of the subsets Γ_{l} = {β_{1}, …, β_{l}}, 1 ≤ l ≤ r, of Γ. When 𝔤 is of Hermitian type, this type of grading corresponds to the realizations of the bounded symmetric domain (corresponding to 𝔤) as a Siegel domain of the third kind for 1 ≤ l ≤ r − 1 and that of the second (or first) kind for l = r. Let 1 ≤ l ≤ r, and put

Δ_{2}(l) = \{ α ∈ Δ_{1} : ϖ(α) = \frac{1}{2}(β_{i} + β_{j}), 1 ≤ i ≤ j ≤ l \},
Δ_{1}(l) = \{α ∈ Δ : \begin{matrix}ϖ(α) = \frac{1}{2}(β_{i} ± β_{j}), 1 ≤ i ≤ l, l + 1 ≤ j ≤ r, or \\ ϖ(α) = \frac{1}{2}β_{i}, 1 ≤ i ≤ l \end{matrix} \} ,

Δ_{0}(l) = \{α ∈ Δ : \begin{matrix}ϖ(α) = 0, or \\ ϖ(α) = ±\frac{1}{2}(β_{i} − β_{j}), 1 ≤ i < j ≤ l or \\ l + 1 ≤ i < j ≤ r, or \\ ϖ(α) = ±\frac{1}{2}(β_{i} + β_{j}), l + 1 ≤ i ≤ j ≤ r, or \\ ϖ(α) = ±\frac{1}{2}β_{i}, l + 1 ≤ i ≤ r \end{matrix}\} ,

Δ_{−1}(l) = −Δ_{1}(l),
Δ_{−2}(l) = −Δ_{2}(l).

(4.1)

Then, for a fixed 1 ≤ l ≤ r, we have a partition of Δ:

Δ = ∐_{k=−2}^{2} Δ_{k}(l).  (4.2)

By using (3.3) and (3.4) we easily have

Proposition 4.1. Let 1 ≤ l ≤ r, and let 𝔠(𝔞) be the centralizer of 𝔞 in 𝔤. If we put

\begin{aligned}𝔤_{0}(l) & = 𝔠(𝔞) + ∑_{α∈Δ_{0}(l)} 𝔤^{α}, \\ 𝔤_{k}(l) & = ∑_{α∈Δ_{k}(l)} 𝔤^{α},  k = ±1, ±2, \end{aligned}  (4.3)

then we have the grading of 𝔤 of the second kind

𝔤 = ∑_{k=−2}^{2} 𝔤_{k}(l),  (4.4)

whose characteristic element is Z_{l} = ∑_{k=1}^{l} β̌_{i}.

Remark 4.2. Gyoja and Yamashita[3] obtained the above gradings for 𝔤 complex simple, in which case there are no roots α ∈ Δ such that ϖ(α) = 0.

Let s_{β_{i}} be the reflection on 𝔞 corresponding to the root β_{i} (1 ≤ i ≤ r), and let s_{l} = s_{β_{1}}s_{β_{2}} … s_{β_{l}} (1 ≤ l ≤ r) and s_{0} = 1. It is known [13] that Ad_{𝔞} a_{l} = s_{l}, 0 ≤ l ≤ r. Let Q_{l} (0 ≤ l ≤ r) be the isotropy subgroup of G at (0^{−}, a_{l} 0^{+}). Then the G-orbit M_{r−l} can be expressed as

M_{r−l} = G/Q_{l}, 0 ≤ l ≤ r,

where Q_{l} = U^{−} ∩ a_{l} U^{+}a_{l}^{−1}. The Lie algebra 𝔮_{l} := Lie Q_{l} can be written as

𝔮_{l} = 𝔠(𝔞) + ∑_{α∈Ψ_{l}} 𝔤^{α},  0 ≤ l ≤ r,  (4.5)

where

Ψ_{l} := { α ∈ Δ_{0} ∪ Δ_{−1} : s_{l}(α) ∈ Δ_{0} ∪ Δ_{1} },  0 ≤ l ≤ r.  (4.6)

Lemma 4.3.

Ψ_{l} = Δ_{0} ∩ Δ_{0}(l) ⨿ Δ_{−1}(l) ⨿ Δ_{−2}(l).  (4.7)

Proof. First we will show the inclusion ⊃ in (4.7). Let α ∈ Δ. Then we have

\begin{aligned}(s_{l}(α), Z) & = (α, Z) − ∑_{k=1}^{l}(α, β̌_{k})(β_{k}, Z) \\ & = (α, Z) − ∑_{k=1}^{l} 2(ϖ(α), β_{k})(β_{k}, β_{k})^{−1}. \end{aligned}  (4.8)

Now let α ∈ Δ_{0} ∩ Δ_{0}(l). By using (4.8) it follows from (3.4) and (4.1) that (s_{l}(α), Z) = 0, or equivalently s_{l}(α) ∈ Δ_{0} and hence α ∈ Ψ_{l}. Suppose next that α ∈ Δ_{−1}(l). Then, by (4.1), there are three possibilities: ϖ(α) = −\frac{1}{2}(β_{i} + β_{j}) or −\frac{1}{2}(β_{i} − β_{j}) for 1 ≤ i ≤ l, l + 1 ≤ j ≤ r, or ϖ(α) = −\frac{1}{2}β_{i} for 1 ≤ i ≤ l. In view of (3.4) and (4.8), we have α ∈ Δ_{−1} and (s_{l}(α), Z) = 0 for the first case, and α ∈ Δ_{0} and (s_{l}(α), Z) = 1 for the second case. For the third case, there are two possibilities (cf. (3.4)): α ∈ Δ_{0} or α ∈ Δ_{−1}. Then we have from (4.8) that (s_{l}(α), Z) = 1 or 0, according as α ∈ Δ_{0} or α ∈ Δ_{−1}, respectively. Consequently s_{l}(α) ∈ Δ_{0} ∪ Δ_{1} for α ∈ Δ_{−1}(l). Suppose α ∈ Δ_{−2}(l). Then by (3.4) and (4.8) we have α ∈ Δ_{−1} and (s_{l}(α), Z) = 1.

To prove the converse inclusion ⊂ in (4.7), let α ∈ Ψ_{l} and suppose that α does not belong to the right-hand side of (4.7). Then the following three cases occur:

(i) α ∈ Δ_{2}(l),

(ii) α ∈ Δ_{1}(l) and

(iii) α ∈ Δ_{0}(l) − Δ_{0}.

For (i), we have α ∈ Δ_{1}, contradicting the assumption that α ∈ Ψ_{l}. For (ii), we have three possibilities: ϖ(α) = \frac{1}{2}(β_{i} + β_{j}) or \frac{1}{2}(β_{i} − β_{j}) both for 1 ≤ i ≤ l, l + 1 ≤ j ≤ r or ϖ(α) = \frac{1}{2}β_{i} for 1 ≤ i ≤ l. For the first case we have α ∈ Δ_{1}, which contradicts α ∈ Ψ_{l}. For the second case, we have α ∈ Δ_{0}. Consequently, by using (4.8), we have that (s_{l}(α), Z) = −1, that is, s_{l}(α) ∈ Δ_{−1}. This contradicts the assumption α ∈ Ψ_{l}. For the third case, we have either α ∈ Δ_{1} or α ∈ Δ_{0}. In view of the condition α ∈ Ψ_{l}, we have the only choice α ∈ Δ_{0}, in which case (s_{l}(α), Z) = −1, still contradicting the assumption α ∈ Ψ_{l}. Let us consider the case (iii) finally. Since α lies in Ψ_{l} ∩ (Δ_{0}(l) − Δ_{0}), we have that ϖ(α) = −\frac{1}{2}(β_{i} + β_{j}), l + 1 ≤ i ≤ j ≤ r, or ϖ(α) = −\frac{1}{2}β_{i}, l + 1 ≤ i ≤ r. In particular α ∈ Δ_{−1}. Therefore, in both cases, we have (s_{l}(α), Z) = −1, which contradicts the assumption α ∈ Ψ_{l}.

Proposition 4.4. The isotropy subalgebra 𝔮_{l} of 𝔤 at the point (0^{−}, a_{l} 0^{+}) is given by

𝔮_{l} = 𝔤_{−2}(l) + 𝔤_{−1}(l) + 𝔤_{0}(l) ∩ 𝔤_{0}, 0 ≤ l ≤ r.

Proof. This follows immediately from Lemma 4.3 and (4.5).

In this paragraph we will determine the isotropy subgroup Q_{r} of G at the point (0^{−}, a_{r} 0^{+}) ∈ M_{0}. Let us consider the special case l = r. Then (4.1) has the following simple form:

Δ_{2}(r) = \{ α ∈ Δ_{1} : ϖ(α) = \frac{1}{2}(β_{i} + β_{j}), 1 ≤ i ≤ j ≤ r \},
Δ_{1}(r) = \{ α ∈ Δ_{0}^{+} ∪ Δ_{1} : ϖ(α) = \frac{1}{2}β_{i}, 1 ≤ i ≤ r \},
Δ_{0}(r) = \{ α ∈ Δ : ϖ(α) = 0 or = ±\frac{1}{2}(β_{i} − β_{j}), 1 ≤ i < j ≤ r \},
Δ_{−k}(r) = −Δ_{k}(r),  k = 1, 2.

(4.9)

We have the grading of the second kind

𝔤 = ∑_{k=−2}^{2} 𝔤_{k}(r).  (4.10)

Remark 4.5. In the case of C_{r} -type, we have Δ_{1}(r) = Ø, Δ_{2}(r) = Δ_{1} and Δ_{0}(r) = Δ_{0}. Hence the grading (4.10) is reduced to the grading (1.1).

Lemma 4.6. 𝔮_{r} = 𝔤_{−2}(r) + 𝔤_{−1}(r) + 𝔤_{0}(r). In particular, 𝔮_{r} is a parabolic subalgebra of 𝔤 of the second kind.

Proof. Since Δ_{0}(r) ⊂ Δ_{0}, we have the inclusion 𝔤_{0}(r) ⊂ 𝔤_{0}.

Now we put

\begin{aligned}Δ_{1}^{+}(r) & = Δ_{1}(r) ∩ Δ_{0}^{+}, \\ Δ_{−1}^{+}(r) & = Δ_{−1}(r) ∩ Δ_{−1}, \end{aligned}  \begin{aligned}Δ_{1}^{−}(r) & = Δ_{1}(r) ∩ Δ_{1}, \\ Δ_{−1}^{−}(r) & = Δ_{−1}(r) ∩ Δ_{0}^{−}. \end{aligned}  (4.11)

Then we have

Δ_{±1}(r) = Δ_{±1}^{+}(r) ⨿ Δ_{±1}^{−}(r).  (4.12)

We define the following four subspaces of 𝔤:

𝔤_{±1}^{+}(r) = ∑_{α∈Δ_{±1}^{+}(r)} 𝔤^{α},  𝔤_{±1}^{−}(r) = ∑_{α∈Δ_{±1}^{−}(r)} 𝔤^{α}.  (4.13)

Those four subspaces are equi-dimensional and abelian ([6]). We have the decompositions

𝔤_{1}(r) = 𝔤_{1}^{−}(r) + 𝔤_{1}^{+}(r),  𝔤_{−1}(r) = 𝔤_{−1}^{+}(r) + 𝔤_{−1}^{−}(r).  (4.14)

The original grading (1.1) of 𝔤 can be reconstructed as

\begin{aligned}𝔤_{−1} & = 𝔤_{−2}(r) + 𝔤_{−1}^{+}(r), \\ 𝔤_{0} & = 𝔤_{−1}^{−}(r) + 𝔤_{0}(r) + 𝔤_{1}^{+}(r), \\ 𝔤_{1} & = 𝔤_{1}^{−}(r) + 𝔤_{2}(r). \end{aligned}  (4.15)

Let C(Z_{r}) be the centralizer of Z_{r} in Aut 𝔤. Then the normalizer N(𝔮_{r}) in Aut 𝔤 of 𝔮_{r} can be written as

N(𝔮_{r}) = C(Z_{r}) · exp(𝔤_{−2}(r) + 𝔤_{−1}(r)). (semi-direct)  (4.16)

Q_{r} is a subgroup of U^{−} ∩ N(𝔮_{r}) = N_{U^{−}}(𝔮_{r}), the normalizer of 𝔮_{r} in U^{−}. (4.15) implies that

N_{U^{−}}(𝔮_{r}) = (C(Z_{r}) ∩ U^{−}) · exp(𝔤_{−2}(r) + 𝔤_{−1}(r)).  (4.17)

Lemma 4.7. Let C(Z, Z_{r}) be the centralizer of both elements Z and Z_{r} in Aut 𝔤. Then we have C(Z_{r}) ∩ U^{−} = C(Z, Z_{r}).

Proof. Let a ∈ C(Z_{r}) ∩ U^{−}. We write a = b exp X, b ∈ C(Z), X ∈ 𝔤_{−1}. Then [X, Z_{r}] ∈ 𝔤_{−1} and hence [X, [X, Z_{r}]] = 0. Therefore we have

Z_{r} = (Ad a)Z_{r} = (Ad b)(Ad exp X)Z_{r} = (Ad b)Z_{r} + (Ad b)[X, Z_{r}] ⊂ 𝔤_{0} + 𝔤_{−1}.

Hence (Ad b)[X, Z_{r}] = 0. Ad b being invertible on 𝔤_{−1}, we have [X, Z_{r}] = 0. Consequently X ∈ 𝔤_{0}(r) ∩ 𝔤_{−1} = (0), and a = b ∈ C(Z).

Proposition 4.8. The isotropy subgroup Q_{r} of G at (0^{−}, a_{r} 0^{+}) ∈ M_{0} is given by

Q_{r} = C(Z, Z_{r}) exp(𝔤_{−2}(r) + 𝔤_{−1}(r)).

Proof. By Lemma 4.5 and (4.17) we have

N_{U^{−}}(𝔮_{r}) = C(Z, Z_{r}) exp(𝔤_{−2}(r) + 𝔤_{−1}(r)).

Recall Q_{r} ⊂ N_{U^{−}}(𝔮_{r}). To prove the converse inclusion, it suffices to show that C(Z, Z_{r}) ⊂ Q_{r}. By using (4.8), one sees s_{r}(Z) = Z − Z_{r}, and consequently

a_{r}^{−1}C(Z)a_{r} = C((Ad a_{r}^{−1})Z) = C(s_{r}(Z)) = C(Z − Z_{r}),
a_{r}^{−1}C(Z_{r})a_{r} = C((Ad a_{r}^{−1})Z_{r}) = C(s_{r}(Z_{r})) = C(Z_{r}).

As a result, a_{r}^{−1}C(Z, Z_{r})a_{r} = C(Z − Z_{r}) ∩ C(Z_{r}) = C(Z, Z_{r}) ⊂ U^{+}. Hence we have C(Z, Z_{r}) ⊂ U^{−} ∩ a_{r} U^{+}a_{r}^{−1} = Q_{r}.

Corollary 4.9. G = C(Z, Z_{r})G^{0}.

Proof. We have M_{0} = G/Q_{r} = G^{0}/Q_{r} ∩ G^{0} = G^{0}Q_{r}/Q_{r}, which implies G = Q_{r}G^{0} = C(Z, Z_{r})G^{0}.

Corollary 4.10. Suppose that M is of C_{r} -type. Then Q_{r} = U^{−} = a_{r} U^{+}a_{r}^{−1} and M_{0} = G/U^{−} = M^{−}.

Proof. By Remark 4.5 and Proposition 4.8, we see that Q_{r} = C(Z) exp 𝔤_{−1} = U^{−}, and hence U^{−} = Q_{r} = U^{−} ∩ a_{r} U^{+}a_{r}^{−1} ⊂ a_{r} U^{+}a_{r}^{−1}. Hence we have U^{−} = a_{r} U^{+}a_{r}^{−1}.

5. Siegel-type realization of orbits

Lemma 5.1.

s_{r}(Δ_{k}(r)) = Δ_{−k}(r),  k = 0, ±1, ±2,  (5.1)
s_{r}(Δ_{−1}^{±}(r)) = Δ_{1}^{±}(r).  (5.2)

Proof. Let α ∈ Δ_{k}(r). Then (s_{r}(α), Z_{r}) = (α, s_{r}(Z_{r})) = −(α, Z_{r}) = −k, which implies that s_{r}(α) ∈ Δ_{−k}(r). Let α ∈ Δ_{−1}^{+}(r). Then, by (4.8) we have (s_{r}(α), Z) = 0, and hence s_{r}(α) ∈ Δ_{0}. One can write ϖ(α) = −\frac{1}{2}β_{i} for some i. Hence we have ϖ(s_{r}(α)) = s_{r}ϖ(α) = s_{r}(−\frac{1}{2}β_{i}) = \frac{1}{2}β_{i}, proving that s_{r}(α) ∈ Δ_{1}^{+}(r).

Lemma 5.2. The operator Ad a_{r} is grade-reversing with respect to the grading (4.10). Moreover Ad a_{r} interchanges 𝔤_{−1}^{±}(r) with 𝔤_{1}^{±}(r), respectively.

Proof. Since Ad a_{r} induces s_{r} on 𝔞 (cf. 4.1), the lemma is immediate from Lemma 5.1.

Up to the present, we have expressed M̃ as M^{−} × M^{+}. Here M^{±} are just the leaves of the product foliation through the origin (0^{−}, 0^{+}). In order to get the Siegel-type realization of G-orbits, we choose the point (0^{−}, a_{r} 0^{+}) ∈ M_{0} as the new origin of M̃. Then M̃ can be expressed as

M̃ = M^{−}(0^{−}, a_{r} 0^{+}) × M^{+}(0^{−}, a_{r} 0^{+}) = G/U^{−} × G/a_{r} U^{+}a_{r}^{−1}.  (5.3)

For simplicity we write (M^{+})_{r} for G/a_{r} U^{+}a_{r}^{−1}. We identify the tangent space T_{0^{−}}(G/U^{−}) with 𝔤_{1} = 𝔤_{2}(r) + 𝔤_{1}^{−}(r) (cf. (4.15)), and T_{0^{+}}(G/U^{+}) with 𝔤_{−1} = 𝔤_{−2}(r) + 𝔤_{−1}^{+}(r). Then the tangent space T_{a_{r} 0^{+}}(G/a_{r} U^{+}a_{r}^{−1}) can be identified with (Ad a_{r})𝔤_{−1} = 𝔤_{2}(r) + 𝔤_{1}^{+}(r) by Lemma 5.2. We will denote (Ad a_{r})𝔤_{−1} by 𝔤′_{1}. Let us consider the exterior direct sum of the vector spaces 𝔤_{1} and 𝔤′_{1}

𝔤_{1} ⊕ 𝔤′_{1} = (𝔤_{2}(r) + 𝔤_{1}^{−}(r)) ⊕ (𝔤_{2}(r) + 𝔤_{1}^{+}(r)).  (5.4)

We define the map ξ of 𝔤_{1} ⊕ 𝔤′_{1} into M̃ = M^{−} × (M^{+})_{r} by

ξ(X, X′) = ((exp X)0^{−}, (exp X′)a_{r} 0^{+}).  (5.5)

Then ξ is an open dense imbedding of 𝔤_{1} ⊕ 𝔤′_{1}. We will always identify 𝔤_{1} ⊕ 𝔤′_{1} with its ξ -image.

Lemma 5.3. Let a_{l}^{±} = exp(∑_{i=1}^{l} E_{±i}) (1 ≤ l ≤ r). Then

a_{r}^{−1}(a_{l}^{−})^{−1}a_{r} = a_{l}^{+},  1 ≤ l ≤ r.  (5.6)

Proof. Consider the elements in 𝔰𝔩(2, ℝ)

e_{+} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , e_{−} = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) .

By an easy computation we have

exp(\frac{π}{2} (e_{+} − e_{−})) exp(−e_{−}) exp(−\frac{π}{2} (e_{+} − e_{−})) = exp e_{+}.

Let φ_{i}: 𝔰𝔩(2, ℝ) → 𝔤 (1 ≤ i ≤ r) be the maps defined by φ_{i}(e_{±}) = E_{±i}. By the strong orthogonality of the β_{i}, we have [φ_{i}, φ_{j}] = 0 (i ≠ j). φ_{i} can be extended to the homomorphism of SL(2, ℝ) to G, denoted again by φ_{i}.

LHS of (5.6)

= exp(\frac{π}{2} ∑_{i=1}^{r}(φ_{i}(e_{+}) − φ_{i}(e_{−}))) exp(− ∑_{i=1}^{l} φ_{i}(e_{−})) exp(−\frac{π}{2} ∑_{i=1}^{r}(φ_{i}(e_{+}) − φ_{i}(e_{−})))

= ∏_{i=1}^{r} φ_{i}(exp(\frac{π}{2} (e_{+} − e_{−}))) ∏_{i=1}^{l} φ_{i}(exp(−e_{−})) ∏_{i=1}^{r} φ_{i}(exp(−\frac{π}{2} (e_{+} − e_{−})))

= ∏_{i=1}^{l} φ_{i}(exp(\frac{π}{2} (e_{+} − e_{−})) exp(−e_{−}) exp(−\frac{π}{2} (e_{+} − e_{−})))

= ∏_{i=1}^{l} φ_{i}(exp e_{+}) = exp(∑_{i=1}^{l} φ_{i}(e_{+})) = exp(∑_{i=1}^{l} E_{i}).

The following lemma was proved in [9]. Note that we do not use there the assumption that Δ(𝔤, 𝔠) is of C_{r} -type.

Lemma 5.4. a_{l}^{−}a_{l}^{−1}a_{l}^{−} = a_{l}^{+}.

Lemma 5.5. (0^{−}, a_{l}a_{r} 0^{+}) ≡ (a_{l}^{+} 0^{−}, (a_{l}^{+})^{2}a_{r} 0^{+}) mod G.

Proof. First note that a_{l} 0^{±} = a_{l}^{−1} 0^{±}. In fact, Ad a_{l}^{2} is the identity on 𝔞, which implies that a_{l}^{2} lies in the centralizer C(Z) = U^{+} ∩ U^{−}. Also note that a_{l}^{−1}a_{r} = a_{r}a_{l}^{−1}, since a_{l} and a_{r} commute. Consequently, in view of Lemmas 5.4 and 5.3, we have

(0^{−}, a_{l}a_{r} 0^{+}) = (0^{−}, a_{r}a_{l} 0^{+}) = (0^{−}, a_{r}a_{l}^{−1} 0^{+}) = (0^{−}, a_{l}^{−1}a_{r} 0^{+})
≡ (a_{l}^{+}a_{l}^{−} 0^{−}, a_{l}^{+}a_{l}^{−}a_{l}^{−1}a_{r} 0^{+}) = (a_{l}^{+} 0^{−}, (a_{l}^{+})^{2}(a_{l}^{−})^{−1}a_{r} 0^{+})
= (a_{l}^{+} 0^{−}, (a_{l}^{+})^{2}a_{r}a_{l}^{+} 0^{+}) = (a_{l}^{+} 0^{−}, (a_{l}^{+})^{2}a_{r} 0^{+}) mod G.
Let us put 0_{l} = ∑_{i=1}^{l} E_{i} ∈ 𝔤_{2}(r), 1 ≤ l ≤ r, 0_{0} = 1.

Proposition 5.6. The point (0_{l}, 2 0_{l}) ∈ 𝔤_{1} ⊕ 𝔤′_{1} (identified with its ξ -image) is a representative of the G-orbit M_{l} for 0 ≤ l ≤ r.

Proof. We have

M_{l} = G(0^{−}, a_{r−l} 0^{+}) = G(0^{−}, a_{l}a_{r} 0^{+}) = G(a_{l}^{+} 0^{−}, (a_{l}^{+})^{2}a_{r} 0^{+})
= G((exp 0_{l}) 0^{−}, (exp 0_{l})^{2}a_{r} 0^{+}) = G((exp 0_{l}) 0^{−}, (exp 2 0_{l})a_{r} 0^{+})
= G(ξ(0_{l}, 2 0_{l})).

A preliminary step for Siegel-type realization of orbits was done by Tanaka [15], which is needed for later consideration. Let Q̂_{r} be the parabolic subgroup of G opposite to Q_{r}, that is, Q̂_{r} = C(Z, Z_{r}) · N, where N = exp 𝔫 and 𝔫 = 𝔤_{2}(r)+𝔤_{1}(r) = 𝔤_{2}(r)+𝔤_{1}^{+}(r)+𝔤_{1}^{−}(r). The group G acts on the vector space 𝔤_{1}⊕𝔤′_{1} birationally through ξ. But the subgroup Q̂_{r} acts on it as affine transformations.

Proposition 5.7. (Tanaka [15]). Let a, x, y ∈ 𝔤_{2}(r), b^{+}, v^{+} ∈ 𝔤_{1}^{+}(r), b^{−}, u^{−} ∈ 𝔤_{1}^{−}(r) and h ∈ C(Z, Z_{r}). Then the ξ -equivariant action of Q̂_{r} is given by

\begin{matrix}exp(a + b^{+} + b^{−})((x, u^{−}) ⊕ (y, v^{+})) \\ = (x + a + [b^{+}, u^{−}] + \frac{1}{2}[b^{+}, b^{−}], u^{−} + b^{−}) \\ ⊕ (y + a + [b^{−}, v^{+}] + \frac{1}{2}[b^{−}, b^{+}], v^{+} + b^{+}), \end{matrix}  (5.7)

h((x, u^{−}) ⊕ (y, v^{+}))

= ((Ad_{𝔤_{2}(r)} h)x, (Ad_{𝔤_{1}^{−}(r)} h)u^{−}) ⊕ ((Ad_{𝔤_{2}(r)} h)y, (Ad_{𝔤_{1}^{+}(r)} h)v^{+}). (5.8)

Definition 5.8. We define a surjective submersion Φ of 𝔤_{1} ⊕ 𝔤′_{1} onto 𝔤_{2}(r) as follows: For X = (x, u^{−}) ∈ 𝔤_{1}, Y = (y, v^{+}) ∈ 𝔤′_{1},

Φ(X ⊕ Y) = y − x + [v^{+}, u^{−}].

Φ has the following (Q̂_{r}, C(Z, Z_{r}))-equivariance property.

Proposition 5.9. ([15]). Φ is invariant under the action of N. Moreover let h ∈ C(Z, Z_{r}) and let X′ ⊕ Y′ = h(X ⊕ Y). Then

Φ(X′ ⊕ Y′) = (Ad_{𝔤_{2}(r)} h)Φ(X ⊕ Y).

We restate Lemma 3.8 [15] as follows:

Lemma 5.10. N acts on 𝔤_{1} ⊕ 𝔤′_{1} freely. Moreover let X ⊕ Y ∈ 𝔤_{1} ⊕ 𝔤′_{1} and let X = (x, u^{−}), Y = (y, v^{+}). Then we have

exp(−v^{+}) exp(−x − u^{−})(X ⊕ Y) = (0, 0) ⊕ (Φ(X, Y), 0).  (5.9)

Note that the group C(Z_{r}) is the group of grade-preserving automorphisms with respect to the grading (4.10) and Lie C(Z_{r}) = 𝔤_{0}(r). C(Z, Z_{r}) is an open subgroup of C(Z_{r}).

Let Q̂_{r}^{0} and C^{0}(Z_{r}) be the identity components of Q̂_{r} and C(Z_{r}), respectively. The following proposition follows from Proposition 5.9 (cf. [15]).

Proposition 5.11. There exists a bijection between the set of Q̂_{r}^{0}-orbits in 𝔤_{1} ⊕ 𝔤′_{1} and the set of C^{0}(Z_{r})-orbits in 𝔤_{2}(r). More precisely, the Φ-image of a Q̂_{r}^{0}-orbit is a C^{0}(Z_{r})-orbit, and the complete inverse image by Φ of a C^{0}(Z_{r})- orbit is a Q̂_{r}^{0}-orbit.

We are interested in the intersection of a G-orbit with 𝔤_{1} ⊕ 𝔤′_{1}. Let M_{l}^{∗} = M_{l} ∩ (𝔤_{1} ⊕ 𝔤′_{1}), 0 ≤ l ≤ r, which is a dense open set in M_{l}. M_{l}^{∗} is stable under Q̂_{r}, and hence it can be expressed as the union of Q̂_{r}^{0}-orbits contained in M_{l}^{∗}. Those Q̂_{r}^{0}-orbits are open in M_{l}^{∗} ([15]). This fact can also be proved by using Proposition 4.4. Let us consider the (reductive) graded subalgebra of 𝔤

𝔤_{ev}(r) = 𝔤_{−2}(r) + 𝔤_{0}(r) + 𝔤_{2}(r),  (5.10)

which contains the simple graded ideal

𝔤′_{ev}(r) = 𝔤_{−2}(r) + [𝔤_{−2}(r), 𝔤_{2}(r)] + 𝔤_{2}(r).  (5.11)

By the table of (𝔤, 𝔤_{ev}) in [7], or by the property of roots forming Δ_{2}(r), it turns out that 𝔤_{2}(r) has the structure of a real simple Jordan algebra and the adjoint action of C^{0}(Z_{r}) on 𝔤_{2}(r) coincides with the identity component of the structure group of this Jordan algebra. Therefore we have the rank decomposition ([2])

𝔤_{2}(r) = V_{r} ⨿ V_{r−1} ⨿ ⋯ ⨿ V_{0},  (5.12)

where V_{l} is the union of equi-dimensional C^{0}(Z_{r})-orbits. V_{r} is open dense in 𝔤_{2}(r), dim V_{k} > dim V_{k−1}, and V_{0} = (0).

Lemma 5.12. Let p = X ⊕ Y ∈ 𝔤_{1} ⊕ 𝔤′_{1}. Then Φ^{−1}(Φ(p)) is the N -orbit through the point p.

Proof. This is an easy consequence of Proposition 5.9 and Lemma 5.10.

Proposition 5.13. M_{l}^{∗} = Φ^{−1}(V_{l}), 0 ≤ l ≤ r.

Proof. By Proposition 5.6, we have M_{l}^{∗} = M_{l}∩(𝔤_{1}⊕𝔤′_{1}) = G^{0}(0_{l}, 2 0_{l})∩(𝔤_{1}⊕𝔤′_{1}), which contains the orbit Q̂_{r}^{0}(0_{l}, 2 0_{l}) of the same dimension. On the other hand

Q̂_{r}^{0}(0_{l}, 2 0_{l}) = Φ^{−1}(C^{0}(Z_{r})Φ(0_{l}, 2 0_{l})) = Φ^{−1}(C^{0}(Z_{r}) 0_{l}).

Let

0_{p,q} = ∑_{i=1}^{p} E_{i} − ∑_{j=p+1}^{p+q} E_{j}.

Note that 0_{l} = 0_{l,0}. Let V_{p,q} = C^{0}(Z_{r}) 0_{p,q}. It is known [8] that V_{l} = ∐_{p+q=l} V_{p,q}. Those spaces V_{p,q} in the right-hand side exhaust all C^{0}(Z_{r})-orbits of the dimension equal to dim C^{0}(Z_{r}) 0_{l}. By Lemma 5.12, Φ^{−1}(V_{p,q}), p + q = l, are the Q̂_{r}^{0}-orbits of the same dimension. Therefore, by Proposition 5.11, we have

Φ^{−1}(V_{l}) = Φ^{−1}( ∐_{p+q=l} V_{p,q}) = ∐_{p+q=l} Φ^{−1}(V_{p,q}) = M_{l}^{∗}.

We say that Φ^{−1}(V_{l}) is the Siegel-type realization of the G-orbit M_{l}. Let P: 𝔤_{2}(r) → End 𝔤_{2}(r) be the quadratic operator of the Jordan algebra 𝔤_{2}(r). Then we have Theorem 5.14. The Siegel-type realization of the G-orbit M_{l}, 0 ≤ l ≤ r, is given by

M_{l}^{∗} = Φ^{−1}(V_{l}) = { (x, u^{−})⊕(y, v^{+}) ∈ 𝔤_{1} ⊕𝔤′_{1} : rk P (y −x+[v^{+}, u^{−}]) = i_{l} }, (5.13)

where i_{l} = rk P (0_{l}). In particular, when l = r, the Siegel-type realization of the parahermitian symmetric space M = G/G_{0} is given by

M_{r}^{∗} = Φ^{−1}(V_{r}) = { (x, u^{−}) ⊕ (y, v^{+}) ∈ 𝔤_{1} ⊕ 𝔤′_{1} : ν(y − x + [v^{+}, u^{−}]) ≠ 0 }, (5.14)

where ν denotes the generic norm of the Jordan algebra 𝔤_{2}(r).

Proof. By Proposition 5.13, we have

M_{l}^{∗} = Φ^{−1}(V_{l}) = { (x, u^{−}) ⊕ (y, v^{+}) ∈ 𝔤_{1} ⊕ 𝔤′_{1} : y − x + [v^{+}, u^{−}] ∈ V_{l} }. (5.15)

Also we have [2] that

V_{l} = { x ∈ 𝔤_{2}(r) : rk P (x) = i_{l} }.  (5.16)

Note that the condition rk P (x) = i_{r} is equivalent to the condition ν(x) ≠ 0.

Remark 5.15. (5.14) is an analogue of the Siegel domain realization of a bounded symmetric domain. In the case of C_{r} -type, (5.13) was obtained in [9], in which case v^{+} = u^{−} = 0.

The closure \overline{V_{l}} of V_{l} in 𝔤_{2}(r), was given by V_{≤l} := ∐_{i=0}^{l} V_{i} ([2]). Therefore, from (5.16) it follows that \overline{V_{l}} = V_{≤l} is an algebraic variety in 𝔤_{2}(r), which we call a generalized determinantal variety. In the case of 𝔤_{2}(r) = M_{n}(ℂ), Sym_{n}(ℂ) (resp. Alt_{2n}(ℂ)), the number l is just the rank (resp. one-half of the rank) of a matrix for M_{n}(ℂ) and Sym_{n}(ℂ) (resp. Alt_{2n}(ℂ)). In those cases, V_{≤l} is a usual determinantal variety. By the definition of M_{l}^{∗}, we have the following decomposition

𝔤_{1} ⊕ 𝔤′_{1} = ∐_{l=0}^{r} M_{l}^{∗},

which is viewed as the rank decomposition of 𝔤_{1} ⊕ 𝔤′_{1} by Theorem 5.14.

6. Stratifications of M̃

We wish to construct a polynomial map Ψ of 𝔤_{1} ⊕ 𝔤′_{1} to 𝔤_{2}(r) × 𝔫 (cf.5.3). Choose a point p = (x, u^{−}) ⊕ (y, v^{+}) ∈ 𝔤_{1} ⊕ 𝔤′_{1}, and consider the element n_{p} = exp(−v^{+}) exp(−x − u^{−}) ∈ N. n_{p} can be written as

n_{p} = exp(−x + \frac{1}{2}[v^{+}, u^{−}] − v^{+} − u^{−}).

Since exp : 𝔫 → N is diffeomorphic, one can define Ψ to be

Ψ(p) = (Φ(p), log n_{p}) = (y − x + [v^{+}, u^{−}], −x + \frac{1}{2}[v^{+}, u^{−}] − v^{+} − u^{−}). (6.1)

Lemma 6.1. The polynomial map Ψ is a diffeomorphism of 𝔤_{1} ⊕ 𝔤′_{1} onto 𝔤_{2}(r) × 𝔫. Ψ^{−1} is also a polynomial map.

Proof. Let p, q ∈ 𝔤_{1} ⊕ 𝔤′_{1}, and suppose Ψ(p) = Ψ(q). Then we have Φ(p) = Φ(q) and n_{p} = n_{q}. By Lemma 5.10, we have that n_{p}(p) = (0, 0) ⊕ (Φ(p), 0) = (0, 0)⊕(Φ(q), 0) = n_{q}(q) = n_{p}(q), which implies that p = q, proving the injectivity of Ψ. Now let (a, X) ∈ 𝔤_{2}(r)×𝔫, and let p := (exp −X)((0, 0)⊕(a, 0)) ∈ 𝔤_{1} ⊕𝔤′_{1}. Then, by Lemma 5.12,Φ(p) = Φ((0, 0) ⊕ (a, 0)) = a, and hence, by Lemma 5.10, we have (exp X)p = (0, 0) ⊕ (Φ(p), 0) = n_{p}(p). Since N acts freely on 𝔤_{1} ⊕ 𝔤′_{1}, it follows that (a, X) = (Φ(p), log n_{p}), proving the surjectivity of Ψ. On the other hand, Ψ^{−1} is given by Ψ^{−1}(a, X) = (exp −X)((0, 0) ⊕ (a, 0)), which is a polynomial in a and X by (5.7).

From the expression of Ψ^{−1} in the above proof, we have

Lemma 6.2. Let V be a C^{0}(Z_{r})-orbit in 𝔤_{2}(r). Then, for the corresponding Q̂_{r}^{0}-orbit Φ^{−1}(V), we have Ψ(Φ^{−1}(V)) = V × 𝔫.

Let M_{≤k} and M_{≤k}^{∗} denote the unions ∐_{i=0}^{k} M_{i} and ∐_{i=0}^{k} M_{i}^{∗}, respectively. By Theorem 3.1 and Proposition 5.13, the closure \overline{M_{k}^{∗}} of M_{k}^{∗} in 𝔤_{1} ⊕ 𝔤′_{1} is given by \overline{M_{k}^{∗}} = M_{≤k}^{∗} = Φ^{−1}(V_{≤k}), which is an algebraic variety in 𝔤_{1} ⊕ 𝔤′_{1} (cf. Theorem 5.14). Since V_{≤k} is an algebraic variety in 𝔤_{2}(r) ([2]), V_{≤k} × 𝔫 is an algebraic variety in 𝔤_{2}(r) × 𝔫. We will denote the singular locus and the regular locus of an algebraic variety A by Sing(A) and Reg(A), respectively.

Proposition 6.3. The algebraic variety M_{≤k}^{∗} is isomorphic to the algebraic variety V_{≤k} × 𝔫, for 0 ≤ k ≤ r − 1. We have

Ψ(Sing(M_{≤k}^{∗})) = Sing(V_{≤k}) × 𝔫, 0 ≤ k ≤ r − 1.

In other words,

Sing(M_{≤k}^{∗}) = Φ^{−1}(Sing(V_{≤k})), 0 ≤ k ≤ r − 1.

Proof. The first assertion is an immediate consequence of Lemmas 6.1 and 6.2. The other assertions follow from the first one.

We wish to find the singular locus Sing(V_{≤k}) of a generalized determinantal variety V_{≤k} in 𝔤_{2}(r). In the case where (𝔤, 𝔤_{0}) is of BC_{r} -type, V_{≤k} is determined by the graded subalgebra (5.10) or (5.11) of the first kind. As is seen from the classification of simple GLA’s of the 2nd kind([8]), the simple GLA (5.11) is of C_{r} -type in this case. As for the case where (𝔤, 𝔤_{0}) is of C_{r} -type, the GLA (5.11) coincides with the original GLA (1.1), more precisely, we have 𝔤_{±2}(r) = 𝔤_{±1} and 𝔤_{0}(r) = 𝔤_{0}. Therefore one has only to consider the generalized determinantal varieties arising from a simple GLA (1.1) of C_{r} -type.

Consider a simple GLA (1.1) of C_{r} -type. In this case 𝔤_{1} is a simple Jordan algebra on which G_{0} acts as the structure group. As for (5.12) we have the rank decomposition

𝔤_{1} = V_{r} ⨿ V_{r−1} ⨿ ⋯ ⨿ V_{0},  (6.2)

where V_{r} is an open subset and V_{0} = (0). If we denote by G_{0}^{0} the identity component of G_{0}, then V_{k} is a union of the equidimensional orbits [8]:

V_{k} = ⨿_{p+q=k}G_{0}^{0}0_{p,q}, 0 ≤ k ≤ r,  (6.3)

where 0_{p,q} is the same as in the proof of Proposition 5.13. (5.16) is still valid by replacing 𝔤_{2}(r) by 𝔤_{1}. Therefore V_{≤k} is an algebraic variety in 𝔤_{1} defined over ℝ. We have V_{≤r−1} = {x ∈ 𝔤_{1} : det P (x) = 0}. Since det P (x) is a power of the generic norm ν of the Jordan algebra 𝔤_{1}, the defining ideal I(V_{≤r−1}) of V_{≤r−1} is generated by the irreducible polynomial ν. The variety V_{≤k} is a conic variety, since G_{0}^{0} contains the one-dimensional center acting on 𝔤_{1} as homotheties. Therefore the defining ideal I(V_{≤k}) of V_{≤k} is a homogeneous ideal. Let I(V_{≤k})_{m} denote the totality of homogeneous polynomials in I(V_{≤k}) of degree m.

Proposition 6.4. For a simple GLA (1.1) with r ≥ 2, the singular locus Sing(V_{≤1}) of the generalized determinantal variety V_{≤1} in 𝔤_{1} coincides with V_{0} = (0).

Proof. Let 𝔞_{1} be the linear span of E_{1}, ⋯, E_{r} in 𝔤_{1}. Then it is known [1,8] that

𝔤_{1} = G_{0}^{0}𝔞_{1}.  (6.4)

First we claim that I(V_{≤1})_{1} = Ø. Suppose the contrary. One can then choose a nonzero linear form f on 𝔤_{1} such that f(V_{1}) = 0. Since r ≥ 2, there exists a point x_{0} ∈ 𝔤_{1} such that f(x_{0}) ≠ 0. By (6.4) one can assume that x_{0} lies in 𝔞_{1}. Since E_{i} is conjugate to E_{1} under G_{0}^{0}, E_{1}, ⋯, E_{r} belong to V_{1}. By the assumption for f we have that f(E_{i}) = 0, 1 ≤ i ≤ r, which implies that f is identically zero on 𝔞_{1}. This contradicts the fact that f(x_{0}) ≠ 0, which shows the claim that I(V_{≤1})_{1} = Ø. Recall that the variety V_{≤1} is defined over ℝ (cf.6.2). One can choose a generator {f_{1}, ⋯, f_{s}} of the ideal I(V_{≤1}) such that each polynomial f_{i} is homogeneous and defined over ℝ. From the above argument, it follows that deg f_{i} ≥ 2. Consequently (df_{i})_{0} = 0, 1 ≤ i ≤ s, which shows that 0 is a singularity of V_{≤1}. Obviously we have that Reg(V_{≤1}) ⊃ V_{1}. Therefore we conclude Sing(V_{≤1}) = V_{0}.

In this paragraph we treat the case where the GLA (1.1) is complex simple of C_{r} -type. The subspace 𝔤_{1} is then a complex simple Jordan algebra. The following is a list of complex simple Jordan algebras.

Type 𝔤_{1} r
I  M_{n}(ℂ) n
II Alt_{2n}(ℂ) n
III Sym_{n}(ℂ) n
IV ℂ^{n} 2
VI H_{3}(𝕆^{ℂ}) 3

Here H_{3}(𝕆^{ℂ}) denotes the exceptional simple Jordan algebra of 3 × 3 Hermitian matrices with entries in complex octonions 𝕆^{ℂ}.

Proposition 6.5. For any complex simple Jordan algebra 𝔤_{1} with r ≥ 2, we have

Sing(V_{≤k}) = V_{≤k−1},  1 ≤ k ≤ r − 1.  (6.5)

Proof. (i) For the case of types I or III, X ∈ V_{≤k} if and only if rk X ≤ k. For the case of type II, X ∈ V_{≤k} if and only if rk X ≤ 2k. In those three cases, (6.5) is well-known (see for example [11]).

(ii) Consider the case of type IV. In this case we have r = 2 and 𝔤_{1} = V_{2} ⨿ V_{1} ⨿ V_{0}. Therefore (6.5) follows from Proposition 6.4.

(iii) Now we consider the case of type VI. In this case 𝔤_{1} can be identified with H_{3}(𝕆^{ℂ}) in a such a way that E_{i}(i = 1, 2, 3) is sent to the diagonal matrix (δ_{i1}, δ_{i2}, δ_{i3}). An element x ∈ H_{3}(𝕆^{ℂ}) is expressed as

x = (\begin{matrix}ξ_{1} & c & b̄ \\ c̄ & ξ_{2} & a \\ b & ā & ξ_{3} \end{matrix}) ,  ξ_{i} ∈ ℂ, a, b, c ∈ 𝕆^{ℂ}.  (6.6)

The generic norm ν(x) of x is given by

ν(x) = ξ_{1}ξ_{2}ξ_{3} − ξ_{1}n(a) − ξ_{2}n(b) − ξ_{3}n(c) + t(abc),

where n and t denote respectively the norm and the trace of an octonion. We express an element a ∈ 𝕆^{ℂ} as a = ∑_{i=0}^{7} a_{i}e_{i}, where {e_{i}} is the canonical basis of 𝕆^{ℂ}. The variety V_{≤2} is defined by the single equation ν(x) = 0, x ∈ 𝔤_{1}. We then have

\begin{aligned}dν & = (ξ_{2}ξ_{3} − n(a)) dξ_{1} + (ξ_{1}ξ_{3} − n(b)) dξ_{2} + (ξ_{1}ξ_{2} − n(c)) dξ_{3} \\ & + ∑_{i=0}^{7}(−2ξ_{1}a_{i} + \frac{∂}{∂a_{i}} t(abc)) da_{i} + ∑_{i=0}^{7}(−2ξ_{2}b_{i} + \frac{∂}{∂b_{i}} t(abc)) db_{i} \\ & + ∑_{i=0}^{7}(−2ξ_{3}c_{i} + \frac{∂}{∂c_{i}} t(abc)) dc_{i}. \end{aligned} (6.7)

Now let x ∈ V_{≤1}. From (6.4) and (6.3) it follows that there exists an element g ∈ G_{0}^{0} such that gx ∈ 𝔞_{1} ∩ V_{≤1}, in other words, gx is a diagonal matrix diag(ξ_{1}, ξ_{2}, ξ_{3}) ∈ V_{≤1}, which implies that at least two of ξ_{1}, ξ_{2}, ξ_{3} are zero. Therefore we have from (6.7) that

(dν)_{gx} = ξ_{2}ξ_{3} dξ_{1} + ξ_{1}ξ_{3} dξ_{2} + ξ_{1}ξ_{2} dξ_{3} = 0

Therefore, in view of the relative invariance of ν under G_{0}, we have (dν)_{x} = 0. By the Jacobian criterion, we obtain V_{≤1} ⊂ Sing(V_{≤2}). On the other hand, clearly we have V_{2} ⊂ Reg(V_{≤2}). Consequently we conclude that V_{≤1} = Sing(V_{≤2}). The equality V_{0} = Sing(V_{≤1}) follows from Proposition 6.4.

We denote the defining ideal of an algebraic variety A by I(A).

Corollary 6.6. Let V_{≤k}(1 ≤ k ≤ r − 1) be a generalized determinantal variety in a complex simple Jordan algebra 𝔤_{1}. Then there exists a basis {f_{1}, …, f_{s_{k}}} of I(V_{≤k}) such that each f_{i} is defined over ℝ and that df_{i} ∈ I(Sing(V_{≤k})), 1 ≤ i ≤ s_{k}, in other words, (df_{i})_{p} = 0, 1 ≤ i ≤ s_{k} for each point p ∈ Sing(V_{≤k}).

Proof. Note that V_{≤k} is defined over ℝ (cf.6.2). For types I and III, we choose, as a generator of I(V_{≤k}), the totality of (k + 1)-minors of a generic element of 𝔤_{1}. For Type II, we choose, as a generator of I(V_{≤k}), the totality of the Pfaffians of principal (2k + 2)-submatrices of a generic element of 𝔤_{1}. Then the assertion is well-known for those cases (cf.[11]). For the remaining two cases, the assertion was shown in the proof of Propositions 6.4 and 6.5.

In this paragraph we wish to show that Proposition 6.5 is valid for a real simple Jordan algebra. Let us consider a real simple but not complex simple GLA(1.1) of C_{r} -type: 𝔤 = 𝔤_{−1} + 𝔤_{0} + 𝔤_{1}. r is the split rank of the symmetric pair (𝔤, 𝔤_{0}). In this case 𝔤_{1} is a real simple but not complex simple Jordan algebra. Consider the complexification of the GLA 𝔤:

𝔤^{c} = 𝔤_{−1}^{c} + 𝔤_{0}^{c} + 𝔤_{1}^{c}

Let r̄ be the split rank of the symmetric pair (𝔤^{c}, 𝔤_{0}^{c}). The following is a list of real simple GLAs of C_{r} -type and their complexifications:

Type I

\begin{matrix}\{ & \begin{aligned}(𝔤^{c}, 𝔤_{0}^{c}, 𝔤_{1}^{c}) & = (𝔰𝔩(2n, ℂ), 𝔰𝔩(n, ℂ) + 𝔰𝔩(n, ℂ) + ℂ, M_{n}(ℂ)),  r̄ = n, \\ (𝔤, 𝔤_{0}, 𝔤_{1}) & = \{ \begin{matrix}(𝔰𝔩(2n, ℝ), 𝔰𝔩(n, ℝ) + 𝔰𝔩(n, ℝ) + ℝ, M_{n}(ℝ)), r = n, \\ (𝔰𝔲(n, n), 𝔰𝔩(n, ℂ) + ℝ, H_{n}(ℂ)),  r = n, \end{matrix} \end{aligned} \\ \{ & \begin{aligned}(𝔤^{c}, 𝔤_{0}^{c}, 𝔤_{1}^{c}) & = (𝔰𝔩(4n, ℂ), 𝔰𝔩(2n, ℂ) + 𝔰𝔩(2n, ℂ) + ℂ, M_{2n}(ℂ)), r̄ = 2n, \\ (𝔤, 𝔤_{0}, 𝔤_{1}) & = (𝔰𝔩(2n, ℍ), 𝔰𝔩(n, ℍ) + 𝔰𝔩(n, ℍ) + ℝ, M_{n}(ℍ)),  r = n, \end{aligned} \end{matrix}

Type II

\{ \begin{matrix}(𝔤^{c}, 𝔤_{0}^{c}, 𝔤_{1}^{c}) & = (𝔰𝔬(4n, ℂ), 𝔤𝔩(2n, ℂ), Alt_{2n}(ℂ)),  r̄ = n, \\ (𝔤, 𝔤_{0}, 𝔤_{1}) & = \{ \begin{matrix}(𝔰𝔬(2n, 2n), 𝔤𝔩(2n, ℝ), Alt_{2n}(ℝ)), r = n, \\ (𝔰𝔬^{∗}(4n), 𝔤𝔩(n, ℍ), H_{n}(ℍ)),  r = n, \end{matrix} \end{matrix}

Type III

\{ \begin{matrix}(𝔤^{c}, 𝔤_{0}^{c}, 𝔤_{1}^{c}) & = (𝔰𝔭(n, ℂ), 𝔤𝔩(n, ℂ), Sym_{n}(ℂ)),  r̄ = n, \\ (𝔤, 𝔤_{0}, 𝔤_{1}) & = (𝔰𝔭(n, ℝ), 𝔤𝔩(n, ℝ), Sym_{n}(ℝ)),  r = n, \end{matrix}

\{ \begin{matrix}(𝔤^{c}, 𝔤_{0}^{c}, 𝔤_{1}^{c}) & = (𝔰𝔭(2n, ℂ), 𝔤𝔩(2n, ℂ), Sym_{2n}(ℂ)),  r̄ = 2n, \\ (𝔤, 𝔤_{0}, 𝔤_{1}) & = (𝔰𝔭(n, n), 𝔤𝔩(n, ℍ), SH_{n}(ℍ)),  r = n, \end{matrix}

Type IV

\{ \begin{matrix}(𝔤^{c}, 𝔤_{0}^{c}, 𝔤_{1}^{c}) & = (𝔰𝔬(n + 2, ℂ), 𝔰𝔬(n, ℂ) + ℂ, ℂ^{n}),  r̄ = 2, \\ (𝔤, 𝔤_{0}, 𝔤_{1}) & = (𝔰𝔬(p + 1, q + 1), 𝔰𝔬(p, q) + ℝ, ℝ^{n}),  r = \{ \begin{matrix}1 (p = 0), \\ 2 (p ≥ 1), \end{matrix} \\ & p ≤ q, p + q = n, \end{matrix}

Type VI

\{ \begin{matrix}(𝔤^{c}, 𝔤_{0}^{c}, 𝔤_{1}^{c}) & = (E_{7}^{ℂ}, E_{6}^{ℂ} + ℂ, H_{3}(𝕆^{ℂ})),  r̄ = 3, \\ (𝔤, 𝔤_{0}, 𝔤_{1}) & = \{ \begin{matrix}(E_{7(7)}, E_{6(6)} + ℝ, H_{3}(𝕆′)),  r = 3, \\ (E_{7(−25)}, E_{6(−26)} + ℝ, H_{3}(𝕆)),  r = 3. \end{matrix} \end{matrix}

In the above list, ℍ denotes the quaternion algebra. 𝕆, (resp. 𝕆′) denotes the octonion (resp. split octonion) algebra. H_{n}(𝔽), 𝔽 = ℂ, ℍ, 𝕆, 𝕆′, denotes the Jordan algebra of n × n 𝔽 -Hermitian matrices. SH_{n}(ℍ) denotes the Jordan algebra of n × n skew-Hermitian quaternion matrices. Note that the generalized determinantal varieties in a complex simple Jordan algebra are defined over ℝ.

From a result of Takeuchi [14] we have

Proposition 6.7. Let 𝔤_{1} and 𝔤_{1}^{c} be as above, and let 𝔤_{1} = ⨿_{k=0}^{r}V_{k} and 𝔤_{1}^{c} = ⨿r̄ ; k=0Ṽ_{k} be the rank decomposition (cf.(6.2)) of 𝔤_{1} and 𝔤_{1}^{c}, respectively. Suppose that r̄ = r. Then V_{≤k}(1 ≤ k ≤ r −1) coincides with the set of ℝ -rational points of the complex algebraic variety Ṽ_{≤k}. Suppose that r̄ = 2r. Then V_{≤k}(1 ≤ k ≤ r − 1) coincides with the sets of ℝ -rational points of the algebraic varieties Ṽ_{≤2k} and of Ṽ_{≤2k+1}.

From Proposition 6.7, we obtain

Lemma 6.8. Let f_{1}, …, f_{s_{k}} be real polynomials on 𝔤_{1}, and let f̃_{i}(1 ≤ i ≤ s_{k}) be the natural extension of f_{i} to 𝔤_{1}^{c}. Then I(V_{≤k})(1 ≤ k ≤ r − 1) is generated by f_{1}, …, f_{s_{k}} if and only if I(Ṽ_{≤k}) (resp. I(Ṽ_{≤2k})) is generated by f̃_{1}, …, f̃_{s_{k}} for r̄ = r (resp. r̄ = 2r.).

Proposition 6.9. For a real simple (not complex simple) Jordan algebra 𝔤_{1}, we have

Sing(V_{≤k}) = V_{≤k−1}, 1 ≤ k ≤ r − 1.

Proof. Let θ be the conjugation of 𝔤_{1}^{c} with respect to 𝔤_{1}. Since Ṽ_{≤k} is θstable, Sing(Ṽ_{≤k}) is also θ-stable. Let (Sing(Ṽ_{≤k}))_{θ} be the set of θ-fixed points in Sing(Ṽ_{≤k}). Suppose first r̄ = r. Since Ṽ_{≤k} is a conic variety defined over ℝ (cf.6.2), one can choose a generator {f̃_{1}, …, f̃_{s_{k}}} of I(Ṽ_{≤k}) such that each f̃_{i} is homogeneous and defined over ℝ. By Corollary 6.6, df̃_{i} ∈ I(Sing(Ṽ_{≤k})). Let f_{i} = f̃_{i}|_{𝔤_{1}}. Then {f_{1}, …, f_{s_{k}}} is a generator of I(V_{≤k}), by Lemma 6.8. Let p ∈ (Sing(Ṽ_{≤k}))_{θ}. Then p ∈ (Ṽ_{≤k})_{θ} = V_{≤k}, by Proposition 6.7. We have (df_{i})_{p} = (df̃_{i})_{p} = 0, which implies that p ∈ Sing(Ṽ_{≤k}). Hence, by Propositions 6.5 and 6.7, we have

V_{≤k−1} = (Ṽ_{≤k−1})_{θ} = (Sing(Ṽ_{≤k}))_{θ} ⊂ Sing(V_{≤k}).  (6.8)

In view of the inclusion V_{k} ⊂ Reg(V_{≤k}), we conclude V_{≤k−1} = Sing(Ṽ_{≤k})_{θ} = Sing(V_{≤k}). As for the case r̄ = 2r, we should replace (6.8) by the equality

V_{≤k−1} = (Ṽ_{≤2k−2})_{θ} = (Ṽ_{≤2k−1})_{θ} = (Sing(Ṽ_{≤2k}))_{θ} ⊂ Sing(V_{≤k}).

Combining Propositions 6.5 and 6.9, we have

Theorem 6.10. Let 𝔤 = 𝔤_{−1} + 𝔤_{0} + 𝔤_{1} be a simple GLA of C_{r} -type, and let 𝔤_{1} = ⨿_{k=0}^{r}V_{k} be the rank decomposition. Then the closure V̄_{k} of V_{k} is the generalized determinantal variety V_{≤k}, and Sing(V_{≤k}) = V_{≤k−1} for 1 ≤ k ≤ r − 1.

From Theorem 6.10 and Proposition 6.3 we have Theorem 6.11. Let 𝔤 = 𝔤_{−1} + 𝔤_{0} + 𝔤_{1} be a simple GLA, and let r be the split rank of the symmetric pair (𝔤, 𝔤_{0}). Let 𝔤_{1}⊕𝔤′_{1} = ⨿_{k=0}^{r}M_{k}^{∗} be the rank decomposition (cf. 5.4) of the vector space 𝔤_{1} ⊕ 𝔤′_{1}. Then the closure \overline{M_{k}^{∗}} of M_{k}^{∗} in 𝔤_{1} ⊕ 𝔤′_{1} is the algebraic variety M_{≤k}^{∗}, and

Sing(M_{≤k}^{∗}) = M_{≤k−1}^{∗},  Reg(M_{≤k}^{∗}) = M_{k}^{∗},  1 ≤ k ≤ r − 1.

Now we go back to the full G-orbits M_{k}.

Lemma 6.12. M_{≤k} (0 ≤ k ≤ r − 1) is a real analytic set in M̃.

Proof. Choose a point p_{0} ∈ M_{≤k}. Then one can find an element g ∈ G such that g(p_{0}) ∈ 𝔤_{1} ⊕ 𝔤′_{1}. Choose a neighborhood U of p_{0} in M̃ in such a way that U′ := g(U) ⊂ 𝔤_{1} ⊕ 𝔤′_{1}. Let p ∈ U. Then we have that p ∈ U ∩ M_{≤k} if and only if g(p) ∈ U′ ∩ M_{≤k}^{∗}. Let {f_{1}, …, f_{s_{k}}} be a basis of the ideal I(M_{≤k}^{∗}). Then M_{≤k} is expressed in U as

U ∩ M_{≤k} = { p ∈ U : (f_{i} ∘ g)(p) = 0, 1 ≤ i ≤ s_{k} },

which implies that M_{≤k} is a real analytic set of M̃.

A point p ∈ M_{≤k} is a regular point of M_{≤k}, if there exists a neighborhood U of p in M̃ such that U ∩M_{≤k} is a smooth manifold of dimension d_{k} := dim M_{k}. Otherwise we say that p is a singular point of M_{≤k}. We denote by Reg(M_{≤k}) (resp. Sing(M_{≤k})) the regular (resp. singular) locus of M_{≤k}. Finally we get the following theorem which gives the stratification of M̃ by G-orbits.

Theorem 6.13. For 1 ≤ k ≤ r−1, we have Reg(M_{≤k}) = M_{k} and Sing(M_{≤k}) = M_{≤k−1}.

Proof. Let p ∈ Reg(M_{≤k}). Choose an element g ∈ G such that g(p) ∈ M_{≤k}^{∗}. Then, since p is a regular point of M_{≤k}, g(p) lies in Reg(M_{≤k}^{∗}) = M_{k}^{∗} by Theorem 6.11. This implies p ∈ M_{k}, or equivalently, Reg(M_{≤k}) = M_{k}. Similarly we can show Sing(M_{≤k}) = M_{≤k−1}.

Corollary 6.14. Suppose that a diffeomorphism f of M̃ leaves the open orbit M_{r} stable. Then f leaves all other orbits M_{k} (0 ≤ k ≤ r − 1) stable.

Proof. When r = 1, the assertion is trivial. Assume that r ≥ 2. By the assumption, f leaves M_{≤r−1} stable. Let k be an integer, 1 ≤ k ≤ r − 1. Then it is enough to prove that if f(M_{≤k}) = M_{≤k}, then f(M_{k}) = M_{k}. Put f(M_{k})^{∗} := f(M_{k}) ∩ (𝔤_{1} ⊕ 𝔤′_{1}). First we want to show f(M_{k})^{∗} ⊂ M_{k}^{∗}. Let p ∈ f(M_{k})^{∗}. Since f(M_{k}) is an open C^{∞}-submanifold of M_{≤k}, f(M_{k})^{∗} is expressed, in a neighborhood of p in 𝔤_{1} ⊕ 𝔤′_{1}, by the same polynomial equations as for the algebraic variety M_{≤k}^{∗}. Consequently the tangent spaces T_{p}(M_{≤k}^{∗}) and T_{p}(f(M_{k})^{∗}) are identical, which implies that

dim T_{p}(M_{≤k}^{∗}) = dim T_{p}(f(M_{k})^{∗}) = dim f(M_{k}) = dim M_{k} = dim M_{≤k}^{∗}.

Therefore it follows that p is a regular point of M_{≤k}^{∗}. By Theorem 6.11, we have p ∈ M_{k}^{∗}, and hence f(M_{k})^{∗} ⊂ M_{k}^{∗}. Now suppose that f(M_{k}) ⊄ M_{k}. Then we have f(M_{k}) ∩ M_{≤k−1} ≠ Ø. Since M_{≤k−1}^{∗} is open dense in M_{≤k−1}, we have f(M_{k})^{∗} ∩ M_{≤k−1}^{∗} = f(M_{k}) ∩ M_{≤k−1}^{∗} ≠ Ø. This contradicts the inclusion f(M_{k})^{∗} ⊂ M_{k}^{∗}. We have thus proved f(M_{k}) ⊂ M_{k}. The converse inclusion can be proved by replacing f by f^{−1} in the assumption f(M_{≤k}) = M_{≤k}.

7. Double foliation on the minimal boundary orbits

In this section, we always assume that M is of BC_{r} -type. In §2, we considered the double foliation on M̃, M^{±} = { M^{±}(g_{1}0^{−}, g_{2}0^{+}) : g_{1}, g_{2} ∈ G }. M^{±} naturally induce a double foliation F_{0}^{±} on the minimal boundary orbit M_{0}. The leaves F_{0}^{±}(p) of F_{0}^{±} through a point p ∈ M_{0} are given by the intersection M^{∓}(p) ∩ M_{0}.

Lemma 7.1. The leaves of F_{0}^{±} through the origin (0^{−}, a_{r} 0^{+}) ∈ M_{0} are given by

F_{0}^{−}(0^{−}, a_{r} 0^{+}) = U^{−}(0^{−}, a_{r} 0^{+}) = U^{−}/Q_{r},
F_{0}^{+}(0^{−}, a_{r} 0^{+}) = a_{r} U^{+}a_{r}^{−1}(0^{−}, a_{r} 0^{+}) = a_{r} U^{+}a_{r}^{−1}/Q_{r}.

Proof. By the definition, F_{0}^{±}(0^{−}, a_{r} 0^{+}) = M^{∓}(0^{−}, a_{r} 0^{+}) ∩ G(0^{−}, a_{r} 0^{+}). Let (g0^{−}, ga_{r} 0^{+}) ∈ F_{0}^{+}(0^{−}, a_{r} 0^{+}), g ∈ G. Then (g0^{−}, ga_{r} 0^{+}) ∈ M^{−}(0^{−}, a_{r} 0^{+}), which implies that ga_{r} 0^{+} = a_{r} 0^{+}, or equivalently, g ∈ a_{r} U^{+}a_{r}^{−1}. Conversely, let u ∈ U^{+}. Then a_{r}ua_{r}^{−1}(0^{−}, a_{r} 0^{+}) = (a_{r}ua_{r}^{−1}0^{−}, a_{r} 0^{+}) ∈ G( 0^{−}, a_{r} 0^{+}) ∩ M^{−}(0^{−}, a_{r} 0^{+}).

Lemma 7.2. The double foliation F_{0}^{±} arises from the subspaces 𝔤_{1}^{±}(r) of the GLA (4.10).

Proof. Let 𝔲^{±} = Lie U^{±}. By Lemma 7.1, the tangent spaces at (0^{−}, a_{r} 0^{+}) to the leaves F_{0}^{±}(0^{−}, a_{r} 0^{+}) are identified with the factor spaces 𝔲^{−}/𝔮_{r} and (Ad a_{r})𝔲^{+}/𝔮_{r}. By (4.15) and Lemma 4.6, we have

𝔲^{−} = 𝔤_{−1} + 𝔤_{0} = 𝔤_{−2}(r) + 𝔤_{−1}(r) + 𝔤_{0}(r) + 𝔤_{1}^{+}(r) = 𝔮_{r} + 𝔤_{1}^{+}(r).  (7.1)

Also, by (4.15) and Lemma 5.2, we have

\begin{aligned}(Ad a_{r})𝔲^{+} & = (Ad a_{r})(𝔤_{−1}^{−}(r) + 𝔤_{0}(r) + 𝔤_{1}^{+}(r) + 𝔤_{1}^{−}(r) + 𝔤_{2}(r)) \\ & = 𝔤_{1}^{−}(r) + 𝔤_{0}(r) + 𝔤_{−1}^{+}(r) + 𝔤_{−1}^{−}(r) + 𝔤_{−2}(r) \\ & = 𝔮_{r} + 𝔤_{1}^{−}(r). \end{aligned}  (7.2)

Therefore 𝔲^{−}/𝔮_{r} and (Ad a_{r})𝔲^{+}/𝔮_{r} can be identified with 𝔤_{1}^{+}(r) and 𝔤_{1}^{−}(r), respectively.

Let E := Z_{r} − 2Z. Then E is a central element of 𝔤_{0}(r). It follows from (4.14) and (4.15) that

ad E = \{ \begin{matrix}0 & on 𝔤_{ev}(r), \\ 1 & on 𝔤_{±1}^{+}(r), \\ −1 & on 𝔤_{±1}^{−}(r). \end{matrix}  (7.3)

Lemma 7.3. Let g ∈ C(Z_{r}) and let I = ad_{𝔤_{1}(r)} E. Then the following three conditions are equivalent:

(i) g(𝔤_{1}^{±}(r)) = 𝔤_{1}^{±}(r),

(ii) gI = Ig on 𝔤_{1}(r),

(iii) g(E) = E.

Proof. The only non-trivial assertion is the implication (ii) → (iii). Suppose (ii). Since Z, Z_{r} ∈ 𝔞, we have τ(Z_{r}) = −Z_{r} and τ(Z) = −Z, and hence τ(E) = −E. This implies that

τ(ad_{𝔤_{1}(r)} E)τ = − ad_{𝔤_{−1}(r)} E.  (7.4)

Consider the inner product < , > on 𝔤_{1}(r) defined by < X, Y >= −(X, τY). Let us denote by g_{±} the restrictions of the actions of g to 𝔤_{±1}(r), and denote by g_{+}^{∗} the adjoint operator of g_{+} with respect to < , >. Then we have that I is self-adjoint with respect to < , >, and hence, by (ii) we have (g_{+}^{∗})^{−1}I = I(g_{+}^{∗})^{−1}. We also have g_{−} = τ(g_{+}^{∗})^{−1}τ. Therefore it follows from (7.4) that

g_{−}(ad_{𝔤_{−1}(r)} E)(g_{−})^{−1} = τ(g_{+}^{∗})^{−1}τ(ad_{𝔤_{−1}(r)} E)τ(g_{+}^{∗})τ
= −τ(g_{+}^{∗})^{−1}(ad_{𝔤_{1}(r)} E)(g_{+}^{∗})τ = −τIτ = ad_{𝔤_{−1}(r)} E,

which implies that g_{−} commutes with ad_{𝔤_{−1}(r)} E. Combining this with (7.3) and (ii), we have that g commutes with ad E on the whole 𝔤. This implies (iii).

Lemma 7.4. C(Z, Z_{r}) = { g ∈ C(Z_{r}) : g(𝔤_{1}^{±}(r)) = 𝔤_{1}^{±}(r) }.

Proof. Let g ∈ C(Z_{r}). Then g ∈ C(Z, Z_{r}) if and only if g(E) = E. Hence the assertion follows from Lemma 7.3.

Remark 7.5. Lemmas 7.2, 7.4, Proposition 4.8 and Corollary 4.9 imply that our flag manifold (M_{0} = G/Q_{r}, F_{0}^{±}) with double foliation F_{0}^{±} is a so-called pseudo-product manifold associated to the simple GLA (4.10) with decomposition (4.14), in the sense of Tanaka [15].

We give a list of simple parahermitian symmetric spaces M of BC_{r} -type and the corresponding minimal boundary orbits M_{0}. The list is obtained by extracting those ones satisfying (4.14) among all simple GLAs of the second kind classified in [7].

Type I (r = p).

M = SL(n, 𝔽)/ S(GL(p, 𝔽) × GL(n − p, 𝔽)),  𝔽 = ℝ, ℍ, ℂ,
1 ≤ p < n − p,
M_{0} = \{ \begin{matrix}SO(n)/ S(O(p) × O(n − 2p) × O(p)), & 𝔽 = ℝ, \\ Sp(n)/ Sp(p) × Sp(n − 2p) × Sp(p), & 𝔽 = ℍ, \\ SU(n)/ S(U(p) × U(n − 2p) × U(p)), & 𝔽 = ℂ. \end{matrix}

Type II (r = n).

\{ \begin{matrix}M = SO^{0}(2n + 1, 2n + 1)/ GL^{0}(2n + 1, ℝ), \\ M_{0} = SO(2n + 1) × SO(2n + 1)/ SO(2n), \end{matrix}
\{ \begin{matrix}M = SO(4n + 2, ℂ)/ GL(2n + 1, ℂ), \\ M_{0} = SO(4n + 2)/ U(2n) · 𝕋^{1}. \end{matrix}

Type V

\{ \begin{matrix}M = E_{6(6)}/ Spin(5, 5) · ℝ^{+}, & (r = 2), \\ M_{0} = Sp(4)/ Spin(4) × Spin(4), \end{matrix}
\{ \begin{matrix}M = E_{6(−26)}/ Spin(1, 9) · ℝ^{+}, & (r = 1), \\ M_{0} = F_{4}/ Spin(8), \end{matrix}
\{ \begin{matrix}M = E_{6}^{ℂ}/ Spin(10, ℂ) · ℂ^{∗}, & (r = 2), \\ M_{0} = E_{6}/ Spin(8) · 𝕋^{2}. \end{matrix}

8. Determination of the automorphism groups of M

Let (M = G/G_{0}, F^{±}) be the parahermitian symmetric space associated with a simple GLA (1.1). In this paragraph we assume M to be of BC_{r} -type. For the minimal boundary orbit (M_{0} = G/Q_{r}, F_{0}^{±}) with double foliation F_{0}^{±}, we define the automorphism group by

Aut(M_{0}, F_{0}^{±}) = { g ∈ Diffeo(M_{0}) : g_{∗}F_{0}^{±} = F_{0}^{±} }.

Tanaka [15] determined this group by establishing a Cartan connection on M_{0} and by showing that (M_{0} = G/Q_{r}, F_{0}^{±}) is the model space for the Cartan connection. Therefore, taking Remark 7.5 into account, we have

Aut(M_{0}, F_{0}^{±}) = G.  (8.1)

Theorem 8.1. Let (M = G/G_{0}, F^{±}) be a parahermitian symmetric space of BC_{r} -type associated with a simple GLA (1.1). Then

Aut(M, F^{±}) = Aut(M_{0}, F_{0}^{±}) = G.

Proof. We identify M with its φ-image in M̃. Since G acts on M effectively and F^{±} are G-invariant, the inclusion G ⊂ Aut(M, F^{±}) is clear. Now let f ∈ Aut(M, F^{±}). Then, by Lemma 2.4, f preserves the fibers of the double fibration M^{−} ⟵^{π^{−}} M ⟶^{π^{+}} M^{+}. Hence f induces the diffeomorphisms f^{±} of M^{±} such that π^{±} ∘ f = f^{±} ∘ π^{±}. Let f̃:= f^{−} × f^{+}. Clearly, the diffeomorphism f̃ preserves the product structure of M̃. We claim that f̃|_{M} = f. In fact, let p ∈ M, and let q = f(p) ∈ M. We write p = (p^{−}, p^{+}) and q = (q^{−}, q^{+}), where p^{±}, q^{±} ∈ M^{±}. The relation ϖ^{±}·φ = π^{±} (cf. §2) implies that q^{±} = ϖ^{±}(q) = ϖ^{±}(f(p)) = f^{±}(π^{±}(p)) = f^{±}(p^{±}). Hence f(p) = (q^{−}, q^{+}) = (f^{−}(p^{−}), f^{+}(p^{+})) = (f^{−} × f^{+})(p^{−}, p^{+}) = f̃(p). Since f̃ leaves M invariant, by Corollary 6.14 f̃ leaves M_{0} invariant. Obviously f_{0} := f̃|_{M_{0}} belongs to Aut(M_{0}, F_{0}^{±}). We wish to show that f̃ can be uniquely recovered by its restriction f_{0}. Corresponding to the expression M̃ = M^{−}×(M^{+})_{r}, one can express f̃ as f̃ = f_{1} × f_{2}, where f_{1} and f_{2} are diffeomorphisms of M^{−} and (M^{+})_{r}, respectively. It follows from Lemma 7.1 that the leaves of its double

foliation F_{0}^{±} arise as the fibers of the double fibration M^{−} ⟵^{π_{0}^{−}} M_{0} ⟶^{π_{0}^{+}} (M^{+})_{r} given in Theorem 3.1 (iv). Moreover this double fibration of M_{0} is just the

restriction of the trivial double fibration M^{−} ⟵^{ϖ_{0}^{−}} M̃ ⟶^{ϖ_{0}^{+}} (M^{+})_{r} (cf. (5.3)). Therefore, if we denote by f_{0}^{−} and f_{0}^{+} the diffeomorphisms of M^{−} and (M^{+})_{r} induced by f_{0}, then it follows that f_{0}^{−} = f_{1} and f_{0}^{+} = f_{1}. We have thus shown that f̃ is uniquely recovered from f_{0}. As a result, the correspondence f ↦ f_{0} is an injective homomorphism of Aut(M, F^{±}) into Aut(M_{0}, F_{0}^{±}). Consequently, in view of (8.1) we have that Aut(M, F^{±}) = G.

In this paragraph we are concerned with C_{r} -type. Under this assumption, for the GLA (4.10) we have 𝔤_{±1}(r) = (0), 𝔤_{±2}(r) = 𝔤_{±1}, 𝔤_{0}(r) = 𝔤_{0}, Z_{r} = Z and hence C(Z_{r}) = C(Z) = G_{0}. The rank decomposition (5.12) becomes 𝔤_{1} = ∐_{k=0}^{r} V_{k}, where V_{k} is a union of equi-dimensional G_{0}-orbits. Now consider the G_{0}-stable conic algebraic set V_{≤r−1}, which is the boundary ∂V_{r} of V_{r}. One can extend the cone ∂V_{r} to a cone field on the whole M^{−} by using the G-action on M^{−}. We call the cone field a generalized conformal structure K ([2]). One can consider the automorphism group Aut(M^{−}, K), the totality of diffeomorphisms leaving the cone field K invariant. This group was determined for each symmetric R-space M^{−} ([2]):

Aut(M^{−}, K) = \{ \begin{matrix}G, & r ≥ 2, \\ Diffeo(M^{−}), & r = 1. \end{matrix}  (8.2)

Recall that U^{−} = a_{r} U^{+}a_{r}^{−1} for C_{r} -type (cf. Theorem 3.1). This is equivalent to the condition a_{r} 0^{+} = 0^{−}, and the new origin (0^{−}, a_{r} 0^{+}) becomes (0^{−}, 0^{−}). By (5.3), M̃ takes the form M̃ = M^{−}(0^{−}, 0^{−}) × M^{+}(0^{−}, 0^{−}) = M^{−} × M^{−}. Further the minimal boundary orbit M_{0} becomes M_{0} = G(0^{−}, 0^{−}) = G/U^{−} = M^{−}, the diagonal set of M̃ = M^{−} × M^{−}.

Now let f ∈ Aut(M, F^{±}), and let f̃ = f^{−} × f^{+} be the extension of f to M̃ given in 8.1. Let us express f̃ as f̃ = f_{1} × f_{2} corresponding to the expression M̃ = M^{−} × M^{−}. By Corollary 6.14, f̃ leaves M_{0}, the diagonal of M̃, invariant, from which we have f_{1} = f_{2}, that is, f̃ = f_{1} × f_{1}. Thus it follows that the correspondence f ↦ f_{1} is an injective homomorphism of Aut(M, F^{±}) into Diffeo(M^{−}). The following lemma is essentially due to Tanaka [15].

Lemma 8.2. Suppose that (0^{−}, 0^{−}) is a fixed point of f̃. Then the differential (f_{1})_{∗} at 0^{−} leaves the cone ∂V_{r} stable.

Proof. Let Y ∈ ∂V_{r}. Then (0, tY) is a path in M_{≤r−1}^{∗}. By the assumption, the curve f̃(0, tY) lies in M_{≤r−1}^{∗} for |t| sufficiently small. Therefore Φ(f̃(0, tY)) = Φ(f_{1}(0), f_{1}(tY)) = f_{1}(tY) − f_{1}(0) = f_{1}(tY) lies in ∂V_{r}. Hence (f_{1})_{∗0^{−}}(Y) = lim_{t→0} \frac{1}{t}f_{1}(tY) ∈ ∂V_{r}.

Lemma 8.3. Let f̃ = f_{1} × f_{1} be the extension of f ∈ Aut(M, F^{±}) to M̃ = M^{−} × M^{−}. Then f_{1} ∈ Aut(M^{−}, K).

Proof. Let K = {(∂V_{r})_{p}}_{p∈M^{−}}, where (∂V_{r})_{p} denotes the cone at a point p ∈ M^{−} belonging to the field K. Note that, if p = b · 0^{−}, b ∈ G, then (∂V_{r})_{p} is just the cone b_{∗}(∂V_{r}). We have to show that (f_{1})_{∗}(∂V_{r})_{p} = (∂V_{r})_{f_{1}(p)}. Choose an element a ∈ G such that a^{−1}f_{1}b(0^{−}) = 0^{−}. Then the transformation a^{−1}f̃b on M̃ is the extension of a^{−1}fb ∈ Aut(M, F^{±}). Decompose a^{−1}fb as a^{−1}f_{1}b × a^{−1}f_{1}b corresponding to the decomposition M̃ = M^{−} × M^{−}. By Lemma 8.3, we see that (a^{−1}f_{1}b)_{∗0^{−}} leaves ∂V_{r} invariant. Consequently we have

(f_{1})_{∗p}(∂V_{r})_{p} = (f_{1})_{∗p}(∂V_{r})_{b·0^{−}} = (f_{1})_{∗p}b_{∗0^{−}}(∂V_{r}) = a_{∗0^{−}}(∂V_{r}) = (∂V_{r})_{a·0^{−}} = (∂V_{r})_{f_{1}(p)}.

This implies that f_{1} ∈ Aut(M^{−}, K).

Theorem 8.4. Let (M = G/G_{0}, F^{±}) be the parahermitian symmetric space of C_{r} -type associated with a simple GLA (1.1), and let K be the above generalized conformal structure on the symmetric R-space M^{−} = G/U^{−}. Then

Aut(M, F^{±}) = Aut(M^{−}, K) = \{ \begin{matrix}G, & r ≥ 2, \\ Diffeo(M^{−}), & r = 1. \end{matrix}

Proof. Suppose r ≥ 2. As we noted before Lemma 8.2, the correspondence Aut(M, F^{±}) ∋ f ↦ f_{1} ∈ Diffeo(M^{−}) is injective. But, by Lemma 8.3, the image f_{1} lies in Aut(M^{−}, K). Hence we have the injective homomorphism Aut(M, F^{±}) ↪ Aut(M^{−}, K). Since G is a subgroup of Aut(M, F^{±}), it follows from (8.2) that G ⊂ Aut(M, F^{±}) ≃ Aut(M^{−}, K) = G. Suppose next r = 1. Then the G-orbit decomposition of M̃ leaves M̃ = M ⨿ M^{−}. So, for any diffeomorphism f_{1} of M^{−}, (f_{1} ×f_{1})|_{M} is an element of Aut(M, F^{±}). Therefore we have Aut(M, F^{±}) ≃ Aut(M^{−}, K) = Diffeo(M^{−}) (cf. (8.27)).

Remark 8.5. As is seen in the table in 6.4, the parahermitian symmetric space M of C_{1}-type is SO^{0}(1, q + 1)/ SO(q) · ℝ^{+}, and the corresponding symmetric Rspace M^{−} is the conformal q-sphere.

Remark 8.6. In case where Aut(M, F^{±}) = G in Theorems 8.1 and 8.4, we also have

Aut(M, F^{±}) = Aut(M, F^{±}, ω).

Remark 8.7. A parahermitian symmetric space M = G/H associated to a simple GLA (1.1) is diffeomorphic to the cotangent bundle of the associated symmetric R-space M^{−} = G/U^{−}. Let M^{−} be the quaternionic Grassmannian Gr_{2}(ℍ^{4}) of quaternionic 2-planes in ℍ^{4}. There are two parahermitian symmetric spaces:

SL(4, ℍ)/ SL(2, ℍ) × SL(2, ℍ) × ℝ^{+} and E_{6(6)}/ Spin(5, 5) · ℝ^{+},

which have Gr_{2}(ℍ^{4}) as the associated symmetric R-spaces. The first one is of C_{2}-type; the second one is of BC_{2}-type. Theorems 8.1 and 8.4 tell us that the cotangent bundle of Gr_{2}(ℍ^{4}) has two different paracomplex structures which are both homogeneous.

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Soji Kaneyuki Department of Mathematics Nihon Institute of Technology Miyashiro-cho, Saitama 345-8501 Japan kaneyuki@hoffman.cc.sophia.ac.jp Received August 5, 2002 and in final form December 5, 2002