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\title[Stratifications and Automorphism Groups]
{Compactification of Parahermitian Symmetric Spaces And Its
Applications, II: Stratifications and Automorphism Groups}

\author{Soji Kaneyuki}

\address{Department of Mathematics\\
        Nihon Institute of Technology\\
        Miyashiro-cho, Saitama 345-8501\\
        Japan}
\email{kaneyuki@hoffman.cc.sophia.ac.jp}

\thanks{Research at MSRI is supported in part by NSF grant DMS-9810361.}

% end of page 73

\begin{document}

\begin{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.
\end{abstract}

\maketitle

% end of page 1

\section*{Introduction}

Let $M$ be a smooth manifold. A pair $(F^\pm, \omega)$ is called a
\textit{parak\"ahler structure} (or \textit{bi-Lagrangian structure})
on $M$ if $\omega$ is a symplectic form on $M$ and $F^\pm$ are two
Lagrangian foliations.  A significant property of
parak\"ahler structures is that a coadjoint orbit of a semisimple Lie
group is hyperbolic if and only if it admits an invariant parak\"ahler
structure (\cite{4}). A symmetric space $G/H$ of a Lie group $G$ is
called a \textit{parahermitian symmetric space} (or
\textit{bi-Lagrangian symmetric space})(\cite{5}) if $G/H$ admits a
$G$-invariant parak\"ahler structure $(F^\pm, \omega)$. 
The simplest example of parahermitian symmetric spaces is the
symmetric space $\SL(2,\bbb{R})/\bbb{R}^*$, realized as the one-sheeted
hyperboloid  $x^{2}+y^{2}-z^{2}=1$ in  $\bbb{R}^{3}= \Lie \SL(2,\bbb{R})$.
The Lagrangian foliations $F^\pm$ 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 \cite{5,6}. 
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
$\mathfrak{g}=\mathfrak{g}_{-1}+\mathfrak{g}_0+\mathfrak{g}_1$,
in such a way that $\mathfrak{g}= \Lie G$ and $\mathfrak{g}_0=
\Lie H$. For the explicit forms of simple parahermitian symmetric
pairs, see the tables in 6.4 and 7.2.

Now let $\mathfrak{g}=\mathfrak{g}_{-1}+\mathfrak{g}_0+\mathfrak{g}_1$
be a simple GLA, and let $M=G/G_0$ be the parahermitian symmetric space
 corresponding to the symmetric pair $(\mathfrak{g},\mathfrak{g}_0)$, 
where $G$ is the largest possible open subgroup of the automorphism
group of $\mathfrak{g}$ such that $G/G_0$ is realized as an 
$\Ad \mathfrak{g}$-orbit in  $\mathfrak{g}$.  The subgroup 
$U^\pm:= G_0\exp\mathfrak{g}_{\pm 1}$ are the parabolic subgroups 
with $\Lie U^\pm = \mathfrak{g}_0 + \mathfrak{g}_{\pm 1}$.
The flag manifolds $M^\pm = G/U^\pm$ are symmetric $R$-spaces.
Let $r$ be the ranks of $M^\pm$.  Then there are exactly $r$ numbers
of  $\mathfrak{sl}(2,\bbb{R})$-triplets in  $\mathfrak{g}$ which are
pairwise commutative and whose direct sum is expressed as a graded
subalgebra 
$\mathfrak{a}_{-1}+ \mathfrak{a}_0 +  \mathfrak{a}_1$ 
in  $\mathfrak{g}$ (cf. \cite{9}).  One has the root system 
$\Delta(\mathfrak{g}, \mathfrak{a}_0 )$ of  $\mathfrak{g}$
with respect to the abelian subspace $\mathfrak{a}_0$.
 $\Delta(\mathfrak{g}, \mathfrak{a}_0 )$ is of  $BC_r$-type or
$C_r$-type (\cite{9,1}).  We say that $G/G_0$ and the GLA  $\mathfrak{g}$  
are of $BC_r$-type or $C_r$-type, if $\Delta(\mathfrak{g}, \mathfrak{a}_0 )$ is.

%\cite{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
%They are in
%one-to-one correspondence (up to covering) with semisimple graded Lie
%%%% algebras (shortly, GLAs) of the first kind $\mathfrak{g} =
%\mathfrak{g}_{-1} + \mathfrak{g}_0 + \mathfrak{g}_1$. 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 of $G$ with
%abelian nilradical.

% end of page 2 (I-1)

%The class of parahermitian symmetric spaces of simple Lie groups
%contains two interesting classes of symmetric spaces:
%\begin{enumerate}
%\item[(i)]
%\textit{causal parahermitian symmetric spaces} or \textit{symmetric
%spaces of Cayley type}, and
%\item[(ii)]
%\textit{simple complex parahermitian symmetric spaces}.
%\end{enumerate}
%Let $G/H$ be a simple symmetric space. Then it is a symmetric space of
%Cayley type if $G$ is of Hermitian type. $G/H$ has an invariant causal
%structure. A typical example is 
%$\SU(n,n)/\SL(n,\bbb{C}) \cdot {\bbb{R}}^*$, the causal
%structure of which comes from the direct sum of the two copies of the
%cone of positive semi-definite $n\times n$ Hermitian matrices. $G/H$
%is a simple complex parahermitian symmetric space if $G$ is complex
%simple and $H$ is the centralizer of $\bbb{C}^*$. It is a Stein
%symmetric space and a complexification of an irreducible Hermitian
%symmetric space. A typical example is
%$\SL(n,\bbb{C})/\operatorname{S}(\GL(p,\bbb{C})\times
%\GL(n-p,\bbb{C}))$, which is a complexification of the complex
%Grassmannian $\SU(n)/\operatorname{S}(\U(p)\times \U(n-p))$. Note that
%the class of simple complex parahermitian symmetric spaces is the same
%as the class
% end of page 3 (I-2)
%of simple reducible pseudo-Hermitian symmetric spaces, treated in
%\cite{1}. We will come back again to those two types of symmetric
%spaces at the last part of the Introduction. Simple parahermitian
%symmetric spaces break up into two classes---those of $BC$-type and
%those of $C$-type.      May 6, 2002 up to here.

A fundamental problem of the geometry of parahermitian symmetric
spaces ($M=G/H$, $F^\pm$, $\omega$) is to determine the automorphism
group $\Aut(M, F^\pm)$---the group consisting of diffeomorphisms of
$M$ leaving the double foliation $F^\pm$ 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  \cite{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  $\tilde{M} = M^-\times M^+$, which is the natural
$G$-equivariant compactification of $M$ (cf. \cite{7} and Sections
2 and 3).  The second step is to show that the $G$-orbit 
decomposition gives $\tilde{M}$ 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 
$\Delta(\mathfrak{g}, \mathfrak{a}_0 )$, we construct a grading
$\mathfrak{g}=\sum_{k=-2}^2 \mathfrak{g}_k(r)$ of the 2nd kind
having the property that 
$\mathfrak{g}_{\pm 1}(r)$ are expressed as the direct sum of
two equi-dimensional abelian subspaces, 
$\mathfrak{g}_{\pm 1}^+(r) + \mathfrak{g}_{\pm 1}^-(r)$  (cf. (4.10),(4.14)).
Such a grading is called a \textit{pseudo-product grading} of  $\mathfrak{g}$
(Tanaka \cite{15}).  Let $Q_r$ be the isotropy subgroup of $G$ at the base 
point of the minimal dimensional $G$-orbit $M_0$.  The third step is to
show that $Q_r$ is the parabolic subgroup with $\Lie Q_r =
\mathfrak{g}_{-2}(r) + \mathfrak{g}_{-1}(r) + \mathfrak{g}_0(r)$
(Proposition 4.6) 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 $F_0^\pm$ induced from the product structure of $\tilde{M}$.
We denote by $\Aut(M_0, F_0^\pm)$  the group of diffeomorphisms of
$M_0$ leaving $F_0^\pm$ invariant.  Then the Tanaka's theory \cite{15}
of Cartan connections for pseudo-product manifolds, together with the
third step assertion guarantees the validity of the relation
$\Aut(M_0, F_0^\pm) = G$.

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 $F_0^\pm$
becomes trivial.  But, in turn, $M^-$ has the generalized conformal
structure $\mathcal{K}$, which is obtained from the cone defined as the
union of singular $G_0$-orbits in  $\mathfrak{g}_1$ (\cite{2}).  
We determined in \cite{2} the conformal automorphism group 
$\Aut(M^-, \mathcal{K})$ for each symmetric $R$-space $M^-$.

The fourth or the last step is to obtain the injective homomorphism of
$\Aut(M, F^\pm)$ to  $\Aut(M_0, F_0^\pm)$  or to  $\Aut(M^-, \mathcal{K})$.
For this purpose, the stratification of $\tilde{M}$ is essential.
Let  $f \in \Aut(M, F^\pm)$.  Then $f$ extends to $\tilde{M}$ as an
automorphism $\tilde{f}$ of the product structure of $\tilde{M}$.
It follows that $\tilde{f}$ is an automorphism of the stratification of
$\tilde{M}$ (Corollary 6.14).  In particular $\tilde{f}$ leaves $M_0$ stable.
It is shown that the assignment 
$f\mapsto\tilde{f}\rest{M_0}$ gives the injective homomorphism
as desired.  The main results are Theorems 8.1 and 8.4.  


         
%    This problem was
%first treated by Tanaka \cite{15} and solved for parahermitian
%symmetric spaces $G/H$ with $G$ \textit{classical} simple. He starts
%with a special type of simple GLA $\mathfrak{g}$ of the second kind,
%which he calls a pseudo-product simple GLA. Then, by using this
%grading, he constructs two flag manifolds $M$ and $N$, in his
%notation, of a group $G$ with Lie $G=\mathfrak{g}$. $G$ acts on the
%product manifold $M\times N$ diagonally. There are a single open orbit
%$\Omega$ and a single closed minimal dimensional orbit $R$.

% end of page 4 (I-3)

%$\Omega$ turns out to be a parahermitian symmetric space and $R$ is a
%flag manifold with double foliation which he calls a pseudo-product
%manifold. The method of determining the automorphism group
%$\Aut(\Omega$) of the double foliation consists of the two steps:
%First he determines the automorphism group $\Aut(R)$ of the double
%foliation by his favorite method of constructing a Cartan connection
%on $R$. The second step is to obtain the injective homomorphism of
%$\Aut(\Omega$) into $\Aut(R$). This is done by extending an
%automorphism $f \in \Aut(\Omega$) to the compactification $M\times N$
%and by checking a criterion for $f$ to leave $R$ stable. He checks the
%criterion case-by-case for each classical simple parahermitian
%symmetric space.

%The final goal of this paper is to determine $\Aut(M,F^\pm)$ for all
%simple parahermitian symmetric spaces $M$. Theorems 8.1 and 8.4 are
%our main theorems. Our results contain Tanaka's classical case. Our
%approach is more Lie- and Jordan-theoretic. Let $M=G/H$ be a
%parahermitian
% end of page 5 (I-4)
%symmetric space corresponding to a simple GLA
%$\mathfrak{g}=\mathfrak{g}_{-1}+ \mathfrak{g}_0+ \mathfrak{g}_1$.
%First of all, in our previous paper \cite{7}, we obtained a natural
%equivariant open dense imbedding of $M=G/H$ into the product manifold
%$\tilde{M}=M^- \times M^+$ of the flag manifold $M^\pm=G/U^\pm$
%(symmetric $R$-space), where $U^\pm$ are parabolic subgroups with
%$\Lie U^\pm = \mathfrak{g}_0+ \mathfrak{g}_{\pm 1}$. We obtained the
%orbit decomposition of $\tilde M$ under the diagonal $G^0$-action,
%where $G^0$ is the identity component of $G$. Let $r$ be the rank of
%the symmetric $R$-spaces $M^\pm$. Then there are exactly $r+1$ \
%$G^0$-orbits $M_r$, $M_{r-1}$, $\ldots$, $M_0$ satisfying the
%condition $\dim\tilde{M} = \dim M_r > \dim M_{r-1} > \cdots > \dim
%M_0$. Note that $M$ is realized as $M_r$. In \S3 in the present paper,
%we will show that each $G^0$-orbit coincides with a $G$-orbit (cf.\
%Theorem 3.1). The direct sum of the graded subspaces $\mathfrak{g}_1
%\oplus \mathfrak{g}_{-1}$ is viewed as the tangent space to $\tilde{M}
%= M^- \times M^+$ at the origin, and it is naturally imbedded in
%$\tilde{M}$ as a Zariski open subset. The intersection
%$M^b_k:=M_k\cap(\mathfrak{g}_1 \oplus \mathfrak{g}_{-1})$ can be
%described in $\mathfrak{g}_1 \oplus \mathfrak{g}_{-1}$ in terms of the
%rank of the Bergman operator $B$
% end of page 6 (I-5)
%of the Jordan pair of $\mathfrak{g}_1 \oplus \mathfrak{g}_{-1}$ (cf.\
%Theorem 4.12 \cite{7}). For example, let
%$M=\SL(n,\bbb{C})/\operatorname{S}(\GL(r,\bbb{C})\times
%\GL(n-r,\bbb{C}))$, \ $1\leq r \leq n-r$. Then the realization $M^b_k$
%of the orbit $M_k$ is given by the set
%\[\{\,(X,Y)\in M_{r,n-r}(\bbb{C}) \times M_{n-r,r}(\bbb{C}):
%\rk(1-XY)=k\,\}, \qquad 0\leq k\leq r.\] By using the above
%expression, we see that the closure $\overline{M}_k$ of $M_k$ is given
%by $M_{\leq k} :=\coprod^k_{i=0} M_i$. As is seen from the above
%example, the realization $M^b_k$ is viewed as a generalization of a
%bounded model of a symmetric bounded domain. However this realization
%of $G$-orbits in $\mathfrak{g}_1 \oplus \mathfrak{g}_{-1}$ is not
%suitable for determining the singular locus of a $G$-orbit since the
%fibers of the map $B$ are rather complicated.

%The first aim of this paper is to obtain the stratification of
%$\tilde{M}$ by $G$-orbits (cf.\ Theorem 6.11). For a simple GLA
%$\mathfrak{g}=\mathfrak{g}_{-1} + \mathfrak{g}_0+ \mathfrak{g}_1$, one
%has a root system $\Delta$ of $\mathfrak{g}$ admitting the partition
%$\Delta= \Delta_{-1} \cup \Delta_0 \cup \Delta_1$ such that the root
%vectors in $\Delta_k$ belong to the subspace $\mathfrak{g}_k$ (cf.\
%\S3). Furthermore one can choose a maximal system of strongly
%orthogonal roots ${\beta_1, \dots, \beta_r}$
% end of page 7 (I-6)
%in $\Delta_1$ consisting of the longest roots. In \S4, by using the
%explicit form of the orthogonal projection of roots in $\Delta$ onto
%the real span of $\beta_1$, $\dots$, $\beta_r$, we construct $r$
%gradings of $\mathfrak{g}$, \ $\mathfrak{g}= \sum_{k=-2}^2
%\mathfrak{g}_k(l)$ for each $1\le l\le r$. The isotropy subalgebra of
%$\mathfrak{g}$ at a point in the orbit $M_l$ can be nicely described
%in the terms of this grading (Propositions 4.3 and 4.6). Especially
%when $l=r$, the grading $\mathfrak{g}=\sum_{k=-2}^2 \mathfrak{g}_k(r)$
%turns out to be just Tanaka's pseudo-product simple GLA. The subspaces
%$\mathfrak{g}_{\pm 1}(r)$ admit the decompositions into abelian
%subspaces $\mathfrak{g}_{\pm 1}(r) = \mathfrak{g}_{\pm 1}^+(r) +
%\mathfrak{g}_{\pm 1}^-(r)$ (cf.\ (4.10), (4.14)). It turns out that
%the changing procedure from our compactification $\tilde{M}=M^- \times
%M^+$ to Tanaka's is done just by changing the origin of $\tilde{M}$ to
%a certain point lying on the minimal boundary orbit $M_0$ (cf.\
%\S5). The original grading $\mathfrak{g} = \mathfrak{g}_{-1} +
%\mathfrak{g}_0 + \mathfrak{g}_1$ is carried over to a new grading
%$\mathfrak{g}=\mathfrak{g}'_{-1} +\mathfrak{g}'_0 +\mathfrak{g}'_1$ by
%the automorphism of $\mathfrak{g}$ corresponding to the change of the
%origin. This process is viewed as the Cayley transform. The tangent
%space to $\tilde{M}$ at the new
% end of page 8 (I-7)


%(origin is given by $\mathfrak{g}_1\oplus \mathfrak{g}'_1 =
%(\bigl(\mathfrak{g}_2(r)+\mathfrak{g}_1^-(r)\bigr)\oplus
%\bigl(\mathfrak{g}_2(r)+\mathfrak{g}_1^+(r)\bigr)$, which is imbedded
%in $\tilde{M}$ as a Zariski open subset.

%Consider now the (reductive) graded subalgebra
%$\mathfrak{g}_{\mathrm{ev}}(r) = \mathfrak{g}_{-2}(r) +
%\mathfrak{g}_0(r) + \mathfrak{g}_2(r)$ of $\mathfrak{g}$. From the
%classification of simple GLAs of the second kind \cite{8}, it follows
%that $\mathfrak{g}_2(r)$ has the structure of a simple Jordan algebra
%and that the adjoint action of $\mathfrak{g}_0(r)$ on
%$\mathfrak{g}_2(r)$ coincides with the structure algebra. Consequently
%one has the ``rank decomposition'' $\mathfrak{g}_2(r)=\coprod
%_{l=0}^rV_l$, where $V_l$ is a union of equi-dimensional orbit under
%the structure group (\cite{2}). The closure $\overline{V_l}$ is given
%by $V_{\leq k} := \coprod_{i=0}^l V_i$. $V_{\leq k}$ is a
%determinantal variety of classical type or exceptional type. Now we go
%back to our main concern. Let us consider the realization of a
%$G$-orbit $M_k$ in $\mathfrak{g}_1\oplus \mathfrak{g}'_1$, that is,
%$M_k^* := M_k \cap (\mathfrak{g}_1\oplus \mathfrak{g}'_1)$.  Put
%$M_{\leq k}^* := \coprod_{i=0}^k M_i^*$. In order to determine
%$\Sing(M_{\leq k})$, it is enough to determine the singular locus
%$\Sing(M_{\leq k}^*)$ of the algebraic variety $M_{\leq k}^*$ in
%$\mathfrak{g}_1\oplus \mathfrak{g}'_1$. By using the connecting map
%$\Phi$ of $\mathfrak{g}_1\oplus \mathfrak{g}'_1$
% end of page 9 (I-8)
%onto $\mathfrak{g}_2(r)$ due to Tanaka \cite{15} (cf.\ Definition
%5.1), we can show that each $M_k^*$ is expressed as the complete
%inverse image $\Phi^{-1}(V_k)$. (Proposition 5.12). By using this, we
%have $\Sing(M_{\leq k}^*) = \Phi^{-1}\bigl(\Sing(V_{\leq k})\bigr)$
%(cf.\ Proposition 6.3), which reduces the problem to finding the
%singular locus of the determinantal varieties. But this is known for
%the classical case (\cite{11}). We carry out here the remaining two
%cases and the real cases (\S6). In the course of the proof of Theorem
%6.11, we get, as a byproduct, the Siegel-type realization $M_k^*$ of
%the $G$-orbit $M_k$ (cf.\ Theorem 5.13). For the case of
%$M=\SL(n,\bbb{C})/\operatorname{S}(\GL(r,\bbb{C})\times
%\GL(n-r,\bbb{C}))$, \ $1\leq r\leq n-r$, we have that $\mathfrak{g}_1
%= M_p(\bbb{C}) + M_{n-2p,p}(\bbb{C})$ and $\mathfrak{g}'_1 =
%M_p(\bbb{C}) + M_{p,n-2p(\bbb{C}})$, and that $M_p^*=\{\,(x,u)\oplus
%(y,v)\in \mathfrak{g}_1\oplus\mathfrak{g}'_1: \rk(y-x+vu)=k\,\}$. As a
%corollary, one has that a diffeomorphism $f$ of $\tilde{M}$ leaving
%the open orbit $M_r$ stable leaves all other orbits $M_k$ stable
%(Corollary 5.14).

We want to supplement some details on the stratification of $\tilde{M}$,
since it is a rather independent topic.  By a stratification of a real
analytic manifold $X$, we mean a partition
$X = \coprod_{k=0}^s A_k$ 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 
$A_{\leq k} := \coprod_{i=0}^k A_i$   $(0\leq k\leq s)$, and (S3) the
singular locus $\Sing(\overline{A_k})$ is given by $A_{\leq{k-1}}$
$(1\leq k\leq {s-1})$. Let $\tilde{M} = \coprod_{k=0}^r M_k$ be the
$G$-orbit decomposition ($G$ acts on $\tilde{M}$ diagonally), where
$\dim M_k > \dim M_{k-1}$ and $M_r = M$ is open dense in $\tilde{M}$.
For the $G$-orbit decomposition of $\tilde{M}$, the properties (S1) and
(S2) were already proved in \cite{7}. We will verify (S3) in this paper.

Suppose first that the $GLA$ $\mathfrak{g}$ is of $BC_r$-type.  
We consider the two abelian subspaces of $\mathfrak{g}$:
$\mathfrak{g}_1 = \mathfrak{g}_2(r) + \mathfrak{g}_1^-(r)$,
$\mathfrak{g}'_1 := \mathfrak{g}_2(r)  + \mathfrak{g}_1^+(r)$.
The direct sum $\mathfrak{g}_1 \oplus\mathfrak{g}'_1$ is imbedded in
$\tilde{M}$ as an open subset.  We identify 
$\mathfrak{g}_1 \oplus\mathfrak{g}'_1$ with its image in $\tilde{M}$.
Let $M_k^* := M_k \cap(\mathfrak{g}_1 \oplus\mathfrak{g}'_1)$,
which is open dense in $M_k$.  The closure $\overline{M_k^*}$ of
$M_k^*$ in $\mathfrak{g}_1 \oplus\mathfrak{g}'_1$ coincides with
$M_{\leq k}^* := \coprod_{i=0}^k M_i^*$ and it is an algebraic variety
in $\mathfrak{g}_1 \oplus\mathfrak{g}'_1$ (Theorem 5.13).  Obviously
we have $\Sing(M_{\leq k}) = G(\Sing(M_{\leq k}^*))$.  Thus, in order to
find the singular locus of  $M_{\leq k}$, it is enough to find that of 
$M_{\leq k}^*$.  Now we look at $\mathfrak{g}_2(r)$.  The Levi subgroup
$L$ of $Q_r$ corresponding to $\mathfrak{g}_0(r)$ acts on 
$\mathfrak{g}_2(r)$.  We have a partition $\mathfrak{g}_2(r) =
\coprod_{k=0}^r V_k$, 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 $\mathfrak{g}_2(r)$.
The validity of (S1) and (S2) was proved in \cite{9}; as was shown in \cite{9},
$V_{\leq k}$ $(0\leq k \leq r-1)$ is an algebraic variety in 
$\mathfrak{g}_2(r)$.
%Compiled up to here 5/7
By Proposition 6.3, the problem of finding the singular locus of 
$M_{\leq k}^*$ is reduced to finding that of $V_{\leq k}$.  In the case 
where the $GLA$  $\mathfrak{g}$ is of $C_r$-type, we have 
$\mathfrak{g}_1 = \mathfrak{g}'_1 = \mathfrak{g}_2(r)$.
Therefore it is enough to concern 
the determinantal varieties $V_{\leq k}$ in $\mathfrak{g}_1$
for the case where the GLA $\mathfrak{g}$ is of  $C_r$-type.
In the realization of $\mathfrak{g}_1$ as a matrix space,
$V_{\leq k}$ is a complex or real determinantal variety of classical or
exceptional type.  The determination of  $\Sing(V_{\leq k})$ will be 
carried out in Section 6.  Levasseur-Stafford \cite{11} is a good reference
for the complex classical case.  The final result on the stratification of
 $\tilde{M}$ is given by Theorem 6.13.
%Compiled up to here 5/7

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, $\tilde{M}$ is the causal compactification of $M$
(cf. \cite{12}).  As an application of the results of  the present paper,
one can determine the full causal automorphism group of $M$.
In the forth-coming paper, we will treat this topic.
%Cmpld up to here 5/7

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 causal compactification of a symmetric space of
% end of page 10 (I-9)
%Cayley type was considered in the recent paper by \'Olafsson-\O{}rsted
%\cite{12}. The causal compactification can also be understood in our
%context. We will take up this problem in the forthcoming paper, and as
%an application of the present paper we will give the determination of
%the full causal automorphism group of a symmetric space of Cayley
%type. For simple complex parahermitian symmetric space, everything can
%be done in the holomorphic category, which was done in the previous
%work \cite{1}.

%The author would like to express his hearty thanks to Simon Gindikin
%who invited him to participate in the Integral Geometry Program at
%MSRI in the fall of 2001. The author also would like to thank MSRI for
%hospitality and for providing pleasant surroundings for research
%during his stay there. The author is grateful to H.\ Asano and T.\
%Nagano for frequent valuable conversations.

% end of page 11 (I-10)

The paper is organized as follows:
\begin{enumerate}
\item[1.] Preliminaries on parahermitian symmetric spaces.
\item[2.] Double foliation of $M$.
\item[3.] Orbit structure of $\tilde{M}$.
\item[4.] Isotropy subgroups for boundary orbits.
\item[5.] Siegel-type realization of orbits.
\item[6.] Stratification of $\tilde{M}$.
\item[7.] Double foliation on the minimal boundary orbit.
\item[8.] Determination of automorphism groups of $M$.
\end{enumerate}
%Cmpld up to here 5/7
\section{Preliminaries on parahermitian symmetric spaces}

\subsection{} %1.1

Let $M$ be a connected $2n$-dimensional smooth manifold, and let
$F^\pm$ be two $n$-dimen\-sion\-al completely integrable distributions
on $M$. $(F^\pm)$ is called a \textit{paracomplex structure} on $M$
(\cite{5}) if the tangent bundle $TM$ of $M$ can be expressed as the
Whitney sum $F^+\oplus F^-$. In this case $(M,F^\pm)$ is called a
\textit{paracomplex manifold}. A paracomplex manifold $(M, F^\pm)$ is
called a \textit{parak\"ahler manifold} (\cite{5}) if there exists a
symplectic form $\omega$ on $M$ with respect to which $F^\pm$ are
Lagrangian subbundles. For a parak\"ahler manifold $(M,F^\pm,\omega)$,
one can consider the two kinds of automorphisms: By a
\textit{paracomplex automorphism} of $M$ we mean a diffeomorphism of
$M$ which leaves $F^\pm$ invariant. By a \textit{paracomplex isometry}
of $M$ we mean a paracomplex automorphism leaving $\omega$
invariant. We denote by $\Aut(M,F^\pm)$ (resp.\
$\Aut(M,F^\pm,\omega)$) the group of paracomplex automorphisms (resp.\
paracomplex isometries) of $M$. The group $\Aut(M,F^\pm,\omega)$ is
always a finite-dimensional Lie group, but $\Aut(M,F^\pm)$ is not in
general.
% end of page 12 (1-1)

\begin{Definition}[\cite{5}] % 1.1
Let $M=G/H$ be an almost effective symmetric coset space of a Lie
group $G$, and let $(F^\pm,\omega)$ be a parak\"ahler structure on
$M$. If $G$ acts on $M$ as paracomplex isometries with respect to
$(F^\pm,\omega)$, then ($M=G/H$, $F^\pm$, $\omega$) is called a
\textit{parahermitian symmetric space}.
\end{Definition}

For each parahermitian symmetric space $M=G/H$, the Lie algebra
$\mathfrak{g} = \Lie G$ has the structure of a GLA of the first kind
$\mathfrak{g} = \mathfrak{g}_{-1} + \mathfrak{g}_0 + \mathfrak{g}_1$
(\cite{5}). Under the assumption of semisimplicity of $\mathfrak{g}$,
the assignment $M\rightsquigarrow
\mathfrak{g}=\mathfrak{g}_{-1}+\mathfrak{g}_0+\mathfrak{g}_1$ 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 (\cite{6}). In this case
the original parak\"ahler structure on $M$ can be recovered by the
grading of $\mathfrak{g}$.

\subsection{} % 1.2

Let us start with a real simple GLA of the first kind
\[\mathfrak{g}=\mathfrak{g}_{-1}+\mathfrak{g}_0+\mathfrak{g}_1. \eqno(1.1)\]
The automorphism group of the Lie algebra $\mathfrak{g}$ is denoted by
$\Aut\mathfrak{g}$. Let $(Z,\tau)$ be
% end of page 13 (1-2)
the associated pair of the GLA: $Z$ is the \textit{characteristic
element} of the GLA, that is, $Z$ is a unique element of
$\mathfrak{g}_0$ satisfying the condition $\ad Z=k1$ on
$\mathfrak{g}_k$, \ $k=0$, $\pm 1$, and $\tau$ is a Cartan involution
of $\mathfrak{g}$ satisfying $\tau(Z)=-Z$. Let $\sigma=\Ad\exp\pi i
Z$.Then $\sigma$ is the involutive automorphism of $\mathfrak{g}$ such
that $\sigma=1$ on $\mathfrak{g}_0$ and $-1$ on $\mathfrak{m} :=
\mathfrak{g}_{-1}+\mathfrak{g}_1$. Thus we have a symmetric triple
$(\mathfrak{g},\mathfrak{g}_0,\sigma)$. Let $G_0$ be the centralizer
of $Z$ in $\Aut\mathfrak{g}$, and let $G$ be the open subgroup of
$\Aut\mathfrak{g}$ generated by $G_0$ and $\Ad\mathfrak{g}$. Note that
$\Lie G_0=\mathfrak{g}_0$ and that $G_0$ coincides the group of
grade-preserving automorphisms of the GLA (1.1).

\begin{Proposition} % 1.2
The coset space $M=G/G_0$ is a parahermitian symmetric space
corresponding to the symmetric triple\/
$(\mathfrak{g},\mathfrak{g}_0,\sigma)$. The group $G$ acts on $M$
effectively as paracomplex isometries.
\end{Proposition}

\begin{proof}
If we put $\tilde{\sigma}(a)=\sigma a\sigma$, \ $a\in G$, then
$\tilde{\sigma}$ is an involutive automorphism of
$\Aut\mathfrak{g}$. $\tilde{\sigma}$ leaves $G$ stable. Let
$G_{\tilde{\sigma}}$ be the subgroup of $G$ consisting of all
$\tilde{\sigma}$-fixed elements of $G$. Then, from the definition of
$\sigma$, it follows that $G_0$ is an open subgroup of
$G_{\tilde{\sigma}}$. So $M=G/G_0$ is a symmetric space. By
% end of page 14 (1-3)
the definition of $G_0$, \ $M$ is realized in $\mathfrak{g}$ as the
adjoint $G$-orbit through $Z\in\mathfrak{g}$. Therefore the
Kirillov-Kostant form
\[\tilde{\omega}(X,Y)=(Z,[X,Y]), \qquad X,Y\in \mathfrak{g} \eqno(1.2)\]
induces a $G$-invariant symplectic form $\omega$ on $M$,where
(   ,   ) denotes the Killing form of $\mathfrak{g}$.  Let $0\in M$
be the origin of $M=G/G_0$. We identify $\mathfrak{m}$ with the
tangent space $T_0 M$ of $M$ at $0$. Then the two $G_0$-invariant
subspaces $\mathfrak{g}_{\pm 1}$ extend to $G$-invariant distributions
$F^\pm$ on $M$, which are Lagrangian with respect to $\omega$ by
(1.2). The complete integrability of $F^\pm$ has been proved in two
ways, one in \cite{5} by 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
$\mathfrak{m}$ can be identified with the linear isotropy group at
$0\in M=G/G_0$. Let $a\in G$ and suppose that $a$ acts on $M$ as the
identity.  Then $a\in G_0$ and $a$ acts on $\mathfrak{m}$ as the
identity. Since $\mathfrak{g}$ is simple, we have $\mathfrak{g}_0 =
[\mathfrak{g}_1,\mathfrak{g}_{-1}]$. So it follows that $a$ acts on
$\mathfrak{g}_0$ as the identity. This implies that $a$
% end of page 15 (1-4)
is the unit element of $G$.
\end{proof}

The parahermitian symmetric space $(M=G/G_0, F^\pm, \omega)$ thus
constructed is called the \textit{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
among the parahermitian symmetric spaces corresponding to the
symmetric triple $(\mathfrak{g}, \mathfrak{g}_0, \sigma)$.

% end of page 16 (1-5)

\section{Double foliations of $M$}

Let $(M=G/G_0, F^\pm, \omega)$ be the parahermitian symmetric space
associated to a simple GLA (1.1). We denote by $F^\pm(p)$ the leaves
of $F^\pm$ through a point $p\in M$. It is easy to see that the leaves
$F^\pm(0)$ through the origin $0\in M$ are given by the orbits
$(\exp\mathfrak{g}_{\pm 1})\cdot 0$. Consider the parabolic subgroups
$U^\pm := G_0\exp\mathfrak{g}_{\pm 1}$ of $G$. Note that
$\mathfrak{u}^\pm := \Lie U^\pm = \mathfrak{g}_0 +
\mathfrak{g}_{\pm1}$. The flag manifolds $G/U^\pm$ are called the
\textit{symmetric $R$-spaces associated with $M$} or with the GLA
(1.1).

\begin{Lemma} % 2.1
Let $M^\pm$ be the sets of leaves of $F^\pm$ on $M$. Then $M^\pm$ are
identified with the flag manifolds $G/U^\pm$.
\end{Lemma}

\begin{proof}
Since $F^\pm$ are $G$-invariant, any element of $G$ induces a
permutation on the sets of leaves of $F^\pm$, which means that $G$
acts on $M^\pm$. The transitivity of $G$ on $M^\pm$ follows from that
of $G$ on $M$. Now let $g\in G$ and suppose for example $g F^+(0) =
F^+(0)$. Then the point $g\cdot 0$ is in $F^+(0)$. One can write
$g\cdot 0 = \exp X\cdot 0$ for some $X\in\mathfrak{g}_1$. Therefore
$g^{-1}\exp X\in G_0$, which implies $g\in U^+$, and $M^+$ is
expressed
% end of page 17 (2-1)
as $G/U^+$.
\end{proof}

\begin{Lemma} % 2.2
For any point $p\in M$, we have $F^+(p)\cap F^-(p) = \{p\}$.
\end{Lemma}

\begin{proof}
One can assume $p$ to be the origin $0$. Any point $q\in F^+(0)\cap
F^-(0)$ can be expressed as $q = \exp X\cdot 0 = \exp Y\cdot 0$ for
$X\in\mathfrak{g}_1$ and $Y\in\mathfrak{g}_{-1}$. This implies that
$\exp X\in (\exp Y)G_0\subset U^-$. Consequently $\exp X\in U^+\cap
U^- = G_0$ and hence $\exp X\in (\exp\mathfrak{g}_1)\cap G_0=(1)$,
which implies $X=0$. Thus we have $q=p$.
\end{proof}

We denote the points $F^\pm(0)\in M^\pm$ by $0^\pm$, and let us
consider the product manifolds
\[\tilde{M} = M^-\times M^+ \eqno(2.1)\]
with the origin $(0^-, 0^+)$. $\tilde{M}$ has a double foliation
arising from the product structure. The leaves through the point
$(g_1 0^-,g_20^+)$, \ $g_1$,
$g_2\in G$, are denoted by $M^\pm(g_1 0^-,
g_2 0^+)$.  $M^-(g_1 0^-,g_2 0^+)$
is called the horizontal leaf and $M^+(g_1 0^-,
g_2 0^+)$ is called the vertical leaf. They are given by
\[\begin{aligned}[c]
M^-(g_1 0^-,g_2 0^+) &= G g_1 0^-\times\{g_2 0^+\} = G/g_1
U^-g_1^{-1}\times\{g_2 0^+\},\\
M^+(g_1 0^-,g_2 0^+) &= \{g_1 0^-\}\times G g_2 0^+ = \{g_1 0^-\}\times
G/g_2 U^+ g_2^{-1}.
\end{aligned} \eqno(2.2)\]
% end of page 18 (2-2)
Let us define a map $\varphi$ of $M$ to $\tilde{M}$ by putting
\[\varphi(p) = \bigl(F^-(p), F^+(p)\bigr), \qquad p\in M. \eqno(2.3)\]

\begin{Lemma} % 2.3
$\varphi$ is a $G$-equivariant open imbedding of $M$ into $\tilde{M}$
and preserves the double foliations on $M$ and $\tilde{M}$\textup{;}
actually we have $\varphi\bigl(F^\mp(p)\bigr)\subset
M^\pm\bigl(\varphi(p)\bigr)$, \ $p\in M$.
\end{Lemma}

\begin{proof}
Let $g\in G$. Then $\varphi(g\cdot 0) = \bigl(F^-(g\cdot 0),
F^+(g\cdot 0)\bigr) = (g\cdot 0^-, g\cdot 0^+)$. From this and Lemma
2.2 it follows that $\varphi$ is $G$-equivariant imbedding. The
openness of $\varphi$ follows from dimension counting. Now let $q\in
F^-(p)$.  Then $F^-(q) = F^-(p)$ and hence $\varphi(q) = \bigl(F^-(p),
F^+(q)\bigr)$, which implies that $\varphi(q)$ lies on the vertical leaf
through $\varphi(p)$.
\end{proof}

Since $U^-\cap U^+=G_0$, \ $M$ has the structure of the double
fibration over $M^\pm$. The projections $\pi^\pm\colon M\to M^\pm$ are
given by
\[\pi^\pm(g\cdot 0) = g\cdot 0^\pm, \qquad g\in G. \eqno(2.4)\]
\begin{diagram}
&&M&&\cr
&\ldTo^{\pi^-}&&\rdTo^{\pi^+}&\cr
M^-&&&&M^+\cr
\end{diagram}

% end of page 19 (2-3)

\begin{Lemma} % 2.4
For each point $p\in M$, we have
\[F^\pm(p) = (\pi^\pm)^{-1}\bigl(\pi^\pm(p)\bigr).\]
\end{Lemma}

\begin{proof}
It is enough to prove the assertion for the case where $p$ is the
origin. Choose a point $(\exp X)0\in F^+(0)$, \
$X\in\mathfrak{g}_1$. Then we have $\pi^+\bigl((\exp X)0\bigr) = (\exp
X)0^+ = 0^+$, and hence $(\exp X)0\in (\pi^+)^{-1}(0^+) =
(\pi^+)^{-1}\bigl(\pi^+(0)\bigr)$.  Conversely, let $p\in
(\pi^+)^{-1}(0^+)$.  Then $\pi^+(p) = 0^+$. Consequently $p$ can be
written as $p=u 0$, where $u \in U^+$. If we write $u=(\exp Y)h$, \
$Y\in\mathfrak{g}_1$, $h\in G_0$, then we have $p=(\exp Y)0\in
F^+(0)$.
\end{proof}

If we denote the projections by $\varpi^\pm\colon \tilde{M}\to M^\pm$,
then we have $\varpi^\pm\cdot\varphi = \pi^\pm$ (cf.\
(2.4)). Therefore, under the identification of $M$ with $\varphi(M)$,
the double fibration of $M$ is the restriction of the trivial double
fibration of $\tilde{M}$ to $M$. Later on we always identify $M$ with
its $\varphi$-image in $\tilde{M}$. As was seen in the proof of Lemma
2.3, \ $M$ is an orbit through the origin $(0^-, 0^+)\in\tilde{M}$
under the diagonal $G$-action.

% end of page 20 (2-4)

\section{Orbit structure of $\tilde{M}$}

\subsection{} % 3.1

We wish to consider the orbit structure of $\tilde{M}$ under the
diagonal $G$-action. We start with simple GLA (1.1). Recall the
decomposition $\mathfrak{g} = \mathfrak{g}_0 + \mathfrak{m}$ by
$\sigma$ (cf.\ \S1). Also we have the Cartan involution $\tau$
satisfying $\tau(Z) = -Z$. The property $\tau(Z)= -Z$ means that
$\tau$ is grade-reversing, i.e., $\tau(\mathfrak{g}_k) =
\mathfrak{g}_{-k}$, \ $k=0$, $\pm 1$. Let $\mathfrak{g} = \mathfrak{k}
+ \mathfrak{p}$ be the Cartan decomposition by $\tau$, where $\tau=1$
on $\mathfrak{k}$ and $-1$ on $\mathfrak{p}$. Since $\sigma$ and
$\tau$ commute, we have the decomposition
\[\mathfrak{g} = \mathfrak{k}_0 + \mathfrak{m}_{\mathfrak{k}} +
\mathfrak{p}_0 + \mathfrak{m}_{\mathfrak{p}}, \eqno(3.1)\] where
$\mathfrak{k}_0 = \mathfrak{g}_0\cap\mathfrak{k}$, \
$\mathfrak{m}_\mathfrak{k} = \mathfrak{m}\cap\mathfrak{k}$, \
$\mathfrak{p}_0 = \mathfrak{g}_0\cap\mathfrak{p}$ and
$\mathfrak{m}_{\mathfrak{p}} = \mathfrak{m}\cap\mathfrak{p}$.  Note
that $Z\in\mathfrak{p}_0$. Choose a maximal abelian subspace
$\mathfrak{a}$ of $\mathfrak{p}$ such that $Z\in\mathfrak{a}$. Then
$\mathfrak{a}$ is contained in $\mathfrak{p}_0$. Let $\Delta$ be the
root system of $\mathfrak{g}$ with respect to $\mathfrak{a}$. Let
$(\phantom{\varphi}, \phantom{\varphi})$ denote the Killing form of
$\mathfrak{g}$. Then we have the partition of $\Delta$ corresponding
to the grading of $\mathfrak{g}$:
\[\begin{aligned}[t]
\Delta &= \Delta_{-1}\amalg\Delta_0\amalg\Delta_1, \\
\Delta_k & = \{\,\alpha\in\Delta: (\alpha,Z)=k\,\}, \qquad k=0, \pm 1.
\end{aligned} \eqno(3.2)\]

% end of page 21 (3-1)

Choose a linear order in $\Delta$ in such a way that
$\Delta_1\subset\Delta^+\subset\Delta_0\cup\Delta_1$, where $\Delta^+$
denotes the positive system of $\Delta$ with respect to that
order. Then choose a maximal system of strongly orthogonal roots,
$\Gamma=\{\beta_1,\dots,\beta_r\}$ in $\Delta_1$,
 such that each $\beta_i$ has the
same length and that $\theta=\beta_1 > \beta_2 > \cdots > \beta_r$, \
$\theta$ being the highest root in $\Delta$. Here the number $r$ is
the split rank of the symmetric pair $(\mathfrak{g},
\mathfrak{g}_0)$. 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
$\mathfrak{g}^{\beta_i}\subset \mathfrak{g}_1$ \ ($1\leq i\leq r$) in
such a way that
\[[E_i,E_{-i}] = \check{\beta}_i = \frac{2}{(\beta_i,\beta_i)}\beta_i,
\qquad 1\leq i\leq r,\] where $E_{-i} =
-\tau(E_i)\in\mathfrak{g}^{-\beta_i}\subset\mathfrak{g}_{-1}$. Put
$X_i := E_i+E_{-i}\in\mathfrak{m}_{\mathfrak{p}}$ and
$Y_i:=E_i-E_{-i}\in\mathfrak{m}_{\mathfrak{k}}$ \ ($1\leq i\leq r$).
Then $\mathfrak{c} = \sum_{i=1}^r \bbb{R} X_i$ is a maximal abelian
subspace of $\mathfrak{m}_{\mathfrak{p}}$. $\mathfrak{c}$ is a split
Cartan subalgebra of the symmetric pair $(\mathfrak{g},
\mathfrak{g}_0)$. Note that $\mathfrak{c}$ is also a Cartan subalgebra
of the noncompact dual of the symmetric $R$-space $G/U^-$. It is
well-known that the root system
% end of page 22 (3-2)
$\Delta(\mathfrak{g},\mathfrak{c})$ of $\mathfrak{g}$ with respect to
$\mathfrak{c}$ 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 $\mathfrak{a}_0$ be the subspace of
$\mathfrak{a}$ spanned by $\beta_1$, $\dots$, $\beta_r$ and $\varpi$
be the orthogonal projection of $\mathfrak{a}$ onto $\mathfrak{a}_0$
with respect to $(\phantom{\varphi}, \phantom{\varphi})$. Then either
one of the following two cases occurs (\cite{13,7}):
\[\left\{\begin{aligned}
\varpi(\Delta_1) &= \bigl\{\,\tfrac12(\beta_i+\beta_j): 1\leq i\leq
j\leq r\,\bigr\}, \hphantom{, \ \tfrac12\beta_i \ (1\leq i\leq r)} \\
\varpi(\Delta_0^+)-(0) &= \bigl\{\,\tfrac12(\beta_i-\beta_j): 1\leq i
< j\leq r\,\bigr\},
\end{aligned}\right. \eqno(3.3)\]
\[\left\{\begin{aligned}
\varpi(\Delta_1) &= \bigl\{\tfrac12(\beta_i+\beta_j) \ (1\leq i\leq
j\leq r), \ \tfrac12\beta_i \ (1\leq i\leq r)\,\bigr\}, \\
\varpi(\Delta_0^+)-(0) &= \bigl\{\tfrac12(\beta_i-\beta_j) \ (1\leq i
< j\leq r), \ \tfrac12\beta_i\ (1\leq i\leq r)\bigr\},
\end{aligned}\right. \eqno(3.4)\]
according as $\Delta(\mathfrak{g}, \mathfrak{c})$ is of $C_r$-type or
$BC_r$-type, respectively. Here $\Delta_0^+ =
\Delta_0\cap\Delta^+$. Now let $K$ be the subgroup of $G$ consisting
of elements which commute with $\tau$. Then $K$ is the maximal compact
subgroup of $G$ with $\Lie K = \mathfrak{k}$.  We denote the identity
component of $K$ by $K^0$, and define the elements $a_l$ \ ($0\leq
l\leq r$) in the normalizer $N_{K^0}(\mathfrak{a})$ of $\mathfrak{a}$
% end of page 23 (3-3)
in $K^0$ by putting 
\[\left\{\begin{aligned}
a_l &= \exp-\frac\pi2\sum_{i=1}^l Y_i, \qquad 1\leq l\leq r, \\
a_0 &= 1.
\end{aligned}\right. \eqno(3.5)\]

The following theorem gives the $G$-orbit structure of $\tilde{M}$.

\begin{Theorem}\hfill\par % 3.1
\begin{enumerate}
\item[(i)] The points $(0^-, a_l\, 0^+)\in\tilde{M}$, \ $0\leq l\leq
r$, are a complete set of representatives of $G$-orbits in
$\tilde{M}$.
\item[(ii)] Let $M_l = G(0^-, a_{r-l}\, 0^+)$, \ $0\leq l\leq r$. Then
the closure $\overline{M_l}$ of $M_l$ in $\tilde{M}$ is given by
\[\overline{M_l} = M_l\amalg M_{l-1}\amalg\cdots\amalg M_0, \qquad
0\leq l\leq r.\]
\item[(iii)] $M_r=M$ is a single open $G$-orbit, and hence $\tilde{M}$
is a $G$-equivariant compactification of $M$.
\item[(iv)] A single closed $G$-orbit $M_0$ has the
property\/\textup{:} 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\/\textup{:}
\[G/U^- = M^-(0^-,a_r\,0^+)\longleftarrow M_0\longrightarrow
M^+(0^-,a_r\,0^+) = G/a_r\,U^+ a_r^{-1}.\]
\end{enumerate}
\end{Theorem}

\begin{proof}
Let us denote by $G^0$ the identity component of $G$. In \cite{7} we
proved the theorem for $G^0$-action. But, by
% end of page 24 (3-4)
Theorem 4.12 in \cite{7}, \ $\dim M_l$ is strictly increasing, as $l$
increases. Let $g\in G$. 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.
\end{proof}

% end of page 25 (3-5)

\section{Isotropy subgroups for boundary orbits}

\subsection{} % 4.1

We go back to a real simple GLA $\mathfrak{g}$ in (1.1). We wish to
construct a certain class of gradings of $\mathfrak{g}$ of the second
kind in terms of the subsets $\Gamma_l = \{\beta_1,\dots,\beta_l\}$, \
$1\leq l\leq r$, of $\Gamma$. When $\mathfrak{g}$ is of Hermitian
type, this type of grading corresponds to the realizations of the
bounded symmetric domain (corresponding to  $\mathfrak{g}$) as a
Siegel domain of the third kind for $1\leq l\leq r-1$ and that of the
second (or first) kind for $l=r$. Let $1\leq l\leq r$, and put
\[\begin{aligned}
\Delta_2(l) &= \bigl\{\,\alpha\in\Delta_1: \varpi(\alpha)=
\tfrac12(\beta_i+\beta_j), \ 1\leq i\leq j\leq l\,\bigr\}, \\
\Delta_1(l) &= \left\{\alpha\in\Delta: \begin{array}{l}
\textup{$\varpi(\alpha) = \tfrac12(\beta_i\pm\beta_j)$, \ $1\leq i\leq
l, l+1\leq j\leq r$, or}\\[4pt]
\varpi(\alpha) = \tfrac12\beta_i, \ 1\leq i\leq l
\end{array}\right\}, \\
\Delta_0(l) &= \left\{\alpha\in\Delta: \begin{array}{l}
\textup{$\varpi(\alpha) = 0$, or} \\[4pt]
\textup{$\varpi(\alpha) = \pm\tfrac12(\beta_i-\beta_j)$, \ $1\leq
i<j\leq l$ or}\\[4pt]
\quad\qquad\qquad\qquad\qquad\textup{$l+1\leq i<j\leq r$, or}\\[4pt]
\textup{$\varpi(\alpha) = \pm\tfrac12(\beta_i+\beta_j)$, \ $l+1\leq
i\leq j\leq r$, or}\\[4pt]
\varpi(\alpha) = \pm\tfrac12\beta_i, \ l+1\leq i\leq r
\end{array}\!\!\right\}, \\
\Delta_{-1}(l) &= -\Delta_1(l), \\
\Delta_{-2}(l) &= -\Delta_2(l).
\end{aligned} \eqno(4.1)\]

% end of page 26 (4-1)

Then, for a fixed $1\leq l\leq r$, we have a partition of $\Delta$:
\[\Delta=\coprod_{k=-2}^2 \Delta_k(l). \eqno(4.2)\]
By using (3.3) and (3.4) we easily have 
 
\begin{Proposition} % 4.1
Let\/ $1\leq l\leq r$, and let\/ $\mathfrak{c}(\mathfrak{a})$ be the
centralizer of\/ $\mathfrak{a}$ in\/ $\mathfrak{g}$. If we put
\[\begin{aligned}[c]
\mathfrak{g}_0(l) &= \mathfrak{c}(\mathfrak{a}) + \sum_{\alpha\in\Delta_0(l)} \mathfrak{g}^\alpha,\\
\mathfrak{g}_k(l) &= \sum_{\alpha\in\Delta_k(l)} \mathfrak{g}^\alpha, \qquad k=\pm 1,
\pm 2,\end{aligned} \eqno(4.3)\]
then we have the grading of $\mathfrak{g}$ of the second kind
\[\mathfrak{g}=\sum_{k=-2}^2 \mathfrak{g}_k(l), \eqno(4.4)\]
whose characteristic element is $Z_l = \sum_{k=1}^l \check{\beta}_i$.
\end{Proposition}

\begin{Remark}
Gyoja and Yamashita \cite{3} obtained the above gradings for
$\mathfrak{g}$ complex simple, in which case there are no roots
$\alpha\in\Delta$ such that $\varpi(\alpha) = 0$.
\end{Remark}

Let $s_{\beta_i}$ be the reflection on $\mathfrak{a}$ corresponding to
the root
% end of page 27 (4-2)
$\beta_i$ \ ($1\leq i\leq r$), and let $s_l = s_{\beta_1} s_{\beta_2}
\dots s_{\beta_l}$ \ ($1\leq l\leq r$) and $s_0 = 1$. It is known
\cite{13} that $\Ad_{\mathfrak{a}} a_l = s_l$, \ $0\leq l\leq r$. Let
$Q_l$ \ ($0\leq l\leq r$) be the isotropy subgroup of $G$ at $(0^-,
a_l\,0^+)$. Then the $G$-orbit $M_{r-l}$ can be expressed as
\[M_{r-l} = G/Q_l, \qquad 0\leq l\leq r,\]
where $Q_l = U^-\cap a_l\,U^+ a_l^{-1}$. The Lie algebra $\mathfrak{q}_l :=
\Lie Q_l$ can be written as
\[\mathfrak{q}_l = \mathfrak{c}(\mathfrak{a}) + \sum_{\alpha\in\Psi_l}
\mathfrak{g}^\alpha, \qquad 0\leq l\leq r, \eqno(4.5)\]
where
\[\Psi_l := \{\,\alpha\in\Delta_0\cup\Delta_{-1}: s_l(\alpha)\in
\Delta_0\cup\Delta_1\,\}, \qquad 0\leq l\leq r. \eqno(4.6)\]

\begin{Lemma} % 4.2
\[\Psi_l = \Delta_0\cap\Delta_0(l)\amalg\Delta_{-1}(l)\amalg
\Delta_{-2}(l). \eqno(4.7)\]
\end{Lemma}

\begin{proof}
First we will show the inclusion $\supset$ in (4.7).  Let
$\alpha\in\Delta$. Then we have
\[\begin{aligned}[c]
\bigl(s_l(\alpha), Z\bigr)
&= (\alpha,Z) - \sum_{k=1}^l (\alpha,\check{\beta}_k)(\beta_k,Z)\\
&= (\alpha,Z) - \sum_{k=1}^l 2(\varpi(\alpha),\beta_k)
(\beta_k,\beta_k)^{-1}.
\end{aligned} \eqno(4.8)\]
Now let $\alpha\in\Delta_0\cap\Delta_0(l)$. By using (4.8) it follows
from (3.4) and (4.1) that
$\bigl(s_l(\alpha),Z\bigr) = 0$, or equivalently
% end of page 28 (4-3)
$s_l(\alpha)\in\Delta_0$ and hence $\alpha\in\Psi_l$. Suppose next
that $\alpha \in \Delta_{-1}(l)$. Then, by (4.1), there are three
possibilities: $\varpi(\alpha) = -\tfrac12(\beta_i+\beta_j)$ or
$-\tfrac12(\beta_i-\beta_j)$ for $1\leq i\leq l$, $l+1\leq j\leq r$,
or $\varpi(\alpha) = -\tfrac12\beta_i$ for $1\leq i\leq l$. In view of
(3.4) and (4.8), we have $\alpha\in\Delta_{-1}$ and
$\bigl(s_l(\alpha),Z\bigr) = 0$ for the first case, and
$\alpha\in\Delta_0$ and $\bigl(s_l(\alpha),Z\bigr) = 1$ for the second
case. For the third case, there are two possibilities (cf.\ (3.4)):
$\alpha\in\Delta_0$ or $\alpha\in\Delta_{-1}$. Then we have from (4.8)
that $\bigl(s_l(\alpha),Z\bigr) = 1$ or $0$, according as
$\alpha\in\Delta_0$ or $\alpha\in\Delta_{-1}$,
respectively. Consequently $s_l(\alpha)\in\Delta_0\cup\Delta_1$ for
$\alpha\in\Delta_{-1}(l)$. Suppose $\alpha\in\Delta_{-2}(l)$. Then by
(3.4) and (4.8) we have $\alpha\in\Delta_{-1}$ and
$\bigl(s_l(\alpha),Z\bigr) = 1$.

To prove the converse inclusion $\subset$ in (4.7), let
$\alpha\in\Psi_l$ and suppose that $\alpha$ does not belong to the
right-hand side of (4.7). Then the following three cases occur:
\begin{enumerate}
\item[(i)] $\alpha\in\Delta_2(l)$,
\item[(ii)] $\alpha\in\Delta_1(l)$ and
\item[(iii)] $\alpha\in\Delta_0(l)-\Delta_0$. 
\end{enumerate}
For (i), we have $\alpha\in\Delta_1$, contradicting the assumption of
$\alpha\in\Psi_l$. For (ii), we have three possibilities:
$\varpi(\alpha) = \tfrac12(\beta_i+\beta_j)$ or
$\tfrac12(\beta_i-\beta_j)$ both for
% end of page 29 (4-4)
$1\leq i\leq l$, $l+1\leq j\leq r$ or $\varpi(\alpha) =
\tfrac12\beta_i$ for $1\leq i\leq l$. For the first case we have
$\alpha\in\Delta_1$, which contradicts $\alpha\in\Psi_l$. For the
second case, we have $\alpha\in\Delta_0$. Consequently, by using
(4.8), we have that $\bigl(s_l(\alpha),Z\bigr) = -1$, that is,
$s_l(\alpha)\in\Delta_{-1}$. This contradicts the assumption
$\alpha\in\Psi_l$. For the third case, we have either
$\alpha\in\Delta_1$ or $\alpha\in\Delta_0$. In view of the condition
$\alpha\in\Psi_l$, we have the only choice $\alpha\in\Delta_0$, in
which case $\bigl(s_l(\alpha),Z\bigr) = -1$, still contradicting the
assumption $\alpha\in\Psi_l$. Let us consider the case (iii)
finally. Since $\alpha$ lies in $\Psi_l\cap \bigl(\Delta_0(l) -
\Delta_0\bigr)$, we have that $\varpi(\alpha) =
-\tfrac12(\beta_i+\beta_j)$, \ $l+1\leq i\leq j\leq r$, or 
$\varpi(\alpha) =-\tfrac12\beta_i$, $l+1\leq i\leq r$.
In particular
$\alpha\in\Delta_{-1}$. Therefore, in both cases, we have 
$\bigl(s_l(\alpha),Z\bigr) =
-1$, which contradicts the assumption $\alpha\in\Psi_l$.
\end{proof}

\begin{Proposition} % 4.3
The isotropy subalgebra\/ $\mathfrak{q}_l$ of\/ $\mathfrak{g}$ at the
point\/ $(0^-, a_l\,0^+)$ is given by
\[\mathfrak{q}_l = \mathfrak{g}_{-2}(l) + \mathfrak{g}_{-1}(l) +
\mathfrak{g}_0(l)\cap\mathfrak{g}_0, \qquad 0\leq l\leq r.\]
\end{Proposition}

\begin{proof}
This follows immediately from Lemma 4.2 and (4.5).
\end{proof}

% end of page 30 (4-5)

\subsection{} % 4.2

In this paragraph we will determine the isotropy subgroup $Q_r$ of $G$
at the point $(0^-, a_r\,0^+)\in M_0$. Let us consider the special
case $l=r$. Then (4.1) has the following simple form:
\[\begin{aligned}[c]
\Delta_2(r) &= \bigl\{\,\alpha\in\Delta_1: \varpi(\alpha) =
\tfrac12(\beta_i+\beta_j), \ 1\leq i\leq j\leq r\,\bigr\}, \\
\Delta_1(r) &= \bigl\{\,\alpha\in\Delta^+_0\cup\Delta_1:
\varpi(\alpha) = \tfrac12\beta_i, \ 1\leq i\leq r\,\bigr\}, \\
\Delta_0(r) &= \bigl\{\,\alpha\in\Delta: \textup{$\varpi(\alpha) = 0$
or $= \pm\tfrac12(\beta_i-\beta_j)$, \ $1\leq i<j\leq r$}\,\bigr\}, \\
\Delta_{-k}(r) &= -\Delta_k(r), \qquad k=1, 2.
\end{aligned}\eqno(4.9)\]

We have the grading of the second kind
\[\mathfrak{g} = \sum_{k=-2}^2 \mathfrak{g}_k(r). \eqno(4.10)\]

\begin{Remark}
In the case of $C_r$-type, we have $\Delta_1(r) = \emptyset$, \ 
$\Delta_2(r) = \Delta_1$ and $\Delta_0(r) = \Delta_0$. Hence the
grading (4.10) is reduced to the grading (1.1).
\end{Remark}

\begin{Lemma} % 4.4
$\mathfrak{q}_r = \mathfrak{g}_{-2}(r) + \mathfrak{g}_{-1}(r) +
\mathfrak{g}_0(r)$. In particular, $\mathfrak{q}_r$ is a parabolic
subalgebra of\/ $\mathfrak{g}$ of the second kind.
\end{Lemma}

\begin{proof}
Since $\Delta_0(r) \subset \Delta_0$, we have the inclusion
$\mathfrak{g}_0(r) \subset \mathfrak{g}_0$.
\end{proof}

% end of page 31 (4-6)
 
Now we put
\[\begin{aligned}[c]
\Delta_1^+(r) &= \Delta_1(r)\cap\Delta_0^+,\\
\Delta_{-1}^+(r) &= \Delta_{-1}(r)\cap\Delta_{-1},
\end{aligned}\qquad\begin{aligned}[c]
\Delta_1^-(r) &= \Delta_1(r)\cap\Delta_1,\\
\Delta_{-1}^-(r) &= \Delta_{-1}(r)\cap\Delta_0^-.
\end{aligned}\eqno(4.11)\]
Then we have
\[\Delta_{\pm 1}(r) = \Delta_{\pm 1}^+(r)\amalg\Delta_{\pm
1}^-(r). \eqno(4.12)\]
We define the four subspaces of $\mathfrak{g}$:
\[\mathfrak{g}_{\pm 1}^+(r) = \sum_{\alpha\in\Delta_{\pm 1}^+(r)}
\mathfrak{g}^\alpha, \qquad \mathfrak{g}_{\pm 1}^-(r) =
\sum_{\alpha\in\Delta_{\pm 1}^-(r)} \mathfrak{g}^\alpha. \eqno(4.13)\]
Those four subspaces are equi-dimensional and abelian (\cite{7}).  We
have the decompositions
\[\mathfrak{g}_1(r) = \mathfrak{g}_1^-(r) + \mathfrak{g}_1^+(r),
\qquad \mathfrak{g}_{-1}(r) = 
\mathfrak{g}_{-1}^+(r) + \mathfrak{g}_{-1}^-(r). \eqno(4.14)\]
The original grading (1.1) of $\mathfrak{g}$ can be reconstructed as
\[\begin{aligned}[c]
\mathfrak{g}_{-1} &= \mathfrak{g}_{-2}(r) + \mathfrak{g}_{-1}^+(r), \\
\mathfrak{g}_0 &= \mathfrak{g}_{-1}^-(r) + \mathfrak{g}_0(r) +
\mathfrak{g}_1^+(r), \\
\mathfrak{g}_1 &= \mathfrak{g}_1^-(r) + \mathfrak{g}_2(r).
\end{aligned} \eqno(4.15)\]

Let $C(Z_r)$ be the centralizer of $Z_r$ in $\Aut\mathfrak{g}$. Then
the normalizer $N(\mathfrak{q}_r)$ in $\Aut\mathfrak{g}$ of
$\mathfrak{q}_r$ can be written as
\[N(\mathfrak{q}_r) =
C(Z_r)\cdot\exp\bigl(\mathfrak{g}_{-2}(r)+\mathfrak{g}_{-1}(r)\bigr).
\qquad\textup{(semi-direct)} \eqno(4.16)\] $Q_r$ is a subgroup of
$U^-\cap N(\mathfrak{q}_r) = N_{U^-}(\mathfrak{q}_r)$, the normalizer
of $\mathfrak{q}_r$ in $U^-$. (4.15) implies that
% end of page 32 (4-7)
\[N_{U^-}(\mathfrak{q}_r) = \bigl(C(Z_r)\cap U^-\bigr)\cdot
\exp\bigl(\mathfrak{g}_{-2}(r) +
\mathfrak{g}_{-1}(r)\bigr). \eqno(4.17)\]

\begin{Lemma} % 4.5
Let $C(Z,Z_r)$ be the centralizer of both elements $Z$ and $Z_r$ in\/
$\Aut\mathfrak{g}$. Then we have $C(Z_r)\cap U^- = C(Z,Z_r)$.
\end{Lemma}

\begin{proof}
Let $a\in C(Z_r)\cap U^-$. We write $a = b\exp X$, \ $b\in C(Z)$, \
$X\in\mathfrak{g}_{-1}$. Then $[X,Z_r]\in\mathfrak{g}_{-1}$ and hence
$\bigl[X,[X,Z_r]\bigr]=0$ Therefore we have
\[Z_r = (\Ad a)Z_r = (\Ad b)(\Ad\exp X)Z_r = (\Ad b)Z_r + (\Ad
b)[X,Z_r]\subset \mathfrak{g}_0 + \mathfrak{g}_{-1}.\] Hence $(\Ad
b)[X,Z_r] = 0$.  $\Ad b$ being invertible on $\mathfrak{g}_{-1}$, we
have $[X,Z_r]=0$. Consequently
$X\in\mathfrak{g}_0(r)\cap\mathfrak{g}_{-1}=(0)$, and $a=b\in C(Z)$.
\end{proof}

\begin{Proposition} % 4.6
The isotropy subgroup $Q_r$ of $G$ at\/ $(0^-, a_r\,0^+)\in M_0$ is
given by
\[Q_r = C(Z,Z_r)\exp\bigl(\mathfrak{g}_{-2}(r) +
\mathfrak{g}_{-1}(r)\bigr).\]
\end{Proposition}

\begin{proof}
By Lemma 4.5 and (4.17) we have
\[N_{U^-}(\mathfrak{q}_r) = C(Z,Z_r)\exp\bigl(\mathfrak{g}_{-2}(r) +
\mathfrak{g}_{-1}(r)\bigr).\] Recall $Q_r\subset
N_{U^-}(\mathfrak{q}_r)$. To prove the converse inclusion, it suffices
to show that $C(Z,Z_r)\subset Q_r$. By using (4.8),
% end of page 33 (4-8)
one sees $s_r(Z) = Z-Z_r$, and consequently
\begin{align*}
a_r^{-1} C(Z) a_r &= C\bigl((\Ad a_r^{-1})Z\bigr) =
C\bigl(s_r(Z)\bigr) = C(Z-Z_r), \\
a_r^{-1} C(Z_r) a_r &= C\bigl((\Ad a_r^{-1})Z_r\bigr) =
C\bigl(s_r(Z_r)\bigr) = C(Z_r).
\end{align*}
As a result, $a_r^{-1} C(Z,Z_r) a_r = C(Z-Z_r)\cap C(Z_r) =
C(Z,Z_r)\subset U^+$. Hence we have $C(Z,Z_r)\subset U^-\cap a_r\,U^+
a_r^{-1} = Q_r$.
\end{proof}

\begin{Corollary} % 4.7
$G = C(Z,Z_r) G^0$.
\end{Corollary}

\begin{proof}
We have $M_0 = G/Q_r = G^0/Q_r\cap G^0 = G^0 Q_r/Q_r$, which implies
$G = Q_r G^0 = C(Z,Z_r) G^0$.
\end{proof}

\begin{Corollary} % 4.8
Suppose that $M$ is of $C_r$-type. Then $Q_r = U^- = a_r\,U^+
a_r^{-1}$ and $M_0 = G/U^- = M^-$.
\end{Corollary}

\begin{proof}
By the Remark before Lemma 4.4 and Proposition 4.6, we see that $Q_r =
C(Z)\exp\mathfrak{g}_{-1} \allowbreak = U^-$, and hence $U^- = Q_r =
U^-\cap a_r\,U^+ a_r^{-1}\subset a_r\,U^+ a_r^{-1}$. Hence we have
$U^- = a_r\,U^+ a_r^{-1}$.
\end{proof}

% end of page 34 (4-9)

\section{Siegel-type realization of orbits}

\subsection{} % 5.1

\begin{Lemma} % 5.1
% need to align these equations (bwg)
\[s_r\bigl(\Delta_k(r)\bigr) = \Delta_{-k}(r), \qquad k=0, \pm 1, \pm
2, \eqno(5.1)\]
\[s_r\bigl(\Delta_{-1}^\pm(r)\bigr) = \Delta_1^\pm(r).
\hphantom{\,\qquad k=0, \pm 1, \pm 2,\,\ \ }
\eqno(5.2)\]
\end{Lemma}

\begin{proof}
Let $\alpha\in\Delta_k(r)$. Then $\bigl(s_r(\alpha),Z_r\bigr) =
\bigl(\alpha,s_r(Z_r)\bigr) = -(\alpha,Z_r) = -k$, which implies that
$s_r(\alpha)\in\Delta_{-k}(r)$. Let $\alpha\in\Delta_{-1}^+(r)$. Then,
by (4.8) we have $\bigl(s_r(\alpha),Z\bigr)=0$, and hence
$s_r(\alpha)\in\Delta_0$. One can write $\varpi(\alpha) =
-\tfrac12\beta_i$ for some $i$. Hence we have
$\varpi\bigl(s_r(\alpha)\bigr) = s_r\varpi(\alpha) =
s_r(-\tfrac12\beta_i) = \tfrac12\beta_i$, proving that
$s_r(\alpha)\in\Delta^+_1(r)$.
\end{proof}

\begin{Lemma} % 5.2
The operator\/ $\Ad a_r$ is grade-reversing with respect to the
grading\/ \textup{(4.10)}. Moreover\/ $\Ad a_r$ interchanges\/
$\mathfrak{g}_{-1}^\pm(r)$ with\/ $\mathfrak{g}_1^\pm(r)$,
respectively.
\end{Lemma}

\begin{proof}
Since $\Ad a_r$ induces $s_r$ on $\mathfrak{a}$ (cf.\ 4.1), the lemma
is immediate from Lemma 5.1.
\end{proof}

\subsection{} % 5.2

Up to the present, we have expressed $\tilde{M}$ as $M^-\times
M^+$. Here $M^\pm$ are just the leaves of the product foliation
through the
% end of page 35 (5-1)
origin $(0^-,0^+)$. In order to get the Siegel-type realization of $G$-orbits,
we
choose the point $(0^-, a_r\,0^+)\in M_0$ as the new origin of
$\tilde{M}$.  Then $\tilde{M}$ can be expressed as
\[\tilde{M} = M^-(0^-, a_r\,0^+)\times M^+(0^-, a_r\,0^+) =
G/U^-\times G/a_r\,U^+ a_r^{-1}. \eqno(5.3)\] For simplicity we write
$(M^+)_r$ for $G/a_r\,U^+ a_r^{-1}$. We identify the tangent space
$T_{0^-}(G/U^-)$ with $\mathfrak{g}_1 = \mathfrak{g}_2(r) +
\mathfrak{g}_1^-(r)$ (cf.\ (4.15)), and $T_{0^+}(G/U^+)$ with
$\mathfrak{g}_{-1} = \mathfrak{g}_{-2}(r) +
\mathfrak{g}_{-1}^+(r)$. Then the tangent space
$T_{a_r\,0^+}(G/a_r\,U^+ a_r^{-1})$ can be identified with $(\Ad
a_r)\mathfrak{g}_{-1} = \mathfrak{g}_2(r) + \mathfrak{g}_1^+(r)$ by
Lemma 5.2. We will denote $(\Ad a_r)\mathfrak{g}_{-1}$ by
$\mathfrak{g}'_1$. Let us consider the direct sum of the vector space
\[\mathfrak{g}_1\oplus\mathfrak{g}'_1 = \bigl(\mathfrak{g}_2(r) +
\mathfrak{g}_1^-(r)\bigr)\oplus \bigl(\mathfrak{g}_2(r) +
\mathfrak{g}_1^+(r)\bigr). \eqno(5.4)\] We define the map $\xi$ of
$\mathfrak{g}_1\oplus\mathfrak{g}'_1$ into $\tilde{M} = M^-\times
(M^+)_r$ by
\[\xi(X,X') = \bigl((\exp X)0^-, (\exp X')a_r\, 0^+\bigr). \eqno(5.5)\]
Then $\xi$ is an open dense imbedding of
$\mathfrak{g}_1\oplus\mathfrak{g}'_1$. We will always identify
$\mathfrak{g}_1\oplus\mathfrak{g}'_1$ with its $\xi$-image.

\begin{Lemma} % 5.3
Let $a_l^\pm = \exp\sum_{i=1}^l E_{\pm i}$ \ \textup{(}$1\leq l\leq
r$\textup{)}. Then
\[a_r^{-1} (a_l^-)^{-1} a_r = a_l^+, \qquad 1\leq l\leq
r. \eqno(5.6)\]
\end{Lemma}

% end of page 36 (5-2)

\begin{proof}
Consider the elements in $\sl(2,\bbb{R})$
\[e_+ = \begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}, \qquad
e_- = \begin{pmatrix} 0 & 0\\ 1 & 0\end{pmatrix}.\]
By easy computation we have 
\[\exp\frac{\pi}{2}(e_+ - e_-) \exp(-e_-) \exp -\frac{\pi}{2}(e_+ -
e_-) = \exp e_+.\] Let $\varphi_i\colon \sl(2,\bbb{R})\to\mathfrak{g}$
\ ($1\leq i\leq r$) be the maps defined by $\varphi_i(e_\pm) = E_{\pm
i}$. By the strong orthogonality of the $\beta_i$, we have
$[\varphi_i, \varphi_j] = 0$ \ ($i\neq j$). $\varphi_i$ can be
extended to the homomorphism of $\SL(2,\bbb{R})$ to $G$, denoted again
be $\varphi_i$.
\[\begin{aligned}[b]
&\textup{LHS of (5.6)} \\
&\quad = \exp\frac{\pi}{2}\sum_{i=1}^r \bigl(\varphi_i(e_+) -
\varphi_i(e_-)\bigr) \exp\biggl(-\sum_{i=1}^l \varphi_i(e_-)\biggr)
\exp\biggl(-\frac{\pi}{2}\sum_{i=1}^r \bigl(\varphi_i(e_+) -
\varphi_i(e_-)\bigr)\biggr)\\
&\quad = \prod_{i=1}^r \varphi_i\Bigl(\exp\frac{\pi}{2}(e_+ -
e_-)\Bigr)
\prod_{i=1}^l \varphi_i(\exp -e_-)
\prod_{i=1}^r \varphi_i\Bigl(\exp\Bigl(-\frac{\pi}{2}(e_+ -
e_-)\Bigr)\Bigr)\\
&\quad = \prod_{i=1}^l \varphi_i\Bigl(\exp\frac{\pi}{2}(e_+ - e_-)
\exp(-e_-) \exp\Bigl(-\frac{\pi}{2}(e_+ - e_-)\Bigr)\Bigr)\\
&\quad = \prod_{i=1}^l \varphi_i(\exp e_+)
= \exp\sum_{i=1}^l \varphi_i(e_+)
= \exp\sum_{i=1}^l E_i.
\end{aligned} \eqno{\Box}\]
\noqed\end{proof}

The following lemma was proved in \cite{10}. Note that we do not use
there the assumption that $\Delta(\mathfrak{g},\mathfrak{c})$ is of
% end of page 37 (5-3)
$C_r$-type.

\begin{Lemma} % 5.4
$a_l^- a_l^{-1} a_l^- = a_l^+$.
\end{Lemma}

\begin{Lemma} % 5.5
$(0^-, a_l a_r\,0^+) \equiv (a_l^+\,0^-, (a_l^+)^2 a_r\,0^+)$
\textup{mod} $G$.
\end{Lemma}

\begin{proof}
First note that $a_l\,0^\pm = a_l^{-1}\,0^\pm$. In fact, $\Ad a_l^2$
is the identity on $\mathfrak{a}$, which implies that $a_l^2$ lies in the
centralizer $C(Z) = U^+\cap U^-$. 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
\[\begin{aligned}[b]
(0^-, a_l a_r\,0^+)
&= (0^-, a_r a_l\,0^+)
= (0^-, a_r a_l^{-1}\,0^+)
= (0^-, a_l^{-1} a_r\,0^+)\\
&\equiv (a_l^+ a_l^-\,0^-, a_l^+ a_l^- a_l^{-1} a_r\,0^+)
= (a_l^+\,0^-, (a_l^+)^2 (a_l^-)^{-1} a_r\,0^+)\\
&= (a_l^+\,0^-, (a_l^+)^2 a_r a_l^+\,0^+)
= (a_l^+\,0^-, (a_l^+)^2 a_r\,0^+) \ \textup{mod $G$}.
\end{aligned} \eqno{\Box}\]
\noqed\end{proof}

Let us put $0_l = \sum_{i=1}^l E_i\in\mathfrak{g}_2(r)$, \ $1\leq
l\leq r$, \ $0_0 = 1$.

\begin{Proposition} % 5.6
The point\/ $(0_l, 2\,0_l)\in\mathfrak{g}_1\oplus\mathfrak{g}'_1$
\textup{(}identified with its $\xi$-image\/\textup{)} is a
representative of the $G$-orbit $M_l$ for\/ $0\leq l\leq r$.
\end{Proposition}

\begin{proof}
\[\begin{aligned}[b]
M_l
&= G(0^-, a_{r-l}\,0^+)
= G(0^-, a_l a_r\,0^+)
= G(a_l^+\,0^-, (a_l^+)^2 a_r\,0^+) \\
% end of page 38 (5-4)
&= G\bigl((\exp 0_l)\,0^-, (\exp 0_l)^2 a_r\,0^+\bigr)
= G\bigl((\exp 0_l)\,0^-, (\exp 2\,0_l) a_r\,0^+\bigr) \\
&= G\bigl(\xi(0_l, 2\,0_l)\bigr).
\end{aligned}\eqno{\Box}\]
\noqed\end{proof}

\subsection{} % 5.3

A preliminary step for Siegel-type realization of orbits was done by
Tanaka \cite{15}, which is needed for later consideration. Let
$\widehat{Q_r}$ be the parabolic subgroup of $G$ opposite to $Q_r$,
that is, $\widehat{Q_r} = C(Z,Z_r)\cdot N$, where $N=\exp\mathfrak{n}$
and $\mathfrak{n} = \mathfrak{g}_2(r) + \mathfrak{g}_1(r) =
\mathfrak{g}_2(r) + \mathfrak{g}_1^+(r) + \mathfrak{g}_1^-(r)$. The
group $G$ acts on the vector space
$\mathfrak{g}_1\oplus\mathfrak{g}'_1$ birationally through $\xi$. But
the subgroup $\widehat{Q}$ acts on it as affine transformations.

\begin{Proposition}[Tanaka \cite{15}] % 5.7
Let $a$, $x$, $y\in\mathfrak{g}_2(r)$, \ $b^+$,
$v^+\in\mathfrak{g}_1^+(r)$, \ $b^-$, $u^-\in\mathfrak{g}_1^-(r)$ and
$h\in C(Z,Z_r)$. Then the $\xi$-equivariant action of $\widehat{Q_r}$
is given by
\[\begin{aligned}[c]
&\exp(a + b^+ + b^-)\bigl((x, u^-)\oplus (y, v^+)\bigr) \\
&\qquad = (x + a + [b^+,u^-] + \tfrac12[b^+, b^-], u^- + b^-)\\
&\qquad\qquad\qquad \oplus (y
+ a + [b^-, v^+] + \tfrac12[b^-, b^+], v^+ + b^+),\end{aligned}
\eqno(5.7)\]
\[\begin{aligned}[c]
&h\bigl((x, u^-)\oplus (y, v^+)\bigr) \\
&\qquad = \bigl((\Ad_{\mathfrak{g}_2(r)} h) x,
(\Ad_{\mathfrak{g}_1^-(r)} h) u^-\bigr)\oplus
\bigl((\Ad_{\mathfrak{g}_2(r)} h) y, (\Ad_{\mathfrak{g}_1^+(r)} h)
v^+\bigr).\end{aligned} \eqno(5.8)\]
\end{Proposition}

% end of page 39 (5-5)

\setcounter{Proposition}{0}

\begin{Definition} % 5.1
We define a surjective submersion $\Phi$ of
$\mathfrak{g}_1\oplus\mathfrak{g}'_1$ onto $\mathfrak{g}_2(r)$ as
follows: For $X = (x, u^-)\in\mathfrak{g}_1$, \ $Y = (y,
v^+)\in\mathfrak{g}'_1$,
\[\Phi(X\oplus Y) = y - x + [v^+, u^-].\]
\end{Definition}

$\Phi$ has the following $\bigl(\widehat{Q_r},
C(Z,Z_r)\bigr)$-equivariancy property.

\setcounter{Proposition}{7}

\begin{Proposition}[\cite{15}] % 5.8
$\Phi$ is invariant under the action of $N$. Moreover let $h\in
C(Z,Z_r)$ and let $X'\oplus Y' = h(X\oplus Y)$.  Then
\[\Phi(X'\oplus Y') = (\Ad_{\mathfrak{g}_2(r)} h)\Phi(X\oplus Y).\]
\end{Proposition}

We restate Lemma 3.8 \cite{15} as follows:

\begin{Lemma} % 5.9
$N$ acts on\/ $\mathfrak{g}_1\oplus\mathfrak{g}'_1$ freely. Moreover
let $X\oplus Y\in\mathfrak{g}_1\oplus\mathfrak{g}'_1$ and let $X = (x,
u^-)$, \ $Y = (y, v^+)$. Then we have
\[\exp(-v^+) \exp(-x-u^-)(X\oplus Y) = (0, 0)\oplus \bigl(\Phi(X,Y),
0\bigr). \eqno(5.9)\]
\end{Lemma}

Note that the group $C(Z_r)$ is the group of grade-preserving
automorphisms with respect to the grading (4.10) and $\Lie C(Z_r) =
\mathfrak{g}_0(r)$. $C(Z,Z_r)$ is an open subgroup of $C(Z_r)$.
% end of page 40 (5-6)

Let $\widehat{Q_r}{}^0$ and $C^0(Z_r)$ be the identity components of
$\widehat{Q_r}$ and $C(Z_r)$, respectively. The following proposition
follows from Proposition 5.8  (cf.\cite{15}).

\begin{Proposition} % 5.10
There exists a bijection between the set of $\widehat{Q_r}{}^0$-orbits
in\/ $\mathfrak{g}_1\oplus\mathfrak{g}'_1$ and the set of
$C^0(Z_r)$-orbits in\/ $\mathfrak{g}_2(r)$. More precisely, the\/
$\Phi$-image of a $\widehat{Q_r}{}^0$-orbit is a $C^0(Z_r)$-orbit, and
the complete inverse image by\/ $\Phi$ of a $C^0(Z_r)$-orbit is a
$\widehat{Q_r}{}^0$-orbit.
\end{Proposition}

\subsection{} % 5.4

We are interested in the intersection of a $G$-orbit with
$\mathfrak{g}_1\oplus\mathfrak{g}'_1$. Let $M_l^* = M_l\cap
(\mathfrak{g}_1\oplus\mathfrak{g}'_1)$, \ $0\leq l\leq r$, which is a
dense open set in $M_l$. $M_l^*$ is stable under $\widehat{Q_r}$, and
hence it can be expressed as the union of $\widehat{Q_r}{}^0$-orbits
contained in $M_l^*$. Those $\widehat{Q_r}{}^0$-orbits are open in
$M_l^*$ (\cite{15}). This fact can also be proved by using Proposition
4.3. Let us consider the (reductive) graded subalgebra of
$\mathfrak{g}$
\[\mathfrak{g}_{\mathrm{ev}}(r) = \mathfrak{g}_{-2}(r) +
\mathfrak{g}_0(r) + \mathfrak{g}_2(r), \eqno(5.10)\]
which contains the simple graded ideal
% end of page 41 (5-7)
\[\mathfrak{g}'_{\mathrm{ev}}(r) = \mathfrak{g}_{-2}(r) +
[\mathfrak{g}_{-2}(r), \mathfrak{g}_2(r)] +
\mathfrak{g}_2(r). \eqno(5.11)\] By the table of $(\mathfrak{g},
\mathfrak{g}_{\mathrm{ev}})$ in \cite{8}, it turns out that
$\mathfrak{g}_2(r)$ has the structure of a real simple Jordan algebra
and the adjoint action of $C^0(Z_r)$ on $\mathfrak{g}_2(r)$ coincides
with the identity component of the structure group of this Jordan
algebra. Therefore we have the rank decomposition (\cite{2})
\[\mathfrak{g}_2(r) = V_r\amalg V_{r-1}\amalg\dots\amalg V_0,
\eqno(5.12)\] where $V_l$ is the union of equi-dimensional
$C^0(Z_r)$-orbits. $V_r$ is open dense in $\mathfrak{g}_2(r)$, \ $\dim
V_k > \dim V_{k-1}$, and $V_0 = (0)$.

\begin{Lemma} % 5.11
Let $p = X\oplus Y\in\mathfrak{g}_1\oplus\mathfrak{g}'_1$. Then\/
$\Phi^{-1}\bigl(\Phi(p)\bigr)$ is the $N$-orbit through the point $p$.
\end{Lemma}

\begin{proof}
This is an easy consequence of Proposition 5.8 and Lemma 5.9.
\end{proof}

\begin{Proposition} % 5.12
$M_l^* = \Phi^{-1}(V_l)$, \ $0\leq l\leq r$.
\end{Proposition}

\begin{proof}
By Proposition 5.6, we have $M_l^* = M_l\cap
(\mathfrak{g}_1\oplus\mathfrak{g}'_1) = G^0(0_l, 2\,0_l)\cap
(\mathfrak{g}_1\oplus\mathfrak{g}'_1)$, which contains the same
dimensional orbit $\widehat{Q_r}{}^0(0_l, 2\,0_l)$. On the other hand
\[\widehat{Q_r}{}^0(0_l, 2\,0_l)
= \Phi^{-1}\bigl(C^0(Z_r) \Phi(0_l, 2\,0_l)\bigr)
= \Phi^{-1}\bigl(C^0(Z_r)\,0_l\bigr).\]
Let
% end of page 42 (5-8)
\[0_{p,q} = \sum_{i=1}^p E_i - \sum_{j=p+1}^{p+q} E_j.\]
Note that $0_l = 0_{l,0}$. Let $V_{p,q} = C^0(Z_r)\,0_{p,q}$. It is
known \cite{9} that $V_l = \coprod_{p+q=l} V_{p,q}$. 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 5.11, \
$\Phi^{-1}(V_{p,q})$, \ $p+q=l$, are the same dimensional
$\widehat{Q_r}{}^0$-orbits. Therefore, by Proposition 5.10, we have
\[\Phi^{-1}(V_l) = \Phi^{-1}\biggl(\coprod_{p+q=l} V_{p,q}\biggr) =
\coprod_{p+q=l} \Phi^{-1}(V_{p,q}) = M_l^*.\eqno{\Box}\]
\noqed\end{proof}

We say that $\Phi^{-1}(V_l)$ is the Siegel-type realization of the
$G$-orbit $M_l$. Let $P\colon\mathfrak{g}_2(r)\to
\End\mathfrak{g}_2(r)$ be the quadratic operator of the Jordan algebra
$\mathfrak{g}_2(r)$. Then we have

\begin{Theorem} % 5.13
The Siegel-type realization of the $G$-orbit $M_l$, \ $0\leq l\leq r$, is
given by
\[M_l^*=\Phi^{-1}(V_l) = \{\,(x, u^-)\oplus (y,
v^+)\in\mathfrak{g}_1\oplus\mathfrak{g}'_1: \rk P(y - x + [v^+, u^-])
= i_l\,\}, \eqno(5.13)\] 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
% end of page 43 (5-9)
\[M_r^* = \Phi^{-1}(V_r)
= \{\,(x, u^-)\oplus (y, v^+)\in\mathfrak{g}_1\oplus\mathfrak{g}'_1:
\nu(y - x + [v^+, u^-])\neq 0\,\}, \eqno(5.14)\] where $\nu$ denotes
the generic norm of the Jordan algebra\/ $\mathfrak{g}_2(r)$.
\end{Theorem}

\begin{proof}
By Proposition 5.12, we have
\[M_l^* = \Phi^{-1}(V_l)
= \{\,(x, u^-)\oplus (y, v^+)\in\mathfrak{g}_1\oplus\mathfrak{g}'_1: y
- x + [v^+, u^-]\in V_l\,\}. \eqno(5.15)\]
Also we have \cite{2} that
\[V_l = \{\,x\in\mathfrak{g}_2(r): \rk P(x) = i_l\,\}. \eqno(5.16)\]
Note that the condition $\rk P(x) = i_r$ is equivalent to the
condition $\nu(x)\neq 0$.
\end{proof}

\begin{Remark}
(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
\cite{10}, in which case $v^+ = u^- = 0$.
\end{Remark}

The closure $\overline{V_l}$ of $V_l$ in $\mathfrak{g}_2(r)$, was
given by $V_{\leq l} := \coprod_{i=0}^l V_i$ (\cite{2}). Therefore,
from (5.16) it follows that $\overline{V_l} = V_{\leq l}$ is an
algebraic variety in $\mathfrak{g}_2(r)$, which we call a
\textit{generalized determinantal variety}. In the case of
$\mathfrak{g}_2(r) = M_n(\bbb{C})$, $\Sym_n(\bbb{C})$ (resp.\
$\Alt_{2n}(\bbb{C})$), the number $l$ is just the rank (resp.\
one-half of the rank) of a matrix for $M_n(\bbb{C})$ and
$\Sym_n(\bbb{C})$ (resp.\ $\Alt_{2n}(\bbb{C})$). In those cases,
$V_{\leq l}$ is a usual determinantal variety.
% end of page 44 (5-10)

\section{Stratifications of $\tilde{M}$}

\subsection{} % 6.1

We wish to construct a polynomial map
$\Psi$ of 
$\mathfrak{g}_1\oplus\mathfrak{g}'_1$ to
$\mathfrak{g}_2(r)\times \mathfrak{n}$ (cf.5.3).
Choose a point 
$p = (x, u^-)\oplus (y,v^+)\in\mathfrak{g}_1\oplus\mathfrak{g}'_1$,
and consider the element
$n_p =
\exp(-v^+) \exp(-x - u^-)\in N$.
$n_p $ can be written as
$n_p =
\exp(-x + \frac12 [v^+, u^-] - v^+ - u^-)$.
Since
$\exp:\mathfrak{n} \to N$
is diffeomorphic, one can define 
$\Psi$ to be
\[\Psi(p) = \bigl(\Phi(p), \log n_p \bigr)
= \bigl( y-x + [v^+, u^-], -x + \frac12 [v^+, u^-] - v^+ - u^- \bigr)
. \eqno(6.1)\]
%
\begin{Lemma} % 6.1
The polynomial map
$\Psi$ is a diffeomorphism of 
$\mathfrak{g}_1\oplus\mathfrak{g}'_1$ onto
$\mathfrak{g}_2(r)\times \mathfrak{n}$.
$\Psi^{-1}$ is also a polynomial map.
%$\Psi$ is a diffeomorphism of\/ $\mathfrak{g}_1\oplus\mathfrak{g}'_1$
%onto\/ $\mathfrak{g}_2(r)\times N$.
\end{Lemma}
%
\begin{proof}
Let $p$, $q\in\mathfrak{g}_1\oplus\mathfrak{g}'_1$, 
and suppose 
$\Psi(p) =\Psi(q)$. 
Then we have $\Phi(p) = \Phi(q)$ and 
$n_p = n_q$. 
By Lemma 5.9, we have 
that 
$n_p(p) = (0,0)\oplus\bigl(\Phi(p), 0\bigr) 
= (0, 0)\oplus\bigl(\Phi(q), 0\bigr) = 
n_q(q) = n_p(q)$, 
which implies that $p = q$, proving the
injectivity of $\Psi$. 
Now let $(a, X)\in\mathfrak{g}_2(r)\times \mathfrak{n}$,
and let $p := (\exp~ -X)
\bigl((0, 0)\oplus (a,0)\bigr)\in\mathfrak{g}_1\oplus\mathfrak{g}'_1$. 
Then, by Lemma 5.11,%%
$\Phi(p) = \Phi\bigl((0, 0)\oplus (a, 0)\bigr) = a$, and hence,
%%
by Lemma 5.9, we have
$(\exp~ X)p = (0, 0)\oplus\bigl(\Phi(p), 0\bigr) = n_p(p)$.
Since $N$ acts freely on $\mathfrak{g}_1\oplus\mathfrak{g}'_1$, 
it follows that 
$(a, X) = (\Phi(p),\log n_p)$,
proving the
surjectivity of $\Psi$.
On the other hand, 
$\Psi^{-1}$ is given by 
$\Psi^{-1}(a, X) =  (\exp -X) \bigl((0, 0)\oplus (a,0)\bigr)$,
which is a polynomial in $a$ and $X$ by (5.7). 
\end{proof}
From the expression of $\Psi^{-1}$ in the above proof, 
we have
%%%%%%%%%%%%%%%
\begin{Lemma} % 6.2
Let $V$ be a $C^0(Z_r)$-orbit in\/ $\mathfrak{g}_2(r)$. 
Then, for the corresponding $\widehat{Q_r}{}^0$-orbit\/ $\Phi^{-1}(V)$,
we have $\Psi(\Phi^{-1}(V)) = V\times \mathfrak{n}$.
\end{Lemma}
Let $M_{\leq k}$ and \ $M_{\leq k}^*$
denote the unions
$\coprod^{k}_{i=0} M_i$ and \ $\coprod^{k}_{i=0} M^*_i$,
respectively. 
By Theorem 3.1 and Proposition 5.12,
the closure $\overline{M^*_k}$ of $M^*_k$
in $\mathfrak{g}_1\oplus\mathfrak{g}'_1$
is given by 
$\overline{M^*_k} = M^*_{\leq k} = \Phi^{-1}(V_{\leq k})$,
which is an algebraic variety in
$\mathfrak{g}_1\oplus\mathfrak{g}'_1$ (cf. Theorem 5.13).
Since $V_{\leq k}$
is an algebraic variety in $\mathfrak{g}_2(r)$ ([2]),
$V_{\leq k} \times \mathfrak{n}$
is an algebraic variety in $\mathfrak{g}_2(r) \times \mathfrak{n}$.
We will denote the singular locus and the regular locus 
of an algebraic variety $A$ by 
$\Sing(A)$ and $\Reg(A)$, respectively.
%
\begin{Proposition} % 6.3
The algebraic variety $M^*_{\leq k}$ is isomorphic to
the algebraic variety $V_{\leq k}\times \mathfrak{n}$,
for $0 \leq k \leq r-1$. We have
$$
\Psi(\Sing(M^*_{\leq k})) 
=  \Sing(V_{\leq k}) \times \mathfrak{n}, ~~0 \leq k \leq r-1.
$$
In other words,
$$
\Sing(M^*_{\leq k}) = \Phi^{-1}(\Sing(V_{\leq k})),
~~0 \leq k \leq r-1.
$$
\end{Proposition}
\begin{proof}
The first assertion is
an immediate consequence of Lemmas 6.1 and 6.2.
The other assertions follow from the first one.
\end{proof} 

\subsection{} % 6.2
 
We wish to find the singular locus 
$\Sing(V_{\leq k})$ of a generalized 
determinantal variety $V_{\leq k}$ in
$\mathfrak{g}_2(r)$.
In the case where
$( \mathfrak{g}, \mathfrak{g}_0)$
is of $BC_r$-type,
$V_{\leq k}$ 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
$( \mathfrak{g}, \mathfrak{g}_0)$
is of $C_r$-type, the GLA (5.11)
coincides with
the original
GLA (1.1), more precisely, we have
$\mathfrak{g}_{\pm 2}(r) = \mathfrak{g}_{\pm 1}$
and 
$\mathfrak{g}_{0}(r) = \mathfrak{g}_{0}$.
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  $\mathfrak{g}_{1}$ is a simple Jordan
algebra on which $G_{0}$ acts as the structure group.
As for (5.12) we have the rank decomposition
\[
\mathfrak{g}_1 = V_r\amalg V_{r-1}\amalg\dots\amalg V_0,
\eqno(6.2)\]
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]:
\[
V_k = \amalg_{p+q=k} G_0^0 0_{p,q}, ~~0 \leq k \leq r,
\eqno(6.3)\]
where $0_{p,q}$ is the same as in the proof of Proposition 5.12.
(5.16) is still valid by replacing 
$\mathfrak{g}_2(r)$ by $\mathfrak{g}_1$.
Therefore $V_{\leq k}$ is an algebraic variety in 
$\mathfrak{g}_1$ defined over $\bbb{R}$.
We have 
$V_{\le r-1}
=\{ x \in \mathfrak{g}_1 : \det P(x) = 0 \}$.
Since $\det P(x)$ is a power of the generic norm $\nu$
of the Jordan algebra $\mathfrak{g}_1$,
the defining ideal 
$I(V_{\leq r-1})$ of  $V_{\leq r-1}$
is generated by the irreducible
polynomial $\nu$.
%%%%%%%%%%%%%%%%%%%%
The variety $V_{\leq k}$ is a conic variety,                           %!
since $G_0^0$ contains the one-dimensional 
center acting on $\mathfrak{g}_1$
as homotheties. Therefore the defining
ideal $I(V_{\leq k})$ of $V_{\leq k}$
is a homogeneous ideal.
Let $I(V_{\leq k})_m$ denote the totality of
homogeneous polynomials in $I(V_{\leq k})$ 
of degree $m$.

\begin{Proposition} % 6.4
For a simple GLA (1.1) with $r \geq 2$,
the singular locus
$\Sing(V_{\leq 1})$ of the generalized determinantal
variety $V_{\leq 1}$ in $\mathfrak{g}_1$                                 %
coincides with $V_0 = (0)$.
\end{Proposition}

\begin{proof}
Let $\mathfrak{a}_1$ be the linear span of
$E_1, \cdots, E_r$ in $\mathfrak{g}_1$.
Then it is known [1,8] that                                                              %
\[
\mathfrak{g}_1 = G_0^0 \mathfrak{a}_1.
\eqno(6.4)\]
First we claim that
$I(V_{\leq 1})_1 = \emptyset$.
Suppose the contrary.
One can then choose a nonzero
linear form $f$ on $\mathfrak{g}_1$
such that $f(V_1)=0$.
Since $r \geq 2$, there exists a point
$x_0 \in \mathfrak{g}_1$ such that
$f(x_0) \ne 0$. By (6.4) one can assume that                         %
$x_0$ lies in $\mathfrak{a}_1$.
Since $E_i$ is conjugate to $E_1$
under $G_0^0$, $E_1, \cdots, E_r$ belong
to $V_1$. By the assumption for $f$
we have that $f(E_i) = 0, 1 \leq i \leq r$,
which implies that
$f$ is identically zero on $\mathfrak{a}_1$.
This contradicts the fact that
$f(x_0) \ne 0$, which shows that 
the claim
$I(V_{\leq 1})_1 = \emptyset$.
Recall that the variety
$V_{\leq 1}$ is defined over
$\bbb{R}$ (cf.6.2).
One can choose a generator
$\{ f_1, \cdots, f_s \}$
of the ideal $I(V_{\leq 1})$
such that each polynomial
$f_i$ is homogeneous and defined over $\bbb{R}$.
From the above argument, 
it follows that $\deg f_i \geq 2$.
Consequently $(df_i)_0 = 0, 1 \leq i \leq s$,
which shows that $0$ is a singularity
of $V_{\leq 1}$. Obviously we have that
$\Reg(V_{\leq 1}) \supset V_1$.
Therefore we conclude 
$\Sing(V_{\leq 1}) = V_0$.
\end{proof}
%%%%%%%%%%%%%%%%%
\subsection{} % 6.3

In this paragraph we treat
the case where the GLA (1.1) is complex
simple of $C_r$-type.
The subspace $\mathfrak{g}_{1}$ is then
a complex simple Jordan algebra.
The following is a list of 
complex simple Jordan algebras.
\begin{table}[h]
\begin{tabular}[H]{ccc}
Type & $\mathfrak{g}_1$               & $r$ \cr\hline I & $M_n(\bbb{C})$           & $n$ \cr
II   & $\Alt_{2n}(\bbb{C})$     & $n$ \cr
III  & $\Sym_n(\bbb{C})$        & $n$ \cr
IV   & $\bbb{C}^n$              & 2 \cr
VI   & $H_3(\bbb{O}^{\bbb{C}})$ & 3 \cr
\end{tabular}
\end{table}
\par\noindent
Here $H_3(\bbb{O}^{\bbb{C}})$ denotes 
the exceptional simple Jordan algebra
of $3\times 3$ Hermitian matrices 
with entries in complex octonions
$\bbb{O}^{\bbb{C}}$.
\begin{Proposition} % 6.5
For any complex simple Jordan algebra
\/ $\mathfrak{g}_1$ with $r \geq 2$, we have
\[\Sing(V_{\leq k}) = V_{\leq k-1}, \qquad 1\leq k\leq r-1.
\eqno(6.5)\]
\end{Proposition}
%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}\hfill\par
(i) For the case of types I or III,
$X \in V_{\leq k}$ if and only if 
$\rk X \leq k$.
For the case of type II,
$X \in V_{\leq k}$ if and only if 
$\rk X \leq 2k$.
In those three cases, (6.5) is 
well-known (see for example [11]).

(ii)\enspace Consider the case of type IV.
In this case we have $r=2$
and $\mathfrak{g}_1 
= V_2\amalg V_1 \amalg V_0$.
Therefore (6.5) follows from Proposition 6.4.

(iii)\enspace Now we consider the case of type VI.
In this case $\mathfrak{g}_1$ can be identified
with $H_3(\bbb{O}^{\bbb{C}})$
in a such a way that
$E_i \/ (i=1,2,3)$ is sent to 
the diagonal matrix 
$(\delta_{i1}, \delta_{i2}, \delta_{i3})$.
An element $x\in H_3(\bbb{O}^{\bbb{C}})$ is
expressed as
\[x = \begin{pmatrix}
\xi_1   & c       & \bar{b} \\
\bar{c} & \xi_2   & a       \\
b       & \bar{a} & \xi_3   \\ 
\end{pmatrix}, \qquad \xi_i\in\bbb{C}, \quad a, b, c\in\bbb{O}^{\bbb{C}}.
\eqno(6.6)\]
The generic norm $\nu(x)$ of $x$ is given by
% end of page 48 (6-4)
\[
\nu(x) = \xi_1 \xi_2 \xi_3 
- \xi_1 n(a) - \xi_2 n(b) - \xi_3 n(c) +
t(abc),
\]
where $n$ and $t$ denote respectively the norm 
and the trace of an octonion. 
We express an element $a \in\bbb{O}^{\bbb{C}}$
as 
$a = \sum^7_{i=0} a_i e_i$, 
where $\{e_i\}$ is the canonical basis of
$\bbb{O}^{\bbb{C}}$.
The variety $V_{\leq 2}$ 
is defined by the single equation
$\nu(x) = 0, \/ x\in \mathfrak{g}_1$.
We then have

\[\begin{aligned}[c]
d\nu &= \bigl(\xi_2 \xi_3 - n(a)\bigr)\,d\xi_1 + \bigl(\xi_1 \xi_3 -
n(b)\bigr)\,d\xi_2 + \bigl(\xi_1 \xi_2 - n(c)\bigr)\,d\xi_3\\ &\qquad
+ \sum^{7}_{i=0} \biggl(-2\xi_1 a_i + \frac{\partial}{\partial a_i}
t(abc)\biggr)\,da_i + \sum^{7}_{i=0} \biggl(-2\xi_2 b_i +
\frac{\partial}{\partial b_i}t(abc)\biggr)\,db_i\\ &\qquad +
\sum^{7}_{i=0} \biggl(-2\xi_3 c_i + \frac{\partial}{\partial c_i}
t(abc)\biggr)\,dc_i.\end{aligned} 
\eqno(6.7)\]

Now let $x\in V_{\leq 1}$.
From (6.4) and (6.3) it follows that 
there exists an element $g \in G_0^0$
such that $gx \in \mathfrak{a}_1 \cap V_{\leq 1}$,
in other words,
$gx$ is a diagonal matrix
diag$(\xi_1, \xi_2, \xi_3)\in V_{\leq 1}$,
which implies that
at least two of $\xi_1$, $\xi_2$, $\xi_3$ are zero.
Therefore we have from (6.7) that
\[
(d\nu)_{gx} = \xi_2 \xi_3\,d\xi_1 + \xi_1
\xi_3\,d\xi_2 + \xi_1 \xi_2\,d\xi_3 = 0
\]
Therefore, in view of the relative
invariance of $\nu$ under $G_0$,
we have $(d\nu)_x = 0$.
By the Jacobian criterion,
we obtain 
$V_{\leq 1}\subset \Sing(V_{\leq 2})$.
On the other hand, clearly we have
$V_2\subset \Reg(V_{\leq 2})$.
Consequently we conclude that
$V_{\leq 1} = \Sing(V_{\leq 2})$.
The equality $V_0 = \Sing(V_{\leq 1})$
follows from Proposition 6.4.
\end{proof}

We denote the defining ideal of an 
algebraic variety $A$ by $I(A)$.%,
%and denoted by $<f_1, \dots, f_s>$
%the ideal generated by polynomials
%$f_1$, $\dots$, $f_{s}$.

\begin{Corollary} % 6.6
Let $V_{\leq k} \/ (1 \leq k \leq r-1)$
be a generalized determinantal variety in 
a complex simple Jordan algebra $\mathfrak{g}_1$.
Then there exists a basis
$\{f_1, \dots, f_{s_k}\}$ of $I(V_{\leq k})$
such that each $f_i$ is defined over $\bbb{R}$
and that $df_i \in I(\Sing (V_{\leq k})), 
\/  1 \leq i \leq s_k$, in other words, 
$(df_i)_p = 0$, $1 \leq i \leq s_k$
for each point $p \in \Sing (V_{\leq k})$.
\end{Corollary}

\begin{proof}\hfill\par
Note that $V_{\leq k}$ is defined over $\bbb{R}$ (cf.6.2).
For types I and III, we choose, as a generator
of $I(V_{\leq k})$, the totality of $(k+1)$-minors
of a generic element of $\mathfrak{g}_1$.
For Type II, we choose, as a generator
of $I(V_{\leq k})$, the totality of 
the Pfaffians of principal
$(2k+2)$-submatrices
of a generic element of $\mathfrak{g}_1$.
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.
\end{proof}

\subsection{} % 6.4
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:
$\mathfrak{g} = 
\mathfrak{g}_{-1} + \mathfrak{g}_0 + \mathfrak{g}_1$.
$r$ is the split rank of the symmetric pair
$(\mathfrak{g}, \mathfrak{g}_0)$.
In this case $\mathfrak{g}_1$ is
a real simple but not
complex simple Jordan algebra.
Consider the complexification of the 
GLA $\mathfrak{g}$:
%
\[
\mathfrak{g}^c = 
\mathfrak{g}^c_{-1} + \mathfrak{g}^c_0 + \mathfrak{g}^c_1
\]
%%%%%%%%%
% end of page 47 (6-3)
%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
% end of page 48 (6-4)
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%
% end of page 49 (6-5)
%%%%%%%%%%%%%%%%%%%%%%%
Let $\bar{r}$ be the split rank of the symmetric pair
$(\mathfrak{g}^c, \mathfrak{g}^c_0)$. 
The following is a list of real
simple GLAs of $C_r$-type 
and their complexifications:

\subsection*{Type I}

\[\begin{array}{l}
\left\{\begin{aligned}
(\mathfrak{g}^c, \mathfrak{g}^c_{0}, \mathfrak{g}^c_1) &=
\makebox[86mm][l]{$\bigl(\sl(2n,\bbb{C}), \sl (n,\bbb{C}) +
\sl(n,\bbb{C}) + \bbb{C}, M_n(\bbb{C})\bigr)$,} \quad \bar{r} = n,\\ 
(\mathfrak{g}, \mathfrak{g}_0, \mathfrak{g}_1) &=
\left\{\begin{aligned}
&\makebox[82mm][l]{$\bigl(\sl(2n, \bbb{R}), \sl(n, \bbb{R}) + \sl(n,
\bbb{R}) + \bbb{R}, M_n(\bbb{R})\bigr)$,} \quad r = n,\\
&\makebox[82mm][l]{$\bigl(\su(n,n), \sl(n, \bbb{C}) + \bbb{R}, H_n
(\bbb{C})\bigr)$,} \quad r = n,\end{aligned}\right.
\end{aligned}\right.\\ \\
\left\{\begin{aligned}
(\mathfrak{g}^c, \mathfrak{g}^c_0, \mathfrak{g}^c_1) &=
\makebox[86mm][l]{$\bigl(\sl(4n, \bbb{C}), \sl(2n, \bbb{C}) + \sl(2n,
\bbb{C}) + \bbb{C}, M_{2n}(\bbb{C})\bigr)$,} \quad \bar{r} = 2n,\\
(\mathfrak{g}, \mathfrak{g}_0, \mathfrak{g}_1) &=
\makebox[86mm][l]{$\bigl(\sl(2n, \bbb{H}), \sl(n, \bbb{H}) + \sl(n,
\bbb{H}) + \bbb{R}, M_n(\bbb{H})\bigr)$,} \quad r = n,
\end{aligned}\right.\end{array}\]

\subsection*{Type II}

\[\left\{\begin{aligned}
(\mathfrak{g}^c, \mathfrak{g}^c_0, \mathfrak{g}^c_1) &=
\makebox[66mm][l]{$\bigl(\so(4n, \bbb{C}), \gl(2n, \bbb{C}),
\Alt_{2n}(\bbb{C})\bigr)$,} \quad \bar{r} = n,\\
(\mathfrak{g}, \mathfrak{g}_0, \mathfrak{g}_1) &=
\left\{\begin{aligned}
&\makebox[62mm][l]{$\bigl(\so(2n, 2n), \gl(2n, \bbb{R}),
\Alt_{2n}(\bbb{R})\bigr)$,} \quad r = n,\\
&\makebox[62mm][l]{$\bigl(\so^*(4n), \gl(n, \bbb{H}),
H_n(\bbb{H})\bigr)$,} \quad r = n\end{aligned}\right.
\end{aligned}\right.\]

\subsection*{Type III}

\[\begin{array}{l}
\left\{\begin{aligned}
(\mathfrak{g}^c, \mathfrak{g}^c_0, \mathfrak{g}^c_{1}) &=
\makebox[68mm][l]{$\bigl(\sp(n, \bbb{C}), \gl(n,\bbb{C}),
\Sym_n(\bbb{C})\bigr)$,} \quad \bar{r} = n,\\
(\mathfrak{g}, \mathfrak{g}_0, \mathfrak{g}_1) &=
\makebox[68mm][l]{$\bigl(\sp(n, \bbb{R}), \gl(n, \bbb{R}),
\Sym_n(\bbb{R})\bigr)$,} \quad r = n,\end{aligned}\right.\\ \\
\left\{\begin{aligned}
(\mathfrak{g}^c, \mathfrak{g}^c_0, \mathfrak{g}^c_1) &=
\makebox[68mm][l]{$\bigl(\sp(2n, \bbb{C}), \gl(2n, \bbb{C}),
\Sym_{2n}(\bbb{C})\bigr)$,} \quad \bar{r} = 2n,\\
(\mathfrak{g}, \mathfrak{g}_0, \mathfrak{g}_1) &=
\makebox[68mm][l]{$\bigl(\sp(n,n), \gl(n, \bbb{H}),
\operatorname{SH}_n(\bbb{H})\bigr)$,} \quad r = n,
\end{aligned}\right.\end{array}\]

\subsection*{Type IV}

\[\left\{\begin{aligned}
(\mathfrak{g}^c, \mathfrak{g}_0^c, \mathfrak{g}_1^c) &=
\makebox[70mm][l]{$\bigl(\so(n+2, \bbb{C}), \so(n, \bbb{C}) + \bbb{C},
\bbb{C}^n\bigr)$,} \quad \bar{r} = 2,\\ 
(\mathfrak{g}, \mathfrak{g}_0, \mathfrak{g}_1) &=
\makebox[70mm][l]{$\bigl(\so(p+1, q+1), \so(p, q) + \bbb{R},
\bbb{R}^n\bigr)$,} \quad r = \left\{\begin{aligned}
& 1 \quad (p=0),\\
& 2 \quad (p\geq 1),\end{aligned}\right.\\
& \qquad\qquad p\leq q, \ p + q = n,
\end{aligned}\right.\]

\subsection*{Type VI}

\[\left\{\begin{aligned}
(\mathfrak{g}^c, \mathfrak{g}_0^c, \mathfrak{g}_1^c) &=
\makebox[64mm][l]{$\bigl(E_7^{\bbb{C}}, E_6^{\bbb{C}} + \bbb{C},
H_3(\bbb{O}^{\bbb{C}})\bigr)$,} \quad \bar{r} = 3,\\
(\mathfrak{g}, \mathfrak{g}_0, \mathfrak{g}_1) &=
\left\{\begin{aligned}
& \makebox[60mm][l]{$\bigl(E_{7(7)}, E_{6(6)} + \bbb{R},
H_3(\bbb{O}')\bigr)$,} \quad r = 3,\\ 
& \makebox[60mm][l]{$\bigl(E_{7(-25)}, E_{6(-26)} + \bbb{R},
H_3(\bbb{O})\bigr)$,} \quad r = 3. 
\end{aligned}\right.\end{aligned}\right.\]

%%%%%%%%%%%%%%%%%%%%
% end of page 51 (6-7)

In the above list, $\bbb{H}$ denotes the quaternion algebra.
$\bbb{O}$, (resp. $\bbb{O}'$) denotes the octonion
(resp. split octonion) algebra.
$H_n(\bbb{F}), \bbb{F} = \bbb{C}, \bbb{H}, \bbb{O},
 \bbb{O}',$ denotes the Jordan algebra of
$n \times n$ $\bbb{F}$-Hermitian matrices.
$\operatorname{SH}_n(\bbb{H})$
denotes the Jordan algebra of
$n \times n$ skew-Hermitian quaternion matrices.
Note that the generalized determinantal varieties
in a complex simple Jordan algebra are
defined over $\bbb{R}$.
From a result of Takeuchi[14]
we have
%
\begin{Proposition}% 6.7
Let $\mathfrak{g}_1$ and $\mathfrak{g}_1^c$ be as above,
and let $\mathfrak{g}_1 = \amalg_{k=0}^{r} V_k$ 
and $\mathfrak{g}_1^c = \amalg_{k=0}^{\bar{r}}\tilde{V}_k$
be the rank decomposition (cf.(6.2)) of 
$\mathfrak{g}_1$ and $\mathfrak{g}_1^c$,
respectively. Suppose that $\bar{r} = r$.
Then $V_{\leq k} (1\leq k \leq r-1)$
coincides with the set of
$\bbb{R}$-rational points of the complex algebraic variety
$\tilde{V}_{\leq k}$.
Suppose that $\bar{r} = 2r$.
Then $V_{\leq k} (1\leq k \leq r-1)$
coincides with the sets of
$\bbb{R}$-rational points of the algebraic varieties
$\tilde{V}_{\leq 2k}$ and of $\tilde{V}_{\leq 2k+1}$.
\end{Proposition}
%
From Proposition 6.7, we obtain
\begin{Lemma} % 6.8
Let $f_1$, $\dots$, $f_{s_k}$ be real polynomials 
on $\mathfrak{g}_1$,
and let $\tilde{f}_i (1 \leq i \leq s_k)$ be the natural extension of $f_i$ to
$\mathfrak{g}_1^c$. 
Then $I(V_{\leq k}) (1 \leq k \leq r-1)$
is generated by $f_1$, $\dots$, $f_{s_k}$ 
if and only if 
$I(\tilde{V}_{\leq k})$ (resp. $I(\tilde{V}_{\leq 2k}))$ 
is generated by 
$\tilde{f}_1$, $\dots$, $\tilde{f}_{s_k}$ 
for $\bar{r} = r $ (resp. $\bar{r} = 2r.$). 
\end{Lemma}
%
\begin{Proposition} % 6.9
For a real simple\/ \textup{(}not complex
simple\/\textup{)} Jordan algebra $\mathfrak{g}_1$, we have
\[\Sing(V_{\leq k}) = V_{\leq k-1}, \qquad 1\leq k\leq r-1.\]
\end{Proposition}

\begin{proof}
Let $\theta$ be the conjugation of
$\mathfrak{g}_1^c$ with respect to $\mathfrak{g}_1$.
Since $\tilde{V}_{\leq k}$ is $\theta$-stable,
$\Sing(\tilde{V}_{\leq k})$ is also $\theta$-stable.
Let $\bigl(\Sing(\tilde{V}_{\leq k})\bigr)_\theta$ 
be the set of $\theta$-fixed points in
$\Sing(\tilde{V}_{\leq k})$.
Suppose  first $\bar{r} = r$.
Since $\tilde{V}_{\leq k}$ is a conic variety 
defined over $\bbb{R}$ (cf.6.2), 
one can choose a generator
$\{\tilde{f}_1, \dots, \tilde{f}_{s_k} \}$ of 
$I(\tilde{V}_{\leq k})$ such that each
$\tilde{f}_i$ is homogeneous and defined over
$\bbb{R}$. By Corollary 6.6, 
$d\tilde{f}_i \in I( \Sing(\tilde{V}_{\leq k}) )$.
Let $f_i = \tilde{f}_i |_{\mathfrak{g}_1}$.
Then
$\{ f_1,\dots, f_{s_k}\}$
is a generator of
$I(V_{\leq k})$, by Lemma 6.8.
Let
$p\in \bigl(\Sing(\tilde{V}_{\leq k})\bigr)_\theta$.
Then
$p\in \bigl(\tilde{V}_{\leq k}\bigr)_\theta
= V_{\leq k}$, by Proposition 6.7.
We have  $(df_i)_p = (d\tilde{f}_i)_p = 0$,
which implies that
$p\in \Sing(\tilde{V}_{\leq k})$.
Hence, by Propositions 6.5 and 6.7, 
we have
\[
V_{\leq k-1} = (\tilde{V}_{\leq k-1})_\theta
= \bigl(\Sing(\tilde{V}_{\leq k})\bigr)_\theta \subset 
\Sing(V_{\leq k}).
\eqno(6.8)\]
In view of the inclusion $V_k\subset\Reg(V_{\leq k})$,
we conclude 
$V_{\leq k-1} = \Sing (\tilde{V}_{\leq k})_\theta
= \Sing(V_{\leq k}) 
$.
As for the case $\bar{r} = 2r$, we should
replace
(6.8) by the equality
\[
V_{\leq k-1} = (\tilde{V}_{\leq 2k-2})_\theta
= (\tilde{V}_{\leq 2k-1})_\theta
= \bigl(\Sing(\tilde{V}_{\leq 2k})\bigr)_\theta \subset 
\Sing(V_{\leq k}).
\]
\end{proof}

\subsection{} % 6.5
Combining Propositions 6.5 and 6.9, we have

\begin{Theorem} % 6.10
Let $\mathfrak{g} = 
\mathfrak{g}_{-1} + \mathfrak{g}_0 + \mathfrak{g}_1$
be a simple GLA of $C_r$-type,
and let $\mathfrak{g}_1 = \amalg_{k=0}^{r} V_k$
be the rank decomposition.
Then the closure $\bar{V}_{k}$ of $V_{k}$
is the generalized determinantal variety
$V_{\leq k}$, and  $\Sing(V_{\leq k}) = V_{\leq k-1}$
for $1 \leq k \leq r-1$.
\end{Theorem}

From Theorem 6.10 and Proposition 6.3 we have

\begin{Theorem} % 6.11
Let $\mathfrak{g} = 
\mathfrak{g}_{-1} + \mathfrak{g}_0 + \mathfrak{g}_1$
be a simple GLA,
and let $r$ be the split rank of the symmetric pair
$(\mathfrak{g}, \mathfrak{g}_0 )$.
Let $\mathfrak{g}_1 \oplus \mathfrak{g}_1' 
= \amalg_{k=0}^{r} M_k^*$
be the rank decomposition (cf. 5.4)
of the Jordan pair $\mathfrak{g}_1 \oplus \mathfrak{g}_1'$.
Then the closure $\bar{M}_k^*$ of $M_k^*$
in $\mathfrak{g}_1 \oplus \mathfrak{g}_1'$
is the algebraic variety $M_{\leq k}^*$, and
\[\Sing(M_{\leq k}^*) = M_{\leq k-1}^*, 
\qquad \Reg(M_{\leq k}^*) = M_k^*,
\qquad 1\leq k\leq r-1. \]
\end{Theorem}


\subsection{} % 6.6
Now we go back to the full $G$-orbits $M_k$.

\begin{Lemma} % 6.12
$M_{\leq k}$ \ \textup{(}$0\leq k\leq r-1$\textup{)} is a real
analytic set in $\tilde{M}$.
\end{Lemma}

\begin{proof}
Choose a point $p_0\in M_{\leq k}$. Then one can find an element $g\in
G$ such that $g(p_0)\in\mathfrak{g}_1\oplus\mathfrak{g}'_1$.
Choose a neighborhood
$U$ of $p_0$ in $\tilde{M}$ in such a way that $U' :=
g(U)\subset\mathfrak{g}_1\oplus\mathfrak{g}'_1$. Let $p\in U$. 
Then we have that $p\in U\cap M_{\leq k}$ if and only if $g(p)\in U'\cap
M^*_{\leq k}$.
 Let $\{ f_1$, $\dots$, $f_{s_k} \}$ be a basis of
the ideal $I(M^*_{\leq k})$. 
Then $M_{\leq k}$ is expressed in $U$ as
\[U\cap M_{\leq k} = \{\,p\in U: (f_i\circ g)(p) = 0, \ 1\leq i\leq
s_k\,\},\]
which implies that $M_{\leq k}$ is a real analytic set of $\tilde{M}$.
\end{proof}

A point $p\in M_{\leq k}$ is a regular point of $M_{\leq k}$, if there
exists a neighborhood $U$ of $p$ in $\tilde{M}$ such that $U\cap
M_{\leq k}$ is a smooth manifold of dimension $d_k := \dim
M_k$. Otherwise we say that $p$ is a singular point of $M_{\leq k}$. 
We denote by
$\Reg(M_{\leq k})$ (resp.\ $\Sing(M_{\leq k})$) the regular (resp.\
singular) locus of $M_{\leq k}$. 
Finally we get the following theorem which gives the
stratification of $\tilde{M}$ by $G$-orbits.

\begin{Theorem} % 6.13
For \/ $1\leq k\leq r-1$, we have
$\Reg(M_{\leq k}) = M_k$ and\/ $\Sing(M_{\leq k}) = M_{\leq k-1}$.
\end{Theorem}

% end of page 55 (6-11)

\begin{proof}
Let $p\in\Reg(M_{\leq k})$. Choose an element $g\in G$ such
that $g(p)\in M^*_{\leq k}$. Then, since $p$ is a regular point
of $M_{\leq k}$, \ $g(p)$ lies in $\Reg(M^*_{\leq k}) = M^*_k$ by Theorem
6.11. 
This implies $p\in M_k$, or equivalently, $\Reg(M_{\leq k}) = M_k$.
Similarly we can show $\Sing(M_{\leq k}) = M_{\leq k-1}$.
\end{proof}

\begin{Corollary} % 6.14
Suppose that a diffeomorphism $f$ of $\tilde{M}$ leaves 
the open orbit $M_r$ stable. 
Then $f$ leaves all other orbits 
$M_k ~~ (0\leq k \leq r-1)$ stable.
\end{Corollary}

\begin{proof}
By the assumption, $f$ leaves $M_{\leq r-1}$ stable, and hence
$\Reg(M_{\leq r-1})$ is stable under $f$, 
that is, $f$ leaves $M_{r-1}$
stable, by Theorem 6.13. Therefore it leaves $M_{\leq r-2}$
stable. Repeating this procedure, one concludes that all
$G$-orbits $M_k$ are stable under $f$.
\end{proof}

% end of page 56 (6-12)

\section{Double foliation on the minimal boundary orbits}

7.1. In this section, we always assume that $M$ is of $BC_r$-type.

In \S2, we considered the double foliation on $\tilde{M}$, \
$\mathcal{M}^\pm = \{\,M^\pm(g_1 0^-, g_2 0^+): g_1, g_2\in
G\,\}$. $\mathcal{M}^\pm$ naturally induce a double foliation
$F_0^\pm$ on the minimal boundary orbit $M_0$. The leaves $F_0^\pm(p)$
of $F_0^\pm$ through a point $p\in M_0$ are given by the intersection
$M^\mp(p)\cap M_0$.

\begin{Lemma} % 7.1
The leaves of $F_0^\pm$ through the origin\/ $(0^-, a_r\,0^+)\in M_0$
are given by
\begin{align*}
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.
\end{align*}
\end{Lemma}

\begin{proof}
By the definition, $F_0^\pm(0^-, a_r\,0^+) = M^\mp(0^-, a_r\,0^+)\cap
G(0^-, a_r\,0^+)$. Let $(g 0^-, g a_r\,0^+)\in F_0^+(0^-, a_r\,0^+)$,
\ $g\in G$. Then $(g 0^-, g a_r\,0^+)\in M^-(0^-, a_r\,0^+)$, which
implies that $g a_r\,0^+ = a_r\,0^+$, or equivalently, $g\in a_r\,U^+
a_r^{-1}$. Conversely, let $u\in U^+$. Then $a_r u a_r^{-1}(0^-,
a_r\,0^+) = (a_r u a_r^{-1} 0^-, a_r\,0^+)\in G(\,0^-, a_r\,0^+)\cap
M^-(0^-, a_r\,0^+)$.
\end{proof}

\begin{Lemma} % 7.2
The double foliation $F_0^\pm$ arises from the subspaces
$\mathfrak{g}_1^\pm(r)$ of the GLA\/ \textup{(4.10)}.
\end{Lemma}

% end of page 57 (7-1)

\begin{proof}
Let $\mathfrak{u}^\pm = \Lie U^\pm$. By Lemma 7.1, the tangent spaces
at $(0^-, a_r\,0^+)$ to the leaves $F_0^\pm(0^-, a_r\,0^+)$ are
identified with the factor spaces $\mathfrak{u}^-/\mathfrak{q}_r$ and
$(\Ad a_r)\mathfrak{u}^+/\mathfrak{q}_r$. By (4.15) and Lemma 4.4, we
have
\[\mathfrak{u}^- = \mathfrak{g}_{-1} + \mathfrak{g}_0 =
\mathfrak{g}_{-2}(r) + \mathfrak{g}_{-1}(r) + \mathfrak{g}_0(r) +
\mathfrak{g}_1^+(r) = \mathfrak{q}_r +
\mathfrak{g}_1^+(r). \eqno(7.1)\] Also, by (4.15) and Lemma 5.2, we
have
\[\begin{aligned}[t]
(\Ad a_r)\mathfrak{u}^+
&= (\Ad a_r)\bigl(\mathfrak{g}_{-1}^-(r) + \mathfrak{g}_0(r) +
\mathfrak{g}_1^+(r) + \mathfrak{g}_1^-(r) + \mathfrak{g}_2(r)\bigr) \\
&= \mathfrak{g}_1^-(r) + \mathfrak{g}_0(r) + \mathfrak{g}_{-1}^+(r) +
\mathfrak{g}_{-1}^-(r) + \mathfrak{g}_{-2}(r)\\
&= \mathfrak{q}_r + \mathfrak{g}_1^-(r).\end{aligned} \eqno(7.2)\]
Therefore $\mathfrak{u}^-/\mathfrak{q}_r$ and $(\Ad
a_r)\mathfrak{u}^+/\mathfrak{q}_r$ can be identified with
$\mathfrak{g}_1^+(r)$ and $\mathfrak{g}_1^-(r)$, respectively.
\end{proof}

Let $E:= Z_r - 2Z$. Then $E$ is a central element of
$\mathfrak{g}_0(r)$. It follows from (4.14) and (4.15) that
\[\ad E = \left\{\begin{array}{rl}
0 & \textup{on $\mathfrak{g}_{\mathrm{ev}}(r)$}, \\
1 & \textup{on $\mathfrak{g}_{\pm 1}^+(r)$}, \\
-1 & \textup{on $\mathfrak{g}_{\pm 1}^-(r)$}.
\end{array}\right. \eqno(7.3)\]

% end of page 58 (7-2)

\begin{Lemma} % 7.3
Let $g\in C(Z_r)$ and let $I=\ad_{\mathfrak{g}_1(r)} E$. Then the
following three conditions are equivalent\/\textup{:}
\begin{enumerate}
\item[(i)] $g\bigl(\mathfrak{g}_1^\pm(r)\bigr) =
\mathfrak{g}_1^\pm(r)$,
\item[(ii)] $gI = Ig$ on $\mathfrak{g}_1(r)$,
\item[(iii)] $g(E) = E$.
\end{enumerate}
\end{Lemma}

\begin{proof}
The only non-trivial assertion is the implication $\textup{(ii)}\to
\textup{(iii)}$. Suppose (ii). Since $Z$, $Z_r\in\mathfrak{a}$, we
have $\tau(Z_r) = -Z_r$ and $\tau(Z) = -Z$, and hence $\tau(E) =
-E$. This implies that
\[\tau\bigl(\ad_{\mathfrak{g}_1(r)} E\bigr)\tau =
-\ad_{\mathfrak{g}_{-1}(r)} E. \eqno(7.4)\]

Consider the inner product $< \phantom{\varphi}, \phantom{\varphi}>$ 
on $\mathfrak{g}_1(r)$ 
defined by $<X,Y> = -(X,\tau Y)$.  Let us denote by $g_\pm$
the restrictions of  the actions of  $g$ to $\mathfrak{g}_{\pm 1}(r)$,
and denote by $g_{+}^*$ the adjoint operator of $g_+$ with respect to
$<\phantom{\varphi}, \phantom{\varphi} >$.  
Then we have that $I$ is self-adjoint with respect to 
$<\phantom{\varphi}, \phantom{\varphi} >$, and hence, 
by (ii) we have 
$(g_+^*)^{-1}I = I(g_+^*)^{-1}$.  We also have
$g_- = \tau(g_+^*)^{-1}\tau$.  Therefore it follows from (7.4) that
 


%Let us denote by $g\rest{\mathfrak{g}_{\pm 1}(r)}$ the action of $g$
%on $\mathfrak{g}_{\pm1}(r)$. Then we have
%$g\rest{\mathfrak{g}_{-1}(r)} =
%\tau(g\rest{\mathfrak{g}_1(r)})\tau$. Therefore it follows from (7.4)
%and (ii) that 
\begin{align*}
g_-(\ad_{\mathfrak{g}_{-1}(r)}E)(g_-)^{-1} &= \tau(g_+^*)^{-1}\tau
(\ad_{\mathfrak{g}_{-1}(r)}E)\tau(g_+^*)\tau\\ &=
-\tau(g_+^*)^{-1}(\ad_{\mathfrak{g}_1(r)}E)(g_+^*)\tau\\ &= 
-\tau I\tau\\ &=\ad_{\mathfrak{g}_{-1}(r)} E,
\end{align*}
%
%
%(g\rest{\mathfrak{g}_{-1}(r)})(\ad_{\mathfrak{g}_{-1}(r)}
%E)(g\rest{\mathfrak{g}_{-1}(r)})^{-1} &=
%\tau(g\rest{\mathfrak{g}_1(r)})\tau(\ad_{\mathfrak{g}_{-1}(r)} E)
%\tau(g\rest{\mathfrak{g}_1(r)})^{-1}\tau\\ &=
%-\tau(g\rest{\mathfrak{g}_1(r)})(\ad_{\mathfrak{g}_1(r)}
%E)(g\rest{\mathfrak{g}_1(r)})^{-1}\tau\\ &= -\tau I\tau\\ &=
%\ad_{\mathfrak{g}_{-1}(r)} E,
%
which implies that $g_-$ commutes with
$\ad_{\mathfrak{g}_{-1}(r)} E$. Combining this with (7.3) and (ii), we
have that $g$ commutes with $\ad E$ on the whole $\mathfrak{g}$. This
implies (iii).
\end{proof}

\begin{Lemma} % 7.4
$C(Z,Z_r) = \bigl\{\,g\in C(Z_r): g\bigl(\mathfrak{g}_1^\pm(r)\bigr) =
\mathfrak{g}_1^\pm(r)\,\bigr\}$.
\end{Lemma}

% end of page 59 (7-3)

\begin{proof}
Let $g\in C(Z_r)$. Then $g\in C(Z,Z_r)$ if and only if $g(E) =
E$. Hence the assertion follows from Lemma 7.3.
\end{proof}

\begin{Remarkn} % 7.1
Lemmas 7.2, 7.4, Proposition 4.6 and Corollary 4.7 imply that our flag
manifold ($M_0 = G/Q_r$, $F_0^\pm$) with double foliation $F_0^\pm$ is
a so-called pseudo-product manifold associated to the simple GLA
(4.10) with decomposition (4.14), in the sense of Tanaka \cite{15}.
\end{Remarkn}

7.2. 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 \cite{8}.

\subsection*{Type I} ($r=p$).

\begin{align*}
M &= \SL(n,\bbb{F})/\operatorname{S}\bigl(\GL(p,\bbb{F}) \times
\GL(n-p,\bbb{F})\bigr), \qquad\ \bbb{F} = \bbb{R}, \bbb{H}, \bbb{C},\\
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad 1\leq p < n-p, \\
M_0 &= \left\{\begin{array}{ll}
\SO(n)/\operatorname{S}\bigl(\operatorname{O}(p)\times
\operatorname{O}(n-2p)\times \operatorname{O}(p)\bigr), & \quad
\bbb{F} = \bbb{R},\\
\Sp(n)/\Sp(p)\times\Sp(n-2p)\times\Sp(p), & \quad \bbb{F} = \bbb{H},\\
\SU(n)/\operatorname{S}\bigl(\operatorname{U}(p)\times
\operatorname{U}(n-2p)\times \operatorname{U}(p)\bigr), & \quad
\bbb{F} = \bbb{C}.
\end{array}\right.
\end{align*}

\subsection*{Type II} ($r=n$).

% end of page 61 (7-4)

\begin{align*}
&\left\{\begin{array}{l}
\makebox[6mm][r]{$M$} = \SO^0(2n+1,2n+1)/\GL^0(2n+1,\bbb{R}),\\
\makebox[6mm][r]{$M_0$} = \SO(2n+1)\times\SO(2n+1)/\SO(2n),
\end{array}\right.\\
&\left\{\begin{array}{l}
\makebox[6mm][r]{$M$} = \SO(4n+2,\bbb{C})/\GL(2n+1,\bbb{C}),\\
\makebox[6mm][r]{$M_0$} = \SO(4n+2)/\operatorname{U}(2n)\cdot \bbb{T}^1.
\end{array}\right.
\end{align*}

\subsection*{Type V}

\begin{align*}
&\left\{\begin{array}{ll}
\makebox[6mm][r]{$M$} =
\makebox[50mm][l]{$E_{6(6)}/\Spin(5,5)\cdot\bbb{R}^+$,} & (r=2),\\
\makebox[6mm][r]{$M_0$} = \Sp(4)/\Spin(4)\times\Spin(4),
\end{array}\right.\\
&\left\{\begin{array}{ll}
\makebox[6mm][r]{$M$} =
\makebox[50mm][l]{$E_{6(-26)}/\Spin(1,9)\cdot\bbb{R}^+$,} & (r=1),\\ 
\makebox[6mm][r]{$M_0$} = F_4/\Spin(8),
\end{array}\right.\\
&\left\{\begin{array}{ll}
\makebox[6mm][r]{$M$} =
\makebox[50mm][l]{$E_6^{\bbb{C}}/\Spin(10,\bbb{C})\cdot\bbb{C}^*$,} &
(r=2),\\
\makebox[6mm][r]{$M_0$} = E_6/\Spin(8)\cdot \bbb{T}^2.
\end{array}\right.
\end{align*}

% end of page 61 (7-5)

\section{Determination of the automorphism groups of $M$}

8.1. Let $(M=G/G_0, F^\pm)$ 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 $(M_0 = G/Q_r,
F_0^\pm)$ with double foliation $F_0^\pm$, we define the automorphism
group by
\[\Aut(M_0, F_0^\pm) = \{\,g\in\Diffeo(M_0): g_* F_0^\pm =
F_0^\pm\,\}.\]
Tanaka \cite{15} determined this group by establishing a Cartan
connection on $M_0$ and by showing that $(M_0 = G/Q_r, F_0^\pm)$ is
the model space for the Cartan connection. Therefore, taking Remark
7.1 into account, we have
\[\Aut(M_0, F_0^\pm) = G. \eqno(8.1)\]

\begin{Theorem} % 8.1
Let\/ $(M=G/G_0, F^\pm)$ be a parahermitian symmetric space of
$BC_r$-type associated with a simple GLA\/ (1.1). Then\/ $\Aut(M,
F^\pm) = \Aut(M_0, F_0^\pm) = G$.
\end{Theorem}

% end of page 62 (8-1)

\begin{proof}
We identify $M$ with its $\varphi$-image in $\tilde{M}$. Since $G$
acts on $M$ effectively and $F^\pm$ are $G$-invariant, the inclusion
$G\subset\Aut(M, F^\pm)$ is clear. Now let $f\in\Aut(M, F^\pm)$. Then,
by Lemma 2.4, \ $f$ preserves the fibers of the double fibration $M^-
\stackrel{\pi^-}{\longleftarrow} M \stackrel{\pi^+}{\longrightarrow}
M^+$. Hence $f$ induces the diffeomorphisms $f^\pm$ of $M^\pm$ such
that $\pi^\pm\circ f = f^\pm\circ\pi^\pm$. Let $\tilde{f} := f^-\times
f^+$. Clearly, the diffeomorphism $\tilde{f}$ preserves the product
structure of $\tilde{M}$. We claim that $\tilde{f}\rest{M} = f$. In
fact, let $p\in M$, and let $q = f(p)\in M$. We write $p = (p^-, p^+)$
and $q = (q^-, q^+)$, where $p^\pm$, $q^\pm\in M^\pm$. The relation
$\varpi^\pm\cdot\varphi = \pi^\pm$ (cf.\ \S2) implies that $q^\pm =
\varpi^\pm(q) = \varpi^\pm\bigl(f(p)\bigr) =
f^\pm\bigl(\pi^\pm(p)\bigr) = f^\pm(p^\pm)$. Hence $f(p) = (q^-, q^+)
= \bigl(f^-(p^-), f^+(p^+)\bigr) = (f^-\times f^+)(p^-, p^+) =
\tilde{f}(p)$. Since $\tilde{f}$ leaves $M$ invariant, by Corollary
6.14 \ $\tilde{f}$ leaves $M_0$ invariant. Obviously $f_0 :=
\tilde{f}\rest{M_0}$ belongs to $\Aut(M_0, F_0^\pm)$. We wish to show
that $\tilde{f}$ can be uniquely recovered by its restriction
$f_0$. Corresponding to the expression $\tilde{M} = M^-\times
(M^+)_r$, one can express $\tilde{f}$ as $\tilde{f} = f_1\times f_2$,
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 $F_0^\pm$ arise as the fibers of the double fibration $M^-
\stackrel{\pi_0^-}{\longleftarrow} M_0
\stackrel{\pi_0^+}{\longrightarrow} (M^+)_r$ given in Theorem 3.1
(iv). Moreover this double fibration of $M_0$ is just the restriction
of the trivial double fibration $M^-
\stackrel{\varpi_0^-}{\longleftarrow} \tilde{M}
\stackrel{\varpi_0^+}{\longrightarrow} (M^+)_r$ (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 $\tilde{f}$ is uniquely
recovered from $f_0$. As a result, the correspondence $f\mapsto f_0$
is an injective homomorphism of $\Aut(M, F^\pm)$ into $\Aut(M_0,
F_0^\pm)$. Consequently, in view of (8.1) we have that $\Aut(M, F^\pm)
= G$.
\end{proof}

% end of page 63 (8-2)

8.2. In this paragraph we are concerned with $C_r$-type. Under this
assumption, for the GLA (4.10) we have $\mathfrak{g}_{\pm 1}(r) =
(0)$, \ $\mathfrak{g}_{\pm 2}(r) = \mathfrak{g}_{\pm 1}$, \
$\mathfrak{g}_0(r) = \mathfrak{g}_0$, \ $Z_r = Z$ and hence $C(Z_r) =
C(Z) = G_0$. The rank decomposition (5.12) becomes $\mathfrak{g}_1 =
\coprod_{k=0}^r V_k$, where $V_k$ is a union of equi-dimensional
$G_0$-orbits. Now consider the $G_0$-stable conic algebraic set
$V_{\leq r-1}$, which is the boundary $\partial V_r$ of $V_r$. One can
extend the cone $\partial V_r$ to a cone field on the whole $M^-$ by
% end of page 64 (8-3)
using the $G$-action on $M^-$. We call the cone field a generalized
conformal structure $\mathcal{K}$ (\cite{2}). One can consider the
automorphism group $\Aut(M^-, \mathcal{K})$, the totality of
diffeomorphisms leaving the cone field $\mathcal{K}$ invariant. This
group was determined for each symmetric $R$-space $M^-$ (\cite{2}):
\[\Aut(M^-, \mathcal{K}) = \left\{\begin{array}{ll}
G, & r\geq 2,\\ \Diffeo(M^-), & r=1.\end{array}\right. \eqno(8.2)\]
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), \
$\tilde{M}$ takes the form $\tilde{M} = M^-(0^-, 0^-)\times M^+(0^-,
0^-) = M^-\times M^-$. Further the minimal boundary orbit $M_0$
becomes $M_0 = G(0^-, 0^-) = G/U^- = M^-$, the diagonal set of
$\tilde{M} = M^-\times M^-$.

Now let $f\in\Aut(M, F^\pm)$, and let $\tilde{f} = f^-\times f^+$ be
the extension of $f$ to $\tilde{M}$ given in 8.1. Let us express
$\tilde{f}$ as $\tilde{f} = f_1\times f_2$ corresponding to the
expression $\tilde{M} = M^-\times M^-$. By Corollary 6.14, \
$\tilde{f}$ leaves $M_0$, the diagonal of $\tilde{M}$, invariant, from
% end of page 65 (8-4)
which we have $f_1 = f_2$, that is, $\tilde{f} = f_1\times f_1$. Thus
it follows that the correspondence $f\mapsto f_1$ is an injective
homomorphism of $\Aut(M, F^\pm)$ into $\Diffeo(M^-)$. The following
lemma is essentially due to Tanaka \cite{15}.

\begin{Lemma} % 8.2
Suppose that\/ $(0^-, 0^-)$ is a fixed point of $\tilde{f}$. Then the
differential\/ $(f_1)_*$ at\/ $0^-$ leaves the cone $\partial V_r$
stable.
\end{Lemma}

\begin{proof}
Let $Y\in\partial V_r$. Then $(0, tY)$ is a path in $M^*_{\leq
r-1}$. By the assumption, the curve $\tilde{f}(0, tY)$ lies in
$M^*_{\leq r-1}$ for $|t|$ sufficiently small. Therefore
$\Phi\bigl(\tilde{f}(0, tY)\bigr) = \Phi\bigl(f_1(0), f_1(tY)\bigr) =
f_1(tY) - f_1(0) = f_1(tY)$ lies in $\partial V_r$. Hence
$(f_1)_{* 0^-}(Y) = \lim_{t\to 0} \frac{1}{t} f_1(tY)\in\partial
V_r$.
\end{proof}

% end of page 66 (8-5)

\begin{Lemma} % 8.3
Let $\tilde{f} = f_1\times f_1$ be the extension of $f\in\Aut(M,
F^\pm)$ to $\tilde{M} = M^-\times M^-$. Then $f_1\in\Aut(M^-,
\mathcal{K})$.
\end{Lemma}

\begin{proof}
Let $\mathcal{K} = \{(\partial V_r)_p\}_{p\in M^-}$, where $(\partial
V_r)_p$ denotes the cone at a point $p\in M^-$ belonging to the field
$\mathcal{K}$. Note that, if $p = b\cdot 0^-$, \ $b\in G$, then
$(\partial V_r)_p$ is just the cone $b_*(\partial V_r)$. We have to
show that $(f_1)_*(\partial V_r)_p = (\partial V_r)_{f_1(p)}$. Choose
an element $a\in G$ such that $a^{-1} f_1 b(0^-) = 0^-$. Then the
transformation $a^{-1} \tilde{f} b$ on $\tilde{M}$ is the extension of
$a^{-1} f b\in\Aut(M, F^\pm)$. Decompose $a^{-1} f b$ as $a^{-1} f_1
b\times a^{-1} f_1 b$ corresponding to the decomposition $\tilde{M} =
M^-\times M^-$. By Lemma 8.3, we see that $(a^{-1} f_1 b)_{* 0^-}$
leaves $\partial V_r$ invariant. Consequently we have
% end of page 67 (8-6)
\[(f_1)_{*p}(\partial V_r)_p = (f_1)_{*p}(\partial V_r)_{b\cdot 0^-} =
(f_1)_{*p} b_{* 0^-}(\partial V_r) = a_{* 0^-}(\partial V_r) =
(\partial V_r)_{a\cdot 0^-} = (\partial V_r)_{f_1(p)}.\]
This implies that $f_1\in\Aut(M^-, \mathcal{K})$.
\end{proof}

\begin{Theorem} % 8.4
Let\/ $(M = G/G_0, F^\pm)$ be the parahermitian symmetric space of
$C_r$-type associated with a simple GLA\/ \textup{(1.1)}, and let
$\mathcal{K}$ be the above generalized conformal structure on the
symmetric $R$-space $M^- = G/U^-$. Then
\[\Aut(M, F^\pm) = \Aut(M^-, \mathcal{K}) = \left\{\begin{array}{ll}
G, & r\geq 2,\\ \Diffeo(M^-), & r=1.\end{array}\right.\]
\end{Theorem}

\begin{proof}
Suppose $r\geq 2$. As we noted before Lemma 8.2, the correspondence
$\Aut(M, F^\pm)\ni f\mapsto f_1\in\Diffeo(M^-)$ is injective. But, by
Lemma 8.3, the image $f_1$ lies in $\Aut(M^-, \mathcal{K})$. Hence we
have the injective homomorphism $\Aut(M, F^\pm) \hookrightarrow
\Aut(M^-, \mathcal{K})$. Since $G$ is a subgroup of $\Aut(M, F^\pm)$,
it follows from (8.2) that $G\subset\Aut(M, F^\pm) \simeq \Aut(M^-,
\mathcal{K}) = G$. Suppose next $r=1$. Then the $G$-orbit
decomposition of $\tilde{M}$ leaves $\tilde{M} = M\amalg M^-$. So, for
any diffeomorphism $f_1$ of $M^-$, \ $(f_1\times f_1)\rest{M}$ is an
element of
% end of page 68 (8-7)
$\Aut(M, F^\pm)$. Therefore we have $\Aut(M, F^\pm) \simeq \Aut(M^-,
\mathcal{K}) = \Diffeo(M^-)$ (cf.\ (8.27)).
\end{proof}

\begin{Remarkn} % 8.1
As is seen in the table in 6.4, the parahermitian symmetric space $M$
of $C_1$-type is $\SO^0(1, q+1)/\SO(q)\cdot\bbb{R}^+$, and the
corresponding symmetric $R$-space $M^-$ is the conformal $q$-sphere.
\end{Remarkn}

\begin{Remarkn} % 8.2
In case where $\Aut(M, F^\pm) = G$ in Theorems 8.1 and 8.4, we also
have
\[\Aut(M, F^\pm) = \Aut(M, F^\pm, \omega).\]
\end{Remarkn}

\begin{Remarkn} % 8.3
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 $\Gr_2(\bbb{H}^4)$ of quaternionic $2$-planes in
$\bbb{H}^4$. There are two parahermitian symmetric spaces:
\[\SL(4,\bbb{H})/\SL(2,\bbb{H})\times \SL(2,\bbb{H})\times\bbb{R}^+
\quad\textup{and}\quad E_{6(6)}/\Spin(5,5)\cdot\bbb{R}^+,\] which have
$\Gr_2(\bbb{H}^4)$ 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 $\Gr_2(\bbb{H}^4)$ has
% end of page 69 (8-8)
two different paracomplex structures which are both homogeneous.
\end{Remarkn}

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