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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% This is an example of writing a book using LaTeX and Springer's
+% MATHSING.STY style option file.
+%
+% The \documentstyle command specifies the use of the 12pt book style.
+% Do not remove the 12pt option since this would only change the
+% fontsizes but no other dimensions. The "mathsing" entry of the
+% style option list specifies the use of the Springer changes to the
+% LaTeX default layout for books. The "mathdef" entry specifies a
+% user defined macro file with extension .STY that is included before
+% processing the various chapters.
+%
+% To start TeX use
+%
+% $ tex &lplain math
+%
+% or a similar command depending on your operating system.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\documentstyle[12pt,mathsing]{book}
+%\documentstyle{mathsing}
+\newthe{conjecture}{conjecture*}{Conjecture}{lemmacount}{\bf}{\it}
+\def\frak{\rm}
+ \numberlikebook
+% \numberlikearticle
+\begin{document}
+\chapter{Orbits on Flag Manifolds}
+Please note that this is a preliminary example text demonstrating our
+preliminary MathSing \LaTeX style file.
+\section{$H$-Orbits on $X=G/P$}
+
+Let $G$ be a connected real semisimple Lie group
+and $X$ the flag manifold of $G$.
+$X$ is a homogeneous space of $G$ and the isotropy subgroup
+$P=P_x$ of
+each point $x$ of $X$ is called a minimal parabolic subgroup
+of $G$.
+Let $\sigma$ be an involutive automorphism ($\sigma ^2=id.$)
+of $G$
+and $H$ a subgroup of $G^\sigma =\{x\in G\mid \sigma x=x\}$
+containing
+the identity component $G^\sigma_0$ of $G^\sigma$.
+Irreducible pairs $({\frak g}, {\frak h})$ of Lie algebras
+of $G$ and $H$ are
+classified by \cite{Be}.
+
+The following are special cases of $H$-orbit decompositions
+of
+$X=G/P$.\\[5pt]
+\indent
+(i) Let $\sigma$ be a Cartan involution of $G$,
+${\frak g}={\frak k}\oplus
+{\frak s}$ the Cartan decomposition of the Lie algebra
+${\frak g}$ of $G$ for
+$\sigma$ and $K=H=G^\sigma$. Then $P=MAN$ where
+$A=P\cap\exp {\frak s}$, $M=Z_K(A)$ and $N$ is the unipotent
+radical of $P$. The Iwasawa decomposition
+$G=KAN(\cong K\times A\times N)$
+implies that $\#(K\setminus G/P)=1$.
+
+(ii) Let $G=G_1\times G_1$, $P=P_1\times P_1$ and
+$\sigma (x, y)=(y, x)$ for $(x, y)\in G_1\times G_1$.
+Then $H=G^\sigma=\{(x, x)\in G\mid x\in G_1\}$. Since
+$H\setminus G\cong G_1$ by the map $H(x, y)\mapsto x^{-1}y$,
+the
+double coset decomposition $H\setminus G/P$ is
+identified with the Bruhat decomposition
+$P_1\setminus G_1/P_1$.
+
+(iii) When $G$ is a complex semisimple Lie group and $\sigma$
+is a
+conjugation of $G$, $H$-orbits on $X$ are studied in \cite{A}.
+This study
+suggested the formulation for the following general cases.
+
+Let $\theta$ be a Cartan involution of $G$ such that
+$\sigma\theta =\theta\sigma$, ${\frak g}={\frak k}\oplus {\frak
+s}$
+the Cartan decomposition of ${\frak g}$ for $\theta$ and $K=G^\theta$.
+
+\begin{definition} An element $x$ of $X$ is called ``special''
+when $A_x=P_x\cap \exp {\frak s}$ is $\sigma$-stable. Put
+$$U=\{x\in X\mid x \mbox{ is special }\}\enspace .$$
+\end{definition}
+
+\noindent
+\begin{theorem} [{\rm[R, M1]}.] {\em $K\cap H\setminus U\cong H\setminus
+X$ by the inclusion map $U\hookrightarrow X$.}
+There exists a unique subgroup $H^a$ of $G$ such that
+$G^{\sigma\theta}_0 \subset H^a \subset
+G^{\sigma\theta}$ and that $K\cap H^a=K\cap H$. (Rem. $(H^a)^a=H$.)
+\end{theorem}
+\begin{corollary} {\rm\cite{M1}.} There exists a one-to-one
+correspondence
+$D\mapsto D^a$ between $H$-orbits and $H^a$-orbits on $X$ given
+by $K\cap
+H\setminus U\cong H\setminus X$ and $K\cap H\setminus
+U\cong H^a\setminus X$.
+\end{corollary}
+
+\begin{example} {}{}Let $G=SL(2,\bbbc)$. Then
+$X=P^1(\bbbc)=\bbbc \cup \{\infty\}$,
+$${\rm where}\quad
+\left(\matrix{a & b \cr c & d}\right)x={ax+b\over cx+d} \quad
+{\rm for}\
+\left(\matrix{a & b \cr c & d}\right)\in SL(2,\bbbc), \ x\in
+X.$$
+$${\rm Let}\quad
+\sigma\left(\matrix{a & b \cr c & d}\right)=
+\left(\matrix{a & -b \cr -c & d}\right),
+\quad{\rm and}\quad\theta g={}^t\bar g^{-1}.$$
+Then
+$$K=SU(2),\quad H=G^\sigma =
+\biggl\{\left(\matrix{a & 0 \cr 0 &
+a^{-1}}\right)\mid a\in \bbbc^\times\biggr\},$$
+$$H^a=G^{\sigma\theta}=SU(1,1)=
+\biggl\{\left(\matrix{a & b \cr \bar b &
+\bar a}\right)\mid a\bar a-b\bar b=1\biggr\}.$$
+The $H$-orbits on $X$ are $\{0\},\ \bbbc^\times$ and $\{\infty\}$
+and the corresponding $H^a$-orbits are $\{|x|<1\},\ \{|x|=1\}$
+and
+$\{|x|>1\}$, respectively.
+($U=\{0\}\cup \{|x|=1\}\cup \{\infty\}$\,.)
+\end{example}
+
+\section{Expression by Symbols}
+
+\begin{remark}[1] If $H=G^\sigma_0$, then $H\setminus X$ depends
+only on the pair $({\frak g}, \sigma )$ because
+$$X\cong \mbox{ the set of minimal parabolic subalgebras of
+}{\frak g}$$
+and
+$$H\setminus X\cong {\rm Ad}(H)\mbox{-conjugacy classes of minimal
+parabolic subalgebras of }{\frak g}\enspace .$$
+\end{remark}
+
+\noindent
+\begin{theorem} {\rm\cite{MO}.} Let $G$ and $H$ be as in the following
+list
+(complex classical cases). Then we can express $H\setminus X$
+(and
+$H^a\setminus X$) by symbols. ($p+q=n$, $[H: G^\sigma_0]=1$
+or $2$.)
+\end{theorem}
+
+\begin{table}
+\begin{petit}
+\caption{Example of a table}
+ \begin{tabular}{l @{\hspace{8pt}}
+ | @{\hspace{8pt}} c @{\hspace{8pt}}
+ | @{\hspace{8pt}} c @{\hspace{8pt}}
+ | @{\hspace{8pt}} c }
+\rule[-5pt]{0pt}{5pt}
+ Type & $G$ & $H$ & $H^a$ \\ \hline
+\rule[5pt]{0pt}{8pt}
+ AI & $GL(n, \bbbc)$ & $O(n, \bbbc)$ & $GL(n, \bbbr)$ \\
+ AII & $GL(n, \bbbc)$ & $Sp(n/2, \bbbc)$ ($n$ even) & $U^*(n)$
+\\
+ AIII & $GL(n, \bbbc)$ & $GL(p, \bbbc)\times GL(q, \bbbc)$
+& $U(p, q)$ \\
+ BI & $SO(2n+1, \bbbc)$ & $S(O(2p+1, \bbbc)\times O(2q, \bbbc))$
+ & $SO(2p+1, 2q)$ \\
+ CI & $Sp(n, \bbbc)$ & $GL(n, \bbbc)$ & $Sp(n, \bbbr)$ \\
+ CII
+ & $Sp(n, \bbbc)$ & $Sp(p, \bbbc)\times Sp(q, \bbbc)$ & $Sp(p,
+q)$ \\
+ DI & $SO(2n, \bbbc)$ & $S(O(2p, \bbbc)\times O(2q, \bbbc))$
+ & $SO(2p, 2q)$ \\
+ DI' & $SO(2n, \bbbc)$ & $S(O(2p+1, \bbbc)\times O(2q-1, \bbbc))$
+ & $SO(2p+1, 2q-1)$ \\
+ DIII & $SO(2n, \bbbc)$ & $GL(n, \bbbc)$ & $SO^*(2n)$ \\
+ \end{tabular}
+\end{petit}
+\end{table}
+
+\begin{note}
+In \cite{MO} p.155, we should read $GL(n, \bbbc)$
+for $\bbbc^\times\times PSL(n, \bbbc)$ on the line of DIII in
+Table 1.
+\end{note}
+
+Precise description of symbols and many examples are given in
+\cite{MO}. But we can explain shortly the essencial part as
+follows.
+
+Let $x\in U\subset X$. Then
+${\frak a}_x={\rm Lie}(P_x)\cap {\frak s}$ is
+$\sigma$-stable by the definition of $U$. Let
+$\Sigma_x$ be the root system of the pair
+$({\frak g}, {\frak a}_x)$ and $\Sigma_x^+$ the positive system
+of
+$\Sigma_x$ corresponding to $P_x$. Let $\Psi_x$ denote the set
+of
+simple roots in $\Sigma_x^+$. Then we can take an orthogonal
+basis
+$\{e_1,\ldots , e_n\}$ of the dual ${\frak a}_x^*$ of ${\frak
+a}_x$
+such that
+\[ \Psi_x=\left\{ \begin{array}{lc} \{\alpha_1,\ldots , \alpha_{n-1}\}
+ & \mbox{ if }G=GL(n, \bbbc), \\
+ \{\alpha_1,\ldots , \alpha_n\} & \mbox{ otherwise,}
+ \end{array} \right. \]
+where $\alpha_1=e_1-e_2, \ldots , \alpha_{n-1}=e_{n-1}-e_n$
+and
+$\alpha_n=e_n$, $e_{2n}$ or $e_{n-1}+e_n$ if $G=SO(2n+1, \bbbc)$,
+$Sp(n, \bbbc)$ or $SO(2n, \bbbc)$, respectively.
+
+To the left coset $(K\cap H)x$ in $U$, there corresponds a sequence
+$\varepsilon_1\varepsilon_2\ldots \varepsilon_n$ consisting
+of the
+following four kinds of letters.
+
+($\pm$) If $\sigma e_i=e_i$, then $\varepsilon_i=+$ (``a boy'')
+or $-$ (``a girl''). When $\varepsilon_i=\pm$ and $\varepsilon_j=\pm$
+($i\ne j$),
+$$\varepsilon_i=\varepsilon_j \iff
+{\frak g}({\frak a}_x, e_i-e_j)\subset {\rm Lie}(H)\enspace
+.$$
+
+(a) If $\sigma e_i=e_j$ with $i\ne j$, then we put a small letter
+(``a family name'') to the couple $(\varepsilon_i, \varepsilon_j)$.
+
+(A) If $\sigma e_i=-e_j$ with $i\ne j$, then we put a capital
+letter
+to the ``old'' couple $(\varepsilon_i, \varepsilon_j)$.
+
+(O) If $\sigma e_i=-e_i$, then $\varepsilon_i=O$ (``the aged''
+or
+``dead''?).
+
+Let $w_i$ be the reflection with respect to the simple root
+$\alpha_i$
+and $P_i=P\cup Pw_iP$ ($P=P_x$) the parabolic subgroup of $G$
+for
+$\alpha_i$. Let $\pi_i$ denote the projection of $X=G/P$ onto
+$G/P_i$.
+
+\section*{Notation} For two $H$-orbits $D_1$ and $D_2$ on $X$, we
+write
+$$D_1\stackrel{i}{\rightarrow}D_2 \iff \pi_i(D_1)=\pi_i(D_2)
+\mbox{ and } \dim D_1<\dim D_2\enspace .$$
+
+%\vspace{1ex}
+We put here two examples. (You can see 23 figures of examples
+in \cite{MO}.)
+
+\setlength{\unitlength}{1mm}
+\thicklines
+\begin{picture}(115,70)(4,0)
+\put(10,60){\makebox(0,0){$-++$}}
+\put(30,60){\makebox(0,0){$+-+$}}
+\put(50,60){\makebox(0,0){$++-$}}
+\put(20,40){\makebox(0,0){$aa+$}}
+\put(40,40){\makebox(0,0){$+aa$}}
+\put(30,20){\makebox(0,0){$a+a$}}
+\put(11,58){\vector(1,-2){8}}
+\put(31,58){\vector(1,-2){8}}
+\put(21,38){\vector(1,-2){8}}
+\put(29,58){\vector(-1,-2){8}}
+\put(49,58){\vector(-1,-2){8}}
+\put(39,38){\vector(-1,-2){8}}
+\put(12,49){1}
+\put(22,49){1}
+\put(36,49){2}
+\put(46,49){2}
+\put(22,29){2}
+\put(36,29){1}
+\put(30,9){\makebox(0,0){\ixpt{\bf Fig. 1.} $G=GL(3, \bbbc)$}}
+\put(30,4){\makebox(0,0){\ixpt$H=GL(2, \bbbc)\times GL(1, \bbbc)$}}
+
+\put(66,64){\makebox(0,0){$++$}}
+\put(82,64){\makebox(0,0){$+-$}}
+\put(98,64){\makebox(0,0){$-+$}}
+\put(114,64){\makebox(0,0){$--$}}
+\put(74,48){\makebox(0,0){$+O$}}
+\put(90,48){\makebox(0,0){$aa$}}
+\put(106,48){\makebox(0,0){$-O$}}
+\put(74,32){\makebox(0,0){$O+$}}
+\put(90,32){\makebox(0,0){$AA$}}
+\put(106,32){\makebox(0,0){$O-$}}
+\put(90,16){\makebox(0,0){$OO$}}
+\put(67,62){\vector(1,-2){6}}
+\put(83,62){\vector(1,-2){6}}
+\put(99,62){\vector(1,-2){6}}
+\put(81,62){\vector(-1,-2){6}}
+\put(97,62){\vector(-1,-2){6}}
+\put(113,62){\vector(-1,-2){6}}
+\put(74,46){\vector(0,-1){12}}
+\put(90,46){\vector(0,-1){12}}
+\put(106,46){\vector(0,-1){12}}
+\put(76,30){\vector(1,-1){12}}
+\put(90,30){\vector(0,-1){12}}
+\put(104,30){\vector(-1,-1){12}}
+\put(67,55){2}
+\put(76,55){2}
+\put(84,55){1}
+\put(95,55){1}
+\put(103,55){2}
+\put(111,55){2}
+\put(72,39){1}
+\put(88,39){2}
+\put(107,39){1}
+\put(79,23){2}
+\put(88,23){1}
+\put(99,23){2}
+\put(90,4){\makebox(0,0){\ixpt{\bf Fig. 2.} $G=Sp(2, \bbbc)$,
+$H=GL(2, \bbbc)$}}
+\end{picture}
+
+\begin{remark}[2] (\cite{Sp}, \cite{M2}) In complex cases, we
+can
+find all
+the closure relations among $H$-orbits on $X$ from the following
+two
+properties.
+
+(a) $D_1\stackrel{i}{\rightarrow}D_2 \Rightarrow D_1\subset
+D_2^{cl}$.
+
+(b) $D_1\stackrel{i}{\rightarrow}D_2, D_3\stackrel{i}{\rightarrow}D_4$
+and
+$D_1\subset D_3^{cl}\ \Rightarrow\ D_2\subset D_4^{cl}$.
+
+This is proved by the same argument as that of the Bruhat ordering
+since
+$$D_1\stackrel{i}{\rightarrow}D_2 \mbox{ and } D_1\stackrel{i}{\to}D_3\
+\Rightarrow\ D_2=D_3$$ in complex cases. To find all the closure
+relations in
+general real cases, we should follow a rather complicated procedure
+given in
+\cite{M2}.
+\end{remark}
+
+\begin{remark}[3] These diagrams of orbits are useful to the
+study of the
+asymptotic behavior of spherical functions on semisimple symmetric
+spaces
+(\cite{O}) and embeddings of Harish-Chandra modules into principal
+series
+(\cite{MO}).
+\end{remark}
+
+\begin{remark} [4] (Problem) If $\Sigma =\Sigma ({\frak g}, {\frak
+a})$ is
+classical, then there exists (in principle) a similar
+(sometimes the same)
+expression of the $H$-orbits on $X$ as that in a complex case.
+Give a
+complete list of such expressions by symbols. (For example,
+it is proved in
+\cite{M2} that the diagram of $H^a\setminus X$ is upside-down
+to that of
+$H\setminus X$.)
+\end{remark}
+\begin{example} {}{}($=$ Exercise). When $G=GL(n, \bbbf)$ and
+$H=GL(p,
+\bbbf)\times GL(n-p, \bbbf)$ for a division algebra $\bbbf$
+of characteristic
+$\ne 2$, the diagram of the $H$-orbits on $X$ does not depend
+on $\bbbf$.
+\end{example}
+
+\begin{problem*} Give good symbols for $H$-orbits on $X$ when
+$\Sigma$ is exceptional.
+\end{problem*}
+
+%pagestyle{myheadings}
+\newpage
+\markright{Uzawa's Stuff (This is to demonstrate changed headlines.)}
+\section{Uzawa's Function $f$ and Vector Field $v$ on $X$ \protect\\
+(Related to Intersections of \protect
+$H$- and $H^a$-Orbits on $X$)}
+%pagestyle{headings}
+
+Recently, T. Uzawa discovered the following function $f$
+and vector field $v$
+on $X$ which have very nice properties with respect to $H$-orbits
+and
+$H^a$-orbits.
+
+Let $Y_0$ be a generic element of ${\frak s}$.
+Then $Y_0$ defines a minimal parabolic subgroup $P_0$ of $G$
+such that $Y_0\in {\frak a}_0={\rm Lie}(P_0)\cap {\frak s}$
+and
+that $Y_0$ is dominant for the positive system of the root system
+$\Sigma ({\frak g}, {\frak a}_0)$ corresponding to $P_0$.
+By the natural identification
+$$G/P_0\cong K/M_0\cong {\rm Ad}(K)Y_0$$
+($K\cap P_0=M_0=$ the centralizer of $Y_0$ in $K$), $X=G/P_0$
+is embedded into ${\frak s}$. Let $Y_x$ denote the element in
+Ad$(K)Y_0$ corresponding to $x\in X$.
+
+\begin{definition} (i) We define a function $f$ on $X$ by
+$f(x)=|Y_x^+|^2=B(Y_x^+, Y_x^+)$ on $X$ where
+$Y_x^+={1\over 2}(Y_x+\sigma Y_x)$ and $B( , )$ is the Killing
+form on ${\frak g}$.
+
+(ii) A vector field $v$ on $X$ is defined by $v_x=$ the
+(infinitesimal) $Y_x^+$-action at $x$ for $x\in X$.
+
+(iii) $\Phi_t$ ($t\in \bbbr$) is the one-parameter group of
+transformations of $X$ for the vector field $v$.
+
+(iv) $\Phi_{\pm\infty}(x)=\lim_{t\to\pm\infty}\Phi_t(x)$ for
+$x\in X$.
+\end{definition}
+
+\begin{remark}[5] The vector field $v$ is the gradient of the
+function $f$ with respect to the $K$-invariant Riemannian metric
+on $X=K/M_0$ induced from the inner product
+$(Z, Z')=B([Z, Y_0], Z'_{\frak s})$ on ${\frak k}^{\perp{\frak
+m}_0}$
+where $Z'_{\frak s}$ is the element in ${\frak s}$ such
+that $Z'_{\frak s}-Z'\in {\rm Lie}(P_0)$.
+\end{remark}
+
+\begin{remark}[6] If the real rank of $G$ is larger than one,
+then $f$ and $v$ depend essencially (not constant multiple)
+on
+the choice of $Y_0$.
+\end{remark}
+
+\begin{example} {}{}{\rm (continued from Example 1.9)} Take
+$$Y_0=\pmatrix{1 & 0 \cr 0 & -1}\in
+{\frak s}=\biggl\{\pmatrix{z & x+iy \cr x-iy & -z}
+\mid x, y, z\in \bbbr \biggr\}\enspace .$$
+Since $P_0$ is the subgroup of $G$ consisting of upper
+triangular matrices, $eP_0$ corresponds to $\infty$ in
+$P^1(\bbbc)=\bbbc\cup \{\infty\}$ and
+$$kP_0\mapsto \pmatrix{a & b \cr -\bar b & \bar a}\infty =
+{a \over -\bar b}\quad \mbox{ for }\quad k=
+\pmatrix{a & b \cr -\bar b & \bar a}\in K\enspace .$$
+On the other hand,
+\begin{eqnarray*}
+\pmatrix{a & b \cr -\bar b & \bar a}
+\pmatrix{1 & 0 \cr 0 & -1}
+\pmatrix{a & b \cr -\bar b & \bar a}^{-1}
+& = & \pmatrix{a & -b \cr -\bar b & -\bar a}
+\pmatrix{\bar a & -b \cr \bar b & a} \\
+& = &
+\pmatrix{a\bar a -b\bar b & -2ab \cr -2\bar a\bar b &
+ -a\bar a +b\bar b}\enspace .
+\end{eqnarray*}
+So Ad$(K)Y_0$ is the sphere given by $x^2+y^2+z^2=1$
+and the function $f$ is
+given by $z^2$. Two points $\{\infty\}$, $\{0\}$ and
+the unit circle in
+$P^1(\bbbc)$ correspond to $(0, 0, 1)$, $(0, 0, -1)$
+and the circle defined
+by $z=0$, respectively, in Ad$(K)Y_0$.
+\end{example}
+
+\noindent
+\begin{theorem} {\rm\cite{U}} (i) $v$ is tangent to
+$H$-orbits and $H^a$-orbits.
+
+(ii) $(df)_x=0 \iff v_x=0 \iff x$ is special.
+
+(iii) Let $D$ be an $H$-orbit on $X$. Then
+there exists $m=\min_{x\in D}f(x)$ and for $x\in D$,
+$$f(x)=m \iff x \enspace \mbox{is special}\enspace .$$
+
+(iv) $\Phi_{-\infty}(D)=D\cap U$ for $H$-orbits $D$ on
+$X$.
+\end{theorem}
+
+\begin{corollary}(1) {\em \cite{M3} }(a) $D\cap D^a=(K\cap
+H)x$
+for an $x\in U$.
+
+(b) For two $H$-orbits $D$ and $E$ on $X$,
+$$D^{cl}\supset E \iff D\cap E^a\ne \phi
+\iff D^a\subset (E^a)^{cl}\enspace .$$
+\end{corollary}
+
+\begin{proof} (\cite{U}) (a) Let $x\in D\cap D^a$.
+We have only to show that
+$x\in U$ by Theorem 1. Let $m$ be the value of the
+function $f$ at the points
+in $D\cap U$ ($=D^a\cap U$). Suppose that $x\notin U$.
+Then $f(x)>m$ by
+(iii). Since the function for the $H^a$-orbit structure
+is $|Y_0|^2-f(x)$, we
+have also $f(x)<m$ by (iii), a contradiction.
+
+(b) Since $(H^a)^a=H$, we have only to prove the left $\iff$.
+
+The assertion $D^{cl}\supset E \Rightarrow D\cap E^a\ne \phi$
+is clear since
+$$T_x(E)+T_x(E^a)=T_x(X)$$
+for any $x\in E\cap E^a$ (\cite{M3}).
+
+Suppose that $D\cap E^a\ne \phi$ and let $x\in D\cap E^a$. Then
+$$\Phi_\infty(x)=\lim_{t\to \infty}\Phi_t(x)\in D^{cl}\cap E^a\cap
+U=
+D^{cl}\cap E\cap U$$
+by (i) and (iv). Hence $D^{cl}\cap E\ne \phi$ and therefore
+$D^{cl}\supset E$. \qed
+\end{proof}
+
+\begin{corollary}(2) Let $D$ be an $H$-orbit on $X$ and
+$x\in D\cap D^a$. Then
+$$D\cong (K\cap H)\times_L\Phi_{-\infty}^{-1}(x)\leqno(i)$$
+where $L=K\cap H\cap P_x$ and
+$$D\cap E^a\cong (K\cap H)\times_L(\Phi_{-\infty}^{-1}(x)\cap
+E^a)
+\leqno(ii)$$
+for any $H^a$-orbit $E^a$ on $X$. (Moreover it is clear that
+the
+fibers $\Phi_{-\infty}^{-1}(x)$ and $\Phi_{-\infty}^{-1}(x)\cap
+E^a$
+are contractible to the point $x$.)
+\end{corollary}
+
+\section{Remarks on Spherical Subgroups}
+
+Suppose that $G$ is a complex semisimple Lie group. A complex
+Lie
+subgroup
+$H$ of $G$ is called ``spherical'' if there exists an open $H$-orbit
+on $X$.
+Such pairs $(G, H)$ are classified by \cite{K} when $G$ is simple
+and $H$ is
+reductive, and by \cite{Br2} in general.\\[8pt]
+\noindent
+\begin{theorem} [{\rm[Br1, V]}] $H\subset G$ is spherical $\iff
+\#(H\setminus X)$ is finite. (Note that $\Leftarrow$ is clear.)
+\end{theorem}
+There is a simple proof of $\Rightarrow$ using ``rank-one sections''
+as follows.
+
+\begin{proof} We may assume that $HP$ is open in $G$. Write
+$G=P_{\beta_1}P_{\beta_2}\cdots P_{\beta_m}$ where the $\beta_i$'s
+are simple
+roots and $P_{\beta_i}=P\cup Pw_{\beta_i}P$. Put
+$P^{(i)}=P_{\beta_1}P_{\beta_2}\cdots P_{\beta_i}$ ($P^{(0)}=P$).
+We will
+show
+$$ \#(H\setminus HP^{(i)}/P)<\infty \mbox{ for } i=0, 1,\ldots
+, m$$
+by induction on $i$.
+
+By the hypothesis of induction, we may assume that
+$$ HP^{(i-1)}=Hg_1P\cup \cdots \cup Hg_kP\enspace .$$
+Then we have
+$$ HP^{(i)}=Hg_1P_{\beta_i}\cup \cdots \cup Hg_kP_{\beta_i}\enspace
+.$$
+We have only to show that $\#(H\setminus Hg_jP_{\beta_i}/P)<\infty$
+for $j=1,\ldots , k$. Since $HP^{(i-1)}$ is open in $G$,
+$(g_jP_{\beta_i}/P)\cap (HP^{(i-1)}/P)$ is (Zariski) open in
+the one-dimensional subvariety $g_jP_{\beta_i}/P$ of the complex
+alge
+braic variety $X$. Hence the compliment of $(g_jP_{\beta_i}/P)\cap
+(HP^{(i-1)}/P)$ in $g_jP_{\beta_i}/P$ consists of finte points
+and therefore $\#(H\setminus Hg_jP_{\beta_i}/P)<\infty$. \qed
+\end{proof}
+
+Let $G$ be a real semisimple Lie group and $H$ a Lie subgroup
+of $G$.
+
+\begin{conjecture}(1) If the real rank of $G$ is one and there
+exists an open $H$-orbit on $X=G/P$, then $\#(H\setminus X)<\infty$.
+\end{conjecture}
+
+By the same argument as above for spherical subgroups, Conjecture
+1 implies
+the following Conjecture 2.
+
+\begin{conjecture}(2) If there exists an open $H$-orbit on
+$X$, then $\#(H\setminus X)<\infty$.
+\end{conjecture}
+
+\begin{remark}[7] In general, $\#(H\setminus G/P)<\infty$ does
+not imply $\#(H_\bbbc \setminus G_\bbbc /P_\bbbc )<\infty$.
+For example, if $G=SU(n, 1)$ ($n>2$) and $H=\theta N$ (where
+$N$ is the unipotent radical of $P$), then $\#(H\setminus G/P)=2$
+and $
+\#(H_\bbbc \setminus G_\bbbc /P_\bbbc )=\infty$.
+\end{remark}
+
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+ \ No newline at end of file