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\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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\end{document}