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\begin{document}
\title[Tangent spaces to Schubert varieties]{ON TANGENT SPACES TO SCHUBERT
VARIETIES-II}
\author[V. Lakshmibai]{V. Lakshmibai${}^{\dag}$}
\address{Department of Mathematics\\ Northeastern University\\ Boston, MA 02115}
\email{lakshmibai@@neu.edu}
\thanks{${}^{\dag}$Partially supported by NSF Grant DMS 9502942}
\maketitle
\begin{abstract}
We prove results on the tangent spaces to Schubert varieties in $G/B$ for
$G$ classical. We give a description of the tangent space to a  Schubert
variety $X(w)$ at a $T$-fixed point $e_\t$ in terms of the
root system. We also relate this result to multiplicities of certain weights in the
fundamental representations of $G$.
\end{abstract}

%%%%%%%%%%%%%
\section*{Introduction}
 Let $G$ be a
semi simple, simply connected algebraic group over an algebraically closed field $K$
of arbitrary characteristic. Let $T$ be a maximal
torus in $G$, and $W$ the Weyl group. Let $R$ be the system of roots
of $G$ relative to $T$. Let $B$ be a Borel subgroup of $G$, where
$B\supset T$. Let $S$ (resp.$R^+$)  be the set of simple (resp.
positive) roots of $R$ relative to $B$. For $\a \in R$, let $s_\a$
be the reflection, and $X_\a$ the element of the Chevalley basis
for ${\frak g}\, (={\rm Lie}G$), corresponding to $\a$. For $w\in W$, let us
denote the point in $G/B$ corresponding to the coset $wB$ by
$e_{w}$. Then the set of
$T$-fixed points in
$G/B$ for the action given by left multiplication is precisely
$\{e_{w}\ |\  w\in W\}$.
For $w\in W$, let $X(w)$ be the associated Schubert variety (the Zariski closure of
$Be_{w}$  in $G/B$).

For $\t \in W$, let $T(w_0,\t)$ denote the tangent space to $G/B$ at $e_\t$. We
have
$$T(w_0,\t): =\oplus\ _{\b
\in
\t(R^+)}\  {{{\frak g}}}_{-\b}. $$
For $\t\leq w$, let $T(w,\t) $ be the  Zariski
tangent space to $X(w)$ at $e_\t $. Let  $$N(w,\t)=\{ \b \in \t (R^+)\ |\
X_{-\b }\in T(w,\t) \} .$$ Since $T(w,\t)$ is a $T$-stable subspace of
$T(w_0,\t)$, we have that
$T(w,\t) $  is spanned by
$N(w,\t)$. For a dominant
weight $\l$, let $V(\l)$ be the irreducible $G_{\Bbb C}$-module (over ${\Bbb C}$) with
highest weight $\l$ (here, $G_{\Bbb C}$ is the group over ${\Bbb C}$ with the same
root data). Let us fix a highest weight vector
$u$ in
$V(\l)$. For
$w
\in W$, fix a representative $n_w$ for $w$ in $N_T(G)$ (the normalizer of $T$ in $G$),
and set
$u_w= w\cdot u$ (one refers to $u_w$ as the {\em extremal weight vector} in $V(\l)$
of weight $w(\l)$), and
$V_{w,{{\Bbb Z}}}=U^+_{{\Bbb {Z}}}({{\frak g_{\Bbb C}}}) u_w$ (here, $ {\frak g_{\Bbb C}}= \text {Lie } G_{\Bbb
C},\  U^+({{\frak g_{\Bbb C}}})$  is the subalgebra of
$U({{\frak g_{\Bbb C}}})$ (the universal enveloping algebra of ${{\frak g_{\Bbb
C}}}$), generated by
$\{X_\a, \a \in S\}$,  and $U^+_{{\Bbb { Z}}}({{\frak g}})$ is the Kostant
${\Bbb { Z}}$-form of $U^+({{\frak g_{\Bbb C}}})$). For any field $k$, let
$V_w(\l)=V_{w,{\Bbb { Z}}}\otimes k,\  w\in W$. Let $\
V_k(\l)=V_{w_0}(\l)\ (=V_{w_0,{\Bbb { Z}}}\otimes k$, the Weyl
module corresponding to $\l$).

For $G$ classical, we give an explicit description of $T(w,\t)$:
(We index the roots as in \cite{bou}.)

\begin{thm} (cf. Theorem \ref{4.4})
Let $G=SL(n)$, and $w\in S_n$. Then $T(w,\t)$ is spanned by
$\{X_{-\b},\b \in \t (R^+)\ |\ w\ge s_\b\t\}$.
\end{thm}

\begin{thh} (cf. Theorem \ref{5.7})
Let $G=Sp(2n)$. Let $w,\t \in W, w\ge \t$, and $\b \in \t (R^+)$, say $\b = \t(\a)$,
where $\a \in R^+$.
\begin{enumerate}
\item Let $\a=\e_j-\e_k, 1\le j<k \le n$, or $2\e_j, 1\le j\le n$. Then $\b \in
N(w,\t)$ if and only if
$w\ge s_\b \t$.
\item Let $\a=\e_j+\e_k, 1\le j<k \le n$.
\begin{enumerate}
\item Let $\t > \t s_\a\  (=s_\b \t)$. Then $\b \in
N(w,\t)$ necessarily.
\item Let $\t <\t s_\a $.
\begin{enumerate}
\item Let $\t > \t s_{2\e_j}$ or $\t s_{2\e_k}$. Then $\b \in
N(w,\t)$ if and only if
$w\ge s_\b \t$.
\item Let $\t < \t s_{2\e_j}$ and $\t s_{2\e_k}$.

\begin{enumerate}
\item If $\t <\t s_{\e_j-\e_k}$, then $\b \in
N(w,\t)$ if and only if
$w\ge s_\b \t\, (=\t s_{\e_j+\e_k})$ or $\t s_{2\e_j}$.
\item If $\t >\t s_{\e_j-\e_k}$, then $\b \in
N(w,\t)$ if and only if $w\ge \t s_{\e_j-\e_k} s_{2\e_j}$.

\end{enumerate}

\end{enumerate}

\end{enumerate}

\end{enumerate}
\end{thh}
 We give similar descriptions for $G= SO(2n+1),\ SO(2n)$ (cf. Theorems \ref{6.11},
\ref{7.11}).

\ni Let $w,\t \in W,\ w\ge \t,\ \b \in \t(R^+)$. Consider the following
three conditions:

\begin{enumerate}
\item $w\ge s_\b\t$.
\item $m_w(\t(\omega_d)-\b)=m(\t(\omega_d)-\b),\text{ for all } 1\le d\le l,\ l$ being
the rank of $G$.
\item $\b \in N(w,\t)$
\end{enumerate}
where $m(\t(\o_d)-\b)$
(resp.
$m_w(\t(\o_d)-\b)$) denotes the multiplicity of the weight
$\t(\o_d)-\b$ in
$V_K(\o_d)$ (resp.$V_{w,\omega_d}$).

For $G$ of any type, we have (1) implies (3) (cf. \cite{ca}, \cite{po}), and (2)
implies (3) (cf. \cite{po}). We show (cf. Theorem \ref{4.7}) that for $G$ of type
 $\mathbf{A}_n$, the above three
conditions are equivalent. For
$G$ of other types, we do not have the equivalence of the above three conditions
in general.

The above results are proved using a result of Polo (cf. \cite{po}, Theorem 3.2),
the basis $\mathcal{B}_d$ (cf.
\cite{l1}) for $V_K(\o_d)$, and the expression for $X_{-\b} q_{\t^{(d)}}$ as a linear
combination of the vectors in $\mathcal{B}_d$ as given in \cite{g/p-7}, \cite{g/p-8} and
\cite{g/p-9}, where $\b \in \t (R^+)$, $q_{\t^{(d)}},\ 1\le d\le l$ ($l$
being the rank of $G$) is the extremal weight vector in $V_K(\o_d)$ of weight
$\t(\o_d)$.

 Since one knows (see \cite{po}, Corollary 4.1 for example) that $N(w,\t)$ is
independent of $K$, we shall assume throughout that $K=\Bbb C$.

The sections are organized as follows. In \S 1, we recall some generalities on
$G/Q$. In \S 2, we recall some results on $T(w,\t)$. In
\S 3, we recall some results from
\cite{l1}. In sections \S 4,5,6,7, we prove the above results for $G$ of type {\bf
A}$_n$, {\bf C}$_n$, {\bf B}$_n$, and {\bf D}$_n$ respectively.

The author is thankful to the referee for some useful comments and suggestions.



%%%%%%%%%%%
\section {Generalities on $G/Q$}\label{s12}

 Let $G,T,B,W,S,R,R^+$ etc be as in the
Introduction.

\subsection{The Chevalley-Bruhat order}Let $w\in W$. A minimal expression for $w$ as a product of
simple reflections is called a {\it reduced expression} for $w$. We denote by $l(w)$
the length of a reduced expression for $w$ (as a product of simple reflections). We
have a partial order on
$W$, the well-known  Chevalley-Bruhat order, namely $w_1\ge w_2$ if every reduced
expression for
$w_1$ contains a  subexpression which is a reduced expression for $w_2$.

\subsection{The Weyl subgroup $W_Q$} Let $Q$ be a parabolic subgroup of $G$ containing
$B$.\label{W_Q} Associated to
$Q$, there is a subset $S_Q$ of $S$ such that $Q$ is the subgroup of $G$ generated by $B$ and
$\{U_{-\a}\mid \a\in R^+_Q\}$, where $R^+_Q=\{\a\in R^+\mid \a=\sum_{\b\in S_Q} a_\b\b\}$ (here,
for $\b\in R$, $U_\b$ denotes the $1$ dimensional unipotent subgroup of $G$ associated to $\b$).
Let
$W_Q$ be the Weyl group of $Q$ (note that $W_Q$ is simply the subgroup of $W$ generated by
$\{s_\a\mid\a\in S_Q\}$; here, for $\a\in S$, $s_\a$ denotes the simple reflection (considered
as an element of $W$), associated to $\a$).

\subsection{The set $W^Q$ of minimal representatives of $W/W_Q$}
In each coset $wW_Q$, there exists a unique element of minimal length (cf.
\cite{bou}). Let
$W^Q$ be this set of representatives of $W/W_Q$. The set
$W^Q$ is  called the {\em set of minimal representatives of} $W/W_Q$.
We have
$$W^Q=\{w\in W\mid l(ww')=l(w)+l(w'),\text{ for all }w'\in W_Q\}.$$
The set $W^Q$ may be also be characterized as
$$W^Q=\{w\in W\mid w(\a)>0, \text{ for all } \a\in S_Q\}$$
(here by a root $\b$ being $>0$, we mean $\b\in R^+$).

\subsection{Maximal parabolic subgroups}
The set of maximal parabolic subgroups containing $B$ is in one-to-one correpondence
with
$S$, namely given
$\a\in S$, the parabolic subgroup $Q$ where $S_Q=S\setminus\{\a\}$ is a maximal parabolic
subgroup, and conversely. In the sequel, we shall denote $Q$, where
$S_Q=S\setminus\{\a_i\}$ by
$P_{i}$.

\subsection{Schubert varieties in $G/Q$}\label{1.6}
For $w\in W$, let us denote the point in $G/Q$ corresponding to the coset $wQ$ by $e_{w,Q}$. Then
the set of
$T$-fixed points in
$G/Q$ for the action given by left multiplication is presisely $\{e_{w,Q}\mid w\in W\}$.
Let $w\in W$, and let $X_Q(w)$ be the Zariski closure of $Be_{w,Q}$ in $G/Q$. Then
$X_Q(w)$  with the canonical reduced structure is called the Schubert variety in $G/Q$ associated
to $wW_Q$. In particular, we have a bijection between $W^Q$ and the set of Schubert
varieties in $G/Q$.
We have the well-known Bruhat decomposition
$$G/Q=\dot{\cup}_{\{w\in W\}}\ Be_{w,Q},\qquad
X_Q(\te)=\dot{\cup}_{\{w\in W\ |\ w\le \te\}}\ Be_{w,Q},\quad
\te\in W.$$
When $Q=B$, we shall denote $X_B(w)$, by just $X(w)$.


%%%%%%%%%
\section{THE TANGENT SPACE $T(w,\t)$.}
\subsection { Sing $X(w)$}
Let Sing $X(w)$ denote the singular locus of $X(w)$. If $X(w)$ is not
smooth, then Sing $X(w)$ is a non-empty
$B$ - stable closed subvariety of $X(w)$. Given a point $x\in
X(w)$, to decide if it is a smooth point or not, it suffices (in
view of Bruhat decomposition) to determine if the $T$-fixed point $e_\t$ of the
$B$-orbit through $x$ is a smooth point or not. For $\t\le w$, let
$T(w,\t)$ denote the Zariski tangent space to $X(w)$ at $e_\t$.

\subsection{The space $T(w,\t) $}

For $\t\leq w$, let $T(w,\t) $ be the  Zariski
tangent space to $X(w)$ at $e_\t $. Let  $$N(w,\t)=\{ \b \in \t (R^+)\ |\
X_{-\b }\in T(w,\t) \} .$$
Then as remarked in the Introduction, $T(w,\t) $  is spanned by
$N(w,\t)$.

For $\t \in W$, let $\t^{(d)}$ denote the element in $W^{P_d}$ which
represents the coset $wW_{P_d},\ W_{P_d}$ being the Weyl group of the maximal
parabolic subgroup corresponding to the simple root $\a_d$. Let $q_{\t^{(d)}}$ denote
the extremal weight vector in $V_k({\omega_d})$ of weight
$\t (\omega_d)$.


We recall the following result from \cite{po}.

\begin{thm}\label{2.5}
Let $w,\t \in W,\
w\ge \t$. Let $\b \in \t(R^+)$. Then
$\b
\in N(w,\t)$ if and only if $X_{-\b}q_{\t^{(d)}}\in V_w(\o_d),\ \text{ for all }
1\le d\le l,\ l$ being the rank of $G$ .
\end{thm}

We end this section with a Lemma to be used in the discussion in the subsequent sections.

\begin{lem}
Let $w,\t,\b$ be as in Theorem \ref{2.5}. Further let $|(\t({\omega_d}), \b^*)|\le 1$ for all $d$. Then
\begin{enumerate}
\item $X_{-\b}q_{\t^{(d)}}$ is either $0$ or $q_{s_\b\t^{(d)}}$.
\item $\b
\in N(w,\t)$ if and only if $w\ge s_\b\t$.
\end{enumerate}
\end{lem}

\begin{proof}
If $X_{-\b}q_{\t^{(d)}}$ is non-zero, then it is a weight vector of weight $\t(\o_d)-\b$ which is in fact the
weight of $q_{s_\b\t^{(d)}}$ (in view of the hypothesis). This proves assertion (1). Assertion (2) follows
from (1) and Theorem \ref{2.5}.
\end{proof}


%%%%%%%%%
\section{ Bases $\mathcal{B}_d$ and $\mathcal{B}_d^*$ for
$V_k({\omega_d})$ and $H^0(G/B, L_{\omega_d})$.}
 Let $G$ be classical. Let ${\omega_d}, 1\le d\le l$, be the fundamental
weights of
$G$. Note that $({\omega_d}, \b^*)\le 2, \text{ for all } \b \in R^+$.

\ni We recall below some basic results from \cite{l1}.

\subsection{Chevalley multiplicity}
Let $\t,\phi \in W^{P_d} $ be such that $X_{P_d}(\phi)$ is a Schubert
divisor in $X_{P_d}(\t)$. Let $\phi=s_\b \t$, where $\b \in R^+$.
Let $m(\t, \phi)=(\phi(\omega_d), \b^*)$ (note that $(\phi(\omega_d), \b^*)> 0$). We
refer to $m(\t,
\phi)$ as the {\it Chevalley multiplicity of $X_{P_d}(\phi)$ in
$X_{P_d}(\t)$ }.
\subsection{Admissible pairs}
A pair of elements $\t,\phi \in W^{P_d} ,\  \t \ge \phi$ is called {\it an
admissible pair}, if either $\t=\phi$ (in which case we call
$(\t,\t)$ as a {\it trivial admissible pair}), or there exists a
chain
$\t=\t_0>\t_1>\cdots >\t_r=\phi$ such that $X_{P_d}(\t_{i+1})$ is a
divisor in
$X_{P_d}(\t_i)$, and
$m(\t_i,\t_{i+1})=2,\  0\le i\le r-1$.

\subsection{Moving divisors}
Let $\t,\phi \in W^{P_d} $ be such that $X_{P_d}(\phi)$ is a Schubert
divisor in $X_{P_d}(\t)$. Let $\phi=s_\b \t$, where $\b \in R^+$.
Then $X_{P_d}(\phi)$ is said to be a {\it moving divisor} in
$X_{P_d}(\t)$, if $\b$ is simple.
\begin{prop}
Let $\t,\phi \in W^{P_d} $ be such that $X_{P_d}(\phi)$ is a
divisor in $X_{P_d}(\t)$ with $m(\t,\phi)=2$.
Then $X_{P_d}(\phi)$ is a moving
divisor in $X_{P_d}(\t)$.
\end{prop}
\begin{prop}
$\t,\phi \in W^{P_d} , \t \ge \phi$. If there exists one chain
$\t=\t_0>\t_1>\cdots >\t_r=\phi$ such that $X_{P_d}(\t_{i+1})$ is a
divisor in
$X_{P_d}(\t_i)$,
$m(\t_i,\t_{i+1})=2, \ 0\le i\le r-1$, then any other chain
$\t=\t'_0>\t_1>\cdots >\t'_r=\phi$ also has the property that
$m(\t'_i,\t'_{i+1})=2,\  0\le i\le r-1$.
\end{prop}
\begin{cor}
Let $\t,\phi \in W^{P_d} $ be such that $(\t,\phi ) $ is an admissible
pair. Then any chain $\t=\t_0>\t_1>\cdots >\t_r=\phi$ such that $X_{P_d}(\t_{i+1})$ is a
divisor in
$X_{P_d}(\t_i)$ has the property that $m(\t_i,\t_{i+1})=2,\  0\le i\le
r-1$.
\end{cor}
\begin{prop}\label{3.7}
Let $\t,\phi \in W^{P_d} $ be a non-trivial admissible
pair, and  $\t=\t_0>\t_1>\cdots >\t_r=\phi$ any chain (so that
$X_{P_d}(\t_{i+1})$ is a divisor in
$X_{P_d}(\t_i)$). Let $\t_{i+1}=s_{\b_i}\t_i, 0\le i\le r-1$.
Define $v_{\t,\phi}\in V_K(\o_d)$ as $v_{\t,\phi}=X_{-\b_0}X_{-\b_1}\cdots
X_{-\b_{r-1}}q_{\phi}$, where $q_{\phi}$ is the extremal weight
vector in $V_K(\o_d)$ of weight $\phi ({\omega_d})$ (here,
for a root $\b$, $X_\b$ denotes the element in the Chevalley basis
of ${{\frak g}}$ associated to $\b$.) Then
$v_{\t,\phi}$ is independent of the chain chosen and depends only on $\t$ and
$\phi$. Further, $v_{\t,\phi}$ is a weight vector of weight
$\frac {1}{2} (\t(\o_d)+
\phi(\o_d))$.
\end{prop}

\vs.2cm Fix $d,\ 1\le d\le l,\ l$ being the rank of $G$.
\subsection{The sets $\mathcal{B}_d$ and ${mathcalB}_w$}\label{3.9}
Let $\t,\phi \in W^{P_d} $ be such that $(\t,\phi ) $ is an admissible
pair. If $\t=\phi$, then set $q_{\t,\t}$ (or just $q_\t$) as the
extremal weight vector in $V_K(\o_d)$ of weight $\t
({\omega_d})$ (which is unique up to scalars). If $\t>\phi$, then set $q_{\t,\phi}$ as
the vector $v_{\t,\phi}$ as given by Proposition \ref{3.7}. Set $\mathcal{B}_d=\{q_{\t,\phi},(\t,\phi) \text { an admissible pair}\}$. For $w\in
W^{P_d}$, set $\mathcal{B}_w=\{q_{\t,\phi}\in \mathcal{B}_d\ |\ w\ge \t\}$.

\vs.2cm Let $V_w(\o_d),\ V_{K}(\omega_d) $ be as in the Introduction (with
$\l=\o_d$).
\begin{thm}
With notations as above, the set $\mathcal{B}_d$ is a basis for
$V_K(\o_d)$. Further, for $w\in W^{P_d}$, the set $\mathcal{B}_w$
is a basis for $V_w(\o_d)$.
\end{thm}
\subsection{The sets $\mathcal{B}_d^*$ and $\mathcal{B}_w^*$.}
Let $L_{\omega_d}$ be the line bundle on $G/P_d$ associated to $\o_d$ (note that
Pic $G/P_d \simeq {\Bbb Z}$ and $L_{\omega_d}$ is the ample generator of Pic $G/P_d$
). Define
$\mathcal{B}_d^*$ to be the basis of
$H^0(G/P_d, L_{\omega_d})\ (=V_{K}(\omega_d)^*)$ dual to $\mathcal{B}_d$. Let
us deote the elements of $\mathcal{B}_d^*$ by $\{p_{\t,\phi},(\t,\phi)
\text { an admissible pair}\}$. For
$w\in W^{P_d}$, set $\mathcal{B}_w^*=\{p_{\t,\phi} \in \mathcal{B}_d^*\ |\ p_{\t,\phi}|_{X_{P_d}(w)}\not= 0\}$. We
have  (cf. \cite{l1}), $\mathcal{B}_w^*=\{p_{\t,\phi} \in \mathcal{B}_d^*\ |\ w\ge \t\}$
\begin{thm}
For $w\in
W^{P_d}$, the set $\mathcal{B}_w^*$ is a basis of $H^0(X_{P_d}(w),
L_{\omega_d})\, (=V_w(\o_d)^*)$.
\end{thm}




%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SL(n)/B

\section{ THE LINEAR GROUP $SL(n)$}\label{s13}
Let $G=SL(n)$, the special linear group (of rank $n-1$). Let $T$ be the maximal torus
consisting of all the diagonal matrices  in $G$,  and $B$ the Borel subgroup
consisting of all the  upper triangular matrices in
$G$. It is well-known that  $W$ can be identified  with  $S_n$, the symmetric group on $n$
letters. Any $w\in S_n$ is usually written as $(a_1\dots a_n)$, where $a_i=w(i)$.

Following \cite{bou}, we denote the simple roots by $\e_i-\e_{i+1}$, $1\le i\le n-1$ (note that
$\e_i-\e_{i+1}$ is the character sending diag$(t_1,\dots,t_n)$ to $t_it_{i+1}^{-1}$).
Then $R=\{\e_i-\e_j\mid 1\le i,j\le n,\ i\not= j\}$, and the reflection
$s_{\e_i-\e_{j}}$ may be  identified with the transposition $(i,j)$ in $S_n$.


\subsection{The partially ordered set $I_{d,n}$}\label{idn}
Let $Q=P_d$. Then
\begin{align}
Q&=\left\{ A\in G\biggm| A=
\begin{pmatrix}
*&*\\
0_{(n-d)\times d}&*
\end{pmatrix}\right\},\notag\\
W_Q&=S_d\times S_{n-d}.\notag
\end{align}
Hence
$$W^Q=\{(a_1\dots a_n)\in W\mid a_1<\dots <a_d,\quad a_{d+1}<\dots <a_n\}.$$
Thus $W^Q$ may be identified with
$$I_{d,n}:=\{\underline{i}=(i_1,\dots,i_d)\mid 1\le i_1<\dots <i_d\le n\}.$$

Given $\underline{i}$, $\underline{j}\in I_{d,n}$, let $X_{\underline{i}}$,
$X_{\underline{j}}$ be the associated Schubert varieties in $G/P_d$. We define
$\ui\ge\uj\iff X_{\ui}\supseteq X_{\uj}$ (in other words, the partial order $\ge$ on $I_{d,n}$
is induced by the Chevalley-Bruhat order on the set of Schubert varieties, via the bijection
in \S\ref{1.6}. In particular, we have
$$\ui\ge\uj\iff i_t\ge j_t,\text{ for all } 1\le t\le d.$$
(see \cite{g/p-2} for details)). In the sequel, we shall denote an element
$(a_1\dots a_n)\in W^Q$ by
just $(a_1\ldots a_d)$.


\subsection{The Chevalley-Bruhat order on $S_n$}
For $w_1$, $w_2\in W$, we have
$$X(w_1)\subset X(w_2)\iff \pi_d(X(w_1))\subset \pi_d(X(w_2)),\text{ for all } 1\le d\le n-1,$$
where $\pi_d$ is the canonical projection $G/B\to G/P_d$. Hence we obtain that for
 $(a_1\dots a_n)$, $(b_1\dots b_n)\in S_n$,
$$(a_1\dots a_n)\ge (b_1\dots b_n)\iff (a_1\dots a_d)\uparrow\,\ge \, (b_1\dots
b_d)\uparrow,
\text{ for all } 1\le d\le n-1$$
(here, for  a $d$-tuple $(t_1\dots t_d)$ of distinct integers,
$(t_1\dots t_d)\uparrow$ denotes the ordered $d$-tuple obtained from $\{t_1,\dots, t_d\}$ by
arranging its elements in ascending order).

\subsection{The bases $\mathcal{B}_d$ and $\mathcal{B}_d^*$.}
Let $G=SL(n)$, and $V=K^n$. We denote the standard basis for $K^n$
by $\{e_1,\cdots , e_n\}$. We follow \cite{bou} for indexing the roots.
Given a positive root $\b=\e_j -\e_k, 1\le j<k \le n$, the element
$X_{-\b}$ of the Chevalley basis of ${{\frak g}}$ is given by
$X_{-\b}=E_{kj}$, where $E_{kj}$ is the elementary matrix with $1$
at the $(k,j)$-th place, and $0$'s elsewhere.  For
$1\le d\le l\,(=n-1)$, we have,
$V_K(\o_d)=\wedge ^d V$. We have,
$$X_{-\b}e_i=
\begin{cases}0,&\text{if }i\not= j\\
e_k,&\text{if }i=j.
\end{cases}
$$

Let $\t \in W$, say $\t=(a_1\cdots a_n)$; denoting as above by $\t^{(d)}$ the
element in $W^{P_d}$ representing the coset $\t W_{P_d}$, we have, the extremal weight
vector $q_{\t ^{(d)}}=\pm e_{a_1}\w \cdots
\w e_{a_d} $. Let $\b \in \t (R^+)$, say $\b=\t
(\e_j-\e_k), 1\le j< k\le n $. We have $X_{-\b}= E_{a_ka_j}$, and it follows easily
that for $1\le d \le l$,
$X_{-\b}q_{\t}\not= 0$ if and only
if
$a_j\in \{a_1,\cdots a_d\}$,
$a_k\not\in \{a_1,\cdots , a_d\}$, i.e., if and only if $j\le d<k$ ,
in which case

\ni $X_{-\b}q_{\t^{(d)}}=\pm e_{a_1}\wedge \cdots
\wedge e_{a_{j-1}}\wedge e_{a_k}\wedge e_{a_{j+1}}\wedge \cdots \w  e_{a_d} $. Hence
we obtain
$$X_{-\b}q_{\t^{(d)}}=\pm q_{(s_\b \t)^{(d)}}.$$ This implies that for $w\ge \t,\ \b
\in N(w,\t)$ if and only if $w^{(d)}\ge (s_\b \t)^{(d)} $, for all $1\le d \le l$,
i.e., if and only if $w\ge s_\b \t$ (note that for $d<j$, or $d\ge k,\ (s_\b
\t)^{(d)}= \t ^{(d)}$). Hence we obtain
\begin{thm}\label{4.4}
Let $G=SL(n)$, and $w, \t \in S_n,\ w\ge \t$. Then $T(w,\t)$ is spanned by
$\{X_{-\b},\b \in \t (R^+)\ |\ w\ge s_\b\t\}$.
\end{thm}

\begin{thm}\label{4.5}
Let $w,\t \in W,\ w\ge \t$, and $\b\in \t (R^+)$. Then $\b \in N(w,\t)$ if and only
if
$m_w(\t(\omega_d)-\b)=m(\t(\omega_d)-\b),\text{ for all } 1\le d\le l$,
where $m(\t(\omega_d)-\b)$ (resp. $m_w(\t(\omega_d)-\b)$) denotes the multiplicity of
$\t (\o_d)-\b$ in
$V_K(\o_d)$ (resp.$V_w(\o_d)$).
\end{thm}
\begin{proof}
Given $d, 1\le d\le l$, and $\b=\t (\e_j-\e_k), 1\le j< k\le n$, from
our discussion above, we see easily that
$$m(\t (\omega_d)-\b)=
\begin{cases}0,&\text{if }d<j \text{ or }d\ge k\\
1,&\text{if }j\le d<k.
\end{cases}
$$
Hence we obtain that $m_w(\t (\omega_d)-\b)=m(\t (\omega_d)-\b),\text{ for all } 1\le d\le
l$ if and only if for all $d,\  j\le d<k,\ w^{(d)}\ge (s_\b\t)^{(d)}$. This
together with Theorem \ref{4.4} implies the required result.
\end{proof}

\subsection{}\label{4.6} Let $w,\t \in W,\ w\ge \t,\ \b \in \t(R^+)$. Consider the
following three conditions:

\begin{enumerate}
\item $w\ge s_\b\t$.
\item $m_w(\t(\omega_d)-\b)=m(\t(\omega_d)-\b),\text{ for all } 1\le d\le l,\ l$ being
the rank of $G$.
\item $\b \in N(w,\t)$.

\end{enumerate}

\begin{thm}\label{4.7}
Let $G$ be of Type $\bold {A}_l$. Let $w,\t \in W,\ w\ge \t,\ \b \in \t(R^+)$. Then
the three conditions in \S \ref{4.6} are equivalent.

\end{thm}

\begin{proof}
The result follows from Theorems \ref{4.4} and \ref{4.5}.

\end{proof}







%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\section {The Symplectic Group $Sp(2n)$.}

 Let $V=K^{2n}$ together with a nondegenerate, skew-symmetric
bilinear form $( , )$. Let
$H=SL(V)$ and $G=Sp(V)=\{A\in SL(V) \mid A$ leaves the form
$( , )$ invariant
$\}$. Taking the matrix of the form (with respect to the standard basis
$\{ e_1,...,e_{2n} \}$ of $V$ ) to be
$$E=\begin{pmatrix}
           0  &  J  \\
           -J &  0
\end{pmatrix}$$
where $J$ is the anti diagonal $(1,\cdots ,1)$ of size $n\times
n$, we may realize
$Sp(V)$ as the fixed point set of a certain involution
$\sigma$ on $SL(V),$ namely $G=H^{\sigma}$, where $\sigma: H \longrightarrow
H$ is given by $\sigma(A)=E(^t\!\!A)^{-1}E^{-1}$. Thus
$$\begin{aligned}
G=Sp(2n)&=\{A \in SL(2n) \mid {}^t\!\!AEA=E\} \\
          &=\{A \in SL(2n) \mid E^{-1}({}^t\!\!A)^{-1}E=A \} \\
          &=\{A \in SL(2n) \mid E({}^t\!\!A)^{-1}E^{-1}=A \} \\
          &=H^{\sigma}.
\end{aligned}$$
(note that $E^{-1}=-E$). Denoting by $T_H$ (resp. $B_H$ ) the maximal torus in $H$ consisting of
diagonal matrices (resp. the Borel subgroup in $H$ consisting of upper
triangular matrices ) we see easily that $T_H, B_H$ are stable under $\sigma$.
We set $T_G={T_H}^{\sigma} ,  B_G={B_H}^{\sigma}$. Then it can be seen easily that $T_G$ is a maximal torus in $G$ and
$B_G$ is a Borel subgroup in $G$. We note that the following hold
(cf. \cite{l2}):

\vs.2cm
(I) Denoting by $W_G$ the Weyl group of $G$, we have
$$W_G=\{(a_1...a_{2n}) \in S_{2n} \mid a_i=2n+1-a_{2n+1-i},\  1 \leq
i \leq 2n \}.$$
Thus $w=(a_1...a_{2n}) \in W_G$ is known once $(a_1...a_n)$ is known.

\ni In the sequel,
we shall denote an element $(a_1...a_{2n})$ in $ W_G$ by just $(a_1...a_n)$.


\vs.2cm
(II). Denoting $R_G$ (resp. $R_G^+$) the set of roots of $G$ with
respect to
$T_G$ (resp. the set of positive roots with respect to $B_G$ ),
we have
$$R_G=\{ \pm (\varepsilon_i \pm \varepsilon_j),\ 1\leq i<j\leq n,\
\pm 2\varepsilon_i,\ i=1,...,n \},$$
$$R_G^+ = \{  (\varepsilon_i \pm \varepsilon_j),\ 1\leq i<j\leq n,\
 2\varepsilon_i,\ i=1,...,n \}.$$

\vs.2cm
 The simple roots in $R_G^+$ are given by
$$ \{ \varepsilon_i - \varepsilon_{i+1},\ 1 \leq i \leq n-1, \
2\varepsilon_n  \}.$$
 Let us denote the simple reflections in $W_G$ by $\{ s_i,\  1
\leq
 i \leq n \}$, namely, $s_i$=  reflection with respect to
$\varepsilon_i -
\varepsilon_ {i+1},\  1 \leq i \leq n-1$, and
$s_n=$ reflection with respect to $2\varepsilon_n.$
Then we have (cf. \cite{bou}),
$$ s_i=\begin{cases}r_ir_{2n-i}, &\text{if}\ 1 \leq i \leq n-1\\
                 r_n,             &\text{if}\  i=n
\end{cases}
$$
where $r_i$ denotes the transposition $(i,i+1)$ in $S_{2n},\   1 \leq
i \leq 2n-1.$

\vs.2cm
(III). For $1 \leq d \leq n $, we let $P_d$ be the maximal parabolic subgroup
of $G$ with $S\setminus \{\a_d\}$ as the associated set of simple
roots. Then it can be seen easily that $W_G^{P_d}$, the set of
minimal  representatives   of $W_G/W_{P_d}$ can be identified with
$$\left\{
  (a_1 \cdots a_d)
    \left |
        \begin{aligned}
            (1)\  & 1\leq a_1 < a_2 < \cdots < a_d  \leq 2n \hfill \\
            (2)\  & {\rm for  }\,\,\, 1 \leq i \leq 2n,\,\,\,{\rm if }\,\,\,
                        i \in \{a_1,..., a_d \} \hfill   \\
                  & {\rm then }\,\,\, 2n+1-i \notin \{ a_1,..., a_d \} \hfill
        \end{aligned}
    \right.
\right\}
.
$$

\vs.2cm
(IV). For $w_1=(a_1 \cdots a_{2n}),\  w_2=(b_1 \cdots b_{2n}),\
w_1,w_2 \in   W_G,$ we have $ w_2 \geq w_1 \Leftrightarrow $ the
$d$-tuple $\{b_1,...,b_d$   arranged in ascending order $\} \geq$
the $d$-tuple $\{a_1,...,a_d$ arranged  in ascending order$\} ,\  1
\leq d \leq n $ (cf. \cite{pr}). Hence for $w \in W_G,$  denoting by
$w^{(d)}$ the element in $W_G^{P_d}$ which represents the
coset $wW_{P_d}$, we have for $w_1, w_2
\in W_G,$
$$ w_2^{(d)} \geq w_1^{(d)},\  1 \leq d \leq n
\ \Longleftrightarrow \
      \{b_1,\cdots,b_i\}\uparrow \geq \{a_1,\cdots,a_i\}
     \uparrow, \ \text{ for all } i, 1\leq i \leq 2n-1.        $$
(Here $\{a_1,\cdots,a_i\}\!\!\uparrow,\
\{b_1,\cdots,b_i\}\!\!\uparrow $  are the
corresponding $i$-tuples arranged in ascending order).  But now, the latter
condition is equivalent to $w_2 \geq w_1 $ in $W_H$. Thus we obtain that
the partial order on $W_G$ is induced by the partial order on $W_H$ (cf. \cite{pr}).
In  particular, for $w_1=(a_1 \cdots a_d), \ w_2=(b_1 \cdots b_d), w_1,
w_2 \in  W_G^{P_d},$  we have  $w_2 \geq w_1 \Leftrightarrow \{b_1,
\cdots, b_d \}
\geq \{a_1, \cdots, a_d\}.$

\ni In the sequel, we shall denote an element $(a_1 \cdots
a_{2n})$ in $W_G^{P_d}$ by just $(a_1 \cdots a_d)$. Further, for $1\le i\le 2n$, we
shall denote $i'=2n+1-i,\ |i|= \text{ min }\{i, i'\}$.

\subsection{Chevalley Basis}
For $1\le i\le 2n$, set $i'=2n+1-i$. The involution $\s:SL(2n)\rightarrow SL(2n),
A\mapsto E(^tA)^{-1}E^{-1}$, induces an involution $\s:sl(2n)\rightarrow sl(2n),
A\mapsto -E(^tA)E^{-1}(=E(^tA)E$, since
$E^{-1}=-E$). In particular, we have, for $1\le i,j\le 2n$
$$
\s(E_{ij})=
\begin{cases}
-E_{j'i'}, &\text{if } i,j \text{ are both }\le n\text{ or both }>n\\
E_{j'i'}, & \text{ if one of } \{i,j\}\text{ is }\le n \text{ and the other }>n.
\end{cases}$$
where $E_{ij}$ is the elementary matrix with $1$ at the $(i,j)$th place and $0$
elsewhere. Further
$$\text{Lie } G=\{A\in \text{sl}(2n)\ |\ E(^tA)E=A \}.$$
The Chevalley basis $\{H_{\e_i-\e_{i+1}}, 1\le i< n, H_{2\e_n},  X_{\pm 2\e_m}, 1\le
m\le n, X_{\pm {(\e_j-\e_k)}}, X_{\pm {(\e_j+\e_k)}} ,  1\le j<k\le n\}$ for
$\text{Lie } G$ may be given as follows:
$$
\begin{aligned}
H_{\e_i-\e_{i+1}}&=E_{ii}-E_{i+1,i+1}+E_{(i+1)',(i+1)'}-E_{i'i'}\\
H_{2\e_n}&=E_{nn}-E_{n'n'}\\
X_{\e_j-\e_k}&=E_{jk}-E_{k'j'}\\
X_{\e_j+\e_k}&=E_{jk'}+E_{kj'}\\
X_{2\e_m}&=E_{mm'}\\
X_{-(\e_j-\e_k)}&=E_{kj}-E_{j'k'}\\
X_{-(\e_j+\e_k)}&=E_{k'j}+E_{j'k}\\
X_{-2\e_m}&=E_{m'm}.
\end{aligned}
$$


\begin{defn}
Let $\phi=(a_1\cdots a_d)\in W^{P_d}$, and let $i,\  1\le i\le n$ be such that
$i,(i+1)'\in \{a_1,\cdots , a_d\}$. Let $\t=(b_1\cdots b_d)$ be the element of $ W^{P_d}$
obtained from $(a_1\cdots a_d)$ by replacing $i$ by $i+1$, and $(i+1)'$ by $i'$. In
this situation, we say that $\t$ is obtained from $\phi$ by a {\it Type I} operation.
\end{defn}
We recall from \cite{l2} the following
\begin{prop}\label{5.4}
Let $\t,\phi \in W^{P_d}, \t \ge \phi$. Then $(\t, \phi)$ is an admissible pair if and
only if either $\t=\phi$, or $\t$ is obtained from $\phi$ by a sequence of Type I
operations.
\end{prop}
\subsection{The $G$-module $V_K(\o_d)$}
For $1\le d\le n$, we have $\o_d=\e_1+\cdots +\e_d$, where $\{\e_1,\cdots ,
\e_{2n}\}$ is the canonical basis of Hom $(D_{2n},\bold{G}_m)$ ($D_{2n}$ being the
maximal torus in GL $(2n)$ consisting of all the diagonal matrices). If $d=1$,
then $V_K(\o_d)=V (=K^{2n})$. Let us then suppose that $d\ge 2$. Consider the
$2$-form $f\in \wedge^2V$ given by
$$f=e_1\w e_{2n}+ e_2\w e_{2n-1}+\cdots +e_n\w e_{n+1}$$
(here $\{e_1,\cdots ,e_{2n}\}$ is the standard basis in $V$). We have
$$
\begin{aligned}
V_K(\o_d)&=\{\text{ the primitive vectors in } \wedge^d V\}\\
&=\{v\in \wedge^d V\ |\ v\,\w \,f^{n+1-d}=0\}.
\end{aligned}
$$
The extremal weight vectors $\{q_\t,\t \in W^{P_d}\}$, say $\t=(a_1\cdots a_d)$ are
given by
$$q_\t=\pm  e_{a_1}\w \cdots \w e_{a_d}.$$

For the rest of this section we fix $\t \in W$, say $\t=(a_1\cdots a_n)$.

\subsection{The elements $\{\l_{di}\}, \{\mu_{di}\}$ in $W^{P_d}$.}\label{5.5} Let
$\b=\t(\e_j+\e_k),\  j < k\le n$,  $s=\text{min }\{|a_j|, |a_k|\} ,\ r=\text{max
}\{|a_j|, |a_k|\}$. Fix
$d, \ k\le d \le n$.

\ni Let $T_d={s}\cup (\{s+1, s+2,\ldots ,r-1\}\,\setminus \{|a_1|, \cdots |a_d|\}\})$. Let $l_d=\# T_d - 1$;
write $T_d=\{s=t_{d0}<t_{d1}<\cdots <t_{dl_d}\}$. Note that $t_{di}\in \{|a_{d+1}|, \cdots |a_n|\}$, for $i>0$.


\ni For
$0\le i\le l_d$, define $\mu _{di}, \l_{di}$ as the elements in $W^{P_d}$ given by the
$d$-tuples

\ni $\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,  t_{di},
t'_{d\,i+1}\}
\uparrow
,\ \{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,  t_{d\,i+1},
t'_{di}\}
\uparrow $, respectively, where $ t_{d\,l_d+1}=r$ ( here, for a $d$-tuple
$(b_1\cdots b_d)$,
$\{b_1\cdots b_d\}\uparrow $ denotes the $d$-tuple obtained from $\{b_1\cdots
b_d\}$ by arranging the entries in ascending order). Note that $(\l_{di},
\mu_{di}),\ 0\le i\le l_d$, are admissible pairs.



\ni Recall (\cite{g/p-7}) the following
\begin{prop}\label{5.6}
Let $\b \in \t (R^+)$, say $\b = \t(\a)$, where $\a \in R^+$.
\begin{enumerate}
\item Let $\a=\e_j-\e_k, 1\le j<k \le n$. Then
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j \text{ or } d\ge k \\
 \pm q_{(s_\b \t)^{(d)}}, & \text{ if } j\le d<k.
\end{cases}
$$
\item Let $\a=2\e_j, 1\le j \le n$. Then
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j \\
 \pm q_{(s_\b \t)^{(d)}}, & \text{ if } j\le d\le n.
\end{cases}
$$
\item Let $\a=\e_j+\e_k, 1\le j<k \le n$. Then
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j  \\
 \pm q_{(s_\b \t)^{(d)}}, & \text{ if } j\le d<k\\
\sum _{i=0}^{l_d}\ c_i q_{\l_{di},\mu_{di}}, & \text{ if } k\le d\le n
\end{cases}
$$
where $\l_{di},\mu_{di}$ are as in \S \ref{5.5}, and $c_i=\pm 1$.
\end{enumerate}
\end{prop}

\begin{thm}\label{5.7}
Let $w,\t \in W, w\ge \t$, and $\b \in \t (R^+)$, say $\b = \t(\a)$,
where $\a \in R^+$.
\begin{enumerate}
\item Let $\a=\e_j-\e_k, 1\le j<k \le n$, or $2\e_j, 1\le j\le n$. Then $\b \in
N(w,\t)$ if and only if
$w\ge s_\b \t$.
\item Let $\a=\e_j+\e_k, 1\le j<k \le n$.
\begin{enumerate}
\item Let $\t > \t s_\a\  (=s_\b \t)$. Then $\b \in
N(w,\t)$ necessarily.
\item Let $\t <\t s_\a $.
\begin{enumerate}
\item Let $\t > \t s_{2\e_j}$ or $\t s_{2\e_k}$. Then $\b \in
N(w,\t)$ if and only if
$w\ge s_\b \t$.
\item Let $\t < \t s_{2\e_j}$ and $\t s_{2\e_k}$.

\begin{enumerate}
\item If $\t <\t s_{\e_j-\e_k}$, then $\b \in
N(w,\t)$ if and only if
$w\ge s_\b \t\, (=\t s_{\e_j+\e_k})$ or $\t s_{2\e_j}$.
\item If $\t >\t s_{\e_j-\e_k}$, then $\b \in
N(w,\t)$ if and only if $w\ge \t s_{\e_j-\e_k} s_{2\e_j}$.

\end{enumerate}

\end{enumerate}

\end{enumerate}

\end{enumerate}
\end{thm}
\begin{proof}

If $\a=\e_j-\e_k, 1\le j<k \le n$, or $2\e_j, 1\le j\le n$, then the result is
immediate from (1) and (2) of Proposition \ref{5.6} (in view of Theorem \ref{2.5}).

Let then $\b=\e_j+\e_k, 1\le j<k \le n$. We have (from (3) of
Proposition \ref{5.6}),
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j  \\
 q_{(s_\b \t)^{(d)}}, & \text{ if } j\le d<k\\
\sum_{i=0}^{l_d}\ c_i q_{\l_{di},\mu_{di}}, & \text{ if } k\le d\le n
\end{cases}
$$
 Hence we obtain (in view of Theorem \ref{2.5}) that $\b \in N(w,\t)$
if and only if
 $$  w^{(d)}\ge
\begin{cases}(s_\b \t)^{(d)}, & \text{ if } j\le d<k ,\\
 \l_{di}, & \ 0\le i\le l_d,\ \text{ if } k\le d\le n .
\end{cases}
$$
 It is easily seen that for
$k\le d\le n,\  w^{(d)}\ge \l_{di} $, for all $0\le i\le l_d$ if and only if
$w^{(d)}\ge
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r, s'\} \uparrow ,\,
r,s$ being as in \S
\ref{5.5}.
Hence we obtain
\begin{equation*}
\b \in N(w,\t) \Leftrightarrow w^{(d)}\ge
\begin{cases}
\{a_1,\cdots ,{\hat a}_j, \cdots , a_{d}, a'_k\} \uparrow , &
\text{ for } j\le d<k \\
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_k, r, s'\} \uparrow , &
\text{ for } k\le d\le n.
\end{cases} \tag{*}
\end{equation*}
Let  min $\{|a_i|,|a_j|\}=s$, max
$\{|a_i|,|a_j|\}=r$.
\begin{enumerate}
\item Let $\t>\t s_\a$. This implies $w>\t s_\a$, and hence $\b \in N(w,\t)$ (cf.
\cite{ca}, \cite{po}).

\item Let $\t<\t s_\a$ and $\t > \t s_{2\e_j}$ or $\t s_{2\e_k}$. This implies
that either $a_j=s, a_k=r'$ or $a_j=r', a_k=s$; in both cases it is easily
seen (using Proposition \ref{5.6}) that $w$ satisfies condition (*) if and only if
$w\ge
\t s_\a$.

\item Let $\t<\t s_\a, \t s_{2\e_j}$, and $\t s_{2\e_k}$. This implies
that either $a_j=s, a_k=r$ or $a_j=r, a_k=s$. It is easily checked (using Proposition
\ref{5.6}) that in the former case, $\t < \t s_{\e_j-\e_k}$, and $w$ satisfies
condition (*) if and only if
$w\ge
\t s_{\e_j+\e_k}$ or
$\t s_{2\e_j}$, and in the latter case, $\t > \t s_{\e_j-\e_k}$, and $w$ satisfies
condition (*) if and only if
$w\ge
\t s_{\e_j-\e_k} s_{2\e_j}$.
\end{enumerate}

\ni This completes the proof of Theorem \ref{5.7}.

\end{proof}


\begin{rem} Let $\b=\t(\a),\ \a \in R^+$, and $w\in W,\ w\ge \t$. We have (cf. \cite{ca}, \cite{po}) that condition (1) of \S \ref{4.6} that $w\ge s_\b\t$ implies the
condition (3) of \S \ref{4.6}  that $\b \in N(w,\t)$.
Similarly, the condition (2) of \S \ref{4.6}  that
$m_w(\t(\o_d)-\b)=m(\t(\o_d)-\b), \text{ for all } 1\le d\le n$ implies the
condition that $\b \in N(w,\t)$ (in view of Theorem \ref{2.5};
note that if
$m_w(\t(\o_d)-\b)=m(\t(\o_d)-\b)$, then $X_{-\b} q_{\t^{(d)}} \in V_{w, \o_d}$).

If $\a=\e_j-\e_k$ or $2\e_j$, then using Proposition 5.6, it can be seen easily that the three conditions of \S
\ref{4.6} are equivalent.

Let $\a=\e_j+\e_k$.

  \ni The condition that $w\ge s_\b\t$ neither implies nor is implied by the
condition that
$m_w(\t(\o_d)-\b)=m(\t(\o_d)-\b), \text{ for all } 1\le d\le n$, in general. For
example, take $G=Sp(10),\ \t=(13452), \ \a=\e_2 + \e_3,\ \b=\t(\a)$.
\begin{enumerate}
\item Let $w=(14'3'52)\ (=s_\b\t)$. Then $w\ge s_\b\t$, but
$m_w(\o_3-\b)\ne m(\o_3-\b)$.
\item Let $w=(12'543)$. Then $m_w(\o_d-\b)=m(\o_d-\b), \text{ for all } 1\le d\le
5$, but $w\not\ge s_\b\t$.
\end{enumerate}
Note that the above two examples also show that condition
(3) of \S \ref{4.6} need not imply (1) or (2) in general.
\end{rem}

\section{The Orthogonal Group SO($2n+1$)}
 Let $V=K^{2n+1}$ together with
a non degenerate symmetric bilinear form (,). Taking the matrix of
the form (,) (with respect to the standard basis $\{e_1,\cdots
,e_{2n+1} \}$ of $V$) to be $E$, the $2n+1 \times 2n+1 $ anti-diagonal matrix with $1$ all along the
anti-diagonal except at the $n+1\times n+1$-th place where the entry is $2$ (note that the associated quadratic
form $Q$ on $V$ is given by $Q(\sum_{i=1}^{2n+1}\ x_ie_i)=x_{n+1}^2+\sum _{i=1}^n\ x_ix_{2n+2-i}$), we may
realize
$G=SO(V)$ as the fixed point set $SL(V)^\sigma $, where $\sigma :SL(V) \rightarrow SL(V)
$ is given by $\sigma (A)=E^{-1}(^tA)^{-1}E$. Set $H=SL(V)$.

Denoting by $T_H$ (resp. $B_H$ ) the maximal torus in $H$ consisting of
diagonal matrices (resp. the Borel subgroup in $H$ consisting of upper
triangular matrices ) we see easily that $T_H, B_H$ are stable under $\sigma$.
We set $T_G={T_H}^{\sigma} ,  B_G={B_H}^{\sigma}.$
Then it can be seen easily that $T_G$ is a maximal torus in $G$ and
$B_G$ is a Borel subgroup in $G$.
We note that the following hold (cf. \cite{l2}):

\vs.2cm
(I). Denoting by $W_G$ the Weyl group of $G$, we have
$$W_G=\{(a_1...a_{2n+1}) \in S_{2n+1} \mid a_i=2n+2-a_{2n+2-i},\  1
\leq i \leq 2n+1 \}.$$
Thus $w=(a_1...a_{2n+1}) \in W_G$ is known once $(a_1...a_n)$ is
known (note that $a_{n+1}=n+1, \text{ for all } w\in W_G)$.

\ni In the sequel, we shall denote an
element $(a_1...a_{2n+1})$ in $W_G$ by just $(a_1...a_n)$.

\vs.2cm
(II). Denoting $R_G$ (resp. $R_G^+$) the set of roots of $G$ with
respect to
$T_G$ (resp. the set of positive roots with respect to $B_G$ ),
we have,
$$R_G = \{  \pm (\varepsilon_i \pm \varepsilon_j),\ 1\leq i<j\leq n,\
 \pm \varepsilon_i,\ i=1,...,n \},$$
$$R_G^+ = \{  (\varepsilon_i \pm \varepsilon_j),\ 1\leq i<j\leq n,\
 \varepsilon_i,\ i=1,...,n \}.$$

\vs.2cm
 The simple roots in $R_G^+$ are given by
$$ \{ \varepsilon_i - \varepsilon_{i+1},\ 1 \leq i \leq n-1, \
\varepsilon_n  \}.$$
 Let us denote the simple reflections in $W_G$ by $\{ s_i,\  1
\leq
 i \leq n \}$, namely,
$s_i=$ the reflection with respect to $\varepsilon_i -
\varepsilon_ {i+1},\ 1 \leq i \leq n-1,$ and
$s_n=$ the reflection with respect to $\varepsilon_n.$
Then we have (cf. \cite{bou}),
$$ s_i=\begin{cases}r_ir_{2n+1-i}, &1 \leq i \leq n-1,\\
                 r_nr_{n+1}r_n,             & i=n
\end{cases}$$
where $r_i$ denotes the transposition $(i,i+1)$ in $S_{2n+1},\  1\leq i\leq 2n$.



\vs.2cm
(III). For $1 \leq d \leq n $, we let $P_d$ be the maximal parabolic subgroup
of $G$ with $S\setminus \{\a_d\}$ as the associated set of simple
roots. Then it can be seen easily that $W_G^{P_d}$, the set of
minimal  representatives   of $W_G/W_{P_d}$ can be identified with
$$\left\{(a_1 \cdots a_d) \left |
\begin{aligned} (1)\  & 1\leq a_1 < a_2 < \cdots < a_d  \leq 2n+1,\ a_i\not=
n+1, 1\le i\le d
\hfill
\\
         (2)\  & {\rm for  }\,\,\, 1 \leq i \leq 2n+1,\,\,\,{\rm if
}\,\,\,
         i \in \{a_1,..., a_d \} \hfill   \cr
         & {\rm then }\,\,\, 2n+2-i \notin \{ a_1,..., a_d \} \hfill
 \end{aligned}  \right. \right\}.$$

\vs.2cm
(IV). For $w_1=(a_1 \cdots a_{2n+1}),\  w_2=(b_1 \cdots b_{2n+1}),\
w_1,w_2 \in   W_G,$ we have $ w_2 \geq w_1 \Leftrightarrow $ the
$d$-tuple $\{b_1,...,b_d$   arranged in ascending order $\} \geq$
the $d$-tuple $\{a_1,...,a_d$ arranged  in ascending order$\} ,\  1
\leq d \leq n $ (cf. \cite{pr}). Hence for $w \in W_G,$  denoting by
$w^{(d)}$ the element of $W_G^{P_d}$ representing the coset
$wW_{P_d}$, we have for $w_1, w_2
\in W_G,$
$$ w_2^{(d)} \geq w_1^{(d)},\  1 \leq d \leq n
\ \Longleftrightarrow \
      \{b_1,\cdots,b_i\}\uparrow \geq \{a_1,\cdots,a_i\}
     \uparrow, \ \text{ for all } i, 1\leq i \leq 2n+1.        $$
(Here $\{a_1,\cdots,a_i\}\!\!\uparrow,\
\{b_1,\cdots,b_i\}\!\!\uparrow $  are the
corresponding $i$-tuples arranged in ascending order).  But now, the latter
condition is equivalent to $w_2 \geq w_1 $ in $W_H$. Thus we obtain that
the partial order on $W_G$ is induced by the partial order on $W_H$. In
particular, for $w_1=(a_1 \cdots a_d), \ w_2=(b_1 \cdots b_d), w_1,
w_2 \in  W_G^{P_d},$  we have  $w_2 \geq w_1 \Leftrightarrow \{b_1,
\cdots, b_d \}
\geq \{a_1, \cdots, a_d\}.$

\ni In the sequel, we shall denote an element $(a_1 \cdots
a_{2n+1})$ in $W_G^{P_d}$ by just $(a_1\cdots a_d)$. Further, for $1\le i\le 2n+1$, we
shall denote $i'=2n+2-i,\ |i|= \text{ min }\{i, i'\}$.

\subsection{Chevalley Basis}
For $1\le k\le 2n+1,\text{ set }
k'=2n+2-k$. The involution $\s:SL(2n+1)\rightarrow SL(2n+1), A\mapsto E(^tA)^{-1}E$,
induces an involution $\s:sl(2n+1)\rightarrow sl(2n+1), A\mapsto -E^{-1}(^tA)E$. In
particular, we have, $\s(E_{ij})=-E_{j'i'},\ 1\le i,j\le 2n+1$, where $E_{ij}$ is the
elementary matrix with $1$ at the $(i,j)$th place and $0$ elsewhere. Further
$$\text{Lie } G=\{A\in \text{sl}(2n+1)\ |\ E^{-1}(^tA)E=-A \}.$$
The Chevalley basis $\{H_{\e_i-\e_{i+1}}, 1\le i< n, H_{\e_n},  X_{\pm \e_m}, 1\le
m\le n, X_{\pm {(\e_j-\e_k)}}, X_{\pm {(\e_j+\e_k)}} ,  1\le j<k\le n\}$ for
$\text{Lie } G$ may be given as follows:
$$
\begin{aligned}
H_{\e_i-\e_{i+1}}&=E_{ii}-E_{i+1,i+1}+E_{(i+1)',(i+1)'}-E_{i'i'}\\
H_{\e_n}&=2(E_{nn}-E_{n'n'})\\
X_{\e_j-\e_k}&=E_{jk}-E_{k'j'}\\
X_{\e_j+\e_k}&=E_{jk'}-E_{kj'}\\
X_{\e_m}&=2E_{mn+1}-E_{n+1m'}\\
X_{-(\e_j-\e_k)}&=E_{kj}-E_{j'k'}\\
X_{-(\e_j+\e_k)}&=E_{k'j}-E_{j'k}\\
X_{-\e_m}&=2E_{n+1m}-E_{m'n+1}.
\end{aligned}
$$


\begin{defn}
Let $\phi=(a_1\cdots a_d)\in W^{P_d},\ 1\le d\le n-1$, and let $i,\  1\le i\le n$ be such
that
$i,(i+1)'\in \{a_1,\cdots , a_d\}$. Let $\t=(b_1\cdots b_d)$ be the element of $ W^{P_d}$
obtained from $(a_1\cdots a_d)$ by replacing $i$ by $i+1$, and $(i+1)'$ by $i'$. In
this situation, we say that $\t$ is obtained from $\phi$ by a {\it Type I} operation.
\end{defn}
\begin{defn}
Let $\phi=(a_1\cdots a_d)\in W^{P_d},\ 1\le d\le n-1$, and let $n\in \{a_1,\cdots ,
a_d\}$. Let
$\t=(b_1\cdots b_d)$ be the element of $ W^{P_d}$ obtained from $(a_1\cdots a_d)$ by
replacing $n$ by
$n'$. In this situation, we say that $\t$ is obtained from
$\phi$ by a {\it Type II} operation.
\end{defn}
We recall from \cite{l2} the following:
\begin{prop}\label{6.4}
Let $\t,\phi \in W^{P_d},\ 1\le d\le n-1, \t \ge \phi$. Then $(\t, \phi)$ is an
admissible pair if and only if either $\t=\phi$, or $\t$ is obtained from $\phi$ by a
sequence of operations of Type I or II.
\end{prop}
\subsection{The $G$-module $V_K(\o_d)$}
For $d=n$, $V_K(\o_d)$ is the spin representation, and the extremal weight vectors,
$q_\t, \t \in W^{P_d}$ form a basis for $V_K(\o_d)$. For $1\le d< n$, we have
$V_K(\o_d)= \wedge^d V$ (here, $V=K^{2n+1}$).
The extremal weight vectors $\{q_\t.\t \in W^{P_d}\}$, say $\t=(a_1\cdots a_d)$ are
given by
$$q_\t=\pm  e_{a_1}\w \cdots \w e_{a_d}$$.

For the rest of this section we fix $\t \in W$, say $\t=(a_1\cdots a_n)$.

\subsection{The integers $\{t_{di}\},\ \{r_{dv}\}$.} Let
$\b=\t(\e_j+\e_k), \ j \le k\le n-1$,  $s=\text{min }\{|a_j|, |a_k|\}$, $ r=\text{max
}\{|a_j|, |a_k|\}$. Fix $d, k\le d < n$. The
integers $\{t_{di}\},\ \{r_{dv}\}$ are defined in exactly the same way as in \S 5
(where of course, for $1\le i\le 2n+1,\ i'\text{ is to be understood as } 2n+2-i$).

\subsection{The elements $\{\mu_{di}\}, \{\l_{di}\},\{\d_{dv}\},\{\te_{dv}\}, \nu_d, \xi_d
$.}\label{6.7} The elements $\{\mu_{di}, \l_{di},\ 0\le
i\le l_d\} ,\{\d_{dv},\te_{dv},\ 0\le v\le q_d\}$ are again defined in exactly the
same way as in
\S 5, where recall that $q_d=0\text{ or }p_d -1$, according as $p_d=0 \text{ or }>0
$. The elements
$ \nu_d, \xi_d $ are defined as


\ni $\nu_d=\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,  r_{d\,p_d-1},
r_{dp_d}\}\uparrow $,

\ni $\xi_d=\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,
r'_{dp_d},r'_{d\,p_d-1}
\}\uparrow $

\ni (here, if $p_d=0$, then $r_{d\,p_d-1}=t_{dl_d}$).

\ni Note that $( \l_{di},\mu_{di}),(\te_{dv},\d_{dv})$ and $(\xi_d,\nu_d)$ are
admissible pairs.

\ni Recall (\cite{g/p-8}) the following
\begin{prop}\label{6.8}
Let $\b \in \t (R^+)$, say $\b = \t(\a)$, where $\a \in R^+$.
\begin{enumerate}
\item Let $\a=\e_j-\e_k, 1\le j<k \le n$. Then
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j \text{ or } d\ge k \\
\pm  q_{(s_\b \t)^{(d)}}, & \text{ if } j\le d<k.
\end{cases}
$$
\item Let $\a=\e_j, 1\le j \le n$. Then
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j \\
\pm q_{\g (d,j),\s (d,j)}, & \text{ if } j\le d\le n
\end{cases}
$$
where for $j\le d\le n$, if either $d=n$ or $|a_m|<|a_j|$, for all $d<m\le n$, then
$\s(d,j)=\g(d,j)=(s_\b
\t)^{(d)}$; and
$\s(d,j)=\{a_1,\cdots ,{\hat a}_j,
\cdots a_d, u_{dj}\}\uparrow,\
\g(d,j)=\{a_1,\cdots ,{\hat a}_j, \cdots a_d, u'_{dj}\}\uparrow$, $u_{dj}$ being the
largest entry $u$ in
$\{|a_{d+1}|,
\cdots ,|a_n|\}$ such that $u_{dj}> |a_j|$, otherwise  (note that
$(\g (d,j),\s (d,j))$ is an admissible pair).
\item Let $\a=\e_j+\e_k, 1\le j<k \le n$. Then
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j  \\
 q_{(s_\b \t)^{(d)}}, & \text{ if } j\le d<k \text{ or }d=n\\
2(\sum _{v=0}^{p_d -1}\ b_iq_{\te_{dv},\d_{dv}})+aq_{\xi_d,\nu_d }+\sum _{i=0}^{l_d}
\ c_i q_{\l_{di},\mu_{di}}, &
\text{ if } k\le d< n
\end{cases}
$$
where $\mu_{di}, \l_{di},\d_{di},\te_{di}, \nu_d, \xi_d
$ are as in \S \ref{6.7} and $c_i=\pm 1$, while $a,b_i$ are zero if precisely one
of $\{a_j,a_k\}$ is $>n$ and are $\pm 1$ otherwise (and the sum $\sum _{v=0}^{p_d
-1}\ b_iq_{\te_{dv},\d_{dv}}$ is understood to be zero, if $p_d=0$).
\end{enumerate}
\end{prop}

\ni For the statement of our results concerning the membership of
$\b=\t(\a),\a=\e_j,\e_{j} +\e_k$ in
$N(w,\t)$, we need to introduce some specific elements in $W$ which we describe
now.

\subsection{ The elements $\t_S^j,\ 1\leq j <n $.}\label{s1}
Fix $j, 1\leq j <n$.
Let $|a_j|=r$. We first define the set $I$. The set $I$ is defined to be the empty set
if
$|a_t|<r, \ j<t\le n$. In the alternating case, $I$ is defined to be the set
$\{i_1<\cdots <i_l\le n\}$:

\ni Set $i_0=j$, and define $i_t$ inductively so that
$$ |a_{i_t}|\ = \text{ max}\ \{|a_m|>r,\  i_{t-1}<m\leq n\}.$$
(note that $|a_m|<r,\ i_l< m\leq n$). Let $S$
be a subset
of $I$.

\ni If $S =\emptyset $, then set
$$ \t_S^j=s_\b \t$$
(note that this includes the case $I=\emptyset$).

\ni Let then $S \not= \emptyset $, say $S=\{x_1, \cdots ,x_m\}$ (arranged in
ascending order).

\ni Denote $x_0=j(=i_0)$.We define $$\t_S^j=\t
{\overline {(x_0,x_1)}}\ {\overline {(x_1, x_2)}}\ \cdots
{\overline {(x_{m-1},x_m)}}$$
where, for $\s = (c_1\cdots c_n) \in W$, and
$ 1\leq k <l \leq n$, $\s {\overline {(k,l)}} $ is the element in
$W$ obtained from
$\s $ by replacing $c_k, c_l $ respectively by $|c_l|', |c_k|'$.

\subsection{The elements $ \t_S^{j,k},\ \
1\leq j<k \leq n-1$.}\label{s2}
 Let $\a=\e_j +\e_k,\ 1\leq j<k \leq n-1$.

\ni Let $\text{ max}\{|a_j|,|a_k|\}=r,\
\text{ min}\{|a_j|,|a_k|\}=s$. As in \S \ref{s1}, $I$ is defined to be the empty set
if
$|a_t|<r, k<t\le n$. In the alternating case, $I$ is defined to be the set
$\{i_1<\cdots <i_p\le n\}$:

\ni Set $i_0=k$, and define $i_t$ inductively so that
$$ |a_{i_t}|\ = \text{ max}\ \{|a_m|>r,\  i_{t-1}<m\leq n\}.$$
(note that $|a_m|<r,\ i_p< m\leq n$, also that $p\le p_k $, and $p=0\ \Leftrightarrow
p_k=0$). Let
$S$ be a subset
of $I$.

\ni If $S =\emptyset $, then set
$$ \t_S^{j,k}=s_\b \t$$
(note that this includes the case $I=\emptyset$).

\ni Let then $S \not= \emptyset $, say $S=\{x_1,
\cdots ,x_m\}$ (arranged in ascending order). Denote $x_0=k(=i_0)$.
We define
$$\t_S^{j,k}=\t' {\overline {(x_0,x_1)}}\ {\overline {(x_1,
x_2)}}\ \cdots {\overline {(x_{m-1},x_m)}}$$
 where $\t' $ is obtained from $\t$ by
replacing $a_j,a_k$, by $s',r'$ respectively, and ${\overline
{(x_{i-1},x_i )}}$ is as in \S \ref{s1}.





\begin{thm}\label{6.11}
Let $w,\t \in W, w\ge \t$, and $\b \in \t (R^+)$, say $\b = \t(\a)$,
where $\a \in R^+$.
\begin{enumerate}
\item    Let $\a = \e_k - \e_l,\ \e_n,\ \text{ or }\
\e_i+\e_n$. Then $\b \in N(w,\t)
\Longleftrightarrow w \geq s_\b \t $.

\item   Let $\a = \e_j,\ j<n $.

\begin{enumerate}

\item If $\t >s_\b \t $, then $\b \in N_{w, \t}$
(necessarily).

\item  Let $\t <s_\b \t $, and let $|a_j|=r$.
\begin{enumerate}

\item Let $|a_m|<r,\ j<m\leq n$. Then $\b \in N_{w, \t}
\Longleftrightarrow w\geq s_\b \t $.

\item Let $|a_m|>r$, for some $m,\ j<m\leq n$. Then $\b \in N_{w, \t}
\Longleftrightarrow w\geq \t_S^j $
(for some $S$, notations being as in
\S \ref{s1} above).
\end{enumerate}
\end{enumerate}


\item   Let $\a = \e_j + \e_k,\ j<k\le  n-1$.

\begin{enumerate}
\item If $\t >s_\b \t $, then $\b \in N_{w, \t}$
(necessarily).

\item Let $\t <s_\b \t $. If $\t$ is $>$ either
$\t s_{\e_j}$ or $\t s_{\e_k}$, then $\b
\in N(w,\t)\ \Longleftrightarrow w \geq s_\b \t $.

\item  Let $\t <s_\b \t ,
\t s_{\e_j}$, and $\t s_{\e_k}$. Let max $\{|a_j|,|a_k|\}=r$.
\begin{enumerate}
\item Let $|a_m|<r,\ k<m\leq n$. Then $\b \in N_{w, \t}
\Longleftrightarrow w\geq s_\b \t $.

\item Let $|a_m|>r$, for some $m,\ k<m\leq n$.
Then $\b \in N_{w, \t}
\Longleftrightarrow w\geq \t_S^{j,k} $  (for some $S $, notations being
as in \S \ref{s2} above).

\end{enumerate}
\end{enumerate}
\end{enumerate}


\end{thm}
\begin{proof}
Let $\a = \e_j - \e_k, 1\le j<k\le n,\ \e_n$, or $\e_i+\e_n, 1\le i< n $. Then the
result follows from Proposition \ref{6.8} (in view of Theorem \ref{2.5}).

Let $\a = \e_j,j<n $. We have by Theorem \ref{2.5} and Proposition \ref{6.8},(2) that
 $\b \in N(w,\t)$ if and only if $w^{(d)}\ge \g (d,j),\ j\le d\le n$, where
$\g (d,j)$ is as in Proposition \ref{6.8},(2).

\ni Let $\t>s_\b \t$. This implies that $w>s_\b \t$, and hence $\b \in N(w,\t)$ (cf.
\cite{ca}, \cite{po}).

\ni Let  $\t<s_\b \t$. This implies $a_j\le n$. Let us denote $a_j$ by $r$. If
$|a_m|<|a_j|$, for all $j<m\le n$, then we have from above that $\b \in N(w,\t)$ if
and only if $w\ge s_\b \t$. Let then $|a_m|>r$, for some $m,\ j<m\leq n$. It is
easily seen (in view of Theorem \ref{2.5} and Proposition \ref{6.8}) that in this
case $\b \in N_{w, \t}
\Longleftrightarrow w\geq \t_S^j $
(for some $S$, notations being as in
\S \ref{s1} above).

Let $\a = \e_j + \e_k,\ j<k\le  n-1$. Let  min $\{|a_j|,|a_k|\}=s$, max
$\{|a_j|,|a_k|\}=r$.

\begin{enumerate}
\item If $\t>\t s_\a$, then as above $\b \in N(w,\t)$ (in view of
\cite{ca}, \cite{po}).

\item Let then $\t<\t s_\a$, and $\t > \t s_{\e_j}$ or $\t s_{\e_k}$. This implies
that either $a_j=s, a_k=r'$ or $a_j=r', a_k=s$; in both cases we have (cf.
Theorem \ref{2.5} and Proposition \ref{6.8},(3)) that $X_{-\b}q_{\t^{(d)}}$ belongs to
$V_{w,\o_d}$, for all $1\le d\le n$, if and only if

\begin{equation*}
w^{(d)}\ge
\begin{cases} (s_\b \t)^{(d)}, &  j\le d<k \text{ or } d=n,\\
\l_{di}, & 0\le i\le l_d,\  k\le d<n
\end{cases}
\end{equation*}

It is easily seen that for
$k\le d< n,\  w^{(d)}\ge \l_{di} $, for all $0\le i\le l_d$, if and only if
$w^{(d)}\ge
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r, s'\} \uparrow $.
Hence we obtain that $\b \in N(w,\t)$ if and only if
\begin{equation*}
w^{(d)}\ge
\begin{cases} (s_\b \t)^{(d)}, &  j\le d<k \text{ or } d=n,\\
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r, s'\} \uparrow , &\
k\le d<n
\end{cases}
\end{equation*}

\ni Now it is easily checked that the last condition is equivalent to the
condition that $w\ge \t s_\a$.


\item Let $\t<\t s_\a, \t s_{\e_j}$, and $\t s_{\e_k}$. This implies
that either $a_j=s, a_k=r$ or $a_j=r, a_k=s$.

We have (cf. Proposition \ref{6.8},(3)) $\b \in N(w,\t)$ if and only if
\begin{equation*}
w^{(d)}\ge
\begin{cases} (s_\b \t)^{(d)}, &  j\le d<k \text{ or } d=n,\\
\l_{di},  \ \te_{dv},\  \xi_d,\ 0\le i\le l_d,\ 0\le v\le q_d, &
\  k\le d<n
\end{cases}
\tag{*}
\end{equation*}
where recall that $q_d=0\text{ or }p_d -1$, according as $p_d=0 \text{ or }>0
$.

\ni Let $|a_m|<r,\ k<m\leq n$. This implies that $p_d=0$ and

\ni $ \xi_d= \{a_1,\cdots
,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r', t'_{dl_d}\} \uparrow $, for all
$k\le d<n$. As above, we have, for
$k\le d< n,\  w^{(d)}\ge \l_{di},\ 0\le i\le l_d$, if and only if
$w^{(d)}\ge
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r, s'\} \uparrow $.
Hence we obtain that for  $k\le d< n,\ w^{(d)}\ge
\l_{di},
\
\xi_d,\ 0\le i\le l_d,$ if and only if $w^{(d)}\ge \{a_1,\cdots ,{\hat a}_j, \cdots ,
{\hat a}_k,
\cdots ,a_d, r', s'\}\ (=(s_\b \t)^{(d)})$. From this it follows that $\b \in
N(w,\t)$ if and only if
 $(s_\b \t)^{(d)}$.

\vs.2cm \ni Let $|a_m|>r$, for some $m,\  k<m\leq n$.

\ni As above, we have, for
$k\le d< n$, if $p_d=0$, then $w^{(d)}\ge
\l_{di}, \ \xi_d,\ 0\le i\le l_d,$ if and only if  $w^{(d)}\ge (s_\b \t)^{(d)} $.
Let then $d,\ k\le d< n$  be such that $p_d >0$. We have,
$w^{(d)}\ge \l_{di},\ 0\le i\le l_d$, if and only if
$w^{(d)}\ge
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r, s'\} \uparrow $, and
 $w^{(d)}\ge \te_{dv},\ 0\le v\le q_d$, if and only if
$w^{(d)}\ge
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,  r_{dp_d}, r'
\}\uparrow $. Hence it follows that for $k\le d< n$ such that $ p_d >
0$ (recalling that $\xi_d=\{a_1,\cdots ,{\hat a}_j, \cdots
, {\hat a}_k, \cdots ,a_d, r'_{dp_d},  r'_{dp_d-1}
\}\uparrow$ when $p_d>0$ ) we have, $ w^{(d)}\ge\l_{di},
\
\te_{dv},\
\xi_d,\ 0\le i\le l_d,\ 0\le v\le q_d$ if and only if $w^{(d)}\ge
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r'_{dp_d},  s'
\}\uparrow $.

\ni From this, it easily follows
that the condition (*) is equivalent to the condition that $w\geq \t_S^{j,k} $  (for
some
$S
$, notations being as in \S
\ref{s2} above).
 \end{enumerate}
\end{proof}

\begin{rem}\label{bn}
Let $\b = \t(\a),\ \a = \e_j + \e_k,\ j<k\le  n-1 $. Let $\t<\t s_\a, \t s_{\e_j}$,
and $\t s_{\e_k}$. Note that if $p\ne 0$ ($p$ being as in \S \ref{s2}), then the
condition that $w\geq \t_S^{j,k} $ (for some $S$, notations being as in
\S \ref{s2} above) is equivalent to the condition that
\begin{equation*}
w^{(d)}\ge
\begin{cases}
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k,
\cdots ,a_{d}, a'_k\} \uparrow , & \text{ if } j\le d<k, \\
 \{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k,
\cdots ,a_{d}, |a_{i_{t+1}}|', s'\} \uparrow , & \text{ if } i_t \le d<i_{t+1},\
0\le t<p,\\
(s_\b \t)^{(d)} , & \text{ if } i_p\le d\le n.
\end{cases}
\end{equation*}

\end{rem}

\begin{rem} Let $\b=\t(\a),\ \a \in R^+$, and $w\in W,\ w\ge \t$. We have (cf. \cite{ca},
\cite{po}) that condition (1) of \S \ref{4.6} that
$w\ge s_\b\t$ implies the condition (3) of \S \ref{4.6} that $\b \in N(w,\t)$ .
Similarly, the condition (2) of \S \ref{4.6} that
$m_w(\t (\o_d)-\b)=m(\t (\o_d)-\b), \text{ for all } 1\le d\le n$ implies the
condition that $\b \in N(w,\t)$ (in view of Theorem \ref{2.5})).

Let $\a=\e_j-\e_k$ or $\e_n$ or $\e_j+\e_n$. Then using Proposition 6.8 it can be seen easily that the three
conditions of
\S
\ref{4.6} are equivalent.

Let $\a=\e_j+\e_k,\ j<k<n$.


\ni The condition that $w\ge s_\b\t$ neither implies nor is implied by the
condition that
$m_w(\t(\o_d)-\b)=m(\t(\o_d)-\b), \text{ for all } 1\le d\le n$, in general. For
example, take $G=SO(11),\ \t=(13452), \ \a=\e_2 + \e_3,\ \b=\t(\a)$.
\begin{enumerate}
\item Let $w=(14'3'52)\ (=s_\b\t)$. Then $w\ge s_\b\t$, but
$m_w(\o_3-\b)\ne m(\o_3-\b)$.
\item Let $w=(12'5'4'3)$. Then $m_w(\o_d-\b)=m(\o_d-\b), \text{ for all } 1\le d\le
5$, but $w\not\ge s_\b\t$.
\end{enumerate}
Note that the above two examples also show that condition (3) of
\S \ref{4.6} that $\b \in N(w,\t)$ need not imply (1) or (2) in general.
\end{rem}



\section{The Orthogonal Group SO($2n$)}
Let $V=K^{2n}$ together with a
non-degenerate symmetric bilinear form (,). Taking the matrix of
the form (,) (with respect to the standard basis $\{e_1,\cdots
,e_{2n} \}$ of $V$) to be $E$, the anti-diagonal ($1,\cdots ,1$)
of size $2n \times 2n $, we may realize $G=SO(V)$ as the fixed
point set $SL(V)^\sigma $, where $\sigma :SL(V) \rightarrow SL(V)
$ is given by $\sigma (A)=E(^tA)^{-1}E$. Set $H=SL(V)$.

Denoting by $T_H$ (resp. $B_H$ ) the maximal torus in $H$ consisting of
diagonal matrices (resp. the Borel subgroup in $H$ consisting of upper
triangular matrices ) we see easily that $T_H, B_H$ are stable under $\sigma$.
We set $T_G={T_H}^{\sigma} , \   B_G={B_H}^{\sigma}.$ Then it follows that $T_G$ is a
maximal torus in $G$ and $B_G$ is a Borel subgroup in $G$.

We note that the following hold (cf. \cite{l2}):



\vs.2cm
(I). Denoting by $W_G$ the Weyl group of $G$, we have
$$W_G=\left\{(a_1 \cdots a_{2n})\in S_{2n} \left |
\begin{aligned}(1)\  & a_{i}=2n+1- a_{2n+1-i},1\le i\le   2n \hfill
\\
         (2)\  & \#\{i, 1\le i\le n\}\ {\rm is\ even} \hfill \\
\end{aligned} \right. \right\}.
$$
Thus $w=(a_1...a_{2n}) \in W_G$ is known once $(a_1...a_n)$ is known.

\ni In the sequel,
we shall denote an element $(a_1 \cdots a_{2n})$ in $W$ by just $(a_1 \cdots a_{n})$.

\vs.2cm
(II). Denoting $R_G$ (resp. $R_G^+$) the set of roots of $G$ with
respect to
$T_G$ (resp. the set of positive roots with respect to $B_G$ ),
we have,
$$R_G = \{  \pm (\varepsilon_i \pm \varepsilon_j),\ 1\leq i<j\leq n
  \},$$
$$R_G^+ = \{  (\varepsilon_i \pm \varepsilon_j),\ 1\leq i<j\leq n
  \}.$$

\vs.2cm
 The simple roots in $R_G^+$ are given by
$$ \{ \varepsilon_i - \varepsilon_{i+1},\ 1 \leq i \leq n-1, \
\varepsilon_{n-1}+\varepsilon_n  \}.$$
 Let us denote the simple reflections in $W_G$ by $\{ s_i,\  1
\leq
 i \leq n \}$, namely, $s_i=$ the reflection with respect to
$\varepsilon_i -
\varepsilon_ {i+1},\ 1 \leq i \leq n-1,$ and $s_n=$ the reflection
with respect to
$\varepsilon_{n-1}+\varepsilon_n.$ Then we have (cf. \cite{bou}),
$$ s_i=\begin{cases}r_ir_{2n-i}, &1 \leq i \leq n-1, \\
                 r_nr_{n-1}r_{n+1}r_n,             & i=n .
\end{cases}$$
where $r_i$ denotes the transposition $(i,i+1)$ in $S_{2n},\  1 \leq
i\leq 2n-1.$

\vs.2cm
(III). For $1 \leq d \leq n $, we let $P_d$ be the
maximal parabolic subgroup   of $G$ with $S\setminus \{\a_d\}$ as the associated set of simple
roots. For $w\in W$, let $w^{(d)}$ denote the element of
$W_G^{P_d}$. Then it can be seen
easily that
$W_G^{P_d},d\not= n-1$,  can be identified with
\begin{equation*}\left\{(a_1 \cdots a_d) \left |
\begin{aligned} (1) & 1\leq a_1 < a_2 < \cdots < a_d  \leq 2n \hfill \\
         (2) & {\rm for  }\,\,\, 1 \leq i \leq 2n,\,\,\,{\rm if }\,\,\,
         i \in \{a_1,..., a_d \} \hfill   \\
         & {\rm then }\,\,\, 2n+1-i \notin \{ a_1,..., a_d \} \hfill \\
\end{aligned} \right. \right\}\tag{*}.
\end{equation*}
\smallskip For $d=n-1$, if $w\in W_G^{P_d}$, then
$$w\equiv wu_i\ (mod\  W_{P_{n-1}}), \ 0\le i\le n, i\not= n-1,$$
where
$$u_i=\begin{cases}s_{\a_n} , \text{ if } i=n \\
id,\ & {\rm if}\ $i=0$ \cr
s_{\a_i}s_{\a_{i+1}}\cdots s_{\a_{n-2}}s_{\a_n}, &\text{ if }
1\le i\le n-2
                   \end{cases}$$
(note that the set $\{wu_i,\ 0\le i\le n, i\not= n-1 \}$ is totally
ordered under the Bruhat order). Hence for $d= n-1,\ W_G^{P_d}$
gets identified with a certain {\it proper} subset of (*); in
particular, for $w_1=(a_1 \cdots a_{2n}),\  w_2=(b_1 \cdots b_{2n}),\
w_1,w_2 \in   W_G,$ we can have
$w_1^{(n-1)}=w_2^{(n-1)}$,
with$\{a_1,\cdots,a_{n-1}\}\!\!\uparrow,\
\{b_1,\cdots,b_{n-1}\}\!\!\uparrow $ being different. For $w\in
W$, say $w=(a_1 \cdots a_{2n})$, we see easily that
$$w^{(d)}=\{a_1,\cdots,a_{d}\}\!\!\uparrow,\ 1\le d\le n,\ d\not=
n-1$$
and
$$w^{(n-1)}= {\rm the\ least\ (under }\ge )\ {\rm in \ the\
totally\ ordered\ set}\ Y$$
where
$$Y=\{(y_1^{(i)},\cdots,y_{n-1}^{(i)})\!\!\uparrow\ 0\le i\le n,\ i\not=
n-1\}.$$
$y_1^{(i)},\cdots,y_{n-1}^{(i)}$ being the first $(n-1)$ entries in
$wu_i,\ 0\le i\le n,\ i\not=
n-1$. (Here, the partial order $\ge$ is the usual partial order,
namely, $(i_1,\cdots,i_{n-1})\ge (j_1,\cdots,j_{n-1})$, if
$i_t\ge j_t,\ 1\le t\le n-1,\ (i_1,\cdots,i_{n-1}),\
(j_1,\cdots,j_{n-1})$ being two increasing sequence of
$(n-1)$-tuples.)

\vs.2cm
\ni (IV). For $1\le i\le 2n$, let $i'=2n+1-i$, and $|i|={\rm min}\
\{i,i'\}$. We shall denote the Bruhat order on $W(G)$ by $\succeq$.
Given
$w_1=(a_1
\cdots a_{2n}),\  w_2=(b_1
\cdots b_{2n}),\  w_1,w_2 \in   W_G,$ we have $ w_2 \succeq w_1
$ if and only if the following two conditions hold (cf. \cite{pr}).
\begin{enumerate}
\item For $1\le d\le
n$, we have
    $\{b_1,\cdots,b_d\}\uparrow \geq \{a_1,\cdots,a_d\}
     \uparrow, \ \text{ for all } d.        $
\item Let $\{c_1,\cdots ,c_d\}$ (resp. $\{e_1,\cdots ,e_d\}$)
be the set $\{a_1,\cdots,a_d\}
     \uparrow$
(resp. $\{b_1,\cdots,b_d\}\uparrow$). Suppose for some $r,\ 1\le r\le
d$, and some
$i,\ 0\le i\le d-r$, $\{|c_{i+1}|,\cdots
|c_{i+r}|\}=\{|e_{i+1}|,\cdots |e_{i+r}|\}= \{n+1-r,\cdots ,n\}$
(in some order). Then $\#\{j, i+1\le j\le i+r\ |\ c_j>n\}$, and
$\#\{j, i+1\le j\le i+r\ |\ e_j>n\}$ should both be even or both
odd.
\end{enumerate}

\vs.2cm Thus the Bruhat order $\succeq$ on $W_G$ is ${\underline {not}
}$ induced from the Bruhat order $\ge $ on $W_H$. Following the
terminology in \cite{pr}, we shall refer to condition (2) above as

\ni `` if
$\{c_1,\cdots ,c_d\}$ and $\{e_1,\cdots ,e_d\}$ have analogous
parts, then they are {\bf D}-compatible"; we shall refer to
$\{|c_{i+1}|,\cdots |c_{i+r}|\}$ and $\{|e_{i+1}|,\cdots
|e_{i+r}|\}$ as analogous parts.

In the sequel, we shall have occasion to use both of the partial orders $\succeq$ and
$\ge$.

\begin{rem}\label{7.1}
(a) Let $(c_1,\cdots ,c_d),\
(e_1,\cdots ,e_d) \in W_G^{P_d}$, where $(c_1,\cdots ,c_d)\succeq
\{e_1,\cdots ,e_d) $. Suppose $(c_1,\cdots ,c_d),\
(e_1,\cdots ,e_d)$ have analogous parts. Then it is easily seen
that the condition (2) is equivalent to the condition that
$\#\{j,1\le j\le d\ |\ c_j>n\}$ and $\#\{j,1\le j\le d\ |\
e_j>n\}$ are both even or both odd.

\ni (b). Given $\te \in W$, say $\te =(a_1\cdots a_{2n})$, denoting by
$y_1^{(i)},\cdots,y_{n-1}^{(i)}$ the first $(n-1)$ entries in $\te
u_i,\ 0\le i\le n,\ i\not= n-1$, we have
$$(y_1^{(i)},\cdots,y_{n-1}^{(i)})=\begin{cases}(x_1,\cdots ,x_{n-1}),\
1\le i\le n,\ i\not= n-1\\
(a_1,\cdots ,a_{n-1}),\ i=0
                   \end{cases} $$
where for $1\le i\le n-2$, $(x_1,\cdots ,x_{n-1})$ is the $(n-1)$-tuple obtained from
$(a_1,\cdots ,a_{n-1})$ by replacing $a_i$ by $a_n'$, and for $i=n,\ (x_1,\cdots
,x_{n-1})=(a_1,\cdots ,a_{n-2},a_n') $. Further, we have $\te^{(n-1)}$ is the least
(under
$\ge $) in
$\{(y_1^{(i)},\cdots,y_{n-1}^{(i)}\ )\uparrow,0\le i\le n,\
i\not= n-1\}$.

\ni (c). Given $\te, w \in W$, say $\te =(a_1\cdots a_{2n}),\
w=(b_1\cdots b_{2n})$, we have (with notations as in (b) above)
$$  w^{(n-1)} \succeq \te ^{(n-1)} \Leftrightarrow
(b_1,\cdots,b_{n-1  })\uparrow \succeq
(y_1^{(i)},\cdots,y_{n-1}^{(i)}\ )\uparrow\ {\rm for\
some\ }i, 0\le i\le n,\ i\not= n-1.$$
\end{rem}
\subsection{Chevalley Basis}
For $1\le k\le 2n$, set $k'=2n+1-k$. The involution $\s:SL(2n)\rightarrow SL(2n),
A\mapsto E(^tA)^{-1}E$, induces an involution $\s:sl(2n)\rightarrow sl(2n), A\mapsto
-E(^tA)E $
 In particular, we have, for $1\le i,j\le 2n,\   \s (E_{ij})= -E_{j'i'}$,
where $E_{ij}$ is the elementary matrix with$1$ at the $(i,j)$th place and $0$
elsewhere; and for $1\le k\le 2n, k'=2n+1-k$. Further
$$\text{Lie } G=\{A\in \text{sl}(2n)\ |\ E(^tA)E=-A \}.$$
The Chevalley basis $\{H_{\e_i-\e_{i+1}}, 1\le i< n,\, H_{\e_{n-1}+\e_n},
 X_{\pm {(\e_j-\e_k)}}, X_{\pm {(\e_j+\e_k)}} ,  1\le j<k\le n\}$ for $\text{Lie } G$
may be given as follows:
$$
\begin{aligned}
H_{\e_i-\e_{i+1}}&=E_{ii}-E_{i+1,i+1}+E_{(i+1)',(i+1)'}-E_{i'i'}\\
H_{\e_{n-1}-\e_{n}}&=E_{n-1,n-1}+E_{n,n}-E_{n',n'}-E_{(n-1)',(n-1)'}\\
X_{\e_j-\e_k}&=E_{jk}-E_{k'j'}\\
X_{\e_j+\e_k}&=E_{jk'}-E_{kj'}\\
X_{-(\e_j-\e_k)}&=E_{kj}-E_{j'k'}\\
X_{-(\e_j+\e_k)}&=E_{k'j}-E_{j'k}.
\end{aligned}
$$

\begin{defn}
Let $\phi=(a_1\cdots a_d)\in W^{P_d},\ 1\le d\le n-2$, and let $i,\  1\le i\le n$ be such
that
$i,(i+1)'\in \{a_1,\cdots , a_d\}$. Let $\t=(b_1\cdots b_d)$ be the element of $ W^{P_d}$
obtained from $(a_1\cdots a_d)$ by replacing $i$ by $i+1$, and $(i+1)'$ by $i'$. In
this situation, we say that $\t$ is obtained from $\phi$ by a {\it Type I} operation.
\end{defn}
\begin{defn}
Let $\phi=(a_1\cdots a_d)\in W^{P_d},\ 1\le d\le n-2$. Further let $n-1,n\in \{a_1,\cdots
, a_d\}$. Let
$\t=(b_1\cdots b_d)$ be the element of $ W^{P_d}$ obtained from $(a_1\cdots a_d)$ by
replacing $n-1$ by
$n'$, and $n$ by $(n-1)'$. In this situation, we say that $\t$ is obtained from
$\phi$ by a {\it Type II} operation.
\end{defn}
We recall from \cite{l2} the following:
\begin{prop}
Let $\t,\phi \in W^{P_d},\ 1\le d\le n-2,\  \t \ge \phi$. Then $(\t, \phi)$ is an
admissible pair if and only if either $\t=\phi$, or $\t$ is obtained from $\phi$ by a
sequence of operations of Type I or II.
\end{prop}
\subsection{The $G$-module $V_K(\o_d)$}
For $d=n-1,n$, $V_K(\o_d)$ is the spin representation, and the extremal weight
vectors,
$q_\t, \t \in W^{P_d}$ form a basis for $V_K(\o_d)$. For $1\le d\le n-2$, we have
$V_K(\o_d)= \wedge^d V$ (here, $V=K^{2n}$), and
the extremal weight vectors $\{q_\t.\t \in W^{P_d}\}$, say $\t=(a_1\cdots a_d)$ are
given by
$$q_\t=\pm e_{a_1}\w \cdots \w e_{a_d}$$.
\subsection{The integers $\{t_{di}\},\ \{r_{dv}\}$.}\label{7.7} Let
$\b=\t(\e_j+\e_k), j \le k< n-1$,  $s=\text{min }{|a_j|, |a_k|} ,\ r=\text{max
}{|a_j|, |a_k|}$. Fix $d, k\le d < n-1$. The
integers $\{t_{di}\},\ \{r_{dv}\}$ are defined in exactly the same way as in \S
5.


\subsection{The elements $\{\mu_{di}\}, \{\l_{di}\},\{\d_{dv}\},\{\te_{dv}\}, \nu_d,
\xi_d
$.}\label{eltss} With notations as in \S 5, the elements $\{\mu_{di}, \l_{di},\ 0\le
i\le l_d\} ,\ \{\d_{dv},\te_{dv},\ 0\le v\le q_d \}$ are again defined in exactly the
same way as in
\S 5, and the elements
$ \nu_d, \xi_d $ are defined as in \S 6.

\ni Note that $( \l_{di},\mu_{di}),(\te_{dv},\d_{dv})$ and $(\xi_d,\nu_d)$ are
admissible pairs.

\ni Recall (\cite{g/p-9}) the following
\begin{prop}\label{7.9}
Let $\b \in \t (R^+)$, say $\b = \t(\a)$, where $\a \in R^+$.
\begin{enumerate}
\item Let $\a=\e_j-\e_k, 1\le j<k \le n$. Then
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j \text{ or } d\ge k \\
\pm  q_{(s_\b \t)^{(d)}}, & \text{ if } j\le d<k.
\end{cases}
$$
\item Let $\a=\e_j+\e_k,\  1\le j<k \le n$.
\begin{enumerate}
\item If $k=n-1,n$, then
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j  \\
\pm  q_{(s_\b \t)^{(d)}}, & \text{ if } j\le d.
\end{cases}
$$
\item Let $k<n-1$.
\begin{enumerate}
\item Let $(l_d,p_d) \ne (0,0)$. Then
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}0,&\text{ if } d<j  \\
 q_{(s_\b \t)^{(d)}}, & \text{ if }  j\le d<k, \text{ or } d=n-1,n\\
aq_{\xi_d,\nu_d }+ v_d, &
\text{ if } k\le d< n-1
\end{cases}
$$
where $v_d=2(\sum _{i=0}^{q_d }\ b_iq_{\te_{di},\d_{di}})-\sum _{i=0}^{y_d} \
c_i q_{\l_{di},\mu_{di}},\ y_d=l_d \text{ or }l_d-1$ according as $p_d>\text{ or
}=0$, $\mu_{di},
\l_{di},\d_{di},\te_{di},
\nu_d,
\xi_d
$ are as in \S \ref{eltss} and $c_i=\pm 1$, while $a,b_i$ are zero if precisely one
of $\{a_j,a_k\}$ is $>n$ and $a, b_i, i<q_d$ are $\pm 1, \ b_{q_d}=\pm 1\text{ or }\pm 3$, otherwise (and the
sum $\sum _{v=0}^{q_d}\ b_iq_{\te_{dv},\d_{dv}}$ is understood to be zero, if $p_d=0$).
\item  Let $(l_d,p_d)=(0 ,0)$. Then
$$X_{-\b}q_{\t^{(d)}}=\pm q_{\xi_d,\nu_d},$$
where note that $\xi_d= \{a_1,\cdots ,{\hat a}_j,\cdots , {\hat a}_k, \cdots ,a_d,
r', s'\} \uparrow\ (=(s_\b \t)^{(d)})$,

\ni $ \nu_d=\{a_1,\cdots ,{\hat a}_j,\cdots , {\hat
a}_k, \cdots ,a_d, r, s\} \uparrow $, $r,s$ being as in \ref{7.7}.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{prop}


\subsection{The integers $\{i_{tm}\}$.}\label{7.10} Let $\{i_t,\
0\leq t\leq p\}$ be the set of all integers  defined as follows: Define $i_0=k $.

\ni If $|a_m|<r$, for all $k<m\le n$, then define $p=0$.

\ni If $|a_m|>r$, for some $k<m\le n$, then
$i_t$ is defined inductively so that
$$ |a_{i_t}|\ =\ \text{ max}\ \{|a_m|>r,\  i_{t-1}<m\leq n\}$$
(note that $|a_m|<r,\ i_p< m\leq n$; also note that $p\le p_k$, and
$p=0\Leftrightarrow p_k=0$).

\ni If $p>0$, then for
$t, 1\leq t \leq p$, define $\{i_{tl},  0\leq l \leq c_t \}$ inductively as
$i_{t0}= i_{t-1}$, and $i_{t1}< \cdots <i_{tc_t}$ are all the indices
lying between $i_{t-1} $ and $i_t$ (when this set is non-empty) with the property $$
|a_{i_{tl}}| > |a_m|, \ \forall m>i_{tl},\ m\not= i_t,\ \text{such that}\ |a_m|>r
.$$
Note that $i_{10}=i_0\,(=k)$. Set $ i_{p+1\,\,0}=i_p$.



\begin{thm}\label{7.11}
Let $w,\t \in W, w\ge \t$, and $\b \in \t (R^+)$, say $\b = \t(\a)$,
where $\a \in R^+$.
\begin{enumerate}
\item    Let $\a = \e_l - \e_m, \text{ or}\
\e_j+\e_k, k=n-1,n$. Then $\b \in N(w,\t)
\Longleftrightarrow w \geq s_\b \t $.

\item   Let $\a = \e_j + \e_k,\ j<k<  n-1$.

\begin{enumerate}
\item If $\t >s_\b \t $, then $\b \in N_{w, \t}$
(necessarily).

\item Let $\t <s_\b \t $.
\begin{enumerate}
\item If precisely one of $\{a_j, a_k\}$ is $>n$, then $\b
\in N(w,\t)\ \Longleftrightarrow w \geq s_\b \t $.

\item  Let $\t <s_\b \t $, and $a_j, a_k\le n$.
 Let max $\{|a_i|,|a_j|\}=r$. Then $\b \in N(w,\t) $ if and only if
$$w\ge s_\b \t,\text{ if } p=0,$$
and if $p>0$, then $w^{(d)}  \succeq $

$$ \{a_1,\cdots ,{\hat a}_j,
 \cdots ,a_{d}  ,a'_k\} \uparrow ,   j\le d<k ,$$
 $$\{a_1,\cdots ,{\hat a}_j, \cdots ,
{\hat a}_k, \cdots ,a_{d},  s', |a_{i_{t\,m+1}}|'\} \uparrow ,
i_{tm}\le d < i_{t\,m+1},\ 0\le m<c_t,\ 1\le t\le p ,$$
$$\{a_1,\cdots ,{\hat a}_j, \cdots ,
{\hat a}_k, \cdots ,a_{d}, s', |a_{i_{t+1}}|'\} \uparrow ,\    i_{tc_t}\le d< i_t, \
t<p, $$
 $$(s_\b \t)^{(d)}, \  i_{pc_p}\le d\le n.$$
notations being as in \S \ref{7.10}.
\end{enumerate}
\end{enumerate}
\end{enumerate}


\end{thm}
\begin{proof}
Let $\a = \e_j - \e_k, 1\le j<k\le n$, or $\e_i+\e_k, \ k=n-1,n $. Then the
result follows from Proposition \ref{7.9} (in view of Theorem \ref{2.5}).


Let $\a = \e_j + \e_k,\ j<k<  n-1$. Let  min $\{|a_j|,|a_k|\}=s$, max
$\{|a_j|,|a_k|\}=r$.

\begin{enumerate}
\item Let $\t>\t s_\a$. This implies $w>\t s_\a$, and hence $\b \in N_{w, \t}$ (cf.
\cite{ca}, \cite{po}).

\item Let $\t<\t s_\a$. This implies that at least one of $\{a_j,a_k\}$ is $\le n$.

\begin{enumerate}
\item Let precisely one of $\{a_j, a_k\} \text{ be }\le n$.

We have that
 either $a_j=s,\  a_k=r'$ or $a_j=r',\ a_k=s$; in both cases we have (cf. Proposition
\ref{7.9},(2)) that $X_{-\b}q_{\t^{(d)}}$ belongs to $V_{w,\o_d}$, for all $1\le d\le
n$, if and only if

 $$w^{(d)}\succeq
\begin{cases}
(s_\b \t)^{(d)},& \ j\le d<k,\\
 \l_{di}, & \ 0\le i\le l_d,\  k\le d<n-1, \\
(s_\b \t)^{(d)}, & d=n-1,n.
\end{cases}
$$
Now it is easily checked that the above condition is equivalent to the condition
$$ w^{(d)}\succeq
\begin{cases}\{a_1,\cdots ,{\hat a}_j,
\cdots ,a_{d}, a'_k\} \uparrow ,& \text{ if } j\le d<k,\\
 \{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r, s'\} \uparrow ,&
\text{ if } k\le d<n-1,\\
 (s_\b \t)^{(d)}, & \text{ if } d=n-1,n.
\end{cases}.$$

\ni It is easily seen that the last condition is equivalent to the
condition that $w\succeq \t s_\a$.


\item Let both $a_j, a_k$ be $\le n$. This implies
that either $a_j=s, a_k=r$ or $a_j=r, a_k=s$.

\ni Let $p=0$. This implies $|a_m|<r,\ k<m\leq n$, $p_d=0$ and $
\xi_d=
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r', t'_{dl_d}\}
\uparrow,\  k\le d<n-1$. We have (cf. Proposition \ref{7.9},(2)) that
$$X_{-\b}q_{\t^{(d)}}=
\begin{cases}
aq_{\xi_d,\nu_d }-\sum _{i=0}^{l_d-1} \
c_i q_{\l_{di},\mu_{di}}, &
\text{ if } k\le d< n-1,\, l_d \ne 0,\\
\pm q_{\xi_d,\nu_d}, & \text{ if } l_d=0
\end{cases}
$$
From this it follows that $\b \in N(w,\t)$ if and only if
\begin{equation*}
w^{(d)}\succeq
\begin{cases} (s_\b \t)^{(d)}, &  j\le d<k \text{ or } d=n-1,n\\
\l_{di}, \ \xi_d,  \ 0\le i\le l_d-1,\ &\  k\le d<n-1,\ l_d\ne 0,\\
\xi_d, & k\le d<n-1,\ l_d=0.
\end{cases}
\end{equation*}
We have for $k\le d<n-1$, if $l_d=0$, then $\xi_d=(s_\b \t)^{(d)}$.

\ni Let then $l_d>0$. It is easily seen that $w^{(d)}\succeq \l_{di},   \ 0\le i\le
l_d-1$, if and only if $w^{(d)}\succeq \{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k,
\cdots ,a_d, t_{dl_d}, s'\}$. Hence it follows that for $k\le d<n-1$ such that
$l_d> 0$, we have, $w^{(d)}\succeq
\l_{di},\ \xi_d ,   \ 0\le i\le l_d-1$ if and only if
$$w^{(d)}\succeq
\begin{cases}
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, t_{dl_d}, s'\}
\uparrow,\\
\xi_d (=\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r',t'_{dl_d} \}
\uparrow ).
\end{cases}
$$
Now the last condition is seen to be equivalent to the condition that
$$w^{(d)}\succeq \{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,
r', s'\}
\uparrow .$$
  Thus in both cases
we obtain that $\b \in N(w,\t)$ if and only if
$$
w^{(d)}\succeq
\begin{cases}
\{a_1,\cdots , {\hat a}_j,\cdots ,  a_{d}, a'_k\}
\uparrow, & \text{ if } j\le d<k,\\
\{a_1,\cdots ,{\hat a}_j,
\cdots , {\hat a}_k,\cdots , a_{d}, r', s'\} \uparrow  , & \text{ if } k\le d\le n
\end{cases}$$.

\ni It is easily checked that the last condition is equivalent to the
condition that $w\succeq \t s_\a$.

\vs.2cm \ni Let $p>0$. This implies $p_k>0$, and hence $|a_m|>r$, for some $m,\
k<m\leq n$. We have (cf. Proposition \ref{7.9},(2) $\b \in N(w,\t)$ if and only if
\begin{equation*}
w^{(d)}\succeq
\begin{cases} (s_\b \t)^{(d)}, &  j\le d<k \text{ or } d=n-1,n\\
\xi_d\ (=(s_\b \t)^{(d)}), & k\le d<n-1,\ (l_d,p_d)=(0,0), \\
\l_{di}, \ \xi_d,\ 0\le i\le l_d-1,\ & \  k\le d<n-1,\ p_d=0,\ l_d\ne 0,\\
\l_{di},  \ \te_{dv},\ \xi_d,\ 0\le i\le l_d,\ 0\le v\le q_d,\ & \  k\le d<n-1,\
p_d\ne 0.
\end{cases}
\end{equation*}




Let $k\le d<n-1$. As above, we have that if $p_d=0,\ l_d> 0$, then
$\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,
r', s'\} \uparrow \ (=(s_\b \t)^{(d)})$ is the smallest element in
$W^{P_d}$ which is $\succeq \l_{di}, \ \xi_d,  \ 0\le i\le l_d-1$.

\ni Let $p_d> 0$. Then as above we have, $  w^{(d)}\ge \l_{di},\ 0\le i\le l_d$,
if and only if
$w^{(d)}\ge
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r, s'\} \uparrow $, and
$w^{(d)}\ge \te_{dv},\ 0\le v\le q_d$, if and only if
$w^{(d)}\ge
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,  r_{dp_d}, r'
\}\uparrow $. Hence we obtain that $w^{(d)}\succeq \l_{di},  \ \te_{dv},\ \xi_d,\
0\le i\le l_d,\ 0\le v\le q_d$, if and only if
$$ w^{(d)}\succeq
\begin{cases}
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r, s'\} \uparrow, & \\
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r_{dp_d}, r'\} \uparrow, &
\\
\xi_d(=\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r'_{dp_d}, r'_{d\,
p_d-1}\}
\uparrow ),
\end{cases}
$$
i.e., if and only if
$$ w^{(d)}\succeq \{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,
 r'_{d\, p_d-1}, s'\} \uparrow .
$$
Note that $\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,
 r'_{d\, p_d-1}, s'\} \uparrow $ is the smallest element (under $\succeq$) in
$W^{P_d}$ which is $ \succeq \{a_1,\cdots ,{\hat a}_j, \cdots ,
{\hat a}_k, \cdots ,a_d, r, s'\} \uparrow $,

\ni $
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r_{dp_d}, r'\} \uparrow ,
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r'_{dp_d}, r'_{d\,
p_d-1}\}
\uparrow $; eventhough $\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d,
 r'_{d\, p_d}, s'\} \uparrow  $ is the smallest element (under $\ge $ ) in $W^{P_d}$

\ni $\ge \{a_1,\cdots ,{\hat a}_j, \cdots ,
{\hat a}_k, \cdots ,a_d, r, s'\} \uparrow, \
\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r_{dp_d}, r'\} \uparrow
$, and


\ni $\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots ,a_d, r'_{dp_d}, r'_{d\,
p_d-1}\}
\uparrow$, it is $\not\succeq $

\ni $\{a_1,\cdots ,{\hat a}_j, \cdots , {\hat a}_k, \cdots
,a_d, r_{dp_d}, r'\} \uparrow $, since they have non-compatible analogous parts (cf.
(IV) above).

The required result  follows from this.
 \end{enumerate}
 \end{enumerate}
\end{proof}

\begin{rem} Compare the similarity between the condition in 2(b),(ii) of Theorem 7.11)
and the condition stated in Remark
\ref{bn}.

\end{rem}

\begin{rem} Let $\b=\t(\a),\ \a \in R^+$, and $w\in W,\ w\succeq \t$. We have (cf. \cite{ca},\ \cite{po}) that
condition (1) of \S \ref{4.6} that $w\succeq s_\b\t$ implies the condition (3) of \S \ref{4.6} that $\b \in
N(w,\t)$ . Similarly, the condition (2) of \S \ref{4.6} that
$m_w(\t(\o_d)-\b)=m(\t(\o_d)-\b), \text{ for all } 1\le d\le n$ implies the
condition that $\b \in N(w,\t)$ (in view of Theorem \ref{2.5}).

Let $\a=\e_j-\e_k$ or $\e_n$ or $\e_j+\e_n$. Then the three conditions of \S
\ref{4.6} are equivalent (in view of Proposition \ref{7.9}).

Let $\a=\e_j+\e_k,\ j<k<n$.

\ni The condition that $w\succeq s_\b\t$ neither implies nor is implied by the
condition that
$m_w(\t(\o_d)-\b)=m(\t(\o_d)-\b), \text{ for all } 1\le d\le n$, in general. For
example, take $G=SO(10),\ \t=(1345'2'), \ \a=\e_2 + \e_3,\ \b=\t(\a)$.
\begin{enumerate}
\item Let $w=(14'3'52)\ (=s_\b\t)$. Then $w\succeq s_\b\t$, but
$m_w(\o_3-\b)\not\succeq m(\o_3-\b)$.
\item Let $w=(12'5'3'4')$. Then $m_w(\o_d-\b)=m(\o_d-\b), \text{ for all } 1\le d\le
5$, but $w\not\succeq s_\b\t$.
\end{enumerate}
Note that the above two examples show that condition (3) of \S \ref{4.6} implies neither (1) nor (2) of
\S \ref{4.6}.
\end{rem}



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\end{document}