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\begin{document}
\title[Tangent spaces to Schubert varieties]{ON TANGENT SPACES TO SCHUBERT
VARIETIES}
\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 the results on the tangent spaces to Schubert varieties announced in
\cite{l4} for
$G$ classical. We give two descriptions of the tangent space to a  Schubert
variety at {\it id}. The first description is in terms of the root system, and the
second one in terms of 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 characteristic $0$. 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 presisely
$\{e_{w}\ |\  w\in W\}$.
For $w\in W$, let $X(w)$ denote the associated Schubert variety (the Zariski closure
of
$Be_{w}$  in $G/B$). Let $U(\frak{g})$ denote the universal enveloping algebra of
$\frak{g}$, and $U^+(\frak{g})$ the subalgebra of $U(\frak{g})$ generated by $\{X_\a,
\a \in S\}$.

 For $\t \in W$, let $T(w_0,e_\t)$ denote the tangent space to $G/B$ at $e_\t$
($w_0$ being the element of largest length in $W$). We have
$$T(w_0,e_\t): =\oplus\ _{\b
\in
\t(R^+)}\  {{{\frak g}}}_{-\b}. $$
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 $T(w,e_\t)$ is a $T$-stable subspace of
$T(w_0,e_\t)$, we have that
$T(w,e_\t) $  is spanned by
$\{X_{-\b},\,\b \in N(w,\t)\}$.

For a dominant
weight $\l$, let $V(\l)$ be the irreducible $G$-module with
highest weight $\l$. Let us fix a highest weight vector
$u(\l)$ 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
$u_w(\l)= n_w\cdot u(\l)$, $V_w(\l)=U^+(\frak{g}) u_w(\l)$ (note that
$V_{w_0}(\l)=V(\l)$ ).

For $w\in W$, we define $M_w$, a subset of $R^+$ as follows, $G$ being classical:
(We index the roots as in \cite{bou}.)

\begin{enumerate}
\item  Let $G$ be of type $ \bold{A}_n $. Then
$M_w=\{ \b \in R^+\ |\   w \geq s_\b \}$.
\item  Let $G$ be of type $ \bold{ C}_n $.
\begin{enumerate}
\item   Let $\b = \e_i - \e_j, \text{ or } 2\e_i$. Then
$\b \in M_w\ \Longleftrightarrow w \geq s_\b $.
\item   Let $\b = \e_i + \e_j,\ i<j \le n$. Then $\b \in
M_w\ \Longleftrightarrow w \geq  s_{\e_i + \e_j}
\text{ or } s_{2\e_i} $.
\end{enumerate}
\item  Let $G$ be of type $ \bold{ B}_n $.
\begin{enumerate}
\item   Let $\b = \e_i - \e_j,\ \e_n, \text{  or }
\e_i+\e_n $. Then $\b \in M_w\ \Longleftrightarrow w \geq s_\b $.
\item  Let $\b = \e_i,i<n $. Then $\b \in
M_w\ \Longleftrightarrow w \geq   s_{\e_i} \text{  or }
s_{\e_i + \e_n}  $.
\item  Let $\b = \e_i + \e_j, i<j <n $. Then $\b \in
M_w\ \Longleftrightarrow w \geq  s_{\e_i + \e_j}\
\text{ or } s_{\e_i}s_{\e_j + \e_n} $.
\end{enumerate}
\item Let $G$ be of type $ \bold{ D}_n $.
\begin{enumerate}
\item  Let $\b = \e_k - \e_l,\text{ or } \e_i + \e_j,\
j=n-1,n$. Then $\b \in M_w\ \Longleftrightarrow w \geq s_\b $.
\item  Let $\b = \e_i + \e_j,\ i<j<n-1$.

\ni Then
  $\b \in
M_w\ \Longleftrightarrow w \geq  s_{\e_i + \e_j}
\text{ or } s_{\e_i-\e_{n}}s_{\e_i+\e_{n}}s_{\e_j + \e_{n-1}} $.
\end{enumerate}
\end{enumerate}


\vs.2cm  We prove for $G$ classical the following result: (cf. Theorems
3.4,4.6,5.7,6.8)
\begin{th}
Let $w \in W$. Let $e$ denote the identity element in $W$. Then $N(w,e)=M_w$.
\end{th}
In particular, we obtain a criterion for smoothness:

\begin{thh}
Let $w\in W$. Then $X(w)$ is smooth if and only if $\# M_w=l(w)$.
\end{thh}

We further prove (cf. Theorems 3.5,5.9,6.10)

\begin{thhh}\label{three}
Let $G$ be of type $\bold {A}_n$, $\bold {B}_n$,  or $\bold {D}_n$. Let $w\in
W$, and $\b \in R^+$. Then $\b
\in N(w,e)$ if and only if
$m_w(\o_d-\b)=m(\o_d-\b),\text{ for all } 1\le d\le n$, where $\o_d,1\le d\le n $ are
the fundamental weights of $G$ and $m(\o_d-\b)$ (resp.
$m_w(\o_d-\b)$) denotes the multiplicity of
$\o_d-\b$ in
$V(\o_d)$ (resp.$V_w(\omega_d)$).
\end{thhh}



\begin{rem}
It turns out that the above result is not true for Type {\bf C}$_n$ (see \S 4 for
details).
\end{rem}

Let $w \in W,\ \b \in R^+$. Consider the following three
conditions:

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

\end{enumerate}

We show (cf. Theorem \ref{4.7}) that for $G$ of Type $\bold {A}_n$, the above three
conditions are equivalent for all $w \in W,\ \b \in R^+$. For the types $\bold
{B}_n$, $\bold {C}_n$, $\bold {D}_n$, we give precise relationships among the
above three conditions (see sections 4,5,6 for details).


The above results are proved using the basis ${\cal B}_d$ (cf. \cite{l1})
for
$V(\o_d)$, $1\le d\le l$, $l$ being the rank of $G$, and a Theorem of Polo (cf.
\cite{po}, Theorem 3.2). All of the results of this paper hold over arbitrary
characteristics, since one knows (see
\cite{po}, Corollary 4.1 for example) that $N(w,\t)$ is independent of the base
field; in Theorem 3 over an algebraically closed field $K$ of arbitrary
characteristic, one should replace $V(\o_d)$ (resp.$V_w(\omega_d)$) by the
corresponding Weyl (resp. Demazure) module.

The sections are organized as follows. In \S 1, we recall the above mentioned result
of Polo. In
\S 2, we recall the basis ${\cal B}_d$. In sections \S 3,4,5,6 we prove Theorems 1\&3
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.

%%%%%%%%%
\section{THE TANGENT SPACE $T(w,\t)$.}


Let $G$ be a semisimple and simply connected  algebraic group defined over $K$. Let
$T,B,W,S,R,R^+,e_w, X(w)$ etc be as in the Introduction. We have the well-known
Bruhat decomposition
$$G/B=\dot{\cup}_{\{w\in W\}}\ Be_{w},\qquad
X(\te)=\dot{\cup}_{\{w\in W\ |\ w\le \te\}}\ Be_{w},\quad
\te\in W,$$
where $\le $ denotes the Chevalley-Bruhat order. Let $P_d$ be the maximal parabolic
subgroup corresponding to the simple root $\a_d$, and $W_{d}$ be the Weyl group of
$P_d$. We shall denote the {\em set of minimal reperesentatives of} $W/W_{P_d}$ in
$W$ by $W^{d}$, namely
$$W^{d}=\{w\in W\mid l(ww')=l(w)+l(w'),\text{ for all }w'\in W_{d}\}.$$

\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.

\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
$\{X_{-\b},\,\b \in N(w,\t)\}$.

\ni Recall the following:

\begin{thm}\label{2.5}(\cite{po}, Theorem 3.2)
Let $w,\t \in W,\
w\ge \t$. Let $\b \in \t(R^+)$. Then
$\b
\in N(w,\t)$ if and only if $X_{-\b}u_\t(\omega_d)\in V_w(\omega_d),\ \text{ for all }
1\le d\le l,\ l$ being the rank of $G$.
\end{thm}

%%%%%%%%%
\section{ Bases ${\cal B}$ and ${\cal B}*$ for
$V_k({\omega_d})$ and $H^0(G/B, L_{\omega_d})$.}
 Let $G$ be classical. For
$1\le d\le l$ ($l$ being the rank of $G$), let $P_d,W_d, W^d$ be as in \S 1. For
$w\in W^d$, we shall denote the associated Schubert variety in $G/P_d$ by
$X_{P_d}(w)\ (={\overline{BwP_d}}(\text{mod}P_d))$. 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 the basic results from
\cite{l1}.
\subsection{Chevalley multiplicity}
Let $\t,\phi \in W^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 $\t,\phi \in W^d ,\  \t \ge \phi$ 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
$m(\t_i,\t_{i+1})=2,\  0\le i\le r-1$.

\begin{prop}\label{3.7}
Let $\t,\phi \in W^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 $\t_{i+1}=s_{\b_i}\t_i,\ \b_i\in R^+,\  0\le i\le r-1$.
Define $v\in V(\o_d)$ as $v=X_{-\b_0}X_{-\b_1}\cdots
X_{-\b_{r-1}}u_{\phi}(\o_d),\ u_{\phi}(\o_d)$ being as in the introduction with
$\l=\o_d$. Then
$v$ is independent of the chain chosen and depends only on $\t$ and
$\phi$. Further, $v$ is a weight vector of weight $\frac {1}{2} (\t(\o_d)+
\phi(\o_d))$.
\end{prop}
\subsection{The sets ${\cal B}$ and ${\cal B}_w$}\label{3.9}
Let $\t,\phi \in W^d $ be such that $(\t,\phi ) $ is an admissible
pair. If $\t=\phi$, then set $q_{\t,\t}$ (or just $q_\t$) equal to $u_\t (\omega_d)$.
If
$\t>\phi$, then set $q_{\t,\phi}$ as the vector $v$ as given by Proposition
\ref{3.7}. Set ${\cal B}=\{q_{\t,\phi},(\t,\phi) \text { an admissible pair}\}$. For
$w\in W^d$, set ${\cal B}_w=\{q_{\t,\phi}\in {\cal B}\ |\ w\ge \t\}$.
\begin{thm}
With notations as above, the set ${\cal B}$ is a basis for
$V(\o_d)$. Further, for $w\in W^d$, the set ${\cal B}_w$
is a basis for $V_w(\omega_d)$.
\end{thm}
\subsection{The sets ${\cal B}^*$ and ${\cal B}_w^*$}
Define ${\cal B}^*$ to be the basis of
$H^0(G/P_d, L_{\omega_d})\ (=V_{K}(\omega_d)^*)$ dual to ${\cal B}$ (here,
$L_{\omega_d}$ denotes the ample generator of Pic$(G/P_d)\,(\simeq {\Bbb Z})$). Let
us denote the elements of
${\cal B}^*$ by
$\{p_{\t,\phi},(\t,\phi)
\text { an admissible pair}\}$. For
$w\in W^d$, set ${\cal
B}_w^*=\{ p_{\t,\phi}|_{X_{P_d}(w)},\ p_{\t,\phi} \in {\cal B}^*\ |\
p_{\t,\phi}|_{X_{P_d}(w)}\not= 0\}$.
\begin{thm}
For $w\in
W^d$, the set ${\cal B}_w^*$ is a basis of $H^0(X_{P_d}(w),
L_{\omega_d})$.
\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 $(w(1), w(2), \cdots w(n))$.

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}
Fix $d,\ 1\le d\le n-1$. We have,
$$
P_d=\left\{ A\in G\biggm| A=
\begin{pmatrix}
*&*\\
0_{(n-d)\times d}&*
\end{pmatrix}\right\},\ W_d=\cal{S}_d\times \cal{S}_{n-d}.
$$
and $W^{d}$ 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
have
$$X_{\ui}\supseteq X_{\uj}\iff \ui\ge\uj\iff i_t\ge j_t,\text{ for all } 1\le
t\le d.$$ (see \cite{g/p-2} for details)). In the sequel, we shall denote an element
$(a_1\dots a_n)\in W^{d}$ by just $(a_1\dots a_d)$.


\subsection{The Chevalley-Bruhat order on $S_n$}
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 ${\cal B}_d$ and ${\cal B}_d^*$.}
Let $G=SL(n)$, and $V=K^n$. We denote the standard basis for $K^n$
by $\{e_1,\cdots , e_n\}$.
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,
$V(\o_d)=\wedge ^d V$, and $q_{id}(=u({\o_d}))=e_1\wedge \cdots \wedge
e_d$ (more generally, given $w=(a_1\cdots a_n)\in W$, denoting by
$w^{(d)}$ the element in $W^d$ which represents the coset $wW_d$, we
have, $q_{w^{(d)}} (=u_w(\o_d))= e_{a_1}\w \cdots
\w e_{a_d} $ (up to $\pm 1$)). We have,
$$X_{-\b}e_i=
\begin{cases}0,&\text{if }i\not= j\\
e_k,&\text{if }i=j.
\end{cases}
$$
From this it follows that $X_{-\b}q_{id}\not= 0$ if and only if
$j\in \{1,\cdots d\}$, and
$k\not\in \{1,\cdots , d\}$, i.e., if and only if $j\le d<k$;
further, for $j\le d<k$, we have,

\ni $X_{-\b}q_{id}=\pm e_1\wedge \cdots
\wedge e_{j-1}\wedge e_{j+1}\wedge \cdots \w  e_d \wedge e_k$. Hence we obtain
$$X_{-\b}q_{id}=\pm q_{s_\b^{(d)}}.$$ This implies that $\b \in N(w,e)$ if and only
if $w^{(d)}\ge s_\b^{(d)}, j\le d <k $, i.e., if and only if $w\ge s_\b$ (note
that for $d<j$, or $d\ge k,\ s_\b^{(d)}= (1\cdots d)$), where $e$
denotes the identity element in $S_n$. Hence we obtain (cf.\cite{lse})
\begin{thm}\label{4.4}
Let $G=SL(n)$, and $w\in S_n$. Then $T(w,e)$ is spanned by
$\{X_{-\b},\b \in R^+\ |\ w\ge s_\b\}$.
\end{thm}

\begin{thm}\label{4.5}
Let $w \in S_n$, and $\b\in R^+$. Then $\b \in N(w,e)$ if and only if
$m_w(\omega_d-\b)=m(\omega_d-\b),\text{ for all } 1\le d\le l\, (=n-1)$,
where $m(\o_d-\b)$ (resp. $m_w(\o_d-\b)$) denotes the multiplicity of $\o_d-\b$
in
$V(\o_d)$ (resp.$V_w(\omega_d)$).
\end{thm}

\begin{pf}
Given $d, 1\le d\le l$, and $\b=\e_j-\e_k, 1\le j< k\le n$, from
our discussions above, we see easily that
$$m(\omega_d-\b)=
\begin{cases}0,&\text{if }d<j \text{ or }d\ge k\\
1,&\text{if }j\le d<k.
\end{cases}
$$
Hence we obtain that $m_w(\omega_d-\b)=m(\omega_d-\b),\text{ for all } 1\le d\le
l$ if and only if for all $d,\  j\le d<k,\ w\ge s_\b^{(d)}$, i.e., if and only if
$w\ge s_\b$. This together with Theorem \ref{4.4} implies the required result.
\end{pf}

\subsection{}\label{condns} Let $w \in W,\ \b \in R^+$. Consider the following three
conditions:

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

\end{enumerate}

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

\end{thm}

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

\end{pf}

%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\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 $\sigma(A)=E(^t\!\!A)^{-1}E^{-1}$. 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). We shall index the simple roots in $G$ as in \cite{bou}. Let us denote the
simple reflections in $W_G$ by $\{ s_i,\  1\leq  i \leq n \}$. We have (cf.
\cite{bou}),
$$ s_i=\begin{cases}r_ir_{2n-i}, & \a_i=\varepsilon_i -
\varepsilon_{i+1},\ 1
\leq i \leq n-1\\
r_n,&   \a_i=2\varepsilon_n
\end{cases}
$$
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 $, $W_G^{d}$ can be identified with
$$\left\{(a_1 \cdots a_d) \left |
\eqalign{ (1)\  & 1\leq a_1 < a_2 < \cdots < a_d  \leq 2n \hfill \cr
         (2)\  & {\rm for  }\,\,\, 1 \leq i \leq 2n,\,\,\,{\rm if
}\,\,\,
         i \in \{a_1,..., a_d \} \hfill   \cr
         & {\rm then }\,\,\, 2n+1-i \notin \{ a_1,..., a_d \} \hfill \cr }
 \right. \right\}. $$

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

\vs.2cm
(IV). For $w_1=(a_1 \cdots a_{n}),\  w_2=(b_1 \cdots b_{n}),\
w_1,w_2 \in   W_G,$ 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}). In particular,
the partial order on $W_G$ is induced by the partial order on $W_H$ (cf. \cite{pr}).


\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:
$$H_{\e_i-\e_{i+1}}=E_{ii}-E_{i+1,i+1}+E_{(i+1)',(i+1)'}-E_{i'i'},\
H_{2\e_n}=E_{nn}-E_{n'n'},$$
$$ X_{\e_j-\e_k}=E_{jk}-E_{k'j'},\
X_{\e_j+\e_k}=E_{jk'}+E_{kj'},\ X_{2\e_m}=E_{mm'},\ $$
$$X_{-(\e_j-\e_k)}=E_{kj}-E_{j'k'},\
X_{-(\e_j+\e_k)}=E_{k'j}+E_{j'k},\ X_{-2\e_m}=E_{m'm}.$$


\begin{defn}\label{4.2}
Let $\phi=(a_1\cdots a_d)\in W^d$, 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^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{prop}\label{5.4}(cf. \cite{l2})
Let $\t,\phi \in W^d, \t \ge \phi$. 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 $V(\o_d)$}
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 $V(\o_d)=V (=K^{2n})$. 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
$$
\begin{aligned}
V(\o_d)&=\{\text{ the primitive vectors in } \wedge^d V\}\\
&=\{v\in \wedge^d V\ |\ v\,\w \,f^{n+1-d}=0\}.
\end{aligned}
$$
The extremal weight vectors $\{q_\t,\t \in W^d\}$, say $\t=(a_1\cdots a_d)$ are
given by
$$q_\t= e_{a_1}\w \cdots \w e_{a_d}$$.
\begin{prop}\label{5.6}(cf. \cite{l2})
Let $\b \in R^+$.
\begin{enumerate}
\item Let $\b=\e_j-\e_k, 1\le j<k \le n$. Then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j \text{ or } d\ge k \\
 \pm q_{s_\b^{(d)}}, & \text{ if } j\le d<k.
\end{cases}
$$
\item Let $\b=2\e_j, 1\le j \le n$. Then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j \\
\pm q_{s_\b^{(d)}}, & \text{ if } j\le d\le n.
\end{cases}
$$
\item Let $\b=\e_j+\e_k, 1\le j<k \le n$. Then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j  \\
 \pm q_{s_\b^{(d)}}, & \text{ if } j\le d<k\\
\pm q_{\t,\phi}, & \text{ if } k\le d\le n
\end{cases}
$$
where $\t=(12\cdots j-1\, j+1 \cdots d j')$ and $\phi=(12\cdots k-1\, k+1 \cdots d
k')$.
\end{enumerate}
\end{prop}

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

If $\b=\e_j-\e_k, 1\le j<k \le n$, 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}),
\begin{equation*}
X_{-\b}q_{id}=\pm q_{s_\b^{(d)}}, \ j\le d<k .\tag{*}
\end{equation*}
 For $k\le d\le n$, we
have
$X_{-\b}q_{id}=\pm q_{\t, \phi}$, ${\t, \phi}$ being as in Proposition \ref{5.6},(3).
We have
$s_{2\e_j}=(12\cdots j-1\,j'\,j+1\cdots n)$, and hence we obtain
\begin{equation*}
X_{-\b}q_{id}=\pm q_{s_{2\e_j}^{(d)},\phi}, \ k\le d\le n\tag{**}
\end{equation*}
Hence from (*) and (**) we obtain (in view of Theorem \ref{2.5}) that $\b \in N(w,e)$
if and only if $w^{(d)}\ge s_\b^{(d)}, j\le d<k$ and $w^{(d)}\ge s_{2\e_j}^{(d)}, k\le
d\le n$.

\ni {\bf Claim.} $w^{(d)}\ge s_\b^{(d)}, j\le d<k$ and $w^{(d)}\ge s_{2\e_j}^{(d)},
k\le d\le n$ if and only if $w\ge $
either $s_\b$ or $s_{2\e_j}$.

If $w\ge $
either $s_\b$ or $s_{2\e_j}$, then clearly $w^{(d)}\ge s_\b^{(d)}, j\le d<k$ and $w^{(d)}\ge s_{2\e_j}^{(d)}, k\le
d\le n$ (note that $s_\b^{(d)}> s_{2\e_j}^{(d)}, k\le d\le n$, and
$s_{2\e_j}^{(d)}> s_\b^{(d)}, j\le d<k $).

Let now $w$ be such that $w^{(d)}\ge s_\b^{(d)}, j\le d<k$ and $w^{(d)}\ge s_{2\e_j}^{(d)},
k\le d\le n$. Further, let $w\not\ge s_\b $. We shall now show that $w\ge
s_{2\e_j}$. Let $w=(a_1\cdots a_n)$. The facts that $w^{(d)}\ge s_\b^{(d)}, j\le
d<k$, and $w\not\ge s_\b $ imply that
\begin{equation*}
w^{(k)}\not\ge s^{(k)}_\b. \tag{\dag}
\end{equation*} On the other
hand, we have $w^{(k)}\ge s_{2\e_j}^{(k)}$ (since $w^{(d)}\ge s_{2\e_j}^{(d)},
k\le d\le n$). This implies that there is an entry, say $x$, in $\{a_1,\cdots
,a_k\}$ such that $x\ge j'$. In fact, this entry $x$ belongs to $\{a_1,\cdots
,a_j\}$; for, if $x\not\in \{a_1,\cdots
,a_j\}$, then this would imply that there is an entry  say $y$, in $\{a_1,\cdots
,a_j\}$ such that $y\ge k'$ (since $w^{(j)}\ge s_\b^{(j)}$), which in turn would
imply that $w^{(k)}\ge s_\b^{(k)}$ (since $x,y \in \{a_1,\cdots
,a_j\}$, and $x\ge j', y\ge k'$) and this is not true (cf. ($\dag$)). It follows
that $w\ge s_{2\e_j}$. This completes the proof of the Claim and also of Theorem
\ref{5.7}.
\end{pf}

\begin{prop}\label{5.8}
Let $ \b \in R^+$.
\begin{enumerate}
\item Let $\b=\e_j-\e_k, 1\le j<k \le n$. Then
$$ m(\o_d-\b)=\begin{cases} 0,& \text{ if } d<j, \text{ or } d \ge k \\
1, & \text{ if } j\le d<k.
\end{cases}$$

\item Let $\b=2\e_j, 1\le j \le n$. Then
$$m(\o_d-\b)=\begin{cases}0,& \text{ if }d<j\\
1, & \text{ if } j\le d\le n.
\end{cases}
$$
\item Let $\b=\e_j+\e_k, 1\le j<k \le n$. Then
$$m(\o_d-\b)=\begin{cases}0,& \text{ if }d<j \\
1, & \text{ if } j\le d<k\\
n+1-d, & \text{ if } k\le d\le n.
\end{cases}$$
\end{enumerate}
\end{prop}
\begin{pf}
Fix $d,1\le d\le n$.

\ni (1) Let $\b=\e_j-\e_k, 1\le j<k \le n$. If $d<j$ or $d\ge k$, then clearly
$m(\o_d-\b)=0$. Let then
$j\le d<k$. Then $q_{s_\b^{(d)}}$ is the only vector in ${\cal B}_d$ of weight
$\o_d-\b$.

\ni (2) Let $\b=2\e_j, 1\le j \le n$. If $d<j$, then clearly
$m(\o_d-\b)=0$. Let then
$j\le d\le n$. Then $q_{s_\b^{(d)}}$ is the only vector in ${\cal B}_d$ of weight
$\o_d-\b$.

\ni (3) Let $\b=\e_j+\e_k, 1\le j<k \le n$. If $d<j$, then clearly
$m(\o_d-\b)=0$. Let then
$j\le d\le n$. If $d<k$, then clearly $q_{s_\b^{(d)}}$ is the only vector in ${\cal B}_d$ of weight
$\o_d-\b$. If $k\le d \le n$, then using the description of admissible pairs (cf.
Proposition
\ref{5.4}) and the fact that $q_{\te,\d}$ is a weight vector of weight $\frac{1}
{2}(\te(\o_d)+\d(\o_d))$, it is easily checked that there are precisely $n+1-d$
vectors in ${\cal B}_d$ of weight $\o_d-\b$, namely

\ni $q_{\t,\phi},\ q_{\te_i,\d_i},\ 0\le i \le n-d-1$ where

\ni $\t=(1\cdots j-1\, j+1 \cdots d j'), \phi=(1\cdots k-1\, k+1
\cdots d k')$,

\ni $\te_0=(1\cdots j-1\, j+1 \cdots k-1\, k+1\cdots d
\,d+1\,k')$,

\ni $\d_0=(1\cdots j-1\, j+1
\cdots  d (d+1)')$,


\ni $\te_i=(12\cdots j-1\, j+1 \cdots k-1\, k+1 \cdots (d+i+1) (d+i)')$,

\ni $\d_i=(12\cdots j-1\, j+1\, \cdots k-1\, k+1 \cdots (d+i) (d+i+1)'),\ 1\le
i\le n-d-1$.

\end{pf}
\begin{cor}\label{5.9}
Let $w \in W$ and $\b \in R^+$.
\begin{enumerate}
\item Let $\b= \e_j -\e_k,\ j<k \le n,\ 2\e_{m},\ 1\le m\le n$. Then
$m_w(\o_d-\b)=m(\o_d-\b),
\text{ for all } 1\le d\le n$ if and only if $w\ge s_\b $.
\item Let $\b=\e_j+\e_k, 1\le j<k \le n$. Then
$m_w(\o_d-\b)=m(\o_d-\b),
\text{ for all } 1\le d\le n$ if and only if $w\ge s_\b $ or $s_{2\e_j}s_{\e_k-\e_n}\ (=(12\cdots j-1
j'j+1\cdots k-1 n k+1 k+2 \cdots n-1 k))$.
\end{enumerate}
\end{cor}
\begin{pf}


\ni (1) If $\b= \e_j -\e_k,\ j<k \le n,\ 2\e_{m},\ 1\le m\le n$, then the result
follows from (1), (2) in the proof of Proposition 4.7.

\ni (2) Let $\b=\e_j+\e_k, 1\le j<k \le n$. We have that
if
$d<j$, then
$m(\o_d-\b)=0$, and if
$j\le d<k$, then
$m(\o_d-\b)=1$ ($q_{id^{(d)}}$ being the only weight vector in $V(\o_d)$ of
weight
$\o_d-\b$). For $k\le d \le n$, we have from the
proof of (3) in Proposition
\ref{5.8} that the weight space in $V(\o_d)$ of weight
$\o_d-\b$ has a basis consisting of the vectors $q_{\t,\phi},\ q_{\te_i,\d_i},\ 0\le
i \le n-d-1$ (notations being as in the proof of (3) in Proposition
\ref{5.8}). Hence we obtain that $m_w(\o_d-\b)=m(\o_d-\b),
\text{ for all } 1\le d\le n$ if and only if $w^{(j)} \ge \{1,2, \cdots ,j-1,
k'\}  $ and $w^{(d)}\ge \t$ and
$\te_i,\ 0\le i \le n-d-1,\ k\le d\le n $. It is now easily checked
$w^{(d)}\ge q_{\t,\phi},\ q_{\te_i,\d_i},\ 0\le i \le n-d-1,\ k\le d\le n $, if and
only if $w^{(k)}\ge
\{1,2\cdots j-1 j'j+1\cdots k-1 n \}$. The required result now follows from this.
\end{pf}

\begin{rem} Let $w\in W$, and
$\b \in R^+$.

\ni  The condition that
$m_w(\o_d-\b)=m(\o_d-\b)$, for all $1\le d\le n$ need not be equivalent to the
condition that $\b \in N(w,e)$. For example, take $w=s_{2\e_j}$ for some $j<n-1$,
$\b =\e_j +\e_k$ for some $k,\ j<k\le n-1$. We have (cf. Theorem \ref{5.7}) $\b \in
N(w,e)$, but $m_w(\o_d-\b)\not= m(\o_d-\b),\ k\le d<n$ (note that $m_w(\o_d-\b)=1,
\ k\le d\le n$, while $m(\o_d-\b)=n+1-d,\ k\le d\le n$).

\ni Also, the condition that
$w\ge s_\b$ need not be equivalent to the condition that $m_w(\o_d-\b)=m(\o_d-\b)$,
for all $1\le d\le n$. For example, take $\b= \e_j +\e_k$, for some $j<k \le n$, and
$w= s_{2\e_j}s_{\e_k-\e_n}$. We have, $m_w(\o_d-\b)=m(\o_d-\b)$, for
all $1\le d\le n$ (cf. Corollary \ref{5.9}), but $w\not\ge s_\b$.

\ni Of course, for $\b= \e_j -\e_k,\ j<k \le n,\ 2\e_{m},\ 1\le m\le n$, and any $w\in
W$, all the three conditions of \S \ref{condns} are equivalent.
\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 $H=SL(V)$.

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). We shall index the simple roots in $G$ as in \cite{bou}. Let us denote the
simple reflections in $W_G$ by $\{ s_i,\  1\leq  i \leq n \}$. We have (cf.
\cite{bou}),
$$ s_i=\begin{cases}r_ir_{2n+1-i}, &\a_i= \varepsilon_i -\varepsilon_ {i+1},1 \leq i
\leq n-1,\\
r_nr_{n+1}r_n,&\a_i =\varepsilon_n
\end{cases}$$
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 $, $W_G^{d}$ can be identified with
$$\left\{(a_1 \cdots a_d) \left |
\eqalign{ (1)\  & 1\leq a_1 < a_2 < \cdots < a_d  \leq 2n+1,\ a_i\not=
n+1, 1\le i\le d
\hfill
\cr
         (2)\  & {\rm for  }\,\,\, 1 \leq i \leq 2n+1,\,\,\,{\rm if
}\,\,\,
         i \in \{a_1,..., a_d \} \hfill   \cr
         & {\rm then }\,\,\, 2n+2-i \notin \{ a_1,..., a_d \} \hfill
\cr }
 \right. \right\}. $$


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

\vs.2cm
(IV). For $w_1=(a_1 \cdots a_{n}),\  w_2=(b_1 \cdots b_{n}),\
w_1,w_2 \in   W_G,$ 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}). In particular,
the partial order on $W_G$ is induced by the partial order on $W_H$.

\subsection{Chevalley Basis}
For $1\le k\le 2n+1,\text{ set }
k'=2n+2-k$. The involution $\s:SL(2n+1)\rightarrow SL(2n+1), A\mapsto
E^{-1}(^tA)^{-1}E$, 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:
$$H_{\e_i-\e_{i+1}}=E_{ii}-E_{i+1,i+1}+E_{(i+1)',(i+1)'}-E_{i'i'},\
H_{\e_n}=2(E_{nn}-E_{n'n'}),$$
$$X_{\e_j-\e_k}=E_{jk}-E_{k'j'},\
X_{\e_j+\e_k}=E_{jk'}-E_{kj'},\
X_{\e_m}=2E_{mn+1}-E_{n+1m'},$$
$$X_{-(\e_j-\e_k)}=E_{kj}-E_{j'k'},\
X_{-(\e_j+\e_k)}=E_{k'j}-E_{j'k},\
X_{-\e_m}=2E_{n+1m}-E_{m'n+1}.$$


\begin{defn}
Let $\phi=(a_1\cdots a_d)\in W^d,\ 1\le d\le n-1$, and let $n\in \{a_1,\cdots ,
a_d\}$. Let
$\t=(b_1\cdots b_d)$ be the element of $ W^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}
\begin{rem}
{\it Type I} operation is defined exactly as in Definition \ref{4.2}
\end{rem}
\begin{prop}\label{6.4}(cf. \cite{l2})
Let $\t,\phi \in W^d,\ 1\le d\le n-1, \t \ge \phi$. 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 $V(\o_d)$}
For $d=n$, $V(\o_d)$ is the spin representation, and the extremal weight vectors,
$q_\t, \t \in W^d$ form a basis for $V(\o_d)$. For $1\le d< n$, we have
$V(\o_d)= \wedge^d V$ (here, $V=K^{2n+1}$).
The extremal weight vectors $\{q_\t.\t \in W^d\}$, say $\t=(a_1\cdots a_d)$ are
given by
$$q_\t= e_{a_1}\w \cdots \w e_{a_d}$$.

\begin{prop}\label{6.7}(cf. \cite{l2})
 Let $\b \in R^+$.
\begin{enumerate}
\item Let $\b=\e_j-\e_k, 1\le j<k \le n$. Then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j \text{ or } d\ge k \\
 \pm q_{s_\b^{(d)}}, & \text{ if } j\le d<k.
\end{cases}
$$
\item Let $\b=\e_j, 1\le j \le n$.
\begin{enumerate}
\item If $j=n$, then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<n \\
 \pm q_{s_{\b}^{(d)}}, & \text{ if } d= n.
\end{cases}
$$
\item If $j<n$, then

$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j \\
\pm q_{s_{\b}^{(d)}}, & \text{ if }  d=n \\
 \pm q_{s_{\e_j+\e_n}^{(d)},s_{\e_j-\e_n}^{(d)}}, & \text{ if } j\le d< n.
\end{cases}
$$
\end{enumerate}
\item Let $\b=\e_j+\e_k, 1\le j<k \le n$.
\begin{enumerate}
\item If $k=n$, then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j \\
\pm q_{s_{\b}^{(d)}}, & \text{ if } j\le d.
\end{cases}
$$
\item If $k<n$, then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j   \\
 \pm q_{s_\b^{(d)}}, & \text{ if } j\le d<k\text{ or }d=n\\
\pm (\sum_{i=0}^{n-d}\ c_iq_{\te_i,\d_i}+aq_{\t,\phi}), & \text{ if } k\le
d<n
\end{cases}
$$
\end{enumerate}
\end{enumerate}
\end{prop}

\ni where $c_i=\pm 2,\ i<n-d,\ c_{n-d}=\pm 1 = a$,

\ni $\t=(1\cdots j-1\, j+1 \cdots d j'), \phi=(1\cdots k-1\, k+1
\cdots d k')$,

\ni $\te_0=(1\cdots j-1\, j+1 \cdots k-1\, k+1\cdots d
\,d+1\,k')$,

\ni $\d_0=(1\cdots j-1\, j+1
\cdots  d (d+1)')$,


\ni $\te_i=(12\cdots j-1\, j+1 \cdots k-1\, k+1 \cdots (d+i+1) (d+i)')$,

\ni $\d_i=(12\cdots j-1\, j+1\, \cdots k-1\, k+1 \cdots (d+i) (d+i+1)'),\ 1\le
i<n-d$,

\ni $\te_{n-d}=(12\cdots j-1\, j+1 \cdots k-1\, k+1 \cdots n' (n-1)')$,

\ni $\d_{n-d}=(12\cdots j-1\, j+1 \cdots k-1\, k+1 \cdots n-1 n)$.

\begin{thm}\label{6.8}
 Let $w\in W$, and $\b \in R^+$.
\begin{enumerate}
\item   Let $\b = \e_j - \e_k, 1\le j<k\le n,\ \e_n$, or $\e_i+\e_n, 1\le i\le n $.
Then
$\b \in N(w,e)$ if and only if $ w \geq s_\b $.
\item Let $\b = \e_j,j<n $. Then $\b \in
N(w,e)$ if and only if $ w \geq $ either $ s_{\b}$ or
$s_{\e_j + \e_n}  $.
\item Let $\b = \e_j + \e_k, 1\le j<k<n $. Then $\b \in
N(w,e)$ if and only if $ w \geq $ either $ s_\b $
or $ s_{\e_j}s_{\e_k + \e_n} $.
\end{enumerate}
\end{thm}
\begin{pf}
If $\b = \e_j - \e_k, 1\le j<k\le n,\ \e_n$, or $\e_i+\e_n, 1\le i\le n $, then the
result follows from 1, 2(a), and 3(a) of Proposition \ref{6.7} (in view of Theorem
\ref{2.5}).

Let $\b = \e_j,j<n $. If $ w \geq $ either $ s_{\b}$ or
$s_{\e_j + \e_n}  $, then clearly, $X_{-\b} q_{id}\in V_w(\o_d),\  \text{ for all } 1\le d\le
n$, and hence $\b \in
N(w,e)$ (in view of Theorem \ref{2.5}). Let then $w $ be such that $X_{-\b} q_{id}\in
V_w(\o_d),\
\text{ for all } 1\le d\le n$, Further, let $w\not\ge s_\b$. We shall now show that
$w\ge s_{\e_j +
\e_n}  $. Let
$w=(a_1\cdots a_n)$. Now the fact that $w\not\ge s_\b$ implies that $w^{(j)}\not\ge
s_\b^{(j)}$. On the other hand, we have $w^{(j)}\ge s_{\e_j+\e_n}^{(j)}$. Hence we
obtain that there exists an entry, say $x$ in $\{a_1\cdots a_j\}$ such that $n'\le
x<j'$. Also, the fact that
$w^{(n)}\ge s_\b^{(n)}$ implies that there exists an entry, say $y$
in $\{a_1\cdots a_n\}$ such that $y\ge j'$. Denoting $\eta=s_{\e_j + \e_n}$, we obtain
$$w^{(j)}\ge (12\cdots j-1\,j+1\,n')\ (=\eta^{(j)}), $$
$$w^{(n)}\ge (12\cdots j-1\,j+1\cdots n-1\,n'\,j')\ (=\eta^{(n)}). $$
 From this it follows that $w\ge s_{\e_j + \e_n}  $.

Let $\b=\e_j+\e_k, \ 1\le j<k < n$. If $ w \geq $ either $ s_\b $
or $ s_{\e_j}s_{\e_k + \e_n} $, then clearly $X_{-\b} q_{id}\in V_w(\o_d),\text{ for all } d$,
and hence $\b \in N(w,e)$. Let now $\b \in N(w,e)$, and let $w\not\ge s_\b$. We shall
now show that $w\ge s_{\e_j}s_{\e_k + \e_n}  $. We have, $w^{(d)}\ge
s_{\b}^{(d)},j\le d<k$. This implies in particular that $w^{(j)}\ge s_{\b}^{(j)}$.
Hence there exists an entry $p$ in $\{a_1\cdots a_j\}$ such that
$p\ge k'$. Also, the facts that $w\not\ge s_\b,\ w^{(j)}\ge
s_{\b}^{(j)}$ imply that
\begin{equation*}
w^{(k)}\not\ge s_\b^{(k)}.\tag{\dag}
\end{equation*}
On the other hand, since $\b \in N(w,e)$, we have, $X_{-\b} q_{id}\in V_w(\o_d),\
1\le d\le n$ (cf. Theorem \ref{2.5}), and hence we obtain,
$w^{(k)}\ge (12\cdots j-1\,j+1\cdots j')\,(=\t)$ (cf. Proposition \ref{6.7}, 3(b)).
Hence there exists an entry
$q$ in
$\{a_1\cdots a_k\}$ such that
$q\ge j'$. We have in fact $q\in \{a_1\cdots a_j\}$, and $q=p$ (otherwise, we
would obtain $w^{(k)}\ge s_\b^{(k)}$ contradicting ($\dag$)).Further, we have,
there exists an entry $r$ in
$\{a_1\cdots a_k\}$ such that
$r\ge n'$ (in view of Proposition \ref{6.7}, 3(b)), and $r<k'$ (since $w^{(k)}\not\ge
s_\b^{(k)}$). Also, the fact that $w^{(n)}\ge s_{\b}^{(n)}$ implies that
there exists an entry $s$ in
$\{a_1\cdots a_n\}$ such that
$s\ge k'$. Thus we obtain, denoting $\xi=s_{\e_j}s_{\e_k + \e_n}$,
$$w^{(j)}\ge (12\cdots j-1\,j+1\,j')\ (=\xi^{(j)}), $$
$$w^{(k)}\ge (12\cdots j-1\,j+1\cdots k\,n'\,j')\ (=\xi^{(k)}), $$
$$w^{(n)}\ge (12\cdots j-1\,j+1\cdots k-1\,k+1\cdots n-1\,n'\,k'\,j')\ (=\xi^{(n)}).
$$
 From this it follows that $w\ge s_{\e_j}s_{\e_k + \e_n}  $.
\end{pf}

\begin{prop}\label{6.9}
 Let $ \b \in R^+$. Let $\b=\e_j+\e_k, 1\le j<k < n$, and $k\le
d<n$. Then
$m(\o_d-\b)=n-d+2$. In all other cases, we have, $m(\o_d-\b)=0 \text{ or }1$
\end{prop}
\begin{pf}
Fix $d,1\le d\le n$.

\ni (1)Let $\b=\e_j-\e_k, 1\le j<k \le n$. If $d<j$ or $d\ge k$, then clearly
$m(\o_d-\b)=0$. Let then
$j\le d<k$. Then $q_{s_\b^{(d)}}$ is the only vector in ${\cal B}_d$ of weight
$\o_d-\b$.

\ni (2) Let $\b=\e_j, 1\le j \le n$. If $d<j$, then clearly
$m(\o_d-\b)=0$. Let then
$j\le d\le n$. If $d=n$, then $q_{s_\b^{(d)}}$ is the only vector in ${\cal B}_d$ of weight
$\o_d-\b$. If $j\le d<n$, then using the description of admissible pairs (cf.
Proposition \ref{6.4}) and the fact that $q_{\te,\d}$ is a weight vector of weight
$\frac{1} {2}(\te(\o_d)+\d(\o_d))$, it
is easily seen that $q_{s_{\e_j+\e_n}^{(d)},s_{\e_j-\e_n}^{(d)}}$ is the only vector
in
${\cal B}_d$ of weight $\o_d-\b$.

\ni (3) Let $\b=\e_j+\e_k, 1\le j<k \le n$. If $d<j$, then clearly
$m(\o_d-\b)=0$. Let then
$j\le d\le n$. If $d=n$, or $k=n$, or $j\le d<k$, then $q_{s_\b^{(d)}}$ is the only
vector in
${\cal B}_d$ of weight
$\o_d-\b$. Let then $k\le d<n$. Then it is easily checked as in (2) (by weight
considerations) that
$ q_{\te_i,\d_i}, \ 0\le i\le n-d,\text{ and }\ q_{\t,\phi}$ (as in Proposition
\ref{6.7}, 3(b)) are the only vectors in
${\cal B}_d$ of weight
$\o_d-\b$.

\ni The required result follows from (1), (2) and (3).
\end{pf}
\begin{thm}\label{6.10}
Let $w\in W$, and $ \b \in R^+$. Then $\b \in N(w,e)$ if and only if
$m_w(\o_d-\b)=m(\o_d-\b),\text{ for all }    1\le d\le n$.
\end{thm}
 \begin{pf}
The required result follows from Proposition 5.6, Theorem \ref{6.8}, and  (1), (2) and
(3) in the proof of Proposition \ref{6.9}.
\end{pf}

\begin{rem}
Let $\b \in R^+,\ w\in W$.

\ni If $\b= \e_j -\e_k,\ j<k \le n,\ \e_{n},\ \text{or
}\e_m +\e_n,\ 1\le m\le n-1$, then the three conditions in \S \ref{condns} are
equivalent.

\ni For all $\b \in R^+,\ w\in W$, the condition that $\b \in N(w,e)$ is equivalent
to the condition $m_w(\o_d-\b)=m(\o_d-\b),\text{ for all }    1\le d\le n$ (cf.
Theorem \ref{6.10}).

\ni The condition that $w\ge s_\b$ need not be equivalent to the
condition that $\b \in N(w,e)$. For example, take $\b=\e_j$ for some $j<n$ and
$w=s_{\e_j +\e_n}$; we have (cf. Theorem \ref{6.8}, (2)), $\b \in N(w,e)$, but
$w\not\ge s_\b$.

\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 $H=SL(V)$.

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

\vs.2cm
(I). Denoting by $W_G$ the Weyl group of $G$, we have
$$W_G=\left\{(a_1 \cdots a_{2n})\in S_{2n} \left |
\eqalign{ (1)\  & a_{i}=2n+1- a_{2n+1-i},1\le i\le   2n \hfill
\cr
         (2)\  & \#\{i, 1\le i\le n\}\ {\rm is\ even} \hfill \cr }
 \right. \right\}. $$
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). We shall index the simple roots in $G$ as in \cite{bou}. Let us denote the
simple reflections in $W_G$ by $\{ s_i,\  1\leq  i \leq n \}$. We have (cf.
\cite{bou}),
$$ s_i=\begin{cases}r_ir_{2n-i}, &\a_i=\varepsilon_i -\varepsilon_ {i+1},1 \leq i
\leq n-1, \\
r_nr_{n-1}r_{n+1}r_n,& \a_i=\varepsilon_{n-1}+\varepsilon_n .
\end{cases}$$
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,\ d\not= n-1 $, $W_G^{d}$ can be identified with
$$\left\{(a_1 \cdots a_d) \left |
\eqalign{ (1) & 1\leq a_1 < a_2 < \cdots < a_d  \leq 2n \hfill \cr
         (2) & {\rm for  }\,\,\, 1 \leq i \leq 2n,\,\,\,{\rm if }\,\,\,
         i \in \{a_1,..., a_d \} \hfill   \cr
         & {\rm then }\,\,\, 2n+1-i \notin \{ a_1,..., a_d \} \hfill \cr }
 \right. \right\}\leqno{(*)}. $$
For $d= n-1,\ W_G^{d}$
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
$w_1=(a_1
\cdots a_{n}),\  w_2=(b_1
\cdots b_{n}),\  w_1,w_2 \in   W_G,$ 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 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 occsion 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:
$$H_{\e_i-\e_{i+1}}=E_{ii}-E_{i+1,i+1}+E_{(i+1)',(i+1)'}-E_{i'i'},$$
$$H_{\e_{n-1}+\e_{n}}=E_{n-1,n-1}+E_{n,n}-E_{n',n'}-E_{(n-1)',(n-1)'},$$
$$X_{\e_j-\e_k}=E_{jk}-E_{k'j'},\
X_{\e_j+\e_k}=E_{jk'}-E_{kj'},$$
$$X_{-(\e_j-\e_k)}=E_{kj}-E_{j'k'},\
X_{-(\e_j+\e_k)}=E_{k'j}-E_{j'k}.$$

\begin{defn}
Let $\phi=(a_1\cdots a_d)\in W^d,\ 1\le d\le n-2$. Further let $n-1,n\in \{a_1,\cdots
, a_d\}$. Let
$\t=(b_1\cdots b_d)$ be the element of $ W^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}
\begin{rem}
{\it Type I} operation is defined exactly as in Definition \ref{4.2}.
\end{rem}
\begin{prop}(cf. \cite{l2})
Let $\t,\phi \in W^d,\ 1\le d\le n-2,\  \t \ge \phi$. 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 $V(\o_d)$}
For $d=n-1,n$, $V(\o_d)$ is the spin representation, and the extremal weight
vectors,
$q_\t, \t \in W^d$ form a basis for $V(\o_d)$. For $1\le d\le n-2$, we have
$V(\o_d)= \wedge^d V$ (here, $V=K^{2n}$), and
the extremal weight vectors $\{q_\t.\t \in W^d\}$, say $\t=(a_1\cdots a_d)$ are
given by
$$q_\t= e_{a_1}\w \cdots \w e_{a_d}$$.

\begin{prop}\label{7.8}(cf. \cite{l2})
 Let $\b \in R^+$.
\begin{enumerate}
\item Let $\b=\e_j-\e_k, 1\le j<k \le n$. Then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j \text{ or } d\ge k \\
 \pm q_{s_\b^{(d)}}, & \text{ if } j\le d<k.
\end{cases}
$$

\item Let $\b=\e_j+\e_k, 1\le j<k \le n$.
\begin{enumerate}
\item If $k=n-1,n$, then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j \\
 \pm q_{s_{\b}^{(d)}}, & \text{ if } j\le d.
\end{cases}
$$
\item If $k<n-1$, then
$$X_{-\b}q_{id}=
\begin{cases}0,&\text{ if } d<j   \\
 \pm q_{s_\b^{(d)}}, & \text{ if } j\le d<k\text{ or }d=n-1,n\\
\pm (\sum_{i=0}^{n-d}\ c_iq_{\te_i,\d_i}+aq_{\t,\phi}), & \text{ if } k\le
d<n-1
\end{cases}
$$
\end{enumerate}
\end{enumerate}
\end{prop}

\ni where $\t, \phi, \te_i,\d_i, 0\le i\le n-d $ are defined in the same way as in
Proposition \ref{6.7}, and $c_i=\pm 2$ or $\pm 1$ according as $i<\text{ or } \ge
n-d-1$, and $a=1$.

\begin{thm}\label{7.9}
 Let $w\in W$, and $\b \in R^+$.
\begin{enumerate}
\item   Let $\b = \e_j - \e_k, 1\le j<k\le n$, or $\e_j+\e_k,\, k=n-1,n,\ 1\le j<k
$. Then
$\b \in N(w,e)$ if and only if $ w \succeq s_\b $.
\item Let $\b = \e_j + \e_k, \ j<k<n-1 $. Then $\b \in
N(w,e)$ if and only if $ w \succeq $ either $ s_\b $
or $ s_{\e_j - \e_{n}}s_{\e_j + \e_{n}}s_{\e_k + \e_{n-1}} $.
\end{enumerate}
\end{thm}
\begin{pf}
If $\b = \e_j - \e_k,\, 1\le j<k\le n$, or $\e_j+\e_k,\, k=n-1,n,\ 1\le j<k$, then
the result follows from Proposition \ref{7.8}, 1 and 2(a).

Let $\b=\e_j+\e_k,\, 1\le j<k \le n-2$. If $ w \succeq $ either $ s_\b $
or $ s_{\e_j - \e_{n}}s_{\e_j + \e_{n}}s_{\e_k + \e_{n-1}} $, then clearly
$X_{-\b} q_{id}\in V_w(\o_d),\text{ for all } 1\le d\le n$, and hence $\b \in N(w,e)$
(cf. Theorem \ref{2.5}). Let now
$\b
\in N(w,e)$, and let $w\not\succeq s_\b$. We shall now show that $ w\succeq s_{\e_j -
\e_{n}}s_{\e_j + \e_{n}}s_{\e_k + \e_{n-1}} $. We have, $w^{(d)}\succeq
s_{\b}^{(d)},j\le d<k$. This implies in particular that $w^{(j)}\succeq s_{\b}^{(j)}$.
Hence there exists an entry $p$ in $\{a_1\cdots a_j\}$ such that
$p\ge k'$. Also, the facts that $w\not\succeq s_\b,\ w^{(j)}\succeq
s_{\b}^{(j)}$ imply that
\begin{equation*}
w^{(k)}\not\succeq s_\b^{(k)}.\tag{\dag}
\end{equation*}
On the other hand, since $\b \in N(w,e)$, we have $X_{-\b} q_{id}\in V_w(\o_d), \
1\le d\le n$ (cf. Theorem \ref{2.5}) and hence we obtain
$w^{(k)}\succeq \ \t,\te $, and $\d $, where $\t=(12\cdots j-1\,j+1\cdots k\,j'),\
\te=(12\cdots j-1\,j+1 \cdots k-1\,n\,k'),\ \d=(12\cdots j-1\,j+1\cdots
k-1\,n'\,(n-1)') $ (cf. Proposition
\ref{7.8}, 2(b)). In particular, we obtain (since $w^{(k)}\succeq \t$)
that there exists an entry
$q$ in
$\{a_1\cdots a_k\}$ such that
$q\ge j'$. We have in fact $q\in \{a_1\cdots a_j\}$, and $q=p$ (otherwise, we
would obtain $w^{(k)}\succeq s_\b^{(k)}$ contradicting ($\dag$)). Thus
\begin{equation}\label{ONE}
w^{(j)}\succeq (12\cdots j-1\,j+1\,j').
\end{equation}
 Further we
obtain  (since $w^{(k)}\succeq \te,\d$)
\begin{equation}\label{TWO}
w^{(k)}\succeq (12\cdots j-1\,j+1\cdots k-1\, (n-1)'\,j')
\end{equation}
(note that $(12\cdots j-1\,j+1\cdots k-1\,(n-1)'\,j')$ is the smallest (under
$\succeq$) element $\phi$ in $W^k$ for the property that $\phi \succeq \t,\te $, and
$\d $; note that eventhough the $k$-tuple $(12\cdots j-1\,j+1\cdots k-1 n'\,j') \ge
\t,\te $, and $\d $, it is $\not\succeq \te$ (since they have non-compatible
analugous parts (cf. (IV) above))). Also, we have
\begin{equation}\label{THREE}
w^{(d)}\succeq s_\b ^{(d)},\ d=n-1,n.
\end{equation}
Now it is easily seen that $\xi=(12\cdots j-1\,j'j+1\cdots k-1\,
(n-1)'\,k+1\cdots n-2\,k'\,n')$ is the smallest (under
$\succeq$)) element in $W$ for the properties given in (\ref{ONE}), (\ref{TWO}), and
(\ref{THREE}) (note that eventhough $\eta:=(12\cdots j-1\,j'j+1\cdots k-1\,
(n-1)'\,k+1\cdots n-2\,n'\,k') $ is the smallest (under
$\geq$) element in $W$ such that $\eta^{(j)}\ge (12\cdots j-1\,j+1\,j'),\
\eta^{(k)}\ge (12\cdots j-1\,j+1\cdots k-1\, (n-1)'\,j'),\ \eta^{(n-1)}\ge (12\cdots j-1\,j+1\cdots k-1\,k+1\cdots \,n-1\,n'\,k')\
(=(s_\b u_j)^{(n-1)}\ (\text{ cf. Remark \ref{7.1}})),\  \eta^{(n)} \ge
s_\b^{(n)}$, we have,
$  \eta^{(n-1)}\not\succeq
s_\b^{(n-1)}$ (since $\eta^{(n-1)}$ and $(s_\b u_j)^{(n-1)}$ have non-compatible
analugous parts)). From this it follows that $ w\succeq s_{\e_j -
\e_{n}}s_{\e_j + \e_{n}}s_{\e_k + \e_{n-1}} $ (note that $s_{\e_j -
\e_{n}}s_{\e_j + \e_{n}}s_{\e_k + \e_{n-1}}=\xi $).

\end{pf}

\begin{prop}\label{7.10}
 Let $ \b \in R^+$. Let $\b=\e_j+\e_k,\, 1\le j<k \le n$, and
$k\le d\le n-2$. Then
$m(\o_d-\b)=n-d+2$. In all other cases, we have, $m(\o_d-\b)=0 \text{ or }1$.
\end{prop}
\begin{pf}
Fix $d,1\le d\le n$.

\ni (1) Let $\b=\e_j-\e_k, 1\le j<k \le n$. If $d<j$ or $d\ge k$, then clearly
$m(\o_d-\b)=0$. Let then
$j\le d<k$. Then $q_{s_\b^{(d)}}$ is the only vector in ${\cal B}_d$ of weight
$\o_d-\b$.

\ni (2) Let $\b=\e_j+\e_k, 1\le j<k \le n$. If $d<j$, then clearly
$m(\o_d-\b)=0$. Let then
$j\le d\le n$. If $d=n-1,n$, or $k=n-1,n$, or $j\le d<k$, then $q_{s_\b^{(d)}}$ is the
only vector in
${\cal B}_d$ of weight
$\o_d-\b$. Let then $k\le d\le n-2$. Then it is easily checked (by weight
considerations) that
$ q_{\te_i,\d_i}, \ 0\le i\le n-d,\ q_{\t,\phi}$ (as in Proposition \ref{7.8}, 2(b))
are the only vectors in
${\cal B}_d$ of weight
$\o_d-\b$.

\ni The required result follows from (1) and (2).
\end{pf}
\begin{thm}\label{7.11}
 Let $w\in W$, and $ \b \in R^+$. Then $\b \in N(w,e)$ if and
only if
$m_w(\o_d-\b)=m(\o_d-\b),  \text{ for all } 1\le d\le n$.
\end{thm}
 \begin{pf}
The required result follows from Proposition 6.7, Theorem \ref{7.9} and  (1), (2) in
the proof of Proposition \ref{7.10}.
\end{pf}
\begin{rem}
Let $\b \in R^+,\ w\in W$.

\ni If $\b= \e_j -\e_k,\ j<k \le n,\ \ \text{or
}\e_j +\e_k,\ k= n-1,n$, then the three conditions in \S \ref{condns} are
equivalent.

\ni For all $\b \in R^+,\ w\in W$, the condition that $\b \in N(w,e)$ is equivalent
to the condition $m_w(\o_d-\b)=m(\o_d-\b),\text{ for all }    1\le d\le n$ (cf.
Theorem \ref{7.11}).

\ni The condition that $w\ge s_\b$ need not be equivalent to the
condition that $\b \in N(w,e)$. For example, take $\b=\e_j+\e_k,\ j<k<n-1$, and
$w=s_{\e_j - \e_{n}}s_{\e_j + \e_{n}}s_{\e_k + \e_{n-1}}$; we have (cf. Theorem
\ref{7.9}, (2)), $\b \in N(w,e)$, but
$w\not\ge s_\b$.
\end{rem}


\begin{rem}
The statement (in Theorem 1 in \cite{l4}) that $\b \in N(w,e)$ if and only if $m_w(\rho -
\b)=m(\rho - \b)$ is incorrect. Although, there is a similarity in the statement
of Theorem 1 in \cite{l4} and those in Theorems
\ref{4.5}, \ref{6.10} and \ref{7.11}, the property that
$m_w(\rho -
\b)=m(\rho - \b)$ seems to be stronger than the property that $\b \in N(w,e)$. For
example, consider
$G=Sp(6),\ w=s_{2\e_1}$. Then for
$\b= \e_1 +\e_2$, we have, $X_{-\b} q_{id}\in V_{w,\rho} $, but $m_w(\rho -
\b)\not=m(\rho - \b)$ (we have, $m(\rho - \b)=6$, $m_w(\rho - \b)=5$. )
\end{rem}

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\end{thebibliography}


\end{document}


For $\t \in W$, let $\t^{(d)}$ denote the projection of
$\t$ under $W\rightarrow W/W_d$.