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\centerline{\title A Criterion for smoothness of Schubert 
varieties in 
${\bf Sp(2n)/B}$}
\vskip 0.2cm
\centerline{V. LAKSHMIBAI
\footnote{*}{\sl  Partially supported by the NSF Grant DMS 9502942
.} and
M. SONG}
\centerline{\sevenrm Mathematics Department, Northeastern University}
\centerline{\sevenrm Boston, MA 02115, USA}
\vskip 0.3cm
\midinsert
\narrower
{\smrm
\baselineskip=10pt
\noindent
{\sc Abstract.} 
\medskip
In this paper we give a criterion for a Schubert variety $X(w)$  in 
$Sp(2n)/B$ to be smooth. The criterion is given in terms of the
permutation giving $w$.
}
\endinsert

\ms \ni {\bf Introduction.} 

Let $k$ be the base field, which we assume to be algebraically
closed of arbitrary characteristic. Let
$V$ be a
$2n$-dimentional
$k$-vector space 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

$$E=\pmatrix{  
           0  &  J  \cr 
           -J &  0  \cr},\hskip 1.5cm
  J=\pmatrix{
           0        & \cdots  &    1     \cr 
           \vdots   & \adots  &  \vdots  \cr
           1        & \cdots  &   0      \cr }_{n \times n}$$  
we may realize $G=Sp(V)$ 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 $\sigma(A)=E({}^t\!\!A)^{-1}E^{-1}$. Let $T_H$ the
maximal torus in $H$ consisting   of diagonal matrices, and $B$ be
the Borel subgroup of $H$ consisting of upper  triangular matrices.
Then $T_G=T_H^\sigma$ is a maximal torus in $G$, and $B_G=B^\sigma$
is a Borel subgroup of $G$. Let $R_H$ (resp. $R_G$) be the
root system of $H$ (resp. $G$) with respect to $T_H$ (resp.
$T_G$), and $R^+_H$ (resp. $R^+_G$) the
system of positive roots of $R_H$ (resp. $R_G$) with respect to
$B_H$ (resp. $B_G$). Then $\sigma$ leaves $R_H$ and $R_H^+$ stable,  
and $R_G$ (resp. $R_G^+$) 
can be identified with $R_H$ (resp. $R_H^+$) modulo the action of 
$\sigma$. Let
$N(T_H)$ (resp.
$N(T_G)$) be the normalizer   of
$T_H$ (resp. $T_G$) in $H$ (resp. $G$). Let $W_H$ (resp. $W_G$)
denote the Weyl group $N(T_H)/T_H$ of $H$ (resp. $N(T_G)/T_G$
of $G$). Then $W_H$ is $\sigma$-stable, and $W_G=W^\sigma_H$, and
as a subgroup of $W_H (=S_{2n})$, $W_G$ gets identified as 
$$W_G=\{(a_1...a_{2n}) \in S_{2n} \mid a_i=2n+1-a_{2n+1-i},\  1 \leq
i \leq 2n \}.$$
(See \$1 for details.)
For $\a \in W_H$ (resp. $W_G$), we shall denote the
refflection $\in W_H$ (resp. $W_G$) with respect to $\a $ by $r_\a$
(resp. $s_\a$).

For  $w\in W_H\ ({\rm resp.}\  W_G)$, let $e_w$ (resp. $f_w$) denote
the point $wB$ (resp.$wB^\sigma$ ) in $H/B$ (resp.$G/B^\sigma$ );
let $X(w)=\overline{BwB} ({\rm mod}\ B)$ (resp.$Y(w)=\overline
{B^\sigma w B^\sigma}({\rm mod}\ B^\sigma)$) be the associated
Schubert variety in $H/B$ (resp.$G/B^\sigma$). 
For  $w\in W_H\ ({\rm resp.}\  W_G)$, let $l_H(w)$ (resp. $l_G(w)$)
denote the length of $w$ in $W_H\ ({\rm resp.}\  W_G)$, and let
$T(w)$ (resp. $S(w)$) denote the Zariski tangent space to
$X(w)$ ( resp. $Y(w)$) at $e_{id}$ ( resp. $f_{id}$). Let

$$N_H(w)=\{ \a \in R^+_H\ |\ w\geq r_\a\}.$$
Then we have (cf. [L-S.2]), $T(w)$ is spanned by
$\{X_{-\a},\ \a \in N_H(w)\}$ (here, for $\a \in R_H,\ X_\a$ denotes
the element of the Chevalley basis of Lie($H$), associated to $\a
$). In particular we have,
$X(w)$ is smooth if and only if $\# N_H(w)=l_H(w)$. Using this, we
have the following  (cf. [L-Sa] ):

\ms \ni {\bf Criterion for smoothness for
Schubert varieties in $SL(n)/B$.} Let $w \in S_n$, say 
$w=(a_1,...,a_n)$. Then $X(w)$ is singular if and only  
if the following property holds 
$$(*) \quad \left\{ \eqalign{
             & {\rm there\,\,\, exist}\,\,\,i,j,k,l,\   1\le i < j <
k < 
                l\le n \,\,\,{\rm such\,\,\, that} \cr
             & {\rm either \,\,\, (1)}\,\,\, a_k < a_l< a_i < a_j 
                {\rm\,\,\,or \,\,\,(2)}\,\,\, a_l < a_j < a_k < a_i. \cr }
              \right. $$ 

\ni In this paper we give a similar criterion
 for smoothness of Schubert varieties in $Sp(2n)/B$. The main
results of this paper are the two Theorems described below.

\ms \ni Let $w=(a_1 \cdots a_{2n}) \in W_G$ ( where $G=Sp(2n)$ ). Let
$$\eqalign {  
          m_w&=\#\{i,\   1\leq i \leq n \mid \ a_i >n\},  \cr  
          c_w&=\#\{i,\  1\leq i\leq n \mid\  w\geq
s_{2\varepsilon_i}\},    \cr  
    \delta _w&={\rm dim}T(w) - \ell_H(w).     \cr  
    } $$  
\medskip
\proclaim Theorem 1. Let $w\in W_G$. Then $Y(w)$ is
singulsr    if and only if the following property holds   
$$ (**)\quad \left \{ \eqalign {
   {\rm either}\,\,\,&(1).\,\,\, m(w)<c(w),   \cr 
   {\rm or}\,\,\,\,\,\, &(2).\,\,\, m(w)=c(w), \,\,\,{\rm and \,\,\,property
               \,\,\,(*)\,\,\, holds\,\,\, for} \,\,\, w, \cr  
   {\rm or}\,\,\,\,\,\, &(3).\,\,\, m(w)>c(w),\,\,\, {\rm and }\,\,\, \delta(w)
                         >m(w) - c(w).   \cr  }   \right.  $$

\medskip \proclaim Theorem 2. Let $w\in W_G$, and let
$X(w)$ be smooth. Then so is $Y(w)$.

\ni The above two theorems are proved by a comparative study
of $T(w)$, and $S(w)$ using the results of [L.1], [L.3]. Using
[L.1], we obtain that $\s$ induces an involution ( which is also
deoted by just $\s$) on $T(w)$, and $S(w)$ is simply the orbit
space of $T(w)$ modulo the action of $\s$. Let 
$$N_G(w)=\{\a \in R_G^+\ |\ X_{-\a} \in S(w)\}.$$
Then using the
explicit description of $S(w)$ as given in [L.3], we obtain
(cf. Proposition 4.5)
$$n_G(w)={{1}\over {2}}(n_H(w)+c_w) \leqno{(\dagger)}$$
where $n_G(w)=\# N_G(w),\ n_H(w)=\# N_H(w)$. This then
enables us to prove Theorems 1\& 2 above.

\ni (Note the resemblance of the formula $(\dagger)$ to the
length formula $l_G(w)={{1}\over {2}}(l_H(w)+m_w)$ (cf.
[L-S.1].)       


\bigskip
\ni {\bf 1.  Preliminaries on Sp($2n$) }
\medskip
Let $G_{\bf Z}$ be a semi-simple, simply connected Chevalley group scheme 
over the ring of integers ${\bf Z}$. We fix a maximal torus subgroup scheme 
$T_{\bf Z}$ and a Borel subgroup scheme $B_{\bf Z}$ containing $T_{\bf Z}$. 
We talk of roots, weights etc. with respect to $T_{\bf Z}$ and $B_{\bf Z}$. 
The Weyl group scheme $N(T_{\bf Z})/T_{\bf Z}$ ($N(T_{\bf Z})$ =normalizer of 
$T_{\bf Z}$) is a constant group scheme and hence we talk of Weyl group $W$ 
of $G_{\bf Z}$. Let $Q_{\bf Z}$ be a parabolic subgroup scheme of $G_{\bf Z}
$ containing $B_{\bf Z}$ and $W_Q$, the Weyl group of $Q_{\bf Z}$. For any 
element $\tau \in W/W_Q$, let $X_{\bf Z}(\tau)$ denote the Schubert subscheme 
associated to $\tau$, namely, the Zariski closure of $B_{\bf Z}\tau$ in 
$G_{\bf Z}/Q_{\bf Z}$ endowed with
the canonical reduced structure. For any field $k$, the base change of
$X_{\bf Z}(\tau)$ by 

\ni Spec$k$ $\longrightarrow$ Spec${\bf Z}$ coincides 
with the Schubert
variety $X_k(\tau)$, the Zariski closure of $B_k \tau$ in $G_k/Q_k$, endowed
with the canonical reduced structure  (here,
for any field $k$, we denote the objects obtained by the base change
 Spec$k$ $\longrightarrow$ Spec${\bf Z}$ with the suffix $k$).
\smallskip
\ni {\bf 1.1 The Chevalley-Bruhat Order.} 

\ni We have a canonical {\it
partial order} in
$W/W_Q$ defined as follows: for
 $\tau_1 , \tau_2$ in $W/W_Q$, we say that $\tau_1 \geq \tau_2$, if
 $X_k(\tau_1) \supseteq X_k(\tau_2)$. Recall that dim$X_k(\tau)
 =\ell (\tau)$, where $\ell$ denotes the length function on $W/W_Q$.



\ni {\bf 1.2 The set $W^Q$ of minimal representatives.}  

\ni Given a 
parabolic subgroup $Q$, the set $W^Q$ is given by

$$W^Q=\{ w \in W \mid \ell(ws_{\alpha})=\ell(w) + 1,\  \alpha \in S_Q
\}$$ ( here $S_Q$ denotes the set of simple roots $\a $ such that
$s_\a
\in W_Q$). For
$w
\in W^Q$, the Schubert variety $X(w)$ in $G/B$ maps birationally
onto its image under the canonical projection $\pi$ : $G/B
\longrightarrow G/Q.$ In particular,  we have
dim$X(w)$=dim$\pi(X(w)).$ In the sequel, if
$Q$ is the maximal parabolic sub group of $G$ corresponding to
``omitting $\a_j$", then we also denote $W^Q$ by $W^j$.  
\smallskip
\ni {\bf 1.3 The Special Linear Group} $SL(n)$. 

\ni Let $H=SL(n).$  It
is well known  that $W=S_n$. Let $Q$ be the maximal parabolic
subgroup
$P_d$  given by

$$P_d =\Biggl \lbrace {A \in SL(n) \Biggl \vert 
A= \pmatrix {     * & * \cr  
0_{n-d \times d}   &  *  \cr }
       } \Biggl \rbrace.$$
We have

$$W^{P_d}=\{ (i)=( i_1,...,i_d) \mid 1 \leq i_1 < i_2 <...< i_d 
\leq n \}.$$

\smallskip
\ni {\bf 1.4 The Chevalley-Bruhat Order Patial Order} in $W(SL(n)).$ 


\ni For
$w_1 = ( a_1 ... a_n)$ and 
$w_2 = ( b_1... b_n),$  $w_1, w_2 \in S_n,$ it can be easily seen that
 $$w_2 \geq w_1 \Leftrightarrow \   
      \{b_1,\cdots,b_d\}\uparrow \ \geq \  \{a_1,\cdots,a_d\}
     \uparrow, \   \forall \   1 \leq d\leq n-1 ,$$  
 (here $\{a_1,\cdots,a_d\}\!\!\uparrow,\ 
\{b_1,\cdots,b_d\}\!\!\uparrow $  are the 
corresponding $d$-tuples arranged in ascending order; also, for two
$d$-tuples $(x_1,...,x_d),\  (y_1,...,y_d)$ where $1
\leq x_1 < x_2 <...< x_d \leq n$ and $1 \leq y_1 < y_2 <...< y_d
\leq n$, by
$(x_1,...,x_d) \geq (y_1,...,y_d)$  we mean $x_t \geq y_t, 1 \leq t \leq d).$
Note that the above condition is equivalent to the condition that
$\pi_d(X(w_2)) \supseteq \pi_d(X(w_1)), 

\ni 1 \leq d \leq n-1,\pi_d$
being the canonical projection $G/B \longrightarrow G/P_d, 1 \leq d
\leq n-1.$

\smallskip
\ni {\bf 1.5. The Symplectic Group} $Sp(2n)$. 

\ni Let $V$ be a
$2n$-dimentional
$k$-vector space 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

$$E=\pmatrix{  
           0  &  J  \cr 
           -J &  0  \cr},\hskip 1.5cm
  J=\pmatrix{
           0        & \cdots  &    1     \cr 
           \vdots   & \adots  &  \vdots  \cr
           1        & \cdots  &   0      \cr }_{n \times n}$$  
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
$$\eqalign{
G=Sp(2n)&=\{A \in SL(2n) \mid {}^t\!\!AEA=E\} \cr
          &=\{A \in SL(2n) \mid E^{-1}({}^t\!\!A)^{-1}E=A \} \cr
          &=\{A \in SL(2n) \mid E({}^t\!\!A)^{-1}E^{-1}=A \} \cr
          &=H^{\sigma}.\cr }$$

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

$$T_G={T_H}^{\sigma} , \hskip 1.5cm  B_G={B_H}^{\sigma}.$$

Then it follows that $T_G$ is a maximal torus in $G$ and $B_G$ is a Borel
subgroup in $G$. Also, it can be seen easily that
$$Lie\,\, G=\{A \in M_{2n} \mid E\ ^{t}A\ E^{-1} + A =0 \}.$$

We note that the following hold (cf. [L-S.1], [L.1]):
\smallskip
(I). Denoting $N(T_H)$ (resp. $N(T_G)$) the normalizer of $T_H$ in $H$
(resp. $T_G$ in $G$), we have $N(T_H)$ is stable under $\sigma$ and
$$N(T_G)=(N(T_H))^{\sigma}$$
\smallskip
(II). The canonical map
$$N(T_G)/T_G \longrightarrow N(T_H)/T_H$$
is an inclusion, i.e., the Weyl group $W_G$ of $G$ can be identified
canonically as a subgroup of the Weyl group $W_H$ of $H$.
\smallskip
(III). The involution $\sigma$ induces an involution on $W_H$. If
$$w=(a_1...a_{2n}) \in W_H,$$
then
$$\sigma(w)=(c_1...c_{2n}), \quad c_i=2n+1-a_{2n+1-i}.$$
Further
$$W_G={W_H}^{\sigma}$$
and hence
$$W_G=\{(a_1...a_{2n}) \in S_{2n} \mid a_i=2n+1-a_{2n+1-i},\  1 \leq
i \leq 2n \}.$$
Thus $w=(a_1...a_{2n}) \in W_G$ is known once $(a_1...a_n)$ is known.
\smallskip
(IV). Denoting $R_H$ (resp. $R_H^+$) the set of roots of $H$ with respect 
to $T_H$ (resp. the set of positive roots with respect to $B_H$ ), we have 


$$R_H=\{ \varepsilon_i - \varepsilon_j \mid i \neq j , 1 \leq i,j
\leq 2n
\},$$   
$$R_H^+=\{ \varepsilon_i-\varepsilon_j \mid 1\leq i < j \leq 2n
\},$$   where $\{ \varepsilon_i ,\  1\leq i \leq 2n \}$ is the
canonical basis of
$X(T_H)(=   {\rm Hom}(T_H, G_m))$.

\ni Now $\sigma$ induces an involution on $X(T_H)$, namely,    
$$\sigma(\varepsilon_i)=-\varepsilon_{2n+1-i},\quad 1\leq i \leq
2n.$$ 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, $\sigma$ leaves $R_H$ and $R_H^+$ stable and $R_G$ (resp.
$R_G^+$)  can be identified with $R_H$ (resp. $R_H^+$) modulo the
action of $\sigma$. Thus we have, 

$$R_G=\{ \pm (\varepsilon_i \pm \varepsilon_j),\ 1\leq i<j\leq n,\ 
\pm 2\varepsilon_i,\ i=1,...,n \},$$
$$R_G^+ = \{  (\varepsilon_i \pm \varepsilon_j),\ 1\leq i<j\leq n,\ 
 2\varepsilon_i,\ i=1,...,n \}.$$

\smallskip
(V). 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,   
$$s_i=\hbox { reflection with respect to $\varepsilon_i -
\varepsilon_ {i+1}$, \hskip 0.5cm $1 \leq i \leq n-1,$ }$$
$$s_n=\hbox { reflection with respect to $2\varepsilon_n.$}$$
Then we have (cf. [B]),
$$ s_i=\cases{r_ir_{2n-i}, &$1 \leq i \leq n-1$, \cr
                 r_n,             & $i=n$ . \cr }$$ 
where $r_i$ denotes the transposition $(i,i+1),\  1 \leq i \leq
2n-1.$
\smallskip
(VI). For $w \in W_G$, denoting by $\ell_G(w)$ the length of $w$ in $W_G$,   
i.e., the length of a reduced decomposition of $w$ with respect to the  
simple reflections $\{ \theta_i \mid 1 \leq i \leq n \}$ in $W_G$, we have  
$$\ell_G(w)= {1 \over 2} ( \ell_H(w) + m(w) ),$$
where $m(w)= \# \{ i ,\  1 \leq i \leq n \ |\  a_i > n \},$ $w$
being the    permutation $w=(a_1 \cdots a_{2n})$ (here, $\ell_H(w)$
denotes the length   of $w$ in $W_H$. Note that since $W_G \subset
W_H$ and $w \in W_G,$ we have  
$w \in W_H$).  
\smallskip
(VII). For $w \in W_H$, we denote by $C(w,H/B_H)$ the Schubert cell in $H/B_H$  
defined by $w$, i.e., the subset $B_Hwe_{B_H}$ in $H/B_H$ ( here $e_{B_H}$  
denotes the point in $H/B_H$ corresponding to the coset $B_H$). If
$w \in  W_G$, we denote by $C(w, G/B_G)$ the Schubert cell in
$G/B_G$ defined by    
$w$. If $w \in W_G$,  then $C(w, H/B_H)$  is stable under $\sigma$ and we  
have   
$$C(w, G/B_G)=(C(w, H/B_H))^{\sigma}.$$  
\smallskip
(VIII). For $1 \leq d \leq n $, we let $P_d$ be the maximal parabolic subgroup  
of $G$ obtained by `` omitting the simple root $ \alpha_d$". Note that         
$$\alpha_d=\cases{\varepsilon_d- \varepsilon_{d+1} , &$1 \leq d \leq n-1$ \cr   
                  2\varepsilon_n ,                & $d=n$. \cr } $$ 
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  
$$\left\{(a_1 \cdots a_d) \left |      
\eqalign{ (1) & 1\leq a_1 < a_2 < \cdots < a_d  \leq 2n \hfill \cr 
         (2) & {\rm for  }\,\,\, 1 \leq i \leq 2n,\,\,\,{\rm if }\,\,\, 
         i \in \{a_1,..., a_d \} \hfill   \cr  
         & {\rm then }\,\,\, 2n+1-i \notin \{ a_1,..., a_d \} \hfill \cr } 
 \right. \right\}. $$
\smallskip
(IX). 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. [P]). Hence for $w \in W_G,$  denoting the
projection of $w$ under $ W_G 
\longrightarrow W_G/W_{P_d}$ by $w^{(d)}$ , we have for $w_1, w_2
\in W_G,$  
$$ w_2^{(d)} \geq w_1^{(d)},\  1 \leq d \leq n
\ \Longleftrightarrow \     
      \{b_1,\cdots,b_i\}\uparrow \geq \{a_1,\cdots,a_i\}
     \uparrow, \ \forall i, 1\leq i \leq 2n-1.        $$ 
(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  $w_2 \geq w_1 \Leftrightarrow \{b_1,
\cdots, b_d \} 
\geq \{a_1, \cdots, a_d\}.$
 
\bigskip 
(X). The involution $\sigma : SL(2n) \longrightarrow
SL(2n), \ A
\longmapsto   E({}^t\!\!A)^{-1}E^{-1},$ induces an involution 
$$ \sigma : sl(2n) \longrightarrow sl(2n),\  
A \longmapsto -E\ {}^t\!\!A\ E^{-1}=E\ {}^t\!\!A\ E,$$
(note that $E^{-1}=-E$). In particular, we have, for $1 \leq i, j
\leq 2n$  
$$\sigma(E_{ij})=\cases{
   -E_{j'i'} & if $i,j$ are both $\leq n$ or both $ >n$ \cr  
    E_{j'i'} & if one of $\{ i,j \}$ is $ \leq n$ and the other $ >n,$ \cr }
$$    
where $E_{i,j}$ is the elementary matrix with $1$ at $(i,j)$th place and $0$  
elewhere; and $i'=2n+1-i,\,\, j'=2n+1-j.$ Further
$$Lie\,\, G=\{ A \in sl(2n) \mid E\ {}^t\!\!A\ E=A \}.$$  
From the description of the positive roots and the simple roots of
$R_G$  given in (IV), (V) , it  
can be easily seen that the Chevalley basis $ \{ H_{\alpha}, \alpha $ simple, 
$ X_{\pm \beta}, \beta \in R_G^+ $\}$ for $ Lie\,\, G$ is given as 
follows:  
$$\eqalign{
H_{\varepsilon_i - \varepsilon_{i+1}} &=E_{ii}-E_{i+1,i+1}+E_{(i+1)',(i+1)'}-  
E_{i'i'}, \cr    H_{2\varepsilon_n} &=E_{nn}-E_{n'n'},   \cr  
     X_{\varepsilon_i - \varepsilon_j} &=E_{ij}-E_{j'i'},   \cr 
     X_{\varepsilon_i + \varepsilon_J} &=E_{ij'}+E_{ji'},   \cr  
                 X_{2\varepsilon_i} &=E_{ii'}  ,  \cr  
  X_{-(\varepsilon_i - \varepsilon_j)} &=E_{ji}-E_{i'j'} ,  \cr   
  X_{-(\varepsilon_i + \varepsilon_j)} &=E_{j'i}+E_{i'j} ,  \cr  
                X_{-2\varepsilon_i} &=E_{i'i} ,   \cr} $$  
( here, for any $l,\  1 \leq l \leq 2n,\  l'= 2n +1 -l $). 
\bigskip
\ni {\bf \S2. A Z-basis for $H^0(G_{\bf Z}/P_{\bf Z}, L_{\bf
Z})$ }                                 
\smallskip
In this section, we shall suppose that $G$ is classical. With
notation as in
\S1 , let
$P$ be a maximal parabolic subgroup in  
$G$, with associated fundamental weight $\omega$. Let $V_{\omega}$ be the 
irreducible $G$-module (over ${\bf Q}$) with highest weight
$\omega$. Let us fix  a highest weight vector $e$ in $V_{\omega}$.
Let $U$ denote the universal   enveloping algebra of $ Lie\,\, G$
and $U_{\bf Z}$ the canonical ${\bf Z}$-form  in $U$, i.e., the
${\bf Z}$ subalgebra of $U$ spanned by $X_{\alpha}^n/n!,\   
\alpha $  being a root of $G$. Let $U_{\bf Z}^+$ ( resp. $U_{\bf Z}^-)$  be  
the subalgebra of $U_{\bf Z}$ spanned by $X_{\alpha}^n/n!$ (resp.
$X_{-\alpha}^n/n! $), $ n \in {\bf N}, \alpha,$ a positive root.
Let  
$$V_{\bf Z}=U_{\bf Z}e.$$  
Given $\tau \in W,\  \tau$ can be represented by a {\bf Z}-valued
point of $ N(T_{\bf Z})$ and we denote this by the same $\tau$. We
set $e_\tau= \tau
\cdot e \in V_{\bf Z}$. Note that $e_\tau $ is well determined up to a unit in 
${\bf Z},$ i.e., up to $\pm 1$). We call $\{e_\tau \}$ the extremal weight 
vectors in $V_{\bf Z}.$ Set  
$$V_{\bf Z}(\tau)= U_{\bf Z}^+ e_\tau,\qquad  V(\tau)=V_{\bf Z}(\tau)
{\otimes}_{\bf Z} {\bf Q}.$$  


\smallskip \ni {\bf Definition 2.1} (cf. [L-M-S], [L-S.3]). A pair of
elements
$\lambda, \mu
\in W^P$ is said to be {\it admissible} if there exists a
chain $\lambda =\lambda_0 >
\lambda_1 > \cdots > \lambda_r =\mu,$  such that
\item{(1)}$X(\lambda_{i+1})$ is a divisor in $X(\lambda_i),\ 0 \leq i
\leq r-1$
\item{(2)} If $\lambda_i=
\lambda_{i+1}s_{\b_{i+1}},$ for some $\b_{i+1} \in R^+,\ 0 \leq i
\leq r-1$, then 
$(\omega, s_{\b_{i+1}}^*)=2.$

\ms \ni {\bf Definition 2.2} Let $\t \in W^P$. A pair $\l ,\mu $
as in Definition 2.1 is said to be {\it an admissible pair on
$X(\t)$}, if $\t \geq \l$. 

\smallskip
\proclaim Theorem 2.3. (cf. [L-M-S], [L-S.3]) Let $\tau \in W^P$.
Then there exsis a basis
$\{ Q(\lambda, \mu )\}$   of the ${\bf Z}$-module $V_{\bf Z}(\tau)$
indexed by admissible pairs $(\lambda, \mu)$ on $X(\tau)$ having
the following properties :  
\item{(1).}  $Q(\lambda, \mu)$ is a weight vector ( for the $T$
action ) of weight  $ {1 \over 2}(\lambda(\omega) + \mu (\omega))$.  
\item{(2).}  $\{ Q(\lambda,\mu), \  \tau \geq \theta \geq
\lambda\}$ is a ${\bf Z}$-basis for $V_{\bf Z}(\theta)$.

\noindent{}(if $\lambda=\mu,$ we denote $Q(\mu,\mu)$ by just $Q(\mu)$).  
\medskip
\noindent{}{\bf Remark 2.4.}(cf. [L-S.3]) Once a choice
of the  extremal weight  
vectors  $Q(\mu)$'s has been made, the construction of $ Q(\lambda,
\mu)$   is canonical; to be very precise, $Q(\lambda,\mu)$ is
uniquely determined by   the admissible pair $(\lambda, \mu)$ and is
given by  
$$ Q(\lambda,\mu)=X_{-\alpha_1} \cdots X_{-\alpha_r}Q(\mu),$$  
where $\lambda =\lambda_0 > \lambda_1 > \cdots > \lambda_r =\mu,$  $X(
\lambda_{i+1})$ is a divisor in $X(\lambda_i)$, $\lambda_i=s_{\alpha_{i+1}}
\lambda_{i+1},$  $0 \leq i \leq r-1,$  $\lambda= \lambda_0 > \lambda_1 > 
\cdots > \lambda_r = \mu $ being ${\underline {{\rm any}}}$ chain
from
$\lambda$ to
$\mu$. 
\smallskip
\ni {\bf 2.5. The elements $\{P(\lambda, \mu)\}$.} 

\ni Let $\{P(\lambda,
\mu)\}$  be the basis in 
$V_{\bf Z}^\vee$  dual to $\{Q(\lambda, \mu)\}$ ( $V_{\bf Z}^\vee =
H^0 (G_{\bf Z}/B_{\bf Z}, L_{\bf Z}), L_{\bf Z}$ being the ample
generator of   Pic$(G_{\bf Z}/P_{\bf Z})).$ Then we recall the
following theorem  (cf. [L-M-S], [L-S.3]).

\smallskip
\proclaim Theorem 2.6.  The set $\{P ( \lambda, \mu)\}$ is  
a basis of $H^0(G_{\bf Z}/P_{\bf Z}, L_{\bf Z})$ having the following  
properties : 
\item{(1).} $P(\lambda, \mu)$ is a weight vector ( for the $T_{\bf
Z}$-action) and   is of weight   
$$- {1\over2}(\lambda(\omega) + \mu(\omega)).$$  
\item{(2).} Let $k$ be any field. Set $p(\lambda, \mu)= P(\lambda,
\mu)\otimes 1,  
\,\,p(\lambda,\mu)$ being the canonical image in $H^0(G_{\bf Z}\otimes k/P_
{\bf Z}\otimes k, L_{\bf Z} \otimes k)$ of $P(\lambda,\mu)$. Then the  
restriction of $p(\lambda, \mu)$ to $X_{\bf Z}(\tau)\otimes k ( \tau \in
W^P)$ is not identically zero if and only if $\tau \geq \lambda.$ 
\item{(3).} For $\tau \in W^P,\  \{p(\lambda,\mu) \mid \tau \geq
\lambda \}$ is a basis  for $H^0(X(\tau), L)$. 

\smallskip
\ni {\bf \S3. Tangent spaces to Schubert varieties}

\ni Let $G$ be of classical type and let $n=$ rank $(G)$. We
recall below some results from [L-S.2], [L-S.3], and [L.1].  

\smallskip \ni {\bf 3.1. The ideal sheaf ${\cal I}(w)$  of $X(w)$ in
$G/B$}.

\ni For $w\in W,$ let $X(w^{(j)})$ denote the projection of $X(w)$
under $G/B \longrightarrow   G/P_j, 1 \leq j \leq n $ (here, $ P_j$
denotes the maximal parabolic    subgroup associated to the simple
root $\alpha_j, 1\leq j \leq n$). Then we have ( cf. [L-S.3]), 
the ideal sheaf ${\cal I}(w)$  of $X(w)$ in $G/B$   is generated
by   
$$I(w)= \left\{ p(\lambda, \mu) \left |
\eqalign{ (1) &(\lambda, \mu)\,\,\, {\rm is\,\,\,\, an\,\,\, admissible\,\,\, 
              pair} \cr
         { } & {\rm in}\,\,\, W^j,\,\,\,{\rm
for\,\,\,some\,\,\,}j,\,\, 
               1 \leq j \leq n            \cr  
       (2) & w^{(j)} \not \geq \lambda\,\,\, {\rm (\,in}\,\,\,
W^j\,) \cr } 
 \right. \right\}$$
$(p(\lambda,\mu)$ being as in Theorem 2.6). 

\smallskip \ni {\bf 3.2. the Jacobian matrix $J_{w,\tau} $}.

\ni Let $\tau, w \in W $ be such that $w \geq \tau $ so that the
point $e_\tau (=\t B\ {\rm   in}\   G/B) \in X(w)$. Denoting by
$U^-$ the opposite big cell in $G/B$, we have, $(\tau U^- \tau^{-1})
\tau (= U^-_\tau \tau)$ gives an affine neighbourhood  of $e_\tau$.
Let $J_{w,\tau}$ be the Jacobian matrix   
$$J_{w,\tau} = \parallel X_{-\alpha} p(\lambda, \mu) \parallel, \qquad 
   \alpha \in \tau(R^+), \qquad  p(\lambda, \mu) \in I(w)$$  

\smallskip \ni We have (cf. [L-S.2]),  
\proclaim Theorem 3.3. Let $w,\t $ be as above. Let   
$$R(w, \tau) = \left\{ \beta \in \tau(R^+) \left|   
\eqalign{ &\,\,{\rm there\,\,\, exists\,\,\, a}\,\,\, p(\lambda, \mu)\in 
            I(w)\,\,\, {\rm such }\,\,\,\hfill \cr  
         &\,\, {\rm that}\,\,\, X_{-\beta} p(\lambda, \mu)=cp(\tau),
c\in 
         k^* \hfill  \cr } \right.   \right\}.$$
Then $$\hbox { rank of
$J_{w,\tau}(e_\tau)=\# R(w,
\tau)$}.$$


\smallskip \ni {\bf 3.4. The sets $M(w,\t),\ N(w,\t) $}:  

\ni Let $w,\t \in W,\ w\ge \t $. It is easily seen that
$X_{-\beta}p(\lambda,\mu)=cp(\tau), c\in k^*$   if and only if
$X_{-\alpha}Q(\tau)$ when written as a ${\bf Z}$-linear 
combination  of the vectors  $Q(\theta,\delta)$  involves 
$Q(\lambda, \mu)$    with a coefficient that is nozero in
$k$. Let now $H=SL(2n), \ G=Sp(2n) $. Let $w, \tau \in W_G $ be such
that $w \geq \tau$. Let $e_\t$ (resp. $f_\t$) be the point $\t
B_H$ (resp. $\t B_G$) in $H/B_H$ (resp. $G/B_G$). Let $ X(w)$ (resp.
$Y(w)$) be the Schubert variety in $H/B_H$ (resp. $G/B_G$)
associated to $\ e_\tau
$ (resp. $f_\tau$). Let  
$$M(w,\tau)= \{ \varepsilon_{a_j} - \varepsilon_{a_k},\  1 \leq j < k
\leq 2n \mid   w \geq (a_j, a_k) \tau \,\,\,\hbox{in $S_{2n}$ } \}.$$

\ni Let $N(w,\tau)$ be the set of all $\beta \in
\tau(R_G^+)$  with  the property  that  for   all $(\lambda
,\mu)$ such that $Q(\lambda, \mu)$ occurs with nozero  coefficient
in the ${\bf Z}$-linear combination for $X_{-\beta}Q(\tau)$ (in   
terms of $Q(\theta, \delta)$' s), $ p(\lambda,
\mu)\mid_{Y(w)}\not\equiv 0 $.  

\ni  Let $T(w,\tau)$( resp. $S(w,\tau)$) be the Zariski tangent
space of $X(w)$  (resp.  $Y(w)$) at $e_\tau$ (resp. $f_\tau$). Then
we have (cf. [L-S.2],[L.1])   

\proclaim Theorem 3.5.
\item{(i)} $T(w, \tau)$ is spanned by  
$$\{ X_{-\beta},\  \beta \in M(w, \tau)\}. $$
In particular, $X(w)$ is smooth at 
$e_ \tau \Leftrightarrow \# M(w,\tau)=   \ell_H(w)$.  
\item{(ii)} $S(w, \tau)$ is spanned by  
$$\{ X_{-\beta},\  \beta \in N(w, \tau)\}.$$   In particular,
$Y(w)$ is smooth at $f_\tau \Leftrightarrow
\# N(w,\tau)=   \ell_G(w)$.  

\proclaim Proposition 3.6. (cf. [L.1])  With notations as above, let
$\beta=\tau(\alpha).$ 
\item{(1)}  Let $\alpha=\varepsilon_j - \varepsilon_k,\quad 1\leq j
< k \leq n $ or $\alpha   =2\varepsilon_j,\quad 1\leq j \leq n.$
Then  
$$\beta \in N(w,\tau) \Leftrightarrow w\geq s_\beta \tau.$$  
\item{(2)} Let $\alpha=\varepsilon_j + \varepsilon_k,\quad 1 \leq j
< k \leq n;$  let $s$   (resp. $r$) be the minimum (resp. maximum)
of $\{
\mid a_j \mid , \mid   a_k \mid \}.$ Then  
$$\beta \in N(w,\tau) \Leftrightarrow \left\{
\matrix {& w^{(j)} \geq  \{a_1,...,a_{j-1},{a_k}'\}, \hfill \cr 
         & w^{(k)} \geq  \{a_1,...,a_{k-1},r,s'\}(\hat a_j, \hat a_k).
           \hfill \cr } \right. $$                                                     

\ni (Here, for $d, \ 1 \leq d \leq n ,\  w^{(d)}$ denotes the
projection of $w$ under $W \longrightarrow W/W_{P_d}$; and for $1\le
m\le 2n,\ m'=2n+1-m,\ |m|= min \{m,2n+1-m\}$).  



\smallskip
\noindent {\bf 3.7.} Let $\t,w$ be as above. For $1 \leq l,m\leq
2n$, we have $w\geq (l,m)\t \Longleftrightarrow w\geq (l',m')\t$,
where $l'=2n+1-l,\ m'=2n+1-m$. This observation together with
Proposition 3.6. yields

\proclaim Proposition 3.8. (cf. [L.1]) For a root $\alpha \in R_H$, 
let $r_\alpha $ denote the  reflection in $S_{2n}$ with respect to
$\alpha.$ Then   

$$N(w,\t)=\left \{ \beta=\tau (\alpha), \alpha\in  R_G^+ \left |   
\eqalign{
(1)\,\, & w \geq r_\beta \tau \,\,\,{\rm  ( in  }\,\,\,
S_{2n}),\,\,\,{\rm   if} \,\,\,\alpha = \varepsilon_j -
\varepsilon_k, 1 \leq j < k \leq n  \cr   &({\rm and
\,\,\,hence\,\,\,} \beta=\varepsilon_{a_j}-\varepsilon_{a_k})\cr
(2)\,\, & w \geq r_\gamma \tau\,\,\, {\rm  ( in }\,\,\,
S_{2n}),\,\,\,{\rm if}
\,\,\, \alpha= \varepsilon_j + \varepsilon_k, 1 \leq j < k \leq n  \cr  
&{\rm ( where}\,\,\, \gamma = \varepsilon_{a_j} - \varepsilon_{a_k'}). \cr
(3)\,\, & w \geq r_\gamma \tau\,\,\, {\rm ( in}\,\,\, S_{2n}), \,\,\,
{\rm if}\,\,\, \alpha= 2\varepsilon_j,  1 \leq j \leq n \hfill    \cr  
& ({\rm  where }\,\,\, \gamma = \varepsilon_{a_j} - \varepsilon_{a_j'}). \cr }  
   \right. \right \} $$    

\ni As a consequence of Proposition 3.8., we obtain 

\proclaim Proposition 3.9. (cf. [L.1]) Notations being as above,
$M(w, \tau)$  is stable under $\sigma$ and $N(w,\tau)$ can be
identified with the orbit space $M(w,\tau)$ modulo the action of
$\sigma.$

\proclaim Theorem 3.10. (cf. [L.1])  Let $w \in W_G$. Let $X(w)$ (
resp. $Y(w)$) be the Schubert  variety in $ H/B_H $ (resp.
$G/B_G$) associated to $w$. Then    
$$ Y(w)= (X(w))^\sigma \qquad \hbox { (scheme-theoretically).}$$   

\bigskip
\ni{\bf \S4. Criterion for smoothness of Schubert varieties in  
${\bf Sp(2n)/B^\sigma}$ }  
\medskip
We keep the notations of Sections 1, 2. We shall
denote $SL(2n)$ by $H$, and $Sp(2n)$ by $G$. Further, $B_H,\ W_H$
etc will denote objects on $H$, while $B_G,\ W_G$
etc will denote objects on $G$. For $\a \in R^+_H $ (resp.
$R^+_G$), $r_\a$ (resp. $s_\a$) shall denote the reflection
with respect to $\a$. As in Section 2,  for $w\in
W_G$, $X(w)$ (resp. $Y(w)$) shall denote the Schubert variety in
$H/B_H$ (resp. $G/B_G$). We have (cf. \S3, and [L.3]) 

\proclaim Theorem 4.1. Let $w \in W_G$. The tangent space to $X(w)$
at $e_{id}$ (resp.
$Y(w)$ at $f_{id}$) is spaned by $\{X_{-\a},\ \a \in N_H (w)\}$
(resp.
$\{X_{-\a},\ \a \in N_G (w)\}$) where
 $$ N_H(w)=\{\a \in R^+_H\ |\ w\geq r_\a\},$$
$$N_G(w) = \Biggl\lbrace{ \alpha \in R_G^+ \Biggl\vert
\eqalign {
   \,{\rm if}\,\,\, & \alpha = \varepsilon_i - \varepsilon_j\,\,\, {\rm or} 
    \,\,\,2\varepsilon_i, \,\,\,
                {\rm then}\,\,\, w \ge s_{\alpha} ,\hfill \cr  
   \,{\rm if}\,\,\, & \alpha = \varepsilon_i + \varepsilon_j,\,\,\,
{\rm then}\,\,\, 
   w \ge s_{\alpha}\,\,\, {\rm or}\,\,\, s_{2\varepsilon_i} \hfill \cr   }
           }\Biggl\rbrace.$$

\vs .2cm \ni {\bf 4.2. Some facts about $W_H$ and
$W_G$.}

\ni The following are easily checked:

\ni  1. For $ 1\leq i<j\leq 2n,\
r_{e_i-e_j}=(i,j)$.


\ni 2. For $1\leq i<j\leq n$, 
$$s_{\varepsilon_i-\varepsilon_j}=(i,j)(i',j'),$$  
 $$s_{\varepsilon_i+\varepsilon_j}=(i,j')(i',j).$$
For $1\le i\le n$,
 $$s_{2\varepsilon_i}=(i,i').$$

  \ni 3. Let $w=(a_1 \cdots a_{2n}) \in W_G$. Then we have:
 
 a. For $1 \leq i <j\leq 2n,\ w\geq (i,j)$ if and only
if 
$w\geq (i',j').$ 

Thus $w\geq s_{\varepsilon_i - \varepsilon_j} 
\Leftrightarrow w\geq (i,j)$.
  
    b. For $1\leq i<j \leq n,\ w \geq $ both $(i',j)$ and  
$w \geq (i,j') \Leftrightarrow w \geq (i',j)(i,j')$ or $w \geq (i,i').$  
\smallskip


\medskip
\ni {\bf 4.3. The integers $c_w$ and $d_w$}:  Let $w=(a_1 \cdots
a_{2n})
\in W_G$. We define  
$$c(w)=\#\{i, 1\leq i\leq n\mid w\geq s_{2\varepsilon_i}\}. $$   
Note that $$c(w)=0 \Leftrightarrow m(w)=0 $$ 
where recall (cf. \S1.5,VI) that
$m_w=\#\{i,\  1\leq i\leq n\ |\ a_i>n\}$. 

\ni When
$c_w \not=0$, we define         
$$d_w={\rm min}\{i,\,\,1\leq i\leq n \mid \exists j\leq i 
          \,\,\,{\rm such \,\,\,that}\,\,\, a_j \geq i'\}.$$ 
Note that 
$$c_w=\#\{\varepsilon_i-\varepsilon_{i'}\in R_H^+\mid
\varepsilon_i-\varepsilon_{i'}\in N_H(w),\  1\leq  i\leq n\}.$$
Also note that 

\ni a. $d_w$ is the least integer $i\leq n $ such that
max$\{a_1,\cdots ,a_i\}\geq i'$.

\ni b. $d_w$ is the least integer $i\leq n $ such that $w\geq
s_{2\varepsilon_i}$.

\proclaim Lemma 4.4 Let $w=(a_1 \cdots a_{2n}) \in W_G$. We have  
$c_w=n+1-d_w$.

\ni The proof is immediate from the definitions of $c_w$ and
$d_w$.  


\proclaim Proposition 4.5. Let $n_H(w)$  (resp.$n_G(w)$) =
$\#N_H(w)$ (resp. $\#N_G(w)$). We have $n_G(w)\ =\ {{1}\over
{2}}(n_H(w)+c_w)$.

\ni (Note the resemblance of this with the length formula (cf.
\S1.5,VI),  
$l_G(w)\ =\ {{1}\over
{2}}(l_H(w)+m_w)$.)

\ni {\sl Proof.} We have (cf. Theorem 4.1)  $$ N_H(w)=\{\a \in R^+_H\
|\ w\geq r_\a\},$$
$$N_G(w) = \Biggl\lbrace{ \alpha \in R_G^+ \Biggl\vert
\eqalign {
   \,{\rm if}\,\,\, & \alpha = \varepsilon_i - \varepsilon_j\,\,\, {\rm or} 
    \,\,\,2\varepsilon_i \,\,\,
                {\rm then}\,\,\, w \ge s_{\alpha} ,\hfill \cr  
   \,{\rm if}\,\,\, & \alpha = \varepsilon_i + \varepsilon_j\,\,\, {\rm then}\,\,\, 
   w \ge s_{\alpha}\,\,\, {\rm or}\,\,\, s_{2\varepsilon_i} \hfill \cr   }
           }\Biggl\rbrace.$$
We have (cf.\S1) $R^+_H $ is $\s $ -stable, and Proposition 3.9. 
implies that $N_H(w)$ is $\s $ -stable, and $N_G(w)$ is the orbit
space of $N_H(w)$ modulo the action of $\s$. The result follows
from this noting that the fixed points in $N_H(w)$ are precisely 
$\varepsilon_j-\varepsilon_{j'}$ such that $w \geq (j,j')$.

\medskip 
\ni {\bf 4.6 The element $w_1$:} 

\ni Let $w=(a_1 \cdots a_{2n}) \in W_G$.
For $1 \le i\le n$, the entry $i$ will be denoted by just $i$,
while its $Sp(2n)$-{\it conjugate} $2n+1-i$ will be denoted by $i'$.
Also, for $1 \le k\le 2n$, we shall denote $|k|=$ min$\{k,k'\}$.
Let $t\le n$ be the integer such that $|a_t|=n$. We define $w_1$
as an element of $W(Sp(2n-2))$, by $w_1=(b_1 \cdots b_{2n-2})$
where the set $\{b_1, \cdots , b_{2n-2}\}$ is simply the set 
$\{a_1, \cdots , a_{2n}\}$ with $a_t,\ a_{t'}$ deleted, and an
entry $i'$ in $w_1$ is to be understood as the
$Sp(2n-2)$-{\it conjugate} of $i$, namely $2(n-1)+1-i\  (=2n-1-i)$.
To make it very precise, we define a bijective map 
$$ \pi: \{1 , \cdots , (n-1) \} \longrightarrow \{ 1 , \cdots , n \}\
\backslash \ \{t\}$$ given by 
$$   \pi ( i ) = 
     \cases { i     &  if  $i < t,$  \cr  
              i+1 &  if  $i \geq t$ . \cr } $$
Then $(b_1\cdots  b_{(n-1)}) $ is given by
$$b_i=\cases{ a_{\pi(i)}     & if $a_{\pi(i)} < n,$  \cr 
           a_{\pi(i)} - 2 & if $a_{\pi(i)}> n,$ \cr }  $$
(note that $(b_1 \cdots  b_{2n-2})$, as an element of
$W(Sp(2n-2))$ is completely known by knowing $\{b_1,\cdots ,
b_{(n-1)} \}$).

\medskip
\proclaim Lemma 4.7. Let $w=(a_1 \cdots a_{2n}) \in W_G$. Further,
let $m_w\not=0$. Let $r$ be an integer $\le n$. The following are
equivalent.
\item (1) $d_w=r$.
\item (2) $a_i\leq r',\ i<r$, and if $a_i<r',\ i<r$, then $a_r\geq
r'$.

\ni {\sl Proof.} Let $d_w=r$. Then the assertion (2) follows from
the definition of $d_w$.

\ni Let now (2) hold. This implies that $a_k\geq r'$, for some
$k\le r$; further, for $i<r$, max$\{a_1, \cdots ,a_i\} \le
r'<i'$. Hence we obtain that $r$ is the smallest integer $i$ with the
property  that max$\{a_1, \cdots ,a_i\}\ge i'$. Hence we obtain (cf.
\S4.3) that $d_w=r$.

\ms \ni {\bf 4.8 The integer $j_w$.}

\ni Let the notations be as in
Lemma 4.7. Let $m_w\not=0$.

If $a_i<r',\ i<r$, then we define $j_w=r$.

In the alternating case, we define $j_w$ to be the unique integer
$i<r$ such that $a_i=r'$.

\ni Note that $j_w\le d_w \ (=r)$.  


\ms \ni We recall below the criterion for smoothness for Schubert
varieties in
$SL(n)/B$ as given in [L-Sa].

\proclaim Theorem 4.9. Let $w=(a_1,...,a_n)\in
W(SL(n))$. Then $X(w)$ is singular if and only   if the following
property holds 
$$(*) \quad \left\{ \eqalign{
             & {\rm there\,\,\, exist}\,\,\,i,j,k,l,\   1\le i < j <
k < 
                l\le 2n \,\,\,{\rm such\,\,\, that} \cr
             & {\rm either \,\,\, (1)}\,\,\, a_k < a_l< a_i < a_j 
                {\rm\,\,\,or \,\,\,(2)}\,\,\, a_l < a_j < a_k < a_i. \cr }
              \right. $$

\smallskip
\ni As an immediate consequence of the above Theorem, we obtain
\vskip 0.2cm
\proclaim Theorem 4.10. Let $w=(a_1 \cdots a_{2n}) \in W_G$. Then,
if $X(w)$ is smooth, then so is $X(w_1)$. 


\smallskip \ni Given $w=(a_1 \cdots a_{2n}) \in W_G$, let $w_1$
be as in \S4.6. The following Proposition spells out the
relationship between $m_w$ and $m_{w_1}$ (resp. $c_w$ and
$c_{w_1}$).

\proclaim Proposition 4.11. Let $w=(a_1 \cdots a_{2n}) \in W_G$.
Then 
\item (1.)  $m_w=m_{w_1}$ or $m_{w_1}+1$.   
\item (2) $c_w=c_{w_1}$ or $c_{w_1}+1$. 

\noindent{\sl Proof.} The assertion (1) is immediate from the
definition of $m_w$ (cf.\S1.5,\ VI).

\ni Let $s_1=c_{w_1}$. 

\ni If $s_1=0$, then 
$m_{w_1}=0$, and hence $m_w=0$ or $1$. Hence, we obtain, $c_w=0$ or
$1$, according as $m_w=0$ or $1$. 

\ni Let then $s_1>0$. Let $d_{w_1}=r_1,\  j_{w_1}=j_1$. Note that 
$r_1=n-s_1$ (cf. Lemma 4.4); also, $\ j_1 \le r_1$ (cf. \S4.8). In
the proof below, we will be required to consider both $W(Sp(2n)),\
W(Sp(2n-2))$, simultaneously. For an integer $i,\ 1\leq i\le n, \ i'$
shall denote
$2n+1-i$ (the Sp($2n$)-{\it conjugate} of $i$), while for an integer
$j,\ 1\leq j\le n-1,\ j^!$ shall denote
$2(n-1)+1-j\  (=2n-1-j)$ (the Sp($2n-2$)-{\it conjugate} of $j$). Let
$t\le n$ be the integer such that
$|a_t|=n$ (cf.\S4.6). We divide the proof into the following cases.

\ni {\bf Case 1.} $j_1 = r_1$.

\ni Writing $w_1=(b_1 \cdots b_{2n-2}) $ (cf. \S4.6), we have (cf.
Lemma 4.7)
$$b_{j_1}\ge r^!_1,\ b_i< r^!_1,\ i<r_1. \leqno{(*)}$$
We further divide this case into the following two subcases.

\ni {\bf Subcase 1(a).} $a_t$ comes before $a_{\pi(j_1)}$ (here,
$\pi$ is as in \S4.6$).

\ni This implies $\pi(j_1)=j_1+1,\ a_{\pi(j_1)}=a_{j_1+1}\
(=a_{r_1+1})$. Hence we obtain (in view of (*))
$$a_{r_1+1}=b_{r_1}+2\ge r'_1>(r_1+1)'. \leqno{(1)}$$
Also for $i<r_1+1,\ i\not= t$, we have 
$$   a_{\pi(i)} = 
     \cases { b_i+2,     &  if  $b_i >n-1,$  \cr  
              b_i, &  if  $b_i \leq n-1$ . \cr } $$
Hence we obtain (using (*))
$$ a_i <r'_1,\ i<r_1+1,$$
i.e.,
$$ a_i \le (r_1+1)',\ i<r_1+1. \leqno{(2)}$$
From (1) and (2), we obtain (in view of Lemma 4.7)
$$d_w=r_1+1.$$
Hence
$$c_w(=n+1-d_w)=n-r_1=s_1=c_{w_1}.$$

\ni {\bf Subcase 1(b).} $a_t$ comes after $a_{\pi(j_1)}$.

\ni In this case we have (in view of (*))
$$\pi(j_1)=j_1(=r_1),\ a_{r_1}= b_{r_1}+2\ge r'_1,\leqno{(3)}$$
and
$$ {\rm for}\  i<r_1,\ a_i<r'_1.\leqno{(4)}$$
From (3) and (4), we obtain (in view of Lemma 4.7)
$$d_w=r_1.$$
Hence
$$c_w(=n+1-d_w)=n+1-r_1=s_1+1=c_{w_1}+1.$$

\ni {\bf Case 2.} $j_1 < r_1$.

\ni In this case,  we have (cf. Lemma 4.7)
$$b_{j_1}= r^!_1,\ b_i< r^!_1,\ i<r_1,\ i\not= j_1. \leqno{(**)}$$
Here again, we further divide this case into the following two
subcases.

\ni {\bf Subcase 2(a).} $a_t$ comes before $a_{\pi(j_1)}$. 

\ni This implies $\pi(j_1)=j_1+1,\ a_{\pi(j_1)}=a_{j_1+1}=r'_1$.
Hence as above we obtain (in view of (**))
$$a_{j_1+1}=b_{j_1}+2= r'_1, \leqno{(5)}$$
$$ {\rm for}\  i<r_1,\ i\not= j_1+1,\ a_i<r'_1.\leqno{(6)}$$
From (5) and (6), we obtain (in view of Lemma 4.7)
$$d_w=r_1.$$
(note that $j_1+1\le r_1$). Hence
$$c_w(=n+1-d_w)=n+1-r_1=s_1+1=c_{w_1}+1.$$

\ni {\bf Subcase 2(b).} $a_t$ comes after $a_{\pi(j_1)}$.

\ni In this case we have (in view of (**))
$$\pi(j_1)=j_1,\ a_{j_1}= b_{j_1}+2=r'_1,\leqno{(7)}$$
and
$$ {\rm for}\  i<r_1,\ i\not= j_1,\  a_i<r'_1.\leqno{(8)}$$
From (7) and (8), we obtain (in view of Lemma 4.7)
$$d_w=r_1.$$
Hence
$$c_w(=n+1-d_w)=n+1-r_1=s_1+1=c_{w_1}+1.$$

\ni This completes the proof of Proposition 4.11.
\ms
\ni {\bf 4.12. The integer $\d_w$.} 

\ni For $w \in W_H$, we define 
$\delta(w) = n_{H} (w) -\ell_{H} (w)$ . 

\ni (note that  in view of Theorem 4.1, $\delta(w) \ge 0 ,\ \forall w
\in W(SL(2n))$ with equlity  iff $ X(w)$ is smooth.) 
\vskip 0.2cm
\proclaim Lemma 4.13. Let $w \in W_G$. Then
 $$ n_{G} (w) - \ell_{G} (w) = {{1}\over{2}} ( \delta(w) + c_w -
 m_w) .$$

\ni The proof follows from \S1.5,VI and Proposition 4.5.
\vskip 0.2cm
\proclaim Proposition 4.14 Let $w \in W_G$. If $X(w)$ is smooth then
$m_w=c_w$ .

\noindent{\sl Proof.} (by induction on $n$.)

\ni {\bf Starting point of induction} $n=2$. We have $H=SL(4)$. The
smooth $X(w)$'s with $w \in Sp(4)$ are precisely
$$(1234),(2143),(1324),(2413),(3142),(4321).$$
For all of these $w$'s, it is easily seen that $m_w=c_w$.

\ni Let now $n \ge 3$. Let $w_1$ be as in \S4.6. Then we have
$X(w_1)$ is smooth (cf. Theorem 4.10). Further, by induction
hypothesis, $m_{w_1}=c_{w_1}$. We have (cf. Proposition 4.11) 
$m_w=m_{w_1}+1$ or $m_w=m_{w_1}$. We may suppose (cf. \S4.3),  
 $m_w>0$. We divide the proof into the following two
cases.

\ms \ni {\bf Case 1} $m_w=m_{w_1}+1$. 

\ni By Proposition 4.11, $c_w = c_{w_1} + 1$ or $c_{w_1}$. 

\ni {\bf Claim 1.} $c_w = c_{w_1} + 1$

\ni If possible, let us assume that $c_w =c_{w_1}$. Then assumption
implies that $ m_{w_1}=c_{w_1}=c_w$, while $m_w=m_{w_1}+1$. Hence
we obtain $m(w) > c(w)$. This together with Lemma 4.13 implies $\d_w
(\ =2(n_G(w)-\ell_G(w))+(m_w-c_w)\ ) >0 $. Hence we obtain that
$X(w)$ is singular contradicting the hypothesis that $X(w)$ is
smooth. Hence our assumption is wrong, and Claim 1 follows. Now
Claim 1 implies
$$c_w = c_{w_1} + 1= m_{w_1}+1=m_w.$$
This proves the required result in this case.
\ms \ni {\bf Case 2} $m_w=m_{w_1}$. 

\ni Let us denote $m_w (= m_{w_1})$ by $ s$. We have $s>0$. 
Let $w=(a_1 \cdots a_{2n})$. Let $t\le n$ be the integer such that
$|a_t|=n$. The hypothesis $m_w=m_{w_1}$ implies that $a_t
= n$. Let $w_1=(b_1 \cdots b_{2n-2})$ (cf. \S4.6). Let
$b_m =$ max $\{ b_i \mid i \le n - 1 $, $b_i \ge n \}$ (note that
$b_m$ exists, since $m_{w_1}>0$.) 

\ms \ni {\bf Claim 2.} $b_i \le n - 1 $, $\forall i < m$ .

 \ni If possible, let us assume that there exists a $p < m$ such that
$b_p \ge n$. As in the proof of Proposition 4.11, let $j^!=
2n-1-j$ (the Sp($2n-2$)-conjugate of $j$), $ 1\le j\le n-1$ , and
$i'=2n+1-i$ (the Sp($2n$)-conjugate of $i$), $1\le i\le n$ , we have
$b_p < b_m ,\  b_{2n-1-p}(=b^!_p) > b_{2n-1-m}(=b^!_m)$. This implies
that
$w_1$ has the form
\vskip 0.2cm
    $$(..., b_p , ... , b_m , ... ,b_{2n-1-m} , ... , b_{2n-1-p} ,... ).$$
\vskip 0.2cm
\ni where $p < m < 2n - 1 - m < 2n - 1 - p$, and $b_{2n-1-m} <
b_{2n-1-p} < b_p < b_m$ . This implies (cf. Theorem 4.9) that
$X(w_1)$ is singular, a contradiction. Hence Claim 2 follows.

\ms \ni {\bf Claim 3.} $a_t$ appears before $a_{\pi(m)},\ \pi$
being as in \S4.6.

\ni If possible, let us assume that $a_t$ appears after
$a_{\pi(m)}$ . Then $\pi(m)=m$, and $w$ has the form

$$(... a_m  , ... , a_t , ... , a_{2n+1-t} , ... , a_{2n+1-m}  ,
... ) ,$$
where $a_t = n , a_{2n+1-t} = n + 1 , a_m  = b_m + 2
\ge n + 2$ and $a_{2n+1-m}   \le n - 1$ . Thus we obtain
$m<t<2n+1-t<2n+1-m,\  a_{2n+1-m}  < a_t < a_{2n+1-t} <
 a_m $ . This implies (cf. Theorem 4.9) that $X(w)$ is
singular, a contradiction. 

\ni Hence Claim 3 follows.

\ms \ni {\bf Claim 4.} $b_i\ge n$,  $\forall i$,  $m\le i\le
n-1$.

\ni If possible, let us assume that there exists a $j ,\  m < j\le n
- 1$ such that $b_j\le n - 1$ .

(1). If $b_j > b_{2n-1-m }(=b^!_m)$, then $w_1$ has the form
 $$ ( ... , b_m , ... , b_j , ... , b_{2n-1-j} , ... , b_{2n-1-m} ,
... ) ,$$
where $m < j < 2n - 1 - j < 2n - 1 - m$, and $b_{2n-1-m} < b_j <
b_{2n-1-j} < b_m $. This implies (cf. Theorem 4.9) that $X(w_1)$ is
singular, a contradiction. 

\vs .1cm (2). If $b_j < b_{2n-1-m}$ , then w has the form
$$( ... , a_t , ... , a_{m+1}  , ... , a_{j+1}  ,
... ,a_{2n-j}  , ... , a_{2n-m}, ... , a_{2n+1-t} ,
... ) ,$$
where $a_{m+1}  = b_m + 2,\ a_{j+1}  = b_j(\leq n-1) $ (note that
in view of Claim 3, $\pi(i)=i+1,\ i\ge m$). We have (since
$b_j<b_{2n-1-m}$)

$$a_{j+1}  < a_{2n-m}  \le n - 1 ,$$
(note that $b_m\ge n$ implies $a_{m+1}\ge n+2(=(n-1)'$), and hence
$a_{2n-m}(=a'_{m+1})\le n-1$). This implies $w$ has the form
$$( ... , a_t , ... , a_{m+1}  , ... , a_{j+1}  , ... , a_{2n-m},
... ) ,$$
 where $t<m+1<j+1<2n-m$, and $ a_{j+1}  < a_{2n-m } <a_t <
a_{m+1}$. This implies (cf. Theorem 4.9) that $X(w)$ is singular, a
contradiction.

\ni Thus our assumption is wrong, and Claim 4 follows.

\ni Now Claims (2) and (4) imply that $m_{w_1}=n-m$, and hence
$m=n-s$ (recall that $m_{w_1}=s$).  Also, $m_{w_1}=s$ implies that
$b_m\ge (n-s)^!\ (=n-1+s)$. Hence we obtain
$$a_{m+1} \  (= b_m + 2)\ \ge n+1+s=2n+1-m>(m+1)',$$
$${\rm for}\  i< m+1,\ a_i \le n <(m+1)'.$$
Hence we obtain (cf. Lemma 4.7), $d_w=m+1=n+1-s$, and hence
$c_w=s=m_{w_1}=m_w$. Thus we obtain $c_w=m_w$, as  required.

\ni This completes the proof of Proposition 4.14.

\vskip 0.2cm
\noindent{\bf Remark 4.15.} The converse of Proposition 4.14 need
not be true. For   example, let $w=(356124)\in W(Sp(6))$. Then
$m(w)=c(w)=2,$ but $X(w)$ is singular.

\proclaim Theorem 4.16. Let $w \in W_G$. If  $X(w)$ is smooth,
then so is $Y(w)$.

\noindent{\sl Proof.} We have (cf. Lemma 4.13)
 $$ n_{G} (w) - \ell_{G} (w) = {{1}\over{2}} ( \delta(w) + c_w -
 m_w) .$$
The hypothesis that $X(w)$ is smooth implies (cf. Theorem 4.1,
Proposition 4.14) $\d_w=0,\ m_w=c_w$. Hence we obtain $ n_{G} (w) =
\ell_{G} (w)$, and hence $Y(w)$ is smooth ( note that $n_{G} (w)$
is the dimension of the tangent space to $Y(w)$ at $Id$ (cf.
Theorem 4.1)).

\ms \ni {\bf Remark 4.17.} We observe that our proof of Theorem 4.16 
being characteristic-free, Theorem 4.16 holds in ${\underline
{all}}$ characteristics. In characteristic zero, the above
Theorem may also be proved using Theorem 3.10, and the fact that  
$\sigma: X(w) \longrightarrow X(w)$ is an automorphism of finite
order.

\vskip 0.2cm
\proclaim Corollary 4.18. Let $w \in W_G$, and let $X(w)$ be
singular, $Y(w)$ be smooth. Then $m(w) > c(w)$.


\noindent{\sl Proof.} The hypotheses together with Lemma 4.13 imply
$\delta(w)=m(w)-c(w)$, and $\delta(w) > 0$. Hence we
obtain $m(w)>c(w)$.

\medskip

\proclaim Theorem 4.19  Let $w \in W_G$. Then we have the following:
\item{(i)} Let $m(w) = c(w)$. Then $Y(w)$ is singular if and only if
$X(w)$ is  singular.
\item{(ii)} Let $ m(w) < c(w)$. Then $Y(w)$ is singular. 
\item{(iii)} Let $ m(w) > c(w)$. Then $Y(w)$ is singular if and only
if $\delta  (w) > m(w) - c(w) $.
\medskip

\ni{\sl Proof.} All of the assertions are immediate consequences of
Lemma 4.13.

\ms \ni Using Theorem 4.19, we give below a criterion for smoothness
of Schubert varieties in $G/B_G$  in terms of the corresponding
permutations (in the same  spirit as in [L-Sa]).
                                                                                            
\medskip  
\proclaim Theorem 4.20. Let $w\in W_G,\ w=(a_i \cdots
a_{2n})$. Then the following are equivalent.
\item{(1)} $Y(w)$ is singular     
\item{(2)}$w$ has the following property.
$$(**)\quad \left\{  
\eqalign{    
            {\rm either}\,\,\,  &(1).\,\,\,m(w)<c(w), \cr  
           {\rm or}\,\,\,\,\,\, &(2).\,\,\,m(w)=c(w),\,\,
{\rm and \,\,\,property
                 \,\,\,(*)\ of\ Theorem\  4.9 \,\,\, holds\ for 
\,\,\,} w,  
\cr  
           {\rm or}\,\,\,\,\,\, &(3).\,\,\,m(w)>c(w),\,\,{\rm and}\,\,\,\delta(w)
                                 > m(w)-c(w).     \cr } \right.  $$
\bigskip 


\centerline{{\bf References}}
\vskip .2cm
{\parindent=1cm 
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\par}



\bye