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\centerline{\bigbf Tangent spaces to Schubert varieties }
\vskip .4cm
\centerline{V. LAKSHMIBAI
\footnote{*}{\sl  Partially supported by the NSF Grant DMS 9103129 .}
}

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\vskip .4cm

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\noindent
 
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In this note, we announce a criterion for smoothness of a Schubert
variety in the flag variety $G/B$. Let $G$ be a semi simple, simply connected
algebraic group, which we assume for simplicity to be defined over an algebraically
closed field $k$ of characteristic $0$. ( The following discussion
is valid in any characteristic, in fact even over {\bf Z}). 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. For $\a \in R^+$, let $E_\a,\ F_\a$ denote the
elements of the Chevalley basis for $\gg (={\rm Lie}G$), corresponding to $\a$. 
For $w \in W$, let $e_w$ be the point in $G/B$, and $X(w)\ (={\overline
{BwB}}$ (mod $B$)), the Schubert variety, associated to $w$. Let
$T(w,e_{id}) $ be the  the tangent space to $X(w)$ at $e_{id} $.
Let  $$N_w=\{ \b \in R^+\ |\ F_\b }\in T(w,e_{id}) \}. \leqno
{(1)} $$
 Now, $T(w,e_{id})$ being a $T$-submodule of the $T$-module $\sum_{\a \in
R^-}\gg_\a$ (here, $R^-$ denotes the set of negative roots in $R$, and $\gg_\a$
denotes the root space $kF_\a$), we have
  $$ T(w,e_{id})\ =\ {\rm the\ span\ of\ }  \{ F_\b,\ \b \in N_w
 \}. \leqno{(2)} $$ For a dominant
weight $\l$, let $V(\l)$ be the irreducible $G$-module (over {\bf
C}) with highest weight $\l$.  Let us fix a highest weight vector $u$
in $V(\l)$. For $w \in W$, let $u_w= w\cdot u$, and $V_w=U^+(\gg)
u_w$ (here, $U^+(\gg)$  is the subalgebra of $U(\gg)$, the
universal enveloping algebra of $\gg$, generated by $\{E_\a, \a \in
S\}$). For a weight $\mu $ in $V(\l)$, let $ m(\mu )\ ({\rm resp.}\
m_w(\mu ))$ denote the multiplicity of $\mu $ in $V(\l )$ ( resp. $
V_w $ ).

\proclaim Theorem 1. Let $\b \in R^+$, and $\rho =  {1 \over 2 }$ the sum 
of positive roots. Then  $\b \in N_w $ if and only if
$ m_w(\rho - \b)\ =\ m(\rho - \b ) $.
 
\ni As a consequence, we obtain a criterion for the smoothness of a Schubert
variety as given by the following

\proclaim Corollary. Let $w\in W$, and $M_w= \{\b \in R^+\ |\  m_w(\rho - \b)\ =\
m(\rho - \b )\}$ . Then $X(w)$ is smooth if and only if
$\#M_w=l(w)
$ , where $l_w$ denotes the length of $w$ (= dim $X(w)$). 

\ni The proof is immediate, since  $X(w)$ is smooth if and only if it is smooth at
$e_{ id}$.

\vfill\eject  
 \ni {\bf Outline of Proof of Theorem 1:}  

\ni For
generalities on algebraic groups, one may refer to [B].

\ni Let us fix a dominant, regular weight $\l$. 
Let $V(\l),\ u,\ u_w,\ V_w $ be as above. We have (cf. [P])
$$ T(w,e_{id})\ =\ {\rm the\ span\ of\ }  \{ F_\b,\ \b \in R^+\
|\ F_\b u \in V_w \}. \leqno{(3)} $$
From (2) and (3), we obtain
$$N_w=\{\b \in R^+\ |\ F_\b u \in V_w \}. \leqno{(4)} $$ 
In [L]$_3$, we constructed a basis
$\Bb (\l) $ for $V(\l)$ which is compatible with the Bruhat order,
i.e., $ V_w\cap \Bb (\l)=\Bb_w (\l)$, say, is a basis for $V_w$. Further, this basis
consists of elements of the form $Du$, where $D$ is either 1 or 
$F_{\g_1}^{(n_1)}\cdots F_{\g_r}^{(n_r)},\ \g_i$ simple, $n_i>0$ 

\ni (for some suitable
$n_i$'s), and $s_{\g_r}\cdots s_{\g_1}$ is reduced ( here,
$F_\g^{(n)}={{F_\g^n}\over {n!}}$).  To be more precise, let 
$$I=\{ {\rm Lakshmibai\_Seshadri\  paths\ of\ shape\ }\l\}, \leqno{(5)} $$ 
$$I_w=\{\pi
\in I\ |\ w\geq \phi(\pi)\}\leqno{(6)}$$
 notations being as in [Li]. Then it is shown
in [Li], 
$$\char V(\lambda)=\sum_{\pi\in I}e^{\nu(\pi)} \leqno{(7)}$$
$$\char V_w=\sum_{\pi\in I_w}e^{\nu(\pi)}\leqno{(8)}$$
In particular, using (7), we obtain a formula for $m(\mu),\ \mu \in X$, the
weight lattice, namely 
$$  m(\mu )=\#\{\pi \in I\ |\ \nu(\pi)=\mu\}\leqno{(9)}$$ 
 Fixing a reduced expression 
s_{i_1}s_{i_2} \cdots s_{i_r} $ for $w $, we have (cf. [L]$_4$)
$$I_w=\{f_{i_1}^{n_1}\cdots f_{i_r}^{n_r} \pi_0,\ {\rm for \ suitable}\ n_i \in {\bf
Z}^+ \}, \leqno{(10)}$$
$$ \Bb_w (\l)=\{F_{i_1}^{(n_1)}\cdots F_{i_r}^{(n_r)} u\ |\ 
f_{i_1}^{n_1}\cdots f_{i_r}^{n_r} \pi_0 \in I_w\}.\leqno{(11)}$$  
Here, $\pi_0$ is the Lakshmibai-Seshadri path given by the line segment in
$X\otimes  {\bf R}$ joining origin  and $\lambda $, and $f_i$
are the operators on $I$ as defined in [Li]).

\ni Let us write $\Bb (\l) =\{Q_\pi,\ \pi \in I\}$. For $\l=\rho $, we are able to
write down (cf.[L]$_4$) very precisely the expression for $F_\b u$ as a linear
combination of the elements in $ \Bb (\l)$, namely,
$$F_\b u=\sum_{I^{\rho, \b}}\ c_\pi Q_\pi,\ \ c_\pi \in k^* \leqno{(12)} $$
where
$$I^{\rho, \b}=\{\pi \in I\ |\ \nu(\pi)=\rho-\b\} \leqno{(13)}$$
We have (cf.(9))
$$ m(\rho -\b)=\# I^{\rho, \b} \leqno{(14)}$$
By Bruhat order compatibility of $ \Bb (\l)$, we have (cf.(12))
$$F_\b u \in V_w \Longleftrightarrow Q_\pi \in V_w,\ \forall \pi \in I^{\rho,
\b}\leqno{(15)}$$
Now (14) and (15) imply that
$$F_\b u \in V_w \Longleftrightarrow m_w(\rho -\b)=m(\rho -\b)\leqno{(16)}$$
Hence, from (4) and (16), we obtain
$$ \b \in N_w \Longleftrightarrow m_w(\rho -\b)=m(\rho -\b)$$ 
 
\vs .2cm \ni  {\bf Further Consequences.} 

Given $\b \in R^+$, using the expression for $F_\b u$ as a linear combination of the
elements in $ \Bb (\rho)$ as given by (12) above, we are able to describe $N_w$ in a
very elegant form for classical groups as described in Theorem 2 below. We shall
follow the notation in [Bou] to denote the elements of $R^$. We shall denote the
Bruhat order in $W$ by $\geq$. 

\vskip .2cm
\proclaim Theorem 2: Let $\b \in R^+$. 

 
\item {(a)} {\it Let $G$ be of type $ {\bf A}_n $. Then 
$ \b \in N_w\ \Longleftrightarrow w \geq s_\b $.}

\vs .2cm

\item {(b)}  {\it Let $G$ be of type $ {\bf C}_n $.} 
\item {(1)} {\it  Let $\b = \e_i - \e_j, \ {\rm or\ } 2\e_i$. Then
$\b \in N_w\ \Longleftrightarrow w \geq s_\b $.}
\item {(2)} {\it  Let $\b = \e_i + \e_j$. Then $\b \in
N_w\ \Longleftrightarrow w \geq \ {\it either\ } s_{\e_i + \e_j}\
{\it or\ } s_{2\e_i} $.}


\vskip .2cm
\item {(c)}  {\it Let $G$ be of type $ {\bf B}_n $.} 
\item {(1)} {\it  Let $\b = \e_i - \e_j,\ \e_n,\ {\rm or}\
\e_i+\e_n $. Then $\b \in N_w\ \Longleftrightarrow w \geq s_\b $.}
\item {(2)} {\it Let $\b = \e_i,i<n $. Then} $\b \in
N_w\ \Longleftrightarrow w \geq \ {\it either\ } s_{\e_i} {\it or\ }
s_{\e_i + \e_n}  $.
\item {(3)} {\it Let $\b = \e_i + \e_j, j <n $. Then $\b \in
N_w\ \Longleftrightarrow w \geq \ {\it either\ } s_{\e_i + \e_j}\
{\it or\ } s_{\e_i}s_{\e_j + \e_n} $.}


\vskip .2cm
\item {(d)} {\it  Let $G$ be of type $ {\bf D}_n $.}  
\item {(1)} {\it  Let $\b = \e_k - \e_l,\ {\rm or}\ \e_i + \e_j,\
j=n-1,n$. Then $\b \in N_w\ \Longleftrightarrow w \geq s_\b $.}
\item {(2)}  {\it Let $\b = \e_i + \e_j,\ j<n-1$. Then 

\ni  $\b \in
N_w\ \Longleftrightarrow w \geq \ {\rm either\ } s_{\e_i + \e_j}\
{\rm or\ } s_{\e_i+\e_n}s_{\e_i-\e_n}s_{\e_j + \e_{n-1}} $.}
\vskip .2cm
\ni {\bf Remark 1 }: The result in Theorem 2 for type {\bf A}$_n$ is
contained in [L-S] also. In [L]$_1$, [L]$_2$, [L-R], results were obtained towards the
determination of $ T(w,e_{id})$ for types {\bf C}$_n$, {\bf B}$_n$, {\bf D}$_n$
respectively. The formulation of $ T(w,e_{id})$ as given in Theorem 2 is a nice
refinement of the formulations in loc.cit. Moreover, the method of proof outlined
above is much more simple and straightforward than the proofs in loc.cit.

\vskip .2cm
\ni {\bf Remark 2 }: In [S]$_2$ ( see also [S]$_1$ ), the authour gives a criterion
for smoothness of Schubert varieties in terms of the nil Hecke ring.

\vs .2cm \ni {\bf Acknowledgement.} The authour expresses her thanks to the referee
for important comments and suggestions.
 
\vskip .4cm
\centerline{\bigsc References}

\vskip .4cm
\ni [B] A.Borel    {\it Linear algebraic groups}, W.A. Benjamin, New
York, 1969.
\vskip .2 cm
\ni [Bou] N. Bourbaki {\it Groupes et albebres  de Lie}, Chapitres
4,5, et 6, Hermann, Paris, 1968.


\vskip .2 cm
\ni [L]$_1$ V. Lakshmibai, {\it Geometry of G/P-VII}, J.Alg, 108
(1987), 403-434.

\vskip .2 cm
\ni [L]$_2$ V. Lakshmibai, {\it Geometry of G/P-VIII}, J.Alg, 108
(1987), 435-471

\vskip .2 cm
\ni
[L]$_3$ V. Lakshmibai, {\it Bases for Quantum Demazure modules}, to appear in CMS 
proceedings of the conference on {\it Representations of groups-
Finite, Algebraic, Lie, and Quantum}, Banff, 1994.

\vskip .2cm
\ni [L]$_4$ V. Lakshmibai, {\it On tangent spaces to Schubert
varieties } (in preparation).

\vskip .2 cm
\ni [L-R] V. Lakshmibai, K.N. Rajeswari {\it Geometry of G/P-IX}, J.Alg, 130
(1990), 122-165.


\vskip .2cm
\ni [L-S] V. Lakshmibai and C.S. Seshadri, {\it Singular locus of a
Schubert variety}, Bull. A.M.S., 2 (1984), 363-366.


\vskip .2cm
\noindent
[Li] P.Littelmann, {\it Littlewood-Richardson rule for symmetrizable
Kac-Moody algebras}, Inv. Math., 116 (1994), 329-346.

\vskip .2cm
\ni [P] P. Polo, {\it On Zariski tangent spaces of Schubert varieties,
and a proof of a conjecture of Deodhar}, Indag. Mathem., N.S.,5 (1994), 483-493.

\vskip .2cm 
\ni [S]$_1$ S. Kumar, {\it The nil Hecke ring and singularities of Schubert
varieties}, Proceedings of the conference in honor of Kostant's 60th birthday,
Birkha\"{u}ser, 1994.

\vskip .2cm \ni [S]$_2$ S. Kumar, {\it The nil Hecke ring and singularities of
Schubert varieties}, preprint, 1995 (detailed version of the above). 





\vskip .6cm








\ni Mathematics Department\hfill\break
Northeastern University\hfill\break
Boston, MA 02115\hfill\break
e-mail address: lakshmibai@neu.edu