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\bigskip
\centerline {\bf {G\'eom\'etrie Alg\'ebrique- Multiplicit\'es sur une vari\'et\'e de
Schubert}}


\vskip .2cm
\centerline{{\bf  (Algebraic Geometry- Multiplicities of points on a Schubert variety
)}} 

\vskip .4cm
\centerline{Venkatramani Lakshmibai
\footnote{*}{\sl  Partially supported by the NSF Grant DMS 9103129 .}
}
\centerline { Northeastern University}
\centerline {Mathematics Department}
\centerline{Boston, MA-02115, USA}


\vskip .4cm
\vskip .4cm

\midinsert
\narrower
{\smrm
\baselineskip=10pt
\noindent
 
\endinsert
\vfill\eject

\centerline{{\bf  Algebraic Geometry- Multiplicities of points on a Schubert variety
}}

\vs .2cm \ni

{\bf R\'esum\'e.} Dans cette Note, nous pr\'esentons une formule pour les
multiplicit\'es des vari\'et\'es de Schubert dans 
$G/Q$, o\`u $G$ est un groupe alg\'ebrique semi-simple, et 
$Q$ un sous-groupe parabolique. Cette formule est particuli\`erement simple
lorsque $G=SL(n)$.


\vs .4cm  {\bf Abstract.} In this note, we give a formula for the
multiplicity of a point on a Schubert variety in $G/Q$, $G$ being a semi simple
algebraic group, and $Q$ a parabolic sub group. This formula takes a particularly
simple form for $G=SL(n)$.





\vfill\eject

\ni {\bf Version Fran\c caise abr\'eg\'ee.} Soit $G$ un groupe alg\'ebrique
semi-simple et simplement connexe, que nous supposerons, pour simplifier, d\'efini
sur un corps alg\'ebriquement clos de caract\'eristique nulle (bien que les 
r\'esultats suivants soient en fait valables en toute caract\'eristique, et
m\^eme sur {\bf Z}). Soient $T$ un tore maximal de $G$, et $W$
son groupe de Weyl. Soit (,) un produit scalaire invariant par $W$,
sur $X(T)\otimes {\bf R}$ (on note $X(T)$ le groupe des caract\`eres
de $T$). Soit $R$ le syst\`eme de racines de $G$ relatif \`a $T$.
Soit $B$ un sous-groupe de Borel de $G$,  avec $B\supset T$.
Soit $S$ (resp.$R^+$)  l'ensemble des racines simples
(resp. positives) de $R$ associ\'ees \`a $B$. Pour $\a \in
R$, soit $s_\a$  la sym\'etrie d\'efinie par $\a$, et $X_\a$
l'\'el\'ement de la base de Chevalley de
$\gg (={\rm Lie}G$) de poids $\a$.
Pour $w \in W$, soit $e_w$ le point de
$G/B$, et $X(w)\ (={\overline {BwB}}$ (mod $B$)), la vari\'et\'e de
Schubert, associ\'es \`a $w$. L'ordre de Bruhat sur $W$ sera not\'e $\leq $.

Soit $\l$ un poids dominant, et $V(\l)$ le $G$-module simple
(sur {\bf C}) de plus grand poids $\l$. Fixons un vecteur $u$
de plus grand poids dans
$V(\l)$. Posons $u_w= w\cdot u$, et
$V_w=U^+(\gg) u_w$ (ici $U^+(\gg)$  est la sous-alg\`ebre de
$U(\gg)$, l'alg\`ebre enveloppante de $\gg$, engendr\'ee par
$\{X_\a, \a \in S\}$). Soit
$V(\l)^*$ (resp. $V_w^*$) l'espace vectoriel dual de $V(\l)$ (resp.
$V_w$)). 
 
Dans [L]$_1$ (voir aussi [L]$_2$), nous avons construit une base
$\Bb (\l) $ de $V(\l)$ qui est compatible avec l'ordre de Bruhat,
c'est-\`a-dire: $\Bb (\l)\cap  V_w$ est une base de
$V_w$. De plus, $\Bb (\l) $ est form\'ee d'\'el\'ements
$Du$, o\`u $D$ est \'egal \`a 1 ou \`a $X_{\g_r}^{n_r}\cdots
X_{\g_1}^{n_1},\ -\g_i$ simple, $n_i>0$ (pour des $n_i$ convenables), et
la d\'ecomposition $s_{\g_r}\cdots s_{\g_1}$ est r\'eduite. Notons
$\Bb^*(\l )$ la base de $V(\l)^* $  duale  de $\Bb (\l )$.

Soit $U^- _\t$ le radical unipotent du sous-groupe de Borel $B^-
_\t\ $, oppos\'e \`a $B_\t\ (=\ \t B\t^{-1})$. On peut identifier
$U^- _\t e_\t$ \`a un ouvert affine de $G/B$ qui contient $e_\t$.
Rappelons que $$U^- _\t = \prod _{\b \in \t (R^-)}\ U_\b,\ U_\b \approx
{\bf G}_a. \leqno {(\dagger )}$$ (ici $R^-$ d\'esigne l'ensemble des racines
n\'egatives). Notons $\{  x_\b, \b \in \t (R^-)\}$ le syst\`eme de coordonn\'ees
canonique, donn\'e par ($\dagger $). Notons $V=V(\l)^* \ (= V=H^0(G/B, L_\l))$, on
peut identifier $V$ aussi
$$  V= \{f:G \rightarrow k\ | \
f(gb)= \l (b) f(g),\ b\in B,\ g \in G\}$$
Si $f\in V$, et
si $f'$ d\'esigne la restriction de $f$ \`a $U^- _\t$, observons que les
\'evaluations de $\partial f' \over \partial x_\b $ et de $X_\b f,\  \b
\in \t (R^- )$ en $e_\t $ co\"incident.

En utilisant l'expression explicite des \'el\'ements de
$\Bb (\l) $, nous pouvons \'ecrire $f', \ f\in \Bb^*(\l)$,
en fonction des coordonn\'ees
$x_\b$'s. Ceci nous permet d'obtenir une formule r\'ecursive pour
$m_\t (w)$, la multiplicit\'e de $X(w)$ en $e_\t$ (ici $\t \leq w$).
Dans le cas o\`u $G/P$ est minuscule, de telles formules ont \'et\'e
obtenues dans [L-W]$_1$, [L-W]$_2$.  Dans les autres cas,
aucune formule (pour les multiplicit\'es des vari\'etes de Schubert)
ne se trouve dans la litt\'erature \`a notre connaissance,
m\^eme lorsque $G$ est le groupe lin\'eaire.

  
\vs .4cm \ni Let $G,\ B,\ T,\ W,\ X(w),\ e_w,\ $ etc be as above. (For generalities
on algebraic groups, one may refer to [B].)

Let $\lambda$ be a dominant weight, say $\lambda=\sum d_i\omega_i$,
($\omega_i$ being the fundamental weights). If $d_t=0$ for $t=j_1,\ldots,j_k$, then
we shall work with $W^Q$, the set of minimal representatives of $W_Q$ in $W$,
$W_Q$ being the subgroup of $W$ generated by the set of simple reflections
$\{s_t, t=j_1,\ldots,j_k\}$). 

\ni {\bf Definition 1.} Let $\t,\ver \in W$ such that $\ver <\t$, and $l(\ver )=
l(\t )-1$. Let $\ver = \t s_\b $, for some positive root  $\b$. 
We define $m_\l(\ver ,\t)$, {\it the} $\l$-{\it multiplicity of} $\ver $ {\it
in} $\t$ as the positive integer $(\l,\b ^*)\ (={{2(\l,\b)} \over {(\b,\b)}})$.

\vs .2cm \ni For $\t \leq w$,
let $T(w,e_\t) $ be the  the tangent space to $X(w)$ at
$e_\t $. Let  $$N_{w,\t}=\{ \b \in \t (R^+)\ |\ X_{-\b }\in
T(w,e_\t) \}. \leqno {(*)} $$

\ni {\bf Definition 2.} For $\t \in W$, let 
$p_\t$ denote the extremal weight vector in 
$H^0(G/B, L_\l) $ of weight $-\t (\l)$. Given $\t,\
w \in W,\ \t \leq w $, we define $${\rm deg}_\l^\t p_w:\ =\ {\rm the \ smallest\ } t\
{\rm such\ that }\ Dp_w=cp_\t ,\ c\in k^* $$ where $D=X_{\beta_r}^{n_r}\cdots
X_{\beta_1}^{n_1},\ -\beta_i \in N_{w,\t}$, $\sum n_i=t$.

\proclaim Theorem 1. Let $\l$ be dominant and regular. Let $\t \leq w $, and let
$\{X(w_i), i=1,\cdots ,m\}$ be all the Schubert divisors in $X(w)$ such that $e_\t
\in X(w_i) $ . Let $m_\t (w)$ denote the multiplicity of $X(w)$ at $e_\t$. We have,
$$ m_\t (w)\ {\rm deg}_\l^\t  p_w = \sum_{i=1}^m\  m_\lambda(w_i,w)\ m_{\t}
(w_i).$$

\vs .2cm \ni The formula in Theorem 1 takes a particularly simple form for $G=SL(n)$,
and $\t={\rm id}$ as described below.

\vs .2cm \ni {\bf The linear group.}

\ni Let $G=SL(n)$. It is well known that $W$ can be identified with $S_n$.

\vs .2cm \ni {\bf Definition 3.} Given $w \in S_n$, say $w = (a_1\cdots
a_n)$, let $I_w=\{i\ |\ a_i>i\}$. We define {\it the content } $c_w $ of $w $ as the
integer $\sum_{i\in I_w}\ (a_i-i)$.

\ni Recall that given a Schubert divisor $X(\t)$ in $X(w)$, we can write $\t=w (k,l)$,
for some suitable transposition $(k,l)$. In this situation, we shall denote $X(\t)$
by  $X(w_{ (k,l)})$.
 
\proclaim Theorem 2. Let $G=SL(n)$, and $w \in W$. Then
(taking $\l = \rho ), \ m_{{\rm id }} (w)$ is given by  $$m_{{\rm id }} (w)\ c_w
=\sum_{(k,l)}\  (l-k) m_{{\rm id }} (w(k,l)) $$ 
where the summation runs over all the Schubert divisors $X(w_{
(k,l)})$ in $X(w)$. 

 \vs.2cm  
 \ni {\bf Outline of Proof:}  

Given a dominant weight $\l$, let us denote $V=H^0 (G/B,L_\l)$. Then one
knows that $V$ can be identified as  $$  V= \{f:G \rightarrow k\ | \
f(gb)= \l (b) f(g),\ b\in B,\ g \in G\}. \leqno {(1)}$$

Now $V$ is also a $\gg$-module. Given $X \in  \gg$, we identify 
$X$ with the corresponding right invariant vector field $D_X$ on $G$.
Thus, if $v\in V $ corresponds to a function $f$ on $G$ as
above, then we have $D_X f = Xf$.

Let $U^- _\t$ be the unipotent part of the Borel sub group $B^-
_\t\ $, opposite to $B_\t\ (=\ \t B\t^{-1})$. We can identify
$U^- _\t e_\t$ as an affine open subset  containing $e_\t$.
Recall that $$U^- _\t = \prod _{\b \in \t (R^-)}\ U_\b,\ U_\b \approx
{\bf G}_a. \leqno {(2)}$$ (here, $R^-$ denotes the set of negative
roots). We denote by $\{  x_\b, \b \in \t (R^-)\}$, the canonical
coordinate system given by (2). If $f\in V$, then denoting the
restriction of $f$ to $U^- _\t$ by  $f'$, we note that the
evaluations of $\partial f' \over \partial x_\b $ and $X_\b f,\  \b
\in \t (R^- )$, at $e_\t $ coincide. 


\vs .2cm \ni {\bf Expression for the elements of $H^0(G/B, L_\l)$) in the local
co-ordinates $x_\b$'s}

  \ni Let $<,>$ denote the canonical pairing on $H^0(G/B,
L_\l)\times  H^0(G/B, L_\l)^*$ (here, 

\ni $H^0(G/B, L_\l)^* $ denotes the linear dual of
$H^0(G/B, L_\l)$). Let us fix a total order on $ \t (R^-) $. Then in the power series
expansion for $f'$ in the local coordinates $x_\b,\ \b \in \t (R^-) $, a typical
monomial $x_{\b_1}^{n_1}\cdots x_{\b_r}^{n_r} $ will occur with a non zero
coefficient if and only if  $X_{\b_1}^{n_1}\cdots X_{\b_r}^{n_r}f=cp_\t ,\ c\in k^*
$. Let us denote  by $Q_\t$ the extremal weight vector in $H^0(G/B, L_\l)^\vee $ of
weight  $\t (\l)$. We have, 

\ni $X_{\b_1}^{n_1}\cdots
X_{\b_r}^{n_r}f=cp_\t,\ c\in k^* $ 

\ni $\Longleftrightarrow \ <X_{\b_1}^{n_1}\cdots X_{\b_r}^{n_r}f, Q_\t> \not=
0$

\ni  $\Longleftrightarrow \ <f,X_{\b_r}^{n_r}\cdots X_{\b_1}^{n_1}Q_\t > \not=
0$ ( by $\gg$- invariance of $<,>$)

 Let us fix a dominant, regular weight $\l$.
Let $V(\l)$ be the irreducible $G$-module (over {\bf C}) with highest weight $\l$.
As above, let us fix a highest weight vector $u$ in $V(\l)$, set
$u_w= w\cdot u$, and $V_w=U^+(\gg) u_w$ (here, $U^+(\gg)$  is the subalgebra of $U(\gg)$, the
universal enveloping algebra of $\gg$, generated by $\{X_\a, \a \in
S\}$). Let
$V(\l)^*$ (resp. $V_w^*$) be the linear dual of $V(\l)$ (resp.
$V_w$)). 
 
In [L]$_1$ (see also [L]$_2$), we constructed a basis
$\Bb (\l) $ for $V(\l)$ which is compatible with the Bruhat order,
i.e., $\Bb (\l)\cap  V_w$ is a basis for $V_w$. This basis is indexed
by the so called L-S paths, which we recall below.


Let $\cc=\{\mu_0, \mu_1, \ldots, \mu_r\}$ be a $\lambda$-chain in $W$,
i.e. $\mu_i\ge
\mu_{i-1},\ l(\mu_i)=l(\mu_{i-1})+1$ (if $\lambda=\sum d_i\omega_i$, ($\omega_i$ being
the fundamental weights), and
$d_t=0$ for $t=j_1,\ldots,j_k$, then
we shall work with $W^Q$, the set of minimal representatives of $W_Q$ in $W$,
$W_Q$ being the subgroup of $W$ generated by the set of simple reflections
$\{s_t, t=j_1,\ldots,j_k\}$). Let $\mu_{i-1}=\mu_is_{\beta_i},\ \beta_i\in
R^+$. We shall denote $m_\l(\mu_{i-1},\mu_i)$ by just $m_i$.   

\vskip .2cm \noindent
{\bf Definition 4}: By a {\it weighted $\lambda$-chain} , we shall mean
$(\cc , \nn)$ where $\cc=\{\mu_0,\ldots,\mu_r\}$ is a $\lambda$-chain
and $\nn=\{n_1,\ldots,n_r\} ,\ n_i\in \zz^+$.
\vskip .2cm
\noindent
{\bf Definition 5}: A weighted $\lambda$-chain $(\cc,\nn)$ is said to be
{\it admissible} if $1\ge {n_1\over m_1}\ge {n_2\over m_2}\geq \ldots\ge
{n_r\over m_r}\ge 0$.

Let $(\cc,\nn)$ be admissible. Let us denote the unequal values in
$\{{n_1\over m_1},\ldots,{n_r\over m_r}\}$ by\hfill\break
$a_1, \ldots, a_s$ so
that $1\ge a_1>a_2>\ldots>a_s\ge0$. Let $i_0,\ldots,i_s$ be defined by
$$i_0=0,\ i_s=r,\ {n_j\over m_j}=a_t,\ i_{t-1}+1\le j\le i_t.$$
We set
$$D_{\cc,\nn}=\{(a_1,\ldots,a_s); (\mu_{i_0},\ldots,\mu_{i_s})\}$$
\vskip .2cm
\noindent
{\bf Definition 6}: Let $(\cc,\nn),\ (\cc',\nn')$ be two admissible weighted
$\lambda$-chains. We say $(\cc,\nn)\sim(\cc',\nn')$, if
$D_{\cc,\nn}=D_{\cc',\nn'}$
\vskip .2cm
\noindent
 Let $C_\lambda=\{$all admissible weighted $\lambda$-chains$\}$, and
$I(\lambda)=C_\lambda/\sim$. Each element in $I(\l)$ can be thought of as a
certain piece-wise linear path ( called an {\it L-S path of shape $\l$} (cf.[Li])) in 
$X\otimes {\bf R}\ (X$ being the weight lattice),  $$\Pi: [0,1] \rightarrow  X\otimes
{\bf R}$$ namely  , say, $a_i \leq t \leq a_{i+1},\  0\leq i\leq s$ 

\ni (here, $a_0=0,\ a_{s+1}=1$), then 
$$\Pi (t)= \sum_{j=0}^{i-1} \ (a_{j+1}-a_j)\ \mu_j(\l)\ +\ (t- a_i)\ \mu_i(\l).$$ 

\vskip .2cm
\noindent
 With notation as above, let $\pi\in I(\lambda)$, and let
$(\cc,\nn)$ be a representative of $\pi$. We denote $$i(\pi)=\mu_{i_s},\ 
j(\pi)=\mu_{i_0},\   
\nu(\pi)=\sum_{t=0}^s(a_t-a_{t+1}) \mu_{i_t}(\lambda),$$ 
(note that $i(\pi),\ j(\pi)$, and  $\nu(\pi)$ depend only on $\pi$ and not on the
representative chosen. Also note that $i(\pi)\geq  j(\pi) ). 

\vskip .2cm
\noindent
Let
$$I_w(\lambda)=\{\pi\in I(\lambda)\mid w\ge i(\pi)\}.$$
Then $\Bb (\l) $ is indexed by $I(\l)$ (and $\Bb_w (\l)=(\{v \in \Bb (\l)\ |\ v \in
V_w\})  $ is indexed by $I_w(\l)$). Further, $\Bb (\l) $ consists of
elements of the form $De$, where $D$ is either 1 or  $X_{\g_r}^{n_r}\cdots
X_{\g_1}^{n_1},\ -\g_i$ simple, $n_i>0$ (for some suitable $n_i$'s), and
$s_{\g_r}\cdots s_{\g_1}$ is reduced. Let us denote $\Bb (\l) = \{Q_\pi,\ \pi
\in I(\l)\}$, and $\Bb^* (\l )= \{p_\pi,\ \pi
\in I(\l)\}$, $\Bb^* (\l )$ being the basis of $V(\l)^* $  dual to $\Bb
(\l )$.
\vs .2cm \ni {\bf Definition 7.} A monomial $p_{\pi_1}\cdots p_{\pi_r}$ is said to be
{\it standard on} $V_w$ if
$$w\geq i(\pi_1) \geq j(\pi_1)\geq \cdots \geq i(\pi_r) \geq j(\pi_r).$$
We have (cf.[L]$_3$)

\ni \proclaim Theorem. Monomials in $p_\pi$'s of degree $m$ standard on $X(w)$
form a basis of 

\ni $H^0(X(w), L_\l^m)$.

\vs .2cm \ni This theorem together with the discussion above (regarding the
expression for the elements of $H^0(G/B, L_\l)$ in the local
co-ordinates $x_\b$'s ) enables us to obtain an inductive formula for $m_\t
(w)$ as given by Theorem 1 above.

\vskip .2cm
\ni {\bf Remark }: The details of the proofs of Theorems 1\& 2
will appear in [L]$_4$.

     


\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 [L]$_1$ V. Lakshmibai, {\it Bases for Demazure modules
for symmetrizable Kac-Moody algebras}, Contemporary Math., vol 153, 1993, 59-78.

\vskip .2 cm
\ni
[L]$_2$ 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]$_3$ V. Lakshmibai, {\it Completion of Standard Monomial Theory} (in
preparation)

\vskip .2cm
\ni [L]$_4$ V. Lakshmibai, {\it  Tangent cones at singular points on a  Schubert
variety } (in preparation).


\vs .2cm \ni 
[L-W]$_1$ V. Lakshmibai and J. Weyman {\it Multiplicites sur une variete de Schubert
dans un minuscule G/P},  
  Comptes Rendus, t.307, Serie I (1988), 993-996.

\vs .2cm \ni 
[L-W]$_2$ V. Lakshmibai and J. Weyman {\it Multiplicities of points on a Schubert
variety in a minuscule G/P}, Advances in Math, Vol. 84 (1990), 179-208.

\vskip .2cm
\noindent
[Li] P.Littelmann, {\it Littlewood-Richardson rule for symmetrizable
Kac-Moody algebras}, Inv. Math., 1993.



 
   








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