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\centerline{\bigbf Bases for quantum Demazure modules }

\centerline{V. LAKSHMIBAI
\footnote{*}{\sl  Partially supported by the NSF Grant DMS 9103129 and the Faculty 
Developement Fund (Northeastern University).}
\footnote{}{1991 Mathematics subject classification:  Primary
20G05, 20G10, Secondary 14F05, 14M15}
}


\vskip .4cm

\midinsert
\narrower
{\smrm
\baselineskip=10pt
\noindent

{\sc Abstract}. Let $\gg$ be a symmetrizable Kac-Moody algebra, and $U_q(\gg)
$ the quantized enveloping algebra of $\gg$. Let $\lambda$ be a dominant,
integral weight, and $V(\lambda)$ the corresponding simple $U_q(\gg)$-module
($q$ being generic). Let $V_A(\lambda)$ be the canonical
$A$-form of $V(\lambda)$ where $A=\zz[q,q^{-1}]$. Let $W$ be the Weyl group
of $\gg$. For $w\in W$, let $V_{A,w}$ be the (quantum) Demazure submodule of
$V_A(\lambda)$. We construct an $A$-basis for $V_A(\lambda)$ compatible with
$\{V_{A,w},\ w\in W\}$, consisting of $\{De\}$, where $e$ is the highest
weight vector in $V(\lambda)$, and $D$ is either 1 or of the form
$F_{\beta_r}^{(n_r)}\cdots F_{\beta_1}^{(n_1)},\ \beta_i$ simple, $n_i>0$
(for some suitable $n_i$'s), and $s_{\beta_r}\cdots s_{\beta_1}$ is reduced.
We also show that for $w\in W$, the transition matrix from our basis for $V_{A,w}$
to Kashiwara's global basis is upper triangular with diagonal entries equal to 1 (for
a suitable indexing). We also give an explicit expression for the crystal base
$B(\lambda)$. Given $w\in W$, and $\a $ a simple root such that $w<s_\a w $ (=$\t $,
say), we exhibit an unique ``Demazure" $U_q(sl_2)$ structure on $V_\t/V_w$.   }
\endinsert

 


\beginsection \S1 Introduction.

Let $\gg$ be a symmetrizable Kac-Moody algebra, and $U_q(\gg)$ the quantized
enveloping algebra of $\gg$ as constructed by Drinfeld (cf. [D]) and Jimbo
(cf. [J]). In the sequel, we shall denote $U_q(\gg)$ by just $U$. For
notation of objects in $U$, we follow [K]$_1$, (with one difference, namely, the
$e_i,\ f_i$ of [K]$_1$, will be denoted by $E_i,\ F_i$ respectively, since we
shall use $e_i,\ f_i$ for some other objects. Also, if $\alpha_i=\alpha$ then
we often denote $E_i,\ F_i$ by $E_\alpha,\ F_\alpha$ respectively). Let
$U^+$ (resp. $U^-$) be the subalgebra of $U$ generated by $E_\alpha$
(resp. $F_\alpha$), $\alpha$ simple. Let $A=\zz[q,q^{-1}]$, and $U_A^\pm$ the
$A$-subalgebra of $U$ generated by $E_\alpha^{(r)}$ (resp. $F_\alpha^{(r)}$)
, $\alpha$ simple, $r\in\zz^+$.

Let $\lambda$ be a dominant, integral weight, and $V(\lambda)$ the simple
$U$-module with highest weight $\lambda$ ($q$ being generic). Let us fix a
highest weight vector $e$ in $V(\lambda)$. Let $V_A(\lambda)=U_A^-e$. Let
$W$ be the Weyl group of $\gg$. For $w\in W$, let $e_w$ be the extremal
weight vector in $V(\lambda)$ of weight $\lambda(w)$. Let $V_w=U^+e_w$,
$V_{A,w}=U_A^+e_w$. We refer to $V_{A,w}$'s (resp. $V_w$'s) as the {\it
quantum Demazure modules}. In [L]$_1$, we conjectured an $A$-basis
$\Bb(\lambda)$ for $V_A(\lambda)$ compatible with $\{V_{A,w},\ w\in W\}$. In
[L]$_3$ ( see also [L]$_2$ ), we carried out a construction of $\Bb(\lambda)$. It
turns out that there is a gap in that construction ( as was pointed out by
Littelmann- the authour is thankful to Littelmann for this). In this paper, we
construct $\Bb(\lambda)$ using Kashiwara's global base (cf.[K]$_1$, [K]$_2$), and the
isomorphism of the (oriented, colored) graph $G(B(\l))$ of crystals with {\cal
G}($I(\l))$ of L-S paths of shape $\l$ ( see \S5\& \S6 for details ). An
element in our basis $\Bb(\lambda)$ looks like $De$, where $D$ is either 1 or of the
form $F_{\beta_r}^{(n_r)}\cdots F_{\beta_1}^{(n_1)} ,\ \beta_i$ simple, $n_i>0$ (for
some suitable $n_i$'s), and $s_{\beta_r} \cdots s_{\beta_1}$ is reduced. In
particular our method gives a very constructive basis for $V_A(\lambda)$. We also
give an explicit expression for the crystal base $B(\lambda)$ in terms of the
elements in our basis. In fact, we completely describe  (cf. \S9)  $\Bb(\lambda)$,
and $B(\l)$,  for $U_q(sl_3)$. 

\ni We also show that for $w\in W$ the transition matrix from $\Bb_w(\lambda)$ to
$\bb_w(\lambda)$ ($=$ the basis for $V_w$ as constructed in [K]$_2$) is upper
triangular with diagonal entries equal to 1 (for a suitable indexing of the bases)
(cf Theorem 6.6). 


\ni Let $\t \in W$, and let $\a$ be a simple root such that $\t>s_\a\t$. Let us
denote $s_\a\t$ by $w$. Let $U_q(\a)$ denote the copy of $U_q(sl(2))$ in $U$. 
Let $(\t (\lambda ),\a^*)$
= $-n$ ( note that $(\t (\lambda ),\a^*)<0 )$. Following
[De] (for the case $q=1$), we show (cf.\S8) that there exists an unique representation
$\rho$ of $U_q(\a)$ in $V_\tau /V_w$  such that $$ \rho (F_\a ){\overline e_\t} =0,\
\rho (K^\epsilon _\a ){\overline e_\t}= q^{-\epsilon (n-1)}{\overline e_\t},\ \rho
(E_\a ){\overline u}= E_\a {\overline u},\ \forall u \in V_\t $$
where $\epsilon = \pm 1$ ( we refer to this as the 
{\it  Demazure $U_q(\a)$-structure
on $V_\tau /V_w$}). Using our basis $\Bb_\t(\lambda)$ for $V_\t$, we give an
explicit realization of the Demazure $U_q(\a)$-structure
on $V_\tau /V_w$.

\vs .2cm The authour would like to thank the referee for many important suggestions.


\beginsection \S2. Preliminaries.

We preserve the notations of \S1. For notations on objects related to $U$
we follow [K]$_1$.

\proclaim Lemma 2.1. {\rm (cf. [L-S])}. Let $w,\tau\in W$. Further, let
$w=s_\alpha\tau,\ l(w)=l(\tau)-1$, and $\alpha$ simple. Let $\theta\in W,
\ \theta\le\tau$. Then, either $\theta\le w$, or $\theta=s_\alpha\theta'$,
and $\theta' \leq w$.

\vskip .2cm
\noindent {\bf 2.2}. For a simple root $\alpha$, let $U_\alpha$ (resp.
$U_{-\alpha}$) be the subalgebra of $U$ generated by $E_\alpha$ (resp.
$F_\alpha$). Let $U_{\alpha,A}$ (resp. $U_{\alpha,A}^-$) denote the
$A$-submodule of $U$ spanned by $E_\alpha^{(n)}$ (resp. $F_\alpha^{(n)}$)
, $n\in\zz^+$. Let $U_q(\alpha)$ (resp. $U_A(\alpha)$) denote the copy of
$U_q(sl(2))$ (resp. $U_A(sl(2))$ in $U$ associated to $\alpha$.

\proclaim Lemma 2.3. Let $w,\tau$ be as in Lemma 2.1. Then $V_{A,\tau}=
U^-_{\alpha,A}V_{A,w}$.

\vskip .2cm
\noindent {\sl Proof.} The result follows from the fact that $V_{A,\tau}$ is
the smallest $U^-_{\alpha,A}$-stable submodule of $V_A(\lambda)$ containing
$V_{A,w}$.

\vskip .2cm
\noindent {\bf 2.4}. In the sequel, we will be often using the following
relations in $U$ (cf [K]$_1$)
$$E_i^{(n)}F_i^{(m)}=\sum_{k\ge0}F_i^{(m-k)}E_i^{(n-k)}{t_iq_i^{n-m}\brace
k}_i$$
$$F_i^{(n)}E_i^{(m)}=\sum_{k\ge0}E_i^{(m-k)}F_i^{(n-k)}{t_i^{-1}
q_i^{n-m}\brace k}_i$$
where
$${x\brace n}_i={\prod_{k=1}^n\{q_i^{1-k}x\}_i\over[n]_i!},$$
$$\{x\}_i={x-x^{-1}\over q_i-q_i^{-1}}.$$
(here, $t_i,\ q_i,\ [n]_i$ are as in [K]$_1$).

\vskip .2cm
\noindent {\bf 2.5 Quantum Demazure character formula}

For a simple root $\alpha$, let $L_\alpha$ be the Demazure operator on
$\zz[P]$ ($P$ being the weight lattice) given by
$$L_\alpha(e^\mu)={e^{\mu+\rho}-e^{s_\alpha(\mu+\rho)}\over 1-e^{-\alpha}}\ e^{-\rho}$$
For $w\in W$ with a reduced expression $w=s_{i_1}\cdots s_{i_r}$, let
$L_w=L_{i_1}L_{i_2}\cdots L_{i_r}$, where $L_{i_t}=L_{\alpha_{i_t}}$. Then
we have (cf [K]$_2$)
$$\char V_w=L_w(e^\lambda)\leqno{(1)}$$
Let $\tau=s_\alpha w$, for some simple root $\alpha$, where $\tau>w$. Then
we have (cf [K]$_2$)
$$\char V_\tau=L_\alpha(\char V_w)\leqno{(2)}$$
The following is easily checked:
$$L_\alpha(e^\mu) \quad = \quad \cases{ \quad\sum_{i=0}^ne^{\mu-i\alpha},
& if $(\mu,\alpha^*)=n\ge0$\cr\cr
\quad 0, & if $(\mu,\alpha^*)=-1$\cr\cr
\quad-\sum_{i=1}^me^{\mu+i\alpha}, & if $(\mu,\alpha^*)=-(m+1)\le-2$\cr}
\leqno{(3)}$$
\vskip .2cm
\noindent 
{\bf 2.6}. Let $E$, $F$, $q^h$ denote the usual elements in $U_q (sl_2)$. Let 
$U_q (\underline b)$ be subalgebra generated by $\{q^h,E\}$. Let
$N$ be a finite dimensional $U_q (\underline b)$-module. For a weight vector  
$u$ in $N$, we shall
denote its weight by $r_u$ ( note that $r_u$ is given by  $q^hu=q^{r_{u}}u$). 

\vskip .2cm
\noindent  {\bf \S3. L-S paths of shape $I(\lambda)$ }

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 $\mu_{i-1}=s_{\beta_i}\mu_i,\ \beta_i\in
R^+$. Let $m_i=(\mu_i,\beta_i^*)$.
\vskip .2cm
\noindent
{\bf Definition 3.1}: 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 3.2}: A weighted $\lambda$-chain $(\cc,\nn)$ is said to be
{\it admissible} if $1\ge {n_1\over m_1}\ge {n_2\over m_2}\ldots\ge
{n_r\over m_r}\ge 0$.

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 3.3}: 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
{\bf 3.4.} Let $C_\lambda=\{$all admissible weighted $\lambda$-chains$\}$, and
$I(\lambda)=C_\lambda/\sim$. An element $\pi $ in $I(\lambda)$ can be thought of as
a certain piecewise linear path in the convex hull in $X\otimes\qq$ of the orbit
$W\cdot\lambda$ , namely, for $ a_{k+1}\leq t \leq a_k, \ 0\leq k\leq s $
 $$ \pi (t)= \sum_{j=k+1}^s(a_j-a_{j+1}) \mu_{i_j}(\lambda) \ +\ 
(t-a_{k+1}) \mu_{i_k}(\lambda) $$
(here $a_0=1, a_{s+1}=0$, $(\cc,\nn) = \{(a_1,\ldots,a_s); (\mu_{i_0},\ldots,\mu_{i_s})\}$
is a representative of $\pi$, and $X=\zz[P]$). 

\vs .2cm In [Li]$_2$, the elements of
$I(\lambda)$ are called the {\it Lakshmibai-Seshadri paths of shape} $\lambda$. Hence
keeping up with the terminology in [Li]$_2$, in the sequel we shall refer to the
elements of $I(\lambda)$ as L-S paths of shape $\lambda$. 
 \vskip .2cm
\noindent
{\bf 3.5.} With notation as above, let $\pi\in I(\lambda)$, and let
$(\cc,\nn)$ be a representative of $\pi$. Keeping up with the notation in
[Li]$_2$, we denote $$\phi(\pi)=\mu_{i_s},\  \nu(\pi)=\sum_{t=0}^s(a_t-a_{t+1})
\mu_{i_t}(\lambda),$$ 
 (note that $\phi(\pi)$ and
$\nu(\pi)$ depend only on $\pi$ and not on the representative chosen).

\vs .2cm For $w \in W$, set
$$I_w(\lambda)=\{\pi\in I(\lambda)\mid w\ge\phi(\pi)\}.\leqno{(*)}$$
\vskip .2cm
\noindent
{\bf 3.6 L-S character formula.}

\ni We have (cf 2.5, (1))
$$\char V_w=L_w(e^\lambda)\leqno{(1)}$$
On the other hand in [Li]$_2$ it is shown that
$$L_w(e^\lambda)=\sum_{\pi\in I_w(\lambda)}e^{\nu(\pi)}\leqno{(2)}$$
where $I_w(\lambda)$ as in (*) ( cf. \S 3.5 above ).
Hence we obtain
$$\char V_w=\sum_{\pi\in I_w(\lambda)}e^{\nu(\pi)}\leqno{(3)}$$
$$\char V(\lambda)=\sum_{\pi\in I(\lambda)}e^{\nu(\pi)}\leqno{(4)}$$

\vs .2cm \ni For a simple root $\a$, let $e_\a, f_\a $ be the operators on
$I(\lambda )$ as defined in [Li]$_2$. We recall the following result from [Li]$_2$.

\proclaim Lemma 3.7.  Let $\pi $ be an L-S path, and $\a $ a  simple root. 
\item{(a)} If $s_\a \phi (\pi) < \phi (\pi) $, then $e_\a \pi \not= 0.$
\item {(b)} Suppose $f_\a\pi \not= 0$, then either $\phi (f_\a\pi)= \phi (\pi) $, or
$\phi (f_\a\pi)=s_\a \phi (\pi) > \phi (\pi) $, and $e_\a\pi = 0$ 

\proclaim Lemma 3.8. Let $\pi $ be an L-S path, and let $\phi (\pi) = \tau $. Let
$\tau = s_{i_1}s_{i_2} \cdots s_{i_r} $ be a reduced expression for $\tau $. Then
there exist uniquely determined  integers $ n_{i_1},\ n_{i_2},\  \cdots ,n_{i_r}\ 
>0$, such that $ \pi = f_{i_1}^{n_1}\cdots f_{i_r}^{n_r} \pi_0 $, $\pi_0$ being the
L-S path given by the line segment joining origin and $\lambda $.

\ni {\sl Proof}: Denote $\a_{i_1} $ by just $\a $. The fact that 
$s_\a \phi (\pi) < \phi (\pi) $, implies that  $e_\a \pi \not= 0$ (cf. Lemma
3.7,(a)).  Let $n_1$ be the largest integer such that $ e_\a^{n_1}\pi  \not= 0 $ .
Let $\pi' = e_\a^{n_1}\pi $. Then  
$$\pi = f_\a^{n_1}\pi',\  \ e_\a \pi' =0 .\leqno {(1)}$$ 
We have (cf. Lemma 3.7,(b)) 
$$\phi (\pi) = \phi (\pi ') \ {\rm or \ } s_\a\phi (\pi '). \leqno {(2)}$$
The fact that $e_\a \pi' =0$ implies (cf. Lemma 3.7,(a))
$$\phi (\pi ') <s_\a\phi (\pi ').\leqno {(3)}$$
Now (2) and (3) together with the fact that $\phi (\pi ) >s_\a\phi (\pi )$ imply
$$\phi (\pi ') =s_\a\phi (\pi ).\leqno {(4)}$$
Thus 
$$ \pi = f_\a^{n_1}\pi',\ \phi (\pi') = s_{i_2} \cdots s_{i_r}.\leqno {(5)}$$  
The result now follows by induction on $l(\tau)$.
    
\vs .2cm \ni {\bf \S4. Kashiwara's crystal base}

\noindent {\bf 4.1} Let $B(\lambda)$ be the (lower) crystal  base for
$V(\lambda)$ as constructed in [K]$_1$. For $b\in B(\lambda)$, we shall
denote (cf [K]$_1$)
$$\epsilon_i(b)=\max\{n\mid\tilde e_i^nb\not=0\}\leqno{(1)}$$
$$\varphi_i(b)=\max\{n\mid\tilde f_i^nb\not=0\}\leqno{(2)}$$
Let
$$\xi_i(b)=\epsilon_i(b)+\varphi_i(b)\leqno{(3)}$$
$$\wt_i(b)=<h_i, \wt(b)>\leqno{(4)}$$
If $\epsilon_ib=0$, then we denote
$$S_i(b)=\{\tilde f_i^kb,\ 0\le k\le\varphi_i(b)\}\leqno{(5)}$$
In the sequel for a simple root $\alpha$, say $\alpha=\alpha_i$, we shall
denote $\epsilon_i(b),\ \varphi_i(b),\ \xi_i(b),\ \wt_i(b),\hfill\break
 \ S_i(b)$, by respectively $\epsilon_\alpha(b),\ \varphi_\alpha(b),
\ \xi_\alpha(b),\ \wt_\alpha(b),\ S_\alpha(b)$ also.

\noindent {\bf 4.2}  It is shown in [K]$_2$ that for $w\in W$, there exists
a subset $B_w(\lambda)$ of the lower crystal base $B(\lambda)$ such that
\item{1.} $\{G_\lambda(b),\ b\in B_w(\lambda)\}$ is a $\qq(q)$-basis for
$V_w$.
 
\noindent
(We shall denote this basis by $\bb_w(\lambda) $; here, $ G_\lambda(b) $ 
is as in [K]$_2$).
\item{2.} $\tilde e_iB_w(\lambda)\subset B_w(\lambda)\cup\{0\}$.
\item{3.} Let $b\in B(\lambda)$ such that $\epsilon_i(b)=0$ for some $i$.
Then $S_i(b)\cap B_w(\lambda)$ is either $\emptyset$ or $S_i(b)$ or $\{b\}$.
\item{4.} If $s_iw<w$, then
$$B_w(\lambda)=\{\tilde f_i^kb,\ k\ge 0,\ b\in B_{s_iw}(\lambda),\ \tilde
e_ib=0\}\setminus \{0\}.$$

\ni {\bf Remark} $\{G_\lambda(b),\ b\in B_w(\lambda)\}$ is in fact an $A$-basis
for $V_{w,A}$ (cf. [H]).

\vs .2cm \noindent {\bf 4.3} Let $b\in B(\lambda)$. Then there exists an unique
minimal $x$ (minimal under B\"{r}uhat order),  such that $b\in B_x(
\lambda)$ ( by B\"{r}uhat-order compatibility of $B(\lambda)$). We set 
$$\phi (b) = x. $$
\vs .2cm
\proclaim Lemma 4.4. Let $ b \in B(\lambda)$, and $\a $ a simple root such that 
$s_\a \phi( b) < \phi( b) $. Then $\tilde e_\a b \not= 0$.

\ni {\sl Proof}: If possible, let us assume $\tilde e_\a b = 0$. Let $\phi( b)=\t $. 
Then since $ \t > s_\a \t $, we have,
$$B_\t(\lambda)=\{\tilde f_i^kb',\ k\ge 0,\ b'\in B_{s_\a \t}(\lambda),\ \tilde
e_ib'=0\}\setminus \{0\}.$$
Hence taking $b'=b$, we obtain $\phi( b)= \phi( b') \leq s_\a \t < \t (=\phi( b))$, 
which is not possible. Hence our assumption is wrong, and the result follows.



\vs .2cm
\proclaim Corollary 4.5. Let $ b \in B(\lambda)$, and $\a $ a simple root such that 
$\tilde e_\a b = 0$. Then $\phi( b) < s_\a \phi( b) $. 

\vs .2cm
\proclaim Lemma 4.6. Let $ b \in B(\lambda)$, and let $\phi (b) = \tau $. Let
$\tau = s_{i_1}s_{i_2} \cdots s_{i_r} $ be a reduced expression for $\tau $. Then
there exist uniquely determined  integers $ n_{i_1}, n_{i_2},  \cdots , n_{i_r} 
>0$, such that $b  = \tilde f_{i_1}^{n_1}\cdots \tilde f_{i_r}^{n_r} e $,
$e$ being the highest weight vector in $V_\lambda $.

\ni {\sl Proof}: Denote $\a_{i_1} $ by just $\a $. The fact that 
$s_\a \phi (b) < \phi (b) $, implies that  $\tilde e_\a b \not= 0$ (cf. Lemma
4.4).  Let $n_1$ be the largest integer such that $ \tilde e_\a^{n_1}b  \not= 0 $
. Let $b' = \tilde e_\a^{n_1}b $. Then  $b = \tilde f_\a^{n_1}b',\  \ \tilde e_\a
b' =0 $.  This implies (cf. Corollary 4.5)
$$ \phi (b') <s_\a \phi (b') \leqno {(1)} $$
Also, by hypothesis, we have
$$ \phi (b) >s_\a \phi (b) \leqno {(2)} $$
The fact that $ b=\tilde f_\a^{n_1}b'$ implies
$$ \phi (b)= s_\a \phi (b') \ {\rm or\ } \phi (b'). \leqno {(3)}$$
 Hence, from (1), (2), and (3), we obtain
$$\phi (b')=s_\a \phi (b). \leqno {(4)} $$ 
Thus
$$ b = \tilde f_\a^{n_1}b',\ \phi (b')= s_{i_2} \cdots s_{i_r}. \leqno {(5)} $$  
The result now follows by induction on $l(\tau)$. 


\vs .2cm \noindent {\bf \S5. The bijection  $\Phi_{x,\lambda}$}. 

\vs .2cm \noindent {\bf 5.1 Crystal graph $G(B(\lambda))$ of $B(\lambda)$} (cf.
[K]$_1$).

\noindent
$G(B(\lambda))$ is the oriented, colored (by simple roots) graph with
$B(\lambda)$ as the set of vertices, and $b\buildrel \a \over\rightarrow b'$,
if $b'=\tilde f_\a b$.

\noindent {\bf 5.2 The graph of $\Gg(I(\lambda))$ of L-S paths of shape
$\lambda$} (cf. [Li]$_2$)

\noindent
$\Gg(I(\lambda))$ is the oriented, colored (by simple roots) graph with
$I(\lambda)\ (=\{$ L-S paths of shape $\lambda\}$) as the set of vertices and
$\pi\buildrel \a \over\rightarrow\pi'$, if $\pi'=f_{\alpha}\pi$. 

\vs .2cm \noindent {\bf 5.3. Littelmann's conjecture}: $\Gg(\pi(\lambda))$ and
$G(B(\lambda))$ are isomorphic.

\ni  This conjecture has been proved to be true independently by
Kashiwara (cf.[K]$_3$),  and Joseph (cf. [Jo]$_2$).



\vs .2cm
\noindent {\bf 5.4. } Let $\pi \in I(\lambda)$. We shall denote wt$_\a \pi$ =
$<\nu(\pi), h_i>$.  For $\pi  $ such that $\ e_\a \pi=0$, let $\ S_\a \pi =\{f_\a^k
\pi,\ 0\leq k\leq  {\rm wt}_\a \pi \}  $. Let $w\in W$. Let $I_w(\lambda)$  
be as in \S3.5. Recall (cf.[Li]$_2$) \item{1.} $ e_\a I_w(\lambda)\subset
I_w(\lambda)\cup\{0\}$, for $\a$ simple. \item{2.} Let $\pi\in I(\lambda)$ such that
$\ e_\a \pi=0$ for some $\a$ simple. Then $S_\a \pi \cap I_w(\lambda)$ is either
$\emptyset$ or $S_\a\pi$ or $\{\pi\}$. 
\item{3.} If $s_\a w<w$, then
$$I_w(\lambda)=\{ f_\a^k\pi,\  0\leq k\leq  {\rm wt}_\a \pi,\ \pi \in
I_{s_\a w}(\lambda),\  e_\a \pi =0\}.$$

  
\vs .2cm
\noindent Let $w\in W$. We set
$$B'_w (\lambda) = \{b \in  B_w(\lambda)\  |\ \phi( b)= w \}$$
$$I'_w(\lambda)=\{\pi\in I_w(\lambda)\mid\phi(\pi)=w\}$$


\vs .2cm \ni {\bf 5.5 A convention on reduced expressions.} Given $\t$, and a reduced
expression $\tau = s_{i_1}s_{i_2} \cdots s_{i_r} $ , for any $x\leq \t$, a reduced
exprssion for $x$ corresponds to a subsequnce  $\{j_1,j_2, \cdots j_s\} $ of
$\{i_1,i_2, \cdots i_r\} $. There is a canonical reduced expression for $x$, namely
the one which is lexicographically the least ( here, the lexicographic ordering on 

\ni $\{{\rm subsequnces}\ {\underline j}= \{j_1,j_2, \cdots j_s\}\   {\rm of} 
\ \{i_1,i_2, \cdots i_r\}\ {\rm of\  a\  given\  length}\  s \}$ is the obvious
one, namely ${\underline j}<{\underline k}$, if there exists a $t \leq s $ such
that $j_p =k_p,\ p<t$, and $j_t <k_t$  ). In the sequel, the elements of
$I'_x(\lambda) $ (respectively $ B'_x(\lambda)$), $x \leq \t$ will be expressed as  $
f_{j_1}^{m_1}\cdots f_{j_s}^{m_s} \pi_0 $ (respectively   $\tilde
f_{j_1}^{(n_1)}\cdots \tilde f_{j_s}^{(n_s)} e $) using this canonical reduced
expression for $x$ ( cf. Lemmas 3.8 \& 4.6).   

\vs .2cm
\noindent {\bf 5.6. The bijections  $\Phi_{x,\lambda},\ \Phi'_{x,
\lambda}$.} Let notations be as in 5.5. In view of the isomorphism of 
$\Gg(\pi(\lambda))$ and
$G(B(\lambda))$, we have a canonical bijection $\Phi_{\tau, \lambda}:I_\t(\lambda)
\rightarrow B_\t(\lambda) $ such that if $\pi \in I_\t(\lambda)$, say $\pi=
f_{i_1}^{n_1}\cdots f_{i_r}^{n_r} \pi_0 $, for suitable $n_i \geq 0$, then    
$\Phi_{\tau, \lambda}(\pi)=   \tilde f_{i_1}^{n_1}\cdots \tilde f_{i_r}^{n_r} e
$. Further, for each $x\le\tau$, the restrictions of $\Phi_{\tau,
\lambda}$ to $I_x(\lambda)$, and $I'_x(\lambda)$ induce bijections
$$\Phi_{x,\lambda}:I_x(\lambda)\buildrel{\rm bij}\over\longrightarrow B_x(
\lambda)$$
$$\Phi'_{x,\lambda}:I'_x(\lambda)\buildrel{\rm bij}\over\longrightarrow B'_x(
\lambda)$$
In the sequel, given $\te \in I_\t(\lambda) $, we shall denote the corresponding
element in $B_\t(\lambda)$ by $b_\te$. 

\vfill\eject








 \noindent {\bf \S 6. The L-S basis $\Bb_\t(\lambda)$  }

\ni {\bf 6.1 An indexing of $I_\tau(\lambda)$.}


\noindent Let $\pi \in I_\tau(\lambda) $, and let $\phi(\pi)= x.$ Then $\pi $
 has an unique expression
$$\pi=f_{j_1}^{n_1}\cdots f_{j_s}^{n_s}\pi_0,\ \ n_i>0,$$
and $x=s_{j_1}\cdots s_{j_s}$ is the canonical reduced expression (cf. Lemma 3.8,
 \S 5.5) . Also,
$$I_\tau(\lambda)=\dot\cup_{y\le\tau}\ I'_y(\lambda)$$




\noindent {\bf I. A total order $\succ$ on $I'_x(\lambda)$}. Let ${\pi_1}
,\ {\pi_2} \in I'_x(\lambda)$, say
$$\pi_1=f_{j_1}^{m_1}\cdots f_{j_s}^{m_s}\pi_0,\ m_i>0$$
$$\pi_2=f_{j_1}^{l_1}\cdots f_{j_s}^{l_s}\pi_0,\ l_i>0$$
We say that $\pi_1 \succ \pi_2$, if $(m_1,\ldots,m_s)$ is
lexicographically $\ge(l_1,\ldots, l_s)$, i.e., there exists a $t,\ 1\le t\le
s$ such that $m_i=l_i,\ i<t$, and $m_t>l_t$. We label the elements of
$\Bb'_x(\lambda)$ as $v_{1},\ v_{2},\ \ldots,\ v_{i_x},\ i_x=\#I'_x
(\lambda)$, where $v_{i}\succ v_{j}$, if $i>j$.

\noindent {\bf II. An indexing $J_\tau$ of $I_\tau(\lambda)$}.

\noindent
We first take a total ordering $\buildrel t\over>$ on $\{x \in W,\ x\le \tau\}$ in
such a way that if $x_1>x_2$ (in the B\"{r}uhat order), then $x_2\buildrel t\over>x_1$
. We now give an indexing $J_\tau$ of $I_\tau(\lambda)$ by putting $I'
_{x_1}(\lambda)$ before $I'_{x_2}(\lambda)$, if $x_2\buildrel t\over>x_1$,
and taking the total order $\succ$ within each $I'_x(\lambda)$ as described
in I. We shall denote the corresponding total order on $I_\tau(\lambda)$ 
by $\succ$.


\noindent {\bf 6.2 An indexing $K_\tau$ of $B_\tau(\lambda)$.}

\noindent
Through the bijection $\Phi_{\tau,\lambda}:I_\tau(\lambda)\buildrel{\rm
bij}\over\rightarrow B_\tau(\lambda)$,  the total order $\succ$ on $I'_x(\lambda)$
induces a total order  $\succ$ on $B'_x(\lambda)$, $x\leq \tau$; the indexing
$J_\tau$ of $I_\tau(\lambda)$  induces an indexing $K_\tau$ of $B_\tau(\lambda)$. We
shall denote the corresponding  total order on $B_\tau(\lambda)$ also  
by $\succ$.

\ni For the rest of this section  
we fix a  $\tau \in W$, and a reduced expression 
$\tau = s_{i_1}s_{i_2} \cdots s_{i_r} $ for $\tau $. We shall denote $\a_{i_1}$ by
$\a $, and  $ s_\alpha \tau$ by $w$ (note that $\tau >w$). 
    
\proclaim Lemma 6.3. Let $b \in B_w(\lambda)$, and 
 $k>0$. Let  $$F_\alpha^{(k)}G_\lambda(b)={{
\epsilon_\alpha(b) + k \atopwithdelims [] k}}_\alpha G_\lambda (\tilde
f_\alpha^kb)+\sum_{b'\in I}d_{b'}G_\lambda(\tilde f_\alpha^{n_{b'}}b')
\leqno{(*)}$$   where $I\subseteq B_w(\lambda)$, and $\tilde e_\alpha b'=0$  (cf.
[K]$_2$).  Then
$n_{b'}> k , \  b' \in I$. 

\ni {\sl Proof}: We have (cf.  [K]$_2$), $$\xi_\alpha(b') > \xi_\alpha(b),\ b' \in I
.$$  Now this fact together with the fact $\tilde e_\alpha b'=0$
  implies $n_{b'}>\epsilon_\alpha(b)+k$ ( by weight  considerations in (*)).
 The required result now follows from
this. 
\vs .2cm
\ni {\bf Remark 6.4} The relation (*) holds even over $A$ (cf. [H]).



\proclaim Theorem 6.5. Let $\pi \in I(\lambda )$. Let $\phi (\pi) =\tau$. Let $ \pi =
 f_{i_1}^{n_1}\cdots f_{i_r}^{n_r} \pi_0 $ (notations being as in Lemma 3.8). Set 
$Q_\pi
 =
 F_{i_1}^{(n_1)}\cdots F_{i_r}^{(n_r)} e $. (Note that $Q_\pi  \in V_{A,\t}$.) Let
$$Q_\pi=\sum_{b\in I}a_bG_\lambda(b),\  \ a_b \in A,\ a_b \not= 0, \leqno {(*)}$$

\noindent where $I$ is some subset of $B_\tau(\lambda)$. Then
\item{(1)} $G_\lambda (b_\pi)$ occurs with coefficient $1$ in (*).
\item {(2)} $b \succ b_\pi,\ b \in I. $

\noindent {\sl Proof.} We prove the result by induction on $l(\tau)$. If $l(\tau) =
0$, then  the result is obvious. Let then $l(\tau) \geq 1$.
We have 
$$Q_\pi\ = F_\alpha^{(k)}Q_\theta \leqno{(1)} $$
where $k= n_1 $, $\a=\a_{i_1}$,  
 and  $\theta  =f_{i_2}^{n_1}\cdots f_{i_r}^{n_r}\pi_0 $. We have $$
\theta\in I'_w (\lambda) ,\ {\rm and}\  e_\a \theta =0 \leqno{(2)}$$ (cf. Proof of
Lemma 3.8).
 We have (by induction hypothesis)
$$Q_\theta\ =\ G_\lambda (b_\theta)\ +\ \sum_{b\in I_1}c_bG_\lambda(b),\ c_b \in A,\
\ b\succ b_\theta \leqno{(3)}$$

\noindent where $I_1\subseteq B_w (\lambda)$. 


\noindent We have (cf. Lemma 6.3) for $ b\in B_w (\lambda)$, 
$$F_\alpha^{(k)}G_\lambda(b)\ =\ {{ \epsilon_\alpha(b) + k \atopwithdelims []
k}}_\alpha G_\lambda (\tilde f_\alpha^kb)\ +\ \sum_{b'\in I_2}d_{b'}G_\lambda(\tilde
f_\alpha^{n_{b'}}b'),\ d_{b'} \in A \leqno{(4)}$$
where $I_2\subseteq B_w (\lambda),\ \tilde e_\alpha b'=0 $, and  
$$ n_{b'}>k ,\   b' \in I_2 \leqno{(5)}$$
Hence, we obtain   
$$Q_\pi\ =\ G(\tilde f_\alpha^k b_\theta)\ +\ \sum_{b\in I_1}s_b G_\lambda(
\tilde f_\alpha^kb)+\sum_{b\in I_3}t_bG_\lambda(\tilde f_\alpha^{m_b}b),\ s_b,t_b
\in A \leqno{(6)}$$ 

\noindent
where $I_3\subseteq B_w (\lambda)$, and
$$ m_b>k,\ {\rm for}\ b\in I_3\leqno{(7)}$$
(note that $\tilde e_\alpha b_\theta =0,$ since $ e_\alpha  \theta =0 (cf.(2))$.
 Hence (6)
takes the form $$Q_\pi\ =\ G(\tilde f_\alpha^k b_\theta)\ +\ \sum_{c\in
I_4}l_cG_\lambda(c),\ l_c \in A\leqno{(8)}$$

\noindent where $I_4\subseteq B_\tau(\lambda)$, $c$  
 = either $\tilde f_\alpha^kb $,  $b \in B_w (\lambda)$, $\ b\succ b_\theta$, or 
$=\tilde f_\alpha^{m_b}b,\ b \in B_w (\lambda),\  
m_b>k$. We have $\phi (c) \leq \t $. If $\phi (c) < \t $, then $ c\succ b_\pi $ (
clearly).  If $\phi (c) = \t $, then we have 
$c$  
 = either $\tilde f_\alpha^kb $,  $b \in B'_w (\lambda)$, $\ b\succ b_\theta$, or 
$=\tilde f_\alpha^{m_b}b,\ b \in B'_w (\lambda),\  
m_b>k$. Hence we obtain $ c\succ b_\pi $. This completes the proof of 
Theorem 6.5 (note that $\pi=f_\a^k \te$, and $\tilde f_\alpha^kb_\theta = b_\pi$). 


\proclaim Theorem 6.6. Let $\tau\in W$. The $A$-module $V_{A,\tau}$\ has a
basis $\Bb_\tau(\lambda)=\{Q_\pi,\ \pi\in I_\tau(\lambda)\}$ with the
following properties:
\item{{\rm (i)}} $Q_\pi$ is a weight vector of weight $\nu(\pi)$.
\item{{\rm (ii)}} $\Bb_\tau(\lambda)=\{e\}\cup\{F_{\gamma_1}^{(m_1)}\dots
F_{\gamma_s}^{(m_s)}e,\ \gamma_i$ simple, $m_i>0$ (for some suitable
 $m_i$'s), and $s_{\gamma_1}\cdots s_{\gamma_s}$ is a reduced expression
$\le\tau\}$.
\item{{\rm (iii)}} For $\tau'\le\tau, \{Q_\pi\mid\tau'\ge\phi(\pi)\}$ is an
$A$-basis for $V_{A,\tau'}$.
\item{{\rm (iv)}} If $Q_\pi = F_{\gamma_1}^{(m_1)}\dots
F_{\gamma_s}^{(m_s)}e $, then $\pi = f_{\gamma_1}^{m_1}\dots
f_{\gamma_s}^{m_s}\pi_0$, and $\phi(\pi) =  
s_{\gamma_1}\cdots s_{\gamma_s}$.

\item{{\rm (v)}} The transition
matrix  from the basis $\Bb_\tau(\lambda)$ to $\bb_\tau(\lambda)$ of
$V_{A,\tau}$, with respect to the indexing $J_\tau$ (resp. $K_\tau$) of $\Bb_\tau
(\lambda)$ (resp. $\bb_\tau(\lambda)$) is upper triangular with diagonal 
entries $=1$ ( here, $\bb_\tau(\lambda)$ is the lower global base for $V_{A,\tau}$).

\ni {\sl Proof}: Fix a a reduced expression  
$\tau = s_{i_1}s_{i_2} \cdots s_{i_r} $  for $\tau .$ Let $\pi\in I_\tau(\lambda)$,
and $\phi(\pi) = x$. Let $x= 
s_{\gamma_1}\cdots s_{\gamma_s}$  be the canonical reduced expression for $x$ (cf
\S 5.5). Then $\pi $ has an unique expression $ \pi = f_{j_1}^{m_1}\cdots
f_{j_s}^{m_s} \pi_0 $.  Set $Q_\pi = F_{\gamma_1}^{(m_1)}\dots
F_{\gamma_s}^{(m_s)}e $, 
$\Bb_\tau(\lambda)=\{Q_\pi,\ \pi\in I_\tau(\lambda)\}$. We have $\{G_\lambda(b),\
b\in B_\tau(\lambda)\}$ is an $A$-basis for $V_{A,\tau}$ (cf. [K]$_2$). Hence we
obtain that $\Bb_\tau(\lambda) $ is an $A$- basis for $V_{A,\tau}$  (note that in
view of Theorem 6.5, the transition matrix from $\{Q_\pi, \pi \in I_\tau(\lambda)\}$
to  $\{G_\lambda(b),\ b\in B_\tau(\lambda)\}$ is upper triangular with diagonal
entries equal to $1$). Assertion (i) follows, since weight of $Q_\pi= \lambda \ -\
\sum _{1\leq t \leq s}\ m_t\g_{j_t}$, and $\nu(\pi) =\lambda \ -\ \sum _{1\leq t \leq
s}\ m_t\g_{j_t}$ (cf. [Li]$_2$). The remaining assertions are immediate from the
definition of $Q_\pi$ (in view of Theorem 6.5).

\beginsection \S7. The filtration $F^l(V_\t)$.

\ni In this section we fix a  $\tau \in W$, and  a reduced expression  
$\tau = s_{i_1}s_{i_2} \cdots s_{i_r} $  for $\tau .$ For $x\leq \t$, we follow
the convention in \S5.5 for denoting the elements of $I'_x(\lambda) $. Let $w
 = s_{i_2} \cdots s_{i_r} $. For simplicity of notations, we shall denote $\a_{i_1}$
by just $\a$.  Let
$U_q(\a)$ denote the copy of $U_q(sl_2)$ in $U$ corresponding to $\a$. Let $F^l(V_\t)=$
the sum of all irreducible $U_q(\a)$-submodules of $V_\t$ of dimension $\geq
l+1$. Let  $F^l(V_w)=F^l(V_\t) \cap V_w.$

Let
$$A=\{\te \in I_w (\lambda)\ |\ e_\a \te = 0\},$$ 
$$B=\{\te \in A\ |\ S_\a \te \not\subseteq I_w(\lambda)\},$$ 
$$C=\{\te \in A\ |\ S_\a \te \subseteq I_w(\lambda)\},$$
(here, for $\te \in A,\ S_\a \te  $ is as in \S5.4 ).
\vs ,2cm
\ni {\bf 7.1} Let $b \in B_\tau(\lambda)$. Then we have (cf. [K]$_2$),
$$F_\alpha^{(k)}G_\lambda(b)={{
\epsilon_\alpha(b) + k \atopwithdelims [] k}}_\alpha G_\lambda (\tilde
f_\alpha^kb)+\sum_{b'\in B_1}c_{b'}G_\lambda( b'),
\leqno{(1)}$$
$$E_\alpha^{(k)}G_\lambda(b)={{
\varphi_\alpha(b) + k \atopwithdelims [] k}}_\alpha G_\lambda (\tilde
e_\alpha^kb)+\sum_{b'\in B_2}d_{b'}G_\lambda( b'),
\leqno{(2)}$$
where $B_i \subseteq B_\tau(\lambda),\ i=1,2$, and $\xi_\alpha(b') > \xi_\alpha(b),\
b' \in B_i,\ i=1,2$.


\ni   
 For $\te \in
A$, we shall denote $r_\te = {\rm wt}_\a \te,\ \Te=f_\a^{r_\te}  \te $(
${\underline {\rm the \ corresponding \ capital \ greek \ letter}}$). Let $\te \in
A$. As a special case of (1), we have
$$F_\alpha^{(k)}G_\lambda(b_\te)= G_\lambda (\tilde
f_\alpha^kb_\te)+\sum_{b'\in B'}c_{b'}G_\lambda(b'),
\leqno{(3)}$$
where $B' \subseteq B_\tau(\lambda)$, and $\xi_\alpha(b') > \xi_\alpha(b_\te),\ b'
\in B'$ (note that $\epsilon_\alpha(b)=0$, since $\tilde e_\a (b_\te)=0$).
 
\ni Let $\te \in A$, and $\xi_\a (b_\te)=l$. Then, $$\tilde f_\alpha^k b_\theta =
\tilde e_\alpha^{l  - k} b_\Theta , 0\leq k \leq l \leqno{(4)}$$
 $$   G(\tilde f_\alpha^k b_\theta) \equiv 
 F_\alpha^{(k)} G(  b_\theta)\ ({\rm mod}\ F^l V_\t/F^{l+1} V_\t) \leqno {(5)}$$
$$    G(\tilde e_\alpha^k b_\Theta) = 
 E_\alpha^{(k)} G(  b_\Theta)\ ({\rm mod}\ F^l V_\t/F^{l+1} V_\t) \leqno {(6)}$$


\vs ,2cm 
\proclaim Lemma 7.2. (cf.[K]$_2$). We have
\item {(1)} $\{ G(\tilde f_\alpha^k b_\theta),\ 0\leq k \leq r_\te,\  r_\te \geq l,
\ \te \in A \}$ is a basis for $F^l (V_\t)$.

\vs ,2cm \ni
\item {(2)} $\{ G(\tilde f_\alpha^k b_\theta),\ 0\leq k \leq r_\te,\  
r_\te \geq l,\ \te \in C \} \ \cup \ 
\{ G( b_\theta), \ \te \in B \}$
{\it is a basis for} $F^l (V_w)$.


\vs ,2cm \ni
\item {(3)} $\{ \overline {G(\tilde f_\alpha^k b_\theta)},\ 0\leq k \leq r_\te,\ 
 r_\te = l, \ \te \in A
\}$ {\it is a basis for} $F^l (V_\t)/F^{l+1} (V_\t)$.


\vs ,2cm \ni
\item {(4)} $\{ \overline {G(\tilde e_\alpha^k b_\Theta)},\ 0\leq k \leq r_\te,\ 
 r_\te = l, \ \te \in A
\}$ {\it is a basis for} $F^l (V_\t)/F^{l+1} (V_\t)$.
\vskip .2cm
\ni ( (4) follows from (3) in view of \S7.1.)

\vskip .2cm
\proclaim Lemma 7.3. Let $\te \in A$. Then, for $l\leq r_\te $, we have
\item{(1)} $F_\alpha^{(k)}Q_\te \in F^lV_\t,\ 0\leq k \leq r_\te .$ 
\vs ,2cm \item{(2)} $\{\overline {F_\alpha^{(k)}Q_\te},\ 0\leq k \leq r_\te,\ r_\te =
l,\ \te \in A\} $ {\it is a basis for} $F^l (V_\t)/F^{l+1} (V_\t)$.

\ni {\sl Proof:} Let
$$ Q_\theta = \ G_\lambda (b_\theta)\ +\ \sum_{b \in A_1}a_b 
G_\lambda (b) \leqno {(a)}$$
where $A_1 \subseteq B_w(\lambda)$, and $b\succ b_\te,\ b \in A_1$ ( cf. Theorem
6.5). Now $e_\a \te = 0 $ implies $\xi_\a (b_\te)={\rm wt}_\a(b_\te)={\rm wt}_\a(b)
\leq \xi_\a (b), b \in A_1$. Hence in $$ F_\a ^{(k)} Q_\theta \  = \  F_\a ^{(k)}
G_\lambda (b_\theta)\ +\ \sum_{b \in A_1}a_b \  F_\a ^{(k)}  G_\lambda(
b) \leqno {(b)}$$
we have $\xi_\a (b) \geq \xi_\a (b_\te)\ (=r_\te)$.
Hence (1) follows from this, and Lemma 7.2,(1). 

\ni Let us denote $r_\te $ by $l$. Then for $0\leq k \leq l$, we
have 
$$ F_\a ^{(k)} Q_\theta \  \equiv \ G(\tilde f_\alpha^k b_\theta)\ +\ \sum_{b\in
A_2}c_b G_\lambda( \tilde f_\alpha^kb)\ \ ({\rm mod}\ F^l V_\t/F^{l+1} V_\t) \leqno
{(c)} $$ where $A_2\subseteq A_1$, and $\xi (b)=l, \ b \in A_2$. For $b \in A_2$, if $
\tilde e_\a b \not= 0$, then replacing $b$ by $\tilde f_\a^{t_b} b '$, where 
$\tilde e_\a b ' = 0$, we may rewrite (c) as
$$ F_\a ^{(k)} Q_\theta \  \equiv \ G(\tilde f_\alpha^k b_\theta)\ +\ \sum_{b '\in
A_3}d_b G_\lambda( \tilde f_\alpha^{s_b}b ')\ \ ({\rm mod}\ F^l V_\t/F^{l+1} V_\t)
\leqno {(d)} $$ where $A_3\subseteq A,\ \xi_\a (b ')=l, \ b' \in A_3$, and for
each $b' \in A_3$, either $s>k$ or $s=k$, and $b' \succ b_\te$. Hence, taking the
total ordering on $\{(m,\pi)\}$, given by the lexicographic ordering, namely,
$(m_1,\pi _1) > (m_2,\pi _2)$, if either $m_1>m_2$ or $m_1=m_2$, and $\pi _1 \succ
\pi _2 $, we obtain that the transition matrix
from $\{\overline {  F_\alpha^{(k)} Q_\theta}},\ 0\leq k \leq r_\te,\ 
 r_\te = l, \te \in A \}$ to $ \{\overline {  F_\alpha^{(k)} G(  b_\theta)}},\ 0\leq k
\leq r_\te,\ 
 r_\te = l, \te \in A
\}$ is upper triangular with the diagonal entries = $1$. From this (2) follows ( in
view of Lemma 7.2,(3), and \S7.1,(5)). 



\vskip .2cm
\proclaim Lemma 7.4. Let $\te, \g \in B$. Suppose for some $ b= e_\a^{n_b}b_ \G $, we
have $b \succ b_\Te $, and wt$_\a b $= wt$_\a \Te$. Then $b_\G\succ b_\Te $.

\ni (Here the convention for the capital and small greek letters is as in \S7.1.)

\ni {\sl Proof:} If $n_b=0$, then there is nothing to be checked. Let then $n_b>0$.
This implies
$$r_\g >r_\te. \leqno{(1)} $$
The hyrothesis that $b \succ b_\Te $ implies
$$ \phi (b) \leq \phi (\Te) \ (=s_\a\phi (\te)) \leqno{(2)} $$
( note that $\phi (\Te) \ =s_\a\phi (\te) $, since $\te \in B$). Also, $\te \in B$
implies that
$$ \phi (\te) < s_\a \phi (\te). \leqno{(3)} $$  
The hyrothesis that $ b= e_\a^{n_b}b_ \G $ implies
$$\phi (\G)=\phi (b)\ {\rm or  \ } s_\a \phi (b) . \leqno{(4)}$$
Hence we obtain
$$\phi (\G) \leq \phi (\Te). \leqno {(5)} $$ 
(This is clear if $\phi (\G)= \phi (b)$, in view of (2). If $\phi (\G)=s_\a \phi (b)$,
 then we have $s_\a \phi (b) \leq s_\a\phi (\te)$, in view of (2)\&(3).)  
Since $\te, \g \in B$, we have 
$$ \phi (\G)=s_\a \phi (\g)  \not\leq w, \leqno {(6)} $$
$$ \phi (\Te)=s_\a \phi (\te)  \not\leq w. \leqno {(7)} $$
We have ( in view of (5),(6)\&(7)), $s_\a \phi (\g) \leq s_\a \phi (\te)$.  
 If $s_\a \phi (\g) < s_\a \phi (\te)$, then $ \phi (\G) < 
\phi (\Te)$ (obviously), and hence $\G \succ \Te$. Let then $s_\a \phi (\g) = s_\a
\phi (\te)$. Then (6) and (7) imply that the canonical reduced expressions for $s_\a
\phi (\g),\ s_\a \phi (\te)$ start with $s_{i_1} (=s_\a)$. This together with the
facts that $ \G=f_\a^{r_\g}\g,\ \Te=f_\a^{r_\te}\te $ imply that $\G \succ \Te$
(since $r_\g >r_\te $ (cf.(1)).   

\proclaim Proposition 7.5. $\{ {\overline {E_\alpha^{(k)} Q_\Theta}}, \ 0\leq k <
r_\te,\ \te \in B\}$ is a basis for $V_\t/V_w $.

\ni
{\sl Proof.}. Denote $\overline V_\t = V_\t/V_w$, and define a filtration  
$\{F^l \overline V_\t \}$, on $\overline V_\t $ by $F^l \overline V_\t =
F^lV_\t/F^lV_w $. Then in view of Lemma 7.2, and the fact that in $F^l V_\t/F^{l+1} V_\t,\ 
\overline {G(\tilde e_\alpha^k b_\Theta)} =  \overline {E_\alpha^{(k)} G(  b_\Theta)}
$, we obtain that  $$ \{ \overline {E_\alpha^k G(  b_\Theta)},\ 0\leq k < r_\te,\ 
 r_\te = l, \te \in B
\} {\rm \ is\  a\  basis\  for}\  F^l \overline V_\t/F^{l+1} \overline V_\t. \leqno
{(1)} $$

\ni Let $\te \in A$, and let
$$ Q_\Theta = \ G_\lambda (b_\Theta)\ +\ \sum_{\g \in A_1}c_\G 
G_\lambda(\tilde e_\alpha^{n_\g} b_\G),\ \tilde e_\alpha^{n_\g} b_\G \succ
b_\Theta \leqno {(2)} $$
where $A_1$ is a subset of $A$  (here, the convention regarding capital and small
greek letters is as  in 7.1). Taking $b=e_\alpha^{n_\g} b_\G$, in
Lemma 7.4, we obtain
$$\G\succ \Te,\ \forall \g \in A_1. \leqno{(3)}$$ Let $r_\te = l$. Then in 
$F^l \overline V_\t/F^{l+1} \overline V_\t $, we have (in view of 7.1,(6)) $$\overline {E_\a
^{(k)} Q_\Theta }= \ \overline {E_\a ^{(k)} G_\lambda (b_\Theta)}\ +\ \sum_{\g \in
A_2}a_\G \ \overline {E_\a ^{(m_\g)}  G_\lambda( b_\G)},\leqno {(4)} $$
where $A_2$ is a subset of $B$,  $r_\g = l, $, and $m_\g = k+n_r  ,\ \g \in A_2
$. Now $m_r  >\ {\rm or}\ k$, according as  $n_r >\ {\rm or}\   =0$. Hence taking the
total ordering on $\{(m,\G)\}$, given by the lexicographic ordering, namely,
 $(m_1,\G_1) > (m_2,\G_2)$, if either $m_1>m_2$, or $m_1=m_2$, and $\G_1 \succ \G_2
$, we obtain that the transition matrix from $\{ \overline {E_\alpha^{(k)}
Q_\Theta)},\ 0\leq k < r_\te,\ 
 r_\te = l, \te \in B
\}$ to $ \{ \overline {E_\alpha^{(k)} G(  b_\Theta)},\ 0\leq k < r_\te,\ 
 r_\te = l, \te \in B\}$ is upper triangular with the diagonal entries = $1$. Hence
we obtain (cf.(1) above) that $\{ \overline {E_\alpha^{(k)} Q_\Theta},\ 0\leq k <
r_\te,\ 
 r_\te = l, \te \in B
\} {\rm \ is\  a\  basis\  for}\  F^l \overline V_\t/F^{l+1} \overline V_\t $. The
required result now follows from this.




\vs .2cm \ni {\bf \S8  The Demazure $U_q(\a)$-structure on $V_\t /V_w$}

\ni We preserve the notations of the previous sections. Also, we shall denote $e_\t$
( the extremal weight vector in $V_\t$ of weight $\t (\lambda )$ ) by just $e$, and $(\t
(\lambda ),\a^*)$ by $-n$ ( note that $(\t (\lambda ),\a^*)<0 )$.

\vskip .2cm
\ni {\bf Definition 8.1} A Demazure $U_q(\a)$-structure on $V_\t /V_w$}
is a representation $\rho $ of $U_q(\a)$ in $V_\t /V_w$} such that
$$ \rho (F_\a ){\overline e} =0,\ \rho (K^\epsilon _\a ){\overline e}=
q^{-\epsilon (n-1)}{\overline e},\ \rho (E_\a ){\overline u}= E_\a {\overline u},\ \forall u \in
V_\t $$
where $\epsilon = \pm 1$. ${\underline {\rm Throughout\  this\  section}}\ 
\epsilon\  {\underline {\rm {shall\  denote\ }}\pm 1}$. Also we shall denote 

\ni $E_\a, F_\a, K_\a
^\epsilon $ by just $E,F,K^ \epsilon $ respectively. 

\vskip .2cm 
\ni {\bf 8.2} Recall (cf.   [K]$_1$ for example) the defining relations
for $U_q(\a)$ : $$[E,F]={{K-K^{-1}} \over {q-q^{-1}}},\ K^\e E=q^{2\e}EK^\e,\ K^\e
F=q^{-2\e}FK^\e. \leqno{(1)} $$ We will also have occasion to use the following
relations in $U$ (cf.[Lu]):
$$ [F, E^{(m)}]=E^{(m-1)}K_{m-1},\ \ K^\e E_\beta^{(m)}=q^{a_{\a \beta}\e
m}E_\beta^{(m)}K^\e,\ K^\e F_\beta^{(m)}=q^{-a_{\a \beta}\e m}F_\beta^{(m)}K^\e 
\leqno{(2)}  $$
where $K_{m-1}={{K^{-1}q^{1-m}-Kq^{m-1}} \over {q-q^{-1}}}$.
 

In this section, we shall prove the existence and uniqueness of a 
Demazure $U_q(\a)$-structure on $V_\t /V_w$} (in the same spirit as in [De] for
the case $q=1$). Let $U^0$  be the subalgebra of $U^+$ generated by $\{E_\beta,\
\beta$ simple,  $\beta\not=\alpha \}$. Let $i_{\e }$ denote the automorphism of
$U^+$ given by inner conjugation by $K^\e$.
  
\proclaim Proposition 8.3. Let $I$ be a left ideal in  $U^0$ of finite codimension.
Further, let $I$ be stable under  ad$F$, and $i_{\e }$. Fix $n \in {\bf N}$.
Let $M = {{U^+}\over {U^+I\  +\   U^+ E^{n+1}}}$. Let $e$ denote the class of $1$.
Then there exists an unique representation $\r$ of $U_q(\a)$ in $M$ such that
$$ \rho (F ) e =0,\ \rho (K^\epsilon  ) e=
q^{-\epsilon n}e,\ \rho (E ) u= E_\a  u,\ \forall u \in
M .$$

\ni {\sl Proof:} Define ${\bf Q}(q)$-linear maps $x,y,k^\e \ :U^+ \rightarrow U^+ $
as follows: Let  $v \in U^+$. We may suppose $v=uE^{(m)}, \ u \in U_0$. We set
$$ x(v)=Ev \leqno{(1)}$$
$$y(uE^{(m)})=[F,u]E^{(m)}+[n-m+1]uE^{(m-1)} \leqno{(2)} $$
$$ k^\e(uE^{(m)})=q^{(-n+2m)\e} K^\e u K^{-\e}E^{(m)} \leqno{(3)}$$
(here, $[s]={{q^s-q^{-s}}\over {q-q^{-1}}}$. Also note that the map $x$ is simply
the left multiplication by $E$). We have (in view of 8.2,(2))
$$ y(v)\equiv [F,v]\ ({\rm mod}\  U^+((K-q^{-n}),(K^{-1}-q^n)), \leqno{(4)}$$

$$k^\e (v)=q^{-\e n} i_\e(v). \leqno {(5)} $$
 
\ni
{\bf Claim:} $ U^+I +  U^+ E^{n+1}$ is stable under $x,y,k^\e$.

\ni
{\bf Proof of Claim:} The stability of  $ U^+I $, and $  U^+ E^{n+1}$ under $x,\
i_\e $ is clear. Also, $  U^+ E^{n+1}$ is generated by $uE^{(m)},\ m\geq n+1 $.
Hence the stability of $  U^+ E^{n+1}$ under $y$ follows in view of (2). Finally,
it remains to check the stability of $ U^+I $ under $y$. Let $v \in U^+I$. We may
suppose, $v=E_\beta^{(m)} u$, where $u \in I$ (and $\beta $ is simple). 

\ni For $\beta \not= \a$, we have modulo $U^+((K-q^{-n}),(K^{-1}-q^n)) + U^+I $,

\ni $y(E_\beta^{(m)} u)\ \equiv [F,E_\beta^{(m)} u]\ \equiv\ E_\beta^{(m)}({\rm ad}F)
u\ \equiv\ 0$ (since $I$ is stable under ad$F$).

\ni
For $\beta = \a $,  we have modulo $U^+((K-q^{-n}),(K^{-1}-q^n)) + U^+I $,


\ni $ y(E^{(m)} u)\ \equiv  \ [F,E^{(m)} u]\ \equiv\ [F,E^{(m)} ]u\ \equiv 
E^{(m-1)}K_{m-1}u$ (cf.8.2) 

\ni $\equiv\ [n-m+1]E^{(m-1)}u\ \equiv \ 0 $. 

\ni This completes
the proof of the claim. Now Claim implies that $x,y,k^\e $ induce $\bf Q
(q)$-linear maps on $M$, which we shall denote by  $\underline x,\underline y,
\underline k^\e $ respectively. We now set $\r (E)=\underline x,\ \r (F)=\underline
y,\ \r (K^\e)=\underline k^\e $. Then it is easily checked that $\underline
x,\underline y, \underline k^\e $ satisfy the $U_q (\a)$-relations (given by 8.2).
Hence we obtain a representation of $U_q(\a)$ on $M$. Further,we have, $ \rho (F ) e
=0,\ \rho (K^\epsilon  ) e= q^{-\epsilon n}e,\ \rho (E ) u= E_\a  u,\ \forall u \in
M .$ This proves the existence of $\r$. The uniqueness is clear.

\ni This completes the proof of Proposition 8.3.

\proclaim Lemma 8.4.  Let $e=e_\t$ ( the extremal weight vector in $V_\t$ of weight
$\t (\lambda )$ ),   and $(\t (\lambda ),\a^*)$ = $-n$ ( note that $(\t (\lambda
),\a^*)<0 )$. Then $$ {\rm Ann}\ _{U^+}\ e\ =\ U^+   {\rm Ann}\ _{U^0}\ e\  +\ 
U^+E^{n+1} .$$


\ni The proof is carried out in the same spirit as in [Jo]$_1$ (for the case $q=1$).

\proclaim Lemma 8.5. There exists a unique Demazure $U_q(\a)$-structure on 
$V_\t /V_w$.

\ni {\sl Proof:} In view of Lemma 8.4, we have an isomorphism of $U^+$-modules
$$V_\t \buildrel \sim \over \longrightarrow {{U^+}\over {{\rm Ann}\ _{U^+}\ e}}$$
$$ e \mapsto {\overline 1}.$$
Under the above isomorphism, the $U^+$-submodule $V_w(=U^+E^{(n)}e)$  gets identified
with ${{U^+I\  +\   U^+ E^n}\over {U^+I\  +\   U^+ E^{n+1}}} $, where $I={\rm
Ann}\ _{U^0}\ e$ (cf. Lemma 8.4). Hence we  obtain an isomorphism
$$V_\t/V_w\approx {{U^+}\over {U^+I\  +\   U^+ E^{n}}}.$$ Hence by Proposition 8.3,
there exists  an unique representation $\r$ of $U_q(\a)$ in $V_\t/V_w$ such that
$$ \rho (F_\a ){\overline e} =0,\ \rho (K^\epsilon _\a ){\overline e}=
q^{-\epsilon (n-1)}{\overline e},\ \rho (E_\a ){\overline u}= E_\a {\overline u},\ \forall u \in
V_\t .$$

\vs .2 cm
\ni {\bf 8.6 An explicit realization of the
Demazure $U_q(\a)$-structure on  $V_\t /V_w.$}  

\ni Recall (cf. Proposition 7.5) that
$\{ E_\alpha^{(k)} Q_\Theta, \ 0\leq k < r_\te,\ \te \in B\}$ is a basis for
$V_\t/V_w $. Let us denote $Q_\theta,\ Q_\Theta,\ r_\te $ by
$u,\ v,\ r_u $ respectively.  Let us define linear maps 

\ni $\eta (E_\a), \eta (F_\a),\
\eta (K_\a^\e)\ : V_\t /V_w \rightarrow V_\t /V_w$ as follows:  $$ \eta (E_\alpha) {\overline
{E_\alpha^{(j) }v}}= [j+1]{\overline  {E_\alpha^{(j+1) }v}}, \leqno{(1)}$$ $$\eta (
F_\alpha) {\overline {E_\alpha^{(j) }v}}= [r_u -j]{\overline  {E_\alpha^{(j-1) }v}},
\leqno {(2)}$$ $$\eta ( K^\e_\alpha) {\overline {E_\alpha^{(j) }v}}= q^{\e (-r_u
+1+2j)}{\overline  {E_\alpha^{(j) }v}}. \leqno {(3)}$$
\vs ,2cm
\proclaim Proposition 8.7. With notations as above, $\eta $ defines a
$U_q(\a)$-structure on  $V_\t /V_w.$ Further the following hold:
$$ \eta (F_\a ){\overline e} =0,\ \eta (K^\epsilon _\a ){\overline e}=
q^{-\epsilon (n-1)}{\overline e},\ \eta (E_\a ){\overline u}= E_\a {\overline u},\ \forall u \in
V_\t.$$

\ni {\sl Proof:} If we denote $\eta (E_\a), \eta (F_\a),\ \eta (K_\a^\e) $ by $x,\
y,\ k^\e $ respectively, then it is easily checked that $x,\
y,\ k^\e $ satisfy the defining relations of $U_q(\a)$ (cf. 8.2). Thus $\eta $ defines
a $U_q(\a)$-structure on  $V_\t /V_w$}. That this structure is in fact the 
Demazure $U_q(\a)$-structure on  $V_\t /V_w$} is also easily checked.

\vskip .2cm
\noindent {\bf Remark 8.8} Proceeding as in [Jo]$_1$, one can show that the following
are equivalent   
\item{(1).}  $Ann\ _{U^+}\ e\ =\  Ann\ _{U^0}\ e\  +\  U^+\ E^{n+1}$
\item{(2).}  The Demazure $U_q(\alpha)$-representation 
on $V_\t/V_w$ exists.
\item{(3).} char$V_\t\ =\ L_\alpha$(char$V_w$)

\ni (here $L_\a$ is as in 2.5).

\vs ,2cm \ni {\bf \S9 Description of  $I(\l),\ B(\l),\ \Bb(\lambda) $  for
$U_q(sl_3)$.}

\ni Let $\gg =sl_3$. Let us denote 
$$W=\{\t_i,\ i=0,1,2,3,\ \phi_j,\ j=1,2\}$$
where $\t_0=s_1s_2s_1,\ \t_1=s_2s_1,\ \t_2=s_1,\ \t_3=$id, and $\phi_1=s_1s_2,\ 
\phi_2=s_2$.

\ni Let $\l=m\omega_1+n\omega_2$. We shall work with the reduced expression 
$s_1s_2s_1$ for $w_0$.  There are four kinds of L-S paths of shape $\l$.

\vs ,2cm \ni {\bf I} $\pi=\{\{\t_3\leq \t_2 \leq \t_1 \leq \t_0\};\ \{1\geq { p\over
m}\geq {{q}\over {m+n}}\geq {r\over n}\geq 0\}\}$.

\ni Then

 $$\pi=f_1^{r}f_2^{q}f_1^{p}\pi_0,$$
 $$b_\pi=\tilde f_1^{r}\tilde f_2^{q}\tilde f_1^{p}e,$$
 $$Q_\pi=F_1^{(r)}F_2^{(q)}F_1^{(p)}e.$$


\vs ,2cm \ni 
\ni {\bf II} $\pi=\{\{\t_3\leq \phi_2 \leq \t_1 \leq \t_0\};\ \{1\geq{ p\over n}\geq
{{q}\over {m}}\geq {r\over n}\geq 0\}\}$.

\ni Then

 $$\pi=f_1^{r}f_2^{p+q}f_1^{q}\pi_0,$$
 $$b_\pi=\tilde f_1^{r}\tilde f_2^{p+q}\tilde f_1^{q}e,$$
$$Q_\pi=F_1^{(r)}F_2^{(p+q)}F_1^{(q)}e.$$

\vs ,2cm \ni {\bf III} $\pi=\{\{\t_3\leq \t_2 \leq \phi_1 \leq \t_0\};\ \{1\geq {
p\over m}\geq {{q}\over {n}}\geq {r\over m}\geq 0\}\}$.

\ni Then

 $$\pi=f_1^{q}f_2^{q+r}f_1^{p}\pi_0,$$
 $$b_\pi=\tilde f_1^{q}\tilde f_2^{q+r}\tilde f_1^{p}e,$$
 $$Q_\pi=F_1^{(q)}F_2^{(q+r)}F_1^{(p)}e.$$

\vs ,2cm \ni {\bf IV} $\pi=\{\{\t_3\leq \phi_2 \leq \phi_1 \leq \t_0\};\ \{1\geq {
p\over n}\geq {{q}\over {m+n}}\geq {r\over m}\geq 0\}\}$.

\ni Then

 $$\pi=f_1^{q-r}f_2^{p+r}f_1^{r}\pi_0,$$
 $$b_\pi=\tilde f_1^{q-r}\tilde f_2^{p+r}\tilde f_1^{r}e,$$
 $$Q_\pi=F_1^{(q-r)}F_2^{(p+r)}F_1^{(r)}e.$$

\ni (note that $q>r$).


    
    


         
            \vs .4cm
\centerline{\bigsc References}
\vskip .6cm
\noindent
[D] V.G.Drinfeld, {\it Hopf Algebra and the Yang-Buxter equation}
, Soviet math. Dokl., {\bf 32} (1985), 254-258
\vskip .2cm
\noindent
[De] M.Demazure, {\it Desingularisation des vari\'eti\'es de Schubert
generalise\'es}, Ann. Sci. \'Ecole. Norm. Sup. {\bf t.7} (1974), 53-88

\vskip .2cm
\noindent
[H] S.R. Hansen, {\it A q-analogue of Kempf's vanishing Theorem} (1994 preprint)
 \vskip .2cm
\noindent
[J] M.Jimbo, {\it A $q$-difference analog of $U(\gg)$ and the Yang-Buxter
equation}, Lett. math. Phys, {\bf 10} (1985), 63-69

\vs ,2cm \ni [Jo]$_1$ A.Joseph, {\it On Demazure character formula}, Ann. Sci.
\'Ecole. Norm. Sup.,  

\ni {\bf t.18}(1985), 389-419.
\vskip .2cm
\noindent
[Jo]$_2$ A.Joseph, {\it Quantum groups and their primitive ideals}, Springer-Verlog
(1994). 



\vskip .2cm
\noindent
[K]$_1$ M. Kashiwara, {\it Crystalizing the $q$-analog of Universal
Enveloping Algebras}, Comm. math. Phys., {\bf 133} (1990), 249-260
\vskip .2cm
\noindent
[K]$_2$ M. Kashiwara, {\it Crystal Base and Littelmann's Refined Demazure
CharacterFormula}, (preprint, RIMS-880, 1992)
\vskip .2cm
\noindent
[K]$_3$ M. Kashiwara, {\it Crystal Bases of modified quantized enveloping algebras},
(preprint, RIMS-917, 1993). \vskip .2cm
\noindent
[L]$_1$ V. Lakshmibai, {\it Bases for Quantum Demazure modules -I}
(to appear in the volume dedicated to Profs. M.S.Narasimhan and C.S.Seshadri
on their 60$^{th}$ birthdays)
\vskip .2cm
\noindent
[L]$_2$ V. Lakshmibai, {\it Bases for Demazure modules for symmetrizable
Kac-moody algebras}, Contemporary math., vol 153, 1993, 59-78
(volume dedicated to Prof. R.Steinberg on his 70$^{th}$
birthday)

\vskip .2cm
\noindent
[L]$_3$ V. Lakshmibai, {\it Bases for Quantum Demazure modules -II}, Proc. Symp.
Pure. Math., vol 56, Part 2, 149-168.

\vskip .2cm
\noindent
[L-S] V.Lakshmibai-C.S.Seshadri, {\it Geometry of $G/P-V$}, J. Algebra
{\bf 100} (1986) , 462-557
\vskip .2cm
\noindent
[Li]$_1$ P.Littelmann, {\it Young tableaux and crystal bases}, J. Alg., 1992
\vskip .2cm
\noindent
[Li]$_2$ P.Littelmann, {\it Littlewood-Richardson rule for symmetrizable
Kac-moody algebras}, Inv. math., 1993


\vskip .2cm
\noindent
[Lu] G.Lusztig, {\it Finite dimensional Hopf algebras arising from Quantum groups},
Journal. A.M.S., 3 (1990), 257-296.

\vskip .6cm
\noindent
{\sc
V. Lakshmibai\hfill\break
mathematics Department\hfill\break
Northeastern University\hfill\break
Boston, mA 02115\hfill\break
e-mail address: lakshmibai@northeastern.edu
}
\bye