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\centerline{\bigbf Frobenius Splittings \& Blow-ups  }
\vskip .4cm
\centerline{V. LAKSHMIBAI
\footnote{*}{\sl  Partially supported by the NSF Grant DMS 9103129 .}
}

\centerline {V.B. MEHTA}
\centerline {and}
\centerline {A.J. PARAMESWARAN }

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\ni {\bf Introduction} The notion of Frobenius splitting for
algebraic varieties was introduced by Mehta-Ramanathan in [M-R]
where they proved that the {\it flag variety} $G/B$ ( and more
generally $G/Q$) and its Schubert varieties are Frobenius split. As
a consequence they obtained the vanishing of $H^i(X,L),\ i>0$,
where $X$ is a Schubert variety in $G/B$, and $L$ an ample line
bundle on $G/B$. Further geometric results were deduced
(cf.[M-S],[R]) as consequences of Frobenius splitting. In [M-R], to
prove results for $G/B$, one is required to use the Bott-Samelson
scheme of $G$ (cf.[D]). 

In this paper, we prove the Frobenius-split
property for $G/B$ without requiring the Bott-Samelson
scheme. We then consider blow ups \& Frobenius splitting, and
deduce that $G/Q$ is Frobenius-split, $Q$ being any parabolic
subgroup. We further show the connection to Wahl's conjecture.
Below, we make these more precise.  

First, we recall one key result
from [M-R] which has much relevance to this paper. Let $X$ be a
smooth projective variety over an algebraically closed field of
characteritic $p>0$. Let $K_X$ denote its canonical class.

\ni {\bf A key result from [M-R]} (cf. [M-R], Proposition 7)
{\it Suppose $s$ is a section of $K_X^{-1}$ which is a normal crossing
divisor at some closed point $P\in X$. Then $s^{p-1}$ gives rise
to a splitting of ${\cal O}_X \rightarrow F_*{\cal O}_X $, where $F:X
\rightarrow X $ is the absolute Frobenius morphism induced by $f
\mapsto f^p,\ f \in {\cal O}_X$.}

\ni In [M-R], such a section is exhibited for the Bott-Samelson
scheme $Z$ of $G$. Then using the proper birational morphism $Z
\rightarrow G/B $, it is proved that $G/B$ is Frobenius-split.

In this paper, we first introduce the (weaker) notion of residual
normal crossing at a closed point (cf.\S1). We then observe that the
proofs of Propositions 7\&8 in [M-R] imply the validity of the
above key result if $s$ is just a residual normal
crossing at a closed point. We then exhibit (cf.\S4) a
section of $K_{G/B}^{-1}$ which is a residual normal crossing
at $e_{id}$ (here, $e_{id}$ is the point of $G/B$
corresponding to the coset  $B$), $G$ being classical or
simple of type $G_2$ ( using the results of [L-S]$_2$, [L]$_2$), 
and conclude 
the Frobenius-split property for $G/B$ (cf. Therem 4.3). In fact,
this section has maximum order of vanishing at $e_{id}$ equal to


\ni dim $G/B$.

The Frobenius-split property for any $G/Q$ is deduced by
considering blow-ups \& Frobenius splittings. To be more
precise, starting with a section $s$ of $H^0(X,K_X^{1-p})$
which gives a splitting ($X$ being smooth), we give a
necessary and sufficient condition for $s$ to extend to a
splitting of $B_Y(X)$ ( resp. a splitting which compatibly
splits the exceptional divisor $E$ ), $B_Y(X)$ being the
blow-up of $X$ along a non singular subvariety $Y$ of
codimension $\geq 2$. As a consequence, we deduce 
the Frobenius-split property for any $G/Q$. (We also 
give (cf.\S5) a section
of $K_{G/Q}^{-1}$ with the said property in the Key result above.
From this also one can deduce the Frobenius-split property for 
$G/Q$).

Finally, we give one application of Frobenius splittings \& blow
ups to Wahl's conjecture in arbitrary characteristics (Wahl's 
conjecture in characteristic $0$ was proved by Kumar
(cf.[K]) using representation-theoretic techniques). To be
more precise,  let $X$ be a homogeneous space in
characteristic $p>0$. Let $\Delta $ denote the diagonal in
$X\times X$. Let $B(X\times X)$ be the blow-up of $X\times
X$ along $\Delta $ with exceptional divisor $E$.


\ni {\bf Conjecture.} $E$ is compatibly split in $B(X\times
X)$, or equivalently there exists a splitting of $X\times X$ which
has maximum multiplicity along $\Delta $.

 \ni (see \S 2 for details.) Here, by ``maximum multiplicity" we
mean the following: Given a subvariety $Y$ ( of codimension $d$ ) 
of a smooth projective variety $X$, if $\sigma$ is a section of
$K_{X}^{1-p}$ vanishing generically along $Y$ to order $>d(p-1)$,
then it can be seen (by local computations) that $\sigma
$ is not a splitting of $X$. Hence we say that {\it a subvariety
$Y$ is compatibly split by $\sigma $ with maximum multiplicity }
if $\sigma $ vanishes to order $d(p-1)$ generically along $Y$.

Note that for $X=G/B,\ G $ semi simple, the above Conjecture is a
strengthening of Ramanathan's result (cf. [R]) that $\Delta $ is
compatibly split in $X\times X$. We have been able to verify
the above Conjecture (by hand computations) for $G=SL(n),\ n\le 6 $.
The main result of this paper ( namely, there is a section of
$K_{G/B}^{-1}$ which induces a splitting of $G/B$, and which has
maximum multiplicity at $e_{id}$) seems to provide a positive
evidence to the above Conjecture.     

In \S 3, we show that the above Conjecture implies Wahl's
conjecture on the surjectivity of the Gaussian map.

The sections are organised as follows. In \S1, we recall 
some basic results on
Frobenius splitting. In \S2, we discuss Frobenius 
splittings \& blow-ups. As an
application of Frobenius splittings \& blow-ups, 
we indicate in \S3, a proof of Wahl's
conjecture ( assuming the truth of the above conjecture). 
In \S4, we constrct a
section of $H^0(G/B,K_{G/B}^{-1})$ which is a residual 
normal crossing at 
$e_{id}$. In \S5, we give an independent proof of the 
Frobenius-split property for any $G/Q$ by exhibiting a section of 
$H^0(G/Q,K_{G/Q}^{-1})$ which is a residual normal crossing at 
$e_{id}$.   

\ni {\bf Acknowledgement} The authors would like to thank
ICTP, Italy for the {\it Advanced workshop on Algebraic
Geometry} where the core of this work was done. The last 
two authors would
like to thank University of Kaiserslautern for hospitality
during when some
of this work was done. The last author would also like to 
thank the
Alexander von Humboldt Stiftung for support during that
period. The authors wish to thank the referees for some
useful, and important suggestions.          



\vs .2cm \ni {\bf \S1 Frobenius splitting}

We first recall the definition of ( and the basic facts on ) 
Frobenius splitting (cf.[M-R]). Let
$X$ be a smooth projective variety over an algebraically 
closed field $k$ of
characteristic $p>0$. Let $F:X\to X$ be the absolute 
Frobnius morphism induced by
$f\mapsto f^p$ for all $f\in {\cal O}_X$. This induces an 
injection $i_*:{\cal O}_X\to
F_*{\cal O}_X$.

\vs .2cm \ni {\bf Definition 1.1.}  $X$ is called a 
{\it Frobenius split}
variety if $i_*$ splits, i.e., if there exists a morphism
$\sigma:F_*{\cal O}_X\to {\cal O}_X$ such that $\sigma\circ
i_*=Identity$. 

\vs .2cm \ni {\bf Definition 1.2.} Let ${\cal I}$ be an ideal 
sheaf in ${\cal O}_X$
corresponding to a subvariety $Y$ of $X$. Then $Y$ 
is said to be {\it
compatibly split} in $X$ if $\sigma(F_*({\cal I}))\subset 
{\cal
I}$. 

\proclaim Proposition 1.3. (cf. [M-R], Proposition 4) 
Let $f:X\to Y$ be a morphism of
varieties with $f_*{\cal O}_X={\cal O}_Y$. If $X$ is 
Frobenius split then so is $Y$.
If $X_1$ is a compatible split subvariety of $X$, then 
$Y_1=f(X_1)$
is compatible split in $Y$. 

\vs .2cm \ni {\bf 1.4.} In [M-R], a local criterion for 
Frobenius splitting is given;
we now recall this briefly. A section 
$\sigma:F_*{\cal O}_X\to {\cal O}_X$ is a
global section of $F_*{\cal O}_X^*=Hom(F_*{\cal O}_X,{\cal
O}_X)$. But note that we have by Serre duality, 
$H^0(X, F_*{\cal
O}_X^*)$ is isomorphic to 
$H^n(X, K_X\otimes F_*{\cal O}_X)^*$.
By the projection formula the latter is isomorphic to
$H^n(X,F_*F^*K_X)^*$. As $F_*$ is an affine morphism 
we obtain a
semilinear isomorphism of the latter with
$H^n(X,F^*K_X)^*=H^n(X,K_X^p)^*$ (since $F^*K_X$ is 
naturally
isomorphic to $K_X^p$). Then by Serre duality again we obtain
that the latter is isomorphic to $H^0(X,K_X^{1-p})$.

In fact Proposition 5 of [M-R] gives a functorial isomorphism
between the sheaves $F_*K_X^{1-p}$ and $F_*{\cal O}_X^*$. Hence
a criterion for a variety $X$ to be Frobenius can be translated
in terms of global sections of $F_*K_X^{1-p}$ as given by the 
following

\vs .2cm \ni {\bf Proposition 1.5.} (cf. [M-R], Proposition 6) {\it Let
$X$ be a smooth projective variety over an algebraically closed 
field $k$ of
characteristic $p>0$. Then $X$ is Frobenius split if and only if 
there exists a
section $s \in H^0(X,K_X^{1-p})$ whose image in 
$K^{1-p} \otimes  _{{\cal O}_P} 
{{\cal O}}\hat  _P$ ( for some $P\in X$), has a power series 
expansion with respect to
some local parameters in which the term 
${ {x^{p-1}} \over {(dx)^{p-1}} }$ occurs
with a non-zero coeeficient.}

    

\vs .2cm Proposition 7 of [M-R] ( which is alluded as 
`` A key result from [M-R]" in
the introduction ) asserts that if $K_X^{-1}$ has a section 
$s$ which is a normal
crossing divisor (here by a normal crossing divisor one means 
that all coordinate
planes are involved in the divisor unlike the classical 
definition where just a part
of the coordinate planes is sufficient) at some point $P\in X$,
then $s^{p-1}$ is a splitting (upto a nonzero constant factor).

\vfill\eject In the following we strengthen this by proving that if
$K_X^{-1}$ has a section which is a residually normal crossing
divisor (definition appears below) at a point $P\in X$,
then the $(p-1)^{th}$ power of this section gives a splitting.

\vs .2cm \ni {\bf Definition 1.6.}  A divisor $D$ defined by $f_0=0$ on
an open neighbourhood of $0\in {\bf k}^n$ is said to have
{\it residually normal crossing} at $0$ if there exists a
coordinate system $\{x_1,\cdots, x_n\}$ and functions
$f_1,\cdots,f_{n-1}\in k[[x_1,\cdots, x_n]]$ such that
$f_i\equiv x_{i+1}f_{i+1}\ ({\rm mod}\ (x_1,\cdots, x_i) 
)$ for $i=0,1,\cdots,n-1$, where $f_n=1$.

\vs .2cm \ni {\bf Remark 1.7.} (i) A normal crossing divisor is a residually
normal crossing divisor if and only if it has multiplicity $n$,
i.e., the components of the divisor form a regular system of
parameters.

(ii) For $n=2$, residual normal crossing is equivalent to normal
crossing.

(iii) For $n\geq 3$, there are divisors which are residually normal
crossing but not normal crossing. A simple example is provided
by the divisor $x(x-yz)=0$ in $k^3$.

(iv) One can define the residually normal crossing divisors in a
more general way. In the definition above one may require 
$f_k$ to be a unit for some $k\leq n$. This way one can extend
the concept of normal crossing divisor to residually 
normal crossing divisor. Since we work only with residually
normal crossing divisors of ``maximal type" (exhausting all the 
variables) we defined it in this fashion.

As a straight forward consequence of the proofs of Propositions 7
and 8 of [M-R], we obtain:

\ni {\bf Proposition 1.8} {\it Let $X$ be a smooth 
projective variety and
$D\subset X$ be an effective divisor corresponding to 
a section $s$ 
of $K_X^{-1}$. If $P\in D$ is a point of $X$ such that
$D$ has residually normal crossing at $P$, then 
$s^{p-1}$ gives
a splitting of $X$.}

\vs 1cm \ni  {\bf \S2 Blow-ups and Frobenius splittings}

Let $X$ be a nonsingular variety and $Y$ a nonsingular
subvariety of codimension $d\geq 2$. Let $B_Y(X)$ (denoted by
$B(X)$, if no confusion is possible) be the blow up of $X$ along
$Y$. Given a section $s$ of $H^0(X,K_X^{1-p})$
which gives a splitting, in this section, 
we give a
necessary and sufficient condition for $s$ to extend to a
splitting of $B_Y(X)$ ( resp. a splitting which compatibly
splits the exceptional divisor $E$ ).

\vs .2cm \ni {\bf Proposition 2.1.} {\it Let $X$ be Frobenius split 
by an element
$\sigma\in H^0(X, K_X^{1-p})$.

(a) $\sigma$ vanishes to order at least $(d-1)(p-1)$ generically
along
$Y$ if and only if $\sigma$ extends to a splitting $\sigma_1$ of
$B(X)$.

(b) $\sigma$ vanishes to order $d(p-1)$ generically along $Y$ if
and only if $E$ is compatibly split by $\sigma_1$ in $B(X)$.}

\vfill\eject
\ni {\bf Proof:} Let $\pi:B(X)\to X$ be the birational blowing up
morphism. Then the cannonical bundles of $B(X)$ and $X$ are
related by 
$$ K_{B(X)}=\pi^*(K_X)+(d-1)E$$
Dually,
$$c_1(B(X))^{p-1}=\pi^*(c_1(X)^{p-1})-(p-1)(d-1)E$$
Hence an element of $c_1(X)^{p-1}$ remains regular as a section
of $c_1(B(X))^{p-1}$ if and only if it vanishes to order at least
$(d-1)(p-1)$ generically along $Y$. Similarly for $E$ to be
compatibly split in $B(X)$, it is necessary and sufficient for
the original section to vanish upto order $d(p-1)$ along $Y$.

\vs .2cm \ni {\bf Remark 2.2.} If $\sigma\in H^0(X,K_X^{1-p})$
vanishes to order
$>d(p-1)$ along a subvariety $Y$ of codimension $d$ for some 
$1\leq d\leq
n-1$, then $\sigma$ is not a splitting of $X$. This can be easily
seen by local computations. Hence we say that {\it a subvariety is 
 compatibly split by $\sigma $ with maximum multiplicity} if 
$\sigma$ 
vanishes to order $d(p-1)$ generically along $Y$. 

\vs .2cm \ni {\bf Proposition 2.3} {\it Let $f:X\to Y$ be a proper
morphism of smooth
varieties with $f_*{\cal O}_X={\cal O}_Y$. Let $X_1\subset X$ be
a smooth subvariety such that $f$ is smooth (submersive) along
$X_1$. If $X$ splits $X_1$ compatibly with maximum multiplicity,
then the induced splitting of $Y$ has maximum multiplicity
along $f(X_1)$.} 

\ni {\bf Proof}: By restricting to the complement of a codimension
$\geq 2$ subset of $Y$, we may assume that $Y_1:=f(X_1)$ is
smooth. By the previous proposition, it suffices to prove that
the blow up $B(Y)$ of $Y$ along $Y_1$ is split compatibly with
the exceptional $E(Y)$. By assumption, the blow up $B(X)$ of $X$
along $X_1$ splits the exceptional $E(X)$ compatibly. Hence it
suffices to show that the rational map $B(f):B(X)\to B(Y)$
induces an equality, $B(f)_*{\cal O}_{B(X)}={\cal O}_{B(Y)}$.
After localizing at the generic points of the exceptional
divisors we obtain a morphism $S\to R$ where $S\subset
k(B(Y))$ is the local ring of $B(Y)$ along $E(Y)$ and $R$ is the
local ring of $B(X)$ along $E(X)$. The hypotheses on $f$ implies
that the morphism $S\to R$ is a local morphism of 
discrete valuation rings. As $k(B(Y))\cap R=S$, any 
function on $B(Y)$ which is
generically regular along $E(X)$ is generaically regular 
along $E(Y)$.
Hence $B(f)_*{\cal O}_{B(X)}={\cal O}_{B(Y)}$.

\vs .2cm \ni {\bf 2.4.} Let $G$ be a semisimple, simply 
connected algebraic group and
$B$ a Borel subgroup of $G$ and $X=G/B$. In this paper we
prove that there is a section $F$ of $K_X^{-1}$ such that this
section has residual normal crossing at the coset of the
identity $e_{id}$. Moreover this section of $K_X^{-1}$
has multiplicity equal to the dimension of $X$ at $e_{id}$;
hence the blow up of a point of $X$ is also Frobenius split. 
  Hence one can obtain a straight forward
proof of the fact that that the homogeneous spaces $G/Q$ are
Frobenius split for any parabolic subgroup $Q$.  All the known
algebro-geometric proofs use the Bott-Samelson-Demazure
desingularization of $G/B$. 

\vs .2cm \ni {\bf 2.5.} Let $X$ be a homogeneous space in char 
$p>0$ and consider
$X\times X$ with diagonal $\Delta$. Let $B(X\times X)$ be the
blow-up of $X\times X$ along $\Delta$ with exceptional divisor
$E$.

\ni {\bf Conjecture:} $E$ is compatibly split in $B(X\times X)$, or equivalently there exists a splitting of $X\times X$ which
has maximum multiplicity along $\Delta $.



\vs .4cm \ni {\bf \S3 An application}

As an application of the above conjecture we prove the
conjecture of J. Wahl (cf. [W]) below on the surjectivity of the
Gaussian map. This has been proved in characteristic $0$ by S.
Kumar (cf. [K]) using representation-theoretic techniques. 
Let $X=G/P$, where $G$ is a complex semi simple group, 
and $P$ is a
parabolic sub group. Let ${\cal L}_1$
and ${\cal L}_2$ be ample line bundles on $X$. Let 
${\cal L}_1\otimes {\cal L}_2$ 
denote the external tensor product $p_1^*{\cal L}_1 \otimes 
p_2^*{\cal L}_2$ on $X\times X$. Let
${\cal I}_{\Delta}$ be the idal sheaf of the diagonal 
$\Delta\subset
X\times X$. Then Wahl (cf. [W]) considered the natural map (he reffered to
it as the Gaussian map):
$$\phi_{{\cal L}_1,{\cal L}_2}:H^0(X\times X,{\cal I}_{\Delta}
\otimes ({\cal L}_1\otimes{\cal L}_2))\to H^0(X,\Omega^1_X\otimes
{\cal L}_1\otimes {\cal L}_2)$$
induced by the projection 
$${\cal I}_{\Delta}\to {\cal I}_{\Delta}/{\cal I}_{\Delta}^2
\cong \Omega^1_X$$
In [W], Wahl had conjectured the following:

{\bf Wahl's Conjecture:} The Gaussian map
 $\phi_{{\cal L}_1,{\cal
L}_2}$ is surjective.

We sketch a proof of this assuming the conjecture of the
previous section. Let $\pi:B(X\times X)\rightarrow X\times X $ be
the blow up of the diagonal $\Delta$ with the exceptional divisor
$E$. Recall the following facts:

(i) $H^0(X\times X, {\cal I}_{\Delta}\otimes ({\cal L}_1\otimes
{\cal L}_2))\cong H^0(B(X\times X), \pi^*({\cal L}_1\otimes{\cal
L}_2)\otimes{\cal O}_{B(X\times X)}(-E))$

(ii) $H^0(X,\Omega^1_X\otimes{\cal L}_1\otimes{\cal L}_2)\cong H^0(E,
\pi^*({\cal L}_1\otimes{\cal L}_2)\otimes{\cal O}_{B(X\times X)}(-E)
\mid_E)$ 

\ni Thus the surjectivity of $H^0(X\times X, {\cal I}_{\Delta}
\otimes ({\cal L}_1\otimes{\cal
L}_2))\rightarrow H^0(X,\Omega^1_X\otimes{\cal L}_1\otimes{\cal
L}_2)$ is equivalent to the surjectivity of $H^0(B(X\times X),
\pi^*({\cal L}_1\otimes{\cal L}_2)\otimes{\cal O}_{B(X\times
X)}(-E))\rightarrow  H^0(E,
\pi^*({\cal L}_1\otimes{\cal L}_2)\otimes{\cal O}_{B(X\times X)}(-E)
\mid_E)$.  

\vs .2cm \ni {\bf Claim:} $\pi^*({\cal L}_1\otimes{\cal
L}_2)\otimes{\cal O}_{B(X\times X)}(-E)$ is generated by its
global sections.

For, we have
$$\pi_*(\pi^*({\cal L}_1\otimes{\cal
L}_2)\otimes{\cal O}_{B(X\times X)}(-E))=
({\cal L}_1\otimes{\cal
L}_2)\otimes\pi_*({\cal O}_{B(X\times X)}(-E))= ({\cal
L}_1\otimes{\cal L}_2)\otimes {\cal I}_{\Delta}.$$ Now,
$H^0(X\times X,  ({\cal
L}_1\otimes{\cal L}_2)\otimes {\cal I}_{\Delta})$ is the kernel of
the multiplication map $H^0(X, {\cal L}_1)\otimes H^0(X, {\cal
L}_2)\rightarrow  H^0(X, {\cal L}_1\otimes{\cal L}_2)$. On the
other hand, we have (cf.[K-R]) that the kernel of the surjective map 
$SH^0(X, {\cal
L}_1)\otimes SH^0(X, {\cal L}_2)  
\rightarrow\bigoplus _{m_1,m_2 \in {\bf Z}^+}\ H^0(X, {\cal
L}_1^{m_1}\otimes{\cal L}_2^{m_2})$ (here, for a vector space $V$,
$SV$ denotes its symmetric algebra) is generated (as an ideal) by
the kernel of $H^0(X, {\cal L}_1)\otimes H^0(X, {\cal
L}_2)\rightarrow  H^0(X, {\cal L}_1\otimes{\cal L}_2)$.
The Claim now follows from this.  By the Conjecture of the previous
section,
$E$ is compatibly split in
$B(X\times X)$. This together with [R], Proposition 1.13,(ii) and
 the Claim above implies the surjectivity of 

\ni$H^0(B(X\times X),
\pi^*({\cal L}_1\otimes{\cal L}_2)\otimes{\cal O}_{B(X\times
X)}(-E))\rightarrow  H^0(E,
\pi^*({\cal L}_1\otimes{\cal L}_2)\otimes{\cal O}_{B(X\times X)}(-E)
\mid_E)$.  


\vs 1cm \ni {\bf \S4 The section $F\in H^0(G/B, L_{2\rho} $)} 
 
Let $G$ be a semi simple, simply connected algebraic group
defined over an algebraically closed field $k$. 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 with respect to $\a$. 
For $\b \in R^+$, let $E_\b,\ F_\b$ the
elements of the Chevalley basis for 
$\gg (={\rm Lie}G$), corresponding to $\b.\ -\b$
respectively. 

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

It is well known that
$$K_{G/B}^{-1}= L_{2\rho}$$
where
$$2\rho = \sum _{\b \in R^+}\ \b$$


Let $\l$ be a character of $T$ (or $B$) which is dominant. Let $L_\l $ be the
associated line bundle on $G/B$. Let $V$ be the $G$-module $H^0 (G/B,L_\l)$. 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 $U^-$ be the unipotent part of the Borel sub group $B^-
 $, opposite to $B$. We can identify
$U^-$ as an affine open subset  containing $e_{{\rm id}}$.
Recall that $$U^- = \prod _{\b \in  R^+}\ U_{-\b},\ U_{-\b} \approx
{\bf G}_a. \leqno {(2)}$$  We denote by $\{  x_\b,\  \b \in  R^+\}$, the canonical
coordinate system given by (2). If $f\in V$, then denoting the
restriction of $f$ to $U^-$  by $f'$, we note that the
evaluations of $\partial f' \over \partial x_\b $ and $F_\b f,\  \b
\in  R^+ $, at $e_{{\rm id}} $ coincide. 

\vs .2cm \ni {\bf 4.1.} Let us fix a dominant weight $\l$. Let $<,>$ denote the
canonical pairing on 

\ni $H^0(G/B, L_\l)\times  H^0(G/B, L_\l)^\vee$ (here, $H^0(G/B, L_\l)^\vee $
denotes the linear dual of 

\ni $H^0(G/B, L_\l)$). For $G$ classical (resp. $G_2$), a
basis for $H^0(G/B, L_{\omega_d})$ ( $\omega_d}$ being a fundamental weight)
consisting of weight vectors ( for the action of $T$) is constructed in [L-S]$_2$
(resp. [L]$_1$). This basis is indexed by certain pairs of Weyl group elements (called
{\it admissible pairs}). Following the notation in [L-S]$_2$, we shall denote this
basis by $\{p_{\t,\phi}\}$. This basis includes the extremal weight vectors (of
weight $-w(\omega_d),\ w\in W)$, the corresponding admissible pairs being simply
$(w,w),\ w \in W$. In the sequel, we shall denote an extremal weight vector $p_{w,w}$
by just $p_w$. We shall denote by $\{Q_{\t,\phi}\}$ the basis of  $H^0(G/B,
L_\l)^\vee $ dual to $\{p_{\t,\phi}\}$. For $f \in H^0(G/B, L_\l)$, let us denote
$f'=f\ |\ _{U^{-}} $. Let us fix a total order on $R^+$. Then in the power series
expansion for $f'$ in the local coordinates $x_\b,\ \b \in 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  $F_{\b_1}^{n_1}\cdots F_{\b_r}^{n_r}f=cp_{{\rm id}},\ c\in k^* $. 

\ni Now 

\ni $F_{\b_1}^{n_1}\cdots
F_{\b_r}^{n_r}f=cp_{{\rm id}},\ c\in k^* $ 

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

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

\ni Hence, taking $f=p_{\t,\phi}$, we obtain

\ni  $<F_{\b_1}^{n_1}\cdots F_{\b_r}^{n_r}f,\  Q_{{\rm id}}> \not=
0$

\ni  $\Longleftrightarrow \ Q_{\t,\phi}$ occurs with a nonzero coefficient in the
expression for $F_{\b_r}^{n_r}\cdots F_{\b_1}^{n_1}Q_{{\rm id}}$ as a 
linear combination of the $Q_{\te,\delta}$'s.

\ni Thus we obtain

\proclaim Proposition 4.2. Given $p_{\t,\phi}$, a monomial $x_{\b_1}^{n_1}\cdots
x_{\b_r}^{n_r} $ occurs with a non zero coefficient in $p'_{\t,\phi}$, if and only
if $Q_{\t,\phi}$ occurs with a nonzero coefficient in the
expression for $F_{\b_r}^{n_r}\cdots F_{\b_1}^{n_1}Q_{{\rm id}}$ as a 
linear combination of the $Q_{\te,\delta}$'s.  

\vs .2cm \ni {\bf 4.3} In the following, we are going to exhibit an $F \in H^0(G/B,
L_{2\r})$ such that $F$ has residual normal crossing at $e_{{\rm id}}$; further, it
has maximum multiplicity at $e_{{\rm id}}$. In the discussion below, we shall follow the notations
in [Bou] to denote the elements of $R$.

\vs .2cm \ni {\bf Type A$_l$.}

\ni Let $V=k^n$ ($n=l+1$), and $G=$ SL($V$). It is well known that $W$ may be
identified with the symmetric group $S_n$. For $1\leq d\leq n-1$, let $P_d$ be the
maximal parabolic subgroup corresponding to ``omitting $\a_d$", and $W_{P_d}$ the
Weyl group of $P_d$. Then $W^{P_d}$, ``the set of minimal representatives of
$W_{P_d}$ in $W$" may be identified with $\{(i_1,\cdots ,i_d),\ 1\leq i_1<i_2<\cdots
<i_d\leq n\}$. 

\vs .2cm \ni {\bf A.1 The elements $F_\b,\ \b \in R^+$, in the Chevalley Basis for
$\gg$.}

\ni  It is well known that
$$\gg={\rm sl}(n)$$
For $\b=\e_i-\e_j,\ F_\b$ is simply the elementary matrix
$E_{ji}$, with $1$ at the $(j,i)^{{\rm th}} $ place, and $0$ elsewhere.

 \vs .2cm \ni {\bf A.2 Total order on $R^+$. }

\ni We shall take the total order on $\{\e_i-\e_j,\ 1\leq i<j \leq n\}$ given by the
lexicographic order on $\{(i,j),\ 1\leq i<j \leq n\}$, namely, $(i,j) < (k,l)$, if
either $i<k$, or $i=k$, and $j<l$.

\vs .2cm \ni \vs .2cm \ni {\bf A.3 The section $F\in H^0(G/B, L_{2\rho}) $} 


\ni For $w \in W$, we shall denote $\pi_d (w)$ by $w^{(d)}$, where $\pi_d$ is 
the canonical map $W \rightarrow W/W_{P_d}$. For $w \in W$, we shall denote by 
$p_w^{(d)}$
the extremal weight vector in $H^0(G/B, L_{\omega_d})$ of weight $-w(\omega_d)$. In
fact, in the linear case, our basis $\{p_{\t,\phi}\}$ is simply $\{p_w^{(d)},\ w\in
W^{P_d}\}$. 

\ni For $\b \in R^+$, say $\b=\e_i-\e_j$, in the sequel we shall denote $s_\b$ by
just ($i,j$), and the affine co-ordinate $x_\b $ by $x_{(i,j)}$.

\ni We are going to
choose two elements $f_1^{(d)},\ f_2^{(d)}$ in $ H^0(G/B, L_{\omega_d}),\ 1\leq d
<n-1$, and one element $f_1^{(n-1)}$ in $ H^0(G/B, L_{\omega_{n-1}})$ so that $F= 
\prod_{l,d}\ f_l^{(d)}$ will have residual normal crossing at $e_{{\rm id}}$. (Here, $f_2^{(n-1)}=p_{{\rm
id}}$ (which of course is $\equiv 1$ on $U^-$)). Set 
$$ f_1^{(d)}= p^{(d)}_{w_{1,d}},\ f_2^{(d)}= p^{(d)}_{w_{2,d}}$$ where $w_{1,d},\
w_{2,d}$ are given as follows. 


\vs .2cm \ni {\bf Case 1.} $n$ is even, say $n=2m$.

\ni For $d< m$   
$$w_{1,d}=(1,2d)(2,2d-1)\cdots (d,d+1)$$
$$w_{2,d}=(1,2d+1)(2,2d)\cdots (d,d+2)$$
and for $d\geq  m$, say $d=m+r$, 
$$w_{1,d}= (2r+1,2m)(2r+2,2m-1)\cdots (d,d+1)$$
$$w_{2,d}= (2r+2,2m)(2r+3,2m-1)\cdots (d,d+2)$$


\vs .2cm \ni {\bf Case 2.} $n$ is odd, say $n=2m+1$.

\ni For $d\leq m$   
$$w_{1,d}=(1,2d)(2,2d-1)\cdots (d,d+1)$$
$$w_{2,d}=(1,2d+1)(2,2d)\cdots (d,d+2)$$
and for $d> m$, say $d=m+r$, 
$$w_{1,d}= (2r,2m+1)(2r+1,2m)\cdots (d,d+1)$$
$$w_{2,d}= (2r+1,2m+1)(2r+2,2m)\cdots (d,d+2)$$

\vs .2cm \ni {\bf The expression for  $f_l^{(d)}\ |\ _{U^-}$ in the local
co-ordinates $x_\b$'s.}

\ni  As above, for $f \in
{\rm H}^0(G/B, L_\l)$, let $f'=f\ |\ _{U^{-} $. Let $p^{(d)}_w \in {\rm H}^0(G/B,
L_{\omega_d})$. Let $w^{(d)}=(a_1,\cdots ,a_d)$. Let $\{ b_1,\cdots ,b_r\}$ be the
set of all $a_k$ such that $a_k \leq d$. Let $\{ c_1,\cdots ,c_s\}$ (resp. $\{
h_1,\cdots ,h_s\}$) be the complement of $\{ b_1,\cdots ,b_r\}$ in $\{1,\cdots ,d \}$
(resp. $\{a_1,\cdots ,a_d \}$). We have $$(H^0(G/B, L_{\omega_d}))^\vee =
\wedge^dV.$$ Taking the monomial differential operator $D=F_{(c_s,h_1)}\cdots
F_{(c_1,h_s)} $, the fact that $ e_{b_1}\wedge \cdots \wedge
e_{b_r}\wedge e_{ h_1}\wedge \cdots \wedge e_{h_s} $ is present in the
expression for $DQ_{{\rm id}}$ as a linear combination   of the $\{Q_\delta \}$'s
(=$\{e_{i_1}\wedge \cdots \wedge e_{i_d},\ (i_1,\cdots ,i_d)\in  W^{P_d}\}$) implies
that $x_{(c_1,h_s)}\cdots x_{(c_s,h_1)}$ occurs with a  non zero
coefficient in $(p^{(d)}_w)'$ (cf. Proposition 4.2). Any other monomial in
$(p^{(d)}_w)'$ may be obtained as follows. Fix a pair $\t,\s \in S_s$. Choose
$Y=\{y_{tj},\ 1\leq j\leq p_t,\ \ 1\leq t\leq s \}$ such that $$y_{tj}\not\in
\{h_{\s(m)}, \ 1\leq m\leq t-1 \},\ c_{\t(t)}<y_{t1}<\cdots <y_{tp_t}< h_{\s (t)} .$$
The corresponding monomial in $p_w'$ is given by   $$ f_w^{\t,\s,Y}= \prod _{1\leq t
\leq s} \  x_{(y_{tp_t},h_{\s (t)})}x_{(y_{tp_t-1},{y_{tp_t}})} \cdots
x_{(y_{t1},y_{t2})}x_{(c_{\t (t)},y_{t1})}$$ (note that any $c_l <$ any $h_k$).

\ni This is because a typical monomial differential operator $D$ such that in
the expression for $D Q_{{\rm id}}$ as a linear combination of the $Q_\delta $'s,
$Q_w$ occurs with a non zero coefficient is of the form

\ni $F_{(y_{tp_t},h_{\s (t)})}F_{(y_{tp_t-1},{y_{tp_t}})} \cdots
F_{(y_{t1},y_{t2})}F_{(c_{\t (t)},y_{t1})}$ (notations being as above).

\ni We now apply these considerations to $f_l^{(d)},\ l=1,2 $. For $ f= f_l^{(d)}$, we
shall denote $f'$ by just $f$. We now distinguish the following two cases. 



\vs .2cm \ni {\bf Case 1.} $n$ is even, say $n=2m$.


\ni For $\te = w_i^{(d)},\ d<m,\ i=1,2$, we have $s=d,\ 
\{ c_1,\cdots ,c_s\}=\{1,\cdots ,d \}$.   


\ni For $\te = w_1^{(d)},\ d\geq m$, say $d=m+r$, we have $s=d-2r,\ 
\{ c_1,\cdots ,c_s\}=\{2r+1,\cdots ,d \}$ 
\ni For $\te = w_2^{(d)},\ d\geq m$, say $d=m+r$, we have $s=d-2r-1,\ 
\{ c_1,\cdots ,c_s\}=\{2r+2,\cdots ,d \}$    

\ni If $d<m$, then we have, modulo $(x_{(i,j)},\   i+j\leq 2d)$
$$f_1^{(d)}\equiv  x_{(1,2d)}x_{(2,2d-1)}\cdots x_{(d,d+1)}$$ 
and modulo $(x_{(i,j)},\   i+j\leq 2d+1)$
$$f_2^{(d)}\equiv  x_{(1,2d+1)}x_{(2,2d)}\cdots x_{(d,d+2)}$$ 
If $d\geq m$. then we have, modulo $(x_{(i,j)},\   i+j\leq 2d)$
$$f_1^{(d)}\equiv x_{(2r+1,2m)}x_{(2r+2,2m-1)}\cdots x_{(d,d+1)}$$
and modulo $(x_{(i,j)},\   i+j\leq 2d+1)$
$$f_2^{(d)}\equiv x_{(2r+2,2m)}x_{(2r+3,2m-1)}\cdots x_{(d,d+2)}$$

\vs .2cm \ni {\bf Case 2.} $n$ is odd, say $n=2m+1$.

\ni For $\te = w_i^{(d)},\ d\leq m,\ i=1,2$, we have $s=d,\ 
\{ c_1,\cdots ,c_s\}=\{1,\cdots ,d \}$.   

\ni For $\te = w_1^{(d)},\ d> m$, say $d=m+r$, we have $s=d-2r+1,\ 
\{ c_1,\cdots ,c_s\}=\{2r,\cdots ,d \}$ 
\ni For $\te = w_2^{(d)},\ d> m$, say $d=m+r$, we have $s=d-2r,\ 
\{ c_1,\cdots ,c_s\}=\{2r+1,\cdots ,d \}$    

\ni If $d\leq m$, then modulo $(x_{(i,j)},\   i+j\leq 2d)$
$$f_1^{(d)}\equiv  x_{(1,2d)}x_{(2,2d-1)}\cdots x_{(d,d+1)}$$
and modulo $(x_{(i,j)},\   i+j\leq 2d+1)$
$$f_2^{(d)}\equiv  x_{(1,2d+1)}x_{(2,2d)}\cdots x_{(d,d+2)}$$ 

\ni If $d> m$, then modulo $(x_{(i,j)},\   i+j\leq 2d)$
$$f_1^{(d)}\equiv x_{(2r,2m+1)}x_{(2r+1,2m)}}\cdots x_{(d,d+1)}$$
and modulo $(x_{(i,j)},\   i+j\leq 2d+1)$
$$f_2^{(d)}\equiv x_{(2r+1,2m+1)}x_{(2r+2,2m)}}\cdots x_{(d,d+2)}$$

\vs .2 cm \ni {\bf Remark $A_n$.} (i) Considering the lower tringular matrix of
variables $(x_{ij})_{1\leq i,j \leq n}, \ x_{ij}=0, \ i<j,\ x_{ii}=1$, we observe that
in the above construction, starting with $x_{21}$, at each step,
the variables along the diagonal are picked up (here, by a diagonal we mean the
set of all entries $x_{kl}$ such that the sum $k+l$ is the same).

\ni (ii) Note that in the above construction of $F$, we have one degree of freedom (
since choosing just one section from $ H^0(G/B, L_{\omega_{n-1}})$ was enough for
constructing the desired section $F$ in H^0(G/B, L_{2\r})$).   

\vs .2cm \ni {\bf Type C$_n$}

\ni  Let $V=k^{2n}$ together with a non-degenerate skew-symmetric bilinear form (,),
and $G=$ Sp($V$). Taking the matrix of the form (,) (with respect to the standard
basis $\{e_1,\cdots ,e_{2n} \}$ of $V$) to be $E$, the anti-diagonal ($d_1,\cdots
,d_{2n}$) where $d_i=\pm 1$ according as $i\leq $ or $>n$, we may realize $G={\rm
Sp}(V)$ as the fixed point set $SL(V)^\sigma $, where $\sigma :SL(V) \rightarrow SL(V)
$ is the involution given by $\sigma (A)=E(^tA)^{-1}E^{-1}$, and $W$ may be identified
with
$$  \{(a_1\cdots a_{2n})\in S_{2n}\ |\ a_i=2n+1-a_{2n+1-i},\  1\leq i\leq 2n \}$$ 
For $1 \leq i \leq 2n  $, we shall denote $$ i'=2n+1-i .$$
For $1\leq d\leq n$, $W^{P_d}$ may be identified with 
$$\{(a_1\cdots ,a_d),\ 1\leq a_1<a_2<\cdots <a_d\leq 2n \ |  
 \ {\rm if\ }i \in \{a_1\cdots ,a_d  \}, \ {\rm then\ }i'\not\in 
\{a_1\cdots ,a_d  \}\}$

\ni Denoting $H={\rm SL}(V)$, let $T_{2n}$ be the maximal torus in $H$ consisting of
diagonal matrices. Then $\s $ induces an involution on the character group
$X(T_{2n})$ (= Hom($T_{2n},{\bf G_m}$)), namely, $$ \s (\e_i)=-\e_{2n+1-i}.\ 1 \leq i
\leq 2n.$$ Further $\s $ leaves $R(H)$, and $R^+(H)$ stable; and $R(G)$ (resp.
$R^+(G)$) may be identified with the orbit space $R(H)$ (resp. $R^+(H)$) modulo the
action of $\s$ (see [L-S]$_1$ for details). We have
$$R^+(G) \quad = \quad \cases{\quad \e_i-\e_j, & if $1\leq i<j\leq n$\cr\cr
\quad \e_i+\e_j, & if $1\leq i<j\leq n$\cr\cr
\quad 2\e_i, & if $1\leq i\leq n$\cr}$$
For $1\leq i,j\leq 2n,\
i\not= j,\  (i,j)$ (considered as an element in $S_{2n}$) shall denote the
transposition of $i$ and $j$.  
\vs .2cm \ni {\bf C.1 The elements $F_\b,\ \b \in R^+$, in the Chevalley Basis for
$\gg$.}

\ni The involution $\s $ induces an involution
$$\s :{\rm sl}(2n) \rightarrow {\rm sl}(2n),\ A\mapsto -E(^tA)E^{-1}\
(=E(^tA)E)$$
(note that $E^{-1}=-E$). In particular, we have
$$\s (E_{ij}) \quad= \quad \cases {\quad -E_{j'i'}, & if $i,j$ are both $\leq n$ or
both $>n$\cr\cr
\quad E_{j'i'}, & if one of $\{i,j\}$ is $\leq n$ and the other
$>n$\cr}$$
We have,
$$\gg=\{A \in {\rm sl}(2n)\ |\ E(^tA)E=A\}.$$
Under the above identification, we have,
$$F_{\e_i-\e_j}=E_{ji}-E_{i'j'},\ F_{\e_i+\e_j}=E_{j'i}+E_{i'j},\ F_{2\e_i}=
E_{i'i}.$$
(see [L]$_2$ for details).

\vs .2cm \ni {\bf C.2 Total order on $R^+$. }

\ni Each $\b \in R^+$ may be identified as $\e_i -\e_j\ |\ _{T(G)} $ for suitable
$i,j, \ 1\leq i\leq n,\ j \leq 2n, \ i<j$ ( here, $T(G)= G \cap T_{2n}$, and for
$j>n,\ \e_j$ as an element of $X(T)$ is to be understood as $-\e _{j'}$. See
[L-S]$_1$ for details). With this identification, we take (as in the
SL($n$)-case) the total order on $R^+$ to be the one induced by the lexicographic
order on $\{(i,j)\}$.

 \vs .2cm \ni \vs .2cm \ni {\bf C.3 The section $F\in H^0(G/B, L_{2\rho}) $} 

\ni Here again we are going to define elements $f_1^{(d)},\ f_2^{(d)}$ in $ {\rm
H}^0(G/B, L_{\omega_d}),\ 1\leq d <n$, and  $f_1^{(n)}$ in $ {\rm H}^0(G/B,
L_{\omega_n})$ so that $F=  \prod_{l,d}\ f_l^{(d)}$ will have the required property.
(Here, $f_2^{(n)}=p_{{\rm id}}$ (which of course is $\equiv 1$ on $U^-$)). These are
defined similar to Type A_n.

\ni Let $1\leq d \leq n$. Set
$$w_{1,d}=(1,2d)(2,2d-1)\cdots (d,d+1)$$
$$w_{2,d}=(1,2d+1)(2,2d)\cdots (d,d+2)$$
$$ f_l^{(d)}= p_{w_{l,d}}^{(d)}}\ |\ _{G} ,\ l=1,2$$
(for $d=n$, we define only $w_{1,d},f_1^{(d)}$). 

\ni Here, $w_{l,d},\ l=1,2 $ need not
belong to $W$. We look at them as elements in $W(H)$, and  $p_{w_{l,d}}^{(d)}},\
l=1,2$ will have the same meaning as in the SL($n$)-case. We have in fact 
 $f_l^{(d)}=p_{\t_l,\phi_l},\ l=1,2$ for suitable admissible pairs $\t_l,\phi_l$
which we now describe. Consider $f\in \wedge ^2V$ given by
$$f=e_1\wedge e_{2n}+e_2\wedge e_{2n-1}+ \cdots + e_n\wedge
e_{n+1}.$$
Then we have
$$ {\rm H}^0(G/B, L_{\omega_d})^\vee =\{v\in \wedge ^d V\ |\ v\wedge
f^{n+1-d}=0\}.$$
In particular we have
$$Q_{{\rm id}}=e_1\wedge \cdots \wedge e_{d}.$$
Set
$$t\quad=\quad\cases{\quad 2d, & if $l=1$ \cr\cr
\quad 2d+1,  & if $l=2$ \cr }$$
$$s\quad=\quad\cases{\quad d+1, & if $l=1$ \cr\cr
\quad d+2,  & if $l=2$ \cr }$$

$$q=n+s-t$$
(note that $q=\ (n+1-d)$ is the same in both cases).

\ni  For $1\leq i \leq n,\ 1\leq j\leq 2n,\
i< j,\ x_{(i,j)}$ shall denote the affine co-ordinate $x_{\e_i-\e_j}$ (here, for
$j>n,\ \e_j$ as an element of $X(T)$ is to be understood as $-\e _{j'}$).

\ni A typical monomial in $(f_l^{(d)})'$ may be described as follows. As in the
SL($n$)-case, let us write $w_{l,d}^{(d)}=(a_1,\cdots ,a_d)$. Note that with notations
as in the SL($n$)-case, we have, $\{c_1,\cdots ,c_d\}=\{1,\cdots ,d\}$. Fix a pair
$\te,\s \in S_d$. Choose 

\ni $Y=\{y_{uj},\ 1\leq j\leq p_u,\ \ 1\leq u\leq d \}$ such that
$$y_{uj}\not\in \{a_{\s(m)},\ a'_{\s(m)} \ 1\leq m\leq
u-1 \},\ \te (u)<y_{u1}<\cdots <y_{up_u}<  a_{\s (u)} .$$ 
The corresponding monomial in $(f_l^{(d)})'$ is given by  
$$ f_l^{\te,\s,Y}= \prod_{1\leq u\leq d}\ x_{(y_{up_u},a_{\s
(u)})}x_{(y_{up_u-1},{y_{up_u}})} \cdots x_{(y_{u1},y_{u2})}x_{({\te (u)},y_{u1})}$$
If $d=n$, or $t\leq n$, then $f_l^{(d)}$ are
extremal weight vectors, and as in the SL($n$) case, we have, 
modulo $(x_{(i,j)},\   i+j\leq 2d)$
$$f_1^{(d)}\equiv  x_{(1,2d)}x_{(2,2d-1)}\cdots x_{(d,d+1)}$$
and modulo $(x_{(i,j)},\   i+j\leq 2d+1)$
 $$f_2^{(d)}\equiv  x_{(1,2d+1)}x_{(2,2d)}\cdots x_{(d,d+2)}$$ 

\vs .2cm \ni Let then $d \leq n-1$, and $t>n$. Let us write $t=k'$, for
some $k\leq n$. We distinguish the following two cases.

\ni {\bf Case 1.} $k<s$

\ni We have
$$ w_{l,d}^{(d)}=( s,s+1,\cdots  ,n ,n' ,(n-1)',\cdots ,s',\cdots ,k' )$$
Further 
$$f_l^{(d)}=p_{\t,\phi}$$ 
(cf.[L]$_2$) where $\{\t,\phi \}$ is the admissible pair given by
$$\phi = (q,q+1,\cdots ,k-1,n',(n-1)',\cdots ,k'),$$
$$\t=(s,s+1\cdots ,n,(s-1)',(s-2)',\cdots ,k',(k-1)'\cdots ,q')$$
Hence as in the SL($n$)-case, we obtain, modulo $(x_{(i,j)},\   i+j\leq t)$
$$f_l^{(d)}\equiv  x_{(1,t)}x_{(2,t-1)}\cdots x_{(d,s)}$$ 
(note that $x_{(1,t)}x_{(2,t-1)}\cdots x_{(d,s)}$ is simply the monomial 
$f_l^{\te ,\s,Y}$, with $\te =\ {\rm id},\ \s=(d\cdots 1) $, and $Y=\emptyset$).

\vs .2cm \ni {\bf Case 2.} $k\geq s$

\ni We have
$$ w_{l,d}^{(d)}=( s, s+1,\cdots ,k, \cdots ,n, n',(n-1)',\cdots ,k' ),$$
$$f_l^{(d)}=p_{\t,\phi}$$ 
where $\{\t,\phi \}$ is the admissible pair given by
 $$\phi = (q,q+1, ,\cdots ,k-1,n',(n-1)'\cdots 
,k'),$$ 
$$\t=(s,s+1,\cdots ,k-1,k,\cdots ,n ,(s-1)',(s-2)',\cdots ,q')$$
As above, we have, modulo $(x_{(i,j)},\   i+j\leq t)$ 

\ni $f_l^{(d)}\equiv  x_{(1,t)}x_{(2,t-1)}\cdots x_{(d,s)}$.              

\vs .2cm \ni {\bf Remark $C_n$.} Here again we have one degree of freedom in the
construction of $F$.

\vs .2cm \ni {\bf Type B$_n$}

\ni Let $V=k^{2n+1}$  together with a non degenerate symmetric bilinear form (,), 
and $G=$ SO($V$). Taking the matrix of the form (,) (with respect to the standard
basis 

\ni $\{e_1,\cdots ,e_{2n+1} \}$ of $V$) to be $E$, the anti-diagonal ($1,\cdots ,1$)
of size $2n+1 \times 2n+1 $, we may realize $G=SO(V)$ as the fixed
point set $SL(V)^\sigma $, where $\sigma :SL(V) \rightarrow SL(V)
$ is the involution given by $\sigma (A)=E(^tA)^{-1}E$, and $W$ may be identified
as $$  W=\{(a_1\cdots a_{2n+1})\in S_{2n+1}\ |\
a_i=2n+2-a_{2n+2-i},\  1\leq i\leq 2n+1 \}$$ (note that $a_{n+1}=n+1
$). 

\ni For $1 \leq i \leq 2n+1  $, we shall denote
$$ i'=2n+2-i .$$

As in the case of Sp($2n$), we have
$$R^+(G) \quad = \quad \cases{\quad \e_i-\e_j, & if $1\leq i<j\leq n$\cr\cr
\quad \e_i+\e_j, & if $1\leq i<j\leq n$\cr\cr
\quad \e_i, & if $1\leq i\leq n$\cr}$$
For $1\leq i,j\leq 2n+1,\
i\not= j,\  (i,j)$ (considered as an element in $S_{2n+1}$) shall denote the
transposition of $i$ and $j$.  


\vs .2cm \ni {\bf B.1 The elements $F_\b,\ \b \in R^+$, in 
the Chevalley Basis for $\gg$.}

\ni The involution $\s $ induces an involution
$$\s :{\rm sl}(2n+1) \rightarrow {\rm sl}(2n+1),\ A\mapsto -E(^tA)E $$
 In particular, we have
$$\s (E_{ij}) = -E_{j'i'}, \ 1\leq i,j\leq 2n+1 .$$
We have,
$$\gg=\{A \in {\rm sl}(2n+1)\ |\ E(^tA)E=-A\}.$$

Under the above identification, we have,
$$F_{\e_i-\e_j}=E_{ji}-E_{i'j'},\ F_{\e_i+\e_j}=E_{j'i}-E_{i'j},\ F_{\e_i}=
E_{n+1i}-E_{i'n+1}.$$ 
(see [L]$_2$ for details). 

\vs .2cm \ni {\bf B.2 Total order on $R^+$. }

\ni As in the Sp($2n$)-case, each $\b \in R^+$ may be identified as $\e_i -\e_j\ |\
_{T(G)} $ for suitable $i,j, \ 1\leq i\leq n,\ j \leq 2n+1, \ i<j,\ j\not= i'$ ( here,
$T(G)= G \cap T_{2n+1}$; further, for $j>n+1,\ \e_j$ as an element of $X(T)$ is to be
understood as $-\e _{j'}$, and $\e_{n+1}$ is to be understood as the trivial
character. See [L-S]$_1$ for details). With this identification, we take (as in the
SL($n$)-case) the total order on $R^+$ to be the one induced by the lexicographic
order on $\{(i,j)\}$.


 \vs .2cm \ni {\bf B.3 The section $F\in H^0(G/B, L_{2\rho}) $}


\ni Here again we are going to define elements $f_1^{(d)},\ f_2^{(d)}$ 
in $ {\rm
H}^0(G/B, L_{\omega_d}),\ 1\leq d <n$, and  $f_1^{(n)}$ in $ 
{\rm H}^0(G/B,
L_{\omega_n})$ so that $F=  \prod_{l,d}\ f_l^{(d)}$ will have 
the required property.
(Here, $f_2^{(n)}=p_{{\rm id}}$ (which of course is $\equiv 1$ 
on $U^-$)). 

\ni For $d=n$, we set
$w_{1,n}=s_{\e_1}\cdots s_{\e_n}$, and define
$f_1^{(n)}=p_{w_{1,n}}$.

\ni For $1\le d\le n-1$, the elements
$f_l^{(d)},\ l=1,2$ are defined similar to Type C$_n$. We
distinguish the following two cases. 
   
\vs .2cm \ni {\bf Case 1.} $n$ is even, say $n=2m$.

\ni For $d< m$, set   
$$w_{1,d}=(1,2d)(2,2d-1)\cdots (d,d+1)$$
$$w_{2,d}=(1,2d+1)(2,2d)\cdots (d,d+2)$$
For $d>  m, \ d\not= n$, set 
$$w_{1,d}= (1,2d+1)(2,2d)\cdots 
(2r,n+2)(2r+1, n)(2r+2, n-1)\cdots 
(d,d+1)$$ 
$$w_{2,d}= (1,2d+2)(2,2d+1)\cdots 
(2r+1, n+2)(2r+2, n)(2r+3, n-1)\cdots
(d,d+2)$$ 
(here $r$ is given by $d=m+r$). For $d=m$
$$w_{1,d}=(1,2d)(2,2d-1)\cdots (d,d+1)$$
$$w_{2,d}= (1,n+2)(2,n)(3, n-1)\cdots (d,d+2)$$ 

\vs .2cm \ni {\bf Case 2.} $n$ is odd, say $n=2m+1$.

\ni For $d\leq m$, set   
$$w_{1,d}=(1,2d)(2,2d-1)\cdots (d,d+1)$$
$$w_{2,d}=(1,2d+1)(2,2d)\cdots (d,d+2)$$
For $d> m,\ d\not= n$, say $d=m+r$, where $r\le m$, set 
$$w_{1,d}= (1,2d+1)(2,2d)\cdots 
(2r-1,n+2)(2r, n)(2r+1, n-1)\cdots 
(d,d+1)$$ 
$$w_{2,d}= (1,2d+2)(2,2d+1)\cdots 
(2r, n+2)(2r+1, n)(2r+2, n-1)\cdots
(d,d+2)$$ 

\ni Set
$$ f_l^{(d)}= p_{w_{l,d}}^{(d)}}\ |\ _{G} ,\ l=1,2,\ d<n,$$
$$ f_1^{(d)}= p_{w_{1,d}}^{(d)}},\ d=n$$
(note that $ f_l^{(d)} \in $ {\rm
H}^0(G/B, L_{\omega_d})). 

\ni  For $1\leq i \leq n,\ 1\leq j\leq 2n+1,\
i< j,\ j\not= i',\ x_{(i,j)}$ shall denote the 
affine co-ordinate $x_{\e_i-\e_j}$
(the convention for $\e_j,\ j>n$ as an element of $X(T)$ being
as above). 

 \ni As in the Sp($2n$)-case, a typical monomial in $(f_l^{(d)})',
1\le d\le n-1$ may be described as follows. Write
$w_{l,d}^{(d)}=(a_1,\cdots ,a_d)$. Note that with notations as in
the SL($n$)-case, we have,
$\{c_1,\cdots ,c_d\}=\{1,\cdots ,d\}$. Fix a pair $\te,\s \in S_d$.
Choose 

\ni $Y=\{y_{uj},\ 1\leq j\leq p_u,\ \ 1\leq u\leq d \}$ such that
$$y_{uj}\not\in \{a_{\s(m)},\ a'_{\s(m)} \ 1\leq m\leq
u-1 \},\ y_{uj}\not= y'_{uj-1},\  \te (u)<y_{u1}<\cdots <y_{up_u}<  a_{\s (u)} .$$ 
(here, $y_{u0}=\te (u)$). The corresponding monomial in $(f_l^{(d)})'$ is given by  
$$ f_l^{\te,\s,Y}= \prod_{1\leq u\leq d}\ x_{(y_{up_u},a_{\s
(u)})}x_{(y_{up_u-1},{y_{up_u}})} \cdots x_{(y_{u1},y_{u2})}x_{({\te (u)},y_{u1})}$$

\vs .2cm \ni Let $n=2m$ (resp.$2m+1$). If  $d<m$,
or $d=m,\ n=2m, \ {\rm and }\ l\not= 2$, then $f_l^{(d)}$ 
are extremal weight
vectors, and as in the SL($n$) case, we have, modulo $(x_{(i,j)},\  
i+j\leq 2d)$,
$$f_1^{(d)}\equiv  x_{(1,2d)}x_{(2,2d-1)}\cdots x_{(d,d+1)}$$
and modulo $(x_{(i,j)},\   i+j\leq 2d+1)$,
 $$f_2^{(d)}\equiv  x_{(1,2d+1)}x_{(2,2d)}\cdots x_{(d,d+2)}$$


\vs .2cm \ni Let $d \leq n-1$, and $d\geq m$ (if $d=m$, 
and $n=2m$, then $l=
2$).

\ni  Set
$$t\quad=\quad\cases{\quad 2d+1, & if $l=1$ \cr\cr
\quad 2d+2,  & if $l=2$ \cr }$$
$$s\quad=\quad\cases{\quad d+1, & if $l=1$ \cr\cr
\quad d+2,  & if $l=2$ \cr }$$

$$q=n+1+s-t$$
(note that $q=\ (n+1-d)$ is the same in both cases). We have, $t\geq n+2$. Let us
write $t=k'$, for some $k\leq n$.

\ni As in the Sp($2n$)-case, we distinguish the following two cases.

\ni {\bf Case 1.} $k<s$

\ni We have
$$ w_{l,d}^{(d)}=( s,s+1,\cdots  ,n ,n',\cdots ,(n-1)',
\cdots ,s',\cdots ,k' ),$$
 $$f_l^{(d)}=p_{\t,\phi},$$ 
(cf. [L]$_2$) where $\{\t,\phi \}$ is the admissible pair given by
$$\phi = (q,q+1,\cdots ,k-1,n',(n-1)',\cdots ,k')$$
$$\t=(s,s+1\cdots ,n,(s-1)',(s-2)',\cdots ,k',(k-1)'\cdots ,q')$$
Hence as in the SL($n$)-case, we obtain, modulo $(x_{(i,j)},\   i+j\leq t)$
$$f_l^{(d)}\equiv  x_{(1,t)}x_{(2,t-1)}\cdots x_{(d,s)}$$ 
(note that $x_{(1,t)}x_{(2,t-1)}\cdots x_{(d,s)}$ is simply the monomial 
$f_l^{\te ,\s,Y}$, with $\te =\ {\rm id},\ \s=(d\cdots 1) $, and $Y=\emptyset$).

\ni {\bf Case 2.} $k\geq s$

\ni We have
$$ w_{l,d}^{(d)}=( s, s+1,\cdots ,k, \cdots ,n, n',(n-1)',
\cdots ,k' )$$
 $$f_l^{(d)}=p_{\t,\phi}$$ 
where $\{\t,\phi \}$ is the admissible pair given by
 $$\phi = (q,q+1, ,\cdots ,k-1,n',(n-1)'\cdots 
,k'),$$ 
$$\t=(s,s+1,\cdots ,k-1,k,\cdots ,n ,(s-1)',(s-2)',
\cdots ,q')$$
As in the Sp($2n$)-case, we have, for $d<n$, modulo $(x_{(i,j)},\  
i+j\leq t)$
  $$f_l^{(d)}\equiv  x_{(1,t)}x_{(2,t-1)}\cdots x_{(d,s)}.$$
And for $d=n$, we have, modulo $(x_{(i,j)},\   i+j\leq 2n+1)$
$$f_1^{(n)}\equiv  x_{\e_1}x_{\e_2}\cdots x_{\e_n}$$
(note that $\{x_{(i,j)},\   i+j\leq 2n+1\}$ corresponds to
$\{ x_\b,\ \b\in
\{\e_l-\e_m,\ 1\leq l<m\leq n,\ \e_l+\e_m,\ 1\leq l<
m\leq n\}\}$ ; note also that
$F_{\e_1}\cdots F_{\e_n}(p_{w_{1,n}})=cp_{id},c\in k^*$). 

\vs .2cm \ni {\bf Remark $B_n$.} Here again we have one degree of freedom in the
construction of $F$.
 


\vs .2cm \ni {\bf Type D$_l$}

\ni  Let $V=k^{2n}$ together with a non-degenerate 
symmetric bilinear form (,), 
and $G=$ SO($V$). Taking the matrix of the form (,) 
(with respect to the standard
basis $\{e_1,\cdots ,e_{2n} \}$ of $V$) to be $E$, 
the anti-diagonal ($1,\cdots ,1$),
we may realize $G={\rm SO}(V)$ as the fixed point set 
$SL(V)^\sigma $, where $\sigma
:SL(V) \rightarrow SL(V) $ is the involution given by 
$\sigma (A)=E(^tA)^{-1}E$, and $W$ may be
identified with

\ni $  \{(a_1\cdots a_{2n})\in S_{2n}\ 
|\ a_i=2n+1-a_{2n+1-i}, 1\leq i\leq 2n\ \
{\rm and}\ \# \{i, 1\leq i\leq n\ |\ a_i>n}\}$ is 
${\underline {\rm even}}\} \}$.  
For $1 \leq i \leq 2n  $, we shall denote $$ i'=2n+1-i.$$

\ni As in the case of Sp($2n$),we have
$$R^+(G) \quad = \quad \cases{\quad \e_i-\e_j, & 
if $1\leq i<j\leq n$\cr\cr
\quad \e_i+\e_j, & if $1\leq i<j\leq n$\cr}$$

\ni For $1\leq i,j\leq 2n,\
i\not= j,\  (i,j)$ (considered as an element in 
$S_{2n}$) shall denote the
transposition of $i$ and $j$. 

\vs .2cm \ni {\bf D.1 The elements $F_\b,\ \b \in R^+$, 
in the Chevalley Basis for
$\gg$.}

\ni The involution $\s $ induces an involution
$$\s :{\rm sl}(2n) \rightarrow {\rm sl}(2n),\ A\mapsto -E(^tA)E
$$
 In particular, we have
$$\s (E_{ij}) = -E_{j'i'}$$
We have,
$$\gg=\{A \in {\rm sl}(2n)\ |\ E(^tA)E=-A\}.$$
Under the above identification, we have,
$$F_{\e_i-\e_j}=E_{ji}-E_{i'j'},\ F_{\e_i+\e_j}=
E_{j'i}-E_{i'j}.$$
(see [L]$_2$ for details). 
 
\vs .2cm \ni {\bf D.2 Total order on $R^+$. }

\ni Each $\b \in R^+$ may be identified as $\e_i -\e_j\ |\ _{T(G)} $ for suitable
$i,j, \ 1\leq i\leq n,\ j \leq 2n, \ i<j,\ j\not= i'$ ( here, $T(G)= G \cap T_{2n}$,
and for $j>n,\ \e_j$ as an element of $X(T)$ is to be understood as $-\e _{j'}$. See
[L-S]$_1$ for details). With this identification, we take (as in the
SL($n$)-case) the total order on $R^+$ to be the one induced by the lexicographic
order on $\{(i,j)\}$.

\vs .2cm \ni \vs .2cm \ni {\bf D.3 The section 
$F\in H^0(G/B, L_{2\rho} )$} 

\ni Here again we are going to define elements $f_1^{(d)},\ f_2^{(d)}$ 
in $ {\rm
H}^0(G/B, L_{\omega_d}),\ 1\leq d <n$, and  $f_1^{(n)}$ in $ 
{\rm H}^0(G/B,
L_{\omega_n})$ so that $F=  \prod_{l,d}\ f_l^{(d)}$ will have 
the required property.
(Here, $f_2^{(n)}=p_{{\rm id}}$ (which of course is $\equiv 1$ 
on $U^-$)). These are
defined similar to Type C$_n$.

\ni Let $1\leq d\leq n-2$. Set
$$w_{1,d}=(1,2d)(2,2d-1)\cdots (d,d+1)$$
$$w_{2,d}=(1,2d+1)(2,2d)\cdots (d,d+2)$$
For $d=n-1$, set
$$w_{1,d}=s_{\e_1+\e_3}s_{\e_2+\e_4}\cdots s_{\e_{n-2}+\e_n}$$
$$w_{2,d}=s_{\e_1+\e_2}s_{\e_2+\e_3}\cdots
s_{\e_{n-2}+\e_{n-1}}s_{\e_{n-1}-\e_n}.$$
For $d=n$, set
$$w_{1,d}=s_{\e_{n-1}+\e_n},\ w_{2,d}=id.$$
Set $$ f_l^{(d)}= p_{w_{l,d}}^{(d)}}\ |\ _{G} ,\ l=1,2,\ d<n-1.$$
$$ f_l^{(d)}= p_{w_{l,d}}^{(d)}} ,\ l=1,2,\ d=n-1,n.$$


\ni As in the Sp($2n$)-case, we set, for $1\le d\le n-2$,

$$t\quad=\quad\cases{\quad 2d, & if $l=1$ \cr\cr
\quad 2d+1,  & if $l=2$ \cr }$$
$$s\quad=\quad\cases{\quad d+1, & if $l=1$ \cr\cr
\quad d+2,  & if $l=2$ \cr }$$

$$q=n+s-t$$
(note that $q=\ (n+1-d)$ is the same in both cases).



\ni For $1\leq i \leq n,\ j\leq 2n,\
i< j,\ j\not= i',\ x_{(i,j)}$ shall denote the 
affine co-ordinate $x_{\e_i-\e_j}$
(here, for $j>n,\ \e_j$ as an element of $X(T)$ 
is to be understood as $-\e
_{j'}$).

 \ni As in the Sp($2n$)-case, a typical monomial in $(f_l^{(d)})'$,
$1\le d\le n-2$, may be described as follows. Write
$w_{l,d}^{(d)}=(a_1,\cdots ,a_d)$. Note that with notations as in
the SL($n$)-case, we have,
$\{c_1,\cdots ,c_d\}=\{1,\cdots ,d\}$. Fix a pair $\te,\s \in S_d$.
Choose 

\ni $Y=\{y_{uj},\ 1\leq j\leq p_u,\ \ 1\leq u\leq d \}$ such that
$$y_{uj}\not\in \{a_{\s(m)},\ a'_{\s(m)} \ 1\leq m\leq
u-1 \},\ y_{uj}\not= y'_{uj-1},\  \te (u)<y_{u1}<\cdots <y_{up_u}<  a_{\s (u)} .$$ 
(here, $y_{u0}=\te (u)$). The corresponding monomial in $(f_l^{(d)})'$ is given by  
$$ f_i^{\te,\s,Y}= \prod_{1\leq u\leq d}\ x_{(y_{up_u},a_{\s
(u)})}x_{(y_{up_u-1},{y_{up_u}})} \cdots x_{(y_{u1},y_{u2})}x_{({\te (u)},y_{u1})}$$
   
\ni For $d \le n-2$, if $t\leq n$, then $f_l^{(d)}$ are
extremal weight vectors, and as in the SL($n$) case, we have, 
modulo $(x_{(i,j)},\   i+j\leq 2d)$, 
$$f_1^{(d)}\equiv  x_{(1,2d)}x_{(2,2d-1)}\cdots x_{(d,d+1)}$$
and modulo $(x_{(i,j)},\   i+j\leq 2d+1)$,
 $$f_2^{(d)}\equiv  x_{(1,2d+1)}x_{(2,2d)}\cdots x_{(d,d+2)}
.$$

\ni Let then $d \le n-2$, and $t>n$. Let us write $t=k'$, for
some $k\leq n$. Then we have (cf.[L]$_2$),
$$f_l^{(d)}=p_{\t,\phi}$$
where $\t,\phi $ are given as follows. 

\ni If $k<s$, then
$$\phi = (q,q+1,\cdots ,k-1,n',(n-1)',\cdots ,k'),$$
$$\t=(s,s+1\cdots ,n,(s-1)',(s-2)',
\cdots ,k',(k-1)'\cdots ,q')$$

\ni If $k\geq s$, then
 $$\phi = (q,q+1, ,\cdots ,k-1,n',(n-1)'\cdots 
,k'),$$ 
$$\t=(s,s+1,\cdots ,k-1,k,\cdots ,n ,(s-1)',(s-2)',
\cdots ,q')$$
We have ( as in the 
Sp($2n$)-case), for $d\le n-2$, 
modulo $(x_{(i,j)},\   i+j\leq t)$,               
$$f_l^{(d)}\equiv  x_{(1,t)}x_{(2,t-1)}\cdots x_{(d,s)}$$
Now, $\{x_{ij}, i+j\leq 2n-2\}$ corresponds to $\{ x_\b,\ \b\in
 \{\e_l-\e_m,\  1\leq l<m\leq n,\ l\leq n-2,\ 

\ni \e_l+\e_m, 1\leq l<m\leq n,\ l+3\leq
m\}\}$. Hence we obtain that modulo

\ni $\{ x_\b,\ \b\in
\{\e_l-\e_m,\  1\leq l<m\leq n,\ l\leq n-2,\ \e_l+\e_m,\  1\leq
l<m\leq n,\ l+3\leq m\}\}$,
$$f_1^{(n-1)}\equiv x_{\e_1+\e_3}x_{\e_2+\e_4}\cdots
x_{\e_{n-2}+\e_n},$$
and modulo $\{ x_\b,\ \b\in
\{\e_l-\e_m,\  1\leq l<m\leq n,\ l\leq n-2,\ \e_l+\e_m,\ 1\leq
l<m\leq n,\ l+2\leq m\}\}$
$$f_2^{(n-1)}\equiv s_{\e_1+\e_2}s_{\e_2+\e_3}\cdots
s_{\e_{n-2}+\e_{n-1}}s_{\e_{n-1}-\e_n}.$$
Now going modulo $\{ x_\b,\ \b\in
\{\e_l-\e_m,\ 1\leq l<m\leq n,\ \e_l+\e_m,\ 1\leq l<
m\leq n,\ l\leq n-2\}\}$, we get   
$$f_1^{(n)}\equiv s_{\e_{n-1}+\e_n}.$$ 

\vs .2cm \ni {\bf Remark $D_n$.} Here again we have one degree of
freedom in the construction of $F$.


\vs .2cm \ni {\bf Type G$_2$}

\ni In the discussion below, we follow the notations in [L]$_1$. 
 
\ni Let $G$ be of type G$_2$. We take the total order 
on $R^+$ as $\{\a_1,\
\a_1+\a_2,\ \a_2,\ 2\a_1+\a_2,\ 3\a_1+\a_2,\ 3\a_1+2\a_2 \}$ 
(actually, one can
work with any total order). 

\ni Define $f_i \in {\rm H}^0(G/B, L_{\omega_1}),\ i=1,2$
 as follows:
$$f_1=p_{s_1},\ f_2= p_{s_2s_1}$$

\ni  Define $g_i \in {\rm H}^0(G/B, L_{\omega_2}),\ i=1,2$
 as follows:
$$g_1=p_{{s_2s_1s_2},\ s_2},\ g_2= p_{s_1s_2s_1s_2}$$
 Set
$$F=\Pi_{i,j}\ f_ig_j $$

\ni For a monomial differential operator $D=F_{\b_1}^{n_1}\cdots
 F_{\b_r}^{n_r},\ 
DQ_{{\rm id}}=Q_{s_1} \ ({\rm in\ }V_{\omega_1})$ 

\ni $\Longleftrightarrow \sum_{1\leq i\leq r}\ 
\b_in_i=\a_1 $ (by weight considerations; note that wt$f_1=-(\a_1+\a_2)$). Hence we
obtain $$f_1=x_{\a _1}$$
Similarly, we have
$$f_2\equiv x_{\a_1+\a_2}\ ({\rm mod }\ (x_{a_1}))$$

$$g_1\equiv x_{\a_2}x_{2\a_1+\a_2}\   ({\rm mod }(x_{a_1},\ x_{\a_1+\a_2}))$$    

$$g_2\equiv x_{3\a_1+\a_2}x_{3\a_1+2\a_2}\ ({\rm mod }(x_{a_1},
\ x_{a_2},\ 
x_{\a_1+\a_2},\ x_{2\a_1+\a_2})   )$$
(note that  wt$f_2=-\a_1$, wt$g_1=-\a_1$,  wt$g_2=3\a_1+\a_2$).

\vs .2cm \ni Thus using the results of \S1 \& \S2, we obtain
\proclaim Theorem 4.3. Let $G$ be simple of type $A_n,\ B_n,\ C_n,\ D_n,\ {\rm or}\
G_2$. Then $G/B$ is Frobenius split. Further, for any parabolic sub group $Q$,
$G/Q$ is Frobenius split. 
 

\vs .4cm \ni {\bf \S5. The discussion of $G/Q$}

\ni In this section, we give an alternate proof for the Frobenius-split property 
for $G/Q$, in the same spirit as in \S4, namely, we exhibit a section of
$K_{G/Q}^{-1}$ which has residual normal crossing at $e_{\rm id}$; further, this
has again maximum multiplicity at $e_{\rm id}$.  First of all we recall that
$$K_{G/Q}^{-1}= L_{\l},\ \ \l=\sum _{\b \in R^+-R^+_Q}\ \b$$
Also observe that while writing the power series expansion for any $f$ in $H^0(G/B,
L_\l )$ in terms of the local co-ordinates $\{x_\b \}$'s, in a neighbourhood of
$e_{{\rm id}}$ ( cf. Lemma 4.2), only the $\b$'s belonging to $R^+-R^+_Q$ need be
considered. 

\vs .2cm \ni {\bf Type $A_l$}

\ni Let $G=SL(n),\ (n=l+1)$. Let $Q=P_{i_1} \cap P_{i_2}\cdots \cap P_{i_r}$, where
for $1\leq i\leq l$,  $P_{i}$ denotes the maximal parabolic subgroup corresponding to
``omitting  $\a_i$". Then it can be checked easily that
$$K_{G/Q}^{-1}=\sum_{1\leq t\leq r}\ (i_{t+1}-i_{t-1})\ \omega_{i_t}$$
where $i_0=0$, and $i_{r+1}=n  $. 

\ni As in \S4, we shall denote the local co-ordinates at $e_{{\rm id}}$ by 
$x_{(k,l)}$, where $k<l$, and if $i_{t-1}+1\leq k \leq i_t -1$, then $l \geq i_t$. We
now define $f_j^{(t)} \in H^0(G/B, L_{\omega_{i_t}}),\ 1\leq j\leq m_t $, where
$m_t=i_{t+1}-i_{t-1}$ (resp. i_{t+1}-i_{t-1}-1$), if $t<$ (resp.=)  $r$, as follows.

\ni Let $k_t=i_{t}-i_{t-1}$. For $1\leq i<j\leq n$, we set
$$w_{i,j}= (i,j)(i-1,j+1)\cdots(a_{i,j},b_{i,j})$$
where
$$(a_{i,j},b_{i,j})\quad=\quad \cases{\quad (1, i+j-1), & if $n\geq i+j-1$\cr\cr
\quad (i+j-n,n), & if $n< i+j-1$\cr} \leqno {(*)}$$
(recall that $(i,j)$ denotes the transposition of $i$ and $j$). We define
$$f_j^{(t)}=p^{(i_t)}_{{w_{k_{tj}, l_{tj}}}}$$ where
$$(k_{tj}, l_{tj})\quad=\quad \cases{\quad (i_{t-1}+j, i_t+1), & if $1\leq j\leq
k_t-1$\cr\cr \quad (i_{t}, i_{t-1}+j+1), & if $k_t\leq j\leq m_t$\cr}$$
For $i<j$, set
$$X_{i,j}=x_{(i,j)}x_{(i-1,j+1)}\cdots x_{(a_{ij},b_{ij)}$$
where $(a_{i,j},b_{i,j})$ is given by (*) above.
Then as in \S4, we obtain that 

\ni modulo ($x_{kl},\  k+l\leq i_{t-1}+i_t+j $)
$$f_j^{(t)}\equiv X_{(k_{tj}, l_{tj})}$$
(note that for all $x_{uv}$ appearing in $X_{(k_{tj}, l_{tj})}$, we have 
$u+v=i_{t-1}+i_t+j+1$. Set
$$ F= \Pi _{t,j}\ f_j^{(t)} $$
Then as in \S4, $F$ has residual normal crossing at $e_{{\rm id}}$. 

\vs .2 cm \ni {\bf Remark 5.1} As in \S4, starting with $x_{i_1+1 i_1}$, at each
step all of the variables along the diagonal ( in the same sense as in Remark $A_n$)
are picked up.

\vs .2 cm \ni {\bf Remark 5.2.} Let $Z=\{w_{l,d}, l=1,2,\ 1\leq d\leq n-1\   | {\rm \
for \ at\ least\ one}\ i_t,\ \pi_{i_t}({w_{l,d})\not= {id}\}$ (here ${w_{l,d}$ are
the elements in $S_n$ as constructed in \S4, Type$A_n$). Then the $\{f_j^{(t)}\}$
constructed above is simply 

\ni $\{ p^{(n_{l,d})}_{w_{l,d}},\ {w_{l,d}\in Z\  {\rm and}
\ n_{l,d} {\rm \ is\  the\  largest }\ i_t\ {\rm such\ that\ }
\pi_{i_t}({w_{l,d})\not= id \}$ ( in fact, $n_{l,d}$ is the largest $i_t$
such that $i_t<2d$ (resp. $2d+1$), if $l=1$ (resp. $l=2$).

\vs .2 cm \ni {\bf Remark 5.3.} Here again, we have one degree of freedom in the
construction of $F$, since we chose only $n-i_r-1 $ sections from 
$H^0(G/B, L_{\omega_{i_t}})$.  

\vs .2cm \ni {\bf II Types $B_n, \ C_n,\ D_n$} 

\ni Consider the set $Z$ as described in Remark 5.2 above, 
$w_i^{(d)$ being the elements  as constructed in \S4, Type$B_n,\ C_n,\ D_n$
respectively. We consider an element in $Z$ as an element in the corresponding
symmetric group. As in Remark 5.2, given $w \in Z$, let $n_w$ be the largest 
$i_t$ such that
$\pi_{i_t} (w) \not= {\rm Id}$. Then $F=\Pi_{w\in Z}\  p^{(n_w)}_{w}$ has  
a residual normal crossing at $e_{{\rm id}}$. 


\vs .2cm \ni {\bf III Type $G_2$}

 \ni In the discussion below, we follow the notations in [L]$_1$. 
 

\ni (i) $Q=P_{\a_2}$

\ni We have, $K_{G/Q}^{-1}=5\omega_1$. Set 
$$f_1=p_{s_1},\ f_2=p_{s_2s_1}, \ f_3=p_{s_1s_2s_1}$$ 
$$f_4=p_{s_2s_1s_2s_1},\ f_5=p_{{s_1s_2s_1},{s_2s_1}$$
It is easily checked that 
$$f_1=x_{\a_1},\ f_2\equiv x_{\a_1+\a_2}\ ({\rm mod}\ (x_{\a_1}))$$
$$ f_3\equiv x_{3\a_1+\a_2}\ ({\rm mod}\ (x_{\a_1},\  x_{\a_1+\a_2})) $$ 
$$ f_4\equiv x_{3\a_1+2\a_2}\ ({\rm mod}\ (x_{\a_1},\  x_{\a_1+\a_2},\
x_{3\a_1+\a_2}))$$
$$ f_5\equiv x_{2\a_1+\a_2}\ ({\rm mod}\ (x_{\a_1},\  x_{\a_1+\a_2},\
x_{3\a_1+\a_2},\ x_{3\a_1+2\a_2}))$$ 
(note that wt $p_{s_1}=-(\a_1+\a_2)$, wt $p_{s_2s_1}=-\a_1$, wt
$p_{s_1s_2s_1}=\a_1$, and wt $p_{s_2s_1s_2s_1}=(\a_1+\a_2)$, wt 
$p_{{s_1s_2s_1},{s_2s_1}}=0$).

\vs .2cm \ni (ii) $Q=P_{\a_1}$

\ni We have, $K_{G/Q}^{-1}=3\omega_2$. Set 
$$f_1=q_{s_1s_2,s_2},\ f_2=p_{s_2s_1s_2,s_2}, \ f_3=p_{s_1s_2s_1s_2}$$
It is easily checked that
$$f_1=x_{\a_1+\a_2},\ f_2\equiv x_{2\a_1+\a_2}x_{\a_2}\ ({\rm mod} \
(x_{\a_1+\a_2}))$$
$$f_3\equiv x_{3\a_1+\a_2} x_{3\a_1+2\a_2}\ ({\rm mod} \
(x_{\a_1+\a_2},\ x_{2\a_1+\a_2},\ x_{\a_2}))$$
(note that wt $q_{s_1s_2,s_2}=-(2\a_1+\a_2)$, wt $p_{s_2s_1s_2,s_2}=-\a_1$, and 
wt $p_{s_1s_2s_1s_2}=3\a_1+\a_2$.

Thus we obtain an independent proof for the Frobenius-split property for any $G/Q$, 
where $G$ is simple of type $A_n,\ B_n,\ C_n,\ D_n,\ {\rm or}\
G_2$, and $Q$ is any parabolic sub group.
  
       
    

\vfill\eject

\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.

\noindent
[D] M.Demazure, {\it Desingularisation des vari\'eti\'es de Schubert
generalise\'es}, Ann. Sci. \'Ecole. Norm. Sup. {\bf t.7} (1974), 53-88


\vs .2cm \ni [K-R]  G. Kempf and A. Ramanathan, {\it Multi-cones
over Schubert varieties}, Inv. Math., 87 (1987), 353-363.

\vs .2cm \ni [K]   S. Kumar, {\it Proof of Wahl's conjucture
on surjectivity of the Gaussian map for flag varieties,} Amer.
Jnl. Math. 1201-1220 (1992).



\vskip .2 cm \ni [L]$_1$ V. Lakshmibai {\it Standard Monomial 
Theory For $G_2$}, J.
Alg., vol 98 (1986), 281-318.

\vskip .2 cm \ni [L]$_2$ V. Lakshmibai {\it Geometry of $G/P$-VI},
 J. Alg., vol 108
(1987), 355-402.


\vskip .2 cm \ni [L-S]$_1$ V. Lakshmibai and C.S. Seshadri 
{\it Geometry of
$G/P$-II}, Proc. Indian. Acad. Sciences., A87 (1978), 1-54.

\vskip .2 cm \ni [L-S]$_2$ V. Lakshmibai and C.S. Seshadri 
{\it Geometry of
$G/P$-V}, J. Alg., vol 100 (1986), 462-557.


\vs .2cm \ni [M] O. Mathieu, {\it Filtrations of G-modules}, 
Ann. Sci. Ecole Norm. Sup., 23 (1990), 625-644.

\vs .2cm \ni [M-R]  V.B. Mehta and A. Ramanathan {\it
Frobenius splitting and  cohomology vanishing for Schubert
varieties,} Ann. Math. 122 27-40 (1985).

\vs .2cm \ni [M-S]  V.B. Mehta and  Srinivas {\it Normality of
Schubert varieties} Amer. J. Math. 987-989 (1987)

\vs .2cm \ni [R]  A.  Ramanathan, {\it Equations defining
Schubert varieties and Frobenius splitting of diagonals}, Publ.
Math. I.H.E.S., 65 (1987).

\vs .2cm \ni [W]  J. Wahl {\it Gaussian maps and tensor
products of irreducible representations,} Manus. Math. 73,
229-259 (1991).  




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

\vs .2cm
\ni V.B. Mehta \& A.J. Parameswaran\hfill\break
 School of Mathematics \hfill\break
Tata Institute of Fundamental Research\hfill\break
Bombay-400 005












\bye