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% A CHARACTER FORMULA FOR SINGLY ATYPICAL
% MODULES OF THE LIE SUPERALGEBRA sl(m/n)
% J. Van der Jeugt, J.W.B. Hughes, R.C. King and J. Thierry-Mieg
% Commun. Algebra 18 (1990) 3453-3480
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\def\VK{\overline V(\La)}
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\hsize=5.5in \vsize=8.5in
\hbox{  }
\vskip 1in
\centerline{\sectionfont A CHARACTER FORMULA FOR SINGLY ATYPICAL}
\vskip 5mm
\centerline{\sectionfont MODULES OF THE LIE SUPERALGEBRA sl(m/n)}
\vskip 1.7cm
\sl
\centerline{by}
\centerline{J.~Van der Jeugt$\,^{a,b}$\footnote*{\rm Aangesteld
Navorser N.F.W.O. (National Fund for Scientific Research of Belgium)},
J.W.B. Hughes$^c$, R.C. King$^a$}
\centerline{and J. Thierry-Mieg$^d$}
\vskip 1.2cm
\rm
\item{a)} Faculty of Mathematical Studies, University of
Southampton, Southampton SO9 5NH, U.K.
\item{b)} Laboratorium voor Numerieke Wiskunde en Informatica,
Rijksuniversiteit Gent, Krijgslaan 281-S9, 9000 Gent, BELGIUM
\item{c)} School of Mathematical Sciences, Queen Mary College,
Mile End Road, London E1 4NS, U.K.
\item{d)} Groupe d'Astrophysique Relativiste, CNRS, Observatoire
de Paris Meudon, F-92195 Meudon, FRANCE
\vskip 2cm

\section{1. Introduction}

Lie superalgebras, originating from physics
$[{\underline 3}]$, are $\Zah_2$-graded algebras ($\Zah_2=\Zah/2\Zah$)
with a bracket operation
which is ``supersymmetric'' (equation 2.1b in this paper) and
which satisfies the ``super Jacobi identity'' (equation 2.1c).
A classification of the finite dimensional simple Lie superalgebras
over $\C$ was given over a decade ago by Kac $[{\underline 8},
{\underline 9},{\underline{10}},{\underline{13}}]$.
A subclass of these, closely analogous to the finite dimensional
Lie algebras over $\C$, is the class of so-called basic classical
Lie superalgebras $[{\underline 9}]$.

The problem of classifying the finite dimensional simple modules
of the basic classical Lie superalgebras has also been considered
by Kac $[{\underline 9},{\underline{11}}]$. He showed that, as in the case of finite
dimensional simple modules of the semi-simple Lie algebras, they
are characterised up to equivalence by a highest weight $\La$.
The weight structure of a simple module $V(\La)$ with highest
weight $\La$ of such a Lie superalgebra $G$ is determined by its
character $\cha V(\La)$. For a subclass of these simple modules,
known as ``typical'' modules, Kac was able to derive a character
formula closely analogous to the Weyl character formula for
simple modules of simple Lie algebras. The problem of obtaining
character formulae for the remaining ``atypical'' modules has
been the subject of intense investigation but is still not solved
other than in various special cases. In this paper, we solve this
problem for the singly atypical modules of the Lie superalgebra
$G=sl(m/n)$, where $sl(m/n)$ ($m,n\in\Nat$) is the special linear
Lie superalgebra analogous to the special linear Lie algebra $sl(m)$.

We consider the indecomposable $G$ modules $\VK$, introduced by
Kac $[{\underline{11}}]$, which we refer to as Kac-modules. $\VK$ is
well-defined for every integral dominant weight $\La$ and has the
important property that every finite dimensional simple $G$
module $V(\La)$ is isomorphic to a quotient module of the form
$\VK/M(\La)$, where $M(\La)$ is the unique maximal submodule of
$\VK$. The character of $\VK$ is easy to determine, and has been
given by Kac (equation 3.17 in this paper):
$$
\cha\VK={\displaystyle\prod_{\be\in\De_1^+}\bigl(e^{\be/2}+e^{-\be/2}\bigr)
\over\displaystyle\prod_{\al\in\De_0^+}\bigl(e^{\al/2}-e^{-\al/2}\bigr)}
\sum_{w\in W} \vep(w)e^{w(\La+\rh)}, \eqno(1.1)
$$
where $\De_0^+$ ($\De_1^+$) is the set of even (resp.~odd)
positive roots of $G$, $W$ is the Weyl group (defined to be the
Weyl group of the even subalgebra of $G$), $\vep(w)$ is the
signature of $w\in W$, and $\rh=\rh_0-\rh_1$ where $\rh_0$ (resp.
$\rh_1$) is half the sum of all even (resp. odd) positive roots
of $G$. The integral dominant weight $\La$ and the module $\VK$
are called typical if
$\langle\La+\rh\mid \be\rangle\not=0$ for all $\be\in\De_1^+$,
where $\langle\;\mid\;\rangle$ is a non-degenerate bilinear form
$[{\underline{11}},{\underline{12}}]$. 
In this case, Kac showed that $M(\La)=\{0\}$, and so (1.1)
gives the character of the {\sl simple} $G$ module $V(\La)=\VK$ 
$[{\underline{11}}]$.

If $\langle\La+\rh\mid\be\rangle=0$ for some $\be\in\De_1^+$, then
$M(\La)\not=\{0\}$ and so $\VK\not= V(\La)$. In this case
$\La$, $\VK$ and $V(\La)$ are called {\sl atypical}; in particular if 
there is a unique $\ga\in\De_1^+$ such that 
$\langle\La+\rh\mid\ga\rangle=0$, then
$\La$, $\VK$ and $V(\La)$ are said to be {\sl singly atypical} of
type $\ga$, and $\ga$ is called the corresponding atypical root.
In this paper we give a unique characterisation of
$M(\La)$ for the singly atypical case (Theorem~4.3):
we show that $M(\La)$ is itself a simple (singly
atypical) $G$ module. Using this theorem, we are then able to
derive a character formula for $V(\La)$, first for the case where
the atypical root is the unique odd simple root
$\al_m$ of $G$ (Theorem 5.3). We then proceed to prove various
properties relating a weight $\La$, singly atypical of type
$\ga$, to a weight that is singly atypical of type $\al_m$. Using
these properties, we then derive a character formula for all singly
atypical simple modules of $sl(m/n)$ in Section~7 (Theorem 7.2).
Finally, we make some comments relating to {\sl
multiply} atypical modules.

We conclude this introduction by mentioning that characters of some
atypical $sl(m/n)$ modules have been obtained by Berele and Regev
$[{\underline 1}]$ and Serge'ev $[{\underline{15}}]$. Using Schur's method, they show that the tensor
product $V^{\otimes N}$, where $V$ is the natural
$(m+n)$-dimensional module of $sl(m/n)$, is completely reducible.
The irreducible components are the (simple) {\sl covariant tensor
modules}, the characters of which can be expressed in terms of
Schur functions $[{\underline 1},{\underline{15}}]$. These covariant tensor 
modules can be typical, singly atypical or even multiply
atypical, but they do not by any means exhaust any of these
categories of modules.
Various formulae and conjectures have
been published in order to accommodate the characters of all atypical
simple $sl(m/n)$ modules $[{\underline 2},{\underline 5}]$, 
but counterexamples to all
formulae proposed so far have been found $[{\underline{17}}]$. Realizing the
failure of all these proposals, a new conjecture has been given
in Ref.~{17}, to which no counterexamples are known. This is
described briefly in Section~8.

\section{2. The Lie superalgebra sl(m/n)}

A complex Lie superalgebra $G$ is a $\Zah_2$-graded linear vector
space, $G=G_{\bar 0}\oplus G_{\bar 1}$ over $\C$ with a bracket
$[\;,\;]$ such that $\forall a\in G_\al$, $\forall b\in G_\be$
and $\forall\al,\be\in\Zah_2$ $[{\underline 9},{\underline{13}}]$
$$
\eqalignno{
	[a,b]&\in G_{\al+\be},&(2.1a)\cr
	[a,b]&=-(-1)^{\al\be}[b,a],&(2.1b)\cr
	[a,[b,c]]&=[[a,b],c]+(-1)^{\al\be}[b,[a,c]].&(2.1c)\cr}
$$
Note that the {\sl even} part $G_{\bar 0}$ is a complex Lie 
algebra, and that the {\sl odd} part
$G_{\bar 1}$ is a $G_{\bar 0}$ module under the adjoint action.
The simplest example of a Lie superalgebra is 
given by $gl(m/n)$ with $m,n\in\Nat$. Its
natural matrix realisation takes the form:
$$
\eqalign{
gl(m/n)=\Bigl\lbrace x=\left(\matrix{A&B\cr C&D\cr}\right) \mid
 &A\in M_{m\times m}, B\in M_{m\times n},\cr& C\in M_{n\times m}, D\in
M_{n\times n} \Bigr\rbrace ,\cr} \eqno(2.2)
$$
where $M_{p\times q}$ is the space of all $p\times q$ complex
matrices. The ``even'' subspace $gl(m/n)_{\bar 0}$ has $B=0$ and
$C=0$; the ``odd'' subspace $gl(m/n)_{\bar 1}$ has $A=0$ and
$D=0$. In the case of $G=gl(m/n)$, the bracket is determined in
the natural matrix representation by
$$
[a,b]=ab-(-1)^{\al\be}ba,\qquad\forall a\in G_\al\hbox{ and }
\forall b\in G_\be.  \eqno(2.3)
$$
We denote by $gl(m/n)_{+1}$ the space of matrices
$\bigl({0\atop 0}{B\atop 0}\bigr)$ and by $gl(m/n)_{-1}$ the
space of matrices $\bigl({0\atop C}{0\atop 0}\bigr)$. Then
$G=gl(m/n)$ has a $\Zah$-grading which is consistent with the
$\Zah_2$-grading $[{\underline{13}}]$:
$$
G=G_{-1}\oplus G_0\oplus G_{+1},
\qquad G_{\bar 0}=G_0 \hbox{ and }G_{\bar 1}=G_{-1}\oplus G_{+1}.
\eqno(2.4)
$$
Note that $gl(m/n)_{\bar 0}=gl(m)\oplus gl(n)$.
With the definition of {\sl supertrace} $[{\underline 9}]$ as $\hbox{str}(x)=\hbox{tr}(A)
-\hbox{tr}(D)$ one can define the subalgebra $sl(m/n)$:
$$
sl(m/n)=\lbrace x\in gl(m/n)\mid \hbox{str}(x)=0\rbrace. \eqno(2.5)
$$
If $m\not=n$ then $sl(m/n)$ is a {\sl simple} Lie superalgebra 
$[{\underline 9},{\underline{13}}]$.
If $m=n$ it contains a one-dimensional ideal $\C I_{2m}$ and then
$sl(m/m) /\C I_{2m}$ is simple. In what follows we put
$G=sl(m/n)$. Note that $sl(m/n)_{\bar 0}=sl(m)\oplus\C\oplus
sl(n)$ is a reductive Lie algebra, the simple modules of which
are well known.

A Cartan subalgebra $H$ of $G$ has dimension $m+n-1$ and is
spanned by
$$
\eqalign{
h_i&=E_{ii}-E_{i+1,i+1}\;\;(1\leq i\leq m-1\hbox{ or }m+1\leq i\leq m+n-1),\cr
h_m&=E_{mm}+E_{m+1,m+1},\cr} \eqno(2.6)
$$
where $E_{ij}$ is the matrix with entry $1$ at position $(i,j)$
and $0$ elsewhere. The dual space $H^*$ is described in the basis of
forms $\ep_i$ ($i=1,2,\ldots,m$) and $\de_j$ ($j=1,2,\ldots,n$),
where $\ep_i\colon x\rightarrow A_{ii}$ and $\de_j\colon
x\rightarrow D_{jj}$ for $x=\bigl({A\atop C}{B\atop D}\bigr)$,
and $\sum_{i=1}^m\ep_i+\sum_{j=1}^n\de_j=0$. The roots and
corresponding root vectors of $sl(m/n)$ are given by $[{\underline 9}]$
$$
\eqalign{
  \ep_i-\ep_j\;&\leftrightarrow\; E_{ij}\qquad (1\leq i,j\leq m)\;\hbox{(even)},\cr
  \de_i-\de_j\;&\leftrightarrow\;E_{m+i,m+j}\qquad(1\leq i,j\leq n)\;\hbox{(even)},\cr
  \ep_i-\de_j\;&\leftrightarrow\;E_{i,m+j}\qquad(1\leq i\leq m,\; 1\leq
j\leq n)\;\hbox{(odd)},\cr
  \de_i-\ep_j\;&\leftrightarrow\;E_{m+i,j}\qquad(1\leq i\leq n,\;
1\leq j\leq m)\;\hbox{(odd)}.\cr} \eqno(2.7)
$$
Denote by $\De$ the set of all roots, by $\De_0$ the set of even
roots, by $\De_1$ the set of odd roots, and by $e(\al)$ the root
vector (2.7) corresponding to the root $\al\in\De$. $G$
has the root space decomposition
$$
G=H\oplus\left(\bigoplus_{\al\in\De}\C e(\al)\right). \eqno(2.8)
$$
A set of simple roots of $\De$ may be chosen as follows:
$$
\eqalign{
\al_i&=\ep_i-\ep_{i+1}\;(1\leq i\leq m-1),\;\al_m=\ep_m+\de_1,\;\cr
\al_{m+j}&=\de_j-\de_{j+1}\;(1\leq j\leq n-1);\cr} \eqno(2.9)
$$
this choice is often referred to as the ``distinguished basis'',
for which there is only one odd simple root $\al_m$ $[{\underline{11}}]$.
With this distinguished choice,
the elements of $H^*$ are partially ordered by
$$
\la,\mu\in H^*\;\colon\quad\la\geq\mu\quad\Leftrightarrow\quad\la-\mu
=\sum_{i=1}^{m+n-1}k_i\al_i \hbox{ with }k_i\geq 0.
\eqno(2.10)
$$
This partial ordering $\geq$ can be extended to a total ordering
$\succeq$ compatible with $\geq$, i.e.
$$
\la\geq\mu\quad\Rightarrow\quad\la\succeq\mu;
\eqno(2.11)
$$
the most natural example of such a total ordering is
lexicographical ordering with respect to the simple roots.
The even and odd positive roots of $sl(m/n)$ are given by
$$
\eqalign{
\De_0^+&=\{\ep_i-\ep_j\;(i<j);\;\de_i-\de_j\;(i<j)\},\cr
\De_1^+&=\{\ep_i-\de_j\}.\cr} \eqno(2.12)
$$
It will be convenient to denote the $mn$ odd positive roots by
$$
\be_{ij}=\ep_i-\de_j\qquad 1\leq i\leq m,\;1\leq j\leq n. \eqno(2.13)
$$


The invariant non-degenerate inner product on $G$ is given by
$\langle x|y\rangle =\hbox{str}(xy)$. The restriction of this to $H$ is also
non-degenerate and the pairing of $H$ and $H^*$ then defines a
non-degenerate inner product $\langle\;\mid\;\rangle$ on $H^*$, explicitly
determined by
$$
\langle\ep_i|\ep_j\rangle=\de_{ij},\;\langle\ep_i|\de_j\rangle=0,\;
\langle\de_i|\de_j\rangle=-\de_{ij},
 \eqno(2.14)
$$
where $\de_{ij}$ is the Kronecker-$\de$. An element $\La\in H^*$
with $\La=\sum_i\la_i\ep_i+\sum_j\mu_j\de_j$ can be written in
terms of its {\sl components} in the $\ep\de$-basis as
$\La=(\la_1\la_2 \ldots\la_m|\mu_1\mu_2\ldots\mu_n)$ with
$\sum_i\la_i +\sum_j\mu_j=0$, or in terms of its {\sl Dynkin
labels} $\La=[a_1,\ldots,a_{m-1};a_m;a_{m+1},\ldots,a_{m+n-1}]$
where $a_i=\La(h_i)$ and $h_i$ is given in (2.6). We call $a_i$
with $i\not=m$ an {\sl even} Dynkin label and $a_m$ the {\sl odd}
Dynkin label.

The {\sl Weyl group} $W$ of $G$ is defined to be the Weyl group of
$G_{\bar 0}$ $[{\underline 9}]$. Hence $W=S_m\times S_n$,
the direct product of the Weyl groups
of $sl(m)$ and $sl(n)$.
For $w=\si\times\ta\in W=S_m\times S_n$, the
signature $\vep(w)$ is the product of the signatures of $\si$ and
$\ta$. We denote by $w_0$ the Coxeter element of $W$, i.e.
$w_0=\om_m \times \om_n$, where $\om_m$ (resp. $\om_n$) is the
element of maximal length in $S_m$ (resp. $S_n$).
The {\sl dot action} is defined as usual:
$$
w\cdot\La=w(\La+\rh)-\rh,\hbox{ where }\rh=\rh_0-\rh_1 \eqno(2.15)
$$
with
$$
\rh_0={1\over 2}\sum_{\al\in\De_0^+}\al\quad\hbox{ and }\quad
\rh_1={1\over 2}\sum_{\be\in\De_1^+}\be . \eqno(2.16)
$$
Explicitly,
$$
\eqalign{
\rh_0&={1\over 2}\sum_{i=1}^m(m-2i+1)\ep_i+{1\over 2}\sum_{j=1}^n(n-2j+1)\de_j\cr
\rh_1&={n\over 2}\sum_{i=1}^m\ep_i-{m\over 2}\sum_{j=1}^n\de_j.\cr}
\eqno(2.17)
$$
Note that $\De_1^+$, given in the distinguished basis by (2.12),
is $W$-invariant. It follows from
(2.16) that $w\rh_1=\rh_1$ for all $w\in W$ (a property which can
also be seen from the explicit form (2.17) for $\rh_1$), and hence
$$
w\cdot\La=w(\La+\rh)-\rh=w(\La+\rh_0)-\rh_0. \eqno(2.18)
$$

We set
$$
\eqalign{
N_0^{\pm}&=\hbox{span}\{e(\al)|\al\in\De_0^{\pm}\},\cr
N_1^{\pm}&=\hbox{span}\{e(\be)|\be\in\De_1^{\pm}\},\cr
N^{\pm}&=N_0^{\pm}\oplus N_1^{\pm}.\cr}
\eqno(2.19)
$$
Note that $N_1^{\pm}=G_{\pm 1}$ and, besides the decomposition
(2.4), one has
$$
\eqalign{
G_{\bar 0}&=N_0^-\oplus H\oplus N_0^+,\cr	
G&=N^-\oplus H\oplus N^+.\cr}
\eqno(2.20)
$$
Let $U(G)$ be the universal enveloping algebra of $G$, and
$U(G')$ the enveloping algebra of any one of the subalgebras $G'=H,G_0,N^\pm,
N_0^\pm,N_1^\pm$. The
Poincar\'e-Birkhoff-Witt theorem for Lie algebras can be extended
to the case of Lie superalgebras $[{\underline 9},{\underline{13}}]$:
\proclaim {Theorem 2.1}. Let $x_1,\ldots,x_M$ be a basis of
$G_{\bar 0}$ and $y_1,\ldots,y_N$ be a basis of $G_{\bar 1}$. Then
the elements of the form
$$
(x_1)^{k_1}\ldots (x_M)^{k_M}y_{i_1}\ldots y_{i_s},
\hbox{ where }k_i\geq 0\hbox{ and }1\leq i_1<\ldots<i_s\leq N,\eqno(2.21)
$$
form a basis of $U(G)$.

A similar theorem is true for each $U(G')$ with $G'$
one of the subalgebras given previously. Therefore $U(G')$ is
$H$-diagonalisable and we can
denote by $U(G')_\et$
the subspace of all elements of $U(G')$ of weight $\et$ with
respect to $H$.

Denote by $\si$ the involutive
antiautomorphism of $G$ defined by the
relations $[{\underline{11}}]$
$$
\eqalign{
  &\si(h)=h,\qquad\forall h\in H,\cr
  &\si\bigl( e(\al)\bigr)=e(-\al),\qquad\forall\al\in\De,	\cr}
\eqno(2.22)
$$
where $e(\al)$ is the  root vector corresponding to $\al$. This
antiautomorphism can be extended to $U(G)$ by
$\si(xy)=\si(y)\si(x)$, for $x,y\in U(G)$.

\section{3. The Kac-module}

Let $V=V_{\bar 0}\oplus V_{\bar 1}$ be a $\Zah_2$-graded linear
vector space over $\C$, and denote by $gl(V)$ the space of
endomorphisms of $V$. Then $gl(V)$ is naturally $\Zah_2$-graded:
$gl(V) =gl(V)_{\bar 0}\oplus gl(V)_{\bar 1}$. A representation
$\ph$ is a linear mapping from $G$ to $gl(V)$ such that $\forall
\al,\be \in\{\bar 0,\bar 1\}$:
$$
\eqalign{
&\ph\colon x\rightarrow\ph(x)\hbox{ with }\ph(x)\in gl(V)_\al
 \hbox{ for }x\in G_\al,\cr
&\ph([x,y])=\ph(x)\ph(y)-(-1)^{\al\be}\ph(y)\ph(x)\qquad
 \forall x\in G_\al\hbox{ and }\forall y\in G_\be .\cr}\eqno(3.1)
$$
Then $V$ is a $G$ module with $xv=\ph(x)v$ for $x\in G$
and $v\in V$.
\proclaim{Definition 3.1}. $V$ is called a highest weight module
for $G$ (resp. for $G_{\bar 0}$) with highest weight $\La\in H^*$
if there exists a non-zero vector $v_\La\in V$ such that
$$
\eqalign{
 &N^+v_\La=0 \quad (\hbox{resp. } N_0^+v_\La=0),\cr
 &hv_\La=\La(h)v_\La\qquad\forall h\in H,\cr
 &U(G)v_\La=V\quad (\hbox{resp. }U(G_{\bar 0})v_\La=V).\cr}\eqno(3.2)
$$
Then $v_\La$ is called a $G$ (resp. $G_{\bar 0}$) highest weight vector.

Highest weight modules are $H$-diagonalizable,
$$ 
V=\bigoplus_{\la\leq\La}V_\la,\hbox{ with }
V_\la=\{v\in V\mid hv=\la(h)v,\quad\forall h\in H\}, \eqno(3.3)
$$
and so are all submodules or subquotients of highest weight modules.

\proclaim{Definition 3.2}. Let $V$ be a $G$ highest weight module
with highest weight vector $v_\La$. We
call $v\in V$ a generating vector if and only if $V=U(G)v$ or,
equivalently, if and only if $v_\La\in U(G)v$.

\proclaim{Definition 3.3}. Let $V$ be a $G$ module. A vector
$v\in V$ is called a weakly primitive vector if there exists a
$G$ module $U\subset V$ such that $v\notin U$ and $N^+v\subseteq U$.

If $U=\{0\}$ in Definition 3.3, the vector $v$ is primitive :

\proclaim{Definition 3.4}. Let $V$ be a $G$ module (resp. a
$G_{\bar 0}$ module). A vector $v\in V$ is called a $G$
primitive vector (resp. a $G_{\bar 0}$ primitive vector) if
$N^+v=0$ (resp. $N_0^+v=0$).

A weight $\La\in H^*$ is called {\sl dominant} if 
$a_i=\La(h_i)\geq 0$ for all $i\not=m$, {\sl integral} if 
$a_i\in\Zah$ for all $i\not=m$, and {\sl integral dominant} if 
$a_i\in\Nat$ for all $i\not=m$. From the theory of
reductive Lie algebras it follows that for every  integral
dominant
weight $\La$ there exists a unique (up to isomorphism)
finite dimensional simple $G_{\bar 0}$ module $V_0(\La)$
with highest weight $\La$. Let $v_\La$ be a highest weight vector
for $V_0(\La)$. The $G_0$ module $V_0(\La)$
can be extended to a $G_0\oplus G_{+1}$ module by
putting $G_{+1}V_0(\La)=0$. In this paper we shall make extensive
use of the following $G$ module, first defined by Kac $[{\underline{11}}]$:

\proclaim{Definition 3.5}. For an  integral  dominant $\La\in H^*$, the
Kac-module $\VK$ is the induced module
$$
\VK={\rm Ind}_{G_0\oplus G_{+1}}^G\;V_0(\La)=U(G)
 \otimes_{\lower2pt\hbox{$\scriptstyle G_0\oplus G_{+1}$}} V_0(\La).
$$

From Theorem~2.1 we see that $U(G)=U(G_{-1})\otimes U(G_0)\otimes
U(G_{+1})$. Therefore Definition~3.5 implies that
$$ 
\VK \cong U(G_{-1})\otimes V_0(\La).
\eqno(3.4)
$$
Since $[G_{-1},G_{-1}]=0$, $U(G_{-1})$ is isomorphic to $\wedge(G_{-1})$, the
exterior algebra over $G_{-1}$. The dimension of $G_{-1}$ is
$mn$, thus $\hbox{dim}\bigl(\wedge(G_{-1})\bigr)=2^{mn}$, and
hence $\VK$ is a finite dimensional $G$-module of dimension 
$2^{mn}\hbox{dim}\bigl(V_0(\La)\bigr)$. It follows from the
definition that $\VK$ is
a $G$ highest weight module.
Unfortunately $\VK$ is not
always a simple $G$ module. Since $\VK$ is a $G$ highest weight
module, it contains a unique maximal submodule $M(\La)$:
$$
M(\La)=\{v\in\VK\mid v_\La\notin U(G)v\},  \eqno(3.5)
$$
such that the quotient module
$$
V(\La)=\VK/M(\La) \eqno(3.6)
$$
is a finite dimensional simple $G$ module with highest weight $\La$. 
Kac proved the following theorem $[{\underline{11}}]$:

\proclaim{Theorem 3.6 [Kac]}. Every finite dimensional simple
$G$ module is isomorphic to a module of type (3.6), where $\La$
is  integral dominant. Moreover, every finite dimensional simple $G$ module
is uniquely characterized by its  integral dominant highest weight $\La$.

Let $T_+$ and $T_-$ be the following elements in $U(G)$:
$$
T_\pm=\prod_{\be\in\De_1^+}e(\pm\be), \eqno(3.7)
$$
where the $\be$'s in the product (3.7) (and in all subsequent
products of $e(\be)$'s) appear in the chosen
lexicographical ordering (note that a different ordering can only
lead to a sign change). One can verify that
$$
[e(\al),T_\pm]=0,\qquad\forall\al\in\De_0. \eqno(3.8)
$$
In $\VK$, let
$$
v_{\La_-}=T_-v_{\La},\hbox{ where }\La_-=\La-2\rh_1. \eqno(3.9)
$$
Note that (2.17) implies that $\La_-$ is also integral dominant;
in fact if $a_i$ 
are the Dynkin labels of $\La$, then $[a_1,\ldots,a_{m-1};a_m+m-n;
a_{m+1},\ldots,a_{m+n-1}]$ are the Dynkin labels of $\La_-$.
Since $G_{\bar 0}\subset G$, $\VK$ is also a $G_{\bar 0}$ module.
It follows from (3.8) that the $G_{\bar 0}$ module $\VK$ contains
$T_-V_0(\La)$ as a simple $G_{\bar 0}$ submodule, with highest
weight vector $v_{\La_-}$. This submodule contains a unique (up
to scalar multiplication) lowest weight vector $v_-$ of weight
$w_0\La_-$. From (3.4) it follows that $v_-$ is the unique
(again, up to scalar multiplication) vector of $\VK$ annihilated
by $N^-$.

\proclaim{Lemma 3.7}. $\VK$ is an indecomposable $G$ module,
indeed every non-zero $G$ submodule $Y$ of the $G$ module
$\VK$ contains the $G_{\bar 0}$ module $T_-V_0(\La)$ as a subspace.

This follows from the fact that every submodule of $\VK$ contains
the vector $v_-$ that is annihilated by $N^-$.

The following lemma appears in the work of Gould $[{\underline 4}]$:

\proclaim{Lemma 3.8}. Let $X(\La)=U(G)v_{\La_-}$. Then $X(\La)$
is a simple $G$ submodule of $\VK$, and every non-zero submodule
of $\VK$ contains $X(\La)$.

Indeed, $X(\La)$ is by definition a submodule of $\VK$. Using Lemma
3.7, every non-zero submodule $Y$ of $\VK$ contains $v_{\La_-}$,
and hence contains $X(\La)$. This also implies that $X(\La)$ has
no proper submodules, so $X(\La)$ is simple.

\proclaim{Lemma 3.9}. $\VK$ is a simple $G$ module if and only if
$T_+T_-v_\La\not= 0$.

\noindent {\sl Proof.} The elements in $U(G)v_{\La_-}$ of weight $\La$ must
be multiples of $T_+v_{\La_-}$.
If $T_+T_-v_\La=0$, then it follows 
that $v_\La\notin X(\La)$, so $X(\La)$ is then a proper non-zero
submodule of $\VK$, so $\VK$ is not simple. Conversely, if
$\VK$ were not simple, then $v_\La\notin M(\La)$. But according
to Lemma 3.7 $T_-v_\La\in M(\La)$, hence $T_+T_-v_\La\in M(\La)$,
and since $T_+T_-v_\La$ is of weight $\La$ we conclude
$T_+T_-v_\La \propto v_\La$. Thus $T_+T_-v_\La=0$.\mybox

\proclaim{Lemma 3.10}. Let $Q(\La)$ be the expression
$$
Q(\La)=\prod_{\be\in\De_1^+}\langle\La+\rh\mid\be\rangle. \eqno(3.10)
$$
Then
$$
T_+T_-v_\La=\pm Q(\La)v_\La. \eqno(3.11)
$$

For a proof, see Kac $[{\underline{11}},{\underline{12}}]$; whether the sign in (3.11) is $+$ or
$-$ depends upon the
ordering of the $e(\be)$'s in (3.7) and is unimportant here.

\proclaim{Definition 3.11}. Let $\La$ be an  integral  dominant weight. We
call $\La$ (resp. $\VK$, resp. $V(\La)$) a typical weight (resp.
a typical Kac-module, resp. a typical simple module) if and only
if $\langle\La+\rh\mid\be\rangle\not=0$ for all $\be\in\De_1^+$.
If there exists a $\be\in\De_1^+$ such that
$\langle\La+\rh\mid\be\rangle=0$ then $\La$, $\VK$ and
$V(\La)$ are called atypical, and $\be$ is called an atypical
root for $\La$. If there exists just one atypical root $\be$ for
$\La$, we call $\La$, $\VK$ and $V(\La)$ singly atypical of type $\be$.

The following theorem now follows from Lemmas 3.9 and 3.10 
$[{\underline{11}},{\underline{12}}]$:

\proclaim{Theorem 3.12}. The Kac-module $\VK$ is a simple $G$
module if and only if $\La$ is typical.

The character ch$V$ of a $G$ module $V$ with weight space
decomposition (3.3) is defined as
$$
\cha V=\sum_{\la\in H^*} \hbox{dim}(V_\la)e^\la, \eqno(3.12)
$$
where $e^\la$ is the formal exponential. The
action of the Weyl group $W$ on such formal exponentials is
defined by $w(e^\la)=e^{w\la}$. Let
$$
L_0=\prod_{\al\in\De_0^+}\left(e^{\al/2}-e^{-\al/2}\right)\hbox{
and }
L_1=\prod_{\be\in\De_1^+}\left(e^{\be/2}+e^{-\be/2}\right). \eqno(3.13)
$$
From (3.4) it follows that the Kac-module has character
$$
\cha\VK=\prod_{\be\in\De_1^+}\left(1+e^{-\be}\right)\cha
V_0(\La), \eqno(3.14)
$$
where $\cha V_0(\La)$ is given by Weyl's character formula $[{\underline{18}}]$:
$$
\cha V_0(\La)=L_0^{-1}\sum_{w\in W}\vep(w)e^{w(\La+\rh_0)}. \eqno(3.15)
$$
Using the Weyl invariance of $\rh_1$, we have
$$
\prod_{\be\in\De_1^+}\left(1+e^{-\be}\right)=L_1e^{-\rh_1}=L_1e^{-w\rh_1},
 \quad\forall w\in W, \eqno(3.16)
$$
and hence we obtain Kac's character formula 
$[{\underline{11}},{\underline{12}}]$:
$$
\cha\VK={L_1\over L_0}\sum_{w\in W}\vep(w)e^{w(\La+\rh)}. \eqno(3.17)
$$
Due to the Weyl invariance of $\De_1^+$ and of $L_1$, (3.17) can
be rewritten in the form
$$
\cha\VK=L_0^{-1}\sum_{w\in W}\vep(w)w\left\lbrace e^{\La+\rh_0}
\prod_{\be\in\De_1^+}\left(1+e^{-\be}\right)\right\rbrace. \eqno(3.18)
$$
Using Theorem 3.12, (3.18) gives the character of all typical simple
modules of $G$. The problem of finding the characters of atypical
simple $G$ modules is unsolved so far. In this paper we shall
deduce a character formula for singly atypical simple $G$ modules.

Finally, let $\la\in H^*$ be integral. We define the ``formal characters'':
$$
\ch_K(\la)=L_0^{-1}\sum_{w\in W}\vep(w)w\left\lbrace e^{\la+\rh_0}
\prod_{\be\in\De_1^+}\left(1+e^{-\be}\right)\right\rbrace; \eqno(3.19)
$$
$$
\ch_W(\la)=L_0^{-1}\sum_{w\in W}\vep(w)e^{w(\la+\rh_0)}. \eqno(3.20)
$$
If $\la$ is  integral dominant the expressions (3.19) and (3.20) coincide with
Kac's character 
$\cha\overline V(\la)$ and Weyl's character $\cha V_0(\la)$ respectively. It is
easy to verify that the formal characters satisfy the following properties:
$$
\ch_K(\la)=e^{-\rh_1}L_1\,\ch_W(\la);\eqno(3.21)
$$
$$
\ch_W(w\cdot\la)=\vep(w)\ch_W(\la),\;\hbox{ and }\;
\ch_K(w\cdot\la)=\vep(w)\ch_K(\la),\;\;\forall w\in W.\eqno(3.22)
$$

\section{4. The maximal submodule of the Kac-module}

Let $\La\in H^*$ be an integral dominant weight. In this section,
we shall consider the even Dynkin labels $a_i=\La(h_i)$ ($i\not=m$) of
$\La$ as fixed integers, and the odd Dynkin label $a_m$ as a
complex variable.
Let $V_0(\La)$ be the
(finite dimensional) simple $G_{\bar 0}$ module with highest
weight $\La$ and highest weight vector $v_\La$. 
$V_0(\La)$ has the following weight space decomposition:
$$
V_0(\La)=\bigoplus_\la V_0(\La)_\la. \eqno(4.1)
$$
Let $P_0(\La)$ be the set of weights $\la$ for which
$V_0(\La)_\la\not= \{0\}$, and denote by $m_0(\La,\la)$ 
the dimension of $V_0(\La)_\la$. 
The following lemma is a well known property of simple modules of
semi-simple Lie algebras, and it is applicable here in the case
of $G_{\bar 0}=sl(m)\oplus\C\oplus sl(n)$.

\proclaim {Lemma 4.1}.
For $\la\in P_0(\La)$ there exists a set of
elements $g_i(\la)\in U(N_0^-)_{\la-\La}$, $(i=1,2,\ldots,
m_0(\La,\la))$, such that $\{g_i(\la)v_\La\}$ forms a basis for
$V_0(\La)_\la$, and moreover, such that
$$
\si\bigl( g_i(\la)\bigr)g_j(\la)v_\La
=\de_{ij}v_\La,  \eqno(4.2)
$$
where $\si$ is the antiautomorphism (2.22),
and $\de_{ij}$ is the usual Kronecker symbol.

\noindent {\sl Proof.} From the results concerning the symmetric bilinear
contravariant form associated with $\si$ (see $[{\underline 7}]$), it
follows that there exists a set of monomials $z_i(\la)\in
U(N_0^-)_{\la-\La}$ ($i=1,\ldots,m_0(\La,\la)$) (i.e. every
$z_i(\la)$ is of the form
$\prod_{\al\in\De_0^+}\left(e(-\al)\right)^{k_\al}$ with
$\sum_{\al} k_\al\al=\La-\la$) such that $z_i(\la)v_\La$ forms a
basis for $V_0(\La)_\la$ and such that the matrix $Z$ of elements
$Z_{ij}$ in $\si(z_i(\la))z_j(\la)v_\La=Z_{ij}v_\La$ is
non-singular [the elements $Z_{ij}$ depend only upon the even
Dynkin labels, and hence are numbers independent of $a_m$]. 
From the properties of $\si$ 
and the real basis (2.7), it follows that $Z$ is a real symmetric
matrix. Diagonalising $Z$ then also gives rise to a new basis
$g_i'(\la)v_\La$ with every $g_i'(\la)$ a linear combination of
the $z_j(\la)$, and such that the matrix $Z'$ of coefficients $Z_{ij}'$
in $\si(g_i'(\la))g_j'(\la)v_\La=Z_{ij}'v_\La$ is diagonal with
real non-zero entries. Rescaling the $g_i'(\la)$ gives $g_i(\la)$.
\mybox

Consider the weight space decomposition (3.3) for the Kac-module
$\VK$, 
$$
\VK = \bigoplus_\mu\VK_\mu, \eqno(4.3)
$$
and let $P(\La)$ be the set of all weights $\mu$
such that $\VK_\mu \not=\{0\}$. Let $\k$ be a
sequence of numbers $k_\be$ ($\be\in\De_1^+$) such that every
$k_\be \in\{0,1\}$.
For $\mu\in P(\La)$,
consider all partitions $(\k,\la)$ of $\mu$ of the form
$$
\mu=\la-\sum_{\be\in\De_1^+}k_\be \be \eqno(4.4)
$$
with $\la\in P_0(\La)$. 
Then it follows from (3.4) that the dimension of $\VK_\mu$,
$m(\La,\mu)$, is given by
$$
m(\La,\mu)=\sum_{(\k,\la)} m_0(\La,\la),  \eqno(4.5)
$$
where the summation in
(4.5) is over all partitions $(\k,\la)$ of $\mu$
of the form (4.4). Moreover, it is easy to give a basis for
$\VK_\mu$, namely
$$ 
\eqalign{
\prod_{\be\in\De_1^+}e(-\be)^{k_\be}
 g_i(\la)v_\La,\;\;&\hbox{ with } (\k,\la) \hbox{ a partition}
\hbox{ of type (4.4)}\cr&\hbox{ and } i=1,2,\ldots,m_0(\La,\la).\cr} \eqno(4.6)
$$
We let
$$
x_{\k,i}=\prod_{\be\in\De_1^+}e(-\be)^{k_\be} g_i(\la)
\quad\in\quad U(N_1^-)U(N_0^-). \eqno(4.7)
$$
For convenience,
 we have dropped the dependence of $x_{\k,i}$ upon $\La$
and $\la$ in the notation. Denote by
$\tilde\k$ the sequence complementary to $\k$, consisting of
numbers $\tilde k_\be=1-k_\be$. Associated with $x_{\k,i}$, we define
$$
\tilde x_{\k,i}=\si\bigl(g_i(\la)\bigr)
\prod_{\be\in\De_1^+}e(-\be)^{\tilde k_\be}.  \eqno(4.8)
$$
Using $e(\be)^2=0$ for $\be\in\De_1$, (3.8), and (4.2),
one obtains the following properties:
$$
\eqalignno{
\tilde x_{\k',i'}x_{\k,i}v_\La &= \de_{\k'\k}\de_{i'i}
v_{\La_-},&(4.9a)\cr
\si(x_{\k,i})\si(\tilde x_{\k',i'})v_{\La_-} &= 
 \pm\de_{\k\k'}\de_{ii'}Q(\La)v_\La,&(4.9b)\cr}
$$
where a $\pm$-sign appears because in general a reordering of
the $e(+\be)$'s is necessary to recover $T_+$ in (4.9b). Note
that $Q(\La)$ is considered as a polynomial of degree $mn$ in the
odd Dynkin label $a_m$.

Finally, let $A$ be the matrix of size $m(\La,\mu)\times
m(\La,\mu)$ defined by
$$
\si(x_{\k,i})x_{\k',i'}v_\La = A_{\k i,\k'i'}v_\La. \eqno(4.10)
$$
This matrix is the Kac-module analogue of the Shapovalov matrix
$[{\underline{14}}]$ for Verma modules of complex semi-simple Lie algebras 
$[{\underline{14}},{\underline 7}]$.
From Definition 3.2 and (4.10) it follows that the rank of $A$ 
is equal to the number of linearly independent
generating vectors $v_\mu$ in $\VK_\mu$. Hence,
using (3.5) and (3.6):
$$
\hbox{rank}(A)=\hbox{dim}\bigl(V(\La)_\mu\bigr). \eqno(4.11)
$$
Similarly, let $B$ be the $m(\La,\mu)\times m(\La,\mu)$ matrix
defined by
$$
\tilde x_{\k,i}\si(\tilde x_{\k',i'})v_{\La_-}=B_{\k i,\k'i'}v_{\La_-}.
\eqno(4.12)
$$
Since $U(H)v_{\La_-}=\C v_{\La_-}$, $N_0^+v_{\La_-}=0$ and
$N_1^-v_{\La_-}=0$ (see equations (3.8)--(3.9)), $X(\La)=U(G)v_{\La_-}=
U(N_1^+)U(N_0^-)v_{\La_-}$. But $U(N_0^-)v_{\La_-}=V_0(\La_-)$ is isomorphic
to $V_0(\La)$ as an $sl(m)\oplus sl(n)$ module. Therefore
$X(\La)_\mu$ is
spanned by the vectors of type $\si(\tilde x_{\k',i'})v_{\La_-}$.
Hence any maximal subset
of linearly independent vectors of the set
$\{\si(\tilde x_{\k',i'})v_{\La_-}\}$ of $m(\La,\mu)$ elements
forms a basis for $X(\La)_\mu$=$\left(U(G)v_{\La_-}\right)_\mu$.
It follows from (4.12) and the structure of the Kac-module 
that the rank of $B$ is equal to the maximal number of linearly
independent vectors of weight $\mu$ in $\VK_\mu$ that
belong to $U(G)v_{\La_-}$=$X(\La)$. Thus
$$
\hbox{rank}(B)=\hbox{dim}\bigl(X(\La)_\mu\bigr). \eqno(4.13)
$$

Now we can prove the main result of this section:
\proclaim {Lemma 4.2}. Let $A$ and $B$ be defined as in (4.10)
and (4.12). Then
$$
\hbox{det}(A)\hbox{det}(B)=\pm\bigl(Q(\La)\bigr)^{m(\La,\mu)},
\eqno(4.14) $$
where $Q(\La)$ is given in (3.10).

\noindent {\sl Proof.} The vector $\si(\tilde x_{\k',i'})v_{\La_-}$ is of
weight $\mu$ in $\VK$, so it can be expressed as a linear
combination of the basis vectors (4.6) of $\VK_\mu$. Thus
$$
\si(\tilde x_{\k',i'})v_{\La_-}=\sum_{\k''i''}C_{\k''i'',\k'i'}
 x_{\k'',i''}v_\La, \eqno(4.15)
 $$
where $C_{\k''i'',\k'i'}$ is the matrix of coefficients of the
linear combinations. Acting on (4.15) with $\si(x_{\k,i})$ and
using (4.9b) yields:
$$
\pm\de_{\k\k'}\de_{ii'}Q(\La)v_\La =
\sum_{\k''i''}A_{\k i,\k''i''}C_{\k''i'',\k'i'}v_\La. \eqno(4.16)
$$
Thus $AC$ is a diagonal matrix, and in particular,
$$
\hbox{det}(A)\hbox{det}(C) =
\pm\bigl(Q(\La)\bigr)^{m(\La,\mu)}.  \eqno(4.17)
$$
Acting on (4.15) with $\tilde x_{\k,i}$, and using (4.9a), yields
$$
B_{\k i,\k'i'}v_{\La_-}=\sum_{\k''i''}C_{\k''i'',\k'i'}
\de_{\k\k''}\de_{ii''}v_{\La_-},
\eqno(4.18)
$$
hence $B=C$, and in particular
$$
\hbox{det}(B)=\hbox{det}(C).  \eqno(4.19)
$$
The lemma now follows from (4.17) and (4.19).
\mybox

\proclaim {Theorem 4.3}. If $\La$ is singly atypical then $M(\La)=X(\La)$.

\noindent {\sl Proof.} Let $\La$ be singly atypical of type $\be$. 
Then the polynomial $Q(\La)$ in (4.14) has a zero of multiplicity
$m(\La,\mu)$ for $a_m=\La(h_m)$.
Now we use the following property: let $M(t)$ be a
$N\times N$-matrix over $\C[t]$ (i.e.~the entries of $M(t)$ are
polynomials in the variable $t$); if $t=t_0$ is a zero of
multiplicity $k$ of $\hbox{det}(M(t))$, then $\hbox{rank}(M(t_0))\geq
N-k$ (this property can be
proved using elementary matrix operations).
Applying this to $A$ and $B$ in (4.14), 
for $a_m=\La(h_m)$, leads to
$$
\hbox{rank}(A)+\hbox{rank}(B)\geq 2m(\La,\mu)-m(\La,\mu)=m(\La,\mu), \eqno(4.20)
$$
or, using (4.11) and (4.13),
$$
\hbox{dim}V(\La)_\mu + \hbox{dim}X(\La)_\mu \geq m(\La,\mu). \eqno(4.21)
$$
But since $V(\La)\cong\VK/M(\La)$ and $X(\La)\subseteq M(\La)$,
$$
\hbox{dim}V(\La)_\mu + \hbox{dim}X(\La)_\mu\leq\hbox{dim}
\VK_\mu = m(\La,\mu). \eqno(4.22)
$$
Hence
$$
\hbox{dim}V(\La)_\mu + \hbox{dim}X(\La)_\mu = \hbox{dim}
\VK_\mu ,\qquad\forall\mu\in P(\La). \eqno(4.23)
$$
This shows that $X(\La)$ is the maximal submodule of $\VK$.\mybox

\section {5. Singly atypical modules of type $\al_m$}

In this section we shall 
consider the special case of a singly atypical $\La$
of type $\al_m$, where $\al_m$ is the unique odd simple root
given in (2.9). In this case it turns out to be rather easy to
determine the highest weight of $X(\La)$.

\proclaim {Lemma 5.1}. Let $\La$ be atypical of type $\al_m$.
Then $v=e(-\al_m)v_\La$ is a $G$ primitive vector in $\VK$.

\noindent {\sl Proof.} For $\al\in\De_0^+$, we have
$e(\al)v=[e(\al),e(-\al_m)]v_\La+e(-\al_m)e(\al)v_\La$. But
$[e(\al),e(-\al_m)]=0$ for all $\al\in\De_0^+$, and
$N^+v_\La=0$,  hence
$$
e(\al)v=0,\qquad\forall\al\in\De_0^+.\eqno(5.1)
$$
Then, using $e(+\al_m)v_\La=0$, one finds
$$
\eqalign{
e(+\al_m)v&=[e(+\al_m),e(-\al_m)]v_\La=h_mv_\La=\La(h_m)v_\la\cr
&=\langle\La\mid\al_m\rangle v_\La
=\langle\La+\rh\mid\al_m\rangle v_\La=0,\cr} \eqno(5.2)
$$
since $\langle\rh\mid\al_m\rangle=0$ and $\La$ is atypical of
type $\al_m$. Then (5.1) and (5.2) imply
$$
e(\al_i)v=0,\qquad i=1,2,\ldots,m+n-1, \eqno(5.3)
$$
where $\al_i$ are the simple roots introduced in (2.9). Since
$N^+$ is generated by the $m+n-1$ elements $e(\al_i)$, it follows
that $N^+v=0$.\mybox

In the case of Lemma 5.1, $U(G)v$ is a proper submodule of $\VK$,
hence Lemma 3.8 implies $X(\La)\subseteq U(G)v\subseteq M(\La)$. But $\La$ is
singly atypical of type $\al_m$, so by Theorem 4.3:
$$
M(\La)=X(\La)=U(G)v, \eqno(5.4)
$$
where $v=e(-\al_m)v_\La$ is a vector of weight $\La-\al_m$. Since
$U(G)v$ is a highest weight module with highest weight vector
$v$, and since $X(\La)$ is simple (see Lemma 3.8), we have the following

\proclaim {Corollary 5.2}. Let $\La$ be singly atypical of type
$\al_m$. Then $X(\La)=U(G)v_{\La_-}$ is the maximal proper submodule of $\,\VK$,
and $X(\La)$ is isomorphic to the simple $G$ module
$V(\La-\al_m)$. Consequently,
$$
\cha\VK=\cha V(\La)+\cha V(\La-\al_m). \eqno(5.5)
$$

Note that if $\La$ is dominant and singly atypical of type $\al_m$, then
$\La-\al_m$ is also dominant and singly atypical of type $\al_m$. Now
we can prove a character formula for this particular case.

\proclaim {Theorem 5.3}. Let $\La$ be singly atypical of type
$\al_m$. Then
$$
\cha V(\La)=L_0^{-1}\sum_{w\in W}\vep(w)w\Biggl\lbrace
e^{\La+\rh_0}\prod_{{\scriptstyle
\be\in\De_1^+}\atop{\scriptstyle \be\not=\al_m}}(1+e^{-\be})\Biggr\rbrace.
\eqno(5.6)
$$

\noindent {\sl Proof.} Using (5.5) as a recursion relation, we find
$$
\eqalignno{
&\cha V(\La)=\cha\VK -\cha V(\La-\al_m)\cr
 &=\cha\VK-\left(\cha\overline V(\La-\al_m)-
\cha V(\La-2\al_m)\right)\cr
 &=\cha\VK-\cha\overline V(\La-\al_m)+
 \left(\cha\overline V(\La-2\al_m)-\cha V(\La-3\al_m)\right)=\ldots\cr
 &=\cha\VK-\cha\overline V(\La-\al_m)+
 \cha\overline V(\La-2\al_m)-\cha\overline V(\La-3\al_m)+\ldots&(5.7)\cr}
$$
which becomes a formal infinite series expression since (5.5)
can be applied for every $\La-k\al_m$ ($k\in\Nat$). Then we can
substitute (3.18) for the characters of the Kac-modules appearing
in (5.7), and sum over the formal series:
$$
\eqalignno{
\cha V(\La)&=L_0^{-1}\sum_{w\in W}\vep(w)w\Bigl\lbrace
e^{\rh_0}\left(
e^{\La}-e^{\La-\al_m}+e^{\La-2\al_m}-e^{\La-3\al_m}+\ldots\right)\cr
&\hskip1.5in\times\prod_{\be\in\De_1^+}(1+e^{-\be})\Bigr\rbrace\cr
 &=L_0^{-1}\sum_{w\in W}\vep(w)w\Bigl\lbrace
e^{\La+\rh_0}\left(1+e^{-\al_m}\right)^{-1}
\prod_{\be\in\De_1^+}(1+e^{-\be})\Bigr\rbrace.&(5.8)\cr}
$$
This proves the theorem.\mybox

Let $\la$ be an integral weight, and $\ga\in\De_1^+$. We define
the formal character
$$
\ch_\ga(\la)=L_0^{-1}\sum_{w\in W}\vep(w)w\Biggl\lbrace
e^{\la+\rh_0}\prod_{{\scriptstyle
\be\in\De_1^+}\atop{\scriptstyle \be\not=\ga}}(1+e^{-\be})\Biggr\rbrace.
\eqno(5.9)
$$
Theorem 5.3 shows that if $\La$ is singly atypical of type
$\al_m$, then $\cha V(\La)=\ch_{\al_m}(\La)$. We shall show in
Section~7 that if $\La$ is singly atypical of type $\ga$, then
$\cha V(\La)=\ch_\ga(\La)$. Note that the formal character
(5.9) satisfies the property:
$$
\ch_{w(\ga)}(w\cdot\la)=\vep(w)\ch_\ga(\la),\qquad\forall w\in W.\eqno(5.10)
$$
Finally, one sees from (3.19) that
$$
\ch_K(\la)=\ch_\ga(\la)+\ch_\ga(\la-\ga),\qquad\forall\ga\in\De_1^+.\eqno(5.11)
$$
{\bf Remark 5.4.} Let $\La$ be integral dominant. In Section~3 we have seen that
$\VK$ has a unique (up to scalar multiplication) vector of weight
$w_0(\La_-)=w_0(\La-2\rh_1)$ that is annihilated by $N^-$;
$w_0(\La_-)$ is the lowest weight of $\VK$, and it also
characterises the Kac-module uniquely. Then $\ch_K(\La)=\cha\VK$
contains a unique lowest term $e^{w_0(\La_-)}$, where the terms
$e^\la$ are partially ordered according to $e^\la\geq
e^\mu\,\Leftrightarrow\,\la\geq\mu$. It follows from 
(5.9) that $e^{w_0(\La_-)}$ is a term of $\ch_\ga(\La-\ga)$ and
not of $\ch_\ga(\La)$; in particular it is the unique lowest term
appearing in $\ch_\ga(\La-\ga)$.

\section{6. The atypicality matrix}

The atypicality of an  integral  dominant weight $\La$ is determined
by the value of the $mn$ numbers $\langle\La+\rh\mid\be\rangle$
with $\be\in\De_1^+$. In this section we shall study some of the
properties of a matrix consisting of these $mn$ numbers 
$[{\underline{17}}]$, and in
particular we prove some crucial lemmas concerning a singly
atypical $\La$.

\proclaim {Definition 6.1}. Let $\La\in H^*$. The atypicality
matrix $A(\La)$ is the $m\times n$ complex matrix with entries
$A(\La)_{ij} = \langle\La+\rh\mid\be_{ij}\rangle$, where
$i=1,\ldots,m$ and $j=1,\ldots,n$, and $\be_{ij}$ is defined in (2.13).

In terms of the $\ep\de$-components of $\La$, one has:
$$
A(\La)_{ij}=\la_i+\mu_j+m-i-j+1. \eqno(6.1)
$$
The properties of this matrix have been studied in another paper 
$[{\underline{17}}]$,
and can be summarized as follows:
\item{a)} Let $w=\si\times\ta\in W=S_m\times S_n$, then
$$
A(w\cdot\La)_{ij}=A(\La)_{\si^{-1}(i),\ta^{-1}(j)}, \eqno(6.2a)
$$
where $w\cdot\La$ is determined by (2.15) or (2.18).
\item{b)} Let $a_i$ be the Dynkin labels of $\La$, then
$$
\eqalign{
	&A(\La)_{ij}-A(\La)_{i+1,j}=a_i+1,\qquad(1\leq i<m)\cr
        &A(\La)_{m1}=a_m,\cr
        &A(\La)_{ij}-A(\La)_{i,j+1}=a_{m+j}+1.\qquad(1\leq j<n)\cr}
        \eqno(6.2b)
$$
\item{c)} Any atypicality matrix $A(\La)$ satisfies:
$$
A(\La)_{ij}+A(\La)_{kl}=A(\La)_{il}+A(\La)_{kj}\; ;\eqno(6.2c)
$$
vice versa, any $m\times n$ matrix satisfying (6.2c) for all
pairs $(i,j)$ and $(k,l)$ with $1\leq i,k\leq m$ and $1\leq
j,l\leq n$ is the atypicality matrix of a unique element $\La\in H^*$.
\item{d)} $\La$ is dominant if and only if         
$$
\eqalign{
&A(\La)_{ij}-A(\La)_{i+1,j}-1\geq 0\qquad
(1\leq i<m,\;1\leq j \leq n)\quad\hbox{and}\cr
&A(\La)_{ij}-A(\La)_{i,j+1}-1\geq 0\qquad
(1\leq i\leq m,\;1\leq j<n).\cr}\eqno(6.2d)
$$
Moreover, $\La$ is integral dominant if the expressions on the
l.h.s. of (6.2d) are all integers.

\proclaim {Lemma 6.2}. Let $\la$ be any integral element of $H^*$. Then
the following statements are equivalent:
$$
\eqalignno{
 &(1)\quad \ch_W(\la)=0;\cr
 &(2)\quad \ch_K(\la)=0;\cr
 &(3)\quad \exists w\in W\hbox{ with }\vep(w)=-1\hbox{ such that
}w\cdot \la=\la;\cr
 &(4)\quad \forall w\in W, w\cdot\la\hbox{ is not dominant};\cr
 &(5)\quad A(\la)\hbox{ has two equal columns or two equal rows.}\cr}
$$

\noindent {\sl Proof.} The equivalence of (1), (3) and (4) is a classical
property of the Weyl group of a semi-simple Lie algebra $[{\underline 6}]$. From
(3.21) it follows that (2) is equivalent to (1). Finally, if
$A(\la)$ has two equal rows or columns, then (6.2a) implies that
there exists a $w\in W$ with $\vep(w)=-1$ such that $A(w\cdot\la)=
A(\la)$ and hence $w\cdot\la=\la$, so that (5)$\Rightarrow$(3).
Conversely, if $A(\la)$ has no equal rows or columns, (6.2a)
together with (6.2b)
implies that there exists a $w\in W$ such that in the matrix
$A(w\cdot\la)$ the elements in every row are strictly decreasing
from left to right and the elements in every column are strictly
decreasing from top to bottom; then (6.2d) is satisfied for
$A(w\cdot\La)$ and implies that
$w\cdot\la$ is dominant, contradicting (4).\mybox

\proclaim {Definition 6.3}. An integral element $\la\in H^*$ is
said to be vanishing if one of the statements (1)--(5) of Lemma 6.2 are
satisfied. Otherwise, $\la$ is said to be non-vanishing.

In the rest of this section, $\La$ is an  integral  dominant weight.
Note that if $\La$ is integral and atypical, then (6.2b) implies
that all entries in the atypicality matrix $A(\La)$ are integers.

\proclaim {Lemma 6.4}. Let $\La$ be singly atypical of type
$\ga=\be_{ij}$. Then
$$
\{-A(\La)_{il}\mid 1\leq l\leq n\}\cap\{A(\La)_{kj}\mid 1\leq
k\leq m\}=\{0\}. \eqno(6.3)
$$

\noindent {\sl Proof.} Since $A(\La)_{ij}=0$, (6.2c) implies
$$
A(\La)_{kl}=A(\La)_{il}+A(\La)_{kj}.
$$
But $\La$ is singly atypical, so $A(\La)_{kl}\not=0$ for
$(k,l)\not=(i,j)$. This implies (6.3).\mybox

\proclaim {Definition 6.5}. Let $\La$ be singly atypical of type
$\ga=\be_{ij}$. Let 
$$
r(\La)=\{-A(\La)_{il}\mid 1\leq l\leq n\}\cup
\{A(\La)_{kj}\mid 1\leq k\leq m\}. \eqno(6.4)
$$
Let $s(\La)$ be the maximal subset of $r(\La)$ consisting of
consecutive integers $\{-q,\ldots,p\}$ with $q,p\in\Nat$ and such
that $0\in s(\La)$. Let $\{(i_t,j_t),\, -q\leq t\leq p\}$ be the
sequence of matrix-positions defined by 
$(i_0,j_0)=(i,j)$ and 
\item{(a)} for $t\geq 0$ $(i_{t+1},j_{t+1})=(i_t,j_t+1)$ if
$-(t+1)$ belongs to the $i$th row of $A(\La)$, and
$(i_{t+1},j_{t+1}) =(i_t-1,j_t)$ if $t+1$ belongs to the $j$th
column of $A(\La)$;
\item{(b)} for $t\leq 0$, $(i_{t-1},j_{t-1})=(i_t,j_t-1)$ if $-(t-1)$
belongs to the $i$th row of $A(\La)$, and
$(i_{t-1},j_{t-1})=(i_t+1,j_t)$ if $t-1$ belongs to the $j$th
column of $A(\La)$.

This sequence of matrix-positions is well defined
thanks to Lemma 6.4. It is useful to introduce a notation for the
subsequences:
$$
\eqalign{
	S_+(\La)&=\{(i_0,j_0),\ldots,(i_p,j_p)\}\cr
	S_-(\La)&=\{(i_{-q},j_{-q}),\ldots,(i_0,j_0)\}\cr}\eqno(6.5)
$$	
Also, we let
$$
\tilde S_\pm(\La)=\{\be_{kl}\mid (k,l)\in S_\pm(\La)\}. \eqno(6.6)
$$

{\bf Example.} Let $G=sl(6/8)$ and $$\La=(7,7,5,4,4,1\mid
0,-2,-2,-4,-4,-4,-4,-7)$$ in the $\ep\de$-basis or
$\La=[02103;1;2020003]$ in Dynkin labels. Then $\La$ is singly
atypical of type $\be_{3,5}$ and the atypicality
matrix $A(\La)$ is given in (6.7), where it is bordered at the
top with the negatives of the third row and at the left with the
fifth column. The numbers actually belonging to $s(\La)$ are in
{\it italic}, and these determine the sequences $S_+(\La)$ and
$S_-(\La)$, also represented in (6.7) by $+$ and $-$ signs,
respectively, in the
table of matrix-positions.

\tenpoint 
$$
A(\La)=\bordermatrix{
	&\hfill-8&\hfill{\it -5}&\hfill{\it -4}&\hfill{\it -1}&\hfill{\it 0}&\hfill{\it 1}&\hfill{\it 2}&\hfill6\cr
	\hfill{\it 4}&\hfill12&\hfill9&\hfill8&\hfill5&\hfill4&\hfill3&\hfill2&\hfill-2\cr
	\hfill{\it 3}&\hfill11&\hfill8&\hfill7&\hfill4&\hfill3&\hfill2&\hfill1&\hfill-3\cr
	\hfill{\it 0}&\hfill8&\hfill5&\hfill4&\hfill1&\hfill0&\hfill-1&\hfill-2&\hfill-6\cr
	\hfill{\it -2}&\hfill6&\hfill3&\hfill2&\hfill-1&\hfill-2&\hfill-3&\hfill-4&\hfill-8\cr
	\hfill{\it -3}&\hfill5&\hfill2&\hfill1&\hfill-2&\hfill-3&\hfill-4&\hfill-5&\hfill-9\cr
	\hfill-7&\hfill1&\hfill-2&\hfill-3&\hfill-6&\hfill-7&\hfill-8&\hfill-9&\hfill-13\cr}
\qquad	
\hbox to 4cm{$
\vcenter{\offinterlineskip
 \halign{&\strut\vrule\hbox to 11.6pt{\hss$#$\hss}\cr
   \noalign{\hrule width 96pt\hfill}
    & & & & & &+& &\cr
   \noalign{\hrule width 96pt\hfill}
    & & & & & &+& &\cr
   \noalign{\hrule width 96pt\hfill}
    & & &-&\pm&+&+& &\cr
   \noalign{\hrule width 96pt\hfill}
    & & &-& & & & &\cr
   \noalign{\hrule width 96pt\hfill}
    &-&-&-& & & & &\cr
   \noalign{\hrule width 96pt\hfill}
    & & & & & & & &\cr
   \noalign{\hrule width 96pt\hfill}
   }}   $}
\eqno(6.7)
$$ 
\twelvepoint
Explicitly, 
$$
S_-(\La)=\{(5,2),(5,3),(5,4),(4,4),(3,4),(3,5)\}
$$
and
$$
S_+(\La)=\{(3,5),(3,6),(3,7),(2,7),(1,7)\}.
$$

\proclaim {Lemma 6.6}. Let $\La$ be singly atypical of type
$\ga=\be_{ij}$. Then
there exists a unique sequence of distinct elements
$\be_{-q}<\be_{-q+1}<\cdots<\be_0=\ga$
from $\De_1^+$ such that the sequence of weights $\nu_{-q-1},\nu_{-q},
\ldots,\nu_0=\La$,
where $\nu_{t-1}=\nu_t-\be_t$, satisfies
$$
\eqalignno{
 &\langle\nu_t+\rh\mid\be_t\rangle=0,\qquad -q\leq t\leq 0;&(6.8a)\cr
 &\nu_t\hbox{ is vanishing for }-q\leq t<0;&(6.8b)\cr
 &q(\La)=\nu_{-q-1}\hbox{ is integral dominant and singly atypical}\cr
&\indent\hbox{ of type }\be_{-q};&(6.8c)\cr
 &\exists w\in W\hbox{ such that }\nu_t=w\cdot(\La+t\ga)
 \hbox{ with }\be_{t+1}=w(\ga)\cr
 &\indent\hbox{ and }\vep(w)=(-1)^{t+1},
 \; -q-1\leq t<0;&(6.8d)\cr
 &\be_t=\be_{i_t,j_t},\hbox{ where }(i_t,j_t)\hbox{ is given in
Definition 6.5}.&(6.8e)\cr}
$$

\noindent {\sl Proof.} 
From the inner product (2.14) one deduces
$$
\langle\be_{ab}\mid\be_{kl}\rangle=\de_{ak}-\de_{bl}. \eqno(6.9)
$$
Using Definition 6.1 this implies that $A(\La-\be_{ab})$ is
obtained from $A(\La)$ by decreasing the elements in row $a$ by one unit and
simultaneously increasing the elements in column $b$ by one unit. Hence the
matrices $A_0=A(\La)$, $A_{-1}=A(\La-\be_{i_0,j_0})$,
$A_{-2}=A(\La-\be_{i_0,j_0} -\be_{i_{-1},j_{-1}})$, $\ldots$, where
$(i_t,j_t)$ is the sequence of Definition 6.5, satisfy
$$
\eqalignno{
 &A_t\hbox{ has two zeroes, at positions }(i_{t+1},j_{t+1})\hbox{
and }\cr&\indent(i_t,j_t)\hbox{ for }-q\leq t\leq -1;&(6.10a)\cr
 &A_{-q-1}\hbox{ has one zero at position }(i_{-q},j_{-q});&(6.10b)\cr
 &A_t\hbox{ is obtained from }A(\La+t\ga)\hbox{ by }-t-1\hbox{
transpositions}\cr&\indent \hbox{of rows and columns}.&(6.10c)\cr}
$$
Thus the existence and the uniqueness of the sequence $\be_0,\be_{-1},
\ldots,\be_{-q}$,
and (6.8e), follow from the properties of the sequence
$S_-(\La)$, which also implies that $\be_0>\cdots>\be_{-q}$.
Then (6.8a) is a consequence of (6.10a). Moreover, from (6.10a)
it follows that $A_t=A(\nu_t)$ ($-q\leq t\leq -1$) has two equal rows or two equal
columns, and then Lemma 6.2 implies (6.8b). The matrix $A_{-q-1}$
has one zero at position $(i_{-q},j_{-q})$, and by construction the
elements in every row are strictly decreasing from left to right
and the elements in every column are strictly decreasing from top
to bottom; thus (6.2d) implies that $q(\La)=\nu_{-q-1}$ is dominant,
proving (6.8c). Finally, (6.10c) and (6.2a) imply (6.8d).\mybox

\proclaim {Lemma 6.7}. Let $\La$ be singly atypical of type
$\ga=\be_{ij}$. Then
there exists a unique sequence of distinct elements
$\be_0=\ga<\be_1<\cdots<\be_p$
from $\De_1^+$ such that the sequence of weights $\nu_0=\La,\nu_1,\ldots,
\nu_{p+1}$, where $\nu_{t+1}=\nu_t+\be_t$, satisfies
$$
\eqalignno{
 &\langle\nu_t+\rh\mid\be_t\rangle=0,\qquad 0\leq t\leq p;&(6.11a)\cr
 &\nu_t\hbox{ is vanishing for }0<t\leq p;&(6.11b)\cr
 &p(\La)=\nu_{p+1}\hbox{ is integral dominant and singly atypical}\cr
&\indent\hbox{ of type }\be_p;&(6.11c)\cr
 &\exists w\in W\hbox{ such that }\nu_t=w\cdot(\La+t\ga)
 \hbox{ with }\be_{t-1}=w(\ga)\cr
 &\indent\hbox{ and }\vep(w)=(-1)^{t-1},
 \; 0<t\leq p+1;&(6.11d)\cr
 &\be_t=\be_{i_t,j_t},\hbox{ where }(i_t,j_t)\hbox{ is given in
Definition 6.5}.&(6.11e)\cr}
$$


The proof of Lemma 6.7 is similar to the proof of Lemma 6.6,
using $S_+(\La)$ instead of $S_-(\La)$.

\section{7. The character formula}

Using the lemmas of Section 6, we are now able to prove a character
formula for $V(\La)$, where $\La$ is a singly atypical integral dominant
weight.

\proclaim {Lemma 7.1}. Let $\La$ be singly atypical of type $\ga$ with
$S_-(\La)$ given by (6.5). Let $\ga'=\be_{i_{-q},j_{-q}}$ and
$q(\La)=\La-\sum_{\be\in\tilde S_-(\La)}\be$ be the dominant weight defined 
in Lemma 6.6,
which is singly atypical of type $\ga'$.
Then, using the notation (5.9):
$$
\ch_\ga(\La-\ga)=\ch_{\ga'}(q(\La)). \eqno(7.1)
$$

\noindent {\sl Proof.} As in the proof of Theorem 5.3, we can expand
$\ch_\ga(\La-\ga)$ in a series of $\ch_K(\la)$-terms:
$$
\eqalign{
\ch_\ga(\La-\ga)&=\ch_K(\La-\ga)-\ch_K(\La-2\ga)+\ch_K(\La-3\ga)-\cdots\cr
&+(-1)^q\ch_K(\La-(q+1)\ga)+\cdots\cr}\eqno(7.2)
$$
But for $-q\leq t\leq -1$, (6.8b) and (6.8d) imply that $\La+t\ga$
is vanishing, hence $\ch_K(\La+t\ga)=0$. Then (7.2) becomes:
$$
\eqalignno{
\ch_\ga(\La-\ga)&=(-1)^q\left(\ch_K(\La-(q+1)\ga)
-\ch_K(\La-(q+2)\ga)+\cdots\right)\cr
 &=(-1)^q\ch_\ga(\La-(q+1)\ga).&(7.3)\cr}
$$
According to (6.8d), there exists a $w\in W$ such that
$w(\La-(q+1)\ga+\rh)=q(\La)+\rh$ with $\ga'=w(\ga)$ and
$\vep(w)=(-1)^q$. Using (5.10) this implies that
$$
\ch_{\ga'}(q(\La))=(-1)^q\ch_\ga(\La-(q+1)\ga). \eqno(7.4)
$$
Then the lemma follows from (7.3) and (7.4).\mybox

\proclaim {Theorem 7.2}. Let $\La$ be singly atypical of type
$\ga$. Then
$$
\cha V(\La)=\ch_\ga(\La). \eqno(7.5)
$$

\noindent {\sl Proof.} In the case that $\ga=\al_m$, the statement follows
from Theorem 5.3. Suppose now that $\ga>\al_m$. Let $\La_0=\La$
and $\ga_0=\ga$, and using the notation of Lemma 7.1 we define a
sequence of dominant weights and elements of $\De_1^+$ by
$$
\La_{k+1}=q(\La_k),\qquad \ga_{k+1}=\ga_k',\qquad(k\geq 0). \eqno(7.6)
$$
Clearly, every $\ga_{k+1}\leq\ga_k$, with equality if and only if
$\# S_-(\La_k)=1$, i.e. if and only if $\La_k-\ga_k$
is dominant. So $\ga_{k+1}=\ga_k$ can happen only a finite number of times if
$\ga_k>\al_m$. 
Therefore, there exists an $s$ such that $\ga_{s-1}>\ga_s=\al_m$,
$\al_m$ being the smallest element of $\De_1^+$ according to the
partial ordering (2.10). Since every $\La_k$ is dominant, we
find, using (3.18)--(3.19) and (5.11):
$$
\cha\overline
V(\La_k)=\ch_{\ga_k}(\La_k)+\ch_{\ga_k}(\La_k-\ga_k). 
\eqno(7.7)
$$
Using Lemma 7.1, this becomes:
$$
\cha\overline V(\La_k)=\ch_{\ga_k}(\La_k)+\ch_{\ga_{k+1}}(\La_{k+1}). 
\eqno(7.8)
$$
On the other hand, since every $\La_k$ is singly atypical,
Theorem 4.3 implies:
$$
\cha\overline V(\La_k)=\cha V(\La_k)+\cha X(\La_k), \eqno(7.9)
$$
where both $V(\La_k)$ and $X(\La_k)$ are simple $G$ modules.
Applying (7.8) and (7.9) for $k=s-1$ and using Theorem 5.3, leads to
$$
\eqalignno{
\cha\overline V(\La_{s-1})&=\ch_{\ga_{s-1}}(\La_{s-1})+\cha V(\La_s),&(7.10a)\cr
\cha\overline V(\La_{s-1})&=\cha V(\La_{s-1})+\cha X(\La_{s-1}).&(7.10b)\cr}
$$
From Remark~5.4, (7.7) and (7.10a), we see that $V(\La_s)$ has
$w_0(\La_{s-1} -2\rh_1)$ as lowest weight. Since a
simple $G$ module is characterised by its lowest weight, it
follows from Lemma 3.7 and Lemma 3.8 that $V(\La_s)$ is
isomorphic to $X(\La_{s-1})$,
and (7.10) implies:
$$
\cha V(\La_{s-1})=\ch_{\ga_{s-1}}(\La_{s-1}). \eqno(7.11)
$$
By iteration one finds, for all $k$ with $0\leq k\leq s$:
$$
\cha V(\La_k)=\ch_{\ga_k}(\La_k). \eqno(7.12)
$$
The theorem follows by putting $k=0$ in (7.12).\mybox

\proclaim {Corollary 7.3}. Let $\La$ be singly atypical. Then the
lowest weight of $V(\La)$ is given by
$$
w_0\Bigl(\La-\sum_{\be\notin\tilde S_+(\La)}\be\Bigr)
=w_0\Bigl(\La_-+\sum_{\be\in\tilde S_+(\La)}\be\Bigr). \eqno(7.13)
$$

\noindent {\sl Proof.} For given singly atypical $\La$, let 
$$
\Om=p(\La)=\La+\sum_{\be\in\tilde S_+(\La)}\be. \eqno(7.14)
$$
Then it is a combinatorial exercise to see that
$S_-(\Om)=S_+(\La)$, hence
$$
\La=q(\Om)=\Om-\sum_{\be\in\tilde S_-(\Om)}\be. \eqno(7.15)
$$
From the proof of Theorem 7.2 it follows that $\La$ is the
highest weight of $X(\Om)$, or $X(\Om)\cong V(\La)$. But
$w_0(\Om_-)$ is the lowest weight of $\overline V(\Om)$ (see
Section~3), and therefore also of $X(\Om)$. So the lowest weight
of $V(\La)$ is given by
$$
\eqalignno{
w_0(\Om_-)&=w_0(\Om-2\rh_1)=w_0\Bigl(\La-2\rh_1+
 \sum_{\be\in\tilde S_+(\La)}\be\Bigr)\cr
  &=w_0\Bigl(\La-\sum_{\be\notin\tilde S_+(\La)}\be\Bigr), &(7.16)\cr}
$$
since $2\rh_1=\sum_{\be\in\De_1^+}\be$.\mybox  

\section{8. Some remarks}

\noindent 1. In order to accommodate the characters of all simple
modules of $G=sl(m/n)$, Bernstein and Leites proposed the
following formula $[{\underline 2}]$:
$$
\ch_L(\la)=L_0^{-1}\sum_{w\in W}\vep(w)w\Biggl\lbrace
e^{\la+\rh_0}\prod_{{\scriptstyle
\be\in\De_1^+}\atop{\scriptstyle \langle\la+\rh\mid\be\rangle
\not=0}}(1+e^{-\be})\Biggr\rbrace.
\eqno(8.1)
$$
Although for any integral dominant $\La$, $\cha
V(\La)=\ch_L(\La)$ if $\La$ is typical, as proved by Kac (Theorem 3.12 and
equation (3.18) in
this paper), and $\cha V(\La)=\ch_L(\La)$ if $\La$ is singly
atypical, as proved here in Theorem 7.2, it is certainly not true
in general that $\cha V(\La)=\ch_L(\La)$. A simple counterexample
is the identity module $V({\bf 0})$ which has character $\cha
V({\bf 0})=1$, and $\ch_L({\bf 0})\not=1$ if $m>1$ and $n>1$. 
\vskip 2mm
\noindent 2. In Kac's classification of classical simple Lie
superalgebras over $\C$ $[{\underline 9}]$, the {\sl Type I} Lie superalgebras are
$A(m,n)$ and $C(n)$, where $A(m,n)=sl(m+1/n+1)$ if $m\not=n$ and
$A(m,m)=sl(m+1/m+1)/\C I_{2m}$, and $C(n)=osp(2,2n-2)$. It is not
too difficult to verify that many lemmas given here for $sl(m/n)$
are also valid for $osp(2,2n-2)$: the proofs in Section~4 can
almost literally be transferred to the case of $osp(2,2n-2)$; the
notions in Sections~5--7 need to be slightly changed. This leads
us to a proof of a character formula for singly atypical modules
of $C(n)$ $[{\underline{16}}]$. But $C(n)$ has only typical or singly atypical
modules. We conclude that, for all integral dominant
$\La$ for $C(n)$, $\cha V(\La)=\ch_L(\La)$, given by (8.1)
but with all symbols defined for $C(n)$.
\vskip 2mm
\noindent 3. Let us return to the case $G=sl(m/n)$. We say that an
integral dominant weight $\La$ is {\sl atypical of degree} $d$ if
there are $d$ distinct elements $\be$ in $\De_1^+$ for which
$\La$ is atypical. We shall try to give the reader an idea of the
complications which arise in identifying the maximal submodule $M(\La)$
if $d>1$ by concentrating on the case of $d=2$.
For $d=1$, Theorem 4.3 shows that $\VK$ always contains 2
composition factors. For $d=2$, for example, we have calculated
the composition factors of some Kac-modules in $sl(2/3)$, and
their number varies : $\overline V([0;0;0,0])$ has 3
composition factors, $\overline V([1;0;1,0])$ has 5 composition
factors, and $\overline V([2;0;2,0])$ has 4 composition factors.

We also have at least one example of a {\sl doubly} atypical
Kac-module that contains {\sl weakly primitive} vectors (see
Definition 3.3), a situation that cannot occur for typical or
singly atypical Kac-modules. The example is the following:
$G=sl(2/2)$ and $\La=[1;0;1]$ (so $V(\La)$ is the adjoint
module). Using the notation of Section~3, $X(\La)=U(G)v_{\La_-}$
is a simple submodule of $\VK$. Using the basis $E_{ij}$
described in Section~2, let $v$ be the following vector of $\VK$:
$$
\eqalign{
v=\bigl(&E_{31}E_{32}E_{41}E_{43}+E_{31}E_{32}E_{42}E_{21}E_{43}\cr
&+E_{32}E_{41}E_{42}E_{21}+E_{31}E_{41}E_{42}\bigr)v_{\La}.\cr} \eqno(8.2)
$$
One can check that $v\notin X(\La)$. However $E_{14}v\not=0$ is
proportional to the highest weight vector of $X(\La)$; in fact
$\{0\}\not=N^+v\subseteq X(\La)$, showing that $v$ is
a weakly primitive vector in $\VK$.
\vskip 2mm
\noindent 4. Despite the difficulties for {\sl multiply atypical}
modules, we have recently given a conjecture $[{\underline{17}}]$ for the character of
all simple $G$ modules with integral dominant highest weight
$\La$, and we shall briefly describe this conjecture here. 
Formula (8.1) can be re-expressed as an infinite alternating sum
of $\ch_K(\mu)$-terms, just as in (7.2). Indeed, if $\La$ is
atypical of degree $d$ with respect to $\be_1,\ldots,\be_d$, then
one defines the {\sl cone} ${\it C}_\La$ with vertex at $\La$ as
the set of lattice points
$$
{\it C}_\La=\{\La-\sum_{i=1}^d k_i\be_i\,|\,k_i\in\Nat\,(i=1,\ldots,d)\}.
\eqno(8.3)
$$
The expansion becomes
$$
\ch_L(\La)=\sum_{\mu\in {\tenit C}_\La} (-1)^{|\La-\mu|}\ch_K(\mu), \eqno(8.4)
$$
where $(-1)^{|\La-\mu|}=(-1)^{k_1+\cdots+k_d}$ for
$\mu=\La-\sum_{i=1}^d k_i\be_i$.
The new formula is of type (8.4) with a restriction on the 
summation such that all terms $\ch_K(\mu)$
for which $\mu$ is a weight beyond certain
truncation planes in the weight space are excluded. These truncation
planes are uniquely determined, for each $\La$, as symmetry
planes $p_{ij}$ under the {\sl dot action} of elements
$w_{ij}$ ($1\leq i<j\leq d$) of the Weyl group $W$, where $w_{ij}$  
is the unique element such that
$w_{ij}(\be_i)=\be_j$ and such that $w_{ij}=1$ when restricted to
the subspace of $H^*$ orthogonal to $\be_i$ and $\be_j$. The
hyperplane $p_{ij}$ divides the weight space $H^*$ into two.
We denote by $H^*_{ij}$ the open half-space of $H^*$ containing
$\La$. The {\sl truncated cone} is defined to be
$$
{\it C}^+_\La={\it
C}_\La{\textstyle\bigcap}\Bigl(\bigcap_{{\tenrm critical }(i,j)} H^*_{ij}\Bigr),
\eqno(8.5)
$$
where the intersection is taken only with those $H^*_{ij}$ for
which $(i,j)$ is {\sl critical}, and the new formula becomes
$$
\ch_T(\La)=\sum_{\mu\in {\tenit C}^+_\La} (-1)^{|\La-\mu|}\ch_K(\mu). \eqno(8.6)
$$
The notion of criticality is
defined elsewhere $[\underline{17}]$, and we shall content ourselves
by describing it merely
for doubly atypical weights. Let $\La$ be
doubly atypical of type $\be_1$ and $\be_2$, with
$\be_1>\be_2$. Then $(1,2)$ is critical if and only if the
weights in the finite set
$H^*_{12}\cap \{\La-t\be_1|\,t=1,2,3,\ldots\}$ are all vanishing,
or equivalently, those in
$H^*_{12}\cap \{\La+t\be_2|\,t=1,2,3,\ldots\}$ are all vanishing.
If $\La$ is not critical, no truncations occur and our
conjectured character formula (8.6) coincides with (8.1). For more
details concerning this conjecture and some arguments in its favour,
we refer to $[{\underline{17}}]$.
\vskip 2cm
\section {Acknowledgements}

We would like to thank S.~Donkin (Queen Mary College, London) for
stimulating discussions. NATO (Belgium), the Royal Society
European Exchange Programme and SERC (U.K.) are acknowledged for
Research Fellowships, and CNRS (France) for supporting some of us
on a research visit to Paris.
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