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% Character formulae for the Lie superalgebra C(n)
% Joris Van der Jeugt
% Commun. Algebra 19 (1991) 199-222.
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\centerline{\titlefont Character formulae for the Lie
superalgebra C(n)}
\vskip 2cm
\centerline{by}
\centerline{Joris Van der Jeugt\footnote*{Aangesteld Navorser
N.F.W.O. (National Fund for Scientific Research of Belgium)}}
\vskip 1cm
\noindent Laboratorium voor Numerieke Wiskunde en Informatica,
Rijksuniversiteit Gent, Krijgslaan 281--S9, 9000~Gent, BELGIUM

and

\noindent Faculty of Mathematical Studies, University of
Southampton, Southampton SO9 5NH, U.K.

\vskip 3cm
\noindent {\bf Abstract.}

\noindent {\sl Using a recent result on the structure of the
maximal submodule of the Kac-module for type~I Lie superalgebras,
we derive two character formulae for all finite-dimensional 
simple modules (i.e.~all irreducible representations) of the Lie
superalgebra $C(n)=osp(2,2n-2)$. This is the first time a
character formula is proven for a series of simple Lie
superalgebras covering both the typical and atypical modules.}
\vfill

\section{1. Introduction}

Lie superalgebras and their representations play an important
role in the understanding and exploitation of supersymmetry in
physical systems in the areas of particle physics, nuclear
physics and condensed matter physics. An early review of their
use was given by Corwin {\sl et al} (1975); a comprehensive
review with the emphasis on their applications in these fields is
given in a paper by Bars (1985).

A complete classification of the finite-dimensional simple Lie
superalgebras over $\C$ has been given by Kac (1975, 1977a) and by
Scheunert {\sl et al} (1976, 1979). Kac showed that the basic
classical Lie superalgebras are closely analogous to the
finite-dimensional simple Lie algebras over $\C$. In particular
they can also be constructed from a (super) Cartan matrix or,
equivalently, from a Kac-Dynkin diagram (Kac 1977a, Van de Leur
1986). 

In his seminal paper on the representations of the basic
classical Lie superalgebras (Kac 1977b), Kac proved that all
inequivalent finite-dimensional irreducible representations (or,
in the mathematicians terminology, all {\sl simple modules}) may
be labelled by means of Kac-Dynkin labels which serve to specify
the highest weight, $\La$, of the corresponding simple module
$V(\La)$. He showed that the simple modules of a basic classical
Lie superalgebra $G=G_{\bar 0}\oplus G_{\bar 1}$ fall into two
classes, which were called {\sl typical} and {\sl atypical}. If
the even, resp.~odd, positive roots of $G$ are denoted by
$\De_0^+$, resp.~$\De_1^+$, and ${\overline\De}_1^+=\{\be\in\De_1^+|
2\be\notin\De_0^+\}$, then the highest weight $\La$ or the
module $V(\La)$ is said to be typical if and only if
$$
\langle\La+\rh\mid\be\rangle\not=0,\quad\forall\be\in{\overline\De}_1^+,
 \eqno(1.1)
$$ 
where $\langle\;\mid\;\rangle$ is a non-degenerate bilinear form,
and $\rh=\rh_0-\rh_1$, with $\rh_0$ and $\rh_1$ half the sum of
the even and odd positive roots, respectively, of $G$.
Conversely, if
$$
\langle\La+\rh\mid\be\rangle =0,\quad\hbox{for any }\be\in{\overline\De}_1^+,
 \eqno(1.2)
$$ 
then $\La$ and the module $V(\La)$ are said to be atypical.

The weight structure of a $G$ module is completely determined by
its character. For typical modules, Kac gave a character formula
for all basic classical Lie superalgebras (Kac 1977b). For
atypical modules no general character formula has been given so
far, although many partial results have been obtained. The Lie
superalgebra that has been studied most extensively is $sl(m/n)$, or
$A(m-1,n-1)$ in Kac's notation. Some progress has been made there
in understanding the properties of atypical modules through the
introduction of supertableaux methods (Balantekin 1984, Berele
and Regev 1987, Cummins and King 1987, Dondi and Jarvis 1981,
Hurni 1987), and the use of shift operators and weight space
techniques (Hurni and Morel 1983, Palev 1986, Van der Jeugt
1987), but to date these methods have not provided character
formulae for all atypical modules of $sl(m/n)$. Bernstein and
Leites (1980, Leites 1980) published a generalisation of Kac's
character formula, but it was soon afterwards observed
(Thierry-Mieg 1983) that their
formula was not correct for all atypical modules (for example, in
general it failed to give the correct character for the identity
representation) although it seemed to work for many particular
cases. Realizing the failure of the Bernstein--Leites formula,
Hughes and King (1987) conjectured another character formula for
$sl(m/n)$, and yet another formula is found in the work of 
Serganova and Serge'ev (1987). In a recent paper, it is shown
that none of the formulae published so far covers all atypical
modules of $sl(m/n)$ (Hughes {\sl et al} 1989a). However, an
important step forward was recently obtained by Hughes {\sl et
al} (1989b) in the case of {\sl singly atypical} modules of
Type~I Lie superalgebras (i.e.~for the series $A(m,n)$ and
$C(n)$). The module $V(\La)$ or the weight $\La$ is called singly
atypical if the condition (1.2) is satisfied for just one
$\be\in\De_1^+$. In that paper, a unique characterisation  was
given of the maximal proper submodule of an indecomposable module
introduced by Kac, the so-called {\sl Kac-module} $\VK$, when
$\La$ is singly atypical. Using this and various combinatorial
properties of the so-called {\sl atypicality matrix}, a character
formula for all singly atypical simple modules of $sl(m/n)$ was
deduced (Hughes {\sl et al} 1989b).

In the present paper we use the theorem proved by Hughes {\sl et
al} (1989b) for singly atypical modules of type~I Lie
superalgebras (Theorem 3.8 in this paper) and apply it to the
case of $G=C(n)$. The series $C(n)$ contains only typical or
singly atypical modules, and hence our analysis covers {\sl all}
simple modules of $C(n)$. In Section~2, we introduce the Lie
superalgebra $C(n)$, and fix some notation. In Section~3, we give
the definition and some properties of the Kac-module, including
the theorem on the maximal proper submodule of a singly atypical
Kac-module. The particular case of a $C(n)$ module which is
singly atypical of type $\al_1$ (where $\al_1$ is the unique odd
simple root of $C(n)$) is considered in Section~4, and it turns
out to be rather easy to obtain a character formula in this
situation. Then we study some properties of a matrix consisting
of the numbers $\langle\La+\rh\mid\be\rangle$, with
$\be\in\De_1^+$; this is the atypicality matrix (Section~5).
Using these properties, we obtain a character formula for all
atypical modules of $C(n)$ in Section~6 (Theorem~6.2 or
Corollary~6.3). Then, exploiting Weyl symmetries of the character
formula obtained, we prove that it is equivalent to yet another
character formula (Theorem~6.6) which is superior to the previous
one from the computational point of view. Finally, some examples
are given to illustrate our results.

\section{2. The Lie superalgebra C(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 $\forall\al,\be\in\Zah_2$ (Kac 1977a, Scheunert 1979)
$$
\eqalign{
	[a,b]&\in G_{\al+\be},\cr
	[a,b]&=-(-1)^{\al\be}[b,a],\cr
	[a,[b,c]]&=[[a,b],c]+(-1)^{\al\be}[b,[a,c]].\cr}\eqno(2.1)
$$
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:
$$
gl(m/n)=\left\lbrace x=\left(\matrix{A&B\cr C&D\cr}\right) \mid
 A\in M_{m\times m}, B\in M_{m\times n}, C\in M_{n\times m}, D\in
M_{n\times n} \right\rbrace , \eqno(2.2)
$$
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
$$
[a,b]=ab-(-1)^{\al\be}ba,\qquad\forall a\in G_\al\hbox{ and }
\forall b\in G_\be.  \eqno(2.3)
$$
One defines the {\sl supertrace} $\hbox{str}(x)$ and the {\sl
supertranspose} $x^T$ of $x=\bigl({A\atop C}{B\atop D}\bigr)$ as 
(Kac 1977a, Scheunert 1979)
$$
\eqalign{
\hbox{str}(x)&=\hbox{tr}(A)-\hbox{tr}(D),\cr
x^T&=\left(\matrix{A^t&-C^t\cr B^t&D^t\cr}\right),\cr}\eqno(2.4)
$$
where {\sl tr} is the ordinary trace and $A^t$ is the transpose of $A$.
Using the supertrace and the supertranspose, one can define
various subalgebras of $gl(m/n)$. The {\sl special linear Lie
superalgebra} $sl(m/n)$ is:
$$
sl(m/n)=\lbrace x\in gl(m/n)\mid \hbox{str}(x)=0\rbrace. \eqno(2.5)
$$
In Kac's notation, $A(m-1,n-1)=sl(m/n)$ for $m\not=n$.
If $m=2p$ and $n=2q$ are both even, one can define the following 
subalgebra of $gl(m/n)$ (Kac 1977a, Scheunert 1979):
$$
\eqalign{
\hbox{osp}(2p,2q)=\Bigl\lbrace x&=\left(\matrix{A&B\cr
C&D\cr}\right)\in sl(2p/2q) \mid
x_\al^TJ+(-1)^\al J x_\al=0,\cr
&\forall x_\al\in sl(2p/2q)_\al\;(\al=\bar 0,\bar 1)\Bigr\rbrace,\cr}
 \eqno(2.6)
$$
where
$$
J=\left(\matrix{0&I_m& & \cr
                I_m&0& & \cr
                 & &0&I_n\cr
                 & &-I_n&0\cr}\right) \eqno(2.7)
$$
and $I_r$ is the $r\times r$ identity matrix. In the notation of
Kac, $D(p,q)=\hbox{osp}(2p,2q)$ for $p\not=1$ and
$C(q+1)=\hbox{osp}(2,2q)$. In what follows, we put $G=C(n)$
($n>1$). Kac (1977a)
showed that $C(n)$ is a {\sl simple} Lie superalgebra of {\sl
type~I}. Note that $C(n)_{\bar 0}=\hbox{so}(2)\oplus\hbox{sp}(2n-2)
\cong\C\oplus\hbox{sp}(2n-2)$, where $\hbox{sp}(2n-2)$ is the
simple symplectic Lie algebra in $2n-2$ dimensions,
is a reductive Lie algebra, the
simple modules of which are well known.

A Cartan subalgebra $H$ of $G$ has dimension $n$ and is the space of
diagonal matrices in (2.6). The dual space $H^*$ is described in
the basis of forms $\ep$ and $\de_j$ ($j=1,\ldots,n-1$), where
$\ep\colon x\rightarrow A_{1,1}$ and $\de_j\colon x\rightarrow
D_{j,j}$ for $x=\bigl({A\atop C}{B\atop D}\bigr)$.
Using the notation $E_{ij}$ for the $2n\times 2n$ matrix with
entry 1 at position $(i,j)$ and 0 elsewhere, the roots and
corresponding root vectors of $C(n)$ are given by
$$
\eqalign{\hbox{even }&\hbox{roots:}\cr
\de_j-\de_k\;&\leftrightarrow\;
 E_{2+j,2+k}-E_{n+1+k,n+1+j}\;(1\leq j,k\leq n-1;j\not=k);\cr
\de_j+\de_k\;&\leftrightarrow\;
 E_{2+j,n+1+k}+E_{2+k,n+1+j}\;(1\leq j,k\leq n-1)\cr
-\de_j-\de_k\;&\leftrightarrow\;
 E_{n+1+j,2+k} +E_{n+1+k,2+j}\;(1\leq j,k\leq n-1)\cr}
 \eqno(2.8)
$$ 
$$
\eqalign{\hbox{ odd }&\hbox{roots:}\cr
\ep-\de_k\;&\leftrightarrow\;E_{1,2+k}-E_{n+1+k,2}\cr
\ep+\de_k\;&\leftrightarrow\;E_{1,n+1+k}+E_{2+k,2}\cr
-\ep-\de_k\;&\leftrightarrow\;E_{2,2+k}-E_{n+1+k,1}\cr
-\ep+\de_k\;&\leftrightarrow\;E_{2,n+1+k}+E_{k+2,1}\;(1\leq k\leq n-1).\cr}
\eqno(2.9)
$$
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 (2.8)--(2.9) corresponding to the root $\al\in\De$. $G$
has the root space decomposition
$$
G=H\oplus\left(\bigoplus_{\al\in\De}\C e(\al)\right). \eqno(2.10)
$$
A set of simple roots $\{\al_1,\al_2,\ldots,\al_n\}$ 
of $\De$ may be chosen as follows (Kac 1977b):
$$
\al_1=\ep-\de_1,\;\al_i=\de_{i-1}-\de_i\;(2\leq i\leq n-1),\;
\al_n=2\de_{n-1}. \eqno(2.11)
$$
Note that $\al_1$ is the only odd simple root. Let $e_i$ (resp.
$f_i$) be root vectors for $\al_i$ (resp. $-\al_i$), and $h_i=[e_i,f_i]$:
$$
\eqalign{
h_1&=E_{1,1}-E_{2,2}+E_{3,3}-E_{n+2,n+2},\cr
h_i&=E_{i+1,i+1}-E_{i+2,i+2}-E_{n+i,n+i}+E_{n+i+1,n+i+1},\;(2\leq
i\leq n-1)\cr
h_n&=E_{n+1,n+1}-E_{2n,2n}.\cr}\eqno(2.12)
$$
These elements form a basis for $H$. 
With the choice of a
simple root system (2.11), the elements of $H^*$ are partially ordered by
$$
\la,\mu\in H^*\;\colon\quad\la\geq\mu\quad\Leftrightarrow\quad\la-\mu
=\sum_{i=1}^{n}k_i\al_i \hbox{ with }k_i\geq 0.
\eqno(2.13)
$$
This partial ordering $\geq$ can be extended to a total ordering
$\succeq$ compatible with $\geq$, i.e.
$$
\la\geq\mu\quad\Rightarrow\quad\la\succeq\mu;
\eqno(2.14)
$$
the most natural example of such a total ordering is
lexicographical ordering with respect to the simple roots.
The even and odd {\sl positive} roots of $C(n)$ are given by
$$
\eqalign{
\De_0^+&=\{\de_i-\de_j\;(i<j),\;\de_i+\de_j\},\cr
\De_1^+&=\{\ep\pm\de_j\}.\cr} \eqno(2.15)
$$
Note that $G=C(n)$ admits a $\Zah$-grading,
$$
G=G_{-1}\oplus G_0\oplus G_{+1}, \eqno(2.16)
$$
where $G_{\pm 1}=\hbox{span}\{e(\al)\mid\al\in\De_1^\pm\}$ and
$G_0 =G_{\bar 0}$. This $\Zah$-grading is consistent with the
$\Zah_2$-grading in the sense that $G_{\bar
0}=\bigoplus_{k\in\Zah}G_{2k}$ and $G_{\bar
1}=\bigoplus_{k\in\Zah}G_{2k+1}$. 

The invariant non-degenerate inner product on $G$ is given by
$\langle x|y\rangle =2\,\hbox{str}(xy)$. 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
$$
\langle\ep|\ep\rangle=1,\;\langle\ep|\de_j\rangle=0,\;
\langle\de_i|\de_j\rangle=-\de_{ij},
 \eqno(2.17)
$$
where $\de_{ij}$ is the Kronecker-$\de$. An element $\La\in H^*$
with $\La=\la_0\ep+\sum_j\la_j\de_j$ can be written in
terms of its {\sl components} in the $\ep\de$-basis as
$\La=(\la_0|\la_1\la_2\ldots\la_{n-1})$,
or in terms of its {\sl Dynkin
labels} $\La=[a_1;a_2,\ldots,a_n]$
where $a_i=\La(h_i)$ and $h_i$ is given in (2.6). Explicitly, one
sees that
$$
a_1=\la_0+\la_1,\;\;a_i=\la_{i-1}-\la_i\;(2\leq i\leq n-1),\;\;a_n=\la_{n-1}.
\eqno(2.18)
$$

The {\sl Weyl group} $W$ of $G$ is defined to be the Weyl group of
$G_{\bar 0}$. Hence $W$ is the Weyl group of the symplectic Lie
algebra $\hbox{sp}(2n-2)$. As usual, $\vep(w)$ denotes the
signature of $w$. We denote by $w_0$ the Coxeter element of $W$, i.e.
the element of maximal length in $W$. Explicitly, $w_0(\la_0\mid\la_1,\ldots,
\la_{n-1})=(\la_0\mid -\la_1,\ldots,-\la_{n-1})$.
The {\sl dot action} is defined as usual:
$$
w\cdot\La=w(\La+\rh)-\rh,\hbox{ where }\rh=\rh_0-\rh_1 \eqno(2.19)
$$
with
$$
\rh_0={1\over 2}\sum_{\al\in\De_0^+}\al\quad\hbox{ and }\quad
\rh_1={1\over 2}\sum_{\be\in\De_1^+}\be . \eqno(2.20)
$$
Explicitly,
$$
\rh_0=\sum_{i=1}^{n-1}(n-i)\de_i
\quad\hbox{ and }\quad
\rh_1=(n-1)\ep.
\eqno(2.21)
$$
Note that $w\rh_1=\rh_1$ and hence
$$
w\cdot\La=w(\La+\rh)-\rh=w(\La+\rh_0)-\rh_0. \eqno(2.22)
$$

We set
$$
\eqalign{
N_0^{\pm}&=\hbox{span}\{e(\al)|\al\in\De_0^{\pm}\},\cr
N_1^{\pm}&=\hbox{span}\{e(\be)|\be\in\De_1^{\pm}\},\cr
N^{\pm}&=N_0^{\pm}\oplus N_1^{\pm}.\cr}
\eqno(2.23)
$$
Note that $N_1^{\pm 1}=G_{\pm 1}$ and, besides the decomposition
(2.4), one has
$$
\eqalign{
G_{\bar 0}&=N_0^-\oplus H\oplus N_0^+,\cr	
G&=N^-\oplus H\oplus N^+\cr}
\eqno(2.24)
$$
Let $U(G)$ be the universal enveloping algebra of $G$, and
$U(G')$ the enveloping algebra of one of the subalgebras $G'=H,N^+,N^-,
N_0^+,N_0^-,N_1^+,N_1^-,G_0,G_{-1}$ or $G_{+1}$. The
Poincar\'e-Birkhoff-Witt theorem for Lie algebras can be extended
to the case of Lie superalgebras (Scheunert 1979):

\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
$$
(x_1)^{k_1}\ldots (x_M)^{k_M}y_{i_1}\ldots y_{i_s},
\hbox{ where }k_i\geq 0\hbox{ and }1\leq i_1<\ldots<i_s\leq N,\eqno(2.25)
$$
form a basis of $U(G)$.

It is clear that a similar theorem is true for $U(G')$ with $G'$
one of the subalgebras given previously.

\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.
Every highest weight module $V$
contains a proper maximal submodule $M$, and then the quotient
module $V/M$ is a simple module. We shall now describe a
particular class of highest weight modules for $G$, introduced by
Kac (1977b).

A weight $\La\in H^*$ is called {\sl dominant} if all
$a_i=\La(h_i)\geq 0$ for $i\not=1$, {\sl integral} if all
$a_i\in\Zah$ for $i\not=1$, and {\sl integral dominant} if all
$a_i\in\Nat$ for $i\not=1$. From the theory of
reductive Lie algebras it follows that for every integral
dominant 
weight $\La$ there exists a unique
finite dimensional simple $G_{\bar 0}$ module $V_0(\La)$
with highest weight $\La$ and highest weight vector $v_\La$.
This can be extended to a $G_0\oplus G_{+1}$ module by
putting $G_{+1}V_0(\La)=0$.

\proclaim{Definition 3.2 {\rm(Kac 1977b)}}. For an  
integral dominant $\La\in H^*$, the
Kac-module $\VK$ is the induced module
$$
\VK={\rm Ind}_{G_0\oplus G_{+1}}^G\;V_0(\La)=
U(G)\otimes_{G_0\oplus G_{+1}}V_0(\La) \cong
U(G_{-1})\otimes V_0(\La).
\eqno(3.4)
$$

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
$2n-2$, thus $\hbox{dim}\bigl(\wedge(G_{-1})\bigr)=2^{2n-2}$, and
hence $\VK$ is a finite dimensional $G$-module of dimension 
$2^{2n-2}\hbox{dim}\bigl(V_0(\La)\bigr)$. It follows from the
definition that $\VK$ is
an indecomposable $G$ highest weight module. Unfortunately $\VK$ is not
always a simple $G$ module. Since $\VK$ is a $G$ highest weight
module, it contains a unique proper maximal submodule $M(\La)$.
The quotient module
$$
V(\La)=\VK/M(\La) \eqno(3.5)
$$
is a finite dimensional simple $G$ module. Kac proved the
following theorem:

\proclaim{Theorem 3.3 {\rm(Kac 1977b)}}. Every finite dimensional simple
$G$ module is isomorphic to a module of type (3.5), 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 $U(G)$:
$$
T_\pm=\prod_{\be\in\De_1^+}e(\pm\be), \eqno(3.6)
$$
where the $\be$'s in the product (3.6) (and in all subsequent
products of $e(\be)$'s) appear in the chosen
lexicographical ordering (note that a different ordering can lead
only to a sign change). One can verify that
$$
[e(\al),T_\pm]=0,\qquad\forall\al\in\De_0. \eqno(3.7)
$$
In $\VK$, let
$$
v_{\La_-}=T_-v_{\La},\hbox{ with }\La_-=\La-2\rh_1. \eqno(3.8)
$$
Note that $\La_-$ is also integral dominant, since $\La_-=[a_1-2n+2;a_2,
\ldots,a_n]$ if $a_i$ are the Dynkin labels of $\La$.
Since $G_{\bar 0}\subset G$, $\VK$ is also a $G_{\bar 0}$ module.
It follows from (3.7) that the $G_{\bar 0}$ module $\VK$ contains
$T_-V_0(\La)$ as a simple $G_{\bar 0}$ submodule. Clearly, $\VK$
has a unique (up to scalar multiplication) vector $v_-$ that is
annihilated by $N^-$. This vector has weight $w_0\La_-$, and both
$v_-$ and $v_{\La_-}$ belong to the simple $G_{\bar 0}$ module
$T_-V_0(\La)$. 
Since every non-zero submodule of $\VK$ contains
the vector $v_-$, it follows that
every non-zero $G$ submodule $X$ of the $G$ module
$\VK$ contains the $G_{\bar 0}$ module $T_-V_0(\La)$ as a subspace.
Therefore, we let
$$
X(\La)=U(G)v_{\La_-}. \eqno(3.9)
$$
Then $X(\La)$
is a {\sl simple} $G$ submodule of $\VK$.

\proclaim{Lemma 3.4}. $\VK$ is a simple $G$ module if and only if
$\,T_+T_-v_\La\not= 0$.

\noindent {\sl Proof.} If $T_+T_-v_\La=0$, then it follows from (3.9)
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 
$T_-v_\La\in M(\La)$, 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 3.5}. 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 (1977b).

\proclaim{Definition 3.6}. 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 $\be$.
We shall see that all atypical modules of $C(n)$ are singly atypical. 
The following theorem now follows from Lemmas 3.4 and 3.5:

\proclaim{Theorem 3.7}. The Kac-module $\VK$ is a simple $G$
module if and only if $\La$ is typical.

The following theorem was proven for all singly atypical
Kac-modules of type~I Lie superalgebras (Hughes {\sl et al} 1989b):

\proclaim {Theorem 3.8}. Let $\La$ be singly atypical. Then
$U(G)v_{\La_-}$ is the proper maximal submodule of $\VK$, i.e.
$$
M(\La)=X(\La). \eqno(3.12)
$$

The character ch$V$ of a $G$ module $V$ with weight space
decomposition (3.3) is defined as
$$
\cha V=\sum_{\la\in H^*} \hbox{dim}(V_\la)e^\la, \eqno(3.13)
$$
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
$$
L_0=\prod_{\al\in\De_0^+}\left(e^{\al/2}-e^{-\al/2}\right)\hbox{
and }
L_1=\prod_{\be\in\De_1^+}\left(e^{\be/2}+e^{-\be/2}\right). \eqno(3.14)
$$
From (3.4) it follows that the Kac-module has character
$$
\cha\VK=\prod_{\be\in\De_1^+}\left(1+e^{-\be}\right)\cha
V_0(\La), \eqno(3.15)
$$
where $\cha V_0(\La)$ is given by Weyl's character formula (Weyl 1926):
$$
\cha V_0(\La)=L_0^{-1}\sum_{w\in W}\vep(w)e^{w(\La+\rh_0)}. \eqno(3.16)
$$
Using the Weyl invariance of $\rh_1$, we have
$$
\prod_{\be\in\De_1^+}\left(1+e^{-\be}\right)=L_1e^{-\rh_1}=L_1e^{-w\rh_1},
 \quad\forall w\in W, \eqno(3.17)
$$
and hence we obtain Kac's character formula (Kac 1977b):
$$
\cha\VK={L_1\over L_0}\sum_{w\in W}\vep(w)e^{w(\La+\rh)}. \eqno(3.18)
$$
Due to the Weyl invariance of $\De_1^+$ and of $L_1$, (3.18) can
be rewritten in the form
$$
\cha\VK=L_0^{-1}\sum_{w\in W}\vep(w)w\left\lbrace e^{\La+\rh_0}
\prod_{\be\in\De_1^+}\left(1+e^{-\be}\right)\right\rbrace. \eqno(3.19)
$$
Using Theorem 3.7, (3.19) gives the character of all typical simple
modules of $G$. In this paper we shall
deduce a character formula for singly atypical simple $G$
modules, and hence for all simple $G$ modules.

Finally, let $\la\in H^*$ be integral. We define the ``formal characters'':
$$
\ch_K(\la)=L_0^{-1}\sum_{w\in W}\vep(w)w\left\lbrace e^{\la+\rh_0}
\prod_{\be\in\De_1^+}\left(1+e^{-\be}\right)\right\rbrace; \eqno(3.20)
$$
$$
\ch_W(\la)=L_0^{-1}\sum_{w\in W}\vep(w)e^{w(\la+\rh_0)}. \eqno(3.21)
$$
If $\la$ is  integral dominant these expressions 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:
$$
\ch_K(\la)=e^{-\rh_1}L_1\,\ch_W(\la);\eqno(3.22)
$$
$$
\ch_W(w\cdot\la)=\vep(w)\ch_W(\la),\qquad
\ch_K(w\cdot\la)=\vep(w)\ch_K(\la),\qquad \forall w\in W.\eqno(3.23)
$$

\section{4. A special case}

In this section we shall consider the special case of a singly atypical $\La$
of type $\al_1$, where $\al_1=\ep-\de_1$ 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 4.1}. Let $\La$ be atypical of type $\al_1$ and
$v=e(-\al_1)v_\La$. Then $N^+v=0$ in $\VK$.

\noindent {\sl Proof.} For $\al\in\De_0^+$, we have
$e(\al)v=[e(\al),e(-\al_1)]v_\La+e(-\al_1)e(\al)v_\La$. But
$N^+v_\La=0$, and $[e(\al),e(-\al_1)]=0$ for all $\al\in\De_0^+$, hence
$$
e(\al)v=0,\qquad\forall\al\in\De_0^+.\eqno(4.1)
$$
Then, using $e(+\al_1)v_\La=0$, one finds
$$
e(+\al_1)v=[e(+\al_1),e(-\al_1)]v_\La=h_1v_\La=\La(h_1)v_\la
=\langle\La\mid\al_1\rangle v_\La
=\langle\La+\rh\mid\al_1\rangle v_\La=0, \eqno(4.2)
$$
since $\langle\rh\mid\al_1\rangle=0$ and $\La$ is atypical of
type $\al_1$. Finally, let $\be\in\De_1^+$ with
$\be\not=\al_1$. Then, $\be=\ep-\de_i$ ($i\not=1$) or $\be=\ep+\de_i$.
Using $e(\be)v_\La=0$ and the (anti)commutation relations
(2.3) for the root vectors (2.7), one obtains
$$
\eqalign{
e(\ep-\de_i)v&=[e(\ep-\de_i),e(-\ep+\de_1)]v_\La=e(\de_1-\de_i)v_\La,
\;(i\not=1),\cr
e(\ep+\de_i)v&=[e(\ep+\de_i),e(-\ep+\de_1)]v_\La=e(\de_1+\de_i)v_\La.\cr}
\eqno(4.3)
$$
But (4.1) implies that the right hand sides of (4.3) vanish.
Hence (4.1)--(4.3) shows that $e(\al)v=0$, for all
$\al\in\De^+$, or $N^+v=0$.\mybox

In the case of Lemma 4.1, $U(G)v$ is a proper submodule of $\VK$,
hence $X(\La)\subseteq U(G)v\subseteq M(\La)$. But if $\La$ is
singly atypical of type $\al_1$, Theorem 3.8 implies
$$
M(\La)=X(\La)=U(G)v, \eqno(4.4)
$$
where $v=e(-\al_1)v_\La$ is a vector of weight $\La-\al_1$. Since
$U(G)v$ is a highest weight module with highest weight vector
$v$, and since $X(\La)$ is simple, it follows that $X(\La)$ is
isomorphic to $V(\La-\al_1)$. Since in general $\cha V=\cha
(V/M)+\cha M$, for any submodule $M$ of a module $V$, applying
this to $V=\VK$ and $M=M(\La)=X(\La)\cong V(\La-\al_1)$ leads to

\proclaim {Corollary 4.2}. Let $\La$ be singly atypical of type
$\al_1$. Then 
$$
\cha\VK=\cha V(\La)+\cha V(\La-\al_1). \eqno(4.5)
$$

Note that if $\La$ is dominant and singly atypical of type $\al_1$, also
$\La-\al_1$ is dominant and singly atypical of type $\al_1$. Now
we can prove a character formula for this particular case.

\proclaim {Theorem 4.3}. Let $\La$ be singly atypical of type
$\al_1$. Then
$$
\cha V(\La)=L_0^{-1}\sum_{w\in W}\vep(w)w\Biggl\lbrace
e^{\La+\rh_0}\prod_{{\scriptstyle
\be\in\De_1^+}\atop{\scriptstyle \be\not=\al_1}}(1+e^{-\be})\Biggr\rbrace.
\eqno(4.6)
$$

\noindent {\sl Proof.} Using (4.5) as a recursion relation, we find
$$
\eqalignno{
\cha V(\La)&=\cha\VK -
\cha V(\La-\al_1)=\cha\VK-\left(\cha\overline V(\La-\al_1)-
\cha V(\La-2\al_1)\right)\cr
 &=\cha\VK-\cha\overline V(\La-\al_1)+
 \left(\cha\overline V(\La-2\al_1)-\cha V(\La-3\al_1)\right)=\ldots\cr
 &=\cha\VK-\cha\overline V(\La-\al_1)+
 \cha\overline V(\La-2\al_1)-\cha\overline V(\La-3\al_1)+\ldots&(4.7)\cr}
$$
which becomes a formal infinite series expression since (4.5)
can be applied for every $\La-k\al_1$ ($k\in\Nat$). Then we can
substitute (3.19) for the characters of the Kac-modules appearing
in (4.7), and sum over the formal series:
$$
\eqalignno{
\cha V(\La)&=L_0^{-1}\sum_{w\in W}\vep(w)w\left\lbrace
\prod_{\be\in\De_1^+}(1+e^{-\be})e^{\rh_0}\left(
e^{\La}-e^{\La-\al_1}+e^{\La-2\al_1}-e^{\La-3\al_1}+\ldots\right)\right\rbrace\cr
 &=L_0^{-1}\sum_{w\in W}\vep(w)w\left\lbrace
e^{\La+\rh_0}\prod_{\be\in\De_1^+}(1+e^{-\be})e^{\La+\rh_0}
\left(1+e^{-\al_1}\right)^{-1}\right\rbrace.&(4.8)\cr}
$$
This proves the theorem.\mybox

Let $\la$ be an integral weight, and $\ga\in\De_1^+$. We define
the formal expression
$$
\ch_\ga(\la)=L_0^{-1}\sum_{w\in W}\vep(w)w\Biggl\lbrace
e^{\la+\rh_0}\prod_{{\scriptstyle
\be\in\De_1^+}\atop{\scriptstyle \be\not=\ga}}(1+e^{-\be})\Biggr\rbrace.
\eqno(4.9)
$$
Theorem 4.3 shows that if $\La$ is singly atypical of type
$\al_1$, then $\cha V(\La)=\ch_{\al_1}(\La)$. We shall show in
Section~6 that if $\La$ is singly atypical of type $\ga$, then
$\cha V(\La)=\ch_\ga(\La)$. Note that the formal expression
(4.9) satisfies the following property:
$$
\ch_{w(\ga)}(w\cdot\la)=\vep(w)\ch_\ga(\la),\qquad\forall w\in W.\eqno(4.10)
$$
Finally, one sees from (3.20) that
$$
\ch_K(\la)=\ch_\ga(\la)+\ch_\ga(\la-\ga),\qquad\forall\ga\in\De_1^+.\eqno(4.11)
$$
{\bf Remark 4.4.} 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 (3.19) and (4.11)
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{5. The atypicality matrix}

The atypicality of an integral dominant weight $\La$ is
determined by the value of the $2n-2$ numbers $\langle\La+\rh\mid
\be\rangle$ with $\be\in\De_1^+$. In this section we shall study
some properties of a matrix consisting of these $2n-2$ numbers
(Hughes {\sl et al} 1989b).

\proclaim {Definition 5.1}. Let $\La=(\la_0\mid\la_1\ldots\la_{n-1})\in H^*$. 
The atypicality matrix $A(\La)$ is the $1\times(2n-2)$ complex 
matrix consisting of the numbers
$$
\eqalignno{
A(\La)_i&=\langle\La+\rh\mid\ep-\de_i\rangle=
 \la_0+\la_i-i+1,\quad(i=1,\ldots,n-1),&(5.1a)\cr
A(\La)_{2n-i-1}&=\langle\La+\rh\mid\ep+\de_i\rangle=
 \la_0-\la_i+i-2n+1,\quad(i=1,\ldots,n-1).&(5.1b)\cr}
$$

In other words,
$$
A(\La)=\Bigl(\langle\La+\rh|\ep-\de_1\rangle\cdots
 \langle\La+\rh|\ep-\de_{n-1}\rangle \parallel
 \langle\La+\rh|\ep+\de_{n-1}\rangle\cdots 
 \langle\La+\rh|\ep+\de_1\rangle\Bigr).\eqno(5.2)
$$
The symbol $\parallel$ is used to denote the ``middle'' of
$A(\La)$. The submatrix $A(\La)_i$ with $i=1,\ldots,n-1$ is
called the {\sl left part} of $A(\La)$, and $A(\La)_i$ with
$i=n,\ldots,2n-2$ is called the {\sl right part} of $A(\La)$.
Using (2.18), the atypicality matrix can be expressed in terms of
the Dynkin labels $a_i$ of $\La$:
$$
\eqalignno{
&A(\La)_1=a_1, \qquad A(\La)_n=A(\La)_{n-1}-2a_n-2,&(5.3a)\cr
&A(\La)_i-A(\La)_{i-1}=A(\La)_{2n-i}-A(\La)_{2n-i-1}=-a_i-1,
 \quad (i=2,3,\ldots,n-1).&(5.3b)\cr}
$$
In this form it is easy to see that $A(\La)_{j+1}<A(\La)_j$ for
all $j\in\{1,\ldots,2n-3\}$ if $\La$ is dominant. Hence there is
at most one $\be\in\De_1^+$ for which an integral dominant $\La$
is atypical, i.e.~all atypical weights $\La$ are {\sl singly
atypical}. The following properties follow from (5.1) and (5.3):

\proclaim {Lemma 5.2}.
\item{a.} Every $1\times(2n-2)$ matrix $A$ satisfying
$$
A_i-A_{i-1}=A_{2n-i}-A_{2n-i-1},\qquad(i=2,3,\ldots,n-1)\eqno(5.4)
$$
is the atypicality matrix of a unique $\La\in H^*$.
\item{b.} $\La$ is dominant if and only if $A(\La)$ satisfies
$$
\eqalignno{
&A(\La)_n-A(\La)_{n-1}+2\leq 0,\quad\hbox{ and}&(5.5a)\cr
&A(\La)_i-A(\La)_{i-1}+1=A(\La)_{2n-i}-A(\La)_{2n-i-1}+1\leq 0,
 \quad (i=2,3,\ldots,n-1);&(5.5b)\cr}
$$
moreover, $\La$ is integral dominant if and only if the
expressions in the l.h.s.~of (5.5) are all integers.

\proclaim {Lemma 5.3}. Let $\la$ be an integral element of $H^*$.
Then the following statements are equivalent:
$$
\eqalign{
&(1)\;\ch_W(\la)=0;\cr
&(2)\;\ch_K(\la)=0;\cr
&(3)\;\exists w\in W\hbox{ with }\vep(w)=-1\hbox{ such that }w\cdot\la=\la;\cr
&(4)\;\forall w\in W,\,w\cdot\la\hbox{ is not dominant}.\cr}
$$

The equivalence of (1), (3) and (4) is a classical property of
the Weyl group of a semi-simple Lie algebra (Humphreys 1972). The equivalence of
(1) and (2) follows from (3.22).

\proclaim {Definition 5.4}. An integral element $\la\in H^*$ is
called vanishing if one of the statements (1)--(4) of Lemma~5.3
is satisfied. Otherwise, $\la$ is called non-vanishing.

\proclaim {Lemma 5.5}. Each of the following conditions is sufficient
for $\la$ to be vanishing:
\item{a)} the left part of $A(\la)$ has two equal elements;
\item{b)} the right part of $A(\la)$ has two equal elements;
\item{c)} $A(\la)_{n-1}=A(\la)_n$.

\noindent {\sl Proof.} If $A(\la)_i=A(\la)_j$ for $1\leq i<j\leq
n-1$, let $w=p_{ij}$, where $p_{ij}$ is the mapping interchanging
the components $\la_i$ and $\la_j$ in
$\La=(\la_0|\la_1,\ldots,\la_{n-1})$. Then, $w\in W$ with
$\vep(w)=-1$, and $A(w\cdot\la)=A(\la)$, so Lemma~5.2a implies
$w\cdot\la=\la$, so condition (3) in Lemma~5.3 is satisfied. From
(5.3b) it follows that b) is equivalent to a). Finally, if c) is
satisfied, then (5.3a) implies $a_n=-1$, or according to (2.18),
$\la_{n-1}=-1$. But then $w\cdot\la=\la$, with $w=r_{\de_{n-1}}$
the reflection with respect to the root $\de_{n-1}$. Again $w\in
W$ and $\vep(w)=-1$, so Lemma~5.3(3) is satisfied.\mybox

In the rest of this section, $\La$ is an integral dominant
weight. Note that if $\La$ is atypical, then (5.3) implies that
all entries in the atypicality matrix are integers.

\proclaim {Lemma 5.6}. Let $\La$ be atypical of type $\be$. Then
there exists a unique sequence of distinct elements $\be_0=\be>\be_1
>\cdots>\be_q$ from $\De_1^+$ such that the sequence of weights $\nu_0=\La,
\nu_1,\ldots,\nu_q,\nu_{q+1}$ with $\nu_{t+1}=\nu_t-\be_t$ satisfies
$$
\eqalignno{
&\langle\nu_t+\rh|\be_t\rangle=0,\quad 0\leq t\leq q;&(5.6a)\cr
&\nu_t\hbox{ is vanishing for }0<t\leq q;&(5.6b)\cr
&q(\La)=\nu_{q+1}\hbox{ is integral dominant and singly atypical
of type }\be_q;&(5.6c)\cr
&\exists w\in W\hbox{ such that }\nu_t=w\cdot (\La-t\be)\hbox{
with }\be_{t-1}=w(\be)\cr
&\hbox{ and }\vep(w)=(-1)^{t-1},\;0<t\leq q+1.&(5.6d)\cr}
$$

\noindent {\sl Proof.} Since $\La$ is atypical, $A(\La)$ has one
zero. First, consider the case where the zero appears in the left
part of $A(\La)$. Notice that the relation between $A(\la)$ and
$A(\la-(\ep-\de_i))$ is given by
\tenpoint
$$
\matrix{
A(\la)&=&\Bigl(A_1&\cdots&A_i&\cdots&A_{n-1}&\parallel&A_n&\cdots
 &A_{2n-1-i}&\cdots&A_{2n-2}\Bigr)\cr
A(\la-\ep+\de_i)&=&\Bigl(A_1-1&\cdots&A_i&\cdots&A_{n-1}-1&\parallel&A_n-1&\cdots
 &A_{2n-1-i}-2&\cdots&A_{2n-2}-1\Bigr)\cr} \eqno(5.7)
$$ 
\twelvepoint
Let $\be_0=\be=\ep-\de_i$. There are two possibilities: if
$A(\La)_{i-1}>A(\La)_i+1$ then $q=0$ and all statements of the
lemma are satisfied. If $A(\La)_{i-1}=A(\La)_i+1$, then
$A(\La-\be_0)$ has two zeros at positions $i-1$ and $i$; then
$\be_1=\ep-\de_{i-1}$. The sequence is now constructed by iteration:
if $A(\La)_{i-2}>A(\La)_{i-1}+1$ then $q=1$, and if
$A(\La)_{i-2}=A(\La)_{i-1}+1$ then $\be_2=\ep-\de_{i-2}$, etc.
From this construction the existence and uniqueness of the
sequence follows, and the matrices 
${\cal A}_0=A(\La)$, ${\cal A}_1=A(\La-\be_0)$,
${\cal A}_2=A(\La-\be_0-\be_1)$, $\ldots$ satisfy
$$
\eqalignno{
&{\cal A}_t\hbox{ has two zeros, at positions }i-t\hbox{ and
}i-t+1\hbox{ for }1\leq t\leq q;&(5.8a)\cr
&{\cal A}_{q+1}\hbox{ has one zero at position }i-q+1;&(5.8b)\cr
&{\cal A}_t\hbox{ is obtained from }A(\La-t\be)\hbox{ by
}t-1\hbox{ transpositions in the}\cr
&\hbox{ left part of }A(\La)\hbox{ (and with (5.3b) also in the
right part).}&(5.8c)\cr}
$$
Now (5.6a) follows from (5.8a), and (5.6b) follows from Lemma~5.5
and (5.8a). By constuction, ${\cal A}_{q+1}$ satisfies the
conditions of Lemma~5.2b, so using this and (5.8b) leads to (5.6c).
Finally, (5.8c) implies (5.6d).

Now consider the case where the zero of $A(\La)$ appears in the
right part of $A(\La)$. Then the counterpart of (5.7) is given by
\tenpoint
$$
\matrix{
A(\la)&=&\Bigl(A_1&\cdots&A_i&\cdots&A_{n-1}&\parallel&A_n&\cdots
 &A_{2n-1-i}&\cdots&A_{2n-2}\Bigr)\cr
A(\la-\ep-\de_i)&=&\Bigl(A_1-1&\cdots&A_i-2&\cdots&A_{n-1}-1&\parallel&A_n-1&\cdots
 &A_{2n-1-i}&\cdots&A_{2n-2}-1\Bigr)\cr} \eqno(5.9)
$$ 
\twelvepoint
Hence, if $a_n\not=0$, the analysis of the sequence is completely
analogous to the previous case, and the lemma follows. There is,
however, one situation that needs more attention. This is the
case with $\be_0=\be=\ep+\de_i$ and $a_{i+1}=a_{i+2}=\cdots=a_n=0$.
In this case $\be_1=\ep+\de_{i+1}$, $\ldots$,
$\be_{n-i-1}=\ep+\de_{n-1}$, and since $a_n=0$,
$\be_{n-i}=\ep-\de_{n-1}$. In other words, the positions of the
zeros in ${\cal A}_t$ appear in the left part of the atypicality
matrix as $t\geq n-i$. Making use of (5.9) and (5.7), one can
verify that in this case $\be_q=\ep-\de_i$. 
All the properties (5.8) are still true, but for (5.8c) one of
the transpositions is the interchanging of the $(n-1)^{th}$ and
$n^{th}$ element in the atypicality matrix if $t\geq n-i$; this
corresponds to the dot action of $r_{\de_{i-1}}$ discussed previously.
The lemma is then proven similarly.\mybox

\noindent {\bf Example.} The following three weights illustrate the
three distinct cases in the proof of Lemma~5.6. Let $G=C(5)$; the
sequence $\be_t$ of Lemma~5.6 is denoted by underlining the
corresponding elements $\langle\La+\rh|\be_t\rangle$ in the
atypicality matrix in (5.10):
$$
\eqalign{
\La=[2;0,0,0,0]\quad&A(\La)=\bigl(\underline 2,\underline 1,
 \underline 0,-1\parallel -3,-4,-5,-6\bigr),\cr	
\La=[10;0,0,1,0]\quad&A(\La)=\bigl(10,9,8,6\parallel 4,
 \underline 2,\underline 1,\underline 0\bigr),\cr
\La=[6;0,0,0,0]\quad&A(\La)=\bigl(6,5,\underline 4,\underline
3\parallel \underline 1,\underline 0,-1,-2\bigr).\cr} \eqno(5.10)
$$

\section{6. The character formulas}

Using the lemmas of Section~5, we are now able to prove a character
formula for $V(\La)$, where $\La$ is a singly atypical integral dominant
weight. The proofs given here are similar to the proofs given for
$sl(m/n)$ in a previous paper (Hughes {\sl et al} 1989b).

\proclaim {Lemma 6.1}. Let $\La$ be singly atypical of type $\ga$
and let $\be_0=\ga,\be_1,\ldots,\be_q=\ga'$ be the sequence of Lemma~5.6.
Let $q(\La)=\La-\be_0-\cdots-\be_q$ be the dominant weight defined 
in Lemma~5.6, which is singly atypical of type $\ga'$.
Then, using the notation (4.9):
$$
\ch_\ga(\La-\ga)=\ch_{\ga'}(q(\La)). \eqno(6.1)
$$

\noindent {\sl Proof.} As in the proof of Theorem 4.3, we can expand
$\ch_\ga(\La-\ga)$ in a series of $\ch_K(\la)$-terms:
$$
\ch_\ga(\La-\ga)=\ch_K(\La-\ga)-\ch_K(\La-2\ga)+\ch_K(\La-3\ga)-\cdots
+(-1)^q\ch_K(\La-(q+1)\ga)+\cdots\eqno(6.2)
$$
But for $0<t\leq q$, (5.6b) and (5.6d) imply that $\La-t\ga$
is vanishing, hence $\ch_K(\La-t\ga)=0$. Then (6.2) is equal to:
$$
\eqalignno{
\ch_\ga(\La-\ga)&=(-1)^q\biggl(\ch_K(\La-(q+1)\ga)
-\ch_K(\La-(q+2)\ga)+\cdots\biggr)\cr
 &=(-1)^q\ch_\ga(\La-(q+1)\ga).&(6.3)\cr}
$$
According to (5.6d), 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 (4.10) this implies that
$$
\ch_{\ga'}(q(\La))=(-1)^q\ch_\ga(\La-(q+1)\ga). \eqno(6.4)
$$
Then the lemma follows from (6.3) and (6.4).\mybox

\proclaim {Theorem 6.2}. Let $\La$ be singly atypical of type
$\ga$. Then
$$
\cha V(\La)=\ch_\ga(\La). \eqno(6.5)
$$

\noindent {\sl Proof.} In the case that $\ga=\al_1$, the statement follows
from Theorem 4.3. Suppose now that $\ga>\al_1$. Let $\La_0=\La$
and $\ga_0=\ga$, and using the notation of Lemma~6.1 we define a
sequence of dominant weights and elements of $\De_1^+$ by
$$
\La_{k+1}=q(\La_k),\qquad \ga_{k+1}=\ga_k',\qquad(k\geq 0). \eqno(6.6)
$$
Clearly, every $\ga_{k+1}\leq\ga_k$, with equality if and only if
$q(\La_k)=\La_k-\ga_k$, i.e. if and only if $\La_k-\ga_k$
is dominant. But $\ga_{k+1}=\ga_k$ can happen only a finite number of times if
$\ga_k>\al_1$. 
Therefore, there exists an $s$ such that $\ga_{s-1}>\ga_s=\al_1$,
$\al_1$ being the smallest element of $\De_1^+$ according to the
partial ordering (2.13). Since every $\La_k$ is dominant, we
find, using (3.18)--(3.19) and (4.11):
$$
\cha\overline
V(\La_k)=\ch_{\ga_k}(\La_k)+\ch_{\ga_k}(\La_k-\ga_k). 
\eqno(6.7)
$$
Using Lemma 6.1, this becomes:
$$
\cha\overline V(\La_k)=\ch_{\ga_k}(\La_k)+\ch_{\ga_{k+1}}(\La_{k+1}). 
\eqno(6.8)
$$
On the other hand, since every $\La_k$ is singly atypical,
Theorem~3.8 implies:
$$
\cha\overline V(\La_k)=\cha V(\La_k)+\cha X(\La_k), \eqno(6.9)
$$
where both $V(\La_k)$ and $X(\La_k)$ are simple $G$ modules.
Applying (6.8) and (6.9) for $k=s-1$, using Theorem~4.3, leads to
$$
\eqalignno{
\cha\overline V(\La_{s-1})&=\ch_{\ga_{s-1}}(\La_{s-1})+\cha V(\La_s),&(6.10a)\cr
\cha\overline V(\La_{s-1})&=\cha V(\La_{s-1})+\cha X(\La_{s-1}).&(6.10b)\cr}
$$
From Remark 4.4, (6.7) and (6.10a), 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 that
$V(\La_s)$ is isomorphic to $X(\La_{s-1})$, and (6.10) implies:
$$
\cha V(\La_{s-1})=\ch_{\ga_{s-1}}(\La_{s-1}). \eqno(6.11)
$$
By iteration one finds:
$$
\cha V(\La_k)=\ch_{\ga_k}(\La_k),\qquad k=s,s-1,\ldots,1,0, \eqno(6.12)
$$
proving the theorem.\mybox

From Theorem~3.7, (3.19),
Theorem~6.2, and the fact that $C(n)$ has only typical or singly
atypical modules, one obtains the following result:

\proclaim {Corollary 6.3}. Let $G=C(n)$ and $\La$ be integral
dominant. Then
$$
\cha V(\La)=L_0^{-1}\sum_{w\in W}\vep(w)w\Bigl\lbrace e^{\La+\rh_0}
 \prod_{{\scriptstyle\be\in\De_1^+}\atop{\scriptstyle\langle\La+\rh|\be\rangle
  \not=0}}(1+e^{-\be})\Bigr\rbrace. \eqno(6.13)
$$

Note that this formula is in fact the Bernstein--Leites formula
(1980, Leites 1980), wrongly claimed to be true for all simple
Lie superalgebras but proven here to be correct for the series $C(n)$.
Finally, we shall derive an alternative character formula for the
atypical case. The validity of this alternative formula is proven
by means of the following lemma:

\proclaim {Lemma 6.4}. Let $\La$ be atypical of type $\be$. Then
there exists a unique sequence of distinct elements $\be_0=\be<\be_1
<\cdots<\be_p$ from $\De_1^+$ such that the sequence of weights $\nu_0=\La,
\nu_1,\ldots,\nu_p,\nu_{p+1}$ with $\nu_{t+1}=\nu_t+\be_t$ satisfies
$$
\eqalignno{
&\langle\nu_t+\rh|\be_t\rangle=0,\quad 0\leq t\leq p;&(6.14a)\cr
&\nu_t\hbox{ is vanishing for }0<t\leq p;&(6.14b)\cr
&p(\La)=\nu_{p+1}\hbox{ is integral dominant and singly atypical
of type }\be_p.&(6.14c)\cr}
$$

\noindent The proof of Lemma~6.4 is analogous to the proof of
Lemma~5.6 (the equivalent of (5.6d) is also true, but is omitted
since we do not need that property in what follows). 

\proclaim {Definition 6.5}. Let $\La$ be atypical of type $\be$,
and let $S(\La)=\{\be_0=\be,\be_1,\ldots,\be_p\}$ be the set
consisting of the elements of the sequence of Lemma~6.4. Then,
$$
\ch_{S(\La)}(\la)=
L_0^{-1}\sum_{w\in W}\vep(w)w\Bigl\lbrace e^{\la+\rh_0}
 \prod_{\be\in\De_1^+\backslash S(\La)}(1+e^{-\be})\Bigr\rbrace. \eqno(6.15)
$$

The following property is analogous to (3.23) and (4.10):
$$
w\,S(\La)=S(\La)\quad\Rightarrow\quad \ch_{S(\La)}(w\cdot\la)=
\vep(w)\ch_{S(\La)}(\la). \eqno(6.16)
$$
Now we can prove an alternative character formula:

\proclaim {Theorem 6.6}. Let $\La$ be atypical of type $\be$, and
let $S(\La)=\{\be_0=\be,\be_1,\ldots,\be_p\}$ be the set
consisting of the elements of the sequence of Lemma~6.4. Then,
$$
\cha V(\La)=\ch_{S(\La)}(\La).\eqno(6.17)
$$

\noindent {\sl Proof.} Just as in the proof of Lemma~5.6, there
are three distinct cases to consider:
\item{1.} The zero in $A(\La)$ appears in the right part of
$A(\La)$. Then $\be=\be_0=\ep+\de_i$, $\be_1=\ep+\de_{i-1}$,
$\ldots$, $\be_p=\ep+\de_{i-p}$, with $1\leq i-p\leq i\leq n-1$,
and $\La=(\la_0|\la_1,\ldots,\la_{n-1})$ satisfies 
$$
\la_{i-p}=\cdots=\la_i\equiv\et\;\hbox{ and }\;
\la_{i-p-1}>\la_{i-p}\hbox{ if }i-p>1. \eqno(6.18a)
$$
\item{2.} The zero in $A(\La)$ appears in the left part of
$A(\La)$, and $a_n\not=0$. Then $\be=\be_0=\ep-\de_i$,
$\be_1=\ep-\de_{i+1}$, $\ldots$, $\be_p=\ep-\de_{i+p}$, with
$1\leq i\leq i+p\leq n-1$, and $\La$ satisfies
$$
\et\equiv\la_i=\la_{i+1}=\cdots=\la_{i+p}\;\hbox{ and }
\la_{i+p}>\la_{i+p+1}\hbox{ if }i+p<n-1.\eqno(6.18b)
$$
\item{3.} The zero in $A(\La)$ appears in the left part of
$A(\La)$, say $\be=\ep-\de_i$, and $a_{i+1}=\cdots=a_n=0$.
Then $\be_0=\ep-\de_i$, $\be_1=\ep-\de_{i+1}$, $\ldots$,
$\be_{n-i-1}=\ep-\de_{n-1}$, $\be_{n-i}=\ep+\de_{n-1}$, $\ldots$,
$\be_p=\be_{2n-2i-1}=\ep+\de_i$, and $\La$ satisfies
$$
\la_i=\cdots=\la_{n-1}=0. \eqno(6.18c)
$$

\noindent We shall give some details of the proof for case (1), and leave
cases (2) and (3) for the reader to verify. In case (1),
$$
S(\La)=\{\ep+\de_i,\ep+\de_{i-1},\ldots,\ep+\de_{i-p}\}.\eqno(6.19)
$$
Now we consider $\ch_\ga(\La)$, with $\ga=\ep+\de_i$. Expanding
the products $(1+e^{-\be})$ for
$\be=\ep+\de_{i-1},\ldots,\ep+\de_{i-p}$ leads to
$$
\ch_\ga(\La)=\sum_{k_{i-1}=0}^1\,\sum_{k_{i-2}=0}^1\cdots\sum_{k_{i-p}=0}^1
 \ch_{S(\La)} \Bigl(\La-\sum_{j=i-p}^{i-1}k_j(\ep+\de_j)\Bigr).\eqno(6.20)
$$
The r.h.s.~of (6.20) has $2^p$ terms, the first of which is
$\ch_{S(\La)}(\La)$, and the others of the form
$\ch_{S(\La)}(\Om)$ with $\Om\not=\La$. But since the components
of $\La$ satisfy (6.18a), it follows that every $\Om=(\om_0|\om_1,
\ldots,\om_{n-1})$ with $\Om\not=\La$ is vanishing, since there 
is at least one $k$ ($i-p\leq k\leq i-1$) such that $\om_k+1=\om_{k+1}$. In this case,
let $w$ be the element of $W$ permuting the components with
indices $k$ and $k+1$. Then $w(\Om+\rh_0)=\Om+\rh_0$, in other
words, $w\cdot\Om=\Om$, with $\vep(w)=-1$, and moreover
$w\,S(\La)=S(\La)$. Using (6.16) implies that all terms in (6.20) with
$\Om\not=\La$ vanish, hence, using (6.5):
$$
\cha V(\La)=\ch_\ga(\La)=\ch_{S(\La)}(\La). \eqno(6.21)
$$
\mybox

\noindent {\bf Examples.} Let $g$ be a $1\times(2n-2)$ matrix
with $g_i\in\{0,1\}$, and denote by $\ch_g(\la)$ the expression
$$
\ch_g(\la)=L_0^{-1}\sum_{w\in W}\vep(w)\,w\Bigl\lbrace e^{\la+\rh_0}
\prod_{i=1}^{n-1}(1+g_i e^{-\ep+\de_i})
\prod_{i=1}^{n-1}(1+g_{2n-i-1}e^{-\ep-\de_i})\Bigr\rbrace.\eqno(6.22)
$$
Let $G=C(5)$, and consider the following weights:
$$
\La_1=[6;0,0,0,0],\quad\La_2=[2;1,0,1,0],\quad\La_3=[2;0,0,0,0],\eqno(6.23)
$$
with atypicality matrices
$$
\eqalign{
A(\La_1)&=\bigl(6,5,4,3\parallel 1,0,-1,-2\bigr),\cr
A(\La_2)&=\bigl(2,0,-1,-3\parallel -5,-7,-8,-10\bigr),\cr
A(\La_3)&=\bigl(2,1,0,-1\parallel -3,-4,-5,-6\bigr).\cr} \eqno(6.24)
$$
Then, using the notation (6.23), the character formulae (6.21)
implies that
$$
\eqalign{
\cha V(\La_1)&=\ch_{(1\,1\,1\,1\parallel 1\,0\,1\,1)}(\La_1)
 =\ch_{(1\,1\,1\,1\parallel 1\,0\,0\,0)}(\La_1),\cr
\cha V(\La_2)&=\ch_{(1\,0\,1\,1\parallel 1\,1\,1\,1)}(\La_2)
 =\ch_{(1\,0\,0\,1\parallel 1\,1\,1\,1)}(\La_2),\cr
\cha V(\La_3)&=\ch_{(1\,1\,0\,1\parallel 1\,1\,1\,1)}(\La_3)
 =\ch_{(1\,1\,0\,0\parallel 0\,0\,1\,1)}(\La_3).\cr} \eqno(6.25)
$$
In (6.25) the first formula is $\ch_\ga(\La)$, the second formula
is $\ch_{S(\La)}(\La)$.
Note that from the computational point of view the formula
$\ch_{S(\La)}(\La)$ is superior to $\ch_\ga(\La)$, since in
general the corresponding matrix $g$ in (6.22) would contain more
zeros for $\ch_{S(\La)}(\La)$, compared to the matrix $g$ for
$\ch_\ga(\La)$ which contains only one zero.

\section {Acknowledgements}

NATO (Belgium) is acknowledged for a Research Fellowship.

\section {References}

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\bye

