math0304037.tex
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% Harish-Chandra modules over the Higher Rank super-Virasoro algebras
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\def\a{\alpha}
\def\b{\beta}
\def\d{\delta}
\def\D{\Delta}
\def\g{\gamma}
\def\G{\Gamma}
\def\l{\lambda}
\def\L{\Lambda}
\def\o{\omiga}
\def\p{\psi}
\def\Si{\Sigma}
\def\si{\sigma}
\def\v{\varphi}
\def\SA{\hbox{$S\!A$}}
\def\SB{\hbox{$S\!B$}}
\def\Vir{\hbox{\bf\sl Vir}}
\def\HVir{{}}
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\def\SVio{\hbox{{\bf\sl SVir}${\ssc\,}_0$}}
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\def\HVir{\hbox{{\bf\sl SVir}[$M$]}}
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\begin{document}
\def\TIT{SIMPLE MODULES OVER
THE HIGHER RANK SUPER-VIRASORO ALGEBRAS}
\def\ABS
{It is proved that uniformly bounded simple modules over higher rank
super-Virasoro algebras are modules of the intermediate series, and
that simple modules with finite dimensional weight spaces are either
modules of the intermediate series or generalized highest weight modules.}
\def\KEYW{higher rank super-Virasoro algebra,
uniformly bounded module,
module of the intermediate series, generalized highest weight module.}
\def\cover{{\bf\par\ \par\ \par\ni\hi5ex\ha1
Article title:\nl \TIT
\par\ \par\ \par\ni\hi5ex\ha1
Mailing address:\nl
  Dr. Yucai Su \nl
  Department of Applied Mathematics \nl
  Shanghai Jiaotong University \nl
  1954 Huashan Road, Shanghai 200030 \nl
  China \nl
  Email: kfimmi@public1.sta.net.cn
\par\ \par\ \par\ni\hi5ex\ha1
Running title:\nl
  Modules over higher rank super-Virasoro algebras
\par\ \par\ \par\ni\hi5ex\ha1
Abstract:\nl \ABS
\par\ \par\ni\hi5ex\ha1 Keywords:\nl
\hs{5ex}\KEYW
}\setcounter{page}{0}\pagebreak}
%\cover
%\par\
%\par\
%\par\
%\vs{-3pt}
%\par
\cl{\bf SIMPLE MODULES OVER THE HIGHER RANK}
\cl{\bf SUPER-VIRASORO ALGEBRAS}
%\par\
%\vs{-7pt}
{\small\vs{4pt}\cl{(Appeared in {\it Lett.Math.Phys.} {\bf53}
(2000), 263-272)}
\par\vs{-7pt}\ \par
\cl{Yucai Su\footnote{Supported by a grant from National
Educational Department of China}} \vs{-12pt}
\par\ \par
\cl{\small\it Department of Mathematics, Shanghai Jiaotong
University, Shanghai 200030, China} \cl{ Email: ycsu@sjtu.edu.cn}
%\par\
\vs{-7pt} \ \par \ni{\bf Abstract.} \ABS
\par\ni
{\bf Mathematics Subject Classification (2000):} 17B68, 17B70, 17B65
\par\ni{\bf Keywords:} \KEYW
%\par\
}\vs{-7pt}
\par\
\par
\cl{\bf \S1. Introduction}\par
The Virasoro algebra $\Vir$, playing a fundamental role in two dimensional
conformal quantum field theory, is the universal central extension of the
complex Lie algebra of the polynomial vector fields on the circle [1].
The notion of higher rank Virasoro algebras and the notion of higher rank
super-Virasoro algebras were introduced in [2,3] and the notions
of generalized Virasoro and super-Virasoro algebras were introduced in [4].
Kac [5] conjectured a theorem that
a simple $\Vir$-module with finite dimensional weight spaces is either
a module of the intermediate series or else a highest or lowest weight
module. This theorem was partially proved in [6,7,8] and fully proved
in [9] and then generalized to the super-Virasoro algebras in [10], and
was in some sense generalized to the higher rank Virasoro algebras in [11].
In this paper, we shall obtain a similar result for
the higher rank super-Virasoro algebras, thus completing the proof of
the theorem conjectured by the author in [3].
%
In [12,] by using the above mentioned theorem, we gave a theorem on
indecomposable $\Vir$-modules.
%
%In [12], by using the above mentioned theorem, we gave in some extent
%a classification
%of simple $\Vir$-modules with the dimensions of the weight spaces uniformly
%bounded (such modules are called {\it uniformly bounded modules}).
%
Using our result here, we shall have a better understanding on
the simple modules over the higher rank super-Virasoro algebras and
it may be possible to give in some extent a classification of the
uniformly bounded modules over the higher rank super-Virasoro algebras.
This is also our motivation to present the result here.
\par
Let $n$ be a positive integer.
Let $M$ be an $n$-dimensional $\Z$-submodule of $\C$, and let
$s\in\C$ such that $2s\in M$. The {\it rank $n$
super-Virasoro algebra} (or a {\it higher rank super-Virasoro algebra}
if $n\ge2$) is the Lie superalgebra $\SVir=\SVio\oplus\SVii$,
where $\SVio$ has a basis $\{L_\mu,c\,|\,\mu\in M\}$ and
$\SVii$ has a basis $\{G_\eta\,|\,\eta\in s+M\}$, with the commutation
%\vs{-3pt}
relations
%\par
$$
\matrix{
[L_\mu,L_\nu]=(\nu-\mu)L_{\mu+\nu}-\d_{\mu+\nu,0}\OVER1{12}(\mu^3-\mu)c,
\hfill&
[L_\mu,G_\eta]=(\eta-\OVER{\mu}2)G_{\mu+\eta},
\vs{4pt}\hfill\cr
[G_\eta,G_\l]=2L_{\eta+\l}-\d_{\eta+\l,0}\OVER13(\eta^2-\OVER14)c,
\hfill&
[L_\mu,c]=[G_\eta,c]=0,
\hfill\cr}
\eqno(1.1)$$
for $\mu,\nu\in M,\,\eta,\l\in s+M.$
Then $\SVio$ is the higher rank Virasoro algebra $\HVir$.
It is proved in [4] that
a {\it module of the intermediate series} (i.e., a module with all the
dimensions of the weight spaces of ``even'' or ``odd'' part
are $\le1$) over $\SVir$ is one of the three
series of the modules $\SA_{a,b},\SA_{a'},\SB_{a'}$ or their quotient modules
for suitable $a,b,a'\in\C$, where $\SA_{a,b},\SA_{a'}$ have basis
$\{x_\mu\,|\,\mu\in M\}\cup\{y_\eta\,|\,\eta\in s+M\}$ and $\SB_{a'}$
has basis $\{x_\eta\,|\,\eta\in s+M\}\cup\{y_\mu\,|\,\mu\in M\}$ such that
$c$ acts trivially and such
%\vsp
that
$$
\matrix{
\matrix{
\SA_{a,b}\!:\!\!\!\!\hfill&
L_\mu x_\nu=(a+\nu+\mu b)x_{\mu+\nu},\hfill&
L_\mu y_\eta=(a+\eta+\mu(b-\OVER12))y_{\mu+\eta}
\vs{4pt}\hfill\cr&
G_\l x_\nu=y_{\l+\nu},\hfill&
G_\l y_\eta=(a+\eta+2\l(b-\OVER12))x_{\l+\eta}
\hfill\cr
}\vs{8pt}\hfill\cr\matrix{
\SA_{a'}\!:\!\!\!\!\hfill&
L_\mu x_\nu=(\nu+\mu)x_{\mu+\nu},\,\nu\ne0,\hfill&
L_\mu v_0=\mu(\mu+a')v_\mu,\hfill&
L_\mu y_\eta=(\eta+\OVER{\mu}2)y_{\mu+\eta}
\vs{4pt}\hfill\cr&
G_\l x_\nu=y_{\l+\nu},\,\nu\ne0,\hfill&
G_\l x_0=(2\l+a')y_\l,\hfill&
G_\l y_\eta=(\eta+\l)x_{\l+\eta}
\hfill\cr
}\vs{8pt}\hfill\cr\matrix{
\SB_{a'}\!:\hfill&
L_\mu x_\eta=(\eta+\OVER{\mu}2)x_{\mu+\eta},\hfill&
L_\mu y_\nu=\nu y_{\mu+\nu},\,\nu\ne-\mu,\hfill&
L_\mu y_{-\mu}=-\mu(\mu+a')y_0,
\vs{4pt}\hfill\cr&
G_\l x_\eta=y_{\l+\eta},\,\eta\ne-\l,\hfill&
G_\l x_{-\l}=(2\l+a')y_0,\hfill&
G_\l y_\nu=\nu x_{\l+\nu},
\hfill\cr}
\hfill\cr}
%\vs{-3pt}
\eqno\matrix{(1.2{\rm a})\cr\vs{2pt}\cr\cr
(1.2{\rm b})\cr\vs{2pt}\cr\cr(1.2{\rm c})\cr}$$
for $\mu,\nu\in M,\,\l,\eta\in s+M$.
%Here and throughout this paper, when $a$ is fixed, for any $\nu\in\C$,
%we use the notation $\ol\nu$ to denote $a+\nu$.
In this paper, we shall prove the following theorem which was conjectured
by the author in [3].
\par\ni
{\bf Theorem 1.1}. A uniformly bounded simple $\SVir$-module is
a module of the intermediate series.
\par
A $\SVir$-module $V$ is called a {\it generalized highest module} if
it is generated by a weight vector $v$ such that
there exists a \Z-basis $B=\{d_1,...,d_n\}$
of $M$ such that
$$L_\mu v=G_\l v=0,\ \forall\,
\mu=\sum_{i=1}^n m_id_i\in M\bs\{0\},\
\l=\sum_{i=1}^n n_id_i\in s+M\bs\{0\},
\eqno(1.3)$$
with all coefficients $m_i\in\Z_+,\,n_i\in\OVER12\Z_+.$
The vector $v$ is called a {\it generalized highest weight vector}.
Observe that a lowest weight module over the super-Virasoro algebras
(i.e., when $n=1$) is a generalized highest weight module by this definition.
In general, if a module $V$ has a weight
space decomposition with all the dimensions of weight spaces being finite,
then we shall call $V$ a {\it weight module}.
The following theorem generalizes Kac's conjecture in some sense.
\par\ni
{\bf Theorem 1.2}. A weight $\SVir$-module is either
a module of the intermediate series or a generalized highest weight module.
\par
Thus the problem of the classification of simple weight modules
is reduced to the problem of the classification of the generalized
highest weight modules.
%\vs{-7pt}
\par\
%\par\
\vs{-3pt}
\par
\cl{\bf\S2. Uniformly bounded modules}
%\vs{-1pt}
\par
The goal of this section is to prove Theorem 1.1. We shall use induction
on the rank of $M$, the results of [11],
the structure of modules of the intermediate series
defined in (1.2), and the defining relations (1.1)
% to obtain some conditions
%which must be satisfied by a \SVir-module $V$, then use the straightforward
%computation
to obtain our result.
\par
Thus suppose that $V$ is a uniformly bounded \SVir-module.
Obviously $c$ must act trivially on such a module [3]. Thus we shall
ALWAYS omit $c$ in this section.
Decompose $V=V^{(0)}\oplus V^{(1)}$ such that
$$
\mbox{\bf\sl SVir}{\ssc\,}_iV^{(j)}
\subset V^{(i+j)},\
V^{(i)}=\bigoplus_{\mu\in\sC}V^{(i)}_\mu,
\mbox{ where }
V^{(i)}_\mu=\{v\in V^{(i)}\,|\,L_0v=\mu v\},
$$
for $i,j\in\Z/2\Z$.
For $a\in\C$, define a submodule
$$
V(a)=\sum_{\mu\in M}V^{(0)}_{\mu+a}\bigoplus\sum_{\l\in s+M}V^{(1)}_{\l+a},
$$
for $a\in\C$. Then $V$ is a direct sum of different $V(a)$. Since
we assume that $V$ is simple, we have
$V=V(a)$
for some $a\in\C.$
Since $V^{(i)},i\in\Z/2\Z$ are $\SVio$-modules, by [11, Lemma 2.1], there
exist nonnegative integers $N^{(0)},N^{(1)}$ such that
$$
{\rm dim{\sc\,}}V^{(i)}_{a+is+\mu}=N^{(i)}
\mbox{ for all $\mu\in M$ with }a+is+\mu\ne0,\,i=0,1.$$
By interchanging $V^{(0)}$ with
$V^{(1)}$ if necessary, we always suppose that $N^{(0)}\le N^{(1)}$.
\par\ni
{\bf Proof of Theorem 1.1}.
The result is obvious if $N^{(1)}=0$. Suppose that $N^{(1)}\ge1$.
If $n=1$, the result follows from [10]. So assume that $n\ge 2$.
Observe that
$${\sl SVir}[M,s]\cong{\sl SVir}[M',s']\ \Lra\
\exists\,\a\in\C\mbox{ such that }M'=\a M
\mbox{ and }s'-\a s\in M'.
$$
Thus we can always suppose that $1\in M$ and
furthermore we can suppose that $1$ is a basis element of $M$
whenever necessary.
So, we can take a \Z-basis $B=\{d_1=1,\,d_2,\cdots,d_{n-1},d_n=d\}$
of $M$ such that
$$
M=M_1\bigoplus \Z d,\ \ M_1=\Z\bigoplus\sum_{i=2}^{n-1}\Z d_i,
\ \ 2s\in M_1.
\eqno(2.1)$$
Let \SVira\, be the rank $n-1$ super-Virasoro algebra generated by
$$
\{L_\mu,G_\l\,|\,\mu\in M_1,\l\in s+M_1\}.
$$
For $k\in\Z$, let
$$
V[k]=\sum_{\mu\in M_1}V^{(0)}_{a+kd+\mu}\bigoplus
\sum_{\l\in s+M_1}V^{(1)}_{a+kd+\l},
$$
which is a uniformly bounded \SVira-module.
Since $\Z d\cap M_1=\{0\}$, there exists at most one
$k\in\Z$ such that
$a+kd\in M_1$. Fix a $k_0$ such that
$$
a+kd\not\in M_1,\ \ \forall\,k\ge k_0,
\eqno(2.2)$$
i.e., $a+kd+\nu\ne0$ for all $k\ge k_0,\,\nu\in M_1$.
First we take $k=k_0$. By inductive assumption, we can suppose that
Theorem 1.1 holds for the rank $n-1$ super-Virasoro algebra \SVira.
Thus there exists a simple \SVira-submodule $V[k,1]$ of $V[k]$ such
that $V[k,1]$ has the
form $\SA_{a+kd,b}$ in (1.2) (by (2.2), $\SA_{a+kd,b}$ is simple),
thus, there exists a basis
$$\{x_\mu\,|\,\mu\in M_1\}\cup
\{y_\l\,|\,\l\in s+M_1\},
\eqno(2.3)$$
of $V[k,1]$,
such that
$$
\mbox{\it(1.2a) holds for
all $\mu,\nu\in M_1,\,\l,\eta\in s+M_1$ with $a$ replaced by
$a+kd$.}
\eqno(2.4)$$
Furthermore, among all simple submodules of $V[k]$, we choose
$V[k,1]$ to be one such that the real part ${\rm Re}(b)$ of $b$
is minimum.
Take a composition series of the \SVira-module $V[k+1]$:
$$
0=V[k+1,0]\subset V[k+1,1]\subset\cdots\subset V[k+1,N]=
V[k+1],
\eqno(2.5)$$
for some $N$. From this, we can observe that $N=N^{(0)}=N^{(1)}$.
We shall choose the composition series (2.5) such that
$$
V[k+1,i]/V[k+1,i-1]\cong \SA_{a+(k+1)d,b'},
\eqno(2.6)$$
for some $b'\in\C,\ i=1,\cdots N,$ where $b'$ may depend on $i$.
Then, we have
$$
L_d V[k,1]^{(0)}\subset L_d V[k,1]\subset L_d V[k]\subset V[k+1],
\eqno(2.7)$$
where $V[k,1]^{(0)}$ is the ``even'' part of $V[k,1]$.
\par
We shall divide the proof into some claims.
\par
{\bf Claim 1}. $L_d V[k,1]^{(0)}\ne0.$
\par
Suppose $L_d V[k,1]^{(0)}=0.$ Using the fact
%\vsp
that
$$
L_{\mu+md}=
\prod_{i=-1}^{m-2}(\mu+id)^{-1}
({\rm ad\sc\,}L_d)^mL_\mu,
\ \
G_{s+\mu+md}=
\prod_{i=-1}^{m-2}(s+\mu+\OVER{i}2d)^{-1}
({\rm ad\sc\,}L_d)^m G_{s+\mu},
%\vsp
$$
we obtain
$$
L_{\mu+md}V[k,1]^{(0)}=G_{s+\mu+md}V[k,1]^{(0)}=0,\ \
\forall\,\mu\in M_1,\,m>0.
$$
Let ${\cal L}_1$ and ${\cal L}_2$
be the Lie super-subalgebras of $\SVir$
generated by
$$
\SVira\cup\{L_{-d}\}\mbox{ \ and \ }
\{L_{\mu+md},G_{s+\mu+md}\,|\,\mu\in M_1,\, m\in\Z_+\bs\{0\}\},
$$
respectively.
Then as space, we have $\SVir={\cal L}_1\oplus {\cal L}_2$.
By decomposing the universal enveloping algebra
$U(\SVir)=U({\cal L}_1)U({\cal L}_2)$, we obtain that
$$
V=U({\cal L}_1)U({\cal L}_2)V[k,1]^{(0)}=U({\cal L}_1)V[k,1]^{(0)}.
$$
{}From this,
we obtain that $V^{(i)}_{\ol\mu+is+md}=0$
for all $\mu\in M_1,\,m>0,\,i=0,1$. Thus $N=0$, contradicting the assumption
that $N=N^{(1)}\ge1$. Hence Claim 1 follows.
\par
Now from (2.5), (2.7), we can take $K\ge0$ to be the integer such that
$$L_d V[k,1]\not\subset V[k+1,K],\mbox{ but } L_d V[k,1]\subset
V[k+1,K+1].
\eqno(2.8)$$
Furthermore, we can choose (2.5) to be a composition series such that
$K$, defined by (2.8), is minimal possible among all composition series of
$V[k+1]$. Then, we can take
$$
\{x'_\mu\in V[k+1,K+1]^{(0)}\,|\,\mu\in M_1\},
\eqno(2.9)$$
such that
$$
(V[k+1,K+1]/V[k+1,K])^{(0)}={\rm span}
\{x'_\mu+V[k+1,K]^{(0)}\,|\,\mu\in M_1\},
$$
and such that there exist some $b'\in\C$ and some $a_\nu\in\C$ with
$$
L_\mu x'_\nu\equiv(a+(k+1)d+\nu+\mu b')x'_{\mu+\nu}, \ \
L_d x_\nu\equiv a_\nu x'_\nu\ \,({\rm mod\,} V[k+1,K]^{(0)}),
\eqno(2.10)$$
for all $\mu,\nu\in M_1.$
\par
Now [11, Lemmas 2.3-6] shows that $b'=b$ and that by rescaling
$x'_\nu$ if necessary, we can suppose that
$a_\nu=a+kd+\nu+bd$, and furthermore $K=0$, i.e.,
$V[k+1,K]=0$. Thus (2.10) becomes
$$
L_\mu x'_\nu=(a+(k+1)d+\nu+\mu b')x'_{\mu+\nu}, \ \
L_d x_\nu=(a+kd+\nu+bd) x'_\nu.
\eqno(2.11)$$
Since $V[k+1,1]$ is a $\SVira$-module of the intermediate series,
there exists basis $\{y'_\l\,|\,\l\in s+M_1\}$ of $V[k+1,1]^{(1)}$ such that
$$
\matrix{
\mbox{\it(1.2a) holds for all $\mu,\nu\in M_1,\,\l,\eta\in s+M_1$ with the symbols
$x,y$ replaced}
\vs{4pt}\hfill\cr
\mbox{\it by symbols $x',y'$ respectively,
and with $a$ replaced by $a+(k+1)d$.}
\hfill\cr}
\eqno(2.12)$$
By (2.8), $L_dV[k,1]^{(1)}\subset V[k+1,1]^{(1)}$.
Thus we can suppose
$$
L_d y_\l=c_\l y'_\l\mbox{ for some }c_\l\in\C,\ \l\in s+M_1.
\eqno(2.13)$$
Then by [11, Lemma 2.3], there exists nonzero scalar $c\in\C$ such that
$$
c_\l=c(a+kd+\l+d(b-\OVER12)),\ \ \forall\,\l\in s+M_1.
$$
\par
{\bf Claim 2}. One can choose a suitable simple
$\SVira$-submodule $V[k,1]$ of $V[k]$
such that $L_{-d}L_d V[k,1]\subset V[k,1]$.
\par
Let $W[k]$ be the direct sum of
ALL simple \SVira-submodules $V[k,1]$ of $V[k]$ of the same type:
$V[k,1]\cong \SA_{a+kd,b}$ and let
$W[k+1]$ be the direct sum of
ALL simple \SVira-submodules $V[k+1,1]$ of $V[k+1]$ of the same type:
$V[k+1,1]\cong \SA_{a+(k+1)d,b}$.
Then the above discussion shows that
$L_d W[k]\subset W[k+1]$, and so we also have
$L_{-d} W[k+1]\subset W[k]$ by replacing $d$ by $-d$ in the above arguments.
By Claim 1, $W[k],W[k+1]$ must have the same number, say, $r$,
of composition factors.
Choose a suitable basis
$\{x^{(j)}_\nu,y^{(j)}_{s+\nu}\,|\,\nu\in M_1, j=1,...,r\}$ of $W[k]$
such that $x^{(1)}_0$ is an eigenvector of $L_{-d}L_d$ and
$$
\matrix{
L_\mu X_\nu=(a+kd+\nu+\mu b)X_{\mu+\nu},\hfill&
L_\mu Y_\eta=(a+kd+\eta+\mu(b-\OVER12))Y_{\mu+\eta}
\vs{4pt}\hfill\cr
G_\l X_\nu=Y_{\l+\nu},\hfill&
G_\l Y_\eta=(a+kd+\eta+2\l(b-\OVER12))X_{\l+\eta},
\hfill\cr}
\eqno(2.14)$$
where
$$
X_\nu=\pmatrix{x_\nu^{(1)}\cr\vdots\cr x_\nu^{(r)}\cr},\ \ \
Y_\eta=\pmatrix{y_\eta^{(1)}\cr\vdots\cr y_\eta^{(r)}\cr},\ \ \
\forall\,\nu\in M_1,\ \eta\in s+M_1.
\eqno(2.15)$$
Then Claim 1, (2.11) and (2.13) allow us to choose a basis
$\{x'^{(j)}_\nu,y'_{s+\nu}\,|\,\nu\in M_1,j=1,...,r\}$ of $W[k+1]$ such
that
$$
L_d X_\nu=(a+kd+\nu+bd)X'_\nu,\ \ \
L_d Y_\eta=c(a+kd+\eta+d(b-\OVER12))Y'_\eta,
$$
where $X'_\nu,Y'_\eta$ are the similar notations as in (2.15),
and such that
(2.14) holds for $k$ replaced by $k+1$ and $X,Y$ replaced by $X',Y'$.
Similarly, with $d$ replaced by $-d$ in the above arguments, we have
$$
L_{-d} X'_\nu=(a+(k+1)d+\nu-bd)A X_\nu,\ \ \
L_{-d} Y'_\eta=c'(a+(k+1)d+\eta-d(b-\OVER12))AY_\eta,
$$
where $A$ is some constant $r\times r$ matrix and $c'\in\C\bs\{0\}$.
Thus we have
$$
L_{-d}L_d X_\nu=(a+kd+\nu+bd)(a+(k+1)d+\nu-bd)AX_\nu.
\eqno(2.16)$$
Since $x^{(1)}_0$ is an eigenvector of $L_{-d}L_d$, on the first row of
$A$, by (2.16),
there is only one possible nonzero entry in the first position.
If we choose
$$
V[k,1]={\rm span}_\sC\{x^{(1)}_\nu,y^{(1)}_{s+\nu}\,|\,\nu\in M_1\},
$$
then we see that $L_{-d}L_d V[k,1]\subset V[k,1]$.
This proves our Claim 2.
\par
Now we can complete the proof of Theorem 1.1 as follows.
Starting from $k=k_0$ and choosing $V[k_0,1]$ as in Claim 2,
by induction on $k>k_0$, we can choose $V[k,1]$ to be the simple
\SVira-submodule generated by $L_d V[k-1,1]$.
As in the proof of Claim 1, using
$L_{-d}L_d V[k_0,1]\subset V[k_0,1]$, by induction on $m$,
we have
$$
L_{\nu+md}V[k,1]\subset V[k+m,1],\
G_{\l+md}V[k,1]\subset V[k+m,1],
\eqno(2.17)$$
for all $\nu\in M_1,\,\l\in s+M_1,\,m,k\in\Z$
such that $k\ge k_0,\,k+m\ge k_0$.
Since $V$ is a simple \SVir-module, it must be generated by
$V[k_0,1]$, thus by (2.17),
$$
{\rm dim\sc\,}V^{(0)}_{a+\nu+kd}=
{\rm dim\sc\,}V^{(1)}_{a+\l+kd}=1,
\ \ \forall\,\nu\in M_1,\,\l\in s+M_1,\,k\ge k_0.
$$
This shows that $N=1$, thus
${\rm dim\sc\,}V^{(0)}_{a+\nu}=1=
{\rm dim\sc\,}V^{(1)}_{a+\l}$
for all $\nu\in M,\,\l\in s+M$ with
$a+\nu\ne0\ne a+\l$, from this one sees that
we must also have
${\rm dim\sc\,}V^{(0)}_{a+\nu}\le1,\,
{\rm dim\sc\,}V^{(1)}_{a+\l}\le1$ if $a+\nu=0=a+\l$.
Thus $V$ must be a module of the intermediate series.
\par\
%\par\
\vs{-3pt}
\par
\cl{\bf\S3. Generalized highest weight modules}
\vs{-1pt}
\par
The aim of this section is to prove Theorem 1.2. This can be done
by three lemmas below. We suppose $n\ge2$ as for $n=1$, the result
follows from [10].
\par\ni{\bf Lemma 3.1}.
Let $V$ be a simple \SVir-module and let $B=\{d_1,...,d_n\}$ be
any \Z-basis of $M$.
If $0\ne v\in V$ is a weight vector and $k\in\Z_+$ such that
$L_\mu v=0=G_\l v$ for all
$$
\matrix{
\dis\mu\in M_{\ssc B}^{(k)}=
\{\mu\in M\,|\,\mu=\sum_{i=1}^n m_id_i,
\mbox{ all }k\le m_i\in\Z_+\},
\vs{4pt}\hfill\cr
\dis\l\in M'^{(k)}_{\ssc B}=\{\l\in s+M\,|\,
\l=\sum_{i=1}^n n_id_i,\mbox{ all }k\le m_i\in\OVER12\Z_+\},
\hfill\cr}
$$
then $V$ is a generalized highest weight module.
\par\ni{\bf Proof}. Take
$$
d'_i=\sum_{j=1}^i(k+i-j+1)d_j+k\sum_{j=i+1}^nd_j,\,i=1,...,n.
$$
It is straightforward to check that the determinant of coefficients of
$\{d_1,...,d_n\}$ is 1, thus
$B'=\{d'_1,...,d'_n\}$ is also a \Z-basis of $M$.
Now for any
$$
0\ne\mu'=\sum_{i=1}^n m'_i d'_i\in M_{\ssc{B'}}^{(0)}
\mbox{ (so all $m'_i\ge0$, at least one $m'_i>0$),}
$$
we have $\mu'=\sum_{i=1}^n m_id_i$ with all coefficients
$$
m_i=k\sum_{j=1}^{i-1}m'_j+
\sum_{j=i}^n(k+j+1-i)m'_j\ge k,
$$
i.e., $\mu'\in M_{\ssc B}^{(k)}$,
thus $L_{\mu'\ssc\,}v=0$.
Similarly, for any $0\ne\l'\in M'^{(0)}_{\ssc{B'}}$, we have
$G_{\l'\ssc\,} v=0$. Hence by definition (cf. (1.3)$\ssc\,$),
$v$ is a generalized highest weight vector and $V$ is a
generalized highest weight module.
\qed\par\ni
{\bf Lemma 3.2}.
Fix a \Z-basis $B=\{d_1,...,d_n\}$ of $M$.
Let
$$
\matrix{
\dis A=\{\mu\in M\,|\,\mu=\sum_{i=1}^nm_id_i,\mbox{ all }m_i\in\{-1,0,1\}\},
\vs{4pt}\hfill\cr
\dis A'=\{\l\in s+M\,|\,\l=\sum_{i=1}^nn_id_i,\mbox{ all }
n_i\in\{-1,-\OVER12,0,\OVER12,1\}\},
\hfill\cr}
$$
be two finite sets.
For any $\mu=\sum_{i=1}^nm_id_i\in M$,
let
$$
A_\mu=\{\nu\in M\,|\,\nu-\mu\in A\},\ \
A'_\mu=\{\l\in s+M\,|\,\l-\mu\in A'\},
$$
\def\LL{\mbox{$\cal L$}}and
let $\LL$ be the Lie super-subalgebra of $\SVir$ generated by
$\{L_\nu\,|\,\nu\in A_\mu\}\cup\{G_\l\,|\,\l\in A'_\mu\}$.
Then there exists a \Z-basis $B'=\{d'_1,...,d'_n\}$ of $M$
such that
$$
\LL\supset \SVir_{\ssc{B'}}^{(k)}=
\{L_\nu,\,G_\l\,|\,\nu\in M_{\ssc{B'}}^{(k)},\,\l\in
 M'^{(k)}_{\ssc{B'}}\}\mbox{ for some }
k\in\Z_+.
$$
\par\ni{\bf Proof}.
If necessary, by replacing $d_i$ by $-d_i$ (this does not change the
sets $A,A'$, neither do $\LL$), we can suppose $m_i\ge0,$ $i=1,...,n$.
First suppose that $m_1\ne0,m_2\ne0$. Take
%\par
$$
\matrix{
d'_1=\hfill&\!\!\!m_2\mu\!+\!d_1\hfill&\!\!\!=\hfill&\!\!\!
\dis(\mu+d_1)+\sum_{i=1}^{m_2-1}\mu=\hfill&\!\!\!(m_1m_2+1)d_1+m^2_2d_2+
\dis\sum_{i=3}^nm_2m_id_i,
\hfill\cr}
$$
$$
\matrix{
%\vs{5pt}\hfill\cr
d'_2=\hfill&\!\!\!m_1\mu\!-\!d_2\hfill&\!\!\!=\hfill&\!\!\!
\dis(\mu-d_2)+\sum_{i=1}^{m_1-1}\mu=\hfill&\!\!\!m^2_1d_1+(m_1m_2-1)d_2+
\dis\sum_{i=3}^nm_1m_id_i,\vs{5pt}\hfill\cr
d'_i=\hfill&\!\!\!d'_1+d_i\hfill&\!\!\!=\hfill&\!\!\!
\dis(\mu\!+\!d_1\!+\!d_i)+\sum_{i=1}^{m_2-1}\mu,\!\!\!\!\hfill&i=3,...,n.\hfill
\cr}$$
We can easily solve $d_i,i\!=\!1,...,n,$ as an
integral linear combination of $B'\!=\!\{d'_1,...,d'_n\}$, so $B'$ is a
\Z-basis of $M$. From the second equality of $d'_i$, we see
$L_{d'_i}\!\in\!\LL,\,i\!=\!1,...,n$. For example,
$$
[L_\mu,...,[L_\mu,L_{\mu+d_1}]...]\ (m_2-1
\mbox{ copies of }L_\mu)= aL_{d'_1},
\mbox{ where }a=\prod_{i=0}^{m_2-2}(i\mu+d_1)\ne0,
$$
so that the left-hand side is in $\LL$, thus $L_{d'_1}\in \LL$ and
so
$\{L_\nu\,|\,\nu\in M_{\ssc{B'}}^{(0)}\}\subset\LL$. Similarly,
$\{G_\l\,|\,\l\in M'^{(0)}_{\ssc{B'}}\}\subset\LL$.
Next suppose that some $m_i=0$, $1\le i\le n$, say, $m_1=0$. Take
$$
d'_1=\mu+d_1,\ \ d'_i=d'_1+d_i=\mu+d_1+d_i, \ i=2,...,n.
$$
Then as above $B'=\{d'_1,...,d'_n\}$ is a \Z-basis of $M$ and
$\HVir_{_{B'}}^{(0)}\subset\LL$.
\qed\par
\ni{\bf Lemma 3.3}.
A simple weight \SVir-module $V$ with dimensions of weight spaces
not uniformly bounded is a generalized highest weight module.
\par\ni
{\bf Proof}.
Suppose conversely that $V$ is not a
generalized highest weight module.
For any $\mu\in M$, let $A,\,A',\,A_\mu,\,A'_\mu,\,\LL$
and $B'$ be as in Lemma 3.2. By Lemmas 3.1 and 3.2, for any
$0\ne v\in V$, we have $\SVir_{\ssc{B'}}^{(k)}v\ne0$, thus,
$L_\nu v\ne0$ for some $\nu\in A_\mu$ or
$G_\l v\ne0$ for some $\l\in A'_\mu$, i.e.,
$$
\bigcap_{\nu\in A_\mu}{\rm ker}(L_\nu|_V)
\bigcap\bigcap_{\l\in A'_\mu}{\rm ker}(G_\l|_V)=0.
$$
In particular,
$$
(\bigoplus_{\nu\in A_\mu}L_\nu\bigoplus\bigoplus_{\l\in A'_\mu}G\l)
|_{V_{a-\mu}}:\
V_{a-\mu}\rar\bigoplus_{\nu\in A}V_{a+\nu}
\bigoplus\bigoplus_{\l\in A'}V_{a+\l},
$$
is an injection.
Therefore, ${\rm dim\sc\,}V_{a-\mu}\le N$, where
$$
N=
\sum_{\nu\in A}{\rm dim\sc\,}V_{a+\nu}
+\sum_{\l\in A'}{\rm dim\sc\,}V_{a+\l},
$$
is a fixed finite integer since both $A$ and $A'$ are finite sets.
As $\mu\in M$ is arbitrary,
this proves that $V$ is uniformly bounded, a
contradiction.
\qed\par
Now Theorem 1.2 follows from Lemma 3.3.
\par\
%\par\
\vs{-3pt}
\par
\cl{\bf References}
\par\small
\parskip .03 truein
\baselineskip 4pt%\baselineskip 15pt
\lineskip 4pt%\lineskip 15pt
\ni\hi3ex\ha1
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%
%12 Su,~Y.: Classification of uniformly bounded modules over the
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