# Zero level modules over twisted affine Lie superalgebras

###### Abstract.

In this paper, we characterize a class of simple finite weight modules over
twisted affine Lie superalgebras. This is an essential step toward completing
the classification of simple finite weight modules over these Lie
superalgebras.

⋆ Corresponding author.

2010 Mathematics Subject Classification: 17B10, 17B67.

Key Words: Finite weight modules, twisted affine Lie superalgebras

This work is based upon research funded by Iran National Science Foundation
INSF (No. 4001480).

Asghar Daneshvara, Hajar Kiamehra, Malihe Yousofzadeha,b,⋆

a Department of Pure Mathematics, Faculty of Mathematics and Statistics,
University of Isfahan,

Isfahan, Iran, P.O.Box 81746-73441 and School of Mathematics.

b Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746,
Tehran, Iran.

Email addresses<EMAIL_ADDRESS>(Asghar Daneshvar<EMAIL_ADDRESS>(Hajar Kiamehr),

<EMAIL_ADDRESS>&<EMAIL_ADDRESS>(Malihe Yousofzadeh).

##  1\. Introduction

The study of Lie (super)algebras is an active research area, uncovering
numerous connections and applications in various domains of both mathematics
and physics (see for instance [12] and [17]). More specifically, Lie
(super)algebras $\mathcal{L}$ having a root space decomposition with respect
to an abelian subalgebra ${\mathcal{H}}$, such as basic classical simple Lie
superalgebras, affine Lie (super)algebras, Virasoro algebra and etc., play a
crucial role. For example, affine Lie (super)algebras have an important role
in advancing investigations in number theory, group theory, knot theory, and
quantum field theory ([4] can be considered as an exemplary reference for knot
theory, and [15, 16] can be cited as examples in the context of number
theory). They hold considerable importance, especially the classification of
their modules is a topic of significance.

After studying finite dimensional $\mathcal{L}$-modules, the next step is to
study finite weight $\mathcal{L}$-modules, i.e., $\mathcal{L}$-modules having
weight space decomposition with respect to ${\mathcal{H}}$ with finite weight
multiplicities. There have been a vast amount of research in this regard. In
the case that real root vectors (root vectors corresponding to the roots which
are not self-orthogonal) act locally nilpotently on $\mathcal{L}$ (e.g., in
affine and finite theory), the key point in these researchs is the behaviour
of real root vectors acting on the module, more precisely, in this case, real
root vectors act on a simple module either injevtively or locally nilpotently.
This behaviour leads the module structure. The simplest case (in some sense),
is the case that all real root vectors act locally nilpotently on a module.
Such a module is called an integrable module. A simple integrable finite
weight module over a finite dimensional reductive Lie algebra is finite
dimensional but in infinite dimensional level, the situation is completely
different.

By reviewing the literature, all investigations on finite weight moduls go
back to fabulous works by S.L. Fernando [11] and O. Mathieu [19]; S.L.
Fernando started studying finite weight modules over finite dimensional
reductive Lie algebras and showed that such a module is parabolically induced
from a cuspidal module; in the context of finite theory, it means a module on
which all nonzero root vectors act injectively. He also proved that cuspidal
modules exist only for those reductive Lie algebras whose simple constituents
are just simple Lie algebras of type $A$ and $C$. Then, Mathieu [19] continued
Fernando’s work by classifying cuspidal modules. His main idea to classify
these modules is the notion of coherent families.

In affine case, the study is more complicated due to the existence of
imaginary roots i.e., roots which are orthogonal to all other roots. This
study was initiated by works of V. Chari and A. Pressley [3, 2]; the focus of
their papers is the study of simple integrable finite weight modules over
affine Lie algebras. They show that these modules are either highest-lowest
weight modules or so-called loop modules. For the result about non-integrable
modules over affine Lie algebras and on some other generalizations of these
Lie algebras, see [5], [8], [9], [10], [13] and [14] among the others.

During the time that the research on module theory of affine Lie algebras was
going on, the study of simple finite weight modules over finite dimensional
Lie superalgebras was started by I. Dimitrov and his colleagues in [6];
according to their results, the characterization of these modules is reduced
to the study of cuspidal modules over certain Lie superalgebras. Their study
was linked, by S. Eswara Rao and V. Futorny [7], to module theory of affine
Lie superalgebras; affine Lie superalgebras had been introduced and classified
by J.W. Van de Leur in 1986. According to the classification, affine Lie
superalgebras are divided into untwisted and twisted types, see [20] or [23,
Appendix A] for the details. An affine Lie superalgebra contains a canonical
central element acting on a simple finite weight module as a scalar called the
level of the module. In 2001, V.G. Kac and M. Wakimoto [16] studied weakly
integrable111Since there is no simple integrable finite weight module over
almost all affine Lie superalgebras, V.G. Kac and M. Wakimoto defined a weaker
version of integrability. highest weight modules over untwisted affine Lie
superalgebras and their study was followed in [7] for nonzero level weakly
integrable modules (not necessarily highest weight ones) again over untwisted
affine Lie superalgebras. Also, the third author of the current paper has
studied general finite weight modules over twisted affine Lie superalgebras in
[23, 21, 22]. It has been shown that the study of the modules under
consideration in [7, 23, 21] is reduced to the study of modules investigated
in [6].

Although, there have been significant amount of research on module theory of
affine Lie superalgebras, the case that all real root vectors act injectively
has been abandoned. In the preprint [24], we study this case; we first notice
that in this case, the level of a simple finite weight module over an affine
Lie superalgebra $\mathcal{L}$ is zero and then show that the study of these
modules is reduced to the study of modules $\Omega$ over a subalgebra
$\mathcal{G}$ of $\mathcal{L},$ containing the Cartan subalgebra of
$\mathcal{L},$ whose root system $T$ contains two closed subsets $T(1)$ and
$T(2)$ such that all real root vectors of $T(1)$ act locally nilpotently on
$\Omega$ while all real roots of $T(2)$ are hybrid (see Theorem 3.1 for the
definition) and also satisfy some other mild and natural conditions. This
means that the classification problem goes back to find the classification of
such modules $\Omega.$ This is the main purpose of this paper. In our main
theorem, we show that such a $T$ contains a parabolic subset $P$ such that
$P\cap-P$ is finite and there is a nonzero weight vector $v$ with

${\mathcal{G}}^{\alpha}v=\\{0\\}\quad\hbox{ for
all}\quad\alpha\in(P\setminus(-P\cup{\mathbb{Z}}\delta))\cup((P\setminus\\{0\\})\cap
T(2)\cap{\mathbb{Z}}\delta),$

here ${\mathcal{G}}^{\alpha}$ is the root space corresponding to the root
$\alpha$ and ${\mathbb{Z}}\delta$ is the set of imaginary roots. In
particular, if the set $T(2)_{im}$ of imaginary roots of $T(2)$ is
${\mathbb{Z}}\delta$, then there is a nonzero weight vector $v$ such that

${\mathcal{G}}^{\alpha}v=\\{0\\}\quad(\hbox{for all }\alpha\in P\setminus-P).$

This immediately implies that $\Omega$ is parabolically induced from a module
over a finite direct sum of basic classical simple Lie superalgebras and we
are done. When $T(2)_{im}\not={\mathbb{Z}}\delta,$ the situation is different
and needs a separate investigation which has been done in the preprint [18].

Here is a summary of the content covered in this paper. Section 2 serves as a
brief introduction, providing definitions and some facts using in the main
section. It is structured into two subsections. The first subsection is
dedicated to definitions, revisiting essential concepts, and summarizing
previous findings. In the second subsection, a fundamental result concerning
zero level integrable modules over split central extensions of affine Lie
algebras is presented, a crucial building block for the main portion of the
paper (see Theorem 2.1). The concluding section presents our main findings and
results (see Theorem 3.6).

##  2\. Preliminaries

###  2.1. Generic Information

In this short subsection, to unify the notations, we gather some generic
definitions and notations. The underlying field, for all vector (super)spaces,
is the field of complex numbers $\mathbb{C}$. Suppose that
$\mathfrak{k}=\mathfrak{k}_{0}\oplus\mathfrak{k}_{1}$
(${\mathbb{Z}}_{2}=\\{0,1\\}$) is a Lie superalgebra equipped with a
nondegenerate supersymmetric invariant even bilinear form $(\cdot,\cdot)$ and
$\mathfrak{h}$ is a finite dimensional subalgebra of the even part
$\mathfrak{k}_{0}$ of $\mathfrak{k}$. A superspace ${V}={V}_{0}\oplus{V}_{1}$
is called an $\mathfrak{h}$-weight $\mathfrak{k}$-module (or simply a weight
$\mathfrak{k}$-module if there is no ambiguity) if

  * (1)

$[x,y]v=x(yv)-(-1)^{\mid x\mid\mid y\mid}y(xv)$ for all $v\in{V}$ and
$x,y\in\mathfrak{k}$,

  * (2)

$\mathfrak{k}_{i}{V}_{j}\subseteq{V}_{i+j}$ for all $i,j\in\\{0,1\\}$,

  * (3)

${V}=\oplus_{\lambda\in\mathfrak{h}^{\ast}}{V}^{\lambda}$ with
${V}^{\lambda}:=\\{v\in{V}\mid hv=\lambda(h)v\;\;(h\in\mathfrak{h})\\}$ for
each $\lambda\in\mathfrak{h}^{\ast}(\hbox{\rm dual space of $\mathfrak{h}$}).$

In this setting, an element $\lambda$ of the support of $V$, defined by

$\hbox{\rm
supp}({V}):=\\{\lambda\in\mathfrak{h}^{*}\mid{V}^{\lambda}\neq\\{0\\}\\},$

is called a weight of $V$, and the corresponding ${V}^{\lambda}$ is called a
weight space. Elements of a weight space are called weight vectors. If all
weight spaces are finite dimensional, then the module ${V}$ is called a finite
weight module.

If $\mathfrak{k}$ is an $\mathfrak{h}$-weight module via the adjoint
representation, we say that $\mathfrak{k}$ has a root space decomposition with
respect to $\mathfrak{h}$; the set of weights of $\mathfrak{k}$ is said to be
the root system and weights, weight vectors and weight spaces are termed
respectively roots, root vectors and root spaces.

Next assume that $\mathfrak{k}$ has a root space decomposition with respect to
$\mathfrak{h}$ with root system ${\Phi}$ such that
$\mathfrak{k}^{0}=\mathfrak{h}$. Since the form on $\mathfrak{k}$ restricted
to $\mathfrak{h}$ is a nondegenerate symmetric bilinear form, we conclude that

(2.1) for each $\alpha\in\mathfrak{h}^{*},$ there is a unique
$t_{\alpha}\in\mathfrak{h}$ with $\alpha(h)=(t_{\alpha},h)$ for all
$h\in\mathfrak{h}.$

Therefore, the form on $\mathfrak{h}$ naturally induces a nondegenerate
symmetric bilinear form on the dual space $\mathfrak{h}^{*}$ of
$\mathfrak{h}$, which we denote again by $(\cdot,\cdot)$. Now, we assume
$S\subseteq\mathfrak{h}^{*}$ and define

(2.2) $\begin{array}[]{ll}\hbox{\footnotesize$S_{re}=\\{\alpha\in
S\mid(\alpha,\alpha)\neq 0\\},$}&\hbox{\footnotesize$S_{im}=\\{\alpha\in
S\mid(\alpha,\Phi)=\\{0\\}\\}$},\\\ \hbox{\footnotesize$S_{ns}=\\{\alpha\in
S\setminus
S_{im}\mid(\alpha,\alpha)=0\\},$}&\hbox{\footnotesize$S^{\times}=S\setminus{S_{im}}.$}\end{array}$

Elements of $S_{im}$ (resp. $S_{re}$ and $S_{ns}$) are called imaginary roots
(resp. real roots and nonsingular roots). Next assume ${V}$ is a
$\mathfrak{k}$-module and $S\subseteq\Phi,$

(2.3) we denote by ${S}^{ln}({V})$ (resp. ${S}^{in}({V})$), the set of all
real roots $\alpha\in S$ for which $0\neq x\in\mathfrak{k}^{\alpha}$ acts on
${V}$ locally nilpotently (resp. injectively).

If moreover, $V=\oplus_{\lambda\in\mathfrak{h}^{*}}V^{\lambda}$ is an
$\mathfrak{h}$-weight $\mathfrak{k}$-module and $W$ is an
$\mathfrak{h}$-submodule of $V,$ then, we have
$W=\oplus_{\lambda\in\mathfrak{h}^{*}}(W\cap V^{\lambda});$ we next set222We
use $\\#$ to indicate the cardinal number of a set.

(2.4)
$\begin{array}[]{l}\mathfrak{B}_{W}:=\\{\alpha\in\hbox{span}_{\mathbb{Z}}\Phi\mid\\#\\{k\in{\mathbb{Z}}^{>0}\mid\lambda+k\alpha\in\hbox{\rm
supp}(W)\\}<\infty\;(\forall\lambda\in\hbox{\rm supp}(W))\\},\\\
\mathfrak{C}_{W}:=\\{\alpha\in\hbox{span}_{\mathbb{Z}}\Phi\mid\alpha+\hbox{\rm
supp}(W)\subseteq\hbox{\rm supp}(W)\\}.\end{array}$

We say that the $\mathfrak{h}$-weight $\mathfrak{k}$-module $V$ has shadow if

  * (s1)

${\Phi}_{re}=\\{\alpha\in{\Phi}\mid(\alpha,\alpha)\neq
0\\}={\Phi}^{in}(V)\cup{\Phi}^{ln}(V),$

  * (s2)

${\Phi}^{ln}(V)=\mathfrak{B}_{V}\cap{\Phi}_{re}$ and
${\Phi}^{in}(V)=\mathfrak{C}_{V}\cap{\Phi}_{re}.$

We note that, the behavior of root vectors corresponding to real roots,
whether they act locally nilpotently or injectively, depends only on $W$.
Thus, in the case of a fixed $W$, we can simplify the notation $T^{\star}(W)$
to $T^{\star}$ where $\star=in,ln$.

If we have a subset $S$ of $\Phi$ and functional
$\bm{f}:\hbox{span}_{\mathbb{R}}S\longrightarrow{\mathbb{R}}$, then the
decomposition

(2.5) $S=S^{+}\cup S^{\circ}\cup S^{-}$

where

$S^{\pm}=S^{\pm}_{\bm{f}}:={\\{\alpha\in
S\mid\pm\bm{f}(\alpha)>0\\}}\quad\hbox{and}\quad
S^{\circ}=S^{\circ}_{\bm{f}}:=\\{\alpha\in S\mid\bm{f}(\alpha)=0\\}$

is said to be a triangular decomposition for $S$. This situation is referred
to as trivial when $S=S^{\circ}_{\bm{f}}$. When $P$ is a subset of the root
system $\Phi$, it is called a parabolic subset of $\Phi$ if it meets the
following conditions:

${\Phi}=P\cup-P\quad\hbox{and}\quad(P+P)\cap{\Phi}\subseteq P.$

For a parabolic set $P$, we define the subalgebras as follows:

$\mathfrak{k}^{0}=\bigoplus_{\alpha\in
P\cap-P}\mathfrak{k}^{\alpha},~{}~{}\mathfrak{k}^{+}=\bigoplus_{\alpha\in
P\setminus-P}\mathfrak{k}^{\alpha},~{}~{}\mathfrak{k}^{-}=\bigoplus_{\alpha\in-P\setminus
P}\mathfrak{k}^{\alpha},~{}~{}{\rm
and}~{}~{}\mathfrak{b}=\mathfrak{k}^{0}\oplus\mathfrak{k}^{+}.$

We can convert each $\mathfrak{k}^{0}$-module $N$ into a $\mathfrak{b}$-module
with the trivial action of $\mathfrak{k}^{+}$. It follows that

$\bar{N}:=U(\mathfrak{k})\otimes_{U(\mathfrak{b})}N$

is an $\mathfrak{k}$-module; here $U(-)$ denotes the universal enveloping
algebra of $-$. If the $\mathfrak{k}$-module $N$ contains a unique maximal
submodule $Z$ that intersects $N$ trivially, then the quotient module

${\rm Ind}_{\mathfrak{k}}(N):=\bar{N}/Z$

is said to be a parabolically induced module. Furthermore, a cuspidal module
is defined as a simple $\mathfrak{k}$-module that is not parabolically induced
from a simple module over $\mathfrak{k}^{0}$ corresponding to a parabolic
subset of $\Phi$.

###  2.2. Level zero integrable modules on split central extensions of affine
Lie algebras

Throughout this subsection, assume $\mathfrak{t}$ is an affine Lie algebra
with Cartan subalgebra $\mathfrak{h}$ and $Z$ is an abelian Lie algebra. For

$\mathfrak{k}:=\mathfrak{t}\oplus Z,$

we assume that

$V$ is a nonzero finite $(\mathfrak{h}\oplus Z=)\mathfrak{H}$-weight
$\mathfrak{k}$-module on which the canonical central element of $\mathfrak{t}$
acts trivially and for all real roots $\alpha$ of the root system $\phi$ of
$\mathfrak{t}$ (and so of $\mathfrak{k}$), $x\in\mathfrak{k}^{\alpha}$ acts
locally nilpotently on $V$.

One knows that there is an irreducible finite root system ${\dot{\phi}}$ and a
free abelian group ${\mathbb{Z}}\delta$ of rank $1$ such that the root system
$\phi$ of $\mathfrak{k}$ is of one of the following forms:

$\phi=\left\\{\begin{array}[]{ll}{\dot{\phi}}+{\mathbb{Z}}\delta&X=Y^{(1)},\\\
({\dot{\phi}}+2{\mathbb{Z}}\delta)\cup(({\dot{\phi}}_{sh}\cup\\{0\\})+2{\mathbb{Z}}\delta+\delta)&X=A_{2\ell-1}^{(2)},D_{\ell}^{(2)},E_{6}^{(2)},\\\
({\dot{\phi}}+3{\mathbb{Z}}\delta)\cup(({\dot{\phi}}_{sh}\cup\\{0\\})+3{\mathbb{Z}}\delta\pm\delta)&X=D_{4}^{(3)},\\\
({\dot{\phi}_{ind}}+2{\mathbb{Z}}\delta)\cup({\dot{\phi}}+2{\mathbb{Z}}\delta+\delta)&X=A_{2\ell}^{(2)},\end{array}\right.$

in which $Y$ is the type of a finite dimensional simple Lie algebra,

${\dot{\phi}_{ind}}:={\dot{\phi}}\setminus{\dot{\phi}}_{ex}$

and “sh”, “lg” and “ex” stand for “short”, “long” and “extra long” roots
respectively (Table 1.24 of [1] potentially holds useful information for the
reader.). Set

$\mathfrak{g}:=\raisebox{0.86108pt}{\scalebox{0.8}{$\displaystyle\bigoplus_{\dot{\alpha}\in{\dot{\phi}_{ind}}}\;$}}\mathfrak{k}^{\dot{\alpha}}.$

Then, $\mathfrak{g}$ is a subalgebra of $\mathfrak{k}$ and $V$ is a finite
weight module over the reductive Lie algebra $\mathfrak{g}$. Using
$\mathfrak{sl}_{2}$-module theory and the fact that all root vectors
corresponding to the real roots act locally nilpotently on $V$, we can deduce
that,

(2.6) for $\alpha\in\phi_{re}$ and $\lambda\in\hbox{\rm supp}(V)$,
$\frac{2(\lambda,\alpha)}{(\alpha,\alpha)}\in{\mathbb{Z}};$ moreover, if
$\lambda\in\hbox{\rm supp}(V)$ and
$\frac{2(\lambda,\alpha)}{(\alpha,\alpha)}\in{\mathbb{Z}}^{>0}$ (resp.
$\in{\mathbb{Z}}^{<0}$), then $\lambda-\alpha\in\hbox{\rm supp}(V)$ (resp.
$\lambda+\alpha\in\hbox{\rm supp}(V)$).

Fix a base $\Delta$ of ${\dot{\phi}_{ind}}$. The same argument as stated in
the proof of [8, Lemma 2.6] together with (2.6), implies that

(2.7) there is $\lambda\in\hbox{\rm supp}(V)$ such that
$\lambda+\dot{\alpha}\not\in\hbox{\rm supp}(V)$ for all
$0\neq\dot{\alpha}\in\hbox{\rm span}_{\mathbb{Z}^{\geq 0}}\Delta,$ in
particular,
$2(\lambda,\dot{\alpha})/(\dot{\alpha},\dot{\alpha})\in{\mathbb{Z}}^{\geq 0}$
for all
$\dot{\alpha}\in{\dot{\phi}_{ind}}^{+}(\Delta):=({\dot{\phi}_{ind}}\setminus\\{0\\})\cap\hbox{\rm
span}_{\mathbb{Z}^{\geq 0}}\Delta.$

The following theorem is crucial for our main result in the next section.

######  Theorem 2.1.

Keep the same notations as above and define a functional

$\bm{\zeta}:\hbox{\rm span}_{\mathbb{R}}\Delta(=\hbox{\rm
span}_{\mathbb{R}}\dot{\phi})\longrightarrow\mathbb{R}$

such that $\bm{\zeta}(\Delta)>0$. Then, there is a nonzero weight vector $v$,
say e.g., of weight $\mu$ such that
$\mathfrak{k}^{\dot{\alpha}+n\delta}v=\\{0\\}$ for all $n\in\mathbb{Z}$ and
$\dot{\alpha}\in\dot{\phi}$ with $\bm{\zeta}(\dot{\alpha})>0$.

###### Proof.

Set

$r:=\left\\{\begin{array}[]{ll}1&X=Y^{(1)},\\\ 3&X=D_{4}^{(3)},\\\
2&\hbox{otherwise.}\end{array}\right.$

Using (2.6) and (2.7) as well as the same argument as in [3, Theorem 2.4(ii)],
we get $\mu\in\hbox{\rm supp}(V)$ with

(2.8) ${\mu+\dot{\gamma}+rn\delta\not\in\hbox{\rm supp}(V)\quad\hbox{and}\quad
2(\mu,\dot{\gamma})/(\dot{\gamma},\dot{\gamma})\in{\mathbb{Z}}^{\geq
0}\quad(\dot{\gamma}\in{\dot{\phi}_{ind}}^{+}(\Delta),n\in{\mathbb{Z}}).}$

In particular,

(2.9) $\mathfrak{k}^{\dot{\alpha}+n\delta}v=\\{0\\}\quad(v\in
V^{\mu},~{}\dot{\alpha}\in{\dot{\phi}_{ind}}^{+}(\Delta),~{}n\in
r{\mathbb{Z}}).$

If there is a nonzero weight vector $v\in V^{\mu}$ with
$\mathfrak{k}^{\dot{\alpha}+n\delta}v=\\{0\\}$ for all $n\in\mathbb{Z}$ and
$\dot{\alpha}\in\dot{\phi}$ with $\bm{\zeta}(\dot{\alpha})>0$, then we are
done. Otherwise, we pick $0\neq v\in V^{\mu}$ and note that by (2.9),

$\mathcal{A}:=\\{\dot{\alpha}\in\dot{\phi}\mid\bm{\zeta}(\dot{\alpha})>0,\mathfrak{k}^{\dot{\alpha}+m\delta}v\neq\\{0\\}\hbox{
for some $m\in{\mathbb{Z}}$}\\}$

is not empty. Next pick $\dot{\alpha}_{*}\in\mathcal{A}$ with
$\bm{\zeta}(\dot{\alpha}_{*})={\rm
max}\\{\bm{\zeta}(\dot{\alpha})\mid\dot{\alpha}\in\mathcal{A}\\}.$ Since
$\dot{\alpha}_{*}\in\mathcal{A},$ there is $m_{*}\in\mathbb{Z}$ with

$\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v\neq\\{0\\}.$

We let $0\neq w\in\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v$ and show that
$\mathfrak{k}^{\dot{\alpha}+n\delta}w=\\{0\\}$ for all $n\in\mathbb{Z}$ and
$\dot{\alpha}\in\dot{\phi}$ with $\dot{\alpha}+n\delta\in\phi$ and
$\bm{\zeta}(\dot{\alpha})>0$. We break the proof into the following cases.

$\bullet$ $\dot{\alpha}\in{\dot{\phi}_{ind}},~{}n\in r{\mathbb{Z}}$. In this
case, using (2.9), we have

$\displaystyle\mathfrak{k}^{\dot{\alpha}+n\delta}w\subseteq\mathfrak{k}^{\dot{\alpha}+n\delta}\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v\subseteq[\mathfrak{k}^{\dot{\alpha}+n\delta},\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}]v+\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}\underbrace{\mathfrak{k}^{\dot{\alpha}+n\delta}v}_{\\{0\\}}\subseteq\mathfrak{k}^{\dot{\alpha}+\dot{\alpha}_{*}+(n+m_{*}\delta)\delta}v.$

If $\dot{\alpha}+\dot{\alpha}_{*}+(n+m_{*})\delta$ is not a root, then
$\mathfrak{k}^{\dot{\alpha}+\dot{\alpha}_{*}+n\delta+m_{*}\delta}=\\{0\\}$ and
there is nothing to prove. If $\dot{\alpha}+\dot{\alpha}_{*}+(n+m_{*})\delta$
is a root, then since
$\zeta(\dot{\alpha}+\dot{\alpha}_{*})>\zeta(\dot{\alpha}_{*})$, due to the
choice of $\dot{\alpha}_{*}$, we have
$\mathfrak{k}^{\dot{\alpha}+\dot{\alpha}_{*}+(n+m_{*})\delta}v=\\{0\\}$ which
in turn implies that $\mathfrak{k}^{\dot{\alpha}+n\delta}w=\\{0\\}$ and so we
are done.

$\bullet$ $X=A_{2\ell-1}^{(2)},D_{\ell}^{(2)},E_{6}^{(2)},D_{4}^{(3)}$,
$\dot{\alpha}\in\dot{\phi}_{sh},~{}n\in r{\mathbb{Z}}\pm 1$. Note that as
$\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v\neq\\{0\\}$, by (2.9), we have
$\dot{\alpha}_{*}\in\dot{\phi}_{sh}$ and $m_{*}\in r{\mathbb{Z}}\pm 1.$ In
particular, we have

(2.10) $\dot{\alpha}_{*}+(m_{*}+n)\delta\in\phi\quad\hbox{and}\quad
2(\dot{\alpha}+\dot{\alpha}_{*},\dot{\alpha}_{*})/(\dot{\alpha}_{*},\dot{\alpha}_{*})>0.$

On the other hand, we have

$\displaystyle\mathfrak{k}^{\dot{\alpha}+n\delta}w\subseteq\mathfrak{k}^{\dot{\alpha}+n\delta}\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v\subseteq
V^{\mu+(\dot{\alpha}+\dot{\alpha}_{*})+(m_{*}+n)\delta}.$

If
$\mathfrak{k}^{\dot{\alpha}+n\delta}\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v\neq\\{0\\},$
we have $\mu+(\dot{\alpha}+\dot{\alpha}_{*})+(m_{*}+n)\delta\in\hbox{\rm
supp}(V).$ This together with (2.6), (2.8) and (2.10) implies that

$\mu+\dot{\alpha}=\mu+(\dot{\alpha}+\dot{\alpha}_{*})+(m_{*}+n)\delta-(\dot{\alpha}_{*}+(m_{*}+n)\delta)\in\hbox{\rm
supp}(V),$

which contradicts (2.8). Therefore,
$\mathfrak{k}^{\dot{\alpha}+n\delta}\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v=\\{0\\}$
and so, we are done.

$\bullet$ $X=A_{2\ell}^{(2)}$, $\dot{\alpha}\in\dot{\phi},~{}n\in
2{\mathbb{Z}}+1$. In this case, we have $r=2.$ Since
$\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v\neq\\{0\\}$, we have $m_{*}\in
2{\mathbb{Z}}+1.$ So

(2.11) $\dot{\phi}_{ind}+(m_{*}+n)\delta\subseteq\phi.$

We shall show $\mathfrak{k}^{\dot{\alpha}+n\delta}w=\\{0\\}.$ To this end, we
note that

$\displaystyle\mathfrak{k}^{\dot{\alpha}+n\delta}w\subseteq\mathfrak{k}^{\dot{\alpha}+n\delta}\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v\subseteq
V^{\mu+(\dot{\alpha}+\dot{\alpha}_{*})+(m_{*}+n)\delta}.$

If
$\mathfrak{k}^{\dot{\alpha}+n\delta}\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v=\\{0\\},$
then, we are done. So we assume
$\mathfrak{k}^{\dot{\alpha}+n\delta}\mathfrak{k}^{\dot{\alpha}_{*}+m_{*}\delta}v\neq\\{0\\}$
and show that it leads a contradiction. We have
$\mu+(\dot{\alpha}+\dot{\alpha}_{*})+(m_{*}+n)\delta\in\hbox{\rm supp}(V).$
Hence, we can contemplate the following scenarios.

  * (i)

If $\dot{\alpha},\dot{\alpha}_{*}\in\dot{\phi}_{ind},$ then since either
$2\frac{(\dot{\alpha}_{*}+\dot{\alpha},\dot{\alpha})}{(\dot{\alpha},\dot{\alpha})}\in{\mathbb{Z}}^{>0}$
or
$2\frac{(\dot{\alpha}_{*}+\dot{\alpha},\dot{\alpha}_{*})}{(\dot{\alpha}_{*},\dot{\alpha}_{*})}\in{\mathbb{Z}}^{>0}$,
we get using (2.6), (2.8) and (2.11) that $\mu+\dot{\alpha}\in\hbox{\rm
supp}(V)$ or $\mu+\dot{\alpha}_{*}\in\hbox{\rm supp}(V)$ contradicting (2.8).

  * (ii)

If exactly one of $\dot{\alpha},\dot{\alpha}_{*}$ belongs to
$\dot{\phi}_{ind},$ then, without loss of generality, we assume
$\dot{\alpha}=2\dot{\beta}$ for some short root $\dot{\beta}$ and
$\dot{\alpha}_{*}\in\dot{\phi}_{ind}.$ We have
$2\frac{(\mu+2\dot{\beta}+\dot{\alpha}_{*},\dot{\beta})}{(\dot{\beta},\dot{\beta})}>0$.
Using (2.6), this implies that
$\mu+\dot{\beta}+\dot{\alpha}_{*}+(m_{*}+n)\delta\in\hbox{\rm supp}(V)$. We
get a contradiction as in the previous case.

  * (iii)

If $\dot{\alpha},\dot{\alpha}_{*}\in\dot{\phi}_{ex},$ then, we have
$\dot{\alpha}=2\dot{\beta}$ and $\dot{\alpha}_{*}=2\dot{\beta}^{\prime}$ for
some $\dot{\beta},\dot{\beta}^{\prime}\in{\dot{\phi}}_{sh}.$ In particular, we
have

(2.12) $\dot{\beta}^{\prime}+(m_{*}+n)\delta\in\phi\quad\hbox{and}\quad
2(\dot{\beta}+\dot{\beta}^{\prime},\dot{\beta}^{\prime})/(\dot{\beta}^{\prime},\dot{\beta}^{\prime})>0.$

This together with (2.6) implies that

$\mu+2\dot{\beta}+\dot{\beta}^{\prime}=\mu+2(\dot{\beta}+\dot{\beta}^{\prime})+(m_{*}+n)\delta-(\dot{\beta}^{\prime}+(m_{*}+n)\delta)\in\hbox{\rm
supp}(V).$

Similarly, we get

$\mu+\dot{\beta}+\dot{\beta}^{\prime}=\mu+2\dot{\beta}+\dot{\beta}^{\prime}-\dot{\beta}\in\hbox{\rm
supp}(V),$

and finally, as
$2(\mu+\dot{\beta}+\dot{\beta}^{\prime},\dot{\beta}^{\prime})/(\dot{\beta}^{\prime},\dot{\beta}^{\prime})>0,$
we have

$\mu+\dot{\beta}=\mu+\dot{\beta}+\dot{\beta}^{\prime}-\dot{\beta}^{\prime}\in\hbox{\rm
supp}(V)$

which is a contradiction due to (2.8).

The proof is now complete. ∎

##  3\. Main results

Starting from this point until the conclusion of this section, we will adhere
to the following notations and conventions. Suppose that $\mathcal{L}$ is a
twisted affine Lie superalgebra with $\mathcal{L}_{1}\neq\\{0\\},$ standard
Cartan subalgebra ${\mathcal{H}}$ and corresponding root system $R=R_{0}\cup
R_{1}$. One knows that there exists an invariant even nondegenerate
supersymmetric $\mathbb{C}$-valued bilinear form $(\cdot,\cdot)$ on
$\mathcal{L}$ which is also nondegenerate on ${\mathcal{H}}$. The root system
$R$ of $\mathcal{L}$ with nonzero odd part can be identified in the following
table:

Table 1. Root systems of twisted affine Lie superalgebras Type |  $R$
---|---
$\stackrel{{\scriptstyle A(2m,2n-1)^{(2)}}}{{{\hbox{\tiny$(m,n\in{\mathbb{Z}}^{\geq 0},n\neq 0)$}}}}$ | $\begin{array}[]{rcl}{\mathbb{Z}}\delta&\cup&{\mathbb{Z}}\delta\pm\\{\epsilon_{i},\delta_{p},\epsilon_{i}\pm\epsilon_{j},\delta_{p}\pm\delta_{q},\epsilon_{i}\pm\delta_{p}\mid i\neq j,p\neq q\\}\\\ &\cup&(2{\mathbb{Z}}+1)\delta\pm\\{2\epsilon_{i}\mid 1\leq i\leq m\\}\\\ &\cup&2{\mathbb{Z}}\delta\pm\\{2\delta_{p}\mid 1\leq p\leq n\\}\end{array}$
$\stackrel{{\scriptstyle A(2m-1,2n-1)^{(2)}}}{{{\hbox{\tiny$(m,n\in{\mathbb{Z}}^{>0},(m,n)\neq(1,1))$}}}}$ | $\begin{array}[]{rcl}{\mathbb{Z}}\delta&\cup&{\mathbb{Z}}\delta\pm\\{\epsilon_{i}\pm\epsilon_{j},\delta_{p}\pm\delta_{q},\delta_{p}\pm\epsilon_{i}\mid i\neq j,p\neq q\\}\\\ &\cup&(2{\mathbb{Z}}+1)\delta\pm\\{2\epsilon_{i}\mid 1\leq i\leq m\\}\\\ &\cup&2{\mathbb{Z}}\delta\pm\\{2\delta_{p}\mid 1\leq p\leq n\\}\end{array}$
$\stackrel{{\scriptstyle A(2m,2n)^{(4)}}}{{{\hbox{\tiny$(m,n\in{\mathbb{Z}}^{\geq 0},(m,n)\neq(0,0))$}}}}$ | $\begin{array}[]{rcl}{\mathbb{Z}}\delta&\cup&{\mathbb{Z}}\delta\pm\\{\epsilon_{i},\delta_{p}\mid 1\leq i\leq m,\;1\leq p\leq n\\}\\\ &\cup&2{\mathbb{Z}}\delta\pm\\{\epsilon_{i}\pm\epsilon_{j},\delta_{p}\pm\delta_{q},\delta_{p}\pm\epsilon_{i}\mid i\neq j,p\neq q\\}\\\ &\cup&(4{\mathbb{Z}}+2)\delta\pm\\{2\epsilon_{i}\mid 1\leq i\leq m\\}\\\ &\cup&4{\mathbb{Z}}\delta\pm\\{2\delta_{p}\mid 1\leq p\leq n\\}\end{array}$
$\stackrel{{\scriptstyle D(m+1,n)^{(2)}}}{{{\hbox{\tiny$(m,n\in{\mathbb{Z}}^{\geq 0},n\neq 0)$}}}}$ | $\begin{array}[]{rcl}{\mathbb{Z}}\delta&\cup&{\mathbb{Z}}\delta\pm\\{\epsilon_{i},\delta_{p}\mid 1\leq i\leq m,\;1\leq p\leq n\\}\\\ &\cup&2{\mathbb{Z}}\delta\pm\\{2\delta_{p},\epsilon_{i}\pm\epsilon_{j},\delta_{p}\pm\delta_{q},\delta_{p}\pm\epsilon_{i}\mid i\neq j,p\neq q\\}\end{array}$

where

$(\delta,R)=\\{0\\},~{}(\epsilon_{i},\epsilon_{j})=\delta_{i,j},~{}(\delta_{p},\delta_{q})=-\delta_{p,q}.$

It is apparent from Table 1 that there exists $r\in\\{2,4\\}$ satisfying the
following:

(3.1) $R+r{\mathbb{Z}}\delta\subseteq R.$

Set

$\dot{R}:=\\{\dot{\alpha}\mid\dot{\alpha}+n\delta\in R~{}~{}(\hbox{for some
$n\in{\mathbb{Z}}$})\\}.$

By Table 1, we have

$\dot{R}^{\times}=\dot{R}\setminus\\{0\\}=(\dot{R}_{ns}\setminus\\{0\\})\cup\dot{R}_{re}$

in which

$\dot{R}_{\star}:=\\{\dot{\alpha}\in\dot{R}\mid
R\cap(\dot{\alpha}+{\mathbb{Z}}\delta)\subseteq
R_{\star}\\}\quad\quad(\star=ns,re),$

additionally, Table 2 provides details regarding real and nonsingular roots of
$\dot{R}:$

Table 2. Real and nonsingular roots of $\dot{R}$ Type |  $\dot{R}_{re}$ | $\dot{R}_{ns}\setminus\\{0\\}$
---|---|---
$\stackrel{{\scriptstyle A(2m,2n-1)^{(2)}}}{{{\hbox{\tiny$(m,n\in{\mathbb{Z}}^{\geq 0},n\neq 0)$}}}}$ | $\begin{array}[]{l}\\{\pm\epsilon_{i},\pm\delta_{p},\pm 2\epsilon_{i},\pm 2\delta_{p},\epsilon_{i}\pm\epsilon_{j},\delta_{p}\pm\delta_{q}\\\ \mid 1\leq i\neq j\leq m,1\leq p\neq q\leq n\\}\\\ \end{array}$ | $\begin{array}[]{l}\\{\pm\epsilon_{i}\pm\delta_{p}\mid 1\leq i\leq m,1\leq p\leq n\\}\\\ \vspace{-2mm}\\\ \subseteq\dot{R}_{re}+\dot{R}_{re}\end{array}$
$\stackrel{{\scriptstyle A(2m-1,2n-1)^{(2)}}}{{{\hbox{\tiny$(m,n\in{\mathbb{Z}}^{>0},(m,n)\neq(1,1))$}}}}$ | $\begin{array}[]{l}\\{\pm 2\epsilon_{i},\pm 2\delta_{p},\epsilon_{i}\pm\epsilon_{j},\delta_{p}\pm\delta_{q}\\\ \mid 1\leq i\neq j\leq m,1\leq p\neq q\leq n\\}\\\ \end{array}$ | $\begin{array}[]{l}\\{\pm\epsilon_{i}\pm\delta_{p}\mid 1\leq i\leq m,1\leq p\leq n\\}\\\ \vspace{-2mm}\\\ \subseteq\frac{1}{2}(\dot{R}_{re}+\dot{R}_{re})\end{array}$
$\stackrel{{\scriptstyle A(2m,2n)^{(4)}}}{{{\hbox{\tiny$(m,n\in{\mathbb{Z}}^{\geq 0},(m,n)\neq(0,0))$}}}}$ | $\begin{array}[]{l}\\{\pm\epsilon_{i},\pm\delta_{p},\pm 2\epsilon_{i},\pm 2\delta_{p},\epsilon_{i}\pm\epsilon_{j},\delta_{p}\pm\delta_{q}\\\ \mid 1\leq i\neq j\leq m,1\leq p\neq q\leq n\\}\\\ \end{array}$ | $\begin{array}[]{l}\\{\pm\epsilon_{i}\pm\delta_{p}\mid 1\leq i\leq m,1\leq p\leq n\\}\\\ \vspace{-2mm}\\\ \subseteq\dot{R}_{re}+\dot{R}_{re}\end{array}$
$\stackrel{{\scriptstyle D(m+1,n)^{(2)}}}{{{\hbox{\tiny$(m,n\in{\mathbb{Z}}^{\geq 0},n\neq 0)$}}}}$ | $\begin{array}[]{l}\\{\pm\epsilon_{i},\pm\delta_{p},\pm 2\delta_{p},\epsilon_{i}\pm\epsilon_{j},\delta_{p}\pm\delta_{q}\\\ \mid 1\leq i\neq j\leq m,1\leq p\neq q\leq n\\}\\\ \end{array}$ | $\begin{array}[]{l}\\{\pm\epsilon_{i}\pm\delta_{p}\mid 1\leq i\leq m,1\leq p\leq n\\}\\\ \vspace{-2mm}\\\ \subseteq\dot{R}_{re}+\dot{R}_{re}\end{array}$

Moreover, we have

${\mathcal{H}}={\mathbb{C}}c\oplus{\mathbb{C}}d\oplus\sum_{\dot{\alpha}\in\dot{R}}{\mathbb{C}}t_{\dot{\alpha}}$

with

$(c,d)=1,(c,c)=(d,d)=0,(c,\sum_{\dot{\alpha}\in\dot{R}}{\mathbb{C}}t_{\dot{\alpha}})=(d,\sum_{\dot{\alpha}\in\dot{R}}{\mathbb{C}}t_{\dot{\alpha}})=\\{0\\}.$

To arrive at the results, it is essential to recall the following theorem
which grants us effective control over the roots.

######  Theorem 3.1.

([23, Theorem 4.8]) Suppose that $V$ is an ${\mathcal{H}}$-weight
$\mathcal{L}$-module having shadow. Then for each $\beta\in R_{re},$ one of
the following will happen:

  * (i)

$\beta$ is full-locally nilpotent, that is $(\beta+{\mathbb{Z}}\delta)\cap
R\subseteq R^{ln}(V),$

  * (ii)

$\beta$ is full-injective, that is $(\beta+{\mathbb{Z}}\delta)\cap R\subseteq
R^{in}(V),$

  * (iii)

$\beta$ is down-nilpotent hybrid, that is, there exist $m\in{\mathbb{Z}}$ and
$t\in\\{0,1,-1\\}$ such that for $\gamma:=\beta+m\delta,$

$\displaystyle(\gamma+{\mathbb{Z}}^{\geq 1}\delta)\cap R\subseteq
R^{in}(V),\quad(\gamma+{\mathbb{Z}}^{\leq 0}\delta)\cap R\subseteq R^{ln}$
$\displaystyle(-\gamma+{\mathbb{Z}}^{\geq t}\delta)\cap R\subseteq
R^{in}(V),\quad(-\gamma+{\mathbb{Z}}^{\leq t-1}\delta)\cap R\subseteq R^{ln},$

  * (iv)

$\beta$ is up-nilpotent hybrid, that is, there exist $m\in{\mathbb{Z}}$ and
$t\in\\{0,1,-1\\}$ such that for $\eta:=\beta+m\delta,$

$\displaystyle(\eta+{\mathbb{Z}}^{\leq-1}\delta)\cap R\subseteq
R^{in}(V),\quad(\eta+{\mathbb{Z}}^{\geq 0}\delta)\cap R\subseteq R^{ln}(V)$
$\displaystyle(-\eta+{\mathbb{Z}}^{\leq-t}\delta)\cap R\subseteq
R^{in}(V),\quad(-\eta+{\mathbb{Z}}^{\geq 1-t}\delta)\cap R\subseteq
R^{ln}(V).$

Suppose that $\mathscr{G}$ is a subalgebra of $\mathcal{L}$ containing
${\mathcal{H}}$ with corresponding root system $S$ and $V$ is an
${\mathcal{H}}$-weight $\mathscr{G}$-module having shadow. Suppose that for
each $\alpha\in S^{\times},$ $(\alpha+{\mathbb{Z}}\delta)\cap R\subseteq
S.$333This assumption makes sure that the proof of [23, Theorem 4.8] can be
utilized for modules over $\mathscr{G}$ as well. A real root $\alpha$ is said
to be hybrid, if $\alpha$ is either down-nilpotent hybrid or up-nilpotent
hybrid. A subset $T$ of $S$ is called a full-locally nilpotent (resp. full-
injective, down-nilpotent hybrid, up-nilpotent hybrid, hybrid) set if all
elements of $T_{re}=T\cap R_{re}$ are full-locally nilpotent (resp. full-
injective, down-nilpotent hybrid, up-nilpotent hybrid, hybrid).

######  Proposition 3.2.

Suppose that $S$ is a nonempty symmetric closed444It means that $S=-S$ and
$(S+S)\cap R\subseteq S.$ subset of $R$ with $S_{re}=R_{re}\cap
S\neq\emptyset$. Set

$S_{0}:=S\cap R_{0}~{}\hbox{ ~{} as well as
~{}}~{}\dot{S}:=\\{\dot{\alpha}\in{\dot{R}}\mid(\dot{\alpha}+{\mathbb{Z}}\delta)\cap
S_{0}\neq\emptyset\\}$

and assume that

$(\dot{\alpha}+\mathbb{Z}\delta)\cap R_{0}\subseteq
S\quad\quad(\dot{\alpha}\in\dot{S}^{\times}=\dot{S}\setminus\\{0\\}).$

Then the following statements hold:

  * (a)

$\dot{S}$ is a finite root system, say e.g. with irreducible components
$\dot{S}(1),\ldots,\dot{S}({n}).$

  * (b)

For $1\leq i\leq n,$ set $S{(i)}:={(\dot{S}{(i)}}+\mathbb{Z}\delta)\cap R_{0}$
and recall (2.2). Then

$\mathcal{G}{(i)}:={\mathcal{H}}+\raisebox{0.43057pt}{\scalebox{1.0}{$\displaystyle\sum_{\alpha\in
S{(i)}^{\times}}\;$}}\mathcal{L}^{\alpha}+\raisebox{0.43057pt}{\scalebox{1.0}{$\displaystyle\sum_{\alpha,\beta\in
S{(i)}^{\times}}\;$}}[\mathcal{L}^{\alpha},\mathcal{L}^{\beta}]$

is an affine Lie algebra, up to a central space.

###### Proof.

(a) is easily verified; just note that as $2(R_{1}\cap R_{re})\subseteq
R_{0},$ we can deduce that $S_{0}\cap R_{re}\neq\\{0\\}$ and so
$\dot{S}\neq\\{0\\}.$

(b) We first recall that the form $(\cdot,\cdot)$ is nondegenerate on
${\mathbb{C}}c\oplus{\mathbb{C}}d$ as well as on ${\mathcal{H}}$ and on
$\raisebox{0.43057pt}{\scalebox{1.0}{$\displaystyle\sum_{\dot{\alpha}\in\dot{S}(i)^{\times}}\;$}}{\mathbb{C}}t_{\dot{\alpha}}$,
and that
$({\mathbb{C}}c\oplus{\mathbb{C}}d,\raisebox{0.43057pt}{\scalebox{1.0}{$\displaystyle\sum_{\dot{\alpha}\in\dot{S}(i)^{\times}}\;$}}{\mathbb{C}}t_{\dot{\alpha}})=\\{0\\}$.
Setting

$\mathfrak{h}(i):=({\mathbb{C}}c\oplus{\mathbb{C}}d)+\sum_{\alpha\in
S(i)}{\mathbb{C}}t_{\alpha}={\mathbb{C}}c\oplus{\mathbb{C}}d\oplus\sum_{\dot{\alpha}\in\dot{S}(i)^{\times}}{\mathbb{C}}t_{\dot{\alpha}},$

we get that the form is nondegenerate on $\mathfrak{h}(i)$ and so, there is an
orthogonal complement $T_{i}$ for $\mathfrak{h}(i)$ in ${\mathcal{H}};$ in
particular, for $\alpha\in S(i)$, $x\in\mathcal{L}^{\alpha}$ and $h\in T_{i},$
we have

$\displaystyle[h,x]=\alpha(h)x=(t_{\alpha},h)x\in(\mathfrak{h}(i),T_{i})x=\\{0\\},$

that is, $T_{i}$ is contained in the center of ${\mathcal{G}}_{i}.$ We claim
that

$\mathfrak{k}(i):=\mathfrak{h}(i)+\raisebox{0.43057pt}{\scalebox{1.0}{$\displaystyle\sum_{\alpha\in
S{(i)}^{\times}}\;$}}\mathcal{L}^{\alpha}+\raisebox{0.43057pt}{\scalebox{1.0}{$\displaystyle\sum_{\alpha,\beta\in
S{(i)}^{\times}}\;$}}[\mathcal{L}^{\alpha},\mathcal{L}^{\beta}]$

is an affine Lie algebra.

To this end, we first note that for each element $\alpha\in S(i),$ we have
$\alpha(T_{i})=\\{0\\}$. So, for distinct elements $\alpha,\beta\in S(i),$ the
restriction of $\alpha$ to $\mathfrak{h}(i)$ is different from the restriction
of $\beta$ to $\mathfrak{h}(i).$ This, in particular, implies that the
decomposition

$\mathfrak{k}(i)=\mathfrak{h}(i)\oplus\raisebox{0.86108pt}{\scalebox{0.8}{$\displaystyle\bigoplus_{\alpha\in
S{(i)}^{\times}}\;$}}\mathcal{L}^{\alpha}\oplus\raisebox{0.86108pt}{\scalebox{0.8}{$\displaystyle\bigoplus_{0\neq
k\in{\mathbb{Z}}}\;$}}(\sum_{\dot{\alpha}\in\dot{S}(i)^{\times}}\sum_{t\in{\mathbb{Z}}}[\mathcal{L}^{\dot{\alpha}+t\delta},\mathcal{L}^{-\dot{\alpha}+(k-t)\delta}])$

coincides with the root space decomposition of $\mathfrak{k}(i)$ with respect
to $\mathfrak{h}(i).$ It is now easily verified that for

$(\mathfrak{k}(i))_{c}:=\sum_{\alpha\in{S(i)^{\times}}}\mathcal{L}^{\alpha}\oplus\sum_{\dot{\alpha}\in\dot{S}(i)^{\times}}\sum_{m,n\in{\mathbb{Z}}}[\mathcal{L}^{\dot{\alpha}+m\delta},\mathcal{L}^{-\dot{\alpha}+n\delta}],$

we have

$\mathfrak{k}(i)=(\mathfrak{k}(i))_{c}\oplus{\mathbb{C}}d$

and that
$\\{x\in\mathfrak{k}(i)\mid(x,(\mathfrak{k}(i))_{c})=\\{0\\}\\}\subseteq(\mathfrak{k}(i))_{c}.$
This together with [1, Theorem 2.32] implies that $\mathfrak{k}(i)$ is an
affine Lie algebra, as we desired. ∎

To present the main result, we require the following three lemmas.

######  Lemma 3.3.

Keep the same notations as in Proposition 3.2 and its proof. Assume $V$ is a
nonzero finite weight module over

$\mathcal{G}_{S}:=\sum_{\alpha\in S}\mathcal{L}^{\alpha}$

such that $c$ acts trivially on $V$ and $S^{ln}(V)=S_{re}.$ Fix a base
$\Delta$ of the finite root system $\dot{S}$. Assume $\bm{\zeta}$ is a
functional on $\hbox{\rm span}_{\mathbb{R}}\dot{S}$ with
$\bm{\zeta}(\Delta)>0$. Then, there exist $\mu\in\hbox{\rm supp}(V)$ and
$0\neq v\in V^{\mu}$ such that

$\mathcal{L}^{\dot{\alpha}+m\delta}v=\\{0\\}\quad\quad(m\in{\mathbb{Z}},~{}\bm{\zeta}(\dot{\alpha})>0).$

###### Proof.

By Proposition 3.2(b), for each $1\leq i\leq n$, $\mathcal{G}{(i)}$ is an
affine subalgebra of $\mathcal{G}_{S},$ up to a central subspace. Set
$V(0):=V$ and for $1\leq i\leq n,$ define

$V(i):=\text{span}_{\mathbb{C}}\\{v\in
V(i-1)\mid\mathcal{L}^{\dot{\alpha}+m\delta}v=\\{0\\}~{}(\dot{\alpha}\in\dot{S}(i),~{}\bm{\zeta}(\dot{\alpha})>0,~{}m\in{\mathbb{Z}})\\}.$

It is easy to see that for $0\leq i\leq n-1,$ $V(i)$ is a finite weight
${\mathcal{G}}(i+1)$-module. In particular, using Theorem 2.1 together with an
induction process, we get $V(i+1)\neq\\{0\\}.$ In particular
$V(n)\neq\\{0\\}.$ This completes the proof. ∎

Using a modification of the proof of [23, Lemma 5.6], one gets the following
lemma.

######  Lemma 3.4.

Suppose that $\mathscr{G}$ is a subalgebra of $\mathcal{L}$ containing
${\mathcal{H}}$ with corresponding root system $T$. Assume $V$ is a
$\mathscr{G}$-module having shadow. Assume $S$ is a symmetric closed nonempty
subset of $T$ with $S_{re}\neq\emptyset$ and ${(S+{\mathbb{Z}}\delta)\cap
R_{0}}\subseteq S.$ If $S$ is hybrid, then either all real roots of $S$ are
up-nilpotent hybrid or all are down-nilpotent hybrid.

######  Lemma 3.5.

Keep the same notations and assumptions as in Lemma 3.4, and define

(3.2) $\displaystyle
P:=\begin{cases}S^{ln}(V)\cup-S^{in}(V)\cup({\mathbb{Z}}^{\geq 0}\delta\cap
S)&\mbox{if $S$ is up-nilpotent hybrid,}\\\
S^{ln}(V)\cup-S^{in}(V)\cup(\mathbb{Z}^{\leq 0}\delta\cap S)&\mbox{if $S$ is
down-nilpotent hybrid}.\end{cases}$

Then, there exists a functional $\bm{\zeta}:\text{\rm
span}_{\mathbb{R}}({S\setminus R_{ns}})\rightarrow\mathbb{R}$ such that

$P=\\{0,\alpha\in S\setminus R_{ns}\;|\;\bm{\zeta}(\alpha)\geq 0\\}.$

In particular

$\\{\alpha\in S\cap R_{re}\;|\;\bm{\zeta}(\alpha)>0\\}\subseteq
S^{ln}(V)\quad\text{and}\quad\\{\alpha\in S\cap
R_{re}\;|\;\bm{\zeta}(\alpha)<0\\}\subseteq S^{in}(V).$

###### Proof.

We know form [23, Theorem 4.7] that if $\alpha$ is a real odd root, then
$\alpha\in T^{ln}(V)$ if and only if $2\alpha\in T^{ln}(V).$ So, without loss
of generality, we assume $S\subseteq R_{0}.$ We know that
$\dot{S}=\\{\dot{\alpha}\in\dot{R}\mid(\dot{\alpha}+{\mathbb{Z}}\delta)\cap
S\neq\emptyset\\}$ is a finite root system, say with irreducible components
$\dot{S}(1),\ldots,\dot{S}(n).$ For each $i=1,\ldots,n$, let $P_{i}$ denote
the set defined as follows:

$\displaystyle
P_{i}:=\begin{cases}S(i)^{ln}(V)\cup-S(i)^{in}(V)\cup(\mathbb{Z}^{\geq
0}\delta\cap S(i))&\mbox{if $S(i)$ is up-nilpotent hybrid},\\\
S(i)^{ln}(V)\cup-S(i)^{in}(V)\cup(\mathbb{Z}^{\leq 0}\delta\cap S(i))&\mbox{if
$S(i)$ is down-nilpotent hybrid}.\end{cases}$

We have $P=P_{1}\cup\cdots\cup P_{n}$. Recalling Lemma 3.4, without loss of
generality, we assume $S(1),\ldots,S(n)$ are up-nilpotent hybrid. Define
$\mathcal{G}(i)$’s as in Lemma 2.1 and assume $\mathfrak{s}(i)$’s are their
root systems, respectively. We note that for
$\mathfrak{s}:=\cup_{i=1}^{n}\mathfrak{s}(i),$

$\hbox{\rm span}_{\mathbb{R}}{S}=\hbox{\rm span}_{\mathbb{R}}\mathfrak{s}.$

Using affine theory (see [23, Remark 5.5] for the details), there are bases
$\dot{\Pi}_{1},\ldots,\dot{\Pi}_{n}$ of finite root systems $\dot{S}(i)$’s
respectively such that for specific roots $\theta_{i}$’s of $\dot{S}(i)$’s
respectively and $s_{i}$’s defined by

$\mathfrak{s}(i)_{im}=s_{i}\mathbb{Z}\delta\quad(1\leq i\leq n),$

we have

$\Pi_{i}:=\dot{\Pi}_{i}\cup\\{s_{i}\delta-\theta_{i}\\}\subseteq P_{i}.$

Now, the statement is proved by applying a modified version of the same
argument as presented in the proof of [23, Lemma 5.7]. ∎

Now, we are ready to prove our main result.

######  Theorem 3.6.

Recall that $\mathcal{L}$ is a twisted affine Lie superalgebra, and as noted
in Table 2, there is $\kappa\in\\{1,2\\}$ such that
$\dot{R}_{ns}\subseteq(1/\kappa)(\dot{R}_{re}+\dot{R}_{re}).$ Suppose that
$T,T(1)$ and $T(2)$ are nonempty symmetric closed subsets of $R$ with
$T(1)\cup T(2)\subseteq T.$ Assume $V$ is a finite weight module over
$\mathcal{G}_{{}_{T}}:=\sum_{\alpha\in T}\mathcal{L}^{\alpha}$ which has
shadow and that its level is zero, that is, the canonical central element of
$\mathcal{L}$ acts trivially on $V$. Set

$\dot{T}:=\\{\dot{\alpha}\in\dot{R}\mid(\dot{\alpha}+{\mathbb{Z}}\delta)\cap
T\neq\emptyset\\}\quad\hbox{and}\quad\dot{T}(i):=\\{\dot{\alpha}\in\dot{R}\mid(\dot{\alpha}+{\mathbb{Z}}\delta)\cap
T(i)\neq\emptyset\\}\quad(i=1,2).$

Recall (2.2) and assume that

  * •

for each $\dot{\alpha}\in\dot{T},$ $(\dot{\alpha}+{\mathbb{Z}}\delta)\cap
R\subseteq T,$

  * •

for $i=1,2,$ $T(i)_{re}\neq\emptyset$ and $T_{re}=T(1)_{re}\cup T(2)_{re},$

  * •

$(T(2),T(1))=\\{0\\}$,

  * •

$\dot{T}_{ns}\subseteq(1/\kappa)(\dot{T}_{re}+\dot{T}_{re})$ and for $i=1,2,$
$\dot{T}(i)_{ns}\subseteq(1/\kappa)(\dot{T}(i)_{re}+\dot{T}(i)_{re}).$

Then, we have the following:

  * (a)

For $i=1,2,$ $T(i)_{im}=T(i)\cap R_{im}$ is either ${\mathbb{Z}}\delta$ or
$2{\mathbb{Z}}\delta.$

  * (b)

If $T(1)$ is full-locally nilpotent while $T(2)$ is hybrid. Then, there exist
a parabolic subset $P$ of $T$ and a nonzero weight vector $v$ such that

$\mathcal{L}^{\alpha}v=\\{0\\}\quad\hbox{ for
all}\quad\alpha\in(P\setminus(-P\cup{\mathbb{Z}}\delta))\cup((P\setminus\\{0\\})\cap
T(2)\cap{\mathbb{Z}}\delta).$

  * (c)

If $T(2)_{im}={\mathbb{Z}}\delta$, then there is a nonzero weight vector $v$
such that

$\mathcal{L}^{\alpha}v=\\{0\\}\quad(\hbox{for all }\alpha\in P\setminus-P).$

In particular, if $V$ is simple, there is a simple finite weight module
$\Omega$ over
${\mathcal{G}}_{{}_{T}}^{\circ}:=\raisebox{0.86108pt}{\scalebox{0.8}{$\displaystyle\bigoplus_{\alpha\in
P\cap-P}\;$}}\mathcal{L}^{\alpha}$ such that $V={\rm Ind}_{{}_{P}}(\Omega).$

###### Proof.

(a) easily follows from the facts that $T(i)$ is symmetric and closed and that
$(T(i)+{\mathbb{Z}}\delta)\cap R\subseteq T(i).$

(b) We first need to set some notations. We start with recalling that by using
Table 1 and [22, Tables 2 and 5], we have the following:

(3.3) (1) For each $0\neq\dot{\alpha}\in\dot{T},$ there are
$r_{\dot{\alpha}}\in\\{1,2,4\\}$ and $0\leq k_{\dot{\alpha}}<r_{\dot{\alpha}}$
such that $(\dot{\alpha}+{\mathbb{Z}}\delta)\cap
R=(\dot{\alpha}+{\mathbb{Z}}\delta)\cap
T=\dot{\alpha}+k_{\dot{\alpha}}\delta+r_{\dot{\alpha}}{\mathbb{Z}}\delta$ and
if $k_{\dot{\alpha}}\neq 0,$ then $k_{\dot{\alpha}}|r_{\dot{\alpha}}$. (2) If
$\dot{\alpha}\in\dot{T}_{re}$, then $(\dot{\alpha}+{\mathbb{Z}}\delta)\cap
T\cap R_{1}$ is one of the following: $\emptyset$,
$\dot{\alpha}+{\mathbb{Z}}\delta$, $\dot{\alpha}+2{\mathbb{Z}}\delta$,
$\dot{\alpha}+(2{\mathbb{Z}}+1)\delta$.

Using Lemma 3.4, without loss of generality, we can assume that all real roots
of $T(2)$ are up-nilpotent hybrid. Following the same argument as in [23,
Lemma 5.4], one sees that

$P_{2}:=T(2)_{re}^{ln}\cup-T(2)_{re}^{in}\cup({\mathbb{Z}}^{\geq 0}\delta\cap
T(2))$

satisfies

$(P_{2}+P_{2})\cap(T(2)\setminus R_{ns})\subseteq P_{2}\quad\hbox{and}\quad
P_{2}\cup-P_{2}=T(2)\setminus R_{ns},$

that is $P_{2}$ is a parabolic subset of $T(2)\setminus R_{ns}.$ Moreover, by
Lemma 3.5, there is a linear functional $\bm{\zeta_{2}}$ on $\hbox{\rm
span}_{\mathbb{R}}(T(2)\setminus R_{ns})=\hbox{\rm span}_{\mathbb{R}}T(2)$
such that

$P_{2}=\\{0,\alpha\in T(2)\setminus R_{ns}\mid\bm{\zeta_{2}}(\alpha)\geq
0\\}.$

Recall (3.3)(1) and for each $\dot{\alpha}\in\dot{T}(2),$ pick
$p_{\dot{\alpha}}\in{\mathbb{Z}}$ such that for

(3.4)
$\beta_{\dot{\alpha}}:=\dot{\alpha}+k_{\dot{\alpha}}\delta+r_{\dot{\alpha}}p_{\dot{\alpha}}\delta,$

we have

$\bm{\zeta_{2}}(\beta_{\dot{\alpha}})>0\quad\hbox{and}\quad\bm{\zeta_{2}}(\dot{\alpha}+k_{\dot{\alpha}}\delta+r_{\dot{\alpha}}(p_{\dot{\alpha}}-1)\delta)<0.$

We next set

(3.5) $S:=R_{0}\cap
T(1)\quad\hbox{and}\quad\dot{S}:=\\{\dot{\alpha}\in\dot{T}\mid
S\cap(\dot{\alpha}+{\mathbb{Z}}\delta)\neq\emptyset\\}.$

Then, $\dot{S}$ is a finite root system. Fix a base $\Delta$ of $\dot{S}$ and
define the linear functional $\bm{\zeta_{1}}$ on $\hbox{\rm
span}_{\mathbb{R}}\dot{T}(1)_{re}$ such that

${\rm
max}\\{\bm{\zeta_{2}}(\beta_{\dot{\eta}})\mid\dot{\eta}\in\dot{T}(2)_{re}\\}<\frac{1}{2}\bm{\zeta_{1}}(\dot{\gamma})\quad\quad(\dot{\gamma}\in\dot{S}^{+}(\Delta)=\dot{S}\cap\hbox{\rm
span}_{{\mathbb{Z}}^{\geq 0}}\Delta);$

in particular, we have

$-\frac{1}{2}\bm{\zeta_{1}}(\dot{\gamma})<\bm{\zeta_{2}}(\beta_{\dot{\eta}})<\frac{1}{2}\bm{\zeta_{1}}(\dot{\gamma})\quad\quad(\dot{\gamma}\in\dot{S}^{\times},~{}\dot{\eta}\in\dot{T}(2)_{re}).$

This implies that for $\dot{\eta}\in\dot{T}(2)_{re}$ and
$\dot{\gamma}\in\dot{S}^{\times},$

(3.6)
$\bm{\zeta_{1}}(\dot{\gamma})+\bm{\zeta_{2}}(\dot{\beta}_{\dot{\eta}})>\frac{1}{2}\bm{\zeta_{1}}(\dot{\gamma})+\bm{\zeta_{2}}(\dot{\beta}_{\dot{\eta}})>0\Longleftrightarrow\bm{\zeta_{1}}(\dot{\gamma})>0.$

Consider (3.4) and define

$\displaystyle\bm{\zeta}:$
$\displaystyle\\{0\\}\cup(\dot{T}\setminus\dot{T}(2))\longrightarrow{\mathbb{R}}$
$\displaystyle\dot{\alpha}\mapsto\left\\{\begin{array}[]{ll}\bm{\zeta_{1}}(\dot{\alpha})&\dot{\alpha}\in\dot{T}(1)\subseteq\hbox{\rm
span}_{\mathbb{R}}\dot{T}(1)_{re},\\\
\bm{\zeta_{1}}(\dot{\gamma})+\bm{\zeta_{2}}(\beta_{\dot{\eta}})&\dot{\alpha}=\frac{1}{\kappa}(\dot{\gamma}+\dot{\eta}),\dot{\gamma}\in\dot{T}(1)_{re},\dot{\eta}\in\dot{T}(2)_{re}.\end{array}\right.$

Now, we are ready to go through the main body of the proof. Before that, we
mention that we will frequently use the following fact:

(3.7) If $\mu\in\hbox{\rm supp}(V)$ and $\pm\alpha\in T^{ln}(V)$ with
$2(\mu,\alpha)/(\alpha,\alpha)>0,$ then $\mu-\alpha\in\hbox{\rm supp}(V).$

Recall (3.5) and consider $V$ as a module over
${\mathcal{G}}_{{}_{S}}:=\raisebox{0.86108pt}{\scalebox{0.8}{$\displaystyle\bigoplus_{\alpha\in
S}\;$}}\mathcal{L}^{\alpha}$ and use Lemma 3.3 to get that the set

(3.8) $\displaystyle W:=$ $\displaystyle\\{v\in
V\mid\mathcal{L}^{\dot{\alpha}+m\delta}v=\\{0\\}\quad(\dot{\alpha}\in\dot{S}^{+}(\Delta),~{}m\in{\mathbb{Z}},\dot{\alpha}+m\delta\in
S\subseteq R_{0})\\}$ $\displaystyle=$ $\displaystyle\\{v\in
V\mid\mathcal{L}^{\dot{\alpha}+m\delta}v=\\{0\\}\quad(\dot{\alpha}\in\dot{T}(1)_{re},~{}m\in{\mathbb{Z}},\dot{\alpha}+m\delta\in
S,~{}\bm{\zeta_{1}}(\dot{\alpha})>0)\\}$

is a nonzero vector space. For each nonzero weight vector $u\in W,$ set

$\mathcal{A}_{u}:=\\{\dot{\alpha}\in\dot{T}\setminus\dot{T}(2)\mid\bm{\zeta}(\dot{\alpha})>0,\exists
m\in{\mathbb{Z}}\hbox{ s.t.
}~{}\mathcal{L}^{\dot{\alpha}+m\delta}u\neq\\{0\\}\\}.$

We note that

(3.9) $(\mathcal{A}_{u}+{\mathbb{Z}}\delta)\cap R\subseteq R_{1}.$

The proof will be completed through the following claims:

Claim 1. The subset

$P:=\\{\dot{\alpha}+m\delta\in((\dot{T}\setminus\dot{T}(2))+{\mathbb{Z}}\delta)\cap
R\mid\bm{\zeta}(\dot{\alpha})>0\\}\cup\\{\alpha\in
T(2)\cup({\mathbb{Z}}\delta\cap T)\mid\bm{\zeta_{2}}(\alpha)\geq 0\\}$

of $T$ is a parabolic subset of $T$.

Claim 2. Suppose that $u_{0}$ is a nonzero weight vector in $W$ (see (3.8))
such that $\mathcal{A}_{u_{0}}$ is of minimum cardinality. Suppose
$\mathcal{A}_{u_{0}}\not=\emptyset,$ and
$\dot{\alpha}_{*}\in\mathcal{A}_{u_{0}}$ satisfies

$\bm{\zeta}(\dot{\alpha}_{*})={\rm
max}\\{\bm{\zeta}(\dot{\alpha})\mid\dot{\alpha}\in\mathcal{A}_{u_{0}}\\}.$

Assume that $m\in{\mathbb{Z}}$ with
$\mathcal{L}^{\dot{\alpha}_{*}+m\delta}u_{0}\neq\\{0\\}$ and $0\neq
v\in\mathcal{L}^{\dot{\alpha}_{*}+m\delta}u_{0}$. Then we have

  * (1)

$v\in W$,

  * (2)

$\mathcal{A}_{v}=\mathcal{A}_{u_{0}}.$

Claim 3. Suppose that $u_{0}$ and $\dot{\alpha}_{*}$ are as in the statement
of the previous claim. Then there are $m_{i}\in{\mathbb{Z}}$ and weight
vectors $u_{i}$ $(i\geq 0)$ such that

$\dot{\alpha}_{*}+m_{0}\delta,\dot{\alpha}_{*}+m_{i}\delta\in
R_{1}\quad\hbox{and}\quad 0\neq
u_{i}\in\mathcal{L}^{\dot{\alpha}_{*}+m_{i-1}\delta}u_{i-1}\quad(i\geq 1).$

Claim 4. Suppose that $u_{0}$ is a nonzero weight vector in $W$ such that
$\mathcal{A}_{u_{0}}$ is of minimum cardinality. Then,
$\mathcal{A}_{u_{0}}=\emptyset.$

Claim 5. The vector space

$U:=\\{v\in
V\mid\mathcal{L}^{\dot{\alpha}+m\delta}v=\\{0\\}\quad(\dot{\alpha}\in\dot{T}\setminus\dot{T}(2),~{}m\in{\mathbb{Z}},\dot{\alpha}+m\delta\in
R,~{}\bm{\zeta}(\dot{\alpha})>0)\\}$

is a nonzero module over
$\mathcal{L}_{{}_{T(2)}}=\raisebox{0.86108pt}{\scalebox{0.8}{$\displaystyle\bigoplus_{\alpha\in
T(2)}\;$}}\mathcal{L}^{\alpha}.$

Claim 6. There is a nonzero weight vector $u\in U$ with
$\mathcal{L}^{\alpha}u=\\{0\\}$ for $\alpha\in T(2)$ with
$\bm{\zeta_{2}}(\alpha)>0.$

Proof of Claim 1. It is enough to show that for $\beta,\beta^{\prime}\in
P\setminus{\mathbb{Z}}\delta$ with $\beta+\beta^{\prime}\in T,$ we have
$\beta+\beta^{\prime}\in P.$ If $\beta,\beta^{\prime}\in T(2),$ then as $T(2)$
is closed, we get that $\beta+\beta^{\prime}\in T(2)$ and so,
$\bm{\zeta_{2}}(\beta+\beta^{\prime})=\bm{\zeta_{2}}(\beta)+\bm{\zeta_{2}}(\beta^{\prime})>0,$
i.e., $\beta+\beta^{\prime}\in P.$

Next assume $\beta\not\in T(2).$ Then, we have $\beta=\dot{\beta}+k_{1}\delta$
with $k_{1}\in{\mathbb{Z}}$ and $\dot{\beta}=\dot{\gamma}+\dot{\eta}$ in which

  * •

either $0\neq\dot{\gamma}\in\dot{T}(1)$ and $\dot{\eta}=0,$

  * •

or $\dot{\gamma}\in\frac{1}{\kappa}\dot{T}(1)_{re}\setminus\\{0\\}$ and
$\dot{\eta}\in\frac{1}{\kappa}\dot{T}(2)_{re}\setminus\\{0\\}$.

Also, $\beta^{\prime}=\dot{\beta}^{\prime}+k_{2}\delta$ with
$k_{2}\in{\mathbb{Z}}$ and
$\dot{\beta}^{\prime}=\dot{\gamma}^{\prime}+\dot{\eta}^{\prime}$ for which one
of the following happens:

  * •

$0\neq\dot{\gamma}^{\prime}\in\dot{T}(1)$ and $\dot{\eta}^{\prime}=0,$

  * •

$\dot{\gamma}^{\prime}=0$ and $0\neq\dot{\eta}^{\prime}\in\dot{T}(2)$,

  * •

$0\neq\dot{\gamma}^{\prime}\in\frac{1}{\kappa}\dot{T}(1)_{re}$ and
$0\neq\dot{\eta}^{\prime}\in\frac{1}{\kappa}\dot{T}(2)_{re}.$

We have

$\beta+\beta^{\prime}=\dot{\beta}+\dot{\beta}^{\prime}+(k_{1}+k_{2})\delta=(\dot{\gamma}+\dot{\gamma}^{\prime})+(\dot{\eta}+\dot{\eta}^{\prime})+(k_{1}+k_{2})\delta.$

Using (3.6), $\bm{\zeta}_{1}(\dot{\gamma})>0$ and
$\bm{\zeta}_{1}(\dot{\gamma}^{\prime})\geq 0.$ This implies that
$\bm{\zeta}(\dot{\beta}+\dot{\beta}^{\prime})>0$ and so
$\beta+\beta^{\prime}\in\\{\dot{\alpha}+m\delta\in((\dot{T}\setminus\dot{T}(2))+{\mathbb{Z}}\delta)\cap
R\mid\bm{\zeta}(\dot{\alpha})>0\\}\subseteq P$ and we are done.

Proof of Claim 2. Since $\dot{\alpha}_{*}\in\dot{T}\setminus\dot{T}(2),$ it is
either an element of $\dot{T}(1)$ or an element of
$\dot{T}_{ns}\setminus\dot{T}(2)_{ns}.$ So, as
$\dot{T}_{ns}\subseteq\frac{1}{\kappa}(\dot{T}_{re}+\dot{T}_{re}),$ we have

(3.10) $\dot{\alpha}_{*}=\dot{\gamma}_{*}+\dot{\eta}_{*}$ in which either
$\dot{\gamma}_{*}\in\hbox{\rm
span}_{\mathbb{R}}\dot{T}(1)_{re}\setminus\\{0\\}$ and $\dot{\eta}_{*}=0$ or
$\dot{\gamma}_{*}\in\frac{1}{\kappa}\dot{T}(1)_{re}\setminus\\{0\\}$,
$\dot{\eta}_{*}\in\frac{1}{\kappa}\dot{T}(2)_{re}\setminus\\{0\\}$.

To show (1), let $\dot{\alpha}\in\dot{T}(1)_{re},$ $m\in{\mathbb{Z}}$,
$\dot{\alpha}+m\delta\in S$ and $\bm{\zeta_{1}}(\dot{\alpha})>0.$ Since
$u_{0}\in W,$ we have

$\displaystyle\mathcal{L}^{\dot{\alpha}+m\delta}v\subseteq\mathcal{L}^{\dot{\alpha}+m\delta}\mathcal{L}^{\dot{\alpha}_{*}+m_{0}\delta}u_{0}$
$\displaystyle\subseteq[\mathcal{L}^{\dot{\alpha}+m\delta},\mathcal{L}^{\dot{\alpha}_{*}+m_{0}\delta}]u_{0}+\mathcal{L}^{\dot{\alpha}_{*}+m_{0}\delta}\mathcal{L}^{\dot{\alpha}+m\delta}u_{0}$
$\displaystyle\subseteq\mathcal{L}^{\dot{\alpha}+\dot{\alpha}_{*}+(m+m_{0})\delta}u_{0}.$

We claim that
$\mathcal{L}^{\dot{\alpha}+\dot{\alpha}_{*}+(m+m_{0})\delta}u_{0}=\\{0\\}.$ If
$\dot{\alpha}+\dot{\alpha}_{*}+(m+m_{0})\delta\not\in R,$ we are done. So,
suppose

$\dot{\alpha}+\dot{\alpha}_{*}+(m+m_{0})\delta\in R\cap(T+T)\subseteq
T=T(1)_{re}\cup T(2)_{re}\cup T_{ns}\cup(T\cap{\mathbb{Z}}\delta).$

Recalling (3.6), we have

$\displaystyle\bm{\zeta}(\dot{\alpha}_{*})>0{\longrightarrow}\bm{\zeta}_{1}(\dot{\gamma}_{*})>0\stackrel{{\scriptstyle\bm{\zeta}_{1}(\dot{\alpha})>0}}{{\longrightarrow}}\bm{\zeta}_{1}(\dot{\alpha}+\dot{\gamma}_{*})\neq
0\longrightarrow\dot{\alpha}+\dot{\gamma}_{*}\neq 0.$

In particular, if $\dot{\eta}_{*}\not=0,$
$\dot{\alpha}+\dot{\alpha}_{*}=(\dot{\alpha}+\dot{\gamma}_{*})+\dot{\eta}_{*}\in\dot{T}_{ns}.$
Therefore, we have

$\displaystyle\bm{\zeta}(\dot{\alpha}+\dot{\alpha}_{*})=\bm{\zeta}(\dot{\alpha}+\dot{\gamma}_{*}+\dot{\eta}_{*})=$
$\displaystyle\left\\{\begin{array}[]{ll}\bm{\zeta_{1}}(\dot{\alpha}+\dot{\gamma}_{*})&\dot{\eta}_{*}=0,\\\
\bm{\zeta_{1}}(\kappa(\dot{\alpha}+\dot{\gamma}_{*}))+\bm{\zeta_{2}}(\beta_{\kappa\dot{\eta}_{*}})&\dot{\eta}_{*}\not=0,\\\
\end{array}\right.$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\bm{\zeta_{1}}(\dot{\alpha})+\bm{\zeta_{1}}(\dot{\gamma}_{*})&\dot{\eta}_{*}=0,\\\
\bm{\zeta_{1}}(\kappa\dot{\alpha})+\bm{\zeta_{1}}(\kappa\dot{\gamma}_{*})+\bm{\zeta_{2}}(\beta_{\kappa\dot{\eta}_{*}})&\dot{\eta}_{*}\not=0,\\\
\end{array}\right.$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\bm{\zeta_{1}}(\dot{\alpha})+\bm{\zeta_{1}}(\dot{\gamma}_{*})&\dot{\eta}_{*}=0,\\\
\bm{\zeta_{1}}(\kappa\dot{\alpha})+\bm{\zeta}(\dot{\gamma}_{*}+\dot{\eta}_{*})&\dot{\eta}_{*}\not=0,\\\
\end{array}\right.$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\bm{\zeta_{1}}(\dot{\alpha})+\bm{\zeta}(\dot{\alpha}_{*})&\dot{\eta}_{*}=0,\\\
\bm{\zeta_{1}}(\kappa\dot{\alpha})+\bm{\zeta}(\dot{\alpha}_{*})&\dot{\eta}_{*}\not=0.\\\
\end{array}\right.$

So, we get
$\bm{\zeta}(\dot{\alpha}+\dot{\alpha}_{*})>\bm{\zeta}(\dot{\alpha}_{*}),$
which in turn implies that
$\mathcal{L}^{\dot{\alpha}+\dot{\alpha}_{*}+(n+m)\delta}u_{0}=\\{0\\}$ due to
the choice of $\dot{\alpha}_{*}.$ This completes the proof of (1).

To prove (2), it is enough to show that
$\mathcal{A}_{v}\subseteq\mathcal{A}_{u_{0}}.$ So, assume
$\dot{\alpha}\in\mathcal{A}_{v}.$ We shall show
$\mathcal{L}^{\dot{\alpha}+n\delta}u_{0}\neq\\{0\\}$ for some integer $n.$
Since $\dot{\alpha}\in\mathcal{A}_{v},$ there exists $n\in{\mathbb{Z}}$ with
$\mathcal{L}^{\dot{\alpha}+n\delta}v\neq\\{0\\}.$ As we have

$\displaystyle\\{0\\}\neq\mathcal{L}^{\dot{\alpha}+n\delta}v\subseteq\mathcal{L}^{\dot{\alpha}+n\delta}\mathcal{L}^{\dot{\alpha}_{*}+m\delta}u_{0}\subseteq$
$\displaystyle\mathcal{L}^{\dot{\alpha}_{*}+m\delta}\mathcal{L}^{\dot{\alpha}+n\delta}u_{0}+[\mathcal{L}^{\dot{\alpha}+n\delta},\mathcal{L}^{\dot{\alpha}_{*}+m\delta}]u_{0},$

it is enough to prove that

$[\mathcal{L}^{\dot{\alpha}+n\delta},\mathcal{L}^{\dot{\alpha}_{*}+m\delta}]u_{0}\subseteq\mathcal{L}^{\dot{\alpha}+\dot{\alpha}_{*}+(n+m)\delta}u_{0}=\\{0\\}.$

If $\dot{\alpha}+\dot{\alpha}_{*}+(n+m)\delta\not\in R,$ we are done. So,
suppose $\dot{\alpha}+\dot{\alpha}_{*}+(n+m)\delta\in R$. Since
$\dot{\alpha}\in\mathcal{A}_{v}\subseteq\dot{T}\setminus\dot{T}(2),$ we have

(3.11) $\dot{\alpha}=\dot{\gamma}+\dot{\eta}$ in which either
$\dot{\gamma}\in\hbox{\rm span}_{\mathbb{R}}\dot{T}(1)_{re}\setminus\\{0\\}$
and $\dot{\eta}=0$ or
$\dot{\gamma}\in\frac{1}{\kappa}\dot{T}(1)_{re}\setminus\\{0\\}$,
$\dot{\eta}\in\frac{1}{\kappa}\dot{T}(2)_{re}\setminus\\{0\\}$.

Also, by (3.9), we have $\dot{\alpha}+n\delta,\dot{\alpha}_{*}+m\delta\in
R_{1},$ so,

$[\mathcal{L}^{\dot{\alpha}+n\delta},\mathcal{L}^{\dot{\alpha}_{*}+m\delta}]\subseteq\mathcal{L}^{\dot{\alpha}+\dot{\alpha}_{*}+(n+m)\delta}_{0}=\mathcal{L}^{(\dot{\gamma}+\dot{\gamma}_{*})+(\dot{\eta}+\dot{\eta}_{*})+(n+m)\delta}_{0}.$

Since $\bm{\zeta}(\dot{\alpha})>0$ (resp. $\bm{\zeta}(\dot{\alpha}_{*})>0$),
recalling (3.6), we have $\bm{\zeta_{1}}(\dot{\gamma})>0$ (resp.
$\bm{\zeta_{1}}(\dot{\gamma}_{*})>0$). Therefore,

(3.12) $\dot{\gamma}+\dot{\gamma}_{*}\neq 0.$

Since
$(\dot{\gamma}+\dot{\gamma}_{*})+(\dot{\eta}+\dot{\eta}_{*})+(n+m)\delta\in
T\cap R_{0}\subseteq T_{re}=T(1)_{re}\cup T(2)_{re},$ one of the following
occurs:

  * •

$(\dot{\gamma}+\dot{\gamma}_{*})+(\dot{\eta}+\dot{\eta}_{*})+(n+m)\delta\in
T(2)_{re},$

  * •

$(\dot{\gamma}+\dot{\gamma}_{*})+(\dot{\eta}+\dot{\eta}_{*})+(n+m)\delta\in
T(1)_{re},$

If the former case happens, we have $\dot{\gamma}+\dot{\gamma}_{*}\in\hbox{\rm
span}_{\mathbb{R}}\dot{T}(2)_{re}.$ This together with the fact that
$(T(1),T(2))=\\{0\\}$ gives that $(\dot{\gamma}+\dot{\gamma}_{*},\hbox{\rm
span}_{\mathbb{R}}\dot{T}(1)_{re})=\\{0\\}$ which in turn implies that
$\dot{\gamma}+\dot{\gamma}_{*}=0$ as $\dot{T}(1)_{re}$ is a finite root
system. This contradicts (3.12). So, the latter case happens, i.e., we have
$\dot{\eta}+\dot{\eta}_{*}\in\hbox{\rm span}_{\mathbb{R}}\dot{T}(1)_{re}$
which as above gives that $\dot{\eta}+\dot{\eta}_{*}=0.$ Therefore,
$\dot{\alpha}+\dot{\alpha}_{*}=\dot{\gamma}+\dot{\gamma}_{*}\in\dot{T}(1)$ and
so due to the choice of $\dot{\alpha}_{*},$ we have

$[\mathcal{L}^{\dot{\alpha}+n\delta},\mathcal{L}^{\dot{\alpha}_{*}+m\delta}]u_{0}\subseteq\mathcal{L}^{\dot{\alpha}+\dot{\alpha}_{*}+(m+n)\delta}u_{0}=\\{0\\}$

as we desired.

Proof of Claim 3. Since $\dot{\alpha}_{*}\in\mathcal{A}_{u_{0}},$ there is
$m_{0}\in{\mathbb{Z}}$ such that
$\mathcal{L}^{\dot{\alpha}_{*}+m_{0}\delta}u_{0}\neq\\{0\\}.$ We pick $0\neq
u_{1}\in\mathcal{L}^{\dot{\alpha}_{*}+m_{0}\delta}u_{0}.$ Using Claim 2
together with an inductive process, we get the result.

Proof of Claim 4. Assume $A_{u_{0}}\neq\emptyset$ and keep the same notation
as above. Set $\mu_{0}$ to be the weight of $u_{0}$ and assume $m_{i}$’s
($i\geq 1$) are as in Claim 3. We can deduce from Claim 3 that

(3.13)
$\mu_{r}:=\mu_{0}+r\dot{\alpha}_{*}+(m_{0}+\cdots+m_{r-1})\delta\in\hbox{\rm
supp}(V)\quad\quad(r\in{\mathbb{Z}}^{>0}).$

We have the following two cases:

$\diamond$ Case 1. $\dot{\alpha}_{*}$ is real: Then, we have
$\dot{\alpha}_{*}\in\dot{T}(1)_{re}$. Pick $r_{0}\in{\mathbb{Z}}^{>0}$ with

$2(\mu_{r_{0}},\dot{\alpha}_{*})/(\dot{\alpha}_{*},\dot{\alpha}_{*})=2(\mu_{0}+r_{0}\dot{\alpha}_{*},\dot{\alpha}_{*})/(\dot{\alpha}_{*},\dot{\alpha}_{*})>0.$

We have

$\mu_{r_{0}+n}:=\mu_{r_{0}}+n\dot{\alpha}_{*}+(m_{r_{0}}+\cdots+m_{r_{0}+n-1})\delta\in\hbox{\rm
supp}(V)\quad(n\in{\mathbb{Z}}^{>0}).$

Contemplating (3.9), for each $i\geq 0$, $\dot{\alpha}_{*}+m_{i}\delta\in
R_{1}$. So by (3.3)(2), for each odd positive integer $n$, we have
$\dot{\alpha}_{*}+(-(n-2)m_{r_{0}}+m_{r_{0}+1}+\cdots+m_{r_{0}+n-1})\delta\in
T(1)$ and so, we get using (3.7) that

$\displaystyle\mu_{r_{0}}+(n-1)(\dot{\alpha}_{*}+m_{r_{0}}\delta)$
$\displaystyle=$
$\displaystyle\mu_{r_{0}}+n\dot{\alpha}_{*}+(m_{r_{0}}+\cdots+m_{r_{0}+n-1})\delta-(\dot{\alpha}_{*}+(-(n-2)m_{r_{0}}+m_{r_{0}+1}+\cdots+m_{r_{0}+n-1})\delta)$
$\displaystyle\in$ $\displaystyle\hbox{\rm supp}(V)\quad\quad(n\in
2{\mathbb{Z}}^{\geq 0}+1).$

This contradicts the fact that $\dot{\alpha}_{*}+m_{r_{0}}\delta\in
T(1)_{re}\subseteq R^{ln}(V)$; see (2.4).

$\diamond$ Case 2. $\dot{\alpha}_{*}$ is nonsingular: We have

$\dot{\alpha}_{*}\in\dot{T}_{ns}\setminus\dot{T}(2)\subseteq\frac{1}{\kappa}(\dot{T}(1)_{re}+\dot{T}(1)_{re})\cup\frac{1}{\kappa}(\dot{T}(1)_{re}+\dot{T}(2)_{re}),$

so $\dot{\alpha}_{*}=\dot{\xi}_{*}+\dot{\zeta}_{*}$ with
$\kappa\dot{\xi}_{*}\in\dot{T}(1)_{re},\kappa\dot{\zeta}_{*}\in\dot{T}(1)_{re}\cup\dot{T}(2)_{re}$.
Recalling (3.13), we have

$\mu_{{}_{\kappa n}}=\mu_{0}+\kappa n\dot{\alpha}_{*}+(m_{0}+\cdots+m_{\kappa
n-1})\delta\in\hbox{\rm supp}(V)\quad(n\in{\mathbb{Z}}^{>0}).$

We note that $(\dot{\xi}_{*},\dot{\zeta}_{*})=0$ and pick $n_{0}$ such that

(3.14) $\frac{2(\mu_{{}_{\kappa
n_{0}}},\kappa\dot{\xi}_{*})}{(\kappa\dot{\xi}_{*},\kappa\dot{\xi}_{*})}=\frac{2(\mu_{0}+\kappa
n_{0}\dot{\alpha}_{*},\kappa\dot{\xi}_{*})}{(\kappa\dot{\xi}_{*},\kappa\dot{\xi}_{*})}\in{\mathbb{Z}}^{>0}\quad\hbox{and}\quad\frac{2(\mu_{{}_{\kappa
n_{0}}},\kappa\dot{\zeta}_{*})}{(\kappa\dot{\zeta}_{*},\kappa\dot{\zeta}_{*})}=\frac{2(\mu_{0}+\kappa
n_{0}\dot{\alpha}_{*},\kappa\dot{\zeta}_{*})}{(\kappa\dot{\zeta}_{*},\kappa\dot{\zeta}_{*})}\in{\mathbb{Z}}^{>0}.$

We have

$\upsilon_{n}:=\mu_{{}_{\kappa n_{0}+\kappa n}}=\mu_{\kappa n_{0}}+\kappa
n\dot{\alpha}_{*}+(m_{\kappa n_{0}}+\cdots+m_{\kappa n-1})\delta\in\hbox{\rm
supp}(V)\quad(n\in{\mathbb{Z}}^{>0}).$

Since real roots of $T(2)$ are up-nilpotent hybrid and all real roots of
$T(1)$ are locally nilpotent, recalling (3.3)(1), for a large enough
$m^{\prime}\in{\mathbb{Z}}^{>0},$ we have

$\left\\{\begin{array}[]{ll}\kappa\dot{\zeta}_{*}+k_{\kappa\dot{\zeta}_{*}}\delta+r_{\kappa\dot{\zeta}_{*}}m^{\prime}\delta\in
T^{ln}(V),~{}~{}-(\kappa\dot{\zeta}_{*}+k_{\kappa\dot{\zeta}_{*}}\delta+r_{\kappa\dot{\zeta}_{*}}m^{\prime}\delta)\in
T^{in}(V)&\hbox{if $\kappa\dot{\zeta}_{*}\in\dot{T}(2)$}\\\
\kappa\dot{\zeta}_{*}+k_{\kappa\dot{\zeta}_{*}}\delta+r_{\kappa\dot{\zeta}_{*}}m^{\prime}\delta\in
T^{ln}(V),~{}~{}-(\kappa\dot{\zeta}_{*}+k_{\kappa\dot{\zeta}_{*}}\delta+r_{\kappa\dot{\zeta}_{*}}m^{\prime}\delta)\in
T^{ln}(V)&\hbox{if $\kappa\dot{\zeta}_{*}\in\dot{T}(1).$}\end{array}\right.$

This, in particular, together with (3.14) and (3.7) implies that

$\displaystyle\upsilon_{n}-i(\kappa\dot{\zeta}_{*}+k_{\kappa\dot{\zeta}_{*}}\delta+r_{\kappa\dot{\zeta}_{*}}m^{\prime}\delta)\in\hbox{\rm
supp}(V)\quad(n\in{\mathbb{Z}}^{>0},~{}1\leq i\leq n),$

in particular, for

$p_{n}:=(m_{\kappa n_{0}}+\cdots+m_{\kappa
n-1})-n(k_{\kappa\dot{\zeta}_{*}}+r_{\kappa\dot{\zeta}_{*}}m^{\prime})\quad\quad(n\in{\mathbb{Z}}^{>0}),$

we have

$\displaystyle\mu_{\kappa n_{0}}+n(\kappa\dot{\xi}_{*})+p_{n}\delta=$
$\displaystyle\mu_{\kappa n_{0}}+n(\kappa\dot{\xi}_{*})+(m_{\kappa
n_{0}}+\cdots+m_{\kappa
n-1})\delta-n(k_{\kappa\dot{\zeta}_{*}}+r_{\kappa\dot{\zeta}_{*}}m^{\prime})\delta$
$\displaystyle=$
$\displaystyle\upsilon_{n}-n(\kappa\dot{\zeta}_{*}+k_{\kappa\dot{\zeta}_{*}}\delta+r_{\kappa\dot{\zeta}_{*}}m^{\prime}\delta)\in\hbox{\rm
supp}(V)\quad(n\in{\mathbb{Z}}^{>0}).$

Therefore, we have

(3.15) $\mu_{\kappa
n_{0}}+n(\kappa\dot{\xi}_{*}+k_{\kappa\dot{\xi}_{*}}\delta)+(p_{n}-nk_{\kappa\dot{\xi}_{*}})\delta=\mu_{\kappa
n_{0}}+n(\kappa\dot{\xi}_{*})+p_{n}\delta\in\hbox{\rm supp}(V)$

for $n\in{\mathbb{Z}}^{>0}.$ We next note that for each
$n\in{\mathbb{Z}}^{>0},$ using the quotient algorithm, we get
$q_{n}\in{\mathbb{Z}}$ and $0\leq s_{n}<r_{\kappa\dot{\xi}_{*}}$ such that

$p_{n}-nk_{\kappa\dot{\xi}_{*}}=q_{n}r_{\kappa\dot{\xi}_{*}}+s_{n}.$

In particular, (3.15) gives that

(3.16) $\displaystyle(\mu_{\kappa
n_{0}}+s_{n}\delta)+n(\kappa\dot{\xi}_{*}+k_{\kappa\dot{\xi}_{*}}\delta)+q_{n}r_{\kappa\dot{\xi}_{*}}\delta$
$\displaystyle=$ $\displaystyle\mu_{\kappa
n_{0}}+n(\kappa\dot{\xi}_{*}+k_{\kappa\dot{\xi}_{*}}\delta)+(q_{n}r_{\kappa\dot{\xi}_{*}}+s_{n})\delta$
$\displaystyle=$ $\displaystyle\mu_{\kappa
n_{0}}+n(\kappa\dot{\xi}_{*}+k_{\kappa\dot{\xi}_{*}}\delta)+(p_{n}-nk_{\kappa\dot{\xi}_{*}})\delta\in\hbox{\rm
supp}(V).$

Since $T(1)_{re}\subseteq R^{ln}(V),$ using this together with (3.7) and
(3.14) repeatedly, we get

$\displaystyle\mu_{\kappa n_{0}}+s_{n}\delta\in\hbox{\rm
supp}(V)\quad(n\in{\mathbb{Z}}^{>0}).$

Since $s_{n}$’s are equal for infinitely many $n$, we get positive integers
$n_{1},n_{2},\ldots$ with $s:=s_{n_{1}}=s_{n_{2}}=\cdots,$ and so
contemplating (3.16), we have

$\displaystyle(\mu_{\kappa
n_{0}}+s\delta)+n_{i}(\kappa\dot{\xi}_{*}+k_{\kappa\dot{\xi}_{*}}\delta)+(q_{n_{0}}r_{\kappa\dot{\xi}_{*}})\delta\in\hbox{\rm
supp}(V)\quad(i>0).$

Using (3.7), we get that

$\displaystyle(\mu_{\kappa
n_{0}}+s\delta)+(n_{i}-1)(\kappa\dot{\xi}_{*}+k_{\kappa\dot{\xi}_{*}}\delta)\in\hbox{\rm
supp}(V)\quad(i>0)$

which is a contradiction as
$\kappa\dot{\xi}_{*}+k_{\kappa\dot{\xi}_{*}}\delta\in T(1)_{re}\subseteq
R^{ln}(V).$ This completes the proof of Claim 4.

Proof of Claim 5. Using Claim 4, we get that $U$ is nonzero. Suppose that
$v\in U$, $\dot{\beta}\in\dot{T}(2)$ and $n\in{\mathbb{Z}}$ with
$\dot{\beta}+n\delta\in R.$ For $\dot{\alpha}\in\dot{T}\setminus\dot{T}(2)$
with $\bm{\zeta}(\dot{\alpha})>0,$ $m\in{\mathbb{Z}}$ and
$\dot{\alpha}+m\delta\in R$, we have

$\displaystyle\mathcal{L}^{\dot{\alpha}+m\delta}\mathcal{L}^{\dot{\beta}+n\delta}v\subseteq\mathcal{L}^{\dot{\beta}+n\delta}\underbrace{\mathcal{L}^{\dot{\alpha}+m\delta}v}_{=\\{0\\}}+[\mathcal{L}^{\dot{\alpha}+m\delta},\mathcal{L}^{\dot{\beta}+n\delta}]v\subseteq\mathcal{L}^{\dot{\alpha}+\dot{\beta}+(n+m)\delta}v.$

It is enough to show that
$\mathcal{L}^{\dot{\alpha}+\dot{\beta}+(n+m)\delta}v=\\{0\\}$. If
$\dot{\alpha}+\dot{\beta}+(n+m)\delta$ is not a root, then, we are done. So,
assume $\dot{\alpha}+\dot{\beta}+(n+m)\delta\in R$ (and so $\in T$). As in
(3.11), we have $\dot{\alpha}=\dot{\gamma}+\dot{\eta}$ in which
$0\neq\dot{\gamma}\in\hbox{\rm span}_{\mathbb{R}}\dot{T}(1)$ and
$\dot{\eta}\in\hbox{\rm span}_{\mathbb{R}}\dot{T}(2).$ Now, we have the
following cases:

  * •

$\dot{\alpha}+\dot{\beta}=\dot{\gamma}+\dot{\eta}+\dot{\beta}\in\hbox{\rm
span}_{\mathbb{R}}\dot{T}(2)_{re}$. Then, $\dot{\gamma}\in\hbox{\rm
span}_{\mathbb{R}}\dot{T}(1)_{re}\cap\hbox{\rm
span}_{\mathbb{R}}\dot{T}(2)_{re}$ which implies that $\dot{\gamma}=0$. This
is a contradiction.

  * •

$\dot{\alpha}+\dot{\beta}=\dot{\gamma}+\dot{\eta}+\dot{\beta}\in\hbox{\rm
span}_{\mathbb{R}}\dot{T}(1)_{re}$. Then $\dot{\eta}+\dot{\beta}\in\hbox{\rm
span}_{\mathbb{R}}\dot{T}(1)_{re}\cap\hbox{\rm
span}_{\mathbb{R}}\dot{T}(2)_{re}.$ This implies that
$\dot{\eta}+\dot{\beta}=0.$ Therefore, we have
$\dot{\alpha}+\dot{\beta}+(m+n)\delta=\dot{\gamma}+(m+n)\delta$ and so
$\mathcal{L}^{\dot{\alpha}+\dot{\beta}+(n+m)\delta}v=\\{0\\}$ as
$\bm{\zeta}(\dot{\gamma})=\bm{\zeta_{1}}(\dot{\gamma})>0$ by (3.6).

  * •

$\dot{\alpha}+\dot{\beta}\not\in\hbox{\rm
span}_{\mathbb{R}}\dot{T}(1)_{re}\cup\hbox{\rm
span}_{\mathbb{R}}\dot{T}(2)_{re}$. Then,
$\dot{\alpha}+\dot{\beta}\in\dot{T}_{ns}\setminus(\dot{T}(1)_{ns}\cup\dot{T}(2)_{ns}).$
Therefore, there are $\dot{\zeta},\dot{\xi}$ with
$\kappa\dot{\zeta}\in\dot{T}(1)_{re}$ and $\kappa\dot{\xi}\in\dot{T}(2)_{re}$
such that

$\dot{\zeta}+\dot{\xi}=\dot{\alpha}+\dot{\beta}=\dot{\gamma}+\dot{\eta}+\dot{\beta}.$

This implies that

$\dot{\zeta}=\dot{\gamma}\quad\hbox{and}\quad\dot{\xi}=\dot{\eta}+\dot{\beta}.$

Since $\bm{\zeta_{1}}(\dot{\zeta})=\bm{\zeta_{1}}(\dot{\gamma})>0.$ Therefore
by (3.6), we have $\bm{\zeta}(\dot{\zeta}+\dot{\xi})>0$ and so
$\mathcal{L}^{\dot{\alpha}+\dot{\beta}+(n+m)\delta}v=\mathcal{L}^{\dot{\zeta}+\dot{\xi}+(n+m)\delta}v=\\{0\\}$,
as desired.

Proof of Claim 6. By Proposition 3.2, there is $K\subseteq T(2)\cap R_{0}$
such that $\sum_{\alpha\in\\{0\\}\cup
K^{\times}}\mathcal{L}^{\alpha}+\sum_{\alpha,\beta\in
K^{\times}}[\mathcal{L}^{\alpha},\mathcal{L}^{\beta}]$ is an affine Lie
algebra, up to a central space. Now $V$ is a finite weight module over this
Lie algebra and $K=K^{+}\cup K^{\circ}\cup K^{-}$ where

$K^{\circ}:=\\{\alpha\in K\mid\bm{\zeta_{2}}(\alpha)=0\\}\quad\hbox{and}\quad
K^{\pm}:=\\{\alpha\in K\mid\bm{\zeta_{2}}(\alpha)\gtrless 0\\}$

is a triangular decomposition with (see Lemma 3.5)

$K^{+}\cap K_{re}\subseteq K^{ln}(V)\quad\hbox{and}\quad K^{-}\cap
K_{re}\subseteq K^{in}(V).$

This implies that there is $\lambda\in\hbox{\rm supp}(V)$ and positive integer
$p$ such that $\lambda+pm\delta\not\in\hbox{\rm supp}(V)$ for all
$m\in{\mathbb{Z}}^{>0}$; see [23, Lemma 5.1]. Using the same argument as in
[23, Theorem 5.8] for the finite weight module $V$, as a module over
$\mathcal{G}:=\sum_{\alpha\in T(2)}\mathcal{L}^{\alpha},$ we are done.

(c) follows from a well-known process in the literature; see e.g., [23,
Proposition 3.3 (ii)]. ∎

######  Remark 3.7.

The case $T(2)_{im}=2{\mathbb{Z}}\delta$ needs more investigation which has
been done in [18].

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