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\datereceived{2020-04-03}
\dateaccepted{2021-01-08}
\dateepreuves{2021-01-18}

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\begin{document}
\frontmatter
\title{Simplicity of vacuum modules and associated~varieties}

\author[\initial{T.} \lastname{Arakawa}]{\firstname{Tomoyuki} \lastname{Arakawa}}
\address{Research Institute for Mathematical Sciences, Kyoto University\\ Kyoto, 606-8502, Japan}
\email{arakawa@kurims.kyoto-u.ac.jp}
\urladdr{http://www.kurims.kyoto-u.ac.jp/~arakawa/}

\author[\initial{C.} \lastname{Jiang}]{\firstname{Cuipo} \lastname{Jiang}}
\address{School of Mathematical Sciences, Shanghai Jiao Tong University\\ Shanghai, 200240, China}
\email{cpjiang@sjtu.edu.cn}
\urladdr{http://math.sjtu.edu.cn/faculty/cpjiang/}

\author[\initial{A.} \lastname{Moreau}]{\firstname{Anne} \lastname{Moreau}}
\address{Faculté des Sciences d'Orsay, Université Paris-Saclay\\
91405 Orsay, France}
\email{anne.moreau@universite-paris-saclay.fr}
\urladdr{https://www.imo.universite-paris-saclay.fr/~moreau/}

\thanks{T.A.\ is supported by partially supported by JSPS KAKENHI Grant No.\,17H01086 and No.\,17K18724. C.J.\ is supported by CNSF grants 11771281 and 11531004. A.M.\ is supported by the ANR Project GeoLie Grant number ANR-15-CE40-0012, and by the Labex CEMPI (ANR-11-LABX-0007-01)}

\subjclass{17B69}

\keywords{Associated variety, affine Kac-Moody algebra,
affine vertex algebra, singular vector, affine $W$-algebra}

\altkeywords{Variété associée, algèbre de Kac-Moody, algèbre vertex affine, vecteur singulier, $W$-algèbre affine}

\alttitle{Simplicité des algèbres vertex affines et variétés associées}

\begin{abstract}
In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $\mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $\mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.
\end{abstract}

\begin{altabstract}
Dans cet article, nous démontrons que l’algèbre vertex affine universelle associée à une algèbre de Lie simple $\mathfrak{g}$ est simple si et seulement si la variété associée à son unique quotient simple est égale à $\mathfrak{g}^*$. Nous en déduisons un résultat analogue pour la réduction quantique de Drinfeld-Sokolov appliquée à l’algèbre vertex affine universelle.
\end{altabstract}
\maketitle

\vspace*{-\baselineskip}
\tableofcontents
\mainmatter

\vspace*{-2\baselineskip}\vskip0pt
\section{Introduction}
Let $V$ be a vertex algebra,
and let
\[
V\to (\End V)\lcr z,z^{-1}\rcr,\quad a\mto a(z)=\sum_{n\in \Z}a_{(n)}z^{-n-1},
\]
be the state-field correspondence.
The \emph{Zhu $C_2$-algebra} \cite{Zhu96} of $V$ is by definition the quotient space
$R_V=V/C_2(V)$,
where $C_2(V)=\on{span}_{\C}\{a_{(-2)}b\mid a,b\in V\}$,
equipped with the Poisson algebra structure given by
\[
\bar a. \bar b=\overline{a_{(-1)}b},\qquad \{\bar a,\bar b\}=\overline{a_{(0)}b},
\]
for $a,b \in V$ with $\bar a := a+C_2(V)$.
The associated variety $X_V$ of $V$ is the reduced scheme
$X_V=\on{Specm}(R_V)$
corresponding to $R_V$.
It is a fundamental invariant of $V$ that captures important properties
of the vertex algebra $V$ itself (see, for example, \cite{BeiFeiMaz,Zhu96,AbeBuhDon04,Miy04,Ara12,Ara09b,A2012Dec,AM15,AM16,Arakawa-Kawasetsu}).
Moreover, the associated variety $X_V$
conjecturally \cite{BeeRas} coincides with
the Higgs branch of
a 4D $\mc{N}=2$ superconformal field theory $\mc{T}$,
if $V$ corresponds to a theory $\mc{T}$
by the 4D/2D duality discovered in \cite{BeeLemLie15}.
Note that the Higgs branch of a 4D $\mc{N}=2$ superconformal field theory
is a hyperkähler cone, possibly singular.

In the case where $V$ is
the universal affine vertex algebra $V^k(\g)$ at level $k \in\C$
associated with a complex finite-dimensional simple Lie algebra $\g$,
the variety $X_V$ is just the affine space $\g^*$ with Kirillov-Kostant Poisson structure.
In the case where~$V$ is
the unique simple graded quotient $L_k(\g)$
of $V^k(\g)$,
the variety $X_V$ is a Poisson subscheme of $\g^*$
which is $G$-invariant and conic, where $G$ is the adjoint group of $\g$.

Note that if the level $k$ is irrational,
then
$L_k(\g)=V^k(\g)$,
and hence $X_{L_k(\g)}=\g^*$.
More generally, if $L_k(\g)=V^k(\g)$, that is, $V^k(\g)$
is simple, then obviously
\hbox{$X_{L_k(\g)}=\g^*$.}

In this article, we prove that the converse is true.

\begin{theorem}\label{MainTheorem}
\label{Th:main}
The equality $L_k(\g)=V^k(\g)$ holds, that is,
$V^k(\g)$
is simple, if and only if $X_{L_k(\g)} = \g^*$.
\end{theorem}

It is known by Gorelik and Kac \cite{GorKac07}
that $V^k(\g)$
is not simple if and only if
\begin{equation}
r^{\vee}(k+h^{\vee}) \in \Q_{\geq 0}\setminus\{\sfrac{1}{m}\mid m\in \Z_{\geq 1}\},
\label{eq:Gorelik-Kac}
\end{equation}
where
$h^{\vee}$ is the dual Coxeter number and
$r^{\vee}$ is the lacing number of $\g$.
Therefore,
Theorem \ref{MainTheorem} can be rephrased as
\begin{equation}
X_{L_k(\g)}\subsetneq \g^*\iff \text{\eqref{eq:Gorelik-Kac} holds.}
\end{equation}
Let us mention the cases when the variety $X_{L_k(\g)}$
is known
for $k$ satisfying \eqref{eq:Gorelik-Kac}.

First, it is known \cite{Zhu96,DonMas06} that $X_{L_k(\g)}=\{0\}$
if and only if $L_k(\g)$ is integrable, that is,
$k$ is a nonnegative integer.
Next, it is known that if $L_k(\g)$ is \emph{admissible} \cite{KacWak89},
or equivalently,
if
\begin{equation*}
k+h^{\vee}=\frac{p}{q},\quad p,q\in \Z_{\geq 1}, \ (p,q)=1,\
p\geq \begin{cases}h^{\vee}&\text{if }(r^{\vee},q)=1,\\
h&\text{if }(r^{\vee},q)\ne 1,
\end{cases}
\end{equation*}
where $h$ is the Coxeter number of $\g$,
then
$X_{L_k(\g)}$ is the closure of some nilpotent orbit in $\g$ (\cite{Ara09b}).
Further, it was observed in \cite{AM15,AM17}
that
there are cases when $L_k(\g)$ is
non-admissible and
$X_{L_k(\g)}$ is the closure of some nilpotent orbit.
In fact,
it was recently
conjectured in physics \cite{XieYan2019lisse} that,
in view of the 4D/2D duality,
there should be
a large list of non-admissible simple affine vertex algebras
whose associated varieties are the closures of some nilpotent orbits.
Finally,
there are also cases \cite{AM16} where $X_{L_k(\g)}$
is neither $\g^*$ nor contained in the nilpotent cone $\mathcal{N}(\g)$
of $\g$.

In general,
the problem of determining the variety $X_{L_k(\g)}$ is wide open.

Now let us explain
the outline of
the proof of Theorem~\ref{Th:main}.
First, Theorem~\ref{Th:main} is known for the critical level $k=-h^\vee$
(\cite{FeiFre92,FreGai04}). Therefore, since $R_{V^k(\g)}$ is a polynomial ring $\C[\g^*]$,
Theorem~\ref{Th:main} follows from the following fact.
\begin{theorem}\label{theorem:image-of-the-singular}
Suppose that the level is non-critical, that is, $k\ne -h^\vee$.
The image of any nonzero singular vector $v$ of $V^k(\g)$
in the Zhu $C_2$-algebra $R_{V^k(\g)}$ is nonzero.
\end{theorem}

The symbol $\sigma(w)$
of a singular vector $w$ in $V^k(\g)$ is
a singular vector
in the corresponding vertex Poisson algebra
$\mathrm{gr} V^k(\g) \cong S(t^{-1}\g[t^{-1}])\cong \C[J_{\infty}\g^*]$,
where $J_\infty \g^*$ is the arc space of $\g^*$.
Theorem \ref{theorem:image-of-the-singular}
states that
the image of $\sigma(w)$
of a non-trivial singular vector $w$
under the projection
\begin{equation}
\C[J_{\infty}\g^*]\to \C[\g^*]=R_{V^k(\g)}
\label{eq:proj}
\end{equation}
is nonzero,
provided that $k$ is non-critical.
Here the projection \eqref{eq:proj} is defined by identifying $\C[\g^*]$ with the
Zhu $C_2$-algebra of the commutative vertex algebra $\C[J_{\infty}\g^*]$.
Hence,
Theorem \ref{theorem:image-of-the-singular}
would follow if
the image of any nontrivial singular vector
in $\C[J_{\infty}\g^*]$
under the projection \eqref{eq:proj}
is nonzero.
However, this is false
as there are singular vectors in $\C[J_{\infty}\g^*]$
that do not come from singular vectors of $V^k(\g)$
and that belong to the kernel of \eqref{eq:proj}
(see Section \ref{subsection:remark}).
Therefore,
we do need to make use of the fact that
$\sigma(w)$ is the symbol of a singular vector $w$ in $V^k(\g)$.
We~also note that
the statement of Theorem \ref{theorem:image-of-the-singular}
is not true if $k$ is critical (see Section~\ref{subsection:remark}).

For this reason the proof of Theorem \ref{theorem:image-of-the-singular}
is divided roughly into two parts.
First, we work in the commutative setting
to deduce
a first important reduction (Lemma~\ref{lem:wi_-}).
Next, we use the Sugawara construction --
which is available only at non-critical levels -- in the non-commutative setting
in order to complete the proof.

Now, let us consider the \emph{$W$-algebra} $\W^k(\g,f)$ associated with
a nilpotent element~$f$ of~$\g$ at the level $k$
defined by the generalized quantized Drinfeld-Sokolov reduction
\cite{FeiFre90,KacRoaWak03}:
\[
\W^k(\g,f)=H^{0}_{\DS,f}(V^k(\g)).
\]
Here, $H^{\sbullet}_{\DS,f}(M)$ denotes the BRST
cohomology of the generalized quantized Drinfeld-Sokolov reduction
associated with $f \in \mathcal{N}(\g)$ with coefficients in
a $V^k(\g)$-module~$M$.

By the Jacobson-Morosov theorem, $f$ embeds into an $\mathfrak{sl}_2$-triple
$(e,h,f)$. The Slodowy slice $\mathscr{S}_{f}$ at $f$ is
the affine
space $\mathscr{S}_{f}=f+\g^{e}$, where $\g^{e}$
is the centralizer of $e$ in $\g$.
It has a natural Poisson structure induced from that of $\g^*$ (see \cite{GanGin02}),
and we have \cite{DSK06,Ara09b} a natural isomorphism
$R_{\W^k(\g,f)}\cong \C[\mathscr{S}_{f}]$ of Poisson algebras, so that
\begin{equation*}
X_{\W^k(\g,f)}= \mathscr{S}_{f}.
\end{equation*}
The natural surjection
$V^k(\g)\twoheadrightarrow L_k(\g)$
induces a surjection
$\W^k(\g,f)\twoheadrightarrow H_{\DS,f}^0(L_k(\g))$
of vertex algebras (\cite{Ara09b}).
Hence the variety
$X_{H^{0}_{\DS,f}(L_k(\g))}$
is a $\C^*$-invariant Poisson
subvarieties of the Slodowy slice $\mathscr{S}_f$.

Conjecturally \cite{KacRoaWak03,KacWak08},
the vertex algebra
$H^{0}_{\DS,f}(L_k(\g))$ coincides
the unique simple (graded) quotient $\W_k(\g,f)$
of $\W^k(\g,f)$
provided that
\hbox{$H^{0}_{\DS,f}(L_k(\g))\ne 0$.}
(This conjecture has been verified in many cases \cite{Ara05,Ara07,Ara08-a,AEkeren19}.)

As a consequence of Theorem \ref{Th:main}, we obtain the following
result.

\begin{theorem}\label{Th:main2}
Let $f$ be any nilpotent element of $\g$.
The following assertions are equivalent:
\begin{enumerate}
\item\label{Th:main21} $V^k(\g)$ is simple,
\item\label{Th:main22} $\W^k(\g,f)=H^{0}_{\DS,f}(L_k(\g))$,
\item\label{Th:main23} $X_{H^{0}_{\DS,f}(L_k(\g))} = \mathscr{S}_f$.
\end{enumerate}
\end{theorem}

Note that Theorem \ref{Th:main2} implies that
$V^k(\g)$ is simple
if
$X_{\W_k(\g,f)}=\mathscr{S}_f$
and $H^{0}_{\DS,f}(L_k(\g))\ne 0$
since
$ X_{H_{\DS,f}^0(L_k(\g))}\supset X_{\W^k(\g,f)}$.

The remainder of the paper is structured as follows.
In Section \ref{sec:affine_vertex_algebras} we set up notation
in the case of affine vertex algebras that will be the framework of this note.
Section \ref{sec:main_proof} is devoted to the proof of Theorem~\ref{Th:main}.
In Section \ref{sec:W-algebras}, we have compiled some known facts
on Slodowy slices, $W$-algebras and their associated varieties.
Theorem~\ref{Th:main2} is proved in this section.

\subsubsection*{Acknowledgements}
T.A.~and A.M.~like to warmly thank Shanghai Jiao Tong University
for its hospitality during their stay in September 2019.

\section{Universal affine vertex algebras and associated
graded vertex \texorpdfstring{Poisson~algebras}{Poisson algebras}}
\label{sec:affine_vertex_algebras}
Let
$\frg$ be the affine Kac-Moody algebra associated with $\g$, that is,
\begin{equation*}
\frg =\g [t,t^{-1}]\oplus \C K,
\end{equation*}
where the commutation relations are given by
\begin{equation*}
[x\otimes t^m,y\otimes t^n]=[x,y]\otimes t^{m+n}+m(x|y)\delta_{m+n,0}K,\quad
[K,\frg]=0,
\end{equation*}
for $x,y\in \g$ and $m,n\in \Z$.
Here,
\[
(~|~)=\displaystyle{\frac{1}{2h^\vee}\times} \text{ Killing form of }\g
\]
is the usual normalized inner product.
For $x \in \g$ and $m \in \Z$, we shall write $x(m)$ for $x
\otimes t^m$.

\subsection{Universal affine vertex algebras}
\label{sec:Universal affine vertex algebras}
For $k \in \C$, set
\begin{equation*}
V^k(\g)=U(\frg)\otimes _{U(\g [t]\oplus \C K)}\C_k,
\end{equation*}
where $\C_k$ is the one-dimensional representation of $\g [t]\oplus \C K$
on which $K$ acts as multiplication by $k$ and $\g\otimes \C[t]$ acts trivially.

By the Poincaré-Birkhoff-Witt Theorem, the direct sum decomposition,
we have
\begin{equation}
V^k(\g)\cong U(\g \otimes t^{-1}\C[t^{-1}]) = U(t^{-1} \g[t^{-1}]).
\label{eq:PBW}
\end{equation}

The space $V^k(\g)$ is naturally graded,
\begin{equation*}
V^k(\g) =\bigoplus_{\Delta\in \Z_{\geq 0}}V^k(\g) _{\Delta},
\end{equation*}
where the grading is defined by
\[
\deg (x^{i_1}(-n_1)\cdots x^{i_r}(-n_r) \mathbf{1}) = \sum_{i=1}^r n_i,
\quad r \geq 0, \; x^{i_j} \in \g,
\]
with $\mathbf{1}$ the image of $1\otimes 1$ in $V^k(\g)$.
We have $V^k(\g)_0=\C\mathbf{1} $,
and we identify $\g$ with $V^k(\g)_1$ via the linear isomorphism
defined by $x\mto x(-1)\mathbf{1} $.

It is well-known that $V^k(\g)$ has a unique vertex algebra structure
such that $\mathbf{1}$ is the vacuum vector,
\[
x(z) := Y(x\otimes t^{-1},z) =\sum_{n \in \Z} x(n) z^{-n-1},
\]
and
\begin{equation*}
[T,x(z)]=\partial_z x(z)
\end{equation*}
for $x \in \g$,
where $T$ is the translation operator.
Here, $x(n)$ acts on $V^k(\g)$ by left multiplication, and
so, one can view $x(n)$ as an endomorphism of $V^k(\g)$.
The vertex algebra
$V^k(\g)$ is called the \emph{universal affine vertex algebra}
associated with $\g$ at level~$k$ \cite{FZ,Zhu96,LL}.

The vertex algebra
$V^k(\g)$ is a vertex operator algebra, provided that $k+h^\vee\not=0$,
by the \emph{Sugawara construction}.
More specifically, set
\[
S=\displaystyle{\frac{1}{2}} \sum_{i=1}^{d}
x_{i}(-1) x^{i}(-1) \mathbf{1},
\]
where $\{x_{i}\mid i=1,\dots,d\}$ is the dual
basis of a basis $\{x^{i}\mid i=1,\dots,\dim \g\}$ of $\g$
with respect to the bilinear form $(~|~)$, with $d = \dim \g$.
Then for $k \not=-h^\vee$, the vector
$\omega=\displaystyle{\spfrac{S}{k+h^\vee}}$
is a conformal vector of $V^k(\g)$ with central charge
\[
c(k)=\displaystyle{\frac{k \dim \g}{k+h^\vee}}.
\]
Note that, writing $\omega(z) = \sum_{n \in \Z} L_n z^{-n-2}$,
we have
\begin{align*}
L_0&= \dfrac{1}{2(k+h^\vee)}
\biggl(\sum_{i=1}^{d}x_i(0)x^i(0)+\sum_{n=1}^{\infty}\sum_{i=1}^{d}(x_i(-n)x^i(n)+x^i(-n)x_i(n)) \biggr),\\
L_n&=\dfrac{1}{2(k+h^\vee)}
\biggl(\sum_{m=1}^{\infty}\sum_{i=1}^{d}x_i(-m)x^i(m+n)
+\sum_{m=0}^{\infty}\sum_{i=1}^{d}x^i(-m+n)x_i(m)\biggr), \quad
\text{if} \ n\!\neq\! 0.
\end{align*}

\begin{lem}[{\cite{Kac_infinite}}]
\label{lem:sugawara-vs-g}
We have
\[
[L_n,x(m)]= - m x(m+n), \quad
\text{ for }x\in \g,\ m,n\in \Z,
\]
and $L_n\mathbf{1}=0$ for $n\geq -1$.
\end{lem}
We have
$V^k(\g) _{\Delta}=\{v\in V^k(\g)\mid L_0 v=\Delta v\}$
and $T=L_{-1}$ on $V^k(\g)$, provided that $k+h^{\vee}\ne 0$.

Any
graded quotient of $V^k(\g)$
as $\frg$-module
has the structure of a quotient vertex algebra.
In particular,
the unique simple graded quotient $L_k(\g)$
is a vertex algebra,
and is called the \emph{simple affine vertex algebra associated with $\g$ at level $k$.}

\Subsection{Associate graded vertex Poisson algebras of affine vertex algebras}
It is known by Li \cite{Li05}
that any vertex algebra $V$ admits a canonical filtration $F^\sbullet V$,
called the \emph{Li filtration} of $V$.
For a quotient $V$ of $V^k(\g)$,
$F^\sbullet V$ is described as follows.
The subspace
$F^p V$ is spanned by the elements
\[
y_{1}(-n_1-1)\cdots y_{r}(-n_r-1)\mathbf{1}
\]
with $y_{i} \in \g$,
$n_i\in\Z_{\geq 0}$, $ n_1+\cdots +n_r\geq p$.
We have
\begin{equation}
\label{eq:translation}
\begin{aligned}
V&=F^0V\supset F^1V\supset\cdots, \quad \textstyle\bigcap_{p}F^pV=0,\\ 
TF^pV&\subset F^{p+1}V,\\
a_{(n)}F^{q}V&\subset F^{p+q-n-1}V \ for \ a\in F^{p}V, \quad n\in\Z,\\
a_{(n)}F^{q}V&\subset F^{p+q-n}V \ for \ a\in F^{p}V, \quad n\geq 0.
\end{aligned}
\end{equation}
Here we have set $F^pV=V$ for $p<0$.

Let $\mathrm{gr}^FV=\bigoplus_p F^pV/F^{p+1}V$ be the associated graded vector space.
The space $\mathrm{gr}^FV$ is a vertex Poisson algebra by
\begin{align*}
\sigma_{p}(a)\sigma_{q}(b)&=\sigma_{p+q}(a_{(-1)}b),\\
T\sigma_{p}(a)&=\sigma_{p+1}(Ta),\\
\sigma_{p}(a)_{(n)}\sigma_{q}(b)&=\sigma_{p+q- n}(a_{(n)}b)
\end{align*}
for $a,b\in V$,
$n\geq 0$,
where $\sigma_p \colon F^p(V)\to F^pV/F^{p+1}V$ is the principal symbol map.
In~particular,
$\on{gr}^F V$ is a $\g[t]$-module by the correspondence
\begin{equation}
\g[t]\ni x(n)\mto \sigma_0(x)_{(n)}\in \End(\on{gr}^F V)
\label{eq:g[t]-action-in-gr}
\end{equation}
for $x\in \g$, $n\geq 0$.

The filtration $F^\sbullet V$
is compatible with the grading:
$F^pV=\bigoplus_{\Delta\in \Z_{\geq 0}}F^pV_{\Delta}$,
where
$F^p V_\Delta := V_\Delta \cap F^p V$.

Let $U_{\sbullet}(t^{-1}\g[t^{-1}])$ be the PBW filtration of
$U(t^{-1}\g[t^{-1}])$,
that is,
$U_{p}(t^{-1}\g[t^{-1}])$
is the subspace of $U(t^{-1}\g[t^{-1}])$
spanned by monomials $y_{1} y_{2} \dots y_{r}$ with $y_i \in \g$,
$r\leq p$.
Define
\begin{equation*}
G_p V= U_{p}(t^{-1}\g[t^{-1}])\mathbf{1}.
\end{equation*}
Then $G_\sbullet V$ defines an increasing filtration of $V$.
We have
\begin{equation}
F^p V_{\Delta}=G_{\Delta-p}G_{\Delta},
\end{equation}
where
$G_p V_{\Delta}:=G_p V\cap V_{\Delta}$,
see \cite[Prop.\,2.6.1]{Ara12}.
Therefore,
the graded space $\mathrm{gr}^G V = \bigoplus_{p \in \Z_{\geq 0}}
G_p V/G_{p-1}V$ is isomorphic to $\on{gr}^F V$.
In particular,
we have
\[
\gr\cong \on{gr}U_{\sbullet}(t^{-1}\g[t^{-1}])\cong S(t^{-1}\g[t^{-1}]).
\]
The action of $\g[t]$ on $\gr=S(t^{-1}\g[t^{-1}])$
coincides with the one
induced from the action of $\g[t]$ on
$\g[t,t^{-1}]/\g[t] \cong t^{-1}\g[t^{-1}]$.
More precisely,
the element $x(m)$, for $x \in \g$ and $m \in \Z_{\geq 0}$,
acts on $S(t^{-1}\g[t^{-1}])$ as follows:
\begin{align}\nonumber
x(m)\cdot\mathbf{1} &=0, & \\\label{eq:action}
x(m) \cdot v &= \sum_{j=1}^r
\sum_{n_{j}-m >0} y_{1}(-n_1)\cdots
[x,y_{{j}}] (m- n_{j})
\cdots y_{r}(-n_r),
\end{align}
if $v =y_{1}(-n_1)\cdots y_{r}(-n_r)$
with $y_i \in \g$, $n_1,\dots,n_r \in \Z_{>0}$.

\Subsection{Zhu's $C_2$-algebras and associated varieties
of affine vertex algebras}
\label{sub:Zhu affine vertex algebras}

We have \cite[Lem.\,2.9]{Li05}
\[
F^p V = \mathrm{span}_\C\{ a_{(-i-1)} b \mid a \in V,\, i \geq 1,\, b \in F^{p-i} V \}
\]
for all $p \geq 1$. In particular,
\[
F^1 V = C_2(V),
\]
where $C_2(V)=\on{span}_{\C}\{a_{(-2)}b\mid a,b\in V\}$.
Set
\[
R_V = V/C_2(V) = F^0 V / F^1 V \subset \mathrm{gr}^FV.
\]
It is known by Zhu \cite{Zhu96} that $R_V$ is a Poisson algebra.
The Poisson algebra structure can be understood as the restriction of the vertex Poisson structure of $\mathrm{gr}^FV$. It is given by
\[
\bar a \cdot \bar b = \overline{a_{(-1)} b}, \quad \{\bar a, \bar b\}= \overline{a_{(0)}b},
\]
for $a, b \in V$, where $\bar a = a+C_2(V)$.

By definition \cite{Ara12}, the
\emph{associated variety} of $V$
is
the reduced scheme
\[
X_V:= \mathrm{Specm}(R_V).
\]
It is easily seen that
\[
F^1 V^k(\g) = C_2(V^k(\g)) = t^{-2} \g[t^{-1}] V^k(\g).
\]

The following map defines an isomorphism of Poisson algebras
\begin{align*}
\C[\g^*] \cong S(\g) & \to R_{V^k(\g)} \\[0.2em]
\g \ni x & \mto x (-1) \mathbf{1}
+ t^{-2} \g[t^{-1}] V^k(\g).
\end{align*}
Therefore, $R_{V^k(\g)} \cong \C[\g^*]$ and so, $
X_{V^k(\g)}\cong \g^*$.

More generally, if $V$ is a quotient of $V^k(\g)$
by some ideal $N$, then
we have
\begin{equation}
R_{V}\cong \C[\g^*]/I_N
\end{equation}
as Poisson algebras,
where $I_N$ is the image of $N$ in $R_{V^k(\g)}=\C[\g^*]$.
Then $X_V$ is just the zero locus of $I_N$ in
$\g^*$.
It is a closed $G$-invariant conic subset of $\g^*$.

Identifying $\g^*$ with $\g$ through the bilinear form $(~|~)$,
one may view $X_V$ as a subvariety of $\g$.

\subsection{PBW basis}
Let ${\Delta}_{+}=\{\be_1,\dots,\be_q\}$ be the set of positive roots for $\g$ with respect
to a triangular decomposition
$\g= \mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$,
where $q=(d-\ell)/2$ and $\ell=\mathrm{rk}(\g)$.

Form now on, we
fix a basis
\[
\{u^i, e_{\be_j}, f_{\be_j}\mid i=1,\dots,\ell, \, j=1,\dots, q\}
\]
of $\g$ such that $\{u^{i} \mid i=1,\dots,\ell\}$
is an orthonormal basis of $\h$ with respect to $(~|~)$ and
$(e_{\be_i}|f_{\be_i})=1$ for $i=1,2,\dots,q$.
In particular, $[e_{\be_i}, f_{\be_i}]= \be_i$
for $i=1,\dots,q$ (see, for example, \cite[Prop.\,8.3]{Hu72}),
where $\h^*$ and $\h$ are identified through $(~|~)$.
One may also assume that $\mathrm{ht}(\be_i)\leq \mathrm{ht}(\be_j)$ for $i< j$,
where $\mathrm{ht}(\be_i)$ stands for the height of the positive root $\be_i$.

We define the structure constants $c_{\al,\be}$ by
\[
[e_\al,e_{\be}] = c_{\al,\be} e_{\al+\be},
\]
provided that $\al$, $\be$ and $\al+\be$ are in $\Delta$.
Our convention is that $e_{-\al}$ stands for $f_\al$ if $\alpha \in \Delta_+$.
If $\al$, $\be$ and $\al+\be$ are in $\Delta_+$,
then from the equalities,
\begin{equation*}
c_{-\al,\al+\be} = (f_{\be} | [f_{\al},e_{\al+\be}]) = - (f_{\be} | [e_{\al+\be},f_{\al}])
= - ([f_{\be}, e_{\al+\be}]|f_{\al}) = - c_{-\be,\al+\be},
\end{equation*}
we get that
\begin{equation}
\label{eq:structure_constant}
c_{-\al,\al+\be}= - c_{-\be,\al+\be}.
\end{equation}

By \eqref{eq:PBW},
the above basis of $\g$
induces a basis of $V^k(\g)$ consisted of $\mathbf{1}$
and the elements of the form
\begin{equation}
\label{eq:PBW_basis}
z = z^{(+)} z^{(-)} z^{(0)} \mathbf{1},
\end{equation}
with
\begin{align*}
z^{(+)} &:= e_{\be_{1}}(-1)^{a_{1,1}}\cdots
e_{\be_{1}} (- r_1)^{a_{1,r_1}}\cdots
e_{\be_q}(- 1)^{a_{q,1}}\cdots e_{\be_q}(- r_q)^{a_{q,r_q}}, \\
z^{(-)} &:=f_{\be_1}(-1)^{b_{1,1}}\cdots f_{\be_1}(- s_1)^{b_{1,s_1}}\cdots
f_{\be_q}(- 1)^{b_{q,1}}
\cdots f_{\be_q}(- s_q)^{b_{q,s_q}}, \\
z^{(0)} &:= u^1(- 1)^{c_{1,1}}\cdots u^1(- t_1)^{c_{1,t_1}}
\cdots u^\ell (- 1)^{c_{\ell,1}}\cdots u^\ell(- t_\ell)^{c_{\ell, t_\ell}},
\end{align*}
where $r_1,\dots,r_q, s_1,\dots,s_q,,t_1,\dots,t_\ell$
are positive integers, and $a_{l,m},b_{l,n},c_{i,j}$, for $l =1,\dots,q$,
$m=1,\dots,r_l$, $n=1,\dots,s_l$, $i = 1,\dots,\ell$, $j=1,\dots,t_i$
are nonnegative integers such
that at least one of them is nonzero.

\begin{defn}
\label{def:monomial_V}
Each element $x$ of $V^k(\g)$
is a linear combination of elements in the above PBW basis,
each of them will be called a \emph{PBW monomial} of $x$.
\end{defn}

\begin{defn}
\label{def:depth_V}
For a PBW monomial $v$ as in \eqref{eq:PBW_basis},
we call the integer
\begin{equation*}
\dep(v) =\sum_{i=1}^q \biggl(\sum_{j=1}^{r_i}
a_{i,j}(j-1) + \sum_{j=1}^{s_i} b_{i,j} (j-1) \biggr)
+ \sum_{i=1}^\ell \sum_{j=1}^{t_i} c_{i,j}(j-1)
\end{equation*}
the \emph{depth} of $v$.
In other words, a PBW monomial $v$ has depth $p$ means that
$v \in F^{p}V^k(\g)$ and $v \not \in F^{p+1}V^k(\g)$.
By convention, $\dep(\mathbf{1})=0$.

For a PBW monomial $v$ as in \eqref{eq:PBW_basis},
we call \emph{degree} of $v$ the integer
\begin{equation*}
\deg(v) =
\sum_{i=1}^q \biggl(\sum_{j=1}^{r_i}
a_{i,j} + \sum_{j=1}^{s_i} b_{i,j}\biggr)
+ \sum_{i=1}^\ell \sum_{j=1}^{t_i} c_{i,j},
\end{equation*}
In other words, $v$ has degree $p$ means that
$v \in G_{p}V^k(\g)$ and $v \not \in G_{p-1}V^k(\g)$
since the PBW filtration of $V^k(\g)$ coincides with the
standard filtration $G_\sbullet V^k(\g)$.
By convention, $\deg(\mathbf{1})=0$.
\end{defn}

Recall that a \emph{singular vector} of a $\g[t]$-representation $M$
is a vector $m \in M$
such that $e_{\alpha}(0).m=0$, for all $\alpha \in {\Delta}_{+}$,
and $f_\theta(1)\cdot m=0$, where $\theta$ is the highest positive root of $\g$.
From the identity
\begin{multline*}
L_{-1}=
\dfrac{1}{k+h^\vee} \biggl(\sum_{i=1}^{\ell}\sum_{m=0}^{\infty} u^i(-1-m) u^i(m)\\[-8pt]
+\sum_{\al\in{\Delta}_{+}}\sum_{m=0}^{\infty}
(e_{\al}(-1-m)f_{\al}(m)+f_{\al}(-1-m)e_{\al}(m))\biggr),
\end{multline*}
we deduce the following easy observation, which will be useful
in the proof of the main result.

\begin{lem}
\label{lem:Sugawara_singular_vector}
If $w$ is a singular vector of $V^k(\g)$,
then
\[
L_{-1} w=\dfrac{1}{k+h^\vee} \biggl(\sum_{i=1}^{\ell}u^i(-1)u^i(0)
+\sum_{\al\in{\Delta}_{+}}e_{\al}(-1)f_{\al}(0)\biggr) w.
\]
\end{lem}
\subsection{Basis of associated graded vertex Poisson algebras}

Note that $\gr=S(t^{-1}\g[t^{-1}])$ has a basis consisting
of $\mathbf{1}$ and elements of the form \eqref{eq:PBW_basis}.
Similarly to Definition \ref{def:monomial_V}, we have the following
definition.

\begin{defn}
\label{def:monomial_S}
Each element $x$ of $S(t^{-1}\g[t^{-1}])$
is a linear combination of elements in the above basis,
each of them will be called a \emph{monomial} of $x$.
\end{defn}

{As in the case of $V^k(\g)$,}
the space $S(t^{-1}\g[t^{-1}])$ has two natural gradations.
The first one is induced from the degree of elements as
polynomials.
We shall write $\deg(v)$ for the degree of a homogeneous
element $v \in S(t^{-1}\g[t^{-1}])$ with respect to this gradation.

The second one is induced from the Li filtration via the isomorphism
$S(t^{-1}\g[t^{-1}]) \cong \mathrm{gr}^F V^k(\g)$.
The degree of a homogeneous
element $v \in S(t^{-1}\g[t^{-1}])$ with respect to the gradation
induced by Li filtration
will be called the \emph{depth} of $v$, and will be denoted by $\dep(v)$.

Notice that any element $v$ of the form \eqref{eq:PBW_basis} is homogeneous
for both gradations.
By convention, $\deg(\mathbf{1})=\dep(\mathbf{1})=0$.

As a consequence of \eqref{eq:action}, we get that
\begin{equation}
\label{eq:homogeneous}
\deg(x(m)\cdot v) = \deg(v) \quad \text{ and }
\quad \dep(x(m)\cdot v)= \dep(v) - m,
\end{equation}
for
$m\geq 0$, $x\in\g$, and any homogeneous element $v \in S(t^{-1}\g[t^{-1}])$
with respect to both gradations.

In the sequel, we will also
use the following notation, for $v$ of the form \eqref{eq:PBW_basis},
viewed either as an element of $V^k(\g)$ or of $S(t^{-1}\g[t^{-1}])$:
\begin{equation}
\label{eq:deg_-1}
\deg_{-1}^{(0)}(v) :=
\sum_{j=1}^\ell c_{j,1 },
\end{equation}
which corresponds to the degree of the element
obtained from $v^{(0)}$ by keeping only the terms of
depth $0$, that is, the terms $u^{i}(-1)$, $i=1,\dots,\ell$.

Notice that a nonzero depth-homogeneous
element of $S(t^{-1}\g[t^{-1}])$ has depth $0$ if
and only if
its image in
\[
R_{V^k(\g)} = V^k(\g)/t^{-2} \g[t^{-1}] V^k(\g)
\]
is nonzero.
\section{Proof of the main result}
\label{sec:main_proof}

This section is devoted to the proof of Theorem \ref{Th:main}.

\subsection{Strategy}
{Let $N_k$ be the maximal graded submodule of $V^k(\g)$,
so that $L_k(\g)=V^k(\g)/N_k$.
Our aim is to show that if $V^k(\g)$ is not simple, that is,
$N_k\not=\{0\}$, then $X_{L_k(\g)}$
is strictly contained in $\g^*\cong\g$, that is,
the image $I_k:=I_{N_k}$ of $N_k$ in
$R_{V^k(\g)}=\C[\g^*]$ is nonzero.}

For $k=-h^\vee$, it follows from \cite{FreGai04} that
$I_k$ is the defining ideal of the
nilpotent
cone $ \mathcal{N}(\g)$ of $\g$,
and so
$X_{L_k(\g)} = \mathcal{N}(\g)$ (see \cite{A11} or Section \ref{subsection:remark}
below).
Hence, there is no loss of generality in assuming that $k+h^\vee \not=0$.

Henceforth, we suppose that $k+h^\vee \not=0$ and
that $V^k(\g)$ is not simple,
that is,
$N_k\ne \{0\}$.
Then there exists at least
one non-trivial (that is, nonzero and different from
$\mathbf{1}$) singular vector $w$
in $V^k(\g)$.
Theorem \ref{theorem:image-of-the-singular}
states that
the image of $w$ in $I_k$ is nonzero,
and
this proves
Theorem \ref{Th:main}. The rest of this section is devoted
to the proof of Theorem \ref{theorem:image-of-the-singular}.

Let $w$ be a nontrivial singular vector of $V^k(\g)$.
One can assume that $w \in F^p V^k(\g) \setminus F^{p+1} V^k(\g)$ for
some $p\in \Z_{\geq 0}$.

The image
\[
\bar{w}:=\sigma(w)
\]
of this singular vector in $S(t^{-1}\g[t^{-1}]) \cong \mathrm{gr}^F V^k(\g)$
is a nontrivial singular vector of $S(t^{-1}\g[t^{-1}])$.
Here $\sigma \colon V^k(\g) \to \mathrm{gr}^F V^k(\g)$
stands for the principal symbol map.
It follows from \eqref{eq:homogeneous}
that one can assume that $\bar{w}$ is homogeneous with respect to both gradations
on $S(t^{-1}\g[t^{-1}])$.
In particular $\bar w$ has depth $p$. It is enough to show that
$p=0$, that is, $\bar w$ has depth zero.
Write
\[
w = \sum_{j \in J} \lambda_j w^j,
\]
where $J$ is a finite index set, $\lambda_j $ are nonzero
scalar for all $j \in J$,
and $w_j$ are pairwise distinct PBW monomials of the form \eqref{eq:PBW_basis}.
Let $I \subset J$ be the subset of $i \in J$ such that $\dep \bar w^{i} = p = \dep \bar w$.
Since $w \in F^p V^k(\g) \setminus F^{p+1} V^k(\g)$,
the set $I$ is nonempty.
Here, $\bar{w}^{i}$ stands for the image of $w^{i}$ in $\mathrm{gr}^F V^k(\g) \cong S(t^{-1}\g[t^{-1}])$.

More specifically, for any $j \in I$, write
\begin{equation}
\label{eq:PBW_w_i}
w^{j} = (w^{j})^{(+)} (w^{j})^{(-)} (w^{j})^{(0)} \mathbf{1},
\end{equation}
with
\begin{align*}
& (w^{j})^{(+)} := e_{\be_{1}}(-1)^{a_{1,1}^{(j)}}\cdots
e_{\be_{1}} (- r_1)^{a_{1,r_1}^{(j)}}\cdots
e_{\be_q}(- 1)^{a_{q,1}^{(j)}}\cdots e_{\be_q}(- r_q)^{a_{q,r_q}^{(j)}} \\\nonumber
& (w^{j})^{(-)} : =f_{\be_1}(-1)^{b_{1,1}^{(j)}}\cdots f_{\be_1}(- s_1)^{b_{1,s_1}^{(j)}}\cdots
f_{\be_q}(- 1)^{b_{q,1}^{(j)}}
\cdots f_{\be_q}(- s_q)^{b_{q,s_q}^{(j)}}, \\\nonumber
& (w^{j})^{(0)} := u^1(- 1)^{c_{1,1}^{(j)}}\cdots u^1(- t_1)^{c_{1,t_1}^{(j)}}
\cdots u^\ell (- 1)^{c_{\ell,1}^{(j)}}\cdots u^\ell(- t_\ell)^{c_{\ell, t_\ell}^{(j)}},
\end{align*}
where $r_1,\dots,r_q, s_1,\dots,s_q,,t_1,\dots,t_\ell$
are nonnegative integers, and
$a_{l,m}^{(j)},b_{l,n}^{(j)},c_{i,p}^{(j)}$, for $l =1,\dots,q$,
$m=1,\dots,r_l$, $n=1,\dots,s_l$,
$i =1,\dots,\ell$,
$p=1,\dots,t_i$,
are nonnegative integers such
that at least one of them is nonzero.

The integers $r_l$'s, for $l=1,\dots,q$, are chosen so that at least one of the
$a_{l,r_l}^{(j)}$'s is nonzero for $j$ running through $J$
if for some $j \in J$, $(w^{j})^{(+)}\not= 1$.
Otherwise, we just set $(w^{j})^{(+)} := 1$.
Similarly are defined the integers $s_l$'s and $t_m$'s,
for $l =1,\dots,q$ and $m=1,\dots,\ell$. By our assumption, note that for all $i \in I$,
\begin{align*}
& \sum_{n=1}^q \biggl(\sum_{l=1}^{r_n}
a_{n,l}^{(i)} + \sum_{l=1}^{s_n} b_{n,l}^{(i)} \biggr)
+ \sum_{n=1}^\ell \sum_{l=1}^{t_n} c_{n,l}^{(i)} = \deg(\bar w)&\\
& \sum_{n=1}^q \biggl(\sum_{l=1}^{r_n}
a_{n,l}^{(i)} (l-1) + \sum_{l=1}^{s_n} b_{n,l}^{(i)} (l-1) \biggr)
+ \sum_{n=1}^\ell \sum_{l=1}^{t_n} c_{n,l}^{(i)} (l-1)
= \dep(\bar w) = p.&
\end{align*}

\subsection{A technical lemma}
\label{sub:technical}
In this paragraph we remain in the commutative setting, and we only
deal with $\bar w \in S(t^{-1}\g[t^{-1}])$ and its monomials $\bar w^{i}$'s,
for $ i \in I$.

Recall from \eqref{eq:deg_-1} that,
\[
\deg_{-1}^{(0)}(w^i)= \sum_{j=1}^\ell c_{j,1}^{(i)}
\]
for $i \in I$.
Set
\[
d_{-1}^{(0)}(I) := \max\{\deg_{-1}^{(0)}(w^i) \mid i \in I\},
\]
and
\[
I_{-1}^{(0)} := \{ i\in I \mid \deg_{-1}^{(0)}(w^i)= d_{-1}^{(0)}(I)\}.
\]
If $(w^i)^{(0)}=1$ for all $i \in I$, we just set $d_{-1}^{(0)}(I)=0$
and then $I_{-1}^{(0)}=I$.

\begin{lem}
\label{lem:wi_-}
If $i\in I_{-1}^{(0)}$, then $(\bar w^i)^{(-)}=1$. In
other words, for $i \in I_{-1}^{(0)}$, we have
$\bar w^{i}= (\bar w^{i})^{(0)} (\bar w^{i})^{(+)} \mathbf{1}.$
\end{lem}
\begin{proof} Suppose the assertion is false.
Then for some positive roots $\beta_{j_1},\dots,\beta_{j_t} \in \Delta_+$,
one can write for any $i \in I_{-1}^{(0)}$,
\begin{equation}\label{e5}
(\bar w^i)^{(-)}=f_{\be_{j_1}}(-1)^{b_{j_1,1}^{(i)}}
\cdots f_{\be_{j_1}}(-s_{j_1})^{b_{j_1,s_{j_1}}^{(i)}}\cdots
f_{\be_{j_t}}(-1)^{b_{j_t,1}^{(i)}}
\cdots f_{\be_{j_t}}(-s_{j_t})^{b_{j_t,s_{j_t}}^{(i)}},
\end{equation}
so that for any $l \in \{1,\dots,t\}$,
\[
\{b_{j_l,s_{j_l}}^{(i)} \mid i \in I_{-1}^{(0)}\}\neq \{0\}.
\]
Set
\[
K_{-1}^{(0)} = \{i \in I_{-1}^{(0)} \mid b_{j_1,s_{j_1}}^{(i)}> 0 \}.
\]
Since $\bar w$ is a singular vector of $S(t^{-1}\g[t^{-1}])$ and $s_{j_1}-1 \in\Z_{\geq 0}$,
we have
\[
e_{\beta_{j_1}}(s_{j_1}-1)\cdot\bar w=0.
\]
On the other hand, using the action of $\g[t]$ on $S(t^{-1}\g[t^{-1}])$
as described by \eqref{eq:action}, we see that
\begin{equation}
\label{eq:wi_-}
0=e_{\beta_{j_1}}(s_{j_1}-1) \cdot\bar w =
\sum_{i \in K_{-1}^{(0)}} \lambda_ib_{j_1,s_{j_1}}^{(i)} v^{i} + v,
\end{equation}
where for $i \in K_{-1}^{(0)}$,
\begin{multline*}
v^{i}:=
(\bar w^{i})^{(0)} {\be_{j_1}(-1)}
f_{\be_{j_1}}(-1)^{b_{j_1,1}^{(i)}}
\cdots f_{\be_{j_1}}(-s_{j_1})^{b_{j_1,s_{j_1}}^{(i)}-1}\\
\cdots
f_{\be_{j_t}}(-1)^{b_{j_t,1}^{(i)}}
\cdots f_{\be_{j_t}}(-s_{j_t})^{b_{j_t,s_{j_t}}^{(i)}} (w^{i})^{(+)} \mathbf{1},
\end{multline*}
and $v$ is a linear combination of monomials $x$
such that
\[
\deg_{-1}^{(0)}(x) \leq d_{-1}^{(0)}(I).
\]
Indeed, for $i \in K_{-1}^{(0)}$, it is clear that
\begin{equation*}
e_{\beta_{j_1}}(s_{j_1}-1)\cdot w^{i} = b_{j_1,s_{j_1}}^{(i)} v^{i} + y^{i},
\end{equation*}
where $y^{i}$ is a linear combination of monomials $y$
such that
$\deg_{-1}^{(0)}(y) \leq d_{-1}^{(0)}(I)$ because
$\mathrm{ht}(\beta_{j_1}) \leq \mathrm{ht}(\beta_{j_l})$
for all $l \in \{1,\dots,t\}$.
Next, for $i \in I_{-1}^{(0)} \setminus K_{-1}^{(0)}$,
$e_{\beta_{j_1}} (s_{j_1}-1)\cdot\bar w^{i} $ is a
linear combination of monomials $z$ such that
$\deg_{-1}^{(0)}(z) \leq d_{-1}^{(0)}(I)$ because
$b_{j_1,s_{j_1}}^{(i)}=0$.
Finally, for $i \in I \setminus I_{-1}^{(0)}$, we have $\deg_{-1}^{(0)}(\bar w^{i}) < d_{-1}^{(0)}(I) $
and, hence, $e_{\beta_{j_1}}(s_{j_1}-1) \cdot\bar w^{i} $ is a
linear combination of monomials $z$ such that
$\deg_{-1}^{(0)}(z) \leq d_{-1}^{(0)}(I)$ as well.

Now, note that for each $i \in K_{-1}^{(0)}$,
\[
\deg_{-1}^{(0)}(v^{i})=\deg_{-1}^{(0)}(\bar w^{i})+1 = d_{-1}^{(0)}(I) +1.
\]
Hence by \eqref{eq:wi_-} we get a contradiction because all monomials
$v^{i}$, for $i$ running through $K_{-1}^{(0)}$, are linearly independent
while $\lambda_i b_{j_1,s_{j_1}}^{(i)} \not=0$, for $i\in K_{-1}^{(0)}$.
This concludes the proof of the lemma.
\end{proof}

\subsection{Use of Sugawara operators}
\label{sub:Sugawara operators}
Recall that $w = \sum_{j \in J} \lambda_j w^j$.
Let $J_1\subseteq J$ be such that for $i\in J_1$, $(w^i)^{(-)}=1$. Then by Lemma \ref{lem:wi_-},
\[
\emptyset \neq I_{-1}^{(0)}\subseteq J_1.
\]
So $J_1\neq \emptyset.$
Set
\[
{d}_{-1}^{(0)}: = {d}_{-1}^{(0)} (J_1) = \max\{\deg_{-1}^{(0)}(w^i) \mid i \in J_1\},
\]
and
\[
J_{-1}^{(0)} := \{ i\in J_1 \mid \deg_{-1}^{(0)}(w^i)= {d}_{-1}^{(0)}\}.
\]
Then $d_{-1}^{(0)}(I)\leq {d}_{-1}^{(0)}$. Set
\[
d^{+} := \max\{\deg (w^{i})^{(+)} \mid i \in J_{-1}^{(0)} \}
\]
and let
\[
J^+= \{i \in J_{-1}^{(0)} \mid \deg (w^{i})^{(+)}= d^{+} \}\subseteq J_{-1}^{(0)}.
\]
Our next aim is to show that for $i\in J^+$,
$w^{i}$ has depth zero, whence $p=0$ since $p$ is by definition
the smallest depth of the $w^j$'s, and so
the image of $w$ in
$R_{V^k(\g)} = F^0 V^k(\g)/ F^1 V^k(\g)$ is nonzero.

This will be achieved in this paragraph through the use of
the Sugawara construction.

Recall that
by Lemma \ref{lem:Sugawara_singular_vector},
\[
L_{-1} w=\tilde{L}_{-1} w
\]
since $w$ is a singular vector of $V^k(\g)$, where
\begin{equation*}
{\tilde{L}_{-1}:=\dfrac{1}{k+h^\vee} \biggl(\sum_{i=1}^{\ell}u^i(-1)u^i(0)
+\sum_{\al\in{\Delta}_{+}}e_{\al}(-1)f_{\al}(0)\biggr)}.
\end{equation*}

\begin{lem}
\label{lem:Sugawara_monomial}
Let $z$ be a PBW monomial of the form \eqref{eq:PBW_basis}.
Then $\tilde L_{-1} z$
is a linear combination of PBW monomials
$x$ satisfying all the following conditions:
\begin{enumeratea}
\item\label{lem:Sugawara_monomiala}
$\deg(x^{(+)}) \leq \deg(z^{(+)})+1$ and $\deg(x^{(0)}) \leq \deg(z^{(0)})+1$,
\item\label{lem:Sugawara_monomialb}
if $z^{(-)}\neq 1$, then $x^{(-)}\neq 1$.
\item\label{lem:Sugawara_monomialc}
if $x^{(-)}=z^{(-)}$, then either $\deg(x^{(0)}) =\deg(z^{(0)}) +1$, or $x^{(0)} = z^{(0)}$.
\item\label{lem:Sugawara_monomiald}
if $\deg(x^{(0)}) =\deg(z^{(0)}) +1$, then $x^{(-)}= z^{(-)}$ and $\deg(x^{(+)}) \leq \deg(z^{(+)})$.
\end{enumeratea}
\end{lem}
\begin{proof}
Parts \eqref{lem:Sugawara_monomiala}--\eqref{lem:Sugawara_monomialc} are easy to see. We only prove \eqref{lem:Sugawara_monomiald}. Assume that $\deg(x^{(0)}) =\deg(z^{(0)}) +1$.
Either $x$ comes from the term $\sum_{i=1}^{\ell} u^i(-1)u^i(0) z$,
or it comes from a term $e_{\al}(-1)f_{\al}(0)z$ for some $\al \in \Delta_+$.

If $x$ comes from the term $\sum_{i=1}^{\ell} u^i(-1)u^i(0) z$, then it is obvious that
$x^{(-)}= z^{(-)}$ and $x^{(+)} = z^{(+)}$.

Assume that $x$ comes from $e_{\al}(-1)f_{\al}(0)z$ for some $\al \in \Delta_+$.
We have
\begin{multline*}
e_{\al}(-1)f_{\al}(0)z = e_{\al}(-1) [f_{\al}(0), z^{(+)}]z^{(-)}z^{(0)}\mathbf{1} + e_{\al}(-1)z^{(+)}[f_{\al}(0), z^{(-)}]z^{(0)} \mathbf{1} \\
+ e_{\al}(-1)z^{(+)} z^{(-)}[f_{\al}(0),z^{(0)} ] \mathbf{1}.
\end{multline*}
Clearly, any PBW monomials $x$ from
\[
e_{\al}(-1)z^{(+)}[f_{\al}(0), z^{(-)}]z^{(0)} \mathbf{1} \quad \text{ or }
\quad e_{\al}(-1)z^{(+)} z^{(-)}[f_{\al}(0),z^{(0)} ] \mathbf{1}
\]
satisfies that $\deg(x^{(0)}) \leq \deg(z^{(0)})$. Then it is enough to consider PBW monomials in
\[
e_{\al}(-1) [f_{\al}(0), z^{(+)}]z^{(-)}z^{(0)}\mathbf{1}.
\]
The only possibility for a PBW monomial $x$ in $ e_{\al}(-1) [f_{\al}(0), z^{(+)}]z^{(-)}z^{(0)}\mathbf{1}$
to satisfy $\deg(x^{(0)}) =\deg(z^{(0)}) +1$ is that it comes from a term
$ [f_{\al}(0), e_{\al}(-n)]= - \al(-n)$ for some $n \in \Z_{>0}$, where
$e_{\al}(-n)$ is a term in $z^{(+)}$.
But then, for PBW monomials $x$ in
$ e_{\al}(-1)[f_{\al}(0), z^{(+)}]z^{(0)}\mathbf{1}$
such that $\deg(x^{(0)}) =\deg(z^{(0)}) +1$,
we have $x^{(-)}=z^{(-)}$ and $\deg(x^{(+)}) \leq \deg(z^{(+)})$.
\end{proof}

We now consider the action of $\tilde{L}_{-1}$ on particular PBW monomials.
\begin{lem}
\label{lem:Sugawara_bracket}
Let $z$ be a PBW monomial of the form \eqref{eq:PBW_basis}
such that $z^{(-)}=1$ and $\dep(z^{(+)})=0$, that is,
either $z^{(+)}=1$, or for some $j_1,\dots,j_t \in \{1,\dots,q\}$
(with possible repetitions),\vspace*{-3pt}
\begin{equation*}
z = e_{\be_{j_1}}(-1)e_{\be_{j_2}}(-1)\cdots e_{\be_{j_t}}(-1) z ^{(0)} \mathbf{1}.
\end{equation*}
Then $\tilde{L}_{-1}z$
is a linear combination of PBW monomials $y$ satisfying one of the following
conditions:
\begin{enumerate}
\item\label{lem:Sugawara_bracket1}
$\!y^{(-)}=1$, $\dep(y^{(+)})\geq 1$, $\deg(y^{(+)})\leq \deg(z^{(+)})$,
$y^{(0)}=z^{(0)}$,
\item\label{lem:Sugawara_bracket2}
$\!y^{(-)}\!=\!1$, $\dep(y^{(+)})\!=\!0$, $\deg (y^{(+)})\!\leq\! \deg (z^{(+)})-1$, and
$\deg(y^{(0)})\!>\!\deg(z^{(0)})$, $\deg_{-1}^{(0)}(y)=\deg_{-1}^{(0)}(z)$,
\item\label{lem:Sugawara_bracket3}
$\!y^{(-)}\!=\!1$, $\dep(y^{(+)})\!\geq\!1$, $\deg(y^{(+)})\!\leq\!\deg(z^{(+)})-1$, and \hbox{$\deg_{-1}^{(0)}(y)\!=\!\deg_{-1}^{(0)}(z)\!+\!1$},
\item\label{lem:Sugawara_bracket4}
$\!y^{(-)}\neq 1$.
\end{enumerate}
\end{lem}
\begin{proof}
First, we have\vspace*{-3pt}
\begin{equation*}
\sum_{i=1}^{\ell} u^i(-1)u^i(0) z
= \sum_{r=1}^t
e_{\be_{j_1}}(-1) \cdots \biggl[\sum_{i=1}^{\ell}u^i(-1)u^i(0), e_{\be_{j_r}}(-1)\biggr] \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1},
\end{equation*}
and\vspace*{-3pt}
\begin{align*}
\sum_{i=1}^{\ell}u^i(-1)u^i(0), e_{\be_{j_r}}(-1)& = \sum_{i=1}^{\ell}\left(u^i(-1) [u^i(0), e_{\be_{j_r}}(-1)] + [u^i(-1), e_{\be_{j_r}}(-1)] u^i(0) \right)\\
& = \be_{j_r}(-1) e_{\be_{j_r}}(-1) + e_{\be_{j_r}}(-2) \be_{j_r}(0).
\end{align*}
So\vspace*{-5pt}
\begin{multline}\label{e37}
\sum_{i=1}^{\ell} u^i(-1)u^i(0) z\\[-8pt]
= \sum_{r=1}^t
e_{\be_{j_1}}(-1) \cdots (\be_{j_r}(-1) e_{\be_{j_r}}(-1) + e_{\be_{j_r}}(-2) \be_{j_r}(0)) \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}.
\end{multline}
Second, we have\vspace*{-5pt}
\begin{multline*}
\sum_{\al\in{\Delta}_+}e_{\al}(-1) f_{\al}(0) z
= \sum_{\al\in{\Delta}_+} \sum_{r=1}^te_{\al}(-1)
e_{\be_{j_1}}(-1) \cdots [f_{\al}(0),e_{\be_{j_r}}(-1)] \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1} \\
+ \sum_{\al\in{\Delta}_+} e_{\al}(-1)e_{\be_{j_1}}(-1)e_{\be_{j_2}}(-1)\cdots e_{\be_{j_t}}(-1)
[f_{\al}(0),z^{(0)} ] \mathbf{1}.
\end{multline*}
It is clear that
any PBW monomial $y$ in\vspace*{-3pt}
\[
\sum_{\al\in{\Delta}_+}e_{\al}(-1) e_{\be_{j_1}}(-1)e_{\be_{j_2}}(-1)\cdots e_{\be_{j_t}}(-1)
[f_{\al}(0),z^{(0)} ] \mathbf{1}
\]
satisfies\vspace*{-3pt}
\begin{equation}\label{e38}
y^{(-)} \not= 1.
\end{equation}
We now consider
\[
u_r:=\sum_{\al\in{\Delta}_+}
e_{\al}(-1)e_{\be_{j_1}}(-1) \cdots [f_{\al}(0),e_{\be_{j_r}}(-1)] \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}, \text{ for } 1\leq r\leq t.
\]
\begin{itemize}
\item
If $\be_{j_r}=\al+\be$ for some $\al,\be\in\Delta_{+}$, then there is a partial sum of two terms in~$u_r$:
\begin{multline*}
c_{-\al,\al+\be}e_{\al}(-1)e_{\be_{j_1}}(-1) \cdots e_{\be}(-1)\cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}\\
+c_{-\be,\al+\be}e_{\be}(-1)e_{\be_{j_1}}(-1) \cdots e_{\al}(-1)\cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}.
\end{multline*}
Rewriting the above sum to a linear combination of PBW monomials, and noticing that
\[
c_{-\al,\al+\be}e_{\al}(-1)e_{\be}(-1)+c_{-\be,\al+\be}e_{\be}(-1)e_{\al}(-1)=c_{-\al,\al+\be}c_{\al,\be}e_{\al+\be}(-2),
\]
due to \eqref{eq:structure_constant},
we deduce that it is a linear combination of PBW monomials $y$ such that
\begin{equation}\label{e43}y^{(-)}=z^{(-)}=1, \ y^{(0)}=z^{(0)}, \ \dep(y^{(+)})\geq 1, \ \deg(y^{(+)})\leq \deg(z^{(+)}),
\end{equation}
where $c_{-\al,\al+\be}, c_{-\be,\al+\be}, c_{\al,\be}\in {\mathbb R}^*$.

\item
If $\al-\be_{j_r}\in\Delta_+$ for some $\al\in\Delta_+$, then there is a term in $u_r$:
\begin{equation}\label{e35}
c_{-\al,\be_{j_r}}e_{\al}(-1)e_{\be_{j_1}}(-1) \cdots e_{\be_{j_{r-1}}}(-1) f_{\al-\be_{j_r}}(-1)e_{\be_{j_{r+1}}}(-1) \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}.
\end{equation}
It is easy to see that (\ref{e35}) is a linear combination of PBW monomials $y$ such that $y$ satisfies one of the following:
\begin{align}\label{e39}
& y^{(-)}=1, \ \dep(y^{(+)})\geq 1, \ \deg(y^{(+)})\leq \deg(z^{(+)}), \ y^{(0)}=z^{(0)},&\\
& y^{(-)}=1, \ \dep(y^{(+)})=0, \ \deg(y^{(+)})\leq \deg(z^{(+)})-1, &\\ \nonumber & \deg(y^{(0)})>\deg(z^{(0)}), \ \deg_{-1}^{(0)}(y)=\deg_{-1}^{(0)}(z),&\\
& y^{(-)}\neq 1.
\end{align}
Notice also that with $\al=\be_{j_r}$, there is a term in $u_r$:
\begin{equation*}
-e_{\be_{j_r}}(-1)e_{\be_{j_1}}(-1) \cdots e_{\be_{j_{r-1}}}(-1)
\be_{j_r}(-1)e_{\be_{j_{r+1}}}(-1) \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}.
\end{equation*}
Together with (\ref{e37}), we see that
\begin{multline*}
\sum_{i=1}^{\ell} u^i(-1)u^i(0) z+\sum_{r=1}^te_{\be_{j_r}}(-1)e_{\be_{j_1}}(-1)\cdots [f_{\be_{j_r}}(0), e_{\be_{j_r}}(-1)]\cdots \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}\\
= \sum_{r=1}^t
e_{\be_{j_1}}(-1) \cdots (\be_{j_r}(-1) e_{\be_{j_r}}(-1) + e_{\be_{j_r}}(-2) \be_{j_r}(0)) \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}\\
\hspace*{-3cm}-\sum_{r=1}^t\sum_{s=1}^{r-1}e_{\be_{j_1}}(-1) \cdots [e_{\be_{j_r}}(-1), e_{\be_{j_s}}(-1)]\\[-5pt]
\shoveright{\cdots e_{\be_{j_{r-1}}}(-1)\be_{j_r}(-1)e_{\be_{j_{r+1}}}(-1) \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}}\\
-\sum_{r=1}^te_{\be_{j_1}}(-1) \cdots e_{\be_{j_{r-1}}}(-1)e_{\be_{j_r}}(-1)\be_{j_r}(-1)e_{\be_{j_{r+1}}}(-1) \cdots e_{\be_{j_t}}(-1) z^{(0)} \mathbf{1}
\end{multline*}
is a linear combination of PBW monomials $y$ satisfying one of the following:
\begin{align}
\label{e40}
& y^{(-)}=1, \ \dep(y^{(+)})\geq 1, \deg(y^{(+)})\leq \deg(z^{(+)}), \ y^{(0)}=z^{(0)}, \\\label{e41}
& y^{(-)}=1, \ \dep(y^{(+)})\geq 1, \deg(y^{(+)})\leq \deg(z^{(+)})-1, \\
& \ \deg_{-1}^{(0)}(y)=\deg_{-1}^{(0)}(z)+1. \nonumber
\end{align}
Then the lemma follows from (\ref{e38}), (\ref{e43}), (\ref{e39})--(\ref{e41}).\qedhere
\end{itemize}
\end{proof}

\begin{lem}\label{Sugawara1}
Let $z$ be a PBW monomial of the form \eqref{eq:PBW_basis}
such that $z^{(-)}=1$. Then
\begin{equation*}
\tilde{L}_{-1}z
= c z^{(+)}(\gamma-\sum_{j=1}^qa_{j,1}\be_j)(-1)z^{(0)}+y^1,
\end{equation*}
where $c$ is a nonzero constant,
$\gamma=\sum_{j=1}^q\sum_{s=1}^{r_j}a_{j,s}\be_j$, and
$y^1$ is a linear combination of PBW monomials $y$ such that
\[
\deg_{-1}^{(0)}(y)=\deg_{-1}^{(0)}(z)+1, \ \deg(y^{(+)})\leq \deg(z^{(+)})-1,
\]
or
\[
\deg_{-1}^{(0)}(y)\leq \deg_{-1}^{(0)}(z).
\]
\end{lem}
\begin{proof}
Since the proof is similar to that of Lemma \ref{lem:Sugawara_bracket},
we left the verification to the reader.
\end{proof}

{
\begin{lem}\label{lem:wi_+}
For $i\in J^+$, we have that $\dep((w^i)^{(+)})=0$.
\end{lem}
\begin{proof} First we have
\[
w = \sum_{j \in J^+} \lambda_j w^j+\sum_{j \in J_{-1}^{(0)}\setminus J^+} \lambda_j w^j+\sum_{j \in J_1\setminus J_{-1}^{(0)}} \lambda_j w^j+\sum_{j \in J\setminus J_1} \lambda_j w^j.
\]
Then by Lemma \ref{lem:Sugawara_monomial}\eqref{lem:Sugawara_monomialb} and Lemma \ref{Sugawara1}, we have
\begin{multline*}
(k+h^{\vee})\tilde{L}_{-1}w
=\sum_{i\in J^+}(w^i)^{(+)} \biggl(\gamma_i-\sum_{j=1}^q
a_{j,1}^{(i)}\be_j\biggr)(-1)(w^i)^{(0)}\\[-8pt]
+\sum_{i\in J_1\setminus J^+}(w^i)^{(+)}\biggl(\gamma_i-\sum_{j=1}^qa^{(i)}_{j1}\be_j\biggr)(-1)(w^i)^{(0)}+y^1,
\end{multline*}
where
{$\gamma_i=\sum_{j=1}^q\sum_{s=1}^{r^{(i)}_j}a^{(i)}_{j,s}\be_i$},
for $i\in J_1$, and $y^1$ is a linear combination of PBW monomials $y$ satisfying
one of the following conditions:
\begin{align*}
& \deg_{-1}^{(0)}(y)=d_{-1}^{(0)}+1, \ \deg(y^{(+)})\leq d^+-1,\\
& \deg_{-1}^{(0)}(y)\leq d_{-1}^{(0)},\\
& y^{(-)}\neq 1.
\end{align*}
On the other hand, by Lemma \ref{lem:Sugawara_singular_vector}
\[
L_{-1}w=\tilde{L}_{-1}w.
\]
By Lemma \ref{lem:sugawara-vs-g}, there is no PBW monomial $y$ in $L_{-1}w$ such that $\deg(y^{(+)})=d^+$, $y^{(-)}=1$, and $\deg_{-1}^{(0)}(y)=d_{-1}^{(0)}+1$. Then we deduce that
\[
\sum_{i\in J^+}(w^i)^{(+)}\biggl(\gamma_i-\sum_{j=1}^qa_{j,1}^{(i)}\be_j\biggr)(-1)(w^i)^{(0)}=0,
\]
which means that ${(\gamma_i-\sum_{j=1}^qa_{j,1}^{(i)}\be_j)}=0$, for $i\in J^+$, that is, $\dep((w^i)^{(+)})=0$.
\end{proof}}

As explained at the beginning of \S\ref{sub:Sugawara operators}, Theorem
\ref{Th:main} will be a consequence of the following lemma.

\begin{lem}
\label{lem:w_0}
For each $i\in J^+$, we have $\dep(w^{i})=0$.
\end{lem}
\begin{proof}
By definition, for $i\in J^+$, $(w^i)^{(0)}=1$. Moreover, by Lemma \ref{lem:wi_+},
$\dep((w^i)^{(+)})=0$. Hence it suffices to prove that for $i\in J^+$,
\[
(w^{i})^{(0)}=u^{1}(-1)^{c^{(i)}_{1,1}}\cdots u^\ell (-1)^{c^{(i)}_{\ell,1}}.
\]
Suppose the contrary. Then
there exists $i \in J^+$ such that
\begin{multline*}
w^i = e_{\be_1}(-1)^{a^{(i)}_{1,1}}\cdots e_{\be_{q}}(-1)^{a^{(i)}_{q,1}} u^{1}(-1)^{c^{(i)}_{1,1}}
\cdots u^{1}(-m_{1})^{c^{(i)}_{1,m_1}} \\[-5pt]
\cdots u^\ell (-1)^{c^{(i)}_{\ell,1}} \cdots u^\ell(-m_{\ell})^{c^{(i)}_{\ell,m_\ell}} \mathbf{1},
\end{multline*}
with at least one of the $m_j$'s, for $j=1,\dots,\ell$, strictly greater than $1$
and $c^{(i)}_{j,m_j} \not=0$ for such a $j$.
Without loss of generality,
one may assume that $1 \in J^+$, that
\[
m_1=\max\{ m_j\mid j=1,\dots,\ell\} \quad \text{ and }
\quad 0\neq c^{(1)}_{1,m_1} \geq c^{(i)}_{1,m_1}, \text{ for } i\in J^+.
\]
Writing $L_{-1}w$ as
\[
L_{-1}w=\sum_{i \in J^+ }L_{-1}w^i
+\sum_{i\in J_{-1}^{(0)}\setminus J^+}L_{-1}w^i
+\sum_{i\in J_1\setminus J_{-1}^{(0)}}L_{-1}w^i
+\sum_{i \in J \setminus J_1}L_{-1}w^i,
\]
we see by Lemma \ref{lem:sugawara-vs-g} that
\begin{equation}\label{e17}
L_{-1}w=\lambda_1 m_1c^{(1)}_{1,m_1} v^1+\sum_{i\in J^+,i\neq 1}\lambda_im_1c^{(i)}_{1,m_1}v^i+v
+v',
\end{equation}
where for $i\in J^+$, $v^{i}$ is the PBW monomial defined by:
\begin{align}\label{e14}
(v^i)^{(-)}&=(w^i)^{(-)}=1,
\\
\label{e13}
(v^i)^{(+)}&=(w^i)^{(+)}=e_{\be_1}(-1)^{a^{(i)}_{1,1}}\cdots e_{\be_{q}}(-1)^{a^{(i)}_{q,1}},
\\
\label{e33}
(v^i)^{(0)} & = u^{1}(-1)^{c^{(i)}_{1,1}}\cdots
u^{1}(-m_{1})^{c^{(i)}_{1,m_1}-1}u^{1}(-m_{1}-1)\cdots u^\ell (-m_{\ell})^{c^{(i)}_{\ell,m_\ell}},
\end{align}
and so, by definition of $J^+ \subset J_{-1}^{(0)}$,
\begin{equation}\label{e34}
\deg_{-1}^{(0)}(v^i)=d_{-1}^{(0)},
\end{equation}
$v$ is a linear combination of PBW monomials $x$ such that
\[
x^{(0)}=u^{1}(-1)^{c^{(x)}_{1,1}}\cdots u^{1}(- n_{1}^{(x)})^{c^{(x)}_{1,n_{1}^{(x)}}}
\cdots u^\ell (-1)^{c^{(x)}_{\ell,1}}\cdots u^\ell(-n_{\ell}^{(x)})^{c^{(x)}_{\ell,n_{\ell}^{(x)}}}
\]
and
\[
\text{either } n_{1}^{(x)}\leq m_1,\quad
\text{or }
\deg(x^{(+)})\leq d^+ -1,\quad
\text{or }
\deg_{-1}^{(0)}(x)\leq d_{-1}^{(0)}-1,
\]
and $v'$ is a linear combination of PBW monomials $x$ such that $x^{(-)}\neq 1$.
Note that the assumption that $m_1\geq 2$ makes sure that (\ref{e34}) holds,
and that $\dep(v^{i}) =\dep(w^i)+1$ for all $i \in J^+$.

On the other hand, by Lemma \ref{lem:Sugawara_singular_vector},
\begin{equation*}
L_{-1} w={\tilde{L}_{-1} w},
\end{equation*}
since $w$ is a singular vector of $V^k(\g)$.
Hence $v^1$ must be a PBW monomial of {$\tilde{L}_{-1} w$}.
Our strategy to obtain the expected contradiction
is to show that there is no PBW monomial $v^1$ in {$\tilde{L}_{-1} w^i$}
for each $i \in J$.

\begin{itemize}
\item
Assume that $i\in J^+$, and suppose that
$v^{1}$ is a PBW monomial in ${\tilde{L}_{-1} w^{i}}$.
First of all, $\deg((w^{i})^{(+)}) = d^+$
because $i \in J^+$. Moreover,
by the definition of $J_1$ and Lemma~\ref{lem:wi_+},
we have $(w^{i})^{(-)}=1$ and $\dep((w^{i})^{(+)})=0$.
Hence by Lemma~\ref{lem:Sugawara_bracket}\eqref{lem:Sugawara_bracket2},
\[
\deg((v^1)^{(+)}) < \deg((w^{i})^{(+)}) = d^+
\]
because $(v^1)^{(-)}=1$ and $\dep((v^{1})^{(+)})=0$ by \eqref{e14}
and \eqref{e13}.
But $d^+ = \deg((v^1)^{(+)})$ by \eqref{e13},
whence a contradiction.

\item
Assume that $i \in J_{-1}^{(0)} \setminus J^+$.
By the definition of $J^+$ and \eqref{e13},
\begin{equation}
\label{eq:d+}
\deg ((w^{i})^{(+)}) < d^+ =\deg((v^1)^{(+)}).
\end{equation}
Suppose that
$v^{1}$ is a PBW monomial in ${\tilde{L}_{-1} w^{i}}$.
Then
\begin{equation}
\label{e15}
(w^{i})^{(-)}=1 = (v^1)^{(-)}
\end{equation}
by Lemma \ref{lem:wi_-} since $i \in J_{-1}^{(0)}$.
The last equality follows from \eqref{e14}.
Then by Lemma~\ref{lem:Sugawara_monomial}\eqref{lem:Sugawara_monomialc},
either $\deg((v^1)^{(0)})=\deg((w^{i})^{(0)})+1$,
or $(v^1)^{(0)}=(w^{i})^{(0)}.$
But it is impossible that $\deg((v^1)^{(0)})=\deg((w^{i})^{(0)})+1$,
by (d) of Lemma \ref{lem:Sugawara_monomial} because $\deg((v^1)^{(+)}) > \deg ((w^{i})^{(+)})$.
Therefore,
\[
(v^1)^{(0)}=(w^{i})^{(0)}.
\]
Computing $\tilde{L}_{-1} w^{i}$, we deduce from
\[
(v^1)^{(+)}=e_{\be_1}(-1)^{a^{(1)}_{1,1}}\cdots e_{\be_{q}}(-1)^{a^{(1)}_{q,1}},
\]
that
\[
(w^i)^{(+)}=e_{\be_1}(-1)^{a^{(j)}_{1,1}}\cdots e_{\be_{q}}(-1)^{a^{(j)}_{q,1}}.
\]
Since $(v^1)^{(-)}\!=\!(w^{i})^{(-)}\!=\!1$, it
results from Lemma \ref{lem:Sugawara_bracket}
that \hbox{$\deg((v^1)^{(+)})\!\leq\!\deg((w^{i})^{(+)})$},
which contradicts \eqref{eq:d+}.

\item
Assume that $i\in J_1\setminus J_{-1}^{(0)}$.
Then
\begin{equation}
\label{eq:d01}
\deg_{-1}^{(0)} (w^{i})< d_{-1}^{(0)} =\deg_{-1}^{(0)}(v^1)
\end{equation}
by \eqref{e34}.
Suppose that $v^1$ is a PBW monomial in ${\tilde{L}_{-1} w^{i}}$.
By Lemma~\ref{lem:Sugawara_monomial}\eqref{lem:Sugawara_monomialb} and \eqref{lem:Sugawara_monomialc},%
\begin{equation}\label{e16}
(w^{i})^{(-)}=1, \quad \deg_{-1}^{(0)}(v^1)=\deg_{-1}^{(0)}(w^i)+1,
\end{equation}
because $(v^1)^{(-)}=1$ by \eqref{e14}.
Remember that
\begin{equation}\label{e42}
(v^1)^{(+)}=e_{\be_1}(-1)^{a^{(1)}_{1,1}}\cdots e_{\be_{q}}(-1)^{a^{(1)}_{q,1}}.
\end{equation}
Computing $\tilde{L}_{-1} w^{i}$, we deduce that
\[
(w^{i})^{(+)}=e_{\be_1}(-1)^{a^{(i)}_{1,1}}\cdots e_{\be_{q}}(-1)^{a^{(i)}_{q,1}}.
\]
{Since $v^{(-)}=1$ and $\deg_{-1}^{(0)}(v^1)=\deg_{-1}^{(0)}(w^i)+1$,
it results from Lemma \ref{lem:Sugawara_bracket}\eqref{lem:Sugawara_bracket3} that $\dep((v^1)^{(+)})\geq 1$,
which contradicts (\ref{e42})}.

\item
Finally, if $j \in J \setminus J_1$, then by Lemma \ref{lem:Sugawara_monomial}\eqref{lem:Sugawara_monomialb}, any PBW monomial $y$ in $\tilde{L}_{-1}w^j$ satisfies that $y^{(-)}\neq 1$. So $v^1$
cannot be a PBW monomial in $\tilde{L}_{-1}w^j$.
\end{itemize}
This concludes the proof of the lemma.
\end{proof}

As already explained, Lemma \ref{lem:w_0} implies that $w$ has zero depth
and so its image in $R_{V^k(\g)}$ is nonzero,
achieving the proof of Theorem \ref{Th:main}.

\subsection{Remarks}\label{subsection:remark}
The statement of Theorem \ref{theorem:image-of-the-singular}
is not true at the critical level.
Also,
it is not true that
the depth of a depth-homogeneous singular vector of $S(\g[t^{-1}]t^{-1})$ is
always zero.
Indeed,
the $\g\lcr t\rcr$-module
$S(\g[t^{-1}]t^{-1})$
can be naturally identified with $\C[J_{\infty}\g^*]$,
where $J_{\infty}X$ is the arc space of $X$,
and so
$S(\g[t^{-1}]t^{-1})^{\g[t]}\cong \C[J_{\infty}\g^*]^{J_{\infty}G}$.
It is known \cite{RaiTau92,BeiDri,EisFre01} that
\begin{equation*}
\C[J_{\infty}\g^*]^{J_{\infty}G}\cong \C[J_{\infty}(\g^*/\!/G)].
\end{equation*}
This means that
the invariant ring is a polynomial ring with infinitely many
variables
$\partial^j p_i$,
$i=1,\dots, \ell$,
$j\geq 0$,
where
$p_1,\dots, p_{\ell}$ is a set of homogeneous generators of
$S(\g)^\g$
considered as elements of
$S(\g[t^{-1}]t^{-1})$ via the embedding
$S(\g)\hookrightarrow S(\g[t^{-1}]t^{-1})$,
$\g\ni x\mto x(-1)$.
We have $\on{depth}(\partial^j p_i)=j$
although
each $\partial^j p_i$ is a singular vector
of $S(\g[t^{-1}]t^{-1})$.

For $k=-h^{\vee}$,
the maximal submodule $N_k$ of $V^k(\g)$
is generated by Feigin-Frenkel center (\cite{FreGai04}).
Hence \cite{FeiFre92,Fre05},
$\on{gr}N_{k}$ is exactly the
argumentation ideal of
$S(\g[t^{-1}]t^{-1})^{\g[t]}$.
Therefore,
the above argument shows that
the statement of Theorem~\ref{theorem:image-of-the-singular}
is false at the critical level.

\section{\texorpdfstring{$W$}{W}-algebras and proof of Theorem \ref{Th:main2}}
\label{sec:W-algebras}
Let $f$ be a nilpotent element of $\g$. By the Jacobson-Morosov theorem,
it embeds into an $\mathfrak{sl}_2$-triple $(e,h,f)$ of $\g$.
Recall that the Slodowy slice $\mathscr{S}_f$ is the affine space $f+\g^{e}$,
where $\g^{e}$ is the centralizer of $e$ in $\g$.
It has a natural Poisson structure induced from that of $\g^*$ (\cite{GanGin02}).

The embedding $\mathrm{span}_\C\{e,h,f\}
\cong \mathfrak{sl}_2 \hookrightarrow \g$
exponentiates to a homomorphism
\hbox{$\mathrm{SL}_2\!\to\!G$}. By restriction to the one-dimensional
torus consisting of diagonal matrices, we obtain
a one-parameter subgroup $\rho \colon \C^* \to G$.
For $t\in\C^*$ and $x\in\g$, set
\begin{equation*} \label{eq:rho}
\tilde{\rho}(t)x := t^{2} \rho(t)(x).
\end{equation*}
We have
$\tilde{\rho}(t)f=f$, and
the $\C^*$-action of $\tilde{\rho}$ stabilizes $\mathscr{S}_f$.
Moreover, it is contracting to $f$ on $\mathscr{S}_f$, that is, for all $x\in\g^{e}$,
\[
\lim_{t\to 0} \tilde{\rho}(t)(f+x)=f.
\]

The following proposition is well-known.
Since its proof is short, we give below
the argument for the convenience of the reader.

\begin{prop}[{\cite{Slo80,Premet02,Charbonnel-Moreau}}]
\label{pro:Slodowy}
The morphism
\[
\theta_f \colon G \times \mathscr{S}_f \to \g,
\quad (g,x) \mto g\cdot x
\]
is smooth onto a dense open subset of $\g^*$.
\end{prop}
\begin{proof}
Since $\g =\g^{e}+[f,\g]$, the map $\theta_f$ is a submersion at
$(1_{G},f)$. Therefore, $\theta _{f}$~is a submersion at all points of
$G\times (f +\g^{e})$ because it is $G$-equivariant for the left
multiplication in $G$, and
\[
\lim _{t \to \infty } \rho(t)\cdot x = f
\]
for all $x$ in $f+\g^{e}$. So, by~\cite[Ch.\,III, Prop.\,10.4]{Hartshorne},
the map $\theta _{f}$ is a smooth morphism onto a dense open subset of
$\g$, containing $G\cdot f$.
\end{proof}

As in the introduction, let $\W^k(\g,f)$ be the {affine $W$-algebra} associated with
a nilpotent element $f$ of $\g$
defined by the generalized quantized Drinfeld-Sokolov reduction:
\[
\W^k(\g,f)=H^{0}_{\DS,f}(V^k(\g)).
\]
Here, $H^{\sbullet}_{\DS,f}(M)$ denotes the BRST
cohomology of the generalized quantized Drinfeld-Sokolov reduction
associated with $f \in \mathcal{N}(\g)$ with coefficients in
a $V^k(\g)$-module $M$.
Recall that we have \cite{DSK06,Ara09b} a natural isomorphism
$R_{\W^k(\g,f)}\cong \C[\mathscr{S}_{f}]$ of Poisson algebras, so that
\begin{equation*}
X_{\W^k(\g,f)}= \mathscr{S}_{f}.
\end{equation*}
We write $\W_k(\g,f)$ for the unique simple (graded) quotient of
$\W^k(\g,f)$. Then $X_{\W_k(\g,f)}$
is a $\C^*$-invariant Poisson
subvariety of the Slodowy slice $\mathscr{S}_f$.

Let $\mathscr{O}_k$ be the category $\mathscr{O}$ of
$\frg$ at level $k$.
We have a functor
\begin{equation*}
\mathscr{O}_k\to\W^k(\g,f)\on{-Mod},\quad M\mto
H^0_{\DS,f}(M),
\end{equation*}
where
$\W^k(\g,f)\on{-Mod}$ denotes the category
of $\W^k(\g,f)$-modules.

The full subcategory of $\mathscr{O}_k$ consisting of
objects $M$ on which $\g$ acts locally finitely will be denoted by $\on{KL}_k$.
Note that both $V^k(\g)$ and $L_k(\g)$ are objects of $\on{KL}_k$.

\skpt
\begin{theorem}[{\cite{Ara09b}}]
\label{Th:W-algebra-variety}
\begin{enumerate}
\item\label{Th:W-algebra-variety1} $H_{\DS,f}^i(M)=0$ for all $i\ne 0$, $M\in
\on{KL}_k$.
In particular, the functor
\[
\on{KL}_k\to\W^k(\g,f)\on{-Mod},\quad M\mto
H_{\DS,f}^{0}(M),
\]
is exact.
\item\label{Th:W-algebra-variety2}
For any quotient $V$ of $V^k(\g)$,
\begin{equation*}
X_{H^{0}_{\DS,f}(V)}=X_{V}\cap \mathscr{S}_{f}.
\end{equation*}
In particular
$H_{\DS,f}^{0}(V)
\ne 0$ if and only if
$\overline{G\cdot f}\subset X_V$.
\label{item:intersection}
\end{enumerate}
\end{theorem}

By Theorem \ref{Th:W-algebra-variety}\eqref{Th:W-algebra-variety1},
$H^{0}_{\DS,f}(L_k(\g))$ is a quotient vertex algebra of $\W^k(\g,f)$ if it is nonzero.
Conjecturally \cite{KacRoaWak03,KacWak08},
we have
\begin{equation*}
\W_k(\g,f)\cong H^{0}_{\DS,f}(L_k(\g))\quad\text{provided that }H^{0}_{\DS,f}(L_k(\g))\ne 0.
\end{equation*}
(This conjecture has been verified in many cases \cite{Ara05,Ara07,Ara08-a,AEkeren19}.)

\begin{proof}[Proof of Theorem \ref{Th:main2}]
The directions \eqref{Th:main21} $\Rightarrow$ \eqref{Th:main22}
and \eqref{Th:main22} $\Rightarrow$ \eqref{Th:main23} are obvious.
Let us show that \eqref{Th:main23} implies \eqref{Th:main21}.
So suppose that $X_{H^{0}_{\DS,f}(L_k(\g))}=\mathscr{S}_f$.
By Theorem~\ref{MainTheorem}, it is enough to show that $X_{L_k(\g)}=\g^*$.
Assume the contrary. Then $X_{L_k(\g)}$ is contained in a proper
$G$-invariant closed subset of $\g$.
On the other hand, by Theorem \ref{Th:W-algebra-variety} and our hypothesis,
we have
\[
\mathscr{S}_f=X_{H^{0}_{\DS,f}(L_k(\g))}= X_{L_k(\g)} \cap \mathscr{S}_f.
\]
Hence, $ \mathscr{S}_f$ must be contained
in a proper
$G$-invariant closed subset of $\g$. But this contradicts Proposition \ref{pro:Slodowy}.
The proof of the theorem is completed.
\end{proof}

\backmatter
\nocite{Mus01}
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\end{document}