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\begin{document}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\footnote[0]{
This work was supported by Grant-in-Aid for JSPS Fellows (No.040070235Z) and the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.
}
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\maketitle
\begin{abstract}
(Affine) $\W$-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra $\g$ over $\C$, a nilpotent element $f$ in $[\g,\g]$, a good grading $\Gamma$ and a symmetric invariant bilinear form $\kappa$ on $\g$. We introduce free field realizations of $\W$-algebras by using Wakimoto representations of affine Lie algebras, where $\W$-algebras are described as the intersections of kernels of screening operators. We call these Wakimoto free fields realizations of $\W$-algebras. As applications, under certain conditions that are valid in all cases of type $A$, we construct parabolic inductions for $\W$-algebras, which we expect to induce the parabolic inductions of finite $\W$-algebras defined by Premet and Losev. In type $A$, we show that our parabolic inductions are a chiralization of the coproducts for finite $\W$-algebras defined by Brundan-Kleshchev. In type $BCD$, we are able to obtain some generalizations of the coproducts in some special cases.
\end{abstract}

\section{Introduction}\label{Introduction sec}

Let $\g$ be a reductive Lie algebra, $f$ a nilpotent element in $[\g,\g]$, $\kappa$ a symmetric invariant bilinear form on $\g$ and
\begin{align*}
\Gamma\colon\g=\bigoplus_{j\in\frac{1}{2}\Z}\g_j
\end{align*}
a good grading on $\g$ for $f$. We associate with the (affine) $\W$-algebra $\W^\kappa(\g,f;\Gamma)$ that is a $\frac{1}{2}\Z_{\geq0}$-graded conformal vertex algebra defined by means of the (generalized) Drinfeld-Sokolov reduction \cite{FF4, KRW}. The vertex algebra structure of $\W$-algebras doesn't depend on the choice of the good grading $\Gamma$ for fixed $\g,f,\kappa$, although the conformal grading does \cite{BG, AKM}.

In this paper, we construct inclusions
\begin{align*}
\Ind^\g_\lf\colon\W^\kappa(\g,f;\Gamma)\rightarrow\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf)
\end{align*}
for Levi subalgebras $\lf$ of $\g$, nilpotent elements $f_\lf$ in $[\lf,\lf]$ and good gradings $\Gamma_\lf$ on $\lf$ for $f_\lf$ that satisfy some conditions. We call the maps $\Ind^\g_\lf$ parabolic inductions of $\W$-algebras. We expect that our construction gives a chiralization of the parabolic induction for finite $\W$-algebras defined by Premet \cite{P} and Losev \cite{L}. In the case of $\g=\gl_N$, we show that these inclusions induce exactly the coproducts of the finite $\W$-algebras of Brundan-Kleshchev \cite{BK2}. In the case of $\g=\so_N,\spf_N$ with rectangular nilpotent elements, we obtain a generalization of the coproducts of the corresponding finite $\W$-algebras.

To state our results more precisely, let $\Pi$ be a set of simple roots of $\g$ compatible with $\Gamma$, $\Pi_j$ the subset of $\Pi$ consisting of simple roots whose root vectors belong to $\g_j$ for $j\in\frac{1}{2}\Z$. Then $\Pi=\Pi_0\sqcup\Pi_{\frac{1}{2}}\sqcup\Pi_1$ by \cite{EK}. According to Lusztig and Spaltenstein \cite{LS}, for any Levi subalgebra $\lf$ of $\g$, each nilpotent orbit $\Oc_\lf$ in $\lf$ defines a nilpotent orbit
\begin{align*}
\Oc_\g=\ind^\g_\lf\Oc_\lf\quad\mathrm{in}\ \g,
\end{align*}
which is called the induced nilpotent orbit of $\Oc_\lf$. Let $\Pi_\lf\subset\Pi$ be the set of simple roots of $\lf$, $G$ a connected Lie group corresponding to $\g$ and $L$ the Lie subgroup of $G$ such that $\mathrm{Lie}(L)=\lf$.

\begin{lemma}[Lemma \ref{ind good lem}]\label{intro lem}
Suppose that a good grading $\Gamma$ satisfies $\Pi\backslash\Pi_\lf\subset\Pi_1$. Then $\Oc_\g=G\cdot f$ is induced from $\Oc_\lf=L\cdot f_\lf$ for a nilpotent element $f_\lf$ in $[\lf,\lf]$. Moreover, the restriction $\Gamma_\lf$ of $\Gamma$ to $\lf$ is a good grading on $\lf$ for $f_\lf$.
\end{lemma}

We note that the existence of $\Gamma$ in Lemma \ref{intro lem} is valid in all cases of type $A$ (\cite{Kr, OW}), in the cases of rectangular nilpotent elements in type $BCD$ (\cite{Ke,Sp}), and in all cases of type $G$ (\cite{EK,GE}). However, there exist some induced nilpotent orbits $\Oc_\g=\ind^\g_\lf\Oc_\lf$ in type $E$ and $F$ such that no good grading on $\g$ satisfies that $\Pi\backslash\Pi_\lf\subset\Pi_1$, see \cite{EK, GE}.

\begin{thmA}[Theorem \ref{induced thm}, Proposition \ref{induced prop}]
Suppose that a good grading $\Gamma$ satisfies the condition that $\Pi\backslash\Pi_\lf\subset\Pi_1$.
\begin{enumerate}
\item For any symmetric invariant bilinear form $\kappa$ on $\g$, there exists an injective vertex algebra homomorphism
\begin{align*}
\Ind^\g_\lf\colon\W^\kappa(\g,f;\Gamma)\rightarrow\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf),
\end{align*}
where $f_\lf$, $\Gamma_\lf$ are given in Lemma \ref{intro lem}, $\kappa_\lf=\kappa+\frac{1}{2}\killing_\g-\frac{1}{2}\killing_\lf$, and $\killing_\g$, $\killing_\lf$ are the Killing forms on $\g$, $\lf$ respectively.
\item $\Ind^\g_\lf$ is a unique vertex algebra homomorphism that satisfies $\mu=\mu_\lf\circ\Ind^\g_\lf$, where $\mu$, $\mu_\lf$ are the Miura maps \cite{KW1} for $\W^\kappa(\g,f;\Gamma)$, $\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf)$ respectively.
\item Let $\lf'$ be any Levi subalgebra of $\g$ such that $\Pi\backslash\Pi_{\lf'}\subset\Pi_1$ and $\lf\subset\lf'\subset\g$. Then the maps $\Ind^\g_{\lf'}$, $\Ind^{\lf'}_{\lf}$ exist and $\Ind^\g_\lf=\Ind^{\lf'}_\lf\circ\Ind^\g_{\lf'}$.
\end{enumerate}
\end{thmA}

See Section \ref{Miura map sec} for the definition of the Miura map. In the case that $f$ is a principal nilpotent element, the map $\Ind^\g_\lf$has been constructed in Theorem B 7.1 of \cite{BFN}.

For any $\frac{1}{2}\Z_{\geq0}$-graded conformal vertex algebra $V$, we can associate with an associative algebra $\Zhu(V)$, called the (twisted) Zhu algebra \cite{Z, FZ, DK}. It is proved in \cite{A1, DK} that $\Zhu(\W^\kappa(\g,f;\Gamma))$ is the finite $\W$-algebra associated with $\g,f,\Gamma$ \cite{P1, GG}, which we denote by $U(\g,f;\Gamma)$. It is easy to see that any vertex algebra homomorphism $\alpha\colon V\rightarrow W$ induces an algebra homomorphism between the Zhu algebras, which we denote by $\Zhu(\alpha)\colon\Zhu(V)\rightarrow\Zhu(W)$. For an algebra homomorphism $A$, we call a map $\alpha$ a chiralization of the map $A$ if $A=\Zhu(\alpha)$. In the case of $\alpha=\Ind^\g_\lf$, we obtain an algebra homomorphism
\begin{align*}
\Zhu(\Ind^\g_\lf)\colon U(\g,f;\Gamma)\rightarrow U(\lf,f_\lf;\Gamma_\lf),
\end{align*}
which is a unique injective algebra homomorphism that satisfies $\bar{\mu}=\bar{\mu}_\lf\circ\Zhu(\Ind^\g_\lf)$, where $\bar{\mu}$, $\bar{\mu}_\lf$ are the Miura maps for $U(\g,f;\Gamma)$, $U(\lf,f_\lf;\Gamma_\lf)$ respectively (Lemma \ref{Zhu ind lem}). See \cite{Ly} or Section \ref{chiral sec} for the definition of the Miura map for $U(\g,f;\Gamma)$.

Given an induced nilpotent orbit $G\cdot f=\ind^\g_\lf (L\cdot f_\lf)$ in $\g$ with a good grading $\Gamma$ on $\g$ for $f$ and a good grading $\Gamma_\lf$ on $\lf$ for $f_\lf$, Losev proved the existence of an injective algebra homomorphism
\begin{align}\label{intro Losev eq}
U(\g,f;\Gamma)\rightarrow\widetilde{U}(\lf,f_\lf;\Gamma_\lf)
\end{align}
in \cite{L}, where $\widetilde{U}(\lf,f_\lf;\Gamma_\lf)$ is a certain completion of $U(\lf,f_\lf;\Gamma_\lf)$. The map \eqref{intro Losev eq} induces a functor from the category of $U(\lf,f_\lf;\Gamma_\lf)$-modules to the category of $U(\g,f;\Gamma)$-modules, called the parabolic induction that was first introduced by Premet \cite{P}. We conjecture that $\Zhu(\Ind^\g_\lf)$ coincides with \eqref{intro Losev eq}, and this is the reason why we call the map $\Ind^\g_\lf$ the parabolic induction of $\W$-algebras.

In the case of $\gl_N$, any nilpotent element in $\slf_N=[\gl_N,\gl_N]$ admits a good $\Z$-grading. These good $\Z$-gradings on $\gl_N$ are classified by combinatoric objects called (even) pyramids $\Py$ introduced in \cite{EK}, which are sequences of the columns of $1\times1$ boxes such that each of rows in $\Py$ is a single connected strip (see Section \ref{coproduct sec} for details). For a pyramid $\Py$ consisting of $N$ boxes, we associate with a nilpotent element $f_\Py$ in $\gl_N$, a good $\Z$ grading $\Gamma_\Py$ on $\gl_N$ for $f_\Py$, and the finite $\W$-algebra $U(\gl_N,\Py)=U(\gl_N,f_\Py;\Gamma_\Py)$. It was shown by Brundan and Kleshchev in \cite{BK2} that $U(\gl_N,\Py)$ is isomorphic to a truncation of the Yangian $Y(\gl_n)$ for some $n\geq1$ and the coproduct of $Y(\gl_n)$ induces an injective algebra homomorphism between finite $\W$-algebras
\begin{align*}
\bar{\cop}=\bar{\cop}^\Py_{\Py_1, \Py_2}\colon U(\gl_N,\Py)\rightarrow U(\gl_{N_1},\Py_1)\otimes U(\gl_{N_2},\Py_2)
\end{align*}
for a pyramid $\Py$ that splits into sum of $\Py_1$ and $\Py_2$ along a column of $\Py$ (see e.g. Section \ref{cop thm sec}), which we denote by $\Py=\Py_1\oplus\Py_2$. This map $\bar{\cop}$ is called a coproduct of finite $\W$-algebras and satisfies the coassociativity, i.e.
\begin{align*}
(\Id\otimes\bar{\cop}^{\Py_2\oplus\Py_3}_{\Py_2,\Py_3})\circ\bar{\cop}^\Py_{\Py_1,\Py_2\oplus\Py_3}=(\bar{\cop}^{\Py_1\oplus\Py_2}_{\Py_1,\Py_2}\otimes\Id)\circ\bar{\cop}^\Py_{\Py_1\oplus\Py_2, \Py_3}
\end{align*}
for a pyramid $\Py=\Py_1\oplus\Py_2\oplus\Py_3$. The coproduct $\bar{\cop}$ plays a fundamental role to produce representations of finite $\W$-algebras of type $A$, see \cite{BK3}.

Consider a maximal Levi subalgebra $\lf$ in $\gl_N$, that is, $\lf=\gl_{N_1}\oplus\gl_{N_2}$ for some $N_1,N_2\in\Z_{\geq1}$ such that $N=N_1+N_2$. According to \cite{Kr, OW}, it follows that any induced nilpotent orbit in $\gl_N$ takes the form
\begin{align*}
\GL_N\cdot f_\Py=\ind^\g_\lf(\GL_{N_1}\cdot f_{\Py_1}+\GL_{N_2}\cdot f_{\Py_2})
\end{align*}
for some pyramid $\Py=\Py_1\oplus\Py_2$, where $f_{\Py_1}\in\gl_{N_1}$ and $f_{\Py_2}\in\gl_{N_2}$. Therefore, it is expected that $\bar{\cop}$ coincides with the special case of \eqref{intro Losev eq} for $\g=\gl_N$ and $\lf=\gl_{N_1}\oplus\gl_{N_2}$.
\smallskip

For $k\in\C$, let us denote by $\W^k(\gl_N,\Py)=\W^\kappa(\gl_N,f_\Py;\Gamma_\Py)$, where $\kappa$ is a symmetric invariant bilinear form on $\gl_N$ such that $\kappa(u|v)=k\ \mathrm{tr}(uv)$ for all $u,v\in\slf_N$. The following assertion is obtained from Theorem A.

\begin{thmB}[Theorem \ref{coproduct thm}, Proposition \ref{chiral prop}]
Let $\Py$ be a pyramid consisting of $N$ boxes such that $\Py=\Py_1\oplus\Py_2$.
\begin{enumerate}
\item For any $k\in\C$, there exists an injective vertex algebra homomorphism
\begin{align*}
\cop=\cop^\Py_{\Py_1,\Py_2}\colon\W^k(\gl_N,\Py)\rightarrow\W^{k_1}(\gl_{N_1},\Py_1)\otimes\W^{k_2}(\gl_{N_2},\Py_2),
\end{align*}
where $k+N=k_1+N_1=k_2+N_2$ and $N_i$ is a number of boxes in $\Py_i$ for $i=1,2$.
\item $\cop$ is a unique vertex algebra homomorphism that satisfies $\mu=(\mu_1\otimes\mu_2)\circ\cop$, where $\mu$, $\mu_1$, $\mu_2$ are the Miura maps for $\W^k(\gl_N,\Py)$, $\W^{k_1}(\gl_{N_1},\Py_1)$, $\W^{k_2}(\gl_{N_2},\Py_2)$ respectively.
\item $\cop$ is coassociative, i.e. $(\Id\otimes\cop^{\Py_2\oplus\Py_3}_{\Py_2,\Py_3})\circ\cop^\Py_{\Py_1,\Py_2\oplus\Py_3}=(\cop^{\Py_1\oplus\Py_2}_{\Py_1,\Py_2}\otimes\Id)\circ\cop^\Py_{\Py_1\oplus\Py_2, \Py_3}$ for $\Py=\Py_1\oplus\Py_2\oplus\Py_3$.
\item $\cop$ is a chiralization of $\bar{\cop}$, that is, $\Zhu(\cop)=\bar{\cop}$.
\end{enumerate}
\end{thmB}

See Section \ref{examples sec} for some examples of $\cop$. In the case that $f_\Py$ is a principal nilpotent element, the coproduct $\cop$ is an injective map
\begin{align}\label{intro prin eq}
\W_N^k\rightarrow\W_{N_1}^{k_1}\otimes\W_{N_2}^{k_2}
\end{align}
for $N=N_1+N_2$ and $k+N=k_1+N_1=k_2+N_2$, where $\W_N^k$ is the $\W$-algebra of $\gl_N$ with a principal nilpotent element and level $k$ \cite{Za,FL}. It seems  that the existence of the map \eqref{intro prin eq} has been suggested in \cite{FiSt}.

In the case of $\g_N=\so_N$ or $\spf_{N}$, any maximal Levi subalgebra of $\g_N$ takes the form $\lf=\gl_{N_1}\oplus\g_{N_2}$ for some $N_1,N_2\in\Z_{\geq1}$ such that $N=2N_1+N_2$. Applying Theorem A to this setting with rectangular nilpotent elements, we obtain some generalizations of the coproducts for $\W$-algebras of $\g_N$. See Theorem \ref{cop BCD thm} for precise statements. We note that our results suggest the existence of certain coproducts for truncated twisted Yangians, which is obtained as Corollary \ref{cop BCD cor} in some special cases. We refer to \cite{R, Br} for connections between twisted Yangians and finite $\W$-algebras of type $BCD$.

The basic tool for the proof of Theorem A is Wakimoto representations of $\W$-algebras, which we introduce in Section \ref{W-alg Wak sec}. For simplicity, we assume that $\g$ is a simple Lie algebra. Denote by $\W^k(\g,f;\Gamma)=\W^\kappa(\g,f;\Gamma)$ if $k=\kappa(\theta|\theta)/2$ for the highest root $\theta$ of $\g$. Wakimoto representations of the affine Lie algebra $\widehat{\g}$ are introduced by Wakimoto \cite{Wak} in the case of $\widehat{\slf_2}$ and Feigin-Frenkel \cite{FF1} in general case, see also Section \ref{aff Wak sec}. The actions of $\widehat{\g}$ on Wakimoto representations are induced from an embedding of the affine vertex algebra $V^k(\g)$ into the tensor product of the Heisenberg vertex algebra $\Hi$ associated with a Cartan subalgebra $\h$ in $\g$ and $\dim\nil_+$ copies of the $\beta\gamma$-system, where $\nil_+=\mathrm{Lie}(N_+)$ and $N_+$ is the big cell of the flag manifold $G/B_-$. The image of this embedding is the intersection of kernels of screening operators $S_\alpha$ for all $\alpha\in\Pi$ if $k$ is a formal parameter (\cite{F}). As explained in detail in Section \ref{W-alg Wak sec}, applying Drinfeld-Sokolov reductions to Wakimoto representations of $\widehat{\g}$, we obtain free fields realizations of $\W$-algebras $\W^k(\g,f;\Gamma)$, which we call Wakimoto free fields realizations of $\W$-algebras $\W^k(\g,f;\Gamma)$.

When the base ring is $T=\C[\para]$, we replace everywhere the complex number $k$ by a formal parameter $\para$, and denote the corresponding $\W$-algebra and Heisenberg vertex algebra by $\W^T(\g,f;\Gamma)$, $\Hi^T$ instead of $\W^k(\g,f;\Gamma)$, $\Hi$ respectively. Let $\mathcal{N}$ be the nilpotent cone of $\g$, $\mathcal{S}_f$ the Slodowy slice of $\g$ through $f$.

\begin{thmC}[Theorem \ref{Miura thm}]
The $\W$-algebras $\W^T(\g,f;\Gamma)$ over $T$ may be embedded into the tensor products of $\Hi^T$ and $\frac{1}{2}\dim(\mathcal{N}\cap\mathcal{S}_f)$ copies of the $\beta\gamma$-system. These image can be identified with the intersections of kernels of screening operators $Q_\alpha$ induced by $S_\alpha$.
\end{thmC}

See Theorem \ref{main thm} for the precise formulae of $Q_\alpha$. In the case that $f$ is a principal nilpotent element, screening operators $Q_\alpha$ coincide with the ones constructed in \cite{FF3}. In the case that $\g_0$ is a Cartan subalgebra $\h$, screening operators $Q_\alpha$ coincide with the ones constructed in \cite{G}.

Our strategy to prove Theorem A is simple. Under the assumption in Theorem A, we consider the specialization of inclusion maps
\begin{align}\label{intro screening eq}
\bigcap_{\alpha\in\Pi}\Ker Q_\alpha\hookrightarrow\bigcap_{\alpha\in\Pi_\lf}\Ker Q_\alpha\hookrightarrow\bigcap_{\alpha\in\Pi_0}\Ker Q_\alpha.
\end{align}
We show by using Theorem C that the first map is nothing but $\Ind^\g_\lf$, i.e.
\begin{align*}
\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf)\simeq\bigcap_{\alpha\in\Pi_\lf}\Ker Q_\alpha.
\end{align*}
Our assumption ($\Pi\backslash\Pi_\lf\subset\Pi_1$) is used here. Since the Miura map is injective by \cite{F,A,G}, Theorem A therefore follows if we show that the map $\Ind^\g_\lf$ satisfies the formula $\mu=\mu_\lf\circ\Ind^\g_\lf$, which in fact follows from \eqref{intro screening eq} and Theorem D.

\begin{thmD}[Theorem \ref{w-alg Miura thm}]
The specialization
\begin{align*}
\mu\colon\W^k(\g,f;\Gamma)\rightarrow\Vtau\otimes\Fne
\end{align*}
of an inclusion map
\begin{align*}
\bigcap_{\alpha\in\Pi}\Ker Q_\alpha\hookrightarrow\bigcap_{\alpha\in\Pi_0}\Ker Q_\alpha.
\end{align*}
coincides with the Miura map.
\end{thmD}

Let us make some comment on the relationship between $\W^k(\gl_N,\Py)$ and the affine Yangian $Y(\widehat{\gl}_n)$. In the case that $f$ is a principal nilpotent element, an action of $Y(\widehat{\gl}_1)$ on $\W^k_N$ was first suggested by Aldey-Gaiotto-Tachikawa \cite{AGT} and was studied by Maulik-Okounkov \cite{MO} and Schiffmann-Vasserot \cite{SV}, see also \cite{BFN} for the generalizations. The coproduct \eqref{intro prin eq} is expected to be induced by the coproduct of $Y(\widehat{\gl}_1)$ as an analogue of the finite cases. We hope to study the relationship between the coproduct $\cop$ in Theorem B and that of affine Yangian $Y(\widehat{\gl}_n)$ in our future works.

The paper is organized as follows. In Section \ref{W-alg sec}, we review the definitions of $\W$-algebras. In Section \ref{aff Wak sec}, we recall Wakimoto representations of $V^k(\g)$ and screening operators $S_\alpha$. In Section \ref{W-alg Wak sec}, we introduce Wakimoto representations of $\W^k(\g,f;\Gamma)$ and screening operators $Q_\alpha$, and state the precise formulae of $Q_\alpha$ in Theorem \ref{main thm}. In Section \ref{old scr sec}, we recall results in \cite{G}. In Section \ref{Miura map sec}, we recall the Miura map and prove Theorem C and Theorem D by using Lemma \ref{mod prop}. In Section \ref{reductive W-alg sec}, we define $\W$-algebras associated with reductive Lie algebras and conclude some results from Theorem C and Theorem D. In Section \ref{ind nil sec}, we recall the definitions and properties of induced nilpotent orbits. In Section \ref{pre sec}, we prepare some preliminary results in order to prove Theorem A. In Section \ref{induced sec}, we prove Theorem A. In Section \ref{chiral sec}, we derive some results for finite $\W$-algebras from Theorem A. In Section \ref{pyramid sec}, we recall the definitions of pyramids. In Section \ref{cop thm sec}, we prove Theorem B. In Section \ref{cop BCD sec}, we derive some generalizations of the coproducts for the $\W$-algebras of type $BCD$ from Theorem A. In Section \ref{examples sec}, we give examples of Theorem B in the case that $f$ is a principal, rectangular and subregular nilpotent element. In Appendix \ref{proof sec}, we prove Theorem \ref{main thm}. In Appendix \ref{mod prop sec}, we prove Lemma \ref{mod prop}.

\vspace{3mm}

{\it Acknowledgments}\quad The author is grateful to his supervisor Tomoyuki Arakawa for valuable discussions and lots of advice to improve this paper. He thanks to Shigernori Nakatsuka for useful comments and discussions. He wishes to express his gratitude to Boris Feigin for sharing his ideas and useful discussions for Theorem B. He is grateful to Hiraku Nakajima for valuable comments, which triggered Theorem A. Some part of this work was done while he was visiting M.I.T. and National Research University Higher School of Economics in 2016. He is grateful to those institutes for their hospitality.

\section{Affine $\W$-algebras}\label{W-alg sec}

We recall the definitions of the (affine) $\W$-algebras, following \cite{KRW}. Let $\g$ be a finite-dimensional simple Lie algebra over $\C$, $f$ a nilpotent element of $\g$ and $\Gamma$ a good grading of $\g$ for $f$ denoted by
\begin{align*}
\Gamma\colon\g=\bigoplus_{j\in\frac{1}{2}}\g_{j},
\end{align*}
where the $\frac{1}{2}\Z$-grading $\Gamma$ is called good for $f$ if $[\g_i, \g_j]\subset\g_{i+j}$ for all $i,j\in\frac{1}{2}\Z$, $f\in\g_{-1}$ and $\ad f\colon\g_{j}\rightarrow\g_{j-1}$ is injective for $j\geq\frac{1}{2}$, surjective for $j\leq\frac{1}{2}$. Then there exists a semisimple element $h\in\g$ such that the grading $\Gamma$ of $\g$ is the eigenspace decomposition of $\ad(\frac{1}{2}h)$. By Jacobson-Morozov Theorem, there exists an $\slf_2$-triple $(e,h,f)$ in $\g$, and $\ad(\frac{1}{2}h)$ defines a $\frac{1}{2}\Z$-grading on $\g$, which is good for $f$ called the {\em Dynkin grading}. Choose the Cartan subalgebra $\h$ containing $h$ so that $\h\subset\g_0$. Let $\Delta$ be the set of roots, $\Delta_{+}$ the set of positive roots such that $\bigoplus_{\alpha\in\Delta_{+}}\g_{\alpha}\subset\g_{\geq0}$, where $\g_{\alpha}$ is the root space of $\alpha\in\Delta$. Let $\Pi$ be the set of simple roots, $\Delta_{j}=\{\alpha\in\Delta\mid\g_{\alpha}\subset\g_{j}\}$ and $\Pi_{j}=\Pi\cap\Delta_{j}$ for all $j\in\frac{1}{2}\Z$. Set $\Deltazero=\Delta_0\cap\Delta_+$. Then
\begin{align*}
\Delta=\bigsqcup_{j\in\frac{1}{2}\Z}\Delta_{j},\quad
\Delta_{+}=\Deltazero\sqcup\bigsqcup_{j>0}\Delta_j,\quad
\Pi=\Pi_{0}\sqcup\Pi_{\frac{1}{2}}\sqcup\Pi_{1},
\end{align*}
see \cite{EK}. Denote by $\degG\alpha=j$ if $\alpha\in\Delta_j$. Fix a root vector $e_{\alpha}\in\g$ for each $\alpha\in\Delta$ and a non-degenerate symmetric invariant bilinear form $\inv$ on $\g$ such that $(\theta|\theta)=2$ for the highest root $\theta$ of $\g$. Then $\killing_\g(u|v)=2h^{\vee}(u|v)$ for all $u,v\in\g$, where $\killing_{\g}$ is the Killing form on $\g$ and $h^{\vee}$ is the dual Coxeter number of $\g$. Let $\chi\colon\g\rightarrow\C$ be a linear map defined by $\chi(u)=(f|u)$ for $u\in\g$. Denote by $\nil_{\pm}=\bigoplus_{\alpha\in\Delta_{\pm}}\g_{\alpha}$ and $\bo_{\pm}=\h\oplus\nil_{\pm}$.

We follow \cite{FBZ, Ka} for the definitions of vertex algebras. We use the following notations:
\begin{align*}
A(z)=\sum_{n\in\Z}A_{(n)}z^{-n-1},\quad\int A(z)\ dz=A_{(0)}
\end{align*}
for any field $A(z)$, and $\delta(z-w)=\sum_{n\in\Z}z^{-n-1}w^n$. Denote by $:A(z)B(z):$ the normally ordered products and by $A(z)B(w)\sim\sum_{n\geq0}\frac{C_n(w)}{(z-w)^{n+1}}$ the operator product expansion for local fields $A(z), B(z)$, where $[A(z),B(w)]=\sum_{n\geq0}\frac{1}{n!}C_n(w)\der_w^n\delta(z-w)$. If $A(z),B(z)$ are fields on a vertex (super)algebra, $C_n(z)=(A_{(n)}B)(z)$ for $n\geq0$. For $k\in\C$, let $V^k(\g)$ be the affine vertex algebra associated with $\g$ of level $k$, whose generating fields $u(z)$ for $u\in\g$ satisfy
\begin{align*}
u(z)v(w)\sim\frac{[u,v](w)}{z-w}+\frac{k(u|v)}{(z-w)^2}
\end{align*}
for all $u,v\in\g$. Let $\Fch$ be the charged fermion vertex superalgebra associated with $\g_{>0}$, whose generating odd fields $\varphi_{\alpha}(z),\varphi^{\alpha}(z)$ for $\alpha\in\Delta_{>0}$ satisfy
\begin{align*}
\varphi_{\alpha}(z)\varphi^{\beta}(w)\sim\frac{\delta_{\alpha,\beta}}{z-w},\quad
\varphi_{\alpha}(z)\varphi_{\beta}(w)\sim0\sim\varphi^{\alpha}(z)\varphi^{\beta}(w)
\end{align*}
for all $\alpha,\beta\in\Delta_{>0}$. The charged decomposition $\Fch=\bigoplus_{n\in\Z}\Fchn$ is defined by the {\em charged} degree $\charge(\varphi_{\alpha}(z))=-1$ and $\charge(\varphi^{\alpha}(z))=1$ for all $\alpha\in\Delta_{>0}$, where $\Fchn=\{A\in\Fch\mid\charge(A)=n\}$. Let $\Fne$ be the neutral vertex algebra associated with $\g_{\frac{1}{2}}$, whose generating (even) fields $\Phi_{\alpha}(z)$ for $\alpha\in\Deltahalf$ satisfy
\begin{align*}
\Phi_{\alpha}(z)\Phi_{\beta}(w)\sim\frac{\chi([e_{\alpha},e_{\beta}])}{z-w}
\end{align*}
for all $\alpha,\beta\in\Deltahalf$. Set
\begin{align*}
C_{k}=V^k(\g)\otimes\Fch\otimes\Fne
\end{align*}
and $d=\int d(z)\ dz$, where $d(z)=\dst(z)+\dne(z)+\dchi(z)$ is an odd field on $C_k$ defined by
\begin{align*}
\dst(z)&=\sum_{\alpha\in\Delta_{>0}}:e_{\alpha}(z)\varphi^{\alpha}(z):-\frac{1}{2}\sum_{\alpha,\beta,\gamma\in\Delta_{>0}}c_{\alpha,\beta}^{\gamma}:\varphi_{\gamma}(z)\varphi^{\alpha}(z)\varphi^{\beta}(z):,\\
\dne(z)&=\sum_{\alpha\in\Deltahalf}:\varphi^{\alpha}(z)\Phi_{\alpha}(z):,\quad
\dchi(z)=\sum_{\alpha\in\Deltaone}\chi(e_{\alpha})\varphi^{\alpha}(z),
\end{align*}
where $c_{\alpha,\beta}^{\gamma}\in\C$ is the structure constant for $\alpha,\beta,\gamma\in\Delta_{>0}$. The charged decomposition $C_k^\bullet=V^k(\g)\otimes\Fch^\bullet\otimes\Fne$ is induced from that of $\Fch$. Since $d^2=0$ and $d\cdot C_k^p\subset C_k^{p+1}$, an odd vertex operator $d$ defines a differential of a cochain complex on $C_k$. The $\W$-algebra $\W^k(\g,f;\Gamma)$ associated with $\g,f,k,\Gamma$ is defined as the BRST cohomology of the complex $(C_k,d)$:
\begin{align*}
\W^k(\g,f;\Gamma)=H(C_k,d),
\end{align*}
called the (generalized) Drinfeld-Sokolov reduction. There exists a decomposition of the complex $C_k=C_{-}\otimes C_+$ such that $H(C_{-},\C)=\C$ and $C_+$ has only non-negative charged degree. Moreover $\W^k(\g,f;\Gamma)=H^{0}(C_+,d)$ (\cite{KW1}). A vertex algebra structure on $\W^k(\g,f;\Gamma)$ is induced from that of $C_k$ and does not depend on the choice of $\Gamma$ \cite{BG, AKM}. A conformal $\frac{1}{2}\Z$-grading on $C_k$ is defined by $\conf(u)=1-j$ ($u\in\g_j$), $\conf(\varphi_\alpha)=1-\degG\alpha$, $\conf(\varphi^\alpha)=\degG\alpha$ and $\conf(\Phi_\alpha)=\frac{1}{2}$, where $\Delta(A)$ is the conformal weight of $A\in C_k$. This conformal grading is preserved by the differential $d$ and induces a $\frac{1}{2}\Z_{\geq0}$-grading on $\W^k(\g,f;\Gamma)$, which depends on the choice of $\Gamma$.

Let $T=\C[\para]$ and $V^T(\g)$ the affine vertex algebra over $T$, where we replace $k$ by a formal parameter $\para$. Set $\FchT=\Fch\otimes T$ and $\FneT=\Fne\otimes T$. Then $d$ defines a differential on
\begin{align*}
C_T=V^T(\g)\otimes\FchT\otimes\FneT,
\end{align*}
where $\otimes=\otimes_T$. Instead of $\otimes_T$, we use the notation $\otimes$ whenever the base ring is $T$. The $\W$-algebra $\W^T(\g,f;\Gamma)$ over $T$ is defined by the BRST cohomology of the complex $(C_T,d)$. We have
\begin{align*}
\W^T(\g,f;\Gamma)\otimes\C_{k}=\W^{k}(\g,f;\Gamma),
\end{align*}
where $\C_{k}$ is a $1$-dimensional $T$-module defined by $\para=k\in\C$. See e.g. \cite{ACL}. For $k\in\C$, we call the functor $?\otimes\C_{k}$ the {\em specialization}.

\section{Wakimoto representations for Affine Vertex Algebras}\label{aff Wak sec}

We introduce Wakimoto representations of $V^T(\g)$ and the screening operators $S_\alpha$ of $V^T(\g)$. We follow the construction given in \cite{F}.

\subsection{Differential representations of $\g$}
Let $G$ be a connected simply-connected Lie group corresponding to $\g$, $B_{+}$ the Borel subgroup corresponding to $\bo_{+}$, $B_{-}$ the opposite Borel subgroup and $N_{+}$ the unipotent subgroup corresponding to $\nil_{+}$. The left $G$-action on a flag variety $G/B_{-}$ induces a Lie algebra homomorphism $\rho_{G/B_{-}}\colon\g\rightarrow\D_{G/B_{-}}$, where $\D_{G/B_{-}}$ is the ring of differential operators of regular functions on $G/B_{-}$. Let $p\colon G\rightarrow G/B_{-}$ be the canonical projection and $U=N_{+}\cdot p(1)$ an $N_+$-orbit in $G/B_{-}$, where $1$ denotes the unit in $G$. An orbit $U$ is a unique open dense orbit in $G/B_{-}$ called the big cell. Since $N_{+}$ is unipotent, the exponential map $c(\nil_+)\colon\nil_{+}\rightarrow N_{+}$ is an isomorphism. The big cell $U\simeq N_{+}$ is then the affine space of the dimension $|\Delta_{+}|$ and the ring $\C[N_{+}]$ of regular functions on $N_+$ is a polynomial ring. A Lie algebra homomorphism $\rho\colon\g\rightarrow\D_{N_{+}}$ is defined by the restriction of $\rho_{G/B_{-}}$ on $U$. Fix a coordinate system $\{x_{\alpha}\}_{\alpha\in\Delta_{+}}$ on $N_{+}$ by using $c(\nil_+)$ such that $h\cdot x_{\alpha}=-\alpha(h)x_{\alpha}$ for all $h\in\h$ and $\alpha\in\Delta_{+}$. This coordinate is called homogeneous.

To describe the image of $\rho$, we introduce the frameworks in \cite{F}. Fix a root vector $e_\alpha\in\g_\alpha$ for $\alpha\in\Delta$. Denote by $f_\alpha=e_{-\alpha}$ and $h_\alpha=[e_\alpha,f_\alpha]$ for $\alpha\in\Delta_+$. Let $G^{\circ}=p^{-1}(U)=N_{+}\cdot B_{-}$ be a dense open submanifold in $G$. For $a\in\g$, set a smooth curve $\gamma(t)=\exp(-t a)$ on $G$. Given $X\in G^{\circ}$,
\begin{align*}
\gamma(t)X=Z_{+}(t)Z_{-}(t)
\end{align*}
for $|t|\ll1$, where $Z_{+}(t)\in N_{+}$ and $Z_{-}(t)\in B_{-}$. A vector field $\zeta_a$ is then given by the following formula :
\begin{align*}
(\zeta_a f)(p(X))=\frac{d}{d t}f(Z_{+}(t))|_{t=0}
\end{align*}
for any smooth function $f$ defined in a open subset in $U$ around $p(X)$. Choose a faithful representation $V_0$ of $\g$ and consider $X\in N_{+}$ as a matrix in $\GL(V_0)$ whose entries are polynomials in $\C[N_{+}]=\C[x_{\alpha}]_{\alpha\in\Delta_{+}}$. We have
\begin{align*}
(1-t a)X=Z_{+}(t) Z_{-}(t)\quad\mod.\ (t^2).
\end{align*}
Hence $Z_{+}(t)=X+t Z$, $Z_{-}=1+t Z'$ $\mod.(t^2)$, where $Z\in\nil_{+}$ and $Z'\in\bo_{-}$. We have
\begin{align*}
\zeta_a\cdot X=-X(X^{-1} a X)_{+},
\end{align*}
where $(\cdot)_{+}:\g=\nil_{+}\oplus\bo_{-}\rightarrow\nil_{+}$ is the first projection. For $a\in\g$, $\rho(a)$ is a derivation in $\C[N_{+}]$ such that
\begin{align*}
\rho(e_{\alpha})&=\sum_{\beta\in\Delta_{+}}P_{\alpha}^{\beta}(x)\der_\beta=\der_\alpha+\sum_{\beta\in\Delta_{+}\backslash\{\alpha\}}P_{\alpha}^{\beta}(x)\der_\beta,\\
\rho(h_{\alpha})&=-\sum_{\beta\in\Delta_{+}}\beta(h_{\alpha})x_{\beta}\der_\beta,\\
\rho(f_{\alpha})&=\sum_{\beta\in\Delta_{+}}Q_{\alpha}^{\beta}(x)\der_\beta
\end{align*}
for all $\alpha\in\Delta_+$, where $\der_\alpha=\der/\der x_\alpha$, $x=(x_{\alpha})_{\alpha\in\Delta_{+}}$ and $P_{\alpha}^{\beta}(x), Q_{\alpha}^{\beta}(x)\in\C[N_{+}]$. For $\lambda\in\h^*$, we have a twisted Lie algebra homomorphism $\rho_{\lambda}\colon\g\rightarrow\D_{N_{+}}$ by
\begin{align*}
\rho_{\lambda}(e_{\alpha})&=\sum_{\beta\in\Delta_{+}}P_{\alpha}^{\beta}(x)\der_\beta,\\
\rho_{\lambda}(h_{\alpha})&=-\sum_{\beta\in\Delta_{+}}\beta(h_{\alpha})x_{\beta}\der_\beta+\lambda(h_{\alpha}),\\
\rho_{\lambda}(f_{\alpha})&=\sum_{\beta\in\Delta_{+}}Q_{\alpha}^{\beta}(x)\der_\beta+\lambda(h_{\alpha})x_{\alpha}
\end{align*}
for all $\alpha\in\Pi$.

\subsection{Wakimoto representations of $V^T(\g)$}
For any finite set $S$, let $\A_S$ be the (infinite-dimensional) Weyl vertex algebra associated with $S$, whose generating fields $a_{\alpha}(z),a^*_{\alpha}(z)$ for $\alpha\in S$ satisfy
\begin{align*}
a_{\alpha}(z)a^{*}_{\beta}(w)\sim\frac{\delta_{\alpha,\beta}}{z-w},\quad
a_{\alpha}(z)a_{\beta}(w)\sim0\sim a^*_{\alpha}(z)a^{*}_{\beta}(w)
\end{align*}
for all $\alpha,\beta\in S$. For a polynomial $P(x)\in\C[N_+]$, we define a field $P(a^*)(z)$ on $\Anil$ by
\begin{align}\label{P(a) pol eq}
P(a^*)(z):=P(x)|_{x_\alpha=a^*_\alpha(z)\ (\alpha\in\Delta_+)}.
\end{align}
Since $a^*_\alpha(z)$ and $a^*_\beta(z)$ commute for all $\alpha,\beta\in\Delta_+$, $P(a^*)(z)$ is well-defined. Denote by $P(a^*)$ the vector in $\Anil$ corresponding to a field $P(a^*)(z)$. We have
\begin{align}\label{a der eq}
a_\alpha(z)P(a^*)(w)\sim\frac{\der_\alpha P(a^*)(w)}{z-w}.
\end{align}
Let $\Hi=V^{k+h^{\vee}}(\h)$ be the Heisenberg vertex algebra associated with the Cartan subalgebra $\h$ of $\g$, whose generating fields $b_{\alpha}(z)$ for $\alpha\in\Pi$ satisfy
\begin{align*}
b_{\alpha}(z)b_{\beta}(w)\sim\frac{(k+h^{\vee})(\alpha|\beta)}{(z-w)^2}
\end{align*}
for all $\alpha,\beta\in\Pi$. For $\lambda\in\h^*$, denote by $\Hi_\lambda$ the highest weight $\Hi$-module with highest weight $\lambda$. Let $\Hi^T$ be the Heisenberg vertex algebra over $T$ and $\Hi_\lambda^T$ the highest weight $\Hi^T$-module with highest weight $\lambda\in\h^*$, where we replace $k$ by a formal parameter $\para$ in $T$. Set $\A_S^T=\A_S\otimes_\C T$.

\begin{lemma}[\cite{F}]\label{aff def lemma} There exists an injective vertex algebra homomorphism $\hat{\rho}\colon V^T(\g)\rightarrow\Anil^T\otimes\Hi^T$ over $T$ with some $c_\alpha\in\C$ for each $\alpha\in\Pi$ such that
\begin{align*}
\hat{\rho}(e_{\alpha}(z))&=\sum_{\beta\in\Delta_{+}}:P_{\alpha}^{\beta}(a^*)(z)a_\beta(z):\ =a_\alpha(z)+\sum_{\beta\in\Delta_{+}\backslash\{\alpha\}}:P_{\alpha}^{\beta}(a^*)(z)a_\beta(z):,\\
\hat{\rho}(h_{\alpha}(z))&=-\sum_{\beta\in\Delta_{+}}\beta(h_{\alpha}):a^*_{\beta}(z)a_\beta(z):+b_\alpha(z),\\
\hat{\rho}(f_{\alpha}(z))&=\sum_{\beta\in\Delta_{+}}:Q_{\alpha}^{\beta}(a^*)(z)a_\beta(z):+:b_\alpha(z)a^*_\alpha(z):+((e_\alpha|f_\alpha)\para+c_\alpha)\der a^*_\alpha(z)
\end{align*}
for all $\alpha\in\Pi$. For any $\alpha\in\Delta_+$, $\hat{\rho}(e_{\alpha}(z))$, $\hat{\rho}(h_{\alpha}(z))$ also take the same forms.
\end{lemma}

The injective vertex algebra homomorphism $\hat{\rho}$ provides a $V^T(\g)$-module structure on any $\Anil^T\otimes\Hi^T$-module, called a Wakimoto representation of $V^T(\g)$. The specialization of $\hat{\rho}$ induces a vertex algebra homomorphism
\begin{align*}
\hat{\rho}_k=\hat{\rho}\otimes\C_k\colon V^k(\g)\rightarrow\Anil\otimes\Hi,
\end{align*}
which is also injective by \cite{F}.

\subsection{Screening Operators for $V^T(\g)$}
Let $\rho^R\colon\nil_{+}\rightarrow\D_{N_{+}}$ be the Lie algebra anti-homomorphism induced by the right action of $N_{+}$ on itself. Denote by
\begin{align}\label{right rep eq}
\rho^R(e_{\alpha})=\sum_{\beta\in\Delta_{+}}P_{\alpha}^{\beta,R}(x)\der_\beta
\end{align}
for $\alpha\in\Delta_{+}$, where $P_{\alpha}^{\beta,R}(x)$ is a polynomial in $\C[N_{+}]$. Since the left and right actions of $N_+$ on itself commute, we have
\begin{align*}
[\rho(e_\alpha),\rho^R(e_{\beta})]=0
\end{align*}
for all $\alpha,\beta\in\Delta_+$. Let $\Wak^T(\lambda)=\Anil^T\otimes\Hi^T_\lambda$ be a Wakimoto representation of $V^T(\g)$ for $\lambda\in\h^*$ and $\locT=T_{\para+h^\vee}$ the localization of $T$ at a multiplicative set $\{(\para+h^\vee)^i\mid i\in\Z_{\geq0}\}$, where $\para$ is a formal parameter in $T$. Denote by $\Wak(\lambda)=\Wak^T(\lambda)\otimes\C_k$, by $V^\locT(\g)=V^T(\g)\otimes\locT$ and by $\Wak^{\locT}(\lambda)=\Wak^T(\lambda)\otimes\locT$. Set $\Wak^T_\g=\Wak^T(0)$, $\Wak_\g=\Wak(0)$ and $\Wak^{\locT}_\g=\Wak^{\locT}(0)$. By \cite{FF5}, there exists an exact sequence
\begin{align}\label{affine Wakimoto}
0\rightarrow V^T(\g)\xrightarrow{\hat{\rho}}\Wak^T_\g\xrightarrow{\bigoplus S_\alpha}\bigoplus_{\alpha\in\Pi}\Wak^\locT(\halpha),
\end{align}
where $\halpha=-(\para+h^{\vee})^{-1}\alpha\in\h^*\otimes\widetilde{T}$ and
$S_\alpha\colon\Wak^T_\g\rightarrow\Wak^\locT(\halpha)$ is an intertwining operator defined by
\begin{align}\label{scaffine eq}
S_\alpha=\int:\hat{\rho}^{R}(e_\alpha(z))\ \e^{-\frac{1}{\para+h^{\vee}}\int b_\alpha(z)}:dz
\end{align}
for $\alpha\in\Pi$, where
\begin{align}\label{right affine eq}
\hat{\rho}^{R}(e_\alpha(z))=\sum_{\beta\in\Delta_{+}}:P_{\alpha}^{\beta,R}(a^*)(z)a_\beta(z):.
\end{align}
In other words,
\begin{align*}
V^T(\g)\simeq\bigcap_{\alpha\in\Pi}\Ker S_\alpha.
\end{align*}
The intertwining operators $S_\alpha$ are called the screening operators for $V^T(\g)$. We note that these screening operators are only considered for generic $\para=k$ in \cite{FF5} but the same proof also holds when the base ring is $T$.

\section{Wakimoto representations for Affine $\W$-algebras}\label{W-alg Wak sec}

\subsection{Coordinates on $N_{+}$}\label{local sec}
Let $G_{>0}$, $G_0^+$ be the unipotent Lie subgroups in $N_{+}$ corresponding to $\g_{>0}$, $\g_0^+=\g_0\cap\nil_+$ respectively. Since $\g_{>0}$ is an ideal in $\nil_{+}$, a subgroup $G_{>0}$ is normal in $N_{+}$. Hence a set $G_{>0}\times G_0^+$ has a group structure and is isomorphic to $N_{+}$ by $G_{>0}\times G_0^+\ni(a,b)\mapsto a\cdot b\in G_{>0}\cdot G_0^+=N_{+}$. Let $c(\g_{>0})$, $c(\g_0^+)$ be homogeneous coordinates on $G_{>0}$, $G_0^+$ respectively. A coordinate $c(\nil_+)$ on $N_+$ is then defined by $c(\nil_+)=c(\g_{>0})\cdot c(\g_0^+)$, which induces a ring isomorphism $\C[N_{+}]\simeq\C[G_{>0}]\otimes\C[G_0^+]$. We call $c(\nil_+)=c(\g_{>0})\cdot c(\g_0^+)$ a coordinate on $N_+$ {\em compatible with the decomposition $N_{+}=G_{>0}\times G_0^+$}. By construction, we have
\begin{align*}
\rho|_{\g_{>0}}=\rho_{\g_{>0}},\quad
\rho^R|_{\g_0^{+}}=\rho^R_{\g_0^{+}},
\end{align*}
where $\rho_{\g_{>0}}$ is the Lie algebra homomorphism derived from the left action of $G_{>0}$ on $G_{>0}$ and $\rho^R_{\g_0^{+}}$ is the Lie algebra anti-homomorphism derived from the right action of $G_0^{+}$ on $G_0^{+}$. Thus, we obtain:

\begin{lemma}\label{pol0 lemma} Suppose that $c(\nil_+)$ is compatible with the decomposition $N_{+}=G_{>0}\times G_0^+$. Then
\begin{enumerate}
\item $\rho(u)$ belongs to $\D_{G_{>0}}$ for all $u\in\g_{>0}$.
\item $\rho^R(u)$ belongs to $\D_{G_0^{+}}$ for all $u\in\g_0^+$. In particular,
\begin{align*}
\rho^R(e_\alpha)=\sum_{\beta\in\Deltazero}P_\alpha^{\beta,R}(x)\der_\beta
\end{align*}
for all $\alpha\in\Deltazero$.
\end{enumerate}
\end{lemma}

Let $\bQ$ be the root lattice of $\g$ and $\bQ_+\subset\bQ$ the set of linear combination with coefficients in $\Z_{\geq0}$ of elements of $\Pi$. Define a $\bQ$-valued grading on $\D_{N_+}$ by
\begin{align*}
\degQ(\der_\alpha)=-\alpha,\quad
\degQ x_\alpha=\alpha
\end{align*}
for $\alpha\in\Delta_+$, which induces a $\bQ_+$-grading on $\C[N_+]$. We define a $\bQ$-valued grading on $\g$ by $\degQ(\g_\alpha)=\alpha$ and $\degQ(\h)=0$. Then $\rho$ and $\rho^R$ reverse the $\bQ$-gradings, i.e. $\degQ\rho(u)=-\degQ(u)$ for $u\in\g$, and $\degQ\rho^R(u)=-\degQ(u)$ for $u\in\nil_+$. Therefore
\begin{align}
\label{polQ eq}\degQ P_\alpha^\beta(x)&=\degQ P_\alpha^{\beta,R}(x)=\beta-\alpha\in\bQ_+,\\
\degQ Q_\alpha^\beta(x)&=\beta+\alpha\in\bQ_+
\end{align}
unless $P_\alpha^\beta(x)=P_\alpha^{\beta,R}(x)=Q_\alpha^\beta(x)=0$. A $\frac{1}{2}\Z$-grading $\degG$ on $\Delta$ may be extended to $\bQ$ linearly. Then the composition map $\degG\circ\degQ$ defines a $\frac{1}{2}\Z$-grading on $\D_{N_{+}}$, which we denote by $\degG$ by abuse of notations. We have
\begin{align}\label{grD eq}
\degG(\der_\alpha)=-\degG\alpha,\quad
\degG x_\alpha=\degG\alpha
\end{align}
for $\alpha\in\Delta_{+}$, which induces a $\frac{1}{2}\Z_{\geq0}$-grading on $\C[N_+]$. Then $\rho$ and $\rho^R$ reverse the gradings, i.e. $\degG\rho(u)=-\degG(u)$ for $u\in\g$, and $\degG\rho^R(u)=-\degG(u)$ for $u\in\nil_+$. We have
\begin{align}
\label{polG eq}\degG P^\beta_\alpha(x)&=\degG P^{\beta,R}_\alpha(x)=\degG\beta-\degG\alpha\geq0,\\
\degG Q_\alpha^\beta(x)&=\degG\beta+\degG\alpha\geq0
\end{align}
unless $P_\alpha^\beta(x)=P_\alpha^{\beta,R}(x)=Q_\alpha^\beta(x)=0$.

\begin{lemma}\label{pol1 lemma}
Suppose that $c(\nil_+)$ is compatible with the decomposition $N_{+}=G_{>0}\times G_0^+$. If $\degG\alpha=\degG\beta$, polynomials $P^\beta_\alpha(x)$ and $P^{\beta,R}_\alpha(x)$ belong to $\C[G_0^+]$.
\end{lemma}
\begin{proof}
Since $\degG P^\beta_\alpha(x)=\degG P^{\beta,R}_\alpha(x)=0$, they are concentrated in the homogeneous component of $\C[N_+]$ with degree $0$, which coincides with $\C[G_0^+]$. This completes the proof.
\end{proof}

\begin{lemma}\label{pol2 lemma}
Suppose that $c(\nil_+)$ is compatible with the decomposition $N_{+} =G_{>0}\times G_0^+$. For $\alpha\in\Delta_{>0}$,
\begin{align*}
\rho(e_\alpha)=\der_\alpha+\sum_{\begin{subarray}{c} \beta\in\Delta_{>0}\\ \degG\beta>\degG\alpha \end{subarray}}P_\alpha^\beta(x)\der_\beta.
\end{align*}
\end{lemma}
\begin{proof}
Let $\alpha\in\Delta_{>0}$. Then $\rho(e_\alpha)$ belongs to $\D_{G_{>0}}$ by Lemma \ref{pol0 lemma}, so polynomials $P_\alpha^\beta(x)$ are in $\C[G_{>0}]$ for all $\beta\in\Delta_{>0}$. First, we assume that $\degG\beta<\degG\alpha$. If $P_\alpha^\beta(x)\neq0$, we have $\degG P^\beta_\alpha(x)=\degG\beta-\degG\alpha\geq0$, which is contrary to our assumption. Therefore $P_\alpha^\beta(x)=0$.

Next, we assume that $\degG\beta=\degG\alpha$. Then $P_\alpha^\beta(x)\in\C[G_{>0}]\cap\C[G_0^+]=\C$ by Lemma \ref{pol1 lemma}. Therefore, $P_\alpha^\beta(x)$ is a scalar. Hence, $\degQ P_\alpha^\beta(x)=\beta-\alpha$ should be zero unless $P_\alpha^\beta(x)=0$. Now, we have $P_\alpha^\beta(x)=0$ if $\degG\beta\leq\degG\alpha$ except for $\beta=\alpha$. Since $P_\alpha^\alpha(x)=1$ by construction, the lemma follows.
\end{proof}

\subsection{The cohomology $H(\V,d)$}
Let $\V$ be a vertex superalgebra over $T$ defined by
\begin{align*}
\V=\Am^T\otimes\FchT\otimes\FneT.
\end{align*}
Since $\rho(u)$ belongs to $\D_{G_{>0}}$ for all $u\in\g_{>0}$ by Lemma \ref{pol0 lemma}, $\hat{\rho}(u(z))$ is a field on $\Am^T$ for all $u\in\g_{>0}$. Since $\V$ has a $V^T(\g_{>0})$-module structure given by $\hat{\rho}$, $(\V,d)$ defines a cochain complex with repsect to the charge degree on $\FchT$. To compute the cohomology $H(\V,d)$, we introduce a $\frac{1}{2}\Z$-grading on $\V$ by
\begin{align*}
\degM(a_{\alpha})&=\degM(\varphi_{\alpha})=-\degG\alpha,\\
\degM(a^*_{\alpha})&=\degM(\varphi^\alpha)=\degG\alpha,\quad
\degM(\Phi_\alpha)=0
\end{align*}
and $\degM(\der A)=\degM(A)$, $\degM(:AB:)=\degM(A)+\degM(B)$ for all $A,B\in\V$. We associate this grading with the subspaces
\begin{align*}
F_p\V=\{A\in\V\mid2\degM(A)\geq p\}.
\end{align*}
of $\V$ for $p\in\Z$, which satisfy that
\begin{align*}
\bigcup_{p\in\Z}F_p\V=\V,\quad
\bigcap_{p\in\Z}F_p\V=0,\quad
F_{p+1}\V\subset F_{p}\V,\quad
d\cdot F_p\V\subset F_p\V.
\end{align*}
Hence, $\{F_p\V\}_{p\in\Z}$ is a filtration of the cochain complex $(\V,d)$. Let $\{E_q\}_{q=0}^{\infty}$ be the spectral sequence induced by $F_p\V$. Let $\Delta(A)$ be the conformal weight of $A\in\V$ defined by
\begin{align*}
\conf(a_{\alpha})=\conf(\varphi_{\alpha})=1-\degG\alpha,\quad
\conf(a^*_{\alpha})=\conf(\varphi^\alpha)=\degG\alpha,\quad
\conf(\Phi_\alpha)=\frac{1}{2}.
\end{align*}
Set
\begin{align*}
\V(n)=\mathrm{Span}_\C\{A\in\V\mid\conf(A)=n\}
\end{align*}
for $n\in\frac{1}{2}\Z$. Since $d\cdot\V(n)\subset\V(n)$,
\begin{align*}
\V=\bigoplus_{n\in\frac{1}{2}\Z}\V(n)
\end{align*}
is the decomposition as a complex. Denote by $F_p\V(n)$ the induced filtration on $\V(n)$.
\begin{lemma}\label{V lem}
$F_p\V(n)=0$ for $p>2n$.
\end{lemma}
\begin{proof}
The vertex algebra $\V$ is spanned by all vectors of the form
\begin{align*}
A=\ :(D a^*)(D a)(D\varphi^*)(D\varphi)(D\Phi):,
\end{align*}
where
\begin{align*}
D a^*&=(\der^{m^{(1)}_1}a^*_{\alpha^{(1)}_1})\cdots(\der^{m^{(1)}_r}a^*_{\alpha^{(1)}_r}),\quad
D a=(\der^{m^{(2)}_1}a_{\alpha^{(2)}_1})\cdots(\der^{m^{(2)}_s}a_{\alpha^{(2)}_s}),\\
D \varphi^*&=(\der^{m^{(3)}_1}\varphi^{\alpha^{(3)}_1})\cdots(\der^{m^{(3)}_t}\varphi^{\alpha^{(3)}_t}),\quad
D \varphi=(\der^{m^{(4)}_1}\varphi_{\alpha^{(4)}_1})\cdots(\der^{m^{(4)}_u}\varphi_{\alpha^{(4)}_u}),\\
D \Phi&=(\der^{m^{(5)}_1}\Phi_{\alpha^{(5)}_1})\cdots(\der^{m^{(5)}_v}\Phi_{\alpha^{(5)}_v})
\end{align*}
for some $m^{(i)}_j,r,s,t,u,v\in\Z_{\geq0}$, $\alpha^{(i)}_j\in\Delta_{>0}$ ($i=1,2,3,4$) and $\alpha^{(5)}_j\in\Deltahalf$. Therefore, it suffices to see that $2\conf(A)\geq p$ for all vectors $A\in F_p\V$ of the above form. By definition,
\begin{align*}
\conf(A)-\degM(A)=\sum_{i,j}m^{(i)}_j+s+u+\frac{1}{2}v.
\end{align*}
Since $A\in F_p\V$,
\begin{align*}
2\conf(A)\geq2\conf(A)-2(\sum_{i,j}m^{(i)}_j+s+u+\frac{1}{2}v)=2\degM(A)\geq p.
\end{align*}
This completes the proof.
\end{proof}
\begin{prop}\label{V prop}
$H(\V,d)\simeq\FneT$.
\end{prop}
\begin{proof}
The spectral sequence induced by $F_p\V(n)$ converges for each $n\in\frac{1}{2}\Z$ by Lemma \ref{V lem} and so the total spectral sequence $E_n$ does. By definition, it is easy to see that
\begin{align*}
E_1=H(\V,\dst)=H(\Am^T\otimes\FchT,\dst)\otimes\FneT,
\end{align*}
where $\dst=\int\dst(z)\ dz$. The cohomology $H(\Am^T\otimes\FchT,\dst)$ coincides with the semi-infinite cohomology of $\hat{\g}_{>0}$-module with the coefficient of the Wakimoto module $\Am^T$ over $T$, see \cite{Fei}. Hence, the vanishing theorem given in \cite{FF3, V} can be applied to our cases:
\begin{align*}
H(\Am^T\otimes\FchT,\dst)=H^0(\Am^T\otimes\FchT,\dst)=T.
\end{align*}
Therefore,
\begin{align}\label{V eq}
H(\V,d)\simeq E_\infty=E_1=\FneT
\end{align}
as required.
\end{proof}
\begin{cor}\label{hatPhi cor}
The isomorphism in Proposition \ref{V prop} is a vertex algebra isomorphism over $T$.
\end{cor}
\begin{proof}
First, notice that $H(\V,d)$ has a vertex algebra structure inherited from that of $\V$. We will show that the isomorphism \eqref{V eq} is a vertex algebra isomorphism. Set
\begin{align}\label{hatPhi eq}
\hat{\Phi}_\alpha(z)=\Phi_\alpha(z)+\sum_{\beta\in\Deltahalf}\chi([e_\alpha,e_\beta])a^*_\beta(z)
\end{align}
for $\alpha\in\Deltahalf$. Recall that $\degM(\Phi_\alpha)=0$ and $\conf(\Phi_\alpha)=\frac{1}{2}$. Moreover, $\degM(\hat{\Phi}_\alpha-\Phi_\alpha)=\frac{1}{2}$ and $\conf(\hat{\Phi}_\alpha)=\frac{1}{2}$. By Lemma \ref{aff def lemma} and Lemma \ref{pol2 lemma}, we have
\begin{align*}
\hat{\rho}(e_\alpha(z))=a_\alpha(z)+\sum_{\beta\in\Delta_{\geq1}}:P_\alpha^\beta(a^*)(z)a_\beta(z):
\end{align*}
for $\alpha\in\Deltahalf$. Hence,
\begin{align*}
d\cdot a^*_\alpha=\dst\cdot a^*_\alpha=\sum_{\beta\in\Deltahalf}\int:\hat{\rho}(e_\beta(z))\varphi^\beta(z):a^*_\alpha\ dz=\varphi^\alpha.
\end{align*}
for $\alpha\in\Deltahalf$. Therefore,
\begin{align*}
d\cdot\hat{\Phi}_\alpha=\sum_{\beta\in\Deltahalf}\chi([e_\beta,e_\alpha])\varphi^\alpha+\sum_{\beta\in\Deltahalf}\chi([e_\alpha,e_\beta])(d\cdot a^*_\beta)=0.
\end{align*}
This implies that the isomorphism \eqref{V eq} is given by the correspondence $\hat{\Phi}_\alpha\mapsto\Phi_\alpha$. We have
\begin{align*}
\hat{\Phi}_\alpha(z)\hat{\Phi}_\beta(w)\sim\frac{\chi([e_\alpha,e_\beta])}{z-w}\sim\Phi(z)\Phi(w)
\end{align*}
for all $\alpha,\beta\in\Deltahalf$. Hence, the isomorphism \eqref{V eq} is a vertex algebra isomorphism.
\end{proof}

\subsection{Wakimoto free fields realizations of $\W^T(\g,f;\Gamma)$}\label{Wak sec}
Given a $V^T(\g)$-module $M$, define a $C_T$-module $C_T(M)=M\otimes\FchT\otimes\FneT$. Then $(C_T(M),d)$ is a cochain complex whose cohomology
\begin{align*}
H_\chi(M)=H(C_T(M),d)
\end{align*}
has a structure of a $\W^T(\g,f;\Gamma)$-module by construction. Consider the case that $M$ is a Wakimoto representation $\Wak^T(\lambda)$ of $V^T(\g)$. We have a $\W^T(\g,f;\Gamma)$-module
\begin{align*}
H_\chi(\Wak^T(\lambda))=H(\Wak^T(\lambda)\otimes\FchT\otimes\FneT,d)
\end{align*}
for each $\lambda\in\h^*$.

\begin{lemma}\label{Wak vanish lem}
For all $\lambda\in\h^*\otimes T$,
\begin{align*}
H_\chi(\Wak^T(\lambda))=H^0_\chi(\Wak^T(\lambda))\simeq\Azero^T\otimes\FneT\otimes\Hi^T_\lambda.
\end{align*}
\end{lemma}
\begin{proof}
By definition, $d\cdot(\Azero^T\otimes\Hi^T_\lambda)=0$. Hence,
\begin{align*}
H_\chi(\Wak^T(\lambda))=\Azero^T\otimes H(\V,d)\otimes\Hi^T_\lambda.
\end{align*}
By Proposition \ref{V prop}, $H(\V,d)\simeq\Fne$. Therefore the assertion follows.
\end{proof}

Let
\begin{align*}
\sigma_\lambda:H_\chi(\Wak^T(\lambda))\xrightarrow{\sim}\Azero^T\otimes\FneT\otimes\Hi^T_\lambda
\end{align*}
be the isomorphism defined in Lemma \ref{Wak vanish lem} for $\lambda\in\h^*\otimes T$. Denote by $\sigma_\lambda(A)=\sigma_\lambda([A])$ for all $A\in C_T(\Wak^T(\lambda))$, where $[A]$ denotes the cohomology class of $A$ in $H_\chi(\Wak^T(\lambda))$. Set $\sigma=\sigma_0$. Notice that $H_\chi(\Wak^T_\g)$ has a vertex algebra structure inherited from that of $C_T(\Wak^T_\g)$. Then $H_\chi(\Wak^T(\lambda))$ is an $H_\chi(\Wak^T_\g)$-module for all $\lambda\in\h^*\otimes T$.

\begin{cor}\label{coh isom cor}
The map $\sigma$ is an isomorphism of vertex algebras over $T$ defined by $\sigma(A)=A$ for $A=a_\alpha, a^*_\alpha$ ($\alpha\in\Deltazero$), $b_\alpha$ ($\alpha\in\Pi$), and $\sigma(\hat{\Phi}_\alpha)=\Phi_\alpha$ ($\alpha\in\Deltahalf$), and $\sigma_\lambda$ is an isomorphism of $\Azero^T\otimes\FneT\otimes\Hi^T$-modules for all $\lambda\in\h^*$, where $\hat{\Phi}_\alpha$ is defined by \eqref{hatPhi eq}.
\end{cor}
\begin{proof}
The corollary is immediate from by Corollary \ref{hatPhi cor} and the construction of the isomorphism $\sigma_\lambda$ in Lemma \ref{Wak vanish lem}.
\end{proof}

The same argument applies to the case that $M$ is a Wakimoto representation $\Wak^\locT(\lambda)$ of $V^\locT(\g)$. Thus, we obtain:

\begin{lemma}\label{Wak vanish2 lem}
For all $\lambda\in\h^*\otimes\locT$, we have a vertex algebra isomorphism
\begin{align*}
H_\chi(\Wak^\locT(\lambda))=H^0_\chi(\Wak^\locT(\lambda))\simeq\Azero^\locT\otimes\FnelocT\otimes\Hi^\locT_\lambda
\end{align*}
over $\locT$, where $X^\locT=X^T\otimes\locT$ for $X=\Anil, \Fne, \Hi_\lambda$.
\end{lemma}

Recall the exact sequence \eqref{affine Wakimoto}:
\begin{align*}
0\rightarrow V^T(\g)\xrightarrow{\hat{\rho}}\Wak^T_\g\xrightarrow{\bigoplus S_\alpha}\bigoplus_{\alpha\in\Pi}\Wak^\locT(\halpha).
\end{align*}
Let $C_\locT(M)=M\otimes\FchlocT\otimes\FnelocT$. Then we have
\begin{align*}
0\rightarrow C_T\rightarrow C_T(\Wak^T_\g)\rightarrow\bigoplus_{\alpha\in\Pi}C_\locT(\Wak^\locT(\halpha)).
\end{align*}
Recall that $H(C_T,d)=H^0(C_T,d)=\W^T(\g,f;\Gamma)$. According to Lemma \ref{Wak vanish lem} and Lemma \ref{Wak vanish2 lem}, we have an exact sequence
\begin{align}\label{w-alg wak eq}
\W^T(\g,f;\Gamma)\xrightarrow{\omega}\Azero^T\otimes\FneT\otimes\Hi^T\xrightarrow{\bigoplus Q_\alpha}\bigoplus_{\alpha\in\Pi}\Azero^\locT\otimes\FnelocT\otimes\Hi^\locT_{\halpha},
\end{align}
where
\begin{align*}
\omega\colon\W^T(\g,f;\Gamma)\rightarrow\Azero^T\otimes\FneT\otimes\Hi^T
\end{align*}
is the vertex algebra homomorphism over $T$ induced by $\hat{\rho}$ and
\begin{align*}
Q_\alpha\colon\Azero^T\otimes\FneT\otimes\Hi^T\rightarrow\Azero^\locT\otimes\FnelocT\otimes\Hi^\locT_{\widetilde{\alpha}}
\end{align*}
is the screening operator induced by $S_\alpha$ for all $\alpha\in\Pi$. Then the map $\omega$ provides a $\W^T(\g,f;\Gamma)$-module structure on any $\Azero^T\otimes\FneT\otimes\Hi^T$-module, which we call a {\em Wakimoto representation for a $\W$-algebra $\W^T(\g,f;\Gamma)$ over $T$}.

Since ${\rho}(e_\alpha)$ and $\rho^R(e_\beta)$ commutes for all $\alpha,\beta\in\Delta_+$, we have $d\cdot\hat{\rho}^R(e_\alpha)=0$. Hence, the right $V^T(\nil_+)$-action on a Wakimoto representation $\Wak^T_\g$ given by $\hat{\rho}^R$ induces that on $H_\chi(\Wak^T_\g)\simeq\Azero^T\otimes\FneT\otimes\Hi^T$ by $\sigma\circ\hat{\rho}^R$.

\begin{lemma}\label{screening sigma lemma}
For all $\alpha\in\Pi$,
\begin{align*}
Q_\alpha

&=\int:\sigma(\hat{\rho}^R(e_\alpha))(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz.
\end{align*}
\end{lemma}
\begin{proof}
Since $Q_\alpha$ is the intertwining operator induced by $S_\alpha$ through the functor $H_\chi(?)$, the assertion of the lemma follows.
\end{proof}

\begin{lemma}\label{sc0 lemma}
For all $\alpha\in\Pi_0$,
\begin{align*}
Q_\alpha=\int:\hat{\rho}^R(e_\alpha)(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz.
\end{align*}
In particular, $Q_\alpha=S_\alpha$ on $\Azero^T\otimes\Hi^T$ and acts as $0$ on $\FneT$ for all $\alpha\in\Pi_0$.
\end{lemma}
\begin{proof}
By Lemma \ref{pol0 lemma} (2),
\begin{align*}
\hat{\rho}^R(e_\alpha)=\sum_{\alpha\in\Deltazero}P_\alpha^{\beta,R}(a^*)a_\beta
\end{align*}
belongs to $\Azero$. Since $\sigma(A)=A$ for all $A\in\Azero$, the assertion of the lemma follows from Lemma \ref{screening sigma lemma}.
\end{proof}

We recall that $P_\alpha^{\beta,R}(a^*)(z)$ is the field on $\Anil$ defined by \eqref{P(a) pol eq} for the polynomial $P_\alpha^{\beta,R}(x)$ that is defined by \eqref{right rep eq} and depends on the choice of coordinates $c(\nil_+)$ on $N_+$. According to Lemma \ref{pol1 lemma}, it follows that $P_\alpha^{\beta,R}(a^*)(z)$ is a field on $\Azero$ if $\degG\alpha=\degG\beta$.

\begin{theorem}\label{main thm}
Suppose that the coordinate $c(\nil_+)$ on $N_{+}$ is compatible with the decomposition $N_{+}=G_{>0}\times G_0^+$. Then, we have
\begin{align*}
Q_\alpha&=\sum_{\beta\in\Deltazero}\int:P_{\alpha}^{\beta,R}(a^*)(z)a_{\beta}(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_0),\\
Q_\alpha&=\sum_{\beta\in\Deltahalf}\int:P_{\alpha}^{\beta,R}(a^*)(z)\Phi_\beta(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_{\frac{1}{2}}),\\
Q_\alpha&=\sum_{\beta\in\Deltaone}\chi(e_{\beta})\int:P_{\alpha}^{\beta,R}(a^*)(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_1).
\end{align*}
\end{theorem}

We will prove Theorem \ref{main thm} in Appendix \ref{proof sec}.
\smallskip

By Theorem \ref{Miura thm}, it turns out that
\begin{align*}
\W^T(\g,f;\Gamma)\simeq\bigcap_{\alpha\in\Pi}\Ker Q_\alpha.
\end{align*}
In particular, the map $\omega$ is injective (Corollary \ref{w-alg wak cor}). Thus, we have an exact sequence
\begin{align*}
0\rightarrow\W^T(\g,f;\Gamma) \xrightarrow{\omega}\Azero^T\otimes\FneT\otimes\Hi^T\xrightarrow{\bigoplus Q_\alpha}\bigoplus_{\alpha\in\Pi}\Azero^\locT\otimes\FnelocT\otimes\Hi^\locT_{\widetilde{\alpha}}.
\end{align*}
We note that $\Azero^T\otimes\FneT$ coincides with $\frac{1}{2}\dim(\mathcal{N}\cap\mathcal{S}_f)$ copies of the $\beta\gamma$-system, where $\mathcal{N}$ is the nilpotent cone of $\g$, $\mathcal{S}_f$ is the Slodowy slice of $\g$ through $f$, since
\begin{align*}
\frac{1}{2}\dim(\mathcal{N}\cap\mathcal{S}_f)=\frac{1}{2}(\dim\g_0+\dim\g_{\frac{1}{2}}-\dim\h)=\dim\g_0^++\frac{1}{2}\dim\g_{\frac{1}{2}}.
\end{align*}

Let $f=f_{\mathrm{prin}}=\sum_{\alpha\in\Pi}e_{-\alpha}$ be a principal nilpotent element in $\g$ and $\Gamma$ the Dynkin grading on $\g$ for $f$. Then $\Pi=\Pi_1=\Delta_1$ and $\chi(e_\alpha)=1$ for all $\alpha\in\Pi$. By Lemma \ref{pol2 lemma}, $P_\alpha^{\beta,R}(z)=\delta_{\alpha,\beta}$ for all $\alpha,\beta\in\Pi$. By Theorem \ref{main thm} and Theorem \ref{Miura thm},
\begin{align*}
\W^T(\g,f_{\mathrm{prin}};\Gamma)\simeq\bigcap_{\alpha\in\Pi}\Ker \int:\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz,
\end{align*}
which is a well-known result given in \cite{FF4}, see also \cite{FBZ}.

In the case that $\g_0=\h$, we have $\Pi=\Pi_{\frac{1}{2}}\sqcup\Pi_1$ and, by Lemma \ref{pol2 lemma}, $P_\alpha^{\beta,R}(z)=\delta_{\alpha,\beta}$ for all $\alpha\in\Pi_i$, $\beta\in\Delta_i$ and $i=\frac{1}{2},1$. By Theorem \ref{main thm} and Theorem \ref{Miura thm}, $\W^T(\g,f;\Gamma)$ is isomorphic to
\begin{align*}
\bigcap_{\alpha\in\Pi_{\frac{1}{2}}}\Ker \int:\Phi_\alpha(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz
\cap
\bigcap_{\begin{subarray}{c} \alpha\in\Pi_1 \\ \chi(e_\alpha)\neq0 \end{subarray}}\Ker \int:\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz,
\end{align*}
which is a result previously obtained in \cite{G}.
\smallskip

Let $\Vtau$ be the affine vertex algebra associated with $\g_0$ and its invariant bilinear form $\tau_k$ defined by
\begin{align*}
\tau_k(u|v)=k(u|v)+\frac{1}{2}\killing_\g(u|v)-\frac{1}{2}\killing_{\g_0}(u|v)
\end{align*}
for all $u,v\in\g_0$, where $\killing_\g$, $\killing_{\g_0}$ are the Killing forms on $\g$, $\g_0$ respectively. Denote by $\VtauT$ instead of $V^{\tau_\para}(\g_0)$ when the base ring is $T=\C[\para]$. By \cite{F}, there exists an exact sequence
\begin{align*}
0\rightarrow\VtauT\xrightarrow{\hat{\rho}_{\g_0}}\Azero^T\otimes\Hi^T\xrightarrow{\bigoplus S_\alpha}\bigoplus_{\alpha\in\Pi_0}\Azero^\locT\otimes\Hi^\locT_{\widetilde{\alpha}},
\end{align*}
where
\begin{align}\label{para wak eq}
\hat{\rho}_{\g_0}\colon\VtauT\rightarrow\Azero^T\otimes\Hi^T
\end{align}
is an injective vertex homomorphism over $T$, called a Wakimoto representation of $\VtauT$, and is defined by
\begin{align*}
\hat{\rho}_\rf(e_{\alpha}(z))&=\sum_{\beta\in\Deltazero}:P_{\alpha}^{\beta}(a^*)(z)a_\beta(z):,\\
\hat{\rho}_\rf(h_{\alpha'}(z))&=-\sum_{\beta\in\Deltazero}\beta(h_{\alpha'}):a^*_{\beta}(z)a_\beta(z):+b_{\alpha'}(z),\\
\hat{\rho}_\rf(f_{\alpha}(z))&=\sum_{\beta\in\Deltazero}:Q_{\alpha}^{\beta}(a^*)(z)a_\beta(z):+:b_\alpha(z)a^*_\alpha(z):+((e_\alpha|f_\alpha)\para+c'_\alpha)\der a^*_\alpha(z)
\end{align*}
for all $\alpha\in\Pi_0$ and $\alpha'\in\Pi$. Thus, we have
\begin{align}\label{para wak eq2}
\VtauT\simeq\bigcap_{\alpha\in\Pi_0}\Ker S_\alpha|_{\Azero^T\otimes\Hi^T}.
\end{align}

\begin{lemma}\label{Miura const lem}
Suppose that the coordinate $c(\nil_+)$ on $N_{+}$ is compatible with the decomposition $N_{+}=G_{>0}\times G_0^+$. Then
\begin{align*}
\bigcap_{\alpha\in\Pi_0}\Ker Q_\alpha\simeq\VtauT\otimes\Fne.
\end{align*}
\end{lemma}
\begin{proof}
Since $Q_\alpha=S_\alpha$ on $\Azero^T\otimes\Hi^T$ and acts as $0$ on $\FneT$ for all $\alpha\in\Pi_0$ by Lemma \ref{sc0 lemma}, the assertion of the lemma follows from \eqref{para wak eq2}.
\end{proof}

\section{Screening operators and Miura map}\label{remark sec}

In this section we recall the screening operators $\hQ_\alpha$ ($\alpha\in\Pi_{>0}$) introduced in \cite{G} and clarify the relationship between $Q_\alpha$ and $\hQ_\alpha$. As a result, we show that $Q_\alpha=\hQ_\alpha$ for all $\alpha\in\Pi_{>0}$ (Theorem \ref{Miura thm}) and the screening operator $Q_\alpha$'s are compatible with the Miura map $\mu$ (Theorem \ref{w-alg Miura thm}).

\subsection{Screening Operators $\hQ_\alpha$}\label{old scr sec}
We follow the construction given in \cite{G}. Though the construction of $\hQ$ was considered in \cite{G} for generic $\para=k$, the same argument applies when the base ring is $T$. Let $\bQ_0=\bigoplus_{\gamma\in\Pi_0}\Z\gamma$ be the root lattice of $\rf$, $\Pi^\Gamma=\{\alpha\in\Delta_{>0}\mid{}^\nexists\beta,\gamma\in\Delta_{>0}\ \mathrm{s.t.}\ \alpha=\beta+\gamma\}$ a set of indecomposable roots in $\Delta_{>0}$. Define an equivalence relation on $\Delta_{>0}$ by $\alpha\sim\beta\iff\alpha-\beta\in\bQ_0$, which may restrict to $\Pi^\Gamma$. Let $[\Pi^\Gamma]=\Pi^\Gamma/\sim$ be the quotient set and
\begin{align}\label{[a] def eq}
[\alpha]=\{\beta\in\Delta_+\mid\beta-\alpha\in\bQ_0\}
\end{align}
the equivalence class of $\alpha\in\Pi^\Gamma$ in $[\Pi^\Gamma]$. Consider a map $\flat\colon\Pi_{>0}\ni\alpha\mapsto[\alpha]\in[\Pi^\Gamma]$.

\begin{lemma}\label{flat lemma}
The map $\flat$ is bijective.
\end{lemma}
\begin{proof}
Since it is clear that $\flat$ is injective, we will show that $\flat$ is surjective. Let $\beta\in\Pi^\Gamma$ and $n=\rootht\beta$ the height of $\beta$. In the case that $n=1$, we have $\beta\in\Pi_{>0}$ and $\flat(\beta)=[\beta]$. Next, we assume that $n>1$. Then there exist $\beta_1,\beta_2\in\Delta_+$ such that $\beta=\beta_1+\beta_2$. Since $\beta\in\Pi^\Gamma$, we may assume that $\beta_2\in\Deltazero$. Then $\beta-\beta_1=\beta_2\in\bQ_0$ and $[\beta]=[\beta_1]$. We claim that $\beta_1\in\Pi^\Gamma$. The reason is below. If there exist $\gamma_1,\gamma_2\in\Delta_{>0}$ such that $\beta_1=\gamma_1+\gamma_2$, we have
\begin{align*}
\g_\beta=[\g_{\beta_1},\g_{\beta_2}]=[[\g_{\gamma_1},\g_{\gamma_2}],\g_{\beta_2}]=[[\g_{\gamma_1},\g_{\beta_2}],\g_{\gamma_2}]+[\g_{\gamma_1},[\g_{\gamma_2},\g_{\beta_2}]].
\end{align*}
Then $\gamma_1+\beta_2\in\Delta_{>0}$ or $\gamma_2+\beta_2\in\Delta_{>0}$. Hence, it turns out that $\beta=\gamma_1+\gamma_2+\beta_2$ can be decomposed to the sum of two roots in $\Delta_{>0}$, which is contrary to our assumption that $\beta\in\Pi^\Gamma$. Therefore there exists $\beta_1\in\Pi^\Gamma$ such that $[\beta]=[\beta_1]$ and $\rootht(\beta_1)<\rootht(\beta)$. By induction on $n$, it follows that there exists $\alpha\in\Pi_{>0}$ such that $\flat(\alpha)=[\beta]$, that is, $\flat$ is surjective. The proof of the lemma is now complete.
\end{proof}

Thanks to Lemma \ref{flat lemma}, we may identify $[\Pi^\Gamma]$ with $\Pi_{>0}$ through $\flat$. Set a vector space $\C^{[\alpha]}=\bigoplus_{\beta\in[\alpha]}\C \tilde{v}_\beta$ for each $\alpha\in\Pi_{>0}$ and define a $\rf$-action on $\C^{[\alpha]}$ by
\begin{align*}
u\cdot \tilde{v}_\beta=\sum_{\gamma\in[\alpha]}c_{\gamma,u}^\beta \tilde{v}_\gamma
\end{align*}
for all $u\in\rf$ and $\beta\in[\alpha]$, where $c_{\gamma,u}^\beta$ is a structure constant defined by the following formula: $[e_\gamma,u]=\sum_{\beta\in[\alpha]}c_{\gamma,u}^\beta e_\beta$. Then $\C^{[\alpha]}$ is the irreducible highest weight $\g_0$-module with a highest weight vector $\tilde{v}_\alpha$ of highest weight $-\alpha$, see Remark 3.3 in \cite{G}. Let
\begin{align*}
\hat{\g}_0^\locT=\g_0\otimes\locT[t,t^{-1}]\oplus\locT K
\end{align*}
be the $\locT$-form of the affine Lie algebra of $\g_0$ that is the central extension of $\g_0\otimes \locT[t,t^{-1}]$ by $\tau_\para$. Then $\locT^{[\alpha]}=\C^{[\alpha]}\otimes\locT$ is a $\g_0\otimes\locT[t]\oplus\locT K$-module by $\g_0\otimes\locT[t]t=0$ and $K=1$. Let $\hM_\alpha$ be the induced $\VtaulocT$-module from $\locT^{[\alpha]}$ defined by
\begin{align*}
\hM_\alpha=U(\hat{\g}^\locT_0)\underset{U(\g_0\otimes \locT[t]\oplus\locT K)}{\otimes}\locT^{[\alpha]}\simeq\VtaulocT\otimes\bigoplus_{\beta\in[\alpha]}\locT \tilde{v}_\beta,
\end{align*}
where $\VtaulocT=\VtauT\otimes\locT$. Since $\C^{[\alpha]}$ is irreducible, the specialization $\hM_\alpha\otimes\C_k$ of $\hM_\alpha$ is an irreducible $\Vtau$-module for generic $k\in\C$. Therefore, $\hM_\alpha$ is an irreducible $\VtaulocT$-module.
\smallskip

We will introduce the screening operators $\hQ_\alpha$ as intertwining operators. For a vertex algebra $V$ and $V$-modules $L,M$ and $N$, a linear map
\begin{align*}
Y_{L,M}^N(\cdot,z)\colon L\rightarrow\Hom(M,N)\{z\}=\sum_{n\in\mathbb{Q}}\Hom(M,N)\ z^n
\end{align*}
is called an intertwining operator of type $\binom{N}{L\ M}$ if it satisfies the Borcherds identity (see \cite{FHL} for the details). For a $V$-module $M$ and a vector $A\in M$,  we shall call $Y_{M,V}^M(A,z)$ an {\em intertwining operator corresponding to $A$}. In the present paper, we only consider intertwining operators of type $\binom{M}{M\ V}$. Let $\widetilde{V}^\beta(z)=\sum_{n\in\Z}\widetilde{V}^\beta_n z^{-n}$ be an intertwining operator corresponding to $\tilde{v}_\beta$ defined by $\widetilde{V}^\beta_n\cdot 1=\delta_{n,0}\tilde{v}_\beta$ $(n\geq0)$ and
\begin{align*}
[u(z),\widetilde{V}^\beta(w)]=\sum_{\gamma\in[\alpha]}c_{\gamma,u}^\beta \widetilde{V}^\gamma(w)\delta(z-w)
\end{align*}
for all $u\in\rf$, where $1$ denotes the vacuum vector in $\VtauT$. Then $\widetilde{V}^\beta(z)$ is well-defined, see Proposition 3.7 in \cite{G}. We define the screening operator
\begin{align*}
\hQ_\alpha\colon\VtauT\otimes\FneT\rightarrow \hM_\alpha\otimes\FnelocT
\end{align*}
for $\alpha\in\Pi_{>0}$ by
\begin{align}
\label{Q[a] eq1}\hQ_\alpha&=\sum_{\beta\in[\alpha]}\int:\widetilde{V}^\beta(z)\Phi_\beta(z):dz &(\alpha\in\Pi_{\frac{1}{2}}),\\
\label{Q[a] eq2}\hQ_\alpha&=\sum_{\beta\in[\alpha]}\chi(e_\beta)\int \widetilde{V}^\beta(z)dz &(\alpha\in\Pi_1).
\end{align}
By \cite{G}, we have a vertex algebra isomorphism
\begin{align}\label{old main eq}
\W^T(\g,f;\Gamma)\simeq\bigcap_{\alpha\in\Pi_{>0}}\Ker\hQ_\alpha.
\end{align}

\subsection{Miura map}\label{Miura map sec}
As mentioned in Section \ref{W-alg sec}, there exists a subcomplex $C_+$ of $C_k$ such that $C_+$ has only non-negative charged degree and $\W^k(\g,f;\Gamma)=H^0(C_+,d)$. Hence, we have
\begin{align*}
\W^k(\g,f;\Gamma)=\Ker d|_{C_+^0}\subset C_+^0=V^{\tau_k}(\g_{\leq0})\otimes\Fne.
\end{align*}
Consider the projection map $\g_{\leq0}\twoheadrightarrow\g_0$. It induces a surjective vertex algebra homomorphism $V^{\tau_k}(\g_{\leq0})\twoheadrightarrow\Vtau$, giving rise to a map
\begin{align*}
\mu\colon\W^k(\g,f;\Gamma)\rightarrow\Vtau\otimes\Fne,
\end{align*}
which is called {\em the Miura map} for $\W^k(\g,f;\Gamma)$ and injective for all $k\in\C$ by \cite{F,A}, see also \cite{G}. Following \cite{G}, the Miura map $\mu$ coincides with the specialization $\mu^T\otimes\C_k$ of an injective vertex algebra homomorphism
\begin{align*}
\mu^T\colon\W^T(\g,f;\Gamma)\rightarrow\VtauT\otimes\FneT.
\end{align*}
over $T$ induced by the formula \eqref{old main eq}.

\begin{prop}\label{[a] prop}
Suppose that $c(\nil_+)$ is compatible with the decomposition $N_+=G_{>0}\times G_0^+$.
\begin{enumerate}
\item Let $\alpha\in\Pi_{>0}$ and $\beta\in\Delta_+$ such that $\degG\alpha=\degG\beta$. Then $P_\alpha^{\beta,R}(x)=0$ unless $\beta\in[\alpha]$.
\item Let $Q_\alpha$ be a screening operator defined in Theorem \ref{main thm}. Then
\begin{align*}
Q_\alpha&=\sum_{\beta\in[\alpha]}\int:P_{\alpha}^{\beta,R}(a^*)(z)\Phi_\beta(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_{\frac{1}{2}}),\\
Q_\alpha&=\sum_{\beta\in[\alpha]}\chi(e_{\beta})\int:P_{\alpha}^{\beta,R}(a^*)(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_1).
\end{align*}
\end{enumerate}
\end{prop}
\begin{proof}
To prove the assertion (1), we assume that $P_\alpha^{\beta,R}(x)\neq0$ for $\alpha\in\Pi_{>0}$ and $\beta\in\Delta_+$ such that $\degG\alpha=\degG\beta$. We will show that $\beta\in[\alpha]$. Since $P_\alpha^{\beta,R}(x)$ is a polynomial in $\C[G_0^+]$ by Lemma \ref{pol1 lemma} and $\degQ P_\alpha^{\beta,R}(x)=\beta-\alpha$ by \eqref{polQ eq}, we have $\beta-\alpha\in\bQ_0$, that is, $\beta\in[\alpha]$. Thus, $P_\alpha^{\beta,R}(x)=0$ unless $\beta\in[\alpha]$.

Next, we derive (2) from (1). By Theorem \ref{main thm}, we have
\begin{align*}
Q_\alpha&=\sum_{\beta\in\Delta_{\frac{1}{2}}}\int:P_{\alpha}^{\beta,R}(a^*)(z)\Phi_\beta(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_{\frac{1}{2}}),\\
Q_\alpha&=\sum_{\beta\in\Delta_1}\chi(e_{\beta})\int:P_{\alpha}^{\beta,R}(a^*)(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_1).
\end{align*}
Since $P_\alpha^{\beta,R}(a^*)(z)=0$ for all $\alpha\in\Pi_{i}$ and $\beta\in\Delta_i\backslash[\alpha]$ with $i=\frac{1}{2},1$, we may restrict the summation range in $Q_\alpha$ to $\{\beta\in[\alpha]\}$. This completes the proof.
\end{proof}

Set
\begin{align*}
v_\beta=P_\alpha^{\beta,R}(a^*)\otimes\e^{\halpha}\in\Azero^\locT\otimes\Hi_{\halpha}^\locT
\end{align*}
for all $\beta\in[\alpha]$, where $\e^{\halpha}$ is a highest weight vector in $\Hi^\locT_{\halpha}$. Let
\begin{align*}
V^\beta(z)=\sum_{n\in\Z}V^\beta_n z^{-n}=\ :P_\alpha^{\beta,R}(a^*)(z)\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:
\end{align*}
be an intertwining operator corresponding to $v_\beta$. By definition, $V_n^\beta\cdot1=\delta_{n,0}v_\beta$ for $n\geq0$, where $1$ denotes the vacuum vector in $\Azero^T\otimes\Hi^T$. Recall that $\hat{\rho}_{\rf}\colon\VtauT\rightarrow\Azero^T\otimes\Hi^T$ is a Wakimoto representation of $\VtauT$ defined in \eqref{para wak eq} and provides a $\VtaulocT$-module structure on $\Azero^\locT\otimes\Hi^\locT_{\halpha}$. We shall often drop the vertex algebra homomorphism $\hat{\rho}_\rf$ for the $\VtaulocT$-action on $\Azero^\locT\otimes\Hi^\locT_{\halpha}$ (e.g. $u(z)$ in place of $\hat{\rho}_\rf(u(z))$ ) if no confusion may arise.

\begin{lemma}\label{mod prop}
For all $u\in\rf$ and $\beta\in[\alpha]$,
\begin{align}\label{mod eq}
[u(z), V^\beta(w)]=\sum_{\gamma\in[\alpha]}c_{\gamma,u}^\beta V^\gamma(w)\delta(z-w).
\end{align}
\end{lemma}

We will prove Lemma \ref{mod prop} in Appendix \ref{mod prop sec}.
\smallskip

By Lemma \ref{mod prop},
\begin{align*}
[u(z), V^\beta(w)]\cdot1=\sum_{\gamma\in[\alpha]}(c_{\gamma,u}^\beta V^\gamma(w)\cdot1)\delta(z-w),
\end{align*}
where $1$ denotes the vacuum vector in $\Azero^T\otimes\Hi^T$. Computing their formal residues at $z=0$, we have
\begin{align*}
u_{(0)}v_\beta=\sum_{\gamma\in[\alpha]}c_{\gamma,u}^\beta v_\gamma.
\end{align*}
Hence, the free $\locT$-module $\bigoplus_{\beta\in[\alpha]}\locT v_\beta$ has a structure of a $\rf$-module, and is isomorphic to $\locT^{[\alpha]}$ by $\tilde{v}_\beta\mapsto v_\beta$ due to the irreducibility of $\locT^{[\alpha]}$. Let $M_\alpha$ be a $\VtaulocT$-submodule generated by $\bigoplus_{\beta\in[\alpha]}\locT v_\beta$ in $\Azero^\locT\otimes\Hi^\locT_{\halpha}$ for each $\alpha\in\Pi_{>0}$, which is isomorphic to $\hM_\alpha$ by $\tilde{v}_\beta\mapsto v_\beta$ due to the irreducibility of $\hM_\alpha$. Then $V^\beta(z)u$ belongs to $M_\alpha((z))$ for all $u\in\VtauT\subset\Azero^T\otimes\Hi^T$. Hence, we may identify
\begin{align*}
V^\beta(z)\colon\VtauT\rightarrow M_\alpha((z))
\end{align*}
with an intertwining operator corresponding to $v_\beta$ for all $\beta\in[\alpha]$, satisfying that $V_n^\beta\cdot1=\delta_{n,0}v_\beta$ $(n\geq0)$ and \eqref{mod eq} by Lemma \ref{mod prop}. Therefore the isomorphism $\hM_\alpha\simeq M_\alpha$ of $\VtaulocT$-modules implies that $\widetilde{V}^\beta(z)=V^\beta(z)$. Thus, we obtain:

\begin{lemma}\label{MN lemma}
For all $\alpha\in\Pi_{>0}$, $\hM_\alpha\simeq M_\alpha$ as $\VtauT$-modules, which induces that $\widetilde{V}^\beta(z)=V^\beta(z)$ on $\VtauT$ for all $\beta\in[\alpha]$.
\end{lemma}

Since $\bigcap_{\alpha\in\Pi_0}\Ker Q_\alpha=\VtauT\otimes\FneT$ by Lemma \ref{Miura const lem}, we have
\begin{align}\label{Miura obs eq}
\bigcap_{\alpha\in\Pi}\Ker Q_\alpha=\bigcap_{\alpha\in\Pi_{>0}}\Ker Q_\alpha|_{\VtauT\otimes\FneT}.
\end{align}

\begin{theorem}\label{Miura thm}
Suppose that $c(\nil_+)$ is compatible with the decomposition $N_+=G_{>0}\times G_0^+$. Then $Q_\alpha=\hQ_\alpha$ on $\VtauT\otimes\FneT$ for all $\alpha\in\Pi_{>0}$. In particular,
\begin{align*}
\W^T(\g,f;\Gamma)\simeq\bigcap_{\alpha\in\Pi}\Ker Q_\alpha.
\end{align*}
\end{theorem}
\begin{proof}
By Proposition \ref{[a] prop},
\begin{align*}
Q_\alpha&=\sum_{\beta\in[\alpha]}\int:V^\beta(z)\Phi_\beta(z):dz &(\alpha\in\Pi_{\frac{1}{2}}),\\
Q_\alpha&=\sum_{\beta\in[\alpha]}\chi(e_{\beta})\int V^\beta(z)dz &(\alpha\in\Pi_1).
\end{align*}
Hence, $Q_\alpha$ is a map from $\VtauT\otimes\FneT$ to $M_\alpha\otimes\FnelocT$ for all $\alpha\in\Pi_{>0}$. According to Lemma \ref{MN lemma}, we have $\hM_\alpha\simeq M_\alpha$ and $\widetilde{V}^\beta(z)=V^\beta(z)$, which implies that $Q_\alpha=\hQ_\alpha$ on $\VtauT\otimes\FneT$ for all $\alpha\in\Pi_{>0}$ by \eqref{Q[a] eq1} and \eqref{Q[a] eq2}. Therefore,
\begin{align*}
\W^T(\g,f;\Gamma)\simeq\bigcap_{\alpha\in\Pi_{>0}}\Ker\hQ_\alpha=\bigcap_{\alpha\in\Pi_{>0}}\Ker Q_\alpha|_{\VtauT\otimes\FneT}=\bigcap_{\alpha\in\Pi}\Ker Q_\alpha
\end{align*}
by \eqref{old main eq} and \eqref{Miura obs eq}. This completes the proof.
\end{proof}

\begin{theorem}\label{w-alg Miura thm}
Suppose that $c(\nil_+)$ is compatible with the decomposition $N_+=G_{>0}\times G_0^+$. Then the specialization of an inclusion map
\begin{align*}
\bigcap_{\alpha\in\Pi}\Ker Q_\alpha\hookrightarrow\bigcap_{\alpha\in\Pi_0}\Ker Q_\alpha
\end{align*}
coincides with the Miura map $\mu$ for $\W^k(\g,f;\Gamma)$.
\end{theorem}
\begin{proof}
Since $Q_\alpha=\hQ_\alpha$ on $\VtauT\otimes\FneT$ for all $\alpha\in\Pi_{>0}$ by Theorem \ref{Miura thm}, the assertion of the corollary follows from the fact that the specialization of an inclusion map
\begin{align*}
\bigcap_{\alpha\in\Pi_{>0}}\Ker\hQ_\alpha\hookrightarrow\VtauT\otimes\FneT
\end{align*}
coincides with the Miura map $\mu$ for $\W^k(\g,f;\Gamma)$ by \cite{G}.
\end{proof}

\begin{cor}\label{w-alg wak cor}
The map $\omega$ defined in \eqref{w-alg wak eq} and the specialization $\omega_k=\omega\otimes\C_k$ of the map $\omega$ are injective for all $k\in\C$.
\end{cor}
\begin{proof}
By \eqref{w-alg wak eq} and Theorem \ref{Miura thm}, the image of $\omega$ coincides with a $\W$-algebra $\W^T(\g,f;\Gamma)$. Since $\omega$ holds conformal gradings, it induces a surjective endomorphism
\begin{align*}
\omega(n)\colon\W^T(\g.f;\Gamma)(n)\rightarrow\W^T(\g.f;\Gamma)(n),
\end{align*}
where $\W^T(\g.f;\Gamma)(n)$ is the homogeneous subspace of $\W^T(\g.f;\Gamma)$ with conformal weight $n$ for all $n\in\frac{1}{2}\Z_{\geq0}$. Since $\W^T(\g.f;\Gamma)(n)$ is finite-dimensional, $\omega(n)$ is isomorphism for all $n\in\frac{1}{2}\Z_{\geq0}$. Hence, $\omega$ is injective.

By construction and Theorem \ref{w-alg Miura thm}, the specialization of injective maps
\begin{align*}
\bigcap_{\alpha\in\Pi}\Ker Q_\alpha\hookrightarrow\bigcap_{\alpha\in\Pi_0}\Ker Q_\alpha\hookrightarrow\Azero^T\otimes\Hi^T\otimes\FneT
\end{align*}
induces vertex algebra homomorphisms
\begin{align*}
\W^k(\g,f;\Gamma)\xrightarrow{\mu}\Vtau\otimes\Fne\xrightarrow{\iota}\Azero\otimes\Hi\otimes\Fne,
\end{align*}
whose composition map coincides with $\omega_k$. According to the proof of Lemma \ref{Miura const lem}, we have
\begin{align*}
\iota=(\hat{\rho}_\rf)_k\otimes\Id_{\Fne},
\end{align*}
where $(\hat{\rho}_\rf)_k=\hat{\rho}_\rf\otimes\C_k$ is a Wakimoto representation of $\Vtau$ and injective due to \cite{F}. Since $\mu$ is injective, $\omega_k=\iota\circ\mu$ is also injective for all $k\in\C$.
\end{proof}

The following diagram summarizes the correspondence between the screening operators $Q_\alpha$ and $\hQ_\alpha$ that we have discussed above.

\begin{align*}
\xymatrix{
\W^T(\g,f;\Gamma) \ar[rr]^(0.4){\omega} \ar[d]^{\mu^T} & & \Azero^T\otimes\Hi^T\otimes\FneT \ar@{}[d]|{\|} \\
\VtauT\otimes\FneT \ar[rr]^(0.45){\hat{\rho}_\rf\otimes\Id} \ar[d]^{\bigoplus\hQ_\alpha} & & \Azero^T\otimes\Hi^T\otimes\FneT \ar[d]^{\bigoplus Q_\alpha} \\
\displaystyle{\bigoplus_{\alpha\in\Pi_{>0}}\hM_\alpha\otimes\FnelocT} \ar[r]^{\displaystyle{\sim}} & \displaystyle{\bigoplus_{\alpha\in\Pi_{>0}}M_\alpha\otimes\FnelocT} \ar@{}[r]|(0.45){\displaystyle{\subset}} & \displaystyle{\bigoplus_{\alpha\in\Pi_{>0}}\Azero^\locT\otimes\Hi^\locT_{\halpha}\otimes\FnelocT}
}
\end{align*}

\section{Parabolic inductions}

In this section, we state and prove our main theorem (Theorem \ref{induced thm}). From now on, we assume that $\g$ is a reductive Lie algebra.

\subsection{$\W$-algebras for reductive Lie algebras}\label{reductive W-alg sec}
Let $\g$ be a finite-dimensional reductive Lie algebra, $f$ a nilpotent element in $[\g,\g]$, $\kappa$ a symmetric invariant bilinear form on $\g$ and $\Gamma$ a good grading for $f$ on $\g$ satisfying that the center $\z_\g$ of $\g$ lies in $\g_0$. The definition of $\W$-algebras $\W^\kappa(\g,f;\Gamma)$ naturally extends for $\g,f,\Gamma$ and $\kappa$. We use the same notations: $\g_i$, $\Delta_i$, $\Pi_i$, $\Deltazero$, $\h$, $\nil_+$ as in Section \ref{W-alg sec} and \ref{local sec}. Set
\begin{align}\label{reductive dec eq}
\g=\z_\g\oplus\bigoplus_{i=1}^m\g^{i},
\end{align}
where $\g^{i}$ is a simple Lie algebra. Let $\g^i_j=\g^i\cap\g_j$ and $f_i\in\g^i$ such that $f=\sum_{i=1}^m f_i$ corresponding to \eqref{reductive dec eq}. Then
\begin{align*}
\Gamma_i:\g^i=\bigoplus_{j\in\frac{1}{2}\Z}\g^i_j
\end{align*}
is good for $f_i$. We have an isomorphism of vertex algebras
\begin{align}\label{reductive W-alg eq}
\W^\kappa(\g,f;\Gamma)\simeq V^\kappa(\z_\g)\otimes\bigotimes_{i=1}^m\W^{k_i}(\g^i,f_i;\Gamma_i),
\end{align}
where $k_i=\kappa(\theta_i|\theta_i)/2\in\C$ for the highest root $\theta_i$ in $\g^i$. Let $\h^i$ be a Cartan subalgebra of $\g^i$ contained in $\g^i_0$ and $h^\vee_i$ the dual Coxter number of $\g^i$. We have $\h=\z_\g\oplus\bigoplus_{i=1}^m\h^i$. Denote by $\Delta^i$, $\Delta_+^i$, $\Pi^i$ the sets of roots, positive roots and simple roots in $\g^i$ respectively. Let $\nil_+^i=\bigoplus_{\alpha\in\Delta_+^i}\g_\alpha$ and $\g_0^{+,i}=\g_0^i\cap\nil_+^i$. We also have $\nil_+=\bigoplus_{i=1}^m\nil_+^i$ and $\g_0^+=\bigoplus_{i=1}^m\g_0^{+,i}$. Set $\Delta_j^i=\Delta_j\cap\Delta^i$, $\Pi_j^i=\Pi_j\cap\Delta^i$ and $(\Delta_0^i)^+=\Delta_0^i\cap\Delta_+^i$. Set $\Azero=\bigotimes_{i=1}^m\mathcal{A}_{(\Delta_0^i)^+}$. Define
\begin{align*}
\Hi=V^\kappa(\z_\g)\otimes\bigotimes_{i=1}^m\Hi^{k_i+h^\vee_i}(\h^i),\quad
V^{\tau_\kappa}(\rf)=V^\kappa(\z_\g)\otimes\bigotimes_{i=1}^m V^{\tau_{k_i}}(\g_0^i).
\end{align*}
Let $T_i=\C[\para_i]$ and $T=\bigotimes_{i=1}^m T_i$, where $\para_i$ is a formal parameter. By Theorem \ref{Miura thm},
\begin{align*}
\W^{T_i}(\g^i,f_i;\Gamma_i)\simeq\bigcap_{\alpha\in\Pi^i}\Ker Q_\alpha\subset\A_{(\Delta_0^i)^+}^{T_i}\otimes \Phi^{T_i}(\g^i_{\frac{1}{2}})\otimes\Hi^{T_i},
\end{align*}
where $Q_\alpha$ is a screening operator. Since $V^T(\z_\g)$ commutes with all $Q_\alpha$, we have an isomorphism of vertex algebras
\begin{align}\label{reductive screening eq}
\W^T(\g,f;\Gamma)\simeq\bigcap_{\alpha\in\Pi}\Ker Q_\alpha\subset\Azero^T\otimes\FneT\otimes\Hi^T
\end{align}
over $T$. By Theorem \ref{w-alg Miura thm}, the Miura map
\begin{align*}
\mu\colon\W^\kappa(\g,f;\Gamma)\rightarrow V^{\tau_\kappa}(\rf)\otimes\Fne
\end{align*}
coincides with the map induced by the specialization of an inclusion map
\begin{align*}
\bigcap_{\alpha\in\Pi}\Ker Q_\alpha\hookrightarrow\bigcap_{\alpha\in\Pi_0}\Ker Q_\alpha.
\end{align*}

\subsection{Induced nilpotent orbits}\label{ind nil sec}
Let $\mathcal{N}$ be the set of all nilpotent elements in $[\g,\g]$. A Lie group $G$ acts on $\mathcal{N}$ by the adjoint action, which decompose $\mathcal{N}$ into finitely many orbits, called nilpotent orbits in $\g$. See e.g. \cite{CM}. Let $\p$ be a parabolic subalgebra, i.e. $\bo\subset\p$. There exists the Levi decomposition $\p=\lf\oplus\uf$ such that $\lf$ is a reductive Lie subalgebra and $\uf$ is a nilpotent subalgebra. We have a root subsystem $\Delta_\lf\subset\Delta$ such that
\begin{align}\label{Levi root eq}
\lf=\h\oplus\bigoplus_{\alpha\in\Delta_\lf}\g_\alpha.
\end{align}
The reductive Lie subalgebra $\lf$ is called a Levi subalgebra of $\g$ and uniquely determined by simple roots $\Pi_\lf$ of $\Delta_\lf$ up to conjugation. Denote by $P$, $L$ the Lie subgroups of $G$ corresponding to $\p$, $\lf$ respectively. The following results are due to Lusztig and Spaltenstein \cite{LS}.
\begin{propdef}[\cite{LS}]
Let $\Oc_\lf$ be a nilpotent orbit in $\lf$. Then there exists a unique nilpotent orbit $\Oc_\g$ in $\g$ such that $(\Oc_\lf+\uf)\cap\Oc_\g$ is Zariski dense in $\Oc_\lf+\uf$, and $\Oc_\g$ doesn't depend on the choice of $\p$. The orbit $\Oc_\g$ is called the induced nilpotent orbit from $\Oc_\lf$ and denoted by $\ind^\g_\lf\Oc_\lf$.
\end{propdef}

\begin{prop}[\cite{LS}]\label{LS prop}
Let $\Oc_\lf$ be a nilpotent orbit in $\lf$ and $\Oc_\g=\ind^\g_\lf\Oc_\lf$ the induced nilpotent orbit from $\Oc_\lf$.
\begin{enumerate}
\item $\Oc_\g$ is a unique nilpotent orbit that has the dimension $\dim\Oc_\g=\dim\Oc_\lf+2\dim\uf$ and $(\Oc_\lf+\uf)\cap\Oc_\g\neq\phi$.
\item Induced nilpotent orbits are transitive, i.e.
\begin{align*}
\ind^\g_\lf\Oc_\lf=\ind^\g_{\lf'}\ind^{\lf'}_\lf\Oc_\lf
\end{align*}
for any Levi subalgebra $\lf'$ such that $\lf\subset\lf'\subset\g$.
\end{enumerate}
\end{prop}

To prove Lemma \ref{ind good lem}, we recall the properties of (good) gradings in \cite{EK}.
\begin{lemma}[\cite{EK}]\label{good pro lem}
Let $\g$ be a reductive Lie algebra, $f$ a nilpotent element of $[\g,\g]$. Let $\Gamma$ be a $\frac{1}{2}\Z$-grading on $\g$ such that $f\in\g_{-1}$, the center of $\g$ lies in $\g_0$ and $[\g_i,\g_j]\subset\g_{i+j}$ for all $i,j\in\frac{1}{2}\Z$.
\begin{enumerate}
\item The following are equivalent.
\begin{enumerate}
\item $\ad(f)\colon\g_j\rightarrow\g_{j-1}$ is injective for $j\geq\frac{1}{2}$.
\item $\ad(f)\colon\g_j\rightarrow\g_{j-1}$ is surjective for $j\leq\frac{1}{2}$.
\item $\Gamma$ is good for $f$.
\end{enumerate}
\item Suppose that $\Gamma$ is good for $f$. Then $\dim\g^f=\dim\g_0+\dim\g_{\frac{1}{2}}$.
\end{enumerate}
\end{lemma}

\begin{lemma}\label{ind good lem}
Let $\Gamma$ be a good grading for $f$ on $\g$, $G\cdot f$ the nilpotent orbit in $\g$ through $f$, $\lf$ a Levi subalgebra of $\g$ with simple roots $\Pi_\lf$. Suppose that $\degG\alpha=1$ for all $\alpha\in\Pi\backslash\Pi_\lf$. Then there exists a nilpotent element $f_\lf$ in $[\lf,\lf]$ such that $\Gamma_{\lf}$ is a good grading for $f_\lf$ and $G\cdot f=\ind^\g_\lf L\cdot{f_\lf}$, where $\Gamma_{\lf}$ is the restriction of $\Gamma$ to $\lf$ and $L\cdot f_\lf$ is the nilpotent orbit in $\lf$ through $f_\lf$.
\end{lemma}
\begin{proof}
As in Section \ref{reductive W-alg sec}, there exists a root decomposition $\g=\h\oplus\bigoplus_{\alpha\in\Delta}\g_\alpha$ compatible with $\Gamma$. We may choose $\Delta_+$ such that $\nil_+=\bigoplus_{\alpha\in\Delta_+}\g_\alpha\subset\g_{\geq0}$. We have a root subsystem $\Delta_\lf$ of $\Delta$ satisfying \eqref{Levi root eq}. Let $\uf=\bigoplus_{\alpha\in\Delta_{-}\backslash(\Delta_{-}\cap\Delta_\lf)}\g_\alpha$. Then $\p=\lf\oplus\uf$ is a parabolic subalgebra including the opposite Borel subalgebra $\bo_-$ and gives the Levi decomposition of $\p$ whose Levi subalgebra is $\lf$. Denote by $\lf_j=\lf\cap\g_j$ and by $\uf_j=\uf\cap\g_j$. Since $\Pi\backslash\Pi_\lf\subset\Pi_1$, we have $\g_j=\lf_j\oplus\uf_j$ for all $j\leq\frac{1}{2}$. Choose $f_\lf\in\lf_{-1}$ and $f_\uf\in\uf_{-1}$ such that $f=f_\lf+f_\uf$ corresponding to $\g_{-1}=\lf_{-1}\oplus\uf_{-1}$. Since $[f,\g_j]=\g_{j-1}$ for all $j\leq\frac{1}{2}$ and $[\lf,\uf]\subset\uf$, we have $[f_\lf,\lf_j]=\lf_{j-1}$ for all $j\leq\frac{1}{2}$. Note that the center of $\lf$ lies in $\h\subset\g_0$ and the formula $[\lf_i,\lf_j]\subset\lf_{i+j}$ is deduced from $[\g_i,\g_j]\subset\g_{i+j}$. These imply that $\Gamma_\lf$ is good for $f_\lf$ by Lemma \ref{good pro lem} (1). Therefore $\dim\lf^{f_\lf}=\dim\lf_0+\dim\lf_{\frac{1}{2}}$ by Lemma \ref{good pro lem} (2). Since $\g_j=\lf_j$ for $j=0,\frac{1}{2}$, we have $\dim\lf^{f_\lf}=\dim\g^f$. Hence,
\begin{align*}
\dim G\cdot f=\dim\g-\dim\g^f=\dim\lf+2\dim\uf-\dim\lf^{f_\lf}=\dim L\cdot{f_\lf}+2\dim\uf.
\end{align*}
By construction, $f\in G\cdot f\cap(L\cdot{f_\lf}+\uf)\neq\phi$. Therefore $G\cdot f=\ind^\g_\lf L\cdot{f_{\lf}}$ by Proposition \ref{LS prop}.
\end{proof}

\begin{cor}\label{ind chi cor}
Under the conditions in Lemma \ref{ind good lem}, $\chi(u)=(f_\lf|u)$ for all $u\in\lf$.
\end{cor}
\begin{proof}
We use the notations in the proof of Lemma \ref{ind good lem}. Since $(\lf\mid\uf)=0$, we have $\chi(u)=(f_\lf|u)+(f_\uf|u)=(f_\lf|u)$ for all $u\in\lf$.
\end{proof}

\begin{rem}\label{ind rem}
The condition $\Pi\backslash\Pi_\lf\subset\Pi_1$ in Lemma \ref{ind good lem} is valid for all cases of type $A$ by \cite{Kr, OW}, rectangular nilpotent cases of type $BCD$ by \cite{Ke, Sp}, all cases of type $G$ and many cases of other exceptional types by \cite{GE}.
\end{rem}

\subsection{Preliminary results}\label{pre sec}
Continue to use the notations in Section \ref{reductive W-alg sec} and \ref{ind nil sec}. Under the condition $\Pi\backslash\Pi_\lf\subset\Pi_1$, we have a nilpotent element $f_\lf$ in $[\lf,\lf]$ such that $\Gamma_{\lf}$ is good for $f_\lf$ by Lemma \ref{ind good lem}. Set $\nil_+^\lf=\lf\cap\nil_+$ and $\lf_0^+=\lf_0\cap\nil_+$. Denote by $(\Delta_\lf)_+=\Delta_\lf\cap\Delta_+$ and by $(\Delta_\lf)_j=\Delta_\lf\cap\Delta_j$. Let $N_+^\lf$, $L_0^+$, $L_{>0}$ be the Lie subgroup of $L$ corresponding to $\nil_+^\lf$, $\lf_0^+$, $\lf_{>0}$ respectively. Since $\g_0=\lf_0$, we have $L_0^+=G_0^+$. Let $c(\nil_+)=c(\g_{>0})\cdot c(\g_0^+)$ be a coordinate on $N_+$ compatible with the decomposition $N_+=G_{>0}\times G_0^+$. Then $c(\nil_+^\lf)=c(\nil_+)|_{N_+^\lf}=c(\g_{>0})|_{L_{>0}}\cdot c(\g_0^+)$ is a coordinate on $N_+^\lf$ compatible with the decomposition $N_+^\lf=L_{>0}\times L_0^+$. Let $\rho^R_\lf\colon\nil_+^\lf\rightarrow\D_{N_+^\lf}$ be the anti-homomorphism induced by the right action of $N_+^\lf$ on itself. Then $\rho^R_\lf(u)=\rho^R(u)$ for all $u\in\nil_+^\lf$ as differentials on $\C[N_+^\lf]$ by construction. Set
\begin{align*}
\rho^R_\lf(e_\alpha)=\sum_{\beta\in(\Delta_\lf)_+}P_{\alpha,\lf}^{\beta,R}(x)\der_\beta
\end{align*}
for all $\alpha\in(\Delta_\lf)_+$. We have
\begin{align}\label{Ps=P eq}
P_{\alpha,\lf}^{\beta,R}(x)=P_\alpha^{\beta,R}(x)|_{x_\gamma=0\ \mathrm{for}\ \mathrm{all}\ \gamma\in\Delta_+\backslash(\Delta_\lf)_+}.
\end{align}

\begin{lemma}\label{Ps=P lemma}
Suppose that $\Pi\backslash\Pi_\lf\subset\Pi_1$. For $\alpha,\beta\in(\Delta_\lf)_+$, $P_{\alpha,\lf}^{\beta,R}(x)=P_\alpha^{\beta,R}(x)$ if $\degG\alpha=\degG\beta$.
\end{lemma}
\begin{proof}
Since $\Pi\backslash\Pi_\lf\subset\Pi_1$, we have $\Delta_+\backslash(\Delta_\lf)_+\subset\Delta_{\geq1}$. The assertion follows by \eqref{Ps=P eq} and Lemma \ref{pol1 lemma}.
\end{proof}

Let $\Phi(\lf_{\frac{1}{2}})$ be the neutral vertex algebra associated with $\lf_{\frac{1}{2}}$, which is defined by
\begin{align*}
\Phi^\lf_\alpha(z)\Phi^\lf_\beta(w)\sim\frac{(f_\lf|[e_\alpha,e_\beta])}{z-w}
\end{align*}
for generating fields $\Phi_\alpha^\lf(z)$, $\Phi_\beta^\lf(z)$ with $\alpha,\beta\in(\Delta_\lf)_{\frac{1}{2}}$.
\begin{lemma}\label{ind 1/2 lemma}
Suppose that $\Pi\backslash\Pi_\lf\subset\Pi_1$. Then $\Phi(\lf_{\frac{1}{2}})=\Fne$.
\end{lemma}
\begin{proof}
The assertion of the lemma immediately follows from Corollary \ref{ind chi cor}.
\end{proof}

Let $(\Pi_\lf)_j=\Pi_\lf\cap\Delta_j$. Recall that $[\alpha]$ is the subset of $\Delta_{>0}$ defined by \eqref{[a] def eq} for $\alpha\in\Pi_{>0}$.

\begin{lemma}\label{[a]s=[a] lemma}
Suppose that $\Pi\backslash\Pi_\lf\subset\Pi_1$. Then all roots in $[\alpha]$ lie in $\Delta_+^\lf$ for all $\alpha\in(\Pi_\lf)_{>0}$.
\end{lemma}
\begin{proof}
All roots in $[\alpha]$ are spanned by $\alpha$ and simple roots in $\Pi_0=(\Pi_\lf)_0$. Hence, the assertion of the lemma follows.
\end{proof}

\subsection{Parabolic inductions}\label{induced sec}
Set the Killing forms $\killing_\g$, $\killing_\lf$ on $\g$, $\lf$ respectively.

\begin{theorem}\label{induced thm}
Let $\Gamma$ be a good grading for $f$ on $\g$ and $\lf$ a Levi subalgebra of $\g$ with simple roots $\Pi_\lf$. Suppose that $\Pi\backslash\Pi_\lf\subset\Pi_1$. Let $\Gamma_\lf$ be a $\frac{1}{2}\Z$-grading on $\lf$ defined by restriction of $\Gamma$ and $f_\lf$ the nilpotent element of $[\lf,\lf]$ chosen by Lemma \ref{ind good lem}. Then there exists an injective vertex algebra homomorphism
\begin{align*}
\Ind^\g_\lf\colon\W^\kappa(\g,f;\Gamma)\rightarrow\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf),
\end{align*}
where
\begin{align*}
\kappa_\lf=\kappa+\frac{1}{2}\killing_\g-\frac{1}{2}\killing_\lf.
\end{align*}
Moreover, the map $\Ind^\g_\lf$ is a unique vertex algebra homomorphism that satisfies
\begin{align*}
\mu=\mu_\lf\circ\Ind^\g_\lf,
\end{align*}
where $\mu$, $\mu_\lf$ are the Miura maps for $\W^\kappa(\g,f;\Gamma)$, $\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf)$ respectively.
\end{theorem}
\begin{proof}
First, we consider the case that $\g$ is a simple Lie algebra. By Theorem \ref{Miura thm},
\begin{align*}
\W^T(\g,f;\Gamma)\simeq\bigcap_{\alpha\in\Pi}\Ker Q_\alpha,
\end{align*}
where $Q_\alpha$ is a screening operator, which acts on $\Azero^T\otimes\FneT\otimes\Hi^T$. Set
\begin{align}\label{s dec eq}
\lf=\z_\lf\oplus\bigoplus_{i=1}^{m_\lf}\lf^i,
\end{align}
where $\z_\lf$ is the center of $\lf$ and $\lf^i$ is a simple Lie algebra. Note that $\h$ is also a Cartan subalgebra of $\lf$. Let $\theta_i$ be the highest root of $\lf^i$, $h^\vee_i$ the dual Coxeter number of $\lf^i$ and $\bm{\kappa}_\lf=\bm{\kappa}+\frac{1}{2}\killing_\g-\frac{1}{2}\killing_\lf$ a $T$-valued invariant bilinear form on $\g$, where $T=\C[\para]$ and $\bm{\kappa}(u|v)=\para(u|v)$. Then
\begin{align*}
\bm{\kappa}_\lf(u|v)=
\begin{cases}
(\para+h^\vee)(u|v) & (u,v\in\z_\lf)\\
\para_i(u|v) & (u,v\in\lf^i),
\end{cases}
\end{align*}
where $\para_i=\frac{2}{(\theta_i|\theta_i)}(\para+h^\vee)-h^\vee_i$. We shall denote by $\W^T(\lf,f_\lf;\Gamma_\lf)$ instead of $\W^{\bm{\kappa}_\lf}(\lf,f_\lf;\Gamma_\lf)$. Set $(\Pi_\lf)^i=\{\alpha\in\Pi_\lf\mid\lf_\alpha\subset\lf^i\}$ and $(\Pi_\lf)_j^i=(\Pi_\lf)^i\cap(\Pi_\lf)_j$. As in Section \ref{reductive W-alg sec}, we have
\begin{align*}
\W^T(\lf,f_\lf;\Gamma_\lf)\simeq\bigcap_{\alpha\in\Pi_\lf}\Ker Q_\alpha^\lf,
\end{align*}
where $Q_\alpha^\lf$ is a screening operator, which acts on $\A_{(\Delta_\lf)_0^+}^T\otimes \Phi^T(\lf_{\frac{1}{2}})\otimes\Hi^T$. By Theorem \ref{main thm} and Proposition \ref{[a] prop},
\begin{align*}
Q_\alpha^\lf&=\sum_{\beta\in(\Delta_\lf)_0^+}\int:P_{\alpha,\lf}^{\beta,R}(a^*)(z)a_{\beta}(z)\ \e^{-\frac{1}{\para_i+h_i^{\vee}}\int b^\lf_{\alpha}(z)}:dz &(\alpha\in(\Pi_\lf)_0^i),\\
Q_\alpha^\lf&=\sum_{\beta\in(\Delta_\lf)_{\frac{1}{2}}\cap[\alpha]}\int:P_{\alpha,\lf}^{\beta,R}(a^*)(z)\Phi_\beta^\lf(z)\ \e^{-\frac{1}{\para_i+h_i^{\vee}}\int b^\lf_{\alpha}(z)}:dz &(\alpha\in(\Pi_\lf)^i_{\frac{1}{2}}),\\
Q_\alpha^\lf&=\sum_{\beta\in(\Delta_\lf)_1\cap[\alpha]}(f_\lf|e_{\beta})\int:P_{\alpha,\lf}^{\beta,R}(a^*)(z)\ \e^{-\frac{1}{\para_i+h_i^{\vee}}\int b^\lf_{\alpha}(z)}:dz &(\alpha\in(\Pi_\lf)^i_1),
\end{align*}
and $b^\lf_\alpha(z)=\frac{2}{(\theta_i|\theta_i)}b_\alpha(z)$ for all $\alpha\in(\Pi_\lf)^i$. Then
\begin{align*}
:\e^{-\frac{1}{\para_i+h_i^{\vee}}\int b^\lf_{\alpha}(z)}:\ =\ :\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:
\end{align*}
for all $\alpha\in(\Pi_\lf)^i$ by definition of $\para_i$. Hence, we have
\begin{align*}
Q_\alpha^\lf&=\sum_{\beta\in\Deltazero}\int:P_{\alpha}^{\beta,R}(a^*)(z)a_{\beta}(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in(\Pi_\lf)_0),\\
Q_\alpha^\lf&=\sum_{\beta\in[\alpha]}\int:P_{\alpha}^{\beta,R}(a^*)(z)\Phi_\beta(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in(\Pi_\lf)_{\frac{1}{2}}),\\
Q_\alpha^\lf&=\sum_{\beta\in[\alpha]}\chi(e_{\beta})\int:P_{\alpha}^{\beta,R}(a^*)(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in(\Pi_\lf)_1),
\end{align*}
thanks to Lemma \ref{Ps=P lemma}, Lemma \ref{ind 1/2 lemma} and Lemma \ref{[a]s=[a] lemma}. Therefore $Q_\alpha=Q_\alpha^\lf$ for all $\alpha\in\Pi_\lf$ by Theorem \ref{main thm} and Proposition \ref{[a] prop}. The specialization of inclusion maps
\begin{align*}
\bigcap_{\alpha\in\Pi}\Ker Q_\alpha\hookrightarrow\bigcap_{\alpha\in\Pi_\lf}\Ker Q_\alpha\hookrightarrow\bigcap_{\alpha\in\Pi_0}\Ker Q_\alpha
\end{align*}
induces vertex algebra homomorphisms
\begin{align}\label{para prf eq}
\W^k(\g,f;\Gamma)\rightarrow\W^{{\kappa}_\lf}(\lf,f_\lf;\Gamma_\lf)\rightarrow\Vtau\otimes\Fne.
\end{align}
Let us denote by
\begin{align*}
\Ind^\g_\lf\colon\W^k(\g,f;\Gamma)\rightarrow\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf)
\end{align*}
the first map and by $\mu_\lf$ the second map in \eqref{para prf eq}. Then
\begin{align*}
\mu=\mu_\lf\circ\Ind^\g_\lf,
\end{align*}
where $\mu$, $\mu_\lf$ are the Miura maps for $\W^k(\g,f;\Gamma)$, $\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf)$ respectively by Theorem \ref{w-alg Miura thm}. Since $\mu$ is injective, so is $\Ind^\g_\lf$. Therefore the assertion of the theorem follows for any simple Lie algebra $\g$.

Next, consider arbitrary reductive Lie algebra $\g$. We use the decomposition \eqref{reductive dec eq}. Let $f^i_\lf$ be the image of $f_\lf$ by the projection $\lf\twoheadrightarrow\lf\cap\g^i$ and $\Gamma^i_\lf$ the good grading for $f^i_\lf$ inherited from $\Gamma_\lf$ provided that $\lf\cap\g^i\neq0$. Following the argument in the above, since $\g^i$ is a simple Lie algebra, we have an injective homomorphism
\begin{align*}
\Ind^{\g^i}_{\lf\cap\g^i}\colon\W^{k_i}(\g^i,f^i;\Gamma^i)\rightarrow\W^{\kappa_i}(\lf\cap\g^i,f^i_\lf;\Gamma^i_\lf)
\end{align*}
for all $i\in I_\lf:=\{j\in\{1,\ldots,m\}\mid\lf\cap\g^j\neq0\}$, where $k_i=\kappa(\theta^i|\theta^i)/2$ for the highest root $\theta^i$ in $\g^i$ and
\begin{align*}
\kappa_i(u|v)=k_i(u|v)+\frac{1}{2}\killing_{\g^i}-\frac{1}{2}\killing_{\lf\cap\g^i}(u|v)
\end{align*}
for all $u,v\in\lf\cap\g^i$. Since
\begin{align*}
\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf)=V^\kappa(\z_\g)\otimes\bigotimes_{i\in I_\lf}\W^{\kappa_i}(\lf\cap\g^i,f^i_\lf;\Gamma_\lf^i),
\end{align*}
we have an injective homomorphism
\begin{align*}
\Ind^\g_\lf=\Id_{V^\kappa(\z_\g)}\otimes\bigotimes_{i\in I_\lf}\Ind^{\g^i}_{\lf\cap\g^i}\colon\W^\kappa(\g,f;\Gamma)\rightarrow\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf),
\end{align*}
where $\kappa_\lf=\kappa+\frac{1}{2}\killing_\g-\frac{1}{2}\killing_\lf$. Then $\Ind^\g_\lf$ satisfies $\mu=\mu_\lf\circ\Ind^\g_\lf$ by the property of $\Ind^{\g^i}_{\lf\cap\g^i}$. The proof of the theorem is now complete except for the uniqueness of $\Ind^\g_\rf$. Let $\psi\colon\W^k(\g,f;\Gamma)\rightarrow\W^{{\kappa}_\lf}(\lf,f_\lf;\Gamma_\lf)$ be a vertex algebra homomorphism such that $\mu=\mu_\lf\circ\psi$. Since $\mu_\lf\circ\Ind^\g_\lf=\mu=\mu_\lf\circ\psi$ and $\mu_\lf$ is injective, we have $\Ind^\g_\lf=\psi$. This completes the proof.
\end{proof}

\begin{prop}\label{induced prop}
Let $\g$ be a reductive Lie algebra, $f$ a nilpotent element in $\g$, $\Gamma$ a good grading for $f$ on $\g$. Let $\lf$, $\lf'$ be Levi subalgebras of $\g$ such that $\lf\subset\lf'$ and $\Pi\backslash\Pi_\lf, \Pi\backslash\Pi_{\lf'}\subset\Pi_1$. Then
\begin{align*}
\Ind^\g_\lf=\Ind^{\lf'}_\lf\circ\Ind^\g_{\lf'}.
\end{align*}
\end{prop}
\begin{proof}
Since $\Pi_{\lf'}\backslash\Pi_\lf\subset\Pi\backslash\Pi_\lf\subset\Pi_1$, a map $\Ind^{\lf'}_\lf$ exists by Theorem \ref{induced thm}. By the characterization of $\Ind^\g_\rf$, $\Ind^{\lf'}_\lf$ and $\Ind^\g_{\lf'}$ given in Theorem \ref{induced thm}, $\mu_\lf\circ\Ind^\g_\lf=\mu=\mu_{\lf'}\circ\Ind^\g_{\lf'}=(\mu_\lf\circ\Ind^{\lf'}_\lf)\circ\Ind^\g_{\lf'}$. Since $\mu_\lf$ is injective, $\Ind^\g_\lf=\Ind^{\lf'}_\lf\circ\Ind^\g_{\lf'}$. Therefore the assertion follows.
\end{proof}

If $f$ is a principal nilpotent element in $[\g,\g]$, there exists a (unique) good grading $\Gamma$. Then $\Pi=\Pi_1=\Delta_1$. Let $\lf$ be any Levi subalgebra of $\g$. Then $\Pi_\lf=(\Pi_\lf)_1$ and the principal nilpotent element $f_\lf$ in $[\lf,\lf]$ is chosen by Lemma \ref{ind good lem}. By Theorem \ref{induced thm}, we have an injective homomorphism, which is constructed in \cite{BFN}.

\subsection{Chiralizations}\label{chiral sec}
Let $V$ be any $\frac{1}{2}\Z_{\geq0}$-graded vertex algebra. Denote by
\begin{align*}
A\circ B&=\sum_{j=0}^\infty\binom{\conf(A)}{j}A_{(j-2)}B,\\
A*B&=\sum_{j=0}^\infty\binom{\conf(A)}{j}A_{(j-1)}B
\end{align*}
for $A,B\in V$. Then a vector space $\Zhu(V)=V/(V\circ V)$ has a structure of an associative algebra by the multiplication induced by $*$, called the (twisted) Zhu algebra of $V$ \cite{Z,FZ,DK}. We call $V$ a {\em chiralization} of an associative algebra $\Zhu(V)$. Recall that $\W^\kappa(\g,f;\Gamma)=H^0(C_+,d)$, see Section \ref{W-alg sec}. By \cite{A,DK}, we have
\begin{align}\label{Zhu BRST eq}
\Zhu(H^0(C_+,d))=H^0(\Zhu(C_+),\bar{d}),
\end{align}
where $\bar{d}$ is the differential induced by $d$ such that a complex $(\Zhu(C_+^\bullet),\bar{d})$ defines the finite $\W$-algebra associated with $\g,f,\Gamma$, which we denote by $U(\g.f;\Gamma)$, i.e.
\begin{align*}
\Zhu(\W^\kappa(\g,f;\Gamma))=U(\g.f;\Gamma).
\end{align*}
See e.g. \cite{Lo2,Wan} for the definitions and properties of finite $\W$-algebras. Let $\Zhu(\Fne)=\fne$. Note that $\Zhu(V^\kappa(\g))=U(\g)$ and $\Zhu(C^0_+)=U(\g_{\leq0})\otimes\fne$. The projection $\g_{\leq0}\oplus\fne\twoheadrightarrow\g_0\oplus\fne$ induces an algebra homomorphism
\begin{align*}
\bar{\mu}\colon U(\g,f;\Gamma)\rightarrow U(\g_0\oplus\fne),
\end{align*}
called the {\em Miura map for $U(\g.f;\Gamma)$}. The following result is proved by \cite{Ly} in the case that $\Gamma$ is $\Z$-graded but may be also applied in general case. We give the sketch of the proof, following the proof of Proposition 4 in Section 2.6 in \cite{A} (with slight generalization).

\begin{lemma}\label{classical inj lem}
$\bar{\mu}$ is injective.
\end{lemma}
\begin{proof}
Recall that $\mathcal{S}_f$ is the Slodowy slice through $f$ and is isomorphic to the Marsden-Weinstein quotient of a transversal slice $f+\g_{\geq-\frac{1}{2}}$ in $\g\simeq\g^*$ by $G_{\geq\frac{1}{2}}$ (\cite{GG}). There exists a filtration on $U(\g,f;\Gamma)$, called the Kazhdan filtration, such that the induced map
\begin{align*}
\gr\bar{\mu}\colon\gr U(\g,f;\Gamma)\rightarrow\gr U(\g_0)\otimes\gr\fne.
\end{align*}
can be identified with the restriction map
\begin{align*}
\nu\colon\C[\mathcal{S}_f]=\C[f+\g_{\geq-\frac{1}{2}}]^{G_{\geq\frac{1}{2}}}\rightarrow\C[f+\g_0\oplus\g_{-\frac{1}{2}}].
\end{align*}
We will show that the restriction map $\nu$ is injective. If $P\in\C[\mathcal{S}_f]$ lies in the kernel of $\nu$, $P(g\cdot u)=0$ for all $g\in G_{\geq\frac{1}{2}}$ and $u\in f+\g_{\geq-\frac{1}{2}}$. Hence, it suffices to show that a map
\begin{align*}
\xi\colon G_{\geq\frac{1}{2}}\times(f+\g_0\oplus\g_{-\frac{1}{2}})\rightarrow f+\g_{\geq-\frac{1}{2}}
\end{align*}
defined by $\xi(g,u)=g\cdot u$ is dominant (i.e. the image of $\xi$ is Zariski dense). Let
\begin{align*}
d\xi\colon\g_{\geq\frac{1}{2}}\times\g_0\oplus\g_{-\frac{1}{2}}\rightarrow\g_{\geq-\frac{1}{2}}
\end{align*}
be the differential of $\xi$. Then the differential of $\xi$ at $(1,u)\in G_{\geq\frac{1}{2}}\times(f+\g_0\oplus\g_{-\frac{1}{2}})$ is given by
\begin{align*}
d\xi_{(1,u)}(a,b)=[a,u]+b
\end{align*}
and is an isomorphism if $u\in f+(\g_0\oplus\g_{-\frac{1}{2}})_\mathrm{reg}$, where
\begin{align*}
(\g_0\oplus\g_{-\frac{1}{2}})_\mathrm{reg}=\{v\in\g_0\oplus\g_{-\frac{1}{2}}\mid\g_{\geq\frac{1}{2}}^v=0\},\quad
\g_{\geq\frac{1}{2}}^v=\{w\in\g_{\geq\frac{1}{2}}\mid[v,w]=0\}.
\end{align*}
Hence, $\xi$ is dominant. See e.g. \cite{TY}. This completes the proof.
\end{proof}

Let $V,W$ be any $\frac{1}{2}\Z_{\geq0}$-graded vertex algebras and $\psi\colon V\rightarrow W$ any vertex algebra homomorphism. Since $\psi(V\circ V)=\psi(V)\circ \psi(V)\subset W\circ W$, the map $\psi$ induces an algebra homomorphism
\begin{align*}
\Zhu(\psi)\colon\Zhu(V)\rightarrow\Zhu(W).
\end{align*}
We shall say that $\psi$ is a chiralization of $\Zhu(\psi)$. For $\psi=\mu$, we obtain a map
\begin{align*}
\Zhu(\mu)\colon U(\g,f;\Gamma)\rightarrow U(\g_0)\otimes\fne.
\end{align*}
\begin{lemma}\label{classical Miura lem}
$\Zhu(\mu)=\bar{\mu}$. In particular, $\Zhu(\mu)$ is injective.
\end{lemma}
\begin{proof}
The formula $\Zhu(\mu)=\bar{\mu}$ is induced by \eqref{Zhu BRST eq}. The injectivity of $\Zhu(\mu)$ follows from Lemma \ref{classical inj lem}.
\end{proof}

For any map $\Ind^\g_\lf$ given in Theorem \ref{induced thm}, it induces an algebra homomorphism
\begin{align*}
\Zhu(\Ind^\g_\lf)\colon U(\g,f;\Gamma)\rightarrow U(\lf,f_\lf;\Gamma_\lf)
\end{align*}
such that
\begin{align}\label{mu ind eq}
\Zhu(\mu)=\Zhu(\mu_\lf)\circ\Zhu(\Ind^\g_\lf)
\end{align}
by the characterization of $\Ind^\g_\lf$.
\begin{lemma}\label{Zhu ind lem}
$\Zhu(\Ind^\g_\lf)$ is a unique injective algebra homomorphism that satisfies $\bar{\mu}=\bar{\mu}_\lf\circ\Zhu(\Ind^\g_\lf)$.
\end{lemma}
\begin{proof}
The assertion of the lemma immediately follows from \eqref{mu ind eq} and Lemma \ref{classical Miura lem}.
\end{proof}

Losev constructs an injective algebra homomorphism in \cite{L}
\begin{align}\label{Losev eq}
U(\g,f;\Gamma)\rightarrow\widetilde{U}(\lf,f_\lf;\Gamma_\lf)
\end{align}
if $G\cdot f=\ind^\g_\lf L\cdot{f_\lf}$, where $\widetilde{U}(\lf,f_\lf;\Gamma_\lf)$ is a certain completion of $U(\lf,f_\lf;\Gamma_\lf)$.

\begin{conj}\label{Conj}
Under the condition that $\Pi\backslash\Pi_\lf\subset\Pi_1$, $\Zhu(\Ind^\g_\lf)$ coincides with the map \eqref{Losev eq}.
\end{conj}

The pull-back of Losev's map \eqref{Losev eq} gives a functor from $U(\lf,f_\lf;\Gamma_\lf)$-mod to $U(\g,f;\Gamma)$-mod, called a parabolic induction functor and first introduced by Premet in \cite{P}. Motivated by these results and Conjecture \ref{Conj}, we call a map $\Ind^\g_\lf$ given in Theorem \ref{induced thm} a {\em parabolic induction} for $\W$-algebras, which induces a functor from $\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf)$-mod to $\W^{\kappa}(\g,f;\Gamma)$-mod.

\section{Coproducts}\label{coproduct sec}

In this section, we consider parabolic inductions $\Ind^\g_\lf$ in the case that $\g=\gl_N$ and the case that $\g$ is of type $BCD$ and $f$ is a rectangular nilpotent element. In the former case, we show that $\Ind^\g_\lf$ is a chiralization of a coproduct $\bar{\cop}$ of finite $\W$-algebras of type $A$ constructed by \cite{BK2}, which we call coproducts for $\W$-algebras of type $A$. In the latter case, we give a structure of coproducts on $W$-algebras of type $BCD$ with rectangular nilpotent elements, giving rise to coproducts of twisted Yangians of level $l$.

\subsection{Pyramids}\label{pyramid sec}
To describe good gradings of $\gl_N$ combinatorially, we introduce {\em pyramids}, which are first introduced by \cite{EK} to classify all good gradings of simple (classical) Lie algebras. Following \cite{BK2}, we only consider pyramids corresponding to good $\Z$-gradings of $\gl_N$, which should be called even pyramids but shall call just pyramids.

Let $q=(q_1,\ldots,q_l)$ be a sequence of positive integers and $\Py$ a diagram defined by stacking $q_1$ boxes in the first column, $q_2$ boxes in the second column, $\cdots$, $q_l$ boxes in the right-most column. The diagram $\Py$ is called a pyramid if each row of $\Py$ consists of a single connected strip, i.e. $0\leq {}^\exists t\leq l$ such that $q_1\leq\cdots\leq q_t$ and $q_{t+1}\geq\cdots\geq q_l$. For example,
\vspace{3mm}
\begin{align*}
\setlength{\unitlength}{1mm}
\begin{picture}(0, 0)(20,10)
\put(-30,0){\line(0,1){15}}
\put(-30,0){\line(1,0){15}}
\put(-30,5){\line(1,0){15}}
\put(-30,10){\line(1,0){10}}
\put(-30,15){\line(1,0){5}}
\put(-25,0){\line(0,1){15}}
\put(-20,0){\line(0,1){10}}
\put(-15,0){\line(0,1){5}}
\put(-14,0){,}
\put(-5,0){\line(0,1){10}}
\put(-5,0){\line(1,0){15}}
\put(-5,5){\line(1,0){15}}
\put(-5,10){\line(1,0){10}}
\put(0,0){\line(0,1){15}}
\put(0,15){\line(1,0){5}}
\put(5,0){\line(0,1){15}}
\put(10,0){\line(0,1){5}}
\put(11,0){,}
\put(20,0){\line(0,1){5}}
\put(20,0){\line(1,0){15}}
\put(20,5){\line(1,0){15}}
\put(25,0){\line(0,1){15}}
\put(25,10){\line(1,0){10}}
\put(25,15){\line(1,0){5}}
\put(30,0){\line(0,1){15}}
\put(35,0){\line(0,1){10}}
\put(36,0){,}
\put(45,0){\line(0,1){5}}
\put(45,0){\line(1,0){15}}
\put(45,5){\line(1,0){15}}
\put(50,0){\line(0,1){10}}
\put(50,10){\line(1,0){10}}
\put(55,0){\line(0,1){15}}
\put(55,15){\line(1,0){5}}
\put(60,0){\line(0,1){15}}
\put(61,0){.}
\end{picture}\\
\quad\\
\end{align*}
Fix a pyramid $\Py$. Set the height $n=\max(q_1,\ldots,q_l)$ of $\Py$, the sequence $p=(p_1,\ldots,p_n)$ of length of rows of $\Py$ from top to bottom, and the number $N=\sum_{i=1}^l q_i=\sum_{i=1}^n p_i$ of boxes in $\Py$ (e.g. $p=(1,2,3)$ and $N=6$ in the above examples). We fix a numbering of boxes in $\Py$ by $1,\ldots,N$ from top to bottom and from left to right, and denote by $\row(i)$ the row number of the box in which $i$ appears and by $\col(i)$ the column number similarly. For example,\vspace{5mm}
\begin{align}\label{cop diag}
\
\setlength{\unitlength}{1mm}
\begin{picture}(0, 0)(20,10)
\put(-26,6){$\Py\ =$}
\put(-11,16.5){\footnotesize$\row(i)$}
\put(-8,11){1}
\put(-8,6){2}
\put(-8,1){3}
\put(0,0){\line(0,1){5}}
\put(0,0){\line(1,0){20}}
\put(0,5){\line(1,0){20}}
\put(2,1){1}
\put(5,0){\line(0,1){15}}
\put(5,10){\line(1,0){10}}
\put(5,15){\line(1,0){5}}
\put(7,11){2}
\put(7,6){3}
\put(7,1){4}
\put(10,0){\line(0,1){15}}
\put(12,6){5}
\put(12,1){6}
\put(15,0){\line(0,1){10}}
\put(17,1){7}
\put(20,0){\line(0,1){5}}
\put(21,0){,}
\put(2,-6){1}
\put(7,-6){2}
\put(12,-6){3}
\put(17,-6){4}
\put(23,-5.5){\footnotesize$\col(i)$}
\put(40,11){$p=(\ 1,2,4\ )$,}
\put(40,6){$q=(\ 1,3,2,1\ )$,}
\put(40,0){$N=7$.}
\end{picture}
\end{align}

\vspace{15mm}\hspace{-5mm}We have $\row(4)=3$ and $\col(4)=2$ etc. Let $\{v_i\}_{i=1}^{N}$ be the standard basis of $\C^N$ and $\{e_{i,j}\}_{i,j=1}^N$ the standard basis of $\gl_N=\End(\C^N)$ by $e_{i,j}(v_k)=\delta_{j,k}v_i$. We attach to $\Py$ a nilpotent element $f_\Py$ by
\begin{align*}
f_\Py(v_j)=
\begin{cases}
\ v_i & (\ \row(i)=\row(j)\quad \mathrm{and}\quad \col(i)=\col(j)+1\ ),\\
\ 0 & (\ \mathrm{otherwise}\ )
\end{cases}
\end{align*}
and a $\Z$-grading $\Gamma_\Py$ on $\gl_n$ by $\deg_{\Gamma_\Py}(e_{i,j})=\col(j)-\col(i)$. Then $f_\Py$ has the standard Jordan form consisting of $p_1$ Jordan block, $p_2$ Jordan block, $\cdots$, $p_n$ Jordan block with all the diagonal $0$, and $\Gamma_\Py$ is good for $f_\Py$. Set a Cartan subalgebra $\h=\bigoplus_{i=1}^N\C e_{i,i}$, the dual basis $\{\epsilon_i\}_{i=1}^N$ of $\h^*$ by $\epsilon_i(e_{j,j})=\delta_{i,j}$, and the root system
\begin{align*}
\Delta=\{\epsilon_i-\epsilon_j\in\h^*\mid1\leq i\neq j\leq N\},\quad
\Pi=\{\alpha_i:=\epsilon_i-\epsilon_{i+1}\mid i=1,\ldots,N-1\}
\end{align*}
as usual. Since $\deg_{\Gamma_\Py}(\epsilon_i-\epsilon_j)=\deg_{\Gamma_\Py}(e_{i,j})$, we have
\begin{align}\label{pi deg eq}
\Pi_{0}=\{\alpha_i\in\Pi\mid\row(i)<\row(N)\},\quad
\Pi_{1}=\{\alpha_i\in\Pi\mid\row(i)=\row(N)\}.
\end{align}
In the case of \eqref{cop diag},
\begin{align*}
\quad\\
f_\Py=e_{5,3}+e_{7,6}+e_{6,4}+e_{4,1},\hspace{60mm}
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)(20,10)
\put(-29,10){\circle{2}}
\put(-30,13){\footnotesize$1$}
\put(-30,5){\footnotesize$\alpha_1$}
\put(-28,10.3){\line(1,0){8}}
\put(-19,10){\circle{2}}
\put(-20,13){\footnotesize$0$}
\put(-20,5){\footnotesize$\alpha_2$}
\put(-18,10.3){\line(1,0){8}}
\put(-9,10){\circle{2}}
\put(-10,13){\footnotesize$0$}
\put(-10,5){\footnotesize$\alpha_3$}
\put(-8,10.3){\line(1,0){8}}
\put(1,10){\circle{2}}
\put(0,13){\footnotesize$1$}
\put(0,5){\footnotesize$\alpha_4$}
\put(2,10.3){\line(1,0){8}}
\put(11,10){\circle{2}}
\put(10,13){\footnotesize$0$}
\put(10,5){\footnotesize$\alpha_5$}
\put(12,10.3){\line(1,0){8}}
\put(21,10){\circle{2}}
\put(20,13){\footnotesize$1$}
\put(20,5){\footnotesize$\alpha_6$}
\put(23,9){.}
\end{picture}\\
\hspace{10mm}
\end{align*}
We split $\Py$ into two pyramids $\Py_1$, $\Py_2$ along a column, which we denote by $\Py=\Py_1\oplus\Py_2$. For example,
\begin{align*}
\quad\\
\setlength{\unitlength}{1mm}
\begin{picture}(0, 0)(20,10)
\put(-10,0){\line(0,1){5}}
\put(-10,0){\line(1,0){20}}
\put(-10,5){\line(1,0){20}}
\put(-8,1){1}
\put(-5,0){\line(0,1){15}}
\put(-5,10){\line(1,0){10}}
\put(-5,15){\line(1,0){5}}
\put(-3,11){2}
\put(-3,6){3}
\put(-3,1){4}
\put(0,0){\line(0,1){15}}
\put(2,6){5}
\put(2,1){6}
\put(5,0){\line(0,1){10}}
\put(7,1){7}
\put(10,0){\line(0,1){5}}
\put(13.5,6){$=$}
\put(20,0){\line(0,1){5}}
\put(20,0){\line(1,0){10}}
\put(20,5){\line(1,0){10}}
\put(22,1){1}
\put(25,0){\line(0,1){15}}
\put(25,10){\line(1,0){5}}
\put(25,15){\line(1,0){5}}
\put(27,11){2}
\put(27,6){3}
\put(27,1){4}
\put(30,0){\line(0,1){15}}
\put(33.5,6){$\oplus$}
\put(40,0){\line(0,1){10}}
\put(40,0){\line(1,0){10}}
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\quad\\ \quad\\ \quad\\
\end{align*}
For $i=1,2$, let $N_i$ be the number of boxes in $\Py_i$ and $\lf_i=\gl_{N_i}$ the Lie subalgebra of $\gl_N$ spanned by all $e_{j,j'}$, where $j,j'$ run over numbers labeling $\Py_i$. Then $\Gamma_{\Py_i}$ is a good grading on $\lf_i$ for $f_{\Py_i}$ and is the restriction of $\Gamma_{\Py}$ by construction. Denote by $\Oc_\Py$ a nilpotent orbit in $\gl_N$ through $f_\Py$, by $\Oc_{\Py_i}$ a nilpotent orbit in $\lf_i$ through $f_{\Py_i}$ and by $\lf=\lf_1\oplus\lf_2$ a maximal Levi subalgebra of $\gl_N$. A combinatorial description of induced nilpotent orbits in $\gl_N$ given in \cite{Kr, OW} is compatible with our cases:
\begin{align*}
\Oc_\Py=\ind^{\gl_N}_\lf(\Oc_{\Py_1}+\Oc_{\Py_2}).
\end{align*}

\subsection{Coproducts for type $A$}\label{cop thm sec}
Let $\Py$ be a pyramid consisting of $N$ boxes. Set
\begin{align*}
\W^k(\gl_N,\Py)=V^{k+N}(\z_{\gl_N})\otimes\W^k(\slf_N,f_\Py;\Gamma_\Py).
\end{align*}
Set the subset $\Pi^i$ of $\Pi$ consisting of simple roots in $\gl_{N_i}$. Then
\begin{align*}
\Pi^1=\{\alpha_1,\ldots,\alpha_{N_1-1}\},\quad
\Pi^2=\{\alpha_{N_1+1},\ldots,\alpha_{N-1}\}.
\end{align*}
Therefore $\Pi\backslash(\Pi^1\sqcup\Pi^2)=\{\alpha_{N_1}\}$ and $\deg_{\Gamma_\Py}\alpha_{N_1}=1$ by $\row(N_1)=\row(N)$.

\begin{theorem}\label{coproduct thm}
Let $\Py$ be a pyramid split into $\Py_1\oplus\Py_2$. Set the numbers $N$, $N_1$, $N_2$ of boxes in $\Py$, $\Py_1$, $\Py_2$ respectively ($N=N_1+N_2$). Then there exists an injective vertex algebra homomorphism
\begin{align*}
\cop=\cop^\Py_{\Py_1,\Py_2}\colon\W^k(\gl_N,\Py)\rightarrow\W^{k_1}(\gl_{N_1},\Py_1)\otimes\W^{k_2}(\gl_{N_2},\Py_2)
\end{align*}
for all $k\in\C$, where $k+N=k_1+N_1=k_2+N_2$, such that
\begin{enumerate}
\item $\cop$ is a unique vertex algebra homomorphism that satisfies $\mu=(\mu_1\otimes\mu_2)\circ\cop$, where $\mu$, $\mu_1$, $\mu_2$ are the Miura maps for $\W^k(\gl_N,\Py)$, $\W^{k_1}(\gl_{N_1},\Py_1)$, $\W^{k_2}(\gl_{N_2},\Py_2)$ respectively.
\item $\cop$ is coassociative, i.e.
\begin{align*}
(\Id\otimes\cop^{\Py_2\oplus\Py_3}_{\Py_2,\Py_3})\circ\cop^\Py_{\Py_1,\Py_2\oplus\Py_3}=(\cop^{\Py_1\oplus\Py_2}_{\Py_1,\Py_2}\otimes\Id)\circ\cop^\Py_{\Py_1\oplus\Py_2, \Py_3}
\end{align*}
for $\Py=\Py_1\oplus\Py_2\oplus\Py_3$.
\end{enumerate}
\end{theorem}
\begin{proof}
We use the notations in Section \ref{pyramid sec}. Let $\Pi_\lf$ be the set of simple roots in $\lf=\lf_1\oplus\lf_2$. Since $\{\alpha_{N_1}\}=\Pi\backslash\Pi_\lf\subset\Pi_1$, we may apply Theorem \ref{induced thm} for a nilpotent element $f_\Py$ of $\gl_N$. Hence, we have an injective vertex algebra homomorphism
\begin{align*}
\cop=\cop^\Py_{\Py_1,\Py_2}=\Ind^{\gl_N}_{\lf}\colon\W^k(\gl_N,\Py)\rightarrow\W^{\kappa_\lf}(\lf,f_\lf;\Gamma_\lf)=\W^{k_1}(\gl_{N_1},\Py_1)\otimes\W^{k_2}(\gl_{N_2},\Py_2)
\end{align*}
for all $k\in\C$, where $k+N=k_1+N_1=k_2+N_2$, which satisfies the desired properties by the characterization of $\Ind^{\gl_N}_\lf$ and Proposition \ref{induced prop}. This completes the proof.
\end{proof}

We will call $\cop$ a {\em coproduct} for $\W$-algebras of type $A$.
\smallskip

Let $U(\gl_N,\Py)=U(\gl_N,f_\Py;\Gamma_\Py)$ be the finite $\W$-algebra associated with $\gl_N,f_\Py,\Gamma_\Py$ for a pyramid $\Py$ consisting of $N$ boxes. It is known that $U(\gl_N,\Py)$ is isomorphic to a truncation of a shifted Yangian by \cite{BK2}. Following \cite{BK2}, for any pyramid $\Py$ split into $\Py_1\oplus\Py_2$, we have an injective algebra homomorphism
\begin{align*}
\bar{\cop}=\bar{\cop}^\Py_{\Py_1, \Py_2}\colon U(\gl_N,\Py)\rightarrow U(\gl_{N_1},\Py_1)\otimes U(\gl_{N_2},\Py_2),
\end{align*}
called a coproduct for finite $\W$-algebras of type $A$, where $N_i$ is the number of boxes in $\Py_i$.

\begin{prop}\label{chiral prop}
$\Zhu(\cop)=\bar{\cop}$. Therefore $\cop$ is a chiralization of $\bar{\cop}$.
\end{prop}
\begin{proof}
Let $\Py$ be a pyramid split into $\Py_1\oplus\Py_2$, $\cop$ the corresponding coproduct for $\W$-algebras, $\bar{\cop}$ the corresponding coproduct for finite $\W$-algebras, $N$ the number of boxes in $\Py$, $N_i$ the number of boxes in $\Py_i$, $l$ the column length of $\Py$ and $l_i$ the column length of $\Py_i$ ($l=l_1+l_2$). We split $\Py_i$ into individual columns, i.e.
\begin{align}
\label{chiral eq}\Py_i&=\Py_i^1\oplus\cdots\oplus\Py_i^{l_i},\\
\label{chiral eq2}\Py&=\Py_1^1\oplus\cdots\oplus\Py_1^{l_1}\oplus\Py_2^1\oplus\cdots\oplus\Py_2^{l_2}
\end{align}
such that $\Py_i^j$ has only one column for all $i=1,2$ and $j=1,\ldots,l_i$. By \cite{BK2}, the coproducts of finite $\W$-algebras corresponding to \eqref{chiral eq}, \eqref{chiral eq2} are the Miura maps $\bar{\mu}_i$, $\bar{\mu}$ for $U(\gl_{N_i},\Py_i)$, $U(\gl_N,\Py)$ respectively. By coassociativity of $\bar{\cop}$, it satisfies that
\begin{align*}
\bar{\mu}=(\bar{\mu}_1\otimes\bar{\mu}_2)\circ\bar{\cop},
\end{align*}
which implies that $\bar{\cop}=\Zhu(\cop)$ by Lemma \ref{Zhu ind lem}. This completes the proof.
\end{proof}

\subsection{Coproducts for type $BCD$}\label{cop BCD sec}
Let $N$ be a positive integer and $\g_N=\so_N$ or $\spf_N$. If $\g_N=\spf_N$, we assume that $N$ is even. Recall that all nilpotent orbits in $\g_N$ are classified by orthogonal partitions of $N$ if $\g_N=\so_N$ and by symplectic partitions of $N$ if $\g_N=\spf_N$. See e.g. \cite{CM}. In case of $\so_{2M}$, we mean nilpotent orbits under the group $O_{2M}$ not $SO_{2M}$ here. Let $f$ be a rectangular nilpotent element in $\g_N$, corresponding to a partition $p=(l^n)$ of $N$. A rectangular pyramid with the height $n$ and the width $l$ represents a good grading for $f$ on $\g_N$ in the classification of good gradings of $\g_N$ in \cite{EK}, and we denote by $\Py^+$ if $\g_N=\so_N$ and by $\Py^-$ if $\g_N=\spf_N$. We fix a numbering  of boxes in $\Py^\epsilon$ ($\epsilon=\pm$) by $1,\ldots,M$ from top to bottom and from left to right, by $-1,\cdots,-M$ in central symmetry and by $0$ in the central box if the central box exists, where $M=\lfloor\frac{N}{2}\rfloor$. For example,\\

\begin{align*}
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\quad\\ \quad\\
\end{align*}
Denote a basis of $\C^N$ by $\{v_i,v_{-i}\}_{i=1}^{M}$ if $N=2M$, and by $\{v_i\}_{i=-M}^M$ if $N=2M+1$. Then $\End(\C^N)$ has a basis consisting of all $e_{i,j}$ by $e_{i,j}v_k=\delta_{j,k}v_i$. To describe $f$ from $\Py^\epsilon$ explicitly, we fix a basis of $\g_N$ in $\End(\C^N)$ as follows:
\begin{align*}
\so_{2M+1}:\ &e_{i,j}-e_{-j,-i},\ e_{s,-t}-e_{t,-s},\ e_{-s,t}-e_{-t,s},\ e_{i,0}-e_{0,-i},\ e_{0,-i}-e_{-i,0}\\
\so_{2M}:\ &e_{i,j}-e_{-j,-i},\ e_{s,-t}-e_{t,-s},\ e_{-s,t}-e_{-t,s}\\
\spf_{2M}:\ &e_{i,j}-e_{-j,-i},\ e_{s,-t}+e_{t,-s},\ e_{-s,t}+e_{-t,s},\ e_{i,-i},\ e_{-i,i}
\end{align*}
with $1\leq i,j\leq M$ and $1\leq s<t\leq M$. We attach to $\Py^\epsilon$ a nilpotent element $f_{\Py^\epsilon}$ by
\begin{align}\label{BCD nil eq}
f_{\Py^\epsilon}(v_j)=
\begin{cases}
\ \pm v_i & (\ \row(i)=\row(j)\quad \mathrm{and}\quad \col(i)=\col(j)+1\ ),\\
\ 0 & (\ \mathrm{otherwise}\ ),
\end{cases}
\end{align}
where the sign $\pm$ is chosen such that $f_{\Py^\epsilon}\in\g_N$. We split $\Py^\epsilon$ into three rectangular pyramids along two columns in line symmetry, which we denote by $\Py^\epsilon=\Py_1^\epsilon\oplus\Py_2^\epsilon\oplus\Py_1^\epsilon$, such that $\Py^\epsilon_2$ represents a symmetric partition of $N_2$ if $\g_{N_2}=\so_{N_2}$, or an orthogonal partition of $N_2$ if $\g_{N_2}=\spf_{N_2}$. For example,\\
\begin{align*}
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\quad\\ \quad\\
\end{align*}
Let $N_a$ be the number of boxes in $\Py_a^\epsilon$ ($N=2N_1+N_2$), and $\h=\bigoplus_{i=1}^M\C h_i$ a Cartan subalgebra of $\g_N$, where $h_i=e_{i,i}-e_{-i,-i}$. Set the dual basis $\{\epsilon_i\}_{i=1}^M$ of $\h^*$ by $\epsilon_i(h_j)=\delta_{i,j}$, and a set $\Pi=\{\alpha_i\}_{i=1}^M$ of simple roots by $\alpha_i=\epsilon_i-\epsilon_{i+1}$ for $i=1,\ldots,M-1$ and
\begin{align*}
\alpha_M=
\begin{cases}
\epsilon_M & (\g_N=\so_{2M+1}),\\
2\epsilon_M & (\g_N=\spf_{2M}),\\
\epsilon_{M-1}+\epsilon_{M} & (\g_N=\so_{2M}).
\end{cases}
\end{align*}
Let $\lf=\lf_1\oplus\lf_2$ be a maximal Levi subalgebra of $\g_N$ such that $\{\alpha_i\}_{i=1}^{N_1-1}$ is a set of simple roots in $\lf_1=\gl_{N_1}$, and $\{\alpha_i\}_{i=N_1+1}^{M}$ is a set of simple roots in $\lf_2=\g_{N_2}$. We attach to $\Py_a^\epsilon$ a nilpotent element $f_{\Py_a^\epsilon}$ in $\lf_a$ by the same formula in \eqref{BCD nil eq}, where $i,j$ run over the set of numbers labeling $\Py_a^\epsilon$. By \cite{Ke}, we have
\begin{align*}
\Oc_{\Py^\epsilon}=\ind^{\g_N}_{\lf}(\Oc_{\Py_1^\epsilon}+\Oc_{\Py_2^\epsilon}),
\end{align*}
where $\Oc_{\Py^\epsilon}$ denotes a nilpotent orbit in $\g_N$ through $f_{\Py^\epsilon}$, and $\Oc_{\Py_a^\epsilon}$ denotes a nilpotent orbit in $\lf_a$ through $f_{\Py_a^\epsilon}$. Define a good $\Z$-grading $\Gamma_{\Py^\epsilon}$ on $\g_N$ for $f_{\Py^\epsilon}$ by $\deg_{\Gamma_{\Py^\epsilon}}(e_{i,j})=\col(j)-\col(i)$ and a good $\Z$-grading $\Gamma_{\Py_a^\epsilon}$ on $\lf_a$ for $f_{\Py_a^\epsilon}$ similarly. Then $\Gamma_{\Py_a^\epsilon}$ is the restriction of $\Gamma_{\Py^\epsilon}$ on $\lf_a$, and satisfies that $\{\alpha_{N_1}\}=\Pi\backslash\Pi_\lf\subset\Pi_1$. Let
\begin{align*}
\W^k(\g_N,\Py^\epsilon)&=\W^k(\g_N,f_{\Py^\epsilon};\Gamma_{\Py^\epsilon}),\\
\W^k(\gl_{N_1},\Py_1^\epsilon)&=V^{k+N_1}(\z_{\gl_{N_1}})\otimes\W^k(\slf_{N_1},f_{\Py_1^\epsilon};\Gamma_{\Py_1^\epsilon})
\end{align*}
and
\begin{align*}
\gamma(\epsilon)=
\begin{cases}
1 & (\epsilon=+), \\
2 & (\epsilon=-).
\end{cases}
\end{align*}
Recall that the dual Coxter number of $\g_N$ is $N-2$, $M+1$ if $\g_N=\so_{N}, \spf_{2M}$ respectively. The same proof as we use to prove Theorem \ref{coproduct thm} is applicable to the following.

\begin{theorem}\label{cop BCD thm}
In the above settings, there exists an injective vertex algebra homomorphism
\begin{align*}
\cop^\epsilon\colon\W^k(\g_N,\Py^\epsilon)\rightarrow\W^{k_1}(\gl_{N_1},\Py_1^\epsilon)\otimes\W^{k_2}(\g_{N_2},\Py_2^\epsilon),
\end{align*}
where $k+h^\vee=\gamma(\epsilon)(k_1+N_1)=k_2+h^\vee_2$ and $h^\vee,h^\vee_2$ are the dual Coxter numbers of $\g_N$, $\g_{N_2}$ respectively. Moreover, $\cop^\epsilon$ is a unique vertex algebra homomorphism that satisfies that $\mu=(\mu_1\otimes\mu_2)\circ\cop^\epsilon$, where $\mu$, $\mu_1$, $\mu_2$ are the Miura maps for $\W^k(\g_N,\Py^\epsilon)$, $W^{k_1}(\gl_{N_1},\Py_1^\epsilon)$, $\W^{k_2}(\g_{N_2},\Py_2^\epsilon)$ respectively.
\end{theorem}

Suppose that the height $n$ of $\Py^\epsilon$ is even if $\g_N=\spf_N$. Let $l_a$ be the width of $\Py_a^\epsilon$ ($2l_1+l_2=l=N/n$). According to \cite{BK2} and \cite{Br}, it follows that $U(\gl_{N_1}, f_{\Py_1^\epsilon};\Gamma_{\Py_1^\epsilon})$ is isomorphic to the Yangian $Y_{l_1}(\gl_n)$ of level $l_1$, and $U(\g_N,f_{\Py^\epsilon};\Gamma_{\Py^\epsilon})$ is isomorphic to the twisted Yangian $Y_l^\epsilon(\g_n)$ of level $l$.

\begin{cor}\label{cop BCD cor}
Suppose that the height $n$ of $\Py^\epsilon$ is even if $\g_N=\spf_N$. Then there exists an injective algebra homomorphism
\begin{align*}
\bar{\cop}^\epsilon\colon Y_l^\epsilon(\g_n)\rightarrow Y_{l_1}(\gl_{n})\otimes Y_{l_2}^\epsilon(\g_{n}).
\end{align*}
\end{cor}
\begin{proof}
The assertion of the corollary immediately follows from Theorem \ref{cop BCD thm} and Lemma \ref{Zhu ind lem}.
\end{proof}

\section{Examples}\label{examples sec}

We describe $\cop$ in Theorem \ref{coproduct thm} explicitly in some examples.

\subsection{Principal nilpotent}\label{principal sec}
Let $\Py_\prin$ be a pyramid that represents a principal nilpotent element in $\gl_N$, i.e. a pyramid consisting of one row of $N$ boxes. Set a basis $\{h_i=e_{i,i}\}_{i=1}^N$ of a Cartan subalgebra $\h$ of $\gl_N$ and the associated Heisenberg vertex algebra $\Hi=\Hi^{k+N}(\h)$, in which
\begin{align*}
h_i(z)h_j(w)\sim\frac{(k+N)\delta_{i,j}}{(z-w)^2}
\end{align*}
holds for all $i,j=1,\ldots,N$. Consider fields $W_i(z)$ on $\Hi$ defined by the following formal products
\begin{align*}
:(\hat{\der}+h_1(z))\cdot(\hat{\der}+h_2(z))\cdots(\hat{\der}+h_N(z)):\ =\sum_{i=0}^N W_i(z)\hat{\der}^{N-i},
\end{align*}
where $\hat{\der}$ is defined by $[\hat{\der},h_i(z)]=(k+N-1)\der_z h_i(z)$ for all $i$. Then, a vertex subalgebra of $\Hi$ generated by $W_i(z)$ for all $i=1,\ldots,N$ is isomorphic to the $\W$-algebra $\W^k_N=\W^k(\gl_N,\Py_\prin)$ by \cite{FL} (and \cite{FF4}), which coincides with the image of the Miura map for $\W^k_N$. Let $N_1$, $N_2$ be positive integers such that $N_1+N_2=N$ and $W_i^1(z)$, $W_j^2(z)$ fields on $\Hi$ defined by
\begin{align*}
\sum_{i=0}^{N_1} W_i^1(z)\hat{\der}^{N_1-i}&=\ :(\hat{\der}+h_1(z))\cdot(\hat{\der}+h_2(z))\cdots(\hat{\der}+h_{N_1}(z)):,\\
\sum_{i=0}^{N_2} W_i^2(z)\hat{\der}^{N_2-i}&=\ :(\hat{\der}+h_{N_1+1}(z))\cdot(\hat{\der}+h_{N_1+2}(z))\cdots(\hat{\der}+h_{N}(z)):.
\end{align*}
For $j=1,2$, a vertex subalgebra of $\Hi$ generated by $W_i^j(z)$ for all $i=1,\ldots,N_j$ is isomorphic to $\W^{k_j}_{N_j}$, where $k+N=k_1+N_1=k_2+N_2$. By construction,
\begin{align}\label{principal eq}
\sum_{i=0}^N W_i(z)\hat{\der}^{N-i}=\ :\left(\sum_{i=0}^{N_1} W_i^1(z)\hat{\der}^{N_1-i}\right)\left(\sum_{i=0}^{N_2} W_i^2(z)\hat{\der}^{N_2-i}\right):,
\end{align}
which induces an injective vertex algebra homomorphism
\begin{align*}
\cop\colon\W^k_N\rightarrow\W^{k_1}_{N_1}\otimes\W^{k_2}_{N_2}
\end{align*}
for all $k\in\C$. This map $\cop$ is a coproduct for $\W^k_N$ corresponding to a splitting of a pyramid:
\vspace{3mm}
\begin{align*}
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\end{align*}

\subsection{Rectangular cases}\label{rectangular sec}
We generalize the above construction to the case that $f$ is a rectangular nilpotent element in $\gl_N$. We follow the framework in \cite{AM}. Let $\Py$ be a rectangular pyramid of the width $l$ and the height $n$ and $N=n l$. The target space of the Miura map for $\W^k(\gl_{N},\Py)$ is a tensor vertex algebra $V^\kappa(\gl_n)^{\otimes l}$, where $\kappa$ is defined by
\begin{align*}
\kappa(u|v)=
\begin{cases}
(k+n l)\mathrm{tr}(u v) & (u,v\in\z_{\gl_n}) \\
(k+n(l-1))\mathrm{tr}(u v) & (u,v\in\slf_n).
\end{cases}
\end{align*}
Denote by $u^{(t)}(z)$ a field $u(z)$ on the $t$-th component in $V^\kappa(\gl_n)^{\otimes l}$ for all $u\in\gl_n$ and $t=1,\ldots,l$. Set a fields-valued matrix
\begin{align*}
A_t(z)=\left(e_{j,i}^{(t)}(z)\right)_{i,j=1}^n
\end{align*}
for each $t=1,\ldots,l$. Let $W_{i,j,t}(z)$ be a field on $V^\kappa(\gl_n)^{\otimes l}$ defined by the formal product
\begin{align*}
:(\hat{\der}+A_1(z))\cdot(\hat{\der}+A_2(z))\cdots(\hat{\der}+A_l(z)):\ =\sum_{t=0}^l W_t(z)\hat{\der}^{(l-t)}
\end{align*}
and $W_t(z)=\left(W_{i,j,t}(z)\right)_{i,j=1}^n$, where a product (e.g. $A_i(z)\cdot A_j(z)$ etc) is computed by the usual product of matrices and $\hat{\der}$ is defined by
\begin{align*}
[\hat{\der},A_t(z)]=(k+n(l-1))\der_z A_t(z),\quad
\der_z A_t(z)=\left(\der_z e_{j,i}^{(t)}(z)\right)_{i,j=1}^n.
\end{align*}
Then a vertex subalgebra of $V^\kappa(\gl_n)^{\otimes l}$ generated by $W_{i,j,t}(z)$ for all $i,j=1,\ldots,n$ and $t=1,\ldots,l$ is isomorphic to a $\W$-algebra $\W^k(\gl_{N},\Py)$ by \cite{AM}, and coincides with the image of the Miura map for $\W^k(\gl_{N},\Py)$. For a splitting $\Py=\Py_1\oplus\Py_2$ of $\Py$, let $l_i$ be the width of $\Py_i$ and $N_i=n l_i$. Then $\Py_i$ is a $n\times l_i$ rectangular pyramid. Define fields $W_{i,j,t}^{1}(z)$, $W_{i,j,t}^{2}(z)$ on $V^\kappa(\gl_n)^{\otimes l}$ by
\begin{align*}
\sum_{i=0}^{l_1} W_t^1(z)\hat{\der}^{(l_1-t)}&=\ :(\hat{\der}+A_1(z))\cdot(\hat{\der}+A_2(z))\cdots(\hat{\der}+A_{l_1}(z)):,\\
\sum_{i=0}^{l_2} W_t^2(z)\hat{\der}^{(l_2-t)}&=\ :(\hat{\der}+A_{l_1+1}(z))\cdot(\hat{\der}+A_{l_1+2}(z))\cdots(\hat{\der}+A_{l}(z)):,
\end{align*}
where $W_t^1(z)=\left(W_{i,j,t}^{1}(z)\right)_{i,j=1}^n$ and $W_t^2(z)=\left(W_{i,j,t}^{2}(z)\right)_{i,j=1}^n$. By construction,
\begin{align}
\sum_{t=0}^l W_t(z)\hat{\der}^{(l-t)}=\ :\left(\sum_{i=0}^{l_1} W_t^1(z)\hat{\der}^{(l_1-t)}\right)\left(\sum_{i=0}^{l_2} W_t^2(z)\hat{\der}^{(l_2-t)}\right):,
\end{align}
which induces an injective vertex algebra homomorphism
\begin{align*}
\cop\colon\W^k(\gl_{N},\Py)\rightarrow\W^{k_1}(\gl_{N_1},\Py_1)\otimes\W^{k_2}(\gl_{N_2},\Py_2)
\end{align*}
for all $k\in\C$. This map $\cop$ is a coproduct corresponding to $\Py=\Py_1\oplus\Py_2$.

\subsection{Subregular nilpotent}\label{subregular sec}
Let $\Py$ be the pyramid with the sequence of column lengths $(2,1^{N-2})$. Then the nilpotent element $f_\Py=\sum_{i=2}^{N-1}e_{i+1,i}$ is a subregular nilpotent element in $\gl_N$. We have $(\gl_N)_0=\z_{(\gl_N)_0}\oplus\slf_2$ and $\z_{(\gl_N)_0}=\bigoplus_{i=3}^N\C h_i$, where $h_i=e_{i,i}$. The corresponding $\W$-algebra $\W^k(\gl_N,\Py)$ is then isomorphic to the tensor of the Feigin-Semikatov algebra $\W^{(2)}_{N}$ (\cite{FS}) and the Heisenberg vertex algebra $V^{k+N}(\z_{\gl_N})$ if $k+N\neq0$ (\cite{G}). From now on, we assume that $k+N\neq0$. Let $H(z),Z(z),E(z),F(z)$ be fields on $V^{\tau_k}((\gl_N)_0)=V^{k+N}(\z_{(\gl_N)_0})\otimes V^{k+2}(\slf_2)$ defined by
\begin{align*}
H(z)&=h_1(z)-\frac{1}{N}\sum_{i=1}^N h_i(z),\quad
Z(z)=\sum_{i=1}^N h_i(z),\quad
E(z)=e_{1,2}(z),\\
F(z)&=\ :(\hat{\der}+(h_1-h_N)(z))(\hat{\der}+(h_1-h_{N-1})(z))\cdots(\hat{\der}+(h_1-h_3)(z))e_{2,1}(z):,
\end{align*}
where $\hat{\der}=(k+N-1)\der_z$, which generate a vertex subalgebra of $V^{\tau_k}((\gl_N)_0)$ isomorphic to $\W^k(\gl_N,\Py)$ by \cite{FS}. We split $\Py$ as
\vspace{3mm}
\begin{align}\label{subregular eq}
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\end{align}
\vspace{3mm}

\hspace{-5mm}which we denote by $\Py=\Py_1\oplus\Py_2$. Let $\Zc=V^{k+N}(\z_{(\gl_N)_0})$ and $\Zc_1$ (resp. $\Zc_2$) a vertex subalgebra of $\Zc$ generated by $h_i(z)$ with $i=3,\ldots,N_1$ (resp. $i=N_1+1,\ldots,N$). Set $N_2=N-N_1$. We have $\Zc=\Zc_1\otimes\Zc_2$. Let $H_1(z), Z_1(z), E_1(z), F_1(z)$ be fields on $\Zc_1\otimes V^{k+2}(\slf_2)$ defined by
\begin{align*}
H_1(z)&=h_1(z)-\frac{1}{N_1}\sum_{i=1}^{N_1} h_i(z),\quad
Z_1(z)=\sum_{i=1}^{N_1} h_i(z),\quad
E_1(z)=e_{1,2}(z),\\
F_{1}(z)&=\ :(\hat{\der}+(h_1-h_{N_1})(z))(\hat{\der}+(h_1-h_{N_1-1})(z))\cdots(\hat{\der}+(h_1-h_3)(z))e_{2,1}(z):,
\end{align*}
which generate a vertex subalgebra of $\Zc_1\otimes V^{k+2}(\slf_2)$ isomorphic to $\W^{k_1}(\gl_{N_1},\Py_1)$ by construction, where $k+N=k_1+N_1$. For $i=0,\ldots,N_2$, let $W_i(z)$ be fields on $\Zc_2$ defined by
\begin{align*}
:(\hat{\der}-h_N(z))\cdots(\hat{\der}-h_{N_1+1}(z)):\ =\sum_{i=0}^{N_2}W_i(z)\hat{\der}^{N_2-i}.
\end{align*}
Since an automorphism $\tau$ on $\Zc_2$ defined by $h_i(z)\mapsto -h_{N_1+1+N-i}(z)$ ($i=N_1+1,\ldots,N$) implies the formula
\begin{align*}
\sum_{i=0}^{N_2}\tau(W_i(z))

=&\ :(\hat{\der}+h_{N_1+1}(z))\cdots(\hat{\der}+h_{N}(z)):,
\end{align*}
which is a formal product defining generating fields of the $\W^{k_2}_{N_2}$ introduced in Section \ref{principal sec}, the fields $W_i(z)$ with $i=1,\ldots,N_2$ generate a vertex subalgebra of $\Zc_2$ isomorphic to $\W^{k_2}(\gl_{N_2},\Py_2)$, where $k+N=k_2+N_2$. We have
\begin{align}
\label{subreg eq1}H(z)=&H_1(z)+\frac{N_2}{N N_1}Z_1(z)-\frac{1}{N}W_1(z),\\
\label{subreg eq2}Z(z)=&Z_1(z)+W_1(z),\\
\label{subreg eq3}E(z)=&E_1(z),\\
\label{subreg eq4}F(z)

=&\sum_{i=0}^{N_2}\sum_{j=0}^{N_2-i}\binom{N_2-j}{i}:\left(W_j(z)\hat{\der}^{N_2-j-i}\right)P_i(z)F_1(z):,
\end{align}
where
\begin{align*}
P_0(z)=1,\quad
P_i(z)=:(\hat{\der}-h_1(z))^{i-1}h_1(z):
\end{align*}
for $i=1,\ldots,N_2$. Here, we use the following lemma:

\begin{lemma}
\begin{align*}
\sum_{j=0}^{N_2-i}\binom{N_2-j}{i}W_j(z)\hat{\der}^{N_2-j-i}
=\sum_{j_1<\cdots<j_i}:(\hat{\der}-h_N(z))\overset{j_1}{\check{\cdots}}\ \overset{\cdots}{\cdots}\ \overset{j_i}{\check{\cdots}}(\hat{\der}-h_{N_1+1}(z)):
\end{align*}
for all $i=0,\ldots,N_2$.
\end{lemma}
\begin{proof}
For $1\leq n\leq N_2$, $1\leq j\leq n$ and $1\leq t_1<\cdots<t_{n}\leq N_2$, we define fields $W_j^{n}(u_{t_1},\cdots,u_{t_n})$ on $\Zc_2$ by the following formula:
\begin{align*}
:(\hat{\der}-u_{t_1}(z))\cdots(\hat{\der}-u_{t_n}(z)):
\ =\sum_{j=0}^{n}W_j^{n}(u_{t_1},\cdots,u_{t_n})\hat{\der}^{n-j},
\end{align*}
where $u_i(z)=h_{N-i+1}(z)$. Set $W_0^0(\phi)=0$. The assertion of the lemma is equivalent to the formula
\begin{align}\label{subreg lem eq}
\binom{n-j}{i}W_j^n(u_1,\ldots,u_n)=\sum_{1\leq j_1<\cdots<j_i\leq n}W_j^{n-i}(u_1,\overset{j_1}{\check{\cdots}}\ \overset{\cdots}{\cdots}\ \overset{j_i}{\check{\cdots}},u_n)
\end{align}
for $n=N_2$, where $(i,j)$ run over $\{(i,j)\in\Z^2\mid0\leq i,j,i+j\leq n\}$. We will show the formula \eqref{subreg lem eq} by induction on $n$ and $i+j$. If $n=1$ or $i+j=n$, it is easy to check that the formula \eqref{subreg lem eq} follows. If $n>1$ and $i+j<n$, we have
\begin{align*}
&\binom{n-j}{i}W_j^n(u_1,\ldots,u_n)
=\binom{n-j}{i}\binom{n-j}{i+1}^{-1}\binom{n-j}{i+1}W_j^n(u_1,\ldots,u_n)\\
=&\frac{i+1}{n-i-j}\sum_{1\leq j_1<\cdots<j_{i+1}\leq n}W_j^{n-i-1}(u_1,\overset{j_1}{\check{\cdots}}\ \overset{\cdots}{\cdots}\ \overset{j_{i+1}}{\check{\cdots}},u_n)\\
=&\frac{1}{n-i-j}\sum_{1\leq j_1<\cdots<j_{i}\leq n}\ \sum_{t\neq j_1,\ldots,j_{i}}W_j^{n-i-1}(u_1,\overset{j_1}{\check{\cdots}}\ \overset{\cdots}{\cdots}\ \overset{t}{\check{\cdots}}\ \overset{\cdots}{\cdots}\ \overset{j_{i}}{\check{\cdots}},u_n)\\
=&\sum_{1\leq j_1<\cdots<j_i\leq n}W_j^{n-i}(u_1,\overset{j_1}{\check{\cdots}}\ \overset{\cdots}{\cdots}\ \overset{j_i}{\check{\cdots}},u_n)
\end{align*}
by using our inductive assumptions. This completes the proof.
\end{proof}

Since $h_1(z)=H_1(z)+\frac{1}{N_1}Z_1(z)$ is a field on $\W^{k_1}(\gl_{N_1},\Py_1)$, $:P_i(z)F_{1}(z):$ are fields on $\W^{k_1}(\gl_{N_1},\Py_1)$ for all $i=0,\cdots,N_2$. Hence, the formula \eqref{subreg eq1}--\eqref{subreg eq4} induces an injective vertex algebra homomorphism
\begin{align*}
\cop\colon\W^k(\gl_N,\Py)\rightarrow\W^{k_1}(\gl_{N_1},\Py_1)\otimes\W^{k_2}(\gl_{N_2},\Py_2),
\end{align*}
which is the coproduct corresponding to $\Py=\Py_1\oplus\Py_2$.

\appendix\section{Proof of Theorem \ref{main thm}}\label{proof sec}

We assume that the coordinate on $N_{+}$ is compatible with the decomposition $N_+=G_{>0}\times G_{0}^+$. Recall that $\sigma_\lambda\colon H_\chi(\Anil^T\otimes\Hi_\lambda^T)\xrightarrow{\sim}\Azero^T\otimes\FneT\otimes\Hi_\lambda^T$ is the isomorphism of $\Azero^T\otimes\FneT\otimes\Hi^T$-modules, on which the $\Azero^T\otimes\FneT\otimes\Hi^T$-action is defined by $\sigma=\sigma_0$ in Corollary \ref{coh isom cor}. Recall that $\sigma(d\cdot A)=0$ for all $A\in C_k(\Anil^T\otimes\Hi^T)$.

\begin{lemma}\label{var lem}
$\sigma(\varphi^{\alpha})=0$ for all $\alpha\in\Delta_{>0}$.
\end{lemma}
\begin{proof}
Since
\begin{align*}
\hat{\rho}(e_\beta(z))=a_\beta(z)+\sum_{\gamma\in\Delta_{>\degG\beta}}:P_\beta^\gamma(a^*(z))a_\gamma(z):
\end{align*}
for all $\beta\in\Delta_{>0}$ by Lemma \ref{pol2 lemma},
\begin{align*}
d\cdot a^*_\alpha=\dst\cdot a^*_\alpha=\varphi^\alpha+\sum_{\begin{subarray}{c} \beta\in\Delta_{>0}\\ \degG\beta<\degG\alpha\end{subarray}}:P_\beta^\alpha(a^*)\varphi^\beta:
\end{align*}
for all $\alpha\in\Delta_{>0}$. Since $\sigma(d\cdot a^*_\alpha)=0$, we have
\begin{align}\label{da eq}
\sigma(\varphi^\alpha)+\sum_{\begin{subarray}{c} \beta\in\Delta_{>0}\\ \degG\beta<\degG\alpha\end{subarray}}:\sigma(P_\beta^\alpha(a^*))\sigma(\varphi^\beta):\ =0
\end{align}
for all $\alpha\in\Delta_{>0}$. We will show that $\sigma(\varphi^\alpha)=0$ for $\alpha\in\Delta_{>0}$ by the induction on $\degG\alpha$. If $\degG\alpha=\frac{1}{2}$, it follows that $\sigma(\varphi^\alpha)=0$ by \eqref{da eq}. If $\sigma(\varphi^{\beta})=0$ for all $\degG\beta<\degG\alpha$, we also have $\sigma(\varphi^\alpha)=0$ by \eqref{da eq}. This completes the proof.
\end{proof}

\begin{lemma}\label{a>1 lem}
$\sigma(a_{\alpha})=0$ for all $\alpha\in\Delta_{>1}$.
\end{lemma}
\begin{proof}
Let $\theta$ be the highest root in $\Delta$. If $\degG\theta\leq1$, there is nothing to prove. Assume that $\degG\theta>1$. For $\alpha\in\Delta_{>1}$,
\begin{align*}
\sigma(d\cdot\varphi_\alpha)=\sigma(\dst\cdot\varphi_\alpha)=\sigma(\hat{\rho}(e_\alpha))+\sum_{\beta,\gamma\in\Delta_{>0}}c_{\alpha,\beta}^\gamma:\sigma(\varphi_\gamma)\sigma(\varphi^\beta):\ =\sigma(\hat{\rho}(e_\alpha))
\end{align*}
by Lemma \ref{var lem}. Since $\sigma(d\cdot\varphi_\alpha)=0$, we have
\begin{align}\label{var eq}
\sigma(a_\alpha)+\sum_{\beta\in\Delta_{>\degG\alpha}}:\sigma(P_\alpha^\beta(a^*))\sigma(a_\beta):\ =0
\end{align}
for all $\alpha\in\Delta_{>1}$ by Lemma \ref{pol2 lemma}. If $\degG\alpha=\degG\theta$, $\sigma(a_\alpha)=0$ by \eqref{var eq}. If $\sigma(a_\beta)=0$ for all $\degG\beta>\degG\alpha$, we also have $\sigma(a_\alpha)=0$ by \eqref{var eq}. Therefore the assertion of the lemma follows inductively.
\end{proof}

\begin{lemma}\label{a=1 lem}
$\sigma(a_{\alpha})=-\chi(e_{\alpha})$ for all $\alpha\in\Deltaone$.
\end{lemma}
\begin{proof}
For $\alpha\in\Delta_1$,
\begin{align*}
d\cdot\varphi_\alpha=\dst\cdot\varphi_\alpha+\dchi\cdot\varphi_\alpha=\hat{\rho}(e_\alpha)+\sum_{\beta,\gamma\in\Delta_{>0}}c_{\alpha,\beta}^\gamma:\varphi_\gamma\varphi^\beta:+\chi(e_\alpha).
\end{align*}
Since $\sigma(d\cdot\varphi_\alpha)=0$, we have $\sigma(a_\alpha)=-\chi(e_\alpha)$ for all $\alpha\in\Delta_1$ by Lemma \ref{var lem} and Lemma \ref{a>1 lem}. This completes the proof.
\end{proof}

\begin{lemma}\label{sc1 lemma} For $\alpha\in\Delta_1$,
\begin{align*}
\sigma(\hat{\rho}^R(e_\alpha))=-\sum_{\beta\in\Delta_1}\chi(e_\beta)P_\alpha^{\beta,R}(a^*).
\end{align*}
\end{lemma}
\begin{proof}
By Lemma \ref{a>1 lem} and Lemma \ref{a=1 lem},
\begin{align*}
\sigma(\hat{\rho}^R(e_\alpha))=\sum_{\beta\in\Delta_{\geq1}}:\sigma(P_\alpha^{\beta,R}(a^*))\sigma(a_\beta):\ =-\sum_{\beta\in\Delta_{1}}\chi(e_\beta)\sigma(P_\alpha^{\beta,R}(a^*))
\end{align*}
for all $\alpha\in\Delta_1$. Since $P_\alpha^{\beta,R}(a^*)\in\Azero$ for all $\alpha,\beta\in\Delta_1$ by Lemma \ref{pol1 lemma}, $\sigma(P_\alpha^{\beta,R}(a^*))=P_\alpha^{\beta,R}(a^*)$. Therefore the assertion follows.
\end{proof}

\begin{lemma}\label{sc1/2 lemma}
For $\alpha\in\Deltahalf$,
\begin{align*}
\sigma(\hat{\rho}^R(e_\alpha))=-\sum_{\beta\in\Deltahalf}:P_\alpha^{\beta,R}(a^*)\Phi_\beta:.
\end{align*}
\end{lemma}
\begin{proof}
Since $\degG P_\alpha^\gamma(x)=\degG P_\alpha^{\gamma,R}(x)=\frac{1}{2}$ for all $\alpha\in\Deltahalf$ and $\gamma\in\Delta_1$, there exist $P_{\alpha,\beta}^\gamma(x), P_{\alpha,\beta}^{\gamma,R}(x)\in\C[G^+_0]$ for all $\beta\in\Deltahalf$ such that
\begin{align*}
P_\alpha^\gamma(x)=\sum_{\beta\in\Deltahalf}P_{\alpha,\beta}^\gamma(x)x_\beta,\quad
P_\alpha^{\gamma,R}(x)=\sum_{\beta\in\Deltahalf}P_{\alpha,\beta}^{\gamma,R}(x)x_\beta
\end{align*}
for all $\alpha\in\Deltahalf$ and $\gamma\in\Delta_1$. We also have $P_{\alpha,\beta}^\gamma(x)\in\C[G_{>0}]\cap\C[G^+_0]=\C$ by Lemma \ref{pol1 lemma}. Denote by $\lambda_{\alpha,\beta}^\gamma=P_{\alpha,\beta}^\gamma(x)\in\C$. Then
\begin{align*}
\rho(e_\alpha)&=\der_\alpha+\sum_{\substack{\beta\in\Deltahalf \\ \gamma\in\Delta_1}}\lambda_{\alpha,\beta}^{\gamma}x_\beta\der_\gamma+\sum_{\gamma\in\Delta_{>1}}P_\alpha^\gamma(x)\der_\gamma,\\
\rho^R(e_\alpha)&=\sum_{\beta\in\Deltahalf}P_\alpha^{\beta,R}(x)\der_\beta+\sum_{\substack{\beta\in\Deltahalf \\ \gamma\in\Delta_1}}P_{\alpha,\beta}^{\gamma,R}(x)x_\beta\der_\gamma+\sum_{\gamma\in\Delta_{>1}}P_\alpha^{\gamma,R}(x)\der_\gamma
\end{align*}
for $\alpha\in\Deltahalf$. Since $\rho(e_\alpha)$ and $\rho^R(e_{\alpha'})$ commute,
\begin{align*}
0=[\rho(e_\alpha),\rho^R(e_{\alpha'})]=\sum_{\gamma\in\Delta_1}(P_{\alpha',\alpha}^{\gamma,R}(x)-\sum_{\beta\in\Deltahalf}\lambda^{\gamma}_{\alpha,\beta}P_{\alpha'}^{\beta,R}(x))\der_\gamma+\sum_{\gamma\in\Delta_{>1}}(\cdots)\der_\gamma
\end{align*}
for all $\alpha,\alpha'\in\Deltahalf$, where $(\cdots)$ denotes some polynomials in $\C[N_+]$. Hence,
\begin{align*}
P_{\alpha,\beta}^{\gamma,R}(x)=\sum_{\beta'\in\Deltahalf}\lambda^{\gamma}_{\beta,\beta'}P_{\alpha}^{\beta',R}(x)
\end{align*}
for all $\beta\in\Deltahalf$ and $\gamma\in\Delta_1$. Therefore,
\begin{multline}\label{1/2 eq2}
\sigma(\hat{\rho}^R(e_\alpha))

=\sum_{\beta\in\Deltahalf}:\sigma(P_\alpha^{\beta,R}(a^*))\Biggl(\sigma(a_\beta)-\sum_{\substack{\beta'\in\Deltahalf\\ \gamma\in\Delta_1}}\chi(e_\gamma)\lambda_{\beta',\beta}^{\gamma}\sigma(a^*_{\beta'})\Biggr):
\end{multline}
for all $\alpha\in\Deltahalf$ by Lemma \ref{a>1 lem} and Lemma \ref{a=1 lem}. Next,
\begin{align*}
\sigma(d\cdot\varphi^\alpha)=\sigma(\dst\cdot\varphi^\alpha)+\sigma(\dne\cdot\varphi^\alpha)=\sigma(a_\alpha)-\sum_{\substack{\beta\in\Deltahalf \\ \gamma\in\Delta_1}}\chi(e_\gamma)\lambda_{\alpha,\beta}^{\gamma}\sigma(a^*_\beta)+\sigma(\Phi_\alpha)
\end{align*}
for $\alpha\in\Deltahalf$ by Lemma \ref{var lem} and Lemma \ref{a=1 lem}. Since $\sigma(d\cdot\varphi^\alpha)=0$, we have
\begin{align}\label{1/2 eq3}
\sigma(\Phi_\alpha)=-\sigma(a_\alpha)+\sum_{\substack{\beta\in\Deltahalf \\ \gamma\in\Delta_1}}\chi(e_\gamma)\lambda_{\alpha,\beta}^{\gamma}\sigma(a^*_\beta).
\end{align}
By using the formula $[\rho(e_\alpha),\rho(e_\alpha')]-\rho([e_\alpha,e_{\alpha'}])=0$, we obtain that
\begin{align*}
\sum_{\gamma\in\Delta_1}(\lambda_{\alpha',\alpha}^\gamma-\lambda_{\alpha,\alpha'}^\gamma-c_{\alpha,\alpha'}^\gamma)\der_\gamma+\sum_{\gamma\in\Delta_{>1}}(\cdots)\der_\gamma=0
\end{align*}
for all $\alpha,\alpha'\in\Deltahalf$, where $c_{\alpha,\alpha'}^\gamma\in\C$ is the structure constant. Hence,
\begin{align}\label{1/2 eq4}
\lambda_{\beta,\alpha}^\gamma=\lambda_{\alpha,\beta}^\gamma+c_{\alpha,\beta}^\gamma
\end{align}
for all $\beta\in\Deltahalf$ and $\gamma\in\Delta_1$. Therefore,
\begin{align}\label{1/2 eq5}
\sigma(\hat{\Phi}_\alpha)
=\sigma(\Phi_\alpha)+\sum_{\beta\in\Deltahalf}\chi([e_\alpha,e_\beta])\sigma(a^*_\beta)
=-\sigma(a_\alpha)+\sum_{\substack{\beta\in\Deltahalf \\ \gamma\in\Delta_1}}\chi(e_\gamma)\lambda_{\beta,\alpha}^{\gamma}\sigma(a^*_\beta)
\end{align}
by \eqref{1/2 eq3} and \eqref{1/2 eq4}. Finally,
\begin{align*}
\sigma(\hat{\rho}^R(e_\alpha))=-\sum_{\beta\in\Deltahalf}:\sigma(P_\alpha^{\beta,R}(a^*))\sigma(\hat{\Phi}_\beta):
\end{align*}
for all $\alpha\in\Deltahalf$ by \eqref{1/2 eq2} and \eqref{1/2 eq5}. Since $\sigma(\hat{\Phi}_\beta)=\Phi_\beta$ and $\sigma(P_\alpha^{\beta,R}(a^*))=P_\alpha^{\beta,R}(a^*)$ for all $\alpha,\beta\in\Deltahalf$ by Lemma \ref{pol1 lemma}, the assertion of the lemma follows. This completes the proof.
\end{proof}

\hspace{-5mm}{\em Proof of Theorem \ref{main thm}}\quad Recall from Lemma \ref{screening sigma lemma} that $Q_\alpha$ is the intertwining operator induced by $S_\alpha$ through the functor $H_\chi(?)$ and satisfies that
\begin{align*}
Q_\alpha

&=\int:\sigma(\hat{\rho}(e_\alpha))(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz
\end{align*}
for all $\alpha\in\Pi$. Hence,
\begin{align*}
Q_\alpha&=\sum_{\beta\in\Deltazero}\int:P_{\alpha}^{\beta,R}(a^*)(z)a_{\beta}(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_0),\\
Q_\alpha&=-\sum_{\beta\in\Deltahalf}\int:P_{\alpha}^{\beta,R}(a^*)(z)\Phi_\beta(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_{\frac{1}{2}}),\\
Q_\alpha&=-\sum_{\beta\in\Deltaone}\chi(e_{\beta})\int:P_{\alpha}^{\beta,R}(a^*)(z)\ \e^{-\frac{1}{\para+h^{\vee}}\int b_{\alpha}(z)}:dz &(\alpha\in\Pi_1).
\end{align*}
by Lemma \ref{sc0 lemma}, Lemma \ref{sc1 lemma} and Lemma \ref{sc1/2 lemma}. Since $\Ker Q_\alpha=\Ker(-Q_\alpha)$, we may replace $Q_\alpha$ by $-Q_\alpha$ for all $\alpha\in\Pi_{>0}$. This completes the proof.

\section{Proof of Lemma \ref{mod prop}}\label{mod prop sec}

We assume that the coordinate on $N_+$ is compatible with the decomposition $N_+=G_{>0}\times G_0^+$.
\begin{lemma}\label{mod sub1 lemma}
Let $\alpha\in\Pi_0$ and $\beta, \gamma\in\Delta_{>0}$ such that $[\beta]=[\gamma]$. Then
\begin{align*}
\der_\gamma P^\beta_\alpha(x)=c_{\gamma,\alpha}^\beta,\quad
\der_\gamma Q^\beta_\alpha(x)=c_{\gamma,-\alpha}^\beta.
\end{align*}
\end{lemma}
\begin{proof}
Let $\alpha\in\Pi_0$, $\gamma\in\Delta_{>0}$ and $n=\degG\gamma>0$. Then
\begin{align*}
\rho(e_\alpha)=\sum_{\sigma\in\Delta_+}P_\alpha^\sigma(x)\der_\sigma
\end{align*}
by Lemma \ref{pol2 lemma}. Since $[\rho(e_\gamma),\rho(e_\alpha)]=\sum_{\beta\in[\gamma]}c_{\gamma,\alpha}^\beta\rho(e_\beta)$,
we have
\begin{align*}
\sum_{\tau\in\Delta_+}&\der_\gamma P_\alpha^\tau(x)\der_\tau
+\sum_{\sigma\in\Delta_{>n}}\sum_{\tau\in\Delta_+}\left(P^\sigma_\gamma(x)\cdot\der_\sigma P_\alpha^\tau(x)\der_\tau-\der_\tau P_\gamma^\sigma(x)\cdot P_\alpha^\tau(x)\der_\sigma\right)\\
=&\sum_{\beta\in[\gamma]}c_{\gamma,\alpha}^\beta\left(\der_\beta+\sum_{\sigma\in\Delta_{>n}}P_\beta^\sigma(x)\der_\sigma\right).
\end{align*}
For $\beta\in[\gamma]$, comparing with the coefficients of $\der_\beta$ in the above, we have
\begin{align*}
\der_\gamma P_\alpha^\beta(x)+\sum_{\sigma\in\Delta_{>n}}P_\gamma^\sigma(x)\cdot\der_\sigma P_\alpha^\beta(x)=c_{\gamma,\alpha}^\beta.
\end{align*}
Here, we use $\degG\beta=\degG\gamma=n$. Since $\degG P_\alpha^\beta(x)=\degG\beta-\degG\alpha=n$ by \eqref{polG eq}, we have $\der_\sigma P_\alpha^\beta(x)=0$ for all $\sigma\in\Delta_{>n}$. Therefore $\der_\gamma P_\alpha^\beta(x)=c_{\gamma,\alpha}^\beta$.

Next, we apply the formula $\rho(f_\alpha)=\sum_{\tau\in\Delta_+}Q_\alpha^\tau(x)\der_\tau$ to the formula $[\rho(e_\gamma),\rho(f_\alpha)]=\sum_{\beta\in[\gamma]}c_{\gamma,-\alpha}^\beta\rho(e_\beta)$. We have
\begin{align*}
\sum_{\tau\in\Delta_+}&\der_\gamma Q_\alpha^\tau(x)\der_\tau
+\sum_{\sigma\in\Delta_{>n}}\sum_{\tau\in\Delta_+}\left(P^\sigma_\gamma(x)\cdot\der_\sigma Q_\alpha^\tau(x)\der_\tau-\der_\tau P_\gamma^\sigma(x)\cdot Q_\alpha^\tau(x)\der_\sigma\right)\\
=&\sum_{\beta\in[\gamma]}c_{\gamma,-\alpha}^\beta\left(\der_\beta+\sum_{\upsilon\in\Delta_{>n}}P_\beta^\upsilon(x)\der_\upsilon\right).
\end{align*}
Similarly to the proof of the first assertion, we have
\begin{align*}
\der_\gamma Q_\alpha^\beta(x)+\sum_{\sigma\in\Delta_{>n}}P_\gamma^\sigma(x)\der_\sigma Q_\alpha^\beta(x)=c_{\gamma,-\alpha}^\beta
\end{align*}
for all $\beta\in[\gamma]$. Since $\degG Q_\alpha^\beta(x)=\degG\beta+\degG\alpha=n$, it follows that $\der_\sigma Q_\alpha^\beta(x)=0$ for all $\sigma\in\Delta_{>n}$. Hence, $\der_\gamma Q_\alpha^\beta(x)=c_{\gamma,-\alpha}^\beta$.
\end{proof}

\begin{lemma}\label{mod sub2 lemma}
For $\alpha,\beta\in\Pi$,
\begin{align*}
[\rho(f_\alpha),\rho^R(e_\beta)]=(\alpha|\beta)x_\alpha\cdot\rho^R(e_\beta).
\end{align*}
\end{lemma}
\begin{proof}
Recall that $G^\circ=p^{-1}(U)=N_+\cdot B_-$ as in Section \ref{aff Wak sec}. Set smooth curves $\gamma_1(t)=\exp(-t f_\alpha)$ and $\gamma_2(t)=\exp(-t e_\beta)$ on $G$. Given $X\in G^\circ$,
\begin{align*}
\gamma_1(t)X=Z_+(t)Z_-(t),\quad
X\gamma_2(t)=Z_+^R(t)
\end{align*}
for $|t|\ll1$, where $Z_{+}(t), Z_{+}^R(t)\in N_{+}$ and $Z_{-}(t)\in B_{-}$. The vector fields $\zeta_{f_\alpha}, \zeta^R_{e_\beta}$ are then given by the formulae
\begin{align*}
(\zeta_{f_\alpha} f)(p(X))=\frac{d}{d t}f(Z_{+}(t))|_{t=0},\quad
(\zeta^R_{e_\beta} f)(p(X))=\frac{d}{d t}f(Z^R_{+}(t))|_{t=0}
\end{align*}
for any smooth function $f$ defined in a open subset in $U$ around $p(X)$. Choose a faithful representation $V_0$ of $\g$ and consider $X\in N_+$ as a matrix in $\GL(V_0)$ whose entries are polynomials in $\C[N_+]$. We have
\begin{align*}
(1-t f_\alpha)X=Z_{+}(t) Z_{-}(t),\quad
X(1-t e_\beta)=Z^R_{+}(t)\quad\mod.\ (t^2).
\end{align*}
Hence $Z_{+}(t)=X+t Z$, $Z_+^R(t)=X+t Z^R$ and $Z_{-}=1+t Z'$ modulo $(t^2)$, where $Z, Z^R\in\nil_{+}$ and $Z'\in\bo_{-}$. Therefore,
\begin{align}\label{zeta eq}
\zeta_{f_\alpha}\cdot X=-X(X^{-1}f_\alpha X)_+,\quad
\zeta^R_{e_\beta}\cdot X=-X e_\beta,
\end{align}
where $(\cdot)_+\colon\g=\nil_+\oplus\bo_-\rightarrow\nil_+$ is the first projection. We have $-(X^{-1}f_\alpha X)_{\leq0}=Z'$, where $(\cdot)_{\leq0}\colon\g=\nil_+\oplus\bo_-\rightarrow\bo_-$ is the second projection. Since
\begin{align*}
\left(\sum_{n=0}^\infty\frac{1}{n!}(\ad(-x_\beta e_\beta))^n(f_\alpha)\right)_{\leq0}=f_\alpha-x_\beta([e_\beta,f_\alpha])_{\leq0}=f_\alpha-x_\alpha h_\alpha\delta_{\alpha,\beta}
\end{align*}
for all $\beta\in\Delta_+$, we have $(X^{-1}f_\alpha X)_{\leq0}=f_\alpha-x_\alpha h_\alpha$. Therefore
\begin{align}\label{Z' eq}
Z'=-f_\alpha+x_\alpha h_\alpha.
\end{align}
To compute the commutation relation between $\zeta_{f_\alpha}$ and $\zeta^R_{e_\beta}$, we compute $\zeta_{f_\alpha}\circ\zeta^R_{e_\beta}(X)$ and $\zeta^R_{e_\beta}\circ\zeta_{f_\alpha}(X)$. First,
\begin{align}\label{zetaRL eq}
\zeta^R_{e_\beta}\circ\zeta_{f_\alpha}(X)=-(\zeta_{f_\alpha}\cdot X)e_\beta
\end{align}
by \eqref{zeta eq}. Next,
\begin{align*}
\zeta_{f_\alpha}\circ\zeta^R_{e_\beta}(X)=\left.\frac{\der^2}{\der t_1\der t_2}(p(\gamma_1(t_1)X\gamma_2(t_2)))\right|_{t_1=t_2=0}.
\end{align*}
By using the Baker-Campbell-Hausdorff formula, we obtain that
\begin{align*}
\gamma_1(t_1)X\gamma_2(t_2)

=&Z_+(t_1)\exp(-t_2(1+t_1(\alpha|\beta)x_\alpha)e_\beta)Z_{\leq0}(t_1,t_2)\quad\mod.\ (t^2)
\end{align*}
for $|t_1|,|t_2|\ll1$ with some $Z_{\leq0}(t_1,t_2)\in B_-$. Here, we use $[Z', e_\beta]=\delta_{\alpha,\beta}h_\alpha+(\alpha|\beta)x_\alpha e_\beta$ by \eqref{Z' eq}. Therefore,
\begin{align*}
\zeta_{f_\alpha}\circ\zeta^R_{e_\beta}(X)
=&\left.\frac{\der^2}{\der t_1\der t_2}\Bigl(Z_+(t_1)\bigl(1-t_2(1+t_1(\alpha|\beta)x_\alpha)e_\beta\bigr)\Bigr)\right|_{t_1=t_2=0}\\

=&\zeta^R_{e_\beta}\circ\zeta_{f_\alpha}(X)+(\alpha|\beta)x_\alpha(\zeta^R_{e_\beta}\cdot X).
\end{align*}
by \eqref{zeta eq} and \eqref{zetaRL eq}. Hence, $[\zeta_{f_\alpha},\zeta^R_{e_\beta}]=(\alpha|\beta)x_\alpha\cdot\zeta^R_{e_\beta}$. Thus, $[\rho(f_\alpha),\rho^R(e_\beta)]=(\alpha|\beta)x_\alpha\cdot\rho^R(e_\beta)$. This completes the proof.
\end{proof}

\begin{lemma}\label{mod sub lemma}
Let $\alpha\in\Pi_0$, $\alphap\in\Pi_{>0}$ and $\beta\in[\alphap]$. Then
\begin{align*}
(1)\ \sum_{\gamma\in\Deltazero}P_\alpha^\gamma(x)\der_\gamma P_{\alphap}^{\beta,R}(x)&=\sum_{\gamma\in[\alphap]}c_{\gamma,\alpha}^\beta P_{\alphap}^{\gamma,R}(x),\\
(2)\ \sum_{\gamma\in\Deltazero}Q_\alpha^\gamma(x)\der_\gamma P_{\alphap}^{\beta,R}(x)&=\sum_{\gamma\in[\alphap]}c_{\gamma,-\alpha}^\beta P_{\alphap}^{\gamma,R}(x)+(\alpha|\alphap)x_{\alpha}P_{\alphap}^{\beta,R}(x).
\end{align*}
\end{lemma}
\begin{proof}
Let $\alpha\in\Pi_0$, $\alphap\in\Pi_{>0}$. Since $[\rho(e_\alpha),\rho^R(e_{\alphap})]=0$, we have
\begin{align*}
\sum_{\beta,\gamma\in\Delta_+}\left(P_\alpha^\gamma(x)\cdot\der_\gamma P_{\alphap}^{\beta,R}(x)-P_{\alphap}^{\gamma,R}(x)\cdot\der_\gamma P_\alpha^\beta(x)\right)\der_\beta=0.
\end{align*}
Hence
\begin{align}\label{mod sub eq1}
\sum_{\gamma\in\Delta_+}P_\alpha^\gamma(x)\cdot\der_\gamma P_{\alphap}^{\beta,R}(x)=\sum_{\gamma\in\Delta_+}P_{\alphap}^{\gamma,R}(x)\cdot\der_\gamma P_\alpha^\beta(x)
\end{align}
for all $\beta\in\Delta_+$. Let $\beta\in[\alphap]$. We assume that $P_{\alphap}^{\gamma,R}(x)\cdot\der_\gamma P_\alpha^\beta(x)\neq0$. Then
\begin{align*}
\degG P_{\alphap}^{\gamma,R}(x)&=\degG\gamma-\degG\alphap\geq0,\\
\degG\der_\gamma P_\alpha^\beta(x)&=\degG\beta-\degG\alpha-\degG\gamma=\degG\alphap-\degG\gamma\geq0.
\end{align*}
Hence, $\degG\gamma=\degG\alphap$, which implies that $\gamma\in[\alphap]$ by Proposition \ref{[a] prop}. Since $P_\alphap^{\beta,R}(x)\in\C[G_0^+]$ by Lemma \ref{pol1 lemma}, $\der_\gamma P_\alphap^{\beta,R}(x)=0$ for all $\gamma\in\Delta_{>0}$. Hence,
\begin{align*}
\sum_{\gamma\in\Deltazero}P_\alpha^\gamma(x)\cdot\der_\gamma P_{\alphap}^{\beta,R}(x)=\sum_{\gamma\in[\alphap]}P_{\alphap}^{\gamma,R}(x)\cdot\der_\gamma P_\alpha^\beta(x)=\sum_{\gamma\in[\alphap]}c_{\gamma,\alpha}^\beta P_{\alphap}^{\gamma,R}(x)
\end{align*}
for all $\beta\in[\alphap]$ by \eqref{mod sub eq1} and Lemma \ref{mod sub1 lemma}. Therefore the assertion of (1) follows.

Next, since $[\rho(f_\alpha),\rho^R(e_\alphap)]=(\alpha|\alphap)x_\alpha\cdot\rho^R(e_\alphap)$ by Lemma \ref{mod sub2 lemma}, we have
\begin{align*}
\sum_{\beta,\gamma\in\Delta_+}\left(Q_\alpha^\gamma(x)\cdot\der_\gamma P_{\alphap}^{\beta,R}(x)-P_{\alphap}^{\gamma,R}(x)\cdot\der_\gamma Q_\alpha^\beta(x)\right)\der_\beta=(\alpha|\alphap)x_\alpha\sum_{\beta\in\Delta_+}P_{\alphap}^{\beta,R}(x)\der_\beta.
\end{align*}
Hence
\begin{align*}
\sum_{\gamma\in\Delta_+}Q_\alpha^\gamma(x)\cdot\der_\gamma P_{\alphap}^{\beta,R}(x)=\sum_{\gamma\in\Delta_+}P_{\alphap}^{\gamma,R}(x)\cdot\der_\gamma Q_\alpha^\beta(x)+(\alpha|\alphap)x_\alpha P_{\alphap}^{\beta,R}(x)
\end{align*}
for all $\beta\in\Delta_+$. Similarly to the proof of the assertion of (1),
\begin{align*}
\sum_{\gamma\in\Deltazero}Q_\alpha^\gamma(x)\cdot\der_\gamma P_{\alphap}^{\beta,R}(x)

&=\sum_{\gamma\in[\alphap]}c_{\gamma,-\alpha}^\beta P_{\alphap}^{\gamma,R}(x)+(\alpha|\alphap)x_\alpha P_{\alphap}^{\beta,R}(x)
\end{align*}
for all $\beta\in[\alphap]$ by Lemma \ref{mod sub1 lemma}. Therefore the assertion of (2) follows. This completes the proof.
\end{proof}

\hspace{-5mm}{\em Proof of Lemma \ref{mod prop}}\quad Let $\alphap\in\Pi_{>0}$ and $\beta\in[\alphap]$. For $u,v\in\rf$,
\begin{align*}
[u(z),v(w)]=[u,v](w)\delta(z-w)+\tau_\para(u|v)\der_w\delta(z-w).
\end{align*}
on $\VtauT$. Hence,
\begin{align}\label{mod prf eq1}
[[u(y),v(z)],V^\beta(w)]=[[u,v](z),V^\beta(w)]\delta(y-z).
\end{align}
If $u(z)$ and $v(z)$ satisfy \eqref{mod eq},
\begin{align*}
[[u(y),v(z)],V^\beta(w)]

=&\sum_{\gamma,\sigma\in[\alphap]}(c_{\gamma,u}^\sigma c_{\sigma,v}^\beta-c_{\gamma,v}^\sigma c_{\sigma,u}^\beta)V^\gamma(w)\delta(y-w)\delta(z-w).
\end{align*}
First, we have $\sum_{\sigma\in[\alphap]}(c_{\gamma,u}^\sigma c_{\sigma,v}^\beta-c_{\gamma,v}^\sigma c_{\sigma,u}^\beta)=c_{\gamma,[u,v]}^\beta$ for all $\gamma\in[\alphap]$ by the Jacobi identity:
\begin{align*}
[e_\gamma,[u,v]]=[[e_\gamma,u],v]-[[e_\gamma,v],u].
\end{align*}
Next, by using the formula $\delta(y-w)\delta(z-w)=\delta(y-z)\delta(z-w)$, we have
\begin{align}\label{mod prf eq2}
[[u(y),v(z)],W^\beta(w)]=\sum_{\gamma\in[\alphap]}c_{\gamma,[u,v]}^\beta W^\gamma(w)\delta(y-z)\delta(z-w).
\end{align}
Hence, combining \eqref{mod prf eq1} with \eqref{mod prf eq2} and computing the residue at $y=0$, we obtain that
\begin{align*}
[[u,v](z),W^\beta(w)]=\sum_{\gamma\in[\alphap]}c_{\gamma,[u,v]}^\beta W^\gamma(w)\delta(z-w).
\end{align*}
Therefore $[u,v](z)$ satisfies \eqref{mod eq}. Thus, it saffices to show that \eqref{mod eq} follows for $u=e_\alpha, h_{\alpha'},f_\alpha$ for $\alpha\in\Pi_0$, $\alpha'\in\Pi$. Recall that $u(z)=\hat{\rho}_\rf(u(z))$ is defined by \eqref{para wak eq}. First, we consider the case that $u=h_\alpha$. Then
\begin{align*}
[h_\alpha(z),V^\beta(w)]

=&-\sum_{\gamma\in\Deltazero}\gamma(h_\alpha):a^*_\gamma(z)[a_\gamma(z),P_{\alphap}^{\beta,R}(a^*)(w)]\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alphap}(w)}:\\
&+:P_{\alphap}^{\beta,R}(a^*)(w)[b_\alpha(z),\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alphap}(w)}]:.
\end{align*}
Recall that $A(z)\delta(z-w)=A(w)\delta(z-w)$ for any field $A(z)$. By \eqref{a der eq},
\begin{align*}
\sum_{\gamma\in\Deltazero}\gamma(h_\alpha):a^*_\gamma(z)[a_\gamma(z),P_{\alphap}^{\beta,R}(a^*)(w)]:\

=&\sum_{\gamma\in\Deltazero}\gamma(h_\alpha)a^*_\gamma(w)\der_\gamma P_{\alphap}^{\beta,R}(a^*)(w)\delta(z-w).
\end{align*}
Notice that $\sum_{\gamma\in\Deltazero}\gamma\cdot x_\gamma\der_\gamma$ defines the $\bQ_0$-valued Euler operator on $\C[G_0^+]$ with respect to $\degQ$. Since $\degQ P_{\alphap}^{\beta,R}(x)=\beta-\alphap$ by \eqref{polQ eq}, we have
\begin{align*}
\sum_{\gamma\in\Deltazero}\gamma(h_\alpha)a^*_\gamma(w)\der_\gamma P_{\alphap}^{\beta,R}(a^*)(w)
=(\beta(h_\alpha)-\alphap(h_\alpha))P_{\alphap}^{\beta,R}(a^*)(w).
\end{align*}
Since
\begin{align}\label{mod h eq4}
[b_\alpha(z),\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alphap}(w)}]=-\alphap(h_\alpha)\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alphap}(w)}\delta(z-w),
\end{align}
we have
\begin{align*}
[h_\alpha(z),V^\beta(w)]

&=\sum_{\gamma\in[\alphap]}c_{\gamma,h_\alpha}^\beta V^\gamma(w)\delta(z-w).
\end{align*}
Hence, \eqref{mod eq} follows for $u=h_\alpha$. Next, we consider the case that $u=e_\alpha$. Then
\begin{align*}
[e_\alpha(z),V^\beta(w)]

&=\sum_{\gamma\in\Deltazero}:P_\alpha^\gamma(a^*)(w)\der_\gamma P_{\alphap}^{\beta,R}(a^*)(w)\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alphap}(w)}:\delta(z-w)
\end{align*}
by \eqref{a der eq}. By applying Lemma \ref{mod sub lemma} (1) to the above formula, we obtain that
\begin{align*}
[e_\alpha(z),V^\beta(w)]

=&\sum_{\gamma\in[\alphap]}c_{\gamma,\alpha}^\beta V^\gamma(w)\delta(z-w).
\end{align*}
Therefore \eqref{mod eq} follows for $u=e_\alpha$. Finally, we consider the case that $u=f_\alpha$. Then
\begin{align*}
&[f_\alpha(z),V^\beta(w)]\\

=&:\left(\sum_{\gamma\in\Deltazero}Q_\alpha^\gamma(a^*)(w)\der_\gamma P_{\alphap}^{\beta,R}(a^*)(w)-(\alpha|\alphap)a^*_\alpha(w)P_{\alphap}^{\beta,R}(a^*)(w)\right)\e^{-\frac{1}{\para+h^{\vee}}\int b_{\alphap}(w)}:\delta(z-w)
\end{align*}
by \eqref{a der eq} and \eqref{mod h eq4}. By applying Lemma \ref{mod sub lemma} (2) to the above formula, we obtain that
\begin{align*}
[f_\alpha(z),V^\beta(w)]

=&\sum_{\gamma\in[\alphap]}c_{\gamma,-\alpha}^\beta V^\gamma(w)\delta(z-w).
\end{align*}
Therefore \eqref{mod eq} follows for $u=f_\alpha$. This completes the proof.