Reduction-by-stages conjecture for W-algebras

Let $(a_1,\dots,a_s,a_{s+1},\dots,a_r)$ be a partition of $n$, let $f_2$ be a nilpotent element of $\mathfrak{sl}_n$ corresponding to this partition, and set $p\coloneqq a_{s+1}+\cdots+a_r$. Let $f_1$ be a nilpotent element corresponding to $(a_1,\dots,a_s,1^p)$, and let $f_0$ be a nilpotent element of $\mathfrak{sl}_p$ corresponding to $(a_{s+1},\dots,a_r)$. For a module $V$ in $\mathsf{KL}^k(\mathfrak{g})$, reduction-by-stages conjecture. There is a vertex algebra isomorphism

$$\mathrm{H}^0_{f_0}(\mathcal{W}^k(\mathfrak{g},f_1))\cong\mathcal{W}^k(\mathfrak{g},f_2),$$

and a $\mathcal{W}^k(\mathfrak{g},f_2)$-module isomorphism

$$\mathrm{H}^0_{f_0}(\mathrm{H}^0_{f_1}(V))\cong\mathrm{H}^0_{f_2}(V).$$

Consequently, for the level $\mathcal{l}$ supplied by the Kac–Roan–Wakimoto construction, there is a vertex algebra embedding $\mathcal{W}^{\mathcal{l}}(\mathfrak{sl}_p,f_0)\hookrightarrow\mathcal{W}^k(\mathfrak{sl}_n,f_2)$. This is presented as a generalisation of the Kac–Roan–Wakimoto embedding and is not proved in the supplied text.