The covering-duality conjecture for associated varieties of affine vertex algebras

Let $\fg$ be simple of simply-laced type and let $k\in\BQ$ satisfy $k+\check{\operatorname{\mathbbm{h}}}=m/u$ in lowest terms with $m\geq1$. Let $\check{\fg}^{(u)}$ be the metaplectic dual, let $\check\BO(m)^{(u)}=\operatorname{Sat}_{\check L^{(u)}}^{\check G^{(u)}}\BO_{\check L^{(u)}}$ be the associated nilpotent orbit, and let $\fl$ be the corresponding Levi of $\fg$ with parabolic $\fp=\fl\oplus\fu$ and center $\fz(\fl)$. Let $\bd^{(u)}$ and $\bd_L^{(u)}$ be the covering duality maps, and let $\cN$ be the nilpotent cone. Covering-duality conjecture. One has

$$X_{L_k(\fg)}=\overline{\operatorname{Ad}G\cdot(\bd_L^{(u)}\BO_{\check L^{(u)}}\times\fz(\fl)\times\fu)}.$$

In particular,

$$X_{L_k(\fg)}\cap\cN=\overline{\bd^{(u)}\check\BO(m)^{(u)}},$$

and $X_{L_k(\fg)}\subseteq\cN$ if and only if $\check\BO(m)^{(u)}$ is distinguished. This is the main conjecture of the paper, giving a proposed description of associated varieties at rational levels beyond the admissible case; the paper presents evidence but leaves the general assertion open.