Affine Harish-Chandra center in positive characteristic

Let $G$ be a split reductive group defined over a field of characteristic bigger than the Coxeter numbers of its simple factors. We identify the Harish–Chandra centers of the associated Kac–Moody vertex algebras and enveloping algebras at noncritical levels with algebras of functions on moduli spaces of connections for the Langlands dual group $\check{G}$. Namely, given a noncritical level $\kappa$ for $G$, consider the associated Kac–Moody vertex algebra $V_{\kappa }(\mathfrak{g})$ defined on a formal disc $\mathcal{D}$, and its Harish–Chandra center of arc group invariants $$V_{\kappa }(\mathfrak{g}{)}^{\mathcal{J}G}\subset V_{\kappa }(\mathfrak{g})\mathrm{.}$$ Write $\check{\kappa }$ for the dual level for $\check{G}$, ${\check{\kappa }}^p-\check{\kappa }$ for its image under the Artin–Schreier map, and ${\mathrm{Op}}_{\check{G}}({\mathcal{D}}^{(1)}{)}_{{\check{\kappa }}^p-\check{\kappa }}$ for the moduli space of $({\check{\kappa }}^p-\check{\kappa })$-opers on the Frobenius twisted disc ${\mathcal{D}}^{(1)}$. We establish a canonical isomorphism $$\mathrm{Spec}{V}_{\kappa }(\mathfrak{g}{)}^{\mathcal{J}G}\simeq {\mathrm{Op}}_{\check{G}}({\mathcal{D}}^{(1)}{)}_{{\check{\kappa }}^p-\check{\kappa }}\mathrm{.}$$ There is a similar identification for the Harish–Chandra center of the filtered complete enveloping algebra at level $\kappa$, where instead of $\mathcal{D}$ we need to consider the punctured disc ${\mathcal{D}}^{\times }$. The presence of these Harish–Chandra centers is a genuinely new phenomenon for loop groups at noncritical level in positive characteristic: these centers are nontrivial, unlike for loop groups at noncritical level in characteristic zero, and in particular are not the reductions mod $p$ of the characteristic zero Harish–Chandra centers, unlike for finite dimensional reductive groups or loop groups at critical level.