Affine Harish-Chandra center in positive characteristic

Abstract

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 $κ$ for $G$, consider the associated Kac--Moody vertex algebra $V_κ(\mathfrak{g})$ defined on a formal disc $\mathscr{D}$, and its Harish--Chandra center of arc group invariants $$V_κ(\mathfrak{g})^{\mathscr{J}G} \subset V_κ(\mathfrak{g}).$$ Write $\checkκ$ for the dual level for $\check{G}$, $\checkκ^p - \checkκ$ for its image under the Artin--Schreier map, and $\operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\checkκ^p - \checkκ}$ for the moduli space of $(\checkκ^p - \checkκ)$-opers on the Frobenius twisted disc $\mathscr{D}^{(1)}$. We establish a canonical isomorphism $$\operatorname{Spec} V_κ(\mathfrak{g})^{\mathscr{J} G} \simeq \operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\checkκ^p - \checkκ}.$$ There is a similar identification for the Harish--Chandra center of the filtered complete enveloping algebra at level $κ$, where instead of $\mathscr{D}$ we need to consider the punctured disc $\mathscr{D}^\times$. 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.