Tamely Ramified Geometric Langlands Correspondence in Positive Characteristic

Abstract

Abstract We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for $GL_{n}(k)$, where $k$ is an algebraically closed field of characteristic $p> n$. Let $X$ be a smooth projective curve over $k$ with marked points, and fix a parabolic subgroup of $GL_{n}(k)$ at each marked point. We denote by $\operatorname{Bun}_{n,P}$ the moduli stack of (quasi-)parabolic vector bundles on $X$, and by $\mathcal{L}oc_{n,P}$ the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category $D^{b}(\operatorname{QCoh}({\mathcal{L}oc_{n,P}^{0}}))$ of quasi-coherent sheaves on an open substack $\mathcal{L}oc_{n,P}^{0}\subset \mathcal{L}oc_{n,P}$, and the bounded derived category $D^{b}(\mathcal{D}^{0}_{{\operatorname{Bun}}_{n,P}}\operatorname{-mod})$ of $\mathcal{D}^{0}_{{\operatorname{Bun}}_{n,P}}$-modules, where $\mathcal{D}^{0}_{\operatorname{Bun}_{n,P}}$ is a localization of $\mathcal{D}_{\operatorname{Bun}_{n,P}}$ the sheaf of crystalline differential operators on $\operatorname{Bun}_{n,P}$. Thus, we extend the work of Bezrukavnikov–Braverman [ 8] to the tamely ramified case. We also prove a correspondence between flat connections on $X$ with regular singularities and meromorphic Higgs bundles on the Frobenius twist $X^{(1)}$ of $X$ with first-order poles.