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We prove a version of the tamely ramified geometric Langlands correspondencein positive characteristic for $GL_n(k)$. Let $k$ be an algebraically closedfield 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 markedpoint. We denote by $\text{Bun}_{n,P}$ the moduli stack of (quasi-)parabolicvector bundles on $X$, and by $\mathcal{L}oc_{n,P}$ the moduli stack ofparabolic flat connections such that the residue is nilpotent with respect tothe parabolic reduction at each marked point. We construct an equivalencebetween the bounded derived category$D^{b}(\text{Qcoh}({\mathcal{L}oc_{n,P}^{0}}))$ of quasi-coherent sheaves on anopen substack $\mathcal{L}oc_{n,P}^{0}\subset\mathcal{L}oc_{n,P}$, and thebounded derived category$D^{b}(\mathcal{D}^{0}_{{\text{Bun}}_{n,P}}\text{-mod})$ of$\mathcal{D}^{0}_{{\text{Bun}}_{n,P}}$-modules, where$\mathcal{D}^0_{\text{Bun}_{n,P}}$ is a localization of$\mathcal{D}_{\text{Bun}_{n,P}}$ the sheaf of crystalline differentialoperators on $\text{Bun}_{n,P}$. Thus we extend the work ofBezrukavnikov-Braverman to the tamely ramified case. We also prove acorrespondence between flat connections on $X$ with regular singularities andmeromorphic Higgs bundles on the Frobenius twist $X^{(1)}$ of $X$ with firstorder poles .

Tamely ramified geometric Langlands correspondence in positive characteristic