Local types of (Γ,G)$(\Gamma ,G)$ ‐bundles and parahoric group schemes

Let G$G$ be a simple algebraic group over an algebraically closed field k$k$ . Let Γ$\Gamma$ be a finite group acting on G$G$ . We classify and compute the local types of (Γ,G)$(\Gamma , G)$ ‐bundles on a smooth projective Γ$\Gamma$ ‐curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in G$G$ . When char(k)=0$\text{char}(k)=0$ , we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a (Γ,Gad)$(\Gamma ,G_{\mathrm{ad}})$ ‐bundle. We also prove a local version of this theorem, that is, parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings.