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

Abstract

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 false(normalΓ,Gfalse)$(\Gamma , G)$‐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 charfalse(kfalse)=0$\text{char}(k)=0$, we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a false(normalΓ,Gadfalse)$(\Gamma ,G_{\mathrm{ad}})$‐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.