Local types of $(\mathrm{\Gamma },G)$-bundles and parahoric group schemes

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