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

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $k$. Let $\Gamma$ be a finite group acting on $G$. We classify and compute the local types of $(\Gamma ,G)$-bundles on a smooth projective $\Gamma$-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 $(\Gamma ,G_{\text{ad}})$-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.

updated version to appear in the Proceedings of the London Mathematical Society