Let be a simple algebraic group over an algebraically closed field . Let be a finite group acting on . We classify and compute the local types of ‐bundles on a smooth projective ‐curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in . When , we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a ‐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.
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Abstract 
Free, publiclyaccessible full text available April 1, 2025

We study the spaces of twisted conformal blocks attached to a
curve$\Gamma$ with marked$\Sigma$ orbits and an action of$\Gamma$ on a simple Lie algebra$\Gamma$ , where$\mathfrak {g}$ is a finite group. We prove that if$\Gamma$ stabilizes a Borel subalgebra of$\Gamma$ , then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed$\mathfrak {g}$ curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let$\Gamma$ be the parahoric Bruhat–Tits group scheme on the quotient curve$\mathscr {G}$ obtained via the$\Sigma /\Gamma$ invariance of Weil restriction associated to$\Gamma$ and the simply connected simple algebraic group$\Sigma$ with Lie algebra$G$ . We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasiparabolic$\mathfrak {g}$ torsors on$\mathscr {G}$ when the level$\Sigma /\Gamma$ is divisible by$c$ (establishing a conjecture due to Pappas and Rapoport).$\Gamma $ 
We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar–Henderson in the twisted setting. We also get some applications to the geometry of the order 2 nilpotent varieties in certain classical symmetric spaces.more » « less