Let \(\mathcal {G}\) be a parahoric Bruhat–Tits group scheme arising from a \(\Gamma \) -curve C and a certain \(\Gamma \) -action on a simple algebraic group G for some finite cyclic group \(\Gamma \) . We prove the flatness of Beilinson–Drinfeld Schubert varieties of \(\mathcal {G}\) , we determine the rigidified Picard group of the Beilinson–Drinfeld Grassmannian \(\textrm{Gr}_{\mathcal {G}, C^n }\) of \(\mathcal {G}\) , and we establish the factorizable and equivariant structures on rigidified line bundles over \(\textrm{Gr}_{\mathcal {G}, C^n }\) . We develop an algebraic theory of global Demazure modules of twisted current algebras, and using our geometric results we prove that when \(C=\mathbb {A}^1\) , the spaces of global sections of line bundles on BD Schubert varieties of \(\mathcal {G}\) are dual to the twisted global Demazure modules. This generalizes the work of Dumanski–Feigin–Finkelberg in the untwisted setting.
