Beilinson-Drinfeld Schubert varieties of parahoric group schemes and twisted global Demazure modules

Let $\mathcal{G}$ be a parahoric Bruhat-Tits group schemes 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 ${\rm Gr}_{\mathcal{G},C^n}$ of $\mathcal{G}$, and we establish the factorizable and equivariant structures on rigidified line bundles on ${\rm Gr}_{\mathcal{G},C^n}$. 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,