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It is well-known that there is a duality between affine Demazure modules and the spaces of sections of line bundles on affine Schubert varieties in affine Grassmannians. This should be regarded as a local theory. In this talk, I will explain an algebraic theory of global Demazure modules of twisted current algebras. Moreover, these modules are dual to the spaces of sections of line bundles on Beilinson-Drinfeld Schubert varieties of certain parahoric groups schemes. This generalizes the works of Dumanski-Feigin(-Finkelberg) in the untwisted case. In order to establish this duality in the twisted case, following ideas of Zhu we prove the flatness of BD Schubert varieties, and establish factorizable and equivariant structures on the line bundles on BD Grassmannians of these parahoric group schemes. This talk will be based on the joint work with Huanhuan Yu.