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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, Comment: 70pages. To appear in Selecta Mathematica

Unsupervised domain adaptation (UDA) aims to learn the unlabeled target domain by transferring the knowledge of the labeled source domain. To date, most of the existing works focus on the scenario of one source domain and one target domain (1S1T), and just a few works concern the scenario of multiple source domains and one target domain (mS1T). While, to the best of our knowledge, almost no work concerns the scenario of one source domain and multiple target domains (1SmT), in which these unlabeled target domains may not necessarily share the same categories, therefore, contrasting to mS1T, 1SmT is more challenging. Accordingly, for such a new UDA scenario, we propose a UDA framework through the model parameter adaptation (PA-1SmT). A key ingredient of PA-1SmT is to transfer knowledge through adaptive learning of a common model parameter dictionary, which is completely different from existing popular methods for UDA, such as subspace alignment, distribution matching etc., and can also be directly used for DA of privacy protection due to the fact that the knowledge is transferred just via the model parameters rather than data itself. Finally, our experimental results on three domain adaptation benchmark datasets demonstrate the superiority of our framework.