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

TL;DR

This paper studies the geometric and algebraic structures of Beilinson-Drinfeld Schubert varieties associated with parahoric group schemes, proving flatness, describing line bundles, and connecting global sections to twisted Demazure modules.

Contribution

It establishes flatness of BD Schubert varieties for parahoric schemes, characterizes line bundles on their Grassmannians, and links global sections to twisted Demazure modules, extending previous untwisted results.

Findings

Proved flatness of BD Schubert varieties of parahoric group schemes.

Determined the rigidified Picard group of the BD Grassmannian.

Connected global sections of line bundles to twisted Demazure modules.

Abstract

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,