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

Jiuzu Hong
[J. Hong]Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, U.S.A.

Huanhuan Yu
[H. Yu]Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, U.S.A.

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 over ${\rm Gr}_{\mathcal{G}, C^n }$.

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,

§ INTRODUCTION

Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C}$, and let $\mf{g}$ be its Lie algebra. We fix a Borel subgroup $B$ and a maximal torus $T\subseteq B$. Let $\Gr_G$ be the affine Grassmannian of $G$. A (spherical) affine Schubert variety in $\Gr_G$ is parametrized by a dominant coweight $\lambda\in X_*(T)^+$ of $G$, denoted by $\Grb_G^\lambda$. Let $\ms{L}$ be the level one line bundle on $\Gr_G$, and let $\ms{L}^c$ be its $c$-th power for some $c> 0$. Then, the space $H^0(\Grb_G^\lambda, \ms{L}^c)$ of global sections is dual to a level $c$ affine Demazure module $D(c, \lambda)$, cf. [27]. By the work of Fourier-Littelmann [20], when $G$ is of type $A, D,E$, the Demazure module $D(c, \lambda)$ has fusion properties. When $c=1$, $D(1, \lambda)$ is a Weyl module. The Weyl modules and Demazure modules have global counterparts, defined in [18] and [19, 9] respectively. Similarly, the global counterparts of affine Grassmanians and affine Schubert varieties are Beilinson-Drinfeld (BD) Grassmannian and BD Schubert varieties, which play a central role in geometric Langlands program. In [10], Dumanski-Feigin-Finkelberg proved that there is a duality between the space of global sections of level $c$ line bundle on BD Schubert varieties and the level $c$ global Demazure modules of the current algebra $\mf{g}[t]$, where under this duality the factorization of line bundles is compatible with the factorization of global Demazure module.

Let $\sigma$ be a standard automorphism on $G$ of order $m$ preserving $B$ and $T$, where $\sigma$ is simply a diagram automorphism if $G$ is not of type $A_{2n} $, see Section <ref>. We consider the twisted current algebra $\gs$, where $\sigma$ acts on $\mf{g}$ and $\sigma$ acts on $t$ by $\sigma(t)=e^{-\frac{2\pi \rm{i}}{m}} t$. In fact, $\gs$ is isomorphic to a special current algebra or hyperspecial current algebra defined in [8, 28], see <cit.> and Theorem <ref>.
The fusion theory of twisted Demazure modules and twisted Weyl modules are extensively studied, e.g. in [16, 15, 8, 28]. Let $X_*(T)^+_\sigma$ be the set of $\sigma$-coinvariants of all dominant coweights of $G$. For each $\bar{\lambda}\in X_*(T)^+_\sigma$, we denote by $D^\sigma(c, \bar{\lambda})$ the twisted Demazure module of $\gs$. Similar to the untwisted case, $D^\sigma(c, \bar{\lambda})$ is dual to the global section of a level $c$ line bundle on the twisted Schubert variety $\Grb^{\bar{\lambda}}_{\ms{G}}$ of the special parahoric group scheme $\ms{G}={\rm Res}_{\mc{O}/ \bar{\mc{O}}}(G_{\mc{O}})^\sigma$, see Section <ref>. In [4], Besson-Hong determined the smooth locus of $\Grb^{\bar{\lambda}}_{\ms{G}}$, where the Demazure modules of $\gs$ and the global Schubert variety $\Grb_{\mc{G}}^\lambda$ of a parahoric Bruhat-Tits group scheme $\mc{G}:={\rm Res}_{\A^1/\bar{\A}^1}(G_{\A^1})^\sigma $ are crucially used.

The aim of this paper is to develop an algebraic theory of twisted global Demazure modules of $\gs$, and generalize the work [10] to the twisted setting, i.e. to establish a duality between twisted global Demazure modules of $\gs$ and the spaces of global sections of line bundles on BD Schubert varieties of $\mc{G}$. In order to establish this duality, a large part of our work is devoted to proving the flatness of BD Schubert varieties of $\mc{G}$, and establishing factorizable and equivariant structures on BD Grassmannians of $\mc{G}$, for more general parahoric Bruhat-Tits group scheme $\G$. On this part, we closely follow the works [37, 38, 39] of Zhu.
In fact, our work also provides proofs of some geometric ingredients needed in [10], which seem to be missing in literature.

Let $\Gamma$ be the finite group acting on $G$ generated by $\sigma$, and let $C$ be an irreducible smooth algebraic curve over $\mathbb{C}$ with a faithful $\Gamma$-action such that all ramified points in $C$ are totally ramified. Let $\bar{C}$ be the quotient curve $C/\Gamma$. Let $\mc{G}$ be the $\Gamma$-fixed point subgroup scheme of the Weil restriction group $\mr{Res}_{C/\bar C}(G\times C)$. Then $\mc{G}$ is a parahoric Bruhat-Tits group scheme over $\bar C$. In Section <ref>, we define Beilinson-Drinfeld Schubert varieties of $\mc{G}$ and prove their flatness.

Let $\Gr_{\mc{G},C^n}$ (resp. $L^+\mc{G}_{C^n}$) denote the Beilinson-Drinfeld Grassmannian (resp. the jet group scheme) of $\mc{G}$ over $C^n$. For any $n$-tuple of coweights $\la=(\lambda_1,\ldots, \lambda_n)$ of $G$, we first construct a section $s_{\la}:C^n\to \Gr_{\mc{G},C^n}$, and then we define the Beilinson-Drinfeld Schubert variety $\Grb_{\mc{G},C^n}^{\la}$ to be the reduced closure of the orbit of $s_{\la}$ under the action of $L^+\mc{G}_{C^n}$. The construction of $s_{\la}$ is achieved via a construction of a locally trivial $(\Gamma, T)$-torsor attached to $\vec{\lambda}$. By the Tannakian interpretation of $(\Gamma,T)$-torsor (cf. Theorem <ref>), the construction of this $(\Gamma,T)$-torsor is reduced to the construction of a $\Gamma$-equivariant tensor functor, see Section <ref>. The main result of Section <ref> is Theorem <ref>, which asserts that

The BD Schubert variety $\Grb_{\mc{G},C^n}^{\la}$ is flat over $C^n$. Moreover, the schematic fiber $\Grb_{\mc{G},\vec{p}}^{\la}$ of $\Grb_{\mc{G},C^n}^{\la}$ at each $\vec{p}\in C^n$ is reduced. More precisely, $\Grb_{\mc{G},\vec{p}}^{\la}$ is a product of affine Schubert varieties (possibly twisted).

In the untwisted case, this theorem is due to Zhu, see [37] for $n=2$. In the twisted case, the main idea of the proof of Theorem <ref> is similar. When $n=1$, it is also due to Zhu [38]. For $n>1$, we need a new ingredient suggested by Zhu (by private communication), a generalization of cohomology and base change, which is formulated in Appendix <ref>. This new method is of course applicable to the untwisted case. In order to employ Theorem <ref>, in Section <ref> we define the convolution variety $\overline{\Conv}_{\G,C^n}^{\la}$ which admits a proper morphism to $\Gr_{\G,C^n}$. In fact, we give two definitions, one is defined via a sequence of fibrations of global Schubert varieties, and the other one is defined as the reduced closure of $L^+\G_{C^n}$-orbit at a section $\tilde{s}_{\la}$ of the convolution Grassmannian. In Proposition <ref>, we show these two definitions are the same. The first definition tells that $\overline{\Conv}_{\G,C^n}^{\la}$ is flat over $C^n$, and the second definition tells that the image of $\Conv_{\G,C^n}^{\la}$ in $\Gr_{\G,C^n}$ is exactly $\Grb_{\G,C^n}^{\la}$. As an application of Theorem <ref>, we also prove that any BD Schubert variety is normal, see Theorem <ref>. When $n=1$, this is due to Zhu, cf. <cit.>.

Section <ref> is devoted to the study of rigidified line bundles on $\Gr_{\mc{G},C^n}$. Let $\mr{Pic}^e(\Gr_{\mc{G},C^n})$ denote the Picard group of rigidified line bundles on $\Gr_{\mc{G},C^n}$, i.e. the line bundles on $\Gr_{\mc{G},C^n}$ together with a trivialization along a distinguished section $e$ over $C^n$, see Section <ref>. We prove the following result in Theorem <ref>.

Suppose that $G$ is simply-connected.

* For any $n\geq 1$, the central charge map $\mr{Pic}^e(\Gr_{\mc{G},C^n})\to \mb{Z}$ is an isomorphism.

* Any line bundle in $\mr{Pic}^e(\Gr_{\mc{G},C^n})$ has factorizable property.

The proof of the injectivity of the central charge map is similar to the proof in the untwisted case, cf. <cit.>. In the twisted case, one of the key ingredients is the construction of a level one line bundle $\mc{L}_{C^n}$ on $\Gr_{\G,C^n}$. By the construction in [4], there is a level one line bundle $\mc{L}_{\A^n}$ on $\Gr_{\G,\A^n}$. For a general $\Gamma$-curve $C$, given any ramified point $p\in C$, there is a $\Gamma$-stable neighborhood $U$ of $p$ which admits a $\Gamma$-equivariant map to $\A^1$. Then, the pull-back of $\mc{L}_{\A^n}$ gives a level one line bundle on $\Gr_{\G,U^n}$. These line bundles together with the level one line bundle over the unramified part (essentially untwisted case) can be glued and extended to a line bundle $\mc{L}_{C^n}$ on $\Gr_{\G,C^n}$, which generates $\mr{Pic}^e(\Gr_{\mc{G},C^n})$. Moreover, this line bundle satisfies factorization property. When $n>1$, we crucially use a relative version of seesaw principle (cf. Lemma <ref>).

Another main result of Section <ref> is proved in Theorem <ref>, which asserts that

With the same assumption as in Theorem <ref>,	there is a unique $L^+\mc{G}_{C^n}$-equivariant stucture on the level one line bundle $\mc{L}_{C^n}$ on $\Gr_{\mc{G},C^n}$.

The hard part of this theorem is the existence. In the proof, we first construct a $L^+\mc{G}_{C^n}$-equivariant line bundle $\mc{L}_{\rm det}$ by embedding $\Gr_{\mc{G},C^n}$ to a BD Grassmannian of $GL(\mc{V})$ for some vector bundle $\mc{V}$ over $C$. By the étale base change property of BD Grassmannians (cf. Proposition <ref>), we are reduced to the case $C=\A^1$ with the standard $\Gamma$-action defined in (<ref>). In this case, the line bundle $\mc{L}_{\rm det}$ has a rigidified structure, and hence is a power of $\mc{L}_{\A^n}$, where Theorem <ref> is crucially used. This in fact implies that $\mc{L}_{\A^n}$ admits a $L^+\mc{G}_{\A^n}$-equivariant structure. This step is guaranteed by Proposition <ref>, whose proof is quite technical. We first prove this proposition when $n=1$ and $n=2$ separately, and for general case we use Hartogs' Lemma. In our argument, we need to restrict line bundles to BD Schubert varieties so that all arguments are in finite type of geometry.

To prove Theorem <ref> (or Theorem <ref>) in full generality, we need to work with adjoint type group $G$. In this case, it is difficult (or impossible) to construct a factorizable level one line bundle on $\Gr_{\G,C^n}$ for a general $\Gamma$-curve $C$. Nevertheless, when $C=\A^1$ with the standard action of $\Gamma$. We are able to construct level one line bundles on Schubert varieties which admits factorization properties compatible with the factorization of Schubert varieties, see Proposition <ref>. Moreover, using Theorem <ref> we are able to prove that these line bundles admit $L^+\mc{G}'_{C^n}$-equivariant structures, where $L^+\mc{G}'_{C^n}$ is the jet group scheme arising from the simply-connected cover of $G$, see Proposition <ref>.

As mentioned earlier, the main motivation of this work is to establish a relation between twisted global Demazure modules and line bundles on BD Schubert varieties of $\G$.
In Section <ref>, for any $\vec{\lambda}\in (X_*(T)^+)^n$, we define the twisted global Demazure module $\mathbb{D}^\sigma(c,\la)$ of $\gs$. Using the results of Fourier-Littelmann [20] and Kus-Vankatesh [28], we prove the following result in Theorem <ref> and Theorem <ref>.
Given $\vec{\lambda}=(\lambda_1,\cdots, \lambda_n)\in (X_*(T)^+)^n$, let $\lb$ be the image of $\sum \lambda_i$ in $X_*(T)^+_\sigma$.

* The fiber $\Ds(c,\la)_{0}$ at $0\in \A^n$ is the twisted Demazure module $D^\sigma(c, \lb)$.

* The twisted global Demazure module $\Ds(c,\la)$ is a graded free $\As(c,\vec{\lambda})$-module, where $\As(c,\vec{\lambda})$ is the weight algebra associated to the cyclic generator of $\Ds(c,\la)$.

As a consequence of Theorem <ref> and Proposition <ref>, the space $H^0(\Grb^{\la}_{\mc{G},\A^n},\mc{L}_{\A^n})$ of global sections of line bundle $\mc{L}_{\A^n}$ is a $(\gs,\C[\A^n])$-bimodule. In Section <ref>, we further equip a $\mb{G}_m$-equivariant structure on the level one line bundle $\mc{L}_{\A^n}$, which follows that $H^0(\Grb^{\la}_{\mc{G},\A^n},\mc{L}_{\A^n})$ is a graded free $\C[\A^n]$-module, see Theorem <ref>. Finally, we prove the following Borel-Weil type theorem in Theorem <ref>.

There exists an isomorphism of $(\gs, \C[\A^n])$-bimodules

\[ \Ds(c,\la)_{\A^n}\simeq H^0\big( \Grb_{\mc{G},\A^n}^{\la}, \mc{L}_{\A^n}^c\big)^\vee,\]

where $\Ds(c,\la)_{\A^n}=\Ds(c,\la)\otimes_{\As(c,\vec{\lambda})}  \mathbb{C}[\A^n]$.  Moreover, $\Ds(c,\la)_{\A^n}$ has factorization property and it is compatible with the factorization of $\mc{L}$ under the above isomorphism.

Our proof is similar to the proof in [10], while there are some distinctions, see Remark <ref>. As a consequence of this theorem, the fiber $ \Ds(c,\la)_{\vec{p}}$ at any  $\vec{p}\in \A^n$ can be determined, see Corollary <ref> and Remark <ref>. In fact, the fiber $ \Ds(c,\la)_{\vec{p}}$ is a tensor product of affine Demazure modules (possibly twisted) depending on the type of the point $\vec{p}$.

The Borel-Weil type theorem for twisted global Weyl module is proved by Braverman-Finkelberg in <cit.>, where they work with twisted Zastava spaces. In fact, when the level $c=1$, $\Ds(c,\vec{\lambda})$  is isomorphic to a twisted global Weyl module, see Theorem <ref>.

Finally, we make a remark when $G$ is $A_{2\ell}$. In this case,  our automorphism $\sigma$ is chosen to be the standard automorphism of order 4. The twisted current algebra $\gs$ corresponds to a special parahoric subalgebra of $A^{(2)}_{2\ell}$. In fact, there is another special parahoric subalgebra of $A^{(2)}_{2\ell}$, which can be realized as the twisted current algebra $\mf{g}[t]^\tau$ corresponding to the diagram automorphism $\tau$ of order $2$. For this automorophism $\tau$, all results in Section <ref> hold, and in Section <ref> similar results hold except that ${\rm Pic}^e(\Gr_{\G, C^n})$ is generated by a level $2$ rigidified line bundle. Moreover, we expect that Theorem <ref> still hold for even levels.

Acknowledgements: We would like to thank I. Dumanski, E. Feigin, M. Finkelberg, G. Fourier and D. Kus for helpful email correspondences. We also thank R. Travkin and X. Zhu for inspiring discussions on geometric aspects of this work.  J. Hong was partially supported by the NSF grant DMS- 2001365.

§ AFFINE DEMAZURE MODULES OF TWISTED CURRENT ALGEBRA

In this section, we collect some known results of affine Demazure modules, Weyl modules, and their twisted analogues, which will be used in Section <ref>.

§.§ Notations

Let $G$ be an almost simple algebraic group over $\C$. We fix a maximal torus $T$ and a Borel subgroup $B$ containing $T$. Let $X^*(T)$ (resp. $X_*(T)$) denote the weight lattice (resp. the coweight lattice) of $T$. Let $X_*(T)^+$ denote the set of dominant coweights of $G$ with respect to $B$. Let $R$ be the set of roots of $G$. Let $R^+$ denote the set of positive roots of $G$ with respect to $B$. Let $\{\alpha_i \mid i\in I\}$ be the set of simple roots, where $I$ is the set of vertices of the Dynkin diagram of $G$.

Let $\mf{g}$, $\mf{b}$, $\mf{h}$ denote the Lie algebras of $G$, $B$, $T$ respectively. One has the triangular decomposition $\mf{g}=\mf{n}^-\oplus\mf{h}\oplus\mf{n}^+$, where $\mf{n}^+$ is the nilpotent radical of $\mf{b}$ and $\mf{n}^-$ is the nilpotent radical of the opposite Borel subalgebra $\mf{b}^-$. Let $\{e_{\alpha} \mid \alpha \in R \}$ be a Chevalley basis of $\mf{g}$. For simplicity of notations, we also set
$e_i=e_{\alpha_i}$, $f_i=e_{-\alpha_i}$, and $h_i=\alpha_i^\vee$ for any $i\in I$.

In this section, we always assume that $G$ is simply-laced and of adjoint type. Then, the set $X_*(T)^+$ can be identified with set of all dominant coweights of $\mf{g}$. In Section <ref> and Section <ref>, we only deal with Lie algebras, but to be consistent with the notations in later sections on geometry, we keep using the notation $X_*(T)^+$.

§.§ Twisted current algebra

Let $\tau$ be a nontrivial diagram automorphism preserving $\mf{b}, \mf{h}$ and the pinning $\{e_i,f_i; i\in I\}$.
Following <cit.>, let $\sigma$ be a standard automorphism of order $m$ on $\mf{g}$ preserving $\mf{b}$ and $\mf{h}$. More specifically, when $\mf{g}$ is not $A_{2\ell}$, $\sigma=\tau$; when $\mf{g}$ is of type $A_{2\ell}$,

\begin{equation}\label{eq_st_h}
\sigma=\tau\circ \mr{i}^h,\end{equation}

where $\mathrm{i}=\sqrt{-1}$, and $h\in \mf{h}^\tau$ such that

\[\alpha_i(h)=
\begin{cases}
0, & \text{if } i\neq\ell, \ell+1;\\
1, & \text{if } i=\ell,\ell+1.
\end{cases}\]

In particular,

\[m=
\begin{cases}
2, & \text{if } \mf{g} \text{ is } A_{2\ell-1}, D_{\ell+1} \text{ or } E_6;\\
3, & \text{if } \mf{g} \text{ is } D_4;\\
4, & \text{if } \mf{g} \text{ is } A_{2\ell}.
\end{cases}\]

Let $\mf{g}^\sigma$ be the $\sigma$-fixed point Lie subalgebra in $\mf{g}$. We have the following descriptions of $\mf{g^\sigma}$,

\begin{equation}
\label{Fix_table}
\begin{tabular}{|c | c | c |c |c |c | c| c| c|c |c|c|c|c|c |c ||}
\hline
$(\mf{g}, m)$ & $(A_{2\ell-1}, 2 ) $ & $(A_{2\ell}, 4)$ & $(D_{\ell+1} , 2 ) $ & $ (D_4, 3)$ & $ (E_6, 2)$ \\ [0.8ex]
\hline
$ \mf{g}^\sigma $ & $ C_\ell $ & $ C_\ell $ & $B_\ell $ & $G_2 $ & $F_4 $ \\ [0.8ex]
\hline
\end{tabular},
\end{equation}

where by convention $C_1$ is $A_1$ and $\ell\geq 3$ for $D_{\ell+1}$.

Let $\epsilon$ be a primitive $m$-th root of unity.  We extend $\sigma$ to an automorphism on the loop algebra $L(\mf{g})=\mf{g}((t))$ by

\begin{equation}\label{eq_act_1}
\sigma( x\otimes t^j ) =\epsilon^{-j}\sigma ( x) \otimes t^j.\end{equation}

We define the twisted affine algebra $\hat{L}(\mf{g},\sigma):=\mf{g}((t))^\sigma\oplus \C K$ as a central extension of $L(\mf{g})^\sigma$ with the canonical center $K$ as follows:

\[[x[f]+zK, x'[f']+z'K]=[x,x'][ff']+m^{-1}\mr{Res}_{t=0}((df)f')(x,x')K\]

for $x[f], x'[f']\in L(\mf{g})$, $z,z'\in \C$, where $\mr{Res}_{t=0}$ denotes the coefficient of $t^{-1}dt$, and $(,)$ is the normalized Killing form on $\mf{g}$ such that $(\theta,\theta)=2$ for the highest root $\theta$.

Let $\gs$ be the twisted current algebra, i.e. the $\sigma$-fixed point subalgebra of $\mf{g}[t]$. In fact, $\gs$ is a subalgebra of $\hat{L}(\mf{g},\sigma)$.

*  When $\mf{g}$ is of type $A_{2\ell}$, let $\alpha_{i,j}$ be the positive root $\alpha_i+\cdots +\alpha_j$, for $1\leq i\leq j\leq 2\ell$, and let $e_{\pm \alpha}$ be the standard basis in $\mf{g}=sl_{2\ell+1}$. Then $\gs$ has the following basis:

\begin{align}\label{802623}
\begin{split}
\quad
& e_{\pm \alpha,2k}:= e_{\pm\alpha}\otimes t^{2k}+(-1)^{i+j+k}e_{\pm\tau(\alpha)}\otimes t^{2k}, \text{ when } \alpha=\alpha_{i,j} \text{ with } 1\leq i\leq j <\ell\\
& e_{\pm \alpha,2k}:= e_{\pm\alpha}\otimes t^{2k}-(-1)^{i+j+k}e_{\pm\tau(\alpha)}\otimes t^{2k}, \text{ when } \alpha=\alpha_{i,2\ell-j} \text{ with } 1\leq i\leq j <\ell\\
& e_{\pm \alpha,4k}:= e_{\pm\alpha}\otimes t^{4k}, \text{ when } \alpha=\alpha_{i,2\ell+1-i} \text{ with } 1\leq i\leq \ell\\
& e_{\pm \alpha,2k+1}:= e_{\pm\alpha}\otimes t^{2k+1} \pm (-1)^{i+\ell+k}e_{\pm\tau(\alpha)}\otimes t^{2k+1}, \text{ when } \alpha=\alpha_{i,\ell} \text{ with } 1\leq i\leq \ell\\
& h_{i,2k}:= h_i\otimes t^{2k}+(-1)^k h_{2\ell+1-i}\otimes t^{2k}, \text{ for any } 1\leq i\leq \ell
\end{split}
\end{align}

* When $\mf{g}$ is not of type $A_{2\ell}$, let $R^+_s$ (resp. $R^+_l$) be the set of roots $\alpha\in R^+$ such that the restriction $\alpha|_{\mf{h}^\sigma}$ is a short (resp. long) root of $\mf{g}^\sigma$. Then $\gs$ has the following basis:

\begin{align*}
\quad
& e_{\pm \alpha,k}:= \sum_{i=0}^{m-1} e_{\pm\sigma^i(\alpha)}\otimes (\epsilon^{i}t)^{k}, \ \alpha \in R^+_{s}; & e_{\pm \alpha,mk}:= \sum_{i=0}^{m-1} e_{\pm\sigma^i(\alpha)}\otimes t^{mk}, \ \alpha\in R^+_{l};\\
& h_{\alpha,k}:= \sum_{i=0}^{m-1} h_{\sigma^i(\alpha)}\otimes (\epsilon^i t)^{k},\ \alpha\in R^+. &
\end{align*}

When $\mf{g}$ is not $A_{2\ell}$, $\gs$ is isomorphic to the special twisted current algebra of $\hat{L}(\mf{g},\sigma)$, see <cit.>. When $\mf{g}$ is $A_{2\ell}$, we now show $\gs$ is isomorphic to the hyperspecial twisted current algebra $\mf{Cg}$ defined in <cit.> and <cit.>.

Following <cit.> and <cit.>, the hyperspecial current algebra $\mathfrak{C} \mathfrak{g} \subset \mf{g}[t,t^{-1}]^\tau$ consists of the following basis elements:

* $e_{\pm \alpha} \otimes t^k + (-1)^{i+j} e_{\pm \tau(\alpha) } \otimes (-t)^k$, $\alpha=\alpha_{ij}$ with $1 \leq i \leq j <\ell$, and $k\geq 0$;

* $e_{\pm \alpha}\otimes t^{k\pm 1} + (-1)^{i+j} e_{\pm \tau(\alpha)} \otimes (-t)^{k\pm 1}$, $\alpha=\alpha_{i,2\ell-j}$ with $1 \leq i \leq j <\ell$, and $k\geq 0$;

* $e_{\pm \alpha} \otimes t^{2k\pm 1} $, $\alpha=\alpha_{i, 2\ell+1-i}$ with $1 \leq i \leq \ell$, and $k\geq 0$;

* $e_{\pm \alpha} \otimes t^{ (2k+1\pm 1)/2} + (-1)^{\ell+i}e_{\pm \tau(\alpha)} \otimes (-t)^{ (2k+1\pm 1)/2}$, $\alpha=\alpha_{i\ell} $ with $1\leq i \leq \ell $, and $k\geq 0$;

* $h_i\otimes t^k + h_{2\ell+1-i}\otimes (-t)^k $, $1\leq i \leq \ell$, and $k\geq 0$,

where $\alpha_{ij}$ denotes the positive root $\alpha_i+\cdots + \alpha_j$, for any $1\leq i\leq j \leq 2\ell$. Note that $\tau(\alpha_{ij})= \alpha_{ 2\ell+1-j,2\ell+1-i }$.

We have the following formula

\begin{equation}
\sigma (e_{\pm \alpha_{ij}}) = (-1)^{j-i } \mathrm{i}^{\pm \alpha_{ij}(h) }e_{ \alpha_{2\ell+1-j,2\ell+1-i} }, \text{ for any } 1\leq i\leq j \leq 2\ell+1,
\end{equation}

and $\sigma(h_i)= h_{2\ell+1-i}$, for any $1\leq i\leq 2\ell+1$, where $h$ is defined in (<ref>).

In <cit.>, the formula for the action of $\tau$ on every basis vector in $\mf{g}$ is given. Combining this formula and $ \sigma= \tau \mathrm{i}^{h}$, we can deduce the formula for $\sigma$.

There exists an isomorphism $ \eta_k: \hat{L}(\mf{g}, \tau)\simeq \hat{L}(\mf{g}, \sigma)$ due to Kac (cf. <cit.> and <cit.>), which is defined as follows

\[ \eta_k( x[t^j])= x[t^{ 2j+s }], \]

for any $x$ a simultaneous $(-1)^j$-eigenvector of $\tau$, and a $s$-eigenvector of $\mr{ad}h$.
Let $\phi$ be the Cartan involution on $\mf{g}$ such that $\phi(e_i)=-f_i$ and $\phi(h_i)=-h_i$. Notice that for any root $\alpha $, $\phi(e_{ \alpha})=-e_{- \alpha}$. Now, we define an automorphism $\eta_c: \hat{L}(\mf{g}, \tau)\to \hat{L}(\mf{g}, \tau)$ as follows,

\[ \eta_c(x[t^s])=\phi(x)[t^s], \text{ for any } x[t^s]\in \hat{L}(\mf{g}, \tau) . \]

It is easy to verify that $\eta_c$ is well-defined, since $\eta_c\circ \tau=\tau\circ \eta_c$, where $\tau$ is the automorphism on the untwisted affine Lie algebra $\hat{\mf{g}}$ induced from the action in (<ref>) .
We define the following isomorphism of twisted affine algebras

\begin{equation}\label{def_eta}
\eta:= \eta_k\circ \eta_c: \hat{L}(\mf{g}, \tau)\simeq \hat{L}(\mf{g}, \sigma) .
\end{equation}

The following theorem originally appeared in an earlier version of [4], where some calculation mistakes are fixed in this paper.

The map $\eta$ restricts to the following isomorphism of Lie algebras

\[\eta: \mathfrak{C} \mathfrak{g} \simeq \mf{g}[t]^\sigma. \]

From the definition (<ref>), we have the following calculations:

* When $\alpha=\alpha_i+\dots +\alpha_j$ with $1 \leq i \leq j <\ell$, and $k\geq 0$, we have

\[ \eta(e_{\pm \alpha} \otimes t^k + (-1)^{i+j} e_{\pm \tau(\alpha) } \otimes (-t)^k)=-(e_{\mp\alpha}\otimes t^{2k}+(-1)^{i+j+k}e_{\mp\tau(\alpha)}\otimes t^{2k}) . \]

* When $\alpha=\alpha_{i,2\ell-j}$ with $1 \leq i \leq j <\ell$, and $k\geq 0$, we have

\[ \eta(e_{\pm \alpha}\otimes t^{k\pm 1} + (-1)^{i+j} e_{\pm \tau(\alpha)} \otimes (-t)^{k\pm 1} )=- (e_{\mp\alpha}\otimes t^{2k}-(-1)^{i+j+k}e_{\mp\tau(\alpha)}\otimes t^{2k} ). \]

* When $\alpha=\alpha_{i, 2\ell+1-i}$ with $1 \leq i \leq \ell$, and $k\geq 0$, we have

\[ \eta( e_{\pm \alpha} \otimes t^{2k\pm 1} ) =-e_{\mp \alpha} \otimes t^{4k}. \]

* When $\alpha=\alpha_{i\ell} $ with $1\leq i \leq \ell $, and $k\geq 0$, we have

\[ \eta(e_{\pm \alpha} \otimes t^{ (2k+1\pm 1)/2} + (-1)^{\ell+i}e_{\pm \tau(\alpha)} \otimes (-t)^{ (2k+1\pm 1)/2} ) =- (e_{\mp\alpha}\otimes t^{2k+1} \mp (-1)^{i+\ell+k}e_{\mp\tau(\alpha)}\otimes t^{2k+1} ). \]

* For any $1\leq i \leq \ell $, and $k\geq 0$, we have

\[ \eta(h_i\otimes t^k + h_{2\ell+1-i}\otimes (-t)^k ) =-(h_i\otimes t^{2k}+(-1)^k h_{2\ell+1-i}\otimes t^{2k}) .\]

Then, it is easy to see that the image of the described basis of the hyperspecial current algebra $\mf{Cg}$ under the map $\eta$ is exactly the basis of $\mf{g}[t]^\sigma$ described above.

§.§ Weyl modules

We denote by $P$ (resp. $P_\sigma$) the weight lattice of $\mf{g}$ (resp. $\mf{g}^\sigma$), and by $P^+$ (resp. $P^+_\sigma$) the set of dominant weights of $\mf{g}$ (resp. $\mf{g}^\sigma$) with respect to the Borel subalgebra $\mf{b}$ (resp. $\mf{b}^\sigma$). We define the map $\iota: X_*(T)\to P$ given by

\[\iota(\lambda)(h)=(\lambda, h), \ \forall \lambda \in X_*(T), h\in \mf{h},\]

where $\lambda$ is regarded as an element in $\mf{h}$ and $(\ ,\ )$ is the normalized Killing form on $\mf{h}$. By restricting $\iota(\lambda)$ to $\mf{h}^\sigma$, it induces a map $X_*(T)\to P_\sigma$, and it descends to the map $X_*(T)_\sigma \to P_\sigma$. We still denote it by $\iota$.

Following [20], for any dominant weight $\mu\in P^+$ of $\mf{g}$, we define the Weyl module $W(\mu)$ of $\mf{g}[t]$ to be the cyclic module generated by an element $w_{\mu}$ subject to the following relations

\[\mf{n}^-\otimes \mathbb{C}[t]\cdot w_{\mu}=0, \ \mf{h}\otimes t\C[t]\cdot w_{\mu}=0,\ (h\otimes 1)\cdot w_\mu =-\mu(h)w_{\mu},\ (e_{\alpha_i}\otimes 1)^{\mu(\alpha_i^\vee)+1}\cdot w_{\mu}=0,\]

for any simple roots $\alpha_i\in R^+$, and any $h\in \mf{h}$.

Following [28], for each dominant weight $\bar{\mu}\in P^+_\sigma$ of $\mf{g}^\sigma$, we define the twisted Weyl module $W^\sigma(\bar{\mu})$ of $\gs$ to be the cyclic module generated by an element $w_{\bar{\mu}}$ subject to the following relations

\begin{equation}\label{989363}
\mf{n}^-[t]^\sigma\cdot w_{\bar{\mu}}=0,\ (\mf{h}\otimes t\C[t])^\sigma\cdot w_{\bar{\mu}}=0,\ (h\otimes 1)\cdot v =-\bar{\mu}(h)w_{\bar{\mu}},\ (e_{\alpha,0})^{k_\alpha+1}\cdot w_{\bar{\mu}}=0 ,
\end{equation}

for any $h\in \mf{h}^\sigma$, ${\alpha}\in R^+$, and $k_\alpha=\bar{\mu}( \bar{\alpha}^\vee)$, where $ \bar{\alpha}$ is the root ${\alpha}|_{\mf{h}^\sigma}$ of $\mf{g}^\sigma$.

We define the twisted global Weyl module $\mb{W}^\sigma(\bar{\mu})$ to be the cyclic $\U(\gs)$-module generated by an element $w_{\bar{\mu}}$ with the following relations

\begin{equation*}
\mf{n}^-[t]^\sigma\cdot w_{\bar{\mu}}=0,\ (h\otimes 1)\cdot v =-\bar{\mu}(h)w_{\bar{\mu}},\ (e_{\alpha,0})^{k_\alpha+1}\cdot w_{\bar{\mu}}=0 ,
\end{equation*}

where $h\in \mf{h}^\sigma$, ${\alpha}\in R^+$, and $k_\alpha=\bar{\mu}( \bar{\alpha}^\vee)$. Similar to the untwisted case in [7], there is a well-defined right action of $\U(\hs)$ on $\mb{W}^\sigma(\lb)$ given by

\[(u\cdot w_{\bar \mu})\ h:= u\cdot (h\cdot w_{\bar\mu})\]

where $u\in \U(\gs), h\in \U(\hs)$. This action commutes with $\U(\gs)$-action. Let ${A}^\sigma(\bar\mu):=\U(\hs)/\mr{Ann}(w_{\bar\mu})$ be the quotient of $\U(\hs)$ by the annihilator of $w_{\bar\mu}$.

In this paper,  the (twisted) Weyl modules and the twisted global counterparts are defined using the lowest weight cyclic generators. By composing the map $\eta$ (when $\mf{g}$ is $A_{2\ell}$) or a Cartan involution (otherwise), our definitions agree with the usual definitions in literature.  Also see Remark <ref> on the definitions of Demazure modules.

Let $I_\sigma$ be the set of vertices of the Dynkin diagram of $\mf{g}^\sigma$. We follow the Bourbaki labeling of the vertices of the Dynkin diagram, which gives a total order on $I$ and $I_\sigma$. Let $\{\beta_j\}_{j\in I_\sigma}$ be the set of simple roots of $\mf{g}^\sigma$. Then, there is a map $\eta:I\to I_\sigma$ such that

\begin{equation}\label{328480}
\check\beta_i=\sum_{j\in \eta^{-1}(i)} \check\alpha_j
\end{equation}

for any simple coroot $\check\beta_i$, cf. <cit.>. Let $\omega_j$ be the fundamental weight of $\mf{g}$ associated to $j\in I$. For any $i \in I_\sigma$, let $\bar{\omega}_i:=\omega_j|_{\mf{h}^\sigma}$ for some $j\in \eta^{-1}(i)$. Then $\{\bar{\omega}_i\}_{i\in I_\sigma}$ gives the set of fundamental weights of $\mf{g}^\sigma$, cf. <cit.>. Moreover, by <cit.>, we have

\begin{equation}
\label{eq_match_wt}
\iota(\check{\omega}_j)=\bar{\omega}_i, \quad  \forall j\in \eta^{-1}(i).  \end{equation}

For each $i\in I_\sigma$, set $m_i:=\frac{m}{|\eta^{-1}(i)|}$. For example, when $\mf{g}$ is $A_{2\ell}$, we have $m_i=2$ for any $i\in I_\sigma$. The following theorem on twisted global Weyl modules was proven in [8], and it will be used in Section <ref>.

Let $\bar\mu=\sum_{i\in I_\sigma} n_i\bar\omega_i\in P^+_\sigma$.

*  The algebra $A^\sigma(\bar\mu)$ is a graded polynomial algebra in variables $T_{i,r}$ of grade $m_ir$, where $i\in I_\sigma$ and $1\leq r\leq n_i$. In particular, when $\mf{g}$ is $A_{2\ell}$, $m_i=2$ for any $i\in I_\sigma$.

* Let $I_{\bar\mu,0}$ be the unique graded maximal ideal in $A^\sigma(\bar\mu)$. Then, there is a $\gs$-isomorphism

\[\mb{W}^\sigma(\bar\mu)\otimes_{A^\sigma(\bar\mu)} (A^\sigma(\bar\mu)/I_{\bar\mu,0})\simeq W^\sigma(\bar\mu).\]

* $\mb{W}^\sigma(\bar\mu)$ is a free $A^\sigma(\bar\mu)$ module of rank equal to
$\prod_{i\in I_\sigma} \big(\dim W^\sigma(\bar\omega_i)\big)^{n_i}$.

(1) Cf. <cit.> when $\mf{g}$ is $A_{2\ell}$ and <cit.> when $\mf{g}$ is not $A_{2\ell}$. (2) It follows from the definition. (3) Cf. <cit.>.

When $\mf{g}$ is of type $A_{2\ell}$, part (2) of the theorem does not hold for the diagram automorphism. This is one of the reason that we use the `standard' automorphism $\sigma$.

§.§ Affine Demazure modules

The integrable highest weight modules of $\hat{L}(\mf{g},\sigma)$ of level $c$ can be parameterized by a set $P(\sigma,c)$ of certain highest weights of $\mf{g}^\sigma$, see <cit.>. In particular, for any $\kappa \in P(\sigma,1)$, we always have $c\kappa \in P(\sigma, c)$. We denote by $\ms{H}_c(\bar\mu)$ the integrable highest weight module associated to a highest weight $\bar\mu\in P(\sigma, c)$.

Set $\ms{H}_c=\oplus_{\kappa\in P(\sigma,1)} \ms{H}_c(c\kappa)$. For any $\bar{\lambda}\in X_*(T)_\sigma^+$, let $v_{\bar{\lambda}}\in \ms{H}_c$ be the extremal weight vector whose $\mf{h}^\sigma$-weight is $-c\iota(\bar{\lambda})$, see <cit.>. We define the twisted affine Demazure $D^\sigma(c,\bar{\lambda})$ as the following $\gs$-module,

\begin{equation}\label{eq_Demazure}
D^\sigma(c,\bar{\lambda}):=\U(\gs) v_{\bar{\lambda}}.\end{equation}

For any $\lambda\in X_*(T)^+$, we define the untwisted affine Demazure module $D(c,\lambda)$ in a similar way by simply taking $\sigma=\mr{id}$.

The affine Demazure module $D(c,\lambda)$ defined here can be identified with the module $D(c, \lambda)$ defined in <cit.> with a $\mf{g}[t]$-action via the Cartan involution $\eta_c: \mf{g}[t]\to \mf{g}[t]$. Similarly, our twisted affine Demazure module $D^\sigma(c,\bar{\lambda}) $ can be identified with the module $D(c, c \iota(\bar{\lambda}))$ defined in <cit.> via a twist of the Cartan involution $\eta_c:\gs\to\gs$ when $\mf{g}$ is not $A_{2\ell}$, or via a twist of the map $\eta: \mf{Cg}\to \gs$ defined in (<ref>) when $\mf{g}$ is $A_{2\ell}$.

Thanks to the works [20, 28] of Fourier-Littelmann and Kus-Venkatesh, we have the following theorem, which will be used in Theorem <ref>.

Let $\mf{g}$ be a simply-laced simple Lie algebra with a standard automorphism $\sigma$. Let $\lambda$ be a dominant coweight of $\mf{g}$. Denote by $\lb$ the image of $\lambda$ under the projection map $X_*(T)^+\to X_*(T)^+_\sigma$. Then

* The untwisted Demazure module $D(c,\lambda)$ is isomorphic to the quotient of the Weyl module $W(c\iota(\lambda))$ by the submodule generated by

\[\{(e_{\alpha}\otimes t^s)^{k_{\alpha,s}+1}\cdot w_{\lambda}\mid\alpha \in R^+, s\geq 0 \},\]

where $k_{\alpha,s}=c\max\{0,(\lambda,\alpha^\vee)-s\}$, and $w_{\lambda}$ is the generator whose $\mf{h}$-weight is $-c\iota(\lambda)$.

* The twisted Demazure module $D^\sigma(c,\bar{\lambda})$ is isomorphic to the quotient of the twisted Weyl module $W^\sigma(c\iota(\bar{\lambda}))$ by the submodule generated by

\begin{align*}
& \{e_{\alpha,m s_{\alpha}}\cdot w_{\lb} \mid \alpha \in R^+_{l} \} \cup \{ e_{\alpha, s_{\alpha} }\cdot w_{\lb} \mid \alpha \in R^+_{s}\}, \text{ when } \mf{g} \text{ is not } A_{2\ell}; \\[0.5em]
& \{e_{\alpha,4s_\alpha}\cdot w_{\lb}\mid \alpha=\alpha_{i,2\ell+1-i}, 1\leq i<\ell \} \cup \{e_{\alpha, 2s_{2\alpha}+1}\cdot w_{\lb} \mid \alpha=\alpha_{i,\ell}, 1\leq i\leq \ell \}\\
& \cup \{e_{\alpha,2s_\alpha}\cdot w_{\lb} \mid \alpha=\alpha_{i,j} \text{ or } \alpha_{i,2\ell-j}, 1\leq i<\ell\}, \text{ when } \mf{g} \text{ is } A_{2\ell},
\end{align*}

where $s_\alpha=c\cdot (\lambda, \bar{\alpha}^\vee)$, and $w_{\bar \lambda}$ is the generator whose $\mf{h}^\sigma$-weight is $-c\iota(\bar\lambda)$. Here, $(, )$ is the normalized Killing form on $\mf{g}$ and $\bar{\alpha}^\vee$ is regarded as an element in $\mf{h}^\sigma$.

See <cit.> for part (1), and <cit.> for part (2).

With the same setup and assumption as in Theorem <ref>, we have

* For any $\lambda\in X_*(T)^+$, the Demazure module $D(1,\lambda)$ is isomorphic to the Weyl module $W(\iota(\lambda))$.

* For any $\bar{\lambda}\in X_*(T)_\sigma^+$, the twisted Demazure module $D^\sigma(1,\bar{\lambda})$ is isomorphic to the twisted Weyl module $W^\sigma(\iota(\bar{\lambda}))$.

For part (1), see <cit.>. For part (2), see <cit.> and <cit.> when $\mf{g}$ is not type $A_{2\ell}$; see <cit.> and <cit.> when $\mf{g}$ is $A_{2\ell}$.

In view of (<ref>), by this Corollary any Weyl module of $\gs$ is isomorphic to a twisted Demazure.

§ GLOBAL DEMAZURE MODULES OF TWISTED CURRENT ALGEBRA

In this section, we define twisted global Demazure modules and study their basic properties. We also prove that when the level is $1$, the global Demazure modules can be identified with twisted global Weyl modules.

§.§ General twisted global modules

Let $V$ be a cyclic graded finite dimensional $\mf{g}[t]$-module with a cyclic vector $v_\mu$ of weight $\mu\in P^+$ such that

\begin{equation}\label{324482}
t\mf{h}[t]\cdot v_\mu=0,
\end{equation}

and the degree of $t$ is $1$. For any $x\in \mf{g}$, $v\in V$, we denote by $xt^s$ the element $x\otimes t^s$ in $\mf{g}[t]$. We denote by $v[f]$ the vector $v\otimes f$ in the vector space $V[z]:=V\otimes \C[z]$, for any $v\in V$ and $f\in \mathbb{C}[z]$. Following <cit.>, we define a $\mf{g}[t]$-module structure on $V[z]$ as follows,

\begin{equation}\label{554908}
xt^s \bu v[f]=\sum_{j=0}^{s} (-1)^{s-j}\binom{s}{j} ( xt^j\cdot v)[f\cdot z^{s-j}]
\end{equation}

where $l,k\geq 0, x\in \mf{g}, v\in V$. It was shown in <cit.> that $V[z]$ is a cyclic module with the cyclic generator $v[1]$.

Consider the cyclic graded finite dimensional $\mf{g}[t]$-modules $V_1,\ldots, V_n$, where $V_i$ has a cyclic vector $v_i$ of weight $\mu_i$ such that $t\mf{h}[t]\cdot v_i=0.$ Set

\begin{equation}
\label{eq_w}
\textstyle w:=v_1[1]\otimes v_2[1]\otimes \cdots \otimes v_n[1]\in \bigotimes_i V_i[z_i]. \end{equation}

Similar to the untwisted case in [9], we define the twisted global module $R^\sigma(V_1,\ldots, V_n)$ to be a cyclic $\U(\gs)$-module generated by the vector $w$ as follows,

\[R^\sigma(V_1,\ldots, V_n):=\U(\gs) \bu w \subseteq \bigotimes_{i=1}^n V_i[z_i].\]

Moreover, we define a right $\U(\hs)$-action on $R^\sigma(V_1,\ldots, V_n)$ by

\begin{equation}\label{936369}
(u\bu w)\ h=u \bu (h\bu w),
\end{equation}

where $u\in \U(\gs), h\in\U(\hs)$.

This action (<ref>) is well-defined.

It suffices to show that if $u\bu w=0$ for some $u\in \U(\gs)$, then

\[u\bu (ht^{s}\bu w)=0\]

for any $ht^s\in \hs$. It follows from the definition (<ref>) with the following calculations:

\begin{align}\label{423239}
\begin{split}
u\bu (ht^{s}\bu w)
& = u\bu \big(ht^{s}\bu ( v_1[1]\otimes\cdots\otimes v_n[1]) \big)\\
& = u\bu \bigg(\sum_{i=1}^n v_1[1]\otimes \cdots \otimes (ht^{s}\bu v_i[1])\otimes \cdots\otimes v_n[1]\bigg)\\
& = u\bu\bigg(\sum_{i=1}^n (-1)^{s}\mu_i(h) v_1[1]\otimes \cdots \otimes v_i[z_i^{s}]\otimes \cdots\otimes v_n[1]\bigg)\\
& = \sum_{i=1}^n (-1)^{s} \mu_i(h) u\bu \big(v_1[1]\otimes \cdots \otimes v_i[z_i^{s}]\otimes \cdots\otimes v_n[1]\big)\\
& = 0 \qedhere
\end{split}
\end{align}

From the proof of Lemma <ref>, we know the action (<ref>) commutes with the $\gs$-action, and only depends on the weights $\mu_1,\ldots,\mu_n$. Set $\vec{\mu}=(\mu_1,\ldots,\mu_n)$ and we define

\begin{equation}\label{749134}
\As(\vec\mu):=\U(\hs)/\mr{Ann}(w)
\end{equation}

as the quotient of $\U(\hs)$ by the annihilator of $w$ defined in (<ref>). Following <cit.>, we call $\As(\vec\mu)$ a twisted weight algebra associated to $w$. Then, the twisted global module $R^\sigma(V_1,\ldots, V_n)$ is a $(\gs,\As(\vec\mu))$-bimodule.

Consider the following embedding

\begin{equation}\label{234983}
\As(\vec\mu)\simeq \As(\vec\mu)\bu w\hookrightarrow \C[z_1,z_2,\ldots, z_n],
\end{equation}

given by $ht^s\mapsto (-1)^s(\mu_1(h)z_1^s+\cdots +\mu_n(h) z_n^s)$. Then, $\As(\vec\mu)$ can be realized as the algebra generated by the following elements

\[\{\mu_1(h)z_1^{ms+j} +\mu_2(h)z_2^{ms+j} +\cdots +\mu_n(h)z_n^{ms+j} \mid h\in \mf{h}_j, 0\leq j \leq m-1, s\geq 0\},\]

where $\mf{h}_j:=\{h\in \mf{h} \mid \sigma(h)=\epsilon^j h\}$.

Let $v$ be a vector of weight $\omega_j$, where $j\in I$. For any $ht^s\in \hs$, $ht^s\bu v[1]=(-1)^s\omega_j(h)v[z^s]$ is non-zero if and only if $m_j\mid s$, where $m_j$ is the order of the stablizer of the vertex $j$ under the action of the cyclic group $\langle \sigma\rangle$. Thus, $\As(\omega_j)\simeq \C[z^{m_j}]$. In particular, when $\mf{g}$ is $A_{2\ell}$, $m_j=2$ for any $j$.

Similar to the untwisted case in [9], we have the following result.

$\As(\vec\mu)$ is a Noetherian ring.

Consider an element $h\in\mf{h}^\sigma$ such that $(\mu_{i_1}+\cdots+\mu_{i_k})(h)\neq 0$ for any subset $\{i_1,\ldots i_k\}\subset \{1,\ldots, n\}$. Let $\mc{A}^\sigma(\vec\mu)_h$ be the subalgebra of $\mb{C}[z_1,z_2,\ldots, z_n]$ generated by

\[\{\mu_1(h)z_1^{ms} +\mu_2(h)z_2^{ms} +\cdots +\mu_n(h)z_n^{ms} \mid s\geq 0\}.\]

Then we have the following embeddings

\[\mc{A}^\sigma(\vec{\mu})_h \hookrightarrow \As(\vec{\mu}) \hookrightarrow \bigotimes_{i=1}^n \mc{A}^\sigma(\mu_i). \]

It was shown in <cit.> that $\bigotimes_{i=1}^n \mc{A}^\sigma(\mu_i)$ is finite over $\mc{A}^\sigma_h(\vec\mu) $, and hence is finite over $\As(\vec\mu)$. Note that $\mc{A}^\sigma(\vec\mu)_h $ is Noetherian. By Artin-Tate Lemma <cit.>, $\As(\vec\mu)$ is finitely generated over $\mc{A}^\sigma(\vec\mu)_h $. Thus, $\As(\vec\mu)$ is also Noetherian.

The definitions and results for $R^\sigma(V_1,\ldots, V_n)$ and $\As(\vec{\mu})$ in this section also works when every $\mu_i$ is anti-dominant.

In the rest part of this subsection, we discuss the fibers of twisted global modules.

For a point $p\in \A^1-\{0\}$ and a $\mf{g}[t]$-module $V$, we denote by $V_p$ the module $V$ with a new $\mf{g}[t]$-action with a shift by $p$, i.e. it is given by

\begin{equation}\label{342834}
xt^s\bu v:=x(t-p)^s\cdot v,
\end{equation}

where $xt^s\in \mf{g}[t]$, $v\in V$.

Let $V_1$ be a cyclic finite dimensional graded $\gs$-module, $V_2,\ldots, V_n$ be cyclic finite dimensional graded $\mf{g}[t]$-modules. Let $\vec{p}\in \A^n$ such that $p_1=0, p_i^m\neq p_j^m$ for any $i\neq j$. Then, the tensor product $V_1\otimes \bigotimes_{i=2}^n V_{i,{p_i}}$ is a cyclic $\U(\gs)$-module.

Cf. <cit.>. One can also prove this proposition by using a similar method in <cit.>.

When $V_1$ is a trivial module, we get the following special case.

Let $\vec{p}\in \A^n$ such that $p_i^m\neq p_j^m\neq 0$ for any $i\neq j$. The tensor product $\bigotimes_{i=1}^n V_{i,p_i}$ is a cyclic $\U(\gs)$-module.
The $\U(\gs)$-module $ \bigotimes_{i=1}^n V_{i,{p_i}}$ is cyclic, and hence has a filtration induced from the $t$-grading on $\gs$. Let $ \mr{gr}\, \big(\bigotimes_{i=1}^n V_{i,{p_i}}\big)$ denote the associated graded $\gs$-module of $ \bigotimes_{i=1}^n V_{i,{p_i}}$.

Let $\vec{p}\in \A^n$ such that $p_i^m\neq p_j^m\neq 0$ for any $i\neq j$.  For each $1\leq i\leq n$, let $V_i$ be a cyclic graded finite dimensional $\mf{g}[t]$-module with a cyclic vector $v_{i}$ of weight $\mu_i\in P^+$ such that $t\mf{h}[t]\cdot v_i=0$.  There is a surjective morphism of $\gs$-modules

\begin{equation}\label{719856}
R^\sigma(V_1,\ldots,V_n)\otimes_{\As(\vec\mu)} \C_{\vec{p}}\twoheadrightarrow \bigotimes_{i=1}^n V_{i,{p_i}}.
\end{equation}

Consider the composition of morphisms of $\gs$-modules

\[ R^\sigma(V_1,\ldots,V_n)\otimes_{\As(\vec\mu)} \C_{\vec{p}}\to \bigg(\bigotimes_{i=1}^n V_i[z_i]\bigg)\otimes_{\As(\vec\mu)} \C_{\vec{p}} \twoheadrightarrow \bigotimes_{i=1}^n V_{i,{p_i}}.\]

Now the proposition follows from Corollary <ref> which asserts that the right hand side is a cyclic $\gs$-module.

We will show in Theorem <ref> that the map (<ref>) is an isomorphism for Demazure modules. The following theorem is a special case of <cit.>, which will be crucially used in Section <ref>.

Let $\vec{p}\in \A^n$ such that $p_i^m\neq p_j^m\neq0$ for any $i\neq j$. Given $\lambda_1,\ldots,\lambda_n \in X_*(T)^+$, let $\bar{\lambda}$ be the image of $\sum_i\lambda_i $ in $X_*(T)_\sigma$. Then, there is an isomorphism of $\gs$-modules

\[ \pushQED{\qed} D^\sigma(c,\bar{\lambda})\simeq \mr{gr}\,\bigg( \bigotimes_{i=1}^n D(c,\lambda_i)_{p_i}\bigg). \qedhere\popQED \]

§.§ Twisted global Demazure modules

In this subsection, we define the global Demazure modules of twisted current algebras, and study their basic properties.

Given $\la=(\lambda_1,\ldots,\lambda_n)\in (X_*(T)^+)^n$, for each $i$, let $v_i$ be the cyclic generator of the Demazure module $D(c,\lambda_i)$ of $\mf{g}[t]$ which is of weight $-c\iota(\lambda_i)$, as in $(\ref{eq_Demazure})$. Recall Section <ref>, one can define the following twisted global module

\begin{equation*}
R^\sigma\big( D(c,\lambda_1), \ldots, D(c,\lambda_n)\big)=\U(\gs)\bu w,
\end{equation*}

where $w=v_{1}[1]\otimes\cdots\otimes v_{n}[1]\in \bigotimes D(c,\lambda_i)[z_i]$. Moreover, this global module has a right $\mc{A}^\sigma(c,\la)$-action, where $\As(c,\la)=\U(\hs)/\mr{Ann}(w)$ is the twisted weight algebra associated to $w$. We remark that $\As(c,\la)=\As(-c\iota(\la))$, see (<ref>). By (<ref>), we have an embedding $\As(c,\la)\hookrightarrow \C[\A^n]:=\C[z_1,\ldots,z_n]$

We defined the twisted global Demazure module $\Ds(c,\la)$ to be the following $(\gs, \As(c,\la))$-bimodule

\begin{equation}\label{eq_D_tilde}
\Ds(c,\la):= R^\sigma\big( D(c,\lambda_1), \ldots, D(c,\lambda_n)\big).
\end{equation}

We will denote by $\Ds(c,\la)_{\vec{p} }:=\Ds(c,\la)\otimes_{\As(c,\la)} \C_{\vec{p}}$ the fiber at a point $\vec{p}\in \A^n$.

We now determine the generic fibers of the twisted global Demazure module.

With the same setup in Theorem <ref>, there is a surjective $\gs$-morphism

\begin{equation}\label{485770}
\Ds(c,\la)_{\vec{p}} \twoheadrightarrow \bigotimes_{i=1}^n D (c, \lambda_i )_{p_i}.
\end{equation}

As a consequence, there is an inequality

\begin{equation}\label{394520}
\dim \big(\Ds(c,\la)_{\vec{p}}\big)\geq \dim \bigg(\bigotimes_{i=1}^n D (c, \lambda_i )_{p_i} \bigg) = \dim \big( D^{\sigma} (c, \bar{\lambda}) \big) .
\end{equation}

The surjective map is given by Proposition <ref>. The inequality follows from the surjection (<ref>) and Theorem <ref>.

Given $\lambda_1,\ldots,\lambda_n \in X_*(T)^+$, let $\bar{\lambda}$ be the image of $\lambda:=\sum_i\lambda_i $ in $X_*(T)^+_\sigma$. There is a $\gs$-isomorphism

\[\Ds(c,\la)_0 \simeq D^\sigma(c,\lb).\]

Our proof is similar to the proofs in <cit.> and <cit.>.

It is known that the function $\varphi(\vec{p}):=\dim \big(\Ds(c,\la)_{\vec{p}}\big)$ is upper semicontinuous, cf. <cit.>. Note that for any $\vec{p}\neq 0$, the function $\varphi$ is a constant on the punctured line $\{a\vec{p}\mid a\in \C^*\}$. It follows that

\begin{equation}\label{213578}
\varphi({0})\geq \varphi(\vec{p}), \text{ for any } \vec{p}\in \A^n.
\end{equation}

Combining with (<ref>), we get

\begin{equation}\label{634830}
\dim \big(\Ds(c,\la)_0\big)\geq \dim \big(D^\sigma(c,\lb)\big).
\end{equation}

From Theorem <ref>, $D^\sigma( c, \bar{\lambda} )$ is isomorphic to the quotient of twisted Weyl module $W^{\sigma}(c\iota(\bar{\lambda}))$ by a submodule $M$, where $M$ is generated by the elements in Theorem <ref> part (2). To prove the theorem, it suffices to show there is a surjective map

\[ D^\sigma(c,\lb) \simto W^{\sigma}(c\iota(\bar{\lambda}))/M \twoheadrightarrow \Ds(c,\la)_0.\]

We first show that the map

\begin{equation}\label{428476}
\psi: W^{\sigma}(c\iota(\bar{\lambda})) \twoheadrightarrow \Ds(c,\la)\otimes_{\As(c,\la)} \C_0
\end{equation}

sending $w_{\bar{\lambda}}$ to $w\otimes 1$ is well-defined. It is clear that the first three defining relations in (<ref>) for $w_{\bar{\lambda}}$ also holds for $w\otimes 1$. The reason is that $w=v_1[1]\otimes v_2[1]\otimes \cdots \otimes v_n[1]$ and by part (1) of Theorem <ref> $v_i$ satisfies the following relations

\begin{equation}\label{428623}
\mf{n}^-[t]\cdot v_i=0, \ \mf{h}\otimes t\C[t]\cdot v_i=0 ,\ (h\otimes 1)\cdot v =-c(\lambda_i,h)v_i,\ (e_{\alpha}\otimes t^s)^{k_{\alpha,s}+1}\cdot v_i=0,
\end{equation}

where $\alpha \in R^+$, $k_{\alpha,s}=c\max\{0,(\lambda,\alpha^\vee)-s\}$.

We now check the last relation in (<ref>). When $\mf{g}$ is not $A_{2\ell}$, for any $\alpha\in R^+$, if $\alpha$ is not fixed by $\sigma$, then the roots $\alpha, \sigma(\alpha),\ldots, \sigma^{m-1}(\alpha)$ are all distinct and orthogonal, cf. <cit.>. In this case, we have $k_\alpha=c(\lambda,\bar{ \alpha}^\vee)\geq c\sum_{j=0}^{m-1} ({\lambda},\ {\sigma^{j}(\alpha^\vee)})$, and

\begin{equation}\label{eq_last_rel}
(e_{\alpha,0})^{k_\alpha+1}\bu w= \bigg(\sum_{j=0}^{m-1} e_{\sigma^j(\alpha)}\otimes 1\bigg)^{k_\alpha+1}\bu w=\sum_{\sum s_j =k_\alpha+1} \prod_{j=0}^{m-1} (e_{\sigma^j(\alpha)}\otimes 1)^{s_j} \bu w.
\end{equation}

In each term of the right side, there must be an $s_j\geq c(\lambda, \sigma^{j}(\alpha^\vee))+1=k_{\sigma^j(\alpha),0}+1$. For this $s_j$, by the last relation in (<ref>), we have $(e_{\sigma^j(\alpha)}\otimes 1)^{s_j} \bu (v_i[1])=0$, $1\leq i\leq n$. This shows that the right side of (<ref>) is $0$. Hence, the last relation in (<ref>) holds in this case. When $\alpha$ is fixed by $\sigma$, this relation can be checked similarly (actually easier).

When $\mf{g}$ is of type $A_{2\ell}$ and $m=4$, the element $e_{\alpha,0}\in \gs$ if and only if $\alpha\neq\alpha_{i,\ell}$, see Section <ref>. Thus, $\alpha$ and $\sigma(\alpha) $ are either equal or orthogonal. By the same argument as above, we still have $(e_{{\alpha},0})^{k_\alpha+1}\bu w= 0$, where $\alpha\neq\alpha_{i,\ell}$ and $k_\alpha=c(\lambda, \bar{\alpha}^\vee)$. This finishes the checking of all relations in (<ref>). Thus, the map (<ref>) is well-defined.

To complete the proof, we still need to show $\psi(M)=0$. When $\mf{g}$ is not $A_{2\ell}$, the submodule $M$ is generated by the following elements (see Theorem <ref>)

\[\{e_{\alpha,m s_{\alpha}}\cdot w_{\lb} \mid \alpha \in R^+_{l} \} \cup \{ e_{\alpha, s_{\alpha} } \cdot w_{\lb} \mid \alpha \in R^+_{s}\}. \]

By the equalities in <cit.> and (<ref>), we always have $s_\alpha=c(\lambda,\bar{ \alpha}^\vee)\geq c(\lambda, \sigma^j(\alpha^\vee))$ for any $\alpha\in R^+$ and $0\leq j\leq m-1$. When $\alpha\in R^+_l$, by definition $\bar{\alpha}$ is a long root of $\mf{g}^\sigma$. Then, $(\lambda,\sigma^j(\alpha^\vee))-ms_{\alpha}\leq (\lambda,\sigma^j(\alpha^\vee))-mc({\lambda},\ {\sigma^{j}(\alpha^\vee) })\leq 0$, which implies

\begin{equation}\label{eq_psi_M}
e_{\alpha,ms_\alpha}\bu w= \sum_{i=0}^{m-1} e_{\sigma^i(\alpha)}\otimes t^{ms_\alpha}\bu w=0.
\end{equation}

Similarly, one can show that $ e_{\alpha,s_\alpha}\bu w=0$ for any $\alpha\in R^+_s$. Thus, $\psi(M)=0$.

When $\mf{g}$ is of type $A_{2\ell}$, the submodule $M$ is generated by the following elements

\begin{align*}
&\{e_{\alpha,4s_\alpha}\cdot w_{\lb} \mid \alpha=\alpha_{i,2\ell+1-i}, 1\leq i<\ell \} \cup \{e_{\alpha, 2s_{2\alpha}+1}\cdot w_{\lb} \mid \alpha=\alpha_{i,\ell}, 1\leq i\leq \ell \}\\
&\cup \{e_{\alpha,2s_\alpha}\cdot w_{\lb} \mid \alpha=\alpha_{i,j} \text{ or } \alpha_{i,2\ell-j}, 1\leq i<\ell\} .
\end{align*}

When $\alpha\neq \alpha_{i,\ell}$, the relations can be checked similarly as in (<ref>). Now, suppose $\alpha=\alpha_{i,\ell}$, and let $s=s_{2\alpha}=c(\lambda,\alpha^\vee+\tau(\alpha^\vee))$. We have

\[(e_{\alpha, 2s+1})\bu w= (e_{\alpha}\otimes t^{2s+1} + (-1)^{i+\ell+s}e_{\tau(\alpha)}\otimes t^{2s+1})\bu w=0,\]

since $(\lambda,\tau^p(\alpha^\vee))-(2s+1)=(\lambda,\tau^p(\alpha^\vee) )-2 c(\lambda, \alpha^\vee+\tau(\alpha^\vee))-1< 0$ for $p=0$, or $1$. Thus, in this case we also have $\psi(M)=0$.

Now, the map $\psi$ induces a surjective morphism

\[\bar\psi: D^\sigma(c,\lb) \simto W^{\sigma}(c\iota(\bar{\lambda}))/M \twoheadrightarrow \Ds(c,\la)_0.\]

By the dimension inequality (<ref>), $\bar\psi$ must be an isomorphism.

Given $\lambda_1,\ldots,\lambda_n \in X_*(T)^+$, let $\bar{\lambda}$ be the image of $\sum_i\lambda_i $ in $X_*(T)^+_\sigma$.

* For any $\vec{p}\in \A^n$ such that $p_i^m\neq p_j^m\neq0$ for any $i\neq j$, there is an isomorphism of $\gs$-modules

\[\Ds(c,\la)_{\vec{p}} \simeq \bigotimes_{i=1}^n D (c, \lambda_i )_{p_i}.\]

* The twisted global Demazure module $\Ds(c,\la)$ is free over $\As(c,\la)$.

By Theorem <ref>, we have $\dim \big(\Ds(c,\la)_0\big)= \dim \big( D^{\sigma} (c, \bar{\lambda}) \big)$. Combining with (<ref>), we get

\begin{equation}\label{528923}
\dim \big( D^{\sigma} (c, \bar{\lambda}) \big) \geq \dim \big(\Ds(c,\la)_{\vec{p}}\big)
\end{equation}

for any $\vec{p}\in\A^n$. It follows that all three terms in the inequality (<ref>) are equal. Hence the surjection (<ref>) is an isomorphism. This proves part (1).

From part (1), we get $\dim \big(\Ds(c,\la)_{\vec{p}}\big)= \dim \big( D^{\sigma} (c, \bar{\lambda}) \big)$ for any $\vec{p}$ such that $p_i^m\neq p_j^m\neq 0, \forall i\neq j$. By the semicontinuity of the function $\varphi(\vec{p}):=\dim \big(\Ds(c,\la)_{\vec{p}}\big)$, we get

\[ \dim \big(\Ds(c,\la)_{\vec{p}}\big)\geq \dim \big( D^{\sigma} (c, \bar{\lambda}) \big)\]

for any $\vec{p}\in \A^n$. Combining with (<ref>), this shows that $\varphi(\vec{p})$ is a constant function. Thus, $\Ds(c,\la)$ is a graded projective $\As(c,\la)$-module. Using the graded Nakayama lemma (cf. <cit.>), $\Ds(c,\la)$ is a free $\As(c,\la)$-module.

In Section <ref>, we will use the geometry of Beilinson-Drinfeld Schubert varieties to fully describe the fibers of $\Ds(c,\la)$ at any point in $\A^n$, see Corollary <ref>.

We now introduce some notations, which will be frequently used in this paper.

Denote by $[n]$ the finite set $\{1, 2,\ldots, n\}$. Let $\xi=(I_1, I_2,\ldots, I_k)$ be a partition of $[n]$. We define $\A^n_{\xi}$ to be the following open subset in $\A^n$,

\begin{equation}\label{345790}
\A^n_{\xi}=\{ \vec{p}\in\A^n \mid {p}_i^m \neq {p}_j^m,\ \forall i\in I_\alpha, j\in I_\beta, \alpha\neq\beta \}.
\end{equation}

For any subset $I=\{i_1,\ldots,i_s\}\subseteq [n]$, set $\A^{I} ={\rm Spec\,}\mathbb{C}[z_{i_1}, \ldots, z_{i_s}] $.

When $\xi=([n])$ is the trivial partition of $[n]$, we have $\A^n_{\xi}=\A^n$; when $\xi_0=(\{1\},\{2\},\ldots,\{n\})$ is the finest partition, we have $\A^n_{\xi_0}=\{ \vec{p}\in\A^n \mid \bar{p}_i\neq \bar{p}_j,\ \forall i\neq j \}$.

Given a partition $\xi=(I_1, I_2,\ldots, I_k)$ of $[n]$, for any $1\leq j\leq k$, we label elements in $I_j=\{j_1,\ldots,j_s\}$ by the natural order of integers, i.e. $j_1<j_2<\cdots <j_s$, where $s$ is the cardinality of $I_j$.
For any point $\vec{p}=(p_1,\ldots,p_n) \in \A^n_{\xi}$ and $\la\in (X_*(T)^+)^n$, we make the following notations:

\begin{gather}
\label{053076} \vec{p}_{I_j}=(p_{j_1},\ldots,p_{j_s})\in\A^{I_j},\\
\label{249296} \vec{\lambda}_{I_j}=(\lambda_{j_1},\ldots,\lambda_{j_s}),\\
\nonumber w_{I_j}=v_{j_1}[1]\otimes \cdots\otimes v_{j_s}[1]\in \bigotimes_{i=1}^s D(c,\lambda_{j_i})[z_{j_i}].
\end{gather}

For simplicity, we denote by $w_{I_j}[1]$ the generator $w_{I_j}\otimes 1$ of
\[\Ds(c,\la_{I_j})_{\A^{I_j}}:=\Ds(c,\la_{I_j})\otimes_{\As(c,\la_{I_j})} \C[\A^{I_j}].\]

* For any $\lambda\in X_*(T)^+$, there is an embedding of $\gs$-modules:

\[\phi: \Ds(c,\lambda)_{\A^1}\hookrightarrow D(c, \lambda)[z]\]

defined by $(u\bu w)[f]\mapsto u\bu (v[f])$ for any $u\in \gs$ and $f\in\C[z]$, where $w:=v[1]$ and $v$ is the cyclic generator of $D(c, \lambda)$ as in (<ref>).

* Let $\xi=(I,J)$ be a partition of $[n]$. There is an embedding of $\gs$-modules:

\begin{equation}\label{eq_psi}
\psi_\xi:\Ds(c,\la)_{\A^n}\otimes_{\C[\A^n]} \C[\A^n_{\xi}]\hookrightarrow \big(\Ds(c,\vec{\lambda}_I)_{\A^I} \otimes \Ds(c,\vec{\lambda}_J)_{\A^J} \big)\otimes_{\C[\A^I\times A^J]} \C[\A^n_{\xi}]\end{equation}

defined by $(u\bu w)[f]\otimes g\mapsto \big((u\bu w_I)[1]\otimes (u\bu w_J)[1]\big)\otimes (fg)$ for any $u\in\gs$, $f\in \C[\A^n]$, and $g\in \C[\A^n_\xi]$.

(1) By Theorem <ref>, the map $\phi$ restricts to an isomorphism $\Ds(c,\lambda)_p \simeq D(c,\lambda)_p$ at any $p\in \A^1\backslash \{0\}$. Thus, $\phi$ is injective.

(2) The map $\psi_\xi$ is well defined since $u$ acts on $w$ via Leibniz rule. Again, by Theorem <ref>, it restricts to an isomorphism

\[\Ds(c,\la)_{\vec{p}}\simeq \Ds(c,\la_{I})_{\vec{p}_I}\otimes \Ds(c,\la_{J})_{\vec{p}_J},\]

for any $\vec{p}$ such that $p_i^m\neq p_j^m\neq0, \forall i\neq j$. It follows that the map $\psi_\xi$ is injective.

In Section <ref>, we will show the injective map $\psi_\xi$ is actually an isomorphism, see Theorem <ref>.

§.§ Twisted global Weyl modules

In this subsection, we identify the twisted global Weyl module defined in Section <ref> with the level one twisted global Demazure module.

Taking $c=1$ in Theorem <ref> and Theorem <ref> and applying Corollary <ref>, we get the following results.

Given a tuple of weights $\vec\mu=(\mu_1,\ldots,\mu_n)\in (P^+)^n$, let $W(\mu_i)$ be the local Weyl module defined in Section <ref>. Then

* The  global module $R^\sigma(W(\mu_1),\ldots,W(\mu_n))$ is a graded free $\As(-\vec\mu)$-module.

* Let $\bar\mu\in P_\sigma^+$ be the restriction of $\sum \mu_i$ to $\mf{h}^\sigma$. There is a $\gs$-isomorphism

\[\pushQED{\qed} R^\sigma(W(\mu_1),\ldots,W(\mu_n))\otimes_{\As(-\vec\mu)}\C_0\simeq W^\sigma(\bar\mu). \qedhere\popQED \]

Given any fundamental weights $\mu_1,\ldots,\mu_n\in P^+$, let $\bar\mu\in P_\sigma^+$ be the restriction of $\sum \mu_i$ to $\mf{h}^\sigma$. Then, there is a $\gs$-isomorphism

\[\mb{W}^\sigma(\bar\mu)\simeq R^\sigma(W(\mu_1),\ldots,W(\mu_n)),\]

and an isomorphism $A^\sigma(\bar\mu)\simeq \As(-\vec{\mu})$ of $\C$-algebras, where $\vec{\mu}=(\mu_1,\ldots,\mu_n)$.

Let $w_i\in W(\mu_i)$ (resp. $w_{\bar\mu}\in \mb{W}^\sigma(\bar\mu)$) denote the cyclic generator with weight $-\mu_i$ (resp. $-\bar\mu$). Similar to the map (<ref>), we have a surjective $\gs$-morphism

\begin{equation}\label{234571}
\mb{W}^\sigma(\bar\mu)\twoheadrightarrow R^\sigma(W(\mu_1),\ldots,W(\mu_n)),
\end{equation}

given by $w_{\bar{\mu}}\mapsto w:=w_1[1]\otimes\cdots\otimes w_n[1]$. This induces a surjective map

\begin{equation}\label{324821}
A^\sigma(\bar\mu)=\U(\hs)/\mr{Ann}(w_{\bar\mu})\twoheadrightarrow \U(\hs)/\mr{Ann}(w)=\As(-\vec\mu).
\end{equation}

We claim this is an isomorphism. We only prove the case when $\mf{g}$ is of type $A_{2\ell}$. The other types can be checked similarly.

Recall Theorem <ref> that $A^\sigma(\bar\mu)$ is a a graded polynomial algebra in variables $T_{i,r}$ of grade $2r$, where $1\leq i\leq \ell$ and $1\leq r\leq \bar\mu(\beta_i^\vee)$. Set $n_i:=\bar\mu(\beta_i^\vee)$. Then, $n=\sum_{i=1}^\ell n_i$. In view of <cit.>, $T_{i,r}$ is the image of $P_{i,r}\in \U(\hs)$ which is defined as follows. For any $1\leq i\leq \ell$, set $P_{i,1}:=h_{i,2}$, which is defined in (<ref>). For $r\geq 2$, we inductively define

\[P_{i,r}:=\frac{1}{r}\sum_{s=0}^r h_{i,2s+2}P_{i,r-s-1}.\]

Now, we determine the image of $T_{i,r}$ under the map (<ref>) by examining the action of $P_{i,r}$ on $w$, where $1\leq i\leq \ell$ and $1\leq r\leq n_i$. Suppose $n_i\geq 1$, otherwise $P_{i,r}$ will not appear. From <cit.>, we have

\begin{equation}\label{324983}
P_{i,r}\bu w=\sum_{j_1+\cdots+j_n=r} \big(P_{i,j_1}\bu w_1[1]\big)\otimes\cdots\otimes \big(P_{i, j_n}\bu w_n[1]\big),
\end{equation}

where $j_k\geq 0$ and by convention $P_{i,0}=1$. Observe that $h_{i,2s+2}\bu w_k[1]=\mu_k(h_i+(-1)^{s+1}h_{2\ell+1-i})w_k[z_k^{2s+2}]$ is nonzero only when $\mu_k=\omega_i \text{ or } \omega_{2\ell+1-i}$. Without loss of generality, suppose $\mu_k=\omega_i$ for $1\leq k\leq n_i'$, and $\mu=\omega_{2\ell+1-i}$ for $n_i'+1\leq k\leq n_i$, where $n_i'$ is the number of $\omega_i$ appearing in $\vec{\mu}$. Then, $P_{i,j_k}\bu w_k[1]=0$ whenever $j_k>1$, or $k>r$ and $j_k\geq 1$. Thus, the equation (<ref>) becomes

\begin{equation}\label{432983}
P_{i,r}\bu w= \sum_{j_1+\cdots+j_{n_i}=r} (-1)^{a}\big(w_1[z_1^{2j_1}]\otimes\cdots\otimes w_{n_i}[z_{n_i}^{2j_{n_i}}]\otimes w_{n_i+1}[1]\otimes \cdots\otimes w_{n}[1]\big),
\end{equation}

where $0\leq j_k\leq 1$ and $a=j_{n_i'+1}+\cdots+j_{n_i}$.

Recall the embedding $\As(-\vec\mu)\hookrightarrow \C[z_1,\ldots,z_n]$ defined in (<ref>). Composing it with the map (<ref>), we get a morphism

\begin{equation}\label{298232}
A^\sigma(\bar\mu)\to \C[z_1,\ldots,z_n].
\end{equation}

Given $1\leq i\leq \ell$, we define $K_i= \{1\leq k\leq n\mid \mu_k=\omega_i\text{ or } \omega_{2\ell+1-i}\}$, which is of cardinality $n_i$. The symmetric group $S_{n_i}$ permutes $K_i$. Let $S$ be the Young subgroup $S_{n_1}\times\cdots\times S_{n_\ell}$ of  $S_n$. For any $1\leq k\leq n$, set $\delta_k=1$ if $\mu_k=\omega_i$; $\delta_k=\sqrt{-1}$ if $\mu_k=\omega_{2\ell+1-i}$. We define the following polynomial

\[f_{i,r}:=\sum_{k\in K_i,\ {\scriptscriptstyle\sum} j_k=r} (\delta_kz_k)^{2j_k}\in \C\big[(\delta_1z_1)^2,\ldots, (\delta_nz_n)^2\big]^S. \]

From the computation (<ref>), the map (<ref>) sends $T_{i,r}$ to $f_{i,r}$. Note that $\{f_{i,r}\mid 1\leq i\leq \ell, 1\leq r\leq n_i\}$ is exactly a set of  algebraically independent generators of the algebra $\C\big[(\delta_1z_1)^2,\ldots, (\delta_nz_n)^2\big]^S$. We conclude that (<ref>) is an embedding. It follows that the surjective map (<ref>) is an isomorphism.

Now, both sides of (<ref>) are graded free $A^\sigma(\bar\mu)\simeq \As(-\vec\mu)$-modules, see Theorem <ref> and Lemma <ref>. Moreover, their fibers at $0$ are the twisted local Weyl module $W^\sigma(\bar\mu)$. By comparing their graded characters (as graded $\mf{h}^\sigma$-modules), we conclude that (<ref>) is an isomorphism.

Given fundamental coweights $\lambda_1,\ldots, \lambda_n \in X_*(T)^+$ with $\bar{\lambda}$ being the image of $\sum \lambda_i$ in $X_*(T)^+_\sigma$, we have an $\gs$-isomorphism

\[\Ds(1,\vec{\lambda})\simeq \mb{W}^\sigma(\iota(\lb)),\]

and an isomorphism of $\C$-algebras $\As(1,\la)\simeq A^\sigma(\iota(\bar\lambda))$.

It directly follows from Theorem <ref> and Corollary <ref>.

From part (1) of Theorem <ref> and Theorem <ref>, we are able to determine the algebra $\As(c,\vec{\lambda})$ when $c=1$ and $\vec{\lambda}$ is a collection of fundamental coweights.

§ BEILINSON-DRINFELD GRASSMANNIANS OF PARAHORIC BRUHAT-TITS GROUP SCHEMES

In this section, we define Beilinson-Drinfeld Schubert varieties of certain parahoric Bruhat-Tits group schemes, and we prove their flatness over the base.

§.§ Beilinson-Drinfeld Grassmannian

Let $G$ be a reductive group over $\mathbb{C}$. Let $\Gamma$ be a finite group acting on $G$ which preserves a maximal torus $T$. Let $C$ be an irreducible smooth algebraic curve over $\C$ with a faithful $\Gamma$-action. Denote by $\bar C$ the quotient curve $C/\Gamma$. Let $\pi: C\to \bar{C}$ be the quotient map, and denote by $\bar{p}:=\pi(p)$ the image of a point $p\in C$. For each point $p\in C$, denote by $\Gamma_p$ the stabilizer of $p$ in $\Gamma$.

We define $\mc{G}_{\bar C}$ to be the $\Gamma$-fixed point subgroup scheme $\mr{Res}_{C/\bar{C}}\big( G\times C\big)^\Gamma $ of the Weil restriction $\mr{Res}_{C/\bar{C}}\big( G\times C\big)$. We will simply denote it by $\mc{G}$ if there is no confusion. Now, we prove an étale base change property of $\mc{G}$ which will be used in Section <ref>.

Let $C_1$, $C_2$ be irreducible smooth curves over $\C$ with a faithful $\Gamma$-actions. Let $f: C_1\to C_2$ be a $\Gamma$-equivariant étale morphism such that $\Gamma_p=\Gamma_{f(q)}$ for any $p\in C_1$. Then the induced map $\bar{f}: \bar C_1\to \bar C_2$ is also étale, and there is an isomorphism

\begin{equation}\label{320259}
C_1\simeq C_2\times_{\bar C_2} \bar C_1.
\end{equation}

We can assume $C_1$, $C_2$ are affine curves. We denote by $\pi_i: C_i\to \bar{C}_i$ the projection maps for $i=1,2$.
Let $K_1$, $K_2$ be their function fields, and $\ms{O}_1$, $\ms{O}_2$ be the coordinate rings. Then we have a diagram of field extensions

\[\xymatrix{
K_1 & \ar[l] K_2 \\
K_1^\sigma \ar[u] & \ar[l] K_2^\sigma \ar[u]

Observing that $K_2\cap K_1^\sigma= K_2^\sigma$, we have

\[ [K_2K_1^\sigma: K_2^\sigma]=[K_2K_1^\sigma: K_2][K_2: K_2^\sigma]=[K_1^\sigma:K_2^\sigma][K_2: K_2^\sigma]=[K_1:K_2^\sigma]. \]
Thus, $K_1=K_2K_1^\sigma$. Since $K_1/K_1^\sigma$ is finite, we have $K_1=K_1^\sigma(\theta)$ for some $\theta\in K_1$, and $f(\theta)=0$ for some polynomial $f$ over $K_1^\sigma$. By multiplying elements in $\ms{O}^\sigma_1$, all coefficients of $f$ can be made to be in $\ms{O}^\sigma_1$. Moreover, by modifying $\theta$, we can further assume $f$ is monic and $f(\theta)=0$. Thus, $\theta$ is integral over $\ms{O}_1^\sigma$. It follows that $\theta$ is integral over $\ms{O}_1$. Since $\ms{O}_1$ is integrally closed in $K_1$, we must have $\theta\in \ms{O}_1$. Hence, $K_1\simeq K^\sigma_1\otimes_{\ms{O}_1^\sigma}\ms{O}_1$, since $\ms{O}_1$ is a free $\ms{O}_1^\sigma$-module of rank $[K_1: K_1^\sigma]$. Similarly, we have $K_2=K^\sigma_2\otimes_{\ms{O}_2^\sigma}\ms{O}_2$. Therefore, we have

\[ \ms{O}_2\otimes_{\ms{O}_2^\sigma} K_1^\sigma \simeq \ms{O}_2\otimes_{\ms{O}_2^\sigma} \ms{O}_1^\sigma \otimes_{\ms{O}_1^\sigma} K_1^\sigma\simeq K_1\simeq \ms{O}_1\otimes_{\ms{O}_1^\sigma} K_1^\sigma.\]

By assumption, $\Gamma_p=\Gamma_{f(p)}$ for any $p\in C_1$. The map $f$ induces an isomorphism $\mb{D}_p\simeq \mb{D}_{f(p)}$ of formal discs around $p$ and $f(p)$, for any $p\in C_1$. Thus, $f$ induces an étale map $\bar f: \bar C_1\to \bar C_2$. It follows that $C_2\times_{\bar C_2}\bar C_1$ is smooth. For any point $(p, q)\in C_2\times_{\bar C_2}\bar C_1$, we have $\pi_2(p)=\bar f(q)$. We pick a point $x$ in $C_1$ such that $\pi_1(x)=q$. Then $\pi_2(f(x) )=\pi_2(p)$. Thus, $f(x)=\gamma ( p)$ for some $\gamma\in \Gamma$. Set $x'=\gamma^{-1}(x)$. Then, $f(x')=p$ and $\pi_1(x')=q$. This shows the surjectivity of the map $C_1\to C_2\times_{\bar C_2}\bar C_1$. Since $C_2\times_{\bar C_2} \bar C_1$ is reduced and $\ms{O}_2\otimes_{\ms{O}_2^\sigma} K_1^\sigma\simeq K_1$, the curve $C_2\times_{\bar C_2}\bar C_1$ is integral and has the same function field as $C_1$. Therefore, we have an isomorphism $C_1\simeq C_2\times_{\bar C_2}\bar C_1$.

Under the same setup as in Lemma <ref> and let $\Gamma$ act on $G$. We have an isomorphism of group schemes over $\bar C_1$

\[\mc{G}_{\bar C_1}\simeq \mc{G}_{\bar C_2}\times_{\bar C_2} \bar C_1.\]

Given any scheme $\varphi:S\to \bar C_1$ over $\bar C_1$, by definition $\mc{G}_{\bar C_1}(S)=\mr{Map}^\Gamma(C_1\times_{\bar C_1}S,G)$ consists of $\Gamma$-equivariant morphisms from $C_1\times_{\bar C_1} S$ to $G$.
On the other hand, $(\mc{G}_{\bar C_2}\times_{\bar C_2} \bar C_1)(S)$ consists of $\Gamma$-equivariant morphisms from $C_2\times_{\bar C_2}S$ to $G$, where $S$ is regarded as a scheme over $\bar C_2$ via $f\circ \varphi:S\to \bar C_2$. By Lemma <ref>, we have an isomorphism

\[C_1\times_{\bar C_1} S \simeq (C_2\times_{\bar C_2}\bar C_1)\times_{\bar C_1} S\simeq C_2\times_{\bar C_2}S.\]

Thus, $\mc{G}_{\bar C_1}(S)\simeq (\mc{G}_{\bar C_2}\times_{\bar C_2} \bar C_1)(S)$ for any scheme $S$ over $\bar C_1$.

For any $\C$-algebra $R$, and any morphism $p:\mr{Spec}(R)\to C$, we denote by $\Delta_p\subseteq C\times \mr{Spec}(R)$ the graph of $p$. Let $C_R:=C\times \mr{Spec}(R)$. We denote by $\hat{\Delta}_p$ the formal completion of $C_R$ along $\Delta_p$, and by $\hat{\Delta}_p^*$ the punctured formal completion along $\Delta_p$.

The Beilinson-Drinfeld Grassmannian $\Gr_{\mc{G},C^n}$ and the jet group scheme $L^+\mc{G}_{C^n}$ over $C^n$ are defined as follows: for any $\C$-algebra $R$, define

\[\Gr_{\mc{G},C^n}(R):= \left\{ (p_1, \ldots, p_n, \mc{F}, \beta ) \,\middle|\, p_i \in C(R), \mc{F} \text{ a } \mc{G}\text{-torsor on } \bar{C}, \beta: \mc{F}|_{\bar{C}_{R} \setminus \cup\Delta_{\bar{p}_i}}\simeq \mathring{\mc{F}}|_{\bar{C}_{R} \setminus \cup\Delta_{\bar{p}_i}} \right\}\]

\[L^+\mc{G}_{C^n}(R):=\left\{ (p_1, \ldots, p_n,\eta ) \,\middle|\, p_i \in C(R), \eta: \mathring{\mc{F}}|_{ \widehat{ \cup{\Delta}_{\bar{p}_i} } }\simeq \mathring{\mc{F}}|_{ \widehat{ \cup{\Delta}_{\bar{p}_i} } }\right\},\]

where $\mathring{\mc{F}}$ is the trivial $\mc{G}$-torsor on $\bar{C}$, and $ \widehat{ \cup{\Delta}_{\bar{p}_i} } $ is the completion of $\bar C_R$ along $ { \cup{\Delta}_{\bar{p}_i} } $.

The BD Grassmannian $\Gr_{\mc{G},C^n}$ is an ind-scheme, ind-of-finite type and ind-projective over $C^n$, see <cit.>.

One can define the BD Grassmannian $\Gr_{\G, \bar{C}^n }$ over $\bar{C}$, see <cit.>. The BD Grassmannian $\Gr_{\G, C^n}$ defined above is the base change $\Gr_{\G,\bar{C}^n } \times_{\bar{C}^n } C^n$.

There is a left $L^+\mc{G}_{C^n}$-action on $\Gr_{\mc{G},C^n}$ given by the following map

\begin{align*}
L^+\mc{G}_{C^n}\times \mr{Conv}_{\mc{G}, C^n} & \to \mr{Conv}_{\mc{G}, C^n} \\
\big( (\vec{p},\eta), (\vec{p}, \mc{F}, \beta) \big) &\mapsto \big(\vec{p}, \mc{F}', \beta' \big),
\end{align*}

where by Beauville-Laszlo's Lemma, there exists a unique $\mc{G}$-torsor $\mc{F}'$ (up to a unique isomorphism) with isomorphisms $\delta:{\mc{F}}'|_{ \widehat{\cup\Delta_{\bar p_i}} }\simeq \mc{F}|_{ \widehat{\cup\Delta_{\bar p_i}} }$ and $\beta': \mc{F}'|_{\bar C_R\setminus \cup\Delta_{\bar p_i} } \simeq \mathring{\mc{F}}|_{\bar C_R\setminus \cup \Delta_{\bar p_i}}$ such that $\beta'\circ\delta^{-1}|_{{\widehat{\cup\Delta_{\bar p_i}} }^*}=\eta\circ \beta|_{{\widehat{\cup\Delta_{\bar p_i}} }^*}$. Here ${\widehat{\cup\Delta_{\bar p_i}} }^*:=\big({{\widehat{\cup\Delta_{\bar p_i}} }}\big)\cap \big(\bar C_R\setminus \cup\Delta_{\bar p_i}\big)$.

The following proposition follows from <cit.> and <cit.>.

* With the same setup as in Proposition <ref>, there exists an isomorphism

\[ \Gr_{\G_{\bar{C}_2}, C_2^n}\times_{C_2^n} C_1^n\simeq \Gr_{\G_{\bar{C}_1}, C_1^n} . \]

Moreover, the $L^+\mc{G}$-action commutes with this base change.

* Let $C$ be an irreducible smooth curve with a faithful $\Gamma$-action such that all ramified points are totally ramified. Let $\mathring{C}$ be the set of unramified points in $C$. Then,

\[ \Gr_{\G, \mathring{C}^n }\simeq \Gr_{G, \mathring{C}^n } ,\]

where $\Gr_{G, \mathring{C}^n }$ is the BD Grassmannian of $G$ over $\mathring{C}^n$. Moreover, we have $L^+\G_{\mathring{C}^n}\simeq L^+G_{\mathring{C}^n}$.

Let $\xi=(I_1,\ldots, I_k)$ be a partition of $[n]$. Similar to Definition <ref>, we define $C_\xi^n$ to be the following open subset in $C^n$,

\[C_\xi^n:=\{(p_1,\ldots,p_n)\in C^n\mid \bar{p}_i\neq \bar{p}_j, \ \forall i\in I_\alpha, j\in J_\beta, \alpha\neq \beta\}.\]

Then, $\Gr_{\mc{G},C^{n}}$ and $L^+\mc{G}_{C^n}$ have a factorization property over $C_\xi^n$ as follows

\begin{align}
\label{447893} \Gr_{\mc{G},C^{n}_\xi}&:=\Gr_{\mc{G},C^{n}}\times_{C^n} C^{n}_\xi \simeq\bigg(\prod_{j=1}^k\Gr_{\mc{G},C^{I_j}}\bigg)\times_{C^n} C^{n}_\xi, \\
\label{346288} L^+\mc{G}_{C^n_\xi}&:=L^+\mc{G}_{C^n}\times_{C^n} C^{n}_\xi \simeq\bigg(\prod_{j=1}^kL^+\mc{G}_{C^{I_j}}\bigg)\times_{C^n} C^{n}_\xi.
\end{align}

§.§ Beilinson-Drinfeld Schubert varieties

With the same setup as in Section <ref>, we define a $\Gamma$-action on $C^{n+1}=C^n\times C$ by letting $\Gamma$ act on the last copy of $C$. More precisely, this action is given by

\begin{equation}\label{038287}
\gamma(p_1,\ldots, p_n, p_{n+1}):=(p_1,\ldots, p_n, \gamma p_{n+1}),
\end{equation}

for any $\gamma\in \Gamma$, $(p_1,\ldots, p_n, p_{n+1})\in C^{n+1}$.

Let $T$ be a $\Gamma$-stable maximal torus in $G$. Given any $\la=(\lambda_1,\lambda_2,\ldots, \lambda_n)$ with $\lambda_i\in X_*(T)$, we shall construct a $\Gamma$-equivariant tensor functor $F$ from the category $\mr{Rep}(T)$ of finite dimensional representations of $T$ to the category $\mr{Coh}^{\mr{lf}}(C^{n+1})$ of locally free sheaves of finite rank over $C^{n+1}$. For each character $\nu$ of $T$, we define

\begin{equation}\label{214694}
F( \C_{\nu}):= \ms{O}_{C^{n+1}}\bigg(\sum_{i=1}^n\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_i), \nu \rangle \Delta_{\gamma, i}\bigg),
\end{equation}

where $\Delta_{\gamma,i}:=\{(a_1, a_2, \ldots, a_n, a_{n+1})\mid a_{n+1}=\gamma a_{i}\}$
is a divisor in $C^{n+1}$.

This assignment (<ref>) extends to a tensor functor $F:\mr{Rep}(G)\to \mr{Coh}^{\mr{lf}}(C^{n+1})$ which is $\Gamma$-equivariant in the sense of Definition <ref>. As a consequence, by Theorem <ref>, this functor $F$ gives a $(\Gamma, T)$-torsor $\mc{F}$ over $C^{n+1}$.

For each $\eta\in \Gamma$, denote by $\eta_T: T\to T$ the map sending $h$ to $\eta (h)$. Let $\eta_T^*:\mr{Rep}(T)\to \mr{Rep}(T)$ be the pullback functor induced from $\eta_T$. Define the $\Gamma$-action on $X^*(T)$ by

\[\eta(\nu)(t):=\nu(\eta^{-1}(t)),\]

where $\eta\in \Gamma$, $\nu\in X^*(T)$, and $t\in T$. Then, we have $\eta_T^*(\C_\nu)=\C_{\eta^{-1}(\nu)}$, and

\begin{equation}\label{826785}
F\eta_T^*( \C_{\nu})= \ms{O}_{C^{n+1}}\bigg(\sum_{i=1}^n\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_i), \eta^{-1}(\nu) \rangle \Delta_{\gamma, i}\bigg)= \ms{O}_{C^{n+1}}\bigg(\sum_{i=1}^n\sum_{\gamma\in \Gamma} \langle \eta\gamma(\lambda_i), \nu \rangle \Delta_{\gamma, i}\bigg),
\end{equation}

for any $\eta\in \Gamma$ and $\nu\in X^*(T)$.

Let $Y:=C^{n+1}$ be the variety with the $\Gamma$-action given in (<ref>). For each $\eta\in \Gamma$, denote by $\eta_Y: Y\to Y$ the map sending $y$ to $\eta(y)$. Let $\eta_Y^*:\mr{Coh}^{\mr{lf}}(Y)\to \mr{Coh}^{\mr{lf}}(Y)$ be the pullback functor induced from $\eta_Y$. Then, there is an isomorphism

\begin{equation}\label{060044}
\eta_Y^*F(\C_\nu) \simto \ms{O}_{C^{n+1}}\bigg(\sum_{i=1}^n\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_i), \nu \rangle \Delta_{\eta^{-1}\gamma, i}\bigg) = \ms{O}_{C^{n+1}}\bigg(\sum_{i=1}^n\sum_{\gamma\in \Gamma} \langle \eta\gamma(\lambda_i), \nu \rangle \Delta_{\gamma, i}\bigg).
\end{equation}

Combining (<ref>) and (<ref>), for each $\eta\in \Gamma$, we have an isomorphism between tensor functors $\theta_\eta: F\eta_T^*\to \eta_Y^*F$. One can check that $F$ together with the collection of isomorphisms $\{\theta_\eta\}_{\eta\in \Gamma}$ satisfy the axioms of Definition <ref>. This extends to a $\Gamma$-equivariant functor from $\mr{Rep}(T)$ to $\mr{Coh}^{\mr{lf}}(C^{n+1}) $.

For any $\C$-scheme $X$ with a $\Gamma$-action, we denote by $\mathring{\mc{F}}_X:= X\times G$ the trivial $G$-torsor over $X$ with a $\Gamma$-action acting diagonally on $X\times G$, and call it the trivial $(\Gamma, G)$-torsor over $X$. For the general definition of $(\Gamma, G)$-torsors, see Definition <ref>.

A $(\Gamma, G)$-torsor $\mc{P}$ over a scheme $X$ is called locally trivial if for any $x\in X$, there exists an open neighborhood $U_x$ of $x$ such that

* $U_x$ is stable under the stabilizer $\Gamma_x$ of $x$,

* $\mc{P}|_{U_x}$ is isomorphic to the trivial $(\Gamma_x, G)$-torsor $\mathring{\mc{F}}_{U_x}$ over $U_x$.

Note that not every $(\Gamma, G)$-torsor is locally trivial, and in fact this is determined by local types of $(\Gamma, G)$-torsors at ramified points, see [11].

The $(\Gamma, T)$-torsor $\mc{F}$ constructed in Lemma <ref> is locally trivial over $C^{n+1}$.

Let $x$ be any point in $C^{n+1}$. We will find a $\Gamma$-stable neighborhood $U_x$ of $x$ in $C^n$ such that $\mc{F}$ is trivializable over $U_x$ as $(\Gamma,T)$-torsors.

Let $\Gamma_x$ be the stabilizer of $x$ in $\Gamma$. Given $1\leq i\leq n$, suppose that $x\in \Delta_{\eta_i, i}$ for some $\eta_i\in \Gamma$. In this case, we observe that $x \in \Delta_{ \gamma, i}$ if and only if $\gamma\in \eta_i\Gamma_x$. Hence there exists an open neighborhood $U_i$ of $x$ with a regular function $f_i$ over $U_i$ such that $f_i$ defines $\Delta_{\eta_i, i}\cap U_i$. Moreover, we can shrink $U_i$ so that $U_i\cap \Delta_{\gamma, i}=\emptyset$ for any $\gamma\not \in \eta_i\Gamma_x$. For the given $i$, if $x\not\in \Delta_{\eta, i}$ for any $\eta\in \Gamma$, by convention we take $\eta_i=e$, $U_i=C^{n+1}\setminus \bigcup_{\gamma\in \Gamma}\Delta_{\gamma, i}$, and $f_i=1$. Combining the above constructions, for any $x\in C^{n+1}$, we have the following open neighborhood $U_x$ of $x$ in $C^{n+1}$,

\[U_x:=\bigcap_{i=1}^n\bigcap_{\gamma\in \Gamma_x} \gamma U_i,\]

and a collection of functions $\{ f_i \}_{i=1,\cdots,n}$ defined over $U_x$.
Clearly $U_x$ is $\Gamma_x$-stable. Moreover, for any $1\leq i\leq n$ and $\gamma\in \eta_i\Gamma_x$, the regular function $\gamma\eta_i^{-1} f_i$ defines the divisor $\Delta_{\gamma, i}=\gamma\eta_i^{-1}\Delta_{\eta_i, i}$ over $U_x$.

Now, for each character $\nu$ of $T$, denote by $D_\nu$ the divisor $\sum_{i=1}^n\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_i), \nu \rangle \Delta_{\gamma, i}$. Set

\[f_\nu=\prod_{i=1}^n\prod_{\gamma\in\eta_i\Gamma_x} (\gamma\eta_i^{-1} f_i)^{\langle \gamma (\lambda_i), \nu \rangle}.\]

Then $f_\nu$ defines the divisor $D_\nu$ locally in $U_x$, since $U_x\cap \Delta_{\gamma,i}=\emptyset$ for any $\gamma\not\in \eta_i\Gamma_x$. For any characters $\nu, \omega\in X^*(T)$, we have $f_\nu\cdot f_\omega=f_{\nu+\omega}$. Consider the tensor functor $F_x^0:\mr{Rep}(T)\to \mr{Coh}^{\mr{lf}}(U_x)$ sending $\C_{\nu}$ to $\ms{O}_{U_x}\otimes_{\C} \C_{\nu}$. It is easy to see $F_x^0$ is $\Gamma_x$-equivariant and this gives a trivial $(\Gamma_x, T)$-torsor $\mathring{\mc{F}}_{U_x}$ over $U_x$. Moreover, there is an isomorphism

\begin{equation*}
\phi_{\nu}: F|_{U_x}(\C_{\nu})=\ms{O}_{U_x}\bigg(\sum_{i=1}^n\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_i), \nu \rangle \Delta_{\gamma, i}\bigg) \simto \ms{O}_{U_x}\simeq F_x^0(\C_{\nu}),
\end{equation*}

where the first isomorphism is given	by multiplying by $f_\nu$. One can check that these isomorphisms $\{\phi_\nu\}_{\nu\in X^*(T)}$ respect the tensor structure since $\ms{O}(D_{\nu})\otimes\ms{O} (D_{\omega})\simeq \ms{O}(D_{\nu+\omega})$. Thus, there is an isomorphism $\phi:F|_{U_x}\to F_x^0$ between tensor functors. By Theorem <ref>, $\phi$ induces an isomorphism $\tilde\phi:\mc{F}|_{U_x}\to \mathring{\mc{F}}_{U_x}$ between $(\Gamma_x, T)$-torsors over $U_x$. Therefore, $\mc{F}$ is locally trivial following Definition <ref>.

Set $U=C^{n+1}\setminus \bigcup_{\gamma,i}\Delta_{ \gamma, i}$. Let $F^0:\mr{Rep}(T)\to \mr{Coh}^{\mr{lf}}(U)$ be the tensor functor sending $\C_{\nu}$ to $\ms{O}_{U}\otimes_{\C} \C_{\nu}$. Then, $F^0$ is $\Gamma$-equivariant, and it gives a trivial $(\Gamma, T)$-torsor $\mathring{\mc{F}}_{U}$ over $U$. For each $\nu\in X_*(T)$, there is a natural isomorphism
\begin{equation}\label{342573}
\eta_\nu:F|_U(\C_\nu)\simto F^0(\C_\nu).
\end{equation}

Then, $\{\eta_\nu\}_{\nu\in T^*}$ defines an isomorphism $\eta: F|_U\simto F^0$. By Theorem <ref>, this is equivalent to an isomorphism $\mc{F}|_U\simto \mathring{\mc{F}}_U$, which will still be denoted by $\eta$.

Let $\mc{F}_G:=\mc{F}\times^T G$ be the $(\Gamma,G)$-torsor over $C^{n+1}$ induced from $\mc{F}$. Then, $\eta$ induces a trivialization $\eta_G$ of $\mc{F}_G$ outside $\bigcup_{\gamma,i}\Delta_{ \gamma, i}$. By Lemma <ref>, $\mc{F}_G$ is a locally trivial $(\Gamma,G)$-torsor. We define a morphism $\pi:C^{n+1}\to C^n\times \bar C$ by $(p_1,\ldots,p_n,p_{n+1})\mapsto (p_1,\ldots p_n, \bar p_{n+1})$. From the local triviality of $\mc{F}_G$, $\pi_*(\mc{F}_G)^\Gamma$ is a $\mc{G}$-torsor over $C^n\times\bar C$, denoted by $\bar{\mc{F}}$. Moreover, $\eta_G$ induces a trivialization $\bar\eta$ of $\bar{\mc{F}}$ outside $\pi\big(\bigcup_{\gamma,i}\Delta_{ \gamma, i}\big)=\bigcup_i\Delta_{\bar P_i}$,
where $P_i:C^n \to C$ is the projection map sending $\vec{p}$ to $p_i$.

We construct a section $s_{\la}:C^n\to \Gr_{\mc{G},C^n}$ as follows:

\begin{equation}\label{926470}
s_{\vec{\lambda}}=\big(P_1,\ldots, P_n, \bar{\mc{F}}, \bar\eta \big),
\end{equation}

The Beilinson-Drinfeld Schubert variety $\Grb_{\mc{G}, C^n}^{\la}$ of $\mc{G}$ over $C^n$ is defined to be the schematic image of $s_{\la}:L^+\mc{G}_{C^n} \to \Gr_{\mc{G}, {C^n}}$. When $n=1$, $\Grb_{\mc{G}, C}^{\lambda}$ is also called a global Schubert variety of $\G$.

Given any $\la\in (X_*(T)^+)^n$ and any partition $\xi=(I_1,\cdots, I_k)$ of $[n]$, the BD Schubert variety $\Grb_{\mc{G}, C^n}^{\la}$ has a factorization property over $C^n_\xi$, i.e.

\[\Grb_{\mc{G},C^n_{\xi}}^{\la}\simeq\bigg(\prod_{j=1 }^k\Grb_{\mc{G},C^{I_j}}^{\la_{I_j}}\bigg)\times_{C^n} C^n_\xi,\]

where $\la_{I_j}$ is defined in (<ref>).

It suffices to prove the case when $\xi=(I,J)$. The map (<ref>) can be given explicitly as follows:

\begin{align}\label{134631}
\Phi: \Gr_{\mc{G},C^{n}_\xi}& \simeq \big(\Gr_{\mc{G},C^{I}}\times \Gr_{\mc{G},C^{J}} \big)|_{C^{n}_\xi}, \\
\nonumber (\vec{p},\mc{F},\beta)&\mapsto \big((\vec{p}_I,\mc{F}_I,\beta_I),(\vec{p}_J,\mc{F}_J,\beta_J)\big)
\end{align}

where by Beauville-Laszlo's Lemma, there exists a $\mc{G}$-torsor $\mc{F}_I$ with isomorphisms $\delta_I:\mc{F}_I|_{ \widehat{\cup_{i\in I} \Delta_{\bar p_i} }} \simeq {\mc{F}}_{ \widehat{\cup_{i\in I} \Delta_{\bar p_i} }}$ and $\beta_I: \mc{F}_I|_{ \bar C_R\setminus{\cup_{i\in I} \Delta_{\bar p_i} }}\to \mathring{\mc{F}}|_{\bar C_R \setminus{\cup_{ i\in I} \Delta_{\bar p_i} }}$ such that $\beta_I\circ\delta_I^{-1}|_{\widehat{\cup_{i\in I} \Delta_{\bar p_i} }^*}=\beta|_{\widehat{\cup_{i\in I} \Delta_{\bar p_i} }^*}$; $\mc{F}_J,\beta_J$ are defined similarly.

Let $s_{\la_I}:C^I\to \Gr_{\mc{G},C^I}$ (resp. $s_{\la_J}$) be the section defined by (<ref>). We shall show there is an equality of sections over $C^n_\xi$

\begin{equation}
\Phi\circ s_{\la}|_{C^n_\xi}=s_{\la_I}\times s_{\la_J}|_{C^n_\xi}.
\end{equation}

Recall the definition (<ref>) of $\bar{\mc{F}}:=\pi_*(\mc{F}\times^T G)^\Gamma$ and $\bar \eta$, where $\mc{F}$ is a $(\Gamma, T)$-torsor defined via the functor $F$ in (<ref>), and $\eta$ is the map induced from the natural maps $\eta_\nu$ in (<ref>).

Denote by $D_{I,\nu}$ the divisor $\big(\sum_{i\in I}\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_i), \nu \rangle \Delta_{\gamma, i}\big)\cap \big(C^n_\xi\times C\big)$ in $C^n_\xi\times C$, and define $D_{J,\nu}$ similarly. One may check that $D_{I,\nu}\cap D_{J,\nu}=\emptyset$ for any $\nu$. Let $U_I\subseteq C^n_\xi\times C$ be a neighborhood of $D_I:=\big(\sum_{i\in I}\sum_{\gamma\in \Gamma} \Delta_{\gamma, i}\big)\cap \big(C^n_\xi\times C\big)$ such that $U_{I}\cap D_{J,\nu}=\emptyset$ for all $\nu$, and let $\mr{pr}_I:C^n_\xi\times C\to C^I\times C$ be the projection map, then

\[\ms{O}_{C^{n+1}}\bigg(\sum_{i=1}^n\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_i), \nu \rangle \Delta_{\gamma, i}\bigg)\Big|_{U_I}\simeq \ms{O}_{U_{I}}(D_{I,\nu})\simeq \mr{pr}_I^*\Big(\ms{O}_{C^I\times C}\big( \mr{pr}_I(D_{I,\nu}) \big) \Big)\big|_{U_{I}}.\]

By the definition of map (<ref>), we have an equality of sections over $C^n_\xi$

\[\Phi\circ s_{\la}|_{C^n_\xi} = \big((\{P_i\}_{i\in I},\bar{\mc{F}}_{I},\bar{\eta}_I),(\{P_j\}_{j\in J},\bar{\mc{F}}_{J},\bar{\eta}_J)\big)|_{C^n_\xi},\]

where $\bar{\mc{F}}_{I}=\pi_*(\mc{F}_I\times^T G)^\Gamma$ is a $\mc{G}$-torsor, $\mc{F}_I$ is the $(\Gamma,T)$-torsor constructed via the functor $F_I:\mr{Rep}(T)\to\mr{Coh}^{\mr{lf}}(C^I\times C)$ given by
$F_I( \C_{\nu}) = \ms{O}_{C^I\times C}\big(\mr{pr}_I(D_{I,\nu})\big)$,
and $\eta_I$ is the trivialization induced from the natural maps $\eta_{I,\nu}:\ms{O}_{C^I\times C\setminus \mr{pr}_I(D_{I})}\big(\mr{pr}_I(D_{I,\nu})\big) \simto \ms{O}_{C^I\times C\setminus \mr{pr}_I(D_{I})}$, $\nu\in X^*(T)$.
By definition (<ref>), $(\{P_i\}_{j\in I},\bar{\mc{F}}_{I},\bar{\eta}_I)$ is exactly the section $s_{\la_I}:C^I\to \Gr_{\mc{G},C^I}$. Similarly, $(\{P_j\}_{i\in J},\bar{\mc{F}}_{J},\bar{\eta}_J)=s_{\la_J}$. This shows that $\Phi\circ s_{\la}|_{C^n_\xi}=s_{\la_I}\times s_{\la_J}|_{C^n_\xi}$. We also notice that the map $\Phi$ is compatible with the factorization of $L^+\G_{C^n}$.	It follows that the map (<ref>) gives an isomorphism
$\Phi: \Grb^{\la}_{\mc{G},C^{n}_\xi} \simeq \Big(\Grb^{\la_I}_{\mc{G},C^{I}}\times \Grb^{\la_J}_{\mc{G},C^{J}} \Big)\big|_{C^{n}_\xi}$.

§.§ Convolution Grassmannian

With the same setup as in Section <ref>, we define the convolution Grassmannian $\mr{Conv}_{\mc{G},C^n}$ as follows: for each $\C$-algebra $R$,

\[ \mr{Conv}_{\mc{G},C^n}(R):=\left\{ \big( \vec{p}, (\mc{F}_i, \beta_i)_{i=1}^n \big) \,\middle|\,
\begin{aligned}
&\vec{p}=(p_1, \ldots, p_n) \in C^n(R),\\
&\mc{F}_1,\ldots, \mc{F}_n \text{ are } \mc{G}\text{-torsors on } \bar{C}, \\
&\beta_i: \mc{F}_i|_{\bar C_{R} \setminus \Delta_{\bar{p}_i}}\simeq \mc{F}_{i-1}|_{\bar C_{R} \setminus \Delta_{\bar{p}_{i}}}, \mc{F}_0:=\mathring{\mc{F}}.
\end{aligned}\right\}.\]

There is a left $L^+\mc{G}_{C^n}$-action on $\mr{Conv}_{\mc{G},C^n}$ as follows. Given $(\vec{p},\zeta)\in L^+\mc{G}_{C^n}(R)$ and $ \big(\vec{p}, (\mc{F}_i, \beta_i)_{i=1}^n \big)\in \mr{Conv}_{\mc{G},C^n}(R)$, by Beauville-Laszlo's Lemma, there exists a $\mc{G}$-torsor $\mc{F}_i'$ with isomorphisms $\delta_i:{\mc{F}}'_i|_{ \hat{D}_i }\simeq \mc{F}_i|_{ \hat{D}_i }$ and $\alpha_i: \mc{F}_i'|_{\bar C_R\setminus D_i } \simeq \mathring{\mc{F}}|_{\bar C_R\setminus D_i}$ such that $\alpha_i\circ\delta^{-1}_i|_{\hat{D}_i^*}=\zeta\circ\beta_1\circ\cdots\circ \beta_i|_{\hat{D}_i^*}$, where $\hat{D}_i$ is the formal completion of $\bar C_R$ along the divisor $D_i:=\underset{\scriptscriptstyle1\leq j\leq i }{\bigcup}\Delta_{\bar p_j}$ and $\hat{D}_i^* :=\hat{D}_i \setminus D_i$. Thus, we get the following map

\begin{align*}
L^+\mc{G}_{C^n}\times \mr{Conv}_{\mc{G}, C^n} & \to \mr{Conv}_{\mc{G}, C^n} \\
\Big( (\vec{p},\zeta), \big(\vec{p}, (\mc{F}_i, \beta_i)_{i=1}^n \big) \Big) &\mapsto \big(\vec{p}, (\mc{F}'_i, \beta'_i)_{i=1}^n \big),
\end{align*}

where $\beta_i':\mc{F}_i'|_{\bar C_R\setminus\Delta_{\bar p_i}} \to \mc{F}_{i-1}'|_{\bar C_R\setminus\Delta_{\bar p_i}}$ is given by
$ \beta_i' =
\begin{cases}
\delta_{i-1}^{-1}\circ \beta_i\circ\delta_i|_{\hat{D}_{i-1}}\\
\alpha_{i-1}^{-1} \circ\alpha_i|_{\bar C_R\setminus D_i}
\end{cases}$.

Moreover, there is a convolution map

\begin{equation}\label{818710}
\mf{m}: \mr{Conv}_{\mc{G},C^n} \to \Gr_{\mc{G}, C^n},
\end{equation}

sending $\big(\vec{p}, (\mc{F}_i, \beta_i)_{i=1}^n \big) $ to $\big(\vec{p}, \mc{F}_n, \beta_1\circ\cdots\circ\beta_n\big) $. One can check that this map is $L^+\mc{G}_{C^n}$-equivariant. Let $\xi_0=( \{1\},\{2\},\ldots, \{n\})$ be the finest partition of $[n]$. We have the following isomorphism over $C^n_{\xi_0}=\{ \vec{p}\in C^n \mid \bar{p}_i\neq \bar{p}_j, \ \forall i\neq j \}$,

\begin{equation}\label{626689}
\varphi_n:\mr{Conv}_{\mc{G},C^n}|_{C_{\xi_0}^n} \xrightarrow{\mf{m}} \Gr_{\mc{G},{C}^n_{\xi_0}} \simto \big(\Gr_{\mc{G},C}\times\cdots\times \Gr_{\mc{G},C}\big)|_{C_{\xi_0}^n},
\end{equation}

sending $\big(\vec{p}, (\mc{F}_i, \beta_i)_{i=1}^n \big) $ to $\big( ( p_i, \mc{F}_i|_{\hat{\Delta}_{\bar p_i}}, \beta_1\circ\cdots\circ\beta_i|_{\hat{\Delta}^*_{\bar p_i}} )_{i=1}^n \big)$.

Following <cit.>, we define an $L^+\mc{G}_C$-torsor $\mb{E}$ over $\mr{Conv}_{\mc{G},C^{n-1}}\times C$ by

\[ \mb{E}(R):=\left\{ \big( \vec{p}, (\mc{F}_i, \beta_i)_{i=1}^{n-1}, \alpha \big) \,\middle|\,
\begin{aligned}
&\vec{p}=(p_1, \ldots, p_n) \in C^n(R),\\
&\mc{F}_1,\ldots, \mc{F}_{n-1} \text{ are } \mc{G}\text{-torsors on } \bar{C}, \\
&\beta_i: \mc{F}_i|_{\bar C_{R} \setminus \Delta_{\bar{p}_i}}\simeq \mc{F}_{i-1}|_{\bar C_{R} \setminus \Delta_{\bar{p}_{i}}}, \mc{F}_0:=\mathring{\mc{F}},\\
&\alpha: \mc{F}_{n-1}|_{\hat \Delta_{\bar p_n}} \to \mathring{\mc{F}}|_{\hat \Delta_{\bar p_n}}.
\end{aligned}\right\}.\]

Then, there is an isomorphism

\begin{align}
\label{643298} f: \mb{E}\times^{L^+\mc{G}_C} \Gr_{\mc{G}, C} &\simto \mr{Conv}_{\mc{G},C^n},\\
\nonumber \Big( \big( \vec{p}, (\mc{F}_i, \beta_i)_{i=1}^{n-1}, \alpha \big), (\mc{F},\beta) \Big) & \mapsto \big( \vec{p}, (\mc{F}_i, \beta_i)_{i=1}^{n}\big)
\end{align}

where by Beauville-Laszlo's Lemma, there exists a $\mc{G}$-torsor $\mc{F}_n$ with isomorphisms $\beta_n: \mc{F}_n|_{\bar C_R\setminus \Delta_{\bar p_n}}\simeq \mc{F}_{n-1}|_{\bar C_R\setminus \Delta_{\bar p_n}}$ and $\delta:\mc{F}_n|_{\Delta_{\bar p_n}}\simeq \mc{F}|_{\Delta_{\bar p_n}}$ such that $\beta_n\circ\delta^{-1}|_{\hat{\Delta}^*_{\bar p_n}}=\alpha^{-1}\circ\beta|_{\hat{\Delta}^*_{\bar p_n}}$. Composing $f$ with the map (<ref>), we get an isomorphism over $C^n_{\xi_0}$

\begin{align}
\label{086999} (\varphi\circ f)|_{C^n_{\xi_0}}: \big(\mb{E}\times^{L^+\mc{G}_C} \Gr_{\mc{G}, C}\big)|_{C_{\xi_0}^n} &\simto \big(\Gr_{\mc{G},C}\times\cdots\times \Gr_{\mc{G},C}\big)|_{C_{\xi_0}^n} \\
\nonumber \Big( \big( \vec{p}, (\mc{F}_i, \beta_i)_{i=1}^{n-1}, \alpha \big), (\mc{F},\beta) \Big) & \mapsto \Big( \big( p_i, \mc{F}_i|_{\hat{\Delta}_{\bar p_i}}, \beta_1\circ\cdots\circ\beta_i|_{\hat{\Delta}^*_{\bar p_i}} \big)_{i=1}^{n} \Big).
\end{align}

By the constructions of $\mc{F}_n$ and $\beta_n$ above, we have

\[(p_n, \mc{F}_n|_{\hat{\Delta}_{\bar p_n}}, \beta_1\circ\cdots\circ\beta_n|_{\hat{\Delta}^*_{\bar p_n}}) =(p_n, \mc{F}|_{\hat{\Delta}_{\bar p_n}}, \beta_1\circ\cdots\circ\beta_{n-1}\circ\alpha^{-1}\circ\beta|_{\hat{\Delta}^*_{\bar p_n}}).\]

Combining (<ref>) and (<ref>), we get an isomorphism over $C^n_{\xi_0}$

\begin{align}
\label{346788} \psi:\big(\mb{E}\times^{L^+\mc{G}_C} \Gr_{\mc{G}, C}\big)|_{C_{\xi_0}^n} &\simto \big(\mr{Conv}_{\mc{G},C^{n-1}}\times \Gr_{\mc{G},C}\big)|_{C_{\xi_0}^n} \\
\nonumber \Big( \big( \vec{p}, (\mc{F}_i, \beta_i)_{i=1}^{n-1}, \alpha \big), (\mc{F},\beta) \Big) & \mapsto \Big( \big( (p_i, \mc{F}_i, \beta_i)_{i=1}^{n-1}\big), (p_n,\mc{F}|_{\hat{\Delta}_{\bar p_n}}, \zeta\circ\beta|_{\hat{\Delta}^*_{\bar p_n}}) \Big),
\end{align}

where $\zeta=(\beta_1\circ\cdots\circ\beta_{n-1})|_{\hat{\Delta}_{\bar p_n}}\circ\alpha^{-1}\in L^+\mc{G}_{p_n}$.

In summary, we have the following commutative diagram

\begin{equation}\label{129303}
\xymatrixcolsep{3pc}
\xymatrix{
\big(\mb{E}\times^{L^+\mc{G}_C} \Gr_{\mc{G}, C}\big)|_{C_{\xi_0}^n} \ar[d]_{\psi}^[@]{\sim} \ar[r]^-{\sim}_-{f|_{C^n_{\xi_0}}} & \mr{Conv}_{\mc{G},C_{\xi_0}^n} \ar[d]_[@]{\sim}^{\varphi_{n}} \\
\big(\mr{Conv}_{\mc{G},C^{n-1}}\times \Gr_{\mc{G},C}\big)|_{C_{\xi_0}^n} \ar[r]^-{\sim}_-{\varphi_{n-1}\times \mr{id}} & \big(\Gr_{\mc{G},C}\times\cdots\times\Gr_{\mc{G},C}\times \Gr_{\mc{G},C}\big)|_{C_{\xi_0}^n}
\end{equation}

The convolution Schubert variety $\overline{\mr{Conv}}^{\la}_{\mc{G},C^n}$ is defined inductively as follows: when $n=1$, set $\overline{\Conv}_{\mc{G},C}^\lambda:=\Grb_{\mc{G},C}^\lambda$; when $n\geq 2$, define

\[\overline{\Conv}_{\mc{G},C^n}^{\la}:= \mb{E}|_{\overline{\Conv}_{\mc{G},C^{n-1}}^{ (\lambda_1,\ldots,\lambda_{n-1}) }\times C } \times^{L^+\mc{G}_C} \Grb_{\mc{G},C}^{\lambda_n}.\]

Now we would like to give another construction of the convolution Schubert variety. For each character $\nu$ of $T$, we define

\[F_i( \C_{\nu}):= \ms{O}_{C^{n+1}}\bigg(\sum_{k=1}^i\sum_{\gamma\in\Gamma}\langle \gamma(\lambda_k), \nu \rangle \Delta_{\gamma, k}\bigg).\]

This assignment extends to a $\mathbb{C}$-linear tensor functor $F_i:\mr{Rep}(G)\to \mr{Coh}^{\mr{lf}}(C^{n+1})$. By the same argument as in Section <ref>, $F_i$ gives a locally trivial $(\Gamma,T)$-torsor $\mc{F}_i$ over $C^{n+1}$. Moreover, the natural maps $\beta_{\nu,i}: F_i(\C_\nu)|_{C^n\times\bar C-\Delta_{\bar P_i}} \simto F_{i-1} (\C_\nu)|_{C^n\times\bar C-\Delta_{\bar P_i}}$ for all $\nu\in X_*(T)$ induce an isomorhism $\beta_i:\mc{F}_i|_{C^n\times\bar C-\Delta_{\bar P_i}} \to \mc{F}_{i-1}|_{C^n\times\bar C-\Delta_{\bar P_i}} $.

Set $ \mc{F}_{i,G}=\mc{F}_i\times^T G$. Then $\bar{\mc{F}}_{i}:=\pi_*(F_{i,G})^\Gamma$ is a $\mc{G}$-torsor over $C^n\times \bar C$, and $\beta_i$ induces an isomorohism $\bar\beta_i: \bar{\mc{F}}_{i}|_{C^n\times\bar C-\Delta_{\bar P_i}} \simeq\bar{\mc{F}}_{i-1}|_{C^n\times\bar C-\Delta_{\bar P_i}}$, where by convention $\bar{\mc{F}}_{0}$ is the trivial torsor. We construct a section $\tilde s_{\la}:C^n\to \mr{Conv}_{\mc{G},C^n}$ as follows:

\begin{equation}\label{282313}
\tilde s_{\la}=\big(P_1,\ldots, P_n, (\bar{\mc{F}}_{i}, \bar\beta_i)_{i=1}^n\big).
\end{equation}

Denote by $\tilde{\Conv}^{\la}_{C^n} $ the schematic image of $\tilde{s}_{\la}:L^+\mc{G}_{C^n}\to \mr{Conv}_{\mc{G},C^n}$.

For any partition $\xi=(I_1,\ldots,I_k)$ of $[n]$, there is an isomorphism $\Phi: \Conv_{\mc{G},C^n_\xi}\to \big(\prod_{i=1}^k\Conv_{\mc{G},C^{I_i}}\big)|_{C^n_\xi}$ satisfying the following commutative diagram

\begin{equation}\label{998995}
\xymatrix{
\Conv_{\mc{G},C^n_\xi} \ar[d]_{\Phi}^[@]{\sim} \ar[r]^{\mf{m}} & \Gr_{\mc{G},C^n_\xi} \ar[d]_[@]{\sim}^{\eqref{447893}} \\
\big(\prod_{i=1}^k\Conv_{\mc{G},C^{I_i}}\big)|_{C^n_\xi} \ar[r]^-{\prod \mf{m}} & \big(\prod_{i=1}^k \Gr_{\mc{G},C^{I_i}}\big)|_{C^n_\xi} }.
\end{equation}

Moreover, we have $\Phi\circ \tilde{s}_{\la}|_{C^n_\xi}=\prod \tilde{s}_{\la_{I_i}}|_{C^n_\xi}$. It follows that $\Phi$ induces an isomorphism

\[\tilde{\Conv}^{\la}_{\mc{G},C^n_\xi}\simeq \Big(\prod_{i=1}^k\tilde{\Conv}^{\la_{I_i}}_{\mc{G},C^{I_i}}\Big)\big|_{C^n_\xi}.\]

We first define the map $\Phi$ as follows,

\begin{align}\label{466981}
\Phi:\Conv_{\mc{G},C^n_\xi} & \to \Big(\prod_{i=1}^k\Conv_{\mc{G},C^{I_i}}\Big)\big|_{C^n_\xi} \\
\nonumber \big( \vec{p}, (\mc{F}_i, \beta_i)_{i=1}^{n}\big) & \mapsto {\textstyle\prod} \big( \vec{p}_{I_i},(\mc{F}'_j,\beta'_j)_{j\in I_i}\big)
\end{align}

where by Beauville-Laszlo's Lemma, for any $j\in I_i$ and the index $j'$ right before $j$ in the ordered set $I_i$, there exists a $\mc{G}$-torsor $\mc{F}'_j$ with isomorphisms $\alpha_j:\mc{F}'_j|_{\hat\Delta_{\bar p_j}}\to \mc{F}_j|_{\hat\Delta_{\bar p_j}}$ and $\beta_j': {\mc{F}'_j}|_{\bar C_R\setminus \Delta_{\bar p_j}}\to {\mc{F}'_{j'}}|_{\bar C_R\setminus \Delta_{\bar p_j}}$ such that $\beta'_j\circ\alpha^{-1}_j|_{\hat\Delta_{\bar p_j}^*}={\beta'}_{j'}^{-1}\circ\beta_1\circ\cdots\circ\beta_j|_{\hat\Delta_{\bar p_j}^*}$. One may check this is an isomorphism and satisfies the commutative diagram (<ref>).

Let $\tilde s_{\la_I}:C^I\to \Conv_{\mc{G},C^I}$ be the section defined by (<ref>), we shall show $ \textstyle \Phi\circ \tilde{s}_{\la}|_{C^n_\xi}= \prod \tilde{s}_{\la_{I_i}}|_{C^n_\xi}$ over $C^n_\xi$ for any partition $\xi$.
For the simplicity of illustration,	we will only prove a special case for $n=4$ and $\xi=(I,J)=(\{1,3\},\{2,4\})$, which indicates the argument in general. Recall the definition (<ref>) of $\bar{\mc{F}}_{i}:=\pi_*(\mc{F}_i\times^T G)^\Gamma$ and $\bar \eta_i$, where $\mc{F}_i$ is a $(\Gamma, T)$-torsor defined via the functor $F_i: \mr{Rep}(T)\to \mr{Coh}^{\mr{lf}}(C^{n+1})$ given by

$$F_i( \C_{\nu}) = \ms{O}_{C^{n+1}}\bigg(\sum_{j=1}^i\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_j), \nu \rangle \Delta_{\gamma, j}\bigg),$$

and $\eta_i$ is the map induced from the natural isomorphisms $\eta_{\nu,i}:F(\C_\nu)|_{C^n\times C\setminus \bigcup_{j\leq i}\Delta_{ \gamma,j}} \simto F^0 (\C_\nu)|_{C^n\times C\setminus \bigcup_{j\leq i}\Delta_{ \gamma, j}} $, $\nu\in X^*(T)$. Denote by $D_{i,\nu}$ the divisor $\big(\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_i), \nu \rangle \Delta_{\gamma, i}\big)\cap \big(C^n_\xi\times C\big)$ in $C^n_\xi\times C$. One may check that $D_{i,\nu}\cap D_{j,\nu}=\emptyset$ for any $i=1,3$, $j=2,4$. Let $U_{I}\subseteq C^n_\xi\times C$ be a neighborhood of $\big(\bigcup_{i\in I,\gamma\in\Gamma} \Delta_{\gamma, i}\big)\cap \big(C^n_\xi\times C\big)$ such that $U_I\cap (D_{2,\nu}\cup D_{4,\nu})=\emptyset$ for any $\nu$, and let $\mr{pr}_I:C^n_\xi\times C\to C^I\times C$ be the projection map. Define $U_J$ and $\mr{pr}_J$ similarly. Then

\begin{gather*}
\ms{O}_{C^{n+1}}\bigg(\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_1), \nu \rangle \Delta_{\gamma, 1}\bigg)\Big|_{U_I}\simeq \ms{O}_{U_I}(D_{1,\nu})\simeq \mr{pr}_I^*\Big(\ms{O}_{C^I\times C}\big( \mr{pr}_I(D_{1,\nu}) \big) \Big)\big|_{U_I},\\
\ms{O}_{C^{n+1}}\bigg(\sum_{i=1}^3\sum_{\gamma\in \Gamma} \langle \gamma(\lambda_i), \nu \rangle \Delta_{\gamma, i}\bigg)\Big|_{U_I}\simeq \ms{O}_{U_I}(D_{1,\nu}\cup D_{3,\nu})\simeq \mr{pr}_I^*\Big(\ms{O}_{C^I\times C}\big( \mr{pr}_I(D_{1,\nu}\cup D_{3,\nu}) \big) \Big)\big|_{U_I}.
\end{gather*}

By the definition of the map (<ref>), we have an equality of sections over $C^n_\xi$

\[\Phi\circ \tilde s_{\la}|_{C^n_\xi} = \big((P_1,P_3,\bar{\mc{F}}_{1},\bar{\mc{F}}_{3}',\bar{\eta}_1,\bar{\eta}'_3),(P_2,P_4,\bar{\mc{F}}'_{2},\bar{\mc{F}}'_{4},\bar{\eta}'_2,\bar{\eta}'_4)\big)|_{C^n_\xi},\]

where $\bar{\mc{F}}'_{3}=\pi_*(\mc{F}'_3\times^T G)^\Gamma$, and $\mc{F}'_3$ is the $(\Gamma,T)$-torsor constructed via the following functor $F'_3:\mr{Rep}(T)\to\mr{Coh}^{\mr{lf}}(C^I\times C)$

\[F'_3: \C_{\nu}\mapsto \ms{O}_{C^I\times C}\big(\mr{pr}_I(D_{1,\nu}\cup D_{3,\nu})\big),\]

$\eta'_3$ is the map induced form the natural maps

\[\eta'_{3,\nu}:\ms{O}_{C^I\times C\setminus \mr{pr}_I(D_{3,\nu})}\big(\mr{pr}_I(D_{1,\nu}\cup D_{3,\nu})\big) \to \ms{O}_{C^I\times C\setminus \mr{pr}_I(D_3)}(\mr{pr}_I(D_{1,\nu})),\]

for all $\nu\in X^*(T)$.
By definition (<ref>), $(P_1,P_3,\bar{\mc{F}}_{1},\bar{\mc{F}}_{3}',\bar{\eta}_1,\bar{\eta}'_3)$ is exactly the section $\tilde s_{\la_I}:C^I\to \Conv_{\mc{G},C^I}$. Similarly, $(P_2,P_4,\bar{\mc{F}}'_{2},\bar{\mc{F}}'_{4},\bar{\eta}'_2,\bar{\eta}'_4)=\tilde s_{\la_J}$. This shows that
$\Phi\circ \tilde s_{\la}|_{C^n_\xi}=\tilde s_{\la_I}\times \tilde s_{\la_J}|_{C^n_\xi}$.
Since $\Phi$ is $L^+\mc{G}_{C^n_\xi}$-equivariant, it follows that the map (<ref>) induces an isomorphism $\Phi: \tilde\Conv^{\la}_{\mc{G},C^{n}_\xi} \simeq \big(\tilde\Conv^{\la_I}_{\mc{G},C^{I}}\times \tilde\Conv^{\la_J}_{\mc{G},C^{J}} \big)|_{C^{n}_\xi}$.

Recall that there is an isomorphism (<ref>) over $C_{\xi_0}^n$:

\[ \varphi_n:\mr{Conv}_{\mc{G},C^n}|_{C_{\xi_0}^n} \to \big(\Gr_{\mc{G},C}\times\cdots\times \Gr_{\mc{G},C}\big)|_{C_{\xi_0}^n}. \]

By Lemma <ref>, this map $\varphi$ induces an isomorphism

\begin{equation}\label{234926}
\tilde{\Conv}^{\la}_{\mc{G},C^n_{\xi_0}} \simeq \big(\Grb^{\lambda_{1}}_{\mc{G},C}\times\cdots\times \Grb^{\lambda_n}_{\mc{G},C}\big)|_{C^n_{\xi_0}}.
\end{equation}

The morphism $f$ defined in (<ref>) induces an isomorphism

\begin{equation}\label{138990}
\overline{ \Conv}_{\mc{G},C^n}^{\la}\simeq \tilde{\Conv }^{\la}_{C^n}.
\end{equation}

We prove it by induction on $n$. When $n=1$, by definition $\tilde{s}_{\lambda}=s_{\lambda}$. Hence, $\tilde{\Conv}^{\lambda}_{\G,C}=\Grb_{\G,C}^\lambda=\overline{\Conv}_{\G,C}^\lambda$. When $n\geq 2$, by induction we have $\overline{\Conv}_{\mc{G},C^{n-1}}^{\lambda_1,\ldots,\lambda_{n-1}}\simeq \tilde{\Conv }^{\lambda_1,\ldots,\lambda_{n-1}}_{\G,C^n}$. Then, the map $\psi$ in (<ref>) induces an isomorphism

\begin{equation} \label{342960}
\overline{\Conv}_{\mc{G},C^n_{\xi_0}}^{\la}\simeq \big(\tilde{\Conv }^{\lambda_1,\ldots,\lambda_{n-1}}_{\G,C^n}\times \Grb_{\G,C}^{\lambda_n}\big)|_{C^n_{\xi_0}}.
\end{equation}

Combining with the isomorphism (<ref>), we get the following diagram

\begin{equation*}
\xymatrixcolsep{3pc}
\xymatrix{
\overline{\Conv}_{\mc{G},C^n_{\xi_0}}^{\la} \ar[d]_{\eqref{342960}}^[@]{\sim} & \tilde{\Conv}^{\la}_{\mc{G},C^n_{\xi_0}} \ar[d]_[@]{\sim}^{\eqref{234926}} \\
\big(\tilde{\Conv }^{\lambda_1,\ldots,\lambda_{n-1}}_{\G,C^n}\times \Grb_{\G,C}^{\lambda_n}\big)|_{C^n_{\xi_0}} \ar[r]^-{\sim}_-{\eqref{234926}\times \mr{id}} & \big(\Grb^{\lambda_{1}}_{\mc{G},C}\times\cdots\times \Grb^{\lambda_n}_{\mc{G},C}\big)|_{C^n_{\xi_0}}
\end{equation*}

This gives an isomorphism $\overline{\Conv}_{\mc{G},C^n_{\xi_0}}^{\la}\simeq \tilde{\Conv}^{\la}_{\mc{G},C^n_{\xi_0}}$. By the commutativity of the diagram (<ref>), this isomorphism is exactly the restriction of the map $f$ defined in (<ref>). Since both of them are reduced and irreducible, we must have an isomorphism 	$\overline{\Conv}_{\mc{G},C^n}^{\la}\simeq \tilde{\Conv}^{\la}_{\mc{G},C^n}$.

§.§ Flatness of BD Schubert varieties

Let $G$ be a simple algebraic group of adjoint type over $\C$, and let $\sigma$ be the standard automorphism defined in Section <ref>. Let $\mc{K}=\C((t))$ and $\mc{O}=\C[[t]]$. We define the action of $\sigma$ on $\mc{K}$ and $\mc{O}$ by $\sigma(t)=\epsilon^{-1} t$, where $\epsilon$ is a fixed primitive $m$-th root of unity. Denote by $\bar{\mc{K}}=\mc{K^\sigma}$ and $\bar{\mc{O}}=\mc{O}^\sigma$ the $\sigma$-fixed points in $\mc{K}$ and $\mc{O}$ respectively.

Let $\ms{G}$ be the $\sigma$-fixed point subgroup scheme $\mr{Res}_{\mc{O}/\bar{\mc{O}}} ( G_\mc{O})^\sigma$ of the Weil restriction $\mr{Res}_{\mc{O}/\bar{\mc{O}}} ( G_\mc{O}) $. Then $\ms{G}$ is a special parahoric group scheme over $\bar{\mc{O}}$ in the sense of Bruhat-Tits, see <cit.>. Let $L\ms{G}$ be the loop group scheme and $L^+\ms{G}$ be the jet group scheme with $L\ms{G}(R)=\ms{G}(R((t)))$ and $L^+\ms{G}(R)=\ms{G}(R[[t]])$, for any $\C$-algebra $R$. We define the affine grassmannian $\Gr_\ms{G}$ to be the fppf quotient $L\ms{G}/L^+\ms{G}$, and call it a twisted affine Grassmannian of $\ms{G}$. In particular, $\Gr_{\ms{G}}(\C)=G(\mc{K})^\sigma/G(\mc{O})^\sigma$. We denote by $e_0$ the base point in $\Gr_{\ms{G}}(\C)$.

For any coweight $\lambda\in X_*(T)$, we are associated to an element $t^\lambda\in T(\mc{K})$. Let $n^\lambda$ be the norm of $t^\lambda$, i.e. $n^\lambda:=\prod_{i=0}^{m-1} \sigma^i(t^\lambda)\in T(\mc{K})^\sigma$. Let $\bar\lambda\in X_*(T)_{\sigma}$ be the image of $\lambda$ under the projection map $X_*(T)\to X_*(T)_\sigma$, and denote by $e_{\lb}$ the point $n^\lambda e_0\in \Gr_{\ms{G}}(\C)$. One can define a Schubert cell $\Gr_{\ms{G}}^{\lb}:=G(\mc{O})^\sigma e_{\lb}$. We define the Schubert variety $\Grb_\ms{G}^{\lb}$ to be the reduced closure of $\Gr_{\ms{G}}^{\lb}$ in $\Gr_\ms{G}$. There is a level one line bundle $\ms{L}$ on $\Gr_{\ms{G}}$ such that

\begin{equation}\label{eq_sch_dem}
H^0\big(\Grb_{\ms{G}}^{\bar{\lambda}}, \ms{L}^c \big) ^\vee\simeq D^\sigma(c,\bar{\lambda}),
\end{equation}

as $\gs$-modules, where $D^\sigma(c,\bar{\lambda})$ is the twisted affine Demazure module defined in Section <ref>, cf. <cit.>).

Let $\Gamma$ be the cyclic group generated by $\sigma$, and let $C$ be an irreducible smooth curve with a faithful $\Gamma$-action. From now on, we assume that the ramified points of $C$ are totally ramified, i.e. the stabilizer $\Gamma_p$ is equal to $\Gamma$ for any ramified point $p$. Moreover, when $C$ is the affine line $\A^1$, we require the $\Gamma$-action to be the `standard' one given by

\begin{equation}\label{eq_standard}
\sigma(p)=\epsilon p, \quad \text{ for any } p\in \A^1.
\end{equation}

With the assumptions above, the global Schubert variety $\Grb^\lambda_{\mc{G},C}$ is flat over $C$, and for any $p\in C$ the fiber $\Grb^\lambda_{\mc{G},p}$ is reduced. Moreover,

\begin{equation}\label{200864}
\Grb^\lambda_{\mc{G},p}\simeq
\begin{cases}
\Grb_{\ms{G}}^{\lb} & \text{for } p\in R;\\
\Grb_{G}^{\lambda} & \text{for } p\not\in R,
\end{cases}
\end{equation}

where $R$ denotes the ramification locus of the action of $\Gamma$ on $C$.

When $C=\A^1$, the theorem was proven in <cit.>. Now, let $C$ be a general curve whose points are either unramified or totally ramified. Note that the theorem is local, we shall reduce it to the affine case. For any ramified point $x\in C$, there exists an open neighborhood $U$ of $x$ with a $\Gamma$-equivariant étale map $U\to \A^1$ sending $x$ to $0$. By Lemma <ref> and Proposition <ref>, the induced map $\bar U\to \bar \A^1$ is also étale, and there is an isomorphism of group schemes over $U$

\[\mc{G}_{\bar U}\simeq \mc{G}_{\bar \A^1}\times_{\A^1} U.\]

By Proposition <ref>, we have a base change isomorphism

\[\Gr_{\mc{G}_{\bar U}, U} \simeq \Gr_{\mc{G}_{\bar\A^1},\A^1}\times_{\A^1} U,\]

and the left $L^+\mc{G}_U$-action on $\Gr_{\mc{G}_{\bar U}, U}$ commutes with the base change. Thus,

\[\Grb_{\mc{G}_{\bar U}, U}^\lambda \simeq \Grb^\lambda_{\mc{G}_{\bar\A^1},\A^1}\times_{\A^1} U.\]

Hence, the theorem holds over $U$. Similarly, one can show that the theorem also holds over $C\backslash R$. This competes the proof of the theorem.

The convolution variety $ \overline{\mr{Conv}}_{\mc{G},C^n}^{\la} $ is flat over $C^n$. Moreover, the convolution morphism $\mf{m}$ induces a surjective morphism $\mf{m}: \overline{\mr{Conv}}_{\mc{G},C^n}^{\la}\to \Grb_{\mc{G},C^n}^{\la}$.

Recall the Definition <ref> that $ \overline{\mr{Conv}}_{\mc{G},C^n}^{\la} $ is a global Schubert variety when $n=1$ and is a sequence of fibrations of global Schubert varieties when $n\geq 2$. By Theorem <ref>, $\overline{\mr{Conv}}_{\mc{G},C^n}^{\la} $ must be flat over $C^n$.

By Proposition <ref>, $\overline{\mr{Conv}}_{\mc{G},C^n}^{\la} $ can be identified with the schematic image of the map $\tilde{s}_{\la}:L^+\mc{G}_{C^n}\to \mr{Conv}_{\mc{G},C^n}$. By the definition (<ref>) of convolution map $\mf{m}$ and the $L^+\mc{G}_{C^n}$-equivariance of $\mf{m}$, we have a commutative diagram

\begin{equation}\label{015520}
\xymatrix{
L^+\mc{G}_{C^n} \ar[rd]_{s_{\la}} \ar[r]^-{\tilde{s}_{\la}} &\mr{Conv}_{\mc{G},C^n} \ar[d]^-{\mf{m}} \\
& \Gr_{\mc{G},C^n}
\end{equation}

Since $\mf{m}$ is proper, it gives a surjective map $\mf{m}:\overline{\Conv}^{\la}_{\mc{G},C^n} \twoheadrightarrow \Grb^{\la}_{\mc{G},C^n}$.

We define the following subset in $C^n$:

\begin{equation}\label{348782}
\Delta_{ [n]}=\{ \vec{p}\in C^n \mid\bar{p}_i= \bar{p}_j, \ \forall 1\leq i, j \leq n \}.
\end{equation}

With the same assumptions in Theorem <ref>, we have

* For any point $\vec{p}=(p,\gamma_2(p),\ldots, \gamma_n (p) ) \in \Delta_{[n]}$, we have

\[\Big(\Grb_{\mc{G},\vec{p}}^{\la} \Big)_{\mr{red}} \simeq \Grb_{\mc{G},p}^{\lambda_{\vec{p}}},\]

where $\lambda_{\vec{p}}:=\lambda_1+\gamma_2^{-1}(\lambda_2)+\cdots+\gamma_n^{-1}(\lambda_n)$.

* For any point $\vec{p}\in C^n$, we have

\[R^i(\mf{m}_{\vec{p}})_*\Big(\ms{O}_{\mr{Conv}_{\mc{G},\vec{p}}^{\la} } \Big)=
\begin{cases}
\ms{O}_{\big(\Grb_{\mc{G},\vec{p}}^{\la}\big)_{\mr{red}}}, & i=0;\\
0, & \forall i\geq 1.
\end{cases}\]

(1) Taking the reduced fibers of both sides of the morphism $\mf{m}:\overline{\Conv}^{\la}_{\mc{G},C^n} \twoheadrightarrow \Grb^{\la}_{\mc{G},C^n}$ in Lemma <ref>, we get a surjective map for any $\vec{p}\in C^n$,

\[\mf{m}_{\vec{p}}: \overline{\Conv}^{\la}_{\mc{G},\vec{p}}=\Big(\overline{\Conv}^{\la}_{\mc{G},\vec{p}}\Big)_{\mr{red}} \twoheadrightarrow \Big(\Grb^{\la}_{\mc{G},\vec{p}}\Big)_{\mr{red}}.\]

Now, let $\vec{p}=(p, \gamma_2(p), \ldots, \gamma_n(p))$ be any point in $\Delta_{[n]}$. To prove part (1), it suffices to show the image of $\overline{\Conv}^{\la}_{\mc{G},\vec{p}}$ under the convolution map $\mf{m}: \mr{Conv}_{\mc{G},C^n} \to \Gr_{\mc{G}, C^n}$ is $\Grb^{\lambda_{\vec{p}}}_{\mc{G},p}$. When $p\in C$ is unramified, the composition of morphisms (<ref>) (<ref>) restricts to the following map:

\begin{align}
\label{518909} \mb{E}|_{\vec{p}}\times^{L^+\mc{G}_{\gamma_n(p)}} \Gr_{\mc{G}, \gamma_n(p)} & \xrightarrow{\mathmakebox[1.5em]{\sim}} \mr{Conv}_{\mc{G},\vec{p}} \xrightarrow{\mathmakebox[1.5em]{\mf{m}}} \Gr_{\mc{G},\vec{p}} \\
\nonumber \Big( \big( \vec{p}, (\mc{F}_i, \beta_i)_{i=1}^{n-1}, \alpha \big), (\mc{F},\beta) \Big) & \mapsto \big( \vec{p}, (\mc{F}_i, \beta_i)_{i=1}^{n}\big) \mapsto (\vec{p}, \mc{F}|_{\hat{\Delta}_{\bar p_n}}, \zeta\circ\beta|_{\hat{\Delta}^*_{\bar p_n}})
\end{align}

where $p_n:=\gamma_n(p)$, $\zeta=(\beta_1\circ\cdots\circ\beta_{n-1}\circ\alpha^{-1})|_{\hat{\Delta}^*_{\bar p_n}}\in G(\mc{K}_{p_n})$. It factors through the usual convolution map

\[G(\mc{K}_{p_n})\times^{G(\mc{O}_{p_n})}\Gr_{G,p_n}\to \Gr_{G,p_n}\simeq \Gr_{\mc{G},\vec{p}}.\]

By induction on $n$, for $\vec{q}:=(p,\gamma_2(p),\cdots,\gamma_{n-1}(p))\in C^{n-1}$, the image of $\overline{\Conv}_{\mc{G},\vec{q}}^{\lambda_1,\ldots,\lambda_{n-1}}$ under the convolution map is $ \Grb^{\lambda_{\vec{q}}}_{\mc{G},p}$. Thus for any $(\vec{q}, (\mc{F}_i,\beta_i)_i)\in \overline{\Conv}_{\mc{G},\vec{q}}^{\lambda_1,\ldots,\lambda_{n-1}}$, we have $(p,\mc{F}_{n-1},\beta_1\circ\cdots\circ\beta_{n-1})\in \Grb_{G,p}^{\lambda_{\vec{q}}}\simeq \Grb_{G,p_n}^{\gamma_n(\lambda_{\vec{q}})}$. Therefore, the following restriction of (<ref>)

\[\mb{E}|_{\overline{\Conv}_{\mc{G},\vec{q}}^{\lambda_1,\ldots,\lambda_{n-1}},p_n}\times^{L^+\mc{G}_{p_n}} \Grb^{\lambda_n}_{G, p_n}\to \Grb^{\gamma_n(\lambda_{\vec{q}})+\lambda_n}_{G, p_n},\]

factors through $\mf{m}_{p_n}: \widetilde{\mr{Gr}}_{\mc{G},p_n}^{\gamma_n(\lambda_{\vec{q}})}\times^{L^+\mc{G}_{p_n}}\Grb^{\lambda_n}_{G, p_n}\to \Grb^{\gamma_n(\lambda_{\vec{q}})+\lambda_n}_{G, p_n}$, where $\widetilde{\mr{Gr}}_{\mc{G},p_n}^{\gamma_n(\lambda_{\vec{q}})}$ is the $L^+\mc{G}_p$-torsor over $\Grb^{\gamma_n(\lambda_{\vec{q}})+\lambda_n}_{G, p_n}$. This implies that the image of $\overline{\Conv}^{\la}_{\mc{G},\vec{p}}$ under the convolution map is $\Grb^{\gamma_n(\lambda_{\vec{q}})+\lambda_n}_{G, p_n}\simeq \Grb^{\lambda_{\vec{p}}}_{G, p}\simeq \Grb^{\lambda_{\vec{p}}}_{\mc{G},p}$. The case when $p\in C$ is totally ramified can be proved similarly.

For part (2), by Lemma <ref> it suffices to consider the case when $\vec{p}\in \Delta_{[n]}$.	Consider the map

\[\mf{m}_{\vec{p}}: \overline{\mr{Conv}}_{\mc{G},\vec{p}}^{\la}\to \Grb_{\mc{G},\vec{p}}^{\la}. \]

From part (1) and <cit.>, we know $(\mf{m}_{\vec{p}})_*\Big(\ms{O}_{\mr{Conv}_{\mc{G},\vec{p}}^{\la} } \Big)=\ms{O}_{ \Grb^{\lambda_{\vec{p}}}_{\mc{G},p}}=\ms{O}_{\big(\Grb_{\mc{G},\vec{p}}^{\la}\big)_{\mr{red}}}$.

For $i \geq 1$, we shall show $R^i(\mf{m}_{\vec{p}})_*\Big(\ms{O}_{\mr{Conv}_{\mc{G},\vec{p}}^{\la} } \Big)=0$ by induction on $n$.	Clearly, this holds when $n=1$. Now, assume $n\geq 2$, and let $\vec{q}=(p,\gamma_2(p),\ldots,\gamma_{n-1}(p))\in C^{n-1}$. By induction, $R^i(\mf{m}_{\vec{q}})_*\Big(\ms{O}_{\mr{Conv}_{\mc{G},\vec{q}}^{\la} } \Big)=0$.
Consider the following diagram

\[\xymatrix{
\mb{E}':=\mb{E}|_{\overline{\Conv}^{\lambda_1,\ldots, \lambda_{n-1}}_{\mc{G},\vec{q}}\times \{ p_n\}} \ar[d]_{\tilde{\mf{m}}_{\vec{q}}} \ar[r]^-{\mr{pr}} & \overline{\Conv}^{\lambda_1,\ldots, \lambda_{n-1}}_{\mc{G},\vec{q}} \ar[d]^{\mf{m}_{\vec{q}}} \\
\widetilde{\mr{Gr}}_{\mc{G},p}^{\lambda_{\vec{q}}} \ar[r]^{\mr{pr}} & \Grb^{\lambda_{\vec{q}}}_{\mc{G},p}
} .\]

By flat base change <cit.>, we have $R^i(\tilde{\mf{m}}_{\vec{q}})_*\big(\ms{O}_{\mb{E}' } \big)=0$ for $i\geq 1$.
From part (1), there is a commutative diagram

\[\xymatrix{
\overline{\Conv}^{\la}_{\mc{G},\vec{p}}\simeq \mb{E}|_{\overline{\Conv}_{\mc{G},\vec{q}}^{\lambda_1,\ldots,\lambda_{n-1}}\times \{p_n\}}\times^{L^+\mc{G}_{p_n}} \Grb^{\lambda_n}_{\mc{G}, p_n}\ar[d]_f \ar[dr]^-{\mf{m}_{\vec{p}}} & \\
W:=\widetilde{\mr{Gr}}_{\mc{G},p_n}^{\gamma_n(\lambda_{\vec{q}})}\times^{L^+\mc{G}_{p_n}} \Grb^{\lambda_n}_{\mc{G}, p_n} \ar[r]_-{\mf{m}_{p_n}} & \Grb^{\gamma_n(\lambda_{\vec{q}})+\lambda_n}_{\mc{G}, p_n}.
} \]

Thus, $R^if_*\Big(\ms{O}_{\overline{\Conv}^{\la}_{\mc{G},\vec{p}}}\Big)=0$. Since $R^i({\mf{m}_{p_n}})_*\ms{O}_W=0$, we have $R^i(\mf{m}_{\vec{p}})_*\Big(\ms{O}_{\overline{\Conv}^{\la}_{\mc{G},\vec{p}}}\Big)=0$.

With the same assumptions as in Theorem <ref>, the BD Schubert variety $\Grb_{\mc{G},C^n}^{\la}$ is flat over $C^n$. Moreover, $\Grb_{\mc{G},\vec{p}}^{\la}$ is reduced.

Let $X=\overline{\mr{Conv}}_{\mc{G},C^n}^{\la}$ and $Y=\Grb_{\mc{G},C^n}^{\la}$.	By Lemma <ref> and Proposition <ref>, we can apply Corollary <ref> to the morphism $\mf{m}:X\to Y$ and we get $\mf{m}_*(\ms{O}_X)$ is flat over $C^n$. To prove the first part of the theorem, it suffices to show $\mf{m}_*(\ms{O}_X)\simeq \ms{O}_Y$. Consider the natural map $\ms{O}_Y\to \mf{m}_*(\ms{O}_X)$. At every point $s=\vec{p}\in C^n$, the composition of the following maps

\begin{equation}\label{960342}
\ms{O}_{Y_s}\to \mf{m}_*(\ms{O}_X)|_{Y_s}\xrightarrow[\text{part (1) of Corollary \ref{173291}}]{\sim} (\mf{m}_s)_*(\ms{O}_{X_s}) \xrightarrow[\text{Proposition \ref{930915}}]{\sim} \ms{O}_{(Y_s)_{\mr{red}}}
\end{equation}

is surjective. Thus, the morphism $\ms{O}_Y\to \mf{m}_*(\ms{O}_X)$ is surjective.

Recall that $Y$ is the schematic image of $s_{\la}:L^+\mc{G}_{C^n}\to \Gr_{\mc{G},C^n}$. By Proposition <ref>, $X$ is the schematic image of $\tilde{s}_{\la}:L^+\mc{G}_{C^n}\to \mr{Conv}_{\mc{G},C^n}$.
Thus, we have an inclusion $\ms{O}_X\hookrightarrow (\tilde{s}_{\la})_*\big(\ms{O}_{L^+\mc{G}_{C^n}}\big)$. By Proposition <ref>, we have $R^1\mf{m}_*( \ms{O}_X)=0$. It follows that we have the following inclusion

\[\mf{m}_*\ms{O}_X\hookrightarrow \big(\mf{m}_*\circ (\tilde{s}_{\la})_*\big) \big(\ms{O}_{L^+\mc{G}_{C^n}}\big)=(s_{\la})_*\big(\ms{O}_{L^+\mc{G}_{C^n}}\big). \]

From the commutativity of the following diagram

\[\xymatrix{
\mf{m}_*\ms{O}_X \ar@{^{(}->}[r] & (s_{\la})_*\big(\ms{O}_{L^+\mc{G}_{C^n}}\big) \\
\ms{O}_{Y} \ar@{->>}[u] \ar@{^{(}->}[ru]&
} ,\]

we must have $\mf{m}_*\ms{O}_X\simeq \ms{O}_Y$. Thus, $\ms{O}_Y$ is flat over $C^n$. On the other hand, this also implies that the composition map (<ref>) is an isomorphism. In particular, it follows that $Y_s=\Grb_{\mc{G},\vec{p}}^{\la}$ is reduced.

With the same assumptions in Theorem <ref>, we have

* Let $\xi=(I_1,\ldots,I_k)$ be a partition of $[n]$. For any $\vec{p}\in C^n_\xi$, we have

\[\Grb_{\mc{G},\vec{p}}^{\la} \simeq \Grb_{\mc{G},\vec{p}_{I_1}}^{\la_{I_1}}\times \cdots\times \Grb_{\mc{G},\vec{p}_{I_k}}^{\la_{I_k}},\]

where $\vec{p}_{I_j}\in C^{I_j}$ and $\la_{I_j}$ are defined in (<ref>) and (<ref>).

* For any $\vec{p}=(p,\gamma_2(p),\ldots,\gamma_{n}(p))\in \Delta_{[n]}$, we have

\[\Grb_{\mc{G},\vec{p}}^{\la} \simeq \Grb_{\mc{G},p}^{\lambda_{\vec{p}}}, \]

where $\lambda_{\vec{p}}:=\lambda_1+\gamma_2^{-1}(\lambda_2)+\cdots+\gamma_n^{-1}(\lambda_n)$.

Part (1) directly follows from Proposition <ref>. Part (2) follows from Proposition <ref> and Theorem <ref>.

The orbit $\Gr_{\mc{G}, {C^n}}^{\la}:=L^+\mc{G}_{C^n}\cdot s^{\la}$ is smooth over $C^n$.

Observe that $\Gr_{\mc{G}, {C^n}}^{\la}$ is open in $\Grb_{\mc{G}, {C^n}}^{\la}$. By Theorem <ref>, $\Gr_{\mc{G}, {C^n}}^{\la}$ is also flat over $C^n$. Since every fiber $\Gr_{\mc{G},\vec{p}}^{\la}$ is smooth, $\Gr_{\mc{G}, {C^n}}^{\la}$ is smooth over $C^n$, cf. <cit.>.

The BD Schubert variety $\Grb^{\la}_{\mc{G},C^n}$ is normal.

By the étale base change property of BD Grassmannian (cf. Proposition <ref>) and a similar argument in Theorem <ref>, it suffices to prove the case when $C$ is the affine line $\A^1$.

When $n=1$, the theorem was proved in <cit.>. For $n\geq2$, we prove it by induction on $n$. Given a non-trivial partition $\xi=(I,J)$ of $[n]$, by Proposition <ref> there is a factorization over $\A^n_{\xi}$

\[\Grb^{\la}_{\mc{G}, \A^n_{\xi}} \simeq \Big(\Grb^{\la_{I}}_{\mc{G}, \A^I}\times \Grb^{\la_{I}}_{\mc{G}, \A^I}\Big)\big|_{\A^n_{\xi}}.\]

By induction, $\Grb^{\la_{I}}_{\mc{G}, \A^I}$ and $\Grb^{\la_{I}}_{\mc{G}, \A^J}$ are normal, hence $\Grb^{\la}_{\mc{G}, \A^n_{\xi}} $ is normal. Let $X=\Grb^{\la}_{\mc{G},\A^n}$. For any $i\in I$, $j\in J$ and $0\leq k\leq m-1$, we regard $t:=z_i-\sigma^k (z_j)$ as a regular function on $X$. We shall show the local ring $\ms{O}_{X,x}$ is normal for any $x\in X$ such that $t(x)=0$. Denote by $(X_0)_{\mr{red}}$ the reduced fiber of $t:X\to \A^1$ at $0\in \A^1$. By Corollary <ref>, $(X_0)_{\mr{red}}$ is isomorphic to a BD Schubert variety $\Grb^{\vec{\mu}}_{\mc{G},\A^{n-1}}$, where $\vec{\mu}$ consists of $\lambda_a$, $1\leq a\leq n$ with $a\not= i,j$, and $\lambda_i+\sigma^k(\lambda_j)$. Thus, $t\ms{O}_{X,x}$ has a unique minimal associated prime ideal $\mf{p}$. By induction, $(X_0)_{\mr{red}}$ is normal, hence $\ms{O}_{X,x}/\mf{p}$ is normal. By Lemma <ref>, $\Gr_{\mc{G}, {\A^n}}^{\la}$ is regular in codimension one, which implies that the localization $(\ms{O}_{X,x})_\mf{p}$ at the minimal ideal $\mf{p}$ is regular. Thus, $\ms{O}_{X,x}$ is normal, since the three conditions of <cit.> are satisfied. Since $i\in I$, $j\in J$ and $0\leq k\leq m-1$ are arbitrary, we conclude that $\Grb^{\la}_{\mc{G},\A^n}=X$ is normal.

Theorem <ref> and Theorem <ref>  remain true for any algebraically closed field $k$ of characteristic $p$, if $p\not \mid m$.  Moreover, when $G$ is of type $A_{2\ell}$, these two theorems still hold, if $\sigma$ is taken to be a diagram automorphism.

§ LINE BUNDLES ON BEILINSON-DRINFELD GRASSMANNIAN OF $\G$

In this section, we determine the rigidified Picard group of the BD Grassamnnian $\Gr_{\G,C^n}$ of $\G$ when $\G$ is generically simply-connected, along the way we construct the level one line bundle  $\mc{L}_{C^n}$ on $\Gr_{\G,C^n}$. We also establish the factorizable and $L^+\G_{C^n}$-equivariant structure on $\mc{L}_{C^n}$. This allow us to construct factorizable and equivariant level one line bundles on BD Schubert varieties of $\G$  when $C=\mathbb{A}^1$ and $\G$ is of adjoint type generically.

§.§ Picard group of rigidified line bundles on BD Grassmannian of $\G$

We first make a digression to prove a relative version of seesaw principal, which is originally due to Mumford <cit.>.

Let $\phi:X\to S$ and $\psi:Y\to S'$ be projective and flat morphisms of varieties over $\C$ such that, for any $s\in S$ and $s'\in S'$, the schematic fibers $X_{s}$ and $Y_{s'}$ are integral. Given an open subset $U\subset S\times S'$, let $L$ be a line bundle on $(X\times Y)|_U$ such that the restriction $L|_{X_s\times q}$ and $L|_{p\times Y_{s'}}$ are trivial for any $(s,q)\in (S\times Y)|_U$, $(p,s')\in (X\times S')|_U$. Then there is a line bundle $M$ on $U$ such that $L\simeq\pi^*M$, where $\pi=(\phi\times\psi )|_U: (X\times Y)|_U\to U$.

Consider the morphism $\pi_2=(\phi\times \mr{id})|_U: (X\times Y)|_U\to (S\times Y)|_U$. By assumption, for any $(s,q)\in (S\times Y)|_U$, the line bundle $L|_{X_s\times q}$ is trivial. Since $X$ is projective over $S$, we have $\dim\big( H^0(X_s\times q, L|_{X_s\times q})\big)=1$ for any $(s,q)\in (S\times Y)|_U$. It follows that $(\pi_2)_*L$ is a locally free sheaf of rank $1$ over $(S\times Y)|_U$. Let $L'=(\pi_2)_*L$. By adjunction, we have a morphism $(\pi_2)^*L'\to L$. This is an isomorphism since the map $H^0(X_s\times q, L)\to L_{(p,q)}$ is an isomorphism for any $(p,q)\in (X\times Y)|_U$.

Similarly, consider the morphism $\pi_1= (\mr{id}\times \psi)|_U: (S\times Y)|_U\to U$ and note that $L'|_{s\times Y_{s'}}$ is trivial for any $(s,s')\in U$. By the same argument as above, $(\pi_1)_*L'$ is a locally free sheaf of rank $1$ over $U$, and $L'\simeq (\pi_1)^*M$ where $M=(\pi_1)_*L'$. Therefore,

\begin{equation*}
\pushQED{\qed} L\simeq (\pi_2)^*L'\simeq (\pi_2)^*(\pi_1)^*M\simeq \pi^*M. \qedhere
\end{equation*}

Let $\pi: X\to S$ be a projective and flat morphism of varieties such that every fiber is integral. Given a section $e:S\to X$, let $(L,\alpha)$ denote a rigidified line bundle on $X$, that is, a line bundle $L$ with an isomorphism $\alpha: \ms{O}_S \simeq e^*L$. We denote by $\mr{Pic}^e(X)$ the Picard groupoid of these rigidified line bundles on $X$ with respect to $e$.

Let $\pi:X\to S$ be a morphism as above. Let $(L,\alpha)$, $(L',\alpha')$ be rigidified line bundles in $\mr{Pic}^e(X)$. If $\psi:L\simeq L'$ is an isomorphism of line bundles on $X$, then there exists a unique isomorphism $\phi:(L,\alpha)\simeq (L',\alpha')$ in $\mr{Pic}^e(X)$, i.e. there is a unique isomorphism $\phi:L\simeq L'$ such that the following diagram commutes

\begin{equation}\label{355605}
\xymatrix{
e^*L \ar[rr]^{e^*(\phi)} & & e^*L' \\
&\ms{O}_S \ar[lu]^{\alpha} \ar[ru]_{\alpha'}&
\end{equation}

By a similar argument as in the proof of Lemma <ref>, there exists an isomorphism $\pi^*\pi_* L\simeq L$. Applying $e^*$, we get $\pi_*L\simeq e^*L$. Thus,

\begin{equation}\label{eq_iso_L}
{\rm Hom}(L,L)\simeq {\rm Hom}(\pi^*\pi_*L,L)\simeq {\rm Hom}(\pi_*L, \pi_*L)\simeq {\rm Hom}(e^*L,e^*L).
\end{equation}

If there are two isomorphisms $\phi_1, \phi_2:(L,\alpha)\simeq (L',\alpha')$, then $\phi_2^{-1}\circ\phi_1$ is an automorphism of $(L,\alpha)\in \mr{Pic}^e(X)$. In view of the isomorphism (<ref>), this must be an identity. This proves the uniqueness.

Given an isomorphism $\psi: L\simeq L'$,	set $\beta=e^*(\psi)\circ \alpha: \ms{O}_S\simeq e^*L'$, which gives another rigidification of $L'$. By the isomorphism (<ref>), there exists a unique $\eta\in \mr{Aut}(L')$ such that $e^*(\eta)=\alpha'\circ \beta^{-1}$. Then, the isomorphism $\phi:= \eta\circ \psi:L\to L'$ satisfies the commutative diagram (<ref>). Hence, it gives an isomorphism $(L,\alpha)\simeq (L',\alpha')$ in $\mr{Pic}^e(X)$.

By Lemma <ref>, the groupoid $\mr{Pic}^e(X)$ is discrete. Thus, we can identify it with the group of isomorphism classes in $\mr{Pic}^e(X)$, and call it the rigidified Picard group of $X$ along $e$. Given an isomorphism class $[(\mc{L},\alpha)]\in \mr{Pic}^e(X )$, by abuse of notation, we simply denote it by $\mc{L}$.

From now on, we are in the setup of Section <ref>. We further assume that the simple algebraic group $G$ is simply-connected.

Let $\mr{Pic}^e(\Gr_{\mc{G},C^n})$ be the rigidified Picard group of $\Gr_{\mc{G},C^n}$ along the section $e:C^n\to \Gr_{\mc{G},C^n}$. Here, we abuse the notation $e$ for the section $e_{\vec{0}}$ over $C^n$. Given a rigidified line bundle $\mc{L}\in \Gr_{\mc{G},C^n}$, we denote by $\mc{L}|_{\Delta_{C^n}}$ the restriction of $\mc{L}$ to the BD Grassmannian $\Gr_{\mc{G},\Delta_{C^n}}$ over the diagonal.
By <cit.>, the central charge of $\mc{L}|_{\vec{p}}$ for any $\vec{p}\in \Delta_{C^n}$ is constant. This gives a well-defined central charge map

\begin{equation}\label{295279}
c:\mr{Pic}^e(\Gr_{\mc{G},C^n}) \to \mb{Z} .
\end{equation}

We shall show the map $c$ is an isomorphism. We first deal with the case when $C=\A^1$ with the standard $\Gamma$-action in (<ref>). Let $\mc{G}$ be the group scheme over $\mb{P}^1$. Recall that $\mr{Pic}(\mr{Bun}_{\mc{G}})\simeq \mb{Z}$, whose ample generator will be denoted by $\mc{L}$. Consider the projection $\mr{pr}:\Gr_{\mc{G},(\mb{P}^1)^n}\to \mr{Bun}_{\mc{G}}$. Let

\begin{equation}\label{eq_linebun_L}
\mc{L}_{\A^n}:=(\mr{pr}^*\mc{L})|_{\Gr_{\mc{G},\A^n}}
\end{equation}

denote the restriction of the line bundle $\mr{pr}^*\mc{L}$ to $\Gr_{\mc{G},\A^n}$. Clearly, $\mc{L}_{\A^n}$ is in $\mr{Pic}^e(\Gr_{\mc{G},\A^n})$ since $e^*(\mc{L}_{\A^n})$ is a trivial bundle over $\A^n$. Note that the restriction $\mc{L}_{\A^n}|_{\Delta_{\A^n}}$ to the diagonal, regarded as a line bundle over $ \Gr_{\mc{G},\A^1}$, is isomorphic to $\mc{L}_{\A^1}$. Moreover, the restriction of the line bundle $\mc{L}_{\A^1}$ to each fiber is the ample generator in $\mr{Pic}(\Gr_{\mc{G},p})$ for any $p\in \A^1$, cf. <cit.>. Hence $c(\mc{L}_{\A^n})=1$, which implies that $c$ is a surjection. From now on, we call $\mc{L}_{\A^n}$ the level one line bundle over $\Gr_{\G,\A^n}$.

Let $C$ be an irreducible smooth curve with a faithful $\Gamma$-action and assume that all ramified points in $C$ are totally ramified.

* For $n\geq 1$, the central charge map (<ref>) is an isomorphism

\[c:\mr{Pic}^e(\Gr_{\mc{G}, C^n})\simeq \mb{Z}.\]

* For $n\geq 2$, let $\xi=(I,J)$ be a partition of $[n]$. Let $\mc{L}$, $\mc{L}_I$, $\mc{L}_J$ be the regidified line bundles over $\Gr_{\G,C^n}$, $\Gr_{\G,C^I}$, $\Gr_{\G,C^J}$ respectively. Suppose they have the same central charge. Then, we have an isomorphism of the line bundles on $\Gr_{\mc{G}, C^n_\xi}$

\[\mc{L}|_{C^n_{\xi}}\simeq (\mc{L}_{I}\boxtimes\mc{L}_{J})|_{C^n_{\xi}}.\]

Let $R:=\{p_1,\ldots, p_k\}$ be the set of ramified points in $C$. Set $\mathring{C}=C\backslash R$. By part (2) of Proposition <ref>, $\Gr_{\G, \mathring{C}^n }\simeq \Gr_{G, \mathring{C}^n }$ for any $n\geq 1$. Then, by <cit.>, the following central charge map is an isomorphism

\begin{equation}\label{349124}
\mr{Pic}^e(\Gr_{\mc{G}, \mathring{C}^n })\simeq \mb{Z}.
\end{equation}

When $n=1$, we consider the following restriction map

\begin{equation} \label{eq_res_map}
\mr{Pic}^e(\Gr_{\mc{G},C})\to \mr{Pic}^e(\Gr_{\mc{G},\mathring{C}}).
\end{equation}

The composition of (<ref>) for $n=1$ and (<ref>) is the central charge map $c: \mr{Pic}^e(\Gr_{\mc{G},C})\to \mathbb{Z}$ for $C$.

We first show the restriction map (<ref>) is injective. Let $\mc{L}\in \mr{Pic}^e(\Gr_{\mc{G},C})$ such that the restriction $\mc{L}|_{\Gr_{\mc{G},\mathring{C} }}$ is trivial. We shall show $\mc{L}$ is also trivial.	It suffices to show the restriction of $\mc{L}$ to $\Grb^{\lambda}_{\mc{G}, C}$ is trivial for all dominant coweight $\lambda$. We still denote the restriction by $\mc{L}$. By assumption, there is an isomorphism $\theta: \ms{O}_{C}\simeq e^*\mc{L}$, and $\mc{L}|_{\Gr_{\mc{G}, \mathring{C}}}$ is trivial. By Lemma <ref>, there is a unique isomorphism

\[s: \ms{O}_{\Grb_{\G,\mathring{C}}^\lambda} \simeq \mc{L}|_{\Grb_{\G,\mathring{C}}^\lambda}\]

such that $e^*(s)=\theta|_{\mathring{C}}$. Let $D_i$ denote the divisor $\Grb_{\G,p_i}^\lambda$. By Theorem <ref>, there is a unique integer $n$ such that

\[s \in \Gamma(\Grb_{\mc{G},C\setminus\{p_1,\ldots,p_{k-1}\}}^\lambda,\mc{L}(nD_k))\setminus \Gamma(\Grb_{\mc{G},C\setminus\{p_1,\ldots,p_{k-1}\}}^\lambda,\mc{L}((n-1)D_k)). \]

Now, we regard $s$ as a regular section of $\mc{L}(nD_k)$. Then $e^*(s)=\theta|_{\mathring{C}}$ can be regarded a section of $(e^*\mc{L})|_{\mathring{C}}$ with pole order $n$ at $p_k\in C$. Since $\theta$ is regular at $p_k$, we must have
$n=0$, and hence $s$ induces an isomorphism

$$s:\ms{O}_{\Grb_{\G,C\setminus \{p_1,\ldots, p_{k-1}\} }^\lambda} \simeq \mc{L}|_{\Grb_{\G,C\setminus \{p_1,\ldots,p_{k-1}\}}^\lambda}.$$

Repeating this process, we get a trivialization of $\mc{L}$. It follows that the restriction map (<ref>) is injective. Hence, the central charge map (<ref>) is injective.

We now prove the surjectivity. For each $p_i\in R$, there exists a $\Gamma$-stable open neighborhood $U_i$ of $p$ with a $\Gamma$-equivariant étale map $U_i\to \A^1$, where $\A^1$ carries the standard $\Gamma$-action as in (<ref>). Observe that $p_i$ is the only ramified point in $U_i$. By Proposition <ref>, there is a natural morphism $\mf{p}_i:\Gr_{\mc{G}_{\bar{U}_i},U_i}\to\Gr_{\mc{G}_{\bar\A^1}, \A^1} $. Let $\mc{L}_{U_i}$ be the pull-back of the line bundle $\mc{L}_{\A^1}$ via the map $\mf{p}_i$. By the same argument as above, the central charge map $c: \mr{Pic}^e(\Gr_{\mc{G}, U_i})\to \mathbb{Z}$ is injective. As $\mc{L}_{U_i}$ is mapped to $1$, we have $c: \mr{Pic}^e(\Gr_{\mc{G}, U_i})\simeq \mathbb{Z}$. Note that $\{\mathring{C}, U_1,\ldots ,U_r\}$ forms an open covering of $C$. Moreover, by Lemma <ref> there are unique isomorphisms between the restrictions of $\mc{L}_{\mathring{C}}$ and $\mc{L}_{U_i}$ on their intersections, since their restrictions are the level one generator in $\mr{Pic}^e(\Gr_{\mc{G},U_i\setminus\{p_i\}})$ or $\mr{Pic}^e(\Gr_{\mc{G},U_i \cap U_j})$ for any $i,j$. Therefore, we can glue them together to get a line bundle in $\mr{Pic}^e(\Gr_{\mc{G},C}) $, which will be denoted by $\mc{L}_C$. From construction, we have $c(\mc{L}_C)=1$. It follows that the central charge map $c: \mr{Pic}^e(\Gr_{\mc{G},C})\to \mathbb{Z}$ is an isomorphism.

When $n=2$, let $\xi=(\{1\},\{2\})$ be the non-trivial partition of $[2]$. Then, we have a factorization $\Gr_{\mc{G},C^2_\xi}\simeq \big(\Gr_{\mc{G},C}\times \Gr_{\mc{G}, C}\big)|_{C^2_\xi}$ over $C^2_\xi$.	Given a point $p\in C$ and $x,y\in \Gr_{\mc{G},p}$, we consider the following map

\begin{equation}\label{712602}
\mr{Pic}^e(\Gr_{\mc{G},C}\times \Gr_{\mc{G}, C}|_{C^2_\xi})\to \mr{Pic}^e(\Gr_{\mc{G},C-p}) \times	\mr{Pic}^e(\Gr_{\mc{G},C-p})
\end{equation}

given by $\mc{L}\mapsto (\mc{L}_1,\mc{L}_2)$, where $\mc{L}_1:=\mc{L}|_{\Gr_{\mc{G},C-p}\times y}$ and $\mc{L}_2:=\mc{L}|_{x\times\Gr_{\mc{G},C-p}}$ are regarded as line bundles over $\Gr_{\mc{G},C-p}$. We claim (<ref>) is an isomorphism.

We first show the injectivity. Since $c:\mr{Pic}^e(\Gr_{\mc{G},C})\to \mathbb{Z}$ is an isomorphism for any curve $C$ under our assumptions, the following restriction map is an isomorphism,

\begin{equation}\label{eq_rest_iso_1}
\mr{Pic}^e(\Gr_{\mc{G}, C})\simeq \mr{Pic}^e(\Gr_{\mc{G}, C-p}). \end{equation}

For any $\mc{L}\in \mr{Pic}^e(\Gr_{\mc{G},C}\times \Gr_{\mc{G}, C}|_{C^2_\xi})$, by the isomorphism (<ref>),	$\mc{L}_1$ and $\mc{L}_2$ uniquely extend to rigidified line bundles over $\Gr_{\mc{G}, C}$, which will be still denoted by $\mc{L}_1$ and $\mc{L}_2$. Set

\[\mc{L}':=\mc{L}^{-1}\otimes(\mc{L}_1\boxtimes\mc{L}_2)|_{C^2_\xi}\in \mr{Pic}^e(\Gr_{\mc{G},C}\times \Gr_{\mc{G}, C}|_{C^2_\xi}).\]

By Lemma <ref>, $\mc{L}'$ is the pullback of some line bundle $M$ on $C^2_\xi$ via the projection $\pi:(\Gr_{\mc{G},C}\times \Gr_{\mc{G}, C})|_{C^2_\xi}\to C^2_\xi$. Note that $M\simeq e^*\pi^*M\simeq e^*\mc{L}'$ is trivial. Then, $\mc{L}'\simeq \pi^*M$ is trivial. It follows that $\mc{L}\simeq \mc{L}_1\boxtimes\mc{L}_2|_{C^2_\xi}$. This proves the injectivity of (<ref>).

Given any $(\mc{L}_1,\mc{L}_2)\in \mr{Pic}^e(\Gr_{\mc{G},C-p}) \times	\mr{Pic}^e(\Gr_{\mc{G}, C-p})$, we still denote their extensions to $\Gr_{\mc{G},C}$ by $(\mc{L}_1,\mc{L}_2)$. Clearly the line bundle $(\mc{L}_1\boxtimes\mc{L}_2)|_{{C^2_\xi}}\in \mr{Pic}^e(\Gr_{\mc{G},C}\times \Gr_{\mc{G}, C}|_{C^2_\xi})$ is mapped to $(\mc{L}_1,\mc{L}_2)$ via the map (<ref>). This shows that (<ref>) is surjective.

\begin{equation}\label{eq_sea}
\mr{Pic}^e(\Gr_{\mc{G}, C^2_\xi})\simeq \mr{Pic}^e(\Gr_{\mc{G},C}\times \Gr_{\mc{G}, C}|_{C^2_\xi})\simeq \mb{Z}\times\mb{Z}.\end{equation}

By the similar argument as the case when $n=1$ (or the similar proof of the untwisted case when $n=2$, see <cit.>), we can show the following restriction map is injective,

\begin{equation}\label{827216}
\mr{Pic}^e(\Gr_{\mc{G}, C^2})\to \mr{Pic}^e(\Gr_{\mc{G},C^2_\xi})\simeq \mb{Z}\times \mb{Z}.
\end{equation}