Demazure and local Weyl modules for twisted hyper current algebras

TL;DR

This paper establishes an isomorphism between local graded Weyl modules and level 1 Demazure modules for twisted hypercurrent algebras, linking twisted and untwisted cases in the representation theory of hypercurrent algebras.

Contribution

It proves that local graded Weyl modules for twisted hypercurrent algebras are isomorphic to level 1 Demazure modules and are restrictions of untwisted modules, revealing structural connections.

Findings

Local graded Weyl modules are isomorphic to level 1 Demazure modules.

These modules are restrictions of untwisted hypercurrent algebra modules.

The results unify twisted and untwisted hypercurrent algebra representations.

Abstract

In this paper, we study local graded Weyl modules and Demazure modules for twisted hypercurrent algebras. We prove that local graded Weyl modules for a twisted hypercurrent algebra are isomorphic to the corresponding level 1 Demazure modules, and moreover, that they are restrictions of corresponding local graded Weyl modules for the untwisted hypercurrent algebra.

Equations

$$r_{\alpha }^{\vee }=\frac{2}{(\alpha ,\alpha )}=\{\begin{array}{ll}1,& \text{if α is long},\\ r^{\vee },& \text{if α is short.}\end{array}$$

$$\sigma \left(\sum _{i\in I}{n}_i{\alpha }_i\right):=\sum _{i\in I}{n}_i{\alpha }_{\sigma (i)}.$$

$$\text{g}=\underset{\epsilon =0}{\overset{m-1}{⨁}}{\text{g}}_{\epsilon }\text{and}[{\text{g}}_{\epsilon },{\text{g}}_{{\epsilon }^{\prime }}]\subseteq {\text{g}}_{\overline{\epsilon +{\epsilon }^{\prime }}},$$

$$x_{\alpha ,\epsilon }^{\pm }=\sum _{j=0}^{m_{\alpha }-1}{\zeta }^{j\epsilon }{x}_{{\sigma }^j(\alpha )}^{\pm }\text{and}{h}_{\alpha ,\epsilon }=\sum _{j=0}^{m_{\alpha }-1}{\zeta }^{j\epsilon }{h}_{{\sigma }^j(\alpha )}.$$

$$x_{\alpha ,\epsilon }^{\pm }:=\{\begin{array}{ll}{\delta }_{1,\epsilon }{x}_{\alpha }^{\pm },& \text{if }\alpha =\sigma (\alpha ),\\ x_{\alpha }^{\pm }+(-1{)}^{\epsilon }{x}_{\sigma (\alpha )}^{\pm },& \text{if }\alpha \ne \sigma (\alpha ),\alpha {|}_{{\text{h}}_0}\in R_{\lg },\\ \sqrt{2}\left(x_{\alpha }^{\pm }+(-1{)}^{\epsilon }{x}_{\sigma (\alpha )}^{\pm }\right),& \text{if }\alpha \ne \sigma (\alpha ),\alpha {|}_{{\text{h}}_0}\in R_{\mathrm{sh}},\end{array}$$

$$h_{\alpha ,\epsilon }:=\{\begin{array}{ll}{\delta }_{0,\epsilon }{h}_{\alpha },& \text{if }\alpha =\sigma (\alpha ),\\ h_{\alpha }+(-1{)}^{\epsilon }{h}_{\sigma (\alpha )},& \text{if }\alpha \ne \sigma (\alpha ),\alpha {|}_{{\text{h}}_0}\in R_{\lg },\\ 2\left(h_{\alpha }+(-1{)}^{\epsilon }{h}_{\sigma (\alpha )}\right),& \text{if }\alpha \ne \sigma (\alpha ),\alpha {|}_{{\text{h}}_0}\in R_{\mathrm{sh}}.\end{array}$$

$$x_{\beta +\sigma (\beta ),1}^{\pm }=\frac{s}{4}[x_{\beta ,0}^{\pm },x_{\beta ,1}^{\pm }]=-s[x_{\beta }^{\pm },x_{\sigma (\beta )}^{\pm }]\text{ for some s∈{−1,1}}.$$

$${\text{h}}_{\epsilon }={\mathrm{span}}_{\text{C}}\{h_{{\alpha }_i,\epsilon }\mid i\in I\}\text{and}({\text{g}}_{\epsilon }{)}_{\pm \mu }=\mathbb{C}{x}_{\alpha ,\epsilon }^{\pm },\text{where}\alpha {|}_{{\text{h}}_0}=\mu .$$

$${\mathcal{C}}^{\sigma }(\mathcal{O}):=\left\{x_{\mu ,\epsilon }^{\pm },h_{i,\epsilon }\mid \epsilon \in \{0,\dots ,m-1\},\mu \in {\mathcal{R}}_{\epsilon }^{+},i\in {\mathcal{I}}_0\right\}\setminus \{0\}$$

$$[h_{\mu ,0},x_{\nu ,\epsilon }^{\pm }]$$

$$\text{if hν,1=0, then}[h_{\nu ,1},x_{\nu ,\epsilon }^{\pm }]$$

$$[x_{\eta ,\epsilon }^{+},x_{\eta ,{\epsilon }^{\prime }}^{-}]$$

$$[x\otimes f,y\otimes g]=[x,y{]}_{\text{a}}\otimes fg,\text{for all x,y∈a, f,g∈C[t,t−1]},$$

$$\tilde{\sigma }(x\otimes t^j)={\zeta }^j\sigma (x)\otimes t^j\text{for all x∈a and j∈Z}.$$

$${\text{a~}}^{\sigma }=\underset{\epsilon =0}{\overset{m-1}{⨁}}{\text{a~}}_{\epsilon }^{\sigma },{\text{a~}}_{\epsilon }^{\sigma }={\text{a}}_{\epsilon }\otimes t^{m-\epsilon }\mathbb{C}[t^m,t^{-m}],\text{and}\text{a}[t{]}^{\sigma }=\underset{\epsilon =0}{\overset{m-1}{⨁}}\text{a}[t{]}_{\epsilon }^{\sigma },\text{a}[t{]}_{\epsilon }^{\sigma }={\text{a}}_{\epsilon }\otimes t^{m-\epsilon }\mathbb{C}[t^m].$$

$$[x\otimes t^r,y\otimes t^s]=[x,y]\otimes t^{r+s}+r{\delta }_{r,-s}(x,y)c,[c,\text{g^}]=0\text{and}[d,x\otimes t^r]=rx\otimes t^r$$

$$\text{h^}=\text{h}\oplus \text{C}c\oplus \text{C}d,{\text{n^}}^{\pm }={\text{n}}^{\pm }\oplus \text{g}[t{]}_{\pm },{\text{b^}}^{\pm }={\text{n^}}^{\pm }\oplus \text{h^}.$$

$$\alpha (c)=0\text{for all}\alpha \in \hat{R}.$$

$$\hat{\Delta }=\{{\alpha }_i\mid i\in \hat{I}\},{\hat{R}}^{+}=R^{+}\cup \{\alpha +r\delta \mid r\in {\text{Z}}_{>0},\alpha \in R_r\cup \{0\}\},$$

$${\text{g^}}_{\alpha +r\delta }=({\text{g}}_r{)}_{\alpha }\otimes t^r,\text{if r∈Z,α∈Rr},\text{and}{\text{g^}}_{r\delta }=\text{h}\otimes t^r,\text{if r∈Z∖{0}}.$$

$$x_{\pm \alpha +r\delta }^{+}=x_{\alpha ,-r}^{\pm }\otimes t^r,x_{\pm \alpha +r\delta }^{-}=x_{\alpha ,r}^{\mp }\otimes t^{-r},{\hat{r}}_{\alpha }^{\vee }=m{r}_{\alpha }^{\vee }/a_0,$$

$$h_{\pm \alpha +r\delta }=[x_{\pm \alpha +r\delta }^{+},x_{\pm \alpha -r\delta }^{-}]=\{\begin{array}{ll}{\delta }_{r,1}{h}_{\frac{\alpha }{2},0}+r{\hat{r}}_{\alpha }^{\vee }c,& \text{if g≅A2n and α∈2Rsh},\\ \pm h_{\alpha ,0}+r{\hat{r}}_{\alpha }^{\vee }c,& \text{otherwise}.\end{array}$$

$${\Lambda }_0(h)=0\text{iff}h\in \text{h}\oplus \text{C}d\text{and}{\Lambda }_i-{\omega }_i={\omega }_i(h_{\theta }){\Lambda }_0\text{for all}i\in I.$$

$$U(\text{g}[t])\cong U({\text{n}}^{-}[t])\otimes U(\text{h}[t])\otimes U({\text{n}}^{+}[t])\text{and}U(\text{g}[t{]}^{\sigma })\cong U({\text{n}}^{-}[t{]}^{\sigma })\otimes U(\text{h}[t{]}^{\sigma })\otimes U({\text{n}}^{+}[t{]}^{\sigma }).$$

$${\Lambda }_{\alpha ;k}(u)=\sum _{r\ge 0}{\Lambda }_{\alpha ,r;k}{u}^r=\exp \left(-\sum _{s>0}\frac{h_{\alpha }\otimes t^{sk}}{s}{u}^s\right).$$

$${\Lambda }_{\mu }^{\sigma }(u)=\sum _{r=0}^{\infty }{\Lambda }_{\mu ,r}^{\sigma }{u}^r=\text{exp}\left(-\sum _{k=1}^{\infty }\sum _{\epsilon =0}^{m-1}\frac{h_{\mu ,\epsilon }\otimes t^{mk-\epsilon }}{mk-\epsilon }{u}^{mk-\epsilon }\right).$$

$${\Lambda }_{\mu }^{\sigma }(u)=\sum _{r=0}^{\infty }{\Lambda }_{\mu ,r}^{\sigma }{u}^r=\text{exp}\left(-\sum _{k=1}^{\infty }\frac{h_{\mu ,0}\otimes t^{mk}}{k}{u}^k\right).$$

$${\Lambda }_{\mu }^{\sigma ,\pm }(u)=\{\begin{array}{ll}{\prod }_{j=0}^{m-1}{\Lambda }_{{\sigma }^j(\alpha )}^{\pm }({\zeta }^{m-j}u),& \text{if}{\Gamma }_{\alpha }=m,\\ {\Lambda }_{\alpha ;m}^{\pm }(u),& \text{if}{\Gamma }_{\alpha }=1.\end{array}$$

$$\mathcal{M}[t{]}^{\sigma }=\left\{(x_{\mu ,-r}^{\pm }\otimes t^r{)}^{(k)},{\Lambda }_{i,k}^{\sigma },(\genfrac{}{}{0ex}{}{h_{i,0}}{k})\mid \mu \in \mathrm{wt}({\text{g}}_{-r})\cap {\mathcal{Q}}_0^{+}\setminus \{0\},i\in {\mathcal{I}}_0,r\in {\mathbb{Z}}_{\ge 0},k\in \mathbb{N}\right\}\setminus \{0\},$$

$$\mathcal{M}=\left\{(x_{\alpha }^{\pm }{)}^{(k)},(\genfrac{}{}{0ex}{}{h_i}{k})\mid \alpha \in {\mathcal{R}}^{+},i\in \mathcal{I},k\in {\mathbb{Z}}_{\ge 0}\right\}$$

$$\mathcal{M}[t]=\left\{(x_{\alpha }^{\pm }\otimes t^r{)}^{(k)},{\Lambda }_{i,k},(\genfrac{}{}{0ex}{}{h_i}{k})\mid \alpha \in {\mathcal{R}}^{+},i\in \mathcal{I},k\in \mathbb{N},r\in {\mathbb{Z}}_{\ge 0}\right\}.$$

$$U_{\mathbb{Z}}(\text{a}[t{]}^{\sigma })=U(\text{a}[t{]}^{\sigma })\cap U_{\mathbb{Z}}(\text{g}[t{]}^{\sigma })$$

Full text

Partially supported by the FAPESP grant 2015/22040-0 (A.B) and the CNPq grant 462315/2014-2 (A.B. and T.M.)

Introduction

The main goal of this paper is to generalize some results relating local Weyl and Demazure modules for twisted current algebras, which are known to hold over an algebraically closed field of characteristic zero, to the positive characteristic setting. Namely, in [CFS08, Theorem 2.i], it was proved that any local Weyl module for a twisted affine algebra is isomorphic to a certain restriction of a local Weyl module for the corresponding untwisted affine algebra. In [FK13, Theorem 6.0.2], it was proved that any graded local Weyl module for a twisted current algebra is isomorphic to a certain graded restriction of a graded local Weyl module for the corresponding untwisted current algebra. It is interesting to notice that these results were also obtained by [FKKS12, Lemma 3.8] in a more general framework of equivariant map algebras.

In the simply laced case, it was proved in [FL07, Proposition 2] that any local Weyl module for a current algebra is isomorphic to a certain automorphic image of a local Weyl module for the corresponding loop algebra. In [CP01], for \text{\mathfrak{sl}}_{2}, and in [CL06], for type $A$, and [FL07, Theorem 7], for types ADE, both using the \text{\mathfrak{sl}}_{2}-case, it was proved that any local graded Weyl module for a current algebra is isomorphic to a certain restriction of a Demazure module for the corresponding affine algebra. This latter result was shown to be false in the non simply laced case. In fact, it was proved that local graded Weyl modules for current algebras admit Demazure flags in [Nao12, Proposition 4.18]. Finally, it was proved in [FK13, Theorem 5.0.2] that, under some somewhat restrictive conditions, any twisted graded local Weyl module for a current algebra is isomorphic to a certain twisted Demazure module.

Some positive characteristic generalizations of the results mentioned above are already known. For instance, it was proved in [BM14, Theorem 4.1] that any local Weyl module for a twisted hyper loop algebra is isomorphic to a certain restriction of a local Weyl module for the corresponding untwisted hyper loop algebra. In the simply laced case, it was proved in [BMM15, Theorem 1.5.2] that any local Weyl module for a hyper current algebra is isomorphic to a certain automorphic image of a local Weyl module for the corresponding hyper loop algebras, and that any local graded Weyl module for a current algebra is isomorphic to a certain restriction of a Demazure module for the corresponding affine algebra.

It is important to remark that, in the non simply laced case, it was proved in [BMM15, Theorem 1.5.2.b] that any local Weyl module for a hyper current algebra admits a Demazure flag. Since Dynkin diagrams associated to non simply laced simple Lie algebras admit only the trivial automorphism, the following claim is an immediate consequence of [BMM15, Theorem 1.5.2]: in the non simply laced case, every local graded Weyl module for a twisted hyper current algebra admits a Demazure flag, and moreover, it is isomorphic to a certain local Weyl module for the twisted hyperloop algebra.

Hence, our main goal in this paper is to prove the following result.

Theorem**.**

The local graded Weyl module of highest weight $\lambda$ for a twisted hyper current algebra is isomorphic to the corresponding level 1 Demazure module. And, moreover, it is a restriction of the corresponding local graded Weyl module of highest weight $\lambda$ for the untwisted hyper current algebra.

The proof of the first part is given in Theorem 4.2 and the proof of the second part is given in Theorem 4.3. Along the way, we also prove some results about local graded Weyl modules and Demazure modules for twisted hyper current algebras. For instance, in Lemma 3.2, we prove that integral local graded Weyl modules are integrable modules, and in Proposition 3.3, we prove that they are finitely generated. Also, in Proposition 3.8, we prove that the purely algebraic definition of a Demazure module given here reflects the usual geometric definition of Demazure modules.

In terms of equivariant map algebras, \text{\mathfrak{g}}[t]^{\sigma} can be regarded as the Lie algebra of equivariant regular maps \varphi\colon\text{\mathbb{C}}\to\text{\mathfrak{g}} under the cyclic group generated by the automorphism $\sigma$. Here, the cyclic group generated by $\sigma$ acts on $\mathfrak{g}$ as in Section 1.1 and on $\mathbb{C}$ via multiplication by roots of unity. Observe that this latter action is not free, since 0\in\text{\mathbb{C}} is a fixed point. Thus, in the current paper, we are dealing with a case that is not covered by [FKKS12]. Moreover, we generalize some of the results in [FKKS12] to fields of positive characteristic (different from 2 in case \text{\mathfrak{g}}\cong A_{2n}).

Notation

Denote by $\mathbb{C},\mathbb{Z},\mathbb{Z}_{\geq 0}$ and $\mathbb{N}$ the set of complex numbers, the set of integers, non–negative integers and positive integers, respectively. Fix an indeterminate $t$ and let $\mathbb{C}[t]$ (resp. $\mathbb{C}[t,t^{-1}]$) be the corresponding polynomial ring (resp. Laurent polynomial ring) with complex coefficients. Throughout this paper, unless we explicitly state otherwise, all vector spaces and tensor products are assumed to be taken over $\mathbb{C}$.

1. Preliminaries on algebras

1.1. Simple Lie algebras and diagram automorphisms

Let $I$ be the set of vertices of a finite-type connected Dynkin diagram and let $\mathfrak{g}$ be the associated finite-dimensional simple Lie algebra over $\mathbb{C}$. Fix a Cartan subalgebra \text{\mathfrak{h}}\subseteq\text{\mathfrak{g}}, a triangular decomposition \text{\mathfrak{g}}=\text{\mathfrak{n}}^{-}\oplus\text{\mathfrak{h}}\oplus\text{\mathfrak{n}}^{+}, and denote by $R$ (resp. $R^{+}$) the associated root system (resp. set of positive roots). Let $\{\alpha_{i}\mid i\in I\}$ (resp. $\{\omega_{i}\mid i\in I\}$) denote the set of simple roots (resp. fundamental weights), and let $Q=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}$, $Q^{+}=\bigoplus_{i\in I}\mathbb{Z}_{\geq 0}\alpha_{i}$, $P=\bigoplus_{i\in I}\mathbb{Z}\omega_{i}$, $P^{+}=\bigoplus_{i\in I}\mathbb{Z}_{\geq 0}\omega_{i}$. Denote by $\cal W$ the Weyl group of $\mathfrak{g}$. Let \text{\mathfrak{g}}_{\alpha} denote the root space of $\mathfrak{g}$ associated to $\alpha\in R$ and $\theta$ denote the highest root of $\mathfrak{g}$.

Fix a Chevalley basis $\cal C=\{x_{\alpha}^{\pm},h_{i}\mid\alpha\in R^{+},\,i\in I\}$ of $\mathfrak{g}$ and, for each $\alpha\in R^{+}$, let $h_{\alpha}=[x^{+}_{\alpha},x^{-}_{\alpha}]$ (in particular, $h_{\alpha_{i}}=h_{i}$ for all $i\in I$). There exists a unique bilinear form $(\ ,\ )$ on $\mathfrak{g}$ that is symmetric, invariant, nondegenerate, and such that $(h_{\theta},h_{\theta})=2$. Let \nu\colon\text{\mathfrak{h}}\to\text{\mathfrak{h}}^{*} be the linear isomorphism induced by $(\ ,\ )$ and keep denoting by $(\ ,\ )$ the nondegenerate bilinear form induced by $\nu$ on \text{\mathfrak{h}}^{*}. Denote by $r^{\vee}\in\{1,2,3\}$ the lacing number of $\mathfrak{g}$, and let

[TABLE]

Fix an automorphism $\sigma$ of the Dynkin diagram of $\mathfrak{g}$ and let $m\in\{1,2,3\}$ denote the order of $\sigma$. There exists a unique Lie algebra automorphism of $\mathfrak{g}$ (which we keep denoting by $\sigma$) that satisfies $\sigma(x^{\pm}_{i})=x^{\pm}_{\sigma(i)}$ for all $i\in I$. Define a permutation of $R$ (which we also denote by $\sigma$) by

[TABLE]

For each $\alpha\in R^{+}$, define m_{\alpha}:=\#\left\{\sigma^{j}(\alpha)\mid j\in\text{\mathbb{N}}\right\}. Observe that $m_{\alpha}\in\{1,m\}$ and \sigma(\text{\mathfrak{g}}_{\alpha})=\text{\mathfrak{g}}_{\sigma(\alpha)} for all $\alpha\in R$.

Fix a primitive $m$-th root of unity $\zeta$, and let \text{\mathfrak{g}}_{\varepsilon}=\{x\in\text{\mathfrak{g}}\mid\sigma(x)=\zeta^{\varepsilon}x\} for each $\varepsilon\in\{0,\dotsc,m-1\}$. Since every finite order automorphism is diagonalizable over $\mathbb{C}$ (with eigenvalues being roots of unity), we have

[TABLE]

where $\overline{\varepsilon+\varepsilon^{\prime}}$ is the remainder of the division of $\varepsilon+\varepsilon^{\prime}$ by $m$. (Henceforth, we will abuse notation and write $\varepsilon+\varepsilon^{\prime}$ instead of $\overline{\varepsilon+\varepsilon^{\prime}}$.) In particular, \text{\mathfrak{g}}_{0} is a subalgebra of $\mathfrak{g}$ and \text{\mathfrak{g}}_{\varepsilon} is a \text{\mathfrak{g}}_{0}-module for all $\varepsilon\in\{1,\dotsc,m-1\}$. Moreover, it is known that \text{\mathfrak{g}}_{0} is a finite-dimensional simple Lie algebra and \text{\mathfrak{g}}_{\varepsilon} is a finite-dimensional irreducible \text{\mathfrak{g}}_{0}-module for all $\varepsilon\in\{0,\dotsc,m-1\}$ (see [Kac90, Chapter 8]).

If $\mathfrak{a}$ is a subalgebra of $\mathfrak{g}$ and $\varepsilon\in\{0,\dotsc,m-1\}$, let \text{\mathfrak{a}}_{\varepsilon}:=\text{\mathfrak{a}}\cap\text{\mathfrak{g}}_{\varepsilon}. It is known that \text{\mathfrak{h}}_{0} is a Cartan subalgebra of \text{\mathfrak{g}}_{0} and \text{\mathfrak{g}}_{0}=\text{\mathfrak{n}}^{+}_{0}\oplus\text{\mathfrak{h}}_{0}\oplus\text{\mathfrak{n}}^{-}_{0} is a triangular decomposition. Let $R_{0}$ denote the root system determined by \text{\mathfrak{h}}_{0}, and $I_{0}$ be an indexing set for the associated simple roots. The simple roots (resp. fundamental weights) of \text{\mathfrak{g}}_{0} determined by the triangular decomposition \text{\mathfrak{g}}_{0}=\text{\mathfrak{n}}^{+}_{0}\oplus\text{\mathfrak{h}}_{0}\oplus\text{\mathfrak{n}}^{-}_{0} will be denoted by $\alpha_{i}$, $i\in I_{0}$ (resp. $\omega_{i}$, $i\in I_{0}$), as this should cause no confusion. Let $\theta_{0}$ denote the highest root of \text{\mathfrak{g}}_{0}, $Q_{0}=\bigoplus_{i\in I_{0}}\mathbb{Z}\alpha_{i}$, $Q_{0}^{+}=\bigoplus_{i\in I_{0}}\mathbb{Z}_{\geq 0}\alpha_{i}$, $P_{0}=\bigoplus_{i\in I_{0}}\mathbb{Z}\omega_{i}$, $P_{0}^{+}=\bigoplus_{i\in I_{0}}\mathbb{Z}_{\geq 0}\omega_{i}$. Let $R_{\operatorname{sh}}$ (resp. $R_{\operatorname{lg}}$) be the subset of $R_{0}^{+}$ corresponding to short (resp. long) positive roots of \text{\mathfrak{g}}_{0}, and $\cal W_{0}$ be the Weyl group of \text{\mathfrak{g}}_{0}.

For each $\mathfrak{g}$-module (resp. of \text{\mathfrak{g}}_{0}-module) $V$, denote by $V_{\lambda}$, \lambda\in\text{\mathfrak{h}}^{*} (resp. \lambda\in\text{\mathfrak{h}}_{0}^{*}), its weight space of weight $\lambda$. Let $\operatorname{wt}(V)$ (resp. $\operatorname{wt}_{0}(V)$) denote the set of weights of $V$ as an $\mathfrak{h}$-module (resp. \text{\mathfrak{h}}_{0}-module). For each $\varepsilon\in\{0,\dotsc,m-1\}$, denote by $R_{\varepsilon}$ the set of weights of \text{\mathfrak{g}}_{\varepsilon} as a \text{\mathfrak{g}}_{0}-module (via the adjoint representation). It is known that $R_{\varepsilon}\subseteq Q_{0}$, so let $R_{\varepsilon}^{+}=(R_{\varepsilon}\cap Q_{0}^{+})\setminus\{0\}$. Finally, denote by $\theta_{1}$ the highest weight of \text{\mathfrak{g}}_{1} (as a \text{\mathfrak{g}}_{0}-module).

1.2. Chevalley basis of \text{\mathfrak{g}}_{0}

Given $\varepsilon\in\{0,\dotsc,m-1\}$ and $\mu\in R_{\varepsilon}\setminus\{0\}$, it is known that: \mu=\alpha|_{\text{\mathfrak{h}}_{0}} for some $\alpha\in R$, and \alpha|_{\text{\mathfrak{h}}_{0}}=\beta|_{\text{\mathfrak{h}}_{0}} if, and only if, $\beta=\sigma^{j}(\alpha)$ for some $j\in\{0,\dotsc,m_{\alpha}-1\}$.

Suppose $\mathfrak{g}$ is not of type $A_{2n}$. Given $\alpha\in R^{+}$ and $\varepsilon\in\{0,\dotsc,m_{\alpha}-1\}$, let

[TABLE]

When $\varepsilon\geq m_{\alpha}$, let $x^{\pm}_{\alpha,\varepsilon}=h_{\alpha,\varepsilon}=0$. Now, suppose $\mathfrak{g}$ is of type $A_{2n}$. Given $\alpha\in R^{+}$ and $\varepsilon\in\{0,1\}$, let

[TABLE]

and

[TABLE]

Moreover, if $\mathfrak{g}$ is of type $A_{2n}$ and $\beta+\sigma(\beta)\in R^{+}$, then \beta|_{\text{\mathfrak{h}}_{0}}\in R_{\operatorname{sh}} and (\beta+\sigma(\beta))|_{\text{\mathfrak{h}}_{0}}\in 2R_{\operatorname{sh}}. In this case, we have

[TABLE]

Since \text{\mathfrak{g}}_{\varepsilon} is a finite-dimensional irreducible \text{\mathfrak{g}}_{0}-module, for all $\varepsilon\in\{0,\dotsc,m-1\}$ and $\mu\in R_{\varepsilon}\setminus\{0\}$, we have

[TABLE]

Choose a complete set of representatives $O$ of the orbits of the action of $\sigma$ on $R^{+}$. For each $\mu\in R^{+}_{\varepsilon}$, let $x_{\mu,\varepsilon}^{\pm}=x_{\alpha,\varepsilon}^{\pm}$, where $\alpha\in O$ is such that \alpha|_{\text{\mathfrak{h}}_{0}}=\mu. Also, notice that there exists a unique injective map $o\colon I_{0}\to I$ such that $\alpha_{o(i)}\in O$ for all $i\in I_{0}$, and let $h_{i,\varepsilon}=h_{o(i),\varepsilon}$. Thus

[TABLE]

is a basis of $\mathfrak{g}$.

For notational convenience, if $\alpha\in O$ is such that \mu=\alpha|_{\text{\mathfrak{h}}_{0}}, let $h_{\mu,\varepsilon}=h_{\alpha,\varepsilon}$, and if $\mu\notin R_{\varepsilon}$, let $x_{\mu,\varepsilon}^{\pm}=0$. Also, if $\mathfrak{g}$ is of type $A_{2n}$, assume that $O$ is chosen in such a way that $s=1$ in (1.2). Moreover, if $\mathfrak{g}$ is of type $D_{4}$ and $m=3$, let $j\in I$ be the unique vertex fixed by $\sigma$, choose $i\in I$ such that $\sigma(i)\neq i$, and let $O=\{\alpha\in R^{+}\mid\sigma(\alpha)=\alpha\}\cup\{\alpha_{i},\,\alpha_{j}+\alpha_{i},\,\alpha_{j}+\sigma(\alpha_{i})+\sigma^{2}(\alpha_{i})\}$. It has been proven by Kac that the set $\cal C^{\sigma}_{0}(O):=\{x_{\mu,0}^{\pm},h_{i,0}\mid\mu\in R_{0}^{+},i\in I_{0}\}$ is a Chevalley basis of \text{\mathfrak{g}}_{0} (see [Kac90, §8.3]).

Later in this paper, we shall need the following formulas of commutators of elements of $\cal C^{\sigma}(O)$, proved in [Bia12, Lemma 2.2.7]. For all $\varepsilon,\varepsilon^{\prime}\in\{0,\dotsc,m-1\}$, $\mu\in R_{0}^{+}$, $\nu\in R^{+}_{\varepsilon}$, $\eta\in R^{+}_{\varepsilon}\cap R^{+}_{\varepsilon^{\prime}}$:

[TABLE]

1.3. Current, loop and affine Kac-Moody algebras

Given a vector space $\mathfrak{a}$, consider \text{\tilde{\mathfrak{a}}}=\text{\mathfrak{a}}\otimes\mathbb{C}[t,t^{-1}] and \text{\mathfrak{a}}[t]=\text{\mathfrak{a}}\otimes\mathbb{C}[t]. If (\text{\mathfrak{a}},[\cdot,\cdot]_{\text{\mathfrak{a}}}) is a Lie algebra, then there exists a unique bilinear map [\cdot,\cdot]\colon\text{\tilde{\mathfrak{a}}}\times\text{\tilde{\mathfrak{a}}}\to\text{\tilde{\mathfrak{a}}} that satisfies

[TABLE]

and endows $\tilde{\mathfrak{a}}$ with a structure of Lie algebra. Notice that \text{\mathfrak{a}}\otimes\mathbb{C} is a subalgebra of \text{\mathfrak{a}}[t] isomorphic to $\mathfrak{a}$, that \text{\mathfrak{a}}[t] is a subalgebra of $\tilde{\mathfrak{a}}$, and that \text{\mathfrak{b}}[t] (resp. $\tilde{\mathfrak{b}}$) is a subalgebra of \text{\mathfrak{a}}[t] (resp. $\tilde{\mathfrak{a}}$) for all subalgebras \text{\mathfrak{b}}\subseteq\text{\mathfrak{a}}.

Fix an automorphism \sigma\colon\text{\mathfrak{a}}\to\text{\mathfrak{a}}. Denote the order of $\sigma$ by $m$ and an $m$-th primitive root of unity by $\zeta$. There exists a unique automorphism \tilde{\sigma}\colon\text{\tilde{\mathfrak{a}}}\to\text{\tilde{\mathfrak{a}}} satisfying

[TABLE]

Notice that $\tilde{\sigma}$ restricts to an automorphism of \text{\mathfrak{a}}[t] and that the order of $\tilde{\sigma}$ is also $m$. Let \text{\tilde{\mathfrak{a}}}^{\sigma} and \text{\mathfrak{a}}[t]^{\sigma} denote the subalgebras of $\tilde{\sigma}$-fixed points; explicitly

[TABLE]

In particular, when \sigma={\rm id}_{\text{\mathfrak{a}}}, notice that \text{\mathfrak{a}}[t]^{\sigma}=\text{\mathfrak{a}}[t] and \text{\tilde{\mathfrak{a}}}^{\sigma}=\text{\tilde{\mathfrak{a}}}.

Given a finite-dimensional complex simple Lie algebra $\mathfrak{g}$ and a Dynkin diagram automorphism \sigma\colon\text{\mathfrak{g}}\to\text{\mathfrak{g}}, the associated twisted affine Kac-Moody algebra $\hat{\mathfrak{g}}$ is the 2-dimensional extension of \text{\tilde{\mathfrak{g}}}^{\sigma} given by \text{\hat{\mathfrak{g}}}:=\text{\tilde{\mathfrak{g}}}^{\sigma}\oplus\mathbb{C}c\oplus\mathbb{C}d, and endowed with the unique Lie bracket that satisfies

[TABLE]

for all x\otimes t^{r},y\otimes t^{s}\in\text{\tilde{\mathfrak{g}}}^{\sigma}. Denote by \text{\hat{\mathfrak{g}}}^{\prime}=[\text{\hat{\mathfrak{g}}},\text{\hat{\mathfrak{g}}}] the derived subalgebra of $\hat{\mathfrak{g}}$, notice that \text{\hat{\mathfrak{g}}}^{\prime}=\text{\tilde{\mathfrak{g}}}^{\sigma}\oplus\mathbb{C}c and that we have a nonsplit short exact sequence of Lie algebras 0\to\mathbb{C}c\to\text{\hat{\mathfrak{g}}}^{\prime}\to\text{\tilde{\mathfrak{g}}}^{\sigma}\to 0.

Given a triangular decomposition \text{\mathfrak{g}}=\text{\mathfrak{n}}^{-}\oplus\text{\mathfrak{h}}\oplus\text{\mathfrak{n}}^{+} and a Chevalley basis $\cal C=\{x_{i}^{\pm},h_{\alpha}\mid i\in I,\,\alpha\in R^{+}\}$, denote (\text{\mathfrak{h}}\oplus\mathbb{C}c) by \text{\mathfrak{h}}^{\prime}, (\text{\mathfrak{g}}\otimes t^{\pm 1}\mathbb{C}[t^{\pm 1}]) by \text{\mathfrak{g}}[t]_{\pm}, and let

[TABLE]

A triangular decomposition of $\hat{\mathfrak{g}}$ is given by \text{\hat{\mathfrak{g}}}=\text{\hat{\mathfrak{n}}}^{-}\oplus\text{\hat{\mathfrak{h}}}\oplus\text{\hat{\mathfrak{n}}}^{+}. The root system, set of positive and simple roots associated to this triangular decomposition will be denoted by $\hat{R}$, $\hat{R}^{+}$ and $\hat{\Delta}$ respectively. Notice that

[TABLE]

Now, let $\hat{I}=I\sqcup\{0\}$, $a_{0}=2$ for \text{\mathfrak{g}}\cong A_{2n}, $a_{0}=1$ for \text{\mathfrak{g}}\not\cong A_{2n}, $h_{0}=\frac{m}{a_{0}}c-h_{\theta_{1}}$, and notice that $\{h_{i}\mid i\in\hat{I}\}\cup\{d\}$ is a basis of $\hat{\mathfrak{h}}$. Identify \text{\mathfrak{h}}^{*} with the subspace \{\lambda\in\text{\hat{\mathfrak{h}}}^{*}\mid\lambda(c)=\lambda(d)=0\}. Let $\delta$ be the unique linear map in \text{\hat{\mathfrak{h}}}^{*} such that $\delta(d)=1$ and $\delta(h_{i})=0$ for all $i\in\hat{I}$. Define $\alpha_{0}=\delta-\theta_{1}$. Then

[TABLE]

Given $\alpha\in R^{+}$, r\in\text{\mathbb{Z}}_{\geq 0}, let

[TABLE]

For each $i\in\hat{I}$, define $\Lambda_{i}$ to be the unique linear map in \text{\hat{\mathfrak{h}}}^{*} that satisfies $\Lambda_{i}(d)=0$ and $\Lambda_{i}(h_{j})=\delta_{ij}$ for all $i,j\in\hat{I}$. Set $\hat{P}=\mathbb{Z}\delta\oplus\left(\oplus_{i\in\hat{I}}\mathbb{Z}\Lambda_{i}\right)$, $\hat{P}^{+}=\mathbb{Z}\delta\oplus\left(\oplus_{i\in\hat{I}}\mathbb{N}\Lambda_{i}\right)$, $\hat{P}^{\prime}=\oplus_{i\in\hat{I}}\mathbb{Z}\Lambda_{i}$, and $\hat{P}^{\prime+}=\hat{P}^{\prime}\cap\hat{P}^{+}$. Notice that

[TABLE]

Hence, \hat{P}=\text{\mathbb{Z}}\Lambda_{0}\oplus P\oplus\text{\mathbb{Z}}\delta. Given $\Lambda\in\hat{P}$, the number $\Lambda(c)$ is called the level of $\Lambda$. By (1.4), the level of $\Lambda$ depends only on its class modulo the root lattice $\hat{Q}$. Set also $\hat{Q}^{+}=\mathbb{N}\hat{R}^{+}$ and let $\widehat{\mathcal{W}}$ denote the affine Weyl group, which is generated by the simple reflections $s_{i},i\in\hat{I}$. Finally, observe that $\{\Lambda_{0},\delta\}\cup\Delta$ is a basis of \text{\hat{\mathfrak{h}}}^{\ast}.

Moreover, the set $\{x_{\alpha}^{\pm}\otimes t^{r},h_{i}\otimes t^{r}\mid\alpha\in R^{+},\ i\in I,\ r\in\mathbb{N}\}$ forms a basis for \text{\mathfrak{g}}[t] and the set $\{x_{\mu,-r}^{\pm}\otimes t^{r},\,h_{i,-r}\otimes t^{r}\mid r\in\mathbb{N},\,\mu\in R_{-r},\,i\in I_{0}\}\setminus\{0\}$ forms a basis for \text{\mathfrak{g}}[t]^{\sigma}.

1.4. Integral forms and hyperalgebras

Given a Lie algebra $\mathfrak{a}$, let U(\text{\mathfrak{a}}) denote its universal enveloping algebra. The Poincaré-Birkhoff-Witt (PBW) Theorem implies that there are isomorphisms of vector spaces

[TABLE]

Given $\alpha\in R^{+}$ and $k\in\mathbb{Z}_{\geq 0}$, consider the following power series with coefficients in U(\text{\mathfrak{h}}[t])

[TABLE]

For simplicity, denote $\Lambda_{\alpha,r;0}$ by $\Lambda_{\alpha,r}$. Similarly, consider the following power series with coefficients in U(\text{\mathfrak{h}}[t]^{\sigma}) (twisted analogues of the above elements). If either $\mathfrak{g}$ is not of type $A_{2n}$ and $\mu\in R_{\operatorname{sh}}$, or $\mathfrak{g}$ is of type $A_{2n}$ and $\mu\in R_{0}^{+}$, set

[TABLE]

If $\mathfrak{g}$ is not of type $A_{2n}$ and $\mu\in R_{\operatorname{lg}}$, set

[TABLE]

For simplicity, denote $\Lambda_{\alpha_{i},r}$ (resp. $\Lambda^{\sigma}_{\alpha_{i},r}$) by $\Lambda_{i,r}$ (resp. $\Lambda^{\sigma}_{i,r}$).

One can easily check the following relation among twisted and non-twisted versions of the above elements. Given $\mu\in R_{0}^{+}$, let $\alpha\in O$ such that \alpha|_{\text{\mathfrak{h}}_{0}}=\mu. Then:

[TABLE]

Given an order on the Chevalley basis of \text{\mathfrak{g}}[t]^{\sigma} and a PBW monomial with respect to this order, we construct an ordered monomial in the elements of the set

[TABLE]

using the correspondence $(x^{\pm}_{\mu,-r}\otimes t^{r})^{k}\leftrightarrow(x^{\pm}_{\mu,-r}\otimes t^{r})^{(k)}$, $h_{i,0}^{k}\leftrightarrow\binom{h_{i,0}}{k}$ and $(h_{i,-r}\otimes t^{r})^{k}\leftrightarrow(\Lambda_{i,r}^{\sigma})^{k}$ for $r\neq 0$. Using the obvious similar correspondence we consider monomials in U(\text{\mathfrak{g}}) formed by elements of

[TABLE]

and in U(\text{\mathfrak{g}}[t]) formed by elements of

[TABLE]

Notice that $\cal M$ can be naturally regarded as a subset of $\tilde{\cal M}$. The sets of ordered monomials thus obtained are basis of U(\text{\mathfrak{g}}[t]^{\sigma}), U(\text{\mathfrak{g}}), and U(\text{\mathfrak{g}}[t]), respectively.

Let U_{\mathbb{Z}}(\text{\mathfrak{g}})\subseteq U(\text{\mathfrak{g}}), U_{\mathbb{Z}}(\text{\mathfrak{g}}[t])\subseteq U(\text{\mathfrak{g}}[t]), and U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]^{\sigma})\subseteq U(\text{\mathfrak{g}}[t]^{\sigma}) be the $\mathbb{Z}$–subalgebras generated respectively by $\{(x_{\alpha}^{\pm})^{(k)}\mid\alpha\in R^{+},k\in\mathbb{Z}_{\geq 0}\}$, $\{(x^{\pm}_{\alpha}\otimes t^{r})^{(k)}\mid\alpha\in R^{+},r,k\in\mathbb{Z}_{\geq 0}\}$, and \{(x^{\pm}_{\mu,-r}\otimes t^{r})^{(k)}\mid\mu\in\operatorname{wt}(\text{\mathfrak{g}}_{-r})\cap Q_{0}^{+}\setminus\{0\},r,k\in\mathbb{Z}_{\geq 0}\}. It has been proven by Kostant [Kos66] (in the U(\text{\mathfrak{g}}) case), Garland [Gar78] (in the U(\text{\tilde{\mathfrak{g}}}) case), Mitzman [Mit85] and Prevost [Pre92] (in the loop and twisted loop cases) that the subalgebras U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]^{\sigma}), U_{\mathbb{Z}}(\text{\mathfrak{g}}), and U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]) are free $\mathbb{Z}$-modules and the sets of ordered monomials constructed from $\cal M[t]^{\sigma}$, $\cal M$, $\cal M[t]$ are $\mathbb{Z}$-basis of U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]^{\sigma}), U_{\mathbb{Z}}(\text{\mathfrak{g}}), and U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]), respectively.

In particular, it follows from this result that the natural map \mathbb{C}\otimes_{\mathbb{Z}}U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]^{\sigma})\to U(\text{\mathfrak{g}}[t]^{\sigma}) is an isomorphism of vector spaces, i.e., U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]^{\sigma}) is a an integral form of U(\text{\mathfrak{g}}[t]^{\sigma}), and similarly for U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]) and U_{\mathbb{Z}}(\text{\mathfrak{g}}). If $\mathfrak{a}$ is a subalgebra of $\mathfrak{g}$ preserved by $\sigma$, set

[TABLE]

and similarly define U_{\mathbb{Z}}(\text{\mathfrak{a}}) and U_{\mathbb{Z}}(\text{\mathfrak{a}}[t]) for any $\mathfrak{a}$ subalgebra of $\mathfrak{g}$. It also follows that, if $\mathfrak{a}$ is either \text{\mathfrak{g}}_{0}, \text{\mathfrak{n}}_{0}^{\pm}, \text{\mathfrak{h}}_{\varepsilon}, \text{\mathfrak{h}}[t]^{\sigma}_{\varepsilon},\text{\mathfrak{n}}^{\pm}[t]^{\sigma}, \text{\mathfrak{n}}^{\pm}, $\mathfrak{h}$, \text{\mathfrak{n}}^{\pm}[t], or \text{\mathfrak{h}}[t], then U_{\mathbb{Z}}(\text{\mathfrak{a}}) is a free $\mathbb{Z}$-module spanned by monomials formed by elements of $\cal M[t]^{\sigma}$ belonging to U(\text{\mathfrak{a}}), and hence \mathbb{C}\otimes_{\mathbb{Z}}U_{\mathbb{Z}}(\text{\mathfrak{a}})\cong U(\text{\mathfrak{a}}). Moreover, we have

[TABLE]

Given a field $\mathbb{F}$, define the $\mathbb{F}$-hyperalgebra of $\mathfrak{a}$ by

[TABLE]

where $\mathfrak{a}$ is any of the subspaces considered above. We will refer to U_{\mathbb{F}}(\text{\mathfrak{g}}[t]^{\sigma}) as the twisted hyper current algebra of $\mathfrak{g}$ associated to $\sigma$ over $\mathbb{F}$. Clearly, if the characteristic of $\mathbb{F}$ is zero, the algebra U_{\mathbb{F}}(\text{\mathfrak{g}}[t]^{\sigma}) is naturally isomorphic to U(\text{\mathfrak{g}}_{\mathbb{F}}[t]^{\sigma}) where \text{\mathfrak{g}}[t]_{\mathbb{F}}^{\sigma}=\mathbb{F}\otimes_{\mathbb{Z}}\text{\mathfrak{g}}[t]_{\mathbb{Z}}^{\sigma} and \text{\mathfrak{g}}[t]_{\mathbb{Z}}^{\sigma} is the $\mathbb{Z}$-span of the Chevalley basis of \text{\mathfrak{g}}[t]^{\sigma}, and similarly for all algebras $\mathfrak{a}$ we have considered. For fields of positive characteristic we just have an algebra homomorphism U(\text{\mathfrak{a}}_{\mathbb{F}})\to U_{\mathbb{F}}(\text{\mathfrak{a}}) which is neither injective nor surjective. We will keep denoting by $x$ the image of an element x\in U_{\mathbb{Z}}(\text{\mathfrak{a}}) in U_{\mathbb{F}}(\text{\mathfrak{a}}). Notice that we have

[TABLE]

Notice that U_{\mathbb{Z}}(\text{\mathfrak{g}})=U(\text{\mathfrak{g}})\cap U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]), i.e., the integral form of $\mathfrak{g}$ coincides with its intersection with he integral form of U(\text{\mathfrak{g}}[t]) which allows us to regard U_{\mathbb{Z}}(\text{\mathfrak{g}}) as a $\mathbb{Z}$-subalgebra of U_{\mathbb{Z}}(\text{\mathfrak{g}}[t]).

If $\mathfrak{g}$ is of type $A_{2n}$ and the characteristic of $\mathbb{F}$ is 2, then U_{\mathbb{F}}(\text{\mathfrak{g}}_{0}) is not isomorphic to the usual hyperalgebra of \text{\mathfrak{g}}_{0} over $\mathbb{F}$, constructed by using Kostant’s integral form of U(\text{\mathfrak{g}}_{0}). However, if the characteristic of $\mathbb{F}$ is not 2, then U_{\mathbb{F}}(\text{\mathfrak{g}}_{0}) is isomorphic to the usual hyperalgebra of \text{\mathfrak{g}}_{0} over $\mathbb{F}$. Details can be found in [BM14, Remark 1.5] and [Bia12]. This is the reason why we are not working with fields of characteristic 2 when $\mathfrak{g}$ is of type $A_{2n}$.

1.5. A certain automorphism of hyperalgebras

One can check that there exists a unique Lie algebra automorphism \psi\colon\text{\mathfrak{g}}\to\text{\mathfrak{g}} satisfying $\psi(x_{\alpha_{i}}^{\pm})=x_{\alpha_{i}}^{\mp}$ for all $i\in I$. Moreover, $\psi$ admits a unique extension to a Lie algebra automorphism of $\tilde{\mathfrak{g}}$ (which we keep denoting by $\psi$) that satisfies $\psi(x\otimes f)=\psi(x)\otimes f$ for all x\in\text{\mathfrak{g}},f\in\text{\mathbb{C}}[t,t^{-1}]. Notice that this automorphism restricts to an automorphism of \text{\mathfrak{g}}[t]. Also notice that $\psi\circ\sigma=\sigma\circ\psi$. Thus $\psi$ restricts to automorphisms of \text{\tilde{\mathfrak{g}}}^{\sigma} and \text{\mathfrak{g}}[t]^{\sigma}.

Keep denoting by $\psi$ the extension of $\psi$ to an automorphism of U(\text{\tilde{\mathfrak{g}}}). In particular, we have

[TABLE]

for all $\alpha\in R^{+}$, $\varepsilon\in\{0,\dotsc,m-1\}$, r\in\text{\mathbb{Z}}, and $k\geq 0$. It follows that $\psi$ restricts to an automorphism of U_{\text{\mathbb{Z}}}(\text{\mathfrak{a}}) for all \text{\mathfrak{a}}\in\{\text{\mathfrak{g}}_{0},\,\text{\mathfrak{h}}_{0},\,\text{\mathfrak{g}}[t],\,\text{\mathfrak{h}}[t],\,\text{\mathfrak{g}}[t]^{\sigma},\,\text{\mathfrak{h}}[t]^{\sigma},\,\text{\mathfrak{h}}[t]^{\sigma}_{\pm}\}. Moreover, notice that, for every $\mu\in P_{0}$, there exists a unique ring homomorphism \mu\colon U_{\text{\mathbb{Z}}}(\text{\mathfrak{h}}_{0})\to\text{\mathbb{Z}} satisfying

[TABLE]

Therefore,

[TABLE]

1.6. Technical identities

The proof of the following lemma is very similar to that of [Bia12, Lema 2.2.8].

Lemma 1.1**.**

Let $\alpha\in R_{0}^{+}$.

1. (a)

Suppose $\mathfrak{g}$ is not of type $A_{2n}$.

1. (i)

If $\alpha\in R_{\operatorname{lg}}$, then \text{\mathfrak{sl}}_{2,\alpha}[t^{m}]:={\rm span}\{x^{\pm}_{\alpha,0}\otimes t^{mk},h_{\alpha,0}\otimes t^{mk}\mid k\in\mathbb{Z}_{\geq 0}\} is a subalgebra of \text{\mathfrak{g}}[t]^{\sigma} isomorphic to \text{\mathfrak{sl}}_{2}[t]. 2. (ii)

If $\alpha\in R_{\operatorname{sh}}$, then \text{\mathfrak{sl}}_{2,\alpha}[t]:={\rm span}\{x^{\pm}_{\alpha,\varepsilon}\otimes t^{mk-\varepsilon},h_{\alpha,\varepsilon}\otimes t^{mk-\varepsilon}\mid k\in\mathbb{Z}_{\geq 0},\ 0\leq\varepsilon<m\} is a subalgebra of \text{\mathfrak{g}}[t]^{\sigma} isomorphic to \text{\mathfrak{sl}}_{2}[t]. 2. (b)

Suppose $\mathfrak{g}$ is of type $A_{2n}$.

1. (i)

If $\alpha\in R_{\operatorname{lg}}$, then \text{\mathfrak{sl}}_{2,\alpha}[t]:={\rm span}\{x^{\pm}_{\alpha,\varepsilon}\otimes t^{mk-\varepsilon},h_{\alpha,\varepsilon}\otimes t^{mk-\varepsilon}\mid k\in\mathbb{Z}_{\geq 0},\ \varepsilon=0,1\} is a subalgebra of \text{\mathfrak{g}}[t]^{\sigma} isomorphic to \text{\mathfrak{sl}}_{2}[t]. 2. (ii)

If $\alpha\in R_{\operatorname{sh}}$, then ${\rm span}\{x_{\alpha,\varepsilon}^{\pm}\otimes t^{2k+\varepsilon},x_{\alpha+\sigma(\alpha),1}^{\pm}\otimes t^{2k+1},h_{\alpha,\varepsilon}\otimes t^{2k+\varepsilon}\mid k\in\mathbb{Z}_{\geq 0},\ \varepsilon=0,1\}$ is a subalgebra of \text{\mathfrak{g}}[t]^{\sigma} isomorphic to \text{\mathfrak{sl}}_{3}[t]^{\tau}, where $\tau$ is the nontrivial automorphism of $A_{2}$.

Given a monomial of the form $(x^{\pm}_{\mu_{1},-r_{1}}\otimes t^{r_{1}})^{(k_{1})}\cdots(x^{\pm}_{\mu_{j},-r_{j}}\otimes t^{r_{j}})^{(k_{j})}$, with $r_{1},\dots,r_{j}\in\mathbb{Z}_{\geq 0}$, a fixed choice of $\pm$, and $j\geq 1$, define its hyperdegree to be $k_{1}+\cdots+k_{j}$. The following result was proved in [Mit85, Lemma 4.2.13].

Lemma 1.2**.**

Let r,s,j,k\in\text{\mathbb{Z}}_{\geq 0}, \mu\in\operatorname{wt}(\text{\mathfrak{g}}_{-r}) and \nu\in\operatorname{wt}(\text{\mathfrak{g}}_{-s}). Then $(x^{\pm}_{\mu,-r}\otimes t^{r})^{(j)}(x^{\pm}_{\nu,-s}\otimes t^{s})^{(k)}$ is in the $\mathbb{Z}$-submodule generated by $(x^{\pm}_{\nu,-s}\otimes t^{s})^{(k)}(x^{\pm}_{\mu,-r}\otimes t^{r})^{(j)}$ and by monomials of hyperdegree strictly smaller than $j+k$.

Given $\alpha\in R^{+}$ and $r,s\in\mathbb{Z}_{\geq 0}$, consider the following power series with coefficients in U(\text{\mathfrak{g}}[t]):

[TABLE]

Next result was proven in [Gar78] in the loop algebras context and its proof remains valid for current algebras.

Lemma 1.3**.**

Let $\alpha\in R^{+}$, $r,s,\ell,k\in\mathbb{Z}_{\geq 0}$. We have

[TABLE]

where U_{\text{\mathbb{Z}}}^{+}:=U_{\text{\mathbb{Z}}}(\text{\mathfrak{n}}^{-}[t])U_{\text{\mathbb{Z}}}(\text{\mathfrak{h}}[t]_{+})^{0}+U_{\text{\mathbb{Z}}}(\text{\mathfrak{g}}[t])U_{\text{\mathbb{Z}}}(\text{\mathfrak{n}}^{+}[t])^{0}.

We now define certain power series with coefficients in U(\text{\mathfrak{g}}[t]^{\sigma}). Let $r,s\in\mathbb{Z}_{\geq 0}$. When either $\mathfrak{g}$ is not of type $A_{2n}$ and $\mu\in R_{\operatorname{sh}}$, or $\mathfrak{g}$ is of type $A_{2n}$ and $\mu\in R_{0}^{+}$, define

[TABLE]

When $\mathfrak{g}$ is not of type $A_{2n}$ and $\mu\in R_{\operatorname{lg}}$, define

[TABLE]

For simplicity, we may denote

[TABLE]

[TABLE]

We now state a lemma whose proof follows directly from Lemma 1.1 (part (a)), Lemma 1.3 (part (b)) and [Mit85, Lemma 4.4.1(ii)] (part (c)).

Lemma 1.4**.**

Let $l,k,r,s\in\mathbb{Z}_{\geq 0}$ and $0\leq l\leq k$.

1. (a)

If either $\mathfrak{g}$ is not of type $A_{2n}$ and $\mu\in R_{\operatorname{sh}}$, or if $\mathfrak{g}$ is of type $A_{2n}$ and $\mu\in R_{\operatorname{lg}}$, we have

[TABLE] 2. (b)

If $\mathfrak{g}$ is not of type $A_{2n}$ and $\mu\in R_{\operatorname{lg}}$, we have

[TABLE] 3. (c)

If $\mathfrak{g}$ is of type $A_{2n}$ and $\mu\in R_{\operatorname{sh}}$, we have

1. (i)

(x_{\mu,0}^{+}\otimes t^{ms})^{(k)}(x_{\mu,0}^{-}\otimes 1)^{(k+r)}=(X(u)^{(r)})_{k+r}\textup{ mod }{U_{\text{\mathbb{Z}}}^{+}}^{\sigma}* where $X(u)=\sum_{k=1}^{\infty}x_{\mu,0}\otimes t^{msj}u^{j}.$* 2. (ii)

(x_{2\mu,1}^{+}\otimes t)^{(l)}(x_{2\mu,1}^{-}\otimes t)^{(k)}=(-1)^{l}\left(Y(u)^{(k-l)}\right)_{k}\textup{ mod }{U_{\text{\mathbb{Z}}}^{+}}^{\sigma},* where $Y(u)=\sum_{k=1}^{\infty}x_{2\mu,1}\otimes t^{2k-1}u^{k}.$* 3. (iii)

(x_{2\mu,1}^{+}\otimes t)^{(k)}(x_{\mu,0}^{-}\otimes 1)^{(2k+r)}=\displaystyle{\sum_{\begin{subarray}{c}k_{1},k_{2}\\ k_{1}+2k_{2}=r\end{subarray}}\left(X^{\sigma}_{\mu}(u)^{(k_{1})}Z(u)^{(k_{2})}\right)_{k+k_{1}}}\textup{ mod }{U_{\text{\mathbb{Z}}}^{+}}^{\sigma}*, where *$Z(u)=\sum_{k=1}^{\infty}x_{2\mu,1}\otimes t^{2k-1}u^{2k-1}.$

2. Preliminaries on modules

2.1. Integrable modules for hyperalgebras

Let ${\mathbb{F}}$ be either a field or $\mathbb{Z}$. In this subsection we will review the representation theory of U_{\mathbb{F}}(\text{\mathfrak{g}}_{0}).

Let $V$ be a U_{\mathbb{F}}(\text{\mathfrak{g}}_{0})-module. For each \mu\in U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*}, denote the weight space of weight $\mu$ by V_{\mu}=\{v\in V\mid hv=\mu(h)v\ \textup{for all h\in U_{\mathbb{F}}(\text{$\mathfrak{h}$}_{0})}\} . A nonzero vector $v\in V_{\mu}$ is said to be a weight vector. If $V_{\mu}\neq 0$, then $\mu$ is said to be a weight of $V$. Let \operatorname{wt}_{0}(V)=\{\mu\in U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*}\mid V_{\mu}\neq 0\} denote the set of weights of $V$. If $V=\bigoplus_{\mu\in\operatorname{wt}_{0}(V)}V_{\mu}$, then $V$ is said to be a weight module.

Using the inclusion P_{0}\hookrightarrow U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*} induced by (1.8), define a partial order $\leq$ on U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*} given by: $\mu\leq\lambda$ if $\lambda-\mu\in Q_{0}^{+}$. Notice that (x_{\alpha,0}^{\pm})^{(k)}V_{\mu}\subseteq V_{\mu\pm k\alpha}\qquad\textup{for all}\ \alpha\in R_{0}^{+},\ k>0,\ \mu\in U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*}. Now, if $v\in V$ is a weight vector such that $(x_{\alpha,0}^{+})^{(k)}v=0$ for all $\alpha\in R_{0}^{+}$ and $k>0$, then $v$ is said to be a highest-weight vector. If $V$ is generated by a highest-weight vector, then it is said to be a highest-weight module. If, for each $v\in V$, there exists $m>0$ such that $(x_{\alpha,0}^{\pm})^{(k)}v=0$ for all $\alpha\in R_{0}^{+}$ and $k>m$, then $V$ is said to be integrable.

If $V$ is a weight-module whose weight spaces are finitely-generated $\mathbb{F}$-modules, its character is defined to be the function \operatorname{ch}(V)\colon U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*}\to\mathbb{Z} given by $\operatorname{ch}(V)(\mu)={\rm rank}_{\mathbb{F}}(V_{\mu})$. Thus, $\operatorname{ch}(V)$ can be regarded as an element of the group ring \mathbb{Z}[U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*}]. Recall that (1.8) induces an inclusion P_{0}\hookrightarrow U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*}. Thus, the group ring $\mathbb{Z}[P_{0}]$ can be regarded as a subring of \mathbb{Z}[U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*}]. Denote by $e^{\mu}$ the element of \text{\mathbb{Z}}[U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*}] corresponding to \mu\in U_{\mathbb{F}}(\text{\mathfrak{h}}_{0})^{*}. There exists a unique action of $\cal W_{0}$ on $\mathbb{Z}[P_{0}]$ (by ring automorphisms) that satisfy $w\cdot e^{\mu}=e^{w\mu}$ for all $w\in\cal W_{0}$, $\mu\in P_{0}$.

Theorem 2.1**.**

Let $V$ be a U_{\mathbb{F}}(\text{\mathfrak{g}}_{0})-module.

1. (a)

If $V$ is a finitely-generated $\mathbb{F}$-module, then $V$ is a weight module and $\operatorname{wt}_{0}(V)\subseteq P_{0}$. 2. (b)

If $V$ is an integrable weight module, then there are isomorphisms of ${\mathbb{F}}$-modules $V_{\mu}\cong V_{w\mu}$ for all $w\in\cal W_{0}$, $\mu\in P_{0}$. Thus, $\operatorname{ch}(V)\in\mathbb{Z}[P_{0}]^{\cal W_{0}}$. 3. (c)

If $V$ is a highest-weight module of highest weight $\lambda$, then $V_{\lambda}$ is a free $\mathbb{F}$-module of rank $1$ and $V_{\mu}\neq 0$ only if $\mu\leq\lambda$. Moreover, $V$ has a unique maximal proper submodule and, hence, also a unique irreducible quotient. In particular, $V$ is indecomposable. 4. (d)

For each $\lambda\in P_{0}^{+}$, the U_{\mathbb{F}}(\text{\mathfrak{g}}_{0})-module $W_{\mathbb{F}}(\lambda)$ given as the quotient of U_{\mathbb{F}}(\text{\mathfrak{g}}_{0}) by the left ideal generated by

[TABLE]

is a nonzero, free ${\mathbb{F}}$-module of finite rank. Moreover, every free ${\mathbb{F}}$-module of finite rank that is a highest-weight module of highest weight $\lambda$ is a quotient of $W_{\mathbb{F}}(\lambda)$. 5. (e)

If ${\mathbb{F}}$ is a field and $V$ is a finite-dimensional irreducible U_{\mathbb{F}}(\text{\mathfrak{g}}_{0})-module, then there exists a unique $\lambda\in P_{0}^{+}$ such that $V$ is isomorphic to the irreducible quotient $V_{\mathbb{F}}(\lambda)$ of $W_{\mathbb{F}}(\lambda)$. If the characteristic of ${\mathbb{F}}$ is zero, then $W_{\mathbb{F}}(\lambda)$ is irreducible. 6. (f)

For each $\lambda\in P_{0}^{+}$, the character of $W_{\mathbb{F}}(\lambda)$ is given by the Weyl character formula. In particular, $\mu\in\operatorname{wt}_{0}(W_{\mathbb{F}}(\lambda))$ if, and only if, $w\mu\leq\lambda$ for all $w\in\cal W_{0}$.

Proof.

If {\mathbb{F}}=\text{\mathbb{Z}}, then parts a, d, f are proved in [Mac13, Theorem 2.3.6] and part b is proved in [Mac13, Proposition 2.3.5].

If ${\mathbb{F}}$ is a field, then U_{\mathbb{F}}(\text{\mathfrak{g}}_{0}) is the algebra of distributions of an algebraic group of the same Lie type as \text{\mathfrak{g}}_{0}. Thus, this result is proved in [Jan03, Part II]. In the particular case where ${\mathbb{F}}$ is a field of characteristic zero, U_{\mathbb{F}}(\text{\mathfrak{g}}_{0})\cong U((\text{\mathfrak{g}}_{0})_{\mathbb{F}}), where (\text{\mathfrak{g}}_{0})_{\mathbb{F}}={\mathbb{F}}\otimes_{\text{\mathbb{Z}}}(\text{\mathfrak{g}}_{0})_{\text{\mathbb{Z}}} and (\text{\mathfrak{g}}_{0})_{\text{\mathbb{Z}}} is the $\mathbb{Z}$-span of the Chevalley basis given in Section 1.2. In this case, this result is also proved in [Hum90]. In the particular case where ${\mathbb{F}}$ is a field of positive characteristic, this result is also proved in [JM07, Section 2]. ∎

2.2. Demazure and local Weyl modules for twisted current hyperalgebras

Throughout this subsection, let $\mathbb{F}$ be either a field or $\mathbb{Z}$, let $\mathfrak{g}$ be a finite-dimensional simple $\mathbb{C}$-Lie algebra with a triangular decomposition \text{\mathfrak{g}}=\text{\mathfrak{n}}^{-}\oplus\text{\mathfrak{h}}\oplus\text{\mathfrak{n}}^{+}, and let \sigma\colon\text{\mathfrak{g}}\to\text{\mathfrak{g}} be a Dynkin diagram automorphism.

Definition 2.2**.**

Given $\lambda\in P^{+}_{0}$, let the twisted local graded Weyl module W_{\text{\mathbb{F}}}^{c,\sigma}(\lambda) be the U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})-module given as a quotient of U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma}) by the left ideal generated by

[TABLE]

Given $\ell\geq 0$ and $\lambda\in P^{+}_{0}$, let the twisted Demazure module D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda) be the U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})-module given as a quotient of U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma}) by the left ideal generated by

[TABLE]

for all h\in U_{\text{\mathbb{F}}}(\text{\mathfrak{h}}_{0}), $\alpha\in R_{0}^{+}$, s\in\text{\mathbb{Z}}_{\geq 0}, $k>\max\{0,\delta_{s,1}\lambda(h_{\frac{\alpha}{2},0})-\hat{r}_{\alpha}^{\vee}\ell s\}$, in case \text{\mathfrak{g}}\cong A_{2n}, $\alpha\in 2R_{\operatorname{sh}}$, and $k>\max\{0,\lambda(h_{\alpha,0})-\hat{r}_{\alpha}^{\vee}\ell s\}$ in every other case.

Notice that W^{c,\sigma}_{\text{\mathbb{F}}}(\lambda), D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda) are weight modules, and that D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda) is a quotient of W^{c,\sigma}_{\text{\mathbb{F}}}(\lambda). Moreover, the ideal defining W^{c,\sigma}_{\text{\mathbb{F}}}(\lambda) (resp. D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda)) is the image of the ideal defining W^{c,\sigma}_{\text{\mathbb{Z}}}(\lambda) (resp. D_{\text{\mathbb{Z}}}^{\sigma}(\ell,\lambda)) in U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma}). Thus, we have the following isomorphisms of U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})-modules:

[TABLE]

3. Preliminary results

3.1. Finite-dimensionality of Weyl modules

The following lemma will be used in Proposition 3.3.

Lemma 3.1**.**

1. (a)

If $V$ is a finite-dimensional U_{\mathbb{F}}(\text{\mathfrak{g}}[t])-module, $\lambda\in P^{+}$ and $v\in V_{\lambda}$ is such that

[TABLE]

then $(x^{-}_{\alpha}\otimes t^{s})^{(k)}v=0$ for all $\alpha\in R^{+},s\geq\lambda(h_{\alpha})$ and k\in\text{\mathbb{Z}}_{>0}. 2. (b)

If $V$ is a finite-dimensional U_{\mathbb{F}}(\text{\mathfrak{g}}[t]^{\sigma})-module, $\lambda\in P_{0}^{+}$ and $v\in V_{\lambda}$ is such that

[TABLE]

then $(x^{-}_{\mu,-s}\otimes t^{s})^{(k)}v=0$ for all $x^{-}_{\mu,-s}\in C^{\sigma}(O),s\geq d_{\mu}\lambda(h_{\mu,0})$ and k\in\text{\mathbb{Z}}_{>0}, where $d_{\mu}=2$ if $\mathfrak{g}$ is of type $A_{2n}$ and $\mu\in R_{\operatorname{sh}}$, and $d_{\mu}=1$ otherwise.

Proof.

The proof of part a is similar to that of [JM07, Proposition 3.1]. The proof of part b is similar to that of [BM14, Proposition 3.2]. ∎

Lemma 3.2**.**

For any $\lambda\in P^{+}_{0}$, W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda) is an integrable U_{\text{\mathbb{Z}}}(\text{\mathfrak{g}}_{0})-module.

Proof.

This proof is similar to the proof of [JM07, Proposition 3.1.1]. ∎

Proposition 3.3**.**

For any $\lambda\in P^{+}_{0}$, W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda) is a finitely generated abelian group.

Proof.

Let $\Xi$ be the set of functions \phi\colon\mathbb{Z}_{>0}\to Q^{+}\times\text{\mathbb{Z}}_{\geq 0}\times\text{\mathbb{Z}}_{\geq 0}, $\phi(j)=(\beta_{j},r_{j},k_{j})$, such that: $\beta_{j}\in R^{+}_{-r_{j}}$ for all j\in\text{\mathbb{Z}}_{>0}, and $k_{j}=0$ for all $j$ sufficiently large. Let $v$ denote the image of 1\in U_{\text{\mathbb{Z}}}(\text{\mathfrak{g}}[t]^{\sigma}) in W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda), and recall that W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda) is generated (as an abelian group) by elements of the form

[TABLE]

Thus, the subgroup of W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda) generated by elements $v_{\phi}$ with a fixed $\sum_{j>0}r_{j}$ and a fixed \text{\mathfrak{h}}_{0}-weight ($\lambda-\sum_{j>0}k_{j}\beta_{j}$) is finitely generated.

By Lemma 3.2, we know that W^{c,\sigma}_{\text{\mathbb{Z}}}(\lambda) is an integrable U_{\text{\mathbb{Z}}}(\text{\mathfrak{g}}_{0})-module. Thus, Theorem 2.1b implies that the set \operatorname{wt}_{0}\left(W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda)\right) is $\cal W_{0}$-invariant. Since the \text{\mathfrak{h}}_{0}-weights of W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda) are bounded above by $\lambda$, it follows that the set of weights of W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda) is contained in that of W_{\text{\mathbb{Z}}}(\lambda). Since the set of weights of W_{\text{\mathbb{Z}}}(\lambda) is finite, it follows that the set of weights of W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda) is also finite. Hence, the subgroup of W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda) generated by elements $v_{\phi}$ with a fixed $\sum_{j>0}r_{j}$ is finitely generated.

In order to show that W^{c,\sigma}_{\text{\mathbb{Z}}}(\lambda) is finitely generated, we will restrict the possibilities of $\sum_{j>0}r_{j}$ to finitely many distinct ones. Define the exponent of $\phi\in\Xi$ to be $e(\phi)=\max\{k_{j}\mid j>0\}$, observe that $0\leq e(\phi)\leq d(\phi)$, and let

[TABLE]

In order to finish the proof, we will show that W^{\prime}=W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda).

In fact, we will use induction on $d$ and $e$ in order to show that $v_{\phi}\in W^{\prime}$ for all $\phi\in\Xi_{d,e}$, $d,e\geq 0$. If $d(\phi)=e(\phi)=0$, then $v_{\phi}=v\in W^{\prime}$. Assume that $d(\phi)>0$ and that the statement holds for all $\phi^{\prime}\in(\cup_{d<d(\phi)}\Xi_{d})\cup(\cup_{e<e(\phi)}\Xi_{d(\phi),e})$. The proof splits into two cases, according to whether $e(\phi)=d(\phi)$ or not.

Suppose $e(\phi)<d(\phi)$, in which case $v_{\phi}=(x_{\beta_{1},-r_{1}}^{-}\otimes t^{r_{1}})^{(k_{1})}\cdots(x_{\beta_{n},-r_{n}}^{-}\otimes t^{r_{n}})^{(k_{n})}v$ for some $n>1$ and $k_{1},\ldots,k_{n}\neq 0$. By induction hypothesis, $(x_{\beta_{2},-r_{2}}^{-}\otimes t^{r_{2}})^{(k_{2})}\cdots(x_{\beta_{n},-r_{n}}^{-}\otimes t^{r_{n}})^{(k_{n})}v\in W^{\prime}$. Thus, without loss of generality, we will assume that $r_{j}<d_{\beta_{j}}\lambda(h_{\beta_{j},0})$ for all $j\in\{2,\dotsc,n\}$. Using Lemma 1.2 repeatedly to commute $(x_{\beta_{1},-r_{1}}^{-}\otimes t^{r_{1}})^{(k_{1})}$ with $(x_{\beta_{2},-r_{2}}^{-}\otimes t^{r_{2}})^{(k_{2})},\ldots,(x_{\beta_{n},-r_{n}}^{-}\otimes t^{r_{n}})^{(k_{n})}$, and induction hypothesis on the terms of hyperdegree strictly less than $d(\phi)$, we obtain that

[TABLE]

By induction hypothesis, $(x_{\beta_{1},-r_{1}}^{-}\otimes t^{r_{1}})^{(k_{1})}v\in W^{\prime}$. Since $r_{2}<d_{\beta_{2}}\lambda(h_{\beta_{2},0}),\dots,r_{n}<d_{\beta_{j}}\lambda(h_{\beta_{n},0})$, it follows that

[TABLE]

Hence, $v_{\phi}\in W^{\prime}$.

Now, suppose $e(\phi)=d(\phi)$, in which case $v_{\phi}=(x_{\beta,-r}^{-}\otimes t^{r})^{(d)}v$. Also suppose that $r\geq d_{\beta}\lambda(h_{\beta,0})$. We will split the rest of this proof into four cases:

Case 1: Assume \text{\mathfrak{g}}\not\cong A_{2n} and $\beta\in R_{\operatorname{sh}}$ or \text{\mathfrak{g}}\cong A_{2n} and $\beta\in R_{\operatorname{lg}}$. Our goal is to show that $(x_{\beta,-r}^{-}\otimes t^{r})^{(d)}v\in W^{\prime}$ for all $d,r\in\mathbb{N}$ with $r\geq d_{\beta}\lambda(h_{\beta,0})$. Replacing $k=l+d,r=0,s=1$ in Lemma 1.4(a) we get

[TABLE]

Since $(x_{\beta,0}^{-}\otimes 1)^{(\kappa)}v=0$ for all $\kappa>\lambda(h_{\beta,0})$, it follows that this last equality is zero for all $l\geq\lambda(h_{\beta,0})$. Thus

[TABLE]

Set $l=d(\lambda(h_{\beta,0})+l^{\prime}),l^{\prime}\geq 0$. Since, from the case $e(\phi)<d(\phi)$,

[TABLE]

for all $d^{\prime}>1$, it follows, from Equation (3.1), that $(x_{\beta,-r}^{-}\otimes t^{\lambda(h_{\beta,0})+l^{\prime}})^{(d)}v\in W^{\prime}$ for all $l^{\prime}\geq 0$.

Case 2: Assume \text{\mathfrak{g}}\not\cong A_{2n} and $\beta\in R_{\operatorname{lg}}$. Here our goal is to show that $(x_{\beta,0}^{-}\otimes t^{r})^{(d)}v\in W^{\prime}$ for all $d\in\mathbb{N}$ and $r\in m\mathbb{N}$ with $r\geq\lambda(h_{\beta,0})$. Replacing $k=l+d,r=0,s=1$ in Lemma 1.4(b) we get

[TABLE]

Since $(x_{\beta,0}^{-}\otimes 1)^{(\kappa)}v=0$ for all $\kappa>\lambda(h_{\beta,0})$, it follows that this last equality is zero for all $l\geq\lambda(h_{\beta,0})$. Thus

[TABLE]

and the conclusion follows as in the previous case.

Case 3: Assume \text{\mathfrak{g}}\cong A_{2n} and $\beta\in 2R_{\operatorname{sh}}$. For this case our goal is to show that $(x_{\beta,-r}^{-}\otimes t^{r})^{(d)}v\in W^{\prime}$ for all $d,r\in\mathbb{N}$ with $r\geq d_{\beta}\lambda(h_{\beta,0})$ and $r$ an odd number. It is achievable proceeding in a similar way to the previous cases by using Lemma 1.4(c)(ii).

Case 4: Assume \text{\mathfrak{g}}\cong A_{2n} and $\beta\in R_{\operatorname{sh}}$. Our goal is to show that $(x_{\beta,-r}^{-}\otimes t^{r})^{(d)}v\in W^{\prime}$ for all $d,r\in\mathbb{N}$ with $r\geq d_{\beta}\lambda(h_{\beta,0})$. The procedure with $r$ even is done by using Lemma 1.4(c)(i) also in a similar way to the other cases. Finally, the case with $r$ odd follows by Lemma 1.4(c)(iii) and by using the conclusion of Case 3. ∎

Corollary 3.4**.**

For any $\lambda\in P_{0}^{+}$ and $\ell>0$, W^{c,\sigma}_{\text{\mathbb{F}}}(\lambda) and D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda) are finite-dimensional.

Proof.

Recall that W^{c,\sigma}_{\text{\mathbb{F}}}(\lambda)\cong\text{\mathbb{F}}\otimes_{\text{\mathbb{Z}}}W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda). Since W^{c,\sigma}_{\text{\mathbb{Z}}}(\lambda) is finitely generated, the dimension of W^{c,\sigma}_{\text{\mathbb{F}}}(\lambda) is at most the number of generators of W^{c,\sigma}_{\text{\mathbb{Z}}}(\lambda). Now recall that D_{\text{\mathbb{Z}}}^{\sigma}(\ell,\lambda) is a quotient of W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda), for all $\ell>0$ and $\lambda\in P_{0}^{+}$. Since W_{\text{\mathbb{Z}}}^{c,\sigma}(\lambda) is finitely generated, it follows that D_{\text{\mathbb{Z}}}^{\sigma}(\ell,\lambda) is also finitely generated. Finally, recall that D^{\sigma}_{\text{\mathbb{F}}}(\ell,\lambda)\cong\text{\mathbb{F}}\otimes_{\text{\mathbb{Z}}}D_{\text{\mathbb{Z}}}^{\sigma}(\ell,\lambda), and, since D_{\text{\mathbb{Z}}}^{\sigma}(\ell,\lambda) is finitely generated, it follows that D^{\sigma}_{\text{\mathbb{F}}}(\ell,\lambda) is also finite-dimensional. ∎

3.2. The category of $\mathbb{Z}$-graded finite-dimensional U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})-modules

Let \cal G_{\text{\mathbb{F}}}^{\sigma} be the category of $\mathbb{Z}$-graded finite-dimensional U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})-modules. Given a module $V$ in \cal G_{\text{\mathbb{F}}}^{\sigma}, let $V[r]$ denote its $r$-th graded piece, and given s\in\text{\mathbb{Z}}, let $\tau_{s}V$ be the module in \cal G_{\text{\mathbb{F}}}^{\sigma} satisfying $(\tau_{s}V)[r]=V[r-s]$ for all r\in\text{\mathbb{Z}}. For each finite-dimensional U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0})-module $V$, let $\operatorname{ev}_{0}V$ be the module in \cal G_{\text{\mathbb{F}}}^{\sigma} obtained by inflating the action of U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0}) to U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma}) by setting U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma}_{+})V=0. For each r\in\text{\mathbb{Z}}, set $\operatorname{ev}_{r}=\tau_{r}\circ\operatorname{ev}_{0}$, and for each $\lambda\in P_{0}^{+}$, set V_{\text{\mathbb{F}}}(\lambda,r)=\operatorname{ev}_{r}V_{\text{\mathbb{F}}}(\lambda).

Theorem 3.5**.**

1. (a)

If $V$ is a simple module in \cal G_{\text{\mathbb{F}}}^{\sigma}, then $V$ is isomorphic to $V_{\mathbb{F}}(\lambda,r)$ for a unique (\lambda,r)\in P_{0}^{+}\times\text{\mathbb{Z}}. 2. (b)

For every $\lambda\in P^{+}_{0}$, $W^{c,\sigma}_{\mathbb{F}}(\lambda)$ is in \cal G_{\text{\mathbb{F}}}^{\sigma}. 3. (c)

Let $\lambda\in P^{+}_{0}$. If $V$ is a module in \cal G_{\text{\mathbb{F}}}^{\sigma} generated by a nonzero vector $v$ satisfying

[TABLE]

then $V$ is a quotient of W^{c,\sigma}_{\text{\mathbb{F}}}(\lambda).

Proof.

To prove part (a), suppose $V[r],V[s]\neq 0$ for some s<r\in\text{\mathbb{Z}}. In that case, $\oplus_{k\geq r}V[k]$ would be a proper submodule of $V$, contradicting the fact that it is simple. Thus there must exist a unique r\in\text{\mathbb{Z}} such that $V[r]\neq 0$. Since $uv\in V[s]$, $s>r$, for all u\in U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma}_{+})^{0} and $v\in V=V[r]$, it follows that U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma}_{+})^{0}V=0. Thus $V$ is in fact the inflation of a simple U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0})-module; that is, V\cong V_{\text{\mathbb{F}}}(\lambda,r) for some $\lambda\in P_{0}^{+}$ and r\in\text{\mathbb{Z}}.

Part (b) follows directly from Corollary 3.4.

To prove part (c), observe that the U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0}[t])-submodule V^{\prime}=U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0}[t])v\subseteq V is a graded, finite-dimensional, highest-weight module of highest weight $\lambda$. Thus $V^{\prime}$ is a quotient of the graded Weyl module for U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0}[t]) of highest weight $\lambda$. Using Lemma 1.1, the statement follows by comparing the defining relations of W^{c,\sigma}_{\text{\mathbb{F}}}(\lambda) with those in (3.3). ∎

3.3. Joseph-Mathieu-Polo relations for Demazure modules

We now explain the reason why we call $D_{\mathbb{F}}^{\sigma}(\ell,\lambda)$ Demazure modules. In order to do that, we need the concepts of weight vectors, weight spaces, weight modules and integrable modules for U_{\text{\mathbb{F}}}(\text{\hat{\mathfrak{g}}}^{\prime}) which are similar to those for U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0}) (cf. Subsection 2.1) by replacing $P_{0}$ by $\hat{P}^{\prime}$. Also, using an analogue of (1.8), we obtain an inclusion \hat{P}^{\prime}\hookrightarrow U_{\mathbb{F}}(\text{\hat{\mathfrak{h}}}^{\prime})^{*}. Let $V$ be a $\mathbb{Z}$-graded U_{\text{\mathbb{F}}}(\text{\hat{\mathfrak{g}}}^{\prime})-module whose weights are in $\hat{P}^{\prime}$. As before, let $V[r]$ denote the $r$-th graded piece of $V$. For $\mu\in\hat{P}$, say $\mu=\mu^{\prime}+n\delta$ with $\mu^{\prime}\in\hat{P}^{\prime}$, n\in\text{\mathbb{Z}}, denote

[TABLE]

The next result is a partial twisted affine analogue of Theorem 2.1.

Theorem 3.6**.**

Let $V$ be a graded U_{\mathbb{F}}(\text{\hat{\mathfrak{g}}}^{\prime})-module.

1. (a)

If $V$ is integrable, then $V$ is a weight-module and $\operatorname{wt}(V)\subseteq\hat{P}$. Moreover, $\dim V_{\mu}=\dim V_{w\mu}$ for all $w\in\widehat{\cal W}$, $\mu\in\hat{P}$. 2. (b)

If $V$ is a highest-weight module of highest weight $\lambda\in\hat{P}^{+}$, then $\dim(V_{\lambda})=1$ and $V_{\mu}\neq 0$ only if $\mu\leq\lambda$. Moreover, $V$ has a unique maximal proper submodule and, hence, also a unique irreducible quotient. In particular, $V$ is indecomposable. 3. (c)

Let $\Lambda\in\hat{P}^{+}$ and $n=\Lambda(d)$. Then, the U_{\mathbb{F}}(\text{\hat{\mathfrak{g}}}^{\prime})-module \widehat{M}_{\text{\mathbb{F}}}(\Lambda) generated by a vector $v$ of degree $n$ satisfying the defining relations

[TABLE]

is nonzero and integrable. Furthermore, every integrable highest-weight module of highest weight $\Lambda$ is a quotient of \widehat{M}_{\text{\mathbb{F}}}(\Lambda). ∎

Given $\Lambda\in\hat{P}^{+}$, traditionally, a Demazure module is defined to be the U_{\text{\mathbb{F}}}(\text{\hat{\mathfrak{b}}}^{\prime+})-submodule V_{\mathbb{F}}^{w}(\Lambda)\subseteq\widehat{M}_{\text{\mathbb{F}}}(\Lambda) generated by the weight space \widehat{M}_{\text{\mathbb{F}}}(\Lambda)_{w\Lambda} for some $w\in\widehat{\cal W}$ (cf. [Mat89, FL07, Nao12, FK13]). Our focus is on Demazure modules that are stable under the action of U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0}). Since V_{\text{\mathbb{F}}}^{w}(\Lambda) is defined as a U_{\text{\mathbb{F}}}(\text{\hat{\mathfrak{b}}}^{\prime+})-module, it is stable under the action of U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0}) if, and only if,

[TABLE]

In particular, since V_{\text{\mathbb{F}}}^{w}(\Lambda) is an integrable U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0})-module, it follows that $(w\Lambda)(h_{\alpha,0})\leq 0$ for all $\alpha\in R_{0}^{+}$. Conversely, using the exchange condition for Coxeter groups (see [Hum90, Section 5.8]), for all $i\in\hat{I}$, we have

[TABLE]

where $\varepsilon=+$ if $w\Lambda(h_{\alpha_{i}})\geq 0$ and $\varepsilon=-$ if $w\Lambda(h_{\alpha_{i}})\leq 0$. Thus, if $w\Lambda(h_{\alpha,0})\leq 0$ for all $\alpha\in R_{0}^{+}$, then V_{\text{\mathbb{F}}}^{w}(\Lambda) is U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0})-stable.

Henceforth, assume that $(w\Lambda)(h_{\alpha,0})\leq 0$ for all $\alpha\in R^{+}_{0}$, and observe that this implies that $w\Lambda$ must have the form

[TABLE]

Conversely, given \ell\in\text{\mathbb{Z}}_{\geq 0}, $\lambda\in P_{0}^{+}$ and n\in\text{\mathbb{Z}}, since $\widehat{\cal W}$ acts simply transitively on the set of alcoves of \text{\hat{\mathfrak{h}}}^{\ast} (see [Hum90, Theorem 4.5.(c)]), there exists a unique $\Lambda\in\hat{P}^{+}$ such that $\ell\Lambda_{0}-\lambda+n\delta\in\widehat{\cal W}\Lambda$. Thus, if $w\in\widehat{\cal W}$ and $\Lambda\in\hat{P}^{+}$ are such that

[TABLE]

then V_{\text{\mathbb{F}}}^{w}(\Lambda) is U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0})-stable. Henceforth, we fix $w$, $\Lambda$, $\lambda$ and $n$ as in (3.6). Notice that, if $\gamma=\pm\alpha+s\delta\in\hat{R}^{+}$, $\alpha\in R_{0}^{+}$, $s\geq 0$, then

[TABLE]

The following lemma is a rewriting of [Mat89, Lemme 26] using the above fixed notation. (Compare it with [FL07, Theorem 1], [FK13, Proposition 4.8] and [BMM15, Lemma 3.5.3].)

Lemma 3.7**.**

The U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})-module V_{\text{\mathbb{F}}}^{w}(\Lambda) is isomorphic to a U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})-module generated by a vector $v$ of degree $n$ satisfying the following defining relations:

[TABLE]

The following is the main result of this subsection.

Proposition 3.8**.**

The graded U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})-modules V_{\text{\mathbb{F}}}^{w}(\Lambda) and D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda,n) are isomorphic.

Proof.

It suffices to prove the statement for $n=0$ and, thus, for simplicity, we assume that this is the case. First, we will show that V_{\text{\mathbb{F}}}^{w}(\Lambda) is a quotient of D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda). Let $\mu=\ell\Lambda_{0}+\lambda$. Since \widehat{M}_{\text{\mathbb{F}}}(\Lambda) is integrable and V_{\text{\mathbb{F}}}^{w}(\Lambda) is a U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}_{0})-submodule of \widehat{M}_{\text{\mathbb{F}}}(\Lambda), it follows from Theorem 3.6a that

[TABLE]

Hence there exists a nonzero vector v\in V_{\text{\mathbb{F}}}^{w}(\Lambda)_{\mu}, which is extremal and generates V_{\text{\mathbb{F}}}^{w}(\Lambda). Since $v$ is an extremal weight vector of weight $\mu$ in \widehat{M}_{\text{\mathbb{F}}}(\Lambda), we have, for all positive real roots $\gamma\in\hat{R}^{+}$,

[TABLE]

In particular, by taking $\gamma=\alpha+s\delta$ with $\alpha\in R_{-s}^{+}$ and $s\geq 0$, we obtain that

[TABLE]

This implies that $(x_{\alpha,-s}^{+}\otimes t^{s})^{(k)}v=0$ for all $k>0$, $s\geq 0$, $\alpha\in R^{+}_{-s}$. Thus U_{\text{\mathbb{F}}}(\text{\mathfrak{n}}^{+}[t]^{\sigma})^{0}v=0. Similarly, by taking $\gamma=-\alpha+s\delta$ with $\alpha\in R_{-s}^{+}$ and $s\geq 0$, we obtain that

[TABLE]

This implies that $(x_{\alpha,-s}^{-}\otimes t^{s})^{(k)}v=0$ for all $k>\max\{0,\delta_{s,1}\lambda(h_{\frac{\alpha}{2},0})-\ell\hat{r}_{\alpha}^{\vee}s\}$, in case \text{\mathfrak{g}}\cong A_{2n}, $\alpha\in 2R_{\operatorname{sh}}$, and implies that $(x_{\alpha,-s}^{-}\otimes t^{s})^{(k)}v=0$ for all $k>\max\{0,\lambda(h_{\alpha,0})-\ell\hat{r}_{\alpha}^{\vee}s\}$ in every other case. These vanishing conditions for $(x_{\alpha,-s}^{-}\otimes t^{s})^{(k)}v$ and $(x_{\alpha,-s}^{+}\otimes t^{s})^{(k)}v$, together with Lemma 1.4 imply that U_{\text{\mathbb{F}}}(\text{\mathfrak{h}}[t]^{\sigma}_{+})^{0}v=0. Thus, $v$ is a generator of V_{\text{\mathbb{F}}}^{w}(\Lambda) satisfying the defining relations (2.1). This implies that V_{\text{\mathbb{F}}}^{w}(\Lambda) is a quotient of D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda).

It now suffices to show that \dim D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda)\leq\dim V_{\text{\mathbb{F}}}^{w}(\Lambda). In fact, we will show that $D^{\prime}$, the pull-back of D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda) by $\psi$, is a quotient of V_{\text{\mathbb{F}}}^{w}(\Lambda). Let this time $v$ be a nonzero generator in D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda)_{\lambda} and $v^{\prime}$ denote $v$ when regarded as an element of $D^{\prime}$. Since U_{\text{\mathbb{F}}}(\text{\mathfrak{n}}^{+}[t]^{\sigma})^{0}v=0 and \psi\left(U_{\text{\mathbb{F}}}(\text{\mathfrak{n}}^{-}[t]^{\sigma})^{0}\right)=U_{\text{\mathbb{F}}}(\text{\mathfrak{n}}^{+}[t]^{\sigma})^{0}, it follows that U_{\text{\mathbb{F}}}(\text{\mathfrak{n}}^{-}[t]^{\sigma})^{0}v^{\prime}=0. Since $\psi$ restricts to an automorphism of U_{\text{\mathbb{F}}}(\text{\mathfrak{h}}[t]^{\sigma}_{+}) and U_{\text{\mathbb{F}}}(\text{\mathfrak{h}}[t]^{\sigma}_{+})^{0}v=0, then U_{\text{\mathbb{F}}}(\text{\mathfrak{h}}[t]^{\sigma}_{+})^{0}v^{\prime}=0. Moreover, since $hv=\lambda(h)v$ for all h\in U_{\text{\mathbb{F}}}(\text{\mathfrak{h}}), (1.9) implies that $hv^{\prime}=-\lambda(h)v^{\prime}$ for all h\in U_{\text{\mathbb{F}}}(\text{\mathfrak{h}}). Finally, the defining relations (2.1) for $v$ and (1.7) imply that

[TABLE]

in case \text{\mathfrak{g}}\cong A_{2n} and $\alpha\in 2R_{\operatorname{sh}}$, and imply that

[TABLE]

in every other case. Thus $v^{\prime}$ satisfies the defining relations of the generator of V_{\text{\mathbb{F}}}^{w}(\Lambda) given in Lemma 3.7. This shows that $D^{\prime}$ is a quotient of V_{\text{\mathbb{F}}}^{w}(\Lambda). Therefore, \dim D_{\text{\mathbb{F}}}^{\sigma}(\ell,\lambda)=\dim D^{\prime}\leq\dim V_{\text{\mathbb{F}}}^{w}(\Lambda). ∎

4. Main results

4.1. Connection between twisted Weyl modules and twisted Demazure modules

In this section we will show that almost all Weyl modules are isomorphic to certain Demazure modules.

The following lemma will be used in Theorem 4.2.

Lemma 4.1**.**

[JM14, Proposition 2.5.4]**. Let $m\in\mathbb{Z}_{\geq 0}$ and let $v$ be a nonzero vector of weight $m\omega_{1}$ of the U_{\mathbb{F}}(\text{\mathfrak{sl}}_{2}[t])-module $W_{\mathbb{F}}^{c}(m)$. Then, $(x_{\alpha_{1}}^{-}\otimes t^{r})^{(k)}v=0$ for all $r\in\mathbb{Z}_{\geq 0}$, $k>max\{0,m-r\}$. ∎

Theorem 4.2**.**

Suppose $\mathfrak{g}$ is of type $A_{2l-1}^{(2)},D_{l+1}^{(2)},E_{6}^{(2)}$ or $D_{4}^{(3)}$. Then, we have and isomorphism of U_{\mathbb{F}}(\text{\mathfrak{g}}[t]^{\sigma})-modules

[TABLE]

Proof.

It follows from the definitions in Section 2.2 that the Demazure module $D_{\mathbb{F}}^{\sigma}(1,\lambda)$ is a quotient of the the Weyl module $W_{\mathbb{F}}^{c,\sigma}(\lambda)$. By comparing the defining relations in Lemma 3.7 and Definition 2.2, to prove that these modules are isomorphic it is suffices to show that the generator of the Weyl module is subject to the following relations:

[TABLE]

Notice that these relations are equivalent to:

[TABLE]

for all $\alpha\in R_{0}^{+},\ 0\leq j\leq m-1,\ s\geq 0,\ k>\begin{cases}\max\{0,\lambda(h_{\alpha,0})-s\},\mbox{ if }\alpha\mbox{ is long }\\ \max\{0,\lambda(h_{\alpha,0})-(ms+j)\},\mbox{ if }\alpha\mbox{ is short }.\end{cases}$

Let $\alpha\in R_{0}^{+}$ be a long root and V=U_{\mathbb{F}}(\text{\mathfrak{sl}}_{2,\alpha}[t^{m}])\cdot v\subseteq W_{\mathbb{F}}^{c,\sigma}(\lambda). Notice that $v$ is a cyclic generator for $V$ and satisfies the defining relations of of the nontwisted graded Weyl U_{\mathbb{F}}(\text{\mathfrak{sl}}_{2}[t])-module $W_{\mathbb{F}}^{c}(\lambda(h_{\alpha,0})\omega_{1})$. Thus, we conclude that $V$ is a quotient $W_{\mathbb{F}}^{c}(\lambda(h_{\alpha,0})\omega_{1})$. In particular, by Lemma 4.1, $v$ satisfies the relations

[TABLE]

Now, by Lemma 1.1 we obtain

[TABLE]

Finally, suppose $\alpha$ is a short root and consider the U_{\mathbb{F}}(\text{\mathfrak{sl}}_{2,\alpha}[t])-submodule V=U_{\mathbb{F}}(\text{\mathfrak{sl}}_{2,\alpha}[t])\cdot v\subseteq W_{\mathbb{F}}^{c,\sigma}(\lambda). In a similar fashion as above we can use Lemma 1.1 to conclude that $V$ is a quotient of $W_{\mathbb{F}}^{c}(\lambda(h_{\alpha,0})\omega_{1})$ and, therefore, $v$ satisfies the relations in (4.2). So, by using the isomorphism in Lemma 1.1 we obtain

[TABLE]

4.2. Connection between twisted and untwisted Weyl modules

In this section we will show that the twisted Weyl modules can be realized as modules for the hyper current algebra constructed from certain untwisted Weyl modules. The version of this result for hyper loop algebra was proved in [BM14] and the methods are completely different.

Suppose that $\rm{char}(\mathbb{F})\neq m$. Denote by \textrm{res}^{U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t])}_{U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})}W_{\text{\mathbb{F}}}^{c}(\lambda) the module obtained by regarding W_{\text{\mathbb{F}}}^{c}(\lambda) as a U_{\mathbb{F}}(\text{\mathfrak{g}}[t]^{\sigma})-module via restriction of the action of U_{\mathbb{F}}(\text{\mathfrak{g}}[t]) to U_{\mathbb{F}}(\text{\mathfrak{g}}[t]^{\sigma}).

Theorem 4.3**.**

There is an isomorphism \textrm{res}^{U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t])}_{U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})}W_{\text{\mathbb{F}}}^{c}(\lambda)\cong W_{\text{\mathbb{F}}}^{c,\sigma}(\lambda) of U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})-modules.

Proof.

Let v\in W_{\text{\mathbb{F}}}^{c}(\lambda) be a cyclic generator and W=U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})v\subseteq U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t])v=W_{\text{\mathbb{F}}}^{c}(\lambda). Under these assumptions we have W=\textrm{res}^{U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t])}_{U_{\text{\mathbb{F}}}(\text{\mathfrak{g}}[t]^{\sigma})}W_{\text{\mathbb{F}}}^{c}(\lambda). We will show that $W$ is a quotient of W_{\text{\mathbb{F}}}^{c,\sigma}(\lambda).

Firstly, from the defining relations of W_{\text{\mathbb{F}}}^{c}(\lambda) and the construction of $\cal C^{\sigma}(O)$, we easily conclude that

[TABLE]

It remains to show that

[TABLE]

for all h\in U_{\text{\mathbb{F}}}(\text{\mathfrak{h}}_{0}), $\alpha\in R_{0}^{+}$, and $k>\lambda(h_{\alpha,0})$. From Section 1.2, we have

[TABLE]

and

[TABLE]

which leads to $hv=\lambda(h)v$ for all h\in U_{\text{\mathbb{F}}}(\text{\mathfrak{h}}_{0}) due to the defining relations of W_{\text{\mathbb{F}}}^{c}(\lambda). Also from Section 1.2 we have

[TABLE]

and supposing $k>\lambda(h_{\alpha,0})$, from the defining relations of W_{\text{\mathbb{F}}}^{c}(\lambda), we conclude that $(x_{\alpha,0}^{-}\otimes 1)^{(k)}v=0$ for all these cases, since the first case is direct, the second case follows by taking the binomial expansion

[TABLE]

where we observe that either $k_{1}>\lambda(h_{\alpha})$ or $k_{2}>\lambda(h_{\sigma(\alpha)})$ and $x_{\alpha}^{-}$ and $x_{\sigma(\alpha)}^{-}$ commutes, and the third case follows from the expansion

[TABLE]

and the fact that \lambda-2\alpha-2\sigma(\alpha)\notin\operatorname{wt}(W_{\text{\mathbb{F}}}^{c}(\lambda)) for any $\alpha\in R_{0}^{+}$. ∎