2.1 Standard automorphisms

Let $\sigma $ be an automorphism of order m on G preserving B and T. Let $\tau $ be a diagram automorphism preserving $B, T$ and a pinning $\{ x_{\alpha _i}, y_{\alpha _i}\,|\, i\in I \} $ . Let r be the order of $\tau $ .

When $\mathfrak {g}$ is not $A_{2\ell }$ , we take $\sigma $ to be $\tau $ . When $\mathfrak {g}$ is $A_{2\ell }$ , by [Reference KacKa, Theorem 8.6], there exists a unique automorphism $\sigma $ of order $m=4$ such that

(2.1) $$ \begin{align} \begin{cases} \sigma(e_i)=e_{\tau(i)}, \quad \text{ if } i\not= \ell, \ell+1; \\ \sigma(e_i)= \mathrm{i} e_{\tau(i)}, \quad \text{ if } i\in \{\ell, \ell+1\}; \\ \sigma(f_\theta)=f_\theta, \end{cases} \end{align} $$

where $\mathrm {i}$ is a square root of $-1$ . One can check that

(2.2) $$ \begin{align} \begin{cases} \sigma(f_i)= f_{\tau(i)}, \quad \text{ if } i\not= \ell, \ell+1; \\ \sigma(f_i)= -\mathrm{i} f_{\tau(i)}, \quad \text{ if } i\in \{\ell, \ell+1\}; \\ \sigma(e_\theta)=e_\theta \end{cases}\!\!\!\!\!. \end{align} $$

In fact, $\sigma = \tau \circ {\mathrm i}^{ h} $ , where $h\in \mathfrak {h}$ such that

$$\begin{align*}\alpha_i(h)=\begin{cases} 0 , \quad \text{ if } i\not= \ell, \ell+1 \\ 1, \quad \text{ if } i=\ell, \ell+1 \end{cases}\!\!\!\!\!. \end{align*}$$

This automorphism induces a unique automorphism on G. We still call it $\sigma $ .

We call these automorphisms on G or $\mathfrak {g}$ ‘standard’, as the fixed point Lie subalgebra $\mathfrak {g}^\sigma $ is the standard finite part of the associated twisted affine Lie algebra $\hat {L}(\mathfrak {g},\sigma )$ (cf. Section 3.1) in the sense of Kac [Reference KacKa, §6.3]. From $\sigma $ , we will construct a twisted affine Grassmannian and a line bundle of level one on it. There will be no level one line bundle on the twisted affine Grassmannian associated to $\tau $ on G of type $A_{2\ell }$ . Throughout this paper, we will only consider standard automorphisms.

The following table describe the fixed point Lie algebras for all standard automorphisms:

(2.3)

where by convention, $C_1$ is $A_1$ and $\ell \geq 3$ for $D_{\ell +1}$ . When $(\mathfrak {g}, m)\not =(A_{2\ell }, 4) $ , the fixed point Lie algebra $\mathfrak {g}^\sigma $ is well known as listed in the above table. When $(\mathfrak {g}, m)=(A_{2\ell }, 4) $ , the fixed Lie algebra $\mathfrak {g}^\sigma $ is of type $C_\ell $ , which can follow from the twisted Kac-Moody theory; cf. [Reference KacKa, §6.3, §8.4].

Recall that we follow the Bourbaki labelling of the vertices of the Dynkin diagram. Set

(2.4) $$ \begin{align} \begin{cases} \beta_i= {\alpha_i} |_ {\mathfrak{h}^\sigma}, \text{ for } i=1,2,\cdots, \ell, \quad \text{ if } (\mathfrak{g}, m)=(A_{2\ell-1}, 2), \text{ or } (D_{\ell+1}, 2) \\ \beta_1=\alpha_1|_{\mathfrak{h}^\sigma}, \beta_2=\alpha_2|_{\mathfrak{h}^\sigma}, \quad \text{ if } (\mathfrak{g}, m)=(D_4, 3)\\ \beta_1=\alpha_2|_{\mathfrak{h}^\sigma}, \beta_2=\alpha_4|_{\mathfrak{h}^\sigma},\beta_3=\alpha_3|_{\mathfrak{h}^\sigma},\beta_4=\alpha_1|_{\mathfrak{h}^\sigma}, \quad \text{ if } (\mathfrak{g}, m)=(E_6, 2)\\ \beta_i= \alpha_i |_ {\mathfrak{h}^\sigma}, \text{ for } i=1,2,\cdots, \ell-1; \beta_\ell= (\alpha_\ell+\alpha_{\ell+1}) |_ {\mathfrak{h}^\sigma}=2\alpha_\ell|_{\mathfrak{h}^\sigma}, \, \text{if} (\mathfrak{g}, m)=(A_{2\ell}, 4). \end{cases} \end{align} $$

Let $I_\sigma $ be the set of all subscript indices of $\beta _i$ . Then for each case, the set $\{\,\beta _j \,|\, j\in I_\sigma \, \}$ gives rise to the set of simple roots of $\mathfrak {g}^\sigma $ . One can see easily that this labelling will coincide with Bourbaki labelling for nonsimply-laced types Dynkin diagrams.

We now define a map $\eta : I\to I_\sigma $ . When $(\mathfrak {g}, m)\not = (A_{2\ell }, 4)$ , $\eta $ is defined such that $\beta _{\eta (i) }= \alpha _i|_{\mathfrak {h}^\sigma }$ for any $i\in I$ . When $(\mathfrak {g}, m)= (A_{2\ell }, 4)$ , set

$$\begin{align*}\eta(i)=\eta(2\ell+1-i)=i, \text{ for any } 1\leq i\leq \ell. \end{align*}$$

Let $\{ \,\check {\beta }_j \,|\, j\in I_\sigma \}$ be the set of simple coroots of $\mathfrak {g}^\sigma $ . We can describe $\check {\beta }_j $ as follows:

(2.5) $$ \begin{align} \check{\beta}_j= \sum_{i\in \eta^{-1}(j) } \check{\alpha}_{ i }. \end{align} $$

The description of $\check {\beta }_j $ also appears in [Reference HainesHa, Section 3] in a slightly different setting.

Let $\{\, \lambda _j \,|\, j\in I_\sigma \,\}$ be the set of fundamental weights of $\mathfrak {g}^\sigma $ , and let $\{\, \check {\lambda }_j \,|\, j\in I_\sigma \,\}$ be the set of fundamental coweights of $\mathfrak {g}^\sigma $ . The fundamental weights can be described as follows:

(2.6) $$ \begin{align} \lambda_j= \omega_i|_{\mathfrak{h}^\sigma}, \quad \text{ for some } i \text{ with } \eta(i)=j. \end{align} $$

In the case of fundamental coweights, we need to describe them separately. When $(\mathfrak {g}, m)\not = (A_{2\ell }, 4)$ ,

(2.7) $$ \begin{align} \check{\lambda}_j=\sum_{i\in \eta^{-1}(j)} \check{\omega}_i. \end{align} $$

When $(\mathfrak {g},m)=(A_{2\ell }, 4)$ , we have

(2.8) $$ \begin{align} \check{\lambda}_j= \begin{cases} \check{\omega}_j+\check{\omega}_{2\ell+1-j}, \quad \quad j=1,2,\cdots, \ell-1 \\ \frac{1}{2}(\check{\omega}_\ell + \check{\omega}_{\ell+1} ), \quad \quad j=\ell \end{cases}\!\!\!\!\!. \end{align} $$

2.2 Affine Grassmannian of special parahoric group schemes

Let $\mathcal {K}$ denote the field of formal Laurent series in t with coefficients in $\mathbb {C}$ . Let $\mathcal {O} $ denote the ring of formal power series in t with coefficients in $\mathbb {C}$ . By abuse of notation, we still use $\sigma $ to denote the automorphism of order m on $\mathcal {K}$ and $\mathcal {O}$ such that $\sigma $ acts on $\mathbb {C}$ trivially, and $\sigma (t)=\epsilon ^{-1} t $ , where $\epsilon =e^{\frac {2\pi \mathrm {i}}{m}}$ . Set $\bar { \mathcal {K} } =\mathcal {K}^\sigma $ and $\bar { \mathcal {O} } =\mathcal {O}^\sigma $ . Then $\bar { \mathcal {K} } ={\mathbb {C}}((\bar {t})) $ and $\bar { \mathcal {O} } ={\mathbb {C}}[[\bar {t}]]$ , where $\bar {t}=t^m$ .

Let ${\mathscr {G}}$ be the smooth group scheme $\mathrm {Res}_{\mathcal {O}/\bar {\mathcal {O}} } (G_{\mathcal {O}})^\sigma $ over $\bar { \mathcal {O} } $ , which represents the following group functor

$$\begin{align*}R\mapsto G(\mathcal{O}\otimes_{\bar{ \mathcal{O} } } R)^\sigma , \quad \text{ for any } \bar{ \mathcal{O} } -\text{algebra } R, \end{align*}$$

where the $G(\mathcal {O}\otimes _{\bar { \mathcal {O} } } R)$ denotes the group of $\sigma $ -equivariant morphisms from $\mathrm {Spec }\,( \mathcal {O}\otimes _{\bar {\mathcal {O}}} R)$ to G, where $\sigma $ acts on $\mathcal {O}$ as above and acts on G as a standard automorphism defined in Section 2.1. Then, ${\mathscr {G}}$ is a special parahoric group scheme in the sense of Bruhat-Tits, as we choose $\sigma $ to be standard. In fact, up to isomorphism, this construction exhausts all special parahoric subgroups in ${\mathscr {G}}(\mathcal {K})$ when ${\mathscr {G}}$ is not of type $A_{2\ell }^{(2)}$ , and special but not absolutely special for $A_{2\ell }^{(2)}$ in the sense of [Reference Haines and RicharzHR, §5], as in this case the special fiber of ${\mathscr {G}}$ has a quotient isomorphic to $\mathrm {Sp}_{2\ell }$ .

We can similarly define the smooth group scheme ${\mathscr {T}}:=\mathrm {Res}_{\mathcal {O}/\bar {\mathcal {O}} } (T_{\mathcal {O}})^\sigma $ , which has connected fibers (cf. [Reference Bruhat and TitsBrT, Lemma 4.4.16, Lemma 4.4.8]). Note that, for general almost simple algebraic group G, we can still define ${\mathscr {G}}$ and ${\mathscr {T}}$ , but we need to take the neutral components of $\mathrm {Res}_{\mathcal {O}/\bar {\mathcal {O}} } (G_{\mathcal {O}})^\sigma $ and $\mathrm { Res}_{\mathcal {O}/\bar {\mathcal {O}} } (T_{\mathcal {O}})^\sigma $ , respectively. For convenience, throughout this paper, we only work with G being adjoint or simply-connected.

Let $L^+ {\mathscr {G}}$ denote the jet group and $L{\mathscr {G}}$ be the loop group of ${\mathscr {G}}$ over $\mathbb {C}$ ; that is, for all $\mathbb {C}$ -algebras R, we set $L^+{\mathscr {G}}(R)={\mathscr {G}}( R[[t]])$ and $L{\mathscr {G}}(R)= {\mathscr {G}}(R((t)))$ . We denote by $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ the affine Grassmannian of ${\mathscr {G}}$ , which is defined as the fppf quotient $L{\mathscr {G}}/L^+{\mathscr {G}}$ . In particular, we have

$$\begin{align*}\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}}({\mathbb{C}})= G(\mathcal{K})^\sigma/ G(\mathcal{O} ) ^\sigma. \end{align*}$$

It is known that $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ is a projective ind-variety; cf. [Reference Pappas and RapoportPR, Theorem 1.4]. Following [Reference Pappas and RapoportPR, Reference ZhuZh2], we will call it a twisted affine Grassmannian of ${\mathscr {G}}$ . We can also attach the twisted affine Grassmannian $\operatorname {\mathrm {\mathtt {Gr}}}_{ {\mathscr {T}} }:=L {\mathscr {T}}/L^+ {\mathscr {T}}$ of ${\mathscr {T}}$ . This is a highly non-reduced ind-scheme. Moreover,

$$\begin{align*}\operatorname{\mathrm{\mathtt{Gr}}}_{ {\mathscr{T}} }({\mathbb{C}})= T(\mathcal{K})^\sigma/T(\mathcal{O})^\sigma. \end{align*}$$

For any $\lambda \in X_*(T)$ , we can naturally attach an element $t^\lambda \in T(\mathcal {K})$ . We now define the norm $n^\lambda \in T(\mathcal {K} )^\sigma $ of $t^\lambda $ ,

(2.9) $$ \begin{align} n^\lambda:= \prod_{i=0}^{m-1} \sigma^i(t^\lambda)=\epsilon^{\sum_{i=1}^{m-1} i \sigma^i(\lambda) } t^{\sum_{i=0}^{m-1} \sigma^i(\lambda)}. \end{align} $$

There exists a natural bijection

(2.10) $$ \begin{align} T(\mathcal{K})^\sigma/T(\mathcal{O})^\sigma \simeq X_*(T)_\sigma , \end{align} $$

where $X_*(T)_\sigma $ denotes the set of $\sigma $ -coinvariants in $X_*(T)$ . Any $\bar {\lambda }\in X_*(T)_\sigma $ corresponds to the coset $n^\lambda T( \mathcal {O})^\sigma $ , where $\lambda $ is a representative of $\bar {\lambda }$ . By Theorem [Reference Pappas and RapoportPR, Theorem 0.1], the components of $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }$ can be parametrized by elements in $\pi _1(G)_\sigma $ , where $\pi _1(G)\simeq X_*(T)/\check {Q}$ , and $(X_*(T)/\check {Q})_\sigma $ is the the set of coinvariants of $\sigma $ in $X_*(T)/\check {Q}$ .

When G is of adjoint type, we describe $(X_*(T)/\check {Q})_\sigma $ in the following table.

(2.11)

2.3 Twisted affine Schubert varieties

Let $e_0$ be the base point in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}({\mathbb {C}})$ . For any $\bar {\lambda }\in X_*(T) _\sigma $ , let $e_{\bar {\lambda }}$ denote the point $n^\lambda e_0 \in \operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }( {\mathbb {C}})$ . The point $e_{\bar {\lambda }}$ only depends on $\bar {\lambda }\in X_*(T)_\sigma $ . Let $X_*(T)^+_\sigma $ denote the set of images of $X_*(T)^+$ in $X_*(T)_\sigma $ via the projection $X_*(T)\to X_*(T)_\sigma $ . Then, we have the following Cartan decomposition for $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ (cf. [Reference RicharzRi1, Proposition 2.8]):

(2.12) $$ \begin{align} \operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}}({\mathbb{C}})= \bigsqcup_{\bar{ \lambda} \in X_*(T)^+_\sigma } \operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}}^{\bar{\lambda}} , \end{align} $$

where $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}^{\bar {\lambda }}:= G(\mathcal {O})^\sigma e_{\bar {\lambda }} $ . The Schubert variety $\overline { \operatorname {\mathrm {\mathtt {Gr}}} }_{\mathscr {G}}^{\bar {\lambda }} $ is defined to be the reduced closure of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}^{\bar {\lambda }}$ in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ . Moreover,

$$\begin{align*}\dim \overline{ \operatorname{\mathrm{\mathtt{Gr}}} }_{\mathscr{G}}^{\bar{\lambda}} =2\langle \lambda, \rho \rangle ,\end{align*}$$

where $\rho $ is the sum of all fundamental weights of $\mathfrak {g}$ . It is easy to see that the dimension is independent of the choice of $\lambda $ .

For any $\bar {\lambda },\bar {\mu }\in X_*(T)^+_\sigma $ , we write $\bar {\mu }\preceq \bar {\lambda }$ if $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}^{\bar {\mu }} \subseteq \overline { \operatorname {\mathrm {\mathtt {Gr}}} }_{\mathscr {G}}^{\bar {\lambda }} $ . For any $i\in I$ , let $\overline {\check {\alpha }}_i$ denote the image of $\check {\alpha }_i$ in $X_*(T)_\sigma $ . For any $j\in I_\sigma $ , set

(2.13) $$ \begin{align} \gamma_j =\overline{\check{\alpha}}_i, \quad \text{ if } j=\eta(i). \end{align} $$

It is clear that $\gamma _j $ is well defined.

The following lemma follows from [Reference RicharzRi1, Corollary 2.10].

By the ramified geometric correspondence [Reference ZhuZh3, §1], the set $X_*(T)_\sigma $ can be realized as the weight lattice of the reductive group $H:=(\check {G})^\tau $ , where $\check {G}$ is the Langlands dual group of G and $\tau $ is a diagram automorphism on $\check {G}$ corresponding to the one on G, and $\{ \,\gamma _j \,|\, j\in I_\sigma \,\}$ is the set of simple roots for H. Moreover, $X_*(T)^+_\sigma $ is the set of dominant weights of H, and the partial order $\preceq $ is exactly the standard partial order for dominant weights of H.

We now assume G is of adjoint type. From the perspective of the geometric Satake, we can determine the minimal elements in $ X_*(T)^+_\sigma $ , in other words the minimal Schubert variety in each connected component of $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }$ . From the table (2.11), we see that when $(G, m)=(A_{2\ell -1}, 2 )$ , $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }$ has two components, where $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }^{ \overline { \check {\omega } }_1 } $ is the minimal Schubert variety in the non-neutral component, since $ \overline { \check {\omega } }_1$ gives the minuscule dominant weight of $H\simeq \mathrm {Sp}_{2\ell }$ . When $(G, m)=(D_{\ell +1}, 2 )$ , $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }$ also has two components and $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }^{ \overline { \check {\omega } }_\ell } $ is the minimal Schubert variety in the non-neutral component, since $\overline { \check {\omega } }_\ell $ is the minuscule dominant weight of $H\simeq \mathrm {Spin}_{2\ell +1}$ . Otherwise, $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }$ has only one component. In fact, when $(G, m)=(A_{2\ell }, 4)$ , $H\simeq \mathrm {SO}_{2\ell +1}$ , in which case the lattice $X_*(T)_\sigma $ concides with the root lattice of H.

Let S denote the following set:

(2.14) $$ \begin{align} S=\begin{cases}\{ 0\} \quad \quad \text{ if } (G,r)\not= (A_{2\ell-1}, 2 ), (D_{\ell+1}, 2 ) \\ \{ 0, \check{\omega}_1 \} \quad \quad \text{ if } (G,r)= (A_{2\ell-1}, 2 ) \\ \{0, \check{\omega}_\ell \} \quad\quad \text{ if } (G,r)= (D_{\ell+1}, 2 ) \\ \end{cases}\!\!\!\!\!. \end{align} $$

For any $\kappa \in S$ , let $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} , \kappa }$ be the component of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}} $ containing the Schubert variety $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}} ^{ \bar {\kappa } }$ , or equivalently containing the point $e_{\bar {\kappa }}$ . Then,

$$\begin{align*}\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}} = \sqcup_{\kappa\in S} \operatorname{\mathrm{\mathtt{Gr}}}_{{\mathscr{G}} , \kappa}. \end{align*}$$

2.4 Global affine Grassmannian of parahoric Bruhat-Tits group schemes

Let C be a complex projective line $\mathbb {P}^1$ with a coordinate t, and with the action of $\sigma $ such that $t\mapsto \epsilon t$ . Let $\bar {C}$ be the quotient curve $C/\sigma $ , and let $\pi : C\to \bar {C}$ be the projection map. Then $\bar {C}$ is also isomorphic to $\mathbb {P}^1$ . Let $\mathcal {G}=\operatorname {{\mathrm {Res}}}_{C/\bar {C}}(G \times C)^{\sigma }$ be the group scheme over $\bar {C}$ , which is the $\sigma $ -fixed point subgroup scheme of the Weil restriction $\operatorname {{\mathrm {Res}}}_{C/\bar {C}}(G \times C)$ of the constant group scheme $G\times C$ from C to $\bar {C}$ . Then, $\mathcal {G}$ is a parahoric Bruhat-Tits group scheme over $\bar {C}$ in the sense of Heinloth [Reference HeinlothHe, §1]. Let o (resp. $\bar {o}$ ) be the origin of C (resp. $\bar {C}$ ), and let $\infty $ (resp. $\bar {\infty }$ ) be the infinite point in C (resp. $\bar {C}$ ).

The group scheme $\mathcal {G}$ has the following properties:

1. 1. For any $y\in \bar {C}$ , if $y\not =\bar {o}, \bar {\infty }$ , the fiber $\mathcal {G}_{|_y}$ over y is isomorphic to G; the restriction $\mathcal {G}_y$ to the formal disc $\mathbb {D}_y$ around y is isomorphic to the constant group scheme $G_{ \mathbb {D}_y}$ over $\mathbb {D}_y$ .

2. 2. When $y=\bar {o}$ or $\bar {\infty }$ in $\bar {C}$ , $\mathcal {G}_{|_y}$ has a reductive quotient $G^\sigma $ ; the restriction $\mathcal {G}_y$ to $\mathbb {D}_y$ is isomorphic to the parahoric group scheme ${\mathscr {G}}$ .

Similarly, we can define the parahoric Bruhat-Tits group scheme $\mathcal {T}:= \operatorname {{\mathrm {Res}}}_{C/\bar {C}}(T \times C)^{\sigma }$ .

Given an R-point $p \in C(R)$ , we denote by $\Gamma _p\subset C_R$ the graph of p where $C_R:=C\times \mathrm { Spec} (R)$ , and denote by $\hat {\Gamma }_p$ the formal completion of $C_R$ along $\Gamma _p$ , and let $\hat {\Gamma }^\times _{p} $ be the punctured formal completion along $\Gamma _p$ . Let $\bar {p}$ be the image of p in $\bar {C}$ . We similarly define $\bar {C}_R$ , $\Gamma _{\bar {p}}$ , $\hat {\Gamma }_{\bar {p}}$ and $\hat {\Gamma }_{\bar {p}}^\times $ .

For any $\mathbb {C}$ -algebra R, we define

(2.15) $$ \begin{align} \operatorname{\mathrm{\mathtt{Gr}}}_{\mathcal{G}, C}(R):= \left. \left\{ \, (p, \mathcal{P}, \beta) \, \middle | \, \begin{aligned}[m] & \, p \in C(R) \\ & \mathcal{P} \text{ a } \mathcal{G}\text{-torsor on } \bar{C} \\ & \, \beta: \mathcal{P}|_{\bar{C}_R \setminus {\Gamma}_{\bar{p}}} \simeq \mathring{\mathcal{P}}|_{\bar{C}_R \setminus \Gamma_{\bar{p}} }\\ \end{aligned} \right\} \right. , \end{align} $$

where $\mathring {\mathcal {P}}$ is the trivial $\mathcal {G}$ -bundle.

The functor $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C}$ is represented by an ind-scheme which is ind-proper over C. We call it the global affine Grassmannian $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G},C}$ of $\mathcal {G}$ over C.

For any $p\not =o,\infty \in C$ , the fiber $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, p}:=\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C}|_p$ is isomorphic to the usual affine Grassmannian $\operatorname {\mathrm {\mathtt {Gr}}}_G$ , and the fiber $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, p}$ over $p=o,\infty $ is isomorphic to the twisted affine Grassmannian $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ of the parahoric group scheme ${\mathscr {G}}$ .

We can also define the jet group scheme $L_C^+ \mathcal {G}$ over C as follows,

(2.16) $$ \begin{align} L_C^+ \mathcal{G} (R):= \left. \left\{(p, \gamma) \, \middle| \begin{aligned} & \, p \in C(R) \\ & \, \gamma \text{ is a trivialization of the trivial } \mathcal{G}\text{-torsor on } \bar{C} \text{ along } \hat{\Gamma}_{\bar{p}} \\ \end{aligned} \right\} \right. \end{align} $$

Again, $L_C^+ \mathcal {G}$ is the base change of the usual jet group scheme $L^+ \mathcal {G}$ of $\mathcal {G}$ along $\pi : C\to \bar {C}$ . For any $p\not =o,\infty \in C$ , the fiber $L_C^+ \mathcal {G}|_p$ is isomorphic to the jet group scheme $L^+G$ of G, and the fiber $L_C^+ \mathcal {G}|_p$ over $p=o,\infty $ is isomorphic to jet group scheme $L^+ {\mathscr {G}}$ .

We have a left action of $L_C^+ \mathcal {G}$ on $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C}$ given by

(2.17) $$ \begin{align} ( (p, \gamma), (p, \mathcal{P}, \beta) )\mapsto (p, \mathcal{P}', \beta), \end{align} $$

where $\mathcal {P}'$ is obtained by choosing a trivialization of $\mathcal {P}$ along $\hat {\Gamma }_{\bar {p}}$ and then composing this trivialization with $\gamma $ and regluing with $\beta $ .

We also can define the global loop group $L_C\mathcal {G}$ of $\mathcal {G}$ over C,

(2.18) $$ \begin{align} L_C \mathcal{G} (R):= \left. \left\{(p, \gamma) \,\middle| \begin{aligned} & \, p \in C(R) \\ & \, \gamma \text{ is a trivialization of the trivial } \mathcal{G}\text{-torsor on } \bar{C} \text{ along } \hat{\Gamma}_{\bar{p}}^\times \\ \end{aligned} \right\} \right.. \end{align} $$

Then $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G} , C}$ is isomorphic to the fppf quotient $L_C{\mathscr {G}} /L_C^+{\mathscr {G}} $ . We can also define $L_C^+ \mathcal {T}$ and $L _C\mathcal {T}$ similarly. Then,

$$\begin{align*}L_C \mathcal{T}|_p\simeq \begin{cases} T_{\mathcal{K}_p } \quad \text{ if } p\not=o ,\infty \\ {\mathscr{T}} \quad \text{ if } p=o, \infty \end{cases} \!\!\!\!\!, \end{align*}$$

where $\mathcal {K}_p$ is the field of formal Laurant series of C at p.

2.5 Global Schubert varieties

For each $p\in C$ , we can attach a lattice $X_*(T)_p$ ,

$$\begin{align*}X_*(T)_p=\begin{cases} X_*(T) \quad \text{ if } p\not=o ,\infty \\ X_*(T)_\sigma \quad \text{ if } p=o, \infty \end{cases}\!\!\!\!\!. \end{align*}$$

By [Reference ZhuZh2, Proposition 3.4], for any $\lambda \in X_*(T)$ , there exists a section $s^\lambda : C\to L_C\mathcal {T}$ , such that for any $p\in C$ , the image of $s^\lambda (p)$ in $X_*(T)_p$ is given by

$$\begin{align*}\begin{cases} \lambda\in X_*(T) \quad \text{ if } p\not=o,\infty \\ \bar{\lambda}\in X_*(T)_\sigma \quad \text{ if } p=o,\infty \end{cases}\!\!\!\!\!. \end{align*}$$

This naturally gives rise to C-points in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {T}, C}$ and $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G},C}$ , which will still be denoted by $s^\lambda $ . Following [Reference ZhuZh2, Definition 3.1], for each $\lambda \in X_*(T)$ , we define the global Schubert variety $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G},C}^\lambda $ to be the minimal $L_C^+\mathcal {G}$ -stable irreducible closed subvariety of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G},C }$ that contains $s^\lambda $ . Then, [Reference ZhuZh2, Theorem 3] asserts the following.

Theorem 2.4. The global Schubert variety $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G},C}^\lambda $ is flat over C, and for any $p\in C$ , the fiber $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G},p}^\lambda $ is reduced and

$$\begin{align*}\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathcal{G},p}^\lambda\simeq \begin{cases} \overline{\operatorname{\mathrm{\mathtt{Gr}}}}^\lambda_G \quad \text{ if } p\not=o,\infty \\ \overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda} } \quad \text{ if } p=o,\infty \end{cases}\!\!\!\!\!. \end{align*}$$

3.1 Borel-Weil-Bott theorem on $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$

Let $\hat {L}(\mathfrak {g},\sigma ):=\mathfrak {g}(\mathcal {K})^\sigma \oplus \mathbb {C} K$ be the twisted affine algebra as a central extension of the twisted loop algebra $\mathfrak {g}(\mathcal {K})^\sigma $ with the canonical center K, whose Lie bracket is defined as follows:

(3.1) $$ \begin{align} [x[f]+z K, x'[f'] +z' K] = [x,x' ][ff'] +m^{-1}\mathrm{Res}_{t=0} \, \bigl (({df}) f'\bigr) ( x,x') K, \end{align} $$

for $x[f],x'[f']\in \mathfrak {g}(\mathcal {K})^\sigma $ where $x,x'\in \mathfrak {g}$ , $f,f'\in \mathcal {K}$ , and $z, z'\in \mathbb {C}$ , and where $\mathrm {Res}_{t=0}$ denotes the coefficient of $t^{-1}dt$ , and $(,)$ is the normalized Killing form on $\mathfrak {g}$ (i.e., $(\check {\theta }, \check {\theta } )=2$ ).

We use $P( \sigma ,c)$ to denote the set of highest weights of $\mathfrak {g}^\sigma $ which parametrizes the integrable highest weight modules of $\hat {L}(\mathfrak {g}, \sigma )$ of level c; see [Reference Hong and KumarHK, Section 2]. For each $\lambda \in P( \sigma ,c)$ , we denote by $\mathscr {H}_{ c}(\lambda ) $ the associated integrable highest weight module of $\hat {L}(\mathfrak {g}, \sigma )$ .

Recall that $\{ \lambda _i \,|\, i\in I_\sigma \}$ is the set of fundamental weights of $\mathfrak {g}^\sigma $ , where we follow the labellings in (2.4). Also, $\{ \, \check {\beta }_i \,|\, i\in I_\sigma \, \}$ is the set of simple coroots of $\mathfrak {g}^\sigma $ .

Lemma 3.1. For a standard automorphism $\sigma $ , we have

$$\begin{align*}P(\sigma,1)=\begin{cases}\{ 0\} \quad \quad \text{ if } (\mathfrak{g},m)\not= (A_{2\ell-1}, 2 ), (D_{\ell+1}, 2 ) \\ \{ 0, \lambda_1 \} \quad \quad \text{ if } (\mathfrak{g},m)= (A_{2\ell-1}, 2 ) \\ \{0, \lambda_\ell \} \quad\quad \text{ if } (\mathfrak{g},m)= (D_{\ell+1}, 2 ) \\ \end{cases}\!\!\!\!\!.\end{align*}$$

Proof. We first consider the case when $(\mathfrak {g},m)\not = (A_{2\ell }, 4 )$ . We can read from [Reference Hong and KumarHK, Lemma 2.1], for any $\lambda \in (\mathfrak {h}^\sigma )^\vee $ , $\lambda \in P(\sigma ,1)$ if and only if

$$\begin{align*}\langle \lambda, \check{\beta}_i \rangle\in \mathbb{Z}_{\geq 0} \quad \text{ for any } i\in I_\sigma , \end{align*}$$

and $\langle \lambda , \check {\theta }_0 \rangle \leq 1$ , where $\theta _0$ is the highest short root of $\mathfrak {g}^\sigma $ and $\check {\theta }_0$ is the coroot of $\theta _0$ , and hence, $\check {\theta }_0$ is the highest coroot of $\mathfrak {g}^\sigma $ . In this case, $\lambda \in P( \sigma ,1)$ if and only if $\lambda =0$ or a minuscule dominant weight of $\mathfrak {g}^\sigma $ (cf. [Reference Besson and HongBH, Lemma 2.13]). Following the labellings in [Reference KacKa, Table Fin, p53], when $\mathfrak {g}^\sigma $ is of type $C_\ell $ , $\lambda _1$ is the only minuscule weight; when $\mathfrak {g}^\sigma $ is of type $B_\ell $ , $\lambda _\ell $ is the only minuscule weight. Any other nonsimply-laced Lie algebra has no minuscule weight. This finishes the argument of the lemma when $(\mathfrak {g},m)\not = (A_{2\ell }, 4 )$ .

Now, we assume that $(\mathfrak {g},m)= (A_{2\ell }, 4 )$ . In this case, it is more convenient to choose a different set of simple roots for $\mathfrak {g}^\sigma $ rather than the one described in (2.4). Namely, we can choose

$$\begin{align*}\{ \alpha_i|_{\mathfrak{h}^\sigma} \,|\, i=1,2,\cdots, \ell-1 \}\cup \{ -\theta|_{\mathfrak{h}^\sigma} \} \end{align*}$$

as a set of simple roots of $\mathfrak {g}^\sigma $ . With this set of simple roots, we can also read from [Reference Hong and KumarHK, Lemma 2.1], for any $\lambda \in (\mathfrak {h}^\sigma )^\vee $ , $\lambda \in P(\sigma ,1)$ if and only if $\lambda =0$ .

We define the following map:

(3.2) $$ \begin{align} \iota: X_*(T) \to (\mathfrak{h}^\sigma )^\vee , \end{align} $$

such that for any ${\lambda }\in X_*(T) $ , $\iota ({\lambda })(h)=(\lambda , h)$ , where we regard $\lambda $ as an element in $\mathfrak {h}$ and $(,)$ is the normalized Killing form on $\mathfrak {h}$ . It is clear that $\iota (0)=0$ . This map naturally descends to a map $X_*(T)_\sigma \to (\mathfrak {h}^\sigma )^\vee $ . By abuse of notation, we still call it $\iota $ .

Recall some terminology introduced in Section 2.1. $I_\sigma $ is the set parametrizing simple roots of $\mathfrak {g}^\sigma $ , and we also defined a map $\eta : I\to I_\sigma $ . The set $\{ \check {\lambda }_j \,|\, j\in I_\sigma \}$ is the set of fundamental coweights of $\mathfrak {g}^\sigma $ , and $\{ \lambda _j \,|\, j\in I_\sigma \}$ is the set of fundamental weights of $\mathfrak {g}^\sigma $ . We also recall that $\check {\alpha }_i$ is a simple coroot of $\mathfrak {g}$ for each $i\in I$ , and $\gamma _j$ is the image of $\check {\alpha }_i$ in $X_*(T)_\sigma $ . The following lemma already appears in [Reference HainesHa, Lemma 3.2] in a slighly different setting.

Lemma 3.3. For any $j\in I_\sigma $ , we have

$$\begin{align*}\iota( \gamma_j ) = \begin{cases} \beta_j , \quad \text{ if } (\mathfrak{g}, m)\not= (A_{2\ell}, 4), \text{ or }, (\mathfrak{g}, m)= (A_{2\ell}, 4) \text{ and } j\not= \ell \\ \frac{1}{2} \beta_\ell, \quad \text{ if } (\mathfrak{g}, m)= (A_{2\ell}, 4) \text{ and } j= \ell \end{cases}. \end{align*}$$

Proof. By the definition of $\iota $ , for any $\gamma _j=\overline {\check {\alpha }}_i$ with $j=\eta (i)$ , and $k\in I_\sigma $ , we have the following equalities:

$$\begin{align*}\langle \check{\lambda}_k, \gamma_j \rangle= \langle \check{\lambda}_k, \iota( \overline{\check{\alpha}}_i ) \rangle=( \check{\lambda}_k, \check{\alpha}_i )=\langle\check{\lambda}_k, \alpha_i \rangle .\end{align*}$$

Then, this lemma readily follows from the description of fundamental coweights of $\mathfrak {g}^\sigma $ in (2.7) and (2.8).

Recall the set S defined in (2.14).

Proof. For any $i\in I$ and $j\in I_\sigma $ , we have

$$\begin{align*}\langle \iota(\check{\omega }_i), \check{\beta}_j \rangle =\left( \check{\omega}_i, \check{\beta}_j \right)=( \check{\omega}_i, \sum_{a\in \eta^{-1}(i) } \check{\alpha}_a )= \delta_{\eta(i) ,j } .\end{align*}$$

Hence, $\iota (\check {\omega }_i)=\lambda _{\eta (i)}$ .

In view of Lemma 3.1, $\iota $ maps S bijectively into $P(\sigma ,1)$ .

For any $g\in G(\mathcal {K})^\sigma $ , we can define a Lie algebra automorphism

(3.3) $$ \begin{align} \widehat{ \mathrm{Ad} }_{ g} (x[f]):= \mathrm{Ad}_{g}(x[f]) + \frac{1}{m} \mathrm{Res}_{t=0} ( g^{-1}dg, x[f] ) K , \end{align} $$

for any $x[f]\in \mathfrak {g}(\mathcal {K})^\sigma $ , where $(,)$ is the normalized Killing form on $\mathfrak {g}$ . By Lemma 3.4, $\iota (\kappa )\in P(\sigma , 1)$ for any $\kappa \in S$ . Thus, $c\iota (\kappa )\in P(\sigma , c)$ for any level $c\geq 1$ .

Set

(3.4) $$ \begin{align} \mathscr{H}_c := \oplus_{\kappa\in S} \mathscr{H}_c(c \iota(\kappa)).\end{align} $$

Let $\tilde {\mathfrak {g}}:= \mathfrak {g}\otimes \mathcal {K}\oplus \mathbb {C} K'\oplus \mathbb {C} d' $ be the untwisted Kac-Moody algebra associated to $\mathfrak {g}$ , where $K'$ is the canonical center and $d'$ is the scaling element. We may define an automorphism $\sigma $ on $\tilde {\mathfrak {g}}$ as follows:

$$\begin{align*}\sigma( x[f(t)] )=\sigma(x)[f(\epsilon t )], \quad \sigma(K')=K', \quad \sigma(d')=d', \end{align*}$$

for any $x[f]\in \mathfrak {g}\otimes \mathcal {K}$ . Then the fixed point Lie algebra $\tilde {\mathfrak {g}}^{\sigma }$ is exactly the twisted Kac-Moody alegbra $\tilde {L}(\mathfrak {g}, \sigma )$ containing $\hat {L}(\mathfrak {g}, \sigma )$ as the derived algebra. Following from [Reference KacKa, Theorem 8.7,§8], in this realization, the canonical center K in $\tilde {L}(\mathfrak {g}, \sigma )$ is equal to $mK'$ , and the scaling element d in $\tilde {\mathfrak {g}}$ is equal to $d'$ when $\tilde {\mathfrak {g}}^{\sigma }$ is not $A_{2\ell }^{(2)}$ , and $d=2d'$ when $\tilde {\mathfrak {g}}^{\sigma }=A_{2\ell }^{(2)}$ .

For any $g\in G(\mathcal {K})$ , one can define an automorphism $\widehat {\mathrm {Ad}}_g$ on $\tilde {\mathfrak {g}}$ as in [Reference KumarKu, Section 13.2.3]. From the formula $\textit {loc.cit}$ , it is clear that if $g\in G(\mathcal {K})^{\sigma }$ , then $\widehat {\mathrm {Ad}}_g$ commutes with $\sigma $ . In particular, it follows that $\widehat {\mathrm {Ad}}_g$ restricts to an automorphism on $\tilde {L}(\mathfrak {g}, \sigma )$ . One may observe easily that, restricting further to $\hat {L}(\mathfrak {g}, \sigma )$ , this is exactly the automorphism defined in (3.3).

By demanding that $d\cdot v_\kappa =0$ for each $\kappa \in S$ , the action $\hat {L}(\mathfrak {g},\sigma )$ on $\mathscr {H}$ extends uniquely to an action of $\tilde {L}(\mathfrak {g},\sigma )$ .

Lemma 3.6. For any $g\in G(\mathcal {K})^\sigma $ , there exists an intertwining operator $\rho _g: \mathscr {H}_c\simeq \mathscr {H}_c$ such that

(3.5) $$ \begin{align} \rho_g( x[f]\cdot v )= \widehat{ \mathrm{Ad} }_{ g} (x[f]) \cdot \rho_g( v ), \end{align} $$

for any $x[f]\in \mathfrak {g}(\mathcal {K})^\sigma $ and $v\in \mathscr {H}_c$ . In particular, for any $\kappa \in S$ ,

(3.6) $$ \begin{align} \widehat{ \mathrm{Ad} }_{n^{-\kappa}}( \mathscr{H}_c(0) )= \mathscr{H}_c(c\iota(\kappa)), \, \text{ and } \, \widehat{ \mathrm{Ad} }_{n^{-\kappa}}( \mathscr{H}_c(c\iota(\kappa)) ) = \mathscr{H}_c(0). \end{align} $$

Proof. Let $G'$ be the simply-connected cover of G, and let $p: G'(\mathcal {K})^\sigma \to G(\mathcal {K})^\sigma $ be the induced map. Then,

(3.7) $$ \begin{align} G(\mathcal{K})^\sigma=\sqcup_{\kappa \in S} n^{-\kappa} \overline{G'(\mathcal{K})^\sigma } , \end{align} $$

where $\overline {G'(\mathcal {K})^\sigma }=p(G'(\mathcal {K})^\sigma )$ . By twisted analogue of Faltings Lemma (cf. [Reference Hong and KumarHK, Proposition 10.2]), for any element $g\in \overline {G'(\mathcal {K})^\sigma }$ , there exists an operator $\rho _g$ which maps $ \mathscr {H}_c(c\iota (\kappa ))$ to $\mathscr {H}_c(c\iota (\kappa ))$ with the desired property (3.5), for any $\kappa \in S$ . By decomposition (3.7), it suffices to show that, for nonzero $\kappa $ , $n^{-\kappa }$ satisfies property (3.6).

Assume $\kappa \not =0$ in S. From the table (2.11), the group $(X_*(T)/{\check {Q}})_\sigma $ is at most of order $2$ . Therefore, $n^{-2\kappa }\in \overline {G'(\mathcal {K})^\sigma } $ . For each $\mathscr {H}_c(c\iota (\kappa ))$ , we denote the action by $\pi _{c, \kappa }: \hat {L}(\mathfrak {g},\sigma )\to \mathrm {End}( \mathscr {H}_c(c\iota (\kappa )) ) $ . Then the property (3.5) for $n^{-2\kappa }$ is equivalent to the existence of an isomorphism of representations,

(3.8) $$ \begin{align} \rho_{n^{-2\kappa}}: (\mathscr{H}_c(c\iota(\kappa) ), \pi_{c,\kappa}) \simeq (\mathscr{H}_c(c\iota(\kappa) ), \pi_{c, \kappa} \circ \widehat{ \mathrm{Ad} }_{n^{-2\kappa}} ) .\end{align} $$

Let $v_\kappa $ be the highest weight vector in $\mathscr {H}_c(c\iota (\kappa ))$ . Then $v_\kappa $ is of $\mathfrak {h}^\sigma $ -weight $c\iota (\kappa )$ . We regard $\check {\beta }_i$ as elements in $\mathfrak {h}^\sigma $ . By formula (3.3),

$$\begin{align*}\widehat{\mathrm{Ad}}_{n^{-\kappa}}( \check{\beta}_i ) = \check{\beta}_i - (\kappa, \check{\beta}_i ) c= \check{\beta}_i - \langle \iota(\kappa), \check{\beta}_i \rangle c. \end{align*}$$

Hence, $v_\kappa $ is of $\mathfrak {h}^\sigma $ -weight 0 and a highest weight vector in the representation

$$\begin{align*}(\mathscr{H}_c(c\iota(\kappa) ), \pi_{c, \kappa} \circ \widehat{ \mathrm{Ad} }_{n^{-\kappa}} ) .\end{align*}$$

By Schur lemma, there exists an intertwining operator $\rho _{0 \kappa }$ ,

(3.9) $$ \begin{align} \rho_{0 \kappa}: (\mathscr{H}_c(0), \pi_{c, 0} ) \simeq (\mathscr{H}_c(c\iota(\kappa) ), \pi_{c, \kappa} \circ \widehat{ \mathrm{Ad} }_{n^{-\kappa}} ). \end{align} $$

We also can regard $\rho _{0 \kappa }$ as the following intertwining operator

(3.10) $$ \begin{align} \rho_{0 \kappa}: (\mathscr{H}_c(0), \pi_{c, 0} \circ \widehat{ \mathrm{Ad} }_{n^{-\kappa}} ) \simeq (\mathscr{H}_c(c\iota(\kappa) ), \pi_{c, \kappa} \circ \widehat{ \mathrm{Ad} }_{n^{-2\kappa}} ). \end{align} $$

Combining isomorphisms (3.8),(3.10), we get

$$\begin{align*}(\mathscr{H}_c(c\iota(\kappa)) ,\pi_{c,\kappa}) \xrightarrow{ \rho_{n^{-2\kappa}} } (\mathscr{H}_c(c\iota(\kappa) ) , \pi_{c, \kappa} \circ \widehat{ \mathrm{Ad} }_{n^{-2\kappa}} )\xrightarrow{ (\rho_{0 \kappa} )^{-1} } (\mathscr{H}_{c}(0) , \pi_{c, 0} \circ \widehat{ \mathrm{Ad} }_{n^{-\kappa}} ). \end{align*}$$

We define $ \rho _{n^{-\kappa }}$ to be the following operator:

$$\begin{align*}\rho_{n^{-\kappa}}=(\rho_{0 \kappa}, (\rho_{0 \kappa} )^{-1} \circ \rho_{n^{-2\kappa}} ): \mathscr{H}_c(0) \oplus \mathscr{H}_c(c\iota(\kappa) )\simeq \mathscr{H}_c(0 ) \oplus \mathscr{H}_c(c\iota(\kappa)). \end{align*}$$

The map $\rho _{n^{-\kappa }}$ satisfies property (3.5).

As discussed in Section 2.2, the components of $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }$ are parametrized by elements in $(X_*(T)/{\check {Q}})_\sigma $ . Moreover, $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}} = \sqcup _{\kappa \in S} \operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} , \kappa } $ , where S is defined in (2.14).

Let ${\mathscr {G}}'$ be the parahoric group scheme $\mathrm {Res}_{\mathcal {O}/\bar {\mathcal {O} }}(G^{\prime }_{\mathcal {O}})^\sigma $ ( $G'$ is the simply-connected cover of G), and let $L^+{\mathscr {G}}'$ (resp. $L{\mathscr {G}}' $ ) denote the jet group scheme (resp. loop group scheme) of ${\mathscr {G}}'$ . The group $L{\mathscr {G}}$ acts on $L{\mathscr {G}}'$ by conjugation. Set

$$\begin{align*}L^+{\mathscr{G}}^{\prime}_\kappa:=\mathrm{Ad}_{n^{-\kappa}}( L^+{\mathscr{G}} ').\end{align*}$$

Then, $L^+{\mathscr {G}} ^{\prime }_\kappa $ is a subgroup scheme of $L{\mathscr {G}} '$ . We have

(3.11) $$ \begin{align} \operatorname{\mathrm{\mathtt{Gr}}}_{{\mathscr{G}} , \kappa}\simeq L{\mathscr{G}} '/L^+{\mathscr{G}} _\kappa'. \end{align} $$

By the twisted analogue of Faltings lemma (cf. [Reference Hong and KumarHK, Proposition 10.2]), there exists a group homomorphism $ L {{\mathscr {G}}'} \to \mathrm {PGL} ( \mathscr {H}_1(0) ) $ . Consider the central extension

(3.12) $$ \begin{align} 1\to \mathbb{G}_m \to \mathrm{GL}( \mathscr{H}_1(0) ) \to \mathrm{PGL} ( \mathscr{H}_1(0) ) \to 1. \end{align} $$

The pull-back of (3.12) to $L{ {\mathscr {G}}}'$ defines the following canonical central extension of $L{\mathscr {G}}'$ :

(3.13) $$ \begin{align} 1\to \mathbb{G}_m \to \widehat{L { {\mathscr{G}}' } } \to L{{\mathscr{G}}'} \to 1. \end{align} $$

It is known that $\widehat {L{{\mathscr {G}}'} }$ is a Kac-Moody group of twisted type (up to a scaling multiplicative group) in the sense of Kumar and Mathieu; see [Reference Pappas and RapoportPR, §9f]. Let $\widehat {L^+{\mathscr {G}} ^{\prime }_\kappa }$ denote the preimage of $L^+{\mathscr {G}} ^{\prime }_\kappa $ in $\widehat {L{{\mathscr {G}}'} }$ via the projection map $\widehat {L{{\mathscr {G}}'} } \to L{{\mathscr {G}}'} $ . As the same proof as in [Reference Besson and HongBH, Lemma 2.19], $\widehat {L^+{\mathscr {G}} ^{\prime }_\kappa } $ is a parabolic subgroup in $\widehat {L{{\mathscr {G}}'} }$ ; moreover,

(3.14) $$ \begin{align} \operatorname{\mathrm{\mathtt{Gr}}}_{{\mathscr{G}} , \kappa} \simeq \widehat{L{{\mathscr{G}}'} }/ \widehat{L^+{\mathscr{G}} ^{\prime}_\kappa } \end{align} $$

(i.e., $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} , \kappa } $ is a partial flag variety of the Kac-Moody group $\widehat {L{{\mathscr {G}}'} }$ ).

The line bundle $\mathscr {L}$ on $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}} }$ naturally has a $\widehat {L{{\mathscr {G}}'} }$ -equivariant structure, since $\mathscr {L}$ admits a unique $\widehat {L{{\mathscr {G}}'} }$ -equivariant structure on each component of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ as a partial flag variety of $\widehat {L{{\mathscr {G}}'} }$ .

Let $v_0$ be the highest weight vector in $\mathscr {H}_c $ . For any $\bar {\lambda }\in X_*(T)_\sigma $ , we define

(3.15) $$ \begin{align} v_{\bar{\lambda}}:= \rho_{n^{\lambda} }(v_0) , \end{align} $$

where $ \rho _{n^{\lambda } }$ in defined in Lemma 3.6. Then $ v_{\bar {\lambda }}$ is independent of the choice of the representative $\lambda $ in $X_*(T)$ and is well defined up to a nonzero scalar.

Proof. For any $h\in \mathfrak {h}^\sigma $ , by Lemma 3.6,

$$\begin{align*}h\cdot v_{\bar{\lambda}}= h\cdot \rho_{n^\lambda} (v_0)= \rho_{n^\lambda} ( \widehat{ \mathrm{Ad} }_{ n^{-\lambda}} ( h ) v_0 ). \end{align*}$$

By the formula (3.3), we have

$$\begin{align*}\widehat{ \mathrm{Ad} }_{ n^{-\lambda}} ( h ) = h-\langle \lambda, h \rangle K. \end{align*}$$

It follows that

$$\begin{align*}h\cdot v_{\bar{\lambda}}= - \langle \lambda, h\rangle c v_{\bar{\lambda}}= -c\iota(\lambda)(h)v_{\bar{\lambda}} .\end{align*}$$

This concludes the proof of the lemma.

Definition 3.10. For any dominant $\bar {\lambda }\in X_*(T)^+_\sigma $ , we define the twisted affine Demazure module $D(c, \bar {\lambda })$ as the following $\mathfrak {g}[t]^\sigma $ -module,

$$\begin{align*}D(c, \bar{\lambda}):=U(\mathfrak{g}[t]^\sigma)v_{\bar{\lambda}}. \end{align*}$$

In view of Lemma 3.9, $D(c, \bar {\lambda })$ contains an irreducible representation $V({-c\iota (\lambda )})$ of $\mathfrak {g}^\sigma $ of lowest weight $-c\iota (\lambda )$ . The following theorem follows from [Reference KumarKu, Theorem 8.2.2 (a)].

3.2 Construction of level one line bundles on $\mathrm {Bun}_{\mathcal {G}}$

In this subsection, we consider the parahoric Bruhat-Tits group scheme $\mathcal {G}:= \mathrm {Res}_{C / \bar {C} }(G\times C )^\Gamma $ over $\bar {C}$ as in the setting of Section 2.4.

Let $\mathrm {Bun}_{\mathcal {G}}$ be the moduli stack of $\mathcal {G}$ -torsors on $\bar {C}$ . It is known that $\mathrm {Bun}_{\mathcal {G}}$ is a smooth Artin stack (cf. [Reference HeinlothHe, Theorem 1]). By [Reference HeinlothHe, Theorem 3], the Picard group $ \mathrm {Pic}(\mathrm {Bun}_{\mathcal {G}} ) $ of $\mathrm {Bun}_{\mathcal {G}}$ is isomorphic to $\mathbb {Z}$ , since the group $X^*(\mathcal {G}|_{y})$ of characters for $\mathcal {G}|_{y}$ is trivial for any $y\in \bar {C}$ . In this subsection, we will construct the ample generator $\mathcal {L}\in \mathrm {Pic}(\mathrm {Bun}_{\mathcal {G}} ) $ when G is simply-connected, and we will construct a level one line bundle on every component of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C }$ when G is of adjoint type.

By Lemma 3.1, we have $0\in P(\sigma ,1)$ for any standard automorphism $\sigma $ . Recall that $\mathscr {H}_1(0)$ is the basic representation of level one associated to $0\in P(\sigma ,1)$ .

We now define the following space of twisted covacua of level one,

(3.16) $$ \begin{align} \mathscr{V}_{C, \sigma }( 0):= \frac{ \mathscr{H}_{1}(0) } { \mathfrak{g}[ t^{-1} ]^\sigma \cdot \mathscr{H}_{1} (0) } , \end{align} $$

where $\mathfrak {g}[ t^{-1} ]^\sigma $ is the Lie subalgebra of $\hat {L}(\mathfrak {g},\sigma )$ .

Proof. Let $v_0$ be the highest weight vector in $\mathscr {H}_{1} (0)$ . Then

$$\begin{align*}\mathscr{H}_{1}(0)=U( (t^{-1}\mathfrak{g}[t^{-1}] )^\sigma ) \cdot v_0 =U( (t^{-1}\mathfrak{g}[t^{-1}] )^\sigma ) (t^{-1}\mathfrak{g}[t^{-1}] )^\sigma v_0\oplus \mathbb{C} v_0 ,\end{align*}$$

where $U( (t^{-1}\mathfrak {g}[t^{-1}] )^\sigma ) $ denotes the universal enveloping algebra of $(t^{-1}\mathfrak {g}[t^{-1}] )^\sigma $ . We can write $\mathfrak {g}[t^{-1}]^\sigma =\mathfrak {g}^\sigma \oplus (t^{-1}\mathfrak {g}[t^{-1}] )^\sigma $ . Hence,

$$ \begin{align*} \mathfrak{g}[t^{-1}]^\sigma\cdot \mathscr{H}_{1}(0) &= \mathfrak{g}^\sigma\cdot U( (t^{-1}\mathfrak{g}[t^{-1}] )^\sigma ) (t^{-1}\mathfrak{g}[t^{-1}] )^\sigma v_0 + U( (t^{-1}\mathfrak{g}[t^{-1}] )^\sigma ) (t^{-1}\mathfrak{g}[t^{-1}] )^\sigma v_0 \\ &= U( (t^{-1}\mathfrak{g}[t^{-1}] )^\sigma ) (t^{-1}\mathfrak{g}[t^{-1}] )^\sigma v_0 , \end{align*} $$

where the first equality holds since $\mathfrak {g}^\sigma \cdot v_0=0$ , and the second equality holds since $\mathfrak {g}^\sigma $ normalizes $(t^{-1}\mathfrak {g}[t^{-1}] )^\sigma $ under the Lie bracket. Therefore, $\dim \mathscr {V}_{C, \sigma }( 0)=1$ .

Let $G' $ be the simply-connected cover of G. Recall the Heinloth uniformization theorem for ${ \mathcal {G}'} := \mathrm {Res}_{C/ \bar {C} }(G' \times C)^\Gamma $ over the affine line $\bar {C}\backslash \bar {o}$ (cf. [Reference HeinlothHe]),

$$\begin{align*}\mathrm{Bun}_{{ \mathcal{G}'} } \simeq G' [t^{-1}]^\sigma \backslash \operatorname{\mathrm{\mathtt{Gr}}}_{{ {\mathscr{G}} '} } ,\end{align*}$$

where $ \operatorname {\mathrm {\mathtt {Gr}}}_{{ {\mathscr {G}} '} }$ denotes the affine Grassmannian of ${\mathscr {G}} ':= \mathrm {Res}_{ \mathcal {O}/ \bar { \mathcal {O} } } (G^{\prime }_{\mathcal {O}})^\sigma $ , and $ G'[t^{-1}]^\sigma \backslash \operatorname {\mathrm {\mathtt {Gr}}}_{ { {\mathscr {G}}'} }$ denotes the fppf quotient.

We consider the projection map $\mathrm {pr}: \operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}', C }\to \mathrm {Bun}_{\mathcal {G}'} $ . By abuse of notation, we still denote by $\mathcal {L}$ the line bundle on $ \operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}', C }$ pulling-back from $\mathcal {L}$ on $\mathrm {Bun}_{\mathcal {G}'} $ .

The following theorem is interesting by itself but will not be used in this paper.

Theorem 3.15. There is a natural isomorphism

$$\begin{align*}H^0(\mathrm{Bun}_{\mathcal{G}'}, \mathcal{L} )\simeq \mathscr{V}_{C, \sigma }( 0)^\vee, \end{align*}$$

where $\mathscr {V}_{C, \sigma }( 0)^\vee $ denotes the dual of $\mathscr {V}_{C, \sigma }( 0)$ . In particular,

$$\begin{align*}\dim H^0(\mathrm{Bun}_{\mathcal{G}'}, \mathcal{L} )=1.\end{align*}$$

Now, we would like to construct the line bundle $\cal {L}$ of level one on $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G} , C}$ , where $\mathcal {G} =\mathrm {Res}_{C/\bar {C} }(G_C)^\sigma $ with G of adjoint type.

Proof. Set $\mathring {C}=C\backslash \{o, \infty \}$ . Then, $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, \mathring {C}}:=\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C}|_{\mathring {C}} \simeq \operatorname {\mathrm {\mathtt {Gr}}}_G\times \mathring {C}$ . Let M be the set consisting of $0$ and miniscule coweights of G. Then, components of $\operatorname {\mathrm {\mathtt {Gr}}}_G$ can be parametrized by $M\simeq X_*(T)/\check {Q}$ . For each $\kappa \in M$ , let $\operatorname {\mathrm {\mathtt {Gr}}}_{G, \kappa }$ denote the component of $\operatorname {\mathrm {\mathtt {Gr}}}_G$ containing $e_{\kappa }$ . Let $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C, \kappa }$ be the closure of $\operatorname {\mathrm {\mathtt {Gr}}}_{G,\kappa } \times \mathring {C}$ in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C}$ . We call $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C, \kappa }$ a $\kappa $ -component of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C}$ .

The neutral component $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C, 0}$ is naturally isomorphic to $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}'}$ (cf. [Reference Hong and YuHY, Lemma 5.16]). Thus, by Theorem 3.13, we naturally get the level one line bundle $\mathcal {L}_0$ on the neutral component $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G} , C ,0}$ . For any $\kappa \in M$ , let $s^{\kappa }$ be a C-point in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G} , C} $ as defined in Section 2.5. Then, the translation by $s^\kappa $ gives rise to an isomorphism

$$\begin{align*}\operatorname{\mathrm{\mathtt{Gr}}}_{\mathcal{G}, C, 0} \simeq \operatorname{\mathrm{\mathtt{Gr}}}_{\mathcal{G}, C, \kappa} .\end{align*}$$

Accordingly, the line bundle $\mathcal {L}_0$ can be translated to the level one line bundle $\mathcal {L}_\kappa $ on $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C, \kappa }$ . Note that given any two elements $\kappa ,\kappa '\in M$ such that $\bar {\kappa }=\bar {\kappa '}$ in $(X_*(T)/\check {Q})_\sigma $ , $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C, \kappa } $ and $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C, \kappa '} $ share the same component $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}}, \bar {\kappa }}$ of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ at o and $\infty $ . Then $\mathcal {L}_\kappa $ and $\mathcal {L}_{\kappa '}$ agree on $\operatorname {\mathrm {\mathtt {Gr}}}_{{\mathscr {G}}, \bar {\kappa }}$ as they have the same levels. Thus, $\{ \mathcal {L}_\kappa \}_{\kappa \in M}$ glues to be a line bundle $\mathcal {L}$ on $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C}$ whose restriction to $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, p}$ is of level one, for any $p\in C$ .

4.1 $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}}$ as a fixed-point ind-subscheme of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$

We first recall a theorem in [Reference ZhuZh1, Theorem 1.3.4].

The original proof of this theorem is not correct (communicated to us by Richarz and Zhu independently); also, see [Reference Haines and RicharzHR2, Remark 3.5]. A correct proof can be found in [Reference Haines and RicharzHR2, Proposition 3.4], and a similar proof was known to Zhu earlier.

It is clear that $T^\sigma $ is a subgroup scheme of $L{\mathscr {T}}$ and $L{\mathscr {G}} $ . Hence, there is a natural action of $T^\sigma $ on $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}} $ . We now prove an analogue of Theorem 4.1 in the setting of special parahoric group schemes.

Proof. Let $L^{--}G$ be the ind-group scheme represented by the following functor, for any $\mathbb {C}$ -algebra R,

$$\begin{align*}L^{--}G(R): = \ker (\mathrm{ev}_{\infty}: G(R[t^{-1}]) \to G(R) ) , \end{align*}$$

where $\mathrm {ev}_{\infty }$ is the evaluation map sending $t^{-1}$ to $0$ . Let $L^{--}{\mathscr {G}}$ be the ind-group scheme which represents the following functor, for any $\mathbb {C}$ -algebra R,

(4.1) $$ \begin{align} L^{--}{\mathscr{G}}(R): = \ker (\mathrm{ev}_{\infty}: G(R[t^{-1}])^\sigma \to G(R)^\sigma ). \end{align} $$

We can similarly define $L^{--}T$ and $L^{--}{\mathscr {T}}$ .

By the similar argument as in [Reference ZhuZh4, Lemma 2.3.5] or [Reference Haines and RicharzHR2, Lemma 3.1], we have an open embedding

(4.2) $$ \begin{align} L^{--}{\mathscr{G}}\hookrightarrow \operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}} \end{align} $$

given by $g\mapsto ge_0$ , where $e_0$ is the base point in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ . Let ${I}$ be the Iwahori subgroup of $L^+{\mathscr {G}}$ , which is the preimage of $B^\sigma $ via the evaluation map $\mathrm {ev}: L^+{\mathscr {G}}\to G^\sigma $ for a $\sigma $ -stable Borel subgroup B in G. We have the following decomposition:

(4.3) $$ \begin{align} \operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}}= \bigsqcup_{\bar{ \lambda} \in X_*(T)_\sigma } I e_{\bar{\lambda}}. \end{align} $$

For each $\bar {\lambda }\in X_*(T)_\sigma $ , we choose a representative $\lambda \in X_*(T)$ . The twisted Iwahori Schubert cell

$$\begin{align*}I e_{\bar{\lambda}}=n^{\lambda} \mathrm{Ad}_{ n^{-\lambda} } (I )e_0\end{align*}$$

is contained in $n^{\lambda } L^{--}{\mathscr {G}} e_0$ . Then by the decomposition (4.3), $\bigcup _{\bar {\lambda }\in X_*(T)_* } n^\lambda L^{--} {\mathscr {G}}e_0$ is an open covering of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}} $ . We may naturally regard $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}}$ as an ind-subscheme of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ . Hence, we may regard $e_0$ as the base point in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}}$ . Under this convention,

$$\begin{align*}\bigcup_{\lambda\in X_*(T)_\sigma} n^\lambda L^{--} {\mathscr{T}}e_0 = \bigcup_{\lambda\in X_*(T)_\sigma} L^{--} {\mathscr{T}} n^\lambda e_0 \end{align*}$$

is an open covering of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}}$ . Therefore, it suffices to show that for each $\bar {\lambda }\in X_*(T)_\sigma $ ,

$$\begin{align*}(n^\lambda L^{--} {\mathscr{G}}e_0)^{T^\sigma}\simeq n^\lambda L^{--} {\mathscr{T}}e_0. \end{align*}$$

Further, it suffices to show that $(L^{--} {\mathscr {G}})^{T^\sigma } \simeq L^{--}{\mathscr {T}}$ , where the action of $T^\sigma $ on $L^{--} {\mathscr {G}}$ is by conjugation. From the proof of [Reference Haines and RicharzHR2, Proposition 3.4], one may see that $(L^{--}G)^{T^\sigma }\simeq L^{--}T$ . This actually implies that $(L^{--} {\mathscr {G}})^{T^\sigma } \simeq L^{--}{\mathscr {T}}$ . Hence, this finishes the proof of the theorem.

An immediate consequence of Theorem 4.2 is the following corollary.

4.2 A duality isomorphism for twisted Schubert varieties

Let $\operatorname {\mathrm {\mathtt {Gr}}}_G$ be the affine Grassmannian of G, and let $\mathrm {L}$ be the line bundle on $\operatorname {\mathrm {\mathtt {Gr}}}_G$ that is of level one on every component of $\operatorname {\mathrm {\mathtt {Gr}}}_G$ . For any $\lambda \in X_*(T)$ , let $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_G^\lambda $ denote the closure of $G(\mathcal {O})$ -orbit at $L_\lambda :=t^\lambda G(\mathcal {O})\in \operatorname {\mathrm {\mathtt {Gr}}}_G$ . Let $(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}^\lambda _G)^T$ denote the T-fixed point subscheme of $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_G^\lambda $ . Zhu [Reference ZhuZh1, Theorem 0.2.2] proved the following.

Theorem 4.4. When G is simply-laced and not of type E, the restriction map

$$\begin{align*}H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{{G}}^{{\lambda}}, \mathrm{L}) \rightarrow H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{{G}}^{{\lambda}})^{T}, \mathrm{L}|_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{ {G}}^{{\lambda}})^{T}}) \end{align*}$$

is an isomorphism.

In Section 5, we will show that this theorem also holds for $E_6$ . It was proved by Evens-Mirković [Reference Evens and MirkovićEM, Thereorem 0.1b] and Malkin-Ostrik-Vybornov [Reference Malkin, Ostrik and VybornovMOV, Corollary B] that the smooth locus of $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_G^\lambda $ is the open cell $\operatorname {\mathrm {\mathtt {Gr}}}_G^\lambda $ for any reductive group G. In fact, this theorem can also be deduced from Theorem 4.4 in the simply- laced type.

We will prove a twisted version of Theorem 4.4 in full generality, and as a consequence we get the similar result of Evans-Mirković and Malkin-Ostrik-Vybornov in twisted setting. In particular, this confirms a conjecture of Haines-Richarz [Reference Haines and RicharzHR].

From Theorem 4.2, we have the identification $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}} \xrightarrow {\simeq } \operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}^{T^{\sigma }}$ . Let $\mathscr {I}^{\bar {\lambda }}$ denote the ideal sheaf of the $T^{\sigma }$ -fixed subscheme $(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }})^{T^\sigma }$ of $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }}$ . Then we have a short exact sequence of sheaves

(4.4) $$ \begin{align} 0 \rightarrow \mathscr{I}^{\bar{\lambda}} \rightarrow \mathscr{O}_{\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}} \rightarrow \mathscr{O}_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})^{T^{\sigma}}} \rightarrow 0. \end{align} $$

Recall that $\mathscr {L}$ is the line bundle on $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ which is of level one on every component. Tensoring the above short exact sequence with $\mathscr {L}$ and taking the functor of global sections, we obtain the following exact sequence:

(4.5) $$ \begin{align} \begin{aligned} 0 \rightarrow & H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{I}^{\bar{\lambda}}\otimes \mathscr{L} ) \rightarrow H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{L}) \xrightarrow{r} H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})^{T^{\sigma}}, \mathscr{L}|_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})^{T^{\sigma}}}) \rightarrow \cdots, \end{aligned} \end{align} $$

where r is the restriction map.

Theorem 4.5. For any special parahoric group scheme ${\mathscr {G}}$ induced from a standard automorphism $\sigma $ , the restriction map

$$\begin{align*}H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{L}) \xrightarrow{r} H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})^{T^{\sigma}}, \mathscr{L}|_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})^{T^{\sigma}}}) \end{align*}$$

is an isomorphism, where $\mathscr {L}$ is the level one line bundle on $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ .

This theorem will follow from the following proposition and Lemma 4.8.

Proof. It is well known that any twisted affine Schubert variety $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }}$ is a usual Schubert variety in a partial affine flag variety of Kac-Moody group. See the identification (3.11) and an argument for untwisted case in [Reference Besson and HongBH, Proposition 2.21]. By [Reference KumarKu, Theorem 8.2.2 (d)], we have that for any $\bar {\lambda } \succeq \bar {\mu }$ in $X_*(T)^+_\sigma $ , the following restriction map

(4.6) $$ \begin{align} H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{L}) \rightarrow H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\mu}}, \mathscr{L}) \end{align} $$

is surjective, and

(4.7) $$ \begin{align} H^0(\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}}, \mathscr{L})= \varprojlim H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}^{\bar{\lambda}}_{\mathscr{G}}, \mathscr{L}|_{\overline{\operatorname{\mathrm{\mathtt{Gr}}}}^{\bar{\lambda}}_{\mathscr{G}}}). \end{align} $$

We also have the following surjective map

(4.8) $$ \begin{align} H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})^{T^{\sigma}}, \mathscr{L}) \rightarrow H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\mu}})^{T^{\sigma}}, \mathscr{L}) \end{align} $$

for all $\bar {\lambda } \succeq \bar {\mu }$ , since these $T^\sigma $ -fixed closed subschemes are affine and the morphism is a closed embedding. Moreover,

$$\begin{align*}H^0((\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}})^{T^\sigma} , \mathscr{L}|_{ (\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}})^{T^\sigma} })= \varprojlim H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}^{\bar{\lambda}}_{{\mathscr{G}} })^{T^{\sigma}}, \mathscr{L}|_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}^{\bar{\lambda}}_{\mathscr{G}})^{T^{\sigma}}}).\end{align*}$$

Therefore, for any $\bar {\lambda } \in X_*(T)^+_\sigma $ , we have the following surjective maps:

$$\begin{align*}H^0({\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}, \mathscr{L}) \rightarrow H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{L}), \quad H^0(({\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}})^{T^{\sigma}}, \mathscr{L}) \rightarrow H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})^{T^{\sigma}}, \mathscr{L}). \end{align*}$$

Then to prove the map

$$\begin{align*}H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{L}) \rightarrow H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})^{T^{\sigma}}, \mathscr{L}|_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})^{T^{\sigma}}}) \end{align*}$$

is surjective, it is sufficient to prove that the map

(4.9) $$ \begin{align} H^0(\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}}, \mathscr{L}) \rightarrow H^0( (\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}})^{T^\sigma} , \mathscr{L}|_{ (\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}})^{T^\sigma} }) \end{align} $$

is surjective, since we will have the following commutative diagram, for all $\bar {\lambda }$ :

(4.10)

By Theorem 4.2, we have $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}}\simeq ( \operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}})^{T^\sigma } $ . Therefore, the surjectivity of the map (4.9) follows from the following Lemma 4.7.

We first make a digression on Heisenberg algebras and their representations. Recall that $\mathfrak {h}$ is a fixed Cartan subalgebra in $\mathfrak {g}$ . The subspace $\hat {\mathfrak {h}}^{\sigma }:=( \mathfrak {h}_{\mathcal {K}})^{\sigma } \oplus \mathbb {C} K \hookrightarrow \hat {L}(\mathfrak {g}, \sigma )$ is a Lie subalgebra. In fact, $\hat {\mathfrak {h}}^{\sigma }$ is an extended (completed) Heisenberg algebra with center $\mathfrak {h}^\sigma \oplus \mathbb {C} K$ . Therefore, any integrable irreducible highest weight representation of $\hat {\mathfrak {h}}^{\sigma }$ is parametrized by an element $\mu \in (\mathfrak {h}^\sigma )^\vee $ and the level c (i.e., K acts by the scalar c on this representation). We denote this representation by $ \pi _{\mu , c}$ . By the standard construction,

(4.11) $$ \begin{align} \pi_{\mu, c}=\mathrm{ind}_{(\mathfrak{h}_{\mathcal{O}})^\sigma\oplus \mathbb{C} K }^{\hat{\mathfrak{h}}^{\sigma}} \mathbb{C}_{\mu, c} ,\end{align} $$

where $\mathrm {ind}$ is the induced representation in the sense of universal enveloping algebras, and $ \mathbb {C}_{\mu , c}$ is the 1-dimensional module over $(\mathfrak {h}_{O})^\sigma \oplus \mathbb {C} K$ where the action of $(\mathfrak {h}_{O})^\sigma $ factors through $\mathfrak {h}^\sigma $ .

Proof. Proving surjectivity here is equivalent to proving injectivity for the dual modules,

$$\begin{align*}0 \rightarrow H^0(\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{T}}, \mathscr{L}^c|_{\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{T}}})^\vee \rightarrow H^0(\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}}, \mathscr{L}^c)^\vee. \end{align*}$$

Note that both of these spaces are modules for the Heisenberg algebra $\hat {\mathfrak {h}}^{\sigma }$ ; the morphism is a $\hat {\mathfrak {h}}^{\sigma }$ -morphism. Since ${\mathscr {T}}$ is discrete, we naturally have the following decomposition:

$$\begin{align*}H^0(\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{T}}, \mathscr{L}^c|_{\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{T}}})\simeq \bigoplus_{\bar{\lambda} \in X_*(T)_\sigma } \mathscr{O}_{ \operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{T}}, e_{\bar{\lambda}} }\otimes \mathscr{L}^c|_{ e_{\bar{\lambda} } }, \end{align*}$$

where $\mathscr {O}_{ \operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}}, e_{\bar {\lambda }} }$ is the structure sheaf of the component of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}}$ containing $e_{\bar {\lambda }}$ . We also notice that the identity component of $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}}$ is naturally the formal group with Lie algebra $(\mathfrak {h}_{\mathcal {K}})^\sigma /(\mathfrak {h}_{\mathcal {O}})^\sigma $ . In view of the construction (4.11), we have

$$\begin{align*}H^0(\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{T}}, \mathscr{L}^c|_{\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{T}}})^\vee= \bigoplus_{\bar{\lambda} \in X_*(T)_\sigma } \pi_{-c\iota(\bar{\lambda}) , c },\end{align*}$$

where the map $\iota : X_*(T)_{\sigma } \rightarrow (\mathfrak {h}^\sigma )^\vee $ is defined in (3.2). Since each $\pi _{-c\iota (\bar {\lambda }) , c }$ is irreducible, and generated by a $-c\iota (\bar {\lambda })$ -weight vector $w_{-c\iota (\bar {\lambda })}$ , it suffices to show that the morphism

$$\begin{align*}\pi_{-c\iota(\bar{\lambda}) , c } \rightarrow H^0(\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}}, \mathscr{L}^c)^\vee\end{align*}$$

sends $w_{-c\iota (\bar {\lambda })}$ to a nonzero vector.

By Theorem 3.8, we may define a Plücker embedding

$$\begin{align*}\phi: \operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}} \rightarrow \mathbb{P}( \mathscr{H}_c )\end{align*}$$

given by $ge_0\mapsto [\rho _g ( v_0 )]$ for any $ge_0\in \operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}} $ , where $\rho _g$ is defined in Lemma 3.6, and $ [\rho _g ( v_0) ]$ represents the line in $\mathscr {H}_c $ that contains $ \rho _g ( v_0) $ , where $v_0$ is the highest weight vector in $\mathscr {H}_c$ . Then we may pick a linear form $f_{\bar {\lambda }}$ on $\mathscr {H}_c$ which is nonzero on the line $[v_{\bar {\lambda }}]$ containing the extremal weight vector $v_{\bar {\lambda }}$ , and which is 0 on other weight vectors, where $v_{\bar {\lambda }}$ is defined in (3.15). The restriction $f_{\bar {\lambda }}|_{\phi (\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}})}$ produces a nontrivial element in $H^0(\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}, \mathscr {L})$ , since $\phi (e_{\bar {\lambda }})=v_{\bar {\lambda }}$ .

Observe that the map $ \pi _{-c\iota (\bar {\lambda }) , c } \rightarrow H^0(\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}, \mathscr {L}^c)^\vee $ sends $w_{-c\iota (\bar {\lambda })}$ to a nonzero scalar of $v_{\bar {\lambda } }$ . Thus, the map $ \pi _{-c\iota (\bar {\lambda }) , c } \rightarrow H^0(\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}, \mathscr {L}^c)^\vee $ is nontrivial and thus injective.

By Lemma 4.7, we obtain the following short exact sequence:

$$\begin{align*}0 \rightarrow H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{I}^{\bar{\lambda}}\otimes \mathscr{L}) \rightarrow H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{L}) \xrightarrow{r} H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{L} \otimes \mathscr{O}_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}})}) \rightarrow 0.\end{align*}$$

Thus, the obstruction to the map r being an isomorphism is the vanishing of the first term $ H^0(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }}, \mathscr {I}^{\bar {\lambda }} \otimes \mathscr {L} )$ .

Let $\mathrm {I}^{\lambda }$ denote the ideal sheaf of the T-fixed subscheme on $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_G^{\lambda }$ . We will show that the vanishing of the first term can be deduced from the vanishing of $H^0(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_G^{\lambda }, \mathrm {I}^{\lambda } \otimes \mathrm {L})$ .

Recall that $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C}^{\lambda }$ is a global Schubert variety defined in Section 2.5. The constant group scheme $T^\sigma \times C$ over C is naturally a closed subgroup scheme of $\mathcal {T}$ . Hence, $T^\sigma $ acts on $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C}^{\lambda }$ naturally. Let $(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C}^{\lambda })^{T^\sigma }$ be the $T^\sigma $ -fixed subscheme of $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C}^{\lambda }$ , and let $\mathcal {I}^{\lambda }$ be the ideal sheaf of $(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C}^{\lambda })^{T^\sigma }$ . Then, $\mathcal {I}^\lambda |_p$ is the ideal sheaf of $(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C}^{\lambda }|_{p})^{T^\sigma }$ . Recall that

$$\begin{align*}\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{ {\mathcal{G}},o}^{\lambda} = \overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \quad \overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{ {\mathcal{G}},\infty}^{\lambda}\simeq \overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \quad \overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathcal{G},p \neq o,\infty }^{\lambda} \simeq \overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{G}^{\lambda}.\end{align*}$$

In particular, we have

$$\begin{align*}\mathcal{I}^\lambda|_o= \mathscr{I}^\lambda, \quad \mathcal{I}^\lambda|_\infty\simeq \mathscr{I}^\lambda, \quad \mathcal{I}^\lambda|_{p\not= o,\infty}\simeq \mathrm{I}^\lambda.\end{align*}$$

Proof. Consider $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, {C} \setminus \{o,\infty \}}^{\lambda }$ and the $T^{\sigma }$ -fixed subscheme $(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, {C} \setminus \{ o,\infty \}}^{\lambda })^{T^{\sigma }}$ . We denote by $Z^{\lambda }$ the flat closure of $(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C \setminus \{o,\infty \}}^{\lambda })^{T^{\sigma }}$ in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathcal {G}, C}$ . Since Z is the closure of a $T^{\sigma }$ -fixed subscheme, we see that $Z^{\lambda }|_o \subset \overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\lambda }_{\mathcal {G}, C}|_o$ , and $Z^{\lambda }|_\infty \subset \overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\lambda }_{\mathcal {G}, C}|_\infty .$

To show $\mathcal {I}^\lambda $ is flat over C, it is sufficient to show that $(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C}^{\lambda })^{T^\sigma }$ is flat over C. This is equivalent to showing $Z^\lambda =(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C}^{\lambda })^{T^\sigma }$ . In particular, it suffices to show the fibers $Z^{\lambda }|_o$ and $Z^{\lambda }|_\infty $ are isomorphic to $(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }})^{T^{\sigma }}$ . Since the fiber $Z^{\lambda }|_\infty $ at $\infty $ is similar to the fiber $Z^{\lambda }|_o$ at o, it suffices to show that $Z^{\lambda }|_o=(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }})^{T^{\sigma }}$ . Note that both of these are finite schemes; we can compare the dimensions of their structure sheaves as follows:

$$ \begin{align*} \dim \mathscr{O}_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{{\mathscr{G}} }^{\bar{\lambda}})^{T^{\sigma}}} \geq \dim \mathscr{O}_{Z^{\lambda}|_o} &= \dim \mathscr{O}_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathcal{G},p \neq o ,\infty}^{\lambda} )^{T^\sigma}} \\ & = \dim \mathscr{O}_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathcal{G},p \neq o,\infty }^{\lambda} )^{T}} = \dim H^0( \overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathcal{G}}^{\lambda}, \mathcal{L}|_{p\neq o,\infty} ) \\ & = \dim H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}}^{\bar{\lambda}}, \mathscr{L}) \geq \dim \mathscr{O}_{(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{{\mathscr{G}} }^{\bar{\lambda}})^{T^{\sigma}}}, \end{align*} $$

where the first equality follows from the flatness of $Z^\lambda $ over C, the third equality follows from Theorem 4.4 and Theorem 5.1, the fourth equality follows since $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G},C}^{{\lambda }}$ is flat over C (cf. Theorem 2.4), and the last inequality follows from Proposition 4.6. From this comparison, it follows that $\dim \mathscr {O}_{Z^{\lambda }|_o} = \dim \mathscr {O}_{(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G},p \neq o }^{\lambda } )^{T^\sigma }}$ . Hence, $\mathscr {O}_{Z^{\lambda }|_o} = \mathscr {O}_{(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G},p \neq 0 }^{\lambda } )^{T^\sigma }}$ . This concludes the proof of the lemma.

Proof of Theorem 4.5.

By Lemma 4.8 and the properness of $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathcal {G}, C}^{\lambda }$ over C, we have

(4.12) $$ \begin{align} \sum_{i\geq 0} (-1)^i \dim H^i(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{G}^{\lambda}, \mathrm{I}^{\lambda} \otimes \mathrm{L})=\sum_{i\geq 0} (-1)^i \dim H^i(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{\mathscr{G}} ^{\bar{\lambda}}, \mathscr{I}^{\bar{\lambda}} \otimes \mathscr{L}). \end{align} $$

By [Reference Pappas and RapoportPR, 9.h.] and [Reference KumarKu, Theorem 8.2.2], we have $H^i(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{G}^{\lambda }, \mathrm {L})=0$ and $H^i(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}} ^{\bar {\lambda }}, \mathscr {L})=0 $ for any $i>0$ . From [Reference ZhuZh1, Section 2.2] when G is of type A and D and from Section 5 when G is of type $E_6$ , we have the following vanishing $H^i(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{G}^{\lambda }, \mathrm {I}^{\lambda } \otimes \mathrm {L})=0$ for any $\lambda \in X_*(T)^+$ , by considering the exact sequence (4.5) in the untwisted case. From Lemma 4.7 and the long exact sequence (4.5), we can see easily that $H^i(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}} ^{\bar {\lambda }}, \mathscr {I}^{\bar {\lambda }} \otimes \mathscr {L})=0$ for any $i\geq 1$ . Hence, the equality (4.12) implies that $H^0(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}} ^{\bar {\lambda }}, \mathscr {I}^{\bar {\lambda }} \otimes \mathscr {L})=0$ for any $\bar {\lambda }\in X_*(T)^+_\sigma $ . Therefore, the theorem finally follows from the long exact sequence (4.5).

As an application of Theorem 4.5, we get a geometric Frenkel-Kac isomorphism for twisted affine algebras.

Theorem 4.9. For any special parahoric group scheme ${\mathscr {G}}$ induced from a standard automorphism $\sigma $ , the restriction map

$$\begin{align*}H^0(\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{G}}, \mathscr{L} ) \rightarrow H^0(\operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{T}}, \mathscr{L}|_{ \operatorname{\mathrm{\mathtt{Gr}}}_{\mathscr{T}} })\end{align*}$$

is an isomoprhism, via the embedding $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {T}}\to \operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ .

4.3 Application: smooth locus of twisted affine Schubert varieties

We now wish to investigate the smooth locus of the Schubert variety $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{{\mathscr {G}} }^{\bar {\lambda }}$ .

Proof. For any $\bar {\mu }\in X_*(T)^+_\sigma $ , if $e_{\bar {\mu }}=n^\mu e_0$ is a smooth point in $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{{\mathscr {G}} }^{\bar {\lambda }}$ , then by [Reference ZhuZh1, Lemma 2.3.3], $\dim \mathscr {O}_{ (\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{{\mathscr {G}} }^{\bar {\lambda }})^{T^\sigma } , e_{\bar {\mu }} } =1$ .

By Theorem 3.11, we have $ H^0(\overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\bar {\lambda }}_{\mathscr {G}}, \mathscr {L} )^\vee \simeq D(1, \bar {\lambda })$ , where $D(1, \bar {\lambda }) $ is the Demazure module defined in Definition 3.10. Then by Theorem 4.5, we have

$$\begin{align*}\dim D(1,\bar{\lambda})_{- \iota(\bar{\mu}) }= \mathrm{length} \big ( \mathscr{O}_{ (\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{{\mathscr{G}} }^{\bar{\lambda}})^{T^\sigma} , e_{\bar{\mu}} } \big ), \end{align*}$$

where $D(1,\bar {\lambda })_{-\iota (\bar {\mu } ) } $ is the $-\iota (\bar {\mu })$ -weight space in $D(1,\bar {\lambda })$ . We will prove that for any $\bar {\mu }\precneqq \bar {\lambda }$ , $\dim D(1,\bar {\lambda })_{- \iota (\bar {\mu }) }\geq 2$ , which would imply that $e_{\bar {\mu }}$ is not a smooth point in $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{{\mathscr {G}} }^{\bar {\lambda }}$ . From the surjectivity of (4.6), we have an embedding $D(1, \bar {\mu })\hookrightarrow D(1,\bar {\lambda })$ . However, $V(-\iota (\bar {\lambda }))\hookrightarrow D(1, \bar {\lambda })$ , where $V(-\iota (\bar {\lambda }))$ is the irreducible representation of $\mathfrak {g}^\sigma $ of lowest weight $-\iota (\bar {\lambda })$ . In view of Lemma 2.2, Lemma 3.3 and Lemma 3.4, when G is not of type $A_{2\ell }$ , the relation $\bar {\mu }\precneqq \bar {\lambda }$ implies that $\iota (\bar {\mu })\precneqq \iota (\bar {\lambda })$ . Hence, $V(-\iota (\bar {\lambda }))_{ -\iota (\bar {\mu }) }\not =0$ . Furthermore, as subspaces in $D(1, \bar {\lambda })$ ,

$$\begin{align*}D(1, \bar{\mu})\cap V(-\iota(\bar{\lambda}))=0.\end{align*}$$

It follows that $\dim D(1,\bar {\lambda })_{- \iota (\bar {\mu }) }\geq 2$ . This concludes the proof of the theorem.

Now we will deal with the case of $A_{2\ell }^{(2)}$ . Recall the group $H=(\check {G})^\tau $ mentioned in Section 2.3. By the ramified geometric Satake, $(X_*(T)_\sigma , X_*(T)^+_\sigma , \gamma _j, j\in I_\sigma )$ can be regarded as the weight lattice, the set of dominant weights, and simple roots of H. When $(G,m)=(A_{2\ell }, 4)$ , H is $B_\ell $ of adjoint type. Let $\varpi _1,\varpi _2,\cdots , \varpi _\ell $ be the set of fundamental dominant weights of H.

Proof. We first prove the following result: for any $\bar {\lambda }, \bar {\mu }\in X_*(T)_\sigma ^+$ with $\bar {\mu }\precneqq \bar {\lambda }$ , the Schubert cell $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}^{\bar {\mu }}$ is contained in the singular locus of $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }} $ , except when $\bar {\mu } \prec \bar {\lambda } $ is a cover relation and the simple short root $\gamma _\ell $ appears in $\bar {\lambda }-\bar {\mu }$ . We will prove this fact by several steps. Let $c_\ell $ be the coefficient of $\gamma _\ell $ in $\bar {\lambda }-\bar {\mu }$ .

Step 1. Observe that using Lemma 3.3 and by the same proof of Theorem 4.10, when the coefficient $c_\ell $ is even, we have $\dim D(1,\bar {\lambda })_{-\iota (\bar {\mu } )}\geq 2$ . Thus, $e_{\bar {\mu }}$ is singular in $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }} $ .

Step 2. Assume that the coefficient $c_\ell>1$ and $c_\ell $ is odd. There exists sequence of dominant elements in $X_*(T)^+_\sigma $ ,

(4.13) $$ \begin{align} \bar{\mu}= \bar{\lambda}_k \prec \bar{\lambda}_{k-1} \prec \cdots \prec \bar{\lambda}_1 \prec \bar{\lambda}_{0} = \bar{\lambda} , \end{align} $$

such that each $\prec $ is a cover relation. Then, by a theorem of Stembridge [Reference StembridgeSt, Theorem 2.8], for each i, $ \bar {\lambda }_{i} - \bar {\lambda }_{i+1} $ is a positive root of H, for any $0\leq i \leq k-1$ , and the coefficient of $\gamma _\ell $ in each $ \bar {\lambda }_{i} - \bar {\lambda }_{i+1}$ is either 0 or 1. Let j be the least integer such that the coefficient of $\gamma _\ell $ in $\bar {\lambda }_{j-1}-\bar {\lambda }_j$ is 1. Such j exists, since $c_\ell \not =1$ . Then the coefficient of $\gamma _\ell $ in $\bar {\lambda }_j- \bar {\mu }$ is even. By Step 1, we have $\dim D(1,\bar {\lambda }_j )_{-\iota (\bar {\mu } )}\geq 2$ . However, we have the inclusion $D(1,\bar {\lambda }_j )\subset D(1,\bar {\lambda })$ . It follows that $\dim D(1,\bar {\lambda })_{-\iota (\bar {\mu } )}\geq 2$ . Hence, the variety $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\bar {\lambda }}_{\mathscr {G}}$ is singular at the point $e_{\bar {\mu }}$ .

Step 3. We now assume that the coefficient $c_\ell =1$ . By assumption, $\bar {\mu }\prec \bar {\lambda }$ is not a cover relation. Then, in the sequence of cover relations in (4.13), either the coefficient of $\gamma _\ell $ in $ \bar {\lambda }_{k-1} - \bar {\lambda }_{k} $ is 0 or the coefficient of $\gamma _\ell $ in $ \bar {\lambda }_{0} - \bar {\lambda }_{1} $ is 0. If the coefficient of $\gamma _\ell $ in $ \bar {\lambda }_{k-1} - \bar {\lambda }_{k} $ is 0, by Step 1, $\dim D(1,\bar {\lambda }_{k-1} )_{-\iota (\bar {\mu })}\geq 2$ , implying that $\dim D(1,\bar {\lambda })_{-\iota (\bar {\mu })}\geq 2$ . Hence, $e_{\bar {\mu }}$ is singular in $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }}$ . If the coefficient of $\gamma _\ell $ in $ \bar {\lambda }_{0} - \bar {\lambda }_{1} $ is 0, then by Step 1 again, $e_{\bar {\lambda }_1}$ is a singular point in $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }}$ . Since the singular locus of $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }}$ is closed, the point $e_{\bar {\mu }}$ is also singular in $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }}$ .

We now explicitly describe the cover relation $\bar {\mu }\prec \bar {\lambda }$ such that $\gamma _\ell $ appears in $\bar {\lambda }-\bar {\mu }$ . Note that $X_*(T)_\sigma $ is a root lattice of $H\simeq \mathrm {SO}_{2n+1}$ . In fact, the lattice $X_*(T)_\sigma $ is spanned by $\varpi _1,\varpi _2, \cdots , \varpi _{\ell -1}, 2\varpi _\ell $ . Reading more carefully from [Reference StembridgeSt, Theorem 2.8], we can see that $\bar {\mu }\prec \bar {\lambda }$ is a cover relation and $\gamma _\ell $ appears in $ \bar {\lambda }-\bar {\mu }$ , if and only if one of the followings holds:

1. 1. $ \bar {\lambda }-\bar {\mu }=\gamma _\ell $ and $\langle \bar {\mu }, \check {\gamma }_\ell \rangle \neq 0 $ , where $\check {\gamma }_\ell $ is the coroot of $\gamma _\ell $ .

2. 2. $ \bar {\lambda }-\bar {\mu }=\sum _{j=i}^\ell \gamma _j$ and $\mu =\sum _{k=1}^{i-1 } a_k\varpi _k$ , for some $1\leq i\leq \ell $ .

Let G be the simply-connected simple group of type $A_{2\ell }$ with the standard automorphism $\sigma $ considered in this paper. Let $\alpha _1,\alpha _2,\cdots , \alpha _{2\ell }$ is a set of simple roots of G. Let L be the Levi subgroup of G generated by the simple roots

$$\begin{align*}\alpha_{i}, \alpha_{i+1},\cdots, \alpha_{\ell}, \alpha_{\ell+1}, \cdots, \alpha_{\ell+i} , \alpha_{\ell+i+1}.\end{align*}$$

Let M be the derived group $[L,L]$ of L. Then M is simply-connected simple group of type $A_{ 2(\ell -i+1)}$ and $\sigma $ still acts on M as a standard automorphism. Let $\mathscr {M}$ be the parahoric group scheme $\mathrm {Res}_{\mathcal {O}/\bar {\mathcal {O} } } (M_{\mathcal {O}})^\sigma $ , which is of type $A_{2(\ell -i+1)}^{(2)}$ . Let $T'=T\cap M$ be the maximal torus of M. We have the inclusion $X_*(T')\to X_*(T)$ , and this induces an inclusion $X_*(T')_\sigma \to X_*(T)_\sigma $ and $X_*(T')^+_\sigma \to X_*(T)^+_\sigma $ . We write $\bar {\lambda }=\sum _{k=1}^\ell b_k \varpi _k$ and $\bar {\mu }=\sum _{k=1}^\ell c_k \varpi _k$ with $b_k\geq 0, c_k\geq 0$ for any $k=1,\cdots , \ell $ . Set

$$\begin{align*}\bar{\lambda}' =\sum_{k=i}^{\ell} b_k \varpi_k, \quad \bar{\mu}' =\sum_{k=i}^{\ell} c_k \varpi_k .\end{align*}$$

Then, $\bar {\lambda }', \bar {\mu }' \in X_*(T')^+_\sigma $ . Moreover, $\bar {\lambda }'- \bar {\mu }' =\bar {\lambda }-\bar {\mu }$ . We have the following twisted analogue of Levi lemma,

(4.14) $$ \begin{align} L^{--}{\mathscr{G}}\cdot e_{\bar{\mu}} \cap \overline{\operatorname{\mathrm{\mathtt{Gr}}}}^{\bar{\lambda}}_{\mathscr{G}} \simeq L^{--}\mathscr{M}\cdot e_{\bar{\mu}' } \cap \overline{\operatorname{\mathrm{\mathtt{Gr}}}}^{\bar{\lambda}' }_{\mathscr{M}} , \end{align} $$

where $ L^{--}{\mathscr {G}}$ and $L^{--}\mathscr {M}$ are defined as in (4.1). It can be proved by exactly the same argument as in [Reference Malkin, Ostrik and VybornovMOV, Corollary 3.4], relying on the following main ingredients:

1. 1. $L^{--}{\mathscr {G}}\cdot e_{\bar {0}}$ (resp. $L^{--}\mathscr {M}\cdot e_{\bar {0}}$ ) is open in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ (resp. $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {M}}$ ); cf. (4.2).

2. 2. Transversal slice lemma for $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ and $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {M}}$ (cf. [Reference Malkin, Ostrik and VybornovMOV, Lemma 2.5], [Reference Kazhdan and LusztigKLu, Section 1.4]); that is, $ L^{--}{\mathscr {G}}\cdot e_{\bar {\mu }}$ (resp. $ L^{--}\mathscr {M}\cdot e_{\bar {\mu }' }$ ) is a transverse slice to $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\bar {\lambda }}_{\mathscr {G}}$ (resp. $ \overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\bar {\lambda }' }_{\mathscr {M}}$ ).

3. 3. $L^{--}{\mathscr {G}}\cdot e_{\bar {\mu }} \cap \overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\bar {\lambda }}_{\mathscr {G}}$ and $L^{--}\mathscr {M}\cdot e_{\bar {\mu }' } \cap \overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\bar {\lambda }' }_{\mathscr {M}} $ are reduced, irreducible and normal; cf. [Reference Malkin, Ostrik and VybornovMOV, Lemma 2.6]. Here, the main point is that $ \overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\bar {\lambda }}_{\mathscr {G}}$ and $ \overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\bar {\lambda }' }_{\mathscr {M}}$ are normal, cf. [Reference Pappas and RapoportPR, Theorem 0.3].

From the isomorphism (4.14) and the transversal slice lemma, we have the following Levi reduction: $e_{\bar {\mu }' }$ is singular in $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {M}}^{\bar {\lambda }' }$ if and only if $e_{\bar {\mu }}$ is singular in $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}^{\bar {\lambda }}_{\mathscr {G}} $ .

Case (1): $ \bar {\lambda }-\bar {\mu }=\gamma _\ell $ and $\langle \bar {\mu }, \check {\gamma }_\ell \rangle \neq 0 $ . In this case, we are reduced to $A_2^{(2)}$ . It is known from [Reference Haines and RicharzHR, Proposition 7.1] that $e_{\bar {\mu }' }$ is singular in $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {M}}^{\bar {\lambda }' }$ , as $\bar {\lambda }'$ is not quasi-minuscule.

Case (2): $ \bar {\lambda }-\bar {\mu }=\sum _{j=i}^\ell \gamma _j$ and $\mu =\sum _{k=1}^{i-1 } a_k\varpi _k$ with all $a_k>0$ , for some $1\leq i\leq \ell $ . In this case, $\bar {\mu }'=0$ and $\bar {\lambda }'=\sum _{j=i}^\ell \gamma _j$ . Thus, we are reduced to consider the singularity of quasi-minuscule affine Schubert variety $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {M}}^{\bar {\lambda }' }$ . It was observed by Richarz (using [Reference ArzdorfArz, Prop 4.16]) that the variety $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {M}}^{\bar {\lambda }' }$ is smooth. We give a different argument here. We consider the parahoric group scheme ${\mathscr {G}}=\mathrm { Res}_{\mathcal {O}/\bar {\mathcal {O} } } (G_{\mathcal {O}})^\sigma $ , and let $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^ {\bar {\lambda }}$ be the quasi-minuscule Schubert variety. Let $\mathfrak {g}_{\mathrm {i}}$ be the eigenspace of $\sigma $ on the Lie algebra $\mathfrak {g}$ of eigenvalue $\mathrm {i}=\sqrt {-1}$ . The vector space $\mathfrak {g}_{\mathrm {i}}$ consists of two $G^\sigma $ -orbits, as $\mathfrak {g}_{\mathrm {i}}$ is actually the standard representation of $G^\sigma =\mathrm {Sp}_{2\ell }$ which is of dimension $2\ell $ . Thus, any element in $\mathfrak {g}_{\mathrm {i}}$ is nilpotent. Then we consider a $G^\sigma $ -equivariant embedding $\mathfrak {g}_{\mathrm {i}}\to {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}$ given by $x\mapsto \mathrm { exp}(( \mathrm {ad } x)t^{-1}) \cdot e_{0}\in {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}$ , where $e_0$ is the base point in $\operatorname {\mathrm {\mathtt {Gr}}}_{\mathscr {G}}$ , and we regard $\mathrm {ad} x$ as an element in $G=G_{ad}$ . Since $e_\ell + e_{\ell +1}\in \mathfrak {g}_{\mathrm {i}}$ , and $\{ e_\ell + e_{\ell +1}, f_\ell +f_{\ell +1}, h_{\ell }+h_{\ell +1} \}$ form a $sl_2$ -triple, one may check easily that $\mathfrak {g}_{\mathrm {i}}$ is mapped into $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\gamma _\ell }=\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }}$ – in particular, $0\mapsto e_0$ . By comparing their dimensions and $G^\sigma $ -equivariance, one may see this is an open embedding. Thus, $\overline {\operatorname {\mathrm {\mathtt {Gr}}}}_{\mathscr {G}}^{\bar {\lambda }}$ is smooth.

5.1 Some reductions

Let G be a simply-laced simple group of adjoint type. Let T be a maximal torus in G. Let $\mathrm {L}$ be the level one line bundle on $\operatorname {\mathrm {\mathtt {Gr}}}_G$ . For any $\lambda \in X_*(T)$ , a general question is if the following restriction map is an isomorphism:

(5.1) $$ \begin{align} H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{G}^{\lambda}, \mathrm{L}) \to H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_G^{\lambda} )^T, \mathrm{L}). \end{align} $$

Zhu proved that this map is always surjective (cf. [Reference ZhuZh1, Prop.2.1.1]), and he also proved that the map is an isomorphism for type $A, D$ and many cases of E.

Since Zhu has proved this theorem for $\check {\omega }_1, \check {\omega }_2, \check {\omega }_3, \check {\omega }_5, \check {\omega }_6$ , by [Reference ZhuZh1, Prop.2.1.3], we will only need to prove that the theorem holds for $\check {\omega }_4$ .

For convenience, we assume that $\lambda $ is dominant and $\lambda $ is in the coroot lattice, and we set

$$\begin{align*}D(1,\lambda):= H^0(\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_{G}^{\lambda}, \mathrm{L} ) ^\vee, \quad D^T(1,\lambda):= H^0((\overline{\operatorname{\mathrm{\mathtt{Gr}}}}_G^{\lambda} )^T, \mathrm{L}) ^\vee. \end{align*}$$

Then, we can identify $D^T(1,\lambda )$ as a subspace of the affine Demazure module $D(1,\lambda )$ .

Let $\tilde {L}(\mathfrak {g})= \mathfrak {g}((t))\oplus \mathbb {C} K\oplus \mathbb {C} d$ be the affine Kac-Moody algebra associated to $\mathfrak {g}$ with center K and scaling element d. Let $\mathscr {H}(\Lambda _0)$ denote the integrable highest weight representation of $\tilde {L}(\mathfrak {g})$ of highest weight $\Lambda _0$ . Let $v_0$ be the highest weight vector of $\mathscr {H}(\Lambda _0)$ . For any $w\in W$ , set