2.1 Groups, root systems and Weyl groups

We refer to [Reference Chriss and Ginzburg21, Reference Kumar57] for precise expositions of general material presented in this subsection.

Let G be a connected, simply connected simple algebraic group of rank r over an algebraically closed field ${\mathbb K}$, and let B and H be a Borel subgroup and a maximal torus of G such that $H \subset B$. We set $N (= [B,B])$ to be the unipotent radical of B and let $N^-$ be the opposite unipotent subgroup of N with respect to H. We denote the Lie algebra of an algebraic group by the corresponding German (Fraktur) small letter. We have a (finite) Weyl group $W := N_G ( H ) / H$. For an algebraic group E, we denote its set of ${\mathbb K} [z]$-valued points by $E [z]$, its set of ${\mathbb K} [\![z]\!]$-valued points by $E [\![z]\!]$, and its set of ${\mathbb K} (\!(z)\!)$-valued points by $E (\!(z)\!)$, and so on. Let $\mathbf I \subset G [\![z]\!]$ be the preimage of $B \subset G$ via the evaluation at $z = 0$ (the Iwahori subgroup of $G [\![z]\!]$). We set $\mathbf {I}^- \subset G \left [z^{-1}\right ]$ as the opposite Iwahori subgroup of $\mathbf {I}$ in $G (\!(z)\!)$ with respect to H. By abuse of notation, we might consider $\mathbf {I}$ and $G [\![z]\!]$ as proalgebraic groups over ${\mathbb K}$ whose ${\mathbb K}$-valued points are given as these.

Let $P := \mathrm {Hom} _{gr} ( H, \mathbb {G}_m )$ be the weight lattice of H, $\Delta \subset P$ be the set of roots, $\Delta _+ \subset \Delta $ be the set of roots that yield root subspaces in $\mathfrak {b}$ and $\Pi \subset \Delta _+$ be the set of simple roots. We set $\Delta _- := - \Delta _+$. Let $Q^{\vee }$ be the dual lattice of P with a natural pairing $\langle \bullet , \bullet \rangle : Q^{\vee } \times P \rightarrow \mathbb {Z}$. We define $\Pi ^{\vee } \subset Q ^{\vee }$ to be the set of positive simple coroots and let $Q_+^{\vee } \subset Q ^{\vee }$ be the set of nonnegative integer spans of $\Pi ^{\vee }$. For $\beta , \gamma \in Q^{\vee }$, we define $\beta \ge \gamma $ if and only if $\beta - \gamma \in Q^{\vee }_+$. We set $P_+ := \left \{ \lambda \in P \mid \left \langle \alpha ^{\vee }, \lambda \right \rangle \ge 0, \ \forall \alpha ^{\vee } \in \Pi ^{\vee } \right \}$ and $P_{++} := \left \{ \lambda \in P \mid \left \langle \alpha ^{\vee }, \lambda \right \rangle> 0, \ \forall \alpha ^{\vee } \in \Pi ^{\vee } \right \}$. Define $\mathtt I := \{1,2,\ldots ,r\}$. We fix bijections $\mathtt I \cong \Pi \cong \Pi ^{\vee }$ such that $i \in \mathtt I$ corresponds to $\alpha _i \in \Pi $, its coroot $\alpha _i^{\vee } \in \Pi ^{\vee }$ and a simple reflection $s_i \in W$ corresponding to $\alpha _i$. Let $\{\varpi _i\}_{i \in \mathtt I} \subset P_+$ be the set of fundamental weights $\left (\text {i.e., }\left \langle \alpha _i^{\vee }, \varpi _j \right \rangle = \delta _{ij}\right )$ and $\rho := \sum _{i \in \mathtt I} \varpi _i = \frac {1}{2}\sum _{\alpha \in \Delta ^+} \alpha \in P_+$.

For a subset $\mathtt J \subset \mathtt I$, we define $P ( \mathtt J )$ as the standard parabolic subgroup of G corresponding to $\mathtt J$ – that is, we have $\mathfrak {b} \subset \mathfrak p (\mathtt J) \subset \mathfrak {g}$, and $\mathfrak p (\mathtt J)$ contains the root subspace corresponding to $- \alpha _i$ ($i \in \mathtt I$) if and only if $i \in \mathtt J$. Let $H \subset L ( \mathtt J ) \subset P ( \mathtt J )$ be the standard Levi subgroup (which is isomorphic to the quotient of $P ( \mathtt J )$ by its unipotent radical). Then the set of characters of $P ( \mathtt J )$ is identified with $P_{\mathtt J} := \sum _{i \in \mathtt I \setminus \mathtt J} \mathbb {Z} \varpi _i$. We also set $P_{\mathtt J, +} := \sum _{i \in \mathtt I \setminus \mathtt J} \mathbb {Z}_{\ge 0} \varpi _i = P_+ \cap P_{\mathtt J}$ and $P_{\mathtt J, ++} := \sum _{i \in \mathtt I \setminus \mathtt J} \mathbb {Z}_{\ge 1} \varpi _i = P_{++} \cap P_{\mathtt J}$. We define $W_{\mathtt J} \subset W$ to be the reflection subgroup generated by $\{s_i\}_{i \in \mathtt J}$. It is the Weyl group of $[L ( \mathtt J ), L (\mathtt J) ]$ and $L ( \mathtt J )$. We define $\rho _{\mathtt J}$ to be the half-sum of positive roots whose root spaces are contained in the unipotent radical of $\mathfrak p ( \mathtt J )$.

Let $\Delta _{\mathrm {af}} := \Delta \times \mathbb {Z} \delta \cup \{m \delta \}_{m \neq 0}$ be the untwisted affine root system of $\Delta $ with its positive part $\Delta _+ \subset \Delta _{\mathrm {af}, +}$. We set $\alpha _0 := - \vartheta + \delta $, $\Pi _{\mathrm {af}} := \Pi \cup \{ \alpha _0 \}$ and $\mathtt I_{\mathrm {af}} := \mathtt I \cup \{ 0 \}$, where $\vartheta $ is the highest root of $\Delta _+$. We set $W _{\mathrm {af}} := W \ltimes Q^{\vee }$ and call it the affine Weyl group. It is a reflection group generated by $\{s_i \mid i \in \mathtt I_{\mathrm {af}} \}$, where $s_0$ is the reflection with respect to $\alpha _0$. We also have a reflection $s_{\alpha } \in W_{\mathrm {af}}$ corresponding to $\alpha \in \Delta \times \mathbb {Z} \delta \subsetneq \Delta ^{\mathrm {af}}$. Let $\ell : W_{\mathrm {af}} \rightarrow \mathbb {Z}_{\ge 0}$ be the length function and $w_0 \in W$ be the longest element in $W \subset W_{\mathrm {af}}$. Together with the normalisation $t_{- \vartheta ^{\vee }} := s_{\vartheta } s_0$ (for the coroot $\vartheta ^{\vee }$ of $\vartheta $), we introduce the translation element $t_{\beta } \in W _{\mathrm {af}}$ for each $\beta \in Q^{\vee }$.

For each $i \in \mathtt I_{\mathrm {af}}$, we have a connected algebraic group $\mathop {\textit{SL}} ( 2, i )$ that is isomorphic to $\mathop {\textit{SL}} ( 2 )$ equipped with an inclusion $\mathop {\textit{SL}} ( 2, i ) ( {\mathbb K} ) \subset G (\!(z)\!)$ as groups corresponding to $\pm \alpha _i \in \mathtt I_{\mathrm {af}}$. Let $\rho _{\pm \alpha _i} : \mathbb {G}_m \rightarrow \mathop {\textit{SL}} ( 2, i )$ denote the unipotent one-parameter subgroup corresponding to $\pm \alpha _i \in \Delta _{\mathrm {af}}$. We set $B_i := \mathop {\textit{SL}} ( 2, i ) \cap \mathbf I$, which is a Borel subgroup of $\mathop {\textit{SL}} ( 2, i )$. For each $i \in \mathtt I$, we set $P_i := P ( \{ i \} )$. For each $i \in \mathtt I_{\mathrm {af}}$, we set $\mathbf {I} ( i ) := \mathop {\textit{SL}} ( 2, i ) \mathbf {I}$. Each $\mathbf {I} (i)$ can be regarded as a proalgebraic group.

As a variation of [Reference Kumar57, Chapter VI], we say an ind-scheme $\mathfrak X$ over ${\mathbb K}$ admits a $G(\!(z)\!)$-action if it admits an action of $\mathbf {I}$ and $\mathop {\textit{SL}} ( 2, i )$ ($i \in \mathtt I_{\mathrm {af}}$) as (ind-)schemes over ${\mathbb K}$ that coincides on $B_i = ( \mathbf {I} \cap \mathop {\textit{SL}} ( 2, i ) )$ and they generate a $G(\!(z)\!)$-action on the set of closed points of $\mathfrak X$ (the latter is a group action on a set). We consider the notion of $G(\!(z)\!)$-equivariant morphisms accordingly.

We set

$$ \begin{align*}Q^{\vee}_< := \left\{\beta \in Q^{\vee} \mid \left\langle \beta, \alpha_i \right\rangle < 0, \ \forall i \in \mathtt I \right\}.\end{align*} $$

Let $\le $ be the Bruhat order of $W_{\mathrm {af}}$. In other words, $w \le v$ holds if and only if a subexpression of a reduced decomposition of v yields a reduced decomposition of w. We define the generic (semi-infinite) Bruhat order $\le _{\frac {\infty }{2}}$ as

(2.1)$$ \begin{align} w \le_{\frac{\infty}{2}} v \Leftrightarrow w t_{\beta} \le v t_{\beta} \quad \text{for every } \beta \in Q^{\vee} \text{ such that } \left\langle \beta, \alpha_i \right\rangle \ll 0 \text{ for } i \in \mathtt I. \end{align} $$

By [Reference Lusztig64], this defines a preorder on $W_{\mathrm {af}}$. Here we remark that $w \le v$ if and only if $w \ge _{\frac {\infty }{2}} v$ for $w, v \in W$. We also have

(2.2)$$ \begin{align} w \le_{\frac{\infty}{2}} v \Leftrightarrow ww_0 \ge_{\frac{\infty}{2}} vw_0, \quad w,v\in W_{\mathrm{af}}. \end{align} $$

For proofs and related results, we refer to [Reference Kato, Naito and Sagaki51, §2.2] and [Reference Peterson74, Lecture 13].

For each $u \in W$ and $\beta \in Q^{\vee }$, we set

$$ \begin{align*}\ell^{\frac{\infty}{2}} \left( u t_{\beta} \right) := \ell ( u ) + \sum_{\alpha \in \Delta_+} \left\langle \beta, \alpha \right\rangle = \ell ( u ) + 2 \left\langle \beta, \rho \right\rangle.\end{align*} $$

For each $\lambda \in P_+$, we denote the corresponding Weyl module by $V ( \lambda )$ (see, e.g. [Reference Andersen, Polo and Wen1, Proposition 1.22] and [Reference Kashiwara41, Theorem 5]). By convention, $V ( \lambda )$ is a finite-dimensional indecomposable G-module with a cyclic B-eigenvector $\mathbf {v}_\lambda ^0$ (highest weight vector) with H-weight $\lambda $ whose character obeys the Weyl character formula. For a semisimple H-module V, we set

$$ \begin{align*}\mathrm{ch}\, V := \sum_{\lambda \in P} e^\lambda \cdot \dim _{{\mathbb K}} \mathrm{Hom}_H ( {\mathbb K}_\lambda, V ).\end{align*} $$

If V is a $\mathbb {Z}$-graded H-module in addition, then we set

(2.3)$$ \begin{align} \mathrm{gch}\, V := \sum_{\lambda \in P, \; n \in \mathbb{Z}} q^n e^\lambda \cdot \dim _{{\mathbb K}} \mathrm{Hom}_H ( {\mathbb K}_\lambda, V_n ). \end{align} $$

Define $\mathscr {B} := G / B$ and call it the flag manifold of G. We have the Bruhat decomposition

(2.4)$$ \begin{align} \mathscr{B} = \bigsqcup _{w \in W} \mathbb{O}_{\mathscr{B}} ( w ) \end{align} $$

into B-orbits such that $\dim \, \mathbb {O}_{\mathscr {B}} ( w ) = \ell ( w _0 ) - \ell ( w )$ for each $w \in W \subset W_{\mathrm {af}}$. We set $\mathscr {B} ( w ) := \overline {\mathbb O_{\mathscr {B}} ( w )} \subset \mathscr {B}$.

For each $\lambda \in P$, we have a line bundle ${\mathcal O} _{\mathscr {B}} ( \lambda )$ such that

$$ \begin{align*}H ^0 ( \mathscr{B}, {\mathcal O}_{\mathscr{B}} ( \lambda ) ) \cong V ( \lambda )^*, \qquad {\mathcal O}_{\mathscr{B}} ( \lambda ) \otimes_{{\mathcal O}_{\mathscr{B}}} {\mathcal O} _{\mathscr{B}} ( - \mu ) \cong {\mathcal O}_{\mathscr{B}} ( \lambda - \mu ), \quad \lambda, \mu \in P_+.\end{align*} $$

For each $w \in W$, let $p_w \in \mathbb {O}_{\mathscr {B}} ( w )$ be the unique H-fixed point. We normalise $p_w$ (and hence $\mathbb {O}_{\mathscr {B}} ( w )$) so that the restriction of ${\mathcal O}_{\mathscr {B}} ( \lambda )$ to $p_w$ is isomorphic to ${\mathbb K}_{- w w_0 \lambda }$ for every $\lambda \in P_+$. (Here we warn that the convention differs from [Reference Kato47].)

2.2 Representations of affine and current algebras

In the rest of this section, we work over ${\mathbb K} = {\mathbb C}$, the field of complex numbers. Material in this subsection without a reference can be found in [Reference Kac40, Reference Kashiwara41]. Every result in this subsection is transferred to an arbitrary field in Section 3.2.

Let $\widetilde {\mathfrak {g}}$ denote the untwisted affine Kac–Moody algebra associated to $\mathfrak {g}$ – that is, we have

$$ \begin{align*}\widetilde{\mathfrak{g}} = \mathfrak{g} \otimes {\mathbb C} \left[z, z^{-1}\right] \oplus {\mathbb C} K \oplus {\mathbb C} d,\end{align*} $$

where K is central, $[d, X \otimes z^m] = m X \otimes z ^m$ for each $X \in \mathfrak {g}$ and $m \in \mathbb {Z}$ and, for each $X, Y \in \mathfrak {g}$ and $f, g \in {\mathbb C} \left [z^{\pm 1}\right ]$, we have

$$ \begin{align*}[X \otimes f , Y \otimes g ] = [X, Y] \otimes f g + ( X, Y )_{\mathfrak{g}} \cdot K \cdot \mathrm{Res}_{z = 0} f \frac{\partial g}{\partial z},\end{align*} $$

where $(\bullet , \bullet )_{\mathfrak {g}}$ denotes the G-invariant bilinear form such that $\left ( \alpha ^{\vee }, \alpha ^{\vee }\right )_{\mathfrak {g}} = 2$ for a long simple root $\alpha $. Let $E_i, F_i$ ($i \in \mathtt I_{\mathrm {af}}$) denote the Kac–Moody generators of $\widetilde {\mathfrak {g}}$ corresponding to $\alpha _i$. We set $\widetilde {\mathfrak {h}} := \mathfrak {h} \oplus {\mathbb C} K \oplus {\mathbb C} d$. Let $\mathfrak {I}$ be the Lie subalgebra of $\widetilde {\mathfrak {g}}$ generated by $E_i$ ($i \in \mathtt I_{\mathrm {af}}$) and $\widetilde {\mathfrak {h}}$, and $\mathfrak {I}^-$ be the Lie subalgebra of $\widetilde {\mathfrak {g}}$ generated by $F_i$ ($i \in \mathtt I_{\mathrm {af}}$) and $\widetilde {\mathfrak {h}}$. For each $i \in \mathtt I_{\mathrm {af}}$ and $n \ge 0$, we set $E_i^{(n)} := \frac {1}{n!} E_i^n$ and $F_i^{(n)} := \frac {1}{n!} F_i^n$.

We define

$$ \begin{align*}Q^{\mathrm{af}, \vee} := \mathbb{Z} d \oplus \bigoplus_{i \in \mathtt I_{\mathrm{af}}} \mathbb{Z} \alpha^{\vee}_i \subset \widetilde{\mathfrak{h}}, \qquad P^{\mathrm{af}} := \mathbb{Z} \delta \oplus \bigoplus_{i \in \mathtt I_{\mathrm{af}}} \mathbb{Z} \Lambda_i \subset \widetilde{\mathfrak{h}}^*,\end{align*} $$

and a pairing $Q^{\mathrm {af}, \vee } \times P^{\mathrm {af}} \rightarrow \mathbb {Z}$ such that

$$ \begin{align*}\left\langle \alpha_i^{\vee}, \Lambda_j \right\rangle = \delta_{ij} \hskip 2mm (i,j \in \mathtt I_{\mathrm{af}}), \hskip 4mm \left\langle \alpha^{\vee}_i, \delta \right\rangle \equiv 0, \hskip 4mm \left\langle d, \Lambda_i \right\rangle = \delta_{i0} \hskip 2mm (i \in \mathtt I_{\mathrm{af}}), \hskip 4mm \left\langle d, \delta \right\rangle = 1.\end{align*} $$

We have a projection map

$$ \begin{align*}P^{\mathrm{af}} \ni \Lambda = k \delta + \sum_{i \in \mathtt I_{\mathrm{af}}} a_i \Lambda_i \mapsto \overline{\Lambda} = \sum_{i \in \mathtt I} a_i \varpi_i \in P,\end{align*} $$

which has a unique splitting $P \subset P^{\mathrm {af}}$ whose image is orthogonal to $d,K \in \widetilde {\mathfrak {h}}$. We set $P^{\mathrm {af}}_+ := \sum _{i \in \mathtt I_{\mathrm {af}}} \mathbb {Z}_{\ge 0} \Lambda _i$. Each $\Lambda \in P^{\mathrm {af}}_+$ defines an irreducible integrable highest weight module $L ( \Lambda )$ of $\widetilde {\mathfrak {g}}$ with its highest weight vector $\mathbf {v}_{\Lambda }$. In addition, each $\lambda \in P_+$ defines a level $0$ extremal weight module $\mathbb {X} ( \lambda )$ of $\widetilde {\mathfrak {g}}$ by means of the specialisation of the quantum parameter $\mathsf q = 1$ in [Reference Kashiwara43, Proposition 8.2.2] and [Reference Kashiwara44, §5.1], which is integrable, and K acts by $0$. The module $\mathbb {X} ( \lambda )$ carries a cyclic $\widetilde {\mathfrak {h}}$-weight vector $\mathbf {v}_{\lambda }$ such that

$$ \begin{align*}H \mathbf{v}_{\lambda} = \lambda ( H ) \mathbf{v}_{\lambda} \hskip 2mm (H \in \mathfrak{h}), \hskip 2mm K \mathbf{v} _{\lambda} = 0 = d \mathbf{v}_{\lambda}, \hskip 2mm E_i \mathbf{v}_{\lambda} = 0 \hskip 2mm (i \in \mathtt I), \hskip 2mm \text{and} \hskip 2mm F_0 \mathbf{v}_{\lambda} = 0. \end{align*} $$

(We can deduce that $\mathbb {X} ( \lambda )$ is the maximal integrable $\widetilde {\mathfrak {g}}$-module that possesses a cyclic vector with these properties [Reference Kashiwara43, §8.1].) Moreover, each $w = u t_{\beta } \in W_{\mathrm {af}}$ ($u \in W, \beta \in Q^{\vee }$) defines an element $\mathbf {v}_{w \lambda } \in \mathbb {X} ( \lambda )$ such that

$$ \begin{align*}H \mathbf{v}_{w \lambda} = ( w \lambda ) ( H ) \mathbf{v}_{w \lambda} \hskip 3mm (H \in \mathfrak{h}), \hskip 3mm K \mathbf{v} _{w \lambda} = 0, \hskip 3mm d \mathbf{v}_{w \lambda} = - \left\langle \beta, \lambda \right\rangle \mathbf{v}_{w \lambda}\end{align*} $$

up to sign [Reference Kashiwara43, §8.1]. We call a vector in $\{\mathbf {v}_{w\lambda }\}_{w \in W_{\mathrm {af}}}$ an extremal weight vector of $\mathbb {X} ( \lambda )$.

We set $\mathfrak {g} [z] := \mathfrak {g} \otimes _{\mathbb C} {\mathbb C} [z]$ and regard it as a Lie subalgebra of $\widetilde {\mathfrak {g}}$. We have $\mathfrak {I} \subset \mathfrak {g} [z] + {\mathbb C} K + {\mathbb C} d$. The Lie algebra $\mathfrak {g} [z]$ is graded, and its grading is the internal grading of $\widetilde {\mathfrak {g}}$ given by d.

For each $\lambda \in P_+$, we set

$$ \begin{align*}\mathbb{W}_w ( \lambda ) := U ( \mathfrak{I} ) \mathbf{v}_{w\lambda} \subset \mathbb{X} ( \lambda ).\end{align*} $$

These are the $\mathsf q = 1$ cases of the Demazure modules of $\mathbb {X} ( \lambda )$, as well as the generalised global Weyl modules in the sense of [Reference Feigin, Makedonskyi and Orr28]. We set $\mathbb {W} ( \lambda ) := \mathbb {W}_{w_0} ( \lambda )$. By construction, both $\mathbb {X} ( \lambda )$ and $\mathbb {W} _w ( \lambda )$ are semisimple as $( H \times \mathbb {G}_m )$-modules, where $\mathbb {G}_m$ acts on z by $a : z^m \mapsto a^{m} z^m$ ($m \in \mathbb {Z}$).

Theorem 2.3 [Reference Kato46]

Set $\lambda , \mu \in P_+$ and $w \in W$. We have a unique $($up to scalar$)$ injective degree $0\; \mathfrak {I}$-module map

$$ \begin{align*}\mathbb{W}_w ( \lambda + \mu ) \hookrightarrow \mathbb{W}_w ( \lambda ) \otimes \mathbb{W}_w ( \mu ).\end{align*} $$

Proof Sketch of proof.

For each $\lambda , \mu \in P_+$, the projectivity of $\mathbb {W} ( \lambda + \mu )$ in the sense of Theorem 2.2 yields a unique graded $\mathfrak {g} [z]$-module map

$$ \begin{align*}\mathbb{W} ( \lambda + \mu ) \longrightarrow \mathbb{W} ( \lambda ) \otimes \mathbb{W} ( \mu )\end{align*} $$

of degree $0$. This map is injective by examining the specialisations to local Weyl modules (for their definitions, see [Reference Kato46, Theorem 1.4] or Lemma 3.20 and Remark 3.21). Examining the $\mathfrak {I}$-cyclic vectors, it uniquely restricts to a map

$$ \begin{align*}\mathbb{W}_w ( \lambda + \mu ) \longrightarrow \mathbb{W}_w ( \lambda ) \otimes \mathbb{W}_w ( \mu )\end{align*} $$

up to scalar. Because the ambient map is injective, this map must be also.

2.3 Semi-infinite flag manifolds

We work over ${\mathbb C}$ as in the previous subsection. Material in this section is re-proved in the setting of characteristic $\neq 2$ in Sections 3.4 and 4.2 (compare Section 4.3). We define the semi-infinite flag manifold as the reduced ind-scheme such that both of the following are true:

• • We have a closed embedding

  $$ \begin{align*}\mathbf{Q}_G^{\mathrm{rat}} \subset \prod_{i \in \mathtt I} \mathbb{P} ( V ( \varpi_i ) \otimes {\mathbb C} (\!(z)\!) ).\end{align*} $$

• • We have an equality $\mathbf {Q}_G^{\mathrm {rat}} ( {\mathbb C} ) = G (\!(z)\!) / \left ( H ( {\mathbb C} ) \cdot N (\!(z)\!) \right )$.

This is a pure ind-scheme of ind-infinite type [Reference Kato, Naito and Sagaki51]. Note that the group $Q^{\vee } \subset H (\!(z)\!) / H ( {\mathbb C} )$ acts on $\mathbf {Q}_G^{\mathrm {rat}}$ from the right. The ind-scheme $\mathbf {Q}_G^{\mathrm {rat}}$ is equipped with a $G (\!(z)\!)$-equivariant line bundle ${\mathcal O} _{\mathbf {Q}_G^{\mathrm {rat}}} ( \lambda )$ for each $\lambda \in P$. Here we normalised so that $\Gamma \left ( \mathbf {Q}_G^{\mathrm {rat}}, {\mathcal O}_{\mathbf {Q}_G^{\mathrm {rat}}} ( \lambda ) \right )$ is $B^- (\!(z)\!)$-cocyclic to an H-weight vector with its H-weight $- \lambda $. We warn that this convention is twisted by $-w_0$ from that of [Reference Kato47], and complies with [Reference Kato, Naito and Sagaki51].

Theorem 2.4 [Reference Finkelberg and Mirković29, Reference Feigin, Finkelberg, Kuznetsov and Mirković23, Reference Kato, Naito and Sagaki51, Reference Lusztig64]

We have an $\mathbf {I}$-orbit decomposition

$$ \begin{align*}\mathbf{Q}_G^{\mathrm{rat}} = \bigsqcup_{w \in W_{\mathrm{af}}} \mathbb{O} ( w )\end{align*} $$

with the following properties:

1. 1. Each $\mathbb O ( w )$ is isomorphic to ${\mathbb A}^{\infty }$ and has a unique $(H \times \mathbb {G}_m)$-fixed point.

2. 2. The right action of $\gamma \in Q^{\vee }$ on $\mathbf {Q}_G^{\mathrm {rat}}$ yields the translation $\mathbb O ( w ) \mapsto \mathbb O ( w t_{\gamma })$.

3. 3. We have $\mathbb O ( w ) \subset \overline {\mathbb O ( v )}$ if and only if $w \le _{\frac {\infty }{2}} v$.

4. 4. The relative dimension of $\mathbb {O} \left ( u t_{\beta } \right )$ ($u \in W, \beta \in Q^{\vee }$) and $\mathbb {O} ( e )$, counted as the difference of the cardinality of the maximal chain of intermediate $\mathbf {I}$-orbits to a common smaller $\mathbf {I}$-orbit, is $\ell ^{\frac {\infty }{2}} \left ( u t_{\beta } \right )$.

For each $w \in W_{\mathrm {af}}$, let $\mathbf {Q}_G ( w )$ denote the closure of $\mathbb {O} ( w )$. We refer to $\mathbf {Q}_G ( w )$ as a Schubert variety of $\mathbf {Q}_G^{\mathrm {rat}}$ (corresponding to $w \in W_{\mathrm {af}}$).

Let $S = \bigoplus _{\lambda \in P_{\mathtt J,+}} S ( \lambda )$ be a $P_{\mathtt J,+}$-graded commutative ring such that $S ( 0 ) = A$ is a principal ideal domain and S is torsion-free over A and generated by $\bigoplus _{i \in \mathtt I \setminus \mathtt J} S ( \varpi _i )$. We define

(2.5)$$ \begin{align} \mathrm{Proj}\, S = ( \mbox{Spec}\, S \setminus E ) / H \subset \prod_{i \in \mathtt I \setminus \mathtt J} \mathbb{P}_A \left( S ( \varpi_i )^{\vee} \right) \end{align} $$

as the $P_{\mathtt J,+}$-graded proj over $\mbox {Spec}\, A$, where E is the locus where all of $S ( \varpi _i )$ vanishes for some $i \in \mathtt I \setminus \mathtt J$ (the irrelevant locus).

Theorem 2.5 [Reference Kato, Naito and Sagaki51]

For each $w \in W_{\mathrm {af}}$, we have

$$ \begin{align*}\mathbf{Q}_G ( w ) \cong \mathrm{Proj} \bigoplus _{\lambda \in P_+} \mathbb{W} _{ww_0} ( \lambda )^{\vee},\end{align*} $$

where the multiplication of the ring $\bigoplus _\lambda \mathbb {W} _{ww_0} ( \lambda )^{\vee }$ is given by Theorem 2.3.

2.4 Quasi-map spaces and Zastava spaces

We work over ${\mathbb C}$ as in the previous subsection. Here we recall basics of quasi-map spaces from [Reference Finkelberg and Mirković29, Reference Feigin, Finkelberg, Kuznetsov and Mirković23].

We have W-equivariant isomorphisms $H^2 ( \mathscr {B}, \mathbb {Z} ) \cong P$ and $H_2 ( \mathscr {B}, \mathbb {Z} ) \cong Q ^{\vee }$. This identifies the (integral points of the) nef cone of $\mathscr {B}$ with $P_+ \subset P$ and the effective cone of $\mathscr {B}$ with $Q_+^{\vee }$. A quasi-map $( f, D )$ is a map $f : \mathbb {P} ^1 \rightarrow \mathscr {B}$ together with a $\Pi ^{\vee }$-coloured effective divisor

$$ \begin{align*}D = \sum_{\alpha \in \Pi^{\vee}, x \in \mathbb{P}^1 ({\mathbb C})} m_x \left(\alpha^{\vee}\right) \alpha^{\vee} \otimes [x] \in Q^{\vee} \otimes_{\mathbb{Z}} \mathrm{Div} \mathbb{P}^1, \quad m_x \left(\alpha^{\vee}\right) \in \mathbb{Z}_{\ge 0}.\end{align*} $$

For $i \in \mathtt I$, we set $D_i := \left \langle D, \varpi _i \right \rangle \in \mathrm {Div} \, \mathbb {P}^1$. We call D the defect of the quasi-map $(f, D)$. Here we define the (total) degree of the defect by

$$ \begin{align*}\lvert D\rvert := \sum_{\alpha \in \Pi^{\vee}, \; x \in \mathbb{P}^1 ({\mathbb C})} m_x \left(\alpha^{\vee}\right) \alpha^{\vee} \in Q_+^{\vee}.\end{align*} $$

For each $\beta \in Q_+^{\vee }$, we set

$$ \begin{align*}\mathscr{Q} ( \mathscr{B}, \beta ) : = \left\{ f : \mathbb{P} ^1 \rightarrow X \mid \text{quasi-map such that } f _* \left[ \mathbb{P}^1 \right] + \lvert D \rvert = \beta \right\},\end{align*} $$

where $f_* \left [\mathbb {P}^1\right ]$ is the class of the image of $\mathbb {P}^1$ multiplied by the degree of $\mathbb {P}^1 \to \mathrm {Im} f$. We denote $\mathscr {Q} ( \mathscr {B}, \beta )$ by $\mathscr {Q} ( \beta )$ when there is no danger of confusion.

Definition 2.6. Drinfeld–Plücker data

Consider a collection $\mathcal L = \left \{\left ( \psi _{\lambda }, \mathcal L^{\lambda } \right ) \right \}_{\lambda \in P_+}$ of inclusions $\psi _{\lambda } : \mathcal L ^{\lambda } \hookrightarrow V ( \lambda ) \otimes _{{\mathbb C}} \mathcal O _{\mathbb {P}^1}$ of line bundles $\mathcal L ^{\lambda }$ over $\mathbb {P}^1$. The data $\mathcal L$ are called Drinfeld–Plücker data (DP-data) if the canonical inclusion of G-modules

$$ \begin{align*}\eta_{\lambda, \mu} : V ( \lambda + \mu ) \hookrightarrow V ( \lambda ) \otimes V ( \mu )\end{align*} $$

induces an isomorphism

$$ \begin{align*}\eta_{\lambda, \mu} \otimes \mathrm{id} : \psi_{\lambda + \mu} \left( \mathcal L ^{\lambda + \mu} \right) \stackrel{\cong}{\longrightarrow} \psi _{\lambda} \left( \mathcal L^{\lambda} \right) \otimes_{{\mathcal O}_{\mathbb{P}^1}} \psi_{\mu} \left( \mathcal L^{\mu} \right)\end{align*} $$

for every $\lambda , \mu \in P_+$.

For each $w \in W$, let $\mathscr {Z} ( \beta , w ) \subset \mathscr {Q} ( \beta )$ be the locally closed subset consisting of quasi-maps that are defined at $z = 0$, and their values at $z = 0$ are contained in $\mathscr {B} ( w ) \subset \mathscr {B}$. We set $\mathscr {Q} ( \beta , w ) := \overline {\mathscr {Z} ( \beta , w )}$. (Hence, we have $\mathscr {Q} ( \beta ) = \mathscr {Q} ( \beta , e)$.)

For each $\lambda \in P$ and $w \in W$, we have a G-equivariant line bundle ${\mathcal O} _{\mathscr {Q} \left ( \beta , w \right )} ( \lambda )$ (and its pro-object ${\mathcal O} _{\mathscr {Q}} ( \lambda )$) obtained by the (tensor products of the) pullbacks ${\mathcal O} _{\mathscr {Q} \left ( \beta , w \right )}( \varpi _i )$ of the ith ${\mathcal O} ( 1 )$ via the embedding

(2.6)$$ \begin{align} \mathscr{Q} ( \beta, w ) \hookrightarrow \prod_{i \in \mathtt I} \mathbb{P} \left( V ( \varpi_i ) \otimes_{{\mathbb C}} {\mathbb C} [z] _{\le - \left\langle w_0 \beta, \varpi_i \right\rangle} \right) \end{align} $$

for each $\beta \in Q_+^{\vee }$.

We have embeddings $\mathscr {B} \subset \mathscr {Q} ( \beta ) \subset \mathbf {Q}_G ( e )$ such that the line bundles ${\mathcal O} ( \lambda )$ ($\lambda \in P$) correspond to each other by restrictions ([Reference Braverman and Finkelberg10, Reference Kato46, Reference Kato, Naito and Sagaki51]).

3.1 Frobenius splittings

Let $\Bbbk $ be a field and p be a prime. We assume $\mathsf {char}\, \Bbbk = p$, $\Bbbk \subset {\mathbb K}$ and that the pth power map is invertible on $\Bbbk $ (e.g., $\Bbbk = \mathbb F_p$ or $\overline {\mathbb F}_p$) throughout this subsection.

We follow the generality on Frobenius splittings in [Reference Brion and Kumar14], which considers separated schemes of finite type. We sometimes use the assertions from [Reference Brion and Kumar14] without the finite-type assumption when the assertion is independent of that; typical disguises are properness, finite generation and the Serre vanishing theorem.

Definition 3.1. Frobenius splitting of a ring

Let R be a commutative ring over $\Bbbk $, and let $R^{(1)}$ denote the set R equipped with the map

$$ \begin{align*}R \times R^{(1)} \ni (r,m) \mapsto r^p m \in R^{(1)}.\end{align*} $$

This equips $R^{(1)}$ with an R-module structure over $\Bbbk $ (the $\Bbbk $-vector space structure on $R^{(1)}$ is also twisted by the pth power operation), together with a map $\imath : R. 1 \rightarrow R^{(1)}$. An R-module map $\phi : R^{(1)} \to R$ is said to be a Frobenius splitting if $\phi \circ \imath $ is an identity.

Note that $\imath $ in Definition 3.1 must be an inclusion if we have a Frobenius splitting $\phi $. Since the pth power map in $\Bbbk $ is invertible, we can twist the (scalar multiplication part of the) $\Bbbk $-vector space structure of R ($\cong R^{(1)}$ as sets) to make $\imath $ into a $\Bbbk $-linear map without making it into an R-linear map (when $R \neq \Bbbk $).

Theorem 3.5 [Reference Brion and Kumar14]

Let $\mathfrak X$ be a separated scheme of finite type over $\Bbbk $ with semiample line bundles $\mathcal L_1,\ldots , \mathcal L_r$. If $\mathfrak X$ admits a Frobenius splitting, then the multisection ring

$$ \begin{align*}\bigoplus_{n_1,\ldots,n_r \ge 0} \Gamma \left( \mathfrak X, \mathcal L_1 ^{\otimes n_1} \otimes \cdots \otimes \mathcal L_r ^{\otimes n_r} \right)\end{align*} $$

admits a Frobenius splitting $\phi $. Moreover, a closed subscheme $\mathfrak {Y} \subset \mathfrak {X} = \mathrm {Proj}\, R$ admits a compatible Frobenius splitting if and only if the homogeneous ideal $I _{\mathfrak {Y}} \subset R$ that defines $\mathfrak {Y}$ satisfies $\phi \left ( I_{\mathfrak {Y}} \right ) \subset I_{\mathfrak {Y}}$ – that is, the pair $\left (R, R / I _{\mathfrak {Y}}\right )$ admits a compatible Frobenius splitting $\phi $. $\Box $

Definition 3.6. Canonical splitting

Let $\mathfrak {X}$ be a separated scheme over $\Bbbk $ equipped with a B-action. A Frobenius splitting $\phi $ is said to be B-canonical if it is H-fixed and each $i \in \mathtt I$ yields

(3.1)$$ \begin{align} \rho_{\alpha_i} ( z^p ) \phi \left( \rho_{\alpha_i} ( - z ) f \right) = \sum_{j = 0}^{p-1} \frac{z^j}{j!} \phi_{i, j} ( f ), \quad z \in \Bbbk, \end{align} $$

where $\phi _{i, j} \in \mbox {Hom}_{{\mathcal O}_{\mathfrak X}} ( \mathsf {Fr}_{*} {\mathcal O}_{\mathfrak X}, {\mathcal O}_{\mathfrak X} )$. We similarly define the notion of $B^-$-canonical splitting (resp., $\mathbf {I}$-canonical splitting and $\mathbf {I}^-$-canonical splitting) by using $\left \{ \rho _{-\alpha _i} \right \}_{i \in \mathtt I}$ (resp., $\left \{ \rho _{\alpha _i} \right \}_{i \in \mathtt I_{\mathrm {af}}}$ and $\left \{ \rho _{-\alpha _i} \right \}_{i \in \mathtt I_{\mathrm {af}}}$) instead. Canonical splitting of a commutative ring S over $\Bbbk $ is defined through its spectrum.

Proposition 3.7. Compare [Reference Brion and Kumar14]

Let $S = \bigoplus _{m \ge 0} S_m$ be a graded ring with $S_0 = \Bbbk $ such that

• • S is equipped with a degree-preserving $\mathbf {I}$-action,

• • each $S_m$ is a graded $\Bbbk $-vector space compatible with the multiplication and

• • we have an $\mathbf {I}$-canonical Frobenius splitting $\phi : S^{(1)} \to S$.

Then the induced map

$$ \begin{align*}\phi^{\vee} : S_{m}^{\vee} \longrightarrow S_{pm}^{\vee}, \quad m \in \mathbb{Z}_{\ge 0},\end{align*} $$

satisfies

$$ \begin{align*}\phi^{\vee} \left( E_i^{(n)} \mathbf{v} \right) = E_i^{(pn)} \phi^{\vee} ( \mathbf{v} ) \quad \forall i \in \mathtt I_{\mathrm{af}}, n \in \mathbb{Z}_{\ge 0}, \mathbf{v} \in S_m^{\vee}.\end{align*} $$

Similar results hold for the $\mathbf {I}^-$- and $B^{\pm }$-actions.

Proof Proof of Proposition 3.7.

The condition that $S_m$ is a graded vector space implies $S_m \stackrel {\cong }{\longrightarrow }\left (S_m^{\vee }\right )^{\vee }$ for each $m \in \mathbb {Z}_{\ge 0}$. By [Reference Brion and Kumar14, Proposition 4.1.8], each $\mathbf {w} \in S_{pm} \subset S^{(1)}$ satisfies $\phi \left ( E_i^{(pn)} \mathbf {w} \right ) = E_i^{(n)} \phi ( \mathbf {w} )$ for $i \in \mathtt I_{\mathrm {af}}$ and $n \ge 0$. Using the natural nondegenerate invariant pairing $\left \langle \bullet , \bullet \right \rangle $ between $S_m^{\vee }$ and $S_m$, we compute the leftmost term of

$$ \begin{align*}\left\langle \mathbf{v}, \phi \left( E^{(p)}_i \mathbf{w} \right) \right\rangle = \left\langle \mathbf{v}, E_i \phi ( \mathbf{w} ) \right\rangle = - \left\langle \phi^{\vee} ( E_i \mathbf{v} ), \mathbf{w} \right\rangle\end{align*} $$

by using the invariance under the corresponding unipotent action, yielding

$$ \begin{align*} \left\langle \mathbf{v}, \phi \left( E^{(p)}_i \mathbf{w} \right) \right\rangle & = - \sum_{k_1 = 1}^{p} \left\langle E^{\left(k_1\right)}\phi^{\vee} ( \mathbf{v} ), E^{\left(p-k_1\right)}_i \mathbf{w} \right\rangle\\ & = \sum_{m=1}^p \sum_{k_\bullet> 0, \; k_1 + k_2 +\cdots + k_m = p} (-1)^m \left\langle E^{\left(k_1\right)}_i E^{\left(k_2\right)}_i \cdots \phi^{\vee} ( \mathbf{v} ), \mathbf{w} \right\rangle\\ & = - \left\langle E^{(p)} \phi^{\vee} ( \mathbf{v} ), \mathbf{w} \right\rangle, \end{align*} $$

since we have $E^{\left (k_1\right )}_i E^{\left (k_2\right )}_i \cdots E^{\left (k_m\right )}_i \in p \mathbb {Z} E^{(p)}_i$ except for $k_1 = p$, $0 = k_2 = \cdots $. This implies the case when $n = 1$.

Similarly, we have

$$ \begin{align*}\left\langle \mathbf{v}, \phi \left( E^{(pn)}_i \mathbf{w} \right) \right\rangle = \sum_{m=1}^n \sum_{k_\bullet> 0, \; k_1 + k_2 +\cdots + k_m = n} (-1)^m \left\langle E^{\left(pk_1\right)}_i E^{\left(pk_2\right)}_i \cdots \phi^{\vee} ( \mathbf{v} ), \mathbf{w} \right\rangle.\end{align*} $$

Compared with

$$ \begin{align*}\left\langle \mathbf{v}, E^{(n)}_i \phi ( \mathbf{w} ) \right\rangle = \sum_{m=1}^n \sum_{k_\bullet> 0, \; k_1 + k_2 +\cdots + k_m = n} (-1)^m \left\langle \phi^{\vee} \left( E^{\left(k_1\right)}_i E^{\left(k_2\right)}_i \cdots\mathbf{v} \right), \mathbf{w} \right\rangle\end{align*} $$

using induction on n, we conclude the result.

3.2 Representations of affine Lie algebras over $\mathbb {Z}$

In this section, we systematically use global basis theory [Reference Kashiwara41, Reference Kashiwara43, Reference Kashiwara44, Reference Kashiwara45, Reference Lusztig65, Reference Grojnowski and Lusztig35] by specialising the quantum parameter $\mathsf q$ to $1$. Therefore, we might refer to these references without an explicit declaration that we specialise $\mathsf q$.

We consider the Kostant–Lusztig $\mathbb {Z}$-form $U ^+ _{\mathbb {Z}}$ (resp., $U^-_{\mathbb {Z}}$) of $U ( [\mathfrak {I},\mathfrak {I}] )$ (resp., $U ( [\mathfrak {I}^-,\mathfrak {I}^-] )$) obtained as the specialisation $\mathsf q = 1$ of the $\mathbb {Z} \left [\mathsf q,\mathsf q^{-1}\right ]$-integral form of the quantised enveloping algebras [Reference Lusztig66, §23.2].

Note that $U^{\pm }_{\mathbb {Z}}$ are equipped with the $\mathbb {Z}$-bases ${\mathbf B} ( \mp \infty )$ obtained by the specialisation $\mathsf q = 1$ of the lower global basis [Reference Kashiwara41] (see also [Reference Lusztig66, §25]). In view of [Reference Lusztig65, Reference Kashiwara43], we have an idempotent $\mathbb {Z}$-integral form

$$ \begin{align*} \dot{U}_{\mathbb{Z}} = \bigoplus_{\Lambda \in P^{\mathrm{af}}} U ^- _{\mathbb{Z}} U ^+ _{\mathbb{Z}} a_{\Lambda} \end{align*} $$

such that

$$ \begin{align*} a_{\Lambda} a_{\Gamma} = \delta_{\Lambda, \Gamma} a_{\Lambda}, \quad \Lambda, \Gamma \in P^{\mathrm{af}}, \end{align*} $$

and

$$ \begin{align*} E_i^{(m)} a_{\Lambda} = a_{\Lambda + m \alpha_i} E_i^{(m)}, \qquad F_i^{(m)} a_{\Lambda} = a_{\Lambda - m \alpha_i} F_i^{(m)}, \quad i \in \mathtt I_{\mathrm{af}}, m \in \mathbb{Z}_{\ge 0}. \end{align*} $$

We set $\dot {U}^{\ge 0}_{\mathbb {Z}} \subset \dot {U}_{\mathbb {Z}}$ to be the subalgebra generated by $\left \{F_i^{(m)}\right \}_{i \in \mathtt I, m \in \mathbb {Z}_{\ge 0}}$, $\{a _{\Lambda }\}_{\Lambda \in P^{\mathrm {af}}}$ and $U_{\mathbb {Z}}^+$.

If a $\dot {U}_{\mathbb {Z}}$-module M admits a decomposition

$$ \begin{align*}M = \bigoplus_{\Lambda \in P^{\mathrm{af}}} a_{\Lambda} M,\end{align*} $$

then we call this the $P^{\mathrm {af}}$-weight decomposition. If $\Lambda \in P^{\mathrm {af}}$ satisfies $a_{\Lambda } M \neq 0$, then we call $\Lambda $ a $P^{\mathrm {af}}$-weight of M. When M is defined over a field $\Bbbk $, we define the $P^{\mathrm {af}}$-character of M as

$$ \begin{align*}\mathrm{gch}\, M := \sum _{\Lambda \in P^{\mathrm{af}}} e^{\Lambda} \dim_{\Bbbk} a_{\Lambda} M\end{align*} $$

whenever the right-hand side makes sense. For each $n \in \mathbb {Z}$, we set

$$ \begin{align*}M_n := \sum_{\Lambda \in P^{\mathrm{af}}, \; \left\langle d, \Lambda \right\rangle = n} a_\Lambda M \subset M\end{align*} $$

and call it the d-degree n-part of M. Note that these are consistent with formula (2.3) through the identification $q = e^{\delta }$.

For each $\lambda \in P$, we set

(3.2)$$ \begin{align}a_{\lambda}^0 M := \sum_{\Lambda \in P^{\mathrm{af}}, \; \lambda = \bar{\Lambda}} a_{\Lambda} M.\end{align} $$

We call the decomposition

$$\begin{align*}M = \bigoplus_{\lambda \in P} a_{\lambda}^0 M\end{align*}$$

the P-weight decomposition. We call a nonzero element of $a_{\Lambda } M$ (resp., $a^0 _\lambda M$) a $P^{\mathrm {af}}$-weight vector of M (resp., a P-weight vector of M). We also call $\lambda \in P$ with $a_{\lambda }^0 M \neq \{ 0 \}$ a P-weight of M.

Similarly, we have the Kostant–Lusztig $\mathbb {Z}$-form $U ^{0, +} _{\mathbb {Z}}$ (resp., $U^{0,- }_{\mathbb {Z}}$) of $U ( \mathfrak {n} )$ (resp., $U ( \mathfrak {n}^- )$). We have $U^{0,+}_{\mathbb {Z}} \subset U^{+}_{\mathbb {Z}}$ and $U^{0,-}_{\mathbb {Z}} \subset U^{-}_{\mathbb {Z}}$. In view of the characterisation of global bases ([Reference Kashiwara41]), we find that ${\mathbf B}^0 ( \mp \infty ) := {\mathbf B} ( \mp \infty ) \cap U^{0,\pm }_{\mathbb {Z}}$ define $\mathbb {Z}$-bases of $U^{0,\pm }$.

We set $\dot {U}^{0}_{\mathbb {Z}} \subset \dot {U}_{\mathbb {Z}}$ to be the subalgebra of $\dot {U}_{\mathbb {Z}}$ generated by $\left \{E_i^{(m)}, F_i^{(m)}\right \}_{i \in \mathtt I, m \in \mathbb {Z}_{\ge 0}}$, $\{a _{\Lambda } \}_{\Lambda \in P^{\mathrm {af}}}$. For a field $\Bbbk $, a $\dot {U}_{\Bbbk }^{\ge 0}$-module M with a $P^{\mathrm {af}}$-weight decomposition is said to be $\dot {U}^0_\Bbbk $-integrable if its $\left \{ E^{(m)}_i, F^{(m)}_i \right \}_{m \ge 0}$-action induces an $\mathop {\textit{SL}} ( 2, i )_\Bbbk $-action whose $( \mathop {\textit{SL}} ( 2, i ) \cap H )_\Bbbk $-eigenvalues are given by the P-weights for each $i \in \mathtt I$.

Note that if a $U \left ( \widetilde {\mathfrak {g}}_{\mathbb C} \right )$-module V over ${\mathbb C}$ carries a cyclic $\widetilde {\mathfrak {h}}_{\mathbb C}$-weight vector whose weight belongs to $P^{\mathrm {af}}$ and each of its $\widetilde {\mathfrak {h}}_{\mathbb C}$-weight spaces is finite-dimensional, then we have a $\dot {U}_{\mathbb {Z}}$-lattice $V_{\mathbb {Z}}$ inside V. In such a case, the module $V_{\mathbb {Z}} \otimes _{\mathbb {Z}} \Bbbk $ admits $P^{\mathrm {af}}$- or P-weight decompositions.

The Chevalley involution of $\dot {U}_{\mathbb {Z}}$ is defined as

$$ \begin{align*}\theta \left( E_i ^{(m)}\right) = F_i ^{(m)}, \theta \left( F_i ^{(m)}\right) = E_i ^{(m)} \quad \text{and} \quad \theta ( a _{\Lambda} ) = a_{- \Lambda}, \quad i \in \mathtt I_{\mathrm{af}}, m \in \mathbb{Z}_{\ge 0}, \Lambda \in P^{\mathrm{af}}.\end{align*} $$

Definition 3.10 [Reference Kashiwara45]

A $U \left ( \widetilde {\mathfrak {g}}_{{\mathbb C}} \right )$-module V over ${\mathbb C}$ with a cyclic $\widetilde {\mathfrak {h}}_{{\mathbb C}}$-weight vector $\mathbf {v}$ is said to be compatible with the negative global basis if we have

$$ \begin{align*}U ^{-}_{\mathbb{Z}} \mathbf{v} = \bigoplus _{b \in {\mathbf B} ( \infty )} \mathbb{Z} b \mathbf{v} \subset V.\end{align*} $$

If $(V,\mathbf {v})$ is compatible with the negative global basis, then we set

$$ \begin{align*}{\mathbf B} ^- ( V ) = {\mathbf B} ^{-} ( V, \mathbf{v} ):= \{ b\mathbf{v} \mid b \in {\mathbf B} ( \infty ) \text{ such that } b\mathbf{v} \neq 0 \} \subset V\end{align*} $$

and refer to them as the negative global basis of V.

Compatibility with the positive global basis and the positive global basis ${\mathbf B} ^{+} ( V ) = {\mathbf B} ^{+} ( V, \mathbf {v} )$ of V is defined similarly.

We set ${\mathbf B} ( \Lambda ) := {\mathbf B}^- ( L ( \Lambda )_{\mathbb C}, \mathbf {v}_{\Lambda } )$ for each $\Lambda \in P^{\mathrm {af}}_+$.

For each $\Lambda \in P^{\mathrm {af}}_+$ and $\lambda \in P_+$, we set

$$ \begin{align*}L ( \Lambda )_{\mathbb{Z}} := U _{\mathbb{Z}}^- \mathbf{v}_{\Lambda} \subset L ( \Lambda )_{\mathbb C} \quad \text{and} \quad V ( \lambda )_{\mathbb{Z}} := \left( U _{\mathbb{Z}}^{0,-} \right) \mathbf{v}_{\lambda}^0 \subset V ( \lambda )_{\mathbb C}.\end{align*} $$

Here $V ( \lambda )_{\mathbb {Z}}$ acquires the action of $\dot {U}^0_{\mathbb {Z}}$ thanks to the splitting $P \hookrightarrow P^{\mathrm {af}}$.

For each $\lambda \in P_+$, we set

$$ \begin{align*}\mathbb{X} ( \lambda ) _{\mathbb{Z}} := \dot{U}_{\mathbb{Z}} \mathbf{v}_{\lambda} \subset \mathbb{X} ( \lambda )_{\mathbb C}.\end{align*} $$

For each $\lambda \in P_+$ and $w \in W_{\mathrm {af}}$, we define

$$ \begin{align*}\mathbb{W} _w ( \lambda )_{\mathbb{Z}} := U_{\mathbb{Z}}^+ \mathbf{v}_{w \lambda} \subset \mathbb{X} ( \lambda )_{\mathbb C} \quad \text{and} \quad \mathbb{W} _w^- ( \lambda )_{\mathbb{Z}} := U_{\mathbb{Z}}^- \mathbf{v}_{w \lambda} \subset \mathbb{X} ( \lambda )_{\mathbb C} .\end{align*} $$

We set $\mathbb {W} ( \lambda )_{\mathbb {Z}} := \mathbb {W} _{w_0} ( \lambda )_{\mathbb {Z}}$ and $\mathbb {W} ^-( \lambda )_{\mathbb {Z}} := \mathbb {W} _{e} ^- ( \lambda )_{\mathbb {Z}}$.

Proof. We borrow the setting of [Reference Kashiwara43, §8.1 and §8.2].

We prove the first assertion. Since $\mathbf {v}_{\lambda }$ and $\mathbf {v}_{t_{\beta } \lambda }$ obey the same relation, $\tau _\beta $ defines an automorphism as $\widetilde {\mathfrak {g}}_{\mathbb C}$-modules. The latter assertion follows from Theorem 3.14.

We prove the second assertion. The defining equation of $\theta ^* ( \mathbf {v}_{\lambda } )$ is the same as the cyclic vector $\mathbf {v}_{- w_0 \lambda } \in \mathbb {X} ( - w_0 \lambda )_{\mathbb C}$ as $\widetilde {\mathfrak {g}}_{\mathbb C}$-modules. This yields a $\widetilde {\mathfrak {g}}_{\mathbb C}$-module isomorphism $\eta : \theta ^* ( \mathbb {X} ( \lambda )_{\mathbb C} ) \longrightarrow \mathbb {X} ( - w_0 \lambda )_{\mathbb C}$. Since $\theta $ exchanges $U^\pm _{\mathbb {Z}}$ and $\mathbf {v}_\lambda $ is cyclic, we deduce that $\eta ( \theta ^* ( \mathbb {X} ( \lambda )_{\mathbb {Z}} ) ) = \dot {U}_{\mathbb {Z}} \mathbf {v}_{- \lambda } \subset \mathbb {X} ( - w_0 \lambda )_{\mathbb C}$. By Theorem 3.14, we conclude $\theta ^* {\mathbf B} ( \mathbb {X} ( \lambda ) ) = {\mathbf B} ( \mathbb {X} ( - w_0 \lambda ) )$, as required.

We prove the third assertion. By Theorem 3.14, the $\mathbb {Z}$-basis of $\mathbb {W} ( \lambda ) _{\mathbb {Z}}$ is formed by the nonzero elements of ${\mathbf B} ( - \infty ) \mathbf {v}_{w_0 \lambda }$ and forms a direct summand of $\mathbb {X} ( \lambda )_{\mathbb {Z}}$ as $\mathbb {Z}$-modules. Hence, the case where $w = w_0$ follows. For $w \in W$, we apply [Reference Kashiwara43, Lemma 8.2.1] repeatedly to deduce the assertion from the $w = w_0$ case by using ${\mathbf B} ( - \infty ) \mathbf {v}_{w \lambda } \subset {\mathbf B} ( - \infty ) \mathbf {v}_{w_0 \lambda }$. For $w = u t_{\beta } \in W_{\mathrm {af}}$ with $u \in W, \beta \in Q^{\vee }$, we additionally apply $\tau _{w_0 \beta }$ to conclude the assertion.

We prove the fourth assertion. The vector $\theta ^* ( \mathbf {v}_{w \lambda } )$ is a $U_{\mathbb {Z}}^-$-cyclic vector of $\theta ^* (\mathbb {W} _w ( \lambda ) _{\mathbb {Z}} )$, and its weight is

$$ \begin{align*}- w \lambda = ww_0 (- w_0 \lambda ).\end{align*} $$

Hence, we conclude the assertion (using the fact that $\theta $ is an involution).

Theorem 3.18 [Reference Kashiwara44] and [Reference Beck and Nakajima5]

Set $\lambda \in P_+$. The unique (d-degree $0$) $\dot {U}^{\ge 0}_{\mathbb {Z}}$-module map

(3.3)$$ \begin{align} \Psi_\lambda : \mathbb{W} ( \lambda )_{\mathbb{Z}} \hookrightarrow \bigotimes _{i \in \mathtt I} \mathbb{W} ( \varpi_i )_{\mathbb{Z}}^{\otimes \left\langle \alpha_i^{\vee}, \lambda \right\rangle}, \end{align} $$

which sends $\mathbf {v}_\lambda $ to the tensor product of $\left \{\mathbf {v}_{\varpi _i}\right \}_{i \in \mathtt I}$s, is injective and defines a direct summand as $\mathbb {Z}$-modules.

3.3 Projectivity of the module $\mathbb {W} ( \lambda )_{\Bbbk }$

Let $\Bbbk $ be a field. We equip $\mathring {U}^+_\Bbbk := \bigoplus _{\lambda \in P} U^+_\Bbbk a_{\lambda }^0$ with the structure of an algebra by setting $a_{\lambda }^0 a_{\mu }^0 = \delta _{\lambda , \mu } a_{\lambda }^0 (\lambda , \mu \in P)$,

$$ \begin{align*}E_0^{(m)} a_{\lambda}^0 = a_{\lambda - m \vartheta}^0 E_0^{(m)}, \qquad E_i^{(m)} a_{\lambda}^0 = a_{\lambda + m \alpha_i}^0 E_i^{(m)} \quad (i \in \mathtt I, m \in \mathbb{Z}_{\ge 0}, \lambda \in P),\end{align*} $$

and require

$$ \begin{align*}\left( \xi a_{\lambda}^0 \right) \left( \xi' a_{\mu}^0 \right) = \left( \xi \xi' \right) a_{\mu}^0 \quad \text{when} \quad a_{\lambda}^0 \xi' a_{\mu}^0 = \xi' a_{\mu}^0 \quad \left( \lambda, \mu \in P, \xi, \xi' \in U^+_{\Bbbk} \right).\end{align*} $$

Similarly, we define

$$ \begin{align*}\mathring{U}^{\ge 0}_\Bbbk:= \bigoplus_{\lambda \in P} U^{0,-}_{\Bbbk} U^{+}_\Bbbk a_{\lambda}^0 \quad \text{and} \quad \mathring{U}^{0}_\Bbbk:= \bigoplus_{\lambda \in P} U^{0,-}_{\Bbbk} U^{0,+}_\Bbbk a_{\lambda}^0\end{align*} $$

and regard them as subalgebras of the completions of $\dot {U}^{\ge 0}_\Bbbk $ and $\dot {U}^{0}_\Bbbk $ with respect to the idempotents by setting

$$ \begin{align*}\sum_{\lambda = \overline{\Lambda}}a_{\Lambda} = a_{\lambda}^0 \quad \left(\lambda \in P \subset P^{\mathrm{af}}\right).\end{align*} $$

The eigendecomposition with respect to $d \in \widetilde {\mathfrak {h}}$ now becomes an external grading of algebras $\mathring {U}^+_\Bbbk $ and $\mathring {U}^{\ge 0}_\Bbbk $. We have algebra inclusions

$$ \begin{align*}\mathring{U}^+_\Bbbk \subset \mathring{U}^{\ge 0}_\Bbbk \supset \mathring{U}^0_\Bbbk.\end{align*} $$

Note that we have a surjective algebra map $\dot {U}^{\ge 0}_{\mathbb {Z}} \rightarrow \dot {U}^0_{\mathbb {Z}}$, since $a_{\Lambda } \dot {U}^{\ge 0}_{\mathbb {Z}} a_{\Gamma } \neq 0$ implies $\left \langle d, \Lambda \right \rangle \ge \left \langle d, \Gamma \right \rangle $ and $a_{\Lambda } \dot {U}^{0}_{\mathbb {Z}} a_{\Gamma } \neq 0$ implies $\left \langle d, \Lambda \right \rangle = \left \langle d, \Gamma \right \rangle $ for $\Lambda , \Gamma \in P^{\mathrm {af}}$. This induces a surjective algebra map $\mathring {U}^{\ge 0}_{\mathbb {Z}} \rightarrow \mathring {U}^0_{\mathbb {Z}}$, which specialises to $\mathring {U}^{\ge 0}_\Bbbk \rightarrow \mathring {U}^0_\Bbbk $.

We sometimes regard $\mathbb {W} ( \lambda )_{\Bbbk }$ ($\lambda \in P_+$) as a $\mathbb {Z}$-graded $\mathring {U}^+_\Bbbk $-module or a $\mathbb {Z}$-graded $\mathring {U}^{\ge 0}_\Bbbk $-module by means of (3.2), whose gradings are given by the d-degrees.

Proof of Lemma 3.20. Note that the endomorphism $\tau _{\beta }$ in Lemma 3.17 induces an endomorphism of $\mathbf B ( \mathbb X ( \varpi_i ))$ for each $\beta \in Q$ and $i \in \mathtt I$, and hence we know the existence of the injective endomorphisms of $\mathbb {W} ( \lambda )_{\Bbbk }$ that form a polynomial algebra whose graded dimension is equal to

$$ \begin{align*}\mathrm{gdim}\, \mathrm{End}_{\mathfrak{g} [z]_{\mathbb C}} \mathbb{W} ( \lambda )_{\mathbb C} = \prod_{i \in \mathtt I} \prod_{j=1}^{\left\langle \alpha_i^{\vee}, \lambda \right\rangle}\frac{1}{1-q^j},\end{align*} $$

which can be read off from [Reference Beck and Nakajima5, Proposition 4.13] (or [Reference Kato46, Theorem 1.4]), through Theorem 3.18. By [Reference Beck and Nakajima5, Reference Chari and Ion18, Reference Kato46], it exhausts the P-weight $\lambda $-part of $\mathbb {W} ( \lambda )_{\Bbbk }$, which generates $\mathbb {W} ( \lambda )_{\Bbbk }$ as a $\mathring {U}^{\ge 0}_\Bbbk $-module. Therefore, we conclude the result.

The rest of this subsection is devoted to the proof of Proposition 3.19.

We consider the Demazure functor $\mathscr D_{w}$ for $w \in W_{\mathrm {af}}$ with respect to $\mathring {U}^+_\Bbbk $ (compare [Reference Joseph39, Reference Kato46, Reference Cherednik and Kato20]). In view of [Reference Kashiwara45, §2.8], the character part of the calculation in [Reference Kato46, Theorem 4.13] carries over to our setting, and hence we have

$$ \begin{align*}\mathbb L^{\bullet} \mathscr D_{t_{\beta}} ( \mathbb{W} ( \lambda )_{\Bbbk}) = \mathscr D_{t_{\beta}} ( \mathbb{W} ( \lambda )_{\Bbbk}) \cong \mathbb{W} ( \lambda )_{\Bbbk} \otimes_{\Bbbk} \Bbbk_{- \left\langle \beta, w_0 \lambda \right\rangle \delta}, \quad \beta \in Q^{\vee}_<.\end{align*} $$

From this, we also derive

$$ \begin{align*}\mathbb L^{\bullet} \mathscr D_{t_{\beta}} ( W ( \lambda )_{\Bbbk}) = \mathscr D_{t_{\beta}} ( W ( \lambda )_{\Bbbk}) \cong W ( \lambda )_{\Bbbk} \otimes_{\Bbbk} \Bbbk_{- \left\langle \beta, w_0 \lambda \right\rangle \delta}, \quad \beta \in Q^{\vee}_<,\end{align*} $$

using the Koszul resolution as in [Reference Cherednik and Kato20, §5.1.4] by Lemma 3.20.

The d-gradings of $\mathring {U}^+_\Bbbk , \mathbb {W} ( \lambda )$ and $W ( \mu )_{\Bbbk }$ are concentrated in nonnegative degrees. It follows that the d-grading of $\mathrm {Ext}^{\bullet } _{\mathring {U}^+_\Bbbk } \left ( \mathbb {W} ( \lambda ), W ( \mu )_{\Bbbk }^* \right )$ is bounded from above. Moreover, we have

$$ \begin{align*}\mathrm{Ext}^{\bullet} _{\mathring{U}^+_\Bbbk} \left( \mathscr D_{t_{\beta}w_0} ( \mathbb{W} ( \lambda )_{\Bbbk}), W ( \mu )_{\Bbbk}^* \right) \cong \mathrm{Ext}^{\bullet} _{\mathring{U}^+_\Bbbk} \left( \mathbb{W} ( \lambda )_{\Bbbk}, \mathscr D_{t_{-w_0 \beta}w_0} \left( W ( \mu )_{\Bbbk} \right)^* \right)\end{align*} $$

for every $\lambda , \mu \in P_+$ and $\beta \in Q^{\vee }_<$ by repeated application of [Reference Feigin, Kato and Makedonskyi25, Proposition 5.7] (the argument of which carries over to our setting). By varying $\beta $, we conclude that

(3.4)$$ \begin{align} \mathrm{Ext}^{\bullet} _{\mathring{U}^+_\Bbbk} \left( \mathbb{W} ( \lambda )_{\Bbbk}, W ( \mu )_{\Bbbk}^* \right) \equiv \{ 0 \}, \quad \lambda \neq -w_0 \mu, \end{align} $$

since $- \left \langle \beta , w_0 \lambda \right \rangle \neq - \left \langle - w_0 \beta , w_0 \mu \right \rangle = \left \langle \beta , \mu \right \rangle $ for some choice of $\beta $.

Consider the simple integrable $\mathring {U}_{\Bbbk }^0$-module quotient $L ( \lambda )_{\Bbbk }$ of $V ( \lambda )_{\Bbbk }$ for each $\lambda \in P_+$.

We return to the proof of Proposition 3.19. We have

$$ \begin{align*}\mathrm{gch}\, W ( \lambda )_{\Bbbk} \equiv \mathrm{gch}\, V ( \lambda )_{\Bbbk} \equiv \mathrm{gch}\, L ( \lambda )_{\Bbbk} \mod \sum_{\lambda> \mu \in P_+} \mathbb{Z} [q] \mathrm{gch}\, L ( \mu )_{\Bbbk}\end{align*} $$

by [Reference Lenart, Naito, Sagaki, Schilling and Shimozono62] (compare [Reference Chari and Ion18]). Therefore, $W ( \lambda )_{\Bbbk }$ admits a (d-graded) Jordan–Hölder series as a $\mathring {U}^{\ge 0}_\Bbbk $-module whose irreducible constituents are of the form $\{L ( \mu )_{\Bbbk }\}_{\mu \le \lambda }$ (up to d-grading shifts). It follows that

$$ \begin{align*}\mathrm{Ext}^{\bullet} _{\mathring{U}^+_\Bbbk} ( \mathbb{W} ( \lambda )_{\Bbbk}, V ( \mu )_{\Bbbk} ) = \mathrm{Ext}^{\bullet} _{\mathring{U}^+_\Bbbk} ( \mathbb{W} ( \lambda )_{\Bbbk}, L ( \mu )_{\Bbbk} ) \equiv \{ 0 \}, \quad \lambda> \mu \in P_+,\end{align*} $$

by a repeated application of the short exact sequences to formula (3.4). Since both $\mathbb {W} ( \lambda )_{\Bbbk }$ and $L ( \lambda )_{\Bbbk }$ are $\dot {U}^0_\Bbbk $-integrable, we find

$$ \begin{align*}\mathrm{Ext}^{1} _{\mathring{U}^{\ge 0}_\Bbbk} ( \mathbb{W} ( \lambda )_{\Bbbk}, L ( \mu )_{\Bbbk} ) \equiv \{ 0 \}, \quad \lambda> \mu \in P_+.\end{align*} $$

From these, it suffices to prove

(3.5)$$ \begin{align} \mathrm{Ext}^{1} _{\mathring{U}^{\ge 0}_\Bbbk} ( \mathbb{W} ( \lambda )_{\Bbbk}, L ( \lambda )_{\Bbbk} ) \equiv \{ 0 \}, \quad \lambda \in P_+, \end{align} $$

to deduce the assertion.

In view of the fact that $\lambda + \alpha _i$ ($i \in \mathtt I$) is not a P-weight of $\mathbb {W} ( \lambda )_{\Bbbk }$, the Drinfeld presentation of $\mathring {U}^{\ge 0}_\Bbbk $ ([Reference Beck and Nakajima5, §3]) forces the space of $\mathring {U}^{\ge 0}_\Bbbk $-module endomorphisms of $\mathbb {W} ( \lambda )_{\Bbbk }$ to be generated by the images of the imaginary weight vectors $\tilde {P}_{i,m\delta }$ for $i \in \mathtt I$ and $m> 0$ (defined in [Reference Beck and Nakajima5, (3.7)]). In view of its descriptions ([Reference Beck and Nakajima5, Proposition 3.17] or [Reference Chari, Fourier and Khandai17, Lemma 4.5]), we find that $\tilde {P}_{i,m}$ acts on $\mathbf {v}_\lambda \in \mathbb {W} ( \lambda )_{\Bbbk }$ by zero if $\left \langle \alpha ^{\vee }_i, \lambda \right \rangle> m$. From this, we derive that the P-weight $\lambda $-part of $\mathbb {W} ( \lambda )_{\Bbbk }$ is maximal possible as a cyclic module with a cyclic vector of P-weight $\lambda $ in the category of $\mathring {U}_\Bbbk ^{\ge 0}$-modules that are $\mathring {U}_\Bbbk ^0$-integrable and whose P-weights are contained in $\mathrm {Conv} W \lambda \subset P \otimes _{\mathbb {Z}} \mathbb {R}$. Therefore, formula (3.5) vanishes and we conclude Proposition 3.19.

3.4 Frobenius splitting of $\mathbf {Q}_{G,\mathtt J}$

Proposition 3.25. For each $\Lambda , \Gamma \in P^{\mathrm {af}}_+$, we have the following commutative diagram of $U^-_{\mathbb {Z}}$-modules:

Moreover, all the maps define direct summands as $\mathbb {Z}$-modules.

Proof. The injectivity of the top horizontal arrow and the fact that it defines a direct summand as $\mathbb {Z}$-modules are Corollary 3.12.

The surjectivity of the vertical arrows is Theorem 3.23. Since they are obtained by annihilating parts of $\mathbb {Z}$-bases, these maps define direct summands as $\mathbb {Z}$-modules.

Since all the modules are generated by the cyclic vectors $\mathbf {v}_{\Lambda + \Gamma }$ or $\mathbf {v}_{\Lambda } \otimes \mathbf {v}_{\Gamma }$ as $\mathfrak {g} \left [z^{-1}\right ]_{\mathbb C}$-modules or $\mathfrak {g} \left [z^{-1}\right ]^{\oplus 2}_{\mathbb C}$-modules, Theorem 2.3 (twisted by $\theta $) implies the injectivity of $\mathsf {m}$ after extending the scalar to ${\mathbb C}$. Hence we deduce

(3.6)$$ \begin{align} \mathsf{m} \left( \mathbb{W}^- (\overline{\Lambda + \Gamma})_{\mathbb{Z}} \right) \subset \mathbb{W}^- (\overline{\Lambda})_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{W}^- (\overline{\Gamma})_{\mathbb{Z}}. \end{align} $$

Therefore, to complete the proof it suffices to prove that formula (3.6) has torsion-free cokernel (as a $\mathbb {Z}$-module). By a repeated use of formula (3.6), we arrive the setting of Theorem 3.18 in view of Theorem 2.3. Thus, the map $\mathsf {m}$ defines a direct summand of $\mathbb {W} (\overline{\Lambda})_{\mathbb {Z}} \otimes _{\mathbb {Z}} \mathbb {W} (\overline{\Gamma})_{\mathbb {Z}}$ as $\mathbb {Z}$-modules.

Corollary 3.26. For each $\lambda , \mu \in P_+$ and $w \in W$, we have the following commutative diagram of $U_{\mathbb {Z}}^+$-modules:

The vertical inclusions are compatible with positive global basis, and the horizontal inclusions define direct summands as $\mathbb {Z}$-modules. In addition, all the inclusions commute with the automorphism $\tau _{\beta } \left (\beta \in Q^{\vee }\right )$ of $\mathbb {X} ( \lambda )_{\mathbb {Z}}, \mathbb {X} ( \mu )_{\mathbb {Z}}$ and $\mathbb {X} ( \lambda + \mu )_{\mathbb {Z}}$.

Set $w \in W_{\mathrm {af}}$ and $\mathtt J \subset \mathtt I$. We define $P_+$- and $P_{\mathtt J,+}$- graded $\mathbb {Z}$-modules

$$ \begin{align*}R ^{\mathrm{af}} := \bigoplus_{\Lambda \in P_+^{\mathrm{af}}} L ( \Lambda )_{\mathbb{Z}} ^{\vee} \quad \text{and} \quad R _{w}( \mathtt J ) := \bigoplus_{\lambda \in P_{\mathtt J,+}} \mathbb{W}_{ww_0} ( \lambda )_{\mathbb{Z}} ^{\vee}.\end{align*} $$

Proof. The maps in Proposition 3.25 are characterised as the d-degree $0$ maps of cyclic $U^-_{\mathbb {Z}}$-modules, which are unique up to a scalar. Therefore, the composition

$$ \begin{align*}\mathbb{W} ( \lambda + \mu + \gamma )_{\mathbb{Z}} \hookrightarrow \mathbb{W} ( \lambda + \mu )_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{W} ( \gamma )_{\mathbb{Z}} \hookrightarrow \mathbb{W} ( \lambda )_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{W} ( \mu )_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{W} ( \gamma )_{\mathbb{Z}}\end{align*} $$

is the same map as

$$ \begin{align*}\mathbb{W} ( \lambda + \mu + \gamma )_{\mathbb{Z}} \hookrightarrow \mathbb{W} ( \lambda )_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{W} ( \mu + \gamma )_{\mathbb{Z}} \hookrightarrow \mathbb{W} ( \lambda )_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{W} ( \mu )_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{W} ( \gamma )_{\mathbb{Z}}\end{align*} $$

for every $\lambda , \mu , \gamma \in P_+$, as the images of the cyclic vectors are the same. Taking their restricted duals implies the associativity of the multiplication of $R _{e}( \mathtt J )$. In view of Lemma 3.17(1) and Corollary 3.26, we deduce the associativity of the multiplication of $R _{w}( \mathtt J )$ for each $w \in W_{\mathrm {af}}$. The associativity of $R ^{\mathrm {af}}$ is proved similarly (compare [Reference Kato49]). The commutativity of $R ^{\mathrm {af}}$ and $R _{w}( \mathtt J )$ follow, as the $\mathsf {q} = 1$ coproduct of $\dot {U}_{\mathbb {Z}}$ is symmetric ([Reference Lusztig66, Lemma 3.1.4, §23.1.5]).

We set $R := R_e$. Note that $R_{w} ( \mathtt J )\subset R_w$ is a subring. We also define

$$ \begin{align*}R^+ ( \mathtt J ) := \bigoplus_{\lambda \in P_{\mathtt J, +}} \mathrm{Span}_{\mathbb{Z}} \prod_{i \in \mathtt I} \left( \mathbb{X} ( \varpi_i )_{\mathbb{Z}}^{\vee} \right)^{\left\langle \alpha_i^{\vee}, \lambda \right\rangle} \subset \bigoplus_{\lambda \in P_+} \mathbb{X} ( \lambda )_{\mathbb{Z}} ^{\vee} =: \widetilde{R},\end{align*} $$

where the multiplication is defined through the projective limit formed by the duals of Corollary 3.26. Here we warn that the ($\mathbb {Z}$-)rank of some $P^{\mathrm {af}}$-weight space of $\mathbb {X} ( \lambda )_{\mathbb {Z}}$ can be infinity, and hence the inclusion $R^+ \subsetneq \widetilde {R}$ has a huge cokernel. Note that the rank of the $P^{\mathrm {af}}$-weight spaces of $\mathbb {X} ( \varpi _i )_{\mathbb {Z}}$ is bounded for each $i \in \mathtt I$ ([Reference Kashiwara44, Proposition 5.16]), and hence $R^+ ( \mathtt J )$ has only countably many generators of $P_{\mathtt J,+}$-degrees $\{ \varpi _i \}_{i \in \mathtt I \setminus \mathtt J}$.

By construction, the rings $R ^{\mathrm {af}}, R _{w}( \mathtt J )$ and $R^+ ( \mathtt J )$ are free over $\mathbb {Z}$.

For each $\lambda \in P_+$ and $w \in W_{\mathrm {af}}$, we have a unique $P^{\mathrm {af}}$-weight vector

(3.7)$$ \begin{align} \mathbf{v}_{w\lambda}^{\vee} \in \mathbb{W}_w ( \lambda )^{\vee}_{\mathbb{Z}} \end{align} $$

with paring $1$ with $\mathbf {v}_{w\lambda } \in \mathbb {W}_w ( \lambda )_{\mathbb {Z}}$. This vector $\mathbf {v}_{w\lambda }^{\vee }$ yields a $U^+_{\Bbbk }$-cocyclic vector of $\mathbb {W}_w ( \lambda )_{\Bbbk }^{\vee }$ for a field $\Bbbk $. By the construction of the ring structure on $R_w$, we have $\mathbf {v}_{ww_0\lambda }^{\vee } \cdot \mathbf {v}_{ww_0\mu }^{\vee } = \mathbf {v}_{ww_0(\lambda +\mu )}^{\vee }$ in $R_w$ for every $\lambda , \mu \in P_+$.

Lemma 3.30. We have morphisms of rings with $U^+_{\mathbb {Z}}$-actions

$$ \begin{align*}R^+ \longrightarrow\!\!\!\!\! \rightarrow R \hookrightarrow R^{\mathrm{af}},\end{align*} $$

that admit $\mathbb {Z}$-module splittings, where the $\dot {U}_{\mathbb {Z}}$-action on $R^{\mathrm {af}}$ is twisted by $\theta $.

For each $w \in W$ and $\mathtt J \subset \mathtt I$, we set

$$ \begin{align*}\left( \mathbf{Q}_{G, \mathtt J} ( w ) \right)_{\mathbb{Z}} := \mathrm{Proj}\, R_{w}( \mathtt J ) \quad \text{and} \quad \big( \mathbf{Q}_{G, \mathtt J}^{\mathrm{rat}} \big)_{\mathbb{Z}} := \bigcup _{w \in W} \left( \mathbf{Q}_{G, \mathtt J} ( w ) \right)_{\mathbb{Z}}.\end{align*} $$

These schemes and ind-schemes are flat over $\mathbb {Z}$.

Remark 3.31. In view of equation (2.5), we have embeddings

(3.8)

where we set

$$ \begin{align*}\mathbb{X} ( \varpi_i )_{{\mathbb K}}^{\wedge} := \prod_{n \in \mathbb{Z}} \mathbb{X} ( \varpi_i )_{n, {\mathbb K}} \quad \text{and} \quad \mathbb{X} ( \varpi_i )_{{\mathbb K}}^{\flat} := \bigcup_{m \in \mathbb{Z}} \prod_{n> m} \mathbb{X} ( \varpi_i )_{n, {\mathbb K}}\end{align*} $$

for each $i \in \mathtt I$. Here all the spaces in diagram (3.8) admit actions of $\mathop {\textit{SL}}( 2,i )_{{\mathbb K}}$ ($i \in \mathtt I$), while only the bottom two spaces admit actions of $G [\![z]\!]_{{\mathbb K}}$. Nevertheless, the top two spaces have some advantages, since they are $\theta $-stable (unlike the bottom two) and consequently also contain $\theta \left ( \big( \mathbf {Q}_{G}^{\mathrm {rat}} \big)_{{\mathbb K}} \right )$.

Proof. The $\mathbf {I}$-canonical Frobenius splitting $\phi $ of $R^{\mathrm {af}}$ gives rise to the following maps, whose composition is the identity:

$$ \begin{align*}L ( \Lambda )_{\mathbb F_p} \stackrel{\phi^{\vee}}{\longrightarrow} L ( p \Lambda )_{\mathbb F_p} \longrightarrow L ( \Lambda )_{\mathbb F_p}, \quad \Lambda \in P^{\mathrm{af}}_+.\end{align*} $$

In view of Proposition 3.25, this prolongs to

The right square is automatic (and is canonically defined) from the adjunction of the Frobenius push-forward (by taking the restricted dual). In order to show that $\phi $ descends to a Frobenius splitting of $R_{\mathbb F_p}$, it suffices to show that the dotted map $\phi ^{\vee }_{\mathbb {W}}$ is a well-defined linear map (induced from $\phi ^{\vee }$ and such that the left square is commutative).

By Corollary 3.24, $\ker \pi _{\Lambda }$ is generated by the P-weight $\left ( P \setminus \mathrm {Conv} W \bar {\Lambda } \right )$-part of $L ( \Lambda )_{\mathbb F_p}$. By the cyclicity of $L ( \Lambda )_{\mathbb F_p}$ as $U^-_{\mathbb {Z}}$-modules and Proposition 3.7, we deduce that $\phi ^{\vee } ( \ker \, \pi _{\Lambda } )$ is contained in the $U^-_{\mathbb F_p}$-submodule of $L ( p \Lambda )_{\mathbb F_p}$ generated by the P-weight $p \left ( P \setminus \mathrm {Conv} W \bar {\Lambda } \right )$-part of $L ( p \Lambda )_{\mathbb F_p}$. The latter is contained in $\ker \, \pi _{p\Lambda }$ by

$$ \begin{align*}p \left( P \setminus \mathrm{Conv} W \bar{\Lambda} \right) \subset P \setminus p \mathrm{Conv} W \bar{\Lambda}.\end{align*} $$

Therefore, we conclude that $\phi ^{\vee }_{\mathbb {W}}$ is a well-defined linear map, and hence $\theta ^* \left ( R_{\mathbb F_p} \right )$ admits a Frobenius splitting induced from $\phi $. The unipotent part of the $\mathbf {I}^-$-canonical splitting condition is in common with subrings. It remains to twist the grading given by $\alpha _0^{\vee }$ with that given by $- \vartheta ^{\vee }$ and twist the $\mathbf {I}^-$-action into the $\mathbf {I}$-action by $\theta $ to conclude that our Frobenius splitting on $R_{\mathbb F_p}$ is $\mathbf {I}$-canonical.

Proof. Since the case of $\mathtt J \neq \emptyset $ follows by the restriction to a part of the $P_+$-grading, we concentrate on the case where $\mathtt J = \emptyset $.

The ring structure of $R^+_{\mathbb F_p}$ is determined by $R_{\mathbb F_p}$ through the application of $U^-_{\mathbb {Z}}$ before taking duals. By Corollary 3.36 (and its proof), it defines an $\mathbf {I}$-canonical splitting $\phi $ of $R^+_{\mathbb F_p}$ compatible with $( R_w )_{\mathbb F_p}$ for each $w \in W_{\mathrm {af}}$. The ring $R^+_{\mathbb F_p}$ admits an $\mathop {\textit{SL}} ( 2, i )_{\mathbb F_p}$-action that integrates the actions of $E_i^{(n)}$ and $F_i^{(n)}$ ($n \in \mathbb {Z}_{>0}$) for each $i \in \mathtt I_{\mathrm {af}}$. By [Reference Brion and Kumar14, Proposition 2.10], this splitting $\phi $ is also $\mathbf {I}^-$-canonical.

The $\mathbf {I}$-cocyclic $P^{\mathrm {af}}$-weight vector $\mathbf {v}_{w w_0\lambda }^{\vee } \in \mathbb {W} _{ww_0} ( \lambda )^{\vee }_{\mathbb F_p}$ is uniquely characterised by its $P^{\mathrm {af}}$-weight. Hence we obtain a map

$$ \begin{align*}\mathbb{X} ( \lambda )^{\vee}_{\mathbb F_p} \supset \mathrm{Span}_{\mathbb{Z}} \prod_{i \in \mathtt I} \left( \mathbb{X} ( \varpi_i)_{\mathbb F_p}^{\vee} \right)^{\left\langle \alpha_i, \lambda \right\rangle} \longrightarrow \!\!\!\!\! \rightarrow \mathbb{W} _{ww_0} ( \lambda )_{\mathbb F_p}^{\vee} \longrightarrow \!\!\!\!\! \rightarrow \mathbb F_p \mathbf{v}_{ww_0\lambda}^{\vee}.\end{align*} $$

It gives rise to the ring surjections

$$ \begin{align*}R^+_{\mathbb F_p} \longrightarrow \!\!\!\!\! \rightarrow ( R_w )_{\mathbb F_p} \longrightarrow \!\!\!\!\! \rightarrow \bigoplus_{\lambda \in P_+} \mathbb F_p \mathbf{v}_{w w_0 \lambda}^{\vee},\end{align*} $$

which are compatible with $\phi $ by construction in the first surjection and by examining the $P^{\mathrm {af}}$-weights in the second (we denote this composition surjective ring map by $\xi $). Consider the ideal

$$ \begin{align*}I (w) := R^+_{\mathbb F_p} \cap \bigcap _{g \in \mathbf{I}^- \left( \overline{\mathbb F}_p \right)} g \left( \overline{\mathbb F}_p \otimes_{\mathbb F_p} \ker \xi \right)\subset R^+_{\overline{\mathbb F}_p}.\end{align*} $$

(Here the action of $\mathbf {I}^- \big( \overline {\mathbb F}_p \big)$ is obtained by the unipotent one-parameter subgroups $\left \{ \rho _{-\alpha _i}\right \}_{i \in \mathtt I_{\mathrm {af}}}$ defined through the exponentials, which are well defined, as we have all the divided powers. The passage from $\mathbb F_p$ to $\overline {\mathbb F}_p$ is necessary to ensure scheme-theoretic invariance, since we want an intersection that is equivalent to the geometric $\mathbb {G}_a$-actions through $\rho _{- \alpha _i}$ for all $i \in \mathtt I_{\mathrm {af}}$ by finding the Zariski dense subsets of $\mathbb {G}_a$. For this purpose, we want the image of each $\rho _{-\alpha _i}$ to be infinite, which cannot be achieved by sending $\mathbb {G}_a \left ( \mathbb F_p \right ) \cong \mathbb F_p$.) This ideal is the maximal $U_{\mathbb F_p}^-$-invariant ideal of $R^+_{\mathbb F_p}$ that is contained in $\ker \xi $. Let us denote the quotient ring by

$$ \begin{align*}Q = \bigoplus_{\lambda \in P_+} Q ( \lambda ) := R^+_{\mathbb F_p} / I ( w ).\end{align*} $$

By the construction of $I (w)$, we deduce that

$$ \begin{align*}U_{\mathbb F_p}^- \mathbf{v} _{w w_0 \lambda} \subset Q ( \lambda )^{\vee}\subset \mathbb{X} ( \lambda )_{\mathbb F_p}\end{align*} $$

for each $\lambda \in P_+$ (otherwise we can differentiate a vector in $I ( w )$ to obtain a nonzero element of $\bigoplus _{\lambda \in P_+} \mathbb F_p \mathbf {v}_{w w_0 \lambda }^{\vee }$). Since $\theta ^* \left ( \mathbb {W} _{ww_0} ( - w_0 \lambda )_{\mathbb F_p} \right )$ is a cyclic $U_{\mathbb F_p}^-$-submodule of $\mathbb {X} ( \lambda )_{\mathbb F_p}$, we have $U_{\mathbb F_p}^- \mathbf {v} _{w w_0 \lambda } = \theta ^* \left ( \mathbb {W} _{ww_0} ( - w_0 \lambda )_{\mathbb F_p} \right )$. In particular, we deduce a vector space surjection

$$ \begin{align*}R^+_{\mathbb F_p} / I ( w ) \longrightarrow \!\!\!\!\! \rightarrow \theta^* \left( \left( R _{ww_0} \right)_{\mathbb F_p} \right)\end{align*} $$

(compare Corollary 3.26). Since the right-hand side is naturally a ring, we conclude

$$ \begin{align*}R^+_{\mathbb F_p} / I ( w ) \cong \theta^* \left( \left( R _{ww_0} \right)_{\mathbb F_p} \right)\end{align*} $$

by the maximality of $I ( w )$.

The ideal $I (w) \subset R^+_{\mathbb F_p}$ also splits compatibly by $\phi $ (since $\phi $ is $\mathbf {I}^-$-canonical and $\ker \xi $ splits compatibly). In particular, each $\theta ^* \left ( \left ( R_{ww_0} \right )_{\mathbb F_p} \right )$ compatibly splits under $\phi $, as required.