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Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

Published online by Cambridge University Press:  13 July 2021

Syu Kato*
Affiliation:
Department of Mathematics, Kyoto University, Oiwake Kita-Shirakawa, Sakyo, Kyoto 606-8502, Japan; E-mail: syuchan@math.kyoto-u.ac.jp

Abstract

We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field ${\mathbb K}$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a $\mathbb {Z}$-model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. 371 no.2 (2018)]) when $\mathsf {char}\, {\mathbb K} =0$ or $\gg 0$, and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$. Some particular cases of these results play crucial roles in our proof [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.

Type
Algebra
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

1 Introduction

The semi-infinite flag varieties are variants of affine flag varieties that encode the modular representation theory of a semi-simple Lie algebra, representation theory of a quantum group at roots of unity and representation theory of an affine Lie algebra at the critical level. They originate from the ideas of Lusztig [Reference Lusztig64] and Drinfeld, put forward by Feigin and Frenkel [Reference Feigin and Frenkel24] and subsequently polished by the work of Braverman, Finkelberg, and their collaborators [Reference Finkelberg and Mirković29Reference Feigin, Finkelberg, Kuznetsov and Mirković23Reference Arkhipov, Bezrukavnikov, Braverman, Gaitsgory and Mirković2Reference Braverman7Reference Braverman, Feigin, Finkelberg and Rybnikov8Reference Braverman and Finkelberg9Reference Braverman and Finkelberg10Reference Braverman and Finkelberg11]. They (mainly) used the ind-model of semi-infinite flag varieties, and achieved spectacular success on the geometric Langlands correspondence [Reference Arkhipov, Bezrukavnikov, Braverman, Gaitsgory and Mirković2Reference Braverman, Finkelberg, Gaitsgory and Mirković12], the quantum K-groups of flag manifolds [Reference Braverman and Finkelberg9] and their (conjectural) relation to the finite $\mathcal W$-algebras [Reference Braverman, Feigin, Finkelberg and Rybnikov8].

In [Reference Kato, Naito and Sagaki51], we initiated the study of the formal model of a semi-infinite flag variety (over ${\mathbb C}$) that follows the classical description of flag varieties [Reference Kumar57Reference Mathieu69Reference Littelmann63Reference Kumar and Littelmann58] more closely than the works already cited. We refer to this formal model of a semi-infinite flag variety as a ‘semi-infinite flag manifold’, since we hope to justify that it is ‘smooth’ in a sense. However, the analysis in [Reference Kato, Naito and Sagaki51] has two defects: the relation with the ind-models of semi-infinite flag varieties is unclear, and the treatment there is rather ad hoc (it is just an ind-scheme whose set of ${\mathbb C}$-valued points have the desired property, and lacks a characterisation as a functor; compare [Reference Beauville and Laszlo3Reference Faltings22]). The first defect produces difficulty in discussing deep properties on the identification between the equivariant K-group of a semi-infinite flag manifold and the equivariant quantum K-group of a flag manifold [Reference Kato47], which is in turn inspired by the works of Givental and Lee [Reference Givental32Reference Givental and Lee33]. The goal of this paper is to study semi-infinite flag manifolds in characteristic $\neq 2$ from scratch, and resolve these defects. In particular, we verify that the scheme in [Reference Kato, Naito and Sagaki51] is universal one among all the ind-schemes with similar set-theoretic properties, and provide new proofs of the normality of Zastava spaces [Reference Braverman and Finkelberg9] and the semi-infinite flag manifolds [Reference Kato, Naito and Sagaki51].

It is possible to regard our work ([Reference Kato46Reference Kato49Reference Kato, Naito and Sagaki51Reference Kato47Reference Kato48]) as part of catch-up of Peterson’s original construction [Reference Peterson74] of his isomorphism [Reference Lam and Shimozono61] between the quantum cohomology of a flag manifold and the cohomology of an affine Grassmannian in the K-theoretic setting. From this viewpoint, this paper provides some varieties considered in [Reference Peterson74, Lecture 11] with their appropriate compactifications. Hence, though there are still some missing pieces to complete the original program along the lines in [Reference Peterson74], this paper provides a step to fully examine Peterson’s ideas.

To explain our results more precisely, we introduce more notation: Let $\mathfrak {g}$ denote a simple Lie algebra (given in terms of root data and the Chevalley generators) over an algebraically closed field ${\mathbb K}$ of characteristic $\neq 2$. Let G denote the connected simply connected algebraic group over ${\mathbb K}$ such that $\mathfrak {g} = \mathrm {Lie}\, G$. Let $H \subset G$ be a Cartan subgroup and N be an unipotent radical of G that is normalised by H. We set $B:= HN$ and $\mathscr {B} := G / B$ (the flag manifold of G). Let $\mathbf {I}^+ \subset G ( {\mathbb K} [\![ z ]\!] )$ denote the Iwahori subgroup that contains B, and let $\mathbf {I}^- \subset G \left ( {\mathbb K} \left [z^{-1}\right ]\right )$ be its opposite Iwahori subgroup. Let $\widetilde {\mathfrak {g}}$ denote the untwisted affine Kac–Moody algebra associated to $\mathfrak {g}$, and let W and $W_{\mathrm {af}}$ be the finite Weyl group and the affine Weyl group of $\mathfrak {g}$, respectively. The coroot lattice $Q^{\vee }$ of $\mathfrak {g}$ yields a natural subgroup $\left \{ t_{\beta } \right \}_{\beta \in Q^{\vee }} \subset W_{\mathrm {af}}$. Let $w_0 \in W$ be the longest element.

Our first main result is as follows:

Theorem A $\doteq $ Theorem 4.18 and Proposition 4.26

There is an ind-scheme $\mathbf {Q}_G ^{\mathrm {rat}}$ with the following properties:

  1. 1. It is expressed as the union of infinite-type integral schemes flat over $\mathbb {Z}$.

  2. 2. If we set $\big( \mathbf{Q}_G ^{\mathrm{rat}} \big)_{\mathbb K} := \mathbf {Q}_G ^{\mathrm {rat}} \otimes _{\mathbb {Z}} {\mathbb K}$, we have

    $$ \begin{align*}\left( \mathbf{Q}_G ^{\mathrm{rat}} \right)_{\mathbb K} ( {\mathbb K} ) \cong G ( {\mathbb K} (\!(z)\!)) / ( H ( {\mathbb K} ) N ( {\mathbb K} (\!(z)\!)) ),\end{align*} $$
    which intertwines the natural $G ( {\mathbb K} (\!(z)\!) ) \ltimes \mathbb {G}_m ( {\mathbb K} )$-actions on both sides, where $\mathbb {G}_m$ is the loop rotation.
  3. 3. The functor

    $$ \begin{align*}A\!f\!\!f^{op}_{{\mathbb K}} \ni R \mapsto G ( R (\!(z)\!)) / ( H ( R ) N ( R (\!(z)\!)) )\in \mathrm{Sets}\end{align*} $$
    is coarsely ind-representable by $\big( \mathbf{Q}_G ^{\mathrm{rat}} \big)_{\mathbb K}$ (see Section 4.3 for the convention).

One can equip $\big( \mathbf{Q}_G ^{\mathrm{rat}} \big)_{\mathbb K} ( {\mathbb K} )$ with an ind-scheme structure using the arc scheme of the basic affine space $\overline {G/N}$. Such an ind-scheme cannot coincide with ours (in general), by the appearance of the nontrivial nilradicals [Reference Mustata71Reference Feigin and Makedonskyi27Reference Feigin and Makedonskyi26]. In fact, such an ind-scheme defines a radicial thickening of $\big( \mathbf{Q}_G ^{\mathrm{rat}} \big)_{\mathbb K}$.

The set of $H \times \mathbb {G}_m$-fixed points of $\big( \mathbf{Q}_G ^{\mathrm{rat}} \big)_{\mathbb K}$ is in bijection with $W_{\mathrm {af}}$. Let $p_w \in \big( \mathbf{Q}_G ^{\mathrm{rat}} \big)_{\mathbb K}$ be the point corresponding to $w \in W_{\mathrm {af}}$. We set $\mathbb {O} ( w ) := \mathbf {I}^+ p_w$ and $\mathbb {O}^- ( w ) := \mathbf {I}^- p_w$ for each $w \in W_{\mathrm {af}}$.

Theorem A has some applications to the theory of quasi-map spaces from $\mathbb {P}^1$ to $\mathscr {B}$ presented in [Reference Finkelberg and Mirković29Reference Feigin, Finkelberg, Kuznetsov and Mirković23Reference Braverman and Finkelberg9Reference Braverman and Finkelberg10Reference Braverman and Finkelberg11] as follows:

Theorem B. In the foregoing settings, the following hold:

  1. 1. (Corollary 3.38 and Theorem 4.30) If $\mathsf {char}\, {\mathbb K}> 0$, then the scheme $\big( \mathbf{Q}_G ^{\mathrm{rat}} \big)_{\mathbb K}$ admits an $\mathbf {I}^{\pm }$-canonical Frobenius splitting that is compatible with $\overline {\mathbb {O} ( w )}$s and $\overline {\mathbb {O} ^- ( v )}$s $(w, v \in W_{\mathrm {af}})$.

  2. 2. (Corollary 4.10 and Remark 4.13) For each $w, v \in W_{\mathrm {af}}$, the intersection $\mathscr {Q} ( v, w ) := \overline {\mathbb {O} ( w )} \cap \overline {\mathbb {O} ^- ( v )}$ is reduced. It is irreducible when $v = w_0 t_{\beta }$ for some $\beta \in Q^{\vee }$.

  3. 3. (Lemma 4.7 and Corollary 5.24) For each $w, v \in W_{\mathrm {af}}$, the scheme $\mathscr {Q} ( v, w )$ is weakly normal. It is normal (and irreducible) when $\mathsf {char}\, {\mathbb K} = 0$ or $\mathsf {char}\, {\mathbb K} \gg 0$.

  4. 4. (Lemma 4.28) For each $\beta \in Q^{\vee }_+$, the set of ${\mathbb K}$-valued points of the scheme $\mathscr {Q} \left ( w_0 t_{\beta }, e \right )$ is in bijection with the set of (${\mathbb K}$-valued) Drinfeld–Plücker data. In particular, $\mathscr {Q} \left ( w_0 t_{\beta }, e \right )$ is isomorphic to the quasi-map space in [Reference Finkelberg and Mirković29] when ${\mathbb K} = {\mathbb C}$.

Theorem B is a key result at the deepest part (correspondence between natural bases) in our proof ([Reference Kato47]) of a conjecture of Lam, Li, Mihalcea and Shimozono [Reference Lam, Li, Mihalcea and Shimozono60] about the comparison between the equivariant K-group of the affine Grassmannian of G and the equivariant small quantum K-group of $\mathscr {B}$. In [Reference Kato47], we also prove that $\mathscr {Q} \left ( w_0 t_{\beta }, w \right )$ admits only rational singularities (and hence it is Cohen–Macaulay) when ${\mathbb K} = {\mathbb C}$ on the basis of Theorem B. We remark that Theorem B(3) is proved in [Reference Braverman and Finkelberg9Reference Braverman and Finkelberg10] when $v = w_0 t_{\beta }$, $w = e$ and ${\mathbb K} = {\mathbb C}$.

Our proof of Theorem B(1) is not at all standard, and in fact it forms the core of the technical contributions in this paper. To appreciate its contents, let us recall that there are two standard ways to construct a Frobenius splitting of $\mathscr {B}$ ([Reference Brion and Kumar14]): One is to consider the Bott–Samelson–Demazure–Hansen resolution of $\mathscr {B}$, which reduces the assertion to the case of a point, which is a Schubert variety with trivial Frobenius splitting. The other is to analyse the space of global sections of (some power of) the canonical bundle of $\mathscr {B}$.

However, neither of these proof strategies works for $\mathbf {Q}_G^{\mathrm {rat}}$. The first one fails because any Schubert variety of $\mathbf {Q}_G^{\mathrm {rat}}$ is infinite-dimensional, and carries rich internal structure by itself. The second one fails because the canonical bundle of $\mathbf {Q}_G^{\mathrm {rat}}$ simply does not make sense, at least naively. These require some new ideas to prove Theorem B(1). Our idea here is as follows: An interpretation of the filtrations in [Reference Kato and Loktev50] reduces the existence of a Frobenius splitting of $\mathbf {Q}_G^{\mathrm {rat}}$ to a property of the Frobenius splitting of the corresponding thick affine flag manifold [Reference Kato49, Corollary B]. This property can be seen as a special case of some homological property in the representation theory of affine Lie algebras ([Reference Chari and Ion18Reference Cherednik and Kato20]), but it is proved only for characteristic $0$. Thus, we use Kashiwara’s theory of global basis ([Reference Kashiwara43Reference Kashiwara44]) to transfer such a homological property into the positive-characteristic setting (Proposition 3.19).

In the rest of this introduction, we assume ${\mathbb K} = {\mathbb C}$ for the sake of simplicity. Let P be the weight lattice of H, and let $P_+ \subset P$ denote its subset corresponding to dominant weights. For each $\lambda \in P$, we have an equivariant line bundle ${\mathcal O}_{\mathbf {Q}_G^{\mathrm {rat}}} ( \lambda )$ on $\mathbf {Q}_G^{\mathrm {rat}}$, whose restriction to $\mathscr {Q} ( v, w )$ is denoted by ${\mathcal O}_{\mathscr {Q} \left ( v, w \right )} ( \lambda )$. Associated to $\lambda \in P_{+}$, we have a level $0$ extremal weight module $\mathbb {X} ( \lambda )$ of $U \left ( \widetilde {\mathfrak {g}} \right )$ in the sense of Kashiwara [Reference Kashiwara44]. We know that $\mathbb {X} ( \lambda )$ is equipped with two kinds of Demazure modules and a distinguished basis (the global basis).

Corollary C $\doteq $ Theorem 4.33

Set $w, v \in W_{\mathrm {af}}$. For each $\lambda \in P_{+}$, we have

$$ \begin{align*}H^{>0} \left( \mathscr{Q} ( v, w ), {\mathcal O}_{\mathscr{Q} \left( v, w \right)} ( \lambda )\right) = \{ 0 \}.\end{align*} $$

The space $H^0 \left ( \mathscr {Q} ( v, w ), {\mathcal O}_{\mathscr {Q} \left ( v, w \right )} ( \lambda )\right )^{\vee }$ is the intersection of two Demazure modules of $\mathbb {X} ( \lambda )$ spanned by a subset of the global basis of $\mathbb {X} ( \lambda )$ if $\lambda $ is strictly dominant. If we have $w',v' \in W_{\mathrm {af}}$ such that $\mathscr {Q} ( v',w' ) \subset \mathscr {Q} ( v, w )$, then the restriction map

$$ \begin{align*}H^0 \left( \mathscr{Q} ( v, w ), {\mathcal O}_{\mathscr{Q} \left( v, w \right)} ( \lambda )\right) \twoheadrightarrow H^0 \left( \mathscr{Q} ( v', w' ), {\mathcal O}_{\mathscr{Q} \left( v', w' \right)} ( \lambda )\right)\end{align*} $$

is surjective.

Note that Corollary C adds new vanishing region to [Reference Braverman and Finkelberg10, Theorem 3.1 1)]. We also provide parabolic versions of Theorems A and B and Corollary C. We have a description of $H^0 \left ( \mathscr {Q} ( v, w ), {\mathcal O}_{\mathscr {Q} \left ( v, w \right )} ( \lambda )\right )$ for general $\lambda \in P_+$ that is more complicated (Theorem B.6).

Let $\mathscr {B}_{2,\beta }$ be the space of genus $0$ stable maps with two marked points to $\mathscr {B}$ with the class of its image $\beta \in Q^{\vee }_+ \subset Q^{\vee } \cong H_2 ( \mathscr {B}, \mathbb {Z} )$. We have evaluation maps $\mathtt {e}_j : \mathscr {B}_{2,\beta } \to \mathscr {B}$ for $j = 1,2$. The following purely geometric result is a by-product of our proof that may be of independent interest:

Corollary D $\doteq $ Corollary 5.19

Set $\beta \in Q^{\vee }_+$ and $x, y \in \mathscr {B}$. The space $\left ( \mathtt {e}_1^{-1} ( x ) \cap \mathtt {e}_2^{-1} \left ( \overline {B y} \right ) \right )$ is connected if it is nonempty.

Note that Corollary D is contained in [Reference Buch, Chaput, Mihalcea and Perrin16] whenever $\{x\},\overline {By} \subset \mathscr {B}$ are in general position.

The plan of this paper is as follows: In Section 2, we collect basic material needed in what follows. In Section 3, after recalling generalities on Frobenius splitting and representation theory of quantum loop algebras, we construct the ind-scheme $\mathbf {Q}_G^{\mathrm {rat}}$ and equip it with a Frobenius splitting (Corollary 3.38). In Section 4, we first interpret $\mathbf {Q}_G^{\mathrm {rat}}$ as an ind-scheme (coarsely) representing the coset $G ( {\mathbb K} (\!(z)\!) ) / (H ( {\mathbb K} ) N ( {\mathbb K} (\!(z)\!) ))$ (Theorem A). Using this, we identify some Richardson varieties of $\mathbf {Q}_G^{\mathrm {rat}}$ with quasi-map spaces (Theorem 4.30) and present their cohomological properties (Theorem 4.33), and hence prove (large parts of) Theorem B and Corollary C. Since our construction equips quasi-map spaces with Frobenius splittings (Lemma 4.7), they are automatically weakly normal. Moreover, we explain how to connect characteristic $0$ and positive characteristic (Section 4.5). In Section 5, we analyse the fibres of the graph-space resolutions of quasi-map spaces and deduce that Richardson varieties of semi-infinite flag manifolds (over ${\mathbb C}$) are normal based on the weak normality proved in the previous section. This proves the remaining part of Theorem B. Our analysis here contains an inductive proof that the fibres of the evaluation maps of the space of genus $0$ stable maps to flag varieties are connected (Corollary D). In Appendix A, we give a proof of the normality of (the ind-pieces of) $\mathbf {Q}_G^{\mathrm {rat}}$ (which works also in the positive characteristic setting) and present an analogue of the Kempf vanishing theorem [Reference Kempf52] for $\mathbf {Q}_G^{\mathrm {rat}}$. Appendix B exhibits the structure of global sections of nef line bundles of Richardson varieties of $\mathbf {Q}_G^{\mathrm {rat}}$.

Note that Theorem B equips a quasi-map space from $\mathbb {P}^1$ to $\mathscr {B}$ with a Frobenius splitting compatible with the boundaries. However, the notion of a boundary in quasi-map spaces depends on a configuration of points in $\mathbb {P}^1$ (we implicitly set them to $\{0,\infty \} \subset \mathbb {P}^1$ throughout this paper). This makes our analogues of open Richardson varieties not necessarily smooth, contrary to the original case [Reference Richardson76] (see also [Reference Finkelberg and Mirković29, §8.4.1]). We hope to give a further account of this, as well as the factorisation structure ([Reference Finkelberg and Mirković29, §6.3]) from the viewpoint presented in this paper, in future work.

2 Preliminaries

We work over an algebraically closed field ${\mathbb K}$ unless stated otherwise. A vector space is a ${\mathbb K}$-vector space, and a graded vector space refers to a $\mathbb {Z}$-graded vector space whose graded pieces are finite-dimensional and whose grading is bounded from above or from below. Tensor products are taken over ${\mathbb K}$ unless specified otherwise.

Let A be a principal ideal domain. For a graded free A-module $M = \bigoplus _{m \in \mathbb {Z}} M_m$, we set $M^{\vee } := \bigoplus _{m \in \mathbb {Z}} \mathrm {Hom}_A ( M_m, A )$, where $\mathrm {Hom}_A ( M_m, A )$ is understood to have degree $-m$.

As a rule, we suppress $\emptyset $ and associated parentheses from notation. This particularly applies to $\emptyset = \mathtt J \subset \mathtt I$, frequently used to specify parabolic subgroups.

2.1 Groups, root systems and Weyl groups

We refer to [Reference Chriss and Ginzburg21Reference 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

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

Theorem 2.1 [Reference Lusztig64]; compare [Reference Lam and Shimozono61]

For each $w, v \in W_{\mathrm {af}}$ such that $w \le _{\frac {\infty }{2}} v$, there exists $\alpha \in \Delta _+^{\mathrm {af}}$ such that $ w \le _{\frac {\infty }{2}} s_{\alpha } v \le _{\frac {\infty }{2}} v$ and $\ell ^{\frac {\infty }{2}} ( s_{\alpha } v ) = \ell ^{\frac {\infty }{2}} ( v ) + 1$.

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 Kac40Reference 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 2.2 [Reference Lenart, Naito, Sagaki, Schilling and Shimozono62]; compare [Reference Kato46]

For each $\lambda \in P_+$, the $\mathfrak {I}$-action on $\mathbb {W} ( \lambda )$ prolongs to $\mathfrak {g} [z]$ and is isomorphic to the global Weyl module of $\mathfrak {g} [z]$ in the sense of Chari and Pressley [Reference Chari and Pressley19]. Moreover, $\mathbb {W} ( \lambda )$ is a projective module in the category of $\mathfrak {g}[z]$-modules whose restriction to $\mathfrak {g}$ is a direct sum of modules in $\{ V ( \mu )\}_{\mu \le \lambda }$.

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 Mirković29Reference Feigin, Finkelberg, Kuznetsov and Mirković23Reference Kato, Naito and Sagaki51Reference 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 Mirković29Reference 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 $\lambda , \mu \in P_+$.

Theorem 2.7 Drinfeld; see [Reference Finkelberg and Mirković29]

The variety $\mathscr {Q} ( \beta )$ is isomorphic to the variety formed by isomorphism classes of the DP-data $\mathcal L = \left \{\left ( \psi _{\lambda }, \mathcal L^{\lambda } \right ) \right \}_{\lambda \in P_+}$ such that $\deg \mathcal L ^{\lambda } = \left \langle w_0 \beta , \lambda \right \rangle $. In addition, $\mathscr {Q} ( \beta )$ is an irreducible variety of dimension $2 \left \langle \rho , \beta \right \rangle + \ell ( w_0 )$.

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 $\mathscr {Q} ( \beta ) = \mathscr {Q} ( \beta , e)$.)

Theorem 2.8 [Reference Finkelberg and Mirković29]

Let ${\mathbb K}$ be an algebraically closed field (not necessarily of characteristic $0$), and let $\mathscr {Q} ( \beta )_{{\mathbb K}}$ and $\mathscr {Z} ( \beta , w_0 )_{{\mathbb K}}$ be the spaces obtained by replacing the base field ${\mathbb C}$ with ${\mathbb K}$ in Definition 2.6. For each $\beta \in Q^{\vee }_+$, the space $\mathscr {Z} ( \beta , w_0 )_{{\mathbb K}}$ is an irreducible affine scheme equipped with an action of $( B \times \mathbb {G}_m )$ over ${\mathbb K}$. In addition, this action has a unique fixed point.

Proof Remarks on the proof.

Theorem 2.8 is proved in [Reference Finkelberg and Mirković29] for ${\mathbb K} = {\mathbb C}$ using [Reference Mirković and Vilonen70], and in the current setting in [Reference Braverman, Finkelberg, Gaitsgory and Mirković12] using [Reference Braverman and Gaitsgory13]. One can also replace the usage of [Reference Mirković and Vilonen70] with [Reference Zhu81, Corollary 5.3.8] along the lines of [Reference Finkelberg and Mirković29].

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 Finkelberg10Reference Kato46Reference Kato, Naito and Sagaki51]).

3 Semi-infinite flag manifolds over $\mathbb {Z} \left [\frac {1}{2}\right ]$

We keep the settings of the previous section. In this section, we sometimes work over a (commutative) ring or a non-algebraically closed field. For a ring S or a scheme $\mathfrak {X}$, we may write $S_{A}$ or $\mathfrak {X}_{A}$ if it is defined over A. In addition, we may consider their scalar extensions $S_{B} := S_A \otimes _A B$ and $\mathfrak {X}_{B}$ for a ring map $A \to B$.

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 $).

Definition 3.2. Frobenius splitting of a scheme

Let $\mathfrak X$ be a separated scheme defined over $\Bbbk $. Let $\mathsf {Fr}$ be the (relative) Frobenius endomorphism of $\mathfrak X$ (which induces a $\Bbbk $-linear endomorphism). We have a natural inclusion $\imath : {\mathcal O}_{\mathfrak X} \rightarrow \mathsf {Fr}_{*} {\mathcal O}_{\mathfrak X}$. A Frobenius splitting of $\mathfrak X$ is a ${\mathcal O}_{\mathfrak X}$-linear morphism $\phi : \mathsf {Fr}_{*} {\mathcal O}_{\mathfrak X} \rightarrow {\mathcal O}_{\mathfrak X}$ such that the composition $\phi \circ \imath $ is the identity.

Definition 3.3. Compatible splitting

Let $\mathfrak {Y} \subset \mathfrak {X}$ be a closed immersion of separated schemes defined over $\Bbbk $. A Frobenius splitting $\phi $ of $\mathfrak {X}$ is said to be compatible with $\mathfrak {Y}$ if $\phi \left (\mathsf {Fr}_* \mathcal I _{\mathfrak {Y}} \right ) \subset \mathcal I_{\mathfrak {Y}}$, where $\mathcal I _{\mathfrak {Y}} := \ker \left ( {\mathcal O}_{\mathfrak {X}} \to {\mathcal O}_{\mathfrak {Y}} \right )$. Compatible Frobenius splitting of a pair of a commutative ring and its quotient ring is defined through their spectra.

Remark 3.4. A Frobenius splitting of $\mathfrak {X}$ compatible with $\mathfrak {Y}$ induces a Frobenius splitting of $\mathfrak {Y}$ (see, e.g., [Reference Brion and Kumar14, Remark 1.1.4 (ii)]).

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.

Remark 3.8. In our opinion, one merit of Proposition 3.7 over [Reference Brion and Kumar14, Proposition 4.1.8] is that it becomes obvious that a projective variety $\mathfrak X$ with a B-action has at most one B-canonical splitting whenever the space of global sections of all ample line bundles is (or can be made) B-cocyclic compatible with multiplications (compare [Reference Brion and Kumar14, Theorem 4.1.15] and Corollary 3.35).

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 Kashiwara41Reference Kashiwara43Reference Kashiwara44Reference Kashiwara45Reference Lusztig65Reference 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].

Remark 3.9. We remark that $U ^\pm _{\mathbb {Z}}$ are the same integral forms dealt with in [Reference Garland31], and also coincide with the integral forms obtained through the Drinfeld presentation ([Reference Beck, Chari and Pressley4, §2] and [Reference Naoi73, Lemma 2.5]).

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 Lusztig65Reference 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.

Theorem 3.11 [Reference Kashiwara41]

We have the following:

  1. 1. For each $\Lambda \in P^{\mathrm {af}}_+$, the $\widetilde {\mathfrak {g}}_{{\mathbb C}}$-module $L ( \Lambda )_{\mathbb C}$ is compatible with the negative global basis.

  2. 2. For each $\lambda \in P_+$, we have

    $$ \begin{align*}V ( \lambda )_{\mathbb C} = \bigoplus _{b \in {\mathbf B}^0 ( \infty )} {\mathbb C} b \mathbf{v}_{\lambda}^0.\end{align*} $$

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 $P \hookrightarrow P^{\mathrm {af}}$.

Corollary 3.12. We have the following:

  1. 1. For each $\Lambda , \Gamma \in P^{\mathrm {af}}_+$, we have a natural inclusion $L ( \Lambda + \Gamma )_{\mathbb {Z}} \hookrightarrow L ( \Lambda )_{\mathbb {Z}} \otimes _{\mathbb {Z}} L ( \Gamma )_{\mathbb {Z}}$ of $\dot {U}_{\mathbb {Z}}$-modules, which is a direct summand as $\mathbb {Z}$-modules.

  2. 2. For each $\lambda ,\mu \in P_+$, we have a natural inclusion $V ( \lambda + \mu )_{\mathbb {Z}} \hookrightarrow V ( \lambda )_{\mathbb {Z}} \otimes _{\mathbb {Z}} V ( \mu )_{\mathbb {Z}}$ of $\dot {U}_{\mathbb {Z}}^0$-modules, which is a direct summand as $\mathbb {Z}$-modules.

Proof. Since the two cases are completely parallel, we prove only the first case. The $\widetilde {\mathfrak {g}}$-module $L ( \Lambda )_{\mathbb C} \otimes _{\mathbb C} L ( \Gamma )_{\mathbb C}$ decomposes into the direct sum of integrable highest weight modules ([Reference Kac40, Proposition 9.10]), with a direct summand $L ( \Lambda + \Gamma )_{\mathbb C}$. In view of [Reference Kashiwara41, Theorem 3], it gives rise to the $\mathbb {Z} [\mathsf q]$-lattice of the quantised version of $L ( \Lambda ) \otimes L ( \Gamma )$ compatible with those of $L ( \Lambda + \Gamma )$ via the natural embedding. By setting $\mathsf q = 1$, we obtain a direct sum decomposition of $L ( \Lambda )_{\mathbb {Z}} \otimes _{\mathbb {Z}} L ( \Gamma )_{\mathbb {Z}}$ as $\mathbb {Z}$-modules with its direct summand $L ( \Lambda + \Gamma )_{\mathbb {Z}}$.

Theorem 3.13 [Reference Kashiwara43]

For each $\lambda \in P_+$, the $\widetilde {\mathfrak {g}}_{\mathbb C}$-module $\mathbb {X} ( \lambda )_{\mathbb C}$ is compatible with the negative/positive global basis (for every extremal weight vector).

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

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

Theorem 3.14 [Reference Kashiwara45]

Set $\lambda \in P_+$. There exists a $\mathbb {Z}$-basis ${\mathbf B} ( \mathbb {X} ( \lambda ) )$ of $\mathbb {X} ( \lambda )_{\mathbb {Z}}$ that contains the negative/positive global basis of $\mathbb {X} ( \lambda )_{\mathbb {Z}}$ constructed from every extremal weight vector of $\mathbb {X} ( \lambda )_{\mathbb C}$.

Proof. We set ${\mathbf B} ( \mathbb {X} ( \lambda ) )$ to be the specialisation of the global basis of a quantum loop algebra module [Reference Kashiwara43, Proposition 8.2.2]. Then it is compatible with the global basis generated from extremal weight vectors by [Reference Kashiwara45, Theorem 3.3].

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

Lemma 3.15 Naito and Sagaki

For each $\lambda \in P_+$ and $w,v \in W_{\mathrm {af}}$, we have $\mathbb {W} _{ww_0} ( \lambda )_{\mathbb {Z}} \subset \mathbb {W} _{vw_0} ( \lambda )_{\mathbb {Z}}$ if $w \le _{\frac {\infty }{2}} v$. If we have $\lambda \in P_{++}$ in addition, then we have $\mathbb {W} _{ww_0} ( \lambda )_{\mathbb {Z}} \subset \mathbb {W} _{vw_0} ( \lambda )_{\mathbb {Z}}$ if and only if $w \le _{\frac {\infty }{2}} v$.

Proof. Apply the inclusion relation of the (labels of the) global basis in [Reference Naito and Sagaki72, Corollary 5.2.5] (see also [Reference Kashiwara45, §2.8]).

Corollary 3.16. For each $\lambda \in P_+$, $w \in W_{\mathrm {af}}$ and $i \in \mathtt I_{\mathrm {af}}$, we have $\mathbb {W} _{s_i ww_0} ( \lambda )_{{\mathbb K}} \subset \mathbb {W} _{ww_0} ( \lambda )_{{\mathbb K}}$ if $s_i w \le _{\frac {\infty }{2}} w$. In this case, $\mathbb {W} _{ww_0} ( \lambda )_{{\mathbb K}}$ inherits an action of $\mathop {\textit{SL}} (2,i)_{{\mathbb K}}$ from $\mathbb {X} ( \lambda )_{{\mathbb K}}$.

Proof. The first part of the assertion is the special case of Lemma 3.15. Given this, it remains to notice that a lift of $s_i \in W_{\mathrm {af}}$ sends $\mathbf {v}_{ww_0 \lambda }$ to $\pm \mathbf {v}_{s_iww_0 \lambda }$, and hence the Bruhat decomposition of $\mathbf {I} ( i )_{{\mathbb K}}$ (into two $\mathbf {I}_{{\mathbb K}}$-double cosets) implies that $\mathbb {W} _{ww_0} ( \lambda )_{{\mathbb K}}$ is stable under $\mathbf {I} ( i )_{{\mathbb K}}$.

Lemma 3.17. For each $\lambda \in P_+$ and $w \in W_{\mathrm {af}}$, we have the following:

  1. 1. Each $\beta \in Q^{\vee }$ defines a $\dot {U}_{\mathbb {Z}}$-module automorphism $\tau _\beta $ on $\mathbb {X} ( \lambda )_{\mathbb {Z}}$ determined by $\tau _{\beta }( \mathbf {v}_{\lambda } ) := \mathbf {v}_{t_{\beta } \lambda }$. Moreover, $\tau _{\beta }{\mathbf B} ( \mathbb {X} ( \lambda ) ) = {\mathbf B} ( \mathbb {X} ( \lambda ) )$.

  2. 2. $\theta ^* ( \mathbb {X} ( \lambda )_{\mathbb {Z}} ) \cong \mathbb {X} ( - w_0 \lambda )_{\mathbb {Z}}$ as $\dot {U}_{\mathbb {Z}}$-modules. Moreover, $\theta ^* {\mathbf B} ( \mathbb {X} ( \lambda ) ) = {\mathbf B} ( \mathbb {X} ( - w_0 \lambda ) )$.

  3. 3. $\mathbb {W} _w ( \lambda ) _{\mathbb {Z}} = \mathbb {W} _w ( \lambda )_{\mathbb C} \cap \mathbb {X} ( \lambda )_{\mathbb {Z}}$.

  4. 4. There is a $U_{\mathbb {Z}}^-$-cyclic vector of $\theta ^* (\mathbb {W} _w ( \lambda ) _{\mathbb {Z}} )$ with weight $- w \lambda = w w_0 ( - w_0 \lambda )$. In particular,

    $$ \begin{align*}\theta^* ( \mathbb{W}_w ( \lambda )_{\mathbb{Z}} ) \cong \mathbb{W}_{ww_0}^- ( -w_0 \lambda )_{\mathbb{Z}} \quad \text{and} \quad \theta^* \left( \mathbb{W}_w^- ( \lambda )_{\mathbb{Z}} \right) \cong \mathbb{W}_{ww_0} ( -w_0 \lambda )_{\mathbb{Z}}.\end{align*} $$

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.

Proof. We set $\mathbb {X}^{\otimes } := \bigotimes _{i \in \mathtt I} \mathbb {X} ( \varpi _i )_{\mathbb {Z}}^{\otimes \left \langle \alpha _i^{\vee }, \lambda \right \rangle }$. The map $\Psi _\lambda $ exists, as $\bigotimes _{i \in \mathtt I} \mathbf {v}_{\varpi _i}^{\otimes \left \langle \alpha _i^{\vee }, \lambda \right \rangle }$ obeys the same defining condition as the extremal weight vector $\mathbf {v}_{\lambda } \in \mathbb {X} ( \lambda )_{\mathbb {Z}}$, and the comultiplication of $\widetilde {\mathfrak {g}}$ induces an algebra map ${\triangle}{\kern-0.5pt}{:}{\kern1.5pt} \dot {U}_{\mathbb {Z}} \subset \dot {U}_{\mathbb {Z}} \otimes \dot {U}_{\mathbb {Z}}$ ([Reference Lusztig66, §23.1.5]).

The map $\Psi _\lambda $ is injective by [Reference Beck and Nakajima5, Corollary 4.15 and Remark 4.17]. By [Reference Kashiwara44, Theorem 8.5], the $\mathsf {q} = 1$ specialisations of the global bases yield the $\dot {U}_{\mathbb {Z}}$-spans of the extremal weight vectors in $\mathbb {X} ( \lambda )_{\mathbb {Z}}$ and $\mathbb {X}^{\otimes }$ up to the action of some rings of (partially symmetric) polynomials with integer coefficients, respectively. This yields a splitting of $\Psi _\lambda $ as $\mathbb {Z}$-modules, where the $\mathbb {Z}$-module structure of the right-hand side comes from the crystal lattice of (the $\mathsf q$-version of) $\mathbb {X}^{\otimes }$ (as in [Reference Kashiwara44, (8.8)]). We call this crystal lattice