---
abstract: 'We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model ${\mathbf{Q}_{G}}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-theory of ${\mathbf{Q}_{G}}$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over ${\mathbf{Q}_{G}}$. In order to achieve this, we provide a number of fundamental results on ${\mathbf{Q}_{G}}$ and its Schubert subvarieties including the Borel-Weil-Bott theory, whose special case is conjectured in [@BF14c]. One more ingredient of this paper besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai-Seshadri paths. In fact, in our Pieri-Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai-Seshadri paths.'
author:
- |
    Syu Kato\
    Department of Mathematics, Kyoto University,\
    Oiwake Kita-Shirakawa, Sakyo, Kyoto 606-8502, Japan\
    (e-mail: [syuchan@math.kyoto-u.ac.jp]{})\
    Satoshi Naito\
    Department of Mathematics, Tokyo Institute of Technology,\
    2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan\
    (e-mail: [naito@math.titech.ac.jp]{})\
    and\
    Daisuke Sagaki\
    Institute of Mathematics, University of Tsukuba,\
    1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan\
    (e-mail: [sagaki@math.tsukuba.ac.jp]{})
title: |
     **Equivariant $K$-theory of\
    semi-infinite flag manifolds\
    and Pieri-Chevalley formula[^1]**
---

Introduction. {#sec:intro}
=============

Let $G$ be a connected and simply-connected simple algebraic group over ${\mathbb{C}}$, and let $X$ be the flag variety of $G$. The torus-equivariant Grothendieck group $K_H ( X )$ of $X$ affords rich structures from the perspective of geometry and representation theory. One of the highlights there is the positivity of the structure constants of the products among natural classes (called the Schubert classes; see Anderson-Griffeth-Miller [@AGM], and Baldwin-Kumar [@BK]), which serves as a basis of its interaction with the eigenvalue problems [@Kly98] and Gaudin models [@MTV]. There is a variant of this theme (called the Pieri-Chevalley formula), namely the structure constants of the products between Schubert classes and (ample) line bundles in $K_H ( X )$, which is also known to be positive by Mathieu [@Mat00] and Brion [@Br02].

Pittie and Ram [@PR99] initiated a program to describe such a positive structure constant by relating them with the standard monomial theory (SMT for short). In particular, they gave an explicit meaning of each structure coefficient in the Pieri-Chevalley formula in terms of Lakshmibai-Seshadri paths (LS paths for short; see, e.g., [@Lit95]), which carries almost all information about simple $G$-modules. Their program is subsequently completed by Littelmann-Seshadri [@LiSe03] and Lenart-Shimozono [@LeSh14] (see also Lenart-Postnikov [@LeP07]).

Peterson [@Pet97] noticed that the quantum $K$-theory of $X$ should be intimately connected with the $K$-theory of the “affine version" of $X$ (see Lam-Shimozono [@LS10] and Lam-Li-Mihalcea-Shimozono [@LLMS]). In view of Givental-Lee [@GL03] and Braverman-Finkelberg [@BF14a; @BF14c], the quantum $K$-theory of $X$ can be defined through the space of quasi-maps, whose union forms a dense subset of the formal power series model ${\mathbf{Q}}_G$ of semi-infinite flag manifolds (cf. Finkelberg-Mirković [@FM99]).

Therefore, it is quite natural to make some rigorous sense of $K_H ( {\mathbf{Q}_{G}})$ and provide the Pieri-Chevalley formula using SMT, which is compatible with the pictures provided by Pittie-Ram and Peterson. This is what we perform in this paper by affording two new theories: [**1)**]{} the Borel-Weil-Bott theory of ${\mathbf{Q}_{G}}$ that enables us to define and calculate a version of $K_H ( {\mathbf{Q}_{G}})$, and [**2)**]{} the SMT of level-zero modules over quantum affine algebras. We remark that the level-zero modules over quantum affine algebras admit an interpretation through the geometry of affine Grassmannian (of Langlands dual type), which is the “affine version” of $X$ (see, e.g., Lenart-Naito-Sagaki-Schilling-Shimozono [@LNSSS2 Introduction]).

In order to explain our results, theories, and ideas more precisely, we need some notation. Let ${\mathfrak{g}}$ denote the Lie algebra of $G$, and let ${\mathfrak{g}}_{\mathrm{af}}$ denote the associated untwisted affine Lie algebra; we fix a Borel subgroup $B$ of $G$ and a maximal torus $H \subset B$, and set $N := [B,B]$. Let $W = N_{G}(H)/H$ be the Weyl group, which is generated by the simple reflections $s_{i}$, $i \in I$; $W$ can be thought of as acting on the dual space ${\mathfrak{h}}^{\ast}$ of the Cartan subalgebra ${\mathfrak{h}}:= {\mathop{\rm Lie}\nolimits}(H)$. We set $W_{{\mathrm{af}}} := W \ltimes Q^{\vee}$, with $Q^{\vee, +} := \sum_{i \in I} {\mathbb{Z}}_{\geq 0} \alpha_{i}^{\vee} \subset 
Q^{\vee} := \bigoplus_{i \in I} {\mathbb{Z}}\alpha_{i}^{\vee}$ (the coroot lattice). Let $P = \bigoplus_{i \in I} {\mathbb{Z}}{\varpi}_{i} \subset {\mathfrak{h}}^{\ast}$ be the weight lattice generated by the fundamental weights ${\varpi}_{i}$, $i \in I$, and set $P^+ := \sum_{i \in I} {\mathbb{Z}}_{\ge 0} {\varpi}_{i}$.

For an algebraic group $E$ over ${\mathbb{C}}$, we denote by $E{(\hspace{-1pt}(z)\hspace{-1pt})}$ and $E{[\hspace{-1pt}[z]\hspace{-1pt}]}$ the space of ${\mathbb{C}}{(\hspace{-1pt}(z)\hspace{-1pt})}$-valued points and the space of ${\mathbb{C}}{[\hspace{-1pt}[z]\hspace{-1pt}]}$-valued points of $E$, respectively, viewed as an (ind-)scheme over ${\mathbb{C}}$. Let ${\mathop{\tt ev}\nolimits}_{0} : G{[\hspace{-1pt}[z]\hspace{-1pt}]} \rightarrow G$ be the evaluation map at $z = 0$, and set ${\mathbf{I}}:= {\mathop{\tt ev}\nolimits}_{0}^{-1}(B)$, an Iwahori subgroup of $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$; we also set $\tilde{{\mathbf{I}}} := {\mathbf{I}}\rtimes {\mathbb{C}}^{\ast}$, the semi-direct product group, where the group ${\mathbb{C}}^{\ast}$ (of loop rotations) acts on ${\mathbf{I}}$ as the dilation on $z$. Now, we define ${\mathbf{Q}_{G}^{\mathrm{rat}}}:= G{(\hspace{-1pt}(z)\hspace{-1pt})} / H N{(\hspace{-1pt}(z)\hspace{-1pt})}$, which is a pure ind-scheme of infinite type. Then the set of ${\mathbf{I}}$-orbits is in natural bijection with $W_{{\mathrm{af}}}$; let ${\mathbf{Q}_{G}}( x )$ denote the ${\mathbf{I}}$-orbit closure corresponding to $x \in W_{{\mathrm{af}}}$. We define ${\mathbf{Q}_{G}}:= {\mathbf{Q}_{G}}( e )$.

Our first main result is the following:

\[fnormal\] For each $x \in W_{\mathrm{af}}$, the scheme ${\mathbf{Q}_{G}}( x )$ is normal. In addition, there is an explicit $P^{+}$-graded algebra $R_G$ such that ${\mathbf{Q}_{G}}= {\mathop{\rm Proj}\nolimits}R_G$; here our ${\mathop{\rm Proj}\nolimits}$ is the $P^+$-graded one.

We remark that Theorem \[fnormal\] affirmatively answers [@BF14c Conjecture 2.1] and relevant speculations therein. Also, as we see below, the scheme ${\mathbf{Q}_{G}}$ is far from being “compact” (cf. [@Kat17 Theorem A] and [@FGT (7.1)]). In order to prove Theorem \[fnormal\] naturally, we introduce a “semi-infinite" Bott-Samelson-Demazure-Hansen tower that yields a normal ring $R_G$. From the construction, $R_G$ contains the projective coordinate ring of ${\mathbf{Q}_{G}}$. Moreover, on the basis of the fact that $R_G$ is generated by the primitive degree terms, a detailed comparison with the computation for the dense subset in [@BF14c] implies that the inclusion must be an isomorphism.

For each $x \in W_{\mathrm{af}}$ and $\lambda \in P$, we have an associated $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$-equivariant line bundle ${\mathcal{O}}_{{\mathbf{Q}_{G}}(x)}(\lambda)$ over ${\mathbf{Q}_{G}}(x)$. Also, for each $x \in W$ and $\lambda \in P^+$, we have a Demazure submodule $V^{-}_{x} ( \lambda )$, in the sense of [@Kas05], of the level-zero extremal weight module $V ( \lambda )$ (of extremal weight $\lambda$) over the quantum affine algebra $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$ associated to ${\mathfrak{g}}_{{\mathrm{af}}}$.

\[fBWB\] For each $x \in W_{\mathrm{af}}$ and $\lambda \in P$, we have $${\mathop{\rm gch}\nolimits}H^{i} ( {\mathbf{Q}_{G}}( x ), {\mathcal{O}}_{{\mathbf{Q}_{G}}(x)}( \lambda ) ) = 
  \begin{cases} 
    {\mathop{\rm gch}\nolimits}V^{-}_{x} ( - w_{\circ} \lambda ) & \text{\rm if $i = 0$ and $\lambda \in P^{+}$}, \\[1.5mm]
    0 & \text{\rm otherwise},   \end{cases}$$ where ${\mathop{\rm gch}\nolimits}$ denotes the character taking values in $({\mathbb{Z}}{(\hspace{-1pt}(q^{-1})\hspace{-1pt})})[P]$, and $w_{\circ} \in W$ is the longest element.

The higher cohomology vanishing part of Theorem \[fBWB\] is based on the fact that the ring $R_G$ is free over a polynomial ring with infinitely many variables (Theorem \[R-free\]), which is also an interesting result in its own. We should mention that Theorem \[fBWB\] have an ind-model counterpart in [@BF14c], but there are no implications between these and the two proofs are totally different.

\[fLB\] For each $x \in W_{\mathrm{af}}$, every ${\widetilde{{\mathbf{I}}}}$-equivariant line bundle over the scheme ${\mathbf{Q}_{G}}( x )$ is isomorphic to some ${\mathcal{O}}_{{\mathbf{Q}_{G}}(x)}( \lambda )$ up to character twist.

Although $R_G$ itself is highly infinite-dimensional (it is [*not*]{} even finitely generated), it admits a grading such that it is almost like an Artin algebra in a graded sense. Moreover, Proposition \[fLB\] supplies “graded indecomposable projectives” of $R_G$. These two facts, combined with Theorem \[fBWB\], assert that the category of ${\widetilde{{\mathbf{I}}}}$-equivariant sheaves on ${\mathbf{Q}_{G}}$ (and on ${\mathbf{Q}_{G}^{\mathrm{rat}}}$) behaves almost like the category of coherent sheaves on an affine scheme.

This series of observations enables us to define a reasonable variant of an equivariant $K$-group $K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}})$ of ${\mathbf{Q}_{G}}$ (and $K_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}^{\mathrm{rat}}})$ of ${\mathbf{Q}_{G}^{\mathrm{rat}}}$) with respect to $\tilde{{\mathbf{I}}}$; see Section \[sec:K-SiFl\] for details. They are rather involved, partly because we need to specify a class of formal power series that is large enough to afford the Pieri-Chevalley rule, and at the same time is small enough so that the Euler character map is injective. Nevertheless, we can prove that $K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}})$ contains (the classes of) the sheaves $[{\mathcal{O}}_{{\mathbf{Q}_{G}}(y)} ( \lambda )]$ for each $\lambda \in P$ and relevant $y \in W_{{\mathrm{af}}}$. We also prove that our $K_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}}^{\mathrm{rat}})$ is natural enough so that it admits a nil-DAHA action as an analog of Kostant-Kumar [@KK90] for ${\mathbf{Q}_{G}^{\mathrm{rat}}}$ (see Section \[sec:nDAHA\] for details).

Here we recall that in Ishii-Naito-Sagaki [@INS] and Naito-Sagaki [@NS16], the semi-infinite path model of the crystal basis of $V_{x}^{-} ( \lambda )$ is constructed for every $\lambda \in P^+$ and $x \in W_{{\mathrm{af}}}$; it is a specific subset of the set of “semi-infinite” LS paths ${\mathbb{B}^{{\frac{\infty}{2}}}}( \lambda )$ of shape $\lambda$ parametrizing the global crystal basis of $V ( \lambda )$. Note that it is endowed with three functions $$\iota,\,\kappa : {\mathbb{B}^{{\frac{\infty}{2}}}}( \lambda ) \longrightarrow W_{\mathrm{af}}\quad \text{and} \quad
{\mathop{\rm wt}\nolimits}: {\mathbb{B}^{{\frac{\infty}{2}}}}( \lambda ) \longrightarrow P \oplus {\mathbb{Z}}\delta,$$ which are called the initial/final directions and the weight, respectively. We set $${\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x} ( \lambda ) := 
 \bigl\{ \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}( \lambda ) \mid \kappa ( \eta ) {\succeq}x \bigr\}.$$

In order to make use of the path model above to derive the Pieri-Chevalley formula for ${\mathbf{Q}_{G}^{\mathrm{rat}}}$, we additionally need a combinatorial version of the semi-infinite SMT. This consists of the definition of the initial direction ${\iota(\eta,\,x)} \in W_{{\mathrm{af}}}$ of a semi-infinite LS path $\eta$ with respect to $x$ (based on the existence of the semi-infinite analog of the so-called Deodhar lift), and of the description of tensor product decomposition of crystals in terms of ${\iota(\bullet,\,x)}$ (Theorem \[thm:SMT\] and Theorem \[thm:Dem\]). We remark that our ${\iota(\eta,\,x)}$ is an analogue of the one in [@LiSe03; @LeSh14] in the setting of level-zero extremal weight modules over $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$. Using them, we obtain our Pieri-Chevalley formula:

\[fPCf\] For $\lambda \in P^{+}$ and $x \in W_{{\mathrm{af}}}^{\ge 0}:=W \times Q^{\vee,+}$, we have $$\label{fPC}
[{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda )] \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( x )}] = 
 \sum_{\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x} (- w_{\circ} \lambda )} 
 e^{{\mathop{\rm fin}\nolimits}(\eta)}q^{{\mathop{\rm nul}\nolimits}(\eta)} \cdot [ {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\iota(\eta,\,x)} )}] \quad \in K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}}),$$ where ${\mathop{\rm fin}\nolimits}(\eta) \in P$ and ${\mathop{\rm nul}\nolimits}(\eta) \in {\mathbb{Z}}$ for $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(- w_{\circ} \lambda )$ are defined by: $${\mathop{\rm wt}\nolimits}(\eta) = {\mathop{\rm fin}\nolimits}(\eta) + {\mathop{\rm nul}\nolimits}(\eta) \delta.$$

Once generalities on $K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}})$ and the semi-infinite SMT are given, our strategy for the proof of Theorem \[fPCf\] is along the line of [@LiSe03]. Namely, we compare the functionals $$P \ni \lambda \mapsto \sum_{i \ge 0}(-1)^{i} 
  {\mathop{\rm gch}\nolimits}H^{i}( {\mathbf{Q}_{G}}, {\mathcal{E}}\otimes_{{\mathcal{O}}_{{\mathbf{Q}_{G}}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda ) ) \in {\mathbb{C}}[P]{(\hspace{-1pt}(q^{-1})\hspace{-1pt})},$$ where $[{\mathcal{E}}]$ is taken from the both sides of .

This paper is organized as follows. In Section \[sec:notation\], we fix our notation for untwisted affine Lie algebras, and then recall some basic facts about semi-infinite LS paths, extremal weight modules, and their Demazure submodules. In Section \[sec:SMT\], we state a combinatorial version of standard monomial theory for level-zero extremal weight modules, and also its refinement for Demazure submodules; the proofs of these results are given in Sections \[sec:prf-SMT\], \[sec:prf-DC\], and \[sec:prf-Dem\]. In Section \[sec:SiSch\], we first review the formal power series model ${\mathbf{Q}_{G}}$ of semi-infinite flag manifold, and then introduce a semi-infinite version of Bott-Samelson-Demazure-Hansen tower for ${\mathbf{Q}_{G}}$. Then, we study the cohomology spaces of line bundles over ${\mathbf{Q}_{G}}$, and prove the higher cohomology vanishing; also, we describe the spaces of global sections in terms of Demazure submodules of extremal weight modules. As an application, we prove the normality of the semi-infinite Schubert varieties ${\mathbf{Q}_{G}}(x)$, $x \in W_{{\mathrm{af}}}^{\geq 0}$. In Section \[sec:K-SiFl\], after giving a definition of ${\widetilde{{\mathbf{I}}}}$-equivariant $K$-group $K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}})$ of ${\mathbf{Q}_{G}}$ (and $K_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}^{\mathrm{rat}}})$ of ${\mathbf{Q}_{G}^{\mathrm{rat}}}$), we establish the Pieri-Chevalley formula (Theorem \[thm:PC\]) by combining our geometric results with the semi-infinite SMT. Also, in Section \[sec:nDAHA\], we show that our $K$-group $K_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}^{\mathrm{rat}}})$ admits a natural nil-DAHA action. Appendices mainly contain some technical results concerning the semi-infinite Bruhat order; in particular, we prove the existence of analogs of Deodhar lifts for the semi-infinite Bruhat order.

Acknowledgments. {#acknowledgments. .unnumbered}
----------------

We thank Michael Finkelberg for sending us unpublished manuscripts. S.K. was partially supported by JSPS Grant-in-Aid for Scientific Research (B) 26287004 and Kyoto University Jung-Mung program. S.N. was partially supported by JSPS Grant-in-Aid for Scientific Research (B) 16H03920. D.S. was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 15K04803.

Algebraic setting. {#sec:notation}
==================

Affine Lie algebras. {#subsec:liealg}
--------------------

A graded vector space is a ${\mathbb{Z}}$-graded vector space over ${\mathbb{C}}$ all of whose homogeneous subspaces are finite-dimensional. Let $V = \bigoplus_{m \in {\mathbb{Z}}} V_{m}$ be a graded vector space with $V_{m}$ its subspace of degree $m$. We define $${\mathop{\rm gdim}\nolimits}V := \sum_{m \in {\mathbb{Z}}} \bigl( \dim V_{m} \bigr) q^{m}.$$ Also, we denote by $V^{\vee}$ (resp., $V^{\ast}$) the full (resp., restricted) dual of $V$; note that $V^{\ast} := \bigoplus _{m \in {\mathbb{Z}}} ( V^{\ast} )_{m}$, with $(V^{\ast})_{m}:=(V_{-m})^{\ast}$. In addition, we set ${\widehat{V}}:=\prod_{m \in {\mathbb{Z}}} V_{m}$, which is a completion of $V$.

Let $G$ be a connected, simply-connected simple algebraic group over ${\mathbb{C}}$, and $B$ a Borel subgroup with unipotent radical $N$. We fix a maximal torus $H \subset B$, and take the opposite Borel subgroup $B^{-}$ of $G$ that contains $H$. In the following, for an (arbitrary) algebraic group $E$ over ${\mathbb{C}}$, we denote its Lie algebra ${\mathop{\rm Lie}\nolimits}(E)$ by the corresponding German letter ${\mathfrak{e}}$; in particular, we write ${\mathfrak{g}}={\mathop{\rm Lie}\nolimits}(G)$, ${\mathfrak{b}}={\mathop{\rm Lie}\nolimits}(B)$, ${\mathfrak{n}}={\mathop{\rm Lie}\nolimits}(N)$, and ${\mathfrak{h}}={\mathop{\rm Lie}\nolimits}(H)$. Thus, ${\mathfrak{g}}$ is a finite-dimensional simple Lie algebra over ${\mathbb{C}}$ with Cartan subalgebra ${\mathfrak{h}}$. Denote by $\{ \alpha_{i}^{\vee} \}_{i \in I}$ and $\{ \alpha_{i} \}_{i \in I}$ the set of simple coroots and simple roots of ${\mathfrak{g}}$, respectively, and set $Q := \bigoplus_{i \in I} {\mathbb{Z}}\alpha_i$, $Q^{+} := \sum_{i \in I} {\mathbb{Z}}_{\ge 0} \alpha_i$, and $Q^{\vee} := \bigoplus_{i \in I} {\mathbb{Z}}\alpha_i^{\vee}$, $Q^{\vee,+} := \sum_{i \in I} {\mathbb{Z}}_{\ge 0} \alpha_i^{\vee}$; for $\xi,\,\zeta \in Q^{\vee}$, we write $\xi \ge \zeta$ if $\xi-\zeta \in Q^{\vee,+}$. Let $\Delta$ and $\Delta^{+}$ be the set of roots and positive roots of ${\mathfrak{g}}$, respectively, with $\theta \in \Delta^{+}$ the highest root of ${\mathfrak{g}}$. For a root $\alpha \in \Delta$, we denote by $\alpha^{\vee}$ its dual root. We set $\rho:=(1/2) \sum_{\alpha \in \Delta^{+}} \alpha$ and $\rho^{\vee}:= (1/2) \sum_{\alpha \in \Delta^{+}} \alpha^{\vee}$. Also, let ${\varpi}_{i}$, $i \in I$, denote the fundamental weights for ${\mathfrak{g}}$, and set $$\label{eq:P-fin}
P:=\bigoplus_{i \in I} {\mathbb{Z}}{\varpi}_{i}, \qquad 
P^{+} := \sum_{i \in I} {\mathbb{Z}}_{\ge 0} {\varpi}_{i}.$$

Let ${\mathfrak{g}}_{{\mathrm{af}}} = \bigl({\mathfrak{g}}\otimes {\mathbb{C}}[z,z^{-1}]\bigr) \oplus {\mathbb{C}}c \oplus {\mathbb{C}}d$ be the untwisted affine Lie algebra over ${\mathbb{C}}$ associated to ${\mathfrak{g}}$, where $c$ is the canonical central element, and $d$ is the scaling element (or the degree operator), with Cartan subalgebra ${\mathfrak{h}}_{{\mathrm{af}}} = {\mathfrak{h}}\oplus {\mathbb{C}}c \oplus {\mathbb{C}}d$. We regard an element $\mu \in {\mathfrak{h}}^{\ast}:={\mathop{\rm Hom}\nolimits}_{{\mathbb{C}}}({\mathfrak{h}},\,{\mathbb{C}})$ as an element of ${\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ by setting ${\langle \mu,\,c \rangle}={\langle \mu,\,d \rangle}:=0$, where ${\langle \cdot\,,\,\cdot \rangle}:{\mathfrak{h}}_{{\mathrm{af}}}^{\ast} \times {\mathfrak{h}}_{{\mathrm{af}}} \rightarrow {\mathbb{C}}$ is the canonical pairing of ${\mathfrak{h}}_{{\mathrm{af}}}^{\ast}:={\mathop{\rm Hom}\nolimits}_{{\mathbb{C}}}({\mathfrak{h}}_{{\mathrm{af}}},\,{\mathbb{C}})$ and ${\mathfrak{h}}_{{\mathrm{af}}}$. Let $\{ \alpha_{i}^{\vee} \}_{i \in I_{{\mathrm{af}}}} \subset {\mathfrak{h}}_{{\mathrm{af}}}$ and $\{ \alpha_{i} \}_{i \in I_{{\mathrm{af}}}} \subset {\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ be the set of simple coroots and simple roots of ${\mathfrak{g}}_{{\mathrm{af}}}$, respectively, where $I_{{\mathrm{af}}}:=I \sqcup \{0\}$; note that ${\langle \alpha_{i},\,c \rangle}=0$ and ${\langle \alpha_{i},\,d \rangle}=\delta_{i0}$ for $i \in I_{{\mathrm{af}}}$. Denote by $\delta \in {\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ the null root of ${\mathfrak{g}}_{{\mathrm{af}}}$; recall that $\alpha_{0}=\delta-\theta$. Also, let $\Lambda_{i} \in {\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$, $i \in I_{{\mathrm{af}}}$, denote the fundamental weights for ${\mathfrak{g}}_{{\mathrm{af}}}$ such that ${\langle \Lambda_{i},\,d \rangle}=0$, and set $$\label{eq:P}
P_{{\mathrm{af}}} := 
  \left(\bigoplus_{i \in I_{{\mathrm{af}}}} {\mathbb{Z}}\Lambda_{i}\right) \oplus 
   {\mathbb{Z}}\delta \subset {\mathfrak{h}}^{\ast}, \qquad 
P_{{\mathrm{af}}}^{0}:=\bigl\{\mu \in P_{{\mathrm{af}}} \mid {\langle \mu,\,c \rangle}=0\bigr\};$$ notice that $P_{{\mathrm{af}}}^{0}=P \oplus {\mathbb{Z}}\delta$, and that $${\langle \mu,\,\alpha_{0}^{\vee} \rangle} = - {\langle \mu,\,\theta^{\vee} \rangle} \quad 
\text{for $\mu \in P_{{\mathrm{af}}}^{0}$}.$$

Let $W := \langle s_{i} \mid i \in I \rangle$ and $W_{{\mathrm{af}}} := \langle s_{i} \mid i \in I_{{\mathrm{af}}} \rangle$ be the (finite) Weyl group of ${\mathfrak{g}}$ and the (affine) Weyl group of ${\mathfrak{g}}_{{\mathrm{af}}}$, respectively, where $s_{i}$ is the simple reflection with respect to $\alpha_{i}$ for each $i \in I_{{\mathrm{af}}}$, with length function $\ell:W_{{\mathrm{af}}} \rightarrow {\mathbb{Z}}_{\ge 0}$, which gives the one on $W$ by restriction; we denote by $e \in W_{{\mathrm{af}}}$ the identity element, and by $w_{\circ} \in W$ the longest element. For each $\xi \in Q^{\vee}$, let $t_{\xi} \in W_{{\mathrm{af}}}$ denote the translation in ${\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ by $\xi$ (see [@Kac Sect. 6.5]); for $\xi \in Q^{\vee}$, we have $$\label{eq:wtmu}
t_{\xi} \mu = \mu - {\langle \mu,\,\xi \rangle}\delta \quad 
\text{if $\mu \in {\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ satisfies ${\langle \mu,\,c \rangle}=0$}.$$ Then, $\bigl\{ t_{\xi} \mid \xi \in Q^{\vee} \bigr\}$ forms an abelian normal subgroup of $W_{{\mathrm{af}}}$, in which $t_{\xi} t_{\zeta} = t_{\xi + \zeta}$ holds for $\xi,\,\zeta \in Q^{\vee}$. Moreover, we know from [@Kac Proposition 6.5] that $$W_{{\mathrm{af}}} \cong 
 W \ltimes \bigl\{ t_{\xi} \mid \xi \in Q^{\vee} \bigr\} \cong W \ltimes Q^{\vee};$$ we also set $$\label{eq:Wge0}
W_{{\mathrm{af}}}^{\ge 0}:=\bigl\{wt_{\xi} \mid w \in W,\, \xi \in Q^{\vee,+} \bigr\} \subset W_{{\mathrm{af}}}.$$

Denote by ${\Delta_{{\mathrm{af}}}}$ the set of real roots of ${\mathfrak{g}}_{{\mathrm{af}}}$, and by ${\Delta_{{\mathrm{af}}}^{+}}\subset {\Delta_{{\mathrm{af}}}}$ the set of positive real roots; we know from [@Kac Proposition 6.3] that ${\Delta_{{\mathrm{af}}}}= 
\bigl\{ \alpha + n \delta \mid \alpha \in \Delta,\, n \in {\mathbb{Z}}\bigr\}$, and ${\Delta_{{\mathrm{af}}}^{+}}= 
\Delta^{+} \sqcup 
\bigl\{ \alpha + n \delta \mid \alpha \in \Delta,\, n \in {\mathbb{Z}}_{> 0}\bigr\}$. For $\beta \in {\Delta_{{\mathrm{af}}}}$, we denote by $\beta^{\vee} \in {\mathfrak{h}}_{{\mathrm{af}}}$ its dual root, and $s_{\beta} \in W_{{\mathrm{af}}}$ the corresponding reflection; if $\beta \in {\Delta_{{\mathrm{af}}}}$ is of the form $\beta = \alpha + n \delta$ with $\alpha \in \Delta$ and $n \in {\mathbb{Z}}$, then $s_{\beta} =s_{\alpha} t_{n\alpha^{\vee}} \in W \ltimes Q^{\vee}$.

Finally, let $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$ denote the quantized universal enveloping algebra over ${\mathbb{C}}({\mathsf{q}})$ associated to ${\mathfrak{g}}_{{\mathrm{af}}}$, with $E_{i}$ and $F_{i}$, $i \in I_{{\mathrm{af}}}$, the Chevalley generators corresponding to $\alpha_{i}$ and $-\alpha_{i}$, respectively. We denote by $U_{{\mathsf{q}}}^{-}({\mathfrak{g}}_{{\mathrm{af}}})$ the negative part of $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$, that is, the ${\mathbb{C}}({\mathsf{q}})$-subalgebra of $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$ generated by $F_{i}$, $i \in I_{{\mathrm{af}}}$.

Parabolic semi-infinite Bruhat graph. {#subsec:SiBG}
-------------------------------------

In this subsection, we fix a subset $J \subset I$. We set ${Q_{J}}:= \bigoplus_{i \in {J}} {\mathbb{Z}}\alpha_i$, ${Q_{J}^{\vee}}:= \bigoplus_{i \in {J}} {\mathbb{Z}}\alpha_i^{\vee}$, ${Q_{J}^{\vee,+}}:= \sum_{i \in {J}} {\mathbb{Z}}_{\ge 0} \alpha_i^{\vee}$, ${\Delta_{J}}:= \Delta \cap {Q_{J}}$, ${\Delta_{J}}^{+} := \Delta^{+} \cap {Q_{J}}$, and ${W_{J}}:= \langle s_{i} \mid i \in {J}\rangle$. Also, we denote by $$\label{eq:prj}
[\,\cdot\,]_{{J}} : 
Q^{\vee} \twoheadrightarrow {Q_{J}^{\vee}}\quad 
\text{(resp., $[\,\cdot\,]^{{J}} : Q^{\vee} \twoheadrightarrow Q_{I \setminus {J}}^{\vee}$)}$$ the projection from $Q^{\vee}=Q_{I \setminus {J}}^{\vee} \oplus {Q_{J}^{\vee}}$ onto ${Q_{J}^{\vee}}$ (resp., $Q_{I \setminus {J}}^{\vee}$) with kernel $Q_{I \setminus {J}}^{\vee}$ (resp., ${Q_{J}^{\vee}}$). Let ${W^{J}}$ denote the set of minimal(-length) coset representatives for the cosets in $W/{W_{J}}$; we know from [@BB Sect. 2.4] that $$\label{eq:mcr}
{W^{J}}= \bigl\{ w \in W \mid 
\text{$w \alpha \in \Delta^{+}$ for all $\alpha \in {\Delta_{J}}^{+}$}\bigr\}.$$ For $w \in W$, we denote by ${\lfloor w \rfloor}={\lfloor w \rfloor}^{{J}} \in {W^{J}}$ the minimal coset representative for the coset $w {W_{J}}$ in $W/{W_{J}}$. Also, following [@Pet97] (see also [@LS10 Sect. 10]), we set $$\begin{aligned}
({\Delta_{J}})_{{\mathrm{af}}} 
  & := \bigl\{ \alpha + n \delta \mid 
  \alpha \in {\Delta_{J}},\,n \in {\mathbb{Z}}\bigr\} \subset \Delta_{{\mathrm{af}}}, \\
({\Delta_{J}})_{{\mathrm{af}}}^{+}
  &:= ({\Delta_{J}})_{{\mathrm{af}}} \cap {\Delta_{{\mathrm{af}}}^{+}}= 
  {\Delta_{J}}^{+} \sqcup \bigl\{ \alpha + n \delta \mid 
  \alpha \in {\Delta_{J}},\, n \in {\mathbb{Z}}_{> 0} \bigr\}, \\
\label{eq:stabilizer}
({W_{J}})_{{\mathrm{af}}} 
 & := {W_{J}}\ltimes \bigl\{ t_{\xi} \mid \xi \in {Q_{J}^{\vee}}\bigr\}
   = \bigl\langle s_{\beta} \mid \beta \in ({\Delta_{J}})_{{\mathrm{af}}}^{+} \bigr\rangle, \\
\label{eq:Pet}
({W^{J}})_{{\mathrm{af}}}
 &:= \bigl\{ x \in W_{{\mathrm{af}}} \mid 
 \text{$x\beta \in {\Delta_{{\mathrm{af}}}^{+}}$ for all $\beta \in ({\Delta_{J}})_{{\mathrm{af}}}^{+}$} \bigr\};\end{aligned}$$ note that if ${J}= \emptyset$, then $(W^{\emptyset})_{{\mathrm{af}}}=W_{{\mathrm{af}}}$ and $(W_{\emptyset})_{{\mathrm{af}}}=\bigl\{e\bigr\}$. We know from [@Pet97] (see also [@LS10 Lemma 10.6]) that for each $x \in W_{{\mathrm{af}}}$, there exist a unique $x_1 \in ({W^{J}})_{{\mathrm{af}}}$ and a unique $x_2 \in ({W_{J}})_{{\mathrm{af}}}$ such that $x = x_1 x_2$; we define a (surjective) map $$\label{eq:PiJ}
{\Pi^{J}}: W_{{\mathrm{af}}} \twoheadrightarrow ({W^{J}})_{{\mathrm{af}}}, \quad x \mapsto x_{1},$$ where $x= x_1 x_2$ with $x_1 \in ({W^{J}})_{{\mathrm{af}}}$ and $x_2 \in ({W_{J}})_{{\mathrm{af}}}$.

\[dfn:sell\] Let $x \in W_{{\mathrm{af}}}$, and write it as $x = w t_{\xi}$ for $w \in W$ and $\xi \in Q^{\vee}$. We define the semi-infinite length ${\ell^{{\frac{\infty}{2}}}}(x)$ of $x$ by: ${\ell^{{\frac{\infty}{2}}}}(x) = \ell (w) + 2 {\langle \rho,\,\xi \rangle}$.

\[dfn:SiB\]

The (parabolic) semi-infinite Bruhat graph ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ is the ${\Delta_{{\mathrm{af}}}^{+}}$-labeled, directed graph with vertex set $({W^{J}})_{{\mathrm{af}}}$ whose directed edges are of the following form: $x {\xrightarrow{\,\beta\,}} s_{\beta} x$ for $x \in ({W^{J}})_{{\mathrm{af}}}$ and $\beta \in {\Delta_{{\mathrm{af}}}^{+}}$, where $s_{\beta } x \in ({W^{J}})_{{\mathrm{af}}}$ and ${\ell^{{\frac{\infty}{2}}}}(s_{\beta} x) = {\ell^{{\frac{\infty}{2}}}}(x) + 1$. When $J=\emptyset$, we write ${\mathrm{BG}^{{\frac{\infty}{2}}}(W_{{\mathrm{af}}})}$ for $\mathrm{BG}^{{\frac{\infty}{2}}}\bigl((W^{\emptyset})_{{\mathrm{af}}}\bigr)$.

The semi-infinite Bruhat order is a partial order ${\preceq}$ on $({W^{J}})_{{\mathrm{af}}}$ defined as follows: for $x,\,y \in ({W^{J}})_{{\mathrm{af}}}$, we write $x {\preceq}y$ if there exists a directed path from $x$ to $y$ in ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$; we write $x {\prec}y$ if $x {\preceq}y$ and $x \ne y$.

In the case $J = \emptyset$, the semi-infinite Bruhat order on $W_{{\mathrm{af}}}$ is just the generic Bruhat order introduced in [@Lu80]; see [@INS Appendix A.3] for details. Also, for a general $J$, the parabolic semi-infinite Bruhat order on $({W^{J}})_{{\mathrm{af}}}$ is nothing but the partial order on $J$-alcoves introduced in [@Lu97] when we take a special point to be the origin.

In Appendix \[sec:basic\], we recall some of the basic properties of the semi-infinite Bruhat order.

For $x \in ({W^{J}})_{{\mathrm{af}}}$, let ${\mathrm{Lift}}(x)$ denote the set of lifts of $x$ in $W_{{\mathrm{af}}}$ with respect to the map ${\Pi^{J}}:W_{{\mathrm{af}}} \twoheadrightarrow ({W^{J}})_{{\mathrm{af}}}$, that is, $$\label{eq:lift}
{\mathrm{Lift}}(x):=
\bigl\{ x' \in W_{{\mathrm{af}}} \mid {\Pi^{J}}(x') = x \bigr\};$$ for an explicit description of ${\mathrm{Lift}}(x)$, see Lemma \[lem:lift\]. The following proposition will be proved in Appendix \[sec:prf-Deo\].

\[prop:Deo\] If $x \in W_{{\mathrm{af}}}$ and $y \in ({W^{J}})_{{\mathrm{af}}}$ satisfy the condition that $y {\succeq}{\Pi^{J}}(x)$, then the set $${\mathrm{Lift}_{{\succeq}x}(y)}:=
\bigl\{ y' \in {\mathrm{Lift}}(y) \mid y' {\succeq}x \bigr\}$$ has the minimum element with respect to the semi-infinite Bruhat order on $W_{{\mathrm{af}}}$[;]{} we denote this element by $\min {\mathrm{Lift}_{{\succeq}x}(y)}$.

Semi-infinite Lakshmibai-Seshadri paths. {#subsec:SLS}
----------------------------------------

In this subsection, we fix $\lambda \in P^{+} \subset P_{{\mathrm{af}}}^{0}$ (see and ), and set $$\label{eq:J}
J:= \bigl\{ i \in I \mid {\langle \lambda,\,\alpha_i^{\vee} \rangle}=0 \bigr\} \subset I.$$

\[dfn:SBa\] For a rational number $0 < a < 1$, we define ${\mathrm{BG}^{{\frac{\infty}{2}}}_{\lambda,a}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ to be the subgraph of ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ with the same vertex set but having only the edges of the form $x {\xrightarrow{\,\beta\,}} y$ with $a{\langle x\lambda,\,\beta^{\vee} \rangle} \in {\mathbb{Z}}$.

\[dfn:SLS\] A semi-infinite Lakshmibai-Seshadri (LS for short) path of shape $\lambda $ is a pair $$\label{eq:SLS}
\pi = ({\mathbf{x}}\,;\, {\mathbf{a}}) 
     = (x_{1},\,\dots,\,x_{s} \,;\, a_{0},\,a_{1},\,\dots,\,a_{s}), \quad s \ge 1,$$ of a strictly decreasing sequence ${\mathbf{x}}: x_1 {\succ}\cdots {\succ}x_s$ of elements in $({W^{J}})_{{\mathrm{af}}}$ and an increasing sequence ${\mathbf{a}}: 0 = a_0 < a_1 < \cdots  < a_s =1$ of rational numbers satisfying the condition that there exists a directed path from $x_{u+1}$ to $x_{u}$ in ${\mathrm{BG}^{{\frac{\infty}{2}}}_{\lambda,a_{u}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ for each $u = 1,\,2,\,\dots,\,s-1$. We denote by ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ the set of all semi-infinite LS paths of shape $\lambda$.

Following [@INS Sect. 3.1] (see also [@NS16 Sect. 2.4]), we endow the set ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ with a crystal structure with weights in $P_{{\mathrm{af}}}$ by the map ${\mathop{\rm wt}\nolimits}:{\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \rightarrow P_{{\mathrm{af}}}$ and the root operators $e_{i}$, $f_{i}$, $i \in I_{{\mathrm{af}}}$; for details, see Appendix \[sec:crystal\]. We denote by ${\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda)$ the connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ containing $\pi_{\lambda}:=(e\,;\,0,1) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$.

If $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ is of the form , then we set $$\label{eq:dir}
\iota(\pi):=x_{1} \in ({W^{J}})_{{\mathrm{af}}} \qquad 
 \text{(resp., $\kappa(\pi):=x_{s} \in ({W^{J}})_{{\mathrm{af}}}$)};$$ we call $\iota(\pi)$ (resp., $\kappa(\pi)$) the initial (resp., final) direction of $\pi$. For $x \in W_{{\mathrm{af}}}$, we set $$\label{eq:SLS-dem}
{\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda) := 
 \bigl\{
   \pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \mid \kappa(\pi) {\succeq}{\Pi^{J}}(x)
 \bigr\}.$$

Extremal weight modules and their Demazure submodules. {#subsec:extremal}
------------------------------------------------------

In this subsection, we fix $\lambda \in P^{+} \subset P_{{\mathrm{af}}}^{0}$ (see and ). Let $V(\lambda)$ denote the extremal weight module of extremal weight $\lambda$ over $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$, which is an integrable $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$-module generated by a single element $v_{\lambda}$ with the defining relation that $v_{\lambda}$ is an “extremal weight vector” of weight $\lambda$; recall from [@Kas02 Sect. 3.1] and [@Kas05 Sect. 2.6] that $v_{\lambda}$ is an extremal weight vector of weight $\lambda$ if and only if ($v_{\lambda}$ is a weight vector of weight $\lambda$ and) there exists a family $\{ v_{x} \}_{x \in W_{{\mathrm{af}}}}$ of weight vectors in $V(\lambda)$ such that $v_{e}=v_{\lambda}$, and such that for every $i \in I_{{\mathrm{af}}}$ and $x \in W_{{\mathrm{af}}}$ with $n:={\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} \ge 0$ (resp., $\le 0$), the equalities $E_{i}v_{x}=0$ and $F_{i}^{(n)}v_{x}=v_{s_{i}x}$ (resp., $F_{i}v_{x}=0$ and $E_{i}^{(-n)}v_{x}=v_{s_{i}x}$) hold, where for $i \in I_{{\mathrm{af}}}$ and $k \in {\mathbb{Z}}_{\ge 0}$, the $E_{i}^{(k)}$ and $F_{i}^{(k)}$ are the $k$-th divided powers of $E_{i}$ and $F_{i}$, respectively; note that the weight of $v_{x}$ is $x\lambda$. Also, for each $x \in W_{{\mathrm{af}}}$, we define the Demazure submodule $V_{x}^{-}(\lambda)$ of $V(\lambda)$ by $$\label{eq:dem}
V_{x}^{-}(\lambda):=U_{{\mathsf{q}}}^{-}({\mathfrak{g}}_{{\mathrm{af}}})v_{x}.$$

We know from [@Kas94 Proposition 8.2.2] that $V(\lambda)$ has a crystal basis ${\mathcal{B}}(\lambda)$ and the corresponding global basis $\bigl\{G(b) \mid b \in {\mathcal{B}}(\lambda)\bigr\}$; we denote by $u_{\lambda}$ the element of ${\mathcal{B}}(\lambda)$ such that $G(u_{\lambda})=v_{\lambda}$, and by ${\mathcal{B}}_{0}(\lambda)$ the connected component of ${\mathcal{B}}(\lambda)$ containing $u_{\lambda}$. Also, we know from [@Kas05 Sect. 2.8] (see also [@NS16 Sect. 4.1]) that $V_{x}^{-}(\lambda) \subset V(\lambda)$ is compatible with the global basis of $V(\lambda)$, that is, there exists a subset ${\mathcal{B}}_{x}^{-}(\lambda)$ of the crystal basis ${\mathcal{B}}(\lambda)$ such that $$\label{eq:deme}
V_{x}^{-}(\lambda) = 
\bigoplus_{b \in {\mathcal{B}}_{x}^{-}(\lambda)} {\mathbb{C}}({\mathsf{q}}) G(b) 
\subset
V(\lambda) = 
\bigoplus_{b \in {\mathcal{B}}(\lambda)} {\mathbb{C}}({\mathsf{q}}) G(b).$$

For every $x \in W_{{\mathrm{af}}}$, we have $V_{x}^{-}(\lambda) = V_{{\Pi^{J}}(x)}^{-}(\lambda)$ and ${\mathcal{B}}_{x}^{-}(\lambda) = {\mathcal{B}}_{{\Pi^{J}}(x)}^{-}(\lambda)$.

We know the following from [@INS Theorem 3.2.1] and [@NS16 Theorem 4.2.1].

\[thm:isom\] There exists an isomorphism $\Phi_{\lambda}:{\mathcal{B}}(\lambda) \stackrel{\sim}{\rightarrow} {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ of crystals such that $\Phi(u_{\lambda}) = \pi_{\lambda}$ and such that $\Phi_{\lambda}({\mathcal{B}}_{x}^{-}(\lambda)) = {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)$ for all $x \in W_{{\mathrm{af}}}$[;]{} in particular, we have $\Phi_{\lambda}({\mathcal{B}}_{0}(\lambda)) = {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda)$.

Let $x \in W_{{\mathrm{af}}}$. If $x$ is of the form $x = wt_{\xi}$ for some $w \in W$ and $\xi \in Q^{\vee}$, then $v_{x} \in V(\lambda)$ is a weight vector of weight $x\lambda = 
w\lambda-{\langle \lambda,\,\xi \rangle}\delta$; note that $w\lambda \in \lambda-Q^{+}$. Also, for $i \in I$ (resp., $i=0 \in I_{{\mathrm{af}}}$), the Chevalley generator $F_{i}$ (resp., $F_{0}$) of $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$ acts on $V(\lambda)$ as a (linear) operator of weight $-\alpha_{i} \in Q$ (resp., $-\alpha_{0}=\theta-\delta \in Q + {\mathbb{Z}}_{< 0}\delta$). Therefore, the Demazure submodule $V_{x}^{-}(\lambda) = U_{{\mathsf{q}}}^{-}({\mathfrak{g}}_{{\mathrm{af}}})v_{x}$ has the weight space decomposition of the form: $$V_{x}^{-}(\lambda) = 
 \bigoplus_{k \in {\mathbb{Z}}}
\Biggl(
\underbrace{
   \bigoplus_{\gamma \in Q} 
   V_{x}^{-}(\lambda)_{\lambda+\gamma+k\delta}}_{=:V_{x}^{-}(\lambda)_{k}}
\Biggr),$$ where $V_{x}^{-}(\lambda)_{k} = \bigl\{ 0 \bigr\}$ for all $k > - {\langle \lambda,\,\xi \rangle}$; in addition, by Theorem \[thm:isom\], together with the definition of the map ${\mathop{\rm wt}\nolimits}:{\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \rightarrow P_{{\mathrm{af}}}$ (see ), we see that if $\gamma \notin -Q^{+}$, then $V_{x}^{-}(\lambda)_{\lambda+\gamma+k\delta} = \bigl\{0\bigr\}$ for all $k \in {\mathbb{Z}}$, since $W_{{\mathrm{af}}}\lambda \subset \lambda - Q^{+} +{\mathbb{Z}}\delta$ by the assumption that $\lambda \in P^{+}$. Here we claim that $V_{x}^{-}(\lambda)_{k}$ is finite-dimensional for all $k \in {\mathbb{Z}}$ with $k \le -{\langle \lambda,\,\xi \rangle}$; we show this assertion by descending induction on $k$. Let $U_{{\mathsf{q}}}^{-}({\mathfrak{g}})$ denote the ${\mathbb{C}}({\mathsf{q}})$-subalgebra of $U_{{\mathsf{q}}}^{-}({\mathfrak{g}}_{{\mathrm{af}}})$ generated by $F_{i}$, $i \in I$. If $k=-{\langle \lambda,\,\xi \rangle}$, then the assertion is obvious since $V_{x}^{-}(\lambda)_{-{\langle \lambda,\,\xi \rangle}} = U_{{\mathsf{q}}}^{-}({\mathfrak{g}})v_{x}$ and $V(\lambda)$ is an integrable $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$-module. Assume that $k < -{\langle \lambda,\,\xi \rangle}$. Observe that $V_{x}^{-}(\lambda)_{k}$ is a $U_{{\mathsf{q}}}^{-}({\mathfrak{g}})$-module generated by $F_{0}V_{x}^{-}(\lambda)_{k+1}$. Because $F_{0}V_{x}^{-}(\lambda)_{k+1}$ is finite-dimensional by our induction hypothesis, and $V(\lambda)$ is an integrable $U_{q}({\mathfrak{g}}_{{\mathrm{af}}})$-module, we deduce that $V_{x}^{-}(\lambda)_{k} = U_{{\mathsf{q}}}^{-}({\mathfrak{g}})(F_{0}V_{x}^{-}(\lambda)_{k+1})$ is also finite-dimensional, as desired.

Now, we define the graded character ${\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda)$ of $V_{x}^{-}(\lambda)$ to be $$\label{eq:gch}
{\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda) : = 
 \sum_{k \in {\mathbb{Z}}}
\Biggl(
 \sum_{\gamma \in Q} 
 \dim \bigl( V_{x}^{-}(\lambda)_{\lambda+\gamma+k\delta} \bigr) 
 e^{\lambda+\gamma}
\Biggr) q^{k};$$ observe that $$\label{eq:gch1}
{\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda) \in 
\bigl(\underbrace{{\mathbb{Z}}[e^{\nu} \mid \nu \in P]}_{={\mathbb{Z}[P]}}\bigr)
{[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}q^{-{\langle \lambda,\,\xi \rangle}}.$$ For $\gamma \in Q$ and $k \in {\mathbb{Z}}$, we set ${\mathop{\rm fin}\nolimits}(\lambda+\gamma+k\delta):=\lambda+\gamma \in P$ and ${\mathop{\rm nul}\nolimits}(\lambda+\gamma+k\delta):=k \in {\mathbb{Z}}$. Then, by Theorem \[thm:isom\], we have $$\label{eq:gch2}
{\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda) = \sum_{\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)} 
e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))} q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))}.$$

Combinatorial standard monomial theory for semi-infinite LS paths. {#sec:SMT}
==================================================================

In this section, we fix $\lambda,\,\mu \in P^{+} \subset P_{{\mathrm{af}}}^{0}$ (see and ), and set $$J:=\bigl\{i \in I \mid {\langle \lambda,\,\alpha_{i}^{\vee} \rangle}=0 \bigr\}, \qquad 
K:=\bigl\{i \in I \mid {\langle \mu,\,\alpha_{i}^{\vee} \rangle}=0 \bigr\}.$$

Standard paths. {#subsec:SP}
---------------

We consider the following condition on $\pi \otimes \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$: $$\label{eq:SM}
\begin{cases}
\text{there exist $x,\,y \in W_{{\mathrm{af}}}$ such that 
$x {\succeq}y$ in $W_{{\mathrm{af}}}$}, \\[1mm]
\text{and such that ${\Pi^{J}}(x)=\kappa(\pi)$, ${\Pi^{K}}(y)=\iota(\eta)$};
\end{cases} \tag{\sf SP}$$ we set $${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu):=
  \bigl\{\pi \otimes \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu) \mid 
  \text{$\pi \otimes \eta$ satisfies condition \eqref{eq:SM}}
  \bigr\}.$$

\[thm:SMT\] The set ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu) \sqcup \{{\bm{0}}\}$ is stable under the action of the Kashiwara [(]{}or, root[)]{} operators $e_{i}$, $f_{i}$, $i \in I_{{\mathrm{af}}}$, on ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$[;]{} in particular, ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ is a crystal with weights in $P_{{\mathrm{af}}}$. Moreover, ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ is isomorphic as a crystal to ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu)$.

We will give a proof of Theorem \[thm:SMT\] in Section \[sec:prf-SMT\].

Defining chains. {#subsec:DC}
----------------

\[dfn:DC\] Let $\pi=(x_{1},\,\dots,\,x_{s}\,;\,{\mathbf{a}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta=(y_{1},\,\dots,\,y_{p}\,;\,{\mathbf{b}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. A defining chain for $\pi \otimes \eta$ is a sequence $x_{1}',\,\dots,\,x_{s}',\,y_{1}',\,\dots,\,y_{p}'$ of elements in $W_{{\mathrm{af}}}$ satisfying the condition: $$\label{eq:DC}
\begin{cases}
x_{1}' {\succeq}\cdots {\succeq}x_{s}' {\succeq}y_{1}' {\succeq}\cdots {\succeq}y_{p}' 
 \quad \text{\rm in $W_{{\mathrm{af}}}$}; \\[1.5mm]
{\Pi^{J}}(x_{u}')=x_{u} 
 \quad \text{\rm for $1 \le u \le s$}; \\[1.5mm]
{\Pi^{K}}(y_{q}')=y_{q}
 \quad \text{\rm for $1 \le q \le p$};
\end{cases}
\tag{\sf DC}
$$ we call $x_{1}'$ (resp., $y_{p}'$) the initial element (resp., the final element) of this defining chain.

\[prop:DC\] Let $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. Then, $\pi \otimes \eta \in {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ if and only if there exists a defining chain for $\pi \otimes \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$.

We will give a proof of Proposition \[prop:DC\] in Section \[subsec:prf-DC\].

Now, let $\eta=(y_{1},\,\dots,\,y_{p}\,;\,{\mathbf{b}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. For each $x \in W_{{\mathrm{af}}}$ such that $\kappa(\eta) = y_{p} {\succeq}{\Pi^{K}}(x)$, we define a specific lift ${\iota(\eta,\,x)} \in W_{{\mathrm{af}}}$ of $\iota(\eta) = y_{1} \in ({W^{K}})_{{\mathrm{af}}}$ as follows. Since $y_{p} {\succeq}{\Pi^{K}}(x)$ by the assumption, it follows from Proposition \[prop:Deo\] that ${\mathrm{Lift}_{{\succeq}x}(y_{p})}$ has the minimum element $\min {\mathrm{Lift}_{{\succeq}x}(y_{p})}=:{\widetilde{y}}_{p}$. Similarly, since $y_{p-1} {\succeq}y_{p} = {\Pi^{K}}({\widetilde{y}}_{p})$, it follows again from Proposition \[prop:Deo\] that ${\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{p}}(y_{p-1})}$ has the minimum element $\min {\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{p}}(y_{p-1})} = :{\widetilde{y}}_{p-1}$. Continuing in this way, we obtain ${\widetilde{y}}_{p}$, ${\widetilde{y}}_{p-1}$, $\dots$, ${\widetilde{y}}_{1}$. Namely, these elements are defined by the following recursive procedure (from $p$ to $1$): $$\label{eq:lift0}
\begin{cases}
{\widetilde{y}}_{p} := \min {\mathrm{Lift}_{{\succeq}x}(y_{p})}, & \\[1mm]
{\widetilde{y}}_{q} := \min {\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{q+1}}(y_{q})}
& \text{for $1 \le q \le p-1$}.
\end{cases}
$$ Finally, we set $$\label{eq:lift0a}
{\iota(\eta,\,x)}:={\widetilde{y}}_{1};$$ this element is sometimes called the initial direction of $\eta$ with respect to $x$.

\[prop:DC2\] Let $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$. Then, $\pi \otimes \eta \in {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ [(]{}or equivalently, there exists a defining chain for $\pi \otimes \eta \in 
{\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$ by Proposition \[prop:DC\][)]{} if and only if $\kappa(\pi) {\succeq}{\Pi^{J}}({\iota(\eta,\,x)})$ for some $x \in W_{{\mathrm{af}}}$ such that $\kappa(\eta) {\succeq}{\Pi^{K}}(x)$.

We will give a proof of Proposition \[prop:DC2\] in Section \[subsec:prf-DC2\].

Demazure crystals in terms of standard paths. {#subsec:dem-SP}
---------------------------------------------

We set $S:=\bigl\{i \in I \mid {\langle \lambda+\mu,\,\alpha_{i}^{\vee} \rangle}=0 \bigr\}={J}\cap {K}$. For each $x \in W_{{\mathrm{af}}}$, we define ${\mathbb{S}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu) 
\subset {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ to be the image of ${\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda+\mu)=
 \bigl\{ \psi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu) \mid \kappa(\psi) {\succeq}{\Pi^{S}}(x) \bigr\}$ under the isomorphism ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda + \mu) \cong {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ in Theorem \[thm:SMT\].

\[thm:Dem\] Let $x \in W_{{\mathrm{af}}}$. For $\pi \otimes \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$, the following conditions , , and are equivalent[:]{} $$\label{eq:Da}
\pi \otimes \eta \in {\mathbb{S}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu); \tag{\sf D1}$$ $$\label{eq:Db}
\begin{cases}
\text{\rm there exists a defining chain for $\pi \otimes \eta$ whose final element, say $y$, } \\[1mm]
\text{\rm satisfies the condition that ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$};
\end{cases} \tag{\sf D2}
$$ $$\label{eq:Dc}
\kappa(\eta) {\succeq}{\Pi^{K}}(x) \quad \text{\rm and} \quad
\kappa(\pi) {\succeq}{\Pi^{J}}({\iota(\eta,\,x)}). \tag{\sf D3}$$ Therefore, we have $${\mathbb{S}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu) = 
\bigl\{ \pi \otimes \eta \mid 
 \text{\rm $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\mu)$ and 
 $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}{\iota(\eta,\,x)}}(\lambda)$}
\bigr\},$$ and hence [(]{}see $${\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda+\mu) = 
\sum_{ \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\mu) } e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\eta))} q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\eta))} 
\underbrace{\sum_{ \pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}{\iota(\eta,\,x)}}(\lambda) } e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))} q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))}
}_{ \text{\rm $={\mathop{\rm gch}\nolimits}V_{{\iota(\eta,\,x)}}^{-}(\lambda)$ by \eqref{eq:gch2}} }.$$

We will give a proof of Theorem \[thm:Dem\] in Section \[sec:prf-Dem\].

Semi-infinite Schubert varieties and their resolutions. {#sec:SiSch}
=======================================================

Geometric setting. {#subsec:geo}
------------------

An (algebraic) variety is an integral separated scheme of finite type over ${\mathbb{C}}$. Also, a pro-affine space is a product of finitely many copies of ${\mathop{\rm Spec}\nolimits}{\mathbb{C}}[ x_{m} \mid m \ge 0 ]$, equipped with a truncation morphism ${\mathop{\rm Spec}\nolimits}{\mathbb{C}}[ x_{m} \mid m \ge 0 ] \rightarrow {\mathop{\rm Spec}\nolimits}{\mathbb{C}}[ x_{m} \mid 0 \le m \le n ]$ for $n \gg 0$; by a morphism of pro-affine spaces, we mean a morphism of schemes that is also continuous with respect to the topology induced by the truncation morphisms (this topology itself is irrelevant to the Zariski topology). For a ${\mathbb{C}}$-vector space $V$, we set ${\mathbb{P}}(V) := (V \setminus \{ 0 \}) / {\mathbb{C}}^{\times}$. We usually regard ${\mathbb{P}}(V)$ as an algebraic variety over ${\mathbb{C}}$ when $\dim V < \infty$, or as an ind/pro-scheme when $\dim V = \infty$ in accordance with the topology of $V$.

For an algebraic group $E$, let $E{[\hspace{-1pt}[z]\hspace{-1pt}]}$, $E{(\hspace{-1pt}(z)\hspace{-1pt})}$, and $E [z]$ denote the set of ${\mathbb{C}}{[\hspace{-1pt}[z]\hspace{-1pt}]}$-valued points, ${\mathbb{C}}{(\hspace{-1pt}(z)\hspace{-1pt})}$-valued points, and ${\mathbb{C}}[z]$-valued points of $E$, respectively; the corresponding Lie algebras are denoted by ${\mathfrak{e}}{[\hspace{-1pt}[z]\hspace{-1pt}]}$, ${\mathfrak{e}}{(\hspace{-1pt}(z)\hspace{-1pt})}$, and ${\mathfrak{e}}[z]$, respectively, with $E$ replaced by its German letter ${\mathfrak{e}}={\mathop{\rm Lie}\nolimits}(E)$. Also, we denote by $R(E)$ the representation ring of $E$.

Recall that $G$ is a connected, simply-connected simple algebraic group over ${\mathbb{C}}$; concerning the Lie algebra ${\mathfrak{g}}={\mathop{\rm Lie}\nolimits}(G)$ and its untwisted affinization ${\mathfrak{g}}_{{\mathrm{af}}}$, we use the notation of Section \[sec:notation\]. We have an evaluation map ${\mathop{\tt ev}\nolimits}_{0} : G{[\hspace{-1pt}[z]\hspace{-1pt}]} \longrightarrow G$ at $z=0$. Let ${\mathbf{I}}:= {\mathop{\tt ev}\nolimits}_{0}^{-1} ( B )$ be an Iwahori subgroup of $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$. Also, for each $i \in I_{{\mathrm{af}}}$, we have a minimal parahoric subgroup ${\mathbf{I}}\subset {\mathbf{I}}(i) \subset G{[\hspace{-1pt}[z]\hspace{-1pt}]}$ corresponding to $\alpha_{i}$, so that ${\mathbf{I}}(i) / {\mathbf{I}}\cong {\mathbb{P}}^{1}$. Note that both $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$ and ${\mathbf{I}}$ admit an action of ${\mathbb{G}_{m}}$ obtained by the scalar dilation on $z$; we denote the resulting semi-direct product groups by ${\widetilde{G}}{[\hspace{-1pt}[z]\hspace{-1pt}]}$ and ${\widetilde{{\mathbf{I}}}}$, respectively. The (finite) Weyl group $W$ of ${\mathfrak{g}}$ is isomorphic to $N_{G}(H)/H$, and $Q^{\vee}$ is isomorphic to $H{(\hspace{-1pt}(z)\hspace{-1pt})}/H{[\hspace{-1pt}[z]\hspace{-1pt}]}$, both of which fit in the following commutative diagram involving the (affine) Weyl group $W_{{\mathrm{af}}}$ of ${\mathfrak{g}}_{{\mathrm{af}}}$: $$\begin{CD}
0 @>>> Q^{\vee} @>>> W_{{\mathrm{af}}} @>>> W @>>> e \\
@.  @A{\cong}AA  @A{\cong}AA @A{\cong}AA \\
@. H{(\hspace{-1pt}(z)\hspace{-1pt})}/H{[\hspace{-1pt}[z]\hspace{-1pt}]} @>>> N_{G{(\hspace{-1pt}(z)\hspace{-1pt})}}(H)/H{[\hspace{-1pt}[z]\hspace{-1pt}]} @>>> 
   \underbrace{N_{G{(\hspace{-1pt}(z)\hspace{-1pt})}}(H)/H{(\hspace{-1pt}(z)\hspace{-1pt})}}_{= N_{G}(H)/H}, \\
\end{CD}
$$ where the first row is exact, and the rightmost isomorphism in the second row holds since $N_{G{(\hspace{-1pt}(z)\hspace{-1pt})}}(H) \cong 
N_{G{(\hspace{-1pt}(z)\hspace{-1pt})}}(H{(\hspace{-1pt}(z)\hspace{-1pt})}) \cong (N_{G}(H)){(\hspace{-1pt}(z)\hspace{-1pt})}$. In particular, we have a lift $\dot{w} \in N_{G{(\hspace{-1pt}(z)\hspace{-1pt})}}(H)$ for each $w \in W_{{\mathrm{af}}}$. Now, for $x \in W_{{\mathrm{af}}}$ and $i \in I_{{\mathrm{af}}}$, we set $$s_{i} \ast x := 
\begin{cases} 
 x & \text{if $s_{i} x < x$}, \\
 s_{i} x & \text{if $s_{i} x > x$}, 
\end{cases}
$$ where we denote by $>$ the ordinary Bruhat order on $W_{{\mathrm{af}}}$. Then, the set $W_{{\mathrm{af}}}$ becomes a monoid, which we denote by ${\mathbf{W}_{{\mathrm{af}}}}$, under the product $\ast$; this monoid is also obtained as a subset of the generic Hecke algebra associated to $(W_{{\mathrm{af}}},\,I_{{\mathrm{af}}})$ by setting $a_{i} = 1$ and $b_{i} = 0$ for $i \in I_{{\mathrm{af}}}$ in [@Hum90 Sect. 7.1, Theorem].

Semi-infinite flag manifolds. {#subsec:SiFl}
-----------------------------

Here we review two models of semi-infinite flag manifold associated to $G$, for which the basic references are [@FM99] and [@FFKM99].

Let $L(\lambda)$ denote the (finite-dimensional) irreducible highest weight ${\mathfrak{g}}$-module of highest weight $\lambda \in P^{+}$. Recall that for each $\lambda,\,\mu \in P^{+}$, we have a canonical embedding of irreducible highest weight ${\mathfrak{g}}$-modules (and hence of $G$-modules) up to scalars: $$\label{HWtensor}
L(\lambda + \mu) \hookrightarrow L(\lambda) \otimes_{{\mathbb{C}}} L(\mu).$$ The embedding induces an embedding $$\label{eq:emb}
L(\lambda+\mu) \otimes_{{\mathbb{C}}} R \hookrightarrow  
  \bigl( L(\lambda) \otimes_{{\mathbb{C}}} L(\mu) \bigr) \otimes_{{\mathbb{C}}} R \cong 
  (L(\lambda) \otimes_{{\mathbb{C}}} R) \otimes_{R} (L(\mu) \otimes_{{\mathbb{C}}} R)$$ for every commutative, associative ${\mathbb{C}}$-algebra $R$.

\[thm:BG\] Let ${\mathbb{K}}$ be a field containing ${\mathbb{C}}$. The set of collections $\{ \ell^{\lambda} \}_{\lambda \in P^{+}}$ of one-dimensional ${\mathbb{K}}$-vector subspaces $\ell^{\lambda}$ in $L(\lambda) \otimes_{{\mathbb{C}}} {\mathbb{K}}$ such that $\ell^{\lambda} \otimes_{{\mathbb{K}}} \ell^{\mu} = \ell^{\lambda+\mu}$ for every $\lambda,\,\mu \in P^{+}$ [(]{}under the embedding is in bijection with the set of closed ${\mathbb{K}}$-points of $G/B$.

For a ${\mathfrak{g}}$-module $V$, we set $V{[\hspace{-1pt}[z]\hspace{-1pt}]} := V \otimes_{{\mathbb{C}}} {\mathbb{C}}{[\hspace{-1pt}[z]\hspace{-1pt}]}$ and $V{(\hspace{-1pt}(z)\hspace{-1pt})} := V \otimes_{{\mathbb{C}}} {\mathbb{C}}{(\hspace{-1pt}(z)\hspace{-1pt})}$.

\[dfn:DP\] Consider a collection $L = \{ L^{\lambda} \}_{\lambda \in P^{+}}$ of one-dimensional ${\mathbb{C}}$-vector subspaces $L^{\lambda}$ in $L(\lambda){[\hspace{-1pt}[z]\hspace{-1pt}]} = L(\lambda) \otimes_{{\mathbb{C}}} {\mathbb{C}}{[\hspace{-1pt}[z]\hspace{-1pt}]}$ (resp., $L(\lambda){(\hspace{-1pt}(z)\hspace{-1pt})} = L(\lambda) \otimes_{{\mathbb{C}}} {\mathbb{C}}{(\hspace{-1pt}(z)\hspace{-1pt})}$). The datum $L$ is called a formal (resp., rational) Drinfeld-Plücker (DP for short) datum if for every $\lambda,\,\mu \in P^{+}$, the equality $$\label{eq:DP}
L^{\lambda + \mu} = L^{\lambda} \otimes_{{\mathbb{C}}} L^{\mu}$$ holds under the embedding , where $L^{\lambda} \otimes_{{\mathbb{C}}} L^{\mu}$ is considered to be its image under the map $L(\lambda){[\hspace{-1pt}[z]\hspace{-1pt}]} \otimes_{{\mathbb{C}}} L(\mu){[\hspace{-1pt}[z]\hspace{-1pt}]} 
\rightarrow L(\lambda){[\hspace{-1pt}[z]\hspace{-1pt}]} \otimes_{{\mathbb{C}}{[\hspace{-1pt}[z]\hspace{-1pt}]}} L(\mu){[\hspace{-1pt}[z]\hspace{-1pt}]}$ (resp., $L(\lambda){(\hspace{-1pt}(z)\hspace{-1pt})} \otimes_{{\mathbb{C}}} L(\mu){(\hspace{-1pt}(z)\hspace{-1pt})} 
\rightarrow L(\lambda){(\hspace{-1pt}(z)\hspace{-1pt})} \otimes_{{\mathbb{C}}{(\hspace{-1pt}(z)\hspace{-1pt})}} L(\mu){(\hspace{-1pt}(z)\hspace{-1pt})}$); we sometimes refer to a collection $\{u^{\lambda}\}_{\lambda \in P^{+}}$ of nonzero elements $u^{\lambda} \in L^{\lambda}$, $\lambda \in P^{+}$, as a formal (resp., rational) DP datum. Let ${\mathbf{Q}_{G}}$ (resp., ${\mathbf{Q}_{G}^{\mathrm{rat}}}$) denote the set of formal (resp., rational) DP data.

\[rem:DP\] By the compatibility condition , a DP datum $\{ L^{\lambda} \}_{\lambda \in P^{+}}$ is determined completely by a collection $\{u^{i}\}_{i \in I}$ of nonzero elements $u^{i} \in L^{{\varpi}_{i}}$ for $i \in I$. We call this collection $\{u^{i}\}_{i \in I}$ DP coordinates.

Let $L = \{ L^{\lambda} \}_{\lambda \in P^{+}} \in {\mathbf{Q}_{G}}$. We define $\deg L^{\lambda}$ to be the degree of a nonzero element in $L^{\lambda}$, viewed as an $L(\lambda)$-valued formal power series (if it is bounded). For each $\xi \in Q^{\vee,+}$, a DP datum of degree $\xi$ is a formal DP datum $L = \{ L^{\lambda} \}_{\lambda \in P^{+}}$ such that $\deg L^{\lambda} \le {\langle \lambda,\,\xi \rangle}$ for all $\lambda \in P^{+}$. For each $\xi \in Q^{\vee,+}$, let ${\mathfrak{Q}}_{G} (\xi)$ denote the set of formal DP data of degree $\xi$. Here we note that if $\xi,\,\zeta \in Q^{\vee,+}$ satisfy $\xi \le \zeta$, i.e., $\zeta - \xi \in Q^{\vee,+}$, then ${\mathfrak{Q}}_{G}(\xi) \subset {\mathfrak{Q}}_{G}(\zeta)$. We set ${\mathfrak{Q}}_{G} := \bigcup_{\xi \in Q^{\vee,+}} {\mathfrak{Q}}_{G}(\xi)$; observe that $${\mathfrak{Q}}_{G}(\xi) \subset {\mathfrak{Q}}_{G} \subset {\mathbf{Q}_{G}}\quad 
 \text{for each $\xi \in Q^{\vee,+}$}.$$ Also, we remark that ${\mathbf{Q}_{G}}$ has a natural $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$-action, and that its subsets ${\mathfrak{Q}}_{G}(\xi)$, $\xi \in Q^{\vee,+}$, and ${\mathfrak{Q}}_{G}$ are stable under the action of $G$ on ${\mathbf{Q}_{G}}$.

\[defbQ\] We have an embedding $$\label{Pemb}
{\mathbf{Q}_{G}}\ni 
  \{ L^{\lambda} \}_{\lambda \in P^{+}} \mapsto 
  \{ [ L^{{\varpi}_{i}} ] \} _{i \in I} \in 
  \prod_{i \in I} {\mathbb{P}}(L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]}),$$ which gives the set ${\mathbf{Q}_{G}}$ a [(]{}reduced[)]{} structure of an infinite type closed subscheme of $\prod_{i \in I} {\mathbb{P}}(L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]})$. In particular, ${\mathbf{Q}_{G}}$ is separated.

Because a DP datum $\{ L^{\lambda} \}_{\lambda \in P^{+}}$ is determined uniquely by $\{ L^{{\varpi}_{i}} \} _{i \in I}$ (see Remark \[rem:DP\]), the map above is injective. Also, condition is equivalent to saying that the image of $L^{\lambda} \otimes_{{\mathbb{C}}} L^{\mu}$ lies in the ${\mathbb{C}}$-vector subspace $L(\lambda+\mu){[\hspace{-1pt}[z]\hspace{-1pt}]} \subset 
\bigl(L(\lambda) \otimes_{{\mathbb{C}}} L(\mu) \bigr){[\hspace{-1pt}[z]\hspace{-1pt}]}$ for all $\lambda,\,\mu \in P^{+}$. This condition imposes infinitely many equations on $\prod _{i \in I} {\mathbb{P}}(L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]})$ that define ${\mathbf{Q}_{G}}$ as its closed subscheme. Since each ${\mathbb{P}}(L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]})$ is separated, so is ${\mathbf{Q}_{G}}$. This proves the lemma.

\[orbbQ\] The set of $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$-orbits in ${\mathbf{Q}_{G}}$ is labeled by $Q^{\vee,+}$. The codimension of the $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$-orbit corresponding to $\xi \in Q^{\vee,+}$ is equal to $2 {\langle \rho,\,\xi \rangle}$.

\[orbits\] The set of ${\mathbf{I}}$-orbits in ${\mathbf{Q}_{G}}$ is in bijection with the set $W_{{\mathrm{af}}}^{\ge 0} = 
\bigl\{w t_{\xi} \mid w \in W,\,\xi \in Q^{\vee,+}\bigr\}$.

Apply (the consequence of) the Bruhat decomposition $G{[\hspace{-1pt}[z]\hspace{-1pt}]} = \bigsqcup_{w \in W} {\mathbf{I}}\dot{w} {\mathbf{I}}$ to each $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$-orbit in ${\mathbf{Q}_{G}}$ described in Theorem \[orbbQ\].

For each $x = w t_{\xi} \in W_{{\mathrm{af}}}^{\ge 0}$, we denote by ${\mathbf{O}}(x)$ the ${\mathbf{I}}$-orbit of ${\mathbf{Q}_{G}}$ that contains a unique $(H \times {\mathbb{G}_{m}})$-fixed point corresponding to $\{z^{ {\langle {\varpi}_{i},\,\xi \rangle} } v_{w w_{\circ} {\varpi}_{i}} \} _{i \in I}$ (see Lemma \[defbQ\]), where for each $\lambda \in P^{+}$ and $w \in W$, we take and fix a nonzero vector $v_{w\lambda}$ of weight $w\lambda$ in $L(\lambda)$; note that the codimension of ${\mathbf{O}}(x) \subset {\overline{{\mathbf{O}}(e)}}$ is given by ${\ell^{{\frac{\infty}{2}}}}(x)$. We set ${\mathbf{Q}_{G}}( x ) := \overline{{\mathbf{O}}( x )} \subset {\mathbf{Q}_{G}}$. For $x,\,y \in W_{{\mathrm{af}}}^{\ge 0}$, we have $$\begin{aligned}
{\mathbf{Q}_{G}}(x) \subset {\mathbf{Q}_{G}}( y ) 
 & \iff x {\succeq}y \quad \text{(\cite[Sect.~5.1]{FFKM99})} \\
 & \iff x w_{\circ} {\preceq}yw_{\circ} \quad 
   \text{(see \cite[Lecture 13]{Pet97})}.\end{aligned}$$ Also, we have ${\mathbf{Q}_{G}}(e) = {\mathbf{Q}_{G}}$ by inspection; in fact, $e \in W_{{\mathrm{af}}}^{\ge 0}$ is the minimum element in the semi-infinite Bruhat order restricted to $W_{{\mathrm{af}}}^{\ge 0}$.

For $\xi \in Q^{\vee,+}$ and $x \in W_{{\mathrm{af}}}^{\ge 0}$, we set ${\mathfrak{Q}}_G (x,\,\xi) := {\mathfrak{Q}}_G (\xi) \cap {\mathbf{Q}_{G}}(x)$, and for $x \in W_{{\mathrm{af}}}^{\ge 0}$, we set ${\mathfrak{Q}}_G (x) := \bigcup_{\xi \in Q^{\vee,+}} {\mathfrak{Q}}_G (x,\,\xi)$.

For each $\xi \in Q^{\vee,+}$, we have an embedding $$\imath_{\xi} : 
 {\mathbf{Q}_{G}}\ni 
 \{ L^{\lambda} \}_{\lambda \in P^{+}} 
 \bigl(= \{ u^{\lambda} \}_{\lambda \in P^{+}} \bigr) \mapsto 
 \{ z ^{ {\langle \lambda,\,\xi \rangle} } u^{\lambda} \}_{\lambda \in P^{+}} \in {\mathbf{Q}_{G}}.$$ Thus we have its direct limit ${\mathbf{Q}_{G}^{\mathrm{rat}}}\cong \varinjlim_{\xi} {\mathbf{Q}_{G}}$, on which we have an action of $G{(\hspace{-1pt}(z)\hspace{-1pt})}$. By its construction, the embedding $\imath_{\xi}$ is $G {[\hspace{-1pt}[z]\hspace{-1pt}]}$-equivariant, and sends the ${\mathbf{I}}$-orbit ${\mathbf{O}}(x)$ to ${\mathbf{O}}(xt_{\xi})$.

Now, by Lemma \[defbQ\], we have a $G {[\hspace{-1pt}[z]\hspace{-1pt}]}$-equivariant line bundle ${\mathcal{O}}_{{\mathbf{Q}_{G}}}({\varpi}_{i})$ obtained by the pullback of the $i$-th ${\mathcal{O}}(1)$ through . For $\lambda = \sum_{i \in I} m_{i} {\varpi}_{i} \in P$, we set ${\mathcal{O}}_{{\mathbf{Q}_{G}}}(\lambda) := 
\bigotimes_{i \in I} {\mathcal{O}}_{{\mathbf{Q}_{G}}}({\varpi}_{i})^{\otimes m_{i}}$ (as a tensor product of line bundles). Also, for each $x \in W_{{\mathrm{af}}}^{\ge 0}$, we have the restriction ${\mathcal{O}}_{{\mathbf{Q}_{G}}(x)}(\lambda)$ obtained through , which is ${\widetilde{{\mathbf{I}}}}$-equivariant. Similarly, we have $( B \times {\mathbb{G}_{m}})$-equivariant line bundles ${\mathcal{O}}_{{\mathfrak{Q}}_G(x,\xi)} ( \lambda )$ and ${\mathcal{O}}_{{\mathfrak{Q}}_G(x)} ( \lambda )$ by further pullbacks (the latter is $G [z]$-equivariant whenever $x = t_{\zeta}$ for some $\zeta \in Q^{\vee,+}$); we set $$\label{eq:Hn}
H^{n} ( {\mathfrak{Q}}_G(x),\, {\mathcal{O}}_{{\mathfrak{Q}}_G(x)}(\lambda) ) := 
 \varprojlim _{\xi} H ^{n} ( {\mathfrak{Q}}_G(x,\,\xi),\,{\mathcal{O}}_{{\mathfrak{Q}}_G(x,\,\xi)}(\lambda) )
 \quad \text{for $n \in {\mathbb{Z}}_{\ge 0}$}.$$

Let $\lambda \in P^{+}$. As explained in [@Kat16 Theorem 1.6], the restricted dual of the Demazure submodule $V_{e}^{-} (- w_{\circ} \lambda )$ (see ) of the extremal weight $U_{{\mathsf{q}}}({\mathfrak{g}}_{{\mathrm{af}}})$-module $V(- w_{\circ} \lambda )$ of extremal weight $- w_{\circ} \lambda$ gives rise to an integrable ${\mathfrak{g}}[z]$-module (by taking the classical limit ${\mathsf{q}}\to 1$), called the global Weyl module; we denote it by $W(\lambda)$. Here we note that global Weyl modules carry natural gradings arising from the dilation of the $z$-variable.

\[BFmain\] For $\lambda \in P$, we have $$H^{0} ( {\mathring{\mathbf{Q}}_{G}}, \, {\mathcal{O}}_{{\mathbf{Q}_{G}}}(\lambda) ) \cong 
\begin{cases}  
 W(\lambda)^{\ast} & \text{\rm if $\lambda \in P^{+}$}, \\[1mm]
 \{ 0 \} & \text{\rm otherwise}, 
\end{cases}
$$ where ${\mathring{\mathbf{Q}}_{G}}$ denotes the open dense $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$-orbit in ${\mathbf{Q}_{G}}$.

\[Kmain\] For each $\lambda \in P$ and $x \in W_{{\mathrm{af}}}^{\ge 0}$, we have $${\mathop{\rm gch}\nolimits}\dot{H}^{n}( {\mathfrak{Q}}_G ( x ),\,{\mathcal{O}}_{{\mathfrak{Q}}_G(x)} ( \lambda ) ) = 
\begin{cases}
{\mathop{\rm gch}\nolimits}V_{x}^{-}( - w_{\circ} \lambda ) 
  & \text{\rm if $\lambda \in P^{+}$ and $n=0$}, \\
0 & \text{\rm otherwise},
\end{cases}
$$ where $\dot{H}^{n}( {\mathfrak{Q}}_G ( x ),\,{\mathcal{O}}_{{\mathfrak{Q}}_G(x)} ( \lambda ) )$ denotes the space of ${\mathbb{G}_{m}}$-finite vectors in .

\[Kmain-rem\] The assertion of [@Kat16 Theorem 4.12] is only for $x \in W$. The assertion of the form above easily follows from the isomorphism ${\mathbf{Q}_{G}}( x ) \cong {\mathbf{Q}_{G}}( x t_{\xi} )$ for $x \in W$ and $\xi \in Q^{\vee, +}$, which is obtained by means of $\imath_{\xi}$.

\[vample\] For each $\lambda \in P^{++}:=\sum_{i \in I} {\mathbb{Z}}_{> 0} {\varpi}_{i}$, the sheaf ${\mathcal{O}}_{{\mathbf{Q}_{G}}}(\lambda)$ is very ample.

The proof is exactly the same as that of [@Kat16 Corollary 2.7].

Bott-Samelson-Demazure-Hansen towers. {#subsec:BSDH}
-------------------------------------

In this subsection, we construct two kind of (pro-)schemes of infinite type, which can be thought of as “resolutions" of ${\mathbf{Q}_{G}}(x)$ for $x \in W_{{\mathrm{af}}}^{\ge 0}$, and study their properties.

\[lem:Mac\] If $\xi \in Q^{\vee}$ is a strictly antidominant coweight, i.e., ${\langle \alpha_{i},\,\xi \rangle} < 0$ for all $i \in I$, then $\ell(t_{\xi})=-2{\langle \rho,\,\xi \rangle}$, and $\ell(wt_{\xi})=\ell(t_{\xi}) - \ell(w)$ for all $w \in W$[;]{} hence we have $\ell(wt_{\xi})=-{\ell^{{\frac{\infty}{2}}}}(wt_{\xi})$ for all $w \in W$.

\[tweakw\]

${\ell^{{\frac{\infty}{2}}}}( x t_{- 2 m \rho^{\vee}} ) + 
 {\ell^{{\frac{\infty}{2}}}}( t _{2 m \rho^{\vee}} ) = {\ell^{{\frac{\infty}{2}}}}( x )$ for all $x \in W_{{\mathrm{af}}}$ and $m \in {\mathbb{Z}}$.

There exists $m_{0} \ge 0$ such that $-{\ell^{{\frac{\infty}{2}}}}( x t_{- 2 m \rho^{\vee}} ) = 
 \ell ( x t_{- 2 m \rho^{\vee}} )$ for all $m \ge m_{0}$.

Part (1) is obvious from the definition of ${\ell^{{\frac{\infty}{2}}}}(\,\cdot\,)$. For part (2), write $x$ as $x =wt_{\xi} \in W_{{\mathrm{af}}}$ for some $w \in W$ and $\xi \in Q^{\vee}$, and take $m_{0} \ge 0$ such that $\xi-2m_{0}\rho^{\vee}$ is strictly antidominant. Then we see from Lemma \[lem:Mac\] that $-{\ell^{{\frac{\infty}{2}}}}( x t_{- 2 m \rho^{\vee}} ) = 
 \ell ( x t_{- 2 m \rho^{\vee}} )$ for all $m \ge m_{0}$. This proves the lemma.

\[rem:ell\] Keep the setting of Lemma \[tweakw\](2). We have $$\ell ( x t_{- 2 (m_{0}+m) \rho^{\vee}} ) = 
\ell ( x t_{- 2 m_{0} \rho^{\vee}} ) + m\ell(t_{-2\rho^{\vee}}) = 
\ell ( x t_{- 2 m_{0} \rho^{\vee}} ) + \ell(t_{-2m\rho^{\vee}})$$ for all $m \ge 0$.

In what follows, we fix $x \in W_{{\mathrm{af}}}^{\ge 0}$ unless stated otherwise. For this $x$, we take $m_{0} \ge 0$ as in Lemma \[tweakw\](2), and fix reduced expressions $$\label{eq:reduced}
x t_{- 2 m_{0} \rho^{\vee}} = 
 s_{i'_1} s_{i'_2} \cdots s_{i'_{\ell'}} \qquad \text{and} \qquad
t_{- 2 \rho^{\vee}} = 
 s_{i''_1} s_{i''_2} \cdots s_{i''_{\ell}},
$$ where $i'_1,\,\ldots,\,i'_{\ell'},\,i''_1,\,\ldots,\,i''_{\ell} \in I_{{\mathrm{af}}}$, with $\ell'=\ell(x t_{- 2 m_{0} \rho^{\vee}})$ and $\ell=\ell(t_{-2\rho^{\vee}})$. We concatenate these sequences periodically to obtain an infinite sequence $$\label{eq:bi}
{\mathbf{i}}= (
\underbrace{i'_1,\,i'_2,\,i'_3,\,\ldots,\,i'_{\ell'}}_{ \text{for $x t_{- 2 m_{0} \rho^{\vee}}$} },\,
\underbrace{i''_1,\,i''_2,\,\ldots,\,i''_{\ell}}_{\text{for $t_{-2\rho^{\vee}}$}},\,
\underbrace{i''_1,\,i''_2,\,\ldots,\,i''_{\ell}}_{\text{for $t_{-2\rho^{\vee}}$}},\,i''_1,\,\ldots) 
\in I_{{\mathrm{af}}}^{\infty},$$ and write it as: ${\mathbf{i}}= (i_{1},\,i_{2},\,\ldots) \in I_{{\mathrm{af}}}^{\infty}$; remark that $s_{i_{1}}s_{i_{2}} \cdots s_{i_{k}}$ is reduced for all $k \ge 0$. For $k \in {\mathbb{Z}}_{\ge 0}$, we set ${\mathbf{i}}_{k} := (i_1,\,i_2,\,\ldots,\,i_{k})$.

Let $k \in {\mathbb{Z}}_{\ge 0}$, and let ${\mathbf{j}}= (i_{j_1},\,\dots,\,i_{j_t})$ be a subsequence of ${\mathbf{i}}_{k}$, where $1 \le j_{1} < \cdots < j_{t} \le k$. We set $\sigma ({\mathbf{j}}) = \sigma_{k}({\mathbf{j}}) : = 
\bigl\{1,\,2,\,\ldots,\,k\bigr\} \setminus 
\bigl\{j_{1},\,\dots,\,j_{t}\bigr\}$. We identify a subsequence ${\mathbf{j}}$ of ${\mathbf{i}}_{k}$ with a subsequence ${\mathbf{j}}'$ of ${\mathbf{i}}_{k'}$ if and only if $\sigma_{k}({\mathbf{j}}) = \sigma_{k'}({\mathbf{j}}')$ (as subsets of ${\mathbb{Z}}_{>0}$); namely, if $k' \ge k$, then ${\mathbf{j}}= (i_{j_1},\,\dots,\,i_{j_t}) \subset {\mathbf{i}}_{k}$ and ${\mathbf{j}}' = (i_{j_1},\,\dots,\,i_{j_t},\,i_{k+1},\,\dots,\,i_{k'}) 
\subset {\mathbf{i}}_{k'}$ are identified. Thus, we identify a subsequence ${\mathbf{j}}$ of ${\mathbf{i}}_{k}$ with a subsequence of ${\mathbf{i}}$ by taking the limit in $\varinjlim_{k} {\mathbf{i}}_{k} = {\mathbf{i}}$.

Let $k \in {\mathbb{Z}}_{\ge 0}$, and let ${\mathbf{j}}= (i_{j_1},\,i_{j_2},\,\ldots,\,i_{j_{t}})$ be a subsequence of ${\mathbf{i}}_{k}$. We set $$x({\mathbf{j}}; k) := 
s_{i_{j_1}} \ast s_{i_{j_2}} \ast \cdots \ast s_{i_{j_t}} \in {\mathbf{W}_{{\mathrm{af}}}}.$$

Let $k' \in {\mathbb{Z}}_{\ge 0}$ be such that $k' \ge k$. Because the sequence ${\mathbf{j}}$ above is identified with the subsequence $(i_{j_1},\,\dots,\,i_{j_t},\,i_{k+1},\,\dots,\,i_{k'})$ of ${\mathbf{i}}_{k'}$, we have $$x({\mathbf{j}}; k') := 
\underbrace{s_{i_{j_1}} \ast \cdots \ast s_{i_{j_t}}}_{=x({\mathbf{j}}; k)} \ast 
s_{i_{k+1}} \ast \cdots \ast s_{i_{k'}} \in {\mathbf{W}_{{\mathrm{af}}}}.$$

\[stab\] Let $m \in {\mathbb{Z}}_{\ge 0}$, and let ${\mathbf{j}}$ be a subsequence of ${\mathbf{i}}_{ \ell' + m \ell}$. Then, there exists $m_1 \ge m$ such that $x ( {\mathbf{j}}; \ell' + m' \ell ) = 
 x ( {\mathbf{j}}; \ell' + m'' \ell) \cdot t_{-2 (m' -m'') \rho^{\vee}}$ for every $m' \ge m'' \ge m_{1}$. In particular, the element $x ( {\mathbf{j}}) := x ( {\mathbf{j}}; \ell' + m' \ell ) \cdot 
 t_{2(m'+m_{0})\rho^{\vee}}  \in W_{{\mathrm{af}}}$ does not depend on the choice of $m' \ge m_{1}$.

We first note that $$x ( {\mathbf{j}}; \ell' + m' \ell ) = 
x ( {\mathbf{j}}; \ell' + m'' \ell) \ast 
\overbrace{
\underbrace{ ( s_{i''_1} \ast s_{i''_2} \ast \cdots \ast s_{i''_{\ell}} ) }_{\text{corresponding to $t_{-2\rho^{\vee}}$}}
\ast \cdots \ast
\underbrace{ ( s_{i''_1} \ast s_{i''_2} \ast \cdots \ast s_{i''_{\ell}} ) }_{\text{corresponding to $t_{-2\rho^{\vee}}$}}
}^{ \text{$(m'-m'')$ times} }.$$ Since $y \ast s_{i} = y s_{i}$ if and only if $\ell ( y s_{i} ) =  \ell ( y ) + 1$ for $y \in W_{{\mathrm{af}}}$ and $i \in I_{{\mathrm{af}}}$, it suffices to show that there exists $m_{1} \ge m$ such that $$\label{eq:ell-0}
\ell ( x ( {\mathbf{j}}; \ell' + m'' \ell) \cdot t_{-2 n \rho^{\vee}} ) = 
\ell ( x ( {\mathbf{j}}; \ell' + m'' \ell) ) + \ell ( t_{-2 n \rho^{\vee}} )$$ for all $n > 0$ and $m'' \ge m_{1}$. Let $k \in {\mathbb{Z}}_{\ge 0}$ be such that $k \ge m$. Since $x ( {\mathbf{j}}; \ell' + k \ell) = 
x ( {\mathbf{j}}; \ell' + m \ell) \ast t_{-2(k-m)\rho^{\vee}}$ and $\ell(t_{-2(k-m)\rho^{\vee}}) = (k-m)\ell$, we see that $$\label{eq:ell-1}
\ell ( x ( {\mathbf{j}}; \ell' + k \ell) ) \ge (k - m) \ell;$$ note that $\ell (y \ast y') \ge 
\max \bigl\{ \ell ( y ), \, \ell ( y' ) \bigr\}$ for $y,\,y' \in W_{{\mathrm{af}}}$, as is verified by induction. Now, for each integer $k \ge m$, we set $$d_{k}:= \underbrace{\ell ( t_{- 2 \rho^{\vee}} )}_{=\ell} - 
\bigl\{
\ell ( x ( {\mathbf{j}}; \ell' + ( k + 1 ) \ell) ) - 
\ell ( x ( {\mathbf{j}}; \ell' + k \ell) ) \bigr\};$$ observe that $d_{k} \in {\mathbb{Z}}_{\ge 0}$. Also, for $k \ge m$, we have $$(k-m)\ell = 
\ell ( x ( {\mathbf{j}}; \ell' + k \ell) ) - 
\ell ( x ( {\mathbf{j}}; \ell' + m \ell) ) + 
\sum_{k'=m}^{k-1} d_{k'}.$$ If $d_{k} > 0$ for infinitely many $k \ge m$, then $(k-m)\ell > \ell ( x ( {\mathbf{j}}; \ell' + k \ell) )$ for $k \gg m$, which contradicts . Hence we deduce that $d_{k} > 0$ only for finitely many $k \ge m$. Thus, if we set $m_{1}:=\max \bigl\{ k \ge m \mid d_{k} > 0 \bigr\}$, then holds. This proves the lemma.

\[si-refl\] For each $y,\,y' \in W_{{\mathrm{af}}}$ such that $y {\preceq}y'$, there exists $m_{2} \in {\mathbb{Z}}_{\ge 0}$ such that $y t_{-2m\rho^{\vee}} \ge y' t_{-2m\rho^{\vee}}$ in the ordinary Bruhat order on $W_{{\mathrm{af}}}$ for all $m \ge m_{2}$.

It suffices to prove the assertion in the case that $y {\xrightarrow{\,\beta\,}} s_{\beta}y = y'$ for some $\beta \in {\Delta_{{\mathrm{af}}}^{+}}$. Here we see from [@INS Corollary 4.2.2] that $\beta$ is either of the following forms:

\(i) $\beta = \alpha$ with $\alpha \in \Delta^{+}$; (ii) $\beta = \alpha+\delta$ with $-\alpha \in \Delta^{+}$.

Moreover, if $y = w t_{\xi}$ with $w \in W$ and $\xi \in Q^{\vee}$, then $\gamma:=w^{-1}\alpha \in \Delta^{+}$ in both cases above. Also, it follows from [@INS Proposition A.1.2] that $$\label{eq:ell}
\ell(w) = 
 \begin{cases}
  \ell(wr_{\gamma})-1 & \text{in case (i)}, \\[1mm]
  \ell(wr_{\gamma})-1+2 {\langle \rho,\,\gamma^{\vee} \rangle} & \text{in case (ii)}. 
 \end{cases}
$$

If we set $\zeta:=\xi-2m\rho^{\vee}$ for $m \in {\mathbb{Z}}$, then $y t_{- 2 m \rho^{\vee}} = w t _{\xi - 2 m \rho^{\vee}} = w t_{\zeta}$, and $$\begin{aligned}
s_{\beta} (y t_{- 2 m \rho^{\vee}}) & = 
 s_{\alpha+k\delta} w t_{\zeta}, 
 \quad \text{where $k=0$ in case (i), and $k=1$ in case (ii)}, \nonumber \\
 & = wt_{\zeta} s_{t_{-\zeta}w^{-1}\alpha+k\delta} 
   = (y t_{- 2 m \rho^{\vee}}) s_{\gamma + n \delta}, 
   \quad \text{with $n:=k+ {\langle \gamma,\,\zeta \rangle}$}. \label{eq:n}\end{aligned}$$ Therefore, in case (i) (resp., case (ii)), we deduce from [@LNSSS Proposition 5.1(1) (resp., (2)) with $v=e$], together with equalities and that $y't_{-2m\rho^{\vee}} = s_{\beta}y t_{-2m\rho^{\vee}} = 
(y t_{-2m\rho^{\vee}}) s_{\gamma + n \delta} < y t_{-2m \rho^{\vee}}$ for all $m \gg 0$. This proves the lemma.

\[BSDH-cover\] For each $y \in W_{{\mathrm{af}}}$ such that $y {\succeq}x$, there exist $k \in {\mathbb{Z}}_{\ge 0}$ and a subsequence ${\mathbf{j}}$ of ${\mathbf{i}}_{k}$ such that $y = x({\mathbf{j}})$.

By Lemma \[si-refl\], there exists $m \gg 0$ such that the assertion of Lemma \[tweakw\](2) holds for $m_{0}+m$ instead of $m_{0}$ (see also Remark \[rem:ell\]), and such that $$yt_{- 2 (m_{0}+m) \rho^{\vee}} < 
x t _{- 2 (m_{0}+m) \rho^{\vee}} = 
x t _{- 2 m_{0} \rho^{\vee}} \cdot t _{- 2 m \rho^{\vee}};$$ note that ${\mathbf{i}}_{\ell'+m\ell}$ gives a reduced expression for $x t _{- 2 (m_{0}+m) \rho^{\vee}}$. By the Subword Property (see, e.g., [@BB Theorem 2.2.2]), $y t_{- 2 (m_{0}+m) \rho^{\vee}}$ is obtained as a subexpression of the reduced expression for $x t_{- 2 (m_{0}+m) \rho^{\vee}}$ corresponding to ${\mathbf{i}}_{\ell'+m\ell}$. Namely, there exists a subsequence ${\mathbf{j}}=(i_{j_1},\,\dots,\,i_{j_t})$ of ${\mathbf{i}}_{\ell'+m\ell}$ such that $$y t_{- 2 (m_{0}+m) \rho^{\vee}} = 
s_{i_{j_1}} \cdots s_{i_{j_t}} = 
s_{i_{j_1}} \ast \cdots \ast s_{i_{j_t}} = 
x({\mathbf{j}},\ell'+m\ell).$$ Let us take $m_{1} \gg m$ as in Lemma \[stab\]. Then, we see by Remark \[rem:ell\] that for $m' \ge m_{1}$, $$\begin{aligned}
x({\mathbf{j}},\ell'+m'\ell) & = 
 x({\mathbf{j}},\ell'+m\ell) \ast t_{-2(m'-m)\rho^{\vee}} \\
& = 
 x({\mathbf{j}},\ell'+m\ell) \cdot t_{-2(m'-m)\rho^{\vee}} =
 y t_{- 2 (m'+m_{0}) \rho^{\vee}}.\end{aligned}$$ Thus, we obtain $y = x({\mathbf{j}})$, as desired. This proves the lemma.

For each $i \in I_{{\mathrm{af}}}$ and $y \in W_{{\mathrm{af}}}$, we define a map $$\label{eq:qiy}
q_{i,y} : {\mathbf{I}}(i) \times^{{\mathbf{I}}} {\mathbf{Q}_{G}}(y) \rightarrow {\mathbf{Q}_{G}^{\mathrm{rat}}}, \quad 
(p,\,L) \mapsto pL.$$

\[one-step\] Let $y \in W_{{\mathrm{af}}}^{\ge 0}$, and $i \in I_{{\mathrm{af}}}$. If $s_{i} y {\succeq}y$, then the map $q_{i,y}$ induces a ${\mathbb{P}}^{1}$-fibration ${\mathbf{I}}(i) \times^{{\mathbf{I}}} {\mathbf{Q}_{G}}(y) \rightarrow {\mathbf{Q}_{G}}(y)$, which we also denote by $q_{i,y}$. If $e \preceq s_i y \preceq y$, then the map $q_{i,y}$ induces a birational map ${\mathbf{I}}(i) \times^{{\mathbf{I}}} {\mathbf{Q}_{G}}(y) \rightarrow {\mathbf{Q}_{G}}(s_{i}y)$, which we also denote by $q_{i,y}$. In both cases, the map $q_{i,y}$ is proper.

If $s_{i} y {\succeq}y$, then the action of ${\mathbf{I}}(i)$ stabilizes ${\mathbf{O}}(y) \cup {\mathbf{O}}( s_{i} y ) \subset {\mathbf{Q}_{G}}( y )$. Hence taking the closure in ${\mathbf{Q}_{G}}$ implies that ${\mathbf{Q}_{G}}( y )$ admits an ${\mathbf{I}}( i )$-action. Therefore, the assertions hold in this case since ${\mathbf{I}}( i )/ {\mathbf{I}}\cong {\mathbb{P}}^{1}$.

If $e \preceq s_{i} y \preceq y$, then $q_{i,y}^{-1} ( {\mathbf{O}}( s_{i} y ) )$ contains ${\mathbf{I}}\dot{s}_i {\mathbf{I}}\times ^{{\mathbf{I}}} {\mathbf{O}}(y)$. Here we deduce from Lemmas \[lem:si\] and \[lem:dmd\] that if $y' \in W_{{\mathrm{af}}}$ satisfies $y' {\succ}y$, then $s_{i}y' {\succ}s_{i}y$; in particular, we have ${\ell^{{\frac{\infty}{2}}}}( s_i y ) <  {\ell^{{\frac{\infty}{2}}}}( s_i y' )$. Hence we have ${\mathbf{I}}\dot{s}_i {\mathbf{I}}\times^{{\mathbf{I}}} {\mathbf{O}}( y ) = 
q_{i,y}^{-1} ( {\mathbf{O}}( s_{i} y ) )$. In addition, the unipotent one-parameter subgroup of ${\mathbf{I}}$ corresponding to $\alpha_{i}$ gives an isomorphism ${\mathbb{A}}^{1} \times \dot{s}_i {\mathbf{O}}( y ) \cong {\mathbf{O}}( s_{i} y )$. Therefore, $q_{i,y}$ is birational. Also, by applying the same observation for $s_{i}y$ instead of $y$, we deduce that $q_{i,y}$ is obtained as the restriction of the ${\mathbb{P}}^{1}$-fibration $q_{i,s_{i}y}$ to a closed subscheme. Hence $q_{i,y}$ defines a proper map. This proves the lemma.

For each $k \in {\mathbb{Z}}_{\ge 0}$, we set $x(k) := s_{i_{k}} s_{i_{k-1}} \cdots s_{i_{1}} x$. We claim that $$\label{eq:xk}
{\ell^{{\frac{\infty}{2}}}}( x(k+1) ) = {\ell^{{\frac{\infty}{2}}}}(x (k) ) + 1 \quad \text{for all $k \ge 0$}.$$ Indeed, since $x(k+1) = s_{i_{k}}x(k)$ for $k \ge 0$, we see by that ${\ell^{{\frac{\infty}{2}}}}( x(k+1) ) = {\ell^{{\frac{\infty}{2}}}}(x (k) ) \pm 1$ for each $k \ge 0$. Therefore, it suffices to show that $$\label{eq:xm}
{\ell^{{\frac{\infty}{2}}}}(x((m-m_{0})\ell + \ell')) = 
 {\ell^{{\frac{\infty}{2}}}}(x(0)) + (m-m_{0})\ell + \ell' \qquad 
 \text{for all $m \ge m_{0}$};$$ note that $x(0) = x$. We see by the definition that $x((m-m_{0})\ell + \ell') = (xt_{-2m\rho^{\vee}})^{-1}x = 
t_{2m\rho^{\vee}}$. Hence we compute: $$\begin{aligned}
{\ell^{{\frac{\infty}{2}}}}(x((m-m_{0})\ell + \ell')) & 
  = {\ell^{{\frac{\infty}{2}}}}(t_{2m\rho^{\vee}}) = 2{\langle \rho,\,2m\rho^{\vee} \rangle} 
  = -m {\ell^{{\frac{\infty}{2}}}}(t_{-2\rho^{\vee}}) \nonumber \\ 
& = m \ell(t_{-2\rho^{\vee}}) = m\ell \quad \text{by Lemma~\ref{lem:Mac}}. 
\label{eq:xk1}\end{aligned}$$ Also, by Lemma \[tweakw\](1), we have ${\ell^{{\frac{\infty}{2}}}}(xt_{-2m_{0}\rho^{\vee}}) + {\ell^{{\frac{\infty}{2}}}}(t_{2m_{0}\rho^{\vee}}) = {\ell^{{\frac{\infty}{2}}}}(x)$. Here we deduce that ${\ell^{{\frac{\infty}{2}}}}(xt_{-2m_{0}\rho^{\vee}}) = - \ell(xt_{-2m_{0}\rho^{\vee}}) = -\ell'$ by Lemma \[tweakw\](2), and that ${\ell^{{\frac{\infty}{2}}}}(t_{2m_{0}\rho^{\vee}}) = m_{0}\ell$. Hence we obtain ${\ell^{{\frac{\infty}{2}}}}(x) = -\ell' + m_{0}\ell$. Combining this equality with shows , as desired.

Now, we set $$\label{defbQi}
{\mathbf{Q}_{G}}( {\mathbf{i}}_k ) := 
 {\mathbf{I}}( i_1 ) \times^{{\mathbf{I}}} {\mathbf{I}}( i_2 ) \times^{{\mathbf{I}}} \cdots 
 \times^{{\mathbf{I}}} {\mathbf{I}}( i_k ) \times^{{\mathbf{I}}} {\mathbf{O}}( x ( k ) ).$$ We also define its ambient space $${\mathbf{Q}}^{\#}_G ( {\mathbf{i}}_k ) := 
 {\mathbf{I}}( i_1 ) \times^{{\mathbf{I}}} {\mathbf{I}}( i_2 ) \times^{{\mathbf{I}}} \cdots \times^{{\mathbf{I}}} 
 {\mathbf{I}}( i_k ) \times^{{\mathbf{I}}} {\mathbf{Q}_{G}}( x ( k ) ).$$ Since $s_{i_{k}} x ( k ) = x ( k - 1 )$ and $x ( k ) {\succeq}x ( k - 1 )$ for each $k \ge 1$ (see ), we have an ${\widetilde{{\mathbf{I}}}}$-equivariant embedding ${\mathbf{O}}( x ( k - 1 ) ) \hookrightarrow {\mathbf{I}}( i_k ) \times^{{\mathbf{I}}} {\mathbf{O}}( x ( k ) )$ by the latter case of Lemma \[one-step\], and hence an ${\widetilde{{\mathbf{I}}}}$-equivariant embedding ${\mathbf{Q}_{G}}( {\mathbf{i}}_{k-1} ) \hookrightarrow {\mathbf{Q}_{G}}( {\mathbf{i}}_k )$ for each $k \ge 1$. By infinite repetition of these embeddings, we obtain a scheme of infinite type $$\label{Qunion}
{\mathbf{Q}_{G}}( {\mathbf{i}}) := \varinjlim_k {\mathbf{Q}_{G}}( {\mathbf{i}}_k ) = \bigcup_{k \ge 0} {\mathbf{Q}_{G}}( {\mathbf{i}}_k ),$$ endowed with an ${\widetilde{{\mathbf{I}}}}$-action. Also, the multiplication of components yields an ${\widetilde{{\mathbf{I}}}}$-equivariant morphism $${\mathsf{m}}: {\mathbf{Q}_{G}}( {\mathbf{i}}) \longrightarrow {\mathbf{Q}_{G}}( x ).$$ Similarly, we have ${\mathsf{m}}_{k} : {\mathbf{Q}_{G}}^{\#}( {\mathbf{i}}_k ) \rightarrow {\mathbf{Q}_{G}}( x )$ for each $k \ge 0$. The natural inclusion ${\mathbf{Q}_{G}}( {\mathbf{i}}_k ) \hookrightarrow {\mathbf{Q}_{G}}^{\#} ( {\mathbf{i}}_k )$ yields a map ${\mathbf{Q}_{G}}({\mathbf{i}}) \to {\mathbf{Q}_{G}}^{\#} ( {\mathbf{i}}_k )$ by taking the product of factors at position $> k$ from the left in . Namely, we collect the maps $${\mathbf{Q}_{G}}( {\mathbf{i}}_{k'} ) \ni 
( p_1,\,\ldots,\,p_{k'},\,L) 
 \mapsto 
(p_1,\,\ldots,\,p_k,\,p_{k+1} \cdots p_{k'}L) \in {\mathbf{Q}_{G}}^{\#}( {\mathbf{i}}_k )$$ for $k'> k$ through , where $p_{j} \in {\mathbf{I}}( i_j )$, $1 \le j \le k'$, and $L \in {\mathbf{O}}(x(k'))$; here each closed point of ${\mathbf{Q}_{G}}( {\mathbf{i}}_{k'} )$ is an equivalence class with respect to the ${\mathbf{I}}^{k'}$-action, and the map above respects equivalence classes. This yields a factorization of ${\mathsf{m}}$ through arbitrary ${\mathsf{m}}_{k}$ in such a way that $$\label{factorBSDH}
{\mathbf{Q}_{G}}( {\mathbf{i}}) \rightarrow 
{\mathbf{Q}_{G}}^{\#}( {\mathbf{i}}_{k'} ) \rightarrow 
{\mathbf{Q}_{G}}^{\#} ( {\mathbf{i}}_k ) \rightarrow {\mathbf{Q}_{G}}( x )$$ for each $k' \ge k$. This also yields an inclusion $${\mathbf{Q}_{G}}( {\mathbf{i}}) \hookrightarrow 
{\mathbf{Q}_{G}}^{\#} ( {\mathbf{i}}) := \varprojlim_k {\mathbf{Q}}^{\#}_G ( {\mathbf{i}}_k ),$$ which fits into the following commutative diagram of ${\widetilde{{\mathbf{I}}}}$-equivariant morphisms for $k < k'$: $$\xymatrix{
{\mathbf{Q}_{G}}({\mathbf{i}}) \ar@{^{(}->}[rr] \ar[d]_{{\mathsf{m}}} \ar[drr] \ar[drrr] & & 
{\mathbf{Q}_{G}}^{\#}({\mathbf{i}}) \ar[dll] \ar[d] \ar[dr] & \\
{\mathbf{Q}_{G}}(x) & & {\mathbf{Q}_{G}}^{\#}({\mathbf{i}}_k) \ar[ll]^{{\mathsf{m}}_k} & 
{\mathbf{Q}_{G}}^{\#}({\mathbf{i}}_{k'}). \ar[l]
}$$

\[up-normal\] The scheme ${\mathbf{Q}_{G}}( {\mathbf{i}})$ is separated and normal.

The scheme ${\mathbf{Q}_{G}}( {\mathbf{i}})$ is an inclusive union of countably many open subschemes each of which is isomorphic to a pro-affine space bundle over a finite successive ${\mathbb{P}}^{1}$-fibrations. Since each of such a space is separated, we deduce the desired separatedness.

Also, by its construction, each ${\mathbf{Q}_{G}}( {\mathbf{i}}_{k} )$ is a union of pro-affine spaces labeled by subsequences of ${\mathbf{i}}_k$ (so that ${\mathbf{Q}_{G}}({\mathbf{i}}_{k})$ is covered by a total of $2^{k}$-copies of open cover consisting of pro-affine spaces). Thus, ${\mathbf{Q}_{G}}( {\mathbf{i}})$ is a union of countably many pro-affine spaces. Since all of these pieces are normal, we deduce the desired normality. This proves the lemma.

For each subsequence ${\mathbf{j}}\subset {\mathbf{i}}_{k}$, we obtain an ${\mathbf{I}}$-stable pro-affine space ${\mathbf{O}}( {\mathbf{j}}) \subset {\mathbf{Q}_{G}}( {\mathbf{i}}_{k} )$ by replacing ${\mathbf{I}}( i_{j} )$, $1 \le j \le k$, with ${\mathbf{I}}\dot{s}_{i_j} {\mathbf{I}}$ (resp., ${\mathbf{I}}$) if $i_{j} \in {\mathbf{j}}$ (resp., $i_{j} \not\in {\mathbf{j}}$) in ; we refer to ${\mathbf{O}}( {\mathbf{j}})$ as the stratum corresponding to a subsequence ${\mathbf{j}}$ of ${\mathbf{i}}_k$.

\[k-stratum\]

For each $k \in {\mathbb{Z}}_{\ge 1}$, it holds that ${\mathbf{Q}_{G}}( {\mathbf{i}}_k ) = \bigsqcup _{{\mathbf{j}}\subset {\mathbf{i}}_k} {\mathbf{O}}( {\mathbf{j}})$.

Let $k \in {\mathbb{Z}}_{\ge 1}$, and let ${\mathbf{j}}$ and ${\mathbf{j}}'$ be subsequences of ${\mathbf{i}}_k$. Then, ${\mathbf{O}}( {\mathbf{j}}) \subset {\overline{{\mathbf{O}}({\mathbf{j}}')}}$ in ${\mathbf{Q}_{G}}( {\mathbf{i}}_{k} )$ if and only if ${\mathbf{j}}\subset {\mathbf{j}}' (\subset {\mathbf{i}}_k)$.

The proofs of the assertions are straightforward by the definitions.

If we regard ${\mathbf{O}}( {\mathbf{j}})$ as a locally closed subscheme of ${\mathbf{Q}_{G}}( {\mathbf{i}})$ via for ${\mathbf{j}}\subset {\mathbf{i}}_k$, then its images in ${\mathbf{Q}_{G}}^{\#}( {\mathbf{i}}_k )$ and ${\mathbf{Q}_{G}}^{\#} ( {\mathbf{i}}_{k'})$ for $k < k'$ are isomorphic through . Therefore, we can freely replace ${\mathsf{m}}$ with ${\mathsf{m}}_{k}$ when we analyze a single stratum in ${\mathbf{Q}_{G}}({\mathbf{i}})$. We set ${\mathbf{Q}_{G}}({\mathbf{j}}) := \overline{{\mathbf{O}}({\mathbf{j}})}$, where the closure is taken in ${\mathbf{Q}_{G}}( {\mathbf{i}})$.

We have ${\mathbf{Q}_{G}}( {\mathbf{i}}) = \bigsqcup_{{\mathbf{j}}} {\mathbf{O}}( {\mathbf{j}})$, where ${\mathbf{j}}$ runs over all subsequences of ${\mathbf{i}}$ such that $|\sigma({\mathbf{j}})| < \infty$.

Since ${\mathbf{Q}_{G}}({\mathbf{i}}) = \bigcup_{k \ge 1} {\mathbf{Q}_{G}}( {\mathbf{i}}_k )$, the assertion is a consequence of Lemma \[k-stratum\] and the consideration just above.

The map ${\mathsf{m}}$ is surjective.

By Lemma \[BSDH-cover\], each fiber of ${\mathsf{m}}$ along an $(H \times {\mathbb{G}_{m}})$-fixed point is nonempty. Because applying the $\tilde{{\mathbf{I}}}$-action to the set of $(H \times {\mathbb{G}_{m}})$-fixed points exhausts ${\mathbf{Q}_{G}}( x )$ by Corollary \[orbits\], we conclude that ${\mathsf{m}}$ is surjective, as required. This proves the lemma.

The map ${\mathsf{m}}_k$ is ${\widetilde{{\mathbf{I}}}}$-equivariant, birational, and proper for each $k \ge 1$.

Since ${\ell^{{\frac{\infty}{2}}}}( x ( k ) ) - {\ell^{{\frac{\infty}{2}}}}( x ) = k$ for each $k \ge 1$, repeated application of Lemma \[one-step\] shows that ${\mathsf{m}}_k$ is birational and proper for each $k \ge 1$; this is because ${\mathsf{m}}_k$ is obtained as the base change of the composite of morphisms of type $q_{i_j,\,x(j)}$ for $1 \le j \le k$. This proves the lemma.

The map ${\mathsf{m}}$ is also ${\widetilde{{\mathbf{I}}}}$-equivariant and birational since the embedding ${\mathbf{Q}_{G}}({\mathbf{i}}) \subset {\mathbf{Q}_{G}}^{\#} ( {\mathbf{i}})$ is open.

Cohomology of line bundles over ${\mathbf{Q}_{G}}$. {#subsec:coh}
---------------------------------------------------

In this subsection, keeping the setting of the previous subsection, we also assume that $x=e$, the identity element; in this case, we can (and do) take the $m_{0}$ (in Lemma \[tweakw\](2)) to be $0$, so that we have $\ell'=0$ (see ). In particular, we have ${\mathbf{Q}_{G}}( x ) = {\mathbf{Q}_{G}}( e ) = {\mathbf{Q}_{G}}$.

For each $\lambda \in P$, the ${\widetilde{{\mathbf{I}}}}$-equivariant line bundle ${\mathcal{O}}_{{\mathbf{Q}_{G}}( x( k ) )} ( \lambda )$ induces a line bundle over ${\mathbf{Q}_{G}}({\mathbf{i}}_{k})$ for each $k \in {\mathbb{Z}}_{\ge 0}$.

\[H-range\] For each $\lambda \in P$, we have an isomorphism $$H^{0}( {\mathbf{Q}_{G}}({\mathbf{i}}), \,{\mathcal{O}}_{{\mathbf{Q}_{G}}({\mathbf{i}})}(\lambda) ) \cong W(\lambda)^{\ast}$$ of ${\widetilde{{\mathbf{I}}}}$-modules. In particular, the left-hand side carries a natural structure of graded ${\mathfrak{g}}[z]$-module.

We adopt the notation of the previous subsection, with $x=e$, $m_{0}=0$, and $\ell' = 0$. Also, since $\ell ( t_{- 2 \rho^{\vee}} ) = 
\ell ( w_{\circ} ) + \ell ( w_{\circ} t_{- 2 \rho^{\vee}} )$, we can rearrange the reduced expression $(i''_1,\,\ldots,\,i''_{\ell}) = (i_1,\,\ldots,\,i_{\ell})$ for $t_{- 2 \rho^{\vee}}$, if necessary, in such a way that the first $\ell(w_{\circ})$-entries $(i_1,\,\ldots,\,i_{\ell(w_{\circ})})$ give rise to a reduced expression for $w_{\circ}$.

The map ${\mathsf{m}}$ factors as ${\mathsf{m}}' \circ {\mathsf{m}}''$, where ${\mathsf{m}}''$ is the map obtained by collapsing the first $\ell ( w_{\circ} )$-factors in ${\mathbf{Q}_{G}}( {\mathbf{i}})$ as: $$\label{eq:mapm}
{\mathbf{Q}_{G}}( {\mathbf{i}}) = {\mathbf{I}}( i_1 ) \times^{{\mathbf{I}}} \cdots \times^{{\mathbf{I}}} 
{\mathbf{I}}(i_{\ell ( w_{\circ} )}) \times ^{{\mathbf{I}}} X 
\rightarrow G {[\hspace{-1pt}[z]\hspace{-1pt}]} \times ^{{\mathbf{I}}} X,$$ with $X$ a certain scheme admitting an ${\widetilde{{\mathbf{I}}}}$-action, and ${\mathsf{m}}'$ is the natural map $G{[\hspace{-1pt}[z]\hspace{-1pt}]} \times^{{\mathbf{I}}} X \rightarrow {\mathbf{Q}_{G}}$ induced by the action. In particular, we deduce that ${\mathsf{m}}_{\ast} {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\mathbf{i}})} ( \lambda )$ admits a ${\widetilde{G}}{[\hspace{-1pt}[z]\hspace{-1pt}]}$-action, and hence that the space of global sections of ${\mathsf{m}}_{\ast} {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\mathbf{i}})} ( \lambda )$ is a ${\mathfrak{g}}[z]$-module.

Now, for each $m \in {\mathbb{Z}}_{\ge 0}$, we have ${\mathbf{Q}_{G}}( {\mathbf{i}}_{\ell(w_{\circ}) + \ell m} ) \subset {\mathbf{Q}_{G}}({\mathbf{i}})$. Also, for each $k \in {\mathbb{Z}}_{\ge 0}$, we have an inclusion ${\mathbf{Q}_{G}}( {\mathbf{i}}_k ) \hookrightarrow {\mathbf{Q}_{G}}( {\mathbf{i}}_{k + \ell} )$ by sending the stratum ${\mathbf{O}}({\mathbf{j}})$ corresponding to ${\mathbf{j}}\subset {\mathbf{i}}_k$ to the one corresponding to ${\mathbf{j}}_{+} \subset {\mathbf{i}}_{k + \ell}$, which is obtained by shifting all the entries by $\ell$ (so that $\{1,\,2,\,\ldots,\,\ell\} \cap {\mathbf{j}}_{+} = \emptyset$ and $|\sigma_{k+\ell} ({\mathbf{j}}_{+})| = |\sigma_{k} ({\mathbf{j}})| + \ell$); this corresponds to twisting ${\mathop{\rm Image}\nolimits}{\mathsf{m}}= {\mathbf{Q}_{G}}$ to ${\mathbf{Q}_{G}}( t_{2 \rho^{\vee}} )$. It follows that the ${\mathbb{G}_{m}}$-action on the line bundle ${\mathcal{O}}_{{\mathbf{Q}_{G}}( {\mathbf{i}}_k )} ( \lambda )$ is twisted by $t_{2 \rho^{\vee}}$, whose actual effect is of degree $- 2 {\langle \lambda,\,\rho^{\vee} \rangle}$. In particular, the structure map $\pi_{m} : {\mathbf{Q}_{G}}( {\mathbf{i}}_{\ell(w_{\circ})+\ell m} ) 
\rightarrow \bigl\{ \mathrm{pt} \bigr\}$ for $m \in {\mathbb{Z}}_{\ge 0}$ factors as: $$\label{factor-pi}
\pi_{m} \cong 
    \bigl( ( \pi_{1})^{\ast} \bigr)^{m} ( \pi_{0} ),$$ where the maps $\pi_0$ and $\pi_1$ are projections $$\begin{aligned}
\pi_{0} & : 
 {\mathbf{I}}( i_{1} ) \times^{{\mathbf{I}}} {\mathbf{I}}( i_{2} ) \times^{{\mathbf{I}}} \cdots \times^{{\mathbf{I}}} 
 {\mathbf{I}}( i_{\ell(w_{\circ})}) \times^{{\mathbf{I}}} 
 {\mathbf{O}}( w_{\circ} ) \rightarrow \bigl\{ \mathrm{pt} \bigr\}, \nonumber \\
\pi_{1} & : 
 {\mathbf{I}}( i_{1} ) \times^{{\mathbf{I}}} {\mathbf{I}}( i_{2} ) \times^{{\mathbf{I}}} \cdots \times^{{\mathbf{I}}} 
 {\mathbf{I}}( i_{\ell} ) / {\mathbf{I}}\rightarrow \bigl\{ \mathrm{pt} \bigr\}, \label{pi1}\end{aligned}$$ and $(\pi_{1})^{\ast}$ means the pullback obtained by identifying $\bigl\{\mathrm{pt}\bigr\}$ with ${\mathbf{I}}/ {\mathbf{I}}\in {\mathbf{I}}( i_{\ell} ) / {\mathbf{I}}$ in .

For each $i \in I_{{\mathrm{af}}}$ and an ${\mathbf{I}}$-module $M$, we define ${\mathbf{D}}_{i}(M)$ to be the space of $(H \times {\mathbb{G}_{m}})$-finite vectors in $H^{0} ( {\mathbf{I}}(i) / {\mathbf{I}}, \, {\mathbf{I}}( i ) \times^{{\mathbf{I}}} M^{\ast} )$; this can be thought as a left exact endo-functor in the category of $(H \times {\mathbb{G}_{m}})$-semi-simple ${\mathfrak{I}}$-modules. We set $${\mathbf{D}}_{- 2 \rho^{\vee}} := 
{\mathbf{D}}_{s_{i_1}} \circ {\mathbf{D}}_{s_{i_2}} \circ \cdots \circ {\mathbf{D}}_{s_{i_{\ell}}}
\quad \text{and} \quad 
{\mathbf{D}}_{w_{\circ}} := 
{\mathbf{D}}_{s_{i_1}} \circ {\mathbf{D}}_{s_{i_2}} \circ \cdots \circ {\mathbf{D}}_{s_{i_{\ell ( w_{\circ} )}}}.$$ Then, successive application of the Leray spectral sequence to , together with the fact that the ${\mathbb{G}_{m}}$-twist commutes with the whole construction, yields $$H^0 ( {\mathbf{Q}_{G}}({\mathbf{i}}) ,\, {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\mathbf{i}})} ( \lambda )) 
\stackrel{\cong}{\longrightarrow} 
\varprojlim_{m} {\mathbf{D}}_{- 2 \rho^{\vee}}^{m} 
 \left( {\mathbf{D}}_{w_{\circ}} 
   \left( {\mathbb{C}}[ {\mathbf{O}}( w_{\circ} t_{2 m \rho^{\vee}} ) ] \otimes 
          {\mathbb{C}}_{- t_{- 2 m \rho^{\vee}} \lambda} \right) \right),$$ where we have used the equality $w_{\circ} t_{2 m \rho^{\vee}} w_{\circ} 
= t_{- 2 m \rho^{\vee}}$.

In view of with $X$ replaced by ${\mathbf{O}}( w_{\circ} t_{2 m \rho^{\vee}})$, Theorem \[BFmain\] implies $${\mathbf{D}}_{w_{\circ}} \left( 
 {\mathbb{C}}[ {\mathbf{O}}( w_{\circ} t_{2 m \rho^{\vee}} ) ] \otimes 
 {\mathbb{C}}_{- t_{- 2 m \rho^{\vee}} \lambda} \right) \cong W ( \lambda )^{\ast},$$ where the grading of the right hand side is shifted by $- 2m {\langle \lambda,\,\rho^{\vee} \rangle}$. Hence, [@Kat16 Theorem 4.13] (cf. [@Kas05 Lemma 2.8] and [@Kat16 Corollary 4.8]) implies $${\mathbf{D}}_{- 2 \rho^{\vee}}^m \left( {\mathbf{D}}_{w_{\circ}} \left( 
  {\mathbb{C}}[ {\mathbf{O}}( w_{\circ} t_{2 m \rho^{\vee}} ) ] \otimes 
  {\mathbb{C}}_{- t_{- 2 m \rho^{\vee}} \lambda} \right) \right) \cong W(\lambda)^{\ast}$$ without grading shift. Therefore, we conclude $$H^{0} ( {\mathbf{Q}_{G}}( {\mathbf{i}}), {\mathcal{O}}_{{\mathbf{Q}_{G}}({\mathbf{i}})}(\lambda)) \cong W(\lambda)^{\ast}$$ for the choice of ${\mathbf{i}}$ fixed at the beginning of the proof. The assertion for a general ${\mathbf{i}}$ follows from the fact that $${\mathbf{D}}_{- 2 \rho^{\vee}}^m \cong {\mathbf{D}}_{s_{j_1}} \circ \cdots \circ {\mathbf{D}}_{s_{j_{m\ell}}}$$ holds for an arbitrary reduced expression $s_{j_1} \cdots s_{j_{m\ell}}$ for $t_{- 2 m \rho^{\vee}}$. This completes the proof of the proposition.

\[w-approx\] The subset $G[z] \subset G{[\hspace{-1pt}[z]\hspace{-1pt}]}$ is dense.

The original claim is that $G{(\hspace{-1pt}(z)\hspace{-1pt})}$ is generated by the set of ${\mathbb{C}}{(\hspace{-1pt}(z)\hspace{-1pt})}$-valued points of the unipotent groups of $G$. Because the set of ${\mathbb{C}}(z)$-valued points in a one-dimensional unipotent group is dense in the set of ${\mathbb{C}}{(\hspace{-1pt}(z)\hspace{-1pt})}$-valued points in the sense that we can approximate the latter by the former up to an arbitrary order of $z$, it follows that we can approximate $G{(\hspace{-1pt}(z)\hspace{-1pt})}$ by elements of $G(z)$ up to an arbitrary order of $z$. Since such an approximation of an element of $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$ is achieved by elements that are regular at $z = 0$, we conclude the assertion.

\[normal\] We have ${\mathsf{m}}_{\ast} {\mathcal{O}}_{{\mathbf{Q}_{G}}({\mathbf{i}})} \cong {\mathcal{O}}_{{\mathbf{Q}_{G}}}$. In particular, the scheme ${\mathbf{Q}_{G}}$ is normal.

We (can) employ the same reduced expression for $t_{-2\rho^{\vee}}$ as in the proof of Proposition \[H-range\]; recall the last sentence of the proof. The pullback defines a map ${\mathsf{m}}^{\ast} {\mathcal{O}}_{{\mathbf{Q}_{G}}} \rightarrow {\mathcal{O}}_{{\mathbf{Q}_{G}}({\mathbf{i}})}$, whose adjunction in turn yields ${\mathcal{O}}_{{\mathbf{Q}_{G}}} \to {\mathsf{m}}_{\ast} {\mathcal{O}}_{{\mathbf{Q}_{G}}({\mathbf{i}})}$. From this, by twisting by ${\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda )$ for some $\lambda \in P^{+}$, we obtain the following short exact sequence: $$0 \rightarrow {\mathcal{O}}_{{\mathbf{Q}_{G}}} (\lambda) \rightarrow 
  {\mathsf{m}}_{\ast} {\mathcal{O}}_{{\mathbf{Q}_{G}}({\mathbf{i}})} (\lambda) \rightarrow 
  {\mathop{\rm Coker}\nolimits}\rightarrow 0,$$ from which we deduce a ${\mathfrak{g}}[z]$-module inclusion $$H^0 ( {\mathbf{Q}_{G}}, {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda ) ) \hookrightarrow 
H^0 ( {\mathbf{Q}_{G}}, {\mathsf{m}}_{\ast} {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\mathbf{i}})} ( \lambda ) ) \cong 
H^0 ( {\mathbf{Q}_{G}}( {\mathbf{i}}), {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\mathbf{i}})} ( \lambda ) ),$$ by taking their global sections. The rightmost one is isomorphic to $W(\lambda)^{\ast}$ by Proposition \[H-range\]. In particular, we have algebra homomorphisms: $$\begin{aligned}
\bigoplus _{\lambda \in P^{+}} 
  \Gamma ( {\mathbf{Q}_{G}}, {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda ) ) 
& \subset 
\bigoplus _{\lambda \in P^{+}} 
 \Gamma ( {\mathbf{Q}_{G}}, {\mathsf{m}}_{\ast} {\mathcal{O}}_{{\mathbf{Q}_{G}}({\mathbf{i}})} ( \lambda ) ) \\
& \cong \bigoplus _{\lambda \in P^{+}} \Gamma ( {\mathbf{Q}_{G}}({\mathbf{i}}), {\mathcal{O}}_{{\mathbf{Q}_{G}}({\mathbf{i}})} ( \lambda ) ) \\
& \cong \bigoplus _{\lambda \in P^{+}} W ( \lambda )^{\ast};\end{aligned}$$ let us denote the leftmost one by $R'_{G}$ and the rightmost one by $R_{G}$. Since ${\mathbf{Q}_{G}}({\mathbf{i}})$ is normal, we deduce that $R_{G}$ is normal when localized with respect to homogeneous elements in $W({\varpi}_{i})^{\ast}$ for each $i \in I$. For the same reason, $R_{G}$ is an integral domain. The ring structure of $R_{G}$ is induced by the unique (up to scalar) ${\mathfrak{g}}[z]$-module map $$\label{d-mult}
W(\lambda + \mu) \longrightarrow W (\lambda) \otimes_{{\mathbb{C}}} W ( \mu ), \qquad 
\lambda,\,\mu \in P^{+},$$ of degree zero. In view of [@Kat16 Proof of Theorem 3.3], we deduce that the multiplication map $W(\lambda)^{\ast} \otimes W (\mu)^{\ast} \rightarrow 
W (\lambda + \mu)^{\ast}$ is surjective since is injective. Therefore, $R_{G}$ is a normal ring generated by terms of primitive degree (see, e.g., [@Har77 Chap. II, Exerc. 5.14]; cf. [@Kat16 Proof of Theorem 3.3]).

From the above, it suffices to prove $R_{G}' = R_{G}$. For this purpose, it is enough to prove that the associated graded ring of the projective coordinate ring $R''_{G}$ of ${\mathbf{Q}_{G}}$, which is arising from its structure of a closed subscheme of $\prod_{i \in I} {\mathbb{P}}(L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]})$, contains $R_{G}$ (see, e.g., [@EGA1 Sect. 2.6] for convention). Recall that the projective coordinate ring of ${\mathbb{P}}( L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]} )$ is $\bigoplus_{n \ge 0} S^{n} \bigl( L({\varpi}_{i})[z]^{\ast}\bigr)$, where $S^{n}V$ denotes the $n$-th symmetric power of a vector space $V$. Thanks to the surjectivity of multiplication map of $R_G$, it is further reduced to seeing that for each $i \in I$, every element of the part $W({\varpi}_{i})^{\ast}$ of degree ${\varpi}_{i}$ in $R_{G}$ is written as the quotient of an element of $\prod_{i \in I} S^{ {\langle \lambda,\,\alpha_{i}^{\vee} \rangle} } \bigl( L ( {\varpi}_{i} ) [z] ^{\ast} \bigr)$ by some power of monomials in elements of $L ( {\varpi}_{j} )[z]^{\ast} \subset W ( {\varpi}_{j} )^{\ast}$, $j \in I$ (note that this condition is particularly apparent in types $A$ and $C$ since $W ( {\varpi}_{i} ) = L ( {\varpi}_{i} ) [z]$ for each $i \in I$).

By [@BF14c Proof of Theorem 3.1], each ${\mathfrak{Q}}_{G}(\xi)$, $\xi \in Q^{\vee, +}$, is a projective variety with rational singularities. By the Serre vanishing theorem [@EGA3-1 Théorème 2.2.1] applied to the ideal sheaf that defines ${\mathfrak{Q}}_{G}(\xi)$ (inside the product of finite-dimensional projective spaces in ${\mathbb{P}}( L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]} )$ obtained by bounding the degree; cf. [@Kat16 (2.1)]), the restriction map $$\label{surj-rest}
\bigotimes_{i \in I} H^{0} \bigl( {\mathbb{P}}( L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]}) , 
{\mathcal{O}}({\langle \lambda,\,\alpha_{i}^{\vee} \rangle}) \bigr) \rightarrow 
H^{0} ( {\mathfrak{Q}}_{G}(\xi), {\mathcal{O}}_{{\mathfrak{Q}}_{G}(\xi)} (\lambda) )$$ is surjective for sufficiently large $\lambda \in P^{+}$, where we have used the fact that $$\bigotimes _{i \in I} 
  H^{0}( {\mathbb{P}}( L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]} ), \mathcal{F}_{i}) \cong 
  H^{0}\left( \prod_{i \in I} {\mathbb{P}}( L ({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]} ), \boxtimes_{i \in I} \mathcal{F}_{i} \right)$$ holds for vector bundles $\mathcal{F}_{i}$ of finite rank on ${\mathbb{P}}( L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]} )$.

\[H-inj\] For a given degree $n \in {\mathbb{Z}}_{\le 0}$, we can choose $\lambda \in P^{+}$ and $\xi \in Q^{\vee,+}$ sufficiently large in such a way that for every $m \in {\mathbb{Z}}_{>0}$, the restriction map $$W ( m \lambda )^{\ast} \subset W ( m \lambda )^{\vee} = 
 H^{0} ( {\mathfrak{Q}}_{G}, {\mathcal{O}}_{{\mathfrak{Q}}_{G}} (m \lambda ) ) \longrightarrow 
 H^{0} ( {\mathfrak{Q}}_{G} ( \xi ), {\mathcal{O}}_{{\mathfrak{Q}}_{G} (\xi)} ( m \lambda ) )$$ is injective at degree greater than or equal to $n$.

[*Proof of Claim \[H-inj\].*]{} Let us denote by $W ({\varpi}_{i} )^{\ast}_{\ge n} \subset W({\varpi}_{i})^{\ast}$ and $W ({\varpi}_{i})_{\le -n} \subset W ({\varpi}_{i})$ the direct sum of the homogeneous components of $W({\varpi}_{i})^{\ast}$ of degree greater or equal to $n$, and the the direct sum of the homogeneous components of $W ( {\varpi}_{i} )$ of degree less than or equal to $-n$, respectively. Also, let $R_{G}^{n}$ be the subring of $R_{G}$ generated by the $W ({\varpi}_{i} )^{\ast}_{\ge n}$, $i \in I$; every homogeneous component of $R_{G}$ of degree greater than or equal to $n$ is contained in $R_{G}^{n}$ by the surjectivity of multiplication map and the fact that $W(\lambda)^{\ast}$ is concentrated in nonpositive degrees. The value of a section of a line bundle over ${\mathop{\rm Proj}\nolimits}R_{G}$ (our ${\mathop{\rm Proj}\nolimits}$ here is the $P^{+}$-graded proj, by which we mean that the $H$-quotient of the subset of the affine spectrum in $\prod_{i \in I} \bigl( W({\varpi}_{i} ) \setminus \{0\} \bigr)$) arising from $R_{G}^{n}$ at a point is determined completely by its image under the projection $${\mathop{\rm pr}\nolimits}: 
 \prod_{i \in I} {\mathbb{P}}\left( {\widehat{W({\varpi}_{i})}} \right) \setminus Z 
 \longrightarrow 
 \prod_{i \in I} {\mathbb{P}}\bigl( W ({\varpi}_{i} )_{\le - n} \bigr)$$ induced by the ${\mathfrak{g}}[z]$-module surjection $W ({\varpi}_{i}) \to W ({\varpi}_{i})_{\le - n}$, where $Z$ denotes the loci in which ${\mathop{\rm pr}\nolimits}$ is not well-defined; for the notation ${\widehat{W({\varpi}_{i})}}$, see Section \[subsec:liealg\]. Thanks to Theorem \[w-approx\], we deduce that $${\mathop{\rm pr}\nolimits}( {\mathfrak{Q}}_{G} \setminus Z ) = {\mathop{\rm pr}\nolimits}( {\mathbf{Q}_{G}}\setminus Z )$$ as the set of closed points. Since the restriction of ${\mathop{\rm pr}\nolimits}$ to ${\mathfrak{Q}}_{G}(\xi) \setminus Z$ for each $\xi \in Q^{\vee,+}$ is a morphism of Noetherian schemes, it follows that ${\mathop{\rm pr}\nolimits}( {\mathfrak{Q}}_{G}(\xi) \setminus Z )$ is a constructible subset of $\prod_{i \in I} {\mathbb{P}}\bigl( W ({\varpi}_{i} )_{\le -n} \bigr)$. Moreover, the irreducibility of ${\mathfrak{Q}}_{G}( \xi )$ forces $\overline{{\mathop{\rm pr}\nolimits}( {\mathfrak{Q}}_{G} ( \xi ) \setminus Z )}$ to be irreducible. Therefore, the equality $${\mathop{\rm pr}\nolimits}( {\mathfrak{Q}}_{G} \setminus Z ) = \bigcup_{\xi \in Q^{\vee,+}} {\mathop{\rm pr}\nolimits}( {\mathfrak{Q}}_{G}(\xi) \setminus Z )$$ implies that there exists some $\xi \in Q^{\vee,+}$ such that $$\label{Z-dense}
{\mathop{\rm pr}\nolimits}( {\mathfrak{Q}}_{G}(\xi) \setminus Z ) \subset {\mathop{\rm pr}\nolimits}( {\mathbf{Q}_{G}}\setminus Z )$$ is Zariski dense, since $\prod_{i \in I} {\mathbb{P}}\left( W ({\varpi}_{i} )_{\le - n} \right)$ is a Noetherian scheme.

Thanks to [@BF14c Proposition 5.1] (cf. Theorem \[Kmain\]), we can find $\lambda$ (by replacing $\xi$ with a larger one if necessary) such that the assertion holds for $m = 1$. Now we assume the contrary to deduce the assertion for $m > 1$. Then, we have an additional equation on ${\mathop{\rm pr}\nolimits}( {\mathbf{Q}_{G}}\setminus Z ) \subset 
\prod_{i \in I} {\mathbb{P}}\bigl( W ({\varpi}_{i} )_{\le - n} \bigr)$ vanishing along ${\mathfrak{Q}}_{G}(\xi)$ by (taking sum of) the multiplication of $W(\lambda)^{\ast}_{\ge n}$. However, this is impossible in view of since ${\mathop{\rm Proj}\nolimits}R_{G}$ is reduced (and hence $R_{G}^{n}$ is integral). Thus we have proved Claim \[H-inj\].

We return to the proof of Theorem \[normal\]. We fix $n \in {\mathbb{Z}}_{\le 0}$ and $\beta \in Q^{\vee,+}$ such that Claim \[H-inj\] holds. By replacing $\lambda$ if necessary to guarantee the surjectivity of the restriction map with keeping the situation of Claim \[H-inj\], we deduce that all the maps $$\label{diag}
\begin{split}
\xymatrix{
\bigotimes _{i \in I} H^{0} \bigl( {\mathbb{P}}(L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]}), {\mathcal{O}}({\langle \lambda,\,\alpha_{i}^{\vee} \rangle}) \bigr) 
  \ar@{=}[d] \ar[r] 
  & H^{0}({\mathbf{Q}_{G}}, {\mathcal{O}}_{{\mathbf{Q}_{G}}}(\lambda)) 
  \ar@{^{(}->}[d] \\
\bigotimes _{i \in I} S^{ {\langle \lambda,\,\alpha_{i}^{\vee} \rangle} } 
  \bigl( L({\varpi}_{i})[z]^{\ast} \bigr) 
  \ar[r] \ar@{=}[d] 
  & W (\lambda)^{\ast} 
  \ar[d] \\
\bigotimes_{i \in I} H^{0} \bigl( {\mathbb{P}}(L({\varpi}_{i}){[\hspace{-1pt}[z]\hspace{-1pt}]}), {\mathcal{O}}( {\langle \lambda,\,\alpha_{i}^{\vee} \rangle}) \bigr) 
 \ar[r] & 
 H^{0} ( {\mathfrak{Q}}_{G}(\xi), {\mathcal{O}}_{{\mathfrak{Q}}_{G}(\xi)} ( \lambda ) ) 
}
\end{split}
$$ are surjective at degree $n$ from the commutativity of the diagram and the surjectivity of the bottom horizontal map. For a degree $n$ element $f \in W ({\varpi}_{i})^{\ast}$ and degree zero element $h_{j} \in L({\varpi}_{j})^{\ast} \subset W({\varpi}_{j})^{\ast}$ for each $j \in I$, we can choose sufficiently large integers $N_{i}$, $i \in I$, such that $$f \cdot \prod_{j \in I} h_{j}^{N_{j}} \in 
 S^{1+N_{i}} \bigl( L({\varpi}_{i}) [z]^{\ast} \bigr) \cdot 
 \prod_{j \in I,\,j \ne i} S^{N_{j}} \bigl( L ( {\varpi}_{j} ) [z]^{\ast} \bigr) \subset R_{G}$$ as the corresponding claim is true after sending to the bottom line of . This forces $W ( {\varpi}_{i})^{\ast}$ to be contained in the part of degree ${\varpi}_{i}$ of the associated graded ring of $R_{G}''$, as required. This completes the proof of the theorem.

\[pcr\] The projective coordinate ring $R_{G}$ of ${\mathbf{Q}_{G}}$ arising from the embedding by means of the DP-coordinates is isomorphic to $\bigoplus_{\lambda \in P^{+}} W(\lambda)^{\ast}$.

\[R-free\] The projective coordinate ring $R_{G}$ of ${\mathbf{Q}_{G}}$ in Corollary \[pcr\] is free over the polynomial algebra $A_{G}$ given by the lowest weight components with respect to the $H$-action.

During this proof, we denote by $v_{\lambda} \in W ( \lambda )$ the unique ${\mathfrak{h}}_{{\mathrm{af}}}$-eigenvector of weight $\lambda$ (which is determined up to a scalar, and is the specialization of the corresponding vector of $V ( \lambda )$ through $\mathsf{q} \to 1$).

For each $\lambda \in P^{+}$, we set $${\mathbb{C}}[{\mathbb{A}}^{(\lambda)}] := \bigotimes_{i \in I} 
  S^{ {\langle \lambda,\,\alpha_{i}^{\vee} \rangle} } {\mathbb{C}}[z] \cong 
  \bigotimes_{i \in I} 
  {\mathbb{C}}[ z_{1}^{(i)},\,\ldots,\,
  z_{ {\langle \lambda,\,\alpha_{i}^{\vee} \rangle} }^{(i)} ]^{ {\mathfrak{S}}_{{\langle \lambda,\,\alpha_{i}^{\vee} \rangle}} }.$$ By the results [@FL07; @Naoi12], due to Fourier-Littelmann and Naoi, we know that $W (\lambda)$ is a free module over ${\mathbb{C}}[{\mathbb{A}}^{(\lambda)}]$, and the $\lambda$-isotypical component of $W ( \lambda )$ is free of rank one. Here we define the polynomial algebra $A_{G}$ by collecting the $(-\lambda)$-isotypical component of $W ( \lambda )^{\ast}$ for all $\lambda \in P^{+}$. It follows that the ring $A_{G}$ is of the form $$\bigotimes_{i \in I} S^{\bullet} {\mathbb{C}}[z]^{\ast} \cong 
\bigoplus_{\lambda \in P^{+}} 
\bigotimes_{i \in I} S^{ {\langle \lambda,\,\alpha_{i}^{\vee} \rangle} } {\mathbb{C}}[z]^{\ast} \cong 
\bigoplus_{\lambda \in P^{+}} {\mathbb{C}}[{\mathbb{A}}^{(\lambda)}]^{\ast}.$$

Let $\psi \in W ( \mu )^{\ast}$ and $\xi \in {\mathbb{C}}[{\mathbb{A}}^{(\lambda - \mu)}]^{\ast} \subset A_G$, where $\lambda, \mu, \lambda - \mu \in P^+$, and assume that both of them are homogeneous with respect to $P$-weights and degrees. Then, we find a product of ${\mathfrak{h}}_{{\mathrm{af}}}$-eigen PBW basis element $F_1 \in U ( {\mathfrak{n}}^- [z] )$ and monomials $f_1, f_2 \in U ( {\mathfrak{h}}[z] z ) \cong 
S^{\bullet} ( {\mathfrak{h}}[z] z )$ such that $\psi ( F _1 f_1 v_{\mu} ) \neq 0$ and $\xi ( f_2 v_{(\lambda - \mu)}) \neq 0$ by the Poincaré-Birkhoff-Witt theorem. It follows that $( \psi \cdot \xi) ( F_1 f_1 f_2 v_{\lambda} ) \neq 0$, since we need to collect the terms $F_1 m_1 v_{\mu} \otimes m_2 v_{\lambda - \mu}$, with $m_1 m_2 = f_1 f_2$, in the tensor product through the embedding $W ( \lambda ) \subset W ( \mu ) \otimes W ( \lambda - \mu )$. This means that $0 \neq \psi \cdot \xi \in W ( \lambda )^{\ast}$; in particular, the ring $R_G$ is torsion-free as an $A_G$-module.

Now, let us fix $i_0 \in I$ and $\lambda \in P^+$ so that ${\langle \lambda,\,\alpha_{i_0}^{\vee} \rangle} = 0$, and set $\lambda_m := \lambda + m {\varpi}_{i_0}$ for each $m \in {\mathbb{Z}}_{\ge 0}$. We also set $A_G^{i_0} := S^{\bullet} {\mathbb{C}}[z]^{\ast} \subset A_G$ for the fixed $i_0$. For each $m, l \in {\mathbb{Z}}_{\ge 0}$ such that $m \ge l$, we denote by $W ( \lambda; m,l )$ the space $({\mathbb{C}}[{\mathbb{A}}^{(l{\varpi}_{i_0})}]^{\ast} \cdot W ( \lambda_{m-l} )^{\ast} )^{\ast}$, which is a ${\mathfrak{b}}[z]$-submodule of $W ( \lambda_m )$. From this description, we have an inclusion $$W ( \lambda; m,l ) \subset W ( \lambda; m,l-1 )$$ when $l > 0$. In particular, $W ( \lambda; m, l )^{\ast}$ is a quotient of $W ( \lambda; m, l - 1 )^{\ast}$. By repeated use of (the dual of) the surjectivity of the multiplication map of $R_G$, we have an embedding: $$\label{W-emb-fw}
\Phi : W ( \lambda_m ) \hookrightarrow W ( \varpi_{i_0} )^{\otimes m} \otimes W ( \lambda ).$$ For $0 \le l \le m$, let $\mathbb W ( \lambda; m, l )$ denote the linear span of pure tensors $$\label{pure-tensor}
\left( \bigotimes_{j = 1}^{m} v_{i_0,j} \right) \otimes v \in 
 W ( \varpi_{i_0} )^{\otimes m} \otimes W ( \lambda )$$ of ${\mathfrak{h}}_{{\mathrm{af}}}$-eigenvectors in which at most $l$-elements of $\{v_{i_0,j}\}_{j = 1}^{m}$ is of the form $z^e v_{{\varpi}_{i_0}}$ for some $e \in {\mathbb{Z}}_{\ge 0}$. If we denote by $W ( \lambda; m, l )'$ the preimage of ${\mathbb{W}}( \lambda; m, l )$ through $\Phi$, then we have $$W ( \lambda; m, l-1 )' \subset W ( \lambda; m, l )'$$ whenever $l > 0$. By construction, $\{ W ( \lambda; m, l )' \}_{0 \le l \le m}$ forms a ${\mathbb{Z}}$-graded increasing filtration whose associated graded modules $${\mathop{\rm gr}\nolimits}_{l} W( \lambda_m ) := 
 W ( \lambda; m, l )' / W ( \lambda; m, l - 1 )', \qquad l \in {\mathbb{Z}}_{\ge 0},$$ stratify $W ( \lambda_m )$.

Here, $W ( \varpi_{i_0} )^{\otimes m} \otimes W ( \lambda )$ admits a graded decomposition coming from the number of elements in $\{v_{i_0,j}\}_{j = 1}^{m}$ of the form $z^e v_{\varpi_{i_0}}$ for $e \in {\mathbb{Z}}_{\ge 0}$ through and . It follows that the space ${\mathbb{W}}( \lambda; m, l )$ is the annihilator of the subspace $$\label{expand-prod}
\sum_{w \in {\mathfrak{S}}_m} \sum_{a=l}^m w ( {\mathbb{C}}[{\mathbb{A}}^{(\varpi_{i_0})}]^{\otimes a} \otimes 
W ( \varpi_{i_0} )^{\otimes m-a} )^{\ast} \otimes 
W ( \lambda )^\ast \subset ( W ( \varpi_{i_0} ) ^{\otimes m} )^\ast \otimes 
W ( \lambda )^{\ast},$$ where ${\mathfrak{S}}_m$ permutes the tensor factors of $W ( \varpi_{i_0} )^{\otimes m}$. Pulling back by $\Phi$, we deduce that $W ( \lambda; m, l-1 )'$ is the annihilator of the space in $W ( \lambda_m )$ through the embedding . Therefore, $W ( \lambda; m, l-1 )' \subset W ( \lambda_m )$ is exactly the annihilator of ${\mathbb{C}}[{\mathbb{A}}^{(l{\varpi}_{i_0})}]^{\ast} \cdot 
W ( \lambda_{m-l} )^{\ast} \subset W ( \lambda _m )^{*}$. It follows that we have a canonical isomorphism $${\mathop{\rm gr}\nolimits}_{l} W(\lambda_m) \cong W ( \lambda; m, l ) / W ( \lambda; m, l+1).$$ If we define a subquotient $M(\lambda; n)$ of $R_{G}$ for each $n \in {\mathbb{Z}}_{\ge 0}$ by $$M( \lambda; n ) := \bigoplus _{l \ge 0} M ( \lambda; n )_l, \qquad 
M( \lambda; n )_l = \left( {\mathop{\rm gr}\nolimits}_{l} W(\lambda_{n+l}) \right)^{\ast},$$ then $M(\lambda; n)$ admits an $A_G^{i_0}$-action.

From the construction of $M ( \lambda; n )_l$ through $\{W ( \lambda; m, l)\}_{m, l \ge 0}$, we deduce that $M ( \lambda; n )$ is generated by $M ( \lambda; n )_{0}$ as an $A_G^{i_0}$-module. Also, from the construction of $M ( \lambda; n )_l$ through $\{W ( \lambda; m, l)'\}_{m, l \ge 0}$, we deduce that the dual of the multiplication map is the natural map $${\mathbb{C}}[{\mathbb{A}}^{(l {\varpi}_{i_0})}] \otimes {\mathop{\rm gr}\nolimits}_{0} W(\lambda_{n}) \rightarrow {\mathop{\rm gr}\nolimits}_{0} \, W(\lambda_{n+l})$$ of ${\mathbb{C}}[{\mathbb{A}}^{(\lambda_{n+l})}]$-modules. The $({\mathbb{C}}[{\mathbb{A}}^{(m {\varpi}_{i_{0}})}] , {\mathfrak{b}}[z] )$-modules ${\mathop{\rm gr}\nolimits}_{l} W(\lambda_m)$, $0 \le l \le m$, stratify $W ( \lambda_{m} )$. In addition, the maximality of $W ( \lambda; m, l )'$ guarantees that each ${\mathop{\rm gr}\nolimits}_{l} W(\lambda_m)$ is torsion-free as a ${\mathbb{C}}[{\mathbb{A}}^{(\lambda_m)}]$-module.

For each $\lambda \in P^{+}$, we can regard $W ( \lambda )$ as a module corresponding to a vector bundle (or a free sheaf) ${\mathcal{W}}(\lambda)$ over ${\mathbb{A}}^{(\lambda)}$, where ${\mathbb{A}}^{(\lambda)}$ denotes ${\mathop{\rm Spec}\nolimits}{\mathbb{C}}[{\mathbb{A}}^{(\lambda)}]$; its fiber is known to be the tensor product of local Weyl modules $W (\mu,\,x)$, where $\mu \in P^{+}$ and $x \in {\mathbb{C}}$ runs over the configurations of points determined by a point of ${\mathbb{A}}^{(\lambda)}$ (see, e.g., [@Kat16 Theorem 1.4]).

The spaces ${\mathop{\rm gr}\nolimits}_{l} W(\lambda_m)$, $0 \le l \le m$, give torsion-free sheaves ${\mathcal{W}}_l(\lambda_m)$ on ${\mathbb{A}}^{(\lambda_{m})}$ that stratify ${\mathcal{W}}( \lambda_m )$. Hence a section of ${\mathcal{W}}_l (\lambda_m)$ is an equivalence class of the set of sections ${\mathbb{A}}^{(\lambda_m)} \rightarrow {\mathcal{W}}( \lambda_m )$ whose specialization to a general point gives an element of the tensor product of local Weyl modules $$\label{point-division}
v = \sum_{k} \bigotimes_{i,j} v_{i,j}^{(k)} \in 
 \bigotimes_{i \in I} \bigotimes_{j=1}^{{\langle \lambda,\,\alpha_{i}^{\vee} \rangle}} 
 W ( {\varpi}_{i}, x_{i,j} )$$ such that exactly $l$-elements in $\{v_{i_{0},j}^{(k)} \}_{j=1}^{m}$ are highest weight vectors, and the other vectors do not have a highest weight component for each $k$.

Since every two points in $\{x_{i,j}\}_{i,j}$ are generically distinct, each pure tensor in divides $\{1,2,\ldots, m\}$ into two subsets $S_{1}$ and $S_{2}$, with $\# S_{1} = l$, so that $\{x_{i_{0},j}\}_{j\in S_{1}}$ carries a highest weight vector (i.e., $v_{i_{0},j}^{(k)} = z^{e}v_{{\varpi}_{i_{0}}}$ for some $e \in {\mathbb{Z}}_{\ge 0}$) and $\{x_{i_{0},j}\}_{j\in S_{2}}$ carries a vector lying in non-highest-weight components (as a ${\mathfrak{b}}$-module). The coordinates $\{x_{i_{0},j}\}_{j\in S_{1}}$ gives rise to the action of ${\mathbb{C}}[{\mathbb{A}}^{(l {\varpi}_{i_0} )}]$ on ${\mathop{\rm gr}\nolimits}_{l} W(\lambda_m)$, while the coordinates $\{x_{i_{0},j}\}_{j\in S_{2}}$ gives rise to a ${\mathbb{C}}[{\mathbb{A}}^{(\lambda_{m-l})}]$-module structure on ${\mathop{\rm gr}\nolimits}_{l} W(\lambda_m)$, and these two module structures are (mutually commutative and) distinct. It follows that every pair of elements of ${\mathbb{C}}[{\mathbb{A}}^{(l {\varpi}_{i_0} )}]$ and ${\mathop{\rm gr}\nolimits}_{0} W(\lambda_{m-l})$ appears as a section in ${\mathcal{W}}_l ( \lambda_{m} )$ after a generic localization. This particularly gives us the ${\mathfrak{S}}_{m}$-action on ${\mathbb{C}}[{\mathbb{A}}^{(l {\varpi}_{i_0} )}] \boxtimes {\mathbb{C}}[{\mathbb{A}}^{(\lambda_{m-l})}]$ and ${\mathop{\rm gr}\nolimits}_{l} W(\lambda_m)$ that changes the order of the highest weight vectors and the one of non-highest weight vectors (or mixes up $S_{1}$ and $S_{2}$). Therefore, ${\mathop{\rm gr}\nolimits}_{l} W(\lambda_m)$ itself is torsion-free as a ${\mathbb{C}}[{\mathbb{A}}^{(l {\varpi}_{i_0} )}] \boxtimes {\mathbb{C}}[{\mathbb{A}}^{(\lambda_{m-l})}]$-module.

Because $M ( \lambda; n )$ is generated by $M ( \lambda; n )_{0}$ as an $A_{G}^{i_{0}}$-module, we have an injective map $$\eta : \mathrm{gr}_{l} \, W(\lambda_{m}) \hookrightarrow 
  {\mathbb{C}}[{\mathbb{A}}^{(l {\varpi}_{i_0} )}] \otimes \mathrm{gr}_{0} \, W(\lambda_{m-l})$$ of ${\mathbb{C}}[{\mathbb{A}}^{(l {\varpi}_{i_0} )}] \boxtimes {\mathbb{C}}[{\mathbb{A}}^{(\lambda_{m-l})}]$-modules, which is an isomorphism after a localization to some Zariski open subset of ${\mathbb{A}}^{(l {\varpi}_{i_0} )} \times {\mathbb{A}}^{(\lambda_{m-l})}$. Since ${\mathop{\rm gr}\nolimits}_{l} W(\lambda_m)$, $0 \le l \le m$, stratifies $W ( \lambda_m )$, we deduce that the $M( \lambda; n )$’s, with $\lambda$ varying, give a stratification of the $A_{G}$-module $R_{G}$. Hence, in order to prove that $R_G$ is free over $A_{G}^{i_0}$, it suffices to verify that each $M ( \lambda; n )$ is a free $A_{G}^{i_0}$-module. By construction, the image of $\eta$ contains ${\mathbb{C}}\otimes {\mathop{\rm gr}\nolimits}_{0} W(\lambda_{m-l})$, which gives a ${\mathbb{C}}[{\mathbb{A}}^{(l {\varpi}_{i_0})}]$-module generator of ${\mathop{\rm gr}\nolimits}_{l} \, W(\lambda_{m})$. Therefore, the map $\eta$ must be an isomorphism. As a consequence, we conclude that $${\mathbb{C}}[{\mathbb{A}}^{(l {\varpi}_{i_0})}]^{\ast} \otimes M ( \lambda; n )_0 \cong M ( \lambda; n )_l$$ through the multiplication map. In other words, $M ( \lambda; n )$ is a free $A_{G}^{i_0}$-module.

Because the above argument is consistent with the filtrations and their associated graded modules arising from a different choice of $i_0 \in I$, we can vary $i_0 \in I$ and construct the associated graded modules inductively on a fixed total order on $I$. This gives a stratification of $R_{G}$ that is free over $A_{G}$. Hence the ring $R_G$ itself is free over $A_G$. This completes the proof of the theorem.

\[coh-e\] For each $\lambda \in P$, we have $${\mathop{\rm gch}\nolimits}H^{n} ( {\mathbf{Q}_{G}}, {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda ) ) = 
  \begin{cases} 
    {\mathop{\rm gch}\nolimits}V^{-}_{e}( - w_{\circ} \lambda ) & 
      \text{\rm if $\lambda \in P^{+}$ and $n = 0$}, \\[1.5mm]
      0 & 
      \text{\rm otherwise}.
  \end{cases}$$

We know that ${\mathbf{Q}_{G}}$ is a closed subscheme of $\prod_{i \in I} {\mathbb{P}}( L ( {\varpi}_{i} ){[\hspace{-1pt}[z]\hspace{-1pt}]} )$ by . Therefore, we have a countable set ${\Omega}$ of $I$-tuples of $( H \times {\mathbb{G}_{m}})$-eigenvectors of $\bigsqcup_{i \in I} L( {\varpi}_{i} ){[\hspace{-1pt}[z]\hspace{-1pt}]}$, one for each $i \in I$, so that it induces an affine open cover ${\mathcal{U}}:= \{ {\mathcal{U}}_{S} \}_{S \subset {\Omega}}$ of ${\mathbf{Q}_{G}}$ (where ${\mathcal{U}}_{S} := \bigl\{ f \neq 0 \mid f \in S \bigr\}$) that is closed under intersection. Now, the maps $L({\varpi}_i){[\hspace{-1pt}[z]\hspace{-1pt}]} \setminus \{ 0 \} \rightarrow 
{\mathbb{P}}(L({\varpi}_i){[\hspace{-1pt}[z]\hspace{-1pt}]} )$, $i \in I$, induce a (right) $H$-fibration ${\widetilde{\mathbf{Q}}_{G}}$ that defines an open scheme of $\prod_{i \in I} L( {\varpi}_i ){[\hspace{-1pt}[z]\hspace{-1pt}]}$, which corresponds to specifying a nonzero vector $u^{{\varpi}_{i}}$ instead of a one-dimensional ${\mathbb{C}}$-vector subspace $L^{{\varpi}_{i}} \ni u^{{\varpi}_{i}}$ in the definition of DP data; its closure ${\widehat{\mathbf{Q}}_{G}}$, which corresponds to allowing $u^{{\varpi}_{i}} = 0$ in a DP datum, is an affine subscheme of $\prod_{i \in I} L( {\varpi}_i ){[\hspace{-1pt}[z]\hspace{-1pt}]}$ of infinite type. We set $Z := {\widehat{\mathbf{Q}}_{G}}\setminus {\widetilde{\mathbf{Q}}_{G}}$, which is a closed subscheme of ${\widehat{\mathbf{Q}}_{G}}$. Also, the pullback ${\widetilde{{\mathcal{U}}}}_S$ of ${\mathcal{U}}_S$ to ${\widetilde{\mathbf{Q}}_{G}}$ defines an affine open subset of ${\widehat{\mathbf{Q}}_{G}}$. By the finiteness of the defining functions, ${\widetilde{{\mathcal{U}}}}_S \hookrightarrow {\widehat{\mathbf{Q}}_{G}}$ is quasi-compact by [@EGA1 Proposition 1.1.10]. For each finite subset $S \subset {\Omega}$, we set ${\widetilde{{\mathcal{U}}}}^S :=  \{ {\widetilde{{\mathcal{U}}}}_{T} \}_{T \subset S}$, which is again a collection of affine subschemes that is closed under intersections, and ${\widetilde{U}}^S := \bigcup_{T \subset S}{\widetilde{{\mathcal{U}}}}_T$. In addition, we set ${\widetilde{Z}}_S := {\widehat{\mathbf{Q}}_{G}}\setminus {\widetilde{U}}^S$.

Let us denote the natural projection ${\widetilde{\mathbf{Q}}_{G}}\rightarrow {\mathbf{Q}_{G}}$ by $\pi$. Since ${\mathbf{Q}_{G}}$ is a (right) free quotient of ${\widetilde{\mathbf{Q}}_{G}}$ by $H$, we deduce that ${\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda ) = ( \pi_{*} {\mathcal{O}}_{{\widetilde{\mathbf{Q}}_{G}}} )^{(H, \lambda)}$, where $\bullet^{(H, \lambda)}$ denotes the $\lambda$-isotypical component with respect to the right $H$-action. Because discarding open sets (in such a way that the remaining ones are closed under intersection) in the Cech complex defines a projective system of complexes satisfying the Mittag-Leffler condition, [@EGA3-1 Proposition 13.2.3] yields an isomorphism $$\label{cech-ref}
H^{n} ( {\widetilde{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widetilde{\mathbf{Q}}_{G}}} ) \cong
 \varprojlim_{S} H^{n} ( {\widetilde{U}}^S, {\mathcal{O}}_{{\widetilde{\mathbf{Q}}_{G}}} )
 \quad \text{for each $n \in {\mathbb{Z}}_{\ge 0}$}.$$ We have $$\label{des-H}
H^{n} ( {\mathbf{Q}_{G}}, {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda ) ) \cong 
H^{n} ( {\widetilde{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widetilde{\mathbf{Q}}_{G}}} )^{(H, \lambda)}
\quad \text{for $n \in {\mathbb{Z}}_{\ge 0}$},$$ since the right $H$-action on ${\widetilde{\mathbf{Q}}_{G}}$ is free, and it induces a semi-simple action on the level of Cech complex.

The long exact sequence of local cohomologies (see [@SGA2 Exposé I, Corollaire 2.9]) yields: $$\label{l-les}
\cdots \rightarrow H^{n}_{{\widetilde{Z}}_S} ( {\widehat{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) 
  \rightarrow H^{n} ( {\widehat{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) 
  \rightarrow H^{n} ( {\widetilde{U}}^S, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) \rightarrow 
  H^{n+1}_{{\widetilde{Z}}_S} ( {\widehat{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) \rightarrow \cdots. 
$$ Since ${\widehat{\mathbf{Q}}_{G}}$ is affine, this induces $$\label{isom-local}
H^{n} ( {\widetilde{U}}^S, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) 
 \stackrel{\cong}{\longrightarrow} 
H^{n+1}_{{\widetilde{Z}}_S} ( {\widehat{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) 
 \quad \text{for $n \ge 1$}$$ by [@EGA3-1 Théorème 1.3.1]. Here the quasi-compactness of the embedding ${\widetilde{U}}^S \hookrightarrow {\widehat{\mathbf{Q}}_{G}}$, together with [@SGA2 Exposé II, Proposition 5], implies that $$\label{Koszul}
H^{n}_{{\widetilde{Z}}_S} ( {\widehat{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) \cong 
H^{n} ( K_S ( {\mathbb{C}}[ {\widehat{\mathbf{Q}}_{G}}] ) ),$$ where $K_S ( {\mathbb{C}}[ {\widehat{\mathbf{Q}}_{G}}] )$ denotes the (cohomological) ${\mathbb{C}}[ {\widehat{\mathbf{Q}}_{G}}]$-Koszul complex defined through $S \subset {\Omega}$ (see [@EGA3-1 (1.1.2)]).

In view of , the comparison of and via yields an isomorphism $$\label{genuine}
H_{Z}^{n} ( {\widehat{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) \cong 
\varprojlim_S H^{n} ( K_S ( {\mathbb{C}}[ {\widehat{\mathbf{Q}}_{G}}] ) ) \cong 
\varprojlim_S H^{n}_{{\widetilde{Z}}_S} ( {\widehat{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ).$$ We know from [@EGA3-1 Proposition 1.1.4] that the Koszul complex $K_S ( {\mathbb{C}}[ {\widehat{\mathbf{Q}}_{G}}] )$ has trivial cohomology at degree $< n$ if $S$ contains a regular sequence of length $n$. Here we see from Corollary \[pcr\] that ${\mathbb{C}}[{\widehat{\mathbf{Q}}_{G}}] = 
\bigoplus_{\lambda \in P^{+}} W(\lambda)^{\ast}$. Also, by Theorem \[R-free\], we can rearrange ${\Omega}$ if necessary in such a way that for each $i \in I$, the set of the $i$-th components of the elements in ${\Omega}$ contain a regular sequence of arbitrary length. Then, we deduce from that $$H_{Z}^{n} ( {\widehat{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) = \{ 0 \} \quad 
 \text{for all $n \in {\mathbb{Z}}_{\ge 0}$}.$$ Therefore, and the affinity of ${\widehat{\mathbf{Q}}_{G}}$ imply that $$H^{n} ( {\widetilde{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widehat{\mathbf{Q}}_{G}}} ) = 
H^{n} ( {\widetilde{\mathbf{Q}}_{G}}, {\mathcal{O}}_{{\widetilde{\mathbf{Q}}_{G}}} ) = \{ 0 \} \quad
 \text{for all $n > 0$}.$$ In view of and Theorem \[BFmain\], we conclude the assertion of the theorem.

\[coh\] For each $x \in W_{{\mathrm{af}}}^{\ge 0}$, the scheme ${\mathbf{Q}_{G}}(x)$ is normal. Moreover, for each $\lambda \in P$ and $x \in W_{{\mathrm{af}}}^{\ge 0}$, we have $${\mathop{\rm gch}\nolimits}H^{n} ( {\mathbf{Q}_{G}}(x), {\mathcal{O}}_{{\mathbf{Q}_{G}}(x)}(\lambda) ) = 
 \begin{cases} 
  {\mathop{\rm gch}\nolimits}V^{-}_{x} ( - w_{\circ} \lambda ) & 
    \text{\rm if $\lambda \in P^{+}$ and $n = 0$}, \\[1.5mm]
  0 & \text{\rm otherwise}. 
 \end{cases}.$$ In particular, we have $${\mathop{\rm gch}\nolimits}H^{0}( {\mathbf{Q}_{G}}(x), {\mathcal{O}}_{{\mathbf{Q}_{G}}(x)}(\lambda) ) \in 
 ({\mathbb{Z}}{(\hspace{-1pt}(q^{-1})\hspace{-1pt})})[P] \subsetneq ({\mathbb{Z}}[P]){(\hspace{-1pt}(q^{-1})\hspace{-1pt})}.$$

Once we know the normality of ${\mathbf{Q}_{G}}$ and the cohomology vanishing result in Theorem \[coh-e\], the same argument as in [@Kat16 Theorem 4.7] (see Theorem \[Kmain\]) yields all the assertions except for the last one.

The last assertion on the character estimate follows from a result about extremal weight modules ([@Kas02 Corollary 5.2]) and the fact that $U_{{\mathsf{q}}}^{-}({\mathfrak{g}})$ is concentrated on subspaces of $q$-degree $\le 0$.

\[w-push\] For an arbitrary $x \in W_{{\mathrm{af}}}^{\ge 0}$, we take ${\mathbf{i}}$ as in . Then we have ${\mathsf{m}}_{\ast} {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\mathbf{i}})} \cong {\mathcal{O}}_{{\mathbf{Q}_{G}}( x )}$.

We adopt the notation of Section \[subsec:BSDH\]. The map ${\mathsf{m}}$ factors as the composite of the map ${\mathsf{m}}$ for $t_{2 m_0 \rho^{\vee}}$ and a successive composite of the $q_{i_{k}, x ( k )}$ for $1 \le k \le \ell'$. The case $x = t_{\xi}$ for $\xi \in Q^{\vee,+}$ is clear since ${\mathbf{Q}_{G}}( t_{\xi} ) \cong {\mathbf{Q}_{G}}$ (through $\imath_{\xi}$). Therefore, it suffices to show that $( q_{i, x} )_{\ast} {\mathcal{O}}_{{\mathbf{I}}(i) \times^{{\mathbf{I}}} {\mathbf{Q}_{G}}( x )} 
 \cong {\mathcal{O}}_{{\mathbf{Q}_{G}}( s_{i} x )}$ for each $x \in W_{{\mathrm{af}}}^{\ge 0}$ and $i \in I_{{\mathrm{af}}}$ such that $s_{i} x \prec x$. This assertion itself follows from Corollary \[coh\] and [@Kat16 Theorem 4.7] (in view of Lemma \[vample\]). Hence we obtain the assertion of the corollary.

$K$-theory of semi-infinite flag manifolds. {#sec:K-SiFl}
===========================================

We keep the notation and setting of Section \[subsec:BSDH\].

\[line-bundles\] Every ${\widetilde{{\mathbf{I}}}}$-equivariant locally free sheaf of rank one [(]{}i.e., line bundle[)]{} on ${\mathbf{Q}_{G}}( x )$ is of the form $\chi \otimes _{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}( x )} ( \lambda )$ for some $\lambda \in P$ and an ${\widetilde{{\mathbf{I}}}}$-character $\chi$.

For $x = e$, the boundary of the open $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$-orbit ${\mathbf{O}}$ in ${\mathbf{Q}_{G}}$ is of codimension at least two, and the open $G {[\hspace{-1pt}[z]\hspace{-1pt}]}$-orbit ${\mathbf{O}}$ has a structure of pro-affine bundle over $G / B$. In particular, an ${\widetilde{{\mathbf{I}}}}$-equivariant line bundle over ${\mathbf{O}}$ is the pullback of a $B \times {\mathbb{G}_{m}}$-equivariant line bundle over $G / B$. Because every line bundle over $G/B$ carries a unique $G$-equivariant structure by [@KKV Sect. 3.3], and $B$-equivariant structures of the trivial line bundle ${\mathcal{O}}_{G/B}$ are in bijection with $P$ (since $H^{0}(G/B,\,{\mathcal{O}}_{G/B})={\mathbb{C}}$), we deduce that every $B$-equivariant line bundle over $G / B$ is an $H$-character twist of a $G$-equivariant line bundle, which is obtained as the restriction of some ${\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda )$. Consequently, the assertion follows for $x = e$.

Now, for $y_{1},\,y_{2} \in W_{{\mathrm{af}}}^{\ge 0}$ such that $y_{1} \preceq y_{2}$, the restriction map transfers an ${\widetilde{{\mathbf{I}}}}$-equivariant line bundle over ${\mathbf{Q}_{G}}(y_{1})$ to an $\tilde{{\mathbf{I}}}$-equivariant line bundle over ${\mathbf{Q}_{G}}(y_{2})$. Also, for an arbitrary $x \in W_{{\mathrm{af}}}^{\ge 0}$, we can find $\xi \in Q^{\vee,+}$ such that $$\label{eq:DPs}
{\mathbf{Q}_{G}}( xt_{\xi} ) \subset {\mathbf{Q}_{G}}( t_{\xi} ) \subset {\mathbf{Q}_{G}}( x ) \subset {\mathbf{Q}_{G}}(e)={\mathbf{Q}_{G}}$$ by (the proof of) Lemma \[BSDH-cover\], since $x = x ( {\mathbf{j}}) \preceq t_{2m\rho^{\vee}}$ for $m \gg 0$. Because we have ${\mathbf{Q}_{G}}( t_{\xi} ) \cong {\mathbf{Q}_{G}}$ as schemes with an ${\widetilde{{\mathbf{I}}}}$-action, we conclude that a nonisomorphic pair of (${\widetilde{{\mathbf{I}}}}$-equivariant) line bundles over ${\mathbf{Q}_{G}}$ restricts to a nonisomorphic pair of (${\widetilde{{\mathbf{I}}}}$-equivariant) line bundles over ${\mathbf{Q}_{G}}(t_{\xi})$. Since ${\mathbf{Q}_{G}}( x t_{\xi} ) \cong {\mathbf{Q}_{G}}( x )$, the same is true for line bundles over ${\mathbf{Q}_{G}}( x )$. Therefore, by means of , we deduce the assertion of the proposition for an arbitrary $x \in W_{{\mathrm{af}}}^{\ge 0}$ from the case $x = e$. This proves the proposition.

The following is an immediate consequence of Corollary \[coh\] and Proposition \[line-bundles\].

\[cvs\] For each ${\widetilde{{\mathbf{I}}}}$-equivariant line bundle ${\mathcal{L}}$ over ${\mathbf{Q}_{G}}(x)$, we have $$H ^{n} ({\mathbf{Q}_{G}}(x),\,{\mathcal{L}}) = \{ 0 \} \quad \text{\rm for all $n > 0$}.$$

\[faith\] For each ${\widetilde{{\mathbf{I}}}}$-equivariant quasi-coherent sheaf ${\mathcal{E}}$ on ${\mathbf{Q}_{G}}$ such that $$\Gamma ( {\mathbf{Q}_{G}}, \, {\mathcal{E}}( \lambda ) ) = \{ 0 \} \quad 
\text{\rm for all $\lambda \in P$, where 
${\mathcal{E}}( \lambda ) = {\mathcal{E}}\otimes_{{\mathcal{O}}_{{\mathbf{Q}_{G}}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}}(\lambda)$},$$ we have ${\mathcal{E}}= \{ 0 \}$.

By the quasi-coherence and ${\widetilde{{\mathbf{I}}}}$-equivariance, every nonzero section of ${\mathcal{E}}$ has an ${\widetilde{{\mathbf{I}}}}$-stable support, which must be a union of ${\mathbf{I}}$-orbits. In addition, it defines a regular section on a complement of finitely many hyperplanes having poles of finite order around the boundary points. Therefore, Lemma \[vample\] implies the desired result.

\[cvb\] Let ${\mathcal{E}}$ be an ${\widetilde{{\mathbf{I}}}}$-equivariant quasi-coherent ${\mathcal{O}}_{{\mathbf{Q}_{G}}}$-module satisfying the following conditions[:]{} $$\label{cvb1}
{\mathop{\rm gdim}\nolimits}\Gamma ( {\mathbf{Q}_{G}},\, {\mathcal{E}}( \lambda ) )^{\ast} \in {\mathbb{Z}}{[\hspace{-1pt}[q]\hspace{-1pt}]} 
\quad \text{\rm for every $\lambda \in P$};$$ $$\label{cvb2}
\begin{array}{l}
\text{\rm there exists $\lambda_{0} \in P$ such that 
$\Gamma ( {\mathbf{Q}_{G}},\, {\mathcal{E}}( \lambda ) ) = \{ 0 \}$} \\[2mm]
\text{\rm for all $\lambda \in P$ with 
${\langle \lambda,\,\alpha_{i}^{\vee} \rangle} < {\langle \lambda_{0},\,\alpha_{i}^{\vee} \rangle}$ 
for some $i \in I$}. 
\end{array}
$$ Then, we have a resolution $\cdots \rightarrow {\mathcal{P}}^{2}_{{\mathcal{E}}} 
  \rightarrow {\mathcal{P}}^{1}_{{\mathcal{E}}} 
  \rightarrow {\mathcal{P}}^{0}_{{\mathcal{E}}}
  \rightarrow {\mathcal{E}}\rightarrow 0$ of ${\widetilde{{\mathbf{I}}}}$-equivariant ${\mathcal{O}}_{{\mathbf{Q}_{G}}}$-modules such that

${\mathop{\rm gdim}\nolimits}\Gamma ( {\mathbf{Q}_{G}},\,{\mathcal{P}}^{k}_{{\mathcal{E}}}(\lambda) )^{\ast} \in {\mathbb{Z}}{[\hspace{-1pt}[q]\hspace{-1pt}]}$ for every $k \ge 0$ and $\lambda \in P$[;]{}

for each $k \ge 0$, the ${\widetilde{{\mathbf{I}}}}$-equivariant ${\mathcal{O}}_{{\mathbf{Q}_{G}}}$-module ${\mathcal{P}}^{k}_{{\mathcal{E}}}$ is a direct sum of line bundles [(]{}if we forget the ${\mathbf{I}}$-module structure[);]{}

for each $m \in {\mathbb{Z}}$ and $\lambda \in P$, the number of direct summands ${\mathcal{P}}^{k}_{{\mathcal{E}}}(\lambda)$ of $\bigoplus_{k \ge 0} {\mathcal{P}}^{k}_{{\mathcal{E}}}(\lambda)$ contributing to the homogeneous subspace of degree $m$ of $\Gamma ( {\mathbf{Q}_{G}},\,\bigoplus _{k \ge 0} {\mathcal{P}}^{k}_{{\mathcal{E}}}(\lambda) )$ is finite.

Moreover, we have $H^{n} ( {\mathbf{Q}_{G}},\,{\mathcal{E}}) = \{ 0 \}$ for all $n > 0$.

Let $$R_{G} := \bigoplus_{\lambda \in P^{+}} W ( \lambda )^{\ast} 
  = \bigoplus_{\lambda \in P^{+}} H^{0} ( {\mathbf{Q}_{G}},\,{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda ) )$$ be the projective coordinate ring. Thanks to Lemma \[faith\], the sheaf ${\mathcal{E}}$ is determined by the $R_G$-module $$M ( {\mathcal{E}}) := \bigoplus_{\lambda \in P} H^{0} ( {\mathbf{Q}_{G}}, \, {\mathcal{E}}( \lambda ) ).$$ Because $M({\mathcal{E}})$ is nonpositively graded and each homogeneous subspace with respect to the $(P \times {\mathbb{Z}})$-grading is finite-dimensional, we obtain a surjection $P^{0}_{{\mathcal{E}}} \rightarrow M ({\mathcal{E}})$, where $P^{0}_{{\mathcal{E}}}$ is a direct sum of $(P \times {\mathbb{Z}})$-graded projective $R_G$-modules tensored with ${\widetilde{{\mathbf{I}}}}$-modules; indeed, we can construct the desired maps inductively by starting with $\lambda=\lambda_{0} \in P$, and then by adding the ${\varpi}_{i}$, $i \in I$, repeatedly, by means of the projectivity of $R_{G}$. Since a (graded) projective $R_G$-module is obtained from $R_{G}$ by a grading shift and an ${\widetilde{{\mathbf{I}}}}$-module twist, we deduce that $P^{0}_{{\mathcal{E}}} \cong M ( {\mathcal{P}}^{0}_{{\mathcal{E}}} )$ as ($P \times {\mathbb{Z}}$)-graded $R_G$-modules for a certain direct sum ${\mathcal{P}}^{0}_{{\mathcal{E}}}$ of ${\widetilde{{\mathbf{I}}}}$-equivariant line bundles (with some twist of the ${\widetilde{{\mathbf{I}}}}$-equivariant structure). Here the surjectivity of $P^{0}_{{\mathcal{E}}} \rightarrow M ( {\mathcal{E}})$ of $R_{G}$-modules implies that ${\mathcal{P}}^{0}_{{\mathcal{E}}} \rightarrow {\mathcal{E}}$ is also surjective. Also, by our character estimate, we deduce that ${\mathop{\rm gdim}\nolimits}\Gamma ( {\mathbf{Q}_{G}},\,{\mathcal{P}}^{0}_{{\mathcal{E}}} ( \lambda ) )^{\ast} \in {\mathbb{Z}}{[\hspace{-1pt}[q]\hspace{-1pt}]}$. Now, let $\Xi ( {\mathcal{E}})$ be the set of those pairs $(\lambda,\,m) \in P \times {\mathbb{Z}}$ for which $$\bigoplus_{ \mu \in \lambda-P^{+} }
\left(\bigoplus _{n > m} \Gamma ( {\mathbf{Q}_{G}}, \, {\mathcal{E}}( \mu ) )_{n}\right) \oplus 
\bigoplus _{\mu \in \lambda-P^{+}, \, \mu \ne \lambda} 
\Gamma ( {\mathbf{Q}_{G}}, \, {\mathcal{E}}( \mu ) )_{m}= \{0\}.$$ Then, we can rearrange ${\mathcal{P}}^{0}_{{\mathcal{E}}}$, if necessary, to assume that $$\label{eq:sp1}
{\mathop{\rm gdim}\nolimits}\ker \bigl( 
 \Gamma ( {\mathbf{Q}_{G}}, \, {\mathcal{P}}^{0}_{{\mathcal{E}}} ( \lambda ) ) \rightarrow 
 \Gamma ( {\mathbf{Q}_{G}}, {\mathcal{E}}( \lambda ) ) 
\bigr)^{\ast} \in {\mathbb{Z}}{[\hspace{-1pt}[q]\hspace{-1pt}]} \quad 
\text{for all $\lambda \in P$};$$ $$\label{eq:sp2}
\Gamma ( {\mathbf{Q}_{G}}, \, {\mathcal{P}}^{0}_{{\mathcal{E}}} ( \lambda ) )^{\ast} = \{ 0 \} 
  \quad \text{\rm for all $\lambda \in P$ with
  ${\langle \lambda,\,\alpha_{i}^{\vee} \rangle} < {\langle \lambda_{0},\,\alpha_{i}^{\vee} \rangle}$ 
  for some $i \in I$};
$$ $$\label{eq:sp3}
\ker \bigl( 
  \Gamma ( {\mathbf{Q}_{G}}, \, {\mathcal{P}}^0_{{\mathcal{E}}} ( \lambda ) ) \rightarrow 
  \Gamma ( {\mathbf{Q}_{G}}, {\mathcal{E}}( \lambda ) ) 
\bigr)_{m} = \{ 0 \} 
\quad \text{for every $(\lambda, m) \in \Xi ( {\mathcal{E}})$}.$$ Thanks to and , we can replace ${\mathcal{E}}$ with $\ker \bigl( {\mathcal{P}}^{k}_{{\mathcal{E}}} \rightarrow {\mathcal{P}}^{k-1} _{{\mathcal{E}}} \bigr)$ repeatedly (with the convention ${\mathcal{P}}^{-1}_{{\mathcal{E}}} = {\mathcal{E}}$) to apply the procedure above in order to obtain ${\mathcal{P}}^{k+1} _{{\mathcal{E}}}$ for each $k \ge 0$. This yields an ${\widetilde{{\mathbf{I}}}}$-equivariant resolution $$\label{eq:res}
\cdots 
  \rightarrow {\mathcal{P}}^{2}_{{\mathcal{E}}} 
  \rightarrow {\mathcal{P}}^{1}_{{\mathcal{E}}} 
  \rightarrow {\mathcal{P}}^{0}_{{\mathcal{E}}} 
  \rightarrow {\mathcal{E}}\rightarrow 0,$$ in which each ${\mathcal{P}}^{k}_{{\mathcal{E}}}$ is a direct sum of ${\widetilde{{\mathbf{I}}}}$-equivariant line bundles (with some twist by ${\widetilde{{\mathbf{I}}}}$-modules). By the construction, we have $$\emptyset = 
 \bigcap_{k \ge 0} \Xi ( \ker d_{k} ) \subset P \times {\mathbb{Z}},$$ and hence the resolution satisfies the first two of the requirements. Also, taking into account and , we see that the resolution satisfies the third one of the requirements.

Finally, by applying Corollary \[cvs\], we conclude the desired cohomology vanishing. This completes the proof of the theorem.

\[conv\] Keep the setting of Theorem \[cvb\]. We have $$\sum_{k \ge 0} 
    {\mathop{\rm gdim}\nolimits}\Gamma ( {\mathbf{Q}_{G}}, \, {\mathcal{P}}^{k}_{{\mathcal{E}}} ( \lambda ) )^{\ast} 
    \in {\mathbb{Z}}{[\hspace{-1pt}[q]\hspace{-1pt}]} \quad 
\text{\rm for all $\lambda \in P$},$$ and an [(]{}unambiguously defined[)]{} equality $${\mathop{\rm gdim}\nolimits}\Gamma ( {\mathbf{Q}_{G}},\,{\mathcal{E}}( \lambda ) )^{\ast} 
  = \sum_{k \ge 0} 
    (-1)^{k} {\mathop{\rm gdim}\nolimits}\Gamma ( {\mathbf{Q}_{G}}, \, {\mathcal{P}}^{k}_{{\mathcal{E}}} ( \lambda ) )^{\ast} 
    \in {\mathbb{Z}}{[\hspace{-1pt}[q]\hspace{-1pt}]} \quad 
\text{\rm for all $\lambda \in P$}.$$

By Theorem \[cvb\](3), there are only finitely many terms ${\mathcal{P}}^{k}_{{\mathcal{E}}}(\lambda)$ contributing to each homogeneous subspace of a fixed $q$-degree of $\Gamma({\mathbf{Q}_{G}},\,{\mathcal{E}}(\lambda))^{\ast}$. Therefore, the projective resolution afforded in Theorem \[cvb\] implies the desired result. This proves the corollary.

We say that a condition depending on $\lambda \in P$ holds for $\lambda \gg 0$ if there exists $\gamma \in P$ and the condition holds for every $\lambda \in \gamma + P^+$.

For an element $f \in ({\mathbb{Z}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$, we define $|f| \in ({\mathbb{Z}}_{\ge 0}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$ as follows: $$f=\sum_{n \in {\mathbb{Z}}_{\le 0}}
\underbrace{\left(\sum_{\nu \in P} c_{\nu,n} e^{\nu} \right)}_{\in {\mathbb{Z}}[P]} q^{n}
\quad \Rightarrow \quad
|f|:=\sum_{n \in {\mathbb{Z}}_{\le 0}}
\underbrace{\left(\sum_{\nu \in P} |c_{\nu,n}| e^{\nu} \right)}_{\in {\mathbb{Z}}_{\ge 0}[P]} q^{n}.$$ We now define $K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$ to be the following set of formal infinite sums, modulo equivalence relation $\sim$: $$\left\{ f=\sum_{ \lambda \in P } 
  f_{\lambda} \cdot [ {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda )] 
\ \Biggm| \  
\text{$f_{\lambda} \in ({\mathbb{Z}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$, $\lambda \in P$, 
satisfy condition \eqref{eq:K}} \right\}\Biggm/\sim$$ where condition is given by: $$\label{eq:K}
\sum_{ \lambda \in P } 
| f_{\lambda} | \cdot 
{\mathop{\rm gch}\nolimits}H^{0} ( {\mathbf{Q}_{G}}, \, {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda + \mu ) ) \in ({\mathbb{Z}}_{\ge 0}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}
\quad \text{for every $\mu \in P$}, \tag{\#}
$$ and the equivalence relation $\sim$ is: $$\label{eq:finite-support}
f \sim 0 \iff \sum_{ \lambda \in P } f_{\lambda} \cdot 
{\mathop{\rm gch}\nolimits}H^{0} ( {\mathbf{Q}_{G}}, \, {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda + \mu ) ) = 0
\quad \text{if $\mu \gg 0$}. \tag{$\sim$}
$$ By construction, $K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$ is topologically spanned by classes of ${\mathbf{I}}$-equivalent line bundles. Hence we deduce that the following map is well-defined: $$\begin{split}
& {\mathop{\rm Pic}\nolimits}^{{\widetilde{{\mathbf{I}}}}} {\mathbf{Q}_{G}}\times 
  K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}}) \rightarrow K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}}), \\ 
& \left( {\mathcal{L}}, f = \sum f_{\lambda} [{\mathcal{O}}( \lambda)] \right) \mapsto 
  [{\mathcal{L}}] \cdot f =  \sum f_{\lambda} [{\mathcal{L}}\otimes {\mathcal{O}}( \lambda)].
\end{split}
$$

By Corollary \[conv\], each ${\mathcal{E}}$ from Theorem \[cvb\] satisfies $$[{\mathcal{E}}] := \sum_{k \ge 0} (-1)^{k} [ {\mathcal{P}}^{k}_{{\mathcal{E}}}] \in K' _{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}}).$$ In particular, thanks to Corollary \[coh\], we have $[{\mathcal{O}}_{{\mathbf{Q}_{G}}( x )} ( \lambda )] \in K' _{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$ for every $x \in W_{{\mathrm{af}}}^{\ge 0}$ and $\lambda \in P$.

Let ${\mathop{\rm Fun}\nolimits}_{ ({\mathbb{C}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]} } P$ denote the space of $({\mathbb{C}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$-valued functions on $P$, and let $${\mathop{\rm Fun}\nolimits}^f_{ ({\mathbb{C}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]} } P \subset {\mathop{\rm Fun}\nolimits}_{ ({\mathbb{C}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]} } P$$ be the subset consisting of those functions that are zero on $\gamma + P^{+}$ for some $\gamma \in P$. Then we form a $({\mathbb{C}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$-module quotient $${\mathop{\rm Fun}\nolimits}^{\mathrm{ess}}_{ ({\mathbb{C}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]} } P 
:= {\mathop{\rm Fun}\nolimits}_{ ({\mathbb{C}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]} } P / {\mathop{\rm Fun}\nolimits}^f_{ ({\mathbb{C}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]} } P.$$ For each $\mu \in P$, we regard the assignment $$P \ni \lambda \mapsto 
\begin{cases}
{\mathop{\rm gch}\nolimits}V^{-}_{e} ( - w_{\circ} ( \lambda + \mu ) ) 
  & \text{if $\lambda+\mu \in P^{+}$}, \\[1.5mm]
0 & \text{otherwise}, 
\end{cases}
$$ as an element of ${\mathop{\rm Fun}\nolimits}_{ ({\mathbb{C}}[P]){[\hspace{-0.5pt}[q^{-1}]\hspace{-0.5pt}]} } P$, which we denote by $\Psi ( [{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \mu )] )$. Passing to the quotient, we obtain a map $\Psi : \{ [{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \mu )] \}_{\mu \in P} \ni [{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \mu )] 
\mapsto \Psi ( [{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \mu )] ) \in {\mathop{\rm Fun}\nolimits}^{\mathrm{ess}}_{ ({\mathbb{C}}[P]){[\hspace{-0.5pt}[q^{-1}]\hspace{-0.5pt}]} } P$.

\[Windep\] The map $\Psi$ extends to an injective $({\mathbb{Z}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$-linear map $\Psi : K' _{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}}) \rightarrow 
 {\mathop{\rm Fun}\nolimits}^{\mathrm{ess}}_{ ({\mathbb{C}}[P]){[\hspace{-0.5pt}[q^{-1}]\hspace{-0.5pt}]} } P$.

We assume the contrary to deduce a contradiction. Let $C \in K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$, and expand the $C$ as: $$C = \sum_{
  n \in {\mathbb{Z}}_{\le 0},\,\nu,\,\mu \in P}
  c _{n,\nu,\mu} \, q^{n} e^{\nu} \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}}(\mu)], 
  \quad \text{with $c_{n,\nu,\mu} \in {\mathbb{Z}}$},$$ inside $K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$. We have $\Psi ( C ) = 0$ if and only if there exists $\gamma \in P$ such that $$\sum_{
  n \in {\mathbb{Z}}_{\le 0},\,\nu,\,\mu \in P}
  c _{n,\nu,\mu} \, q^{n} e^{\nu} \cdot 
  {\mathop{\rm gch}\nolimits}V_e^- ( - w_{\circ} ( \lambda + \mu ) ) = 0 \qquad \text{for $\mu \in \gamma + P ^+$}.$$ This is exactly the condition $C \sim 0$. Hence the map $\Psi$ defines an injective map. It is $({\mathbb{Z}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$-linear by construction.

For countably many elements ${C_{p}}$, $p \ge 0$, in $K_{{\widetilde{{\mathbf{I}}}}}' ( {\mathbf{Q}_{G}})$ that represent the classes of ${\widetilde{{\mathbf{I}}}}$-equivariant quasi-coherent sheaves, we expand them as: $${C_{p}}= \sum_{ \lambda \in P } a_{\lambda} ( {C_{p}}) [{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda )], 
 \quad \text{with $a_{\lambda} ( {C_{p}}) \in ({\mathbb{Z}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$}$$ by using the procedure of Theorem \[cvb\]; we say that the sum $\sum _{p \ge 0} {C_{p}}$ converges absolutely to an element of $K'_{{\mathbf{I}}} ( {\mathbf{Q}_{G}})$ if there exists some $\lambda_0 \in P$ (uniformly for all $p \ge 0$) such that $a_{\lambda} ( {C_{p}}) = 0$ for all $\lambda \in P$ with ${\langle \lambda,\,\alpha_{i}^{\vee} \rangle} < 
{\langle \lambda_{0},\,\alpha_{i}^{\vee} \rangle}$ for some $i \in I$, and if the number of those $( \lambda, p ) \in 
P \times {\mathbb{Z}}_{\ge 0}$ for which $a_{\lambda} ( {C_{p}})$ has a nonzero term of $q$-degree $m$ is finite for each $m \in {\mathbb{Z}}$. It is straightforward to see that $\sum_{p \ge 0} {C_{p}}$ defines an element of $K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$, which does not depend on the order of the ${C_{p}}$’s.

\[rem:conv\] Since the coefficients for $K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$ are in ${\mathbb{Z}}$, the sum $\sum_{p \ge 0} {C_{p}}$ must “diverge” or “oscillate” when it does not converge absolutely.

\[indep-of-QGy\] Let $f_{y}\in ({\mathbb{Z}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$, $y \in W_{{\mathrm{af}}} ^{\ge 0}$. Then the formal sum $$\label{fsum}
\sum _{y \in W_{{\mathrm{af}}}^{\ge 0}} f_{y} \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}]$$ converges absolutely to an element of $K'_{{\mathbf{I}}} ( {\mathbf{Q}_{G}})$ if and only if $\sum_{y \in W_{{\mathrm{af}}} ^{\ge 0}} | f_{y} | 
\in ({\mathbb{Z}}_{\ge 0}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$. Moreover, in this case, the equation $$\sum _{y \in W_{{\mathrm{af}}}^{\ge 0}} f_{y} \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}] = 0$$ implies $f_{y} = 0$ for all $y \in W_{{\mathrm{af}}}^{\ge 0}$.

First, we remark that $[{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}] \in K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$ for each $y \in W_{{\mathrm{af}}}^{\ge 0}$ by Corollary \[coh\] and Theorem \[cvb\]. More precisely, by means of the cohomology vanishing: $$H^{\ast}({\mathbf{Q}_{G}},\,{\mathcal{O}}_{{\mathbf{Q}_{G}}(y)}(\mu))=\{0\} \quad \text{if $\mu \notin P^{+}$},$$ we can take $\lambda_{0}=0$ in Theorem \[cvb\] by setting ${\mathcal{E}}={\mathcal{O}}_{{\mathbf{Q}_{G}}(y)}$. In addition, we have $H^{0} ( {\mathbf{Q}_{G}},\,{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}) = {\mathbb{C}}$. Hence the construction in Theorem \[cvb\] implies that $$\label{v-exp}
[{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}] = 
 [{\mathcal{O}}_{{\mathbf{Q}_{G}}}] + \sum_{ \lambda \in - P^{+} } 
 a_{y} ( \lambda ) [ {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda ) ] \in K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$$ for some $a_{y}(\lambda) \in ({\mathbb{Z}[P]}){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$. Therefore, the coefficient of $[{\mathcal{O}}_{{\mathbf{Q}_{G}}}]$ in must be the sum $\sum_{y \in W_{{\mathrm{af}}} ^{\ge 0}} f_{y}$, which unambiguously defines an element of $({\mathbb{Z}[P]}){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$ if and only if the coefficient ($\in {\mathbb{Z}}$) of each $q^{n}$, $n \in {\mathbb{Z}}_{\le 0}$, in the sum $\sum_{y \in W_{{\mathrm{af}}}^{\ge 0}} f_{y}$ converges absolutely (see Remark \[rem:conv\]). This proves the first assertion.

We prove the second assertion. Let us assume the contrary to deduce a contradiction. Let $S$ be the set of those $y \in W_{{\mathrm{af}}}^{\ge 0}$ for which $f_{y} \neq 0$; denote by $n_{0}$ the maximal $q$-degree of all $f_{y}$, $y \in S$. Also, let $S_{1}$ be the set of those $y \in S$ for which ${\tau}( y ) \not> {\tau}( y')$ for any $y' \in S$, where for $y \in W_{{\mathrm{af}}}^{\ge 0}$ of the form $y = w t_{\xi}$ with $w \in W$ and $\xi \in Q^{\vee,+}$, we set ${\tau}( y ):=\xi$; since a polynomial ring (of finite variables) is Noetherian, we deduce that $|S_{1}| < \infty$. We choose and fix $y_{0} \in S_{1}$ such that $$P^{\#} := \bigl\{ 
 \lambda \in P^{+} \mid 
 \text{${\langle \lambda,\,{\tau}(y_{0}) \rangle} < {\langle \lambda,\,{\tau}(y) \rangle}$ 
 for all $y \in S$ with $y \ne y_{0}$} \bigr\}$$ is Zariski dense in ${\mathfrak{h}}^{\ast}$. Let $n_{1}$ denote the maximal $q$-degree of those $f_{y}$, $y \in S_{1}$, for which ${\tau}( y ) = {\tau}( y_{0} )$. Then, the subset $$P^{\#\#} := \bigl\{ 
 \lambda \in P^{+} \mid 
 \text{${\langle \lambda,\,{\tau}(y_{0}) \rangle} < {\langle \lambda,\,{\tau}(y) \rangle} + n_{0} - n_{1}$
 for all $y \in S$ with $y \neq y_{0}$}\bigr\}$$ of $P^{\#}$ is still Zariski dense in ${\mathfrak{h}}^{\ast}$.

For each $\lambda \in P^{\#\#}$, the coefficient of the part of degree $\bigl( n_{1} - {\langle \lambda,\,{\tau}( y_{0} ) \rangle} \bigr)$ of $$\Psi \left( 
 \sum _{y \in W_{{\mathrm{af}}}^{\ge 0}} f_{y} \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}] \right) 
( \lambda )$$ is equal to $$\sum _{w \in W} f_{ w t_{{\tau}(y_0)} }^{(n_{1})} \cdot 
{\mathop{\rm ch}\nolimits}L^{-}_{w} ( - w_{\circ} \lambda ),$$ where $f_{y}^{(n_{1})} \in {\mathbb{Z}}[P]$ is the part of degree $n_{1}$ of $f_{y}$; here, for $w \in W$ and $\mu \in P^{+}$, $L^{-}_{w} ( \mu ):=U({\mathfrak{b}}^{-})L(\mu)_{w\mu}$ denotes the (opposite) Demazure submodule of $L(\mu)$. This defines a ${\mathbb{Z}[P]}$-valued function of $\lambda \in P^{\#\#}$; note that the above is a finite sum. Here we have the equality ${\mathop{\rm ch}\nolimits}L^{-}_{w} ( - w_{\circ} \lambda )^{\ast} = 
D_{w w_{\circ}} ( e ^{\lambda} )$ in terms of the Demazure operator $D_{ww_{\circ}}$ for each $w \in W$ (see [@Kum02 Theorem 8.2.9]); recall that the Demazure operator $D_{i}=D_{s_{i}}$, $i \in I$, is defined by $D_{i}(e^{\mu}):=(e^{\mu}-e^{s_{i}\mu-\alpha_{i}})/(1-e^{-\alpha_{i}})$ for $\mu \in P$. Also, we know by [@Mac91 pp.28–29] that the operators $D_{w}$, $w \in W$, form a set of ${\mathbb{Z}[P]}$-linearly independent ${\mathbb{Z}}$-linear operators acting on ${\mathbb{Z}[P]}$. Therefore, we obtain $f_{w t_{{\tau}( y_{0} )}}^{(n_{1})} = 0$ for all $w \in W$. This is a contradiction, and hence we cannot take the $S_{1}$ above from the beginning. Thus, we conclude the desired result. This completes the proof of the proposition.

\[critK\] Let $x \in W_{{\mathrm{af}}}^{\ge 0}$ and $\lambda \in P^{+}$. Consider a collection $f_{y} ( \lambda ) \in ({\mathbb{Z}[P]}){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$, $y \in W_{{\mathrm{af}}} ^{\ge 0}$, such that $\sum_{y \in W_{{\mathrm{af}}}^{\ge 0}} f_{y} ( \lambda ) \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}]$ converges absolutely in $K'_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}})$. Then, $$\label{class-eq}
[{\mathcal{O}}_{{\mathbf{Q}_{G}}( x )} ( \lambda )] = 
\sum _{y \in W_{{\mathrm{af}}}^{\ge 0}} f_{y} ( \lambda ) \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}]$$ if and only if $$\label{char-eq}
{\mathop{\rm gch}\nolimits}V^{-}_{x} ( - w_{\circ} ( \lambda + \mu ) ) = 
  \sum _{y \in W_{{\mathrm{af}}}^{\ge 0}} f_{v} ( \lambda ) \cdot 
  {\mathop{\rm gch}\nolimits}V^{-}_{y} ( - w_{\circ} \mu ) \quad 
  \text{\rm for $\mu \gg 0$}.$$

We have an expansion $$[{\mathcal{O}}_{{\mathbf{Q}_{G}}( x )} ( \lambda )] = 
\sum _{y \in W_{{\mathrm{af}}}^{\ge 0}} f_{y} ( \lambda ) \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}]$$ inside $K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$. From this, by twisting by the line bundle ${\mathcal{O}}( \mu )$ for $\mu \in P^{+}$, we obtain $$[{\mathcal{O}}_{{\mathbf{Q}_{G}}( x )} ( \lambda + \mu )] = 
\sum_{y \in W_{{\mathrm{af}}}^{\ge 0}} 
 f_{y} ( \lambda ) \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )} ( \mu )].$$ By Corollary \[coh\], this equation in turn implies , which proves the “only if” part of the assertion.

We now assume . Then we have $$\Psi ( [{\mathcal{O}}_{{\mathbf{Q}_{G}}( x )} ( \lambda )] )  = 
 \sum _{y \in W_{{\mathrm{af}}}^{\ge 0}} f_{y} ( \lambda ) \cdot \Psi ( [{\mathcal{O}}_{{\mathbf{Q}_{G}}( y )}] )$$ by Corollary \[coh\]. Therefore, by Theorem \[Windep\], we deduce that both sides of represent the same class in $K'_{ {\widetilde{{\mathbf{I}}}} } ( {\mathbf{Q}_{G}})$. Thus, we have proved the “if” part of the assertion. This proves the corollary.

\[thm:PC\] For each $\lambda \in P^{+}$ and $x \in W_{{\mathrm{af}}}^{\ge 0}$, there holds the equality $$[{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda )] \cdot [{\mathcal{O}}_{{\mathbf{Q}_{G}}( x )}] = 
 \sum_{\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x} (- w_{\circ} \lambda )} 
 e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\eta))}q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\eta))} \cdot [ {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\iota(\eta,\,x)} )}]$$ in $K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$.

By Theorem \[thm:Dem\], we have $${\mathop{\rm gch}\nolimits}V^{-}_{x}( - w_{\circ}(\lambda + \mu) ) = 
\sum_{\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x} ( - w_{\circ}\lambda )} 
e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\eta))}q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\eta))} \cdot 
 {\mathop{\rm gch}\nolimits}V^{-}_{{\iota(\eta,\,x)}}( -w_{\circ} \mu )$$ for each $\mu \in P^{+}$. Taking into account the fact that the LHS is zero if $\lambda + \mu \not\in P^{+}$, and the RHS is zero if $\mu \not \in P^{+}$, we conclude the above equation for $\mu \gg 0$. Here we see from Section \[subsec:extremal\] that ${\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\eta)) \in {\mathbb{Z}}_{\le 0}$ for each $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(-w_{\circ}\lambda)$. Also, we deduce from that for each $m \in {\mathbb{Z}}_{\le 0}$, there exist only finitely many $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x} ( - w_{\circ}\lambda )$ such that ${\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\eta)) \ge m$. Because ${\mathop{\rm gdim}\nolimits}H^{0} ( {\mathbf{Q}_{G}}, \, {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\iota(\eta,\,x)} )} ( \mu ) ) 
\in {\mathbb{Z}}{[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$ by Corollary \[coh\], we deduce that $$\sum_{\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x} ( - w_{\circ} \lambda )} 
e ^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\eta))}q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\eta))} \cdot 
[ {\mathcal{O}}_{{\mathbf{Q}_{G}}( {\iota(\eta,\,x)} )}] \in K_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}}).$$ From this, by applying Corollary \[critK\], we conclude the desired result. This proves the theorem.

nil-DAHA action on $K_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}^{\mathrm{rat}}})$. {#sec:nDAHA}
=======================================================================================

The nil-DAHA ${{\mathcal{H}\hspace{-5.6pt}\mathcal{H}}}$ (of adjoint type) is the unital ${\mathbb{Z}}[{\mathbf{q}}^{\pm 1}]$-algebra generated by $T_i$, $i \in I_{{\mathrm{af}}}$, and ${\mathbf{e}}(\nu)$, $\nu \in P$, subject to the following relations: $$\label{eq:nDAHA}
\begin{cases}
T_i ( T_i + 1 ) = 0 & \text{for each $i \in I_{{\mathrm{af}}}$}; \\[3mm]
\text{if 
$\overbrace{s_i s_j \cdots}^{\text{$m_{ij}$ times}} = 
 \overbrace{s_j s_i \cdots}^{\text{$m_{ij}$ times}}$, then
$\overbrace{T_i T_j \cdots}^{\text{$m_{ij}$ times}} = 
 \overbrace{T_j T_i \cdots}^{\text{$m_{ij}$ times}}$} & 
\text{for each $i,\,j \in I_{{\mathrm{af}}}$}; \\[3mm]
\text{${\mathbf{e}}( \nu_{1} ) {\mathbf{e}}( \nu_{2} ) = 
{\mathbf{e}}( \nu_{1} + \nu_{2} )$ and ${\mathbf{e}}( 0 ) = 1$} & 
\text{for each $\nu_{1},\,\nu_{2} \in P$}; \\[3mm]
T_i {\mathbf{e}}( \nu ) - {\mathbf{e}}( s_i \nu ) T_i = 
\dfrac{{\mathbf{e}}( s_i \nu ) - {\mathbf{e}}( \nu )}{1 - {\mathbf{e}}( \alpha_i )}
& \text{for each $\nu \in P$ and $i \in I_{{\mathrm{af}}}$}. 
\end{cases}
$$ We define ${\mathcal{H}}$ to be the ${\mathbb{Z}}[{\mathbf{q}}^{-1}]$-subalgebra of ${{\mathcal{H}\hspace{-5.6pt}\mathcal{H}}}$ generated by $T_i$, $i \in I$, and ${\mathbf{e}}( \nu )$, $\nu \in P$.

\[KK-reint\] The assignment $$\begin{aligned}
{\mathtt{e}}( {\varpi}_{i} ) & : 
[{\mathbb{C}}_{\lambda} \otimes_{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \mu )] \mapsto 
[{\mathbb{C}}_{- {\varpi}_{i} + \lambda} \otimes_{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \mu )], \\
{\mathtt{T}}_{i} & : [{\mathbb{C}}_{\lambda} \otimes _{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \mu )] \mapsto
\frac{{\mathtt{e}}(-\lambda) - {\mathtt{e}}(-s_{i}\lambda+\alpha_{i})}
     {1-{\mathtt{e}}(\alpha_{i})} 
     [{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \mu )],\end{aligned}$$ for each $i \in I$ and $\lambda,\,\mu \in P$, equips $K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$ with an action of the subalgebra ${\mathcal{H}}$ of ${{\mathcal{H}\hspace{-5.6pt}\mathcal{H}}}$ through the identifications[:]{} $${\mathbf{q}}\mapsto q^{-1}, \qquad 
T_{i} \mapsto {\mathtt{T}}_{i} - 1 \quad \text{\rm for $i \in I$}, \qquad 
{\mathbf{e}}( \nu ) \mapsto {\mathtt{e}}( \nu ) \quad \text{\rm for $\nu \in P$}.
$$

By the construction, $K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$ contains a dense subset isomorphic to $({\mathbb{Z}}[P]){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]} 
\otimes_{{\mathbb{Z}}} K _{G} ( G / B )$ (see Proposition \[line-bundles\]). Also, we have a surjection $({\mathbb{Z}}[P])[q^{-1}] \otimes_{{\mathbb{Z}}} K _{G} ( G / B ) 
\twoheadrightarrow 
{\mathbb{Z}}[q^{-1}] \otimes_{{\mathbb{Z}}} K_{B}(G/B)$; see, e.g., [@KK90 (3.17)]. Here, for each $i \in I$, the action of ${\mathtt{T}}_i$ is identical to the action of the Demazure operator $D_{i}=D_{s_{i}}$, and the action of ${\mathtt{e}}({\varpi}_{i})$ corresponds to the twist by the $B$-character $-{\varpi}_{i}$; these define an ${\mathcal{H}}$-action on $K _{B} ( G / B )$ by [@KK90 Sect. 3]. Notice that both of the actions of ${\mathtt{e}}(\,\cdot\,)$ and $T_i$, $i \in I$, are neutral with respect to tensoring with ${\mathcal{O}}_{G/B} ( \lambda )$ for each $\lambda \in P$, and that they also commute with the ${\mathbb{G}_{m}}$-twist corresponding to $q^{-1}$. Therefore, the ${\mathcal{H}}$-action on $K_B ( G / B ) \cong {\mathbb{Z}}[P] [{\mathcal{O}}_{G/B}]$ induces an ${\mathcal{H}}$-action on $({\mathbb{Z}[P]})[q^{-1}] \otimes_{{\mathbb{Z}}} K _{G} ( G / B )$ through $$({\mathbb{Z}[P]})[q^{-1}] \otimes_{{\mathbb{Z}}} K _{G} ( G / B ) \cong 
({\mathbb{Z}[P]})[q^{-1}] \otimes_{{\mathbb{Z}}} {\mathbb{Z}}[P] [{\mathcal{O}}_{G/B}] = 
\bigoplus _{\lambda \in P} ({\mathbb{Z}[P]})[ q^{-1} ] [{\mathcal{O}}_{G/B} ( \lambda )],$$ where the second factor of the leftmost one is responsible for the factors $\{ [{\mathcal{O}}_{G/B} ( \lambda )] \} _{\lambda \in P}$. Finally, we complete $({\mathbb{Z}}[P])[ q^{-1} ] \otimes_{{\mathbb{Z}}} K_{B}(G/B)$ to obtain the desired assertion. This proves the proposition.

The ${\mathcal{H}}$-action in Proposition [\[KK-reint\]]{} is induced by the ${\widetilde{{\mathbf{I}}}}$-character twists and the convolution action of the structure sheaves through $q_{i,e}$ for $i \in I$ [(]{}see .

The assertion holds for the actions of ${\mathcal{H}}$ on $K_B ( G / B )$ and ${\mathbb{Z}}[q^{-1}] \otimes_{{\mathbb{Z}}} K _{B} ( G / B )$ by [@KK90 Sect. 3]. Also, by Lemma \[one-step\], for each $i \in I$, the map $q_{i,e}$ is a ${\mathbb{P}}^{1}$-fibration, and hence $${\mathbb{R}}^{k} (q_{i,e})_{\ast} {\mathcal{O}}_{{\mathbf{I}}( i ) \times ^{{\mathbf{I}}} {\mathbf{Q}_{G}}} \cong 
\begin{cases} 
 {\mathcal{O}}_{{\mathbf{Q}_{G}}} & \text{if $k=0$}, \\[1mm]
 \{ 0 \} & \text{if $k \neq 0$}. 
\end{cases} 
$$ This implies that the convolution action of ${\mathbf{I}}( i ) / {\mathbf{I}}$, $i \in I$, on ${\mathbf{Q}_{G}}$ fixes the classes of $[{\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \lambda )]$ for each $\lambda \in P$ by the projection formula. Taking into account the fact that the twist of $R ( {\widetilde{{\mathbf{I}}}} ) \cong R(B \times {\mathbb{G}_{m}})$ has an effect through the fiber of $q_{i,e}$, we conclude that $T_{i}$, $i \in I$, is identical to the convolution action induced by $q_{i,e}$ through the inclusion ${\mathbb{Z}}[q^{-1}] \otimes_{{\mathbb{Z}}} K _{B} ( G / B ) 
\subset K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$. This proves the corollary.

For each $\xi \in Q^{\vee,+}$, the natural inclusion map $\imath_{\xi} : {\mathbf{Q}_{G}}\hookrightarrow {\mathbf{Q}_{G}}$ induces an inclusion $(\imath_{\xi})_{\ast} : 
K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}}) \hookrightarrow K'_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}})$ of $({\mathbb{Z}[P]}){[\hspace{-1pt}[q^{-1}]\hspace{-1pt}]}$-modules such that $( \imath_{\xi} )_{\ast} [{\mathcal{O}}_{{\mathbf{Q}_{G}}(x)} ( \lambda )] = 
 [{\mathcal{O}}_{{\mathbf{Q}_{G}}(xt_{\xi})} ( \lambda )]$ for each $x \in W_{{\mathrm{af}}}$. We define $$K_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}^{\mathrm{rat}}}) := 
 {\mathbb{Z}}{(\hspace{-1pt}(q^{-1})\hspace{-1pt})} \otimes_{{\mathbb{Z}}{[\hspace{-0.5pt}[q^{-1}]\hspace{-0.5pt}]}} 
 \varinjlim K'_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}}).$$

\[thm:BFU\] The assignment $$\begin{aligned}
{\mathtt{e}}({\varpi}_{i}) & : 
 [{\mathbb{C}}_{\lambda} \otimes_{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi} )} ( \mu )] \mapsto 
 [{\mathbb{C}}_{-{\varpi}_{i} + \lambda} \otimes_{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi} )} ( \mu )]
 \quad \text{\rm for $i \in I$}, \\[1mm]
{\mathtt{T}}_i & : [{\mathbb{C}}_{\lambda} \otimes _{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi} )} ( \mu ) ] \\[3mm]
& \hspace*{5mm} \mapsto 
 \begin{cases} 
   \dfrac{{\mathtt{e}}( - \lambda ) - {\mathtt{e}}( - s_0 \lambda )}
     {1 - {\mathtt{e}}( \alpha_0 )} [{\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi} )} ( \mu )] + 
     {\mathtt{e}}( - s_0 \lambda ) [ {\mathcal{O}}_{{\mathbf{Q}_{G}}( s_0 t_{\xi} )} ( \mu )] 
     & \text{\rm for $i = 0$}, \\[7mm]
   \dfrac{{\mathtt{e}}( - \lambda ) - {\mathtt{e}}( - s_i \lambda + \alpha_i )}
     {1 - {\mathtt{e}}( \alpha_i )} [{\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi} )} ( \mu )]
     & \text{\rm for $i \neq 0$}, 
  \end{cases}
$$ where $\xi \in Q^{\vee,+}$, and $\lambda, \mu \in P$, equips $K_{{\widetilde{{\mathbf{I}}}}} ({\mathbf{Q}_{G}^{\mathrm{rat}}})$ with an action of ${{\mathcal{H}\hspace{-5.6pt}\mathcal{H}}}$ through the identifications[:]{} $${\mathbf{q}}\mapsto q^{-1}, \qquad
T_{i} \mapsto {\mathtt{T}}_{i} - 1 \quad \text{\rm for $i \in I_{{\mathrm{af}}}$}, \qquad
{\mathbf{e}}( \nu ) \mapsto {\mathtt{e}}( \nu ) \quad \text{\rm for $\nu \in P$}.$$

Thanks to [@KK90 Sect. 3] (and Lemma \[one-step\]), for each $i \in I$, the action of ${\mathtt{T}}_i$ is induced by the pushforward of an ${\widetilde{{\mathbf{I}}}}$-equivariant inflated sheaf through $q_{i,e}$ (see Section \[subsec:BSDH\]), and the action of ${\mathtt{e}}({\varpi}_{i})$ is induced by an ${\widetilde{{\mathbf{I}}}}$-character twist. Because these geometric counterparts commute with the pullback through $\imath_{\xi}$ for each $\xi \in Q^{\vee, +}$, our formulas define an action of ${\mathcal{H}}$ on $K_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}^{\mathrm{rat}}})$ induced by Proposition \[KK-reint\].

Now, we have $$\label{D-factor}
\frac{f g - e^{- \alpha_0} s_0 (f g )}{1 - e^{- \alpha_{0}}} = 
\frac{f - s_{0}(f)}{1 - e^{- \alpha_{0}}} g + s_{0} ( f ) 
\frac{g - e^{- \alpha_{0}} s_{0} ( g )}{1 - e^{- \alpha_{0}}} 
\quad \text{for $f,\,g \in ({\mathbb{C}}[P]){(\hspace{-1pt}(q^{-1})\hspace{-1pt})}$}.
$$ Let $p_{0,t_{\xi}}:{\mathbf{I}}(0) \times^{{\mathbf{I}}} {\mathbf{Q}_{G}}(t_{\xi}) \rightarrow {\mathbb{P}}^{1}$ be the inflation of the structure map of ${\mathbf{Q}_{G}}(t_{\xi})$, and let ${\mathcal{E}}(W)$ denote the vector bundle over ${\mathbb{P}}^{1} \cong \tilde{{\mathbf{I}}}(0)/\tilde{{\mathbf{I}}}$ associated to an $\tilde{{\mathbf{I}}}$-module $W$. Then, by taking into account equation , Corollary \[coh\] and Theorem \[cvb\], and [@Kat16 Corollary 4.8], we deduce that for each $\lambda,\,\mu \in P$, $\nu \in P$, and $\xi \in Q^{\vee,+}$ such that $s_{0} t_{\xi} \in W_{{\mathrm{af}}}^{\ge 0}$, $$\begin{aligned}
\sum_{m,\,n \ge 0} (-1)^{m+n} & 
  {\mathop{\rm gch}\nolimits}H^{m} \Bigl( {\mathbf{Q}_{G}}, \, {\mathbb{R}}^{n} ( q_{0,t_{\xi}} ) _{\ast} 
  \bigl( {\mathbb{C}}_{- \nu} \otimes_{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi})} ( \lambda ) \bigr) 
    \otimes_{{\mathcal{O}}_{{\mathbf{Q}_{G}}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}} ( \mu ) \Bigr) \nonumber \\[2mm]
= & \sum_{m \ge 0} (-1)^{m}
  {\mathop{\rm gch}\nolimits}H^{m} \bigl( {\mathbf{I}}( 0 ) \times^{{\mathbf{I}}} {\mathbf{Q}_{G}}, \, 
      {\mathbb{C}}_{- \nu} \otimes_{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi})} ( \lambda + \mu ) \bigr) \nonumber \\[2mm]
= & \sum_{m \ge 0} (-1)^{m} 
  {\mathop{\rm gch}\nolimits}H^{0} \Bigl( {\mathbb{P}}^{1}, \, {\mathbb{R}}^{m} ( p_{0, t_{\xi}} )_{\ast} 
      \bigl( {\mathbb{C}}_{- \nu} \otimes_{{\mathbb{C}}} 
             {\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi})} ( \lambda + \mu ) \bigr) \Bigr) \nonumber \\[2mm]
= & \sum_{m \ge 0} (-1)^{m} 
  {\mathop{\rm gch}\nolimits}H^{0} \Bigl( {\mathbb{P}}^{1}, \, {\mathbb{C}}_{- \nu} \otimes_{{\mathbb{C}}} 
      {\mathcal{E}}\bigl( H^{0} ( {\mathbf{Q}_{G}}( t_{\xi} ), \, 
      {\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi} )} ( \lambda + \mu ) )^{\ast} \bigr) \Bigr) \nonumber \\[2mm]
= & \dfrac{e^{-\nu} \cdot {\mathop{\rm gch}\nolimits}V^{-}_{t_\xi} (-w_{\circ}(\lambda + \mu)) - 
    e^{- \alpha_0} s_0 ( e^{-\nu} \cdot {\mathop{\rm gch}\nolimits}V^{-}_{t_\xi} (-w_{\circ}(\lambda + \mu)) )}
       {1 - e^{- \alpha_0}} \nonumber \\[2mm]
= & \dfrac{ e^{-\nu} - s_0 ( e^{-\nu} )}
          {1 - e^{- \alpha_0}} 
    {\mathop{\rm gch}\nolimits}V^{-}_{t_\xi} (-w_{\circ}(\lambda + \mu)) + 
    e^{- s_0 \nu} {\mathop{\rm gch}\nolimits}V^{-}_{s_0 t_\xi} (-w_{\circ}(\lambda + \mu)) \nonumber \\[2mm]
= & \dfrac{ e^{-\nu} - s_0 ( e^{-\nu} )}{1 - e^{- \alpha_0}} 
    \Psi ( [ {\mathcal{O}}_{{\mathbf{Q}_{G}}( t_{\xi} )} ( \lambda ) ] ) ( \mu ) + 
    e^{- s_0 \nu} \Psi ( [ {\mathcal{O}}_{{\mathbf{Q}_{G}}( s_0 t_{\xi} )} ( \lambda ) ] ) ( \mu ), \label{conv-eqn}\end{aligned}$$ where the first and fourth equalities follow by the Leray spectral sequence. In particular, the term represents the image under $\Psi$ of the convolution of $[{\mathbb{C}}_{-\mu} \otimes_{{\mathbb{C}}} {\mathcal{O}}_{{\mathbf{Q}_{G}}(t_{\xi})} ( \lambda )]$ with respect to $q_{0,t_{\xi}}$. Therefore, from the injectivity of $\Psi$, we conclude that ${\mathtt{T}}_{0}$ is induced by the pushforward of an ${\widetilde{{\mathbf{I}}}}$-equivariant inflated sheaf through $q_{0, t_{\xi}}$ for some $\xi \in Q^{\vee, +}$.

From the above, we deduce that the actions ${\mathtt{T}}_i$, $i \in I_{{\mathrm{af}}}$, and ${\mathbf{e}}(\nu)$, $\nu \in P$, generate the convolution action of Schubert cells and the ${\widetilde{{\mathbf{I}}}}$-character twists of the (thin) affine flag manifold $G{(\hspace{-1pt}(z)\hspace{-1pt})} / {\mathbf{I}}$ on ${\mathbf{Q}_{G}^{\mathrm{rat}}}$ (or rather, on ${\mathbf{Q}_{G}}$). In particular, the $T_{i}$, $i \in I_{{\mathrm{af}}}$, generate the nil-Hecke algebra of affine type by [@KK90 Sect. 3]. Therefore, their commutation relations with ${\mathtt{e}}( {\varpi}_i )$, $i \in I$, imply that the $T_{i}$, $i \in I_{{\mathrm{af}}}$, and the ${\mathtt{e}}( {\varpi}_{i} )$, $i \in I$, satisfy the relations for ${{\mathcal{H}\hspace{-5.6pt}\mathcal{H}}}$ (see also [@BFU Sect. 3.4]); we remark that their convention differs from ours by the twist by the Serre duality and line bundle twist [@BFU Sects. 3.1 and 3.21].

Finally, we complete the proof by observing that ${\mathtt{T}}_0$ preserves $K_{{\widetilde{{\mathbf{I}}}}} ( {\mathbf{Q}_{G}^{\mathrm{rat}}})$ by inspection.

Proof of Theorem \[thm:SMT\]. {#sec:prf-SMT}
=============================

Affine Weyl group action. {#subsec:Weyl}
-------------------------

Let ${\mathcal{B}}$ be a regular crystal in the sense of [@Kas02 Sect. 2.2] (or, a normal crystal in the sense of [@HK p.389]); for example, ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ for $\lambda \in P^{+}$ is a regular crystal by Theorem \[thm:isom\], and hence so is ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$ for $\lambda,\,\mu \in P^{+}$. Then we know from [@Kas94 Sect. 7] that the affine Weyl group $W_{{\mathrm{af}}}$ acts on ${\mathcal{B}}$ as follows: for $b \in {\mathcal{B}}$ and $i \in I_{{\mathrm{af}}}$, $$\label{eq:W-act}
s_{i} \cdot b := 
\begin{cases}
f_{i}^{n}b & \text{if $n:={\langle {\mathop{\rm wt}\nolimits}(b),\,\alpha_{i}^{\vee} \rangle} \ge 0$}, \\[1.5mm]
e_{i}^{-n}b & \text{if $n:={\langle {\mathop{\rm wt}\nolimits}(b),\,\alpha_{i}^{\vee} \rangle} \le 0$}. 
\end{cases}
$$

Also, for $b \in {\mathcal{B}}$ and $i \in I_{{\mathrm{af}}}$, we define $e_{i}^{\max}b=e_{i}^{{\varepsilon}_{i}(b)}b$ and $f_{i}^{\max}b=f_{i}^{{\varphi}_{i}(b)}b$, where ${\varepsilon}_{i}(b):=\max\bigl\{n \ge 0 \mid e_{i}^{n}b \ne {\bm{0}}\bigr\}$ and ${\varphi}_{i}(b):=\max\bigl\{n \ge 0 \mid f_{i}^{n}b \ne {\bm{0}}\bigr\}$; note that if $b \in {\mathcal{B}}$ satisfies $e_{i}b = {\bm{0}}$ (resp., $f_{i}b = {\bm{0}}$), i.e., ${\varepsilon}_{i}(b)=0$ (resp., ${\varphi}_{i}(b)=0$), then $f_{i}^{\max}b = s_{i} \cdot b$ (resp., $e_{i}^{\max}b = s_{i} \cdot b$).

Connected components of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$. {#subsec:conn}
---------------------------------------------------------------------

Let $\lambda \in P^{+}$, and write it as $\lambda = \sum_{i \in I} m_{i} {\varpi}_{i}$, with $m_{i} \in {\mathbb{Z}}_{\ge 0}$; note that $J=\bigl\{i \in I \mid {\langle \lambda,\,\alpha_{i}^{\vee} \rangle}=0\bigr\}= 
\bigl\{i \in I \mid m_{i} = 0\bigr\}$. We define ${\mathop{\rm Par}\nolimits}(\lambda)$ to be the set of $I$-tuples of partitions ${\bm{\rho}}= (\rho^{(i)})_{i \in I}$ such that $\rho^{(i)}$ is a partition of length (strictly) less than $m_{i}$ for each $i \in I$; a partition of length less than $0$ (or $1$) is understood to be the empty partition $\emptyset$. Also, for ${\bm{\rho}}= (\rho^{(i)})_{i \in I} \in {\mathop{\rm Par}\nolimits}(\lambda)$, we set $|{\bm{\rho}}|:=\sum_{i \in I} |\rho^{(i)}|$, where for a partition $\rho = (\rho_{1} \ge \rho_{2} \ge \cdots \ge \rho_{m})$, we set $|\rho| := \rho_{1}+\cdots+\rho_{m}$. We endow the set ${\mathop{\rm Par}\nolimits}(\lambda)$ with a crystal structure as follows: for ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$ and $i \in I_{{\mathrm{af}}}$, $$e_{i} {\bm{\rho}}= f_{i} {\bm{\rho}}:= {\bm{0}}, \quad 
{\varepsilon}_{i} ({\bm{\rho}}) = {\varphi}_{i} ({\bm{\rho}}) := -\infty, \quad  
{\mathop{\rm wt}\nolimits}({\bm{\rho}}) := - |{\bm{\rho}}| \delta.$$

We recall from [@INS Sect. 7] the relation between ${\mathop{\rm Par}\nolimits}(\lambda)$ and the set ${\mathop{\rm Conn}\nolimits}({\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda))$ of connected components of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$. We set ${\mathop{\rm Turn}\nolimits}(\lambda):=
 \bigl\{k/m_{i} \mid i \in I \setminus {J}\text{ and }
 0 \le k \le m_{i}\bigr\}$. By [@INS Proposition 7.1.2], each connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ contains a unique element of the form: $$\label{eq:ext}
\bigl( {\Pi^{J}}(t_{\xi_{1}}),\,\dots,\,{\Pi^{J}}(t_{\xi_{s-1}}),\,e \,;\, 
  a_{0},\,a_{1},\,\dots,\,a_{s} \bigr),$$ where $s \ge 1$, $\xi_{1},\,\dots,\,\xi_{s-1} \in Q^{\vee}_{I \setminus {J}}$ such that $\xi_{1} > \cdots > \xi_{s-1} > 0=:\xi_{s}$, and $a_{u} \in {\mathop{\rm Turn}\nolimits}(\lambda)$ for all $0 \le u \le s$. For each element of the form (or equivalently, each connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$), we define an element ${\bm{\rho}}= (\rho^{(i)})_{i \in I} \in {\mathop{\rm Par}\nolimits}(\lambda)$ as follows. First, let $i \in I \setminus {J}$; note that $m_{i} \ge 1$. For each $1 \le k \le m_{i}$, take $0 \le u \le s$ such that $a_{u}$ is contained in the interval $\bigl( (k-1)/m_{i},\,k/m_{i} \bigr]$. Then we define $\rho^{(i)}_{k}$ to be ${\langle {\varpi}_{i},\,\xi_{u} \rangle}$, the coefficient of $\alpha_{i}^{\vee}$ in $\xi_{u}$; we know from (the proof of) [@INS Proposition 7.2.1] that $\rho^{(i)}_{k}$ does not depend on the choice of $u$ above. Since $\xi_{1} > \cdots > \xi_{s-1} > 0=\xi_{s}$, we see that $\rho^{(i)}_{1} \ge \cdots \ge 
\rho^{(i)}_{m_{i}-1} \ge \rho^{(i)}_{m_{i}}=0$. Thus, for each $i \in I \setminus {J}$, we obtain a partition of length less than $m_{i}$. For $i \in {J}$, we set $\rho^{(i)}:=\emptyset$. Thus we obtain an element ${\bm{\rho}}= (\rho^{(i)})_{i \in I} \in {\mathop{\rm Par}\nolimits}(\lambda)$, and hence a map from ${\mathop{\rm Conn}\nolimits}({\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda))$ to ${\mathop{\rm Par}\nolimits}(\lambda)$. Moreover, we know from [@INS Proposition 7.2.1] that this map is bijective; we denote by $\pi_{{\bm{\rho}}} \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ the element of the form corresponding to ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$ under this bijection.

\[rem:ext\] Let ${\bm{\rho}}= (\rho^{(i)})_{i \in I} \in {\mathop{\rm Par}\nolimits}(\lambda)$, with $\rho^{(i)}=(\rho^{(i)}_{1} \ge \cdots )$ for $i \in I$; note that $\rho^{(i)}_{1}=0$ if $\rho^{(i)}=\emptyset$. It follows from the definition that $$\label{eq:par0}
\iota(\pi_{{\bm{\rho}}}) = {\Pi^{J}}(t_{\xi_{1}}), \quad 
\text{where} \quad \xi_{1} = \sum_{i \in I} 
\rho^{(i)}_{1} \alpha_{i}^{\vee} \in Q^{\vee}_{I \setminus {J}}.$$

For ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, we denote by ${\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\rho}}}(\lambda)$ the connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ containing $\pi_{{\bm{\rho}}}$. Also, we denote by ${\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda)$ the connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ containing $\pi_{\lambda}=(e\,;\,0,1)$; note that $\pi_{\lambda}=\pi_{{\bm{\rho}}}$ for ${\bm{\rho}}=(\emptyset)_{i \in I}$. We know from [@INS Proposition 3.2.4] (and its proof) that for each ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, there exists an isomorphism ${\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\rho}}}(\lambda) \stackrel{\sim}{\rightarrow}
\bigl\{{\bm{\rho}}\bigr\} \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda)$ of crystals, which maps $\pi_{{\bm{\rho}}}$ to ${\bm{\rho}}\otimes \pi_{\lambda}$. Hence we have $$\label{eq:isom}
{\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) = 
\bigsqcup_{{\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)} {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\rho}}}(\lambda) \cong 
\bigsqcup_{{\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)}
\bigl\{{\bm{\rho}}\bigr\} \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda) \quad \text{as crystals}.$$

The following lemma is shown by induction on the (ordinary) length $\ell(x)$ of $x$; for part (1), see also [@NS16 Remark 3.5.2]

\[lem:Weyl\]

Let $\lambda \in P^{+}$. If $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ is of the form , then for $x \in W_{{\mathrm{af}}}$, $$\label{eq:tx0}
x \cdot \pi = 
\bigl( {\Pi^{J}}(xt_{\xi_{1}}),\,\dots,\,{\Pi^{J}}(xt_{\xi_{s-1}}),\,{\Pi^{J}}(x) \,;\, 
  a_{0},\,a_{1},\,\dots,\,a_{s} \bigr).$$

Let $\lambda,\,\mu \in P^{+}$. Let ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, ${\bm{\chi}}\in {\mathop{\rm Par}\nolimits}(\mu)$, and $\xi,\zeta \in Q^{\vee}$. Then, for $x \in W_{{\mathrm{af}}}$, $$\label{eq:Weyl}
x \cdot \bigl( (t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes (t_{\zeta} \cdot \pi_{{\bm{\chi}}}) \bigr)
= (xt_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes (xt_{\zeta} \cdot \pi_{{\bm{\chi}}}).$$

Let $\xi \in Q^{\vee}$. It follows from Lemma \[lem:SiB\](3) that if $\pi = (x_{1},\,\dots,\,x_{s}\,;\,{\mathbf{a}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$, then $$\label{eq:Txi}
T_{\xi}\pi:=
\bigl( {\Pi^{J}}(x_{1}t_{\xi}),\,\dots,\,{\Pi^{J}}(x_{s}t_{\xi})\,;\,{\mathbf{a}}\bigr) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda);$$ the map $T_{\xi} : {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \rightarrow {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ is clearly bijective, with $T_{\xi}^{-1}=T_{-\xi}$. We can verify by the definitions that $$\label{eq:Tx}
\begin{cases}
T_{\xi}e_{i}\pi= e_{i}T_{\xi}\pi, \quad 
T_{\xi}f_{i}\pi= f_{i}T_{\xi}\pi 
 & \text{for $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $i \in I_{{\mathrm{af}}}$}, \\[1.5mm]
{\varepsilon}_{i}(T_{\xi}\pi) = {\varepsilon}_{i}(\pi), \quad 
{\varphi}_{i}(T_{\xi}\pi) = {\varphi}_{i}(\pi)
 & \text{for $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $i \in I_{{\mathrm{af}}}$}, \\[1.5mm]
{\mathop{\rm wt}\nolimits}(T_{\xi}\pi) = {\mathop{\rm wt}\nolimits}(\pi) - {\langle \lambda,\,\xi \rangle}\delta,
\end{cases}
$$ where $T_{\xi}{\bm{0}}$ is understood to be ${\bm{0}}$.

\[rem:Tx\] Let ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, and assume that $\pi_{{\bm{\rho}}}$ is of the form . For $\xi \in Q^{\vee}$, we see from and that $$T_{\xi}\pi_{{\bm{\rho}}} = 
\bigl( {\Pi^{J}}(t_{\xi_{1}+\xi}),\,\dots,\,{\Pi^{J}}(t_{\xi_{s-1}+\xi}),\,{\Pi^{J}}(t_{\xi}) \,;\, 
  a_{0},\,a_{1},\,\dots,\,a_{s} \bigr) = t_{\xi} \cdot \pi_{{\bm{\rho}}},$$ which implies that $T_{\xi}\pi_{{\bm{\rho}}} \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\rho}}}(\lambda)$. Therefore, it follows from that $T_{\xi}({\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\rho}}}(\lambda)) = {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\rho}}}(\lambda)$.

Quantum Lakshmibai-Seshadri paths. {#subsec:QLS}
----------------------------------

Let $\lambda \in P^{+}$. Let ${\mathop{\rm cl}\nolimits}: {\mathbb{R}}\otimes_{{\mathbb{Z}}} P_{{\mathrm{af}}} \twoheadrightarrow 
({\mathbb{R}}\otimes_{{\mathbb{Z}}} P_{{\mathrm{af}}})/{\mathbb{R}}\delta$ denote the canonical projection. For an element $\pi=(x_{1},\,\dots,\,x_{s}\,;\,a_{0},\,a_{1},\,\dots,\,a_{s}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$, we define a piecewise-linear, continuous map ${\mathop{\rm cl}\nolimits}(\pi) : [0,1] \rightarrow ({\mathbb{R}}\otimes_{{\mathbb{Z}}} P_{{\mathrm{af}}})/{\mathbb{R}}\delta$ by $({\mathop{\rm cl}\nolimits}(\pi))(t):={\mathop{\rm cl}\nolimits}({\overline{\pi}}(t))$ for $t \in [0,1]$ (for ${\overline{\pi}}$, see ). As explained in [@NS16 Sect. 6.2], the set $\bigl\{{\mathop{\rm cl}\nolimits}(\pi) \mid \pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)\bigr\}$ is identical to the set ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ of all “projected (by ${\mathop{\rm cl}\nolimits}$)” LS paths of shape $\lambda$, introduced in [@NS05 (3.4)] and [@NS06 page 117] (see also [@LNSSS2 Sect. 2.2]). Also, by [@LNSSS2 Theorem 3.3], ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ is identical to the set ${\mathrm{QLS}}(\lambda)$ of all quantum LS paths of shape $\lambda$, introduced in [@LNSSS2 Sect. 3.2]. We can endow the set ${\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}} = {\mathrm{QLS}}(\lambda)$ with a crystal structure with weights in ${\mathop{\rm cl}\nolimits}(P_{{\mathrm{af}}})$ in such a way that $$\label{eq:cl}
\begin{cases}
e_{i}{\mathop{\rm cl}\nolimits}(\pi) = {\mathop{\rm cl}\nolimits}(e_{i}\pi), \quad 
f_{i}{\mathop{\rm cl}\nolimits}(\pi) = {\mathop{\rm cl}\nolimits}(f_{i}\pi) 
  & \text{for $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $i \in I_{{\mathrm{af}}}$}, \\[1.5mm]
{\mathop{\rm wt}\nolimits}({\mathop{\rm cl}\nolimits}(\pi))={\mathop{\rm cl}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))
  & \text{for $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$}, \\[1.5mm]
{\varepsilon}_{i}({\mathop{\rm cl}\nolimits}(\pi))={\varepsilon}_{i}(\pi), \quad 
{\varphi}_{i}({\mathop{\rm cl}\nolimits}(\pi))={\varphi}_{i}(\pi)
  & \text{for $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $i \in I_{{\mathrm{af}}}$}, 
\end{cases}
$$ where we understand that ${\mathop{\rm cl}\nolimits}({\bm{0}}) = {\bm{0}}$. The next theorem follows from [@NS05 Proposition 3.23 and Theorem 3.2].

\[thm:QLS\]

For every $\lambda \in P^{+}$, the crystal ${\mathrm{QLS}}(\lambda)={\mathbb{B}}(\lambda)_{{\mathop{\rm cl}\nolimits}}$ is connected.

For every $\lambda,\,\mu \in P^{+}$, there exists an isomorphism ${\mathrm{QLS}}(\lambda) \otimes {\mathrm{QLS}}(\mu) \cong {\mathrm{QLS}}(\lambda+\mu)$ of crystals. In particular, ${\mathrm{QLS}}(\lambda) \otimes {\mathrm{QLS}}(\mu)$ is connected.

\[lem:conn\] Let $\lambda,\,\mu \in P^{+}$, and set ${J}:=\bigl\{ i \in I \mid {\langle \lambda,\,\alpha_{i}^{\vee} \rangle}=0\bigr\}$, ${K}:=\bigl\{ i \in I \mid {\langle \mu,\,\alpha_{i}^{\vee} \rangle}=0\bigr\}$. Each connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$ contains an element of the form[:]{} $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ for some $\xi \in Q^{\vee}_{I \setminus ({J}\cup {K})}$, ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, and ${\bm{\chi}}\in {\mathop{\rm Par}\nolimits}(\mu)$.

Let $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. By Theorem \[thm:QLS\](2), there exists a monomial $X$ in root operators on ${\mathrm{QLS}}(\lambda) \otimes {\mathrm{QLS}}(\mu)$ such that $X \bigl( {\mathop{\rm cl}\nolimits}(\pi) \otimes {\mathop{\rm cl}\nolimits}(\eta) \bigr) = 
 {\mathop{\rm cl}\nolimits}(\pi_{\lambda}) \otimes {\mathop{\rm cl}\nolimits}(\pi_{\mu})$; recall that $\pi_{\lambda}=(e\,;\,0,\,1) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\pi_{\mu}=(e\,;\,0,\,1) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. It follows from and the tensor product rule for crystals that $X(\pi \otimes \eta)$ is of the form $\pi_{1} \otimes \eta_{1}$ for some $\pi_{1} \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ such that ${\mathop{\rm cl}\nolimits}(\pi_{1})={\mathop{\rm cl}\nolimits}(\pi_{\lambda})$ and $\eta_{1} \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$ such that ${\mathop{\rm cl}\nolimits}(\eta_{1})={\mathop{\rm cl}\nolimits}(\pi_{\mu})$. Here, we see from [@NS16 Lemma 6.2.2] that $$\begin{split}
& \bigl\{ \pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \mid {\mathop{\rm cl}\nolimits}(\pi)={\mathop{\rm cl}\nolimits}(\pi_{\lambda})\bigr\} = 
  \bigl\{ t_{\xi} \cdot \pi_{{\bm{\rho}}} \mid {\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda),\,\xi \in Q^{\vee}\bigr\}, \\
& \bigl\{ \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu) \mid {\mathop{\rm cl}\nolimits}(\eta)={\mathop{\rm cl}\nolimits}(\pi_{\mu})\bigr\} = 
  \bigl\{ t_{\zeta} \cdot \pi_{{\bm{\chi}}} \mid {\bm{\chi}}\in {\mathop{\rm Par}\nolimits}(\mu),\,\zeta \in Q^{\vee}\bigr\}. 
\end{split}
$$ Therefore, $X(\pi \otimes \eta) = 
\bigl(t_{\xi_1} \cdot \pi_{{\bm{\rho}}}\bigr) \otimes \bigl(t_{\zeta_1} \cdot \pi_{{\bm{\chi}}}\bigr)$ for some ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, $\xi_1 \in Q^{\vee}$ and ${\bm{\chi}}\in {\mathop{\rm Par}\nolimits}(\mu)$, $\zeta_1 \in Q^{\vee}$. Also, by , we have $t_{-\zeta_1} \cdot \bigl( (t_{\xi_1} \cdot \pi_{{\bm{\rho}}}) \otimes 
(t_{\zeta_1} \cdot \pi_{{\bm{\chi}}}) \bigr) = 
(t_{\xi_1-\zeta_1} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$; we deduce from and that $t_{\xi_1-\zeta_1} \cdot \pi_{{\bm{\rho}}} = t_{\xi_2} \cdot \pi_{{\bm{\rho}}}$, with $\xi_2=[\xi_1-\zeta_1]^{{J}}$, where $[\,\cdot\,]^{{J}}:Q^{\vee} \twoheadrightarrow Q^{\vee}_{I \setminus {J}}$ is the projection in . We set $\gamma:=[\xi_{2}]_{{K}} \in Q^{\vee}_{K}$, where $[\,\cdot\,]_{{K}}:Q^{\vee} \twoheadrightarrow Q^{\vee}_{{K}}$ is the projection defined as in ; we deduce from and that $t_{-\gamma} \cdot \pi_{{\bm{\chi}}} = \pi_{{\bm{\chi}}}$. In addition, we set $\xi:=\xi_{2}-\gamma$; notice that $\xi \in Q^{\vee}_{I \setminus ({J}\cup {K})}$. Summarizing the above, we have $$\begin{aligned}
(t_{-\gamma}t_{-\zeta_1}) \cdot X(\pi \otimes \eta) & = 
(t_{-\gamma}t_{-\zeta_1}) \cdot 
  \bigl( (t_{\xi_1} \cdot \pi_{{\bm{\rho}}}) \otimes 
         (t_{\zeta_1} \cdot \pi_{{\bm{\chi}}}) \bigr) = 
t_{-\gamma} \cdot 
  \bigl( (t_{\xi_1-\zeta_1} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}} \bigr) \\
& = t_{-\gamma} \cdot 
  \bigl( (t_{\xi_2} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}} \bigr) 
  = (t_{\xi_2-\gamma} \cdot \pi_{{\bm{\rho}}}) \otimes (t_{-\gamma} \cdot \pi_{{\bm{\chi}}}) \\
& = (t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}. \end{aligned}$$ Because the action of the affine Weyl group $W_{{\mathrm{af}}}$ on ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$ is defined by means of root operators (see ), we conclude that $\pi \otimes \eta$ and $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ above are in the same connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. This proves the lemma.

Proof of Theorem \[thm:SMT\]. {#subsec:prf-SMT}
-----------------------------

Recall that $\lambda,\,\mu \in P^{+}$, and ${J}=\bigl\{ i \in I \mid {\langle \lambda,\,\alpha_{i}^{\vee} \rangle}=0\bigr\}$, ${K}=\bigl\{ i \in I \mid {\langle \mu,\,\alpha_{i}^{\vee} \rangle}=0\bigr\}$.

\[prop:SMT1\] The set ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu) \sqcup \{{\bm{0}}\}$ is stable under the action of the root operators $e_{i}$, $f_{i}$, $i \in I_{{\mathrm{af}}}$, on ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$.

We give a proof of the assertion only for $e_{i}$, $i \in I_{{\mathrm{af}}}$; the proof for $f_{i}$, $i \in I_{{\mathrm{af}}}$, is similar. Let $\pi \otimes \eta \in {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$, and $i \in I_{{\mathrm{af}}}$. We may assume that $e_{i}(\pi \otimes \eta) \ne {\bm{0}}$. Then it follows from the tensor product rule for crystals that $$e_{i}(\pi \otimes \eta) = 
  \begin{cases}
  (e_{i}\pi) \otimes \eta & \text{if ${\varphi}_{i}(\pi) \ge {\varepsilon}_{i}(\eta)$}, \\[1.5mm]
  \pi \otimes (e_{i}\eta) & \text{if ${\varphi}_{i}(\pi) < {\varepsilon}_{i}(\eta)$};
  \end{cases}
$$ recall from Remark \[rem:SLS\] that $$\label{eq:vevp}
{\varepsilon}_{i}(\eta) = - m^{\eta}_{i} \quad \text{and} \quad 
{\varphi}_{i}(\pi) = H^{\pi}_{i}(1)-m^{\pi}_{i}.$$ Let $x,y \in W_{{\mathrm{af}}}$ be such that $x {\succeq}y$ and ${\Pi^{J}}(x) = \kappa(\pi)$, ${\Pi^{K}}(y) = \iota(\eta)$ (see ); we write $x$ and $y$ as: $$\label{eq:xy}
\begin{cases}
x=\kappa(\pi)x_1 & \text{with $x_1 \in ({W_{J}})_{{\mathrm{af}}}$}, \\[1mm]
y=\iota(\eta)y_1 & \text{with $y_1 \in ({W_{K}})_{{\mathrm{af}}}$}.
\end{cases}
$$

#### Case 1.

Assume that ${\varphi}_{i}(\pi) \ge {\varepsilon}_{i}(\eta)$, i.e., $e_{i}(\pi \otimes \eta) = (e_{i}\pi) \otimes \eta$. Note that $\kappa(e_{i}\pi)$ is equal either to $\kappa(\pi)$ or to $s_{i}\kappa(\pi)$ by the definition of the root operator $e_{i}$. If $\kappa(e_{i}\pi)=\kappa(\pi)$, then there is nothing to prove. Hence we may assume that $\kappa(e_{i}\pi) = s_{i} \kappa(\pi)$. Then we deduce from the definition of the root operator $e_{i}$ that the point $t_{1} = \min \bigl\{ t \in [0,1] \mid H^{\pi}_{i}(t)=m^{\pi}_{i} \bigr\}$ is equal to $1$, and hence $$\label{eq:case1a}
H^{\pi}_{i}(1) = m^{\pi}_{i} \quad \text{and} \quad
{\langle \kappa(\pi)\lambda,\,\alpha_{i}^{\vee} \rangle} < 0.$$ By , the equality in , and our assumption that ${\varphi}_{i}(\pi) \ge {\varepsilon}_{i}(\eta)$, we see that $m^{\eta}_{i} \ge 0$, and hence $m^{\eta}_{i}=0$; in particular, we obtain ${\langle \iota(\eta)\mu,\,\alpha_{i}^{\vee} \rangle} \ge 0$. In addition, it follows from that $\kappa(\pi)^{-1}\alpha_{i} \in 
- \bigl(\Delta^{+} \setminus {\Delta_{J}}^{+}\bigr) + {\mathbb{Z}}\delta$. Since $x_{1} \in ({W_{J}})_{{\mathrm{af}}}$ by , we have $$\label{eq:x-}
x^{-1}\alpha_{i}=
x_1^{-1}\kappa(\pi)^{-1}\alpha_{i} \in 
- \bigl(\Delta^{+} \setminus {\Delta_{J}}^{+}\bigr) + {\mathbb{Z}}\delta \subset -\Delta^{+} + {\mathbb{Z}}\delta.$$ Also, since $s_{i}\kappa(\pi) = \kappa(e_{i}\pi) \in ({W^{J}})_{{\mathrm{af}}}$ and $x_{1} \in ({W_{J}})_{{\mathrm{af}}}$, we have $${\Pi^{J}}(s_{i}x) = 
{\Pi^{J}}(s_{i}\kappa(\pi)x_{1}) = 
s_{i}\kappa(\pi) = \kappa(e_{i}\pi).$$ If $y^{-1}\alpha_{i} \in \Delta^{+} + {\mathbb{Z}}\delta$, then we see from Lemma \[lem:dmd\](2) (applied to the case $J=\emptyset$) and that $s_{i}x {\succeq}y$ in $W_{{\mathrm{af}}}$. Therefore, $s_{i}x,\,y \in W_{{\mathrm{af}}}$ satisfy condition for $e_{i}(\pi \otimes \eta) = (e_{i}\pi) \otimes \eta$. If $y^{-1}\alpha_{i} \in -\Delta^{+} + {\mathbb{Z}}\delta$, then we see from Lemma \[lem:dmd\](3) (applied to the case $J=\emptyset$) and that $s_{i}x {\succeq}s_{i}y$ in $W_{{\mathrm{af}}}$. We now claim that ${\Pi^{K}}(s_{i}y)=\iota(\eta)$. Indeed, since ${\langle \iota(\eta)\mu,\,\alpha_{i}^{\vee} \rangle} \ge 0$ as seen above, we have $\iota(\eta)^{-1}\alpha_{i} \in 
\bigl(\Delta^{+} \sqcup (-{\Delta_{K}}^{+})\bigr)+{\mathbb{Z}}\delta$. In addition, since $y_{1} \in ({W_{K}})_{{\mathrm{af}}}$, we deduce that $y^{-1}\alpha_{i} = y_{1}^{-1}\iota(\eta)^{-1}\alpha_{i}$ is contained in $\bigl(\Delta^{+} \sqcup (-{\Delta_{K}}^{+})\bigr)+{\mathbb{Z}}\delta$. However, since $y^{-1}\alpha_{i} \in -\Delta^{+} + {\mathbb{Z}}\delta$ by our assumption, we have $y^{-1}\alpha_{i} \in -\Delta_{K}^{+}+{\mathbb{Z}}\delta$, which implies that $s_{y^{-1}\alpha_{i}} \in ({W_{K}})_{{\mathrm{af}}}$. Therefore, we obtain ${\Pi^{K}}(s_{i}y) = 
{\Pi^{K}}(ys_{y^{-1}\alpha_{i}}) = 
{\Pi^{K}}(y) = \iota(\eta)$, as desired. Thus, $s_{i}x,\,s_{i}y \in W_{{\mathrm{af}}}$ satisfy condition for $e_{i}(\pi \otimes \eta) = (e_{i}\pi) \otimes \eta$.

#### Case 2.

Assume that ${\varphi}_{i}(\pi) < {\varepsilon}_{i}(\eta)$, i.e., $e_{i}(\pi \otimes \eta) = \pi \otimes (e_{i}\eta)$. If $\iota(e_{i}\eta)=\iota(\eta)$, then there is nothing to prove. Hence we may assume that $\iota(e_{i}\eta) = s_{i} \iota(\eta)$. Then we deduce from the definition of the root operator $e_{i}$ that ${\langle \iota(\eta)\mu,\,\alpha_{i}^{\vee} \rangle} < 0$; observe that with notation in and , $t_{0}=0$, and $H^{\eta}_{i}(t)$ is strictly decreasing on $[t_{0},\,t_{1}] = [0,\,t_{1}]$. Thus we obtain $\iota(\eta)^{-1}\alpha_{i} \in 
- \bigl(\Delta^{+} \setminus {\Delta_{K}}^{+}\bigr) + {\mathbb{Z}}\delta$. In addition, since $y_{1} \in ({W_{K}})_{{\mathrm{af}}}$ (see ), we deduce that $$y^{-1}\alpha_{i} = y_1^{-1}\iota(\eta)^{-1}\alpha_{i} \in 
- \bigl(\Delta^{+} \setminus {\Delta_{K}}^{+}\bigr) + {\mathbb{Z}}\delta \subset 
-\Delta^{+} + {\mathbb{Z}}\delta,$$ which implies that $y {\succ}s_{i}y$ by Lemma \[lem:si\]. Since $x {\succeq}y$, we get $x {\succ}s_{i}y$. Also, since $s_{i}\iota(\eta) = \iota(e_{i}\eta) \in ({W^{K}})_{{\mathrm{af}}}$ and $y_{1} \in ({W_{K}})_{{\mathrm{af}}}$, we have $${\Pi^{K}}(s_{i}y) = 
{\Pi^{K}}(s_{i}\iota(\eta)y_{1}) = 
s_{i}\iota(\eta) = \iota(e_{i}\eta).$$ Thus, $x,\,s_{i}y \in W_{{\mathrm{af}}}$ satisfy condition for $e_{i}(\pi \otimes \eta) = \pi \otimes (e_{i}\eta)$.

This completes the proof of the proposition.

By this proposition, ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ is a subcrystal of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. Hence our remaining task is to prove that ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu) \cong {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu)$ as crystals. Recall from Lemma \[lem:conn\] that each connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$ contains an element of the form: $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ for some $\xi \in Q^{\vee}_{I \setminus ({J}\cup {K})}$, ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, and ${\bm{\chi}}\in {\mathop{\rm Par}\nolimits}(\mu)$.

\[prop:tx\] Let ${\bm{\rho}}=(\rho^{(i)}) \in {\mathop{\rm Par}\nolimits}(\lambda)$, ${\bm{\chi}}=(\chi^{(i)}) \in {\mathop{\rm Par}\nolimits}(\mu)$, and $\xi=\sum c_{i}\alpha_{i}^{\vee} 
\in Q^{\vee}_{I \setminus ({J}\cup {K})}$. Then, $$\label{eq:tx1}
(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}} \in {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$$ if and only if $$\label{eq:tx2}
c_{i} \ge \chi^{(i)}_{1} \quad \text{\rm for all 
$i \in I \setminus ({J}\cup {K})$}.$$

We first show the “only if” part; assume that holds. We see that $$\begin{aligned}
& \kappa(t_{\xi} \cdot \pi_{{\bm{\rho}}}) = {\Pi^{J}}(t_{\xi}) \quad \text{by \eqref{eq:tx0}}; \\[1mm]
& \iota(\pi_{{\bm{\chi}}}) = {\Pi^{K}}(t_{\zeta_{1}}), \quad 
\text{with} \quad \zeta_{1} = \sum_{i \in I} 
\chi^{(i)}_{1} \alpha_{i}^{\vee} \in Q^{\vee}_{I \setminus {K}} \quad \text{by \eqref{eq:par0}}. \end{aligned}$$ Since holds, there exist $x,\,y \in W_{{\mathrm{af}}}$ such that $x {\succeq}y$ in $W_{{\mathrm{af}}}$, and such that $$\begin{cases}
{\Pi^{J}}(x) = 
\kappa(t_{\xi} \cdot \pi_{{\bm{\rho}}}) ={\Pi^{J}}(t_{\xi}), & \\[1mm]
{\Pi^{K}}(y) = 
\iota(\pi_{{\bm{\chi}}}) = {\Pi^{K}}(t_{\zeta_{1}});
\end{cases}
$$ we write $x$ and $y$ as: $$\begin{cases}
x={\Pi^{J}}(t_{\xi})x_{1} 
  & \text{with $x_{1} \in ({W_{J}})_{{\mathrm{af}}}$}, \\[1mm]
y={\Pi^{K}}(t_{\zeta_{1}})y_{1} 
  & \text{with $y_{1} \in ({W_{K}})_{{\mathrm{af}}}$}.
\end{cases}
$$ Because $x \in W_{{\mathrm{af}}}$ is a lift of ${\Pi^{J}}(t_{\xi}) \in ({W^{J}})_{{\mathrm{af}}}$, it follows from Lemma \[lem:lift\] that $x = v t_{\xi+\gamma}$ for some $v \in {W_{J}}$ and $\gamma \in {Q_{J}^{\vee}}$. Similarly, we have $y = v't_{\zeta_1+\gamma'}$ for some $v' \in {W_{K}}$ and $\gamma' \in {Q_{K}^{\vee}}$. Because $x {\succeq}y$ in $W_{{\mathrm{af}}}$, we deduce from Lemma \[lem:SiB\](1) (applied to the case $J=\emptyset$) that $\xi+\gamma \ge \zeta_1+\gamma'$, which implies since $\gamma,\,\gamma' \in Q^{\vee}_{{J}\cup {K}}$.

We next show the “if” part; assume that holds. Recall that $$\begin{aligned}
& \kappa(t_{\xi} \cdot \pi_{{\bm{\rho}}}) = {\Pi^{J}}(t_{\xi}) \quad \text{by \eqref{eq:tx0}}; \\[1mm]
& \iota(\pi_{{\bm{\chi}}}) = {\Pi^{K}}(t_{\zeta_{1}}), \quad 
\text{with} \quad \zeta_{1} = \sum_{i \in I} 
\chi^{(i)}_{1} \alpha_{i}^{\vee} \in Q^{\vee}_{I \setminus {K}} \quad \text{by \eqref{eq:par0}}. \end{aligned}$$ We set $\gamma:=\sum_{i \in {J}\setminus {K}} 
\chi^{(i)}_{1} \alpha_{i}^{\vee} \in {Q_{J}^{\vee}}$. Then it follows from that $\xi+\gamma \ge \zeta_{1}$ since $I \setminus K = (I \setminus (J \cup K)) \sqcup (J \setminus K)$. Hence we deduce from Lemma \[lem:SiB\](2) that $x:=t_{\xi+\gamma} {\succeq}t_{\zeta_1}=:y$ in $W_{{\mathrm{af}}}$. It is obvious that ${\Pi^{K}}(y) = \iota(\pi_{{\bm{\chi}}})$. Also, since $\gamma \in {Q_{J}^{\vee}}$, we see from that ${\Pi^{J}}(x) = {\Pi^{J}}(t_{\xi}) = \kappa(t_{\xi} \cdot \pi_{{\bm{\rho}}})$. Thus, $x$ and $y$ satisfy condition for $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$, which implies . This proves the proposition.

\[prop:conn\] Each connected component of ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ contains a unique element of the form[:]{} $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ for some ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, ${\bm{\chi}}\in {\mathop{\rm Par}\nolimits}(\mu)$, and $\xi \in Q^{\vee}_{I \setminus ({J}\cup {K})}$ satisfying condition in Proposition \[prop:tx\]. Therefore, there exists a one-to-one correspondence between the set ${\mathop{\rm Conn}\nolimits}({\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu))$ of connected components of ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ and the set of triples $({\bm{\rho}},\,{\bm{\chi}},\,\xi) \in {\mathop{\rm Par}\nolimits}(\lambda) \times {\mathop{\rm Par}\nolimits}(\mu) \times 
Q^{\vee}_{I \setminus ({J}\cup {K})}$ satisfying condition in Proposition \[prop:tx\].

The “existence” part follows from Lemma \[lem:conn\] and Proposition \[prop:tx\]. Hence it suffices to prove the “uniqueness” part. Let $({\bm{\rho}},\,{\bm{\chi}},\,\xi)$ and $({\bm{\rho}}',\,{\bm{\chi}}',\,\xi')$ be elements in ${\mathop{\rm Par}\nolimits}(\lambda) \times {\mathop{\rm Par}\nolimits}(\mu) \times 
Q^{\vee}_{I \setminus ({J}\cup {K})}$ satisfying condition in Proposition \[prop:tx\], and suppose that $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ and $(t_{\xi'} \cdot \pi_{{\bm{\rho}}'}) \otimes \pi_{{\bm{\chi}}'}$ are contained in the same connected component of ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$. Then there exists a monomial $X$ in root operators such that $X ( (t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}} ) = 
(t_{\xi'} \cdot \pi_{{\bm{\rho}}'}) \otimes \pi_{{\bm{\chi}}'}$. By the tensor product rule for crystals, we see that $X ( (t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}} ) 
= X_{1} (t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes X_{2}\pi_{{\bm{\chi}}}$ for some monomials $X_{1}$, $X_{2}$ in root operators. Then we have $X_{1} (t_{\xi} \cdot \pi_{{\bm{\rho}}}) = t_{\xi'} \cdot \pi_{{\bm{\rho}}'}$, which implies that $t_{\xi} \cdot \pi_{{\bm{\rho}}}$ and $t_{\xi'} \cdot \pi_{{\bm{\rho}}'}$ are contained in the same connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$, and hence so are $\pi_{{\bm{\rho}}}$ and $\pi_{{\bm{\rho}}'}$. Therefore, by the uniqueness of an element of the form in a connected component of ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ (see Section \[subsec:conn\]), we deduce that ${\bm{\rho}}= {\bm{\rho}}'$. Similarly, we obtain ${\bm{\chi}}={\bm{\chi}}'$. Suppose, for a contradiction, that $\xi \ne \xi'$; we may assume that for some $k \in I \setminus ({J}\cup {K})$, the coefficient of $\alpha_{k}^{\vee}$ in $\xi$ is greater than that in $\xi'$, i.e., the coefficient of $\alpha_{k}^{\vee}$ in $\xi'-\xi$ is a negative integer. Because $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ and $(t_{\xi'} \cdot \pi_{{\bm{\rho}}'}) \otimes \pi_{{\bm{\chi}}'} = 
(t_{\xi'} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ are contained in the same connected component, there exists a monomial $Y$ in root operators such that $Y \bigl((t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}\bigr) 
   = (t_{\xi'} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}
   = (t_{\xi+(\xi'-\xi)} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$. Here, the same argument as in the proof of [@INS Lemma 7.1.4] (or, as in the proof of [@INS Proposition 7.1.2]) shows that $$Y^{N} \Bigl((t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}\Bigr) 
 = (t_{\xi+N(\xi'-\xi)} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}
\quad \text{for all $N \in {\mathbb{Z}}_{\ge 1}$}.$$ Since this element is contained in ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ for all $N \in {\mathbb{Z}}_{\ge 1}$ by Proposition \[prop:SMT1\], it follows from Proposition \[prop:tx\] that the coefficient of $\alpha_{k}^{\vee}$ in $\xi+N(\xi'-\xi)$ is greater than or equal to $\chi^{(k)}_{1}$ for all $N \ge 1$. This contradicts the fact that the coefficient of $\alpha_{k}^{\vee}$ in $\xi'-\xi$ is a negative integer. This proves the proposition.

Now, we write $\lambda$ and $\mu$ as: $\lambda=\sum_{i \in I} m_{i} {\varpi}_{i}$ and $\mu=\sum_{i \in I} n_{i} {\varpi}_{i}$. For $({\bm{\rho}},\,{\bm{\chi}},\,\xi) \in {\mathop{\rm Par}\nolimits}(\lambda) \times {\mathop{\rm Par}\nolimits}(\mu) \times 
 Q^{\vee}_{I \setminus ({J}\cup {K})}$ satisfying , define ${\bm{\omega}}= ({\omega}^{(i)})_{i \in I} \in {\mathop{\rm Par}\nolimits}(\lambda+\mu)$ as follows. Write ${\bm{\rho}}$, ${\bm{\chi}}$, and $\xi$ as: $$\begin{aligned}
& {\bm{\rho}}= (\rho^{(i)})_{i \in I}, \quad \text{with} \quad
  \rho^{(i)}=(\rho^{(i)}_{1} \ge \cdots \ge \rho^{(i)}_{m_i-1} \ge 0) \quad
  \text{for $i \in I$}, \\
& {\bm{\chi}}= (\chi^{(i)})_{i \in I}, \quad \text{with} \quad
  \chi^{(i)}=(\chi^{(i)}_{1} \ge \cdots \ge \chi^{(i)}_{n_i-1} \ge 0) \quad
  \text{for $i \in I$}, \\
& \xi=\sum_{i \in I \setminus ({J}\cup {K})} 
  c_{i}\alpha_{i}^{\vee}; \quad \text{recall that $c_{i} \ge \chi^{(i)}_{1}$ 
  for all $i \in I \setminus ({J}\cup {K})$}.\end{aligned}$$ Let $i \in I$.

-   If $i \in {J}\cap {K}$ (note that $m_{i}=n_{i}=0$), we set ${\omega}^{(i)} := \emptyset$;

-   if $i \in {J}\setminus {K}$ (note that $m_{i}=0$), we set ${\omega}^{(i)}:=\chi^{(i)}$, which is a partition of length less than $n_{i} = 0+n_{i}= m_{i}+n_{i}$;

-   if $i \in {K}\setminus {J}$ (note that $n_{i}=0$), we set ${\omega}^{(i)}:=\rho^{(i)}$, which is a partition of length less than $m_{i} =m_{i}+0= m_{i}+n_{i}$;

-   if $i \in I \setminus ({J}\cup {K})$, we set $${\omega}^{(i)}=
    (\rho^{(i)}_{1}+c_{i} \ge \cdots \ge \rho^{(i)}_{m_i-1}+c_{i} \ge c_{i} \ge 
    \chi^{(i)}_{1} \ge \cdots \ge \chi^{(i)}_{n_i-1}),$$ which is a partition of length less than $(m_{i}-1)+1+(n_{i}-1) + 1=m_{i}+n_{i}$; note that $|{\omega}^{(i)}| = | \rho^{(i)} | + |\chi^{(i)} | + m_{i}c_{i}$.

It follows that ${\bm{\omega}}=({\omega}^{(i)})_{i \in I} \in {\mathop{\rm Par}\nolimits}(\lambda+\mu)$. Thus we obtain a map $\Theta$ from the set of those $({\bm{\rho}},\,{\bm{\chi}},\,\xi) \in {\mathop{\rm Par}\nolimits}(\lambda) \times {\mathop{\rm Par}\nolimits}(\mu) \times 
 Q^{\vee}_{I \setminus ({J}\cup {K})}$ satisfying to the set ${\mathop{\rm Par}\nolimits}(\lambda+\mu)$; we can easily deduce that the map $\Theta$ is bijective. Also, by direct calculation, we have $$\begin{aligned}
{\mathop{\rm wt}\nolimits}\Bigl( \underbrace{(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}}_{\text{satisfying \eqref{eq:tx2}}} \Bigr)
 & = {\mathop{\rm wt}\nolimits}(t_{\xi} \cdot \pi_{{\bm{\rho}}}) + {\mathop{\rm wt}\nolimits}(\pi_{{\bm{\chi}}}) 
   = t_{\xi}\bigl(\lambda-|{\bm{\rho}}|\delta\bigr) + \bigl(\mu-|{\bm{\chi}}|\delta\bigr) \\
 & = (\lambda+\mu) - \bigl( |{\bm{\rho}}| + {\langle \lambda,\,\xi \rangle} + |{\bm{\chi}}| \bigr)\delta \\[2mm]
 & = (\lambda+\mu) - 
   \left( |{\bm{\rho}}| + \sum_{i \in I \setminus (J \cup K)} m_{i}c_{i} + |{\bm{\chi}}| \right)\delta \\[2mm]
 & = (\lambda+\mu) - \bigl| \Theta({\bm{\rho}},\,{\bm{\chi}},\,\xi) \bigr|\delta.\end{aligned}$$

We claim that the connected component of ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ containing $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ is isomorphic, as a crystal, to $\bigl\{\Theta({\bm{\rho}},\,{\bm{\chi}},\,\xi)\bigr\} \otimes {\mathbb{S}^{\frac{\infty}{2}}}_{0}(\lambda+\mu)$, where ${\mathbb{S}^{\frac{\infty}{2}}}_{0}(\lambda+\mu)$ denotes the connected component of ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ containing $\pi_{\lambda} \otimes \pi_{\mu} = 
(e\,;\,0,1) \otimes (e\,;\,0,1) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. Indeed, let us consider the composite of the following bijections: $$\begin{aligned}
& {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\rho}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\chi}}}(\mu) 
  \stackrel{T_{-\xi} \otimes {\mathop{\rm id}\nolimits}}{\longrightarrow} 
  {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\rho}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\bm{\chi}}}(\mu) 
  \quad \text{(see \eqref{eq:Tx} and Remark~\ref{rem:Tx})} \\
& \quad \stackrel{\sim}{\rightarrow} 
   \bigl(\bigl\{{\bm{\rho}}\bigr\} \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda)\bigr) \otimes 
   \bigl(\bigl\{{\bm{\chi}}\bigr\} \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\mu)\bigr) \quad
   \text{(see the comment preceding \eqref{eq:isom})} \\
& \quad \stackrel{\sim}{\rightarrow}
   \bigl(\bigl\{{\bm{\rho}}\bigr\} \otimes \bigl\{{\bm{\chi}}\bigr\}\bigr) \otimes 
   \bigl({\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\mu)\bigr) \quad 
   \text{(by the tensor product rule for crystals)} \\
& \quad \stackrel{\sim}{\rightarrow}
   \bigl\{\Theta({\bm{\rho}},\,{\bm{\chi}},\,\xi)\bigr\} \otimes 
   \bigl({\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\mu)\bigr),\end{aligned}$$ where the last map sends $({\bm{\rho}}\otimes {\bm{\chi}}) \otimes (\pi \otimes \eta)$ to $\Theta({\bm{\rho}},\,{\bm{\chi}},\,\xi) \otimes (\pi \otimes \eta)$ for each $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda)$ and $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\mu)$. We deduce by and the tensor product rule for crystals that the composite of these bijections is an isomorphism of crystals, which sends $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ to $\Theta({\bm{\rho}},\,{\bm{\chi}},\,\xi) \otimes (\pi_{\lambda} \otimes \pi_{\mu})$. Therefore, the connected component of ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ containing $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}}$ is mapped to $\bigl\{\Theta({\bm{\rho}},\,{\bm{\chi}},\,\xi)\bigr\} \otimes {\mathbb{S}^{\frac{\infty}{2}}}_{0}(\lambda+\mu)$ under this isomorphism of crystals. It follows from Proposition \[prop:conn\] and the bijectivity of $\Theta$ that $$\label{eq:conn}
{\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu) \cong \bigsqcup_{{\bm{\omega}}\in {\mathop{\rm Par}\nolimits}(\lambda+\mu)} 
\bigl\{{\bm{\omega}}\bigr\} \otimes {\mathbb{S}^{\frac{\infty}{2}}}_{0}(\lambda+\mu).$$

\[prop:BN\] As crystals, ${\mathbb{S}^{\frac{\infty}{2}}}_{0}(\lambda+\mu) \cong {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda+\mu)$.

Write $\lambda$ and $\mu$ as: $\lambda = \sum_{i \in I} m_{i}{\varpi}_{i}$ with $m_{i} \in {\mathbb{Z}}_{\ge 0}$, and $\mu = \sum_{i \in I} n_{i}{\varpi}_{i}$ with $n_{i} \in {\mathbb{Z}}_{\ge 0}$, respectively. We know from [@Kas02 Conjecture 13.1(iii)], which is proved in [@BN Remark 4.17], that there exists an isomorphism ${\mathcal{B}}(\lambda+\mu) \stackrel{\sim}{\rightarrow} 
\bigotimes_{i \in I} {\mathcal{B}}((m_{i}+n_{i}){\varpi}_{i})$ of crystals, which maps $u_{\lambda+\mu}$ to $\bigotimes_{i \in I} u_{(m_{i}+n_{i}){\varpi}_{i}}$; the restriction of this isomorphism to ${\mathcal{B}}_{0}(\lambda+\mu) \subset {\mathcal{B}}(\lambda+\mu)$ gives an embedding ${\mathcal{B}}_{0}(\lambda+\mu) \hookrightarrow 
\bigotimes_{i \in I} {\mathcal{B}}_{0}((m_{i}+n_{i}){\varpi}_{i})$ of crystals. Also, we know from [@Kas02 Conjecture 13.2(iii)], which is proved in [@BN Remark 4.17], that for each $i \in I$, there exists an embedding ${\mathcal{B}}_{0}((m_{i}+n_{i}){\varpi}_{i}) \hookrightarrow 
{\mathcal{B}}({\varpi}_{i})^{\otimes (m_{i}+n_{i})}$ of crystals, which maps $u_{(m_{i}+n_{i}){\varpi}_{i}}$ to $u_{{\varpi}_{i}}^{\otimes (m_{i}+n_{i})}$; recall from [@Kas02 Proposition 5.4] that ${\mathcal{B}}({\varpi}_{i})$ is connected. Thus we obtain an embedding $${\mathcal{B}}_{0}(\lambda+\mu) \hookrightarrow 
\bigotimes_{i \in I} {\mathcal{B}}_{0}((m_{i}+n_{i}){\varpi}_{i})
\hookrightarrow 
\bigotimes_{i \in I} {\mathcal{B}}({\varpi}_{i})^{\otimes (m_{i}+n_{i})}$$ of crystals, which maps $u_{\lambda+\mu}$ to $\bigotimes_{i \in I} u_{{\varpi}_{i}}^{\otimes (m_{i}+n_{i})}$. Here, we recall from [@Kas02 Sect. 10] that for each $j,\,k \in I$, there exists an isomorphism ${\mathcal{B}}({\varpi}_{j}) \otimes {\mathcal{B}}({\varpi}_{k}) 
\stackrel{\sim}{\rightarrow} 
{\mathcal{B}}({\varpi}_{k}) \otimes {\mathcal{B}}({\varpi}_{j})$ of crystals, which maps $u_{{\varpi}_{j}} \otimes u_{{\varpi}_{k}}$ to $u_{{\varpi}_{k}} \otimes u_{{\varpi}_{j}}$. Hence we obtain an isomorphism of crystals $$\bigotimes_{i \in I} {\mathcal{B}}({\varpi}_{i})^{\otimes (m_{i}+n_{i})} 
\stackrel{\sim}{\rightarrow} 
\left(\bigotimes_{i \in I} {\mathcal{B}}({\varpi}_{i})^{\otimes m_{i}}\right)
\otimes 
\left(\bigotimes_{i \in I} {\mathcal{B}}({\varpi}_{i})^{\otimes n_{i}}\right) = : {\mathcal{B}},$$ which maps $\bigotimes_{i \in I} u_{{\varpi}_{i}}^{\otimes (m_{i}+n_{i})}$ to $\left(\bigotimes_{i \in I} u_{{\varpi}_{i}}^{\otimes m_{i}}\right) \otimes 
\left(\bigotimes_{i \in I} u_{{\varpi}_{i}}^{\otimes n_{i}}\right)=:b$. From these, we obtain an embedding ${\mathcal{B}}_{0}(\lambda+\mu) \hookrightarrow {\mathcal{B}}$ of crystals, which maps $u_{\lambda+\mu}$ to $b$. Similarly, we obtain an embedding ${\mathcal{B}}_{0}(\lambda) \otimes {\mathcal{B}}_{0}(\mu) \hookrightarrow {\mathcal{B}}$ of crystals, which maps $u_{\lambda} \otimes u_{\mu}$ to $b$. Consequently, there exists an isomorphism of crystals from ${\mathcal{B}}_{0}(\lambda+\mu)$ to the connected component (denoted by ${\mathcal{S}}_{0}(\lambda+\mu)$) of ${\mathcal{B}}_{0}(\lambda) \otimes {\mathcal{B}}_{0}(\mu)$ containing $u_{\lambda} \otimes u_{\mu}$, which maps $u_{\lambda+\mu}$ to $u_{\lambda} \otimes u_{\mu}$. Now, by Theorem \[thm:isom\], we have an isomorphism ${\mathcal{B}}_{0}(\lambda+\mu) \stackrel{\sim}{\rightarrow} {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda+\mu)$ of crystals, which maps $u_{\lambda+\mu}$ to $\pi_{\lambda+\mu}$. In addition, we have an isomorphism ${\mathcal{B}}_{0}(\lambda) \otimes {\mathcal{B}}_{0}(\mu) \stackrel{\sim}{\rightarrow} 
{\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\mu)$ of crystals, which maps $u_{\lambda} \otimes u_{\mu}$ to $\pi_{\lambda} \otimes \pi_{\mu}$; by restriction, we obtain an isomorphism of crystals from ${\mathcal{S}}_{0}(\lambda+\mu)$ to ${\mathbb{S}^{\frac{\infty}{2}}}_{0}(\lambda+\mu)$. Summarizing, we obtain the following isomorphism of crystals: $$\begin{array}{ccccccc}
{\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda+\mu) & \stackrel{\sim}{\rightarrow} &
{\mathcal{B}}_{0}(\lambda+\mu) & \stackrel{\sim}{\rightarrow} &
{\mathcal{S}}_{0}(\lambda+\mu) & \stackrel{\sim}{\rightarrow} &
{\mathbb{S}^{\frac{\infty}{2}}}_{0}(\lambda+\mu), \\[3mm]
\pi_{\lambda+\mu} & \mapsto & 
u_{\lambda+\mu} & \mapsto & 
u_{\lambda} \otimes u_{\mu} & \mapsto &
\pi_{\lambda} \otimes \pi_{\mu}.
\end{array}
$$ This proves the proposition.

By using , Proposition \[prop:BN\], and (with $\lambda$ replaced by $\lambda+\mu$), we conclude that $$\begin{aligned}
{\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu) & \cong 
\bigsqcup_{{\bm{\omega}}\in {\mathop{\rm Par}\nolimits}(\lambda+\mu)} 
\bigl\{{\bm{\omega}}\bigr\} \otimes {\mathbb{S}^{\frac{\infty}{2}}}_{0}(\lambda+\mu) \\[3mm]
& \cong 
\bigsqcup_{{\bm{\omega}}\in {\mathop{\rm Par}\nolimits}(\lambda+\mu)} 
\bigl\{{\bm{\omega}}\bigr\} \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}_{0}(\lambda+\mu)
\cong {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu)\end{aligned}$$ as crystals. This completes the proof of Theorem \[thm:SMT\].

\[cor:Sext\] For each ${\bm{\omega}}\in {\mathop{\rm Par}\nolimits}(\lambda+\mu)$, the element $\pi_{{\bm{\omega}}} \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu)$ is mapped to $(t_{\xi} \cdot \pi_{{\bm{\rho}}}) \otimes \pi_{{\bm{\chi}}} \in {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ for some $\xi \in Q^{\vee}_{I \setminus ({J}\cup {K})}$ and ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, ${\bm{\chi}}\in {\mathop{\rm Par}\nolimits}(\mu)$ satisfying under the isomorphism ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu) \cong {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ of crystals in Theorem \[thm:SMT\].

Proof of Propositions \[prop:DC\] and \[prop:DC2\]. {#sec:prf-DC}
===================================================

Recall that $\lambda,\,\mu \in P^{+}$, and that ${J}= \bigl\{ i \in I \mid {\langle \lambda,\,\alpha_{i}^{\vee} \rangle}=0\bigr\}$, ${K}= \bigl\{ i \in I \mid {\langle \mu,\,\alpha_{i}^{\vee} \rangle}=0\bigr\}$.

Proof of Proposition \[prop:DC\]. {#subsec:prf-DC}
---------------------------------

The “if” part is obvious from the definition of defining chains and condition . Let us prove the “only if” part. Assume that $\pi \otimes \eta \in {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$, and write $\pi$ and $\eta$ as: $\pi=(x_{1},\,\dots,\,x_{s}\,;\,{\mathbf{a}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta=(y_{1},\,\dots,\,y_{p}\,;\,{\mathbf{b}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$, respectively. It follows from that there exist $x_{s}',\,y_{1}' \in W_{{\mathrm{af}}}$ such that $x_{s}' {\succeq}y_{1}'$ in $W_{{\mathrm{af}}}$, and such that ${\Pi^{J}}(x_{s}')=x_{s}$, ${\Pi^{K}}(y_{1}')=y_{1}$; we write $x_{s}'=x_{s}z_{1}$ for some $z_{1} \in ({W_{J}})_{{\mathrm{af}}}$, and $y_{1}'=y_{1}z_{2}$ for some $z_{2} \in ({W_{K}})_{{\mathrm{af}}}$. Now we set $$\begin{cases}
 x_{u}':=x_{u}z_{1} & \text{for $1 \le u \le s$}, \\[1.5mm]
 y_{q}':=y_{q}z_{2} & \text{for $1 \le q \le p$}. 
\end{cases}
$$ Because $x_{1} {\succeq}x_{2} {\succeq}\cdots {\succeq}x_{s}$ in $({W^{J}})_{{\mathrm{af}}}$ by the definition of semi-infinite LS paths, it follows from Lemma \[lem:SiL\] that $x_{1}' {\succeq}x_{2}' {\succeq}\cdots {\succeq}x_{s}'$ in $W_{{\mathrm{af}}}$. Similarly, we see that $y_{1}' {\succeq}y_{2}' {\succeq}\cdots {\succeq}y_{p}'$ in $W_{{\mathrm{af}}}$. Combining these inequalities with the inequality $x_{s}' {\succeq}y_{1}'$, we obtain $x_{1}' {\succeq}\cdots {\succeq}x_{s}' {\succeq}y_{1}' {\succeq}\cdots {\succeq}y_{p}'$ in $W_{{\mathrm{af}}}$. Since ${\Pi^{J}}(x_{u}')=x_{u}$ for all $1 \le u \le s$, and ${\Pi^{J}}(y_{q}')=y_{q}$ for all $1 \le q \le p$, the sequence $x_{1}',\,\dots,\,x_{s}',\,y_{1}',\,\dots,\,y_{p}'$ is a defining chain for $\pi \otimes \eta$. This completes the proof of Proposition \[prop:DC\].

Proof of Proposition \[prop:DC2\]. {#subsec:prf-DC2}
----------------------------------

We write $\pi$ and $\eta$ as: $\pi=(x_{1},\,\dots,\,x_{s}\,;\,\,{\mathbf{a}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta=(y_{1},\,\dots,\,y_{p}\,;\,\,{\mathbf{b}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$, respectively. First, we prove the “only if” part. Take an (arbitrary) defining chain $x_{1}',\,\dots,\,x_{s}',\,y_{1}',\,\dots,\,y_{p}' \in W_{{\mathrm{af}}}$ for $\pi \otimes \eta$. It suffices to show the following claim.

\[c:DC2-1\] Let $x \in W_{{\mathrm{af}}}$ be such that $y_{p}' {\succeq}x$[;]{} note that $\kappa(\eta)=y_{p}={\Pi^{K}}(y_{p}') {\succeq}{\Pi^{K}}(x)$ by Lemma \[lem:611\]. Then, $\kappa(\pi) {\succeq}{\Pi^{J}}({\iota(\eta,\,x)})$.

[*Proof of Claim \[c:DC2-1\].*]{} For the given $x \in W_{{\mathrm{af}}}$, we define ${\widetilde{y}}_{p},\,{\widetilde{y}}_{p-1},\,\dots,\,{\widetilde{y}}_{1} = {\iota(\eta,\,x)}$ as in . We show by descending induction on $q$ that $$\label{eq:DC2-1}
y_{q}' {\succeq}{\widetilde{y}}_{q} \quad \text{for all $1 \le q \le p$}.$$ Because $y_{p}' {\succeq}x$ and ${\Pi^{K}}(y_{p}')=y_{p}$, it follows that $y_{p}' \in {\mathrm{Lift}_{{\succeq}x}(y_{p})}$, and hence $y_{p}' {\succeq}\min {\mathrm{Lift}_{{\succeq}x}(y_{p})} = {\widetilde{y}}_{p}$. Assume that $q < p$. Since $y_{q}' {\succeq}y_{q+1}'$ by the definition of defining chains, and since $y_{q+1}' {\succeq}{\widetilde{y}}_{q+1}$ by our induction hypothesis, we obtain $y_{q}' {\succeq}{\widetilde{y}}_{q+1}$. In addition, we have ${\Pi^{K}}(y_{q}')=y_{q}$. From these, we deduce that $y_{q}' \in {\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{q+1}}(y_{q})}$, and hence $y_{q}' {\succeq}\min {\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{q+1}}(y_{q})}
= {\widetilde{y}}_{q}$. Thus we have shown . Hence we have $x_{s}' {\succeq}y_{1}' {\succeq}{\widetilde{y}}_{1} = {\iota(\eta,\,x)}$ by the assumption. Therefore, it follows from Lemma \[lem:611\] that $$\label{eq:kappa-Deo}
\kappa(\pi) = x_{s} = {\Pi^{J}}(x_{s}') {\succeq}{\Pi^{J}}({\iota(\eta,\,x)}).$$ This proves Claim \[c:DC2-1\].

Next, we prove the “if” part. We define ${\widetilde{y}}_{p}$, ${\widetilde{y}}_{p-1}$, $\dots$, ${\widetilde{y}}_{1} = {\iota(\eta,\,x)}$ as in . By the definitions, we have $$\label{eq:DC2-2}
\begin{cases}
{\iota(\eta,\,x)}
 = {\widetilde{y}}_{1} {\succeq}{\widetilde{y}}_{2} {\succeq}\cdots {\succeq}{\widetilde{y}}_{p} {\succeq}x, & \\[1mm]
{\Pi^{K}}({\widetilde{y}}_{q}) = y_{q} 
\quad \text{for $1 \le q \le p$}.
\end{cases}
$$ Write ${\iota(\eta,\,x)} \in W_{{\mathrm{af}}}$ as: ${\iota(\eta,\,x)} = {\Pi^{J}}({\iota(\eta,\,x)})z$ with $z \in ({W_{J}})_{{\mathrm{af}}}$. Since $\kappa(\pi) {\succeq}{\Pi^{J}}({\iota(\eta,\,x)})$ by the assumption, we deduce from Lemma \[lem:SiL\] that $x_{s}':=\kappa(\pi)z {\succeq}{\Pi^{J}}({\iota(\eta,\,x)})z = {\iota(\eta,\,x)} = {\widetilde{y}}_{1}$. Similarly, if we set $x_{u}':=x_{u}z$ for $1 \le u \le s$, then we have $$\label{eq:DC2-3}
\begin{cases}
x_{1}' {\succeq}x_{2}' {\succeq}\cdots {\succeq}x_{s}' \ 
 ({\succeq}{\iota(\eta,\,x)} = {\widetilde{y}}_{1}), & \\[1mm]
{\Pi^{J}}(x_{u}') = x_{u} 
\quad \text{for $1 \le u \le s$}.
\end{cases}
$$ Concatenating the sequences in and , we obtain a defining chain $$\label{eq:minDC}
x_{1}' {\succeq}x_{2}' {\succeq}\cdots {\succeq}x_{s}' {\succeq}{\widetilde{y}}_{1} {\succeq}{\widetilde{y}}_{2} {\succeq}\cdots {\succeq}{\widetilde{y}}_{p}$$ for $\pi \otimes \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. This completes the proof of Proposition \[prop:DC2\].

Proof of Theorem \[thm:Dem\]. {#sec:prf-Dem}
=============================

Recall that $\lambda,\,\mu \in P^{+}$, and that ${J}= \bigl\{ i \in I \mid {\langle \lambda,\,\alpha_{i}^{\vee} \rangle}=0\bigr\}$, ${K}= \bigl\{ i \in I \mid {\langle \mu,\,\alpha_{i}^{\vee} \rangle}=0\bigr\}$, and $S = \bigl\{ i \in I \mid {\langle \lambda+\mu,\,\alpha_{i}^{\vee} \rangle}=0\bigr\} = {J}\cap {K}$.

Proof of $\Leftrightarrow$ . {#subsec:23}
----------------------------

We prove the implication $\Rightarrow$ . Let $x_{1}',\,\dots,\,x_{s}',\,y_{1}',\,\dots,\,y_{p}'=:y$ be a defining chain for $\pi \otimes \eta$ such that ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$. Write $x$ as: $x = {\Pi^{S}}(x)z$ for some $z \in ({W_{S}})_{{\mathrm{af}}}$; note that $({W_{S}})_{{\mathrm{af}}} \subset ({W_{J}})_{{\mathrm{af}}} \cap ({W_{K}})_{{\mathrm{af}}}$. We deduce from Lemma \[lem:SiL\] that $${\Pi^{S}}(x_{1}')z,\,\dots,\,{\Pi^{S}}(x_{s}')z,\,
{\Pi^{S}}(y_{1}')z,\,\dots,\,{\Pi^{S}}(y_{p}')={\Pi^{S}}(y)z$$ is also a defining chain for $\pi \otimes \eta$ such that ${\Pi^{S}}({\Pi^{S}}(y)z) = {\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$. Hence we may assume from the beginning that $y {\succeq}x$. We deduce from Lemma \[lem:611\] that $\kappa(\eta) = {\Pi^{K}}(y) {\succeq}{\Pi^{K}}(x)$. Also, the inequality $\kappa(\pi) {\succeq}{\Pi^{J}}({\iota(\eta,\,x)})$ was shown in Claim \[c:DC2-1\] in the proof of Proposition \[prop:DC2\].

The implication  $\Rightarrow$ follows from the fact that the defining chain for $\pi \otimes \eta$ satisfies the desired condition in .

Proof of $\Leftrightarrow$ . {#subsec:12}
----------------------------

Let ${\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$ denote the set of elements in ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$ satisfying condition (or equivalently, condition ); by Proposition \[prop:DC\], we see that ${\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu) \subset {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$.

\[lem:Dem1\]

The set ${\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu) \cup \{{\bm{0}}\}$ is stable under the action of the root operator $f_{i}$ for all $i \in I_{{\mathrm{af}}}$.

The set ${\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu) \cup \{{\bm{0}}\}$ is stable under the action of the root operator $e_{i}$ for those $i \in I_{{\mathrm{af}}}$ such that ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \ge 0$.

Let $i \in I_{{\mathrm{af}}}$ be such that ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \ge 0$. Then, $$\label{eq:Dem1-3a}
{\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu) = \bigl\{e_{i}^{n}(\pi \otimes \eta) \mid 
\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu),\,n \ge 0\bigr\} \setminus \{{\bm{0}}\}.$$

\(1)  Let $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$, and let $i \in I_{{\mathrm{af}}}$; we may assume that $f_{i}(\pi \otimes \eta) \ne {\bm{0}}$. We give a proof only for the case that $f_{i}(\pi \otimes \eta) = \pi \otimes f_{i}\eta$; the proof for the case that $f_{i}(\pi \otimes \eta) = f_{i}\pi \otimes \eta$ is similar. We write $\pi$ and $\eta$ as: $\pi=(x_{1},\,\dots,\,x_{s}\,;\,{\mathbf{a}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta=(y_{1},\,\dots,\,y_{p}\,;\,{\mathbf{b}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$, respectively. Let $x_{1}' {\succeq}\cdots {\succeq}x_{s}' {\succeq}y_{1}' {\succeq}\cdots {\succeq}y_{p}'$ be a defining chain for $\pi \otimes \eta$ such that ${\Pi^{S}}(y_{p}') {\succeq}{\Pi^{S}}(x)$. Take $0 \le t_{0} < t_{1} \le 1$ as in (with $\pi$ replaced by $\eta$); note that $H^{\eta}_{i}(t)$ is strictly increasing on the interval $[t_{0},\,t_{1}]$. We see from that $f_{i}\eta$ is of the form: $$f_{i} \eta := ( y_{1},\,\ldots,\,y_{k},\,s_{i}y_{k+1},\,\dots,\,
  s_{i} y_{m},\,s_{i} y_{m+1},\,y_{m+1},\,\ldots,\,y_{p}\,;\,{\mathbf{b}}')$$ for some $0 \le k \le m \le p-1$ and some increasing sequence ${\mathbf{b}}'$ of rational numbers in $[0,1]$. Here, since $H^{\eta}_{i}(t)$ is strictly increasing on the interval $[t_{0},\,t_{1}]$, it follows that ${\langle y_{n}\mu,\,\alpha_{i}^{\vee} \rangle} > 0$, and hence that $y_{n}^{-1}\alpha_{i} \in (\Delta^{+} \setminus {\Delta_{K}}^{+}) + {\mathbb{Z}}\delta$ for all $k+1 \le n \le m+1$. Hence we deduce that $$\label{eq:yn}
(y_{n}')^{-1}\alpha_{i} 
\in (\Delta^{+} \setminus {\Delta_{K}}^{+}) + {\mathbb{Z}}\delta \subset \Delta^{+} + {\mathbb{Z}}\delta
\quad \text{for all $k+1 \le n \le m+1$}
$$ since $y_{n}' = y_{n}z_{n}$ for some $z_{n} \in ({W_{K}})_{{\mathrm{af}}}$. Therefore, it follows from Lemma \[lem:dmd\](3) (applied to the case $J=\emptyset$) that $s_{i}y_{k+1}' {\succeq}\cdots {\succeq}s_{i}y_{m+1}'$. Also, we see from that $s_{i}y_{m+1}' {\succeq}y_{m+1}'$. Thus we obtain $$\label{eq:Dem1-0}
s_{i}y_{k+1}' {\succeq}\cdots {\succeq}s_{i}y_{m+1}' {\succeq}y_{m+1}' {\succeq}\cdots {\succeq}y_{p}';$$ note that ${\Pi^{K}}(s_{i}y_{n}') = s_{i}y_{n}$ for all $k+1 \le n \le m+1$ by Lemma \[lem:si\] since $s_{i}y_{n} \in ({W^{K}})_{{\mathrm{af}}}$. If $t_{1} \ne 1$, then $\kappa(f_{i}\eta) = \kappa(\eta) = y_{p}$, and the final element of the sequence is $y_{p}'$, which satisfies ${\Pi^{S}}(y_{p}') {\succeq}{\Pi^{S}}(x)$ by our assumption. If $t_{1} = 1$, then $m+1=p$, $\kappa(f_{i}\eta) = s_{i}\kappa(\eta) = s_{i}y_{p}$, and the final element of the sequence is $s_{i}y_{m+1}' = s_{i}y_{p}'$. Since $s_{i}y_{m+1}' {\succeq}y_{m+1}'$ as shown above, we deduce from Lemma \[lem:611\] that ${\Pi^{S}}(s_{i}y_{p}') = {\Pi^{S}}(s_{i}y_{m+1}') 
{\succeq}{\Pi^{S}}(y_{m+1}') = {\Pi^{S}}(y_{p}') {\succeq}{\Pi^{S}}(x)$. In what follows, we will give a defining chain for $f_{i}(\pi \otimes \eta) = \pi \otimes (f_{i}\eta)$ in which the sequence lies at the tail.

#### Case 1.

Assume that the set $\bigl\{1 \le n \le k \mid {\langle y_{n}\mu,\,\alpha_{i}^{\vee} \rangle} \ne 0\bigr\}$ is nonempty, and let $k_{0}$ be the maximum element of this set. Because the function $H^{\eta}_{i}(t)$ attains its minimum value $m^{\eta}_{i}$ at $t=t_{0}$, it follows that ${\langle y_{k}\mu,\,\alpha_{i}^{\vee} \rangle} = 
{\langle y_{k-1}\mu,\,\alpha_{i}^{\vee} \rangle} = \cdots = 
{\langle y_{k_{0}+1}\mu,\,\alpha_{i}^{\vee} \rangle} = 0$, and ${\langle y_{k_{0}}\mu,\,\alpha_{i}^{\vee} \rangle} < 0$, which implies that $y_{n}^{-1}\alpha_{i} \in {\Delta_{K}}+ {\mathbb{Z}}\delta$ for all $k_{0}+1 \le n \le k$, and $y_{k_{0}}^{-1}\alpha_{i} \in -(\Delta^{+} \setminus {\Delta_{K}}^{+}) + {\mathbb{Z}}\delta$. Hence we deduce that $(y_{n}')^{-1}\alpha_{i} \in {\Delta_{K}}+ {\mathbb{Z}}\delta$ for all $k_{0}+1 \le n \le k$, and $(y_{k_{0}}')^{-1}\alpha_{i} \in -(\Delta^{+} \setminus {\Delta_{K}}^{+}) + {\mathbb{Z}}\delta$. Therefore, there exists $k_{0} \le k_{1} \le k$ such that $(y_{n}')^{-1}\alpha_{i} \in {\Delta_{K}}^{+} + {\mathbb{Z}}\delta$ for all $k_{1}+1 \le n \le k$, and such that $(y_{k_{1}}')^{-1}\alpha_{i} \in -\Delta^{+} + {\mathbb{Z}}\delta$; recall from that $(y_{k+1}')^{-1}\alpha_{i} \in \Delta^{+} + {\mathbb{Z}}\delta$. Hence, in this case, we deduce from Lemma \[lem:dmd\](1) and (3) that $y_{k_{1}}' {\succeq}s_{i}y_{k_{1}+1}' {\succeq}\cdots {\succeq}s_{i}y_{k}' {\succeq}s_{i}y_{k+1}'$; since $(y_{n}')^{-1}\alpha_{i} \in {\Delta_{K}}+ {\mathbb{Z}}\delta$ for all $k_{1}+1 \le n \le k$, we see by Remark \[rem:si\] that ${\Pi^{K}}(s_{i}y_{n}') = {\Pi^{K}}(y_{n}') = y_{n}$ for all $k_{1}+1 \le n \le k$. Thus, we obtain a defining chain $$\label{eq:Dem1-1}
x_{1}' {\succeq}\cdots {\succeq}x_{s}' {\succeq}y_{1}' {\succeq}\cdots {\succeq}y_{k_{1}}' {\succeq}s_{i}y_{k_{1}+1}' {\succeq}\cdots {\succeq}s_{i}y_{m+1}' 
{\succeq}y_{m+1}' {\succeq}\cdots {\succeq}y_{p}'$$ for $f_{i}(\pi \otimes \eta) = \pi \otimes (f_{i}\eta)$.

#### Case 2.

Assume that the set $\bigl\{1 \le n \le k \mid {\langle y_{n}\mu,\,\alpha_{i}^{\vee} \rangle} \ne 0\bigr\}$ is empty, i.e., ${\langle y_{n}\mu,\,\alpha_{i}^{\vee} \rangle} = 0$ for all $1 \le n \le k$; note that $(y_{n}')^{-1}\alpha_{i} \in {\Delta_{K}}+ {\mathbb{Z}}\delta$ for all $1 \le n \le k$. If there exists $1 \le k_{1} \le k$ such that $(y_{n}')^{-1}\alpha_{i} \in {\Delta_{K}}^{+} + {\mathbb{Z}}\delta$ for all $k_{1}+1 \le n \le k$, and $(y_{k_{1}}')^{-1}\alpha_{i} \in -{\Delta_{K}}^{+} + {\mathbb{Z}}\delta$, then we obtain a defining chain of the form for $f_{i}(\pi \otimes \eta) = \pi \otimes (f_{i}\eta)$ in exactly the same way as in Case 1. Hence we may assume that $(y_{n}')^{-1}\alpha_{i} \in {\Delta_{K}}^{+} + {\mathbb{Z}}\delta$ for all $1 \le n \le k$. It follows from Lemma \[lem:dmd\](3) and that $$\label{eq:Dem1-2}
s_{i}y_{1}' {\succeq}\cdots {\succeq}s_{i}y_{k}' {\succeq}s_{i}y_{k+1}' {\succeq}\cdots {\succeq}s_{i}y_{m+1}' 
{\succeq}y_{m+1}' {\succeq}\cdots {\succeq}y_{p}';$$ note that by Remark \[rem:si\], ${\Pi^{K}}(s_{i}y_{n}') = {\Pi^{K}}(y_{n}') = y_{n}$ for all $1 \le n \le k$. Now, we define $u_{0}$ to be the maximum element of the set $\bigl\{1 \le u \le s \mid {\langle x_{u}\lambda,\,\alpha_{i}^{\vee} \rangle} \ne 0\bigr\} \cup \{0\}$. We claim that if $u_{0} \ge 1$, then ${\langle x_{s}\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle x_{s-1}\lambda,\,\alpha_{i}^{\vee} \rangle} = \cdots = 
{\langle x_{u_{0}+1}\lambda,\,\alpha_{i}^{\vee} \rangle} = 0$, and that if $u_{0} \ge 1$, then ${\langle x_{u_{0}}\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$; this would imply that $x_{u}^{-1}\alpha_{i} \in {\Delta_{J}}+ {\mathbb{Z}}\delta$ for all $u_{0}+1 \le u \le s$, and that if $u_{0} \ge 1$, then $x_{u_{0}}^{-1}\alpha_{i} \in -(\Delta^{+} \setminus {\Delta_{J}}^{+}) + {\mathbb{Z}}\delta$. Indeed, since ${\langle y_{n}\mu,\,\alpha_{i}^{\vee} \rangle} = 0$ for all $1 \le n \le k$ by our assumption, we see that $H^{\eta}_{i}(t)$ is identically zero on the interval $[0,\,t_{0}]$, and hence $m^{\eta}_{i} = 0$, from which it follows that ${\varepsilon}_{i}(\eta) = - m^{\eta}_{i} = 0$ by Remark \[rem:SLS\]. Here we recall that $f_{i}(\pi \otimes \eta) = \pi \otimes f_{i}\eta$ (if and) only if ${\varphi}_{i}(\pi) \le {\varepsilon}_{i}(\eta)$ by the tensor product rule for crystals. Hence we see that ${\varphi}_{i}(\pi) = H^{\pi}_{i}(1) - m^{\pi}_{i} = 0$ by Remark \[rem:SLS\]. Since ${\langle x_{s}\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle x_{s-1}\lambda,\,\alpha_{i}^{\vee} \rangle} = \cdots = 
{\langle x_{u_{0}+1}\lambda,\,\alpha_{i}^{\vee} \rangle} = 0$ by our assumption, we obtain ${\langle x_{u_{0}}\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$ if $u_{0} \ge 1$, as desired. Therefore, by the same argument as in Case 1, we get $0 \le u_{0} \le u_{1} \le s$ such that $(x_{u}')^{-1}\alpha_{i} \in {\Delta_{J}}^{+} + {\mathbb{Z}}\delta$ for all $u_{1}+1 \le u \le s$, and such that $(x_{u_{1}}')^{-1}\alpha_{i} \in -\Delta^{+} + {\mathbb{Z}}\delta$ if $u_{1} \ge 1$; recall that $(y_{1}')^{-1}\alpha_{i} \in \Delta_{K}^{+} + {\mathbb{Z}}\delta$. Also we note that by Remark \[rem:si\], ${\Pi^{J}}(s_{i}x_{u}') = {\Pi^{J}}(x_{u}') = x_{u}$ for all $u_{1}+1 \le i \le s$. In this case, by Lemma \[lem:dmd\](1) and (3), together with , we obtain a defining chain $$x_{1}' {\succeq}\cdots {\succeq}x_{u_{1}}' {\succeq}s_{i}x_{u_{1}+1}' {\succeq}\cdots {\succeq}s_{i}x_{s}' {\succeq}s_{i}y_{1}' {\succeq}\cdots {\succeq}s_{i}y_{m+1}' {\succeq}y_{m+1}' {\succeq}\cdots {\succeq}y_{p}'$$ for $f_{i}(\pi \otimes \eta) = \pi \otimes (f_{i}\eta)$. This proves part (1).

\(2)  Let $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$, and let $i \in I_{{\mathrm{af}}}$ be such that ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \ge 0$; we may assume that $e_{i}(\pi \otimes \eta) \ne {\bm{0}}$. Since $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$, there exists a defining chain for $\pi \otimes \eta$ whose final element, say $y \in W_{{\mathrm{af}}}$, satisfies ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$. We can show the following claims by arguments similar to those in part (1).

\[c:Dem1-1\]

If $e_{i}(\pi \otimes \eta) = e_{i}\pi \otimes \eta$, or if $e_{i}(\pi \otimes \eta) = \pi \otimes (e_{i}\eta)$ and $\kappa(e_{i}\eta)=\kappa(\eta)$, then there exists a defining chain for $e_{i}(\pi \otimes \eta)$ whose final element is $y$.

If $e_{i}(\pi \otimes \eta) = \pi \otimes (e_{i}\eta)$ and $\kappa(e_{i}\eta)=s_{i}\kappa(\eta)$, then there exists a defining chain for $e_{i}(\pi \otimes \eta)$ whose final element is $s_{i}y$.

In case (i) of Claim \[c:Dem1-1\], it is obvious that $e_{i}(\pi \otimes \eta) \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$. In case (ii) of Claim \[c:Dem1-1\], we see by the definition of the root operator $e_{i}$ that with notation in and , $t_{1} = 1$, and the function $H^{\eta}_{i}(t)$ is strictly decreasing on $[t_{0},t_{1}] = [t_{0},1]$. Hence we have ${\langle \kappa(\eta)\mu,\,\alpha_{i}^{\vee} \rangle} < 0$, which implies that $\kappa(\eta)^{-1}\alpha_{i} \in 
-(\Delta^{+} \setminus {\Delta_{K}}^{+}) + {\mathbb{Z}}\delta$. Since ${\Pi^{K}}({\Pi^{S}}(y)) = {\Pi^{K}}(y) = \kappa(\eta)$, we see that $({\Pi^{S}}(y))^{-1}\alpha_{i} \in - (\Delta^{+} \setminus {\Delta_{K}}^{+}) + {\mathbb{Z}}\delta 
\subset - (\Delta^{+} \setminus {\Delta_{S}}^{+}) + {\mathbb{Z}}\delta$, which implies that ${\langle {\Pi^{S}}(y)(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} < 0$. Also, it follows from Lemma \[lem:si\] that $s_{i}{\Pi^{S}}(y) \in ({W^{S}})_{{\mathrm{af}}}$ and ${\Pi^{S}}(s_{i}y) = s_{i}{\Pi^{S}}(y)$. Here, by the assumption, we have ${\langle {\Pi^{S}}(x)(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} = 
{\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \ge 0$. Therefore, we deduce from Lemma \[lem:dmd\](2), together with ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$, that ${\Pi^{S}}(s_{i}y) = s_{i}{\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$. Thus, we conclude that $e_{i}(\pi \otimes \eta) \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$. This proves part (2).

\(3)  If ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} = 0$, then ${\Pi^{S}}(s_{i}x)={\Pi^{S}}(x)$ by Remark \[rem:si\], and hence ${\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu) = {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$. Hence the assertion is obvious from part (2).

Assume that ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} > 0$. Then we see from Lemma \[lem:si\] that ${\Pi^{S}}(s_{i}x) = s_{i}{\Pi^{S}}(x) \in ({W^{S}})_{{\mathrm{af}}}$ and ${\Pi^{S}}(s_{i}x) {\succeq}{\Pi^{S}}(x)$, which implies that ${\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu) \supset {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu)$. Therefore, by part (2), we obtain the inclusion $\supset$ in . In order to show the opposite inclusion $\subset$ in , it suffices to show that $f_{i}^{\max}(\pi \otimes \eta) \in 
{\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu)$ for all $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$. In view of part (1), this assertion itself follows from the following claim.

\[c:Dem1-2\] Let $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$. If $f_{i}(\pi \otimes \eta) = {\bm{0}}$, i.e., ${\varphi}_{i}(\pi \otimes \eta) = 0$, then $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu)$.

[*Proof of Claim \[c:Dem1-2\].*]{} We write $\pi$ and $\eta$ as: $\pi=(x_{1},\,\dots,\,x_{s}\,;\,{\mathbf{a}})$ and $\eta=(y_{1},\,\dots,\,y_{p}\,;\,{\mathbf{b}})$, respectively. Let $x_{1}' {\succeq}\cdots {\succeq}x_{s}' {\succeq}y_{1}' {\succeq}\cdots {\succeq}y_{p}'$ be a defining chain for $\pi \otimes \eta$ such that ${\Pi^{S}}(y_{p}') {\succeq}{\Pi^{S}}(x)$. We see from Lemma \[lem:611\] that ${\Pi^{S}}(x_{1}') {\succeq}\cdots {\succeq}{\Pi^{S}}(x_{s}') {\succeq}{\Pi^{S}}(y_{1}') {\succeq}\cdots {\succeq}{\Pi^{S}}(y_{p}')$ is also a defining chain for $\pi \otimes \eta$ satisfying ${\Pi^{S}}({\Pi^{S}}(y_{p}')) = {\Pi^{S}}(y_{p}') {\succeq}{\Pi^{S}}(x)$. Hence we may assume from the beginning that $x_{1}',\,\dots,\,x_{s}',\,y_{1}',\,\dots,\,y_{p}' \in ({W^{S}})_{{\mathrm{af}}}$.

Assume first that the set $$\label{eq:set1}
\begin{split}
& \bigl\{1 \le q \le p \mid 
(y_{q}')^{-1}\alpha_{i} \not\in ({\Delta_{K}}^{+} \setminus {\Delta_{S}}^{+}) + {\mathbb{Z}}\delta\bigr\} \\
& \hspace*{20mm}
= \bigl\{1 \le q \le p \mid 
(y_{q}')^{-1}\alpha_{i} \in 
\bigl((\Delta \setminus {\Delta_{K}}^{+}) \sqcup {\Delta_{S}}^{+}\bigr) + {\mathbb{Z}}\delta\bigr\}
\end{split}
$$ is nonempty. Let $q_{1}$ be the maximum element of this set; notice that ${\langle y_{q}'(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} > 0$ and ${\langle y_{q}\mu,\,\alpha_{i}^{\vee} \rangle} = 
{\langle y_{q}'\mu,\,\alpha_{i}^{\vee} \rangle} = 0$ for all $q_{1} < q \le p$. Note that $f_{i}(\pi \otimes \eta)={\bm{0}}$ implies $f_{i}\eta = {\bm{0}}$ by the tensor product rule for crystals, and hence $H^{\eta}_{i}(1) - m^{\eta}_{i}=0$. From this it follows that ${\langle y_{q_{1}}'\mu,\,\alpha_{i}^{\vee} \rangle} = 
{\langle y_{q_{1}}\mu,\,\alpha_{i}^{\vee} \rangle} \le 0$, and hence $(y_{q_{1}}')^{-1}\alpha_{i} \in ((-\Delta^{+}) \sqcup {\Delta_{S}}^{+}) + {\mathbb{Z}}\delta$ by the definition of $q_{1}$, which implies that ${\langle y_{q_{1}}'(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \le 0$. Therefore, we see from Lemma \[lem:dmd\](1) and (3) that $$\label{eq:Dem1-2-1}
x_{1}' {\succeq}\cdots {\succeq}x_{s}' {\succeq}y_{1}' {\succeq}\cdots {\succeq}y_{q_{1}}' {\succeq}s_{i}y_{q_{1}+1}' {\succeq}\cdots {\succeq}s_{i}y_{p}';$$ note that ${\Pi^{K}}(s_{i}y_{q}')= {\Pi^{K}}(y_{q}')$ for all $q_{1} < q \le p$ since ${\langle y_{q}'\mu,\,\alpha_{i}^{\vee} \rangle} =  0$. Thus the sequence is also a defining chain for $\pi \otimes \eta$. If $q_{1} = p$, then the final element of is $y_{p}'$, and ${\langle y_{p}'(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \le 0$. Hence it follows from Lemma \[lem:dmd\](1) that $y_{p}' {\succeq}s_{i}{\Pi^{S}}(x) = {\Pi^{S}}(s_{i}x)$. If $q_{1} < p$, then the final element of is $s_{i}y_{p}'$, and ${\langle y_{p}'(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} > 0$. This implies that $s_{i}y_{p}' \in ({W^{S}})_{{\mathrm{af}}}$ by Lemma \[lem:si\], and that $s_{i}y_{p}' {\succeq}s_{i}{\Pi^{S}}(x) = {\Pi^{S}}(s_{i}x)$ by Lemma \[lem:dmd\](3). Hence we conclude that $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu)$.

Assume next that the set in is empty, that is, $(y_{q}')^{-1}\alpha_{i} \in ({\Delta_{K}}^{+} \setminus {\Delta_{S}}^{+}) + {\mathbb{Z}}\delta$ for all $1 \le q \le p$; notice that ${\langle y_{q}'(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} > 0$ for all $1 \le  q \le p$. Also, since ${\langle y_{q}\mu,\,\alpha_{i}^{\vee} \rangle} = 
{\langle y_{q}'\mu,\,\alpha_{i}^{\vee} \rangle} = 0$ for all $1 \le  q \le p$, we have $H^{\eta}_{i}(t) = 0$ for all $t \in [0,1]$, and hence ${\varepsilon}_{i}(\eta)=0$. Since $f_{i}(\pi \otimes \eta)={\bm{0}}$ by the assumption, we obtain $f_{i}\pi = {\bm{0}}$ by the tensor product rule for crystals. Let $u_{1}$ be the maximum element of the set $\bigl\{ 1 \le u \le s \mid 
(x_{u}')^{-1}\alpha_{i} \not\in ({\Delta_{J}}^{+} \setminus {\Delta_{S}}^{+}) + {\mathbb{Z}}\delta\bigr\} \cup \{0\}$. Then we have ${\langle x_{u}'(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} > 0$ and ${\langle x_{u}'\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle x_{u}\lambda,\,\alpha_{i}^{\vee} \rangle} = 0$ for all $u_{1} < u \le s$. In addition, we can show by the same argument as above that if $u_{1} \ge 1$, then ${\langle x_{u_{1}}'(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \le 0$. Therefore, it follows from Lemma \[lem:dmd\](1) and (3) that $$\label{eq:Dem1-2-2}
x_{1}' {\succeq}\cdots {\succeq}x_{u_{1}}' {\succeq}s_{i}x_{u_{1}+1} {\succeq}\cdots {\succeq}s_{i}x_{s}' {\succeq}s_{i}y_{1}' {\succeq}\cdots {\succeq}s_{i}y_{p}'.$$ In the same way as for , we can verify that the sequence is a defining chain for $\pi \otimes \eta$ satisfying the condition in . This proves Claim \[c:Dem1-2\].

This completes the proof of Lemma \[lem:Dem1\].

\[cor:Dem2\] Let $x \in W_{{\mathrm{af}}}$, and $i \in I_{{\mathrm{af}}}$. For every $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$, we have $f_{i}^{\max}(\pi \otimes \eta) \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu)$.

If ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \ge 0$, then the assertion follows from the proof of Lemma \[lem:Dem1\](3). If ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} < 0$, then we have ${\Pi^{S}}(s_{i}x) = s_{i}{\Pi^{S}}(x) {\preceq}{\Pi^{S}}(x)$ by Lemma \[lem:si\], and hence ${\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu) \supset {\mathbb{D}^{\frac{\infty}{2}}}_{x}(\lambda+\mu)$. Therefore, the assertion follows from Lemma \[lem:Dem1\](1). This proves the corollary.

\[lem:Dem3\] Let $x,\,y \in W_{{\mathrm{af}}}$.

If ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$ in $({W^{S}})_{{\mathrm{af}}}$, then $y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$ for all $({\bm{\rho}},\,{\bm{\chi}},\,\xi) \in {\mathop{\rm Par}\nolimits}(\lambda) \times {\mathop{\rm Par}\nolimits}(\mu) \times 
Q^{\vee}_{I \setminus ({J}\cup {K})}$ satisfying .

If $y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) 
\in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$ for some $({\bm{\rho}},\,{\bm{\chi}},\,\xi) \in 
{\mathop{\rm Par}\nolimits}(\lambda) \times {\mathop{\rm Par}\nolimits}(\mu) \times Q^{\vee}_{I \setminus ({J}\cup {K})}$ satisfying , then ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$.

\(1) By the definitions (see ), $\pi_{{\bm{\rho}}} \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\pi_{{\bm{\chi}}} \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$ are of the form: $$\label{eq:Dem3b}
\begin{cases}
\pi_{{\bm{\rho}}}=({\Pi^{J}}(t_{\xi_1}),\,\dots,\,{\Pi^{J}}(t_{\xi_{s-1}}),\,e\,;\,{\mathbf{a}}), & \\[1mm]
\pi_{{\bm{\chi}}}=({\Pi^{K}}(t_{\zeta_1}),\,\dots,\,{\Pi^{K}}(t_{\zeta_{p-1}}),\,e\,;\,{\mathbf{b}})
\end{cases}
$$ for some $\xi_{1},\,\dots,\,\xi_{s-1} \in Q^{\vee}_{I \setminus {J}}$ such that $\xi_{1} > \cdots > \xi_{s-1} > 0$, and $\zeta_{1},\,\dots,\,\zeta_{p-1} \in Q^{\vee}_{I \setminus {K}}$ such that $\zeta_{1} > \cdots > \zeta_{p-1} > 0$, respectively. Also, recall from that $$t_{\xi} \cdot \pi_{{\bm{\rho}}}=
({\Pi^{J}}(t_{\xi_1+\xi}),\,\dots,\,
 {\Pi^{J}}(t_{\xi_{s-1}+\xi}),\,{\Pi^{J}}(t_{\xi})\,;\,{\mathbf{a}}),$$ and from Lemma \[lem:Weyl\] that $$\label{eq:Dem3c}
\begin{split}
& y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) = 
(y \cdot (t_{\xi} \cdot \pi_{{\bm{\rho}}})) \otimes (y \cdot \pi_{{\bm{\chi}}}), \quad \text{with} \\
& 
\begin{cases}
y \cdot (t_{\xi} \cdot \pi_{{\bm{\rho}}}) = 
 ({\Pi^{J}}(yt_{\xi_1+\xi}),\,\dots,\,{\Pi^{J}}(yt_{\xi_{s-1}+\xi}),\,{\Pi^{J}}(yt_{\xi})\,;\,{\mathbf{a}}), & \\[1mm]
y \cdot \pi_{{\bm{\chi}}}=({\Pi^{K}}(yt_{\zeta_1}),\,\dots,\,{\Pi^{K}}(yt_{\zeta_{p-1}}),\,{\Pi^{K}}(y)\,;\,{\mathbf{b}}). 
\end{cases}
\end{split}
$$

Now, if ${\bm{\chi}}=(\chi^{(i)})_{i \in I} \in {\mathop{\rm Par}\nolimits}(\mu)$, with $\chi^{(i)}=(\chi^{(i)}_{1} \ge \chi^{(i)}_{2} \ge \cdots)$ for $i \in I$, then we have $\zeta_{1} = \sum_{i \in I} \chi_{1}^{(i)} \alpha_{i}^{\vee}$ by ; we set $\gamma:=\sum_{i \in {J}\setminus K}
\chi_{1}^{(i)}\alpha_{i}^{\vee} \in {Q_{J}^{\vee}}$. Since $({\bm{\rho}},\,{\bm{\chi}},\,\xi)$ satisfies , and $\chi_{1}^{(i)}=0$ for all $i \in K$, we deduce that $\xi + \gamma \ge \zeta_{1}$, and hence that $$\xi_{1}+\xi+\gamma > \cdots > \xi_{s-1}+\xi+\gamma > \xi+\gamma \ge 
\zeta_{1} > \cdots > \zeta_{p-1} > 0.$$ Therefore, it follows from Lemma \[lem:SiB\](2) (applied to the case $J=\emptyset$) that $$\label{eq:dc63}
yt_{\xi_{1}+\xi+\gamma} {\succ}\cdots {\succ}yt_{\xi_{s-1}+\xi+\gamma} {\succ}yt_{\xi+\gamma} {\succeq}yt_{\zeta_{1}} {\succ}\cdots {\succ}yt_{\zeta_{p-1}} {\succ}y = yt_{0} \quad \text{in $W_{{\mathrm{af}}}$};$$ note that by Lemma \[lem:PiJ\], ${\Pi^{J}}(yt_{\xi_{u}+\xi+\gamma}) = {\Pi^{J}}(yt_{\xi_{u}+\xi})$ for all $1 \le u \le s$ since $\gamma \in {Q_{J}^{\vee}}$. Hence the sequence is a defining chain for $y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}})$. Since ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$ in $({W^{S}})_{{\mathrm{af}}}$ by the assumption, we conclude that $y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) 
\in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$. This proves part (1).

\(2) We divide the proof into several steps.

#### Step 1.

Assume that $x = t_{\zeta'}$ for some $\zeta' \in Q^{\vee}$, and $y = t_{\xi'}$ for some $\xi' \in Q^{\vee}$; in this case, in order to prove that ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$ in $({W^{S}})_{{\mathrm{af}}}$, it suffices to show that $[\xi']^{S} \ge [\zeta']^{S}$ (see Lemma \[lem:SiB\](2)). In the same way as for , we obtain $$\label{eq:Dem3e}
\begin{split}
& y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) = 
(y \cdot (t_{\xi} \cdot \pi_{{\bm{\rho}}})) \otimes (y \cdot \pi_{{\bm{\chi}}}), \quad \text{with} \\
& 
\begin{cases}
y \cdot (t_{\xi} \cdot \pi_{{\bm{\rho}}}) = 
 ({\Pi^{J}}(t_{\xi_1+\xi+\xi'}),\,\dots,\,{\Pi^{J}}(t_{\xi_{s-1}+\xi+\xi'}),\,{\Pi^{J}}(t_{\xi+\xi'})\,;\,{\mathbf{a}}), & \\[1mm]
y \cdot \pi_{{\bm{\chi}}}=({\Pi^{K}}(t_{\zeta_1+\xi'}),\,\dots,\,{\Pi^{K}}(t_{\zeta_{p-1}+\xi'}),\,{\Pi^{K}}(t_{\xi'})\,;\,{\mathbf{b}}). 
\end{cases}
\end{split}
$$ Here, since $y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) 
\in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$ satisfies condition (or equivalently, condition ; see Section \[subsec:12\]), we have $$\label{eq:Dem3f}
\kappa(y \cdot \pi_{{\bm{\chi}}}) {\succeq}{\Pi^{K}}(x) \quad \text{and} \quad
\kappa \bigl(y \cdot (t_{\xi} \cdot \pi_{{\bm{\rho}}}))\bigr) {\succeq}{\Pi^{J}}({\iota(y \cdot \pi_{{\bm{\chi}}},\,x)}).$$ We deduce from the first inequality in that ${\Pi^{K}}(t_{\xi'}) = \kappa(y \cdot \pi_{{\bm{\chi}}}) {\succeq}{\Pi^{K}}(x) = {\Pi^{K}}(t_{\zeta'})$, which implies that $[\xi']^{{K}} \ge [\zeta']^{{K}}$ by Lemma \[lem:SiB\](2). Since $I \setminus S = (I \setminus {K}) \sqcup ({K}\setminus S)$, it remains to show that $[\xi']_{{K}\setminus S} \ge [\zeta']_{{K}\setminus S}$. We define ${\widetilde{y}}_{p}$, ${\widetilde{y}}_{p-1}$, $\dots$, ${\widetilde{y}}_{1}$ by the recursive procedure , that is, $$\begin{split}
& {\widetilde{y}}_{p}:=\min {\mathrm{Lift}_{{\succeq}x}({\Pi^{K}}(t_{\xi'}))}, \quad 
  \text{with} \quad x = t_{\zeta'}, \\
& {\widetilde{y}}_{p-1}:=\min {\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{p}}({\Pi^{K}}(t_{\zeta_{p-1}+\xi'}))},  \\
& \qquad \vdots \\
& {\widetilde{y}}_{1}:=\min {\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{2}}({\Pi^{K}}(t_{\zeta_{1}+\xi'}))}
  = {\iota(y \cdot \pi_{{\bm{\chi}}},\,x)}.
\end{split}
$$

\[c:Dem3-1\] The elements ${\widetilde{y}}_{q}$, $1 \le q \le p$, are of the form[:]{} ${\widetilde{y}}_{q} = t_{\zeta_{q}+\xi'+\gamma_{q}}$ for some $\gamma_{q} \in {Q_{K}^{\vee}}$, where we set $\zeta_{p}:=0$.

[*Proof of Claim \[c:Dem3-1\].*]{} We show the assertion by descending induction on $1 \le q \le p$. Assume that $q = p$. We see from Lemma \[lem:lift\] that ${\widetilde{y}}_{p} = z_{p} t_{\xi'+\gamma_{p}}$ for some $z_{p} \in {W_{K}}$ and $\gamma_{p} \in {Q_{K}^{\vee}}$. Since $z_{p} t_{\xi'+\gamma_{p}} = {\widetilde{y}}_{p} {\succeq}x = t_{\zeta'}$, it follows from Lemma \[lem:SiB\](1) and(2) that $t_{\xi'+\gamma_{p}} {\succeq}t_{\zeta'} = x$ in $W_{{\mathrm{af}}}$. Also, we have $t_{\xi'+\gamma_{p}} \in {\mathrm{Lift}}({\Pi^{K}}(t_{\xi'}))$ by Lemma \[lem:lift\]. Combining these, we obtain $t_{\xi'+\gamma_{p}} \in 
{\mathrm{Lift}_{{\succeq}x}({\Pi^{K}}(t_{\xi'}))}$. Since ${\widetilde{y}}_{p} = z_{p} t_{\xi'+\gamma_{p}} {\succeq}t_{\xi'+\gamma_{p}}$ in $W_{{\mathrm{af}}}$ by Remark \[rem:SiB\], we deduce that ${\widetilde{y}}_{p} = t_{\xi'+\gamma_{p}}$ by the minimality of ${\widetilde{y}}_{p}$.

Assume that $q < p$; by our induction hypothesis, we have ${\widetilde{y}}_{q+1} = t_{\zeta_{q+1}+\xi'+\gamma_{q+1}}$ for some $\gamma_{q+1} \in {Q_{K}^{\vee}}$. Also, we see from Lemma \[lem:lift\] that ${\widetilde{y}}_{q} = z_{q} t_{\zeta_{q}+\xi'+\gamma_{q}}$ for some $z_{q} \in {W_{K}}$ and $\gamma_{q} \in {Q_{K}^{\vee}}$. Now, the same argument as above shows that $${\widetilde{y}}_{q} = z_{q} t_{\zeta_{q}+\xi'+\gamma_{q}} {\succeq}t_{\zeta_{q}+\xi'+\gamma_{q}} {\succeq}t_{\zeta_{q+1}+\xi'+\gamma_{q+1}} = {\widetilde{y}}_{q+1},$$ and that $t_{\zeta_{q}+\xi'+\gamma_{q}} \in {\mathrm{Lift}}({\Pi^{K}}(t_{\zeta_{q}+\xi'}))$. Hence we obtain ${\widetilde{y}}_{q} = t_{\zeta_{q}+\xi'+\gamma_{q}}$ by the minimality of ${\widetilde{y}}_{q}$. This proves Claim \[c:Dem3-1\].

Because ${\widetilde{y}}_{1} {\succeq}\cdots {\succeq}{\widetilde{y}}_{p} {\succeq}x$ in $W_{{\mathrm{af}}}$, it follows from Lemma \[lem:SiB\](2) and Claim \[c:Dem3-1\] that $$\zeta_{1}+\xi'+\gamma_{1} \ge \zeta_{2}+\xi'+\gamma_{2} \ge \cdots \ge 
\zeta_{p}+\xi'+\gamma_{p} \ge \zeta';$$ in particular, we have $$\label{eq:step1-4}
[\zeta_{1}+\xi'+\gamma_{1}]_{{K}\setminus S} \ge 
[\zeta']_{{K}\setminus S}.$$ Also, we see by the second inequality in and Claim \[c:Dem3-1\] that $${\Pi^{J}}(t_{\xi+\xi'}) = 
\kappa(y \cdot (t_{\xi} \cdot \pi_{{\bm{\rho}}})) {\succeq}{\Pi^{J}}({\iota(y \cdot \pi_{{\bm{\chi}}},\,x)}) = 
{\Pi^{J}}(t_{\zeta_{1}+\xi'+\gamma_{1}}),$$ which implies that $[\xi+\xi']^{{J}} \ge 
[\zeta_{1}+\xi'+\gamma_{1}]^{{J}}$ by Lemma \[lem:SiB\](2); in particular, we have $[\xi+\xi']_{{K}\setminus S} \ge 
[\zeta_{1}+\xi'+\gamma_{1}]_{{K}\setminus S}$. Here, since $\xi \in Q^{\vee}_{I \setminus ({J}\cup {K})}$, we have $[\xi+\xi']_{{K}\setminus S} = 
[\xi']_{{K}\setminus S}$. Therefore, we deduce that $$\label{eq:step1-5}
[\xi']_{{K}\setminus S} \ge 
[\zeta_{1}+\xi'+\gamma_{1}]_{{K}\setminus S}.$$ Combining and , we obtain $[\xi']_{{K}\setminus S} \ge 
[\zeta']_{{K}\setminus S}$, as desired.

#### Step 2.

Assume that $x = t_{\zeta'}$ for some $\zeta' \in Q^{\vee}$, and write $y \in W_{{\mathrm{af}}}$ as $y = v t_{\xi'}$ for some $v \in W$ and $\xi' \in Q^{\vee}$. Let us show the assertion by induction on $\ell(v)$. If $\ell(v) = 0$, i.e., $v = e$, then the assertion follows from Step 1. Assume that $\ell(v) > 0$. We take $i \in I$ such that $\ell(s_{i}v) = \ell(v) -1$; note that $y^{-1}\alpha_{i} \in -\Delta^{+} + {\mathbb{Z}}\delta$. Since ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} \le 0$ and ${\langle y\mu,\,\alpha_{i}^{\vee} \rangle} \le 0$, we see by the definition of the root operator $f_{i}$ and that $f_{i}\bigl(y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}})\bigr) = {\bm{0}}$, and hence that $$\label{eq:step2-1}
e_{i}^{\max} \bigl( y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) \bigr)
= (s_{i}y) \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}).$$ Since $x = t_{\zeta'}$, we have ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} = 
 {\langle \lambda+\mu,\,\alpha_{i}^{\vee} \rangle} \ge 0$. Therefore, by Lemma \[lem:Dem1\](2), together with , we obtain $(s_{i}y) \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) 
\in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$. Hence, by our induction hypothesis, we have ${\Pi^{S}}(s_{i}y) {\succeq}{\Pi^{S}}(x)$. Here we recall that ${\langle y(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \le 0$ since $y^{-1}\alpha_{i} \in -\Delta^{+} + {\mathbb{Z}}\delta$. If ${\langle y(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} < 0$, then ${\Pi^{S}}(y) {\succeq}s_{i}{\Pi^{S}}(y) = {\Pi^{S}}(s_{i}y) {\succeq}{\Pi^{S}}(x)$ by Lemma \[lem:si\] and Remark \[rem:si\]. If ${\langle y(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} = 0$, then ${\Pi^{S}}(y) = {\Pi^{S}}(s_{i}y) {\succeq}{\Pi^{S}}(x)$ by Remark \[rem:si\]. In both cases, we obtain ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$, as desired.

#### Step 3.

Let $x,\,y \in W_{{\mathrm{af}}}$. We see from [@AK] that there exist $i_{1},\,\dots,\,i_{N} \in I_{{\mathrm{af}}}$ such that ${\langle s_{i_{n-1}} \cdots s_{i_{1}}x(\lambda+\mu),\,\alpha_{i_n}^{\vee} \rangle} \ge 0$ for all $1 \le n \le N$, and such that $s_{i_{N}} \cdots s_{i_{1}}x = t_{\zeta'}$ for some $\zeta' \in Q^{\vee}$. Let us show the assertion by induction on $N$. If $N=0$, i.e., $x = t_{\zeta'}$, then the assertion follows from Step 2. Assume that $N > 0$; for simplicity of notation, we set $i:=i_{1}$. It follows from Corollary \[cor:Dem2\] that $$\label{eq:step3-1}
f_{i}^{\max} 
\bigl( y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) \bigr)
\in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu).$$

#### Case 3.1.

Assume that ${\langle y(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \le 0$; note that ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} \le 0$ and ${\langle y\mu,\,\alpha_{i}^{\vee} \rangle} \le 0$. We see by the definition of the root operator $f_{i}$ and that $f_{i}^{\max} 
\bigl( y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) \bigr)= 
y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}})$. Hence, by our induction hypothesis, we have ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(s_{i}x)$. Here we recall that ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \ge 0$. If ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} = 0$, then we have ${\Pi^{S}}(s_{i}x) = {\Pi^{S}}(x)$ by Remark \[rem:si\]. If ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} > 0$, then it follows from Lemma \[lem:si\] that ${\Pi^{S}}(s_{i}x) = s_{i}{\Pi^{S}}(x) {\succeq}{\Pi^{S}}(x)$. In both cases, we obtain ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$, as desired.

#### Case 3.2.

Assume that ${\langle y(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} > 0$; note that ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} \ge 0$ and ${\langle y\mu,\,\alpha_{i}^{\vee} \rangle} \ge 0$. We see by the definition of the root operator $e_{i}$ and that $e_{i} \bigl( y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) \bigr) = {\bm{0}}$, and hence $$f_{i}^{\max} 
\bigl( y \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}) \bigr)= 
(s_{i}y) \cdot \bigl( t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}).$$ Hence, by our induction hypothesis, we have ${\Pi^{S}}(s_{i}y) {\succeq}{\Pi^{S}}(s_{i}x)$. As in Case 3.1, we see that ${\Pi^{S}}(s_{i}x) {\succeq}{\Pi^{S}}(x)$, and hence ${\Pi^{S}}(s_{i}y) {\succeq}{\Pi^{S}}(x)$. Also, since ${\langle y(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} > 0$, it follows from Lemma \[lem:si\] that ${\Pi^{S}}(s_{i}y) = s_{i}{\Pi^{S}}(y)$, and hence from Lemma \[lem:dmd\](2) that ${\Pi^{S}}(y) {\succeq}{\Pi^{S}}(x)$.

This proves part (2), and completes the proof of Lemma \[lem:Dem3\].

Now, the equivalence $\Leftrightarrow$ follows from the next lemma.

\[lem:DemX\] Let $\psi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu)$, and assume that $\psi$ is mapped to $\pi \otimes \eta \in {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ under the isomorphism ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu) \cong {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ in Theorem \[thm:SMT\]. Let $x \in W_{{\mathrm{af}}}$.

If $\psi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda+\mu)$, then $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$.

If $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$, then $\psi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda+\mu)$.

By [@NS16 Lemma 5.4.1], there exist $i_{1},\,i_{2},\,\dots,\,i_{N} \in I_{{\mathrm{af}}}$ satisfying the conditions that $$\begin{cases}
{\langle s_{i_{n-1}}s_{i_{n-2}} \cdots 
  s_{i_{2}}s_{i_{1}}x(\lambda+\mu),\,\alpha_{i_n}^{\vee} \rangle} \ge 0 
  & \text{for all $1 \le n \le N$, and} \\[1.5mm]
f_{i_{N}}^{\max}f_{i_{N-1}}^{\max} \cdots f_{i_{2}}^{\max}f_{i_{1}}^{\max}\psi = 
t_{\xi'} \cdot \pi_{{\bm{\omega}}}
  & \text{for some $\xi' \in Q^{\vee}$ and ${\bm{\omega}}\in {\mathop{\rm Par}\nolimits}(\lambda+\mu)$}. 
\end{cases}
$$ We prove part (1) by induction on $N$. Assume that $N=0$, i.e., $\psi = t_{\xi'} \cdot \pi_{{\bm{\omega}}}$; recall from that $\kappa(\psi) = {\Pi^{S}}(t_{\xi'})$. Since $\psi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda+\mu)$ by the assumption, we have $$\label{eq:DemX-1}
{\Pi^{S}}(t_{\xi'}) = \kappa(\psi) {\succeq}{\Pi^{S}}(x).$$ By Corollary \[cor:Sext\], $\psi = t_{\xi'} \cdot \pi_{{\bm{\omega}}}$ is mapped to $t_{\xi'} \cdot 
\bigl(t_{\xi} \cdot \pi_{{\bm{\rho}}} \otimes \pi_{{\bm{\chi}}}\bigr)$, which is $\pi \otimes \eta$, for some $\xi \in Q^{\vee}_{I \setminus ({J}\cup {K})}$ and ${\bm{\rho}}\in {\mathop{\rm Par}\nolimits}(\lambda)$, ${\bm{\chi}}\in {\mathop{\rm Par}\nolimits}(\mu)$ satisfying under the isomorphism ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu) \cong {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ of crystals in Theorem \[thm:SMT\]. Therefore, we deduce from Lemma \[lem:Dem3\](1), together with , that $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$.

Assume that $N > 0$. For simplicity of notation, we set $i_{1}:=i$; note that ${\langle x(\lambda+\mu),\,\alpha_{i}^{\vee} \rangle} \ge 0$. We see from [@NS16 Corollary 5.3.3] that $f_{i}^{\max}\psi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}s_{i}x}(\lambda+\mu)$. By our induction hypothesis, we have $f_{i}^{\max}(\pi \otimes \eta) 
\in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}s_{i}x}(\lambda+\mu)$. Since $\pi \otimes \eta = e_{i}^{k} f_{i}^{\max}(\pi \otimes \eta)$ for some $k \ge 0$, we deduce from Lemma \[lem:Dem1\](3) that $\pi \otimes \eta \in {\mathbb{D}^{\frac{\infty}{2}}}_{{\succeq}x}(\lambda+\mu)$. This proves part (1).

We can prove part (2) similarly, using Lemma \[lem:Dem3\](2) instead of Lemma \[lem:Dem3\](1).

Appendices. {#appendices. .unnumbered}
===========

Basic properties of the semi-infinite Bruhat order. {#sec:basic}
===================================================

We fix $J \subset I$ and $\lambda \in P^{+} \subset P_{{\mathrm{af}}}^{0}$ (see and ) such that $\big\{ i \in I \mid {\langle \lambda,\,\alpha_{i}^{\vee} \rangle}= 0 \bigr\} = {J}$.

\[lem:PiJ\]

It holds that $$\label{eq:PiJ2}
\begin{cases}
{\Pi^{J}}(w)={\lfloor w \rfloor} 
  & \text{\rm for all $w \in W$}; \\[1mm]
{\Pi^{J}}(xt_{\xi})={\Pi^{J}}(x){\Pi^{J}}(t_{\xi}) 
  & \text{\rm for all $x \in W_{{\mathrm{af}}}$ and $\xi \in Q^{\vee}$};
\end{cases}
$$ in particular, $$\label{eq:PiJ3}
({W^{J}})_{{\mathrm{af}}} 
  = \bigl\{ w {\Pi^{J}}(t_{\xi}) \mid w \in W^J,\,\xi \in Q^{\vee} \bigr\}.$$

For each $\xi \in Q^{\vee}$, the element ${\Pi^{J}}(t_{\xi})$ is of the form[:]{} ${\Pi^{J}}(t_{\xi})=z_{\xi}t_{\xi+\phi_{{J}}(\xi)}$ for [(]{}a unique[)]{} $z_{\xi} \in {W_{J}}$ and $\phi_{{J}}(\xi) \in {Q_{J}^{\vee}}$.

For $\xi,\,\zeta \in Q^{\vee}$, $$\label{eq:PiJ1}
{\Pi^{J}}(t_{\xi}) = {\Pi^{J}}(t_{\zeta}) \iff \xi-\zeta \in {Q_{J}^{\vee}}.$$

\[lem:si\] Let $x \in ({W^{J}})_{{\mathrm{af}}}$, and $i \in I_{{\mathrm{af}}}$. Then, $$\label{eq:si1}
s_{i}x \in ({W^{J}})_{{\mathrm{af}}} \iff 
{\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} \ne 0 \iff 
x^{-1}\alpha_{i} \in (\Delta \setminus {\Delta_{J}})+{\mathbb{Z}}\delta.$$ Moreover, in this case, $$\label{eq:simple}
\begin{cases}
x {\xrightarrow{\,\alpha_{i}\,}} s_{i}x \iff
{\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} > 0 \iff 
x^{-1}\alpha_{i} \in (\Delta^{+} \setminus {\Delta_{J}}^{+})+{\mathbb{Z}}\delta, & \\[1.5mm]
s_{i}x {\xrightarrow{\,\alpha_{i}\,}} x  \iff 
{\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} < 0 \iff 
x^{-1}\alpha_{i} \in -(\Delta^{+} \setminus {\Delta_{J}}^{+})+{\mathbb{Z}}\delta. & 
\end{cases}
$$

\[rem:si\] Keep the setting of Lemma \[lem:si\]. If $x^{-1}\alpha_{i} \in {\Delta_{J}}+{\mathbb{Z}}\delta$, i.e., ${\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} = 0$, then ${\Pi^{J}}(s_{i}x) = x$.

\[lem:dmd\] Let $x,\,y \in ({W^{J}})_{{\mathrm{af}}}$ be such that $x {\preceq}y$, and let $i \in I_{{\mathrm{af}}}$.

If ${\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} > 0$ and ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} \le 0$, then $s_{i}x {\preceq}y$.

If ${\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} \ge 0$ and ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$, then $x {\preceq}s_{i}y$.

If ${\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} > 0$ and ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} > 0$, or if ${\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$ and ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$, then $s_{i}x {\preceq}s_{i}y$.

\[lem:SiB\]

Let $w,\,v \in {W^{J}}$, and $\xi,\,\zeta \in Q^{\vee}$. If $w{\Pi^{J}}(t_{\xi}) {\succeq}v{\Pi^{J}}(t_{\zeta})$, then $[\xi]^{{J}} \ge [\zeta]^{{J}}$, where $[\,\cdot\,]^{{J}}:Q^{\vee} \twoheadrightarrow 
Q^{\vee}_{I \setminus {J}}$ is the projection in .

Let $w \in {W^{J}}$, and $\xi,\,\zeta \in Q^{\vee}$. Then, $w{\Pi^{J}}(t_{\xi}) {\succeq}w{\Pi^{J}}(t_{\zeta})$ if and only if $[\xi]^{{J}} \ge [\zeta]^{{J}}$.

Let $x,\,y \in ({W^{J}})_{{\mathrm{af}}}$ and $\beta \in {\Delta_{{\mathrm{af}}}^{+}}$ be such that $x {\xrightarrow{\,\beta\,}} y$ in ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$. Then, ${\Pi^{J}}(xt_{\xi}) {\xrightarrow{\,\beta\,}} {\Pi^{J}}(yt_{\xi})$ in ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ for all $\xi \in Q^{\vee}$. Therefore, if $x {\succeq}y$, then ${\Pi^{J}}(xt_{\xi}) {\succeq}{\Pi^{J}}(yt_{\xi})$ for all $\xi \in Q^{\vee}$.

\[rem:SiB\] Let $w \in W$. Since $w \ge e$ in the ordinary Bruhat order on $W$, we see that $w {\succeq}e$ in the semi-infinite Bruhat order on $W_{{\mathrm{af}}}$. Hence it follows from Lemma \[lem:SiB\](3) that $wt_{\xi} {\succeq}t_{\xi}$ for all $\xi \in Q^{\vee}$.

\[lem:SiL\] Let $x,\,y \in ({W^{J}})_{{\mathrm{af}}}$ be such that $x {\preceq}y$ in $({W^{J}})_{{\mathrm{af}}}$. Then, $xz {\preceq}yz$ in $W_{{\mathrm{af}}}$ for all $z \in ({W_{J}})_{{\mathrm{af}}}$.

Let $z \in ({W_{J}})_{{\mathrm{af}}}$. We know from [@Pet97] (see also [@A Theorem 3.3]) that ${\ell^{{\frac{\infty}{2}}}}(xz) = {\ell^{{\frac{\infty}{2}}}}(x) + {\ell^{{\frac{\infty}{2}}}}(z)$ and ${\ell^{{\frac{\infty}{2}}}}(yz) = {\ell^{{\frac{\infty}{2}}}}(y) + {\ell^{{\frac{\infty}{2}}}}(z)$. Also, we may assume that $x {\xrightarrow{\,\beta\,}} y$ in ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$ for some $\beta \in {\Delta_{{\mathrm{af}}}^{+}}$; by the definition of ${\mathrm{BG}^{{\frac{\infty}{2}}}\bigl(({W^{J}})_{{\mathrm{af}}}\bigr)}$, we have $y = s_{\beta}x$, with ${\ell^{{\frac{\infty}{2}}}}(y)={\ell^{{\frac{\infty}{2}}}}(x)+1$. Therefore, $yz = s_{\beta}xz$, and $${\ell^{{\frac{\infty}{2}}}}(yz) 
 = {\ell^{{\frac{\infty}{2}}}}(y)+{\ell^{{\frac{\infty}{2}}}}(z)
 = {\ell^{{\frac{\infty}{2}}}}(x)+1+{\ell^{{\frac{\infty}{2}}}}(z) = {\ell^{{\frac{\infty}{2}}}}(xz)+1.$$ Thus, we obtain $xz {\xrightarrow{\,\beta\,}} yz$ in ${\mathrm{BG}^{{\frac{\infty}{2}}}(W_{{\mathrm{af}}})}$, as desired.

\[lem:611\] If $x,\,y \in W_{{\mathrm{af}}}$ satisfy $x {\succeq}y$, then ${\Pi^{J}}(x) {\succeq}{\Pi^{J}}(y)$.

Proof of Proposition \[prop:Deo\]. {#sec:prf-Deo}
==================================

\[lem:lift\] Let $x \in ({W^{J}})_{{\mathrm{af}}}$, and write it as[:]{} $x = w {\Pi^{J}}(t_{\xi}) \in ({W^{J}})_{{\mathrm{af}}}$ for some $w \in {W^{J}}$ and $\xi \in Q^{\vee}$ [(]{}see . Then, ${\mathrm{Lift}}(x) = \bigl\{ w' t_{\xi+\gamma} \mid 
 w' \in w{W_{J}},\, \gamma \in {Q_{J}^{\vee}}\bigr\}$.

We set $L := \bigl\{ w' t_{\xi+\gamma} \mid 
 w' \in w{W_{J}},\, \gamma \in {Q_{J}^{\vee}}\bigr\}$. We first prove that $L \subset {\mathrm{Lift}}(x)$. Let $w' t_{\xi+\gamma} \in L$. Then we see from and that ${\Pi^{J}}(w' t_{\xi+\gamma}) = {\Pi^{J}}(w'){\Pi^{J}}(t_{\xi+\gamma}) 
 = {\lfloor w' \rfloor}\Pi(t_{\xi}) = w \Pi(t_{\xi}) = x$. This proves the inclusion $L \subset {\mathrm{Lift}}(x)$.

We next show that $L \supset {\mathrm{Lift}}(x)$. Each element of ${\mathrm{Lift}}(x)$ is of the form $x z$ for some $z \in ({W_{J}})_{{\mathrm{af}}}={W_{J}}\ltimes {Q_{J}^{\vee}}$; we write $z$ as $z = v_{1}t_{\gamma_{1}}$ for some $v_{1} \in {W_{J}}$ and $\gamma_{1} \in {Q_{J}^{\vee}}$. Since ${\Pi^{J}}(t_{\xi}) = v_{2}t_{\xi + \gamma_{2}}$ for some $v_{2} \in {W_{J}}$ and $\gamma_{2} \in {Q_{J}^{\vee}}$, we have $xz = (w {\Pi^{J}}(t_{\xi})) (v_{1}t_{\gamma_{1}}) = 
w (v_{2}t_{\xi + \gamma_{2}})(v_{1}t_{\gamma_{1}}) = 
wv_{2}v_{1}t_{ v_{1}^{-1}(\xi + \gamma_{2}) + \gamma_{1} }$, which is of the form $w v t_{\xi+\gamma}$ for some $v \in {W_{J}}$ and $\gamma \in {Q_{J}^{\vee}}$. Thus, this element is contained in $L$. This proves the opposite inclusion, and hence the lemma.

Now, we give a proof of Proposition \[prop:Deo\]. If $J=I$, then the assertion is obvious. Hence we may assume that $J \subsetneqq I$.

#### Step 1.

Assume that $x = t_{\xi}$ for some $\xi \in Q^{\vee}$, and $y = {\Pi^{J}}(t_{\zeta})$ for some $\zeta \in Q^{\vee}$; since ${\Pi^{J}}(t_{\xi})=y {\succeq}{\Pi^{J}}(x)={\Pi^{J}}(t_{\xi})$ by the assumption, it follows from Lemma \[lem:SiB\](1) that $[\zeta]^{{J}} \ge [\xi]^{{J}}$, where $[\,\cdot\,]^{{J}}:Q^{\vee} \twoheadrightarrow Q_{I \setminus {J}}^{\vee}$ is the projection in . We set $\gamma:=[\zeta-\xi]_{{J}} \in {Q_{J}^{\vee}}$, where $[\,\cdot\,]_{{J}}:Q^{\vee} \twoheadrightarrow {Q_{J}^{\vee}}$ is the projection in ; note that $[\zeta-\gamma]_{{J}} = [\xi]_{{J}}$. We claim that $t_{\zeta-\gamma}$ is the minimum element of ${\mathrm{Lift}_{{\succeq}x}(y)}$. It is clear by Lemma \[lem:lift\] that $t_{\zeta-\gamma} \in {\mathrm{Lift}}(y)$. In addition, since $[\zeta-\gamma]^{{J}} = [\zeta]^{{J}} \ge [\xi]^{{J}}$ and $[\zeta-\gamma]_{{J}} = [\xi]_{{J}}$, we have $\zeta - \gamma \ge \xi$, and hence $t_{\zeta - \gamma} {\succeq}t_{\xi} = x$ by Lemma \[lem:SiB\](2). Thus, $t_{\zeta - \gamma} \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Now, by Lemma \[lem:lift\], each element $y' \in {\mathrm{Lift}_{{\succeq}x}(y)} \subset {\mathrm{Lift}}(y)$ is of the form $y'=v't_{\zeta-\gamma'}$ for some $v' \in {W_{J}}$ and $\gamma' \in {Q_{J}^{\vee}}$. Since $v' t_{\zeta-\gamma'} = y' {\succeq}x = t_{\xi}$ in $W_{{\mathrm{af}}}$ by the assumption, we deduce from Lemma \[lem:SiB\](1) that $\zeta-\gamma' \ge \xi$; in particular, $[\zeta-\gamma']_{{J}} \ge [\xi]_{{J}}$. Here, since $\gamma=[\zeta-\xi]_{{J}}$ by the definition, we have $[\zeta-\gamma']_{{J}} \ge [\xi]_{{J}}=[\zeta-\gamma]_{{J}}$. Also, since $\gamma,\,\gamma' \in {Q_{J}^{\vee}}$, we have $[\zeta-\gamma']^{{J}} = [\zeta-\gamma]^{{J}}$. Combining these, we obtain $\zeta-\gamma' \ge \zeta-\gamma$. Therefore, by Remark \[rem:SiB\] and Lemma \[lem:SiB\](2), $y'=v' t_{\zeta-\gamma'} {\succeq}t_{\zeta-\gamma'} {\succeq}t_{\zeta-\gamma}$. Thus, $t_{\zeta-\gamma}$ is the minimum element of ${\mathrm{Lift}_{{\succeq}x}(y)}$.

In the following, we fix $\Lambda \in P^{+}$ and $\lambda \in P^{+}$ such that $\bigl\{ i \in I \mid {\langle \Lambda,\,\alpha_{i}^{\vee} \rangle} = 0\bigr\} = \emptyset$ and $\bigl\{ i \in I \mid {\langle \lambda,\,\alpha_{i}^{\vee} \rangle} = 0\bigr\} = {J}$. Note that ${\langle \Lambda,\,\beta^{\vee} \rangle} \ne 0$ for all $\beta \in {\Delta_{{\mathrm{af}}}}$.

#### Step 2.

Let $x \in W_{{\mathrm{af}}}$, and assume that $y = {\Pi^{J}}(t_{\zeta})$ for some $\zeta \in Q^{\vee}$. We deduce from [@AK] that there exist $i_{1},\,\dots,\,i_{N} \in I_{{\mathrm{af}}}$ such that ${\langle s_{i_{n-1}} \cdots s_{i_{1}}x\Lambda,\,\alpha_{i_n}^{\vee} \rangle} > 0$ for all $1 \le n \le N$, and such that $s_{i_{N}} \cdots s_{i_{1}}x = t_{\xi}$ for some $\xi \in Q^{\vee}$. We show the assertion of the proposition by induction on $N$. If $N=0$, then the assertion follows from Step 1. Assume that $N \ge 1$; for simplicity of notation, we set $i:=i_{1} \in I_{{\mathrm{af}}}$. Since $$\label{eq:s2-1}
{\langle x\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$$ by the assumption, it follows that $x^{-1}\alpha_{i} \in \Delta^{+}$, and hence ${\langle {\Pi^{J}}(x)\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} \ge 0$. Also, by Lemma \[lem:si\] and Remark \[rem:si\], $$\label{eq:Pix}
{\Pi^{J}}(s_{i}x) = 
 \begin{cases}
 s_{i}{\Pi^{J}}(x) & \text{if ${\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} > 0$}, \\[1mm]
 {\Pi^{J}}(x) & \text{if ${\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} = 0$}.
 \end{cases}
$$

#### Case 2.1.

Assume that $i \in I \setminus {J}$, and hence ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle}=
{\langle \lambda,\,\alpha_{i}^{\vee} \rangle} > 0$; note that $s_{i}y \in ({W^{J}})_{{\mathrm{af}}}$ by Lemma \[lem:si\]. We first claim that $s_{i}y {\succeq}{\Pi^{J}}(s_{i}x)$. Indeed, if ${\langle {\Pi^{J}}(x)\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} > 0$, then it follows from Lemma \[lem:dmd\](3), , and the assumption $y {\succeq}{\Pi^{J}}(x)$ that $s_{i}y {\succeq}s_{i}{\Pi^{J}}(x) = {\Pi^{J}}(s_{i}x)$. If ${\langle {\Pi^{J}}(x)\lambda,\,\alpha_{i}^{\vee} \rangle} = {\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} = 0$, then it follows from Lemma \[lem:si\], , and the assumption $y {\succeq}{\Pi^{J}}(x)$ that $s_{i}y {\succ}y {\succeq}{\Pi^{J}}(x) = {\Pi^{J}}(s_{i}x)$. In both cases, we obtain $s_{i}y {\succeq}{\Pi^{J}}(s_{i}x)$, as desired. Hence, by our induction hypothesis, $${\mathrm{Lift}_{{\succeq}s_{i}x}(s_{i}y)}=
\bigl\{ y'' \in W_{{\mathrm{af}}} \mid 
 \text{\rm ${\Pi^{J}}(y'') = s_{i}y$ and $y'' {\succeq}s_{i}x$}
\bigr\}$$ has the minimum element $y''_{\min}$. We next claim that $s_{i}y''_{\min} \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Indeed, since ${\langle y''_{\min}\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle {\Pi^{J}}(y''_{\min})\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle s_{i}y\lambda,\,\alpha_{i}^{\vee} \rangle} = - {\langle y\lambda,\,\alpha_{i} \rangle} < 0$, it follows from Lemma \[lem:si\] that ${\Pi^{J}}(s_{i}y''_{\min}) = s_{i} {\Pi^{J}}(y''_{\min}) = s_{i}(s_{i}y) = y$, which implies that $s_{i}y''_{\min} \in {\mathrm{Lift}}(y)$. In addition, the inequality ${\langle y''_{\min}\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$ above implies that ${\langle y''_{\min}\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$. Since ${\langle s_{i}x\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$ by , we deduce from Lemma \[lem:dmd\](3), together with the assumption $y''_{\min} {\succeq}s_{i}x$, that $s_{i}y''_{\min} {\succeq}x$. Thus we get $s_{i}y''_{\min} \in {\mathrm{Lift}_{{\succeq}x}(y)}$, as desired. Finally, we claim that $$\label{eq:min2-1}
\text{$s_{i}y''_{\min}$ is the minimum element of ${\mathrm{Lift}_{{\succeq}x}(y)}$}.$$ Let $y' \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Since ${\langle y'\lambda,\,\alpha_{i}^{\vee} \rangle}=
{\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} = {\langle \lambda,\,\alpha_{i}^{\vee} \rangle} > 0$ by our assumption, we see that $(y')^{-1}\alpha_{i} \in \Delta^{+} + {\mathbb{Z}}\delta$, and hence ${\langle y'\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$. Also, since ${\langle x\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$ by , it follows from Lemma \[lem:dmd\](3) that $s_{i}y' {\succeq}s_{i}x$, which implies that $s_{i}y' \in {\mathrm{Lift}_{{\succeq}s_{i}x}(s_{i}y)}$. Therefore, we obtain $s_{i}y' {\succeq}y''_{\min}$. Since ${\langle y''_{\min}\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$ as seen above, and ${\langle s_{i}y'\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$, we deduce from Lemma \[lem:dmd\](3) that $y' {\succeq}s_{i}y''_{\min}$. This shows .

#### **Case 2.2.**

Assume that $i \in {J}$, and hence ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle}=
{\langle \lambda,\,\alpha_{i}^{\vee} \rangle} = 0$. We first claim that $y {\succeq}{\Pi^{J}}(s_{i}x)$. Indeed, if ${\langle {\Pi^{J}}(x)\lambda,\,\alpha_{i}^{\vee} \rangle} 
 = {\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} > 0$, then it follows from Lemma \[lem:dmd\](1), , and the assumption $y {\succeq}{\Pi^{J}}(x)$ that $y {\succeq}s_{i}{\Pi^{J}}(x) = {\Pi^{J}}(s_{i}x)$. If ${\langle {\Pi^{J}}(x)\lambda,\,\alpha_{i}^{\vee} \rangle} = {\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} = 0$, then $y {\succeq}{\Pi^{J}}(x) = {\Pi^{J}}(s_{i}x)$ by . In both cases, we obtain $y {\succeq}{\Pi^{J}}(s_{i}x)$, as desired. Hence, by our induction hypothesis, $${\mathrm{Lift}_{{\succeq}s_{i}x}(y)}=
\bigl\{ y'' \in W_{{\mathrm{af}}} \mid 
 \text{\rm ${\Pi^{J}}(y'') = y$ and $y'' {\succeq}s_{i}x$}
\bigr\}$$ has the minimum element $y''_{\min}$. We set $$y'_{\min}:=
 \begin{cases}
 y''_{\min} & 
 \text{if ${\langle  y''_{\min}\Lambda ,\, \alpha_{i}^{\vee}  \rangle} > 0$}, \\[1mm]
 s_{i}y''_{\min} & 
 \text{if ${\langle  y''_{\min}\Lambda ,\, \alpha_{i}^{\vee}  \rangle} < 0$}; 
 \end{cases}$$ remark that $y'_{\min} {\preceq}y''_{\min}$ by Lemma \[lem:si\]. First, we show that $y'_{\min} \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Since $y''_{\min} \in {\mathrm{Lift}}(y)$ and $i \in {J}$, it follows from Remark \[rem:si\] that $y'_{\min} \in {\mathrm{Lift}}(y)$. Also, since ${\langle x\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$, we see by Lemma \[lem:si\] that $s_{i}x {\succeq}x$. Hence we have $y_{\min}' = y_{\min}'' {\succeq}s_{i}x {\succeq}x$ if ${\langle  y''_{\min}\Lambda ,\, \alpha_{i}^{\vee}  \rangle} > 0$. If ${\langle  y''_{\min}\Lambda ,\, \alpha_{i}^{\vee}  \rangle} < 0$, then we deduce from Lemma \[lem:dmd\](3) that $y_{\min}' = s_{i}y_{\min}'' {\succeq}s_{i}(s_{i}x) = x$ since $y_{\min}'' {\succeq}s_{i}x$. Thus, in both cases, we obtain $y'_{\min} \in {\mathrm{Lift}_{{\succeq}x}(y)}$, as desired. Next, we show that $$\label{eq:min2-2}
\text{$y'_{\min}$ is the minimal element of ${\mathrm{Lift}_{{\succeq}x}(y)}$}.$$ Let $y' \in {\mathrm{Lift}_{{\succeq}x}(y)}$. If ${\langle y'\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$, then it follows from Lemma \[lem:dmd\](1) that $y' {\succeq}s_{i}x$, and hence $y' \in {\mathrm{Lift}_{{\succeq}s_{i}x}(y)}$. This implies that $y' {\succeq}y_{\min}''$. If ${\langle  y''_{\min}\Lambda ,\, \alpha_{i}^{\vee}  \rangle} > 0$, then we have $y' {\succeq}y_{\min}'' = y_{\min}'$ by the definition. If ${\langle  y''_{\min}\Lambda ,\, \alpha_{i}^{\vee}  \rangle} < 0$, then we see from Lemma \[lem:si\] that $y' {\succeq}y_{\min}'' {\succeq}s_{i}y_{\min}'' = y_{\min}'$. Assume now that ${\langle y'\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$. Since ${\langle x\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$, it follows from Lemma \[lem:dmd\](3) that $s_{i}y' {\succeq}s_{i}x$. In addition, since $i \in {J}$ and $y' \in {\mathrm{Lift}}(y)$, we see from Remark \[rem:si\] that ${\Pi^{J}}(s_{i}y')=y$. Hence we obtain $s_{i}y' \in {\mathrm{Lift}_{{\succeq}s_{i}x}(y)}$, so that $s_{i}y' {\succeq}y''_{\min}$; note that ${\langle s_{i}y'\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$ by our assumption. If ${\langle  y''_{\min}\Lambda ,\, \alpha_{i}^{\vee}  \rangle} > 0$, then we deduce from Lemma \[lem:dmd\](2) that $y' {\succeq}y''_{\min} = y'_{\min}$. If ${\langle  y''_{\min}\Lambda ,\, \alpha_{i}^{\vee}  \rangle} < 0$, then we deduce from Lemma \[lem:dmd\](3) that $y' {\succeq}s_{i}y''_{\min} = y'_{\min}$. Thus, in all cases, we have shown that $y' {\succeq}y'_{\min}$, as desired.

#### **Case 2.3.**

Assume that $i = 0$. In this case, we have ${\langle y\lambda,\,\alpha_{0}^{\vee} \rangle}=
{\langle \lambda,\,\alpha_{0}^{\vee} \rangle} < 0$ since $\alpha_{0}=-\theta+\delta$, where $\theta \in \Delta^{+}$ is the highest root. By the same argument as that at the beginning of Case 2.2, we see that $y {\succeq}{\Pi^{J}}(s_{0}x)$. Hence, by the induction hypothesis, $${\mathrm{Lift}_{{\succeq}s_{0}x}(y)}=
\bigl\{ y'' \in W_{{\mathrm{af}}} \mid 
 \text{\rm ${\Pi^{J}}(y'') = y$ and $y'' {\succeq}s_{0}x$}
\bigr\}$$ has the minimum element $y''_{\min}$. Since ${\langle  x\Lambda ,\, \alpha_{0}^{\vee}  \rangle} > 0$ by , it follows from Lemma \[lem:si\] that $s_{0}x {\succeq}x$, which implies that $y''_{\min} \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Here we claim that $$\text{$y''_{\min}$ is the minimal element of ${\mathrm{Lift}_{{\succeq}x}(y)}$.}$$ Let $y' \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Then we have ${\langle  y'\Lambda ,\, \alpha_{0}^{\vee}  \rangle} < 0$. Indeed, we deduce from Lemma \[lem:lift\] that $y'=z t_{\zeta+\gamma}$ for some $z \in {W_{J}}$ and $\gamma \in {Q_{J}^{\vee}}$. Since $z \in {W_{J}}$ and $\theta \in \Delta^{+} \setminus {\Delta_{J}}^{+}$ (recall that $J \subsetneqq I$), we see that $z^{-1}\theta \in \Delta^{+} \setminus {\Delta_{J}}^{+}$, and hence that ${\langle y'\Lambda,\,\alpha_{0}^{\vee} \rangle} = 
{\langle \Lambda,\,-z^{-1}\theta^{\vee} \rangle} < 0$. Since ${\langle  x\Lambda ,\, \alpha_{0}^{\vee}  \rangle} > 0$ by , it follows from Lemma \[lem:dmd\](1) that $y' {\succeq}s_{0}x$, and hence $y' \in {\mathrm{Lift}_{{\succeq}s_{0}x}(y)}$. This shows that $y' {\succeq}y''_{\min}$.

#### Step 3.

Let $x \in W_{{\mathrm{af}}}$, $y \in ({W^{J}})_{{\mathrm{af}}}$, and write $y$ as $y = v {\Pi^{J}}(t_{\zeta})$ for some $v \in {W^{J}}$ and $\zeta \in Q^{\vee}$. We show the assertion by induction on $\ell(v)$. If $\ell(v)=0$, then the assertion follows from Step 2. Assume that $\ell(v) \ge 1$, and take $i \in I$ such that ${\langle v\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$; note that in this case, $v^{-1}\alpha_{i} \in - (\Delta^{+} \setminus {\Delta_{J}}^{+})$, and $s_{i}v \in W^{J}$ (see, for example, [@LNSSS Lemmas 5.8 and 5.9]), and hence that $s_{i}y \in ({W^{J}})_{{\mathrm{af}}}$. Also, for all $y' \in {\mathrm{Lift}}(y)$, we have ${\langle y'\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle v\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$, which implies that $$\label{eq:S3-1}
{\langle y'\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0.$$

#### Case 3.1.

Assume that ${\langle x\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$; note that ${\langle {\Pi^{J}}(x)\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} \ge 0$. Since ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} = {\langle v\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$, it follows from Lemma \[lem:dmd\](2) that $s_{i}y {\succeq}{\Pi^{J}}(x)$. Hence, by our induction hypothesis, $${\mathrm{Lift}_{{\succeq}x}(s_{i}y)} =
\bigl\{ y'' \in W_{{\mathrm{af}}} \mid 
 \text{\rm ${\Pi^{J}}(y'') = s_{i}y$ and $y'' {\succeq}x$}
\bigr\}$$ has the minimum element $y''_{\min}$. Since ${\langle y''_{\min}\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle s_{i}y\lambda,\,\alpha_{i}^{\vee} \rangle} > 0$, it follows that $(y''_{\min})^{-1}\alpha_{i} \in \Delta^{+}$, and hence ${\langle y''_{\min}\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$. This implies that $s_{i}y''_{\min} {\succeq}y''_{\min} {\succeq}x$ by Lemma \[lem:si\]. In addition, we see by Lemma \[lem:si\] that ${\Pi^{J}}(s_{i}y''_{\min}) = s_{i}{\Pi^{J}}(y''_{\min}) = s_{i}(s_{i}y) = y$. Therefore, we conclude that $s_{i}y''_{\min} \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Here we claim that $$\label{eq:min3-1}
\text{$s_{i}y''_{\min}$ is the minimum element of ${\mathrm{Lift}_{{\succeq}x}(y)}$}.$$ Let $y' \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Since ${\langle x\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$ by the assumption, and ${\langle y'\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$ by , we deduce from Lemma \[lem:dmd\](2) that $s_{i}y' {\succeq}x$. In addition, we see by Lemma \[lem:si\] that ${\Pi^{J}}(s_{i}y') = s_{i} {\Pi^{J}}(y') = s_{i}y$, which implies that $s_{i}y' \in {\mathrm{Lift}_{{\succeq}x}(s_{i}y)}$, and hence $s_{i}y' {\succeq}y''_{\min}$. Because ${\langle y''_{\min}\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$ and ${\langle s_{i}y'\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$, it follows from Lemma \[lem:dmd\](3) that $y' {\succeq}s_{i}y''_{\min}$. This shows .

#### Case 3.2.

Assume that ${\langle x\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$; note that ${\langle x\lambda,\,\alpha_{i}^{\vee} \rangle} \le 0$. Since ${\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$, it follows from Lemma \[lem:dmd\](2) and (3), together with Lemma \[lem:si\] and Remark \[rem:si\], that $s_{i}y {\succeq}{\Pi^{J}}(s_{i}x)$. Hence, by our induction hypothesis, $${\mathrm{Lift}_{{\succeq}s_{i}x}(s_{i}y)} = 
\bigl\{ y'' \in W_{{\mathrm{af}}} \mid 
 \text{\rm ${\Pi^{J}}(y'') = s_{i}y$ and $y'' {\succeq}s_{i}x$}
\bigr\}$$ has the minimum element $y''_{\min}$; as in Case 3.1, we obtain ${\langle y''_{\min}\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$. Since ${\langle s_{i}x\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$, we deduce from Lemma \[lem:dmd\](3) that $s_{i}y''_{\min} {\succeq}x$. In addition, we see by Lemma \[lem:si\] that ${\Pi^{J}}(s_{i}y_{\min}'') = s_{i} {\Pi^{J}}(y_{\min}'') = s_{i} (s_{i}y) = y$, which implies that $s_{i}y''_{\min} \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Here we claim that $$\label{eq:min3-2}
\text{$s_{i}y''_{\min}$ is the minimum element of ${\mathrm{Lift}_{{\succeq}x}(y)}$}.$$ Let $y' \in {\mathrm{Lift}_{{\succeq}x}(y)}$. Since ${\langle y'\lambda,\,\alpha_{i}^{\vee} \rangle} = 
{\langle y\lambda,\,\alpha_{i}^{\vee} \rangle} < 0$, it follows that ${\langle y'\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$. Also, since ${\langle x\Lambda,\,\alpha_{i}^{\vee} \rangle} < 0$ by the assumption, we deduce from Lemma \[lem:dmd\](3) that $s_{i}y' {\succeq}s_{i}x$. In addition, we see by Lemma \[lem:si\] that ${\Pi^{J}}(s_{i}y') = s_{i}{\Pi^{J}}(y') = s_{i}y$, which implies that $s_{i}y' \in {\mathrm{Lift}_{{\succeq}s_{i}x}(s_{i}y)}$, and hence $s_{i}y' {\succeq}y''_{\min}$. Because ${\langle y''_{\min}\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$ and ${\langle s_{i}y'\Lambda,\,\alpha_{i}^{\vee} \rangle} > 0$, it follows from Lemma \[lem:dmd\](3) that $y' {\succeq}s_{i}y''_{\min}$. This shows .

This completes the proof of Proposition \[prop:Deo\].

Crystal structure on ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$. {#sec:crystal}
==================================================================

We fix $\lambda \in P^{+} \subset P_{{\mathrm{af}}}^{0}$ (see and ). Let $$\label{eq:pi2}
\pi = ({\mathbf{x}}\,;\, {\mathbf{a}}) 
     = (x_{1},\,\dots,\,x_{s} \,;\, a_{0},\,a_{1},\,\dots,\,a_{s}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda).$$ Define ${\overline{\pi}}:[0,1] \rightarrow {\mathbb{R}}\otimes_{{\mathbb{Z}}} P_{{\mathrm{af}}}$ to be the piecewise-linear, continuous map whose “direction vector” on the interval $[a_{u-1},\,a_{u}]$ is $x_{u}\lambda \in P_{{\mathrm{af}}}$ for each $1 \le u \le s$, that is, $$\label{eq:olpi}
{\overline{\pi}} (t) := 
\sum_{k = 1}^{u-1}(a_{k} - a_{k-1}) x_{k}\lambda + (t - a_{u-1}) x_{u}\lambda
\quad
\text{for $t \in [a_{u-1},\,a_u]$, $1 \le u \le s$}.$$ We know from [@INS Proposition 3.1.3] that ${\overline{\pi}}$ is an (ordinary) LS path of shape $\lambda$, introduced in [@Lit95 Sect. 4]. We set $$\label{eq:wt}
{\mathop{\rm wt}\nolimits}(\pi):= {\overline{\pi}}(1) = \sum_{u = 1}^{s} (a_{u}-a_{u-1})x_{u}\lambda \in P_{{\mathrm{af}}}.$$

Now, we define root operators $e_{i}$, $f_{i}$, $i \in I_{{\mathrm{af}}}$. Set $$\label{eq:H}
\begin{cases}
H^{\pi}_{i}(t) := {\langle {\overline{\pi}}(t),\,\alpha_{i}^{\vee} \rangle} \quad 
\text{for $t \in [0,1]$}, \\[1.5mm]
m^{\pi}_{i} := 
 \min \bigl\{ H^{\pi}_{i} (t) \mid t \in [0,1] \bigr\}. 
\end{cases}
$$ As explained in [@NS16 Remark 2.4.3], all local minima of the function $H^{\pi}_{i}(t)$, $t \in [0,1]$, are integers; in particular, the minimum value $m^{\pi}_{i}$ is a nonpositive integer (recall that ${\overline{\pi}}(0)=0$, and hence $H^{\pi}_{i}(0)=0$).

We define $e_{i}\pi$ as follows. If $m^{\pi}_{i}=0$, then we set $e_{i} \pi := {\bm{0}}$, where ${\bm{0}}$ is an additional element not contained in any crystal. If $m^{\pi}_{i} \le -1$, then we set $$\label{eq:t-e}
\begin{cases}
t_{1} := 
  \min \bigl\{ t \in [0,\,1] \mid 
    H^{\pi}_{i}(t) = m^{\pi}_{i} \bigr\}, \\[1.5mm]
t_{0} := 
  \max \bigl\{ t \in [0,\,t_{1}] \mid 
    H^{\pi}_{i}(t) = m^{\pi}_{i} + 1 \bigr\}; 
\end{cases}
$$ notice that $H^{\pi}_{i}(t)$ is strictly decreasing on the interval $[t_{0},\,t_{1}]$. Let $1 \le p \le q \le s$ be such that $a_{p-1} \le t_{0} < a_p$ and $t_{1} = a_{q}$. Then we define $e_{i}\pi$ to be $$\label{eq:epi}
\begin{split}
& e_{i} \pi := ( 
  x_{1},\,\ldots,\,x_{p},\,s_{i}x_{p},\,s_{i}x_{p+1},\,\ldots,\,
  s_{i}x_{q},\,x_{q+1},\,\ldots,\,x_{s} ; \\
& \hspace*{40mm}
  a_{0},\,\ldots,\,a_{p-1},\,t_{0},\,a_{p},\,\ldots,\,a_{q}=t_{1},\,
\ldots,\,a_{s});
\end{split}
$$ if $t_{0} = a_{p-1}$, then we drop $x_{p}$ and $a_{p-1}$, and if $s_{i} x_{q} = x_{q+1}$, then we drop $x_{q+1}$ and $a_{q}=t_{1}$.

Similarly, we define $f_{i}\pi$ as follows. Note that $H^{\pi}_{i}(1) - m^{\pi}_{i}$ is a nonnegative integer. If $H^{\pi}_{i}(1) - m^{\pi}_{i} = 0$, then we set $f_{i} \pi := {\bm{0}}$. If $H^{\pi}_{i}(1) - m^{\pi}_{i}  \ge 1$, then we set $$\label{eq:t-f}
\begin{cases}
t_{0} := 
 \max \bigl\{ t \in [0,1] \mid H^{\pi}_{i}(t) = m^{\pi}_{i} \bigr\}, \\[1.5mm]
t_{1} := 
 \min \bigl\{ t \in [t_{0},\,1] \mid H^{\pi}_{i}(t) = m^{\pi}_{i} + 1 \bigr\};
\end{cases}
$$ notice that $H^{\pi}_{i}(t)$ is strictly increasing on the interval $[t_{0},\,t_{1}]$. Let $0 \le p \le q \le s-1$ be such that $t_{0} = a_{p}$ and $a_{q} < t_{1} \le a_{q+1}$. Then we define $f_{i}\pi$ to be $$\label{eq:fpi}
\begin{split}
& f_{i} \pi := ( x_{1},\,\ldots,\,x_{p},\,s_{i}x_{p+1},\,\dots,\,
  s_{i} x_{q},\,s_{i} x_{q+1},\,x_{q+1},\,\ldots,\,x_{s} ; \\
& \hspace{40mm} 
  a_{0},\,\ldots,\,a_{p}=t_{0},\,\ldots,\,a_{q},\,t_{1},\,
  a_{q+1},\,\ldots,\,a_{s});
\end{split}
$$ if $t_{1} = a_{q+1}$, then we drop $x_{q+1}$ and $a_{q+1}$, and if $x_{p} = s_{i} x_{p+1}$, then we drop $x_{p}$ and $a_{p}=t_{0}$. In addition, we set $e_{i} {\bm{0}}= f_{i} {\bm{0}}:= {\bm{0}}$ for all $i \in I_{{\mathrm{af}}}$.

\[thm:SLS\]

The set ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \sqcup \{ {\bm{0}}\}$ is stable under the action of the root operators $e_{i}$ and $f_{i}$, $i \in I_{{\mathrm{af}}}$.

For each $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $i \in I_{{\mathrm{af}}}$, we set $$\begin{cases}
{\varepsilon}_{i} (\pi) := 
 \max \bigl\{ n \ge 0 \mid e_{i}^{n} \pi \neq {\bm{0}}\bigr\}, \\[1.5mm]
{\varphi}_{i} (\pi) := 
 \max \bigl\{ n \ge 0 \mid f_{i}^{n} \pi \neq {\bm{0}}\bigr\}.
\end{cases}
$$ Then, the set ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$, equipped with the maps ${\mathop{\rm wt}\nolimits}$, $e_{i}$, $f_{i}$, $i \in I_{{\mathrm{af}}}$, and ${\varepsilon}_{i}$, ${\varphi}_{i}$, $i \in I_{{\mathrm{af}}}$, defined above, is a crystal with weights in $P_{{\mathrm{af}}}$.

\[rem:SLS\] Let $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$, and $i \in I_{{\mathrm{af}}}$. If $e_{i}\pi \ne {\bm{0}}$, then we deduce from the definition of the root operator $e_{i}$ that $m^{e_{i}\pi}_{i} = m^{\pi}_{i}+1$. Hence it follows that ${\varepsilon}_{i}(\pi)=-m^{\pi}_{i}$. Similarly, we have ${\varphi}_{i}(\pi)=H^{\pi}_{i}(1)-m^{\pi}_{i}$.

A formula for graded characters of Demazure submodules. {#sec:tx-gch}
=======================================================

\[prop:gch-tx\] For each $x \in W_{{\mathrm{af}}}$ and $\xi \in Q^{\vee}$, there holds the equality $${\mathop{\rm gch}\nolimits}V_{xt_{\xi}}^{-}(\lambda) = q^{-{\langle \lambda,\,\xi \rangle}}{\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda).$$

Let $\pi = (x_{1},\,\dots,\,x_{s}\,;\,{\mathbf{a}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$. We see that $$\begin{aligned}
\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda) 
 & \Rightarrow x_{s} {\succeq}{\Pi^{J}}(x) 
   \Rightarrow {\Pi^{J}}(x_{s}t_{\xi}) {\succeq}{\Pi^{J}}({\Pi^{J}}(x)t_{\xi}) 
   \quad \text{by Lemma~\ref{lem:SiB}\,(3)} \\
 & \Rightarrow {\Pi^{J}}(x_{s}t_{\xi}) {\succeq}{\Pi^{J}}(xt_{\xi}) 
   \quad \text{by \eqref{eq:PiJ2}} \\
 & \Rightarrow T_{\xi}(\pi) \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}xt_{\xi}}(\lambda);\end{aligned}$$ for the definition of $T_{\xi}$, see . From this, we conclude that $T_{\xi}({\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)) \subset
{\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}xt_{\xi}}(\lambda)$. Replacing $x$ by $xt_{\xi}$, and $T_{\xi}$ by $T_{-\xi}$, we obtain $T_{-\xi}({\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}xt_{\xi}}(\lambda)) \subset
{\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)$, and hence ${\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}xt_{\xi}}(\lambda) \subset T_{\xi}({\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda))$. Combining these, we conclude that $T_{\xi}({\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)) = {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}xt_{\xi}}(\lambda)$. Therefore, using , we compute: $$\begin{aligned}
{\mathop{\rm gch}\nolimits}V_{xt_{\xi}}^{-}(\lambda) 
 & = \sum_{\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}xt_{\xi}}(\lambda)} e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))} q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))}
   = \sum_{\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)} 
      e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(T_{\xi}(\pi)))} 
      q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(T_{\xi}(\pi)))} \\[3mm]
 & = \sum_{\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)} 
      e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\pi)-{\langle \lambda,\,\xi \rangle}\delta)} 
      q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\pi)-{\langle \lambda,\,\xi \rangle}\delta)} 
      \quad \text{by \eqref{eq:Tx}} \\[3mm]
 & = \sum_{\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x}(\lambda)} 
      e^{{\mathop{\rm fin}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))} 
      q^{{\mathop{\rm nul}\nolimits}({\mathop{\rm wt}\nolimits}(\pi))-{\langle \lambda,\,\xi \rangle}}
   = q^{-{\langle \lambda,\,\xi \rangle}} {\mathop{\rm gch}\nolimits}V_{x}^{-}(\lambda). \end{aligned}$$ This proves the proposition.

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[^1]: Key words and phrases: semi-infinite flag manifold, normality, $K$-theory, Pieri-Chevalley formula, standard monomial theory, semi-infinite Lakshmibai-Seshadri path. Mathematics Subject Classification 2010: Primary 17B37; Secondary 14N15, 14M15, 33D52, 81R10.
