2.1 Notation for root systems

Let G be a connected, simply-connected, simple algebraic group over $\mathbb {C}$ , H a maximal torus of G. We set $\mathfrak {g} := \mathrm {Lie}(G)$ and $\mathfrak {h} := \mathrm {Lie}(H)$ . Thus, $\mathfrak {g}$ is a finite-dimensional simple Lie algebra over $\mathbb {C}$ , and $\mathfrak {h}$ is a Cartan subalgebra of $\mathfrak {g}$ . We denote by $\langle \cdot \,, \cdot \rangle : \mathfrak {h}^{\ast } \times \mathfrak {h} \rightarrow \mathbb {C}$ the canonical pairing, where $\mathfrak {h}^{\ast } = \mathrm {Hom}_{\mathbb {C}}(\mathfrak {h}, \mathbb {C})$ .

Let $\Delta \subset \mathfrak {h}^{\ast }$ be the root system of $\mathfrak {g}$ , $\Delta ^{+} \subset \Delta $ the set of positive roots, and $\{ \alpha _{i} \}_{i \in I} \subset \Delta ^{+}$ the set of simple roots. We denote by $\alpha ^{\vee } \in \mathfrak {h}$ the coroot corresponding to $\alpha \in \Delta $ . Also, we denote by $\theta \in \Delta ^+$ the highest root of $\Delta $ and set $\rho := (1/2) \sum _{\alpha \in \Delta ^{+}} \alpha $ . The root lattice Q and the coroot lattice $Q^{\vee }$ of $\mathfrak {g}$ are defined by $Q := \sum _{i \in I} \mathbb {Z} \alpha _{i}$ and $Q^{\vee } := \sum _{i \in I} \mathbb {Z} \alpha _{i}^{\vee }$ . In addition, for $\alpha \in \Delta $ , we set

$$ \begin{align*} \operatorname{\mathrm{sgn}}(\alpha):= \begin{cases} 1 & \text{if } \alpha \in \Delta^{+}, \\ -1 & \text{if } \alpha \in \Delta^{-}, \end{cases} \qquad |\alpha|:=\operatorname{\mathrm{sgn}}(\alpha)\alpha \in \Delta^{+}. \end{align*} $$

For $i \in I$ , the weight $\varpi _{i} \in \mathfrak {h}^{\ast }$ given by $\langle \varpi _{i}, \alpha _{j}^{\vee } \rangle = \delta _{i,j}$ for all $j \in I$ , where $\delta _{i,j}$ denotes the Kronecker delta, is called the i-th fundamental weight. The weight lattice P of $\mathfrak {g}$ is defined by $P := \sum _{i \in I} \mathbb {Z} \varpi _{i}$ . We denote by $\mathbb {Z}[P]$ the group algebra of P, which is the associative algebra generated by formal elements $\mathbf {e}^{\nu }$ , $\nu \in P$ , where the product is defined by $\mathbf {e}^{\mu } \mathbf {e}^{\nu } := \mathbf {e}^{\mu + \nu }$ for $\mu , \nu \in P$ ; we identify the group algebra $\mathbb {Z}[P]$ with the representation ring $R(H)$ of H.

A reflection $s_{\alpha } \in GL(\mathfrak {h}^{\ast })$ , $\alpha \in \Delta $ , is defined by $s_{\alpha } \mu := \mu - \langle \mu , \alpha ^{\vee } \rangle \alpha $ for $\mu \in \mathfrak {h}^{\ast }$ . We write $s_{i} := s_{\alpha _{i}}$ for $i \in I$ . Then the (finite) Weyl group W of $\mathfrak {g}$ is defined to be the subgroup of $GL(\mathfrak {h}^{\ast })$ generated by $\{ s_{i} \}_{i \in I}$ ; that is, $W := \langle s_{i} \mid i \in I \rangle $ . For $w \in W$ , there exist $i_{1}, \ldots , i_{r} \in I$ such that $w = s_{i_{1}} \cdots s_{i_{r}}$ . If r is minimal, then the product $s_{i_{1}} \cdots s_{i_{r}}$ is called a reduced expression for w, and r is called the length of w; we denote by $\ell (w)$ the length of w. Note that a reduced expression for w is not unique, but the length $\ell (w)$ is uniquely determined. Also, the affine Weyl group $W_{\mathrm {af}}$ of $\mathfrak {g}$ is, by definition, the semi-direct product group $W \ltimes \{ t_{\xi } \mid \xi \in Q^{\vee } \}$ of W and the abelian group $\{ t_{\xi } \mid \xi \in Q^{\vee } \} \cong Q^{\vee }$ , where $t_{\xi }$ denotes the translation in $\mathfrak {h}^{\ast }$ corresponding to $\xi \in Q^{\vee }$ .

For an edge $x \xrightarrow {\hspace {2pt}\alpha \hspace {2pt}} y$ in $\mathrm {QBG}(W)$ , we sometimes write $x \xrightarrow [\mathsf {B}]{\hspace {2pt}\alpha \hspace {2pt}} y$ (resp., $x \xrightarrow [\mathsf {Q}]{\hspace {2pt}\alpha \hspace {2pt}} y$ ) to indicate that the edge is a Bruhat (resp., quantum) edge.

2.2 The root system of type A

We recall the root system of type A. In the rest of this paper, when G is of type A, we use the notation introduced in this subsection. Also, we set $[m]:=\{ 1,2,\dots ,m \}$ for $m \in \mathbb {Z}_{\ge 0}$ .

Assume that G is of type $A_{n}$ (i.e., $G = SL_{n+1}(\mathbb {C})$ ). Then $\mathfrak {g} = \mathfrak {sl}_{n+1}(\mathbb {C})$ , and $\mathfrak {h} := \{ h \in \mathfrak {g} \mid h \text { is a diagonal matrix}\}$ is a Cartan subalgebra of $\mathfrak {g}$ . We let $\{\epsilon _{k} \mid k \in [n+1] \}$ be the standard basis of $\mathbb {Z}^{n+1}$ and realize the weight lattice as $P=\mathbb {Z}^{n+1}/\mathbb {Z}(\epsilon _1+\cdots +\epsilon _{n+1})$ . By abuse of notation, we continue to denote the image of $\epsilon _k$ in P by the same symbol. Thus, $\varpi _k := \epsilon _1+\dotsm +\epsilon _k$ , for $k \in I = [n]$ , are the fundamental weights of $\mathfrak {g}$ , and $\epsilon _1 + \cdots + \epsilon _n + \epsilon _{n+1} = 0$ . We set $\alpha _{i} := \epsilon _{i} - \epsilon _{i+1}$ for $i \in I = [n]$ and $\alpha _{i, j} := \alpha _{i} + \alpha _{i+1} + \cdots + \alpha _{j}=\epsilon _{i}-\epsilon _{j+1}$ for $i, j \in [n]$ with $i \le j$ . Then $\Delta = \{ \pm \alpha _{i,j} \mid 1 \le i \le j \le n \}$ forms a root system of $\mathfrak {g}$ , with the set of positive roots $\Delta ^+ = \{ \alpha _{i,j} \mid 1 \le i \le j \le n \}$ and the set of simple roots $\{ \alpha _{1}, \ldots , \alpha _{n} \}$ .

The Weyl group of $\mathfrak {g}=\mathfrak {sl}_{n+1}(\mathbb {C})$ is, by definition, the subgroup $W = \langle s_{1}, \ldots , s_{n} \rangle $ of $GL(\mathfrak {h}^{\ast })$ , where $s_{1}, \ldots , s_{n}$ are the simple reflections. For $i,j \in [n+1]$ with $i < j$ , we denote by $(i,j)$ the transposition of i and j. It is known that the assignment $s_{1} \mapsto (1, 2),\, s_{2} \mapsto (2, 3),\,\dots ,\, s_{n} \mapsto (n, n+1)$ induces a group isomorphism $W \xrightarrow {\sim } S_{n+1}$ , where $S_{n+1}$ denotes the symmetric group of degree $n+1$ . By this isomorphism, we regard $x \in W$ as a permutation on $[n+1]=\{ 1, \ldots , n+1 \}$ ; in addition, it follows that $x \epsilon _{i} = \epsilon _{x(i)}$ for $i \in [n+1]$ . Also, the longest element of W, denoted by $w_{\circ }$ , is regarded as the permutation

$$ \begin{align*} \begin{pmatrix} 1 & 2 & \cdots & n & n+1 \\ n+1 & n & \cdots & 2 & 1 \end{pmatrix}; \end{align*} $$

that is, $w_{\circ }$ is considered as the permutation defined by $w_{\circ }(k) = n+2-k$ for $k \in [n+1]$ .

We know from [Reference LenartLen, Proposition 3.6] the following.

2.3 Equivariant K-groups of semi-infinite flag manifolds

Let $\mathbf {Q}_{G}^{\mathrm {rat}}$ denote the semi-infinite flag manifold associated to G, which is a (reduced) ind-scheme of ind-infinite type whose set of $\mathbb {C}$ -valued points is $G(\mathbb {C}(\!(z)\!))/(H(\mathbb {C}) \cdot N(\mathbb {C}(\!(z)\!)))$ (see [Reference Kato, Naito and SagakiKaNS, Reference KatoKat2] for details), where G is a connected, simply-connected, simple algebraic group over $\mathbb {C}$ , $B=HN$ is a Borel subgroup, H is a maximal torus, and N is the unipotent radical of B; note that $\mathbf {Q}_{G}^{\mathrm {rat}}$ can be thought of as an inductive limit of copies of the (reduced) closed subscheme $\mathbf {Q}_{G} \subset \prod _{i \in I} \mathbb {P}(L(\varpi _{i}) \otimes _{\mathbb {C}} \mathbb {C}[\![z]\!])$ of infinite type, introduced in [Reference Finkelberg and MirkovicFM, Section 4.1], where $L(\varpi _{i})$ is the irreducible highest weight G-module of highest weight $\varpi _{i}$ . One has the semi-infinite Schubert (sub)variety $\mathbf {Q}_{G}(x) \subset \mathbf {Q}_{G}^{\mathrm {rat}}$ associated to each element x of the affine Weyl group $W_{\mathrm {af}} \cong W \ltimes Q^{\vee }$ , with $W = \langle s_i \mid i \in I \rangle $ the (finite) Weyl group and $Q^{\vee } = \sum _{i \in I} \mathbb {Z} \alpha _i^{\vee }$ the coroot lattice of G; note that $\mathbf {Q}_{G}(x)$ is, by definition, the closure of the orbit under the Iwahori subgroup $\mathbf {I} := (\mathrm {ev}_{z = 0})^{-1}(B) \subset G(\mathbb {C}[\![z]\!])$ through the ( $H \times \mathbb {C}^{*}$ )-fixed point labeled by $x \in W_{\mathrm {af}}$ in exactly the same way as in [Reference Kato, Naito and SagakiKaNS, Section 4.2] and [Reference OrrO, Section 2.3], and that $\mathbf {Q}_{G}(x)$ is contained in $\mathbf {Q}_{G}(e) = \mathbf {Q}_{G}$ for all $x \in W_{\mathrm {af}}^{\geq 0} := \{ x = w t_{\xi } \in W_{\mathrm {af}} \mid w \in W, \xi \in Q^{\vee ,+} \}$ , where $Q^{\vee , +} := \sum _{i \in I} \mathbb {Z}_{\geq 0} \alpha _{i}^{\vee } \subset Q^{\vee }$ . Also, for each weight $\nu = \sum _{i \in I} m_{i} \varpi _{i} \in P$ with $m_{i} \in \mathbb {Z}$ , we have a ( $G(\mathbb {C}[\![z]\!]) \rtimes \mathbb {C}^{*}$ )-equivariant line bundle $\mathcal{O}_{\mathbf {Q}_{G}}(\nu )$ over $\mathbf {Q}_{G}$ , which is given by the restriction of the line bundle $\boxtimes _{i \in I} \mathcal{O}(m_{i})$ over $\prod _{i \in I} \mathbb {P}(L(\varpi _{i}) \otimes _{\mathbb {C}} \mathbb {C}[\![z]\!])$ .

The $(H \times \mathbb {C}^{*})$ -equivariant K-group $K_{H \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ is a module over $\mathbb {Z}[q,q^{-1}][P]$ (equivariant parameters) and has the semi-infinite Schubert classes $[\mathcal{O}_{\mathbf {Q}_{G}(x)}]$ associated to $x \in W_{\mathrm {af}}^{\geq 0} \simeq W \times Q^{\vee ,+}$ as a topological basis (in the sense of [Reference Kato, Naito and SagakiKaNS, Proposition 5.11]) over $\mathbb {Z}[q,q^{-1}][P]$ , where $P = \sum _{i \in I} \mathbb {Z} \varpi _{i}$ is the weight lattice of G whose group algebra $\mathbb {Z}[P] = \bigoplus _{\nu \in P} \mathbb {Z}\mathbf {e}^{\nu }$ is identified with the representation ring $R(H)$ of H, and $q \in R(\mathbb {C}^{*})$ corresponds to the loop rotation action of $\mathbb {C}^{*}$ . More precisely, the K-group $K_{H \times \mathbb {C}^{*}}(\mathbf {Q}_{G})$ is defined to be the $\mathbb {Z}[q, q^{-1}][P]$ -submodule of the equivariant K-group $\widetilde {K}^{\prime }(\mathbf {Q}_{G})$ of $\mathbf {Q}_{G}$ , introduced in [Reference KatoKat1, Section 1.5], consisting of all those convergent (in the sense of [Reference Kato, Naito and SagakiKaNS, Proposition 5.11]) formal infinite linear combinations of the classes $[\mathcal{O}_{\mathbf {Q}_{G}(x)}]$ , $x \in W_{\mathrm {af}}^{\geq 0}$ , of the structure sheaves $\mathcal{O}_{\mathbf {Q}_{G}(x)}$ of the semi-infinite Schubert varieties $\mathbf {Q}_{G}(x) \subset \mathbf {Q}_{G}$ with coefficients $a_{x} \in \mathbb {Z}[q, q^{-1}][P]$ ; briefly speaking, convergence holds if the sum $\sum _{x \in W_{\mathrm {af}}^{\geq 0}} \vert a_{x} \vert $ of the absolute values $|a_{x}|$ lies in $\mathbb {Z}[P](\!( q^{-1} )\!)$ . For each $x \in W_{\mathrm {af}}^{\geq 0}$ and $\nu \in P$ , it follows from [Reference Kouno, Lenart and NaitoKoLN, Theorem 5.16] and (the proof of) [Reference Kato, Naito and SagakiKaNS, Corollary 5.12] that the twisted semi-infinite Schubert class $[\mathcal{O}_{\mathbf {Q}_{G}(x)}(\nu )] := [\mathcal{O}_{\mathbf {Q}_{G}(x)}] \otimes [\mathcal{O}_{\mathbf {Q}_{G}}(\nu )]$ , defined by the tensor product in $\widetilde {K}^{\prime }(\mathbf {Q}_{G})$ (see [Reference KatoKat1, Theorem 1.25]), indeed lies in $K_{H \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ ; in particular, we have $[\mathcal{O}_{\mathbf {Q}_{G}}(\nu )] = [\mathcal{O}_{\mathbf {Q}_{G}}] \otimes [\mathcal{O}_{\mathbf {Q}_{G}}(\nu )] \in K_{H \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ for all $\nu \in P$ .

Also, following [Reference KatoKat1, Section 1.5], we define the H-equivariant K-group $K_{H}(\mathbf {Q}_{G})$ of $\mathbf {Q}_{G}$ to be the specialization (of coefficients) at $q = 1$ of $\widetilde {K}^{\prime }(\mathbf {Q}_{G})$ (or equivalently, of $K_{H \times \mathbb {C}^{*}}(\mathbf {Q}_{G})$ ), which turns out to be (cf. [Reference KatoKat1, Lemma 1.22]) the $\mathbb {Z}[P]$ -module $\prod _{x \in W_{\mathrm {af}}^{\geq 0}} \mathbb {Z}[P][\mathcal{O}_{\mathbf {Q}_{G}(x)}]$ (direct product). Namely, the K-group $K_{H}(\mathbf {Q}_{G})$ consists of all infinite linear combinations of the semi-infinite Schubert classes $[\mathcal{O}_{\mathbf {Q}_{G}(x)}]$ , $x \in W_{\mathrm {af}}^{\geq 0}$ , with coefficients in $\mathbb {Z}[P]$ ; the semi-infinite Schubert classes $[\mathcal{O}_{\mathbf {Q}_{G}(x)}]$ , $x \in W_{\mathrm {af}}^{\geq 0}$ , form a topological basis of $K_{H}(\mathbf {Q}_{G})$ over $\mathbb {Z}[P]$ . Then, for each $\nu \in P$ , a $\mathbb {Z}[P]$ -linear endomorphism $\bullet \otimes [\mathcal{O}_{\mathbf {Q}_{G}}(\nu )]$ of $K_{H}(\mathbf {Q}_{G})$ is induced from the $\mathbb {Z}[q, q^{-1}][P]$ -linear endomorphism $\bullet \otimes [\mathcal{O}_{\mathbf {Q}_{G}}(\nu )]$ of $K_{H \times \mathbb {C}^{*}}(\mathbf {Q}_{G})$ by the specialization (of coefficients) at $q = 1$ (see [Reference KatoKat1, Theorem 1.26]); thus, we have $[\mathcal{O}_{\mathbf {Q}_{G}}(\nu )] = [\mathcal{O}_{\mathbf {Q}_{G}}] \otimes [\mathcal{O}_{\mathbf {Q}_{G}}(\nu )] \in K_{H}(\mathbf {Q}_{G})$ for $\nu \in P$ . In addition, for $\xi \in Q^{\vee ,+}$ , a $\mathbb {Z}[P]$ -linear endomorphism $\mathsf {t}_{\xi }$ on $K_{H}(\mathbf {Q}_{G})$ , given by $\mathsf {t}_{\xi }[\mathcal{O}_{\mathbf {Q}_{G}(x)}] := [\mathcal{O}_{\mathbf {Q}_{G}(x t_{\xi })}]$ for $x \in W_{\mathrm {af}}^{\geq 0}$ , is induced by the right action of $Q^{\vee } \subset H(\mathbb {C}(\!(z)\!))/H(\mathbb {C})$ on $\mathbf {Q}_{G}^{\mathrm {rat}}$ (see [Reference KatoKat1, equation (1.20)]); the $\mathbb {Z}[P]$ -linear endomorphism $\mathsf {t}_{\xi }$ of $K_{H}(\mathbf {Q}_{G})$ is also obtained by the specialization at $q = 1$ from the $\mathbb {Z}[q, q^{-1}]$ -linear endomorphism of $K_{H \times \mathbb {C}^{*}}(\mathbf {Q}_{G})$ given by the same formula. We know from [Reference KatoKat1, Theorem 1.26] (cf. [Reference OrrO, Proposition 2.4])

(2.1) $$ \begin{align} (\mathsf{t}_{\xi}[\mathcal{O}_{\mathbf{Q}_{G}(x)}]) \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\nu)] = \mathsf{t}_{\xi}([\mathcal{O}_{\mathbf{Q}_{G}(x)}] \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\nu)]) \end{align} $$

for $x \in W_{\mathrm {af}}^{\geq 0}$ , $\xi \in Q^{\vee ,+}$ , and $\nu \in P$ ; for $j \in I$ , we set $\mathsf {t}_{j} := \mathsf {t}_{\alpha _j^{\vee }}$ . Since the semi-infinite Schubert classes $[\mathcal{O}_{\mathbf {Q}_{G}(x)}]$ , $x \in W_{\mathrm {af}}^{\geq 0}$ , form a topological basis of $K_{H}(\mathbf {Q}_{G})$ over $\mathbb {Z}[P]$ , it follows that

(2.2) $$ \begin{align} (\mathsf{t}_{\xi} \, \bullet) \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\nu)] = \mathsf{t}_{\xi}(\bullet \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\nu)]) \end{align} $$

for an arbitrary element $\bullet \in K_{H}(\mathbf {Q}_{G})$ and $\xi \in Q^{\vee ,+}$ , $\nu \in P$ .

Remark 2.3. Let $\mathsf {F}_{1},\dots ,\mathsf {F}_{N}$ be polynomials in $\mathsf {t}_{i}$ , $i \in I$ , and let $\nu _{1},\dots ,\nu _{N} \in P$ . When we write $(\mathsf {F}_{1}[\mathcal{O}_{\mathbf {Q}_{G}}( \nu _{1} )]) \otimes (\mathsf {F}_{2}[\mathcal{O}_{\mathbf {Q}_{G}}( \nu _{2} )]) \otimes \cdots \otimes (\mathsf {F}_{N}[\mathcal{O}_{\mathbf {Q}_{G}}( \nu _{N} )]) \otimes \bullet $ for $\bullet \in K_{H}(\mathbf {Q}_{G})$ , it is always understood to be the element $(\mathsf {F}_{1}\mathsf {F}_{2} \cdots \mathsf {F}_{N})([\mathcal{O}_{\mathbf {Q}_{G}}( \nu _{1} + \nu _{2} + \cdots + \nu _{N} )] \otimes \bullet )$ in $K_{H}(\mathbf {Q}_{G})$ . Namely, for $\bullet \in K_{H}(\mathbf {Q}_{G})$ , we always understand that

$$ \begin{align*} \begin{aligned} & (\mathsf{F}_{1}[\mathcal{O}_{\mathbf{Q}_{G}}( \nu_{1} )]) \otimes (\mathsf{F}_{2}[\mathcal{O}_{\mathbf{Q}_{G}}( \nu_{2} )]) \otimes \cdots \otimes (\mathsf{F}_{N}[\mathcal{O}_{\mathbf{Q}_{G}}( \nu_{N} )]) \otimes \bullet \\ &\quad = (\mathsf{F}_{1}\mathsf{F}_{2} \cdots \mathsf{F}_{N}) ([\mathcal{O}_{\mathbf{Q}_{G}}( \nu_{1} + \nu_{2} + \cdots + \nu_{N} )] \otimes \bullet). \end{aligned} \end{align*} $$

2.4 An explicit formula expressing the semi-infinite Schubert class associated to $w_{\circ }$

In this subsection, we assume that $G = SL_{n+1}(\mathbb {C})$ and hence $\mathfrak {g} = \mathfrak {sl}_{n+1}(\mathbb {C})$ , which is of type $A_{n}$ . For $1 \le k \le n$ , we set

$$ \begin{align*} \begin{aligned} w_{k} &:= (s_{1}s_{2} \cdots s_{n}) (s_{1}s_{2} \cdots s_{n-1}) \cdots (s_{1}s_{2} \cdots s_{k+1}) (s_{1}s_{2} \cdots s_{k}) \\ & = (1,2,\dots,n+1)(1,2,\dots,n) \cdots (1,2,\dots,k+2)(1,2,\dots,k+1) \\[1.5mm] & = \begin{pmatrix} 1 & 2 & \cdots & k-1 & k & k+1 & \cdots & n & n+1 \\ n-k+2 & n-k+3 & \cdots & n & n+1 & n-k+1 & \cdots & 2 & 1 \end{pmatrix}, \end{aligned} \end{align*} $$

which is an element of the Weyl group $W = S_{n+1}$ of $\mathfrak {g} = \mathfrak {sl}_{n+1}(\mathbb {C})$ ; we set $w_{n+1}:=e$ by convention. Note that $w_{1} = w_{\circ }$ , the longest element of W. Also, for a subset J of $[n+1]=\big \{1,2,\dots ,n+1\big \}$ , we set

(2.3) $$ \begin{align} \epsilon_{J}:=\sum_{j \in J} \epsilon_{j}. \end{align} $$

We have $\epsilon _{J} \in W\varpi _{|J|}$ , where $\varpi _{0}=\varpi _{n+1}:=0$ by convention. In particular, $\epsilon _{J}$ is a minuscule weight (i.e., $\langle \epsilon _{J}, \alpha _{j}^{\vee } \rangle \in \{-1,0,1\}$ for all $1 \le j \le n$ ). We set

(2.4) $$ \begin{align} J^{-} & := \big\{1 \le j \le n \mid j \notin J,\,j+1 \in J \big\} \nonumber \\ & = \big\{ 1 \le j \le n \mid \langle \epsilon_{J}, \alpha_{j}^{\vee} \rangle < 0 \big\} \nonumber \\ & = \big\{ 1 \le j \le n \mid \langle \epsilon_{J}, \alpha_{j}^{\vee} \rangle = -1 \big\}; \end{align} $$

recall that $\alpha _{j} = \epsilon _{j}-\epsilon _{j+1}$ . For $0 \le p \le k \le n+1$ , we set

(2.5) $$ \begin{align} \mathfrak{F}^{k}_{p} := \sum_{ \begin{subarray}{c} J \subset [k] \\[1mm] |J|=p \end{subarray}} \left(\prod_{ j \in J^{-} } (1-\mathsf{t}_{j})\right)[\mathcal{O}_{\mathbf{Q}_{G}}( w_{\circ} \epsilon_{J} )] \in K_{H}(\mathbf{Q}_{G}), \end{align} $$

where $[k]=\big \{1,2,\dots ,k\big \}$ ; note that $\mathfrak {F}^{k}_{0}=1$ . The proof of the following proposition will be given in Section 5.

Proposition 2.4. For $1 \leq k \leq n$ , the following equality holds in $K_{H}(\mathbf {Q}_{G})$ :

(2.6) $$ \begin{align} \left( \sum_{p=0}^{k}(-1)^{p} \mathbf{e}^{p \epsilon_{n+1-k}} \mathfrak{F}^{k}_{p} \right) \otimes [\mathcal{O}_{\mathbf{Q}_{G}(w_{k+1})}] = [\mathcal{O}_{\mathbf{Q}_{G}(w_{k})}]. \end{align} $$

Noting that $w_{1}=w_{\circ }$ and $w_{n+1}=e$ , we obtain the following.

Corollary 2.5. The following equality holds in $K_{H}(\mathbf {Q}_{G})$ :

(2.7) $$ \begin{align} [\mathcal{O}_{\mathbf{Q}_{G}(w_{\circ})}] = \bigotimes_{k=1}^{n} \left( \sum_{p=0}^{k}(-1)^{p} \mathbf{e}^{p \epsilon_{n+1-k}} \mathfrak{F}^{k}_{p} \right). \end{align} $$

3.1 Relationship between $K_{H}(\mathbf {Q}_{G})$ and $QK_{H}(G/B)$

Let G be a connected, simply-connected, simple algebraic group over $\mathbb {C}$ , with Borel subgroup $B \subset G$ and maximal torus $H \subset B$ . Let $QK_{H}(G/B) := K_{H}(G/B) \otimes _{R(H)} R(H)[\![Q^{\vee ,+}]\!]$ denote the H-equivariant (small) quantum K-theory ring of the ordinary flag manifold $G/B$ , defined by Givental [Reference GiventalGiv] and Lee [Reference LeeLee], where $R(H)[\![Q^{\vee ,+}]\!]$ is the ring of formal power series in the Novikov variables $Q_i = Q^{\alpha _i^{\vee }}$ , $i \in I$ , with coefficients in the representation ring $R(H)$ of H; for $\xi = \sum _{i \in I} k_i \alpha _i^{\vee } \in Q^{\vee ,+} = \sum _{i \in I} \mathbb {Z}_{\geq 0} \alpha _i^{\vee }$ , we set $Q^{\xi } := \prod _{i \in I} Q_i^{k_i} \in R(H)[\![Q^{\vee ,+}]\!]$ . The quantum K-theory ring $QK_{H}(G/B)$ is a free module over $R(H)[\![Q^{\vee ,+}]\!]$ having the (opposite) Schubert classes $[\mathcal{O}^{w}]$ , $w \in W$ , as a basis; also, the quantum multiplication $\star $ in $QK_{H}(G/B)$ is a deformation of the classical tensor product in $K_{H}(G/B)$ and is defined in terms of the $2$ -point and $3$ -point (genus zero, equivariant) K-theoretic Gromov-Witten invariants; see [Reference GiventalGiv] and [Reference LeeLee] for details.

In [Reference KatoKat1, Reference KatoKat3], based on [Reference Givental and LeeGL, Reference Braverman and FinkelbergBF, Reference Iritani, Milano and TonitaIMT] (see also [Reference Anderson, Chen and TsengACT]), Kato established an $R(H)$ -module isomorphism $\Phi $ from $QK_{H}(G/B)$ onto the H-equivariant K-group $K_{H}(\mathbf {Q}_{G})$ of the semi-infinite flag manifold $\mathbf {Q}_{G}$ , in which tensor product operation with an arbitrary line bundle class is induced from that in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ by the specialization at $q = 1$ ; in our notation, the map $\Phi $ sends the (opposite) Schubert class $\mathbf {e}^{\mu }[\mathcal{O}^{w}] Q^{\xi }$ in $QK_{H}(G/B)$ to the corresponding semi-infinite Schubert class $\mathbf {e}^{-\mu }[\mathcal{O}_{\mathbf {Q}_{G}(wt_{\xi })}]$ in $K_{H}(\mathbf {Q}_{G})$ for $\mu \in P$ , $w \in W$ and $\xi \in Q^{\vee ,+}$ . The isomorphism $\Phi $ also respects, in a sense, quantum multiplication $\star $ in $QK_{H}(G/B)$ and tensor product in $K_{H}(\mathbf {Q}_{G})$ . More precisely, one has the commutative diagram

(3.1)

where for $\nu \in P$ , the line bundle $\mathcal{O}_{G/B}(-\nu )$ over $G/B$ denotes the G-equivariant line bundle constructed as the quotient space $G \times ^{B} \mathbb {C}_{\nu }$ of the product space $G \times \mathbb {C}_{\nu }$ by the usual (free) left action of the Borel subgroup B of G corresponding to the positive roots, with $\mathbb {C}_{\nu }$ the one-dimensional B-module of weight $\nu $ . (Here, we warn the reader that the convention of [Reference KatoKat1] differs from that of [Reference Kato, Naito and SagakiKaNS] and this paper, by the twist coming from the involution $-w_{\circ }$ .) Also, we know from [Reference KatoKat1] that for $w \in W$ and $\xi \in Q^{\vee ,+}$ , the equality $\Phi ([\mathcal{O}^{w}] Q^{\xi }) = \mathsf {t}_{\xi } \, \Phi ([\mathcal{O}^{w}])$ holds, and hence that for an arbitrary element $\bullet $ of $QK_{H}(G/B)$ and $\xi \in Q^{\vee ,+}$ , the equality

(3.2) $$ \begin{align} \Phi(\bullet \, Q^{\xi}) = \mathsf{t}_{\xi} \, \Phi(\bullet) \end{align} $$

holds.

We know from [Reference Buch, Chaput, Mihalcea and PerrinBCMP, Corollary 5.14] that the quantum multiplicative structure over $R(H)[\![Q^{\vee ,+}]\!]$ of $QK_{H}(G/B)$ is completely determined by the quantum multiplication with the line bundle classes $[\mathcal{O}_{G/B}(-\varpi _{i})]$ , $i \in I$ ; in fact, by using Nakayama’s Lemma (i.e., [Reference EisenbudE, Corollary 4.8 b]) together with [Reference EisenbudE, Exercise 7.3], we can show that $QK_{H}(G/B)$ is generated as an algebra (with quantum multiplication  $\star $ ) over $R(H)[\![Q^{\vee ,+}]\!]$ by the line bundle classes $[\mathcal{O}_{G/B}(-\varpi _{i})]$ , $i \in I$ , since $K_{H}(G/B)$ is known to be generated by the same line bundle classes as an algebra (with tensor product $\otimes $ ) over $R(H)$ ([Reference MihalceaMi]). Therefore, if G is of type $A_{n}$ (i.e., $G = SL_{n+1}(\mathbb {C})$ ), then we deduce from [Reference Maeno, Naito and SagakiMaNS, Propositions 5.1 and 5.2], together with the comments following them, that $QK_{H}(Fl_{n+1})$ is generated as an algebra over $R(H)[\![Q^{\vee ,+}]\!] = R(H)[\![Q]\!] = R(H)[\![Q_1, \ldots , Q_{n}]\!]$ by the line bundle classes $[\mathcal{O}_{Fl_{n+1}}(-\epsilon _{j})]$ , $1 \leq j \leq n$ ; indeed, for $1 \leq j \leq n$ and an arbitrary element $\bullet \in QK_{H}(Fl_{n+1})$ , we have

$$ \begin{align*} \begin{aligned} \bullet \star [\mathcal{O}_{Fl_{n+1}}(-\varpi_{j})] = \bullet \star [\mathcal{O}_{Fl_{n+1}}(- \epsilon_{1})] \star \frac{1}{1 - Q_{1}}[\mathcal{O}_{Fl_{n+1}}(- \epsilon_{2})] \star \cdots \star \frac{1}{1 - Q_{j-1}} [\mathcal{O}_{Fl_{n+1}}(- \epsilon_{j})]. \end{aligned} \end{align*} $$

3.2 An explicit formula expressing the Schubert class associated to $w_{\circ }$

In this subsection, we assume that G is of type $A_{n}$ (i.e., $G = SL_{n+1}(\mathbb {C})$ ). Now, for $0 \le p \le k \le n+1$ , we set

(3.3) $$ \begin{align} \mathcal{F}^{k}_{p}:= \sum_{ \begin{subarray}{c} J \subset [k] \\[1mm] |J|=p \end{subarray}} \, \prod_{ \begin{subarray}{c} 1 \le j \le k \\[1mm] j,\,j+1 \in J \end{subarray} } \frac{1}{1-Q_{j}} \sideset{}{^\star}\prod_{j \in J} [\mathcal{O}_{Fl_{n+1}}(-\epsilon_{j})] \in QK_{H}(Fl_{n+1}), \end{align} $$

with $[k]=\big \{1,2,\dots ,k\big \}$ , where $\prod ^{\star }$ denotes the product with respect to the quantum multiplication $\star $ ; note that $\mathcal{F}^{k}_{0} = 1$ for $0 \le k \le n+1$ . Also, recall from (2.5) that for $0 \leq p \leq k \leq n+1$ , the element $\mathfrak {F}^{k}_{p} \in K_{H}(\mathbf {Q}_{G})$ is defined by

(3.4) $$ \begin{align} \mathfrak{F}^{k}_{p} = \sum_{ \begin{subarray}{c} J \subset [k] \\[1mm] |J|=p \end{subarray}} \left( \prod_{ \begin{subarray}{c} 1 \le j \le k \\[1mm] j \notin J,\,j+1 \in J \end{subarray}} (1-\mathsf{t}_{j}) \right)[\mathcal{O}_{\mathbf{Q}_{G}}(w_{\circ} \epsilon_{J})]. \end{align} $$

Here, since $- \epsilon _j = \varpi _{j-1} - \varpi _j$ for $1 \leq j \leq n+1$ ( $\varpi _0 = \varpi _{n+1} := 0$ by convention), we see from [Reference Maeno, Naito and SagakiMaNS, Proposition 5.2] that

(3.5) $$ \begin{align} \Phi \left(\bullet \star \frac{1}{1 - Q_{j-1}}[\mathcal{O}_{Fl_{n+1}}(- \epsilon_{j})]\right) = \Phi(\bullet) \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(w_{\circ} \epsilon_{j})] \end{align} $$

for $1 \leq j \leq n+1$ and an arbitrary element $\bullet \in QK_{H}(Fl_{n+1})$ , where $G = SL_{n+1}(\mathbb {C})$ , and $Q_{0} = Q_{n+1} := 0$ by convention.

Proposition 3.1. For $0 \leq p \leq k \leq n+1$ , the following equality holds in $K_{H}(\mathbf {Q}_{G})$ :

(3.6) $$ \begin{align} \Phi(\mathcal{F}^{k}_{p}) = \mathfrak{F}^{k}_{p}. \end{align} $$

Proof. First note that for a subset $J \subset [k]$ , the equality $[\mathcal{O}_{\mathbf {Q}_{G}}(w_{\circ } \epsilon _{J})] = \bigotimes _{j \in J} [\mathcal{O}_{\mathbf {Q}_{G}}(w_{\circ } \epsilon _{j})]$ holds since $\epsilon _{J} = \sum _{j \in J} \epsilon _{j}$ . Therefore, by repeated application of equation (3.5), together with equations (2.2) and (3.2), we deduce that the image $\Phi ^{-1}(\mathfrak {F}^{k}_{p})$ of $\mathfrak {F}^{k}_{p}$ under the inverse mapping $\Phi ^{-1}$ of the $R(H)$ -module isomorphism $\Phi $ is

(3.7) $$ \begin{align} \sum_{ \begin{subarray}{c} J \subset [k] \\[1mm] |J|=p \end{subarray}} \, \prod_{ \begin{subarray}{c} 1 \le j \le k \\[1mm] j \notin J,\,j+1 \in J \end{subarray} } (1-Q_{j}) \prod_{j \in J} \frac{1}{1 - Q_{j-1}} \sideset{}{^\star}\prod_{j \in J} [\mathcal{O}_{Fl_{n+1}}(-\epsilon_{j})], \end{align} $$

which is easily seen to be identical to $\mathcal{F}^{k}_{p}$ . This proves the proposition.

Also, from Corollary 2.5, we obtain the following, again by repeated application of equation (3.5), together with relations (2.2) and (3.2).

Proposition 3.2. Let $w_{\circ } \in W = S_{n+1}$ be the longest element. Then, the following equality holds in $QK_{H}(Fl_{n+1})$ :

(3.8) $$ \begin{align} [\mathcal{O}^{w_{\circ}}] = \sideset{}{^\star}\prod_{k=1}^{n} \Bigg( \sum_{p=0}^{k}(-1)^{p} \mathbf{e}^{-p \epsilon_{n+1-k}} \mathcal{F}^{k}_{p}\Bigg) \in QK_{H}(Fl_{n+1}). \end{align} $$

3.3 Proof of the main result for $w = w_{\circ }$

We know from [Reference Maeno, Naito and SagakiMaNS, Section 6] that there exists an $R(H)[\![Q]\!]$ -algebra isomorphism

(3.9) $$ \begin{align} \Psi^{Q}:(R(H)[\![Q]\!])[x_{1},\dots,x_{n},x_{n+1}]/\mathcal{I}^{Q} \xrightarrow {\hspace {2pt}\cong \hspace {2pt}} QK_{H}(Fl_{n+1}), \end{align} $$

where $\mathcal{I}^{Q}$ denotes the ideal of $(R(H)[\![Q]\!])[x_{1},\dots ,x_{n},x_{n+1}]$ generated by

(3.10) $$ \begin{align} \sum_{ \begin{subarray}{c} J \subset [n+1] \\[1mm] |J|=l \end{subarray}} \, \prod_{j \in J, j+1 \notin J} (1-Q_{j}) \prod_{j \in J} (1-x_{j}) - \sum_{ \begin{subarray}{c} J \subset [n+1] \\[1mm] |J|=l \end{subarray}}\mathbf{e}^{\epsilon_{J}}, \qquad 1 \le l \le n+1, \end{align} $$

with $[n+1]=\big \{1,2,\dots ,n+1\big \}$ ; here, we understand that $Q_{n+1} = 0$ . Also, we know that the isomorphism $\Psi ^{Q}$ maps the residue class $(1-Q_j)(1-x_{j}) \ \mathrm {mod} \ \mathcal{I}^{Q}$ to $[\mathcal{O}_{Fl_{n+1}}(-\epsilon _{j})]$ for $1 \le j \le n$ , and the residue class $(1-x_{n+1}) \ \mathrm {mod} \ \mathcal{I}^{Q}$ to $[\mathcal{O}_{Fl_{n+1}}(-\epsilon _{n+1})]$ ; note that $-\epsilon _{n+1} = \epsilon _1 + \cdots + \epsilon _n$ .

In the following, by $1 - y_{j} = \mathbf {e}^{-\epsilon _{j}}$ for $1 \le j \le n$ , we identify

1. (ID1) $R(H) \cong \mathbb {Z}[\mathbf {e}^{\pm \epsilon _{1}},\dots ,\mathbf {e}^{\pm \epsilon _{n}}]$ with $\mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}]$ , and

2. (ID2) $(R(H)[\![Q]\!])[x_{1},\dots ,x_{n},x_{n+1}]$ with $(\mathbb{Z}[(1-y_{1})^{\pm 1},\dots,(1-y_{n})^{\pm 1}][\![Q]\!])[x_{1},\dots,x_{n},x_{n+1}]$ .

For $0 \le p \le k \le n$ , we define

(3.11) $$ \begin{align} F_{p}^{k}(x_{1},\dots,x_{n},x_{n+1}):= \sum_{ \begin{subarray}{c} J \subset [k] \\[1mm] |J|=p \end{subarray}} \, \prod_{j \in J, j+1 \notin J} (1-Q_{j}) \prod_{j \in J} (1-x_{j}) \in \mathbb{Z}[\![Q]\!][x_{1},\dots,x_{n},x_{n+1}]. \end{align} $$

Then, we can easily verify that for $0 \le p \le k \le n$ ,

$$ \begin{align*} \Psi^{Q}(F_{p}^{k}(x_{1},\dots,x_{n},x_{n+1}) \ \mathrm{mod} \ \mathcal{I}^{Q}) = \mathcal{F}^{k}_{p} \in QK_{H}(Fl_{n+1}). \end{align*} $$

Therefore, we deduce from Proposition 3.2 that the residue class

$$ \begin{align*} \prod_{k=1}^{n} \Bigg( \sum_{p=0}^{k}(-1)^{p}(1-y_{n+1-k})^{p}F_{p}^{k}(x_{1},\dots,x_{n},x_{n+1}) \Bigg)\ \mod \mathcal{I}^{Q} \end{align*} $$

is mapped under the $R(H)[\![Q]\!]$ -algebra isomorphism $\Psi ^{Q}$ to

$$ \begin{align*} \sideset{}{^\star}\prod_{k=1}^{n} \Bigg( \sum_{p=0}^{k}(-1)^{p}\mathbf{e}^{-p \epsilon_{n+1-k}} \mathcal{F}^{k}_{p} \Bigg) = [\mathcal{O}^{w_{\circ}}] \in QK_{H}(Fl_{n+1}); \end{align*} $$

recall the identification $1-y_{j} = \mathbf {e}^{-\epsilon _{j}}$ for $1 \le j \le n$ .

Now, let us set

(3.12) $$ \begin{align} \begin{aligned} \mathfrak{G}_{w_{\circ}}^{Q}(x,y) & := \prod_{k=1}^{n}\Bigg( \sum_{l=0}^{k}(-1)^{l}(1-y_{n+1-k})^{l}F_{l}^{k}(x_{1},\dots,x_{n},x_{n+1}) \Bigg) \\ & \in \mathbb{Z}[\![Q]\!][x_{1},\dots,x_{n},x_{n+1}][(1-y_{1})^{\pm 1},\dots,(1-y_{n})^{\pm 1}], \end{aligned} \end{align} $$

which can be thought of as a subring of $(\mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}][\![Q]\!])[x_{1},\dots ,x_{n},x_{n+1}]$ ; see Section 4.1 below. Then, from the above, we obtain the following.

Theorem 3.3. Let $w_{\circ } \in W=S_{n+1}$ be the longest element. Then, under the identification (ID2), the following equality holds in $QK_{H}(Fl_{n+1})$ :

(3.13) $$ \begin{align} \Psi^{Q}(\mathfrak{G}_{w_{\circ}}^{Q}(x,y) \ \mathrm{mod} \ \mathcal{I}^{Q}) = [\mathcal{O}^{w_{\circ}}] \in QK_{H}(Fl_{n+1}), \end{align} $$

where $\Psi ^{Q}$ is the $R(H)[\![Q]\!]$ -algebra isomorphism in (3.9).

4.1 Quantum double Grothendieck polynomials

First we recall from [Reference Lenart and MaenoLM, Section 8] the Demazure operators $\pi ^{(y)}_{i}$ acting on the y variables; note that we use Demazure operators acting on the y variables, while in [Reference Lenart and MaenoLM], they are acting on the x variables. Let $i \in [n]=\{1,2,\dots ,n\}$ . We define the Demazure operator $\pi ^{(y)}_{i}$ (with respect to the y variables) on $\mathbb {Z}[(1 - y_1)^{\pm 1},\ldots ,(1 - y_n)^{\pm 1}]$ by

$$ \begin{align*} \pi^{(y)}_{i}=\operatorname{\mathrm{id}} + \frac{1-y_{i}}{y_{i}-y_{i+1}}(\operatorname{\mathrm{id}} - s_{i}), \end{align*} $$

where we identify the Laurent polynomial ring $\mathbb {Z}[(1-y_1)^{\pm 1}, \ldots , (1-y_{n})^{\pm 1}]$ with the quotient of the polynomial ring $\mathbb {Z}[(1-y_1), \ldots , (1-y_{n}), (1-y_{n+1})]$ by the relation $(1-y_1) \cdots (1-y_{n})(1-y_{n+1}) = 1$ ; we extend it to $\mathbb {Z}[\![Q]\!][x_{1},\dots ,x_{n},x_{n+1}][(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}]$ by $\mathbb {Z}[\![Q]\!][x_{1},\dots ,x_{n},x_{n+1}]$ -linearity, and also to $(\mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}][\![Q]\!])[x_{1},\dots ,x_{n},x_{n+1}]$ by letting it act trivially on the elements of $\mathbb {Z}[\![Q]\!][x_{1}. \ldots , x_{n}, x_{n+1}]$ . Also, for $w \in W=S_{n+1}$ with reduced expression $w = s_{i_{1}} \cdots s_{i_{l}}$ , we define $\pi ^{(y)}_{w}:=\pi ^{(y)}_{i_1} \cdots \pi ^{(y)}_{i_l}$ ; it is well-known that this definition does not depend on the choice of a reduced expression for w.

Now, let us recall from [Reference Lenart and MaenoLM, Section 8] the definition of quantum double Grothendieck polynomials. For the longest element $w_{\circ } \in W=S_{n+1}$ , we define $\mathfrak {G}_{w_{\circ }}^{Q}(x,y)$ as in (3.12). Note that the polynomial $\mathfrak {G}_{w_{\circ }}^{Q}(x,y)$ is the image under the quantization map $\mathscr {Q}$ (with respect to the x variables) of the ordinary double Grothendieck polynomial

$$ \begin{align*} \mathfrak{G}_{w_{\circ}}(x,y):= \prod_{k=1}^{n}\Bigg( \sum_{l=0}^{k} (-1)^{l}(1-y_{n+1-k})^{l}f_{l}^{k}(x_{1},\dots,x_{n},x_{n+1}) \Bigg), \end{align*} $$

where

$$ \begin{align*} f_{l}^{k}(x_{1},\dots,x_{n},x_{n+1}):= \sum_{ \begin{subarray}{c} J \subset [k] \\[1mm] |J|=l \end{subarray}} \, \prod_{j \in J}(1-x_{j}), \end{align*} $$

with $[k] = \{1, 2, \ldots , k\}$ ; for more details on the quantization map $\mathscr {Q}$ , we refer the reader to [Reference Lenart and MaenoLM, Sections 3 and 5]. For an arbitrary $w \in W=S_{n+1}$ , the quantum double Grothendieck polynomial $\mathfrak {G}_{w}^{Q}(x,y)$ is defined by

$$ \begin{align*} \mathfrak{G}_{w}^{Q}(x,y):=\pi_{ww_{\circ}}^{(y)}\mathfrak{G}_{w_{\circ}}^{Q}(x,y) \in \mathbb{Z}[\![Q]\!][x_{1},\dots,x_{n},x_{n+1}][(1-y_{1})^{\pm 1},\dots,(1-y_{n})^{\pm 1}], \end{align*} $$

which is a subring of $(\mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}][\![Q]\!])[x_{1},\dots ,x_{n},x_{n+1}]$ .

4.2 Quantum left Demazure operators

Let $K_{H}(Fl_{n+1})$ be the H-equivariant (ordinary) K-theory ring of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ . The ring $K_{H}(Fl_{n+1})$ admits a left action of the Weyl group $W=S_{n+1}$ , given by the left multiplication on the flag manifold $Fl_{n+1}= SL_{n+1}(\mathbb {C})/B$ , where B is the Borel subgroup of $G = SL_{n+1}(\mathbb {C})$ consisting of the upper triangular matrices in $G = SL_{n+1}(\mathbb {C})$ ; for each $w \in W = S_{n+1}$ , let $w^{L}$ denote the corresponding ring automorphism of $K_{H}(Fl_{n+1})$ . By using the ring automorphisms $s_{i}^{L}$ , $i \in [n]$ , of $K_{H}(Fl_{n+1})$ , one can define the (dual) left Demazure operators $\delta _{i}^{\vee }$ , $i \in [n]$ , on $K_{H}(Fl_{n+1})$ by

(4.1) $$ \begin{align} \delta_{i}^{\vee}:= \frac{1}{ 1-\mathbf{e}^{-(\epsilon_{i}-\epsilon_{i+1})} } (\operatorname{\mathrm{id}} - \mathbf{e}^{-(\epsilon_{i}-\epsilon_{i+1})}s_{i}^{L}). \end{align} $$

Here, we recall the well-known presentation

$$ \begin{align*} K_{H}(Fl_{n+1}) = (\mathbb{Z}[\mathbf{e}^{\pm \epsilon_1},\dots,\mathbf{e}^{\pm \epsilon_n}] \otimes \mathbb{Z}[x_{1}^{\pm 1},\dots,x_{n}^{\pm 1}])/\mathcal{I}, \end{align*} $$

where $\mathcal{I}$ is the ideal of $\mathbb {Z}[\mathbf {e}^{\pm \epsilon _1},\dots ,\mathbf {e}^{\pm \epsilon _n}] \otimes \mathbb {Z}[x_{1}^{\pm 1},\dots ,x_{n}^{\pm 1}]$ generated by $\iota (f) \otimes 1 - 1 \otimes f$ , $f \in \mathbb {Z}[x_{1}^{\pm 1},\dots ,x_{n}^{\pm 1}]^{S_{n+1}}$ , with $\iota :\mathbb {Z}[x_{1}^{\pm 1},\dots ,x_{n}^{\pm 1}] \rightarrow \mathbb {Z}[\mathbf {e}^{\pm \epsilon _1},\dots ,\mathbf {e}^{\pm \epsilon _n}]$ the natural isomorphism given by $\iota (x_{i}^{\pm 1}) = \mathbf {e}^{\pm \epsilon _i}$ , $1 \le i \le n$ ; we identify the Laurent polynomial ring $\mathbb {Z}[x_{1}^{\pm 1}, \ldots , x_{n}^{\pm 1}]$ with the quotient of $\mathbb {Z}[x_{1}, \ldots , x_{n}, x_{n+1}]$ by the relation $x_{1} \cdots x_{n} x_{n+1} = 1$ , and the Laurent polynomial ring $\mathbb {Z}[\mathbf {e}^{\pm \epsilon _1}, \ldots , \mathbf {e}^{\pm \epsilon _n}]$ with the quotient of $\mathbb {Z}[\mathbf {e}^{\epsilon _1}, \ldots , \mathbf {e}^{\epsilon _n}, \mathbf {e}^{\epsilon _{n+1}}]$ by the relation $\mathbf {e}^{\epsilon _1} \cdots \mathbf {e}^{\epsilon _n} \mathbf {e}^{\epsilon _{n+1}} = 1$ . We know from (the dual version of) [Reference Mihalcea, Naruse and SuMNS, Proposition 9.5] that under the presentation above of $K_{H}(Fl_{n+1})$ , the action of the (dual) left Demazure operator $\delta _{i}^{\vee }$ on $K_{H}(Fl_{n+1})$ coincides with that on the quotient ring $(\mathbb {Z}[\mathbf {e}^{\pm \epsilon _1},\dots ,\mathbf {e}^{\pm \epsilon _n}] \otimes \mathbb {Z}[x_{1}^{\pm 1},\dots ,x_{n}^{\pm 1}])/\mathcal{I}$ induced by the (dual) Demazure operator on the left factor $\mathbb {Z}[\mathbf {e}^{\pm \epsilon _1},\dots ,\mathbf {e}^{\pm \epsilon _n}]$ , given by

(4.2) $$ \begin{align} \frac{1}{ 1-\mathbf{e}^{-(\epsilon_{i}-\epsilon_{i+1})} } (\operatorname{\mathrm{id}} - \mathbf{e}^{-(\epsilon_{i}-\epsilon_{i+1})}s_{i}), \end{align} $$

where $s_{i}$ denotes the natural action of the simple reflection $s_{i} \in W=S_{n+1}$ on $R(H) \cong \mathbb {Z}[\mathbf {e}^{\pm \epsilon _1},\dots ,\mathbf {e}^{\pm \epsilon _n}]$ .

Let $QK_{H}(Fl_{n+1})=K_{H}(Fl_{n+1}) \otimes _{R(H)} R(H)[\![Q]\!]$ be the H-equivariant (small) quantum K-theory ring of $Fl_{n+1}$ . We know from [Reference Mihalcea, Naruse and SuMNS, Section 8] that for each $w \in W = S_{n+1}$ , the ring automorphism $w^{L}$ of $K_{H}(Fl_{n+1})$ extends by $\mathbb {Z}[\![Q]\!]$ -linearity to the ring automorphism of $QK_{H}(Fl_{n+1})$ , and hence, the (dual) left Demazure operators $\delta _{i}^{\vee }$ , $1 \leq i \leq n$ , on $K_{H}(Fl_{n+1})$ extend by $\mathbb {Z}[\![Q]\!]$ -linearity to $QK_{H}(Fl_{n+1})$ . Moreover, we know from [Reference Mihalcea, Naruse and SuMNS, Proposition 8.3] that the following equalities hold in $QK_{H}(Fl_{n+1})$ for $1 \leq i \leq n$ :

(4.3) $$ \begin{align} \delta_{i}^{\vee}(\bullet \star \mathcal{O}_{Fl_{n+1}}(\nu)) = \delta_{i}^{\vee}(\bullet) \star \mathcal{O}_{Fl_{n+1}}(\nu) \end{align} $$

for $\nu \in P$ and $\bullet \in QK_{H}(Fl_{n+1})$ , and

(4.4) $$ \begin{align} \delta_{i}^{\vee}([\mathcal{O}^{w}])= \begin{cases} [\mathcal{O}^{s_iw}] & \text{if } s_{i}w < w, \\ [\mathcal{O}^{w}] & \text{if } s_{i}w> w \end{cases} \end{align} $$

for $w \in W = S_{n+1}$ .

4.3 Proof of the main result for an arbitrary w

Recall the $R(H)[\![Q]\!]$ -algebra isomorphism $\Psi ^{Q}$ from (3.9). Also, recall the identification (ID1) and (ID2) from Section 3.3. Let $i \in [n] = \{1, \ldots , n\}$ . If we extend the Demazure operator $\pi ^{(y)}_{i}$ on $\mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}] \text{ by }\mathbb{Z}[\![Q]\!]$ -linearly to $(\mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}][\![Q]\!])[x_{1},\dots ,x_{n},x_{n+1}]$ by letting it act trivially on the elements of $\mathbb {Z}[x_{1}, \ldots , x_{n}, x_{n+1}]$ , then the (resulting) Demazure operator $\pi ^{(y)}_{i}$ stabilizes the ideal $\mathcal{I}^{Q}$ generated by the elements in (3.10), since the elementary symmetric functions $\sum _{J \subset [n+1], \, |J|=l} \, \mathbf {e}^{\epsilon _{J}}$ , $1 \leq l \leq n+1$ , are invariant under the natural action of $s_i$ and the Demazure operator $\pi ^{(y)}_{i}$ satisfies the (twisted) Leibniz rule. Hence, it induces a $\mathbb {Z}[\![Q]\!]$ -linear operator, also denoted by $\pi_{i}^{(y)}$ , acting trivially on the (residue classes of the) variables $x_{i}$ for $1 \leq i \leq n+1$ on the quotient ring $(\mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}][\![Q]\!])[x_{1},\dots ,x_{n},x_{n+1}]/\mathcal{I}^{Q}$ . Also, by (4.3) and the fact that $\Psi ^{Q}$ maps the residue class of $(1-Q_{j})(1-x_{j})$ modulo $\mathcal{I}^{Q}$ to $[\mathcal{O}_{Fl_{n+1}}(-\epsilon _{j})]$ for $1 \le j \le n+1$ (recall that $Q_{n+1} = 0$ by convention), it follows that the quantum (dual) left Demazure operator $\delta _{i}^{\vee }$ on $QK_{H}(Fl_{n+1})$ is a $\mathbb {Z}[\![Q]\!]$ -linear operator acting trivially on the variables $x_{i}$ for $1 \leq i \leq n+1$ . Since $QK_{H}(Fl_{n+1})$ and the quotient ring $(\mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}][\![Q]\!])[x_{1},\dots ,x_{n},x_{n+1}]/\mathcal{I}^{Q}$ are both generated by the (image under the quotient map of the) elements of $\mathbb {Z}[x_{1}, \ldots , x_{n}, x_{n+1}]$ over $\mathbb {Z}[\mathbf {e}^{\pm \epsilon _{1}},\dots ,\mathbf {e}^{\pm \epsilon _{n}}][\![Q]\!] \cong \mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}][\![Q]\!]$ , and since the (dual) left Demazure operator $\delta _{i}^{\vee }$ and the Demazure operator $\pi ^{(y)}_{i}$ coincide on $\mathbb {Z}[\mathbf {e}^{\pm \epsilon _{1}},\dots ,\mathbf {e}^{\pm \epsilon _{n}}] \cong \mathbb {Z}[(1-y_{1})^{\pm 1},\dots ,(1-y_{n})^{\pm 1}]$ by Remark 4.3, together with the comment on $\delta _{i}^{\vee }$ preceding it, we deduce that the operators $\delta _{i}^{\vee }$ and $\pi ^{(y)}_{i}$ coincide on

$$ \begin{align*} QK_{H}(Fl_{n+1}) \cong (\mathbb{Z}[(1-y_{1})^{\pm 1},\dots,(1-y_{n})^{\pm 1}][\![Q]\!])[x_{1},\dots,x_{n},x_{n+1}]/\mathcal{I}^{Q}; \end{align*} $$

recall the identification (ID2). Namely, we obtain the following commutative diagram for all $1 \leq i \leq n$ :

(4.5)

Now, we are ready to state and prove the main result of this paper.

Theorem 4.4. Let w be an arbitrary element of $W = S_{n+1}$ . Then, under the identification (ID2), the following equality holds in $QK_{H}(Fl_{n+1})$ :

$$ \begin{align*} \Psi^{Q}(\mathfrak{G}_{w}^{Q}(x,y) \ \mathrm{mod} \ \mathcal{I}^{Q}) = [\mathcal{O}^{w}], \end{align*} $$

where $\Psi ^{Q}$ is the $R(H)[\![Q]\!]$ -algebra isomorphism in (3.9).

Proof. The assertion of the theorem is already proved for $w = w_{\circ } \in W=S_{n+1}$ by Theorem 3.3:

(4.6) $$ \begin{align} \Psi^{Q}(\mathfrak{G}_{w_{\circ}}^{Q}(x,y) \ \mathrm{mod} \ \mathcal{I}^{Q}) = [\mathcal{O}^{w_{\circ}}] \in QK_{H}(Fl_{n+1}). \end{align} $$

For an arbitrary $w \in W=S_{n+1}$ , let $ww_{\circ } = s_{i_{1}} \cdots s_{i_{l}}$ be a reduced expression; notice that $l = \ell (w_{\circ })-\ell (w)$ . Then, by the definition of quantum double Grothendieck polynomials, we have

(4.7) $$ \begin{align} \mathfrak{G}_{w}^{Q}(x,y) = \pi_{i_{1}}^{(y)} \cdots \pi_{i_{l}}^{(y)} \mathfrak{G}_{w_{\circ}}^{Q}(x,y). \end{align} $$

Here, note that $w = s_{i_1} \cdots s_{i_l} w_{\circ }$ , and that $w < s_{i_2} \cdots s_{i_l} w_{\circ } < \cdots < s_{i_l}w_{\circ } < w_{\circ }$ in the Bruhat order $<$ on $W = S_{n+1}$ , since $\ell (w) = \ell (w_{\circ }) - l$ . Therefore, by the property (4.4) of quantum (dual) left Demazure operators $\delta _{i}^{\vee }$ , we deduce that

(4.8) $$ \begin{align} (\delta_{i_1}^{\vee} \cdots \delta_{i_l}^{\vee})[\mathcal{O}^{w_{\circ}}] = [\mathcal{O}^{w}] \in QK_{H}(Fl_{n+1}). \end{align} $$

From the equalities (4.6), (4.7) and (4.8), using the commutative diagram (4.5), we conclude that $\Psi ^{Q}(\mathfrak {G}_{w}^{Q}(x,y) \ \mathrm {mod} \ \mathcal{I}^{Q}) = [\mathcal{O}^{w}] \in QK_{H}(Fl_{n+1})$ . This proves the theorem.

5.1 General Chevalley formula

We briefly review the general Chevalley formula for $K_{H}(\mathbf {Q}_{G})$ , following [Reference Lenart, Naito and SagakiLNS] (cf. [Reference Kouno, Lenart and NaitoKoLN, Theorem 5.16]); this formula plays an important role in the proof of Proposition 2.4. We use the notation and setting of Section 2.4. Recall that $\mathfrak {g}$ is an (arbitrary) finite-dimensional simple Lie algebra over $\mathbb {C}$ . Let $\mathfrak {h}^{\ast }_{\mathbb {R}} :=\mathbb {R} \otimes _{\mathbb {Z}} P$ be a real form of $\mathfrak {h}^{\ast }$ , and set

$$ \begin{align*} H_{\beta,l}:=\big\{ \mu \in \mathfrak{h}^{\ast}_{\mathbb{R}} \mid \langle \mu, \beta^\vee \rangle =l\big\} \quad \text{for } \beta \in \Delta \text{ and } l \in \mathbb{Z}. \end{align*} $$

We denote by $s_{\beta ,l}$ the affine reflection in the affine hyperplane $H_{\beta ,l}$ . The affine hyperplanes $H_{\beta ,l}$ , $\beta \in \Delta $ , $l \in \mathbb {Z}$ , divide the real vector space $\mathfrak {h}^{\ast }_{\mathbb {R}}$ into open regions, called alcoves; the fundamental alcove is defined as

$$ \begin{align*} A_{\circ} := \big\{ \mu \in \mathfrak{h}_{\mathbb{R}}^{\ast} \mid 0 < \langle \mu, \alpha^\vee \rangle < 1 \quad \text{for all } \alpha \in \Delta^{+} \big\}. \end{align*} $$

We say that two alcoves are adjacent if they are distinct and have a common wall. Given a pair of adjacent alcoves A and B, we write $A \xrightarrow {\hspace {2pt}\beta \hspace {2pt}} B$ for $\beta \in \Delta $ if the common wall is orthogonal to $\beta $ and $\beta $ points in the direction from A to B.

Let $\lambda \in P$ , and let $A_{\lambda }=A_{\circ }+\lambda $ be the translation of the fundamental alcove $A_{\circ }$ by the weight $\lambda $ .

Definition 5.2 [Reference Lenart and PostnikovLP]

Let $\lambda \in P$ . A sequence $(\beta _1, \beta _2, \dots , \beta _m)$ of roots is called a reduced $\lambda $ -chain (of roots) if

(5.1) $$ \begin{align} A_{\circ}=A_{0} \xrightarrow{\hspace{2pt}-\beta_1\hspace{2pt}} A_1 \xrightarrow{\hspace{2pt}-\beta_2\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}-\beta_m\hspace{2pt}} A_m=A_{-\lambda} \end{align} $$

is a reduced alcove path.

A reduced alcove path $(A_0=A_{\circ },A_1,\ldots ,A_m=A_{-\lambda })$ can be identified with the corresponding total order on the hyperplanes, to be called $\lambda $ -hyperplanes, which separate $A_\circ $ from $A_{-\lambda }$ . This total order is given by the sequence $H_{\beta _i,-l_i}$ for $i=1,\ldots ,m$ , where $H_{\beta _i,-l_i}$ contains the common wall of $A_{i-1}$ and $A_i$ . Note that $\langle \lambda , \beta _i^\vee \rangle \ge 0$ , and that the integers $l_i$ , called heights, have the following ranges:

$$ \begin{align*} \begin{aligned} & 0 \le l_i \le \langle \lambda, \beta_i^\vee \rangle-1 \quad \text{if} \quad \beta_i \in \Delta^+, \\ & 1 \le l_i \le \langle \lambda, \beta_i^\vee \rangle \quad \text{if} \quad \beta_i \in \Delta^- = - \Delta^+. \end{aligned} \end{align*} $$

Note also that a reduced $\lambda $ -chain $(\beta _1, \ldots , \beta _m)$ determines the corresponding reduced alcove path, and hence we can identify them.

Let $\lambda \in P$ , and fix a reduced $\lambda $ -chain, which we denote by $\Gamma (\lambda )=(\beta _1,\,\ldots ,\,\beta _m)$ ; later in Section 5.2, for $\lambda = - \epsilon _{J} \in P$ with $J \subset [k]$ , we make a specific choice of reduced $\lambda $ -chain and denote it by $\Gamma _{J}$ . Let $w \in W$ .

Definition 5.3 [Reference Lenart and LubovskyLL]

A (possibly empty) subset $A=\left \{ j_1 < j_2 < \cdots < j_s \right \}$ of $[m]=\{1,\ldots ,m\}$ is called a w-admissible subset if we have the following directed path in the quantum Bruhat graph $\mathrm {QBG}(W)$ :

(5.2) $$ \begin{align} \begin{aligned} & \Pi(w,A): w \xrightarrow{\hspace{2pt}|\beta_{j_1}|\hspace{2pt}} w s_{\beta_{j_1}} \xrightarrow{\hspace{2pt}|\beta_{j_2}|\hspace{2pt}} ws_{\beta_{j_1}}s_{\beta_{j_2}} \xrightarrow{\hspace{2pt}|\beta_{j_3}|\hspace{2pt}} \cdots \\ & \hspace{30mm} \cdots \xrightarrow{\hspace{2pt}|\beta_{j_s}|\hspace{2pt}} ws_{\beta_{j_1}}s_{\beta_{j_2}} \cdots s_{\beta_{j_s}}=:\operatorname{\mathrm{end}}(A). \end{aligned} \end{align} $$

Let $\mathcal{A}(w,\Gamma (\lambda ))$ denote the collection of all w-admissible subsets of $[m] = \{1, \ldots , m\}$ .

Let $A=\{ j_1 < \cdots < j_s\} \in \mathcal{A}(w,\Gamma (\lambda ))$ . The weight of A is defined by

(5.3) $$ \begin{align} \operatorname{\mathrm{wt}}(A):=-w s_{\beta_{j_1},-l_{j_1}} \cdots s_{\beta_{j_s},-l_{j_s}}(-\lambda).\end{align} $$

Also, we set

(5.4) $$ \begin{align} n(A):=\# \{ \beta_{j_1},\ldots,\beta_{j_s} \} \cap \Delta^{-}, \qquad\qquad\qquad\qquad\qquad\qquad \end{align} $$

(5.5) $$ \begin{align} A^{-}:=\left\{j_i \in A \ \biggm| \ \begin{array}{l} ws_{\beta_{j_1}} \cdots s_{\beta_{j_{i-1}}} \xrightarrow{\hspace{2pt}|\beta_i|\hspace{2pt}} ws_{\beta_{j_1}} \cdots s_{\beta_{j_{i-1}}}s_{\beta_{j_{i}}} \\[2mm] \text{is a quantum edge} \end{array} \right\},\end{align} $$

(5.6) $$ \begin{align} \operatorname{\mathrm{down}}(A):=\sum_{j\in A^-}|\beta_j|^\vee\in Q^{\vee,+}. \qquad\qquad\qquad\qquad\qquad\qquad \end{align} $$

We write $\lambda \in P$ as $\lambda =\sum _{i\in I}\lambda _i\varpi _i \in P$ , with $\lambda _{i} \in \mathbb {Z}$ for $i \in I$ . Following [Reference Lenart, Naito and SagakiLNS, Section 4.1], let $\overline {\mathrm {Par}(\lambda )}$ denote the set of I-tuples of partitions $\boldsymbol {\chi }=(\chi ^{(i)})_{i\in I}$ such that $\chi ^{(i)}$ is a partition of length at most $\max (\lambda _i,0)$ ; in [Reference Naito and SagakiNS1, Section 2.5], for $\lambda \in P^{+}$ , we introduced the set $\mathrm {Par}(\lambda )$ of I-tuples of partitions $\boldsymbol {\chi }=(\chi ^{(i)})_{i\in I}$ such that $\chi ^{(i)}$ is a partition of length less than $\lambda _i$ (which we do not use in this paper). For $\boldsymbol {\chi } = (\chi ^{(i)})_{i \in I} \in \overline {\mathrm {Par}(\lambda )}$ , we set $\iota (\boldsymbol {\chi }) := \sum _{i \in I} \chi ^{(i)}_1 \alpha _i^{\vee } \in Q^{\vee ,+}$ , with $\chi ^{(i)}_1$ the first part of the partition $\chi ^{(i)}$ for each $i \in I$ .

By specializing at $q = 1$ in [Reference Lenart, Naito and SagakiLNS, Theorem 33] (cf. [Reference Kouno, Lenart and NaitoKoLN, Theorem 5.16]), we obtain the following general Chevalley formula for $K_{H}(\mathbf {Q}_{G})$ .

Theorem 5.4. Let $\lambda =\sum _{i\in I}\lambda _i\varpi _i \in P$ be an arbitrary weight, $\Gamma (\lambda )$ an arbitrary reduced $\lambda $ -chain, and $x=wt_{\xi }\in W_{\mathrm {af}}^{\ge 0} \simeq W \times Q^{\vee ,+}$ . Then, the following equality holds in $K_{H}(\mathbf {Q}_{G})$ :

(5.7) $$ \begin{align} & [\mathcal{O}_{\mathbf{Q}_{G}}(-w_{\circ} \lambda)] \otimes [\mathcal{O}_{\mathbf{Q}_{G}(x)}] \nonumber \\[3mm] & \quad = \sum_{A \in \mathcal{A}(w,\Gamma(\lambda))} \sum_{ \boldsymbol{\chi} \in \overline{\mathrm{Par}(\lambda)} } (-1)^{n(A)} \mathbf{e}^{\operatorname{\mathrm{wt}}(A)} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(A)t_{\xi+\operatorname{\mathrm{down}}(A)+\iota(\boldsymbol{\chi})} )}]. \end{align} $$

In order to prove equation (2.6), we compute

(5.8) $$ \begin{align} \left( \left(\prod_{ j \in J^{-} } (1-\mathsf{t}_{j})\right) [\mathcal{O}_{\mathbf{Q}_{G}}( w_{\circ} \epsilon_{J} )] \right) \otimes [\mathcal{O}_{\mathbf{Q}_{G}(w_{k+1})}] \in K_{H}(\mathbf{Q}_{G}) \end{align} $$

for $J = \big \{ j_{1},j_{2},\dots ,j_{p} \big \} \subset [k] =\big \{1,2,\ldots ,k\big \}$ , with $1 \le j_{1} < j_{2} < \cdots < j_{p} \le k$ ; recall Remark 2.3 and the definition of $J^{-}$ from (2.4).

Lemma 5.5. Keep the setting above. The following equality holds in $K_{H}(\mathbf {Q}_{G})$ :

(5.9) $$ \begin{align} & \left( \left(\prod_{ j \in J^{-} } (1-\mathsf{t}_{j})\right) [\mathcal{O}_{\mathbf{Q}_{G}}( w_{\circ} \epsilon_{J} )] \right) \otimes [\mathcal{O}_{\mathbf{Q}_{G}(w_{k+1})}] \nonumber \\[3mm] & \qquad = \sum_{A \in \mathcal{A}(w_{k+1},\Gamma(-\epsilon_{J}))} (-1)^{n(A)} \mathbf{e}^{\operatorname{\mathrm{wt}}(A)} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(A)t_{\operatorname{\mathrm{down}}(A)} )}]. \end{align} $$

Proof. Applying Theorem 5.4 to the case that $\lambda =-\epsilon _{J}$ and $x = w_{k+1}$ , we obtain

$$ \begin{align*} & [\mathcal{O}_{\mathbf{Q}_{G}}(w_{\circ} \epsilon_{J})] \otimes [\mathcal{O}_{\mathbf{Q}_{G}(w_{k+1})}] = \sum_{A \in \mathcal{A}(w_{k+1},\Gamma(-\epsilon_{J}))} \, \sum_{ \boldsymbol{\chi} \in \overline{\mathrm{Par}(-\epsilon_{J})} } (-1)^{n(A)} \mathbf{e}^{\operatorname{\mathrm{wt}}(A)} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(A)t_{\operatorname{\mathrm{down}}(A)+\iota(\boldsymbol{\chi})} )}]. \end{align*} $$

Here, observe that for $1 \le j \le n$ , we have $\langle -\epsilon _{J}, \alpha _{j}^{\vee } \rangle \in \{-1,0,1\}$ , and that the equality $\langle -\epsilon _{J}, \alpha _{j}^{\vee } \rangle = 1$ holds if and only if $j \in J^{-}$ . This implies that $-\epsilon _{J}=\sum _{i \in J^{-}} \varpi _{i}$ . Hence, $\boldsymbol {\chi } = (\chi ^{(i)})_{i \in I} \in \overline {\mathrm {Par}(-\epsilon _{J})}$ if and only if $\chi ^{(i)}$ is a partition of length at most one for all $i \in J^{-}$ and the empty partition for all $i \notin J^{-}$ . By the definition of $\iota (\boldsymbol {\chi })$ , we deduce that the map $\boldsymbol {\chi } \mapsto \iota (\boldsymbol {\chi })$ gives a bijection from $\overline {\mathrm {Par}(-\epsilon _{J})}$ onto $\sum _{j \in J^{-}}\mathbb {Z}_{\ge 0}\alpha _{j}^{\vee }$ . Therefore, we see that

$$ \begin{align*} & \sum_{A \in \mathcal{A}(w_{k+1},\Gamma(-\epsilon_{J}))} \, \sum_{ \boldsymbol{\chi} \in \overline{\mathrm{Par}(-\epsilon_{J})} } (-1)^{n(A)} \mathbf{e}^{\operatorname{\mathrm{wt}}(A)} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(A)t_{\operatorname{\mathrm{down}}(A)+\iota(\boldsymbol{\chi})} )}] \\[3mm] & \quad = \sum_{A \in \mathcal{A}(w_{k+1},\Gamma(-\epsilon_{J}))} \, \sum_{ \zeta \in \sum_{j \in J^{-}}\mathbb{Z}_{\ge 0}\alpha_{j}^{\vee} } (-1)^{n(A)} \mathbf{e}^{\operatorname{\mathrm{wt}}(A)} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(A)t_{\operatorname{\mathrm{down}}(A)+\zeta} )}] \\[3mm] & \quad = \left(\prod_{j \in J^{-}} \frac{1}{1-\mathsf{t}_{j}}\right) \sum_{A \in \mathcal{A}(w_{k+1},\Gamma(-\epsilon_{J}))} (-1)^{n(A)} \mathbf{e}^{\operatorname{\mathrm{wt}}(A)} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(A)t_{\operatorname{\mathrm{down}}(A)} )}]. \end{align*} $$

From this, we deduce that in $K_{H}(\mathbf {Q}_{G})$ ,

$$ \begin{align*} & \left( \left(\prod_{ j \in J^{-} } (1-\mathsf{t}_{j})\right) [\mathcal{O}_{\mathbf{Q}_{G}}( w_{\circ} \epsilon_{J} )] \right) \otimes [\mathcal{O}_{\mathbf{Q}_{G}(w_{k+1})}] \\[3mm] & \hspace{5mm} = \left(\prod_{ j \in J^{-} } (1-\mathsf{t}_{j})\right) \big( [\mathcal{O}_{\mathbf{Q}_{G}}( w_{\circ} \epsilon_{J} )] \otimes [\mathcal{O}_{\mathbf{Q}_{G}(w_{k+1})}] \big) \quad \text{(see Remark~2.3)} \\[3mm] & \hspace{5mm} = \left(\prod_{ j \in J^{-} } (1-\mathsf{t}_{j})\right) \left( \left(\prod_{ j \in J^{-} } \frac{1}{1-\mathsf{t}_{j}}\right) \sum_{A \in \mathcal{A}(w_{k+1},\Gamma(-\epsilon_{J}))} (-1)^{n(A)} \mathbf{e}^{\operatorname{\mathrm{wt}}(A)} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(A)t_{\operatorname{\mathrm{down}}(A)} )}] \right) \\[3mm] & \hspace{5mm} = \sum_{A \in \mathcal{A}(w_{k+1},\Gamma(-\epsilon_{J}))} (-1)^{n(A)} \mathbf{e}^{\operatorname{\mathrm{wt}}(A)} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(A)t_{\operatorname{\mathrm{down}}(A)} )}], \end{align*} $$

as desired. This proves the lemma.

5.2 A special reduced chain (of roots)

Let us fix $J = \big \{ j_{1},j_{2},\dots ,j_{p} \big \} \subset [k]$ , with $1 \le j_{1} < j_{2} < \cdots < j_{p} \le k$ . Following [Reference Lenart, Naito, Orr and SagakiLNOS, Lemma 4.1], we take a special reduced $(-\epsilon _{J})$ -chain (of roots) from $A_{\circ }$ to $A_{\epsilon _{J}}$ (see Definition 5.2) as follows. Notice that $\epsilon _{J} \in W\varpi _{p}$ . Let $x_{J}$ be the (unique) minimal-length element in $\big \{ w \in W \mid w \varpi _{p} = \epsilon _{J} \big \}$ , and let $x_{J} = s_{i_{a}} \cdots s_{i_{1}}$ , with $a:=\ell (x_{J})$ , be the reduced expression of the following form:

(5.10) $$ \begin{align} \begin{aligned} x_{J} & = s_{i_{a}} \cdots s_{i_{1}} \\ & = \underbrace{(s_{j_1-1} \cdots s_{2}s_{1})}_{\text{mapping }\epsilon_{1} \text{ to } \epsilon_{j_{1}}.} \underbrace{(s_{j_2-1} \cdots s_{3}s_{2})}_{\text{mapping } \epsilon_{2} \text{ to } \epsilon_{j_{2}}.} \cdots \\[2mm] & \qquad \cdots \underbrace{(s_{j_{p-1}-1} \cdots s_{p}s_{p-1})}_{\text{mapping } \epsilon_{p-1} \text{ to } \epsilon_{j_{p-1}}.} \underbrace{(s_{j_{p}-1} \cdots s_{p+1}s_{p}).}_{\text{mapping } \epsilon_{p} \text{ to } \epsilon_{j_p}.} \end{aligned} \end{align} $$

Also, let $y_{J}$ be the (unique) element such that $y_{J}x_{J}$ is the (unique) minimal-length element in $\big \{ w \in W \mid w \varpi _{p} = w_{\circ } \varpi _{p} \big \}$ , and let $y_{J} = s_{k_{1}} \cdots s_{k_{b}}$ , with $b:=\ell (y_{J})$ , be the reduced expression of the following form:

(5.11) $$ \begin{align} \begin{aligned} y_{J} & = s_{k_{1}} \cdots s_{k_{b}} \\ & = \underbrace{(s_{n-p+1} \cdots s_{j_{1}+1}s_{j_{1}})}_{\text{mapping } \epsilon_{j_1} \text{ to } \epsilon_{n-p+2}.} \underbrace{(s_{n-p+2} \cdots s_{j_{2}+1}s_{j_{2}})}_{\text{mapping } \epsilon_{j_2} \text{ to } \epsilon_{n-p+3}.} \cdots \\[2mm] & \qquad \cdots \underbrace{(s_{n-1} \cdots s_{j_{p-1}+1}s_{j_{p-1}}) }_{\text{mapping } \epsilon_{j_{p-1}} \text{ to } \epsilon_{n}.} \underbrace{(s_{n} \cdots s_{j_{p}+1}s_{j_{p}}).}_{\text{mapping } \epsilon_{j_p} \text{ to } \epsilon_{n+1}.} \end{aligned} \end{align} $$

We set

(5.12) $$ \begin{align} & \beta_{c}:=s_{i_{a}} \cdots s_{i_{c+1}}\alpha_{i_{c}} \in \Delta^{+} \quad \text{for } 1 \le c \le a, \end{align} $$

(5.13) $$ \begin{align} & \gamma_{d}:=s_{k_{b}} \cdots s_{k_{d+1}}\alpha_{k_{d}} \in \Delta^{+} \quad \text{for } 1 \le d \le b. \end{align} $$

We call the sequences $\boldsymbol {\beta }:=(\beta _{a},\dots ,\beta _{1})$ and $\boldsymbol {\gamma }:=(\gamma _{1},\dots ,\gamma _{b})$ the $\boldsymbol {\beta }$ -sequence and $\boldsymbol {\gamma }$ -sequence for J, respectively.

Proposition 5.6 [Reference Lenart, Naito, Orr and SagakiLNOS, Lemma 4.1]

If we set

(5.14) $$ \begin{align} \Gamma_{J}=(\zeta_{1},\dots,\zeta_{a+b}):= (-\beta_{a},\dots,-\beta_{1},\gamma_{1},\dots,\gamma_{b}), \end{align} $$

then $\Gamma _{J}$ is a reduced $(-\epsilon _{J})$ -chain from $A_{\circ }$ to $A_{\epsilon _{J}}$ . Moreover, for $1 \le c \le a$ , the affine hyperplane between the $(c-1)$ -th alcove and the c-th alcove in $\Gamma _{J}$ is $H_{-\beta _{a-c+1},0}$ , and for $1 \le d \le b$ , the affine hyperplane between the $(a+d-1)$ -th alcove and the $(a+d)$ -th alcove in $\Gamma _{J}$ is $H_{\gamma _{d},1}$ ; remark that $\epsilon _{J} \in H_{\gamma _{d},1}$ , and hence, $s_{\gamma _{d},1}(\epsilon _{J}) = \epsilon _{J}$ for all $1 \le d \le b$ (see also Remark 5.11 below).

Example 5.7. Assume that $n = 5$ , $k=4$ and $J=\{2,4\}$ . In this case, we see that

$$ \begin{align*} x_{J}=s_1s_3s_2, \qquad y_{J}=s_4s_3s_2s_5s_4; \end{align*} $$

in particular, $a=\ell (x_{J})=3$ and $b = \ell (y_{J})=5$ . We have

$$ \begin{align*} \beta_{3} & = \alpha_{1} = (1,2), \\ \beta_{2} & = s_1\alpha_3 = \alpha_{3} = (3,4), \\ \beta_{1} & = s_1s_3\alpha_2 = \alpha_{1}+\alpha_{2}+\alpha_{3} = (1,4), \\ \gamma_{1} & = s_4s_5s_2s_3\alpha_{4} = \alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5} = (2,6), \\ \gamma_{2} & = s_4s_5s_2\alpha_{3} = \alpha_{2}+\alpha_{3}+\alpha_{4} = (2,5), \\ \gamma_{3} & = s_4s_5\alpha_{2} = \alpha_{2} = (2,3), \\ \gamma_{4} & = s_4\alpha_{5} = \alpha_{4} + \alpha_{5} = (4,6), \\ \gamma_{5} & = \alpha_{4} = (4,5). \end{align*} $$

Thus,

$$ \begin{align*} \Gamma_{J} & =(-\alpha_{1}, -\alpha_{3}, -(\alpha_{1}+\alpha_{2}+\alpha_{3}), \\ & \qquad \qquad \alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}, \alpha_{2}+\alpha_{3}+\alpha_{4}, \alpha_{2}, \alpha_{4} + \alpha_{5}, \alpha_{4}) \end{align*} $$

is a reduced $(-\epsilon _{J})$ -chain from $A_{\circ }$ to $A_{\epsilon _{J}}$ .

Let us compute $\beta _{c} \in \Delta ^{+}$ , $1 \le c \le a$ , more explicitly. For $1 \le u \le p$ and $u \le t \le j_{u}-1$ , we set

$$ \begin{align*} \beta_{u,t} & := (s_{j_1-1} \cdots s_{2}s_{1}) \cdots (s_{j_{u-1}-1} \cdots s_{u}s_{u-1}) (s_{j_{u}-1} \cdots s_{t+1})\alpha_{t}; \end{align*} $$

notice that the sequence $\boldsymbol {\beta }=(\beta _{a},\dots ,\beta _{1})$ is identical to

(5.15) $$ \begin{align} \begin{aligned} & (\beta_{1,j_{1}-1},\dots,\beta_{1,2},\beta_{1,1},\ \beta_{2,j_{2}-1},\dots,\beta_{2,3},\beta_{2,2},\ \dots\dots, \\ & \ \beta_{p-1,j_{p-1}-1},\dots,\beta_{p-1,p},\beta_{p-1,p-1},\ \beta_{p,j_{p}-1},\dots,\beta_{p,p+1},\beta_{p,p}). \end{aligned} \end{align} $$

We set $\tau (u,t):= |\big \{1 \le r \le u-1 \mid t-r+1 \le j_{u-r} \big \}|$ .

Proof. For $1 \le u \le p$ and $u \le t \le j_{u}-1$ , we have

$$ \begin{align*} \beta_{u,t} & := (s_{j_1-1} \cdots s_{2}s_{1}) \cdots (s_{j_{u-1}-1} \cdots s_{u}s_{u-1}) (s_{j_{u}-1} \cdots s_{t+1})(\epsilon_{t}-\epsilon_{t+1}) \\ & = (s_{j_1-1} \cdots s_{2}s_{1}) \cdots (s_{j_{u-1}-1} \cdots s_{u}s_{u-1}) (\epsilon_{t}-\epsilon_{j_{u}}) \\ & = \begin{cases} \epsilon_{t}-\epsilon_{j_{u}} & \text{if } t> j_{u-1}, \\ (s_{j_1-1} \cdots s_{2}s_{1}) \cdots (s_{j_{u-2}-1} \cdots s_{u-1}s_{u-2})(\epsilon_{t-1}-\epsilon_{j_{u}}) & \text{otherwise}. \end{cases} \end{align*} $$

In the case $t \le j_{u-1}$ , we have

$$ \begin{align*} & (s_{j_1-1} \cdots s_{2}s_{1}) \cdots (s_{j_{u-2}-1} \cdots s_{u-1}s_{u-2})(\epsilon_{t-1}-\epsilon_{j_{u}}) \\ & = \begin{cases} \epsilon_{t-1}-\epsilon_{j_{u}} & \text{if } t-1> j_{u-2}, \\ (s_{j_1-1} \cdots s_{2}s_{1}) \cdots (s_{j_{u-3}-1} \cdots s_{u-2}s_{u-3})(\epsilon_{t-2}-\epsilon_{j_{u}}) & \text{otherwise}. \end{cases} \end{align*} $$

Repeating this computation, we deduce that $\beta _{u,t}=\epsilon _{t-\tau (u,t)}-\epsilon _{j_{u}}$ . This proves the lemma.

Remark 5.9. Let $1 \le u \le p$ , and write

(5.16) $$ \begin{align} \big\{1,2,\dots,j_{u}-1\big\} \setminus \big\{j_{1},\dots,j_{u-1}\big\} = \big\{t_{u,1}< t_{u,2} < \dots < t_{u,m_{u}}\big\}, \end{align} $$

where $m_{u}=(j_{u}-1)-(u-1) = j_{u}-u$ ; we set $t_{u,m_{u}+1}:=j_{u}$ by convention. Then, we have

$$ \begin{align*} \begin{aligned} & (\beta_{u,j_{u}-1},\dots,\beta_{u,u+1},\beta_{u,u}) = \big( \epsilon_{t_{u,m_u}}-\epsilon_{j_{u}}, \ \epsilon_{t_{u,m_u-1}}-\epsilon_{j_{u}}, \ \dots, \ \epsilon_{t_{u,2}}-\epsilon_{j_{u}}, \ \epsilon_{t_{u,1}}-\epsilon_{j_{u}} \big). \end{aligned} \end{align*} $$

Hence, the sequence $\boldsymbol {\beta }=(\beta _{a},\dots ,\beta _{1})$ is of the form

where $(j_{r},j_{s})$ does not appear in the sequence above for any $1 \le r < s \le p$ .

Similarly, let us compute $\gamma _{d} \in \Delta ^{+}$ , $1 \le d \le b$ , more explicitly. For $1 \le u \le p$ and $j_{u} \le t \le n-p+u$ , we set

$$ \begin{align*} \gamma_{u,t} & := (s_{j_{p}} s_{j_{p}+1} \cdots s_{n}) \cdots (s_{j_{u+1}} s_{j_{u+1}+1} \cdots s_{n-p+u+1}) (s_{j_{u}} s_{j_{u}+1} \cdots s_{t-1})\alpha_{t}; \end{align*} $$

notice that the sequence $\boldsymbol {\gamma }=(\gamma _{1},\dots ,\gamma _{b})$ is identical to

(5.17) $$ \begin{align} \begin{aligned} & (\gamma_{1,n-p+1},\dots,\gamma_{1,j_{1}+1},\gamma_{1,j_{1}}, \ \gamma_{2,n-p+2},\dots,\gamma_{2,j_{2}+1},\gamma_{2,j_{2}}, \ \dots\dots, \\ & \ \gamma_{p-1,n-1},\dots,\gamma_{p-1,j_{p-1}+1},\gamma_{p-1,j_{p-1}}, \ \gamma_{p,n},\dots,\gamma_{p,j_{p}+1},\gamma_{p,j_{p}} ). \end{aligned} \end{align} $$

We set $\sigma (u,t):= |\big \{ u+1 \le r \le p \mid t+r-u \ge j_{r} \big \}|$ . The proof of the following lemma is similar to that of Lemma 5.8.

Remark 5.11. Let $1 \le u \le p$ , and write

$$ \begin{align*} \big\{j_{u}+1,j_{u}+2,\dots,n+1 \big\} \setminus \big\{j_{u+1},\dots,j_{p}\big\} = \big\{t_{1}< t_{2} < \dots < t_{m}\big\}, \end{align*} $$

where $m=(n+1-j_{u})-(p-u) = n + 1 - p - j_{u}+u$ . Then, we have

$$ \begin{align*} \begin{aligned} & (\gamma_{u,n-p+u},\dots,\gamma_{u,j_{u}+1},\gamma_{u,j_{u}}) \\ & = \big( \epsilon_{j_{u}}-\epsilon_{t_{m}}, \ \epsilon_{j_{u}}-\epsilon_{t_{m-1}}, \ \dots, \ \epsilon_{j_{u}}-\epsilon_{t_{2}}, \ \epsilon_{j_{u}}-\epsilon_{t_{1}} \big). \end{aligned} \end{align*} $$

Hence, the sequence $\boldsymbol {\gamma }=(\gamma _{1},\dots ,\gamma _{b})$ is of the form

where $(j_{r},j_{s})$ does not appear in the sequence above for any $1 \le r < s \le p$ .

5.3 Proof of Proposition 2.4

For each $J = \big \{ j_{1},j_{2},\dots ,j_{p} \big \} \subset [k]$ , let $\mathbf {D}_{J}$ be the set of directed paths in $\mathrm {QBG}(W)$ of the form

(5.18) $$ \begin{align} \begin{aligned} & \mathbf{p}:w_{k+1} = \underbrace{ x_{s} \xrightarrow{\hspace{2pt}\beta_{a_s}\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}\beta_{a_1}\hspace{2pt}} x_{0}}_{ =:\,\mathbf{p}_{\beta}, \text{ the } \beta\text{-part of } \mathbf{p}} = \underbrace{ y_{0} \xrightarrow{\hspace{2pt}\gamma_{b_1}\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}\gamma_{b_t}\hspace{2pt}} y_{t}}_{ =:\,\mathbf{p}_{\gamma}, \text{ the } \gamma\text{-part of } \mathbf{p}} = : \operatorname{\mathrm{end}}(\mathbf{p}), \\ & \textrm{with } s \ge 0, a \ge a_{s}> \cdots > a_{1} \ge 1, \\ & \textrm{and } t \ge 0, 1 \le b_{1} < b_{2} < \cdots < b_{t} \le b. \end{aligned} \end{align} $$

We call the directed subpath $w_{k+1} = x_{s} \xrightarrow {\hspace {2pt}\beta _{a_s}\hspace {2pt}} \cdots \xrightarrow {\hspace {2pt}\beta _{a_1}\hspace {2pt}} x_{0}$ of $\mathbf {p}$ the $\boldsymbol {\beta }$ -part of $\mathbf {p}$ and denote it by $\mathbf {p}_{\boldsymbol {\beta }}$ . Notice that the label sequence $(\beta _{a_s},\dots ,\beta _{a_1})$ of $\mathbf {p}_{\boldsymbol {\beta }}$ is a subsequence of $\boldsymbol {\beta }=(\beta _{a},\dots ,\beta _{1})$ , described in Lemma 5.8 and Remark 5.9, with $a = \ell (x_{J})$ ; the subscript $\boldsymbol {\beta }$ of $\mathbf {p}_{\boldsymbol {\beta }}$ indicates that $\mathbf {p}_{\boldsymbol {\beta }}$ is obtained from $\mathbf {p}$ by taking the first consecutive edges whose labels are contained in the $\boldsymbol {\beta }$ -sequence. Similarly, we call the directed subpath $y_{0} \xrightarrow {\hspace {2pt}\gamma _{b_1}\hspace {2pt}} \cdots \xrightarrow {\hspace {2pt}\gamma _{b_t}\hspace {2pt}} y_{t} = \operatorname {\mathrm {end}}(\mathbf {p})$ of $\mathbf {p}$ the $\boldsymbol {\gamma }$ -part of $\mathbf {p}$ and denote it by $\mathbf {p}_{\boldsymbol {\gamma }}$ . Notice that the label sequence $(\gamma _{b_1},\dots ,\gamma _{b_t})$ is a subsequence of $\boldsymbol {\gamma }=(\gamma _{1},\dots ,\gamma _{b})$ , described in Lemma 5.15 and Remark 5.11, with $b = \ell (y_{J})$ ; the subscript $\boldsymbol {\gamma }$ of $\mathbf {p}_{\boldsymbol {\gamma }}$ indicates that $\mathbf {p}_{\boldsymbol {\gamma }}$ is obtained from $\mathbf {p}$ by taking the second consecutive edges whose labels are contained in the $\boldsymbol {\gamma }$ -sequence.

Proposition 5.13. The following equality holds in $K_{H}(\mathbf {Q}_{G})$ :

(5.19) $$ \begin{align} & \overbrace{ \left( \sum_{p=0}^{k}(-1)^{p} \mathbf{e}^{p \epsilon_{n+1-k}} \mathfrak{F}^{k}_{p} \right) \otimes [\mathcal{O}_{\mathbf{Q}_{G}( w_{k+1} )}] }^{ \mathrm{LHS\ of\ (2.6)} } \nonumber \\[3mm] & = \sum_{J \subset [k]} \sum_{\mathbf{p} \in \mathbf{D}_{J}} \mathbf{e}^{|J| \epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})\epsilon_{J}} (-1)^{|J|+\ell(\mathbf{p}_{\boldsymbol{\gamma}})} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(\mathbf{p})t_{\operatorname{\mathrm{wt}}(\mathbf{p})} )}]. \end{align} $$

Proof. First recall Lemma 5.5. It is easy to see that there exists a bijection between the sets $\mathbf {D}_{J}$ and $\mathcal{A}(w_{k+1},\Gamma _{J})$ . Indeed, for a subset $A=\big \{p_{1},\dots ,p_{u}\}$ of $[a+b]=\{1,2,\dots ,a+b\}$ , A is an element of $\mathcal{A}(w_{k+1},\Gamma _{J})$ if and only if

(5.20) $$ \begin{align} \mathbf{p}(A):w_{k+1}=w_{0} \xrightarrow{\hspace{2pt}\zeta_{p_1}\hspace{2pt}} w_{1} \xrightarrow{\hspace{2pt}\zeta_{p_2}\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}\zeta_{p_u}\hspace{2pt}} w_{u} \end{align} $$

is an element of $\mathbf {D}_{J}$ . Notice that $\operatorname {\mathrm {end}}(A)=\operatorname {\mathrm {end}}(\mathbf {p}(A))$ and $\operatorname {\mathrm {down}}(A)=\operatorname {\mathrm {wt}}(\mathbf {p})$ . For $\mathbf {p} \in \mathbf {D}_{J}$ of the form (5.18), we set $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }}):=x_{0}$ and $\ell (\mathbf {p}_{\boldsymbol {\gamma }}):=t$ ; we see that for $A \in \mathcal{A}(w_{k+1},\Gamma _{J})$ , $\operatorname {\mathrm {wt}}(A)=-\operatorname {\mathrm {end}}(\mathbf {p}(A)_{\boldsymbol {\beta }})\epsilon _{J}$ and $n(A)=\ell (\mathbf {p}(A)_{\boldsymbol {\gamma }})$ . Therefore, we deduce that in $K_{H}(\mathbf {Q}_{G})$ ,

$$ \begin{align*} & \left( \left(\prod_{ j \in J^{-} } (1-\mathsf{t}_{j})\right) [\mathcal{O}_{\mathbf{Q}_{G}}( w_{\circ} \epsilon_{J} )] \right) \otimes [\mathcal{O}_{\mathbf{Q}_{G}(w_{k+1})}] \\[3mm] & \hspace{5mm} = \sum_{A \in \mathcal{A}(w_{k+1},\Gamma(-\epsilon_{J}))} (-1)^{n(A)} \mathbf{e}^{\operatorname{\mathrm{wt}}(A)} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(A)t_{\operatorname{\mathrm{down}}(A)} )}] \qquad \text{by (5.9)} \\[3mm] & \hspace{5mm} = \sum_{\mathbf{p} \in \mathbf{D}_{J}} \mathbf{e}^{-\operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})\epsilon_{J}} (-1)^{\ell(\mathbf{p}_{\boldsymbol{\gamma}})} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(\mathbf{p})t_{\operatorname{\mathrm{wt}}(\mathbf{p})} )}], \end{align*} $$

and hence that

$$ \begin{align*} & \left( \sum_{p=0}^{k}(-1)^{p} \mathbf{e}^{p \epsilon_{n+1-k}} \mathfrak{F}^{k}_{p} \right) \otimes [\mathcal{O}_{\mathbf{Q}_{G}( w_{k+1} )}] \nonumber \\[3mm] & = \sum_{p=0}^{k}(-1)^{p} \mathbf{e}^{p \epsilon_{n+1-k}} \sum_{ \begin{subarray}{c} J \subset [k] \\[1mm] |J|=p \end{subarray}} \left( \left(\prod_{ j \in J^{-} } (1-\mathsf{t}_{j})\right)[\mathcal{O}_{\mathbf{Q}_{G}}( w_{\circ} \epsilon_{J} )] \right) \otimes [\mathcal{O}_{\mathbf{Q}_{G}( w_{k+1} )}] \nonumber \\[3mm] & = \sum_{J \subset [k]} (-1)^{|J|} \mathbf{e}^{|J| \epsilon_{n+1-k}} \left( \left(\prod_{ j \in J^{-} } (1-\mathsf{t}_{j})\right)[\mathcal{O}_{\mathbf{Q}_{G}}( w_{\circ} \epsilon_{J} )] \right) \otimes [\mathcal{O}_{\mathbf{Q}_{G}( w_{k+1} )}] \nonumber \\[3mm] & = \sum_{J \subset [k]} \sum_{\mathbf{p} \in \mathbf{D}_{J}} \mathbf{e}^{|J| \epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})\epsilon_{J}} (-1)^{|J|+\ell(\mathbf{p}_{\boldsymbol{\gamma}})} [\mathcal{O}_{\mathbf{Q}_{G}( \operatorname{\mathrm{end}}(\mathbf{p})t_{\operatorname{\mathrm{wt}}(\mathbf{p})} )}]. \end{align*} $$

This proves the proposition.

Recall that

$$ \begin{align*} \begin{aligned} & w_{k+1} := (s_{1}s_{2} \cdots s_{n}) (s_{1}s_{2} \cdots s_{n-1}) \cdots (s_{1}s_{2} \cdots s_{k+1}) \\ & = (1,2,\dots,n+1)(1,2,\dots,n) \cdots (1,2,\dots,k+2) \\[1.5mm] & = \begin{pmatrix} 1 & 2 & \cdots & k & k+1 & k+2 & \cdots & n & n+1 \\ n-k+1 & n-k+2 & \cdots & n & n+1 & n-k & \cdots & 2 & 1 \end{pmatrix}. \end{aligned} \end{align*} $$

Also, recall that $J = \big \{ j_{1},j_{2},\dots ,j_{p} \big \} \subset [k]$ , with $1 \le j_{1} < j_{2} < \cdots < j_{p} \le k$ , and recall from (5.14) that $\Gamma _{J}=(-\beta _{a},\ldots ,-\beta _{1},\gamma _{1},\dots ,\gamma _{b})$ . Let $\mathbf {p} \in \mathbf {D}_{J}$ . We see from Remark 5.9 that the (label sequence of the) $\boldsymbol {\beta }$ -part $\mathbf {p}_{\boldsymbol {\beta }}$ of $\mathbf {p}$ is of the following form:

(5.21) $$ \begin{align} \begin{aligned} w_{k+1} & \xrightarrow{\hspace{2pt}(t_{1,m_{1}},j_{1})\hspace{2pt}} \bullet \xrightarrow{\hspace{2pt}(t_{1,m_{1}-1},j_{1})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(t_{1,m_{1}-r_{1}+1},j_{1})\hspace{2pt}} \bullet \\& \xrightarrow{\hspace{2pt}(t_{2,m_{2}},j_{2})\hspace{2pt}} \bullet \xrightarrow{\hspace{2pt}(t_{2,m_{2}-1},j_{2})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(t_{2,m_{2}-r_{2}+1},j_{2})\hspace{2pt}} \bullet \\& \cdots \cdots\\& \xrightarrow{\hspace{2pt}(t_{p,m_{p}},j_{p})\hspace{2pt}} \bullet \xrightarrow{\hspace{2pt}(t_{p,m_{p}-1},j_{p})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(t_{p,m_{p}-r_{p}+1},j_{p})\hspace{2pt}} \bullet, \end{aligned} \end{align} $$

where $m_{q} \ge r_{q} \ge 0$ for $1 \le q \le p$ ; for each $1 \le q \le p$ such that $r_{q} \ge 1$ , we have

(5.22) $$ \begin{align} \begin{aligned} & j_{q}> t_{q,m_{q}} > t_{q,m_{q}-1} > \cdots > t_{q,m_{q}-r_{q}+1} \ge 1, \\ & \hspace{30mm} \text{with } t_{q,m_{q}-r+1} \ne j_{1},j_{2},\dots,j_{q-1} \text{ for } 1 \le r \le r_{q}. \end{aligned} \end{align} $$

By convention, we set $t_{q,m_{q}+1}:=j_{q}$ for $1 \le q \le p$ .

Proof. It follows from Lemma 2.2 that

(5.23) $$ \begin{align} w_{k+1} = x_{0} \xrightarrow{\hspace{2pt}(k_{1},j_{1})\hspace{2pt}} x_{1} \xrightarrow{\hspace{2pt}(k_{2},j_{1})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(k_{r},j_{1})\hspace{2pt}} x_{r} \end{align} $$

is a directed path in $\mathrm {QBG}(W)$ , where $r \ge 0$ and $j_{1}-1 \ge k_{1}> k_{2} > \cdots > k_{r} \ge 1$ , if and only if $k_{1}=j_{1}-1$ , $k_{2}=j_{1}-2$ , $\dots $ , $k_{r}=j_{1}-r$ ; in this case, all the edges in (5.23) are Bruhat edges.

Also, it follows from Lemma 2.2 that

(5.24) $$ \begin{align} \begin{aligned} w_{k+1} = x_{0} & \xrightarrow{\hspace{2pt}(j_{1}-1,j_{1})\hspace{2pt}} x_{1} \xrightarrow{\hspace{2pt}(j_{1}-2,j_{1})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(j_{1}-r_{1},j_{1})\hspace{2pt}} x_{r_{1}} \\ & \xrightarrow{\hspace{2pt}(k_{1},j_{2})\hspace{2pt}} x_{r_{1}+1} \xrightarrow{\hspace{2pt}(k_{2},j_{2})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(k_{r},j_{2})\hspace{2pt}} x_{r_{1}+r} \end{aligned} \end{align} $$

is a directed path in $\mathrm {QBG}(W)$ , where $r_{1},r \ge 0$ and $j_{2}-1 \ge k_{1}> k_{2} > \cdots > k_{r} \ge 1$ with $k_{r'} \ne j_{1}$ for any $1 \le r ' \le r$ , if and only if $\{k_{1},k_{2},\dots .k_{r}\}$ is identical to the largest r elements in $\{1,2,\dots ,j_{2}-1\} \setminus \{j_{1}\}$ and $k_{r} \ge j_{1}-r_{1}$ ; in this case, all the edges in (5.24) are Bruhat edges.

Similarly, it follows from Lemma 2.2 that

(5.25) $$ \begin{align} \begin{aligned} w_{k+1} = x_{0} & \xrightarrow{\hspace{2pt}(j_{1}-1,j_{1})\hspace{2pt}} x_{1} \xrightarrow{\hspace{2pt}(j_{1}-2,j_{1})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(j_{1}-r_{1},j_{1})\hspace{2pt}} x_{r_{1}} \\ & \xrightarrow{\hspace{2pt}(t_{2,m_{2}},j_{2})\hspace{2pt}} x_{r_{1}+1} \xrightarrow{\hspace{2pt}(t_{2,m_{2}-1},j_{2})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(t_{2,m_{2}-r_{2}+1},j_{2})\hspace{2pt}} x_{r_{1}+r_{2}} \\ & \xrightarrow{\hspace{2pt}(k_{1},j_{3})\hspace{2pt}} x_{r_{1}+r_{2}+1} \xrightarrow{\hspace{2pt}(k_{2},j_{3})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(k_{r},j_{3})\hspace{2pt}} x_{r_{1}+r_{2}+r} \end{aligned} \end{align} $$

is a directed path in $\mathrm {QBG}(W)$ , where $r_{1},r_{2},r \ge 0$ , $t_{2,m_{2}-r_{2}+1} \ge j_{1}-r_{1}$ , and $j_{3}-1 \ge k_{1}> k_{2} > \cdots > k_{r} \ge 1$ with $k_{r'} \ne j_{1},j_{2}$ for any $1 \le r ' \le r$ , if and only if $\{k_{1},k_{2},\dots .k_{r}\}$ is identical to the largest r elements in $\{1,2,\dots ,j_{3}-1\} \setminus \{j_{1},j_{2}\}$ and $k_{r} \ge t_{2,m_{2}-r_{2}+1}$ ; in this case, all the edges in (5.25) are Bruhat edges.

Repeating the argument above, we can verify the assertions of the lemma.

We now define

(5.26) $$ \begin{align} \mathbf{D}:=\bigsqcup_{J \subset [k]} \mathbf{D}_{J}. \end{align} $$

Let $J = \big \{ j_{1},j_{2},\dots ,j_{p} \big \} \subset [k]$ , and let $\mathbf {p} \in \mathbf {D}$ be of the form (5.18); for simplicity of notation, we will henceforth write as

$$ \begin{align*} \mathbf{p} & = [\beta_{a_s},\,\dots,\,\beta_{a_1} \mid \gamma_{b_1},\,\dots,\gamma_{b_t}], \\ \mathbf{p}_{\boldsymbol{\beta}} & = [\beta_{a_s},\,\dots,\,\beta_{a_1}], \quad \mathbf{p}_{\boldsymbol{\gamma}} = [\gamma_{b_1},\,\dots,\gamma_{b_t}]. \end{align*} $$

Then we define subsets $\mathbf {D}_{J}^{\mathrm {A}}$ , $\mathbf {D}_{J}^{\mathrm {B}}$ , and $\mathbf {D}_{J}^{\mathrm {C}}$ of $\mathbf {D}_{J}$ as follows (recall (5.21), (5.22) and Lemma 5.14):

1. (A) When $J \ne \emptyset $ and $j_{1} = \min J \ge 2$ (and hence $1 \notin J$ ), we define $\mathbf {D}_{J}^{\mathrm {A}}$ to be the subset of $\mathbf {D}_{J}$ consisting of all those elements $\mathbf {p} \in \mathbf {D}_{J}$ whose $\boldsymbol {\beta }$ -part $\mathbf {p}_{\boldsymbol {\beta }}$ is of the form

   (5.27) $$ \begin{align} \mathbf{p}_{\boldsymbol{\beta}} & = [ (j_{1}-1,j_{1}), (j_{1}-2,j_{1}),\dots,(2,j_{1}),(1,j_{1}), \nonumber \\ & \underbrace{ (t_{2,m_{2}},j_{2}),\dots,(t_{2,m_{2}-r_{2}+1},j_{2}), \dots\dots, (t_{p,m_{p}},j_{p}),\dots,(t_{p,m_{p}-r_{p}+1},j_{p})}_{=:\,\mathbf{s}}]. \end{align} $$

2. (B) When $J \ne \emptyset $ and $j_{1}=\min J$ is equal to $1$ , we set $\mathbf {D}_{J}^{\mathrm {B}}:=\mathbf {D}_{J}$ ; note that the $\boldsymbol {\beta }$ -part $\mathbf {p}_{\boldsymbol {\beta }}$ of $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {B}}$ is of the form

   (5.28) $$ \begin{align} \mathbf{p}_{\boldsymbol{\beta}} & = [ (t_{2,m_{2}},j_{2}),\dots,(t_{2,m_{2}-r_{2}+1},j_{2}), \nonumber \\ & \qquad \qquad \dots\dots, (t_{p,m_{p}},j_{p}),\dots,(t_{p,m_{p}-r_{p}+1},j_{p})]. \end{align} $$

3. (C) When $J = \emptyset $ , we set $\mathbf {D}_{J}^{\mathrm {C}}:=\mathbf {D}_{J}$ . Also, when $J \ne \emptyset $ and $j_{1}=\min J \ge 2$ , we define $\mathbf {D}_{J}^{\mathrm {C}}$ to be the subset of $\mathbf {D}_{J}$ consisting of all those elements $\mathbf {p} \in \mathbf {D}_{J}$ whose $\boldsymbol {\beta }$ -part $\mathbf {p}_{\boldsymbol {\beta }}$ is of the form

   (5.29) $$ \begin{align} \mathbf{p}_{\boldsymbol{\beta}} & = [ (j_{1}-1,j_{1}), (j_{1}-2,j_{1}),\dots,(j_{1}-r_{1}+1,j_{1}),(j_{1}-r_{1},j_{1}), \nonumber \\ & (t_{2,m_{2}},j_{2}),\dots,(t_{2,m_{2}-r_{2}+1},j_{2}), \dots\dots, (t_{p,m_{p}},j_{p}),\dots,(t_{p,m_{p}-r_{p}+1},j_{p})], \end{align} $$

   where $r_{1} = 0$ , or $r_{1} \ge 1$ and $j_{1} - r_{1} \ge 2$ .

It is easily seen that $\mathbf {D}_{J} = \mathbf {D}_{J}^{\mathrm {A}} \sqcup \mathbf {D}_{J}^{\mathrm {B}} \sqcup \mathbf {D}_{J}^{\mathrm {C}}$ .

Lemma 5.15. Let $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {A}} \sqcup \mathbf {D}_{J}^{\mathrm {B}}$ , and assume that the $\boldsymbol {\gamma }$ -part $\mathbf {p}_{\boldsymbol {\gamma }}$ of $\mathbf {p}$ is of the form

$$ \begin{align*} \begin{aligned} \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}}) & \xrightarrow{\hspace{2pt}(j_{1},k_{r})\hspace{2pt}} \bullet \xrightarrow{\hspace{2pt}(j_{1},k_{r-1})\hspace{2pt}} \cdots \xrightarrow{\hspace{2pt}(j_{1},k_{2})\hspace{2pt}} \bullet \xrightarrow{\hspace{2pt}(j_{1},k_{1})\hspace{2pt}} \bullet \\ & \cdots \cdots, \end{aligned} \end{align*} $$

where $r \ge 0$ , and $n+1 \ge k_{r}> k_{r-1} > \cdots > k_{1} \ge j_{1}+1$ with $k_{u} \notin \{j_{2},\dots ,j_{p}\}$ for any $1 \le u \le r$ . Then, either of the following holds: (i) $r = 0$ , or (ii) $r = 1$ and $k_{1}=j_{1}+1$ ; note that case (ii) does not occur if $j_{1}+1 \in J$ . In case (ii), the edge $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }}) \xrightarrow {\hspace {2pt}(j_{1},j_{1}+1)\hspace {2pt}} \bullet $ is a Bruhat edge.

Proof. Assume that $r> 0$ . Note that $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }})(j_{1}) = w_{k+1}(1) = n-k+1$ ; see (5.27) and (5.28). If $k_{r} \ge k+2$ , then

$$ \begin{align*} \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(k_{r}) = w_{k+1}(k_{r}) = n-k_{r}+2 < n-k+1 = \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(j_{1}). \end{align*} $$

However, $j_{1} < k+1 < k_{r}$ , and $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }})(k+1) = n+1$ is greater than both $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }})(k_{r})$ and $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }})(j_{1})$ , which is a contradiction (see Lemma 2.2). Thus, we see that $k_{r} \le k+1$ .

Suppose, for a contradiction, that $j_{1}+1=j_{2} \in J$ ; note that $j_{2} < k_{r} \le k+1$ and $k_{r} \notin \{j_{3},\dots ,j_{p}\}$ . In this case, it follows rom Lemma 5.14 and (5.27), (5.28) that $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }})(j_{2}) = w_{k+1}(t_1) = n-k+t_1$ for some $1 \le t_{1} \le j_{2}$ , and hence that

$$ \begin{align*} \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(j_{2}) = n-k+t_{1}> n-k+1 = \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(j_{1}). \end{align*} $$

Since $j_{2} < k_{r} \le k+1$ and $k_{r} \notin \{j_{3},\dots ,j_{p}\}$ , we deduce by Lemma 5.14 and (5.27), (5.28) that $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }})(k_{r}) = w_{k+1}(t_{2})=n-k+t_{2}$ for some $j_{2} < t_{2} \le k+1$ . Hence, we obtain

$$ \begin{align*} \underbrace{ \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(j_{1}) }_{=\, n-k+1} < \underbrace{ \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(j_{2}) }_{=\, n-k+t_1} < \underbrace{ \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(k_{r}) }_{=\, n-k+t_2}, \end{align*} $$

which is a contradiction (see Lemma 2.2). Thus, we see that $j_{1}+1 \not \in J$ .

Suppose, for a contradiction, that $j_{1}+1 \notin J$ and $j_{1}+1 < k_{r} \le k+1$ ; recall that $k_{r} \notin \{j_{2},\dots ,j_{p}\}$ . In this case, we deduce by (5.27) and (5.28) that

$$ \begin{align*} \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(j_{1}) < \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(j_{1}+1) < \operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})(k_{r}), \end{align*} $$

which is a contradiction (see Lemma 2.2). Thus, we see that $k_{r}=j_{1}+1$ ; since $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }})(j_{1}) < \operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }})(j_{1}+1)$ , the edge $\operatorname {\mathrm {end}}(\mathbf {p}_{\boldsymbol {\beta }}) \xrightarrow {\hspace {2pt}(j_{1},j_{1}+1)\hspace {2pt}} \bullet $ is a Bruhat edge. This proves the lemma.

Let $\mathbf {D}_{J}^{\mathrm {A}_1}$ (resp., $\mathbf {D}_{J}^{\mathrm {A}_2}$ ) denote the subset of $\mathbf {D}_{J}^{\mathrm {A}}$ consisting of all those elements $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {A}}$ whose initial label of the $\boldsymbol {\gamma }$ -part $\mathbf {p}_{\boldsymbol {\gamma }}$ is (resp., is not) $(j_{1},j_{1}+1)$ ; see Lemma 5.15. Similarly, let $\mathbf {D}_{J}^{\mathrm {B}_1}$ (resp., $\mathbf {D}_{J}^{\mathrm {B}_2}$ ) denote the subset of $\mathbf {D}_{J}^{\mathrm {B}}$ consisting of all those elements $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {B}}$ whose initial label of the $\boldsymbol {\gamma }$ -part $\mathbf {p}_{\boldsymbol {\gamma }}$ is (resp., is not) $(1,2)$ .

We can easily show the following; the element $\mathbf {q}$ in this lemma is a unique element of $\mathbf {D}$ fixed by our ‘sijection’ (i.e., a bijection between signed sets) defined below.

Lemma 5.16. We have

(5.30) $$ \begin{align} \mathbf{q}:=[(k-1,k),(k-2,k),\dots,(1,k) \mid (k,k+1)] \in \mathbf{D}_{\{k\}}^{\mathrm{A}_1}; \end{align} $$

note that $\{k\} \subset [k]$ is a unique subset $J \subset [k]$ such that $\mathbf {q} \in \mathbf {D}_{J}^{\mathrm {A}_1}$ . Also, we have

(5.31) $$ \begin{align} \operatorname{\mathrm{end}}(\mathbf{q}_{\boldsymbol{\beta}})\epsilon_{J} = \epsilon_{n+1-k}, \quad \ell(\mathbf{p}_{\boldsymbol{\gamma}})=1, \quad \operatorname{\mathrm{end}}(\mathbf{p})t_{\operatorname{\mathrm{wt}}(\mathbf{p})} = w_{k}. \end{align} $$

Let us complete the proof of Proposition 2.4. For this purpose, we define a ‘sijection’ (i.e., a bijection between signed sets) $\Phi :\mathbf {D} \rightarrow \mathbf {D}$ as follows. We set $\Phi (\mathbf {q}):=\mathbf {q}$ (see Lemma 5.16). Let $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {A}_2}$ ; note that $j_{1}=\min J \ge 2$ and $j_{1}-1 \notin J$ . We set $\Phi (J):=(J \setminus \{j_{1}\}) \sqcup \{j_{1}-1\}$ and then define $\Phi (\mathbf {p}) \in \mathbf {D}_{\Phi (J)}^{\mathrm {A}_1} \sqcup \mathbf {D}_{\Phi (J)}^{\mathrm {B}_1}$ as follows. Recall that the $\boldsymbol {\beta }$ -part $\mathbf {p}_{\boldsymbol {\beta }}$ of $\mathbf {p}$ is of the form (5.27), and that if the $\boldsymbol {\gamma }$ -part $\mathbf {p}_{\boldsymbol {\gamma }}$ of $\mathbf {p}$ is nontrivial, then it starts at the edge whose label is $(j_{u},t)$ for some $2 \le u \le p$ and $j_{u} \le t \le n+1$ , with $t \notin \{j_{u+1},\dots ,j_{p}\}$ (see Lemma 5.15). We define $\Phi (\mathbf {p})$ by

(5.32) $$ \begin{align} \Phi(\mathbf{p}) = [ (j_{1}-2,j_{1}-1),\dots,(2,j_{1}-1),(1,j_{1}-1), \mathbf{s}' \mid (j_{1}-1,j_{1}), \mathbf{p}_{\boldsymbol{\gamma}} ], \end{align} $$

where $\mathbf {s}'$ is obtained from $\mathbf {s}$ by replacing $(j_{1}-1,j_{u})$ appearing in $\mathbf {p}_{\boldsymbol {\beta }}$ with $(j_{1},j_{u})$ for each $2 \le u \le p$ ; by [Reference Naito and SagakiNS2, Lemma 2.3 (2)] and Lemma 5.14, we deduce that $\Phi (\mathbf {p}) \in \mathbf {D}_{\Phi (J)}^{\mathrm {A}_1} \sqcup \mathbf {D}_{\Phi (J)}^{\mathrm {B}_1}$ (note that $\Phi (\mathbf {p}) \in \mathbf {D}_{\Phi (J)}^{\mathrm {B}_1}$ if and only if $j_{1}=2$ ). Also, it follows that

(5.33) $$ \begin{align} \begin{aligned} & |\Phi(J)|=|J|, \\ & |\Phi(J)|\epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\Phi(\mathbf{p})_{\boldsymbol{\beta}})\epsilon_{\Phi(J)} = |J|\epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\mathbf{p}_{\boldsymbol{\beta}})\epsilon_{J}, \\ & \ell(\Phi(\mathbf{p})_{\boldsymbol{\gamma}})=\ell(\mathbf{p}_{\boldsymbol{\gamma}})+1, \quad \operatorname{\mathrm{end}}(\Phi(\mathbf{p}))t_{\operatorname{\mathrm{wt}}(\Phi(\mathbf{p}))} = \operatorname{\mathrm{end}}(\mathbf{p})t_{\operatorname{\mathrm{wt}}(\mathbf{p})}. \end{aligned} \end{align} $$

Let $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {A}_1} \sqcup \mathbf {D}_{J}^{\mathrm {B}_1}$ , with $J \ne \{k\}$ ; recall that $\mathbf {D}_{ \{ k \} }^{\mathrm {A}_1} \sqcup \mathbf {D}_{ \{ k \} }^{\mathrm {B}_1} = \{\mathbf {q}\}$ , and note that $j_{1}+1 \notin [k] \setminus J$ . We set $\Phi (J):= (J \setminus \{j_{1}\}) \sqcup \{j_{1}+1\}$ and then define $\Phi (\mathbf {p}) \in \mathbf {D}_{\Phi (J)}^{\mathrm {A}_1}$ as follows. Recall that the $\boldsymbol {\beta }$ -part $\mathbf {p}_{\boldsymbol {\beta }}$ of $\mathbf {p}$ is of the form (5.27) or (5.28), and that the $\boldsymbol {\gamma }$ -part $\mathbf {p}_{\boldsymbol {\gamma }}$ of $\mathbf {p}$ starts at the edge whose label is $(j_{1},j_{1}+1)$ (see Lemma 5.15). We define $\Phi (\mathbf {p})$ by

(5.34) $$ \begin{align} \begin{aligned} \Phi(\mathbf{p}) & = [ (j_{1},j_{1}+1), (j_{1}-1,j_{1}+1),\dots,(2,j_{1}+1),(1,j_{1}+1), \mathbf{s}" \mid \mathbf{p}_{\boldsymbol{\gamma}} \setminus (j_{1},j_{1}+1) ], \end{aligned} \end{align} $$

where $\mathbf {s}"$ is obtained from $\mathbf {s}$ by replacing $(j_{1}+1,j_{u})$ appearing in $\mathbf {p}_{\beta }$ with $(j_{1},j_{u})$ for each $2 \le u \le p$ , and $\mathbf {p}_{\gamma } \setminus (j_{1},j_{1}+1)$ is the sequence obtained form $\mathbf {p}_{\gamma }$ by removing the initial label $(j_{1},j_{1}+1)$ ; by [Reference Naito and SagakiNS2, Lemma 2.3 (4)] and Lemma 5.14, we deduce that $\Phi (\mathbf {p}) \in \mathbf {D}_{\Phi (J)}^{\mathrm {A}_2}$ . Also, it follows that

(5.35) $$ \begin{align} \begin{aligned} & |\Phi(J)|=|J|, \\ & |\Phi(J)|\epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\Phi(\mathbf{p})_{\beta})\epsilon_{\Phi(J)} = |J|\epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\mathbf{p}_{\beta})\epsilon_{J}, \\ & \ell(\Phi(\mathbf{p})_{\gamma})=\ell(\mathbf{p}_{\gamma})-1, \quad \operatorname{\mathrm{end}}(\Phi(\mathbf{p}))t_{\operatorname{\mathrm{wt}}(\Phi(\mathbf{p}))} = \operatorname{\mathrm{end}}(\mathbf{p})t_{\operatorname{\mathrm{wt}}(\mathbf{p})}. \end{aligned} \end{align} $$

If $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {B}_2}$ , then $\mathbf {p} \in \mathbf {D}_{\Phi (J)}^{\mathrm {C}}$ with $\Phi (J):=J \setminus \{1\}$ . We define $\Phi (\mathbf {p}):=\mathbf {p} \in \mathbf {D}_{\Phi (J)}^{\mathrm {C}}$ for $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {B}_2}$ . It follows that

(5.36) $$ \begin{align} \begin{aligned} & |\Phi(J)|=|J|-1, \\ & |\Phi(J)|\epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\Phi(\mathbf{p})_{\beta})\epsilon_{\Phi(J)} = |J|\epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\mathbf{p}_{\beta})\epsilon_{J}, \\ & \ell(\Phi(\mathbf{p})_{\gamma})=\ell(\mathbf{p}_{\gamma}), \qquad \operatorname{\mathrm{end}}(\Phi(\mathbf{p}))t_{\operatorname{\mathrm{wt}}(\Phi(\mathbf{p}))} = \operatorname{\mathrm{end}}(\mathbf{p})t_{\operatorname{\mathrm{wt}}(\mathbf{p})}. \end{aligned} \end{align} $$

Similarly, if $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {C}}$ , then $\mathbf {p} \in \mathbf {D}_{\Phi (J)}^{\mathrm {B}_2}$ with $\Phi (J):=J \sqcup \{1\}$ . We define $\Phi (\mathbf {p}):=\mathbf {p} \in \mathbf {D}_{\Phi (J)}^{\mathrm {B}_2}$ for $\mathbf {p} \in \mathbf {D}_{J}^{\mathrm {C}}$ . It follows that

(5.37) $$ \begin{align} \begin{aligned} & |\Phi(J)|=|J|+1, \\ & |\Phi(J)|\epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\Phi(\mathbf{p})_{\beta})\epsilon_{\Phi(J)} = |J|\epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\mathbf{p}_{\beta})\epsilon_{J}, \\ & \ell(\Phi(\mathbf{p})_{\gamma})=\ell(\mathbf{p}_{\gamma}), \qquad \operatorname{\mathrm{end}}(\Phi(\mathbf{p}))t_{\operatorname{\mathrm{wt}}(\Phi(\mathbf{p}))} = \operatorname{\mathrm{end}}(\mathbf{p})t_{\operatorname{\mathrm{wt}}(\mathbf{p})}. \end{aligned} \end{align} $$

Thus, we have obtained a bijection $\Phi :\mathbf {D} \rightarrow \mathbf {D}$ such that (i) $\Phi (\mathbf {q})=\mathbf {q}$ , (ii) $\Phi (\mathbf {p}) \ne \mathbf {p}$ , $\Phi (\Phi (\mathbf {p}))=\mathbf {p}$ , and such that

$$ \begin{align*} \begin{aligned} & \mathbf{e}^{|\Phi(J)| \epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\Phi(\mathbf{p})_{\beta})\epsilon_{\Phi(J)}} (-1)^{|\Phi(J)|+\ell(\Phi(\mathbf{p})_{\gamma})} [\mathcal{O}_{\mathbf{Q}_{G}(\operatorname{\mathrm{end}}(\Phi(\mathbf{p}))t_{\operatorname{\mathrm{wt}}(\Phi(\mathbf{p}))})}] \\ & = - \mathbf{e}^{|J| \epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\mathbf{p}_{\beta})\epsilon_{J}} (-1)^{|J|+\ell(\mathbf{p}_{\gamma})} [\mathcal{O}_{\mathbf{Q}_{G}(\operatorname{\mathrm{end}}(\mathbf{p})t_{\operatorname{\mathrm{wt}}(\mathbf{p})})}] \end{aligned} \end{align*} $$

for all $\mathbf {p} \in \mathbf {D}_{J} \setminus \{\mathbf {q}\}$ with $J \subset [k]$ . Therefore, the right-hand side of equation (5.19) becomes

$$ \begin{align*} & \mathbf{e}^{|\{k\}| \epsilon_{n+1-k}-\operatorname{\mathrm{end}}(\mathbf{q}_{\beta})\epsilon_{\{k\}}} (-1)^{|\{k\}|+\ell(\mathbf{q}_{\gamma})} [\mathcal{O}_{\mathbf{Q}_{G}(\operatorname{\mathrm{end}}(\mathbf{q})t_{\operatorname{\mathrm{wt}}(\mathbf{q})})}] = [\mathcal{O}_{\mathbf{Q}_{G}(w_{k})}] \qquad \text{by Lemma~5.16}, \end{align*} $$

as desired. This completes the proof of Proposition 2.4.