2.1 Notation for root systems

Let G be a connected, simply connected, simple algebraic group over $\mathbb {C}$ , B its Borel subgroup, $T \subset B$ a maximal torus, and I the index set for the simple roots of G. Let $\mathfrak {h}$ be the Lie algebra of T; we denote by $\langle \cdot , \cdot \rangle $ the canonical pairing of $\mathfrak {h}^{\ast } := \operatorname {\mathrm {Hom}}_{\mathbb {C}}(\mathfrak {h}, \mathbb {C})$ and $\mathfrak {h}$ . Let $P = \sum _{i \in I} \mathbb {Z} \varpi _{i}$ be the weight lattice of G, with $\varpi _{i}$ the i-th fundamental weight for $i \in I$ , and $Q^{\vee } = \sum _{i \in I} \mathbb {Z} \alpha _{i}^{\vee }$ the coroot lattice of G, with $\alpha _{i}^{\vee }$ the coroot such that $\langle \varpi_{i}, \alpha_{j}^{\vee} \rangle = \delta _{i,j}$ for $i, j \in I$ ; we set $Q^{\vee , +} := \sum _{i \in I} \mathbb {Z}_{\ge 0} \alpha _{i}^{\vee }$ . In addition, let W be the Weyl group of G, which is generated by the simple reflections $s_{i}$ , $i \in I$ . For $\lambda \in P$ , we denote by $\mathbf {e}^{\lambda }$ the character of the one-dimensional representation of T whose weight is $\lambda $ . Then, the representation ring $R(T)$ of T is written as $R(T) = \bigoplus _{\lambda \in P} \mathbb {Z} \mathbf {e}^{\lambda }$ , with product given by $\mathbf {e}^{\lambda } \cdot \mathbf {e}^{\mu } := \mathbf {e}^{\lambda +\mu }$ for $\lambda , \mu \in P$ ; $R(T)$ is canonically identified with the group algebra $\mathbb {Z}[P]$ of P, on which W acts by $w \cdot \mathbf {e}^{\lambda } := \mathbf {e}^{w \lambda }$ for $w \in W$ and $\lambda \in P$ .

2.2 The quantum K-ring of flag manifolds

Let $K_{T}(G/B)$ denote the (ordinary) T-equivariant K-ring of the flag manifold $G/B$ . The T-equivariant (small) quantum K-ring $QK_{T}(G/B)$ of the flag manifold $G/B$ , defined by Givental [Reference Givental4] and Lee [Reference Lee15], is, as a module,

$$ \begin{align*} QK_{T}(G/B) := K_{T}(G/B) \otimes_{R(T)} R(T)[\![ Q_{1}, \ldots, Q_{n} ]\!], \end{align*} $$

where n is the rank of G; here, $R(T)[\![ Q_{1}, \ldots , Q_{n} ]\!]$ denotes the formal power series ring in the (Novikov) variables $Q_{1}, \ldots , Q_{n}$ with coefficients in $R(T)$ . Under the quantum product $\star $ , defined in terms of the two-point and three-point (genus 0, equivariant) K-theoretic Gromov-Witten invariants, $QK_{T}(G/B)$ becomes an associative and commutative algebra (see [Reference Givental4] and [Reference Lee15] for details). For $\xi = \sum _{i \in I} c_{i} \alpha _{i}^{\vee } \in Q^{\vee , +}$ , we set $Q^{\xi } := \prod _{i \in I} Q_{i}^{c_{i}}$ .

For $w \in W$ , we define the (opposite) Schubert variety $X^{w} \subset G/B$ by $X^{w} := \overline {B^{-}wB/B}$ , where $B^{-}$ is the Borel subgroup (opposite to B) of G such that $B \cap B^{-} = T$ , and $\overline {{}\cdot {}}$ denotes the Zariski closure; let $\mathcal {O}^{w}$ denote the structure sheaf of $X^{w}$ . Then it is well known that the classes $[\mathcal {O}^{w}]$ , $w \in W$ , of $\mathcal {O}^{w}$ in $K_{T}(G/B)$ form a basis of $K_{T}(G/B)$ as an $R(T)$ -module, which are called the (opposite) Schubert classes. It follows that the (opposite) Schubert classes $[\mathcal {O}^{w}]$ , $w \in W$ , form a basis of $QK_{T}(G/B)$ as an $R(T)[\![ Q_{1}, \ldots , Q_{n} ]\!]$ -module.

To each $\lambda \in P$ , we can associate the line bundle $G \times _{B} \mathbb {C}_{-\lambda } \twoheadrightarrow G/B$ over $G/B$ , denoted by $\mathcal {O}_{G/B}(\lambda )$ , where $\mathbb {C}_{-\lambda }$ is the one-dimensional representation of B whose weight is $- \lambda $ ; let $[\mathcal {O}_{G/B}(\lambda )] \in K_{T}(G/B) \subset QK_{T}(G/B)$ denote the class of the line bundle $\mathcal {O}_{G/B}(\lambda )$ .

2.3 The root system of type $C_{n}$

We fix the notation for the root system of type $C_{n}$ . Let $J_{n}$ be the $n \times n$ matrix given by

$$ \begin{align*} J_{n} := \begin{bmatrix} 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \cdots & 1 & 0 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & 1 & \cdots & 0 & 0 \\ 1 & 0 & \cdots & 0 & 0 \end{bmatrix}, \end{align*} $$

and set

$$ \begin{align*} J_{2n}^{-} := \begin{bmatrix} 0 & J_{n} \\ -J_{n} & 0 \end{bmatrix}; \end{align*} $$

the corresponding symplectic form $(\cdot , \cdot )$ on $\mathbb {C}^{2n}$ is given by: $(u, v) = {}^{t}\!u J_{2n}^{-} v$ for $u, v \in \mathbb {C}^{2n}$ . Then, the symplectic group $\mathrm {Sp}_{2n}(\mathbb {C})$ of rank n over $\mathbb {C}$ is defined as:

$$ \begin{align*} \mathrm{Sp}_{2n}(\mathbb{C}) = \{ A \in \mathrm{GL}_{2n}(\mathbb{C}) \mid {}^{t}\!AJ_{2n}^{-}A = J_{2n}^{-} \}. \end{align*} $$

Let T be a maximal torus of $\mathrm {Sp}_{2n}(\mathbb {C})$ given by:

$$ \begin{align*} T = \{ \operatorname{\mathrm{diag}}(x_{1}, \ldots, x_{n}, x_{n}^{-1}, \ldots, x_{1}^{-1}) \mid x_{1}, \ldots, x_{n} \in \mathbb{C}^{\ast} \}, \end{align*} $$

where $\operatorname {\mathrm {diag}}(a_{1}, \ldots , a_{2n})$ denotes the $2n \times 2n$ diagonal matrix with entries $a_{1}, \ldots , a_{2n}$ ; its Lie algebra $\mathfrak {h} = \operatorname {\mathrm {Lie}}(T)$ can be written as:

$$ \begin{align*} \mathfrak{h} = \{ \operatorname{\mathrm{diag}}(a_{1}, \ldots, a_{n}, -a_{n}, \ldots, -a_{1}) \mid a_{1}, \ldots, a_{n} \in \mathbb{C} \}. \end{align*} $$

We define $\varepsilon _{k} \in \mathfrak {h}^{\ast }$ , $1 \le k \le n$ , to be

Then, the set $\Delta ^{+}$ of positive roots of G can be written as:

$$ \begin{align*} \Delta^{+} = \{\varepsilon_{i} \pm \varepsilon_{j} \mid 1 \le i < j \le n\} \cup \{2\varepsilon_{j} \mid 1 \le j \le n\}; \end{align*} $$

the simple roots $\alpha _{1}, \ldots , \alpha _{n}$ are $\alpha _{j} = \varepsilon _{j} - \varepsilon _{j+1}$ , $1 \le j \le n-1$ , and $\alpha _{n} = 2\varepsilon _{n}$ . We set

$$ \begin{align*} (i, j) &:= \varepsilon_{i} - \varepsilon_{j} \quad \text{for } 1 \le i < j \le n, \\ (i, \overline{j}) &:= \varepsilon_{i} + \varepsilon_{j} \quad \text{for } 1 \le i < j \le n, \text{ and} \\ (i, \overline{i}) &:= 2\varepsilon_{i} \quad \text{for } 1 \le i \le n. \end{align*} $$

The fundamental weight $\varpi _{j}$ for $1 \le j \le n$ is of the form $\varpi _{j} = \varepsilon _{1} + \cdots + \varepsilon _{j}$ . It follows that $R(T) = \mathbb {Z}[P] = \mathbb {Z}[\mathbf {e}^{\pm \varepsilon _{1}}, \ldots , \mathbf {e}^{\pm \varepsilon _{n}}]$ .

The Weyl group $W \simeq S_{n} \ltimes (\mathbb {Z} / 2 \mathbb {Z})^{n}$ of $\mathrm {Sp}_{2n}(\mathbb {C})$ can be regarded as the group of signed permutations on $[1, \overline {1}] := \{1 < 2 < \cdots < n < \overline {n} < \cdots < \overline {2} < \overline {1}\}$ . For $1 \le i < j \le n$ , the reflection $s_{(i, j)}$ (resp., $s_{(i, \overline {j})}$ ) corresponding to $(i, j)$ (resp., $(i, \overline {j})$ ) is viewed as the transposition of i and j (resp., i and $\overline {j}$ ). In addition, for $1 \le i \le n$ , the reflection $s_{(i, \overline {i})}$ corresponding to $(i, \overline {i})$ can be regarded as the transposition of i and $\overline {i}$ . The action of W on P is a natural extension of that on $[1, \overline {1}]$ , given by: $w \cdot \varepsilon _{k} = \varepsilon _{w(k)}$ for $w \in W$ and $k = 1, \ldots , n$ , where we set $\varepsilon _{\overline {j}} := -\varepsilon _{j}$ for $1 \le j \le n$ . We sometimes write $w \in W$ as $w = [w(1), w(2), \ldots , w(n)]$ by regarding it as a signed permutation as above; this notation is called the window notation.

4.1 Semi-infinite flag manifolds and the quantum K-ring

Following [Reference Maeno, Naito and Sagaki21, §3.1] (which is based on [Reference Kato9, §1.4 and §1.5]) and [Reference Orr23, §2.1 and §2.3], we recall semi-infinite flag manifolds and their equivariant K-groups (see also [Reference Kato, Naito and Sagaki10, §4.2]).

Let G be of an arbitrary type, and N the unipotent radical of B. The semi-infinite flag manifold $\mathbf {Q}_{G}^{\mathrm {rat}}$ is a (reduced) ind-scheme of ind-infinite type whose set of $\mathbb {C}$ -valued points is $G(\mathbb {C}(\!( z )\!))/(T(\mathbb {C}) \cdot N(\mathbb {C}(\!( z )\!)))$ . The semi-infinite flag manifold $\mathbf {Q}_{G}^{\mathrm {rat}}$ can be thought of as an inductive limit of copies of the scheme $\mathbf {Q}_{G}$ of infinite type, introduced in [Reference Finkelberg and Mirković3, §4.1]. The scheme $\mathbf {Q}_{G}$ can be described as follows. In what follows, for a $\mathbb {C}$ -vector space V, let $\mathbb {P}(V)$ denote the projective space of lines in V. For $\lambda \in P^{+}$ , let $V(\lambda )$ be the irreducible highest weight G-module of highest weight $\lambda $ . The scheme $\mathbf {Q}_{G}$ parametrizes tuples $(\ell _{\lambda })_{\lambda \in P^{+}}$ of $\mathbb {C}$ -lines in $\prod _{\lambda \in P^{+}} \mathbb {P}(V(\lambda ) \otimes _{\mathbb {C}} \mathbb {C}[\![ z ]\!])$ such that for $\lambda , \mu \in P^{+}$ , the line $\ell _{\lambda +\mu }$ is mapped to $\ell _{\lambda } \otimes \ell _{\mu }$ under the embedding

$$ \begin{align*} V(\lambda + \mu) \otimes_{\mathbb{C}} \mathbb{C}[\![ z ]\!] \hookrightarrow (V(\lambda) \otimes_{\mathbb{C}} \mathbb{C}[\![ z ]\!]) \otimes_{\mathbb{C}[\![ z ]\!]} (V(\mu) \otimes_{\mathbb{C}} \mathbb{C}[\![ z ]\!]) \end{align*} $$

of $\mathbb {C}$ -vector spaces induced by the embedding

$$ \begin{align*} V(\lambda + \mu) \hookrightarrow V(\lambda) \otimes_{\mathbb{C}} V(\mu) \end{align*} $$

of G-modules (unique up to scalars). Such a tuple $(\ell _{\lambda })_{\lambda \in P^{+}}$ is uniquely determined by $\ell _{\varpi _{i}}$ for $i \in I$ , and so we have the closed embedding

(4.1) $$ \begin{align} \mathbf{Q}_{G} \hookrightarrow \mathbb{P} := \prod_{i \in I} \mathbb{P}(L(\varpi_{i}) \otimes_{\mathbb{C}} \mathbb{C}[\![ z ]\!]) \end{align} $$

(see [Reference Kato, Naito and Sagaki10, Lemma 4.4]). Also, we define the $(G(\mathbb {C}[\![ z ]\!]) \rtimes \mathbb {C}^{\ast })$ -equivariant line bundle $\mathcal {O}_{\mathbf {Q}_{G}}(\lambda )$ over $\mathbf {Q}_{G}$ for each $\lambda = \sum _{i \in I} m_{i} \varpi _{i} \in P$ as the pullback of the line bundle $\displaystyle\boxtimes _{i \in I} \mathcal {O}(m_{i})$ over $\mathbb {P}$ under the embedding (4.1), where the i-th $\mathcal {O}(m_{i})$ is over $\mathbb {P}(L(\varpi _{i}) \otimes _{\mathbb {C}} \mathbb {C}[\![ z ]\!])$ . Note that for $\lambda , \mu \in P$ , we have $\mathcal {O}_{\mathbf {Q}_{G}}(\lambda ) \otimes \mathcal {O}_{\mathbf {Q}_{G}}(\mu ) = \mathcal {O}_{\mathbf {Q}_{G}}(\lambda + \mu )$ .

Let $\mathbf {I}$ be the Iwahori subgroup of $G(\mathbb {C}[\![ z ]\!])$ , which is the preimage of B under the evaluation map $G(\mathbb {C}[\![ z ]\!]) \rightarrow G$ , . Also, let $W_{\mathrm {af}} = \{wt_{\xi } \mid w \in W, \ \xi \in Q^{\vee }\} \simeq W \ltimes Q^{\vee }$ be the affine Weyl group of G, where $t_{\xi }$ for $\xi \in Q^{\vee }$ denotes the translation element corresponding to $\xi $ . Then, the $(T \times \mathbb {C}^{\ast })$ -fixed points of $\mathbf {Q}_{G}$ are labeled by the subset $W_{\mathrm {af}}^{\geq 0} := \{ wt_{\xi } \mid w \in W, \ \xi \in Q^{\vee ,+} \}$ of $W_{\mathrm {af}}$ ([Reference Kato, Naito and Sagaki10, §4.2], [Reference Orr23, §2.3]); more precisely, the $(T \times \mathbb {C}^{\ast })$ -fixed point of $\mathbf {Q}_{G}$ labeled by $x = wt_{\xi } \in W_{\mathrm {af}}^{\geq 0}$ , with $w \in W$ and $\xi \in Q^{\vee ,+}$ , is the tuple of $\mathbb {C}$ -lines $p_{x} := (z^{-\langle \lambda , w_{\circ }\xi \rangle } V(\lambda )_{ww_{\circ }\lambda })_{\lambda \in P^{+}}$ , where $V(\lambda )_{\mu }$ is the $\mu $ -weight space of $V(\lambda )$ for $\lambda \in P^{+}$ , $\mu \in P$ . We denote by $\mathbf {Q}_{G}(x)$ the closure of the $\mathbf {I}$ -orbit of $p_{x}$ , called the semi-infinite Schubert variety associated to $x \in W_{\mathrm {af}}^{\geq 0}$ ; it is a (reduced) closed subscheme of $\mathbf {Q}_{G}(e) = \mathbf {Q}_{G}$ , where $e \in W_{\mathrm {af}}$ is the identity element. For $x \in W_{\mathrm {af}}^{\geq 0}$ , we denote by $\mathcal {O}_{\mathbf {Q}_{G}(x)}$ the structure sheaf of $\mathbf {Q}_{G}(x)$ .

For $a = \sum _{\lambda \in P} \sum _{k \in \mathbb {Z}} c_{\lambda , k} q^{k} \mathbf {e}^{\lambda } \in \mathbb {Z}[q, q^{-1}][P]$ with $c_{\lambda , k} \in \mathbb {Z}$ , we set $|a| := \sum _{\lambda \in P} \sum _{k \in \mathbb {Z}} |c_{\lambda , k}| q^{k} \mathbf {e}^{\lambda }$ . Let $\widetilde {K}'(\mathbf {Q}_{G})$ denote the $\mathbb {Z}[q, q^{-1}][P]$ -module consisting of all formal sums $\sum _{x \in W_{\mathrm {af}}^{\ge 0}} a_{x} [\mathcal {O}_{\mathbf {Q}_{G}(x)}]$ with coefficients $a_{x} \in \mathbb {Z}[q, q^{-1}][P]$ such that

$$ \begin{align*} \sum_{x \in W_{\mathrm{af}}^{\ge 0}} |a_{x}| \operatorname{\mathrm{gch}} H^{0}(\mathbf{Q}_{G}, \mathcal{O}_{\mathbf{Q}_{G}(x)}(\lambda)) \in \mathbb{Z}[P](\!( q^{-1} )\!) \end{align*} $$

for $\lambda \in P^{++} := \sum _{i \in I} \mathbb {Z}_{> 0} \varpi _{i}$ , where $\operatorname {\mathrm {gch}}$ denotes the character of a weight module over $T \times \mathbb {C}^{\ast }$ and q denotes the character of the loop rotation action $\mathbb {C}^{\ast } \curvearrowright \mathbb {C}[\![ z ]\!]$ ; the classes $[\mathcal {O}_{\mathbf {Q}_{G}(x)}]$ , $x \in W_{\mathrm {af}}^{\ge 0}$ , are called semi-infinite Schubert classes. By [Reference Kato9, Theorem 1.25], we see that the sheaf $\mathcal {O}_{\mathbf {Q}_{G}(x)}(\lambda ) := \mathcal {O}_{\mathbf {Q}_{G}(x)} \otimes \mathcal {O}_{\mathbf {Q}_{G}}(\lambda )$ for $x \in W_{\mathrm {af}}^{\ge 0}$ and $\lambda \in P$ defines the class $[\mathcal {O}_{\mathbf {Q}_{G}(x)}(\lambda )] \in \widetilde {K}'(\mathbf {Q}_{G})$ , called the twisted semi-infinite Schubert class; in particular, for $\lambda \in P$ , $\widetilde {K}'(\mathbf {Q}_{G})$ contains the class $[\mathcal {O}_{\mathbf {Q}_{G}}(\lambda )] = [\mathcal {O}_{\mathbf {Q}_{G}(e)}(\lambda )]$ of the line bundle $\mathcal {O}_{\mathbf {Q}_{G}}(\lambda )$ . Also, $\widetilde {K}'(\mathbf {Q}_{G})$ is stable under the tensor product for $\lambda \in P$ .

Let $K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ denote the $\mathbb {Z}[q, q^{-1}][P]$ -submodule of $\widetilde {K}'(\mathbf {Q}_{G})$ consisting of all “convergent” (in the sense described in [Reference Kato, Naito and Sagaki10, Proposition 5.11]) sums; that is, $K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ consists of all formal infinite linear combinations $\sum _{x \in W_{\mathrm {af}}^{\ge 0}} a_{x} [\mathcal {O}_{\mathbf {Q}_{G}(x)}]$ of the semi-infinite Schubert classes $[\mathcal {O}_{\mathbf {Q}_{G}(x)}]$ , $x \in W_{\mathrm {af}}^{\geq 0}$ , with coefficients $a_{x} \in \mathbb {Z}[q, q^{-1}][P]$ such that

$$ \begin{align*} \sum_{x \in W_{\mathrm{af}}^{\ge 0}} |a_{x}| \in \mathbb{Z}[P](\!( q^{-1} )\!); \end{align*} $$

we see from [Reference Kato, Naito and Sagaki10, Corollary 4.31] that $K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ is indeed a $\mathbb {Z}[q, q^{-1}][P]$ -submodule of $\widetilde {K}'(\mathbf {Q}_{G})$ . Also, it follows from [Reference Kouno, Lenart and Naito12, Theorem 5.16] and (the proof of) [Reference Kato, Naito and Sagaki10, Corollary 5.12] that $[\mathcal {O}_{\mathbf {Q}_{G}(x)}(\lambda )] \in K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ for $x \in W_{\mathrm {af}}^{\ge 0}$ and $\lambda \in P$ ; in particular, we have $[\mathcal {O}_{\mathbf {Q}_{G}}(\lambda )] = [\mathcal {O}_{\mathbf {Q}_{G}(e)}(\lambda )] \in K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ .

By taking the specialization of $K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ at $q = 1$ , we obtain the T-equivariant K-group $K_{T}(\mathbf {Q}_{G})$ of $\mathbf {Q}_{G}$ , which turns out to be the $R(T) (= \mathbb {Z}[P])$ -module consisting of all infinite linear combinations of the semi-infinite Schubert classes $[\mathcal {O}_{\mathbf {Q}_{G}(x)}]$ , $x \in W_{\mathrm {af}}^{\ge 0}$ , with coefficients in $R(T)$ ; namely, we can write each element of $K_{T}(\mathbf {Q}_{G})$ uniquely as an infinite linear combination of the semi-infinite Schubert classes $[\mathcal {O}_{\mathbf {Q}_{G}(x)}]$ , $x \in W_{\mathrm {af}}^{\ge 0}$ , with coefficients in $R(T)$ ([Reference Kato9, Lemma 1.22]). It follows that $[\mathcal {O}_{\mathbf {Q}_{G}(x)}(\lambda )] \in K_{T}(\mathbf {Q}_{G})$ for $\lambda \in P$ and $x \in W_{\mathrm {af}}^{\ge 0}$ (see also [Reference Kato9, Theorem 1.26]); in particular, for $\lambda \in P$ , we have the class $[\mathcal {O}_{\mathbf {Q}_{G}}(\lambda )] = [\mathcal {O}_{\mathbf {Q}_{G}(e)}(\lambda )] \in K_{T}(\mathbf {Q}_{G})$ .

Theorem 4.1 [Reference Kato9, Corollary 3.13 and Theorem 4.17].

There exists an isomorphism

$$ \begin{align*} \Phi: QK_{T}(G/B) \xrightarrow{\sim} K_{T}(\mathbf{Q}_{G}) \end{align*} $$

of $R(T)$ -modules such that

• (i) $\Phi (\mathbf {e}^{\mu }Q^{\xi }[\mathcal {O}^{w}]) = \mathbf {e}^{-\mu }[\mathcal {O}_{\mathbf {Q}_{G}(wt_{\xi })}]$ for $\mu \in P$ , $\xi \in Q^{\vee , +}$ , and $w \in W$ ;

• (ii) for $\mathcal {Z} \in QK_{T}(G/B)$ , we have

  (4.2) $$ \begin{align} \Phi(\mathcal{Z} \star [\mathcal{O}_{G/B}(-\varpi_{i})]) = \Phi(\mathcal{Z}) \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(w_{\circ}\varpi_{i})], \quad i \in I, \end{align} $$

where $w_{\circ }$ denotes the longest element of W.

We warn the reader that the convention for line bundles in this paper is different from that in [Reference Kato9] by the twist coming from the involution $-w_{\circ }$ , and so we need $w_{\circ }$ on the right-hand side of (4.2). If G is of type C, then $w_{\circ }\lambda = -\lambda $ for all $\lambda \in P$ , and hence the right-hand side of (4.2) is $\Phi (\mathcal {Z}) \otimes [\mathcal {O}_{\mathbf {Q}_{G}}(-\varpi _{i})]$ .

Note that for a general $\lambda \in P$ , the $R(T)$ -module isomorphism $\Phi $ is not necessarily compatible with the quantum product $\star $ with $\mathcal {O}_{G/B}(\lambda )$ in $QK_{T}(G/B)$ and the tensor product $\otimes $ with $\mathcal {O}_{\mathbf {Q}_{G}}(-w_{\circ} \lambda)$ in $K_{T}(\mathbf {Q}_{G})$ . However, if G is of type C, that is, if $G = \mathrm {Sp}_{2n}(\mathbb {C})$ , then we have the following, which plays an important role in this paper.

Proposition 4.2 [Reference Maeno, Naito and Sagaki21, Proposition 5.3].

Let $G = \mathrm {Sp}_{2n}(\mathbb {C})$ be the symplectic group of rank n. For $\mathcal {Z} \in QK_{T}(G/B)$ , we have

$$ \begin{align*} \Phi \left( \mathcal{Z} \star \left( \frac{1}{1-Q_{j}} [\mathcal{O}_{G/B}(\varepsilon_{j})] \right) \right) &= \Phi(\mathcal{Z}) \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{j})], \\ \Phi \left( \mathcal{Z} \star \left( \frac{1}{1-Q_{j-1}} [\mathcal{O}_{G/B}(-\varepsilon_{j})] \right) \right) &= \Phi(\mathcal{Z}) \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{j})] \end{align*} $$

for $1 \le j \le n$ ; here we set $Q_{0} := 0$ .

For each $i \in I$ , we define an $R(T)$ -linear endomorphism $\mathsf {t}_{i}$ of $K_{T}(\mathbf {Q}_{G})$ by

$$ \begin{align*} \mathsf{t}_{i}([\mathcal{O}_{\mathbf{Q}_{G}(x)}]) := \left[\mathcal{O}_{\mathbf{Q}_{G}\left(xt_{\alpha_{i}^{\vee}}\right)}\right] \end{align*} $$

for $x \in W_{\mathrm {af}}^{\ge 0}$ . Then, for $i \in I$ , $\mathfrak {Z} \in K_{T}(\mathbf {Q}_{G})$ , and $\lambda \in P$ , we have

$$ \begin{align*} \mathsf{t}_{i}(\mathfrak{Z}) \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\lambda)] = \mathsf{t}_{i}(\mathfrak{Z} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\lambda)]) \end{align*} $$

(see [Reference Maeno, Naito and Sagaki21, (3.2)]).

4.2 Inverse Chevalley formula in type C

The inverse Chevalley formula is a formula expressing the expansion of the product $\mathbf {e}^{\mu } [\mathcal {O}_{\mathbf {Q}_{G}(x)}]$ for $\mu \in P$ and $x \in W_{\mathrm {af}}^{\ge 0}$ as an explicit linear combination of the elements $[\mathcal {O}_{\mathbf {Q}_{G}(y)}] \otimes [\mathcal {O}_{\mathbf {Q}_{G}}(\lambda )]$ , $y \in W_{\mathrm {af}}^{\ge 0}$ , $\lambda \in P$ , with coefficients in $\mathbb {Z}[q, q^{-1}]$ , which is of the following form:

$$ \begin{align*} \mathbf{e}^{\mu} [\mathcal{O}_{\mathbf{Q}_{G}(x)}] = \sum_{y \in W_{\mathrm{af}}^{\ge 0}, \, \lambda \in P} d_{x, \mu}^{y, \mu} [\mathcal{O}_{\mathbf{Q}_{G}(y)}] \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\lambda)], \end{align*} $$

with $d_{x, \mu }^{y, \mu } \in \mathbb {Z}[q, q^{-1}]$ . In types $A, D, E$ , the inverse Chevalley formula in the general case is given in [Reference Kouno, Naito, Orr and Sagaki14] and [Reference Lenart, Naito, Orr and Sagaki19], and in type C, it is given in [Reference Kouno, Naito and Orr13] in some special cases. In this paper, we use the inverse Chevalley formula in type C.

Assume that G is of type $C_{n}$ , that is, $G = \mathrm {Sp}_{2n}(\mathbb {C})$ . From [Reference Kouno, Naito and Orr13, Theorem 4.5], we obtain the following.

Proposition 4.3. Let $1\le k \le n-1$ . The following equality holds in $K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ :

(4.3) $$ \begin{align} \begin{aligned} \mathbf{e}^{\varepsilon_{1}}[\mathcal{O}_{\mathbf{Q}_{G}(s_{1}s_{2} \cdots s_{k})}] &= [\mathcal{O}_{\mathbf{Q}_{G}(s_{1}s_{2} \cdots s_{k})}(\varepsilon_{k+1})] - [\mathcal{O}_{\mathbf{Q}_{G}(s_{1}s_{2} \cdots s_{k+1})}(\varepsilon_{k+1})] \\ & \quad + q \sum_{j = 1}^{k} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1}s_{2} \cdots s_{j-1} t_{\alpha_{j}^{\vee} + \alpha_{j+1}^{\vee} + \cdots + \alpha_{k}^{\vee}}\right)}(\varepsilon_{j})\right] \\ & \quad - q \sum_{j = 1}^{k} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1}s_{2} \cdots s_{j} t_{\alpha_{j}^{\vee} + \alpha_{j+1}^{\vee} + \cdots + \alpha_{k}^{\vee}}\right)} (\varepsilon_{j})\right]. \end{aligned} \end{align} $$

Also, from [Reference Kouno, Naito and Orr13, Theorem 4.3], we have the following.

Proposition 4.4. Let $1 \le k \le n$ . The following equality holds in $K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ :

(4.4) $$ \begin{align} \begin{aligned} & \mathbf{e}^{\varepsilon_{1}}[\mathcal{O}_{\mathbf{Q}_{G}(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{k})}] \\ & = [\mathcal{O}_{\mathbf{Q}_{G}(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{k})}(-\varepsilon_{k})] - [\mathcal{O}_{\mathbf{Q}_{G}(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{k-1})}(-\varepsilon_{k})] \\ & \quad + q \sum_{j = k+1}^{n} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{j} t_{\alpha_{k}^{\vee} + \alpha_{k+1}^{\vee} + \cdots + \alpha_{j-1}^{\vee}}\right)}(-\varepsilon_{j})\right] \\ & \quad - q \sum_{j = k+1}^{n} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{j-1} t_{\alpha_{k}^{\vee} + \alpha_{k+1}^{\vee} + \cdots + \alpha_{j-1}^{\vee}}\right)}(-\varepsilon_{j})\right] \\ & \quad + q \sum_{j = 1}^{k} \left[ \mathcal{O}_{\mathbf{Q}_{G}\left(s_{1}s_{2} \cdots s_{j-1} t_{\alpha_{j}^{\vee} + \alpha_{j+1}^{\vee} + \cdots + \alpha_{n}^{\vee}}\right)}(\varepsilon_{j}) \right] \\ & \quad - q \sum_{j = 1}^{k} \left[ \mathcal{O}_{\mathbf{Q}_{G}\left(s_{1}s_{2} \cdots s_{j} t_{\alpha_{j}^{\vee} + \alpha_{j+1}^{\vee} + \cdots + \alpha_{n}^{\vee}}\right)}(\varepsilon_{j}) \right]; \end{aligned} \end{align} $$

when $k = 1$ , the second term $[\mathcal {O}_{\mathbf {Q}_{G}(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{k-1})}(-\varepsilon _{k})]$ on the right-hand side of the formula above is understood to be $0$ .

We defer the proofs of (4.3) and (4.4) to Appendix A.

4.3 Demazure operators

Following [Reference Maeno, Naito and Sagaki21, §A], we review Demazure operators acting on $K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ and $K_{T}(\mathbf {Q}_{G})$ . Let G be of an arbitrary type. The nil-DAHA $\mathbb {H}_{0}$ is defined to be the $\mathbb {Z}[\mathsf {q}^{\pm 1}]$ -algebra generated by $\mathsf {T}_{i}$ , $i \in I \sqcup \{0\}$ , and $\mathsf {X}^{\nu }$ , $\nu \in P$ , subject to certain defining relations (including the braid relations); see, for example, [Reference Maeno, Naito and Sagaki21, (A.2)–(A.6)]). For $i \in I \sqcup \{0\}$ , we set $\mathsf {D}_{i} := 1 + \mathsf {T}_{i}$ for $i \in I \sqcup \{0\}$ .

By [Reference Kato, Naito and Sagaki10, Theorem 6.5] (see also [Reference Orr23, §2.6] and [Reference Kouno, Naito, Orr and Sagaki14, §3.1.2]), we have a left $\mathbb {H}_{0}$ -action on $K_{T \times \mathbb {C}^{*}}(\mathbf {Q}_{G}^{\mathrm {rat}})$ . In particular, for $i \in I$ , we have an action of $\mathsf {D}_{i}$ on $K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ , which is called a Demazure operator; note that these Demazure operators satisfy the braid relations. Let $D_{i}$ (not to be confused with $\mathsf {D}_{i}$ ) for $i \in I$ be an operator on $\mathbb {Z}[P]$ given by

$$ \begin{align*} D_{i}(\mathbf{e}^{\nu}) := \frac{\mathbf{e}^{\nu} - \mathbf{e}^{\alpha_{i}}\mathbf{e}^{s_{i}\nu}}{1-\mathbf{e}^{\alpha_{i}}} \end{align*} $$

for $\nu \in P$ ; in other words, we have

$$ \begin{align*} D_{i}(\mathbf{e}^{\nu}) = \begin{cases} \mathbf{e}^{\nu}(1 + \mathbf{e}^{\alpha_{i}} + \mathbf{e}^{2\alpha_{i}} + \cdots + \mathbf{e}^{-\langle \nu, \alpha_{i}^{\vee} \rangle\alpha_{i}}) & \text{if } \langle \nu, \alpha_{i}^{\vee} \rangle \le 0, \\ 0 & \text{if } \langle \nu, \alpha_{i}^{\vee} \rangle = 1, \\ -\mathbf{e}^{\nu}(\mathbf{e}^{-\alpha_{i}} + \mathbf{e}^{-2\alpha_{i}^{\vee}} + \cdots + \mathbf{e}^{-(\langle \nu, \alpha_{i}^{\vee} \rangle-1)\alpha_{i}}) & \text{if } \langle \nu, \alpha_{i}^{\vee} \rangle \ge 2. \end{cases} \end{align*} $$

Lemma 4.5 [Reference Maeno, Naito and Sagaki21, Lemma A.2]; [Reference Orr23, §2.6] and [Reference Kouno, Naito, Orr and Sagaki14, §3.1.2].

Let $i \in I$ , and let $\xi \in Q^{\vee ,+}$ , $\nu , \lambda \in P$ . In $K_{T \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ , we have

$$ \begin{align*} \mathsf{D}_{i}(\mathbf{e}^{\nu}[\mathcal{O}_{\mathbf{Q}_{G}(t_{\xi})}(\lambda)]) = D_{i}(\mathbf{e}^{\nu}) [\mathcal{O}_{\mathbf{Q}_{G}(t_{\xi})}(\lambda)]. \end{align*} $$

5.1 “Fundamental” relation

From equations (4.3) and (4.4), which are special cases of the inverse Chevalley formula, we will obtain a “fundamental” relation in $K_{T}(\mathbf {Q}_{G})$ . First, by multiplying both sides of (4.3) by $[\mathcal {O}_{\mathbf {Q}_{G}}(-\varepsilon _{k+1})]$ and specializing q to $1$ , we obtain the following.

Lemma 5.1. Let $1 \le k \le n-1$ . The following equality holds in $K_{T}(\mathbf {Q}_{G})$ :

(5.1) $$ \begin{align} \begin{aligned} [\mathcal{O}_{\mathbf{Q}_{G}(s_{1}s_{2} \cdots s_{k+1})}] &= -\mathbf{e}^{\varepsilon_{1}} [\mathcal{O}_{\mathbf{Q}_{G}(s_{1}s_{2} \cdots s_{k})}(-\varepsilon_{k+1})] + [\mathcal{O}_{\mathbf{Q}_{G}(s_{1}s_{2} \cdots s_{k})}] \\ & \quad + \sum_{j = 1}^{k} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1}s_{2} \cdots s_{j-1} t_{\alpha_{j}^{\vee} + \alpha_{j+1}^{\vee} + \cdots + \alpha_{k}^{\vee}}\right)}(\varepsilon_{j} - \varepsilon_{k+1}) \right] \\ & \quad - \sum_{j = 1}^{k} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1}s_{2} \cdots s_{j} t_{\alpha_{j}^{\vee} + \alpha_{j+1}^{\vee} + \cdots + \alpha_{k}^{\vee}}\right)}(\varepsilon_{j} - \varepsilon_{k+1}) \right]. \end{aligned} \end{align} $$

Also, by multiplying both sides of (4.4) by $[\mathcal {O}_{\mathbf {Q}_{G}}(\varepsilon _{k})]$ and specializing q to $1$ , we obtain the following.

Lemma 5.2. Let $1 \le k \le n$ . The following equality holds in $K_{T}(\mathbf {Q}_{G})$ :

(5.2) $$ \begin{align} &[\mathcal{O}_{\mathbf{Q}_{G}(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{k-1})}] \nonumber\\&= -\mathbf{e}^{\varepsilon_{1}}[\mathcal{O}_{\mathbf{Q}_{G}(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{k})}(\varepsilon_{k})] + [\mathcal{O}_{\mathbf{Q}_{G}(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{k})}] \nonumber\\& \quad + \sum_{j = k+1}^{n} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{j} t_{\alpha_{k}^{\vee} + \alpha_{k+1}^{\vee} + \cdots + \alpha_{j-1}^{\vee}}\right)}(\varepsilon_{k} - \varepsilon_{j})\right] \nonumber\\& \quad - \sum_{j = k+1}^{n} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{j-1} t_{\alpha_{k}^{\vee} + \alpha_{k+1}^{\vee} + \cdots + \alpha_{j-1}^{\vee}}\right)}(\varepsilon_{k} - \varepsilon_{j})\right] \nonumber\\& \quad + \sum_{j = 1}^{k} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1}s_{2} \cdots s_{j-1} t_{\alpha_{j}^{\vee} + \alpha_{j+1}^{\vee} + \cdots + \alpha_{n}^{\vee}}\right)}(\varepsilon_{j}+\varepsilon_{k}) \right]\nonumber\\ & \quad - \sum_{j = 1}^{k} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_{1}s_{2} \cdots s_{j} t_{\alpha_{j}^{\vee} + \alpha_{j+1}^{\vee} + \cdots + \alpha_{n}^{\vee}}\right)}(\varepsilon_{j}+\varepsilon_{k}) \right]; \end{align} $$

when $k = 1$ , the left-hand side of the formula above is understood to be $0$ .

For $1 \le k \le n$ , we set

$$ \begin{align*} \mathfrak{P}_{k} &:= [\mathcal{O}_{\mathbf{Q}_{G}(s_{1}s_{2} \cdots s_{k})}], \\ \mathfrak{Q}_{k} &:= [\mathcal{O}_{\mathbf{Q}_{G}(s_{1} \cdots s_{n-1}s_{n}s_{n-1} \cdots s_{k})}]. \end{align*} $$

Also, we set $\mathfrak {P}_{0} := 1$ and $\mathfrak {Q}_{0} := 0$ . Then, (5.1) and (5.2) can be rewritten as:

(5.3) $$ \begin{align} \begin{aligned} \mathfrak{P}_{k+1} &= -\mathbf{e}^{\varepsilon_{1}}\mathfrak{P}_{k} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{k+1})] + \mathfrak{P}_{k} \\ & \quad + \sum_{j = 1}^{k} \mathsf{t}_{j}\mathsf{t}_{j+1} \cdots \mathsf{t}_{k} \mathfrak{P}_{j-1} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{j} - \varepsilon_{k+1})] \\ & \quad - \sum_{j = 1}^{k} \mathsf{t}_{j}\mathsf{t}_{j+1} \cdots \mathsf{t}_{k} \mathfrak{P}_{j} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{j} - \varepsilon_{k+1})], \end{aligned} \end{align} $$

(5.4) $$ \begin{align} \begin{aligned} \mathfrak{Q}_{k-1} &= -\mathbf{e}^{\varepsilon_{1}}\mathfrak{Q}_{k} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{k})] + \mathfrak{Q}_{k} \\ & \quad + \sum_{j = k+1}^{n} \mathsf{t}_{k}\mathsf{t}_{k+1} \cdots \mathsf{t}_{j-1} \mathfrak{Q}_{j} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{k} - \varepsilon_{j})] \\ & \quad - \sum_{j = k+1}^{n} \mathsf{t}_{k}\mathsf{t}_{k+1} \cdots \mathsf{t}_{j-1} \mathfrak{Q}_{j-1} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{k} - \varepsilon_{j})] \\ & \quad + \sum_{j = 1}^{k} \mathsf{t}_{j}\mathsf{t}_{j+1} \cdots \mathsf{t}_{n} \mathfrak{P}_{j-1} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{j}+\varepsilon_{k})] \\ & \quad - \sum_{j = 1}^{k} \mathsf{t}_{j}\mathsf{t}_{j+1} \cdots \mathsf{t}_{n} \mathfrak{P}_{j} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{j}+\varepsilon_{k})]. \end{aligned} \end{align} $$

Equations (5.3) and (5.4) can be thought of as recurrence relations for $\mathfrak {P}_{0}$ , $\mathfrak {P}_{1}$ , $\mathfrak {P}_{2}$ , $\ldots $ , $\mathfrak {P}_{n}$ and $\mathfrak {Q}_{n}$ , $\mathfrak {Q}_{n-1}$ , $\ldots $ , $\mathfrak {Q}_{1}$ , $\mathfrak {Q}_{0}$ . By solving these recurrence relations, we can write these elements as explicit linear combinations of line bundle classes with coefficients in $\operatorname {\mathrm {End}}_{\mathbb {C}}(K_{T}(\mathbf {Q}_{G}))$ .

Example 5.4. Let $n = 4$ and $J = \{2, 3, \overline {3}, \overline {1}\}$ . Then, the elements $\psi _{J}(j)$ for $1 \le j \le \overline {1}$ are as in Table 2.

Table 2 The list of $\psi _{J}(j)$ for $1 \le j \le \overline {1}$ .

For $J \subset [1, \overline {1}]$ , we set

$$ \begin{align*} \varepsilon_{J} := \sum_{j \in J} \varepsilon_{j}. \end{align*} $$

Proof. Part (1) can be proved by the same argument as for [Reference Maeno, Naito and Sagaki21, Proposition 4.7]. If we set $\mathfrak {F}_{l}^{\overline {0}} := \mathfrak {F}_{l}$ for $0 \le l \le 2n$ , then part (3) can be regarded as the special case $k = 0$ of (5.5). Hence, to prove parts (2) and (3), it suffices to prove (5.5) for $0 \le k \le n$ . We will prove (5.5) by downward induction on $k = n, n-1, \ldots , 1, 0$ . Since $\mathfrak {P}_{n} = \mathfrak {Q}_{n}$ , the case $k = n$ is already proved by part (1). Now assume that (5.5) holds for all k, with $0 < k \le n$ . By (5.4), we see that

(5.7) $$ \begin{align} \mathfrak{Q}_{k-1} &= -\mathbf{e}^{\varepsilon_{1}} \sum_{l = 0}^{2n-k} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \mathfrak{F}_{l}^{\overline{k}} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{k})] + \sum_{l = 0}^{2n-k} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \mathfrak{F}_{l}^{\overline{k}}\nonumber \\& \quad + \sum_{i = k+1}^{n} \mathsf{t}_{k}\mathsf{t}_{k+1} \cdots \mathsf{t}_{i-1} \sum_{l = 0}^{2n-i} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \mathfrak{F}_{l}^{\overline{i}} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{k} - \varepsilon_{i})]\nonumber \\& \quad - \sum_{i = k+1}^{n} \mathsf{t}_{k}\mathsf{t}_{k+1} \cdots \mathsf{t}_{i-1} \sum_{l = 0}^{2n-i+1} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \mathfrak{F}_{l}^{\overline{i-1}} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{k} - \varepsilon_{i})] \nonumber\\& \quad + \sum_{i = 1}^{k} \mathsf{t}_{i}\mathsf{t}_{i+1} \cdots \mathsf{t}_{n} \sum_{l = 0}^{i-1} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \mathfrak{F}_{l}^{i-1} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{i}+\varepsilon_{k})]\nonumber \\& \quad - \sum_{i = 1}^{k} \mathsf{t}_{i}\mathsf{t}_{i+1} \cdots \mathsf{t}_{n} \sum_{l = 0}^{i} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \mathfrak{F}_{l}^{i} \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{i}+\varepsilon_{k})] \nonumber\\&= \sum_{l = 0}^{2n-k} (-1)^{l+1} \mathbf{e}^{(l+1)\varepsilon_{1}} \sum_{\substack{J \subset [1, \overline{k+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J} + \varepsilon_{k})]\nonumber \\& \quad + \sum_{l = 0}^{2n-k} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \sum_{\substack{J \subset [1, \overline{k+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})]\nonumber \\& \quad + \sum_{i = k+1}^{n} \mathsf{t}_{k}\mathsf{t}_{k+1} \cdots \mathsf{t}_{i-1} \sum_{l = 0}^{2n-i} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \sum_{\substack{J \subset [1, \overline{i+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J} + \varepsilon_{k} - \varepsilon_{i})]\nonumber\\& \quad - \sum_{i = k+1}^{n} \mathsf{t}_{k}\mathsf{t}_{k+1} \cdots \mathsf{t}_{i-1} \sum_{l = 0}^{2n-i+1} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \sum_{\substack{J \subset [1, \overline{i}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J} + \varepsilon_{k} - \varepsilon_{i})] \nonumber\\& \quad + \sum_{i = 1}^{k} \mathsf{t}_{i}\mathsf{t}_{i+1} \cdots \mathsf{t}_{n} \sum_{l = 0}^{i-1} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \sum_{\substack{J \subset [1, i-1] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J}+\varepsilon_{i}+\varepsilon_{k})] \nonumber\\& \quad - \sum_{i = 1}^{k} \mathsf{t}_{i}\mathsf{t}_{i+1} \cdots \mathsf{t}_{n} \sum_{l = 0}^{i} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \sum_{\substack{J \subset [1, i] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J}+\varepsilon_{i}+\varepsilon_{k})]\nonumber\\&= \sum_{J \subset [1, \overline{k+1}]} (-1)^{|J|+1} \mathbf{e}^{(|J|+1)\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J}+\varepsilon_{k})]\nonumber\\ & \quad + \sum_{J \subset [1, \overline{k+1}]} (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})]\nonumber \\ & \quad - \sum_{i = k+1}^{n} \mathsf{t}_{k}\mathsf{t}_{k+1} \cdots \mathsf{t}_{i-1} \sum_{\substack{J \subset [1, \overline{i}] \\ \overline{i} \in J}} (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J}+\varepsilon_{k} - \varepsilon_{i})]\nonumber \\ & \quad - \sum_{i = 1}^{k} \mathsf{t}_{i}\mathsf{t}_{i+1} \cdots \mathsf{t}_{n} \sum_{\substack{J \subset [1, i] \\ i \in J}} (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J}+\varepsilon_{i}+\varepsilon_{k})].\end{align} $$

Let us compute the term corresponding to each $J \subset [1, \overline {k}]$ on the RHS of (5.7).

Let $J \subset [1, \overline {k}]$ be such that $\overline {k} \notin J$ ; we set $l := |J|$ . Then we find that the second sum on the RHS of (5.7) contains the term

$$ \begin{align*} & (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})] \\ &= (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})], \end{align*} $$

which is just the term in $\mathfrak {Q}_{k-1}$ corresponding to J.

Next, let $J \subset [1, \overline {k}]$ be such that $\overline {k} \in J$ ; we set $M := \max (J \setminus \{\overline {k}\}) \in [1, \overline {1}]$ . Assume that $M < n$ . By summing up

• ○ the term in the first sum on the RHS of (5.7) corresponding to $J \setminus \{\overline {k}\}$ ,

• ○ the terms in the third sum on the RHS of (5.7) corresponding to $(J \setminus \{\overline {k}\}) \sqcup \{\overline {i}\}$ for $k+1 \le i \le n$ and $j \in J$ ,

• ○ the terms in the fourth sum on the RHS of (5.7) corresponding to $(J \setminus \{\overline {k}\}) \sqcup \{i\}$ for $M+1 \le i \le k$ in the case $M < k$ ,

we deduce that

(5.8) $$ \begin{align} & (-1)^{(|J|-1) + 1} \mathbf{e}^{((|J|-1)+1)\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J \setminus \{\overline{k}\}}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J \setminus \{\overline{k}\}} + \varepsilon_{k})]\nonumber \\&- \sum_{i = k+1}^{n} \mathsf{t}_{k}\cdots \mathsf{t}_{i-1} (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}}+\varepsilon_{k}-\varepsilon_{i})] \nonumber\\&- \sum_{i = M+1}^{k} \mathsf{t}_{i} \cdots \mathsf{t}_{n} (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{ i\}}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{(J \setminus \{\overline{k}\}) \sqcup \{ i \}}+\varepsilon_{i}+\varepsilon_{k})] \nonumber\\&= (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \left( \prod_{1 \le j \le \overline{1}} \psi_{J \setminus \{\overline{k}\}}(j) \right) - \sum_{i = k+1}^{n} \mathsf{t}_{k} \cdots \mathsf{t}_{i-1} \left( \prod_{1 \le j \le \overline{1}} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}}(j) \right) \right. \nonumber\\[-2pt]& \hspace{40mm} \left. - \sum_{i = M+1}^{k} \mathsf{t}_{i} \cdots \mathsf{t}_{n} \left( \prod_{1 \le j \le \overline{1}} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{ i\}}(j) \right) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})]. \nonumber \\[-2pt]&= (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le M} \psi_{J \setminus \{\overline{k}\}}(j) \right) \nonumber\\[-2pt]& \qquad \times \left( 1 - \sum_{i = k+1}^{n} \mathsf{t}_{k} \cdots \mathsf{t}_{i-1} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}} (\overline{i+1}) \right.\nonumber \\[-2pt]& \hspace{30mm} \left. {}-{} \mathsf{t}_{M+1} \cdots \mathsf{t}_{n} - \sum_{i = M+2}^{k} \mathsf{t}_{i} \cdots \mathsf{t}_{n} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{i\}}(i-1) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})]. \end{align} $$

Here, we set

$$ \begin{align*} \varphi := 1 - \sum_{i = k+1}^{n} \mathsf{t}_{k} \cdots \mathsf{t}_{i-1} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}} (\overline{i+1}) - \mathsf{t}_{M+1} \cdots \mathsf{t}_{n} - \sum_{i = M+2}^{k} \mathsf{t}_{i} \cdots \mathsf{t}_{n} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{i\}}(i-1). \end{align*} $$

If $M < k$ , then

$$ \begin{align*} \varphi &= 1 - \sum_{i = k+1}^{n} \mathsf{t}_{k} \cdots \mathsf{t}_{i-1} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}} (\overline{i+1}) \\[-2pt] & \qquad \quad - \mathsf{t}_{M+1} \cdots \mathsf{t}_{n} - \sum_{i = M+2}^{k} \mathsf{t}_{i} \cdots \mathsf{t}_{n} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{i\}}(i-1)\\[-2pt]&= 1 - \sum_{i = k+1}^{n} \mathsf{t}_{k} \cdots \mathsf{t}_{i-1} (1 - \mathsf{t}_{i}) - \mathsf{t}_{M+1} \cdots \mathsf{t}_{n} - \sum_{i = M+2}^{k} \mathsf{t}_{i} \cdots \mathsf{t}_{n} (1 - \mathsf{t}_{i-1}) \\ &= 1-\mathsf{t}_{k} \\ &= \psi_{J}(\overline{k+1}). \end{align*} $$

If $M = k$ , then

$$ \begin{align*} \varphi &= 1 - \sum_{i = k+1}^{n} \mathsf{t}_{k} \cdots \mathsf{t}_{i-1} \underbrace{\psi_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}}(\overline{i+1})}_{= \, 1 - \mathsf{t}_{j}} \\ &= 1 - \mathsf{t}_{k} + \mathsf{t}_{k} \cdots \mathsf{t}_{n} \\ &= \psi_{J}(\overline{k+1}). \end{align*} $$

If $M> k$ , then

From these, we conclude that the RHS of (5.8) equals

$$ \begin{align*} (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le 1} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})], \end{align*} $$

which is just the term in $\mathfrak {Q}_{k-1}$ corresponding to J.

Assume that $M \ge n$ . If $M> n$ , then take $1 \le m \le n$ such that $M = \overline {m}$ . If $M = n$ , then set $m := \overline {n}$ . By summing up

• ○ the term in the first sum on the RHS of (5.7) corresponding to $J \setminus \{\overline {k}\}$ ,

• ○ the terms in the third sum on the RHS of (5.7) corresponding to $(J \setminus \{\overline {k}\}) \sqcup \{\overline {i}\}$ for $k+1 \le i \le m-1$ (if $m = \overline {n}$ , then regard $\overline {n}-1$ as n),

we deduce that (if $m = \overline {n}$ , then regard $\overline {n}-2$ as $n-1$ )

$$ \begin{align*} & (-1)^{(|J|-1)+1} \mathbf{e}^{((|J|-1)+1)\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J \setminus \{\overline{k}\}}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J \setminus \{\overline{k}\}}+\varepsilon_{k})] \\& - \sum_{i = k+1}^{m-1} \mathsf{t}_{k} \cdots \mathsf{t}_{i-1} (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le \overline{1}} \psi_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}}+\varepsilon_{k} - \varepsilon_{i})] \\&= (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \left( \prod_{1 \le j \le M} \psi_{J \setminus \{\overline{k}\}}(j) \right) \left(\vphantom{- \sum_{i = k+1}^{m-2} \mathsf{t}_{k} \cdots \mathsf{t}_{i-1} \underbrace{\psi_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}}(\overline{i+1})}_{= \, 1 - \mathsf{t}_{i}}} 1 - \mathsf{t}_{k} \cdots \mathsf{t}_{m-2} \right. \right. \\& \left. \left. \qquad\qquad\qquad\qquad\qquad - \sum_{i = k+1}^{m-2} \mathsf{t}_{k} \cdots \mathsf{t}_{i-1} \underbrace{\psi_{(J \setminus \{\overline{k}\}) \sqcup \{\overline{i}\}}(\overline{i+1})}_{= \, 1 - \mathsf{t}_{i}} \right) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})] \\&= (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le M} \psi_{J \setminus \{\overline{k}\}}(j) \right) (1 - \mathsf{t}_{k}) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})] \\&= (-1)^{|J|} \mathbf{e}^{|J|\varepsilon_{1}} \left( \prod_{1 \le j \le 1} \psi_{J \setminus \{\overline{k}\}}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})], \end{align*} $$

which is just the term in $\mathfrak {Q}_{k-1}$ corresponding to J. This proves (5.5) for $k-1$ . Thus, by downward induction on k, equation (5.5) is proved. This completes the proof of the proposition.

Also, we can show the following; we defer the proof of this proposition to Appendix B.

From this proposition, by multiplying both sides of (5.6) by $\mathbf {e}^{-n\varepsilon _{1}}$ , we obtain the following “fundamental” relation.

Corollary 5.8. The following equality holds in $K_{T}(\mathbf {Q}_{G})$ :

(5.9) $$ \begin{align} \sum_{l = 0}^{n-1} (-1)^{l} (\mathbf{e}^{-(n-l)\varepsilon_{1}} + \mathbf{e}^{(n-l)\varepsilon_{1}}) \mathfrak{F}_{l} + (-1)^{n}\mathfrak{F}_{n} = 0. \end{align} $$

5.2 The other relations

From the fundamental relation (5.9), we will derive sufficiently many relations needed to obtain a Borel-type presentation of $QK_{T}(G/B)$ . For $k \ge 0$ , let $h_{k}(x_{1}, \ldots , x_{2n})$ be the k-th complete symmetric polynomial in the $2n$ variables $x_{1}, \ldots , x_{2n}$ , that is,

$$ \begin{align*} h_{k}(x_{1}, \ldots, x_{2n}) := \sum_{1 \le i_{1} \le \cdots \le i_{k} \le 2n} x_{i_{1}} x_{i_{2}} \cdots x_{i_{2n}}; \end{align*} $$

by convention, we set $h_{0}(x_{1}, \ldots , x_{2n}) := 0$ . For $l \ge 0$ and $k \ge 1$ , we set

$$ \begin{align*} H_{l}^{k} := h_{l}(\mathbf{e}^{\varepsilon_{1}}, \ldots, \mathbf{e}^{\varepsilon_{k-1}}, \mathbf{e}^{\varepsilon_{k}}, \mathbf{e}^{-\varepsilon_{k}}, \mathbf{e}^{-\varepsilon_{k-1}}, \ldots, \mathbf{e}^{-\varepsilon_{1}}) \in R(T). \end{align*} $$

The aim of this section is to prove the following.

Proposition 5.9. The following recurrence relations hold:

(5.10) $$ \begin{align} \left\{ \begin{aligned} & \mathfrak{F}_{0} & &{}= 1, \\ & \sum_{l = 0}^{n-k} (-1)^{l} (H_{n-l-k}^{k+1} - H_{n-l-k-2}^{k+1}) \mathfrak{F}_{l} & & {}= 0 \quad \text{for } 0 \le k \le n-1. \end{aligned} \right. \end{align} $$

First, by applying the Demazure operator $\mathsf {D}_{1}$ to (5.9), we obtain the following.

Lemma 5.10. We have

(5.11) $$ \begin{align} \sum_{l = 0}^{n-1} (-1)^{l} \mathbf{e}^{-(n-l)\varepsilon_{1}} \left( \sum_{r = 0}^{n-l-1} \mathbf{e}^{r(\varepsilon_{1}-\varepsilon_{2})} \right) \left( \sum_{s = 0}^{n-l-1} \mathbf{e}^{s(\varepsilon_{1}+\varepsilon_{2})} \right) \mathfrak{F}_{l} = 0. \end{align} $$

Proof. By multiplying both sides of (5.9) by $\mathbf {e}^{\varepsilon _{1}}$ , we obtain

$$ \begin{align*} \sum_{l = 0}^{n-1} (-1)^{l} (\mathbf{e}^{-(n-l-1)\varepsilon_{1}} + \mathbf{e}^{(n-l+1)\varepsilon_{1}}) \mathfrak{F}_{l} + (-1)^{n}\mathbf{e}^{\varepsilon_{1}}\mathfrak{F}_{n} = 0. \end{align*} $$

Then, by applying $\mathsf {D}_{1}$ , we see that

$$ \begin{align*} 0 &= \mathsf{D}_{1}\left(\sum_{l = 0}^{n-1} (-1)^{l} (\mathbf{e}^{-(n-l-1)\varepsilon_{1}} + \mathbf{e}^{(n-l+1)\varepsilon_{1}}) \mathfrak{F}_{l} + (-1)^{n}e^{\varepsilon_{1}}\mathfrak{F}_{n}\right) \\&= \sum_{l = 0}^{n-1} (-1)^{l} (D_{1}(\mathbf{e}^{-(n-l-1)\varepsilon_{1}}) + D_{1}(\mathbf{e}^{(n-l+1)\varepsilon_{1}})) \mathfrak{F}_{l} + (-1)^{n}D_{1}(\mathbf{e}^{\varepsilon_{1}})\mathfrak{F}_{n} \\&= \sum_{l = 0}^{n-1} (-1)^{l} \left( \frac{\mathbf{e}^{-(n-l-1)\varepsilon_{1}}-\mathbf{e}^{\alpha_{1}}\mathbf{e}^{s_{1}(-(n-l-1)\varepsilon_{1})}}{1-\mathbf{e}^{\alpha_{1}}} + \frac{\mathbf{e}^{(n-l+1)\varepsilon_{1}}-\mathbf{e}^{\alpha_{1}}\mathbf{e}^{s_{1}((n-l+1)\varepsilon_{1})}}{1-\mathbf{e}^{\alpha_{1}}} \right) \mathfrak{F}_{l} \\& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad + (-1)^{n}\frac{\mathbf{e}^{\varepsilon_{1}} - \mathbf{e}^{\alpha_{1}}\mathbf{e}^{s_{1}\varepsilon_{1}}}{1-\mathbf{e}^{\alpha_{1}}}\mathfrak{F}_{n} \\&= \sum_{l = 0}^{n-1} (-1)^{l} \left( \frac{\mathbf{e}^{-(n-l-1)\varepsilon_{1}}-\mathbf{e}^{\varepsilon_{1}-\varepsilon_{2}}\mathbf{e}^{-(n-l-1)\varepsilon_{2}} + \mathbf{e}^{(n-l+1)\varepsilon_{1}}-\mathbf{e}^{\varepsilon_{1}-\varepsilon_{2}}\mathbf{e}^{(n-l+1)\varepsilon_{2}}}{1-\mathbf{e}^{\alpha_{1}}} \right) \mathfrak{F}_{l} \\& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad + (-1)^{n}\frac{\mathbf{e}^{\varepsilon_{1}} - \mathbf{e}^{\varepsilon_{1}-\varepsilon_{2}}\mathbf{e}^{\varepsilon_{2}}}{1-\mathbf{e}^{\alpha_{1}}}\mathfrak{F}_{n} \\&= \mathbf{e}^{\varepsilon_{1}} \sum_{l = 0}^{n-1} (-1)^{l} \frac{\mathbf{e}^{-(n-l)\varepsilon_{1}}-\mathbf{e}^{-(n-l)\varepsilon_{2}} + \mathbf{e}^{(n-l)\varepsilon_{1}}-\mathbf{e}^{(n-l)\varepsilon_{2}}}{1-\mathbf{e}^{\alpha_{1}}} \mathfrak{F}_{l} \\&= \mathbf{e}^{\varepsilon_{1}} \sum_{l = 0}^{n-1} (-1)^{l} \frac{(\mathbf{e}^{-(n-l)\varepsilon_{1}}-\mathbf{e}^{-(n-l)\varepsilon_{2}})(1-\mathbf{e}^{(n-l)(\varepsilon_{1}+\varepsilon_{2})})}{1-\mathbf{e}^{\alpha_{1}}} \mathfrak{F}_{l}\\&= \mathbf{e}^{\varepsilon_{1}}(1-\mathbf{e}^{\varepsilon_{1}+\varepsilon_{2}}) \sum_{l = 0}^{n-1} (-1)^{l} \left( \sum_{s = 0}^{n-l-1} \mathbf{e}^{s(\varepsilon_{1}+\varepsilon_{2})} \right) \frac{\mathbf{e}^{-(n-l)\varepsilon_{1}}-\mathbf{e}^{-(n-l)\varepsilon_{2}}}{1-\mathbf{e}^{\alpha_{1}}} \mathfrak{F}_{l} \\&= \mathbf{e}^{\varepsilon_{1}}(1-\mathbf{e}^{\varepsilon_{1}+\varepsilon_{2}}) \sum_{l = 0}^{n-1} (-1)^{l} \left( \sum_{s = 0}^{n-l-1} \mathbf{e}^{s(\varepsilon_{1}+\varepsilon_{2})} \right) \frac{\mathbf{e}^{-(n-l)\varepsilon_{1}}(1 -\mathbf{e}^{(n-l)(\varepsilon_{1}-\varepsilon_{2})})}{1-\mathbf{e}^{\varepsilon_{1}-\varepsilon_{2}}} \mathfrak{F}_{l} \\&= \mathbf{e}^{\varepsilon_{1}}(1-\mathbf{e}^{\varepsilon_{1}+\varepsilon_{2}}) \sum_{l = 0}^{n-1} (-1)^{l} \mathbf{e}^{-(n-l)\varepsilon_{1}} \left( \sum_{r=0}^{n-l-1} \mathbf{e}^{r(\varepsilon_{1}-\varepsilon_{2})} \right) \left( \sum_{s = 0}^{n-l-1} \mathbf{e}^{s(\varepsilon_{1}+\varepsilon_{2})} \right) \mathfrak{F}_{l}. \end{align*} $$

Finally, by dividing both sides of this equation by $\mathbf {e}^{\varepsilon _{1}}(1-\mathbf {e}^{\varepsilon _{1}+\varepsilon _{2}})$ , we conclude that

$$ \begin{align*} \sum_{l = 0}^{n-1} (-1)^{l} \mathbf{e}^{-(n-l)\varepsilon_{1}} \left( \sum_{r=0}^{n-l-1} \mathbf{e}^{r(\varepsilon_{1}-\varepsilon_{2})} \right) \left( \sum_{s = 0}^{n-l-1} \mathbf{e}^{s(\varepsilon_{1}+\varepsilon_{2})} \right) \mathfrak{F}_{l} = 0, \end{align*} $$

as desired. This proves the lemma.

Then, by successively applying Demazure operators to (5.11), we obtain the other relations. For this purpose, we use the following easy lemma.

Lemma 5.12. For $2 \le k \le n-1$ , we have

(5.12) $$ \begin{align} \begin{aligned} & \sum_{l = 0}^{n-k} (-1)^{l} \left( \sum_{r_{1}=k-1}^{n-l-1} \sum_{s_{1}=0}^{n-l-1-r_{1}} \sum_{r_{2}=k-2}^{r_{1}-1} \sum_{s_{2}=0}^{r_{1}-1-r_{2}} \cdots \sum_{r_{k-1}=1}^{r_{k-2}-1} \sum_{s_{k-1}=0}^{r_{k-2}-1-r_{k-1}} \right. \\ & \hspace{30mm} \mathbf{e}^{(l+r_{1}+2s_{1})\varepsilon_{1}+(-r_{1}+r_{2}+2s_{2})\varepsilon_{2}+\cdots+(-r_{k-2}+r_{k-1}+2s_{k-1})\varepsilon_{k-1}+(-r_{k-1})\varepsilon_{k}} \\ & \left. \hspace{40mm} \times \sum_{p_{k}=0}^{r_{k-1}-1}\mathbf{e}^{p_{k}(\varepsilon_{k}-\varepsilon_{k+1})} \sum_{q_{k}=0}^{r_{k-1}-1} \mathbf{e}^{q_{k}(\varepsilon_{k}+\varepsilon_{k+1})} \right) \mathfrak{F}_{l} = 0. \end{aligned} \end{align} $$

Proof. We prove the lemma by induction on k. First, we consider the case $k = 2$ ; in this case, $n \ge 3$ . By multiplying both sides of (5.11) by $\mathbf {e}^{(n-1)\varepsilon _{1}}$ , we obtain

(5.13) $$ \begin{align} \sum_{l = 0}^{n-1} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \left( \sum_{r = 0}^{n-l-1} \sum_{s = 0}^{n-l-1} \mathbf{e}^{r(\varepsilon_{1}-\varepsilon_{2}) + s(\varepsilon_{1}+\varepsilon_{2})} \right) \mathfrak{F}_{l} = 0. \end{align} $$

Then, by multiplying both sides of (5.13) by $\mathbf {e}^{\varepsilon _{2}}$ , we obtain

(5.14) $$ \begin{align} \sum_{l = 0}^{n-1} (-1)^{l} \left( \sum_{r = 0}^{n-l-1} \sum_{s = 0}^{n-l-1} \mathbf{e}^{(l+r+s)\varepsilon_{1} + (-r+s+1)\varepsilon_{2}} \right) \mathfrak{F}_{l} = 0. \end{align} $$

Here, note that $\langle (l+r+s)\varepsilon _{1} + (-r+s+1)\varepsilon _{2}, \alpha _{2}^{\vee } \rangle = -r+s+1.$ Hence, by applying the Demazure operator $\mathsf {D}_{2}$ to both sides of (5.14), we deduce that

(5.15) $$ \begin{align} \begin{aligned} \sum_{l = 0}^{n-2} (-1)^{l} & \left( \underbrace{\sum_{r = 1}^{n-l-1} \sum_{s = 0}^{r-1} \mathbf{e}^{(l+r+s)\varepsilon_{1} + (-r+s+1)\varepsilon_{2}} (1 + \mathbf{e}^{\alpha_{2}} + \cdots + \mathbf{e}^{(r-s-1)\alpha_{2}})}_{(*)} \right. \\ & \left. -\underbrace{\sum_{r = 0}^{n-l-2} \sum_{s = r+1}^{n-l-1} \mathbf{e}^{(l+r+s)\varepsilon_{1} + (-r+s+1)\varepsilon_{2}} (\mathbf{e}^{-\alpha_{2}} + \mathbf{e}^{-2\alpha_{2}} + \cdots + \mathbf{e}^{-(-r+s)\alpha_{2}} )}_{(**)} \right) \mathfrak{F}_{l} = 0. \end{aligned} \end{align} $$

In $(*)$ , we put

(5.16) $$ \begin{align} r_{1} = r-s, \quad s_{1} = s, \end{align} $$

while in $(**)$ , we put

(5.17) $$ \begin{align} r_{1} = s-r, \quad s_{1} = r. \end{align} $$

Then, we see that

(5.18) $$ \begin{align} & \text{(LHS) of (5.15)} \nonumber\\ &= \sum_{l = 0}^{n-2} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}} \left( \sum_{r_{1} = 1}^{n-l-1} \sum_{s_{1} = 0}^{n-l-1-r_{1}} \mathbf{e}^{(r_{1}+2s_{1})\varepsilon_{1} + (-r_{1}+1)\varepsilon_{2}} (1 + \mathbf{e}^{\alpha_{2}} + \cdots + \mathbf{e}^{(r_{1}-1)\alpha_{2}}) \right. \nonumber\\ & \hspace{30mm} \left. - \sum_{r_{1}=1}^{n-l-1} \sum_{s_{1}=0}^{n-l-1-r_{1}} \mathbf{e}^{(r_{1}+2s_{1})\varepsilon_{1} + (r_{1}+1)\varepsilon_{2}} (\mathbf{e}^{-\alpha_{2}} + \mathbf{e}^{-2\alpha_{2}} + \cdots + \mathbf{e}^{-r_{1}\alpha_{1}}) \right) \mathfrak{F}_{l} \nonumber\\ &= \sum_{l=0}^{n-2} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}+\varepsilon_{2}} \left( \sum_{r_{1}=1}^{n-l-1} \sum_{s_{1}=0}^{n-l-1-r_{1}} \mathbf{e}^{(r_{1}+2s_{1})\varepsilon_{1}} \right. \nonumber \\ & \hspace{40mm} \left. \times \left( \mathbf{e}^{-r_{1}\varepsilon_{2}} \sum_{t=0}^{r_{1}-1} \mathbf{e}^{t(\varepsilon_{2}-\varepsilon_{3})} - \mathbf{e}^{r_{1}\varepsilon_{2}} \sum_{t=1}^{r_{1}} \mathbf{e}^{t(-\varepsilon_{2}+\varepsilon_{3})} \right) \right) \mathfrak{F}_{l} \nonumber \\ & = \sum_{l=0}^{n-2} (-1)^{l} \mathbf{e}^{l\varepsilon_{1}+\varepsilon_{2}} \left( \sum_{r_{1}=1}^{n-l-1} \sum_{s_{1}=0}^{n-l-1-r_{1}} \mathbf{e}^{(r_{1}+2s_{1})\varepsilon_{1}} \right. \nonumber \\ & \hspace{40mm} \left. \times \left( \mathbf{e}^{-r_{1}\varepsilon_{2}} (1-\mathbf{e}^{\varepsilon_{2}+\varepsilon_{3}}) \sum_{p_{2}=0}^{r_{1}-1}\mathbf{e}^{p_{2}(\varepsilon_{2}-\varepsilon_{3})} \sum_{q_{2}=0}^{r_{1}-1} \mathbf{e}^{q_{2}(\varepsilon_{2}+\varepsilon_{3})} \right) \right) \mathfrak{F}_{l}, \end{align} $$

where, for the last equality, we have used Lemma 5.11 (2). Hence, by dividing the rightmost-hand side of (5.18) by $\mathbf {e}^{\varepsilon _{2}}(1-\mathbf {e}^{\varepsilon _{2}+\varepsilon _{3}})$ , we obtain

$$ \begin{align*} \sum_{l=0}^{n-2} (-1)^{l} \left( \sum_{r_{1}=1}^{n-l-1} \sum_{s_{1}=0}^{n-l-1-r_{1}} \mathbf{e}^{(l+r_{1}+2s_{1})\varepsilon_{1}} \left( \mathbf{e}^{-r_{1}\varepsilon_{2}} \sum_{p_{2}=0}^{r_{1}-1}\mathbf{e}^{p_{2}(\varepsilon_{2}-\varepsilon_{3})} \sum_{q_{2}=0}^{r_{1}-1} \mathbf{e}^{q_{2}(\varepsilon_{2}+\varepsilon_{3})} \right) \right) \mathfrak{F}_{l} = 0. \end{align*} $$

This proves the lemma for $k = 2$ .

Now, let $3 \le k \le n-1$ . We assume that the lemma holds for $k-1$ , and prove the assertion of the lemma for k; note that $n \ge k+1$ in this case. By the induction hypothesis, we have

$$ \begin{align*} \begin{aligned} & \sum_{l = 0}^{n-k+1} (-1)^{l} \Bigg( \sum_{r_{1}=k-2}^{n-l-1} \sum_{s_{1}=0}^{n-l-1-r_{1}} \sum_{r_{2}=k-3}^{r_{1}-1} \sum_{s_{2}=0}^{r_{1}-1-r_{2}} \cdots \sum_{r_{k-2}=1}^{r_{k-3}-1} \sum_{s_{k-2}=0}^{r_{k-3}-1-r_{k-2}} \\ & \hspace{30mm} \mathbf{e}^{(l+r_{1}+2s_{1})\varepsilon_{1}+(-r_{1}+r_{2}+2s_{2})\varepsilon_{2}+\cdots+(-r_{k-3}+r_{k-2}+2s_{k-2})\varepsilon_{k-2}+(-r_{k-2})\varepsilon_{k-1}} \\ & \hspace{40mm} \times \sum_{p_{k-1}=0}^{r_{k-2}-1}\mathbf{e}^{p_{k-1}(\varepsilon_{k-1}-\varepsilon_{k})} \sum_{q_{k-1}=0}^{r_{k-2}-1} \mathbf{e}^{q_{k-1}(\varepsilon_{k-1}+\varepsilon_{k})} \Bigg) \mathfrak{F}_{l} = 0. \end{aligned} \end{align*} $$

By

1. (i) multiplying both sides of this equation by $\mathbf {e}^{\varepsilon _{k}}$ ,

2. (ii) applying the Demazure operator $\mathsf {D}_{k}$ to both sides,

3. (iii) making a change of variables similar to (5.16) and (5.17), and

4. (iv) dividing both sides by $\mathbf {e}^{\varepsilon _{k}}(1-\mathbf {e}^{\varepsilon _{k}+\varepsilon _{k+1}})$ ,

we deduce that

$$ \begin{align*} \begin{aligned} & \sum_{l = 0}^{n-k+1} (-1)^{l} \Bigg( \sum_{r_{1}=k-1}^{n-l-1} \sum_{s_{1}=0}^{n-l-1-r_{1}} \sum_{r_{2}=k-2}^{r_{1}-1} \sum_{s_{2}=0}^{r_{1}-1-r_{2}} \cdots \sum_{r_{k-1}=1}^{r_{k-2}-1} \sum_{s_{k-1}=0}^{r_{k-2}-1-r_{k-1}} \\ & \quad \mathbf{e}^{(l+r_{1}+2s_{1})\varepsilon_{1}+(-r_{1}+r_{2}+2s_{2})\varepsilon_{2}+\cdots+(-r_{k-3}+r_{k-2}+2s_{k-2})\varepsilon_{k-2}+(-r_{k-2}+r_{k-1}+2s_{k-1})\varepsilon_{k-1}+(-r_{k-1})\varepsilon_{k}} \\ & \hspace{70mm} \times \sum_{p_{k} = 0}^{r_{k-1}-1} \mathbf{e}^{p_{k}(\varepsilon_{k}-\varepsilon_{k+1})} \sum_{q_{k} = 0}^{r_{k-1}-1} \mathbf{e}^{q_{k}(\varepsilon_{k}+\varepsilon_{k+1})} \Bigg) \mathfrak{F}_{l} = 0, \end{aligned} \end{align*} $$

as desired. Thus, by induction on k, the lemma is proved.

Also, we can prove the following lemma for complete symmetric polynomials. Since the proof of this lemma is elementary, we leave it to the reader.

Proof of Proposition 5.9.

By multiplying both sides of (5.11) by $\mathbf {e}^{\varepsilon _{1}}$ and then applying Lemma 5.13 (3) with $x_{1} = \mathbf {e}^{\varepsilon _{1}}$ , $x_{2} = \mathbf {e}^{\varepsilon _{2}}$ , and $m = n-l-1$ , we obtain

$$ \begin{align*} \sum_{l = 0}^{n-1} (-1)^{l} (H_{n-l-1}^{2} - H_{n-l-3}^{2}) \mathfrak{F}_{l} = 0. \end{align*} $$

For $2 \le k \le n-1$ , by multiplying both sides of (5.12) by $\mathbf {e}^{-(2n-k-2)\varepsilon _{1} + \varepsilon _{2} + \cdots + \varepsilon _{n-1}}$ and then applying Lemma 5.13 (4) with $x_{1} = \mathbf {e}^{\varepsilon _{1}}, \ldots , x_{k+1} = \mathbf {e}^{\varepsilon _{k+1}}$ , and $m = n-l-k$ , we obtain

$$ \begin{align*} \sum_{l = 0}^{n-k} (-1)^{l} (H_{n-l-k}^{k+1} - H_{n-l-k-2}^{k+1}) \mathfrak{F}_{l} = 0. \end{align*} $$

This proves Proposition 5.9.

5.3 The solution of the recurrence relations

In this subsection, we solve the recurrence relations (5.10) for $\mathfrak {F}_{l}$ , $0 \le l \le n$ , with coefficients in $R(T)$ . Recall the notation $E_{l}$ , $0 \le l \le 2n$ , from (3.1).

Lemma 5.14. We have the following:

(5.19) $$ \begin{align} E_{0} &= 1, \end{align} $$

(5.20) $$ \begin{align} \sum_{l = 0}^{n-k} (-1)^{l} (H_{n-l-k}^{k+1} - H_{n-l-k-2}^{k+1}) E_{l} &= 0, \quad 0 \le k \le n-1. \end{align} $$

Proof. First, equation (5.19) is obvious from the definition of elementary symmetric polynomials. We will prove (5.20). Since

$$ \begin{align*} \sum_{l = 0}^{\infty} h_{l}(x_{1}, \ldots, x_{d}) t^{l} = \prod_{i = 1}^{d} \frac{1}{1-x_{i}t} \end{align*} $$

for $1 \le k+1 \le n$ , we have

(5.21) $$ \begin{align} \sum_{l = 0}^{\infty} H_{l}^{k+1} t^{l} = \left( \prod_{i = 1}^{k+1} \frac{1}{1 - \mathbf{e}^{\varepsilon_{i}}t} \right) \left( \prod_{j = 1}^{k+1} \frac{1}{1 - \mathbf{e}^{-\varepsilon_{j}}t} \right). \end{align} $$

By multiplying both sides of (5.21) by $t^{2}$ and then subtracting the resulting equation from (5.21), we obtain

(5.22) $$ \begin{align} \sum_{l = 0}^{\infty} (H_{l}^{k+1} - H_{l-2}^{k+1}) t^{l} = (1-t^{2}) \left( \prod_{i = 1}^{k+1} \frac{1}{1 - \mathbf{e}^{\varepsilon_{i}}t} \right) \left( \prod_{j = 1}^{k+1} \frac{1}{1 - \mathbf{e}^{-\varepsilon_{j}}t} \right). \end{align} $$

Also, since

$$ \begin{align*} \sum_{l = 0}^{d} e_{l}(x_{1}, \ldots, x_{d}) t^{l} = \prod_{i = 1}^{d} (1 + x_{i}t), \end{align*} $$

it follows that

(5.23) $$ \begin{align} \sum_{l = 0}^{2n} E_{l} t^{d} = \left( \prod_{i = 1}^{n} (1 + \mathbf{e}^{\varepsilon_{i}}t) \right) \left( \prod_{j = 1}^{n} (1 + \mathbf{e}^{-\varepsilon_{j}}t) \right). \end{align} $$

By multiplying (5.23) with the equation obtained from (5.22) by replacing t with $-t$ , we deduce that

(5.24) $$ \begin{align} \begin{aligned} & \left( \sum_{l = 0}^{\infty} (-1)^{l} (H_{l}^{k+1} - H_{l-2}^{k+1}) t^{l} \right) \left( \sum_{m = 0}^{2n} E_{m} t^{m} \right) \\ & = (1-t^{2}) \left( \prod_{i = k+2}^{n} (1 + \mathbf{e}^{\varepsilon_{i}}t) \right) \left( \prod_{j = k+2}^{n} (1 + \mathbf{e}^{-\varepsilon_{j}}t) \right). \end{aligned} \end{align} $$

The LHS of (5.24) can be rewritten as:

(5.25) $$ \begin{align} & \left( \sum_{l = 0}^{\infty} (-1)^{l} (H_{l}^{k+1} - H_{l-2}^{k+1}) t^{l} \right) \left( \sum_{m = 0}^{2n} E_{m} t^{m} \right) \nonumber \\ &= \sum_{l = 0}^{\infty} \left( \sum_{d = 0}^{l} (-1)^{l-d} (H_{l-d}^{k+1} - H_{l-d-2}^{k+1}) E_{d} \right) t^{l}. \end{align} $$

By setting $e_{m}(x_{1}, \ldots , x_{d}) = 0$ if $m < 0$ or $m> d$ , the RHS of (5.24) can be rewritten as:

(5.26) $$ \begin{align} & (1-t^{2}) \left( \prod_{i = k+2}^{n} (1 + \mathbf{e}^{\varepsilon_{i}}t) \right) \left( \prod_{j = k+2}^{n} (1 + \mathbf{e}^{-\varepsilon_{j}}t) \right) \nonumber \\ &= (1-t^{2}) \sum_{m = 0}^{2(n-k-1)} e_{m}(\mathbf{e}^{\varepsilon_{k+2}}, \ldots, \mathbf{e}^{\varepsilon_{n}}, \mathbf{e}^{-\varepsilon_{n}}, \ldots, \mathbf{e}^{-\varepsilon_{k+2}}) t^{m} \nonumber \\ &= \sum_{m = 0}^{2(n-k-1)} (e_{m}(\mathbf{e}^{\varepsilon_{k+2}}, \ldots, \mathbf{e}^{\varepsilon_{n}}, \mathbf{e}^{-\varepsilon_{n}}, \ldots, \mathbf{e}^{-\varepsilon_{k+2}}) - e_{m-2}(\mathbf{e}^{\varepsilon_{k+2}}, \ldots, \mathbf{e}^{\varepsilon_{n}}, \mathbf{e}^{-\varepsilon_{n}}, \ldots, \mathbf{e}^{-\varepsilon_{k+2}})) t^{m}. \end{align} $$

Therefore, by comparing the coefficients of $t^{n-k}$ in (5.25) and (5.26), we see that

$$ \begin{align*} & \sum_{d = 0}^{n-k} (-1)^{n-k-d} (H_{n-k-d}^{k+1} - H_{n-k-d-2}^{k+1}) E_{d} \\ &= e_{n-k}(\mathbf{e}^{\varepsilon_{k+2}}, \ldots, \mathbf{e}^{\varepsilon_{n}}, \mathbf{e}^{-\varepsilon_{n}}, \ldots, \mathbf{e}^{-\varepsilon_{k+2}}) - e_{n-k-2}(\mathbf{e}^{\varepsilon_{k+2}}, \ldots, \mathbf{e}^{\varepsilon_{n}}, \mathbf{e}^{-\varepsilon_{n}}, \ldots, \mathbf{e}^{-\varepsilon_{k+2}}) \\ &= e_{(n-k-1)+1}(\underbrace{\mathbf{e}^{\varepsilon_{k+2}}, \ldots, \mathbf{e}^{\varepsilon_{n}}, \mathbf{e}^{-\varepsilon_{n}}, \ldots, \mathbf{e}^{-\varepsilon_{k+2}}}_{2(n-k-1) \text{ variables}}) - e_{(n-k-1)-1}(\mathbf{e}^{\varepsilon_{k+2}}, \ldots, \mathbf{e}^{\varepsilon_{n}}, \mathbf{e}^{-\varepsilon_{n}}, \ldots, \mathbf{e}^{-\varepsilon_{k+2}}) \\ &= 0 \quad \text{by (3.2)}. \end{align*} $$

Thus, we conclude that

$$ \begin{align*} \sum_{l = 0}^{n-k} (-1)^{l} (H_{n-l-k}^{k+1} - H_{n-l-k-2}^{k+1}) E_{l} = (-1)^{n-k} \sum_{l = 0}^{n-k} (-1)^{n-k-l} (H_{n-l-k}^{k+1} - H_{n-l-k-2}^{k+1}) E_{l} = 0. \end{align*} $$

This proves the Lemma.

6.1 Some relations in $QK_{T}(G/B)$

We will derive some relations in $QK_{T}(G/B)$ from the corresponding ones in $K_{T}(\mathbf {Q}_{G})$ , given in Corollary 5.15.

In the following, ${\prod }^{\star }$ denotes the product with respect to the quantum product $\star $ .

Recall the notation $\Phi $ from Theorem 4.1.

Proof. We prove only (2) since (1) can be proved by the same argument as (2). In this proof, we thought of $1/(1-\mathsf {t}_{i})$ for $1 \le i \le n$ as the infinite sum $\sum _{d = 0}^{\infty } \mathsf {t}_{i}^{d}$ ; this infinite sum is a well-defined operator on $K_{T}(\mathbf {Q}_{G})$ . We define $\varphi _{J}^{\frac {\infty }{2}}(j)$ for $J \subset [1, \overline {1}]$ and $j \in [1, \overline {1}]$ as follows.

1. (1) For $1 \le j \le n$ , we set

   $$ \begin{align*} \varphi_{J}^{\frac{\infty}{2}}(j) := \begin{cases} \dfrac{1}{1-\mathsf{t}_{j}} & \text{if } j, j+1 \in J, \\ 1 & \text{otherwise}. \end{cases} \end{align*} $$

2. (2) For $2 \le j \le n$ , we set

   $$ \begin{align*} \varphi_{J}^{\frac{\infty}{2}}(\overline{j}) := \begin{cases} 1 + \dfrac{\mathsf{t}_{j-1} \mathsf{t}_{j} \cdots \mathsf{t}_{n}}{1-\mathsf{t}_{j-1}} & \text{if } J = \{ \cdots < j-1 < \overline{j-1} < \cdots\}, \\ \dfrac{1}{1-\mathsf{t}_{j-1}} & \text{if } \overline{j}, \overline{j-1} \in J, \\ 1 & \text{otherwise}. \end{cases} \end{align*} $$

3. (3) We set $\varphi _{J}^{\frac {\infty }{2}}(\overline {1}) := 1$ .

Note that $\varphi _{J}^{\frac {\infty }{2}}$ is obtained from $\varphi _{J}^{Q}$ by replacing $Q_{i}$ with $\mathsf {t}_{i}$ for $1 \leq i \leq n$ .

By Proposition 4.2, we see that

$$ \begin{align*} \Phi(\mathcal{F}_{l}^{\overline{k}}) &= \sum_{\substack{J \subset [1, \overline{k+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \varphi_{J}^{\frac{\infty}{2}}(j) \right) \left( \prod_{\substack{1 \le j \le n \\ j \in J}} (1 - \mathsf{t}_{j-1}) \right) \left( \prod_{\substack{1 \le j \le n \\ \overline{j} \in J}} (1-\mathsf{t}_{j}) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})] \\ &= \sum_{\substack{J \subset [1, \overline{k+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \varphi_{J}^{\frac{\infty}{2}}(j) \right) \left( \prod_{\substack{0 \le j \le n-1 \\ j+1 \in J}} (1 - \mathsf{t}_{j}) \right) \left( \prod_{\substack{2 \le j \le n+1 \\ \overline{j-1} \in J}} (1-\mathsf{t}_{j-1}) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})]. \end{align*} $$

Let us take and fix $J \subset [1, \overline {1}]$ . For $1 \le j \le n-1$ , we set

$$ \begin{align*} \theta_{J}(j) := \begin{cases} 1 - \mathsf{t}_{j} & \text{if } j+1 \in J, \\ 1 & \text{otherwise.} \end{cases} \end{align*} $$

Then, for $1 \le j \le n-1$ , it follows that

$$ \begin{align*} \varphi_{J}^{\frac{\infty}{2}}(j) \theta_{J}(j) &= \begin{cases} \dfrac{1}{1-\mathsf{t}_{j}} \cdot (1-\mathsf{t}_{j}) & \text{if } j \in J \text{ and } j+1 \in J, \\ 1 \cdot 1 & \text{if } j \in J \text{ and } j+1 \notin J, \\ 1 \cdot (1-\mathsf{t}_{j}) & \text{if } j \notin J \text{ and } j+1 \in J, \\ 1 \cdot 1 & \text{if } j \notin J \text{ and } j+1 \notin J \end{cases} \\ &= \begin{cases} 1 - \mathsf{t}_{j} & \text{if } j \notin J \text{ and } j+1 \in J, \\ 1 & \text{otherwise} \end{cases} \\ &= \psi_{J}(j). \end{align*} $$

In addition, we set

$$ \begin{align*} \theta_{J}(n) := \begin{cases} 1 - \mathsf{t}_{n} & \text{if } \overline{n} \in J, \\ 1 & \text{otherwise}. \end{cases} \end{align*} $$

Then it follows that

$$ \begin{align*} \varphi_{J}^{\frac{\infty}{2}}(n) \theta_{J}(n) &= \begin{cases} \dfrac{1}{1-\mathsf{t}_{n}} \cdot (1-\mathsf{t}_{n}) & \text{if } n \in J \text{ and } \overline{n} \in J, \\ 1 \cdot 1 & \text{if } n \in J \text{ and } \overline{n} \notin J, \\ 1 \cdot (1-\mathsf{t}_{n}) & \text{if } n \notin J \text{ and } \overline{n} \in J, \\ 1 \cdot 1 & \text{if } n \notin J \text{ and } \overline{n} \notin J \end{cases} \\ &= \begin{cases} 1-\mathsf{t}_{n} & \text{if } n \notin J \text{ and } \overline{n} \in J, \\ 1 & \text{otherwise} \end{cases} \\ &= \psi_{J}(n). \end{align*} $$

Also, for $2 \le j \le n$ , we set

$$ \begin{align*} \theta_{J}(\overline{j}) := \begin{cases} 1 - \mathsf{t}_{j-1} & \text{if } \overline{j-1} \in J, \\ 1 & \text{otherwise.} \end{cases} \end{align*} $$

Then, for $2 \le j \le n$ , it follows that

$$ \begin{align*} \varphi_{J}^{\frac{\infty}{2}}(\overline{j}) \theta_{J}(\overline{j}) &= \begin{cases} \left( 1 + \dfrac{\mathsf{t}_{j-1} \mathsf{t}_{j} \cdots \mathsf{t}_{n}}{1 - \mathsf{t}_{j-1}} \right) \cdot (1 - \mathsf{t}_{j-1}) & \text{if } J = \{ \cdots < j-1 < \overline{j-1} < \cdots \}, \\ 1 \cdot (1-\mathsf{t}_{j-1}) & \text{if } \overline{j} \notin J, \overline{j-1} \in J, \text{ and } J \text{ is not of the above form,} \\ 1 \cdot 1 & \text{if } \overline{j} \notin J \text{ and } \overline{j-1} \notin J, \\ \dfrac{1}{1-\mathsf{t}_{j-1}} \cdot (1-\mathsf{t}_{j-1}) & \text{if } \overline{j} \in J \text{ and } \overline{j-1} \in J, \\ 1 \cdot 1 & \text{if } \overline{j} \in J \text{ and } \overline{j-1} \notin J \end{cases} \\ &= \begin{cases} 1 - \mathsf{t}_{j-1} + \mathsf{t}_{j-1}\mathsf{t}_{j} \cdots \mathsf{t}_{n} & \text{if } J = \{ \cdots < j-1 < \overline{j-1} < \cdots \}, \\ 1 - \mathsf{t}_{j-1} & \text{if } \overline{j} \notin J, \overline{j-1} \in J, \text{ and } J \text{ is not of the above form,} \\ 1 & \text{otherwise} \end{cases} \\ &= \psi_{J}(\overline{j}). \end{align*} $$

Therefore, by setting $\theta _{J}(\overline {1}) := 1$ , we conclude that

$$ \begin{align*} \Phi(\mathcal{F}_{l}^{\overline{k}}) &= \sum_{\substack{J \subset [1, \overline{k+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \varphi_{J}^{\frac{\infty}{2}}(j) \theta_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})] \\ &= \sum_{\substack{J \subset [1, \overline{1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \psi_{J}(j) \right) [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{J})] \\ &= \mathfrak{F}_{l}^{\overline{k}}, \end{align*} $$

as desired. This proves the proposition.

By Proposition 6.3, we obtain the following relations in $QK_{T}(G/B)$ ; recall the notation $E_{l}$ for $0 \le l \le 2n$ from (3.1).

Proof. By Proposition 6.3 and Corollary 5.15, we have

$$ \begin{align*} \Phi(\mathcal{F}_{l}-E_{l}) = \mathfrak{F}_{l}-E_{l} = 0; \end{align*} $$

note that $\Phi (E_{l}) = E_{l}$ , $0 \le l \le 2n$ , because we have

$$ \begin{align*} e_{l}(\mathbf{e}^{-\varepsilon_{1}}, \ldots, \mathbf{e}^{-\varepsilon_{n}}, \mathbf{e}^{\varepsilon_{n}}, \ldots, \mathbf{e}^{\varepsilon_{1}}) = e_{l}(\mathbf{e}^{\varepsilon_{1}}, \ldots, \mathbf{e}^{\varepsilon_{n}}, \mathbf{e}^{-\varepsilon_{n}}, \ldots, \mathbf{e}^{-\varepsilon_{1}}). \end{align*} $$

Since $\Phi $ is injective, we conclude that $\mathcal {F}_{l} = E_{l}$ , as desired. This proves the corollary.

Also, by combining Proposition 6.3 with Proposition 5.6, we obtain an explicit expression, in terms of line bundle classes, of the Schubert classes $[\mathcal {O}^{s_{1}s_{2} \cdots s_{k}}]$ and $[\mathcal {O}^{s_{1}s_{2} \cdots s_{n}s_{n-1} \cdots s_{k}}]$ for $1 \le k \le n$ .

6.2 The existence of a homomorphism

It follows from Corollary 6.4 that there exists a homomorphism $(R(T)[\![ Q ]\!])[z_{1}^{\pm 1}, \ldots , z_{n}^{\pm 1}] \rightarrow QK_{T}(G/B)$ of $R(T)[\![ Q ]\!]$ -algebras, which eventually gives a Borel-type presentation of $QK_{T}(G/B)$ .

Definition 6.6. We define a homomorphism $\widehat {\Psi }^{Q}: (R(T)[\![ Q ]\!])[z_{1}^{\pm 1}, \ldots , z_{n}^{\pm 1}] \rightarrow QK_{T}(G/B)$ of $R(T)[\![ Q ]\!]$ -algebras by:

Let us compute the images $\widehat {\Psi }^{Q}(z_{j}^{-1})$ , $1 \le j \le n$ . By convention, we set $Q_{0} := 0$ .

Lemma 6.7. For $1 \le j \le n$ , the line bundle class $[\mathcal {O}_{G/B}(-\varepsilon _{j})] \in QK_{T}(G/B)$ is invertible with respect to the quantum product $\star $ in $QK_{T}(G/B)$ , and its inverse is given as:

$$ \begin{align*} [\mathcal{O}_{G/B}(-\varepsilon_{j})]^{-1} = \frac{1}{(1-Q_{j})(1-Q_{j-1})} [\mathcal{O}_{G/B}(\varepsilon_{j})]. \end{align*} $$

Proof. We set

$$ \begin{align*} \mathcal{L} &:= \frac{1}{(1-Q_{j})(1-Q_{j-1})} [\mathcal{O}_{G/B}(\varepsilon_{j})] \star [\mathcal{O}_{G/B}(-\varepsilon_{j})] \\ &= \left( \frac{1}{1-Q_{j}} [\mathcal{O}_{G/B}(\varepsilon_{j})] \right) \star \left( \frac{1}{1-Q_{j-1}} [\mathcal{O}_{G/B}(-\varepsilon_{j})] \right). \end{align*} $$

It suffices to show that $\mathcal {L} = 1$ . By using the map $\Phi : QK_{T}(G/B) \xrightarrow {\sim } K_{T}(\mathbf {Q}_{G})$ (see Theorem 4.1), we see that

$$ \begin{align*} \Phi(\mathcal{L}) &= \Phi \left( \left( \frac{1}{1-Q_{j}} [\mathcal{O}_{G/B}(\varepsilon_{j})] \right) \star \left( \frac{1}{1-Q_{j-1}} [\mathcal{O}_{G/B}(-\varepsilon_{j})] \right) \right) \\ &= [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{j})] \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{j})] \quad \text{(by Proposition 4.2)} \\ &= [\mathcal{O}_{\mathbf{Q}_{G}}(\varepsilon_{j})] \otimes [\mathcal{O}_{\mathbf{Q}_{G}}(-\varepsilon_{j})] \\ &= [\mathcal{O}_{\mathbf{Q}_{G}}]. \end{align*} $$

Since $\Phi (1) = [\mathcal {O}_{\mathbf {Q}_{G}}]$ , we conclude that $\mathcal {L} = 1$ , as desired. This proves the lemma.

Corollary 6.8. For $1 \le j \le n$ , we have

$$ \begin{align*} \widehat{\Psi}^{Q}(z_{j}^{-1}) = \frac{1}{1-Q_{j-1}} [\mathcal{O}_{G/B}(\varepsilon_{j})]. \end{align*} $$

Proof. We compute as:

$$ \begin{align*} \widehat{\Psi}^{Q}(z_{j}^{-1}) &= \widehat{\Psi}^{Q}(z_{j})^{-1} \\ &= \left( \frac{1}{1-Q_{j}} [\mathcal{O}_{G/B}(-\varepsilon_{j})] \right)^{-1} \\ &= (1-Q_{j}) \cdot [\mathcal{O}_{G/B}(-\varepsilon_{j})]^{-1} \\ &= (1-Q_{j}) \cdot \frac{1}{(1-Q_{j})(1-Q_{j-1})} [\mathcal{O}_{G/B}(\varepsilon_{j})] \quad \text{(by Lemma 6.7)} \\ &= \frac{1}{1-Q_{j-1}} [\mathcal{O}_{G/B}(\varepsilon_{j})]. \end{align*} $$

This proves the corollary.

Recall that $z_{\overline {j}} = z_{j}^{-1}$ for $1 \le j \le n$ .

Note that if $k = 0$ , then $F_{l}^{\overline {0}} = F_{l}$ for $0 \le l \le 2n$ .

Proof. We prove only (2) since (1) can be proved by the same argument as (2). By Corollary 6.8, we see that

$$ \begin{align*} & \widehat{\Psi}^{Q}(F_{l}^{\overline{k}}) \\ &= \sum_{\substack{J \subset [1, \overline{k+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \zeta_{J}(j) \right) \left( {\prod_{\substack{1 \le j \le n \\ j \in J}}}^{\star} \frac{1}{1-Q_{j}} [\mathcal{O}_{G/B}(-\varepsilon_{j})] \right) \left( {\prod_{\substack{1 \le j \le n \\ \overline{j} \in J}}}^{\star} \frac{1}{1-Q_{j-1}}[\mathcal{O}_{G/B}(\varepsilon_{j})] \right) \\ &= \sum_{\substack{J \subset [1, \overline{k+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \zeta_{J}(j) \right) \left( \prod_{\substack{1 \le j \le n \\ j \in J}} \frac{1}{1-Q_{j}} \right) \left( \prod_{\substack{1 \le j \le n \\ \overline{j} \in J}} \frac{1}{1-Q_{j-1}} \right) \left( {\prod_{j \in J}}^{\star} [\mathcal{O}_{G/B}(-\varepsilon_{j})] \right). \end{align*} $$

Let us take and fix $J \subset [1, \overline {1}]$ . For $1 \le j \le n$ , we set

$$ \begin{align*} \eta_{J}(j) := \begin{cases} \dfrac{1}{1-Q_{j}} & \text{if } j \in J, \\ 1 & \text{otherwise.} \end{cases} \end{align*} $$

Then it follows that

$$ \begin{align*} \zeta_{J}(j) \eta_{J}(j) &= \begin{cases} 1 \cdot \dfrac{1}{1-Q_{j}} & \text{if } j \in J \text{ and } j+1 \in J, \\ (1-Q_{j}) \cdot \dfrac{1}{1-Q_{j}} & \text{if } j \in J \text{ and } j+1 \notin J, \\ 1 \cdot 1 & \text{if } j \notin J \end{cases} \\ &= \begin{cases} \dfrac{1}{1-Q_{j}} & \text{if } j \in J \text{ and } j+1 \in J, \\ 1 & \text{othewise} \end{cases} \\ &= \varphi_{J}^{Q}(j). \end{align*} $$

Also, for $2 \le j \le n$ , we set

$$ \begin{align*} \eta_{J}(\overline{j}) := \begin{cases} \dfrac{1}{1-Q_{j-1}} & \text{if } \overline{j} \in J, \\ 1 & \text{otherwise.} \end{cases} \end{align*} $$

Then it follows that

$$ \begin{align*} \zeta_{J}(\overline{j}) \eta_{J}(\overline{j}) &= \begin{cases} \left( 1 + \dfrac{Q_{j-1} Q_{j} \cdots Q_{n}}{1 - Q_{j-1}} \right) \cdot 1 & \text{if } J = \{ \cdots < j-1 < \overline{j-1} < \cdots \}, \\ 1 \cdot 1 & \text{if } \overline{j} \notin J \text{ and } J \text{ is not of the above form,} \\ 1 \cdot \dfrac{1}{1-Q_{j-1}} & \text{if } \overline{j} \in J \text{ and } \overline{j-1} \in J, \\ (1-Q_{j-1}) \cdot \dfrac{1}{1-Q_{j-1}} & \text{if } \overline{j} \in J \text{ and } \overline{j-1} \notin J \end{cases} \\ &= \begin{cases} 1 + \dfrac{Q_{j-1} Q_{j} \cdots Q_{n}}{1 - Q_{j-1}} & \text{if } J = \{ \cdots < j-1 < \overline{j-1} < \cdots \}, \\ \dfrac{1}{1-Q_{j-1}} & \text{if } \overline{j} \in J \text{ and } \overline{j-1} \in J, \\ 1 & \text{otherwise} \end{cases} \\ &= \varphi_{J}^{Q}(\overline{j}). \end{align*} $$

From these, we conclude that

$$ \begin{align*} \widehat{\Psi}^{Q}(F_{l}^{\overline{k}}) &= \sum_{\substack{J \subset [1, \overline{k+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \zeta_{J}(j) \eta_{J}(j) \right) \left( {\prod_{j \in J}}^{\star} [\mathcal{O}_{G/B}(-\varepsilon_{j})] \right) \\ &= \sum_{\substack{J \subset [1, \overline{k+1}] \\ |J| = l}} \left( \prod_{1 \le j \le \overline{1}} \varphi_{J}^{Q}(j) \right) \left( {\prod_{j \in J}}^{\star} [\mathcal{O}_{G/B}(-\varepsilon_{j})] \right) \\ &= \mathcal{F}_{l}^{\overline{k}}, \end{align*} $$

as desired. This proves the lemma.

By Corollary 6.4, we obtain the following.

Hence, the map $\widehat {\Psi }^{Q}$ induces the $R(T)[\![ Q ]\!]$ -algebra homomorphism (not yet proved to be an isomorphism) $\Psi ^{Q}$ , given by (3.3).

Also, by combining Lemma 6.10 with Corollary 6.5, we obtain an explicit expression, in terms of line bundle classes, of the Schubert classes $[\mathcal {O}^{s_{1} \cdots s_{k}}]$ and $[\mathcal {O}^{s_{1} \cdots s_{n} s_{n-1} \cdots s_{k}}]$ for $1 \le k \le n$ .

6.3 Finishing the proof of Theorem 3.6

Let us complete the proof of Theorem 3.6. It remains to prove that the $R(T)[\![ Q ]\!]$ -algebra homomorphism $\Psi ^{Q}$ , given by (3.3), is an isomorphism. For this, we make use of the following lemma, which follows from Nakayama’s lemma.

We apply this lemma to the case

$$ \begin{align*} \qquad\qquad\qquad& R = R(T)[\![ Q ]\!], \quad I = (Q_{1}, \ldots, Q_{n}), \\ & M = (R(T)[\![ Q ]\!])[z_{1}^{\pm 1}, \ldots, z_{n}^{\pm 1}]/\mathcal{I}^{Q}, \quad N = QK_{T}(G/B), \quad \text{and} \quad f = \Psi^{Q}; \end{align*} $$

note that R is I-adic complete. Recall that N is a free R-module of rank $|W| (< \infty )$ since it admits the Schubert basis $\{[\mathcal {O}_{G/B}^{w}] \mid w \in W\}$ . In addition, we see that $N/IN \simeq K_{T}(G/B)$ is also a free $R/I \simeq R(T)$ -module of rank $|W|$ since it also admits the Schubert basis. Hence, it remains to verify the following:

1. (i) M is a finitely generated R-module.

2. (ii) The induced $R(T)$ -algebra homomorphism $\overline {\Psi ^{Q}}: M/IM \rightarrow N/IN$ is an isomorphism.

Let $\mathcal {I}$ be the ideal of $R(T)[z_{1}^{\pm 1}, \ldots , z_{n}^{\pm 1}]$ generated by:

$$ \begin{align*} e_{l}(z_{1}, \ldots, z_{n}, z_{n}^{-1}, \ldots, z_{1}^{-1}) - E_{l} \quad \text{for } 1 \le l \le n. \end{align*} $$

Then, by Remark 3.4, we see that

(6.1) $$ \begin{align} M/IM \simeq R(T)[z_{1}^{\pm 1}, \ldots, z_{n}^{\pm 1}]/\mathcal{I}. \end{align} $$

It is well-known (see, for example, [Reference Pittie and Ram24]) that there exists an isomorphism

(6.2)

of $R(T)$ -algebras; this is just the classical Borel presentation. Now part (2) follows from (6.2). For part (1), we can apply the following lemma.

Thus, by Lemma 6.13, the proof of Theorem 3.6 is completed.