Inverse K-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

Abstract

We prove an explicit inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb {Z}\left [q^{\pm 1}\right ]$ -linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars $e^{\lambda }$ , where $\lambda $ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type $E_8$ . The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetric q-Toda operators for minuscule weights in ADE type.

1. Introduction

Let $\mathbf {Q}_{G}^{\mathrm {rat}}$ be the semi-infinite flag manifold. This is a reduced ind-scheme whose set of $\mathbb {C}$ -valued points is $G(\mathbb {C}\left (\!\left (z\right )\!\right ))/(H(\mathbb {C}) \cdot N(\mathbb {C}\left (\!\left (z\right )\!\right )))$ (see [11] for details), where G is a simply connected simple algebraic group over $\mathbb {C}$ , $B=HN\subset G$ is a Borel subgroup, H is a maximal torus and N is the unipotent radical of B. For each affine Weyl group element $x\in W_{\mathrm {af}} = W \ltimes Q^{\vee }$ , with $W = \langle s_i \mid i \in I \rangle $ the (finite) Weyl group and $Q^{\vee } = \bigoplus _{i \in I} \mathbb {Z} \alpha _i^{\vee }$ the coroot lattice of G, one has a semi-infinite Schubert variety $\mathbf {Q}_{G}(x)\subset \mathbf {Q}_{G}^{\mathrm {rat}}$ , which is infinite-dimensional and is given as an orbit closure for the Iwahori subgroup $\mathbf {I}\subset G(\mathbb {C}[\![z]\!])$ . We distinguish the semi-infinite Schubert variety $\mathbf {Q}_{G} := \mathbf {Q}_{G}(e) \subset \mathbf {Q}_{G}^{\mathrm {rat}}$ associated to the identity element e of the affine Weyl group, and also call it the semi-infinite flag manifold.

Our main object of study is the equivariant K-group $K_{H \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ –and that of $\mathbf {Q}_{G}^{\mathrm {rat}}$ , denoted by $K_{H \times \mathbb {C}^{\ast }}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ –which is a variant of the K-group $K_{\tilde {\mathbf {I}}}^{\prime }(\mathbf {Q}_{G})$ introduced recently in [13]. Our K-group is a module over the equivariant scalar ring $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ , where P is the weight lattice of G, $\mathbb {Z}[P]=\mathbb {Z}[e^{\mu } : \mu \in P]$ is the character ring of H and $q\in R(\mathbb {C}^*)$ is the character of loop rotation. Therefore, the K-group $K_{H \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ is a $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ -submodule of an extension of scalars of the equivariant K-group $K_{\tilde {\mathbf {I}}}^{\prime }(\mathbf {Q}_{G})$ of [13] with respect to the Iwahori subgroup $\mathbf {I}$ and loop rotation.

A fundamental result of [13] is the combinatorial Chevalley formula for dominant weights in the K-group $K_{\tilde {\mathbf {I}}}^{\prime }(\mathbf {Q}_{G})$ (and hence in $K_{H \times \mathbb {C}^{\ast }}(\mathbf {Q}_{G})$ ). This formula describes, in terms of semi-infinite Lakshmibai–Seshadri paths, the tensor product of the class of the line bundle $\left [\mathcal {O}_{\mathbf {Q}_{G}}(\lambda )\right ]$ associated to a dominant weight $\lambda \in P^+$ with the class of the structure sheaf $\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}\right ]$ of a semi-infinite Schubert variety $\mathbf {Q}_{G}(x) \subset \mathbf {Q}_{G}$ for $x = w t_{\xi } \in W_{\mathrm {af}}^{\geq 0} := W \times Q^{\vee ,+} \subset W_{\mathrm {af}}$ , where $Q^{\vee ,+} := \sum _{i \in I} \mathbb {Z}_{\geq 0} \alpha _{i}^{\vee } \subset Q^{\vee }$ . This was followed up in [24] by another combinatorial Chevalley formula in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ , giving the tensor product of a Schubert class with an antidominant line bundle. The two Chevalley formulas–dominant [13] and antidominant [24]–were unified in [19], giving the general Chevalley formula in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ for arbitrary weights $\lambda \in P$ .

The Chevalley formulas of [13, 24, 19] thus provide the complete analogue for semi-infinite flag manifolds of their previously well-understood K-theory counterparts for the standard Kac–Moody flag varieties [26, 22, 7, 20, 21]. In all such formulas, the objective is to expand the tensor product of a Schubert class with an equivariant line bundle, as a linear combination of Schubert classes with equivariant scalar coefficients. In the case of $\mathbf {Q}_{G}$ , by [19], this takes the form

(1.1) $$ \begin{align} \left[\mathcal{O}_{\mathbf{Q}_{G}(x)}(\lambda)\right] = \sum_{\substack{y\in W_{\mathrm{af}}^{\ge 0}\\\mu\in P}} c_{x,y}^{\lambda,\mu} \cdot e^\mu \cdot \left[\mathcal{O}_{\mathbf{Q}_{G}(y)}\right], \end{align} $$

with $x,y\in W_{\mathrm {af}}^{\ge 0}$ , $\lambda ,\mu \in P$ and $c_{x,y}^{\lambda ,\mu }\in \mathbb {Z}\left [q^{\pm 1}\right ]$ . We note that the sum on the right-hand side of equation (1.1), while generally infinite, satisfies a notion of convergence introduced in [13].

In this paper, we shall study the inverse ¹ expansion in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ :

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

for $x\in W_{\mathrm {af}}^{\ge 0}$ and $\lambda \in P$ . In contrast to equation (1.1), the expansion in equation (1.2) is finite, as established in [25] for simply laced G–namely, $d_{x,y}^{\lambda ,\mu }\in \mathbb {Z}\left [q^{\pm 1}\right ]$ for any $\lambda ,\mu \in P$ and $x,y\in W_{\mathrm {af}}^{\ge 0}$ , and the right-hand side of equation (1.2) is always a finite sum. (These properties are expected to hold for arbitrary G.)

We call any formula for the right-hand side of equation (1.2) an inverse Chevalley formula for $\lambda \in P$ in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ . The analogous expansion for finite-dimensional flag manifolds $G/B$ was studied by Mathieu [23] in the context of filtrations of B-modules (compare [23, p. 239]). In fact, by [10, Proof of Proposition 1.15], one has a surjective map of $\mathbb {Z}[P]$ -modules given by

(1.3) $$ \begin{align} K_{H\times\mathbb{C}^*}(\mathbf{Q}_{G}) &\overset{\mathrm{trunc}}{\longrightarrow} K_H(G/B)\notag\\ q &\longmapsto 1\notag\\ \left[\mathcal{O}_{\mathbf{Q}_{G}(y)}\right] &\longmapsto \begin{cases}\left[\mathcal{O}_{X(y)}\right] &\text{if}\ y\in W,\\0 &\text{otherwise,}\end{cases} \end{align} $$

where $\left [\mathcal {O}_{X(y)}\right ]$ for $y\in W$ denotes the structure sheaf of the Schubert variety

(1.4) $$ \begin{align} X(y)=\overline{Byw_{\circ} B/B}\subset G/B \end{align} $$

and $w_{\circ }$ is the longest element of W. The map $\mathrm {trunc}$ also respects the tensor product by equivariant line bundles. Thus, for $x\in W$ , the image of equation (1.2) under $\mathrm {trunc}$ produces the analogous inverse Chevalley expansion in $K_H(G/B)$ (see also Remark 3.13). In this sense, any inverse Chevalley formula (1.2) must incorporate the necessary ‘corrections’ in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ to the classical inverse Chevalley formulas in $K_H(G/B)$ .

We note that in the case of $K_H(G/B)$ , there is a simple transformation to pass between ordinary and inverse Chevalley formulas (as explained in, e.g., [23, p. 239]). The lack of such a transformation for $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ justifies the independent study of inverse Chevalley formulas in the semi-infinite setting. Moreover, the finiteness of equation (1.2) offers an advantage over equation (1.1).

The purpose of this paper is to prove a completely explicit, combinatorial inverse Chevalley formula in the equivariant K-group $K_{H \times \mathbb {C}^*}(\mathbf {Q}_{G})$ in the case of a simply laced group G and a minuscule weight $\lambda \in P$ . Before stating our results more precisely, let us discuss further motivations for this work.

1.1. Nil-DAHA and q-Heisenberg actions on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$

One approach to understanding equations (1.1) and (1.2) is that they relate two actions of the group algebra $\mathbb {Z}[P]$ on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ , one given by equivariant scalar multiplication (the left-hand side of equation (1.2)) and the other by the tensor product with equivariant line bundles (the left-hand side of equation (1.1)). These actions of $\mathbb {Z}[P]$ extend to that of two distinct algebras on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ : the nil double affine Hecke algebra (nil-DAHA) and a q-Heisenberg algebra.

Multiplication by equivariant scalars extends to a left action of the nil-DAHA $\mathbb {H}_0$ on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ , in a way that is conceptually similar to the action of nil-Hecke algebras on the equivariant K-theory of Kac–Moody flag varieties [8]. On the right, however, instead of a nil-Hecke algebra, one has an action of a q-Heisenberg algebra $\mathfrak {H}$ . This is generated by tensor products with equivariant line bundles $[\mathcal {O}(\lambda )]$ ( $\lambda \in P$ ) and translations $Q^\vee \cong H(\mathbb {C}\left (\!\left (z\right )\!\right ))/H(\mathbb {C}[\![z]\!])$ . The two actions commute, making $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ an $(\mathbb {H}_0,\mathfrak {H})$ -bimodule.

This bimodule structure is a fundamental tool in the study of $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ , as standard methods such as localisation are not available. Furthermore, as explained in [25], it gives a geometric realisation of the nonsymmetric q-Toda system introduced in [4] (see [1, 2, 5] for related results on the usual q-Toda system and [14, 15] for its $(\mathsf {q},\mathsf {t})$ -extension given by Macdonald difference operators in type A). For us, the fact that the $\mathbb {H}_0$ -action on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ includes the operators of multiplication by equivariant scalars is key. We use the limit construction [25] of the $\mathbb {H}_0$ -action on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ to find and prove our inverse Chevalley formula in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ , which is given by Theorem 1.

1.2. Quantum K-theory of $G/B$

In [10, 12], Kato established a $\mathbb {Z}[P]$ -module isomorphism – with $\mathbb {Z}[P]$ acting by equivariant scalars – from the (completed) H-equivariant quantum K-group $QK_{H}(G/B) := K_{H}(G/B) \otimes \mathbb {Z}\left [\!\left [Q^{\vee ,+}\right ]\!\right ]$ of the finite-dimensional flag manifold $G/B$ onto the H-equivariant K-group $K_{H}(\mathbf {Q}_{G})$ obtained by specialising $q=1$ in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ . (Here $\mathbb {Z}\left [\!\left [Q^{\vee ,+}\right ]\!\right ]$ is the ring of formal power series in the (Novikov) variables $Q_{i}$ , $i \in I$ .) Kato’s isomorphism respects Schubert classes and intertwines the quantum multiplication in $QK_{H}(G/B)$ with the tensor product by line bundles in $K_H(\mathbf {Q}_{G})$ . Thus it provides a means to transport formulas from $K_H(\mathbf {Q}_{G})$ to $QK_H(G/B)$ . In the sequel [16] to this paper, we will use Kato’s isomorphism to derive a corresponding inverse Chevalley formula in $QK_{H}(G/B)$ .

1.3. Our results

Let us now explain our results in more detail. Recall that a weight $\lambda \in P$ is called minuscule if $\left \langle \lambda ,\alpha ^\vee \right \rangle \in \{0,\pm 1\}$ for all $\alpha \in \Delta $ . Nonzero minuscule weights exist in all types except $E_8$ , $F_4$ and $G_2$ .

Our results explicitly describe the inverse Chevalley formula in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ for arbitrary minuscule weights $\lambda \in P$ in the case when G is simply laced. By iteration, our formulas completely determine the inverse Chevalley rule for arbitrary weights in ADE type (except in type $E_8$ ). Indeed, a result of Stembridge [27] (as stated in [18, Theorem 2.1]) asserts that in all types except $E_8$ , $F_4$ and $G_2$ , the minuscule weights form a set of generators for the weight lattice. (The full version of Stembridge’s result, which holds in arbitrary type, requires quasi-minuscule weights. We plan to take up the study of our constructions in the case of quasi-minuscule weights elsewhere.)

Any minuscule weight belongs to the Weyl group orbit of a dominant minuscule weight – that is, a minuscule fundamental weight. Suppose $\varpi _k\in P^+$ is a minuscule fundamental weight. Set $J=I\setminus \{k\}$ , and consider the parabolic subgroup $W_J=\left \langle s_j \mid j\in J\right \rangle $ , which is the stabiliser of $\varpi _k$ . Let $W^J$ be the set of minimal coset representatives for $W/W_J$ . Finally, let $\lambda =x\varpi _k\in P$ be an arbitrary minuscule weight, where $x\in W^J$ .

1.3.1. Algebraic formula

For G simply laced, $\lambda $ as before and any $w\in W$ , our first main result gives an algebraic expression for the product $e^\lambda \cdot \left [\mathcal {O}_{\mathbf {Q}_{G}(w)}\right ]\in K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ in terms of the right q-Heisenberg action on $K_{H\times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ . This is given explicitly as a sum over a set $\mathbf {QW}_{\lambda ,w}$ of walks $(w_1,\dotsc ,w_n)$ in the quantum Bruhat graph $\mathrm {QBG}(W)$ [3], beginning at $w_0=w$ and with steps prescribed by a set of positive roots determined by $\lambda $ . (Here n is the length of the minimal representative of the coset $w_{\circ } W_J$ .)

Theorem 1 (= Theorem 3.11)

Assume that G is of $ADE$ type but not of type $E_8$ . For any minuscule weight $\lambda =x\varpi _k\in P$ , where $x\in W^J$ , and any $w\in W$ , we have

(1.5) $$ \begin{align} e^\lambda\cdot\left[\mathcal{O}_{\mathbf{Q}_{G}(w)}\right] = \sum_{\mathbf{w}=(w_1,\dotsc,w_n)\in\mathbf{QW}_{\lambda,w}}\left[\mathcal{O}_{\mathbf{Q}_{G}(w_n)}\right]\cdot \widetilde{g}^+_{\mathbf{w}} X^{-w_{\circ} w_l^{-1}\lambda} \widetilde{g}^-_{\mathbf{w}}, \end{align} $$

where $l=\ell (x)$ and $\widetilde {g}^+_{\mathbf {w}} X^{-w_{\circ } w_l^{-1}\nu } \widetilde {g}^-_{\mathbf {w}}$ is an element of the q-Heisenberg algebra $\mathfrak {H}$ given explicitly by equations (3.63) and (3.64).

Acting by translations from the q-Heisenberg algebra, one immediately obtains a corresponding formula for $e^\lambda \cdot \left [\mathcal {O}_{\mathbf {Q}_{G}\left (wt_\xi \right )}\right ]\in K_{H\times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ , for any $\xi \in Q^\vee $ .

In order to prove Theorem 1, we apply the main result of [25]. The main step in our proof (see Theorem 3.7) is the intricate computation of a limit, as $\mathsf {t}\to 0$ , from the polynomial representation of the double affine Hecke algebra (DAHA). Thus one should also regard Theorem 1 as an explicit determination of all nonsymmetric q-Toda operators, in the sense of [4, 25], for minuscule weights in ADE type.

1.3.2. Combinatorial formula

Our second main result expresses the same product $e^\lambda \cdot \left [\mathcal {O}_{\mathbf {Q}_{G}(w)}\right ]$ combinatorially. To achieve this, we enhance the set of quantum walks $\mathbf {QW}_{\lambda ,w}$ to a set $\widetilde {\mathbf {QW}}_{\lambda ,w}$ of decorated quantum walks. Roughly speaking, a decorated quantum walk $(\mathbf {w},\mathbf {b})\in \widetilde {\mathbf {QW}}_{\lambda ,w}$ consists of a quantum walk $\mathbf {w}=(w_1,\dotsc ,w_n)\in \mathbf {QW}_{\lambda ,w}$ together with a decoration $\mathbf {b}: S(\mathbf {w})\to \{0,1\}$ . The latter is a $\{0,1\}$ -valued function on an explicit subset $S(\mathbf {w})\subset \{t : w_t=w_{t-1}\}$ of the stationary steps in the walk $\mathbf {w}$ . Each $(\mathbf {w},\mathbf {b})\in \widetilde {\mathbf {QW}}_{\lambda ,w}$ carries a sign $(-1)^{(\mathbf {w},\mathbf {b})}\in \{\pm 1\}$ , a weight $\mathrm {wt}(\mathbf {w},\mathbf {b})\in Q^\vee $ and a degree $\deg (\mathbf {w},\mathbf {b})\in \mathbb {Z}$ . For details, see Section 3.4.1.

Our combinatorial inverse Chevalley formula reads as follows:

Theorem 2 (= Theorem 3.14)

Assume that G is of $ADE$ type but not of type $E_8$ . For any minuscule weight $\lambda =x\varpi _k\in P$ , where $x\in W^J$ , and any $w\in W$ , we have

(1.6) $$ \begin{align} &e^{\lambda}\cdot \left[\mathcal{O}_{\mathbf{Q}_{G}(w)}\right]\nonumber\\ &= \sum_{(\mathbf{w},\mathbf{b})\in\widetilde{\mathbf{QW}}_{\lambda,w}}(-1)^{(\mathbf{w},\mathbf{b})}q^{\deg(\mathbf{w},\mathbf{b})}\cdot[\mathcal{O}_{\mathbf{Q}_{G}(w_nt_{-w_\circ(\mathrm{wt}(\mathbf{w},\mathbf{b}))})}(-w_\circ w_l^{-1}\lambda+\mathrm{wt}(\mathbf{w},\mathbf{b}))]. \end{align} $$

Theorem 2 is obtained as an immediate consequence of Theorem 1, by fully expanding the right-hand side of equation (1.5) in the q-Heisenberg algebra. Our combinatorial framework is designed to record the terms in this expansion. We find it satisfying that the DAHA-based limit used to prove equation (1.5) automatically manufactures the combinatorics of decorated quantum walks necessary for both formulas. We mention that equation (1.5) gives the extension to $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ of Lenart’s rule [17, Theorem 3.1], which holds in $K(SL(n+1)/B)$ , for multiplying a Grothendieck polynomial by a variable.

In Appendix A we work out some further details in the type A case, including concrete examples of equation (1.5) in equations (A.1) and (A.2) and the important special case $w=w_\circ $ , $k=1$ in equation (A.7). We also explain, following [25, §5.1], how to obtain the usual q-Toda difference operator by symmetrising Theorem 1.

1.4. The sequel

In [16] we will establish a different, equivalent version of our inverse Chevalley formula (1.6) in terms of paths (instead of walks) in the quantum Bruhat graph. By means of this alternate formula, we will give a separate and logically independent proof of Theorem 2 in type A, based on the Chevalley formulas of [13, 24] and an equivalent set of character identities for Demazure submodules of level $0$ extremal weight modules. Finally, we will use Theorem 2 to derive a corresponding inverse Chevalley formula in the quantum K-ring $QK_H(G/B)$ , by means of Kato’s isomorphism.

2. Basic notation

2.1. Root system

Let G be a simply connected simple algebraic group over $\mathbb {C}$ . As in the introduction, we fix a maximal torus and Borel subgroup $H\subset B\subset G$ . Set $\mathfrak {g} := \mathrm {Lie}(G)$ and $\mathfrak {h} := \mathrm {Lie}(H)$ . We denote by $\left \langle \cdot , \cdot \right \rangle : \mathfrak {h}^{\ast } \times \mathfrak {h} \rightarrow \mathbb {C}$ 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 positive roots (with respect to B) 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 we set $\rho := (1/2) \sum _{\alpha \in \Delta ^{+}} \alpha $ . The root lattice Q and the coroot lattice $Q^{\vee }$ of $\mathfrak {g}$ are $Q := \sum _{i \in I} \mathbb {Z} \alpha _{i}$ and $Q^{\vee } := \sum _{i \in I} \mathbb {Z} \alpha _{i}^{\vee }$ .

For $i \in I$ , let $\varpi _{i} \in \mathfrak {h}^{\ast }$ be the fundamental weight determined by $\left \langle \varpi _{i}, \alpha _{j}^{\vee } \right \rangle = \delta _{i, j}$ for all $j \in I$ , where $\delta _{i, j}$ denotes the Kronecker delta. 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 – that is, the associative algebra generated by formal elements $\left \{ e^{\lambda } \mid \lambda \in P\right \}$ , where the product is defined by $e^{\lambda } e^{\mu } := e^{\lambda + \mu }$ for $\lambda , \mu \in P$ .

A reflection $s_{\alpha } \in GL(\mathfrak {h}^{\ast })$ , $\alpha \in \Delta $ , is defined by $s_{\alpha }(\lambda ) := \lambda - \left \langle \lambda , \alpha ^{\vee } \right \rangle \alpha $ for $\lambda \in \mathfrak {h}^{\ast }$ . We write $s_{i} := s_{\alpha _{i}}$ for $i \in I$ . Then the Weyl group $W := \langle s_{i} \mid i \in I \rangle $ of $\mathfrak {g}$ is the subgroup of $GL(\mathfrak {h}^{\ast })$ generated by $\{ s_{i} \}_{i \in I}$ . We denote by $\ell (w)$ the length of $w\in W$ with respect to $\{ s_{i} \}_{i \in I}$ .

2.2. Quantum Bruhat graph

The quantum Bruhat graph $\mathrm {QBG}(W)$ (compare [3, Definition 6.1]) is the $\Delta ^+$ -labelled directed graph whose vertices are the elements of W and whose edges are of the form $x \xrightarrow {\alpha } y$ , with $x, y \in W$ and $\alpha \in \Delta ^+$ , such that $y = x s_{\alpha }$ and either of the following holds: (B) $\ell (y) = \ell (x) + 1$ or (Q) $\ell (y) = \ell (x) - 2\left \langle \rho , \alpha ^\vee \right \rangle + 1$ . An edge satisfying (B) (resp., (Q)) is called a Bruhat edge (resp., a quantum edge).

2.3. Affine root system

Let $\mathfrak {g}_{\mathrm {af}} := \left (\mathfrak {g} \otimes \mathbb {C}\left [z, z^{-1}\right ]\right ) \oplus \mathbb {C} c \oplus \mathbb {C} d$ be the (untwisted) affine Lie algebra over $\mathbb {C}$ associated to $\mathfrak {g}$ , where c is the canonical central element and d is the degree operator. Then $\mathfrak {h}_{\mathrm {af}} := \mathfrak {h} \oplus \mathbb {C} c \oplus \mathbb {C} d$ is the Cartan subalgebra of $\mathfrak {g}_{\mathrm {af}}$ . We denote by $\left \langle \cdot , \cdot \right \rangle : \mathfrak {h}_{\mathrm {af}}^{\ast } \times \mathfrak {h}_{\mathrm {af}} \rightarrow \mathbb {C}$ the canonical pairing. Regarding $\lambda \in \mathfrak {h}^{\ast }$ as $\lambda \in \mathfrak {h}_{\mathrm {af}}^{\ast } = \mathrm {Hom}_{\mathbb {C}}(\mathfrak {h}_{\mathrm {af}}, \mathbb {C})$ by setting $\left \langle \lambda , c \right \rangle = \left \langle \lambda , d \right \rangle = 0$ , we have $\mathfrak {h}^{\ast } \subset \mathfrak {h}_{\mathrm {af}}^{\ast }$ . In this identification, we see that the canonical pairing $\left \langle \cdot , \cdot \right \rangle $ on $\mathfrak {h}_{\mathrm {af}}^{\ast } \times \mathfrak {h}_{\mathrm {af}}$ extends that on $\mathfrak {h}^{\ast } \times \mathfrak {h}$ .

Let us consider the root system of $\mathfrak {g}_{\mathrm {af}}$ . We define $\delta $ to be the unique element of $\mathfrak {h}_{\mathrm {af}}^{\ast }$ which satisfies $\left \langle \delta , h \right \rangle = 0$ for all $h \in \mathfrak {h}$ , $\left \langle \delta , c \right \rangle = 0$ and $\left \langle \delta , d \right \rangle = 1$ . We set $\alpha _{0} := -\theta + \delta \in \mathfrak {h}_{\mathrm {af}}^{\ast }$ . Then the root system $\Delta _{\mathrm {af}}$ of $\mathfrak {g}_{\mathrm {af}}$ has simple roots $\{ \alpha _{i} \}_{i \in I_{\mathrm {af}}}$ , where $I_{\mathrm {af}} := I \sqcup \{ 0 \}$ .

For each $\alpha \in \Delta _{\mathrm {af}}$ , we have a reflection $s_{\alpha } \in GL(\mathfrak {h}_{\mathrm {af}})$ , defined as for $\mathfrak {g}$ . Note that for $\alpha \in \Delta \subset \Delta _{\mathrm {af}}$ , the restriction of a reflection $s_{\alpha }$ defined on $\mathfrak {h}_{\mathrm {af}}$ to $\mathfrak {h}$ coincides with a reflection $s_{\alpha }$ defined on $\mathfrak {h}$ . Set $s_{i} := s_{\alpha _{i}}$ for $i \in I_{\mathrm {af}}$ . Then the Weyl group $W_{\mathrm {af}}$ of $\mathfrak {g}_{\mathrm {af}}$ (called the affine Weyl group) is defined to be the subgroup of $GL(\mathfrak {h}_{\mathrm {af}})$ generated by $\{ s_{i} \}_{i \in I_{\mathrm {af}}}$ , namely $W_{\mathrm {af}} = \langle s_{i} \mid i \in I_{\mathrm {af}} \rangle $ . In [9, §6.5], it is shown that $W_{\mathrm {af}} \simeq W \ltimes \left \{ t_{\alpha ^{\vee }} \mid \alpha ^{\vee } \in Q^{\vee } \right \} \simeq W \ltimes Q^{\vee }$ , where $t_{\alpha ^{\vee }}$ is the translation element corresponding to $\alpha ^{\vee } \in Q^{\vee }$ .

2.4. Simply laced assumption

While the definitions we have given are completely general, we will in fact assume throughout that G is simply laced. As a result, we almost always identify $\mathfrak {h}^\ast $ with $\mathfrak {h}$ , roots with coroots and so on. We do this by means of the nondegenerate W-invariant symmetric bilinear form $(\cdot ,\cdot ) : \mathfrak {h}^* \times \mathfrak {h}^* \to \mathbb {C}$ , normalised so that $(\alpha ,\alpha )=2$ for all $\alpha \in \Delta $ . We write $\lvert \lambda \rvert ^2=(\lambda ,\lambda )$ for $\lambda \in \mathfrak {h}^*$ .

2.5. Extended affine Weyl group

Let $W_{\mathrm {ex}}=W\ltimes P$ be the extended affine Weyl group, which we let act on $P\oplus \mathbb {Z}\frac {\delta }{e}$ by the level $0$ action

(2.1) $$ \begin{align} wt_\mu (\lambda)=w(\lambda)-(\mu,\lambda)\delta,\qquad wt_\mu(\delta)=\delta. \end{align} $$

Here we choose e to be the smallest positive integer such that $e\cdot (P,P)\subset \mathbb {Z}$ .

We also define the group $\Pi =P/Q$ , which we realise as the subgroup of length $0$ elements in $W_{\mathrm {ex}}$ .

2.6. Parameters

Let us introduce a parameter $q^{1/e}$ such that $\left (q^{1/e}\right )^e=q$ , where $q\in R(\mathbb {C}^*)$ is the equivariant parameter corresponding to loop rotation on $\mathbf {Q}_{G}^{\mathrm {rat}}$ . We will also use a related, but distinct parameter $\mathsf {q}^{1/e}$ , as well as parameters $\mathsf {t},\mathsf {v}$ such that $\mathsf {t}=\mathsf {v}^2$ . Our base field for DAHA constructions will be $\mathsf {K}=\mathbb {Q}\left (\mathsf {q}^{1/e},\mathsf {v}\right )$ .

2.7. Matrices

For any ring R with $1$ , let $\mathrm {Mat}_W(R)$ denote the R-algebra of $W\times W$ matrices with entries in R. Let $\{e_w\}_{w\in W}$ be the standard basis of the free R-module $R^{\lvert W\rvert }$ . For $A\in \mathrm {Mat}_W(R)$ , we denote by $A_{w,\bullet }=\sum _{v\in W}A_{w,v}e_v$ the row of the matrix A indexed by $w\in W$ .

3. Inverse Chevalley formula via DAHA

In this section we use the methods of [25], based on the $(\mathbb {H}_0,\mathfrak {H})$ -bimodule structure of $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ , to prove the inverse Chevalley formulas (1.5) and (1.6).

3.1. K-groups

Let $K_{\tilde {\mathbf {I}}}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ be the equivariant K-group of $\mathbf {Q}_{G}^{\mathrm {rat}}$ introduced in [13, §6], where $\tilde {\mathbf {I}}=\mathbf {I}\rtimes \mathbb {C}^*$ is a semidirect product of the Iwahori subgroup $\mathbf {I}\subset G(\mathbb {C}[\![z]\!])$ and loop rotation $\mathbb {C}^*$ . Correspondingly, $K_{\tilde {\mathbf {I}}}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ is a module over $\mathbb {Z}[P]\left (\!\left (q^{-1}\right )\!\right )$ , which acts by equivariant scalar multiplication.

One has the following classes in $K_{\tilde {\mathbf {I}}}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ , for each $x\in W_{\mathrm {af}}$ and $\lambda \in P$ :

• • Schubert classes $\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}\right ]$

• • Equivariant line bundle classes $[\mathcal {O}(\lambda )]$

• • Classes $\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}(\lambda )\right ]$ corresponding to the tensor product sheaves $\mathcal {O}_{\mathbf {Q}_{G}(x)}\otimes \mathcal {O}(\lambda )$ .

We follow the conventions of [13] for indexing equivariant line bundles and Schubert varieties in $\mathbf {Q}_{G}^{\mathrm {rat}}$ .

Definition 3.1 ( $(H\times \mathbb {C}^*)$ -equivariant K-groups of $\mathbf {Q}_{G}^{\mathrm {rat}}$ and $\mathbf {Q}_{G}$ )

Let $K_{H\times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ be the $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ -submodule of $K_{\tilde {\mathbf {I}}}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ consisting of all convergent infinite $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ -linear combinations of Schubert classes $\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}\right ]$ for $x\in W_{\mathrm {af}}$ , where convergence holds in the sense of [13, Proposition 5.11].

Similarly, we define $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ to be the $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ -submodule of $K_{\tilde {\mathbf {I}}}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ consisting of all convergent infinite $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ -linear combinations of Schubert classes $\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}\right ]$ for $x\in W_{\mathrm {af}}^{\ge 0}$ .

The classes $\left \{\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}\right ]\right \}_{x\in W_{\mathrm {af}}}$ satisfy a notion of topological linear independence in $K_{\tilde {\mathbf {I}}}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ given by [13, Proposition 5.11]; thus they form a topological $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ -basis of $K_{H\times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ . Also, one has $\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}(\lambda )\right ]\in K_{H\times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ for any $x\in W_{\mathrm {af}}$ and $\lambda \in P$ , thanks to the two Chevalley formulas for dominant weights [13] and antidominant weights [24]. Similar (in fact, equivalent) assertions hold for $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ .

Definition 3.2. Define $\mathbb {K}\subset K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ to be the $\mathbb {Z}\left [q^{\pm 1}\right ]$ -submodule consisting of all finite $\mathbb {Z}\left [q^{\pm 1}\right ]$ -linear combinations of the classes $\left \{\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}(\lambda )\right ]\right \}_{x\in W_{\mathrm {af}},\lambda \in P}$ .

By definition, $\mathbb {K}$ is only a $\mathbb {Z}\left [q^{\pm 1}\right ]$ -submodule of $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ . But, as shown in [25, Theorem 5.1], it is indeed a $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ -submodule of $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ . We note that the classes $\left \{\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}(\lambda )\right ]\right \}_{x\in W_{\mathrm {af}},\lambda \in P}$ are linearly independent over $\mathbb {Z}\left [q^{\pm 1}\right ]$ , by the Chevalley formula of [13].

To summarise, we have the following chain of $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ -modules, and $K_{\tilde {\mathbf {I}}}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ is in fact a $\mathbb {Z}[P]\left (\!\left (q^{-1}\right )\!\right )$ -module:

$$ \begin{align*}\mathbb{K}\subset K_{H \times \mathbb{C}^*}\left(\mathbf{Q}_{G}^{\mathrm{rat}}\right)\subset K_{\tilde{\mathbf{I}}}\left(\mathbf{Q}_{G}^{\mathrm{rat}}\right),\end{align*} $$

where $\mathbb {K}$ is equipped with the $\mathbb {Z}\left [q^{\pm 1}\right ]$ -basis $\left \{\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}(\lambda )\right ]\right \}_{x\in W_{\mathrm {af}},\lambda \in P}$ , and $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ with the topological $\mathbb {Z}\left [q^{\pm 1}\right ][P]$ -basis $\left \{\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}\right ]\right \}_{x\in W_{\mathrm {af}}}$ . Next we discuss additional structures on these modules.

3.1.1. q-Heisenberg algebra

Let $\widehat {\mathfrak {H}}$ be the q-Heisenberg algebra defined as a $\mathbb {Z}\left [q^{\pm 1/e}\right ]$ -algebra by generators

(3.1) $$ \begin{align} X^\nu\ (\nu\in P),\quad t_\mu\ (\mu\in P), \end{align} $$

and relations

(3.2) $$ \begin{align} &X^\lambda X^\nu = X^{\lambda+\nu} \end{align} $$

(3.3) $$ \begin{align} &t_\mu t_\xi = t_{\mu+\xi} \end{align} $$

(3.4) $$ \begin{align} &X^0 = t_0 = 1 \end{align} $$

(3.5) $$ \begin{align} &X^\nu t_\mu = q^{\left(\mu,\nu\right)}t_\mu X^\nu \end{align} $$

for all $\lambda ,\mu ,\nu ,\xi \in P$ .

Let $\mathfrak {H}\subset \widehat {\mathfrak {H}}$ be the $\mathbb {Z}\left [q^{\pm 1}\right ]$ -subalgebra generated by $X^\nu $ ( $\nu \in P$ ) and $t_\beta $ ( $\beta \in Q$ ).

Proposition 3.3. There exists a unique right $\mathfrak {H}$ -module structure on $\mathbb {K}$ and $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ such that

(3.6) $$ \begin{align} \left[\mathcal{O}_{\mathbf{Q}_{G}(x)}(\lambda)\right]\cdot X^\nu &= \left[\mathcal{O}_{\mathbf{Q}_{G}(x)}(\lambda+\nu)\right] \end{align} $$

(3.7) $$ \begin{align} \left[\mathcal{O}_{\mathbf{Q}_{G}(x)}(\lambda)\right]\cdot t_\beta &= q^{\left(\beta,\lambda\right)}\cdot\left[\mathcal{O}_{\mathbf{Q}_{G}\left(xt_{-w_\circ\beta}\right)}(\lambda)\right] \end{align} $$

for all $x\in W_{\mathrm {af}}$ , $\lambda ,\nu \in P$ and $\beta \in Q$ . Moreover, $\mathbb {K}$ is free as an $\mathfrak {H}$ -module, with $\mathfrak {H}$ -basis given by $\left \{\left [\mathcal {O}_{\mathbf {Q}_{G}(w)}\right ]\right \}_{w\in W}$ .

Proof. See [10] and [25, Proposition 2.3]. We note that our conventions here differ slightly from those of [25], where the class $\left [\mathcal {O}_{\mathbf {Q}_{G}(x)}\right ]$ is denoted by $\left [\overline {\mathcal {O}}_{x}\right ]$ . This accounts for the twist by $-w_\circ $ in the action of $t_\beta $ .

3.1.2. Nil-DAHA

Next we turn to the nil-DAHA $\mathbb {H}_0$ , which is the $\mathbb {Z}\left [\mathsf {q}^{\pm 1}\right ]$ -algebra (note the notational distinction between $\mathsf {q}$ and q) defined by generators

(3.8) $$ \begin{align} T_i \ (i\in I_{\mathrm{af}}),\quad X^{\nu} \ (\nu\in P), \end{align} $$

and relations

(3.9) $$ \begin{align} &T_i T_j\dotsm = T_j T_i\dotsm \qquad\left(m_{ij}=\left\lvert s_is_j\right\rvert \text{factors on both sides}\right) \end{align} $$

(3.10) $$ \begin{align} &T_i(T_i+1) = 0 \end{align} $$

(3.11) $$ \begin{align} &X^0=1 \end{align} $$

(3.12) $$ \begin{align} &X^{\nu}X^{\mu}=X^{\nu+\mu} \end{align} $$

(3.13) $$ \begin{align} &T_i X^{\nu} = X^{s_i(\nu)}T_i-\frac{X^{\nu}-X^{s_i(\nu)}}{1-X^{\alpha_i}}, \end{align} $$

where it is understood that $X^\delta =\mathsf {q}$ . We also use $D_i=1+T_i$ ( $i\in I_{\mathrm {af}}$ ). These elements satisfy the braid relations as before and quadratic relations $D_i^2=D_i$ .

Proposition 3.4 [13, Theorem 6.5]. There is a left $\mathbb {H}_0$ -action on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ uniquely determined by

(3.14) $$ \begin{align} \mathsf{q}\cdot \left[\mathcal{O}_{\mathbf{Q}_{G}\left(t_\alpha\right)}(\lambda)\right] &= q^{-1}\left[\mathcal{O}_{\mathbf{Q}_{G}\left(t_\alpha\right)}(\lambda)\right] \end{align} $$

(3.15) $$ \begin{align} X^{\nu}\cdot\left[\mathcal{O}_{\mathbf{Q}_{G}\left(t_\alpha\right)}(\lambda)\right] &= e^{-\nu}\left[\mathcal{O}_{\mathbf{Q}_{G}\left(t_\alpha\right)}(\lambda)\right] \end{align} $$

(3.16) $$ \begin{align} D_i\cdot \left[\mathcal{O}_{\mathbf{Q}_{G}\left(t_\alpha\right)}(\lambda)\right] &= \left[\mathcal{O}_{\mathbf{Q}_{G}\left(t_\alpha\right)}(\lambda)\right] \end{align} $$

(3.17) $$ \begin{align} D_0\cdot \left[\mathcal{O}_{\mathbf{Q}_{G}\left(t_\alpha\right)}(\lambda)\right] &= \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_0t_\alpha\right)}(\lambda)\right] \end{align} $$

for all $i\in I$ , $\alpha \in Q$ and $\nu ,\lambda \in P$ .

We warn the reader that $X^\nu $ ( $\nu \in P$ ) can be viewed both as an element of $\mathbb {H}_0$ and as an element of $\mathfrak {H}$ . Its left and right actions on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ differ drastically. From the left, $X^\nu \in \mathbb {H}_0$ acts as equivariant scalar multiplication by $e^{-\nu }$ , whereas from the right, $X^\nu \in \mathfrak {H}$ acts as the line bundle twist by $\mathcal {O}(\nu )$ .

3.1.3. $\mathbb {K}$ as a bimodule

By [25, Theorem 5.1], the $\mathbb {H}_0$ -action on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ leaves $\mathbb {K}$ stable. Thus $\mathbb {K}$ is an $(\mathbb {H}_0,\mathfrak {H})$ -bimodule. Since $\mathbb {K}$ is free as a right $\mathfrak {H}$ -module, the nil-DAHA action on $\mathbb {K}$ and hence on $K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ is therefore characterised by the unique ring homomorphism $\varrho _0 : \mathbb {H}_0 \to \mathrm {Mat}_W(\mathfrak {H})$ satisfying

(3.18) $$ \begin{align} H\cdot \left[\mathcal{O}_{\mathbf{Q}_{G}(w)}\right] = \sum_{v\in W}\left[\mathcal{O}_{\mathbf{Q}_{G}(v)}\right]\cdot \varrho_0(H)_{vw} \end{align} $$

for all $H\in \mathbb {H}_0$ and $w\in W$ .

For us, the key point is that the matrices encoding the inverse Chevalley formula (1.2) are exactly $\varrho _0\left (X^{-\lambda }\right )$ for $\lambda \in P$ .

3.1.4. Further observations

For any $i\in I_{\mathrm {af}}$ and $x\in W_{\mathrm {af}}$ , we have

(3.19) $$ \begin{align} D_i\cdot \left[\mathcal{O}_{\mathbf{Q}_{G}(x)}\right] = \begin{cases} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_ix\right)}\right] &\text{if}\ s_ix\prec x,\\ \left[\mathcal{O}_{\mathbf{Q}_{G}(x)}\right] &\text{if}\ s_ix\succ x,\\ \end{cases} \end{align} $$

where $\prec $ is the semi-infinite Bruhat order on $W_{\mathrm {af}}$ (see [13] or [25]). In the special case where $i\in I$ and $x=w\in W$ , we have

(3.20) $$ \begin{align} D_i\cdot\left[\mathcal{O}_{\mathbf{Q}_{G}(w)}\right]= \begin{cases} \left[\mathcal{O}_{\mathbf{Q}_{G}\left(s_iw\right)}\right] &\text{if}\ s_iw<w,\\ \left[\mathcal{O}_{\mathbf{Q}_{G}(w)}\right] &\text{if}\ s_iw>w, \end{cases} \end{align} $$

where $<$ is the (usual) Bruhat order on W.

It follows from equation (3.20) and Proposition 3.4 (see also [13, Proposition 6.3]) that the nil affine Hecke $\mathbb {Z}\left [\mathsf {q}^{\pm 1}\right ]$ -subalgebra $\mathcal {H}_0\subset \mathbb {H}_0$ generated by

$$ \begin{align*} D_i \ (i\in I),\quad X^{\nu} \ (\nu\in P), \end{align*} $$

leaves $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})\subset K_{H \times \mathbb {C}^*}\big(\mathbf {Q}_{G}^{\mathrm {rat}}\big )$ stable. (Our inverse Chevalley formulas make this explicit.)

We also deduce from equation (3.20) that $\left [\mathcal {O}_{\mathbf {Q}_{G}\left (w_\circ \right )}\right ]$ generates $\mathbb {K}$ as an $(\mathcal {H}_0,\mathfrak {H})$ -bimodule (in fact, the action of $\mathfrak {H}$ and the $D_i$ for $i\in I$ on $\left [\mathcal {O}_{\mathbf {Q}_{G}\left (w_\circ \right )}\right ]$ are sufficient to generate all of $\mathbb {K}$ ).

3.2. DAHA

The main result of [25] gives an algebraic construction of the homomorphism $\varrho _0 : \mathbb {H}_0\to \mathrm {Mat}_W(\mathfrak {H})$ of equation (3.18), as a limit as $\mathsf {t}\to 0$ from the polynomial representation of DAHA. Let us recall the necessary details of this construction.

3.2.1. Definition of DAHA

Recall that $\mathsf {K}=\mathbb {Q}\left (\mathsf {q}^{1/e},\mathsf {v}\right )$ , where $\mathsf {t}=\mathsf {v}^2$ . The (extended) DAHA $\widehat {\mathbb {H}}$ is the $\mathsf {K}$ -algebra defined by generators

(3.21) $$ \begin{align} T_i\ (i\in I_{\mathrm{af}}),\quad X^\nu\ (\nu\in P),\quad \pi\ (\pi\in \Pi), \end{align} $$

and relations

(3.22) $$ \begin{align} &T_i T_j\dotsm = T_j T_i \dotsm \qquad \left(m_{ij}=\left\lvert s_is_j\right\rvert \text{factors on both sides}\right) \end{align} $$

(3.23) $$ \begin{align} &(T_i-\mathsf{t})(T_i+1) = 0 \end{align} $$

(3.24) $$ \begin{align} &X^\nu X^\mu=X^{\nu+\mu} \end{align} $$

(3.25) $$ \begin{align} &X^0=1 \end{align} $$

(3.26) $$ \begin{align} &T_i X^\nu = X^{s_i(\nu)} T_i + (\mathsf{t}-1)\left(1-X^{\alpha_i}\right)^{-1}\left(X^\nu-X^{s_i(\nu)}\right) \end{align} $$

(3.27) $$ \begin{align} &\pi T_i \pi^{-1} = T_j \qquad \left(\pi(\alpha_i)=\alpha_j\right) \end{align} $$

(3.28) $$ \begin{align} &\pi X^\nu \pi^{-1} = X^{\pi(\nu)} \end{align} $$

for all $i,j\in I_{\mathrm {af}}$ , $\nu ,\mu \in P$ and $\pi \in \Pi $ . Here it is understood that $X^{\delta /e}=\mathsf {q}^{1/e}$ .

One has well-defined elements $Y^\nu \in \widehat {\mathbb {H}}$ ( $\nu \in P$ ) given by

(3.29) $$ \begin{align} Y^\nu = \mathsf{v}^{-\sum_{k=1}^\ell \epsilon_k}\pi T_{i_1}^{\epsilon_1}\dotsm T_{i_\ell}^{\epsilon_\ell} \end{align} $$

for any reduced expression $t_\nu = \pi s_{i_1}\dotsm s_{i_\ell }\in P\rtimes W=W_{\mathrm {ex}}$ , where $i_k\in I_{\mathrm {af}}$ and $\pi \in \Pi $ , and where $\epsilon _k\in \{\pm 1\}$ are determined by

(3.30) $$ \begin{align} \epsilon_k = \begin{cases} +1 &\text{if}\ \pi s_{i_1}\dotsm s_{i_{k-1}}\left(\alpha_{i_k}\right)\in-\Delta^++\mathbb{Z}\delta,\\-1 &\text{if}\ \pi s_{i_1}\dotsm s_{i_{k-1}}\left(\alpha_{i_k}\right)\in\Delta^++\mathbb{Z}\delta.\end{cases} \end{align} $$

These elements satisfy

(3.31) $$ \begin{align} &Y^\nu Y^\mu=Y^{\nu+\mu} \end{align} $$

(3.32) $$ \begin{align} &Y^0=1 \end{align} $$

(3.33) $$ \begin{align} &T_i Y^\nu - Y^{s_i(\nu)} T_i = (\mathsf{t}-1)\left(1-Y^{-\alpha_i}\right)^{-1}\left(Y^\nu-Y^{s_i(\nu)}\right) \end{align} $$

for any $i\in I$ and $\nu ,\mu \in P$ . (To extend the latter relation to $i\in I_{\mathrm {af}}$ , see, e.g., [25, §3.2].)

3.2.2. Polynomial representation

Let $\mathsf {K}(P)$ be the field of fractions of $\mathsf {K}[P]=\mathsf {K}[X^\nu : \nu \in P]$ . The extended affine Weyl group $W_{\mathrm {ex}}$ acts on $\mathsf {K}[P]$ by automorphisms as follows: $t_\mu w(X^\nu )=q^{-\left (\mu ,w\nu \right )}X^{w\nu }$ . By a ‘difference-reflection operator’ we mean an element of the smash product

$$ \begin{align*} \mathsf{K}(P)\rtimes W_{\mathrm{ex}}\cong\mathsf{K}(P)\otimes_{\mathsf{K}} \mathsf{K}[W_{\mathrm{ex}}] \quad \text{(}\mathsf{K}\text{-linear isomorphism)} \end{align*} $$

acting from the left on $\mathsf {K}(P)$ by multiplication operators (first tensor factor) and the induced action of $W_{\mathrm {ex}}$ (second tensor factor). We note that $\mathsf {K}[P]\rtimes P$ , where $P\cong \left \{t_\mu : \mu \in P\right \}\subset W_{\mathrm {ex}}$ , is nothing but the q-Heisenberg algebra $\widehat {\mathfrak {H}}$ with scalars extended to $\mathsf {K}$ .

The polynomial representation $\varrho : \widehat {\mathbb {H}}\to \mathrm {End}(\mathsf {K}[P])$ is a faithful representation of $\widehat {\mathbb {H}}$ given as follows: $\varrho (X^\nu )$ is multiplication by $X^\nu $ , the elements $\pi \in \Pi \subset W_{\mathrm {ex}}$ act by the automorphisms already discussed and $T_i$ ( $i\in I_{\mathrm {af}}$ ) act by Demazure–Lusztig operators:

(3.34) $$ \begin{align} \varrho(T_i) &= \frac{\mathsf{t}X^{\alpha_i}-1}{X^{\alpha_i}-1}s_i+\frac{1-\mathsf{t}}{X^{\alpha_i}-1}. \end{align} $$

In general, elements of $\widehat {\mathbb {H}}$ act on $\mathsf {K}[P]$ by difference-reflection operators from $\mathsf {K}(P)\rtimes W_{\mathrm {ex}}$ ; therefore, they are represented uniquely as sums of $ft_\mu u = f\otimes t_\mu u$ , where $f\in \mathsf {K}(P)$ and $t_\mu u\in W_{\mathrm {ex}}$ . Obviously, not all elements of $\mathsf {K}(P)\rtimes W_{\mathrm {ex}}$ leave $\mathsf {K}[P]$ stable, but those from $\widehat {\mathbb {H}}$ do. As in equation (3.34), we will simultaneously regard $\varrho (H)$ for $H\in \widehat {\mathbb {H}}$ as an element of $\mathrm {End}(\mathsf {K}[P])$ and as a difference-reflection operator – that is, an element of $\mathsf {K}(P)\rtimes W_{\mathrm {ex}}$ .

3.2.3. Limiting procedure

Consider the homomorphism

$$ \begin{align*} \varkappa : \mathsf{K}(P)\rtimes W_{\mathrm{ex}} \to \mathrm{Mat}_W(\mathsf{K}(P)\rtimes P), \end{align*} $$

which for $ft_\mu u\in \mathsf {K}(P)\rtimes W_{\mathrm {ex}}$ and $v,w\in W$ is given by

(3.35) $$ \begin{align} \varkappa\left(f t_\mu u\right)_{vw}= \begin{cases} \left.\left(v^{-1}\cdot f\right)\right\rvert_{X^\nu\mapsto \mathsf{t}^{-\left(\rho,\nu\right)}X^\nu}\mathsf{t}^{\left(\rho,v^{-1}\mu\right)}t_{v^{-1}(\mu)} &\text{if}\ v=uw,\\ 0 &\text{otherwise.} \end{cases} \end{align} $$

By [25, Theorem 5.1], the sought-after matrices $\varrho _0\left (X^{-\lambda }\right )$ from equation (3.18) can be obtained by applying a simple automorphism to the result of the entry-wise limit

(3.36) $$ \begin{align} \varrho^{\prime}_0\left(Y^\lambda\right):=\lim_{\mathsf{v}\to 0} \varkappa\left(\varrho\left(Y^{\lambda}\right)\right)\in\mathrm{Mat}_W(\mathfrak{H}). \end{align} $$

The existence of this limit for any $\lambda \in P$ is ensured by [4, Theorem 4.4]. The significance of $Y^{\lambda }$ here is that it is the image of $X^{-\lambda }$ under the DAHA duality anti-involution; see [25, §3.2] for further details.

3.3. Computing limits

Our goal is thus to compute certain $\varrho ^{\prime }_0\left (Y^\lambda \right )$ given by formula (3.36). To do so, we will work with compositions of the auxiliary difference-reflection operators

(3.37) $$ \begin{align} \mathsf{G}_\eta^\pm &= \frac{\mathsf{t}^{\pm 1}-X^{\eta}}{1-X^{\eta}}+\frac{\mathsf{t}^{\pm 1}-1}{1-X^{-\eta}} s_\eta \end{align} $$

for $\eta \in \Delta ^+_{\mathrm {af}}$ . These are cousins of the Demazure–Lusztig operators $\varrho (T_i)$ :

$$ \begin{align*} \mathsf{G}^+_{\alpha_i} &= s_i\varrho(T_i), \qquad \mathsf{G}^-_{\alpha_i} = \left(\mathsf{G}^+_{\alpha_i}\right)^{-1}= \varrho\left(T_i^{-1}\right)s_i. \end{align*} $$

As we wish to take $\mathsf {v}\to 0$ , all rational functions in such operators will be expanded in $\mathbb {Z}\left [\mathsf {q}^{\pm 1/e}\right ][P]\left (\!\left (\mathsf {v}\right )\!\right )$ . In this context, we shall write $A\approx B$ to mean that A and B have the same lowest term – that is, that A and B have the same order with respect to $\mathsf {v}$ and that $A-B$ has strictly greater order than that of A and B.

Set $\eta \in \Delta ^+$ . By equation (3.35), we see that a matrix entry of $\varkappa \left (\mathsf {G}_\eta ^\pm \right )$ vanishes unless it is indexed by $(w,w)$ or $\left (w,s_\eta w\right )$ for some $w\in W$ . The nonvanishing entries are given as follows:

(3.38) $$ \begin{align} \varkappa\left(\mathsf{G}_\eta^+\right)_{w,w} &= \frac{\mathsf{t}-\mathsf{t}^{-\left(\rho,w^{-1}\eta\right)}X^{w^{-1}(\eta)}}{1-\mathsf{t}^{-\left(\rho,w^{-1}\eta\right)}X^{w^{-1}(\eta)}} \approx \begin{cases} 1 & \text{if}\ w^{-1}\eta>0,\\ \mathsf{t}\left(1-X^{w^{-1}\eta}\right) & \text{if}\ \left(\rho,w^{-1}\eta\right)=-1,\\ \mathsf{t} & \text{if}\ \left(\rho,w^{-1}\eta\right)<-1 \end{cases} \end{align} $$

(3.39) $$ \begin{align} \varkappa\left(\mathsf{G}_\eta^+\right)_{w,s_\eta w} &= \frac{\mathsf{t}-1}{1-\mathsf{t}^{\left(\rho,w^{-1}\eta\right)}X^{-w^{-1}\eta}} \approx \begin{cases} -1 &\text{if}\ w^{-1}\eta>0,\\ \mathsf{t}^{-\left(\rho,w^{-1}\eta\right)}X^{w^{-1}\eta} & \text{if}\ w^{-1}\eta<0 \end{cases} \end{align} $$

(3.40) $$ \begin{align} \varkappa\left(\mathsf{G}_\eta^-\right)_{w,w} &= \frac{\mathsf{t}^{-1}-\mathsf{t}^{-\left(\rho,w^{-1}\eta\right)}X^{w^{-1}(\eta)}}{1-\mathsf{t}^{-\left(\rho,w^{-1}\eta\right)}X^{w^{-1}(\eta)}} \approx \begin{cases} 1-X^{-w^{-1}\eta} & \text{if}\ \left(\rho,w^{-1}\eta\right)=1,\\ 1 & \text{if}\ \left(\rho,w^{-1}\eta\right)>1,\\ \mathsf{t}^{-1} & \text{if}\ w^{-1}\eta<0 \end{cases} \end{align} $$

(3.41) $$ \begin{align} \varkappa\left(\mathsf{G}_\eta^-\right)_{w,s_\eta w} &= \frac{\mathsf{t}^{-1}-1}{1-\mathsf{t}^{\left(\rho,w^{-1}\eta\right)}X^{-w^{-1}\eta}}\approx \begin{cases} \mathsf{t}^{-1} &\text{if}\ w^{-1}\eta>0,\\ -\mathsf{t}^{-\left(\rho,w^{-1}\eta\right)-1}X^{w^{-1}\eta} & \text{if}\ w^{-1}\eta<0. \end{cases} \end{align} $$

Remark 3.5. These computations are closely related to those of [3, §4] involving quantum Bruhat operators. The precise connection to the quantum Bruhat graph will be made later.

3.3.1. Expanding products as sums over walks

Using the vanishing of the matrix coefficients, we obtain the following row expansion (see Section 2.7) for any $w\in W$ and any sequence $\vec {\eta }=(\eta _1,\dotsc ,\eta _n)\in \left (\Delta ^+\right )^n$ :

(3.42) $$ \begin{align} \varkappa\left(\mathsf{G}_{\eta_1}^\pm\mathsf{G}_{\eta_2}^\pm\dotsm \mathsf{G}_{\eta_n}^\pm\right)_{w,\bullet} &=\sum_{\mathbf{w}\in\mathbf{W}_{w}^{\vec{\eta}}}\varkappa\left(\mathsf{G}_{\eta_1}^\pm\right)_{w_0,w_1}\varkappa\left(\mathsf{G}_{\eta_2}^\pm\right)_{w_1,w_2}\dotsm\varkappa\left(\mathsf{G}_{\eta_n}^\pm\right)_{w_{n-1},w_n}e_{w_n}, \end{align} $$

where $w_0=w$ and $\mathbf {W}_{w}^{\vec {\eta }}$ is the set of sequences $\mathbf {w}=(w_1,\dotsc ,w_n)$ of elements of W such that

(3.43) $$ \begin{align} w_t\in\left\{w_{t-1},s_{\eta_t}w_{t-1}\right\} \text{ for each } t=1,\dotsc,n. \end{align} $$

We call elements of $\mathbf {W}_{w}^{\vec {\eta }}$ walks in W.

3.3.2. Quantum walks

We will see that the matrices describing our inverse Chevalley formula are given by sums over subsets of walks given as follows:

Definition 3.6. We call $\mathbf {w}=(w_1,\dotsc ,w_n)\in \mathbf {W}_{w}^{\vec {\eta }}$ a quantum walk if it defines a directed walk in the quantum Bruhat graph $\mathrm {QBG}(W)$ – that is, for each $1\le t\le n$ , either $w_t=w_{t-1}$ or $w_{t-1}\to w_t=s_{\eta _t}w_{t-1}$ is an edge in $\mathrm {QBG}(W)$ .

We note that such an edge $w_{t-1}\to w_t$ in $\mathrm {QBG}(W)$ is labelled by $\pm w_{t-1}^{-1}\eta _t$ . In the case when $w_{t-1}>w_t$ , the requirement for an edge is that $\ell (w_{t-1})-\ell (w_t)=\left (2\rho ,-w_{t-1}^{-1}\eta _t\right )-1$ .

3.3.3. Main limit

Now we proceed to the limit computation which will give our first main result. Let $\varpi _k\in P_+$ be a minuscule fundamental weight. (Thus we implicitly assume that our simply laced group G is not of type $E_8$ .) As in the introduction, let $J=I\setminus \{k\}$ , let $W_J=\left \langle s_j \mid j\in J\right \rangle \subset W$ be the corresponding parabolic subgroup, which is the stabiliser of $\varpi _k$ , and let $W^J$ denote the set of minimal coset representatives for $W/W_J$ . For $w\in W$ , let $\left \lfloor w \right \rfloor \in W^J$ be the minimal representative of its coset $wW_J$ .

Fix $x\in W^J$ and let $\lambda =x\varpi _k\in P$ , which is an arbitrary minuscule weight. Let $y \in W$ be the unique element such that $\left \lfloor w_{\circ } \right \rfloor = yx$ . Since $\varpi _{k}$ is minuscule, we have $\ell (\left \lfloor w_{\circ } \right \rfloor ) = \ell (y) + \ell (x)$ ; indeed, on $W^J$ , the Bruhat order coincides with the left weak Bruhat order [6, Lemma 11.1.16].

We fix reduced expressions $x=s_{j_l}\dotsm s_{j_1}$ and $y = s_{i_{1}} \cdots s_{i_{m}}$ , and we define

(3.44) $$ \begin{align} \beta_r &= s_{j_{l}}s_{j_{l-1}} \cdots s_{j_{r+1}}\left(\alpha_{j_{r}}\right)\kern-2pt, \quad 1 \le r \le l, \end{align} $$

(3.45) $$ \begin{align} \gamma_s &=s_{i_{m}}s_{i_{m-1}} \cdots s_{i_{s+1}}\left(\alpha_{i_{s}}\right)\kern-2.5pt, \quad 1 \le s \le m. \end{align} $$

Hence, if we set

(3.46) $$ \begin{align} \mathrm{Inv}(w)=\Delta^+\cap w\left(-\Delta^+\right)\kern-2pt, \end{align} $$

then $\mathrm {Inv}(x)=\{\beta _1,\dots ,\beta _l\}$ and $\mathrm {Inv}\left (y^{-1}\right )=\{\gamma _{1},\dots ,\gamma _{m}\}$ .

Now, for any $w\in W$ , define the set of walks $\mathbf {W}_{\lambda ,w}=\mathbf {W}_{w}^{\vec {\eta }}$ , where $\vec {\eta }$ is given by

(3.47) $$ \begin{align} \vec{\eta}=(\eta_1,\dotsc,\eta_n)=(\beta_l,\dotsc,\beta_1,\gamma_1,\dotsc,\gamma_m) \end{align} $$

and $n=l+m$ . Let $\mathbf {QW}_{\lambda ,w}$ denote the subset of quantum walks in $\mathbf {W}_{w}^{\vec {\eta }}$ . (These sets depend on the choice of reduced expressions for x and y, even though our notation does not indicate this.)

The inverse Chevalley formula for minuscule weights is an immediate consequence of the following, which is the main technical achievement of this paper:

Theorem 3.7. For any minuscule $\lambda \in P$ and $w\in W$ , we have

(3.48) $$ \begin{align} \varrho_0'\left(Y^{\lambda}\right)_{w,\bullet}=\sum_{\mathbf{w}=(w_1,\dotsc,w_n)\in\mathbf{QW}_{\lambda,w}}g^-_{\mathbf{w}} t_{w_{l}^{-1}\lambda} g^+_{\mathbf{w}} e_{w_n}, \end{align} $$

with

(3.49) $$ \begin{align} g^-_{\mathbf{w}} &= \prod_{1\le t\le l} \begin{cases} 1-X^{-w_{t-1}^{-1}\eta_t} & \text{if}\ w_t=w_{t-1}\ \text{and}\ \left(\rho,w_{t-1}^{-1}\eta_t\right)=1,\\ -X^{w_{t-1}^{-1}\eta_t} & \text{if}\ w_t<w_{t-1},\\ 1 & \text{otherwise} \end{cases} \end{align} $$

(3.50) $$ \begin{align} g^+_{\mathbf{w}} &= \prod_{l<t\le n} \begin{cases} 1-X^{w_{t-1}^{-1}\eta_t} & \text{if}\ w_t=w_{t-1}\ \text{and}\ \left(\rho,w_{t-1}^{-1}\eta_t\right)=-1,\\ X^{w_{t-1}^{-1}\eta_t} & \text{if}\ w_t<w_{t-1},\\ -1 & \text{if}\ w_t>w_{t-1},\\ 1 & \text{otherwise}. \end{cases} \end{align} $$

For the proof of Theorem 3.7, we will need the following notions, which rely heavily on $\lambda \in P$ being minuscule. For $w \in W$ , and with all other notation as before, let us define

(3.51) $$ \begin{align} & \mathrm{Inv}(w)_{\lambda}^{+}=\{ \gamma \in \mathrm{Inv}(w) \mid (\gamma,\lambda)=1\}, \qquad \ell_{\lambda}^{+}(w)=\# \mathrm{Inv}(w)_{\lambda}^{+}, \end{align} $$

(3.52) $$ \begin{align} & \mathrm{Inv}(w)_{\lambda}^{-}=\{ \gamma \in \mathrm{Inv}(w) \mid (\gamma,\lambda)=-1\}, \qquad \ell_{\lambda}^{-}(w)=\# \mathrm{Inv}(w)_{\lambda}^{-}, \end{align} $$

and

(3.53) $$ \begin{align} \mathrm{Inv}(w)_{\lambda,r}^{-}&=\mathrm{Inv}(w) \cap \{\beta_{r},\beta_{r+1},\dots,\beta_{l}\}, \qquad \ell_{\lambda,r}^{-}(w)=\#\mathrm{Inv}(w)_{\lambda,r}^{-}, \end{align} $$

(3.54) $$ \begin{align} \mathrm{Inv}(w)_{\lambda,s}^{+}&=\mathrm{Inv}(w) \cap \{\gamma_{s},\gamma_{s+1},\dots,\gamma_{m}\}, \qquad \ell_{\lambda,s}^{+}(w)=\#\mathrm{Inv}(w)_{\lambda,s}^{+}, \end{align} $$

where $1 \le r \le l$ and $1 \le s \le m$ .

The following lemma seems to be well known, but we give a simple proof in Appendix B for the convenience of the reader:

Lemma 3.8. With all notation as before, the following are true:

1. 1. We have

   (3.55) $$ \begin{align} 2\ell(x) - \ell(\left\lfloor w_{\circ} \right\rfloor) &= - 2(\rho,\lambda). \end{align} $$

2. 2. For any $w\in W$ ,

   (3.56) $$ \begin{align} (\rho-w\rho,\lambda) &= \ell_{\lambda}^{+}(w) - \ell_{\lambda}^{-}(w) \end{align} $$

   and

   (3.57) $$ \begin{align} \mathrm{Inv}(w)_{\lambda}^{-} &= \mathrm{Inv}(w) \cap \mathrm{Inv}(x), \end{align} $$
   (3.58) $$ \begin{align} \mathrm{Inv}(w)_{\lambda}^{+} &= \mathrm{Inv}(w) \cap \mathrm{Inv}\left(y^{-1}\right). \end{align} $$

The proof of the following proposition will be given in Appendix B:

Proposition 3.9. With all notation as before, the following are true:

1. 1. If $s_{\beta _{r}}w < w$ for $1 \le r \le l$ , then

   (3.59) $$ \begin{align} \ell(w)-\ell\left(s_{\beta_{r}}w\right) = 2 \left(\ell_{\lambda,r}^{-}(w)-\ell_{\lambda,r}^{-}\left(s_{\beta_{r}}w\right)\right) - 1. \end{align} $$

2. 2. If $s_{\gamma _{s}}w < w$ for $1 \le s \le m$ , then

   (3.60) $$ \begin{align} \ell(w)-\ell\left(s_{\gamma_{s}}w\right) = 2 \left(\ell_{\lambda,s}^{+}(w)-\ell_{\lambda,s}^{+}\left(s_{\gamma_{s}}w\right)\right)- 1. \end{align} $$

Remark 3.10. Equations (3.59) and (3.60) generalise the following well-known formula for length differences in the symmetric group $S_n$ :

$$ \begin{align*} \ell(w)-\ell((ij)w) &= 1+2\cdot\left\lvert\left\{k: i<k<j \text{ and } w^{-1}(i)>w^{-1}(k)>w^{-1}(j)\right\}\right\rvert, \end{align*} $$

where $1\le i < j\le n$ , $w\in S_n$ is a permutation such that $w^{-1}(i)>w^{-1}(j)$ , and $(ij)$ denotes the transposition swapping i and j.

Proof of Theorem 3.7. Let $\pi \in \Pi $ be the length $0$ element of $W_{\mathrm {ex}}$ corresponding to coset $\varpi _k+Q$ in $P/Q$ . We have the formula $t_{\varpi _k} = \pi \left \lfloor w_{\circ } \right \rfloor =\pi yx$ (which may be taken as the definition of $\pi $ ), and hence $t_{\lambda } = x\pi y.$

Using equations (3.33) and (3.55), we find

$$ \begin{align*} Y^{\lambda} &= \mathsf{v}^{2\ell(x)-\ell\left(\left\lfloor w_{\circ} \right\rfloor\right)} T_{x^{-1}}^{-1} \pi T_y\\ &= \mathsf{t}^{-\left(\rho,\lambda\right)}T_{x^{-1}}^{-1} \pi T_y \end{align*} $$

in $\widehat {\mathbb {H}}$ , and

$$ \begin{align*} \varrho\left(Y^{\lambda}\right) &= \mathsf{t}^{-\left(\rho,\lambda\right)} \varrho\left(T_{x^{-1}}^{-1}\right)x^{-1} t_{\lambda}y^{-1} \varrho\left(T_y\right)\\ &= \mathsf{t}^{-\left(\rho,\lambda\right)}\mathsf{G}^-_{\beta_l}\dotsm \mathsf{G}^-_{\beta_1}t_{\lambda}\mathsf{G}^+_{\gamma_1}\dotsm \mathsf{G}^+_{\gamma_m} \end{align*} $$

in the polynomial representation. Thus we need to compute the limit

$$ \begin{align*} \varrho_0'\left(Y^{\lambda}\right)=\lim_{\mathsf{v}\to 0} \varkappa\left(\mathsf{G}^-_{\beta_l}\dotsm \mathsf{G}^-_{\beta_1}\right)\mathsf{t}^{-\left(\rho,\lambda\right)}\varkappa(t_{\lambda})\varkappa\left(\mathsf{G}^+_{\gamma_1}\dotsm \mathsf{G}^+_{\gamma_m}\right). \end{align*} $$

The matrix $\mathsf {t}^{-\left (\rho ,\lambda \right )}\varkappa (t_{\lambda })$ is diagonal with entries

$$ \begin{align*} \mathsf{t}^{-\left(\rho,\lambda\right)}\varkappa(t_{\lambda})_{w,w} = \mathsf{t}^{\left(\rho,w^{-1}\lambda\right)-\left(\rho,\lambda\right)}t_{w^{-1}\lambda}. \end{align*} $$

Using equation (3.56), we can write $\mathsf {t}^{-\left (\rho ,\lambda \right )}\varkappa (t_{\lambda })=\varkappa _\lambda ^-\tau _\lambda \varkappa _\lambda ^+$ , where $\varkappa _\lambda ^\pm $ and $\tau _\lambda $ are the diagonal matrices given by

(3.61) $$ \begin{align} \left(\varkappa_\lambda^-\right)_{w,w}&=\mathsf{t}^{\ell_\lambda^-(w)},\qquad (\tau_\lambda)_{w,w} = t_{w^{-1}\lambda},\qquad \left(\varkappa_\lambda^+\right)_{w,w}=\mathsf{t}^{-\ell_\lambda^+(w)}. \end{align} $$

Our strategy is to commute $\varkappa _\lambda ^-$ to the left past $\varkappa \left (\mathsf {G}^-_{\beta _l}\dotsm \mathsf {G}^-_{\beta _1}\right )$ , and $\varkappa _\lambda ^+$ to the right past $\varkappa \left (\mathsf {G}^+_{\gamma _1}\dotsm \mathsf {G}^+_{\gamma _m}\right )$ . After we have done so, all negative powers of $\mathsf {v}$ will disappear.

For $r=1,\dotsc ,l$ , we have

$$ \begin{align*} \varkappa\left(\mathsf{G}_{\beta_r}^-\right)_{w,w} \cdot \mathsf{t}^{\ell_{\lambda,r}^-(w)} &\approx \mathsf{t}^{\ell_{\lambda,r+1}^-(w)}\times \begin{cases} 1-X^{-w^{-1}\beta_r} & \text{if}\ \left(\rho,w^{-1}\beta_r\right)=1,\\ 1 & \text{if}\ \left(\rho,w^{-1}\beta_r\right)>1,\\ 1 & \text{if}\ w^{-1}\beta_r<0, \end{cases} \end{align*} $$

and

$$ \begin{align*} &\varkappa\left(\mathsf{G}_{\beta_r}^-\right)_{w,s_{\beta_r}w} \cdot \mathsf{t}^{\ell_{\lambda,r}^-\left(s_{\beta_r}w\right)}\\ &\approx \mathsf{t}^{\ell_{\lambda,r+1}^-(w)} \times \begin{cases} \mathsf{t}^{-1+\ell_{\lambda,r}^-\left(s_{\beta_r}w\right)-\ell_{\lambda,r}^-(w)} &\text{if}\ w^{-1}\beta_r>0,\\ -\mathsf{t}^{-\left(\rho,w^{-1}\beta_r\right)+\ell_{\lambda,r}^-\left(s_{\beta_r}w\right)-\ell_{\lambda,r}^-(w)}X^{w^{-1}\beta_r} & \text{if}\ w^{-1}\beta_r<0 \end{cases}\\ &= \mathsf{t}^{\ell_{\lambda,r+1}^-(w)} \times \begin{cases} \mathsf{v}^{-1+\ell\left(s_{\beta_r}w\right)-\ell(w)} &\text{if}\ w^{-1}\beta_r>0,\\ -\mathsf{v}^{-\left(2\rho,w^{-1}\beta_r\right)-1+\ell\left(s_{\beta_r}w\right)-\ell(w)}X^{w^{-1}\beta_r} & \text{if}\ w^{-1}\beta_r<0, \end{cases} \end{align*} $$

where we use equations (3.59) to obtain the last equality. Notice that in all cases, each factor after the bracket involves no negative powers of $\mathsf {v}$ . Moreover, after commuting $\varkappa _\lambda ^-$ all the way to the left, we arrive at $\mathsf {t}^{\ell _{\lambda ,l+1}^-(w)}=1$ , for any $w\in W$ .

Similarly, for $s=1,\dotsc ,m$ , we have

$$ \begin{align*} \mathsf{t}^{-\ell_{\lambda,s}^+(w)}\cdot \varkappa\left(\mathsf{G}_{\gamma_s}^+\right)_{w,w} & \approx \mathsf{t}^{-\ell_{\lambda,s+1}^+(w)}\times \begin{cases} 1 & \text{if}\ w^{-1}\gamma_s>0,\\ 1-X^{w^{-1}\gamma_s} & \text{if}\ \left(\rho,w^{-1}\gamma_s\right)=-1,\\ 1 & \text{if}\ \left(\rho,w^{-1}\gamma_s\right)<-1, \end{cases} \end{align*} $$

and

$$ \begin{align*} &\mathsf{t}^{-\ell_{\lambda,s}^+(w)}\cdot \varkappa\left(\mathsf{G}_{\gamma_s}^+\right)_{w,s_{\gamma_s} w}\\ &\approx \mathsf{t}^{-\ell_{\lambda,s+1}^+(s_{\gamma_s}w)} \times \begin{cases} -\mathsf{t}^{-1+\ell_{\lambda,s}^+\left(s_{\gamma_s}w\right)-\ell_{\lambda,s}^+(w)}&\text{if}\ w^{-1}\gamma_s>0,\\ \mathsf{t}^{-\left(\rho,w^{-1}\gamma_s\right)+\ell_{\lambda,s}^+\left(s_{\gamma_s}w\right)-\ell_{\lambda,s}^+(w)}X^{w^{-1}\gamma_s} & \text{if}\ w^{-1}\gamma_s<0 \end{cases}\\ &= \mathsf{t}^{-\ell_{\lambda,s+1}^+\left(s_{\gamma_s}w\right)} \times \begin{cases} -\mathsf{v}^{-1+\ell\left(s_{\gamma_s}w\right)-\ell(w)} & \text{if}\ w^{-1}\gamma_s>0,\\ \mathsf{v}^{-\left(2\rho,w^{-1}\gamma_s\right)-1+\ell\left(s_{\gamma_s}w\right)-\ell(w)}X^{w^{-1}\gamma_s} & \text{if}\ w^{-1}\gamma_s<0. \end{cases} \end{align*} $$

As before, each factor after the bracket involves no negative powers of $\mathsf {v}$ , and after commuting $\varkappa _\lambda ^+$ all the way to the right, we arrive at $\mathsf {t}^{\ell _{\lambda ,m+1}^+(w)}=1$ , for any $w\in W$ .

To complete the proof, we expand the product of resulting matrices (including $\tau _\lambda $ ) and take $\mathsf {v}\to 0$ . Taking into account the exponents of $\mathsf {v}$ , one sees that the surviving terms are exactly those indexed by quantum walks $\mathbf {QW}_{\lambda ,w}$ .□

3.4. Inverse Chevalley formula

By means of [25, Theorem 5.1], Theorem 3.7 immediately gives the following algebraic form of the inverse Chevalley formula in $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ , which is our first main result:

Theorem 3.11. For any minuscule $\lambda \in P$ and $w\in W$ , we have

(3.62) $$ \begin{align} e^\lambda\cdot\left[\mathcal{O}_{\mathbf{Q}_{G}(w)}\right] = \sum_{\mathbf{w}=(w_1,\dotsc,w_n)\in\mathbf{QW}_{\lambda,w}}\left[\mathcal{O}_{\mathbf{Q}_{G}(w_n)}\right]\cdot \widetilde{g}^+_{\mathbf{w}} X^{-w_{\circ} w_l^{-1}\lambda} \widetilde{g}^-_{\mathbf{w}} \end{align} $$

in terms of the q-Heisenberg action of Proposition 3.3, with

(3.63) $$ \begin{align} \widetilde{g}^-_{\mathbf{w}} &= \prod_{1\le t\le l} \begin{cases} 1-\widetilde{X}^{-w_{\circ} w_{t-1}^{-1}\eta_t} & \text{if}\ w_t=w_{t-1}\ \text{and}\ \left(\rho,w_{t-1}^{-1}\eta_t\right)=1,\\ -\widetilde{X}^{w_{\circ} w_{t-1}^{-1}\eta_t} & \text{if}\ w_t<w_{t-1},\\ 1 & \text{otherwise} \end{cases} \end{align} $$

(3.64) $$ \begin{align} \widetilde{g}^+_{\mathbf{w}} &= \prod_{l< t\le n} \begin{cases} 1-\widetilde{X}^{w_{\circ} w_{t-1}^{-1}\eta_t} & \text{if}\ w_t=w_{t-1}\ \text{and}\ \left(\rho,w_{t-1}^{-1}\eta_t\right)=-1,\\ \widetilde{X}^{w_{\circ} w_{t-1}^{-1}\eta_t} & \text{if}\ w_t<w_{t-1},\\ -1 & \text{if}\ w_t>w_{t-1},\\ 1 & \text{otherwise}, \end{cases} \end{align} $$

and $\widetilde {X}^\beta =q\cdot t_\beta X^\beta $ for $\beta \in \Delta $ .

Remark 3.12. The elements $\widetilde {X}^\beta $ commute, as one easily checks.

Remark 3.13. The image of equation (3.62) under the map $\mathrm {trunc}$ of formula (1.3) recovers the corresponding inverse Chevalley formula in $K_H(G/B)$ given by [23, p. 239]; note that under the map $\mathrm {trunc}$ , the class $\left [\mathcal {O}_{\mathbf {Q}_{G}}(\mu )\right ]$ for $\mu \in P$ corresponds to the class $\left [\mathcal {O}_{G/B}(\mu )\right ]$ , which is denoted by $[\mathcal {L}(-\mu )]$ in [23]. We note also that the image of equation (3.62) under $\mathrm {trunc}$ has positive coefficients for $\lambda \in -P_+$ (i.e., when $l=n$ ), and alternating coefficients for $\lambda \in P_+$ (i.e., when $l=0$ ). Informally, the effect of the map $\mathrm {trunc}$ is achieved by setting $\widetilde {X}^\beta =0$ in equations (3.63) and (3.64), where $\beta \in \Delta ^+$ always.

3.4.1. Decorated quantum walks

We now proceed to further combinatorialise equation (3.62). First we enlarge the summation set, so that each term on the right-hand side of equation (3.62) is a product of monomials in the q-Heisenberg algebra.

Given $\mathbf {w}=(w_1,\dotsc ,w_n)\in \mathbf {QW}_{\lambda ,w}$ , let $S^-(\mathbf {w})$ denote the set of steps t, for $1\le t\le l$ , such that $w_t=w_{t-1}$ and $\left (\rho ,w_{t-1}^{-1}\eta _t\right )=1$ . Similarly, let $S^+(\mathbf {w})$ denote the set of steps t, for $l<t\le n$ such that $w_t=w_{t-1}$ and $\left (\rho ,w_{t-1}^{-1}\eta _t\right )=-1$ .

Let $S(\mathbf {w})=S^-(\mathbf {w})\cup S^+(\mathbf {w})$ , and define the set of decorated quantum walks $\widetilde {\mathbf {QW}}_{\lambda ,w}$ to consist of all pairs $(\mathbf {w},\mathbf {b})$ where $\mathbf {w}\in \mathbf {QW}_{\lambda ,w}$ and $\mathbf {b}$ is a $\{0,1\}$ -valued function on $S(\mathbf {w})$ . Then equation (3.62) can be written as

(3.65) $$ \begin{align} e^{\lambda}\cdot\left[\mathcal{O}_{\mathbf{Q}_{G}(w)}\right] = \sum_{\left(\mathbf{w},\mathbf{b}\right)\in\widetilde{\mathbf{QW}}_{\lambda,w}}\left[\mathcal{O}_{\mathbf{Q}_{G}(w_n)}\right]\cdot \widetilde{g}^+_{\left(\mathbf{w},\mathbf{b}\right)} X^{-w_{\circ} w_l^{-1}\lambda} \widetilde{g}^-_{\left(\mathbf{w},\mathbf{b}\right)}, \end{align} $$

with

(3.66) $$ \begin{align} \widetilde{g}^-_{\left(\mathbf{w},\mathbf{b}\right)} &= \prod_{1\le t\le l} \begin{cases} \left(-\widetilde{X}^{-w_{\circ} w_{t-1}^{-1}\eta_t}\right)^{\mathbf{b}(t)} & \text{if}\ t\in S^-(\mathbf{w}),\\ -\widetilde{X}^{w_{\circ} w_{t-1}^{-1}\eta_t} & \text{if}\ w_t<w_{t-1},\\ 1 & \text{otherwise} \end{cases} \end{align} $$

(3.67) $$ \begin{align} \widetilde{g}^+_{\left(\mathbf{w},\mathbf{b}\right)} &= \prod_{l< t\le n} \begin{cases} \left(-\widetilde{X}^{w_{\circ} w_{t-1}^{-1}\eta_t}\right)^{\mathbf{b}(t)} & \text{if}\ t\in S^+(\mathbf{w}),\\ \widetilde{X}^{w_{\circ} w_{t-1}^{-1}\eta_t} & \text{if}\ w_t<w_{t-1},\\ -1 & \text{if}\ w_t>w_{t-1},\\ 1 & \text{otherwise}. \end{cases} \end{align} $$

Next, to expand equation (3.65) further, let us introduce some more notation. For $(\mathbf {w},\mathbf {b})\in \widetilde {\mathbf {QW}}_{\lambda ,w}$ , define the sign

$$ \begin{align*} (-1)^{\left(\mathbf{w},\mathbf{b}\right)} = \prod_{\substack{1\le t\le l\\w_t<w_{t-1}}}(-1)\prod_{\substack{l< t\le n \\ w_t>w_{t-1}}}(-1)\prod_{t\in S(\mathbf{w})}(-1)^{\mathbf{b}(t)} \end{align*} $$

and partial weights (in Q)

$$ \begin{align*} \mathrm{wt}_0(\mathbf{w},\mathbf{b})&=0\\ \mathrm{wt}_t(\mathbf{w},\mathbf{b})&=\mathrm{wt}_{t-1}(\mathbf{w},\mathbf{b})+ \begin{cases} -\mathbf{b}(t)w_{\circ} w_{t-1}^{-1}\eta_t & \text{if}\ t\in S^-(\mathbf{w}),\\ w_{\circ} w_{t-1}^{-1}\eta_t & \text{if}\ w_t<w_{t-1},\\ 0 & \text{otherwise} \end{cases} \quad \text{for}\ 1\le t\le l\\ \mathrm{wt}_t(\mathbf{w},\mathbf{b})&=\mathrm{wt}_{t-1}(\mathbf{w},\mathbf{b})+ \begin{cases} \mathbf{b}(t)w_{\circ} w_{t-1}^{-1}\eta_t & \text{if}\ t\in S^+(\mathbf{w}),\\ w_{\circ} w_{t-1}^{-1}\eta_t & \text{if}\ w_t<w_{t-1},\\ 0 & \text{otherwise} \end{cases} \quad \text{for}\ l<t\le n. \end{align*} $$

Define the weight of $(\mathbf {w},\mathbf {b})$ to be $\mathrm {wt}(\mathbf {w},\mathbf {b})=\mathrm {wt}_n(\mathbf {w},\mathbf {b})\in Q$ , and set $d_t(\mathbf {w},\mathbf {b})=\mathrm {wt}_t(\mathbf {w},\mathbf {b})-\mathrm {wt}_{t-1}(\mathbf {w},\mathbf {b})$ for $1\le t\le n$ .

Define the partial degrees (in $\mathbb {Z}$ ):

$$ \begin{align*} \deg_0^-(\mathbf{w},\mathbf{b}) &= 0\\ \deg_t^-(\mathbf{w},\mathbf{b}) &= \deg_{t-1}^-(\mathbf{w},\mathbf{b})+\frac{\left\lvert d_t(\mathbf{w},\mathbf{b})\right\rvert^2}{2}+(d_t(\mathbf{w},\mathbf{b}),\mathrm{wt}_{t-1}(\mathbf{w},\mathbf{b})),\quad \text{for}\ 1\le t\le l\\ \deg_{l}^+(\mathbf{w},\mathbf{b}) &= \deg_{l}^-(\mathbf{w},\mathbf{b})+\left(-w_{\circ} w_{l}^{-1}\lambda,\mathrm{wt}_{l}(\mathbf{w},\mathbf{b})\right)\\ \deg_t^+(\mathbf{w},\mathbf{b}) &= \deg_{t-1}^+(\mathbf{w},\mathbf{b})+\frac{\left\lvert d_t(\mathbf{w},\mathbf{b})\right\rvert^2}{2}+(d_t(\mathbf{w},\mathbf{b}),\mathrm{wt}_{t-1}(\mathbf{w},\mathbf{b})), \quad \text{for}\ l<t\le n. \end{align*} $$

Define $\deg (\mathbf {w},\mathbf {b})=\deg _n^+(\mathbf {w},\mathbf {b})\in \mathbb {Z}$ .

Then, from equation (3.65), we obtain our second main result, the combinatorial form of our inverse Chevalley formula:

Theorem 3.14. For any minuscule $\lambda \in P$ and $w\in W$ , we have

(3.68) $$ \begin{align} &e^\lambda\cdot\left[\mathcal{O}_{\mathbf{Q}_{G}(w)}\right]\nonumber\\ &\quad= \sum_{\left(\mathbf{w},\mathbf{b}\right)\in\widetilde{\mathbf{QW}}_{\lambda,w}}(-1)^{\left(\mathbf{w},\mathbf{b}\right)}q^{\deg\left(\mathbf{w},\mathbf{b}\right)}\cdot\left[\mathcal{O}_{\mathbf{Q}_{G}\left(w_nt_{-w_{\circ}\left(\mathrm{wt}\left(\mathbf{w},\mathbf{b}\right)\right)}\right)}\left(-w_{\circ} w_l^{-1}\lambda+\mathrm{wt}(\mathbf{w},\mathbf{b})\right)\right]. \end{align} $$

Proof. Proceed from right to left in equation (3.65), commuting all translations to the left of line bundle twists, and then act on $\left [\mathcal {O}_{\mathbf {Q}_{G}(w_n)}\right ]$ .□