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 [Reference Brenti, Fomin and Postnikov3, 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 [Reference Kac9, §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 $w\in W$ .

3.1. 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 [Reference Kato, Naito and Sagaki13, §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 [Reference Kato, Naito and Sagaki13] for indexing equivariant line bundles and Schubert varieties in $\mathbf {Q}_{G}^{\mathrm {rat}}$ .

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 [Reference Kato, Naito and Sagaki13, 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 [Reference Kato, Naito and Sagaki13] and antidominant weights [Reference Naito, Orr and Sagaki24]. Similar (in fact, equivalent) assertions hold for $K_{H\times \mathbb {C}^*}(\mathbf {Q}_{G})$ .

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 [Reference Orr25, 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 [Reference Kato, Naito and Sagaki13].

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 [Reference Kato10] and [Reference Orr25, Proposition 2.3]. We note that our conventions here differ slightly from those of [Reference Orr25], 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 [Reference Kato, Naito and Sagaki13, 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 [Reference Orr25, 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 [Reference Kato, Naito and Sagaki13] or [Reference Orr25]). 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 [Reference Kato, Naito and Sagaki13, 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 [Reference Orr25] 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., [Reference Orr25, §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 [Reference Orr25, 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 [Reference Cherednik and Orr4, 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 [Reference Orr25, §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) $$ \begin{align} \varkappa\left(\mathsf{G}_\eta^-\right)_{w,s_\eta w} &= \frac{\mathsf{t}^{-1}-1}{1-\mathsf{t}^{\left(\rho,w^{-1}\eta\right)}X^{-w^{-1}\eta}}\approx \begin{cases} \mathsf{t}^{-1} &\text{if}\ w^{-1}\eta>0,\\ -\mathsf{t}^{-\left(\rho,w^{-1}\eta\right)-1}X^{w^{-1}\eta} & \text{if}\ w^{-1}\eta<0. \end{cases} \end{align} $$

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.

3.3.3. 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 [Reference Green6, 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 [Reference Orr25, 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 $ .

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 $\left [\mathcal {O}_{\mathbf {Q}_{G}(w_n)}\right ]$ .□