2.1 Notations

Let $G$ be a connected, semisimple, simply connected, complex linear algebraic group with a Borel subgroup $B$, whose roots are positive. Let $A\subset B$ be a maximal torus of $G$, ${\frak h}$ be the Lie algebra of $A$, and $\Lambda$ (respectively $\Lambda ^\vee$) the group of characters (respectively cocharacters) of $A$. Let $\rho$ denote the half sum of the positive roots. Let $\pm :={\frak C}_\pm \subset \Lambda ^\vee \otimes _{\mathbb {Z}}{\mathbb {R}}$ denote the dominant/anti-dominant Weyl chambers in ${\frak h}$. For any coroot $\alpha ^\vee$, define the hyperplanes in $\mathfrak {h}^*_{\mathbb {R}}$ by

\[ H_{\alpha^\vee,n}=\{\lambda\in \mathfrak{h}_{\mathbb{R}}^*\mid(\lambda, \alpha^{{\vee}})=n\}, \quad n\in {\mathbb{Z}}. \]

We refer the connected components of

\[ \mathfrak{h}_{\mathbb{R}}^* \setminus \Big(\underset{\alpha>0, n\in {\mathbb{Z}}}\bigcup H_{\alpha^\vee,n}\Big) \]

as the alcoves of this hyperplane arrangement. The walls of an alcove are the codimension-$1$ facets in the boundary of this alcove. The following alcove will be referred to as the fundamental alcove

\[ \nabla_+:=\{\lambda\mid0<(\lambda, \alpha^\vee)<1, \text{ for any positive coroot } \alpha^\vee\}. \]

In other words, it contains $\epsilon \rho$ for small $\epsilon >0$. Write $\nabla _-=-\nabla _+$. We say two alcoves $\nabla _1,\nabla _2$ are adjacent if they share a wall on some hyperplane $H_{\alpha ^\vee , n}$.

We will use the same terminologies of alcoves, walls, and fundamental alcoves for their intersections with $\Lambda _\mathbb {Q}\subseteq {\frak h}^*_{{\mathbb {R}}}$.

Denote by $\mathcal {B}\simeq G/B$, the variety of all the Borel groups of $G$, and $T^*\mathcal {B}$ (respectively $T\mathcal {B}$) its cotangent bundle (respectively tangent bundle). For each $\lambda \in \Lambda$, there is an associated line bundle $\mathcal {L}_\lambda$ on $\mathcal {B}$ independent on the choice of $B$ [Reference Chriss and GinzburgCG97, §6.1.11]. Pulling it back onto $T^*\mathcal {B}$, we still denote it by $\mathcal {L}_\lambda$. Therefore, we have identity

(1)\begin{equation} \operatorname{Pic}(T^*\mathcal{B})\otimes_{\mathbb{Z}}{\mathbb{R}}=\mathfrak{h}_{\mathbb{R}}^*. \end{equation}

Let $\mathbb {C}^*$ act on $T^*\mathcal {B}$ by $z\cdot (B', x)=(B', z^{-2}x)$, where $z\in \mathbb {C}^*$ and $(B', x)\in T^*\mathcal {B}$. Let $q^{-1}$ denote the character of cotangent fiber under this action. We refer the readers to [Reference Chriss and GinzburgCG97] for a beautiful account of the equivariant K-theory. Let $T=A\times \mathbb {C}^*$, and $K_{T}(\operatorname {pt})\cong {\mathbb {Z}}[q^{1/2},q^{-1/2}][\Lambda ]$. In this paper, we will consider $K_T(T^*\mathcal {B})$, which is a module over $K_T(\operatorname {pt})$. The $A$-fixed points of $T^*\mathcal {B}$ are indexed by $W$; each $w\in W$ determines the fixed point $wB\in \mathcal {B}\subset T^*\mathcal {B}$. We will just use $w$ to denote the corresponding fixed point $wB$. For each $w \in W$, denote by $1_w\in K_{T}((T^*\mathcal {B})^A)\cong \oplus _{v\in W}K_T(\operatorname {pt})$ the basis corresponding to $w$, and by $\iota _w$ its image via the push-forward of the embedding $i:w\to T^*\mathcal {B}$, that is, $\iota _w=i_*(1)\in K_T(T^*\mathcal {B})$. It follows from the localization theorem [Reference Chriss and GinzburgCG97] that $\{\iota _w|w\in W\}$ forms a basis for the localized equivariant K-theory $K_T(T^*\mathcal {B})_{{loc}}:= K_T(T^*\mathcal {B})\otimes _{K_T(\operatorname {pt})}\operatorname {Frac} K_T(\operatorname {pt})$, where $\operatorname {Frac} K_T(\operatorname {pt})$ denotes the fraction field of $K_T(\operatorname {pt})$. For any vector space $V$ with a $T$-action, write

\[ \bigwedge {}^\bullet V=\sum_k({-}1)^k\wedge^kV^\vee{=}\prod (1-e^{-\alpha})\in K_{T}(\operatorname{pt}), \]

where the last product is over all the $T$-weights in $V$, counted with multiplicities. Note that this is not the standard notation for the wedge product because of the dual $^\vee$.

Recall there is a non-degenerate pairing $\langle \cdot , \cdot \rangle$ on $K_T(T^*\mathcal {B})$ defined via localization:

(2)\begin{equation} \langle\mathcal{F}, \mathcal{G}\rangle:=\sum_{w}\frac{\mathcal{F}|_w\mathcal{G}|_w}{\bigwedge {}^\bullet T_w(T^*\mathcal{B})}=\sum_{w}\frac{\mathcal{F}|_w\mathcal{G}|_w}{\prod_{\alpha>0}(1-e^{w\alpha}) (1-qe^{{-}w\alpha})}\in\operatorname{Frac} K_T(pt), \end{equation}

where $\mathcal {F}, \mathcal {G}\in K_T(T^*\mathcal {B})$, and $\mathcal {F}|_w$ denotes the pullback of $\mathcal {F}$ to the fixed point $wB\in \mathcal {B}\subset T^*\mathcal {B}$.

2.2 Definition of stable bases

We recall the definition of stable bases of Maulik and Okounkov.

Let ${\frak C}$ be a chamber in $\Lambda ^\vee _{\mathbb {R}}$. For any cocharacter $\sigma \in {\frak C}$, the stable leaf (also called the attracting set) of the fixed point $w$ is

\[ \operatorname{Attr}_{\frak C}(w)= \Big\{x\in T^*\mathcal{B}\Bigm|\underset{z\to 0}\lim \sigma(z)\cdot x=w \Big\}. \]

It defines a partial order on $W$ as follows:

\[ w\preceq_{\frak C} v \quad \text{if } \overline{\operatorname{Attr}_{\frak C}(v)}\cap w\neq \emptyset. \]

For example, with ${\frak C}_+$ (respectively ${\frak C}_-$) denoting the dominant chamber (respectively the anti-dominant chamber) where all the positive roots take positive (respectively negative) values, then

\[ u\preceq_{{\frak C}_+} v\iff u\le v, \quad \text{and}\quad u\preceq_{{\frak C}_-}v \iff u\ge v, \]

where $\le$ is the usual Bruhat order of $W$. In this case, $\operatorname {Attr}_{{\frak C}_+}(w)$ (respectively $\operatorname {Attr}_{{\frak C}_-}(w)$) is equal to the conormal bundle of the Schubert cell $BwB/B$ (respectively opposite Schubert cell $B^-wB/B$, where $B^-$ is the opposite Borel subgroup) inside the flag variety $\mathcal {B}$. Denote the full attracting set by

\[ \operatorname{FAttr}_{\frak C}(v)=\bigcup_{w\preceq_{{\frak C}} v}\operatorname{Attr}_{\frak C}(w). \]

Define a polarization $T^{1/2}\in K_T(T^*\mathcal {B})$ to be an equivariant K-theory class such that the following identity holds,

\[ T^{1/2}+q^{{-}1}(T^{1/2})^\vee{=}T(T^*\mathcal{B})\in K_T(T^*\mathcal{B}). \]

Write $T^{1/2}_{\operatorname {opp}}=q^{-1}(T^{1/2})^\vee$. We will mostly use the following two mutually opposite polarizations: $T\mathcal {B}$ and $T^*\mathcal {B}$, which lie in $K_{G\times \mathbb {C}^*}(T^*\mathcal {B})$.

Let $N_w:=T_w(T^*\mathcal {B})$ be the tangent space of $T^*\mathcal {B}$ at the torus fixed point $wB$, and $T^{1/2}_w:=T^{1/2}|_w$. Each chamber ${\frak C}$ determines a decomposition $N_w=N_{w,+}\oplus N_{w,-}$ of $N_w$ into $A$-weight spaces which are positive and negative with respect to ${\frak C}$, and similarly $T^{1/2}_w=T^{1/2}_{w,+}\oplus T^{1/2}_{w,-}$. Then

\[ N_{w,-}=T^{1/2}_{w,-}\oplus q^{{-}1}(T^{1/2}_{w,+})^\vee, \]

and

\[ N_{w,-}\ominus T^{1/2}_w=q^{{-}1}(T^{1/2}_{w,+})^\vee{\ominus} T^{1/2}_{w,+}. \]

Thus,

\[ \bigg(\frac{\det N_{w,-}}{\det T^{1/2}_w}\bigg)^{1/2} =q^{-{\operatorname{rank} T^{1/2}_{w,+}}/{2}}\det (T^{1/2}_{w,+})^\vee{\in} K_T(\operatorname{pt}). \]

For any Laurent polynomial $f=\sum _{\mu \in \Lambda }f_\mu e^\mu \in K_T(\operatorname {pt})$ with $f_\mu \in K_{\mathbb {C}^*}(\operatorname {pt})$, define its Newton polygon to be

\[ \deg_Af=\text{Convex hull }(\{\mu\mid f_\mu\neq 0\})\subset \Lambda\otimes_{\mathbb{Z}}\mathbb{Q}. \]

Definition 2.1 [Reference Okounkov and SmirnovOS16, Reference OkounkovOko17]

For any chamber ${\frak C}$, polarization $T^{1/2}$, alcove $\nabla$, there is a unique map of $K_T(\operatorname {pt})$-modules (called the stable envelope),

\[ \operatorname{stab}: K_{T}((T^*\mathcal{B})^{A})\to K_T(T^*\mathcal{B}), \]

satisfying the following three conditions. Write $\operatorname {stab}^{{\frak C},T^{1/2}, \nabla }_w=\operatorname {stab}(1_w)$. Then we have:

1. (i) (Support) $\operatorname {supp}( \operatorname {stab}^{{\frak C}, T^{1/2},\nabla }_w)\subset \operatorname {FAttr}_{ {\frak C}}(w)$;

2. (ii) (Normalization) $\operatorname {stab}^{{\frak C},T^{1/2}, \nabla }_w|_w=(-1)^{\operatorname {rank} T^{1/2}_{w, +}}({\det N_{w,-}}/{\det T_w^{1/2}})^{1/2} \mathcal {O}_{\operatorname {Attr} _{\frak C}(w)}|_w$;

3. (iii) (Degree) $\deg _A(\operatorname {stab}^{{\frak C}, T^{1/2},\nabla }_w|_v)\subset \deg _A(\operatorname {stab}^{{\frak C},T^{1/2}, \nabla }_v|_v)+\mathcal {L}|_v- \mathcal {L}|_w$ for any $v\prec _{\frak C} w, \mathcal {L}\in \nabla$.

Remark 2.2 (a) Using the identification in (1), $\mathcal {L}\in \nabla$ is the same as a fractional line bundle.

(b) The degree condition only depends on the choice of $\nabla$, not the fractional line bundle $\mathcal {L}\in \nabla$ itself. Moreover, the normalization condition does not depend on the alcove $\nabla$.

(c) We have duality [Reference Okounkov and SmirnovOS16, Proposition 1]

(3)\begin{equation} \langle \operatorname{stab}^{{\frak C}, T^{1/2}, \nabla}_v, \operatorname{stab}^{-{\frak C}, T^{1/2}_{\operatorname{opp}}, -\nabla}_w\rangle =\delta_{v,w}. \end{equation}

(d) The existence of the K-theory stable bases also follows from the existence of the elliptic stable envelopes, see [Reference Aganagic and OkounkovAO21, Reference OkounkovOko20a, Reference OkounkovOko20b].

It follows immediately from the definition that we have the following lemma.

Lemma 2.3 For any $w\in W$, we have

\[ \operatorname{stab}^{{\frak C}, T^{1/2}, \nabla}_w|_w \cdot \operatorname{stab}^{-{\frak C}, T^{1/2}_{\operatorname{opp}}, -\nabla}_w|_w=\bigwedge{}^\bullet T_w(T^*\mathcal{B}). \]

We will mostly consider the following two cases:

\[ \operatorname{stab}^{+, {\nabla}}_{w}:=\operatorname{stab}^{{\frak C}_+, T\mathcal{B}, \nabla}_w, \quad \operatorname{stab}^{-,{\nabla}}_{w}:=\operatorname{stab}^{{\frak C}_-, T^*\mathcal{B}, \nabla}_w. \]

From [Reference Su, Zhao and ZhongSZZ20, Lemma 3.2], which follows immediately from Definition 2.1(ii), we have

(4)\begin{align} \operatorname{stab}^{+, {\nabla}}_{y}|_y&=q^{-\ell(y)/2}\prod_{\beta>0, y\beta<0}(q-e^{y\beta})\prod_{\beta>0,y\beta>0}(1-e^{y\beta}), \end{align}

(5)\begin{align} \operatorname{stab}^{-,{\nabla}}_{y}|_y&=q^{\ell(y)/2}\prod_{\beta>0, y\beta<0}(1-e^{{-}y\beta})\prod_{\beta>0, y\beta>0}(1-qe^{{-}y\beta}). \end{align}

Moreover, it is easy to see from definition that

(6)\begin{equation} \operatorname{stab}^{+, {\nabla}}_{id}=[\mathcal{O}_{T_{id}^*\mathcal{B}}], \quad \text{and}\quad \operatorname{stab}^{-,{\nabla}}_{w_0}=({-}q^{1/2})^{\dim G/B}e^{2\rho}[\mathcal{O}_{T_{w_0}^*\mathcal{B}}], \end{equation}

where $w_0\in W$ is the longest element in the Weyl group.

2.3 The affine Hecke algebra

We give a brief reminder on the Hecke algebra and Demazure–Lusztig operators.

Let ${\mathbb {H}}$ be the affine Hecke algebra, see [Reference Chriss and GinzburgCG97, § 7.1]. Then we have the following well-known Kazhdan–Lusztig and Ginzburg isomorphism [Reference Kazhdan and LusztigKL87, Reference Chriss and GinzburgCG97]

(7)\begin{equation} {\mathbb{H}} \simeq K_{G\times \mathbb{C}^*}(Z), \end{equation}

where $Z=T^*\mathcal {B}\times _\mathcal {N} T^*\mathcal {B}$ is the Steinberg variety and $\mathcal {N}\subset \mathfrak {g}$ is the nilpotent cone. The right-hand side $K_{G\times \mathbb {C}^*}(Z)$ has a convolution algebra structure. Let us recall one construction of this isomorphism [Reference RicheRic08, Reference LusztigLus98].

Let $\triangle (T^*\mathcal {B})\subset T^*\mathcal {B}\times _{\mathcal {N}} T^*\mathcal {B}$ denote the diagonal copy of $T^*\mathcal {B}$ inside the Steinberg variety $Z$. For any torus weight $\lambda$, let $\mathcal {O}_\triangle (\lambda ):=\mathcal {L}_\lambda$, where $\mathcal {L}_\lambda$ is the line bundle on $T^*\mathcal {B}$ introduced in § 2.1. For any simple root $\alpha$, let $P_\alpha$ be the corresponding minimal parabolic subgroup and $\mathcal {P}_\alpha =G/P_\alpha$. Let

\[ Y_\alpha=\mathcal{B}\times_{\mathcal{P}_\alpha}\mathcal{B}\subset \mathcal{B}\times\mathcal{B}. \]

On $Y_\alpha$, the $A$-fixed points are of the form either $(w, w)$ or $(w, ws_\alpha )$ with $w\in W$. Let $T^*_{Y_\alpha }:=N^*_{\mathcal {B}\times \mathcal {B}/Y_\alpha }$ be the conormal bundle of $Y_\alpha$ inside $\mathcal {B}\times \mathcal {B}$. Let $\mathcal {O}_{T_{Y_{\alpha }}^*}(\lambda ,\mu )=\pi _1^*\mathcal {L}_\lambda \otimes \pi _2^*\mathcal {L}_\mu$, where $\pi _i:T^*_{Y_\alpha }\to T^*\mathcal {B}$ are the two projections. Let $T_\alpha$ be the usual generator of ${\mathbb {H}}$ that satisfies $(T_\alpha +1)(T_\alpha -q)=0$. For the purpose of this paper, let us choose the isomorphism (7) as follows:

\[ T_\alpha\mapsto -[\mathcal{O}_{{\bigtriangleup}}]-[\mathcal{O}_{T_{Y_{\alpha}}^*}(0,\alpha)],\quad \text{and} \quad e^\lambda\mapsto \mathcal{O}_\triangle(\lambda). \]

This is conjugate to the one in [Reference RicheRic08, Proposition 6.1.5] by $\mathcal {L}_\rho$, since our $q$ and $T_\alpha$ are the same as $v^2$ and $vT_\alpha$ in [Reference RicheRic08], respectively.

The convolution algebra $K_{G\times \mathbb {C}^*}(Z)$ is a subalgebra of $K_{A\times \mathbb {C}^*}(Z)$, and hence defines two actions of ${\mathbb {H}}$ on $K_{A\times \mathbb {C}^*}(T^*\mathcal {B})$, namely, by convolution from the left, or from the right (which therefore is a right action). The operators on $K_{A\times \mathbb {C}^*}(T^*\mathcal {B})$ corresponding to these two convolutions with $T_\alpha$ will then be denoted by $T_\alpha$ and $T'_\alpha$ respectively, following the notations used in [Reference Su, Zhao and ZhongSZZ20].Footnote ¹ That is, for any $\mathcal {F}\in K_{A\times \mathbb {C}^*}(T^*\mathcal {B})$,

\[ T_\alpha(\mathcal{F})={-}\mathcal{F}-\pi_{1*}(\pi_2^*\mathcal{F}\otimes\pi_2^*\mathcal{L}_\alpha),\quad \text{and}\quad T'_\alpha(\mathcal{F})={-}\mathcal{F}-\pi_{2*}(\pi_1^*\mathcal{F}\otimes \pi_2^*\mathcal{L}_\alpha). \]

Let $D_\alpha :=-T_\alpha -1$ and $D'_\alpha :=-T'_\alpha -1$. Then we have adjointness [Reference Su, Zhao and ZhongSZZ20, Lemma 4.4]

(8)\begin{equation} \langle D_\alpha(\mathcal{F}), \mathcal{G}\rangle =\langle \mathcal{F}, D'_\alpha (\mathcal{G})\rangle, \quad \forall \,\mathcal{F},\mathcal{G}\in K_{T}(T^*\mathcal{B}). \end{equation}

The relation between these two operators $T_\alpha$ and $T'_\alpha$ can be deduced as follows. Since $[\mathcal {O}_{T_{Y_\alpha }^*}(\rho ,-\rho +\alpha )]=[\mathcal {O}_{T_{Y_\alpha }^*}(-\rho +\alpha ,\rho )]$ (see [Reference RicheRic08, Lemma 1.5.1]), the two operators on $K_{A\times \mathbb {C}^*}(T^*\mathcal {B})$, given by left and right convolutions with this sheaf, are equal to each other. In other words, for any $\mathcal {F}\in K_{A\times \mathbb {C}^*}(T^*\mathcal {B})$,

\[ \pi_{1*}(\pi_2^*\mathcal{F}\otimes[\mathcal{O}_{T_{Y_\alpha}^*}(\rho,-\rho+\alpha)]) =\pi_{2*}(\pi_1^*\mathcal{F}\otimes[\mathcal{O}_{T_{Y_\alpha}^*}(\rho,-\rho+\alpha)]). \]

Therefore, we get the following equality as operators on $K_{A\times \mathbb {C}^*}(T^*\mathcal {B})$:

\[ \mathcal{L}_{\rho}T_\alpha\mathcal{L}_{-\rho}=\mathcal{L}_{-\rho} T'_\alpha\mathcal{L}_{\rho}. \]

The operator of the left action is denoted by $T^L_\alpha :=\mathcal {L}_{\rho }T_\alpha \mathcal {L}_{-\rho }$ and the right action is denoted by $T^{\operatorname {R}}_\alpha :=\mathcal {L}_{-\rho } T'_\alpha \mathcal {L}_{\rho }$.

Remark 2.4 In the present paper, compositions of operators are read from right to left, even for right action operators. So for any $K$-theory class $\mathcal {F}\in K_{A\times \mathbb {C}^*}(T^*(G/B))$, $\mathcal {L}_\rho (T_{\alpha }(\mathcal {L}_{-\rho }\otimes \mathcal {F}))= \mathcal {L}_{-\rho }T'_{\alpha }(\mathcal {L}_{\rho }\otimes \mathcal {F})$. For any reduced decomposition $w=s_1\cdots s_k$, we define $T_w^L:=T^L_{s_1}\cdots T^L_{s_k}$ and $T^{\operatorname {R}}_w:=T^{\operatorname {R}}_{s_k}\cdots T^{\operatorname {R}}_{s_1}$. On the other hand, in [Reference Su, Zhao and ZhongSZZ20] a left action of the affine Hecke algebra through right convolution operators were used. In particular, under the notations of [Reference Su, Zhao and ZhongSZZ20] we have $T'_{s_1\cdots s_k}:=T'_{s_1}\cdots T'_{s_k}$. Compared to the notations in the present paper we have, for any $x\in W$,

(9)\begin{equation} T^{\operatorname{R}}_x=\mathcal{L}_{-\rho}T'_{x^{{-}1}}\mathcal{L}_\rho. \end{equation}

The following is one of the main results in [Reference Su, Zhao and ZhongSZZ20].

Theorem 2.5 [Reference Su, Zhao and ZhongSZZ20, Theorem 4.5]

Let $\alpha$ be a simple root. Then

(10)\begin{align} T_\alpha(\operatorname{stab}^{-,{\nabla_+}}_{w})&=\left\{ \begin{array}{@{}ll} (q-1)\operatorname{stab}^{-,{\nabla_+}}_{w}+q^{1/2}\operatorname{stab}^{-,{\nabla_+}}_{ws_\alpha} & \text{if }ws_\alpha< w,\\ q^{1/2}\operatorname{stab}^{-,{\nabla_+}}_{ws_\alpha} & \text{if }ws_\alpha>w.\end{array}\right. \end{align}

(11)\begin{align} T'_\alpha(\operatorname{stab}^{+, {\nabla_-}}_{w})&=\left\{\begin{array}{@{}ll}(q-1)\operatorname{stab}^{+, {\nabla_-}}_{w}+q^{1/2}\operatorname{stab}^{+, {\nabla_-}}_{ws_\alpha} & \text{if }ws_\alpha< w,\\ q^{1/2}\operatorname{stab}^{+, {\nabla_-}}_{ws_\alpha} & \text{if }ws_\alpha>w.\end{array}\right. \end{align}

In particular,

\[ \operatorname{stab}^{-,{\nabla_+}}_{w}=q^{{\ell(w_0w)}/{2}}T^{{-}1}_{w_0w}(\operatorname{stab}^{-,{\nabla_+}}_{w_0}), \quad \text{and}\quad \operatorname{stab}^{+, {\nabla_-}}_{w}=q^{-{\ell(w)}/{2}}T'_{w^{{-}1}}(\operatorname{stab}^{+, {\nabla_-}}_{id}). \]

3.1 Duality of coherent sheaves

Since $T^*\mathcal {B}$ is smooth, $K_{T}(T^*\mathcal {B})$ is generated by vector bundles. For each vector bundle $\mathcal {F}$, there is a dual vector bundle $\mathcal {F}^\vee$. This operation is well defined on $K_{T}(T^*\mathcal {B})$, and gives the duality operation

\begin{align*}\mathcal{F}\mapsto (\mathcal{F})^\vee,\end{align*}

which sends $q$ to $q^{-1}$. Then we have the following relations.

Lemma 3.1 Let $w\in W$. Then

(12)\begin{align} ({-}q)^{\dim \mathcal{B}}\mathcal{L}_{{-}2\rho}\otimes(\operatorname{stab}^{-,{\nabla}}_{w})^\vee &=\operatorname{stab}^{-,{-\nabla}}_{w}\in K_{T}(T^*\mathcal{B}), \end{align}

(13)\begin{align} ({-}1)^{\dim \mathcal{B}}\mathcal{L}_{2\rho}\otimes(\operatorname{stab}^{+, {\nabla}}_{w})^\vee & =\operatorname{stab}^{+, {-\nabla}}_{w}\in K_{T}(T^*\mathcal{B}). \end{align}

Proof. We first show that (12) implies (13). Indeed, by duality (3) and (12), we have

\begin{align*} \delta_{w,z}&=\langle ({-}q)^{\dim \mathcal{B}}\mathcal{L}_{{-}2\rho}\otimes(\operatorname{stab}^{-,{\nabla}}_{w})^\vee, \operatorname{stab}^{+, {\nabla}}_{z}\rangle\\ &=({-}q)^{\dim \mathcal{B}}\sum_{y\in W}\frac{e^{{-}2y\rho}\ast(\operatorname{stab}^{-,{\nabla}}_{w}|_y) \operatorname{stab}^{+, {\nabla}}_{z}|_y}{\prod_{\alpha>0}(1-e^{y\alpha})(1-qe^{{-}y\alpha})}, \end{align*}

where $\ast$ in the second line acts on $K_T(pt)$ by sending $e^\lambda$ to $e^{-\lambda }$ and $q$ to $q^{-1}$. Applying $\ast$ to the above identity, we get

\begin{align*} \delta_{w,z}&=({-}q)^{-\dim \mathcal{B}}\sum_{y\in W}\frac{e^{2y\rho}\operatorname{stab}^{-,{\nabla}}_{w}|_y \ast(\operatorname{stab}^{+, {\nabla}}_{z}|_y)}{\prod_{\alpha>0}(1-e^{{-}y\alpha})(1-q^{{-}1}e^{y\alpha})}\\ &=({-}1)^{\dim \mathcal{B}}\sum_{y\in W}\frac{e^{2y\rho}\operatorname{stab}^{-,{\nabla}}_{w}|_y \ast(\operatorname{stab}^{+, {\nabla}}_{z}|_y)}{\prod_{\alpha>0}(1-e^{y\alpha})(1-qe^{{-}y\alpha})}\\ &=\langle \operatorname{stab}^{-,{\nabla}}_{w}, ({-}1)^{\dim \mathcal{B}}\mathcal{L}_{2\rho}\otimes(\operatorname{stab}^{+, {\nabla}}_{z})^\vee\rangle. \end{align*}

Again by duality (3), we then get (13).

Now we prove (12). It suffices to show that $(-q)^{\dim \mathcal {B}}\mathcal {L}_{-2\rho }\otimes (\operatorname {stab}^{-,{\nabla }}_{w})^\vee$ satisfies the defining properties of $\operatorname {stab}^{-,{-\nabla }}_{w}$. The support condition is obvious. The normalization condition is checked as follows:

\begin{align*} \big(({-}q)^{\dim \mathcal{B}}\mathcal{L}_{{-}2\rho}\otimes(\operatorname{stab}^{-,{\nabla}}_{w})^\vee\big)|_w &= ({-}q)^{\dim \mathcal{B}}e^{{-}2w\rho}\ast(\operatorname{stab}^{-,{\nabla}}_{w}|_w)\\ & \overset{5}= ({-}q)^{\dim \mathcal{B}}q^{-({\ell(w)}/{2})}e^{{-}2w\rho} \prod_{\beta>0, w\beta<0}(1-e^{w\beta})\prod_{\beta>0, w\beta>0}(1-q^{{-}1}e^{w\beta}) \\ &= q^{{\ell(w)}/{2}}\prod_{\beta>0, w\beta<0}(1-e^{{-}w\beta})\prod_{\beta>0, w\beta>0}(1-qe^{{-}w\beta})\\ & \overset{5}= \operatorname{stab}^{-,{-\nabla}}_{w}|_w. \end{align*}

Therefore, the normalization condition is verified.

Finally, we check the degree condition. Pick $\mathcal {L}\in -\nabla$. We need to check that for any $y> w$ the following is true:

\[ \deg_A ({-}q)^{\dim \mathcal{B}}\big(\mathcal{L}_{{-}2\rho}\otimes(\operatorname{stab}^{-,{\nabla}}_{w})^\vee\big)|_y\subset \deg_A ({-}q)^{\dim \mathcal{B}}\big(\mathcal{L}_{{-}2\rho}\otimes(\operatorname{stab}^{-,{\nabla}}_{y})^\vee\big)|_y+\mathcal{L}|_y-\mathcal{L}|_w. \]

Indeed, this in turn is equivalent to

\[ \deg_A \operatorname{stab}^{-,{\nabla}}_{w}|_y\subset \deg_A \operatorname{stab}^{-,{\nabla}}_{y}|_y+\mathcal{L}|_w-\mathcal{L}|_y= \deg_A\operatorname{stab}^{-,{\nabla}}_{y}|_y+\mathcal{L}^{{-}1}|_y-\mathcal{L}^{{-}1}|_{w} . \]

Since $\mathcal {L}^{-1}\in \nabla$, the above identity is precisely the degree condition for the stable basis $\operatorname {stab}^{-,{\nabla }}_{w}$. Therefore, the degree condition also follows.

3.2 Weyl group action

Recall on $K_T(T^*\mathcal {B})$, we have a left Weyl group action induced from the $G$-action on $T^*\mathcal {B}$. Using equivariant localization, $w(\mathcal {F})$ for $\mathcal {F}\in K_T(T^*\mathcal {B})$ and $w\in W$ is determined by its restrictions to the fixed points, which is given by

(14)\begin{equation} w(\mathcal{F})|_v=w(\mathcal{F}|_{w^{{-}1}v}), \quad \mathcal{F}\in K_{T}(T^*\mathcal{B}). \end{equation}

Moreover, the action by $w$ also acts on the base field $K_T(pt)$ by the usual Weyl group action. From the above formula, it is easy to get

\[ w(\iota_{y})|_{v}=\left\{\begin{array}{@{}ll} w(\iota_{y}|_y) & \text{if }v=wy,\\ 0 & \text{otherwise}.\end{array} \right. \]

That is, $w(\iota _{y})=\iota _{wy}$. This action agrees with the $\odot$-action defined in [Reference Lenart, Zainoulline and ZhongLZZ20, Definition 3.2]. The effect of this group action on a stable basis is the following, which can be proved in exactly the same way as the proof of (12) in Lemma 3.1 above.

Lemma 3.3 [Reference Aluffi, Mihalcea, Schürmann and SuAMSS19, Lemma 11.1(a)]

For any $w, y\in W$, we have

\[ w(\operatorname{stab}^{\mathfrak{C},T^{1/2},\nabla}_y)=\operatorname{stab}^{w\mathfrak{C},w(T^{1/2}),\nabla}_{wy}. \]

3.3 Wall-crossing matrix

Our computation of wall-crossings for the K-theoretic stable bases is based on the following.

Lemma 3.5 [Reference Okounkov and SmirnovOS16, Theorem 1]

Suppose the alcoves $\nabla _1$ and $\nabla _2$ are adjacent and separated by $H_{\alpha ^\vee ,n}$. For any $y\in W$, we have

\[ \operatorname{stab}^{{\frak C}, T^{1/2}, \nabla_1}_y=\left\{\begin{array}{@{}ll}\operatorname{stab}^{{\frak C}, T^{1/2}, \nabla_2}_y+f^{\nabla_1\leftarrow \nabla_2}_y\operatorname{stab}_{ys_\alpha}^{{\frak C}, T^{1/2}, \nabla_2} & \text{if }ys_\alpha\prec_{\frak C} y,\\ \operatorname{stab}^{{\frak C}, T^{1/2}, \nabla_2}_y & \text{if } ys_\alpha \succ_{\frak C} y, \end{array}\right. \]

where $f_y^{\nabla _1\leftarrow \nabla _2}\in K_{T}(\operatorname {pt})$. If $n=0$, $f_y^{\nabla _1\leftarrow \nabla _2}\in K_{\mathbb {C}^*}(\operatorname {pt})$, i.e. it does not depend on the equivariant parameters of $A$. In particular,

\[ f_y^{\nabla_1\leftarrow \nabla_2}={-}f_y^{\nabla_2\leftarrow \nabla_1}. \]

Proof. Let us first recall the moment map $\mu :(T^*\mathcal {B})^A\to H_2(T^*\mathcal {B}, {\mathbb {Z}})\otimes \Lambda$ defined in [Reference Okounkov and SmirnovOS16, § 2.1.7]. It is characterized by the following condition. If $p$ and $q$ are two fixed points connected by a torus invariant curve $C$, then

\[ \mu(p)-\mu(q)=[C]\otimes \textit{weight}(T_pC). \]

This is also closely related to the Goresky–Kottwitz–MacPherson (GKM) condition in equivariant cohomology, see [Reference Maulik and OkounkovMO19, § 4.8.5].

Identifying $H_2(T^*\mathcal {B}, {\mathbb {Z}})$ with the coroot lattice, we get

\[ \mu(y)-\mu(ys_\alpha)=\alpha^\vee{\otimes}({-}y\alpha). \]

For any other $z$ different from $y$ and $ys_\alpha$, $\mu (y)-\mu (z)$ will not be a multiple of $\alpha ^\vee$.Footnote ² The conclusion then follows from [Reference Okounkov and SmirnovOS16, Theorem 1].

We are interested in computing these wall-crossing coefficients $f_y^{\nabla _1\leftarrow \nabla _2}$, which form the entries of the so-called wall R-matrix [Reference Okounkov and SmirnovOS16, § 2.2.3]. Similarly as stable bases, these matrices depend on the choice of the alcove, the chamber, and the polarization. For simplicity of notations, we omit the last two choices whenever they are clear from the context.

From the definition, those wall-crossing coefficients depend on the wall of the alcove lying on the hyperplane $H_{\alpha ^\vee ,n}$. However, it follows from Theorem 5.1 below, that the coefficients do not depend on the wall where we cross the hyperplane $H_{\alpha ^\vee ,n}$.

Example 3.6 We consider the case $\operatorname {SL}(2,\mathbb {C})$.Footnote ³ Then $\mathcal {B}\cong {\mathbb {P}}^1$, and $\operatorname {Pic}(T^*{\mathbb {P}}^1)\otimes _{\mathbb {Z}} \mathbb {Q}=\mathbb {Q}\alpha$. The two $A$-fixed points on $T^*{\mathbb {P}}^1$ are denoted by $0$ and $\infty$, corresponding to $e$ and $s_\alpha$ in the Weyl group, respectively. The alcoves are $(({n}/{2})\alpha ,(({n+1})/{2})\alpha ), n\in {\mathbb {Z}}$. The wall $H_{\alpha ^\vee , 0}$ is the origin $0$. We have $\nabla _+= (0,\frac {1}{2}\alpha )$ and $\nabla _-=(\frac {1}{2}\alpha , 0)=\nabla _+-({\alpha }/{2})$. Using the translation formula of Lemma 3.7, we compute $f_{s_\alpha }^{\nabla _-\leftarrow \nabla _+}$ for the dominant chamber and the polarization $T{\mathbb {P}}^1$, in which case $0\prec _{{\frak C}_+}\infty$. The stable basis is given by the following formulas [Reference Su, Zhao and ZhongSZZ20, Example 2.4]:

\begin{gather*} \operatorname{stab}^{+, {\nabla_-}}_{e}=[\mathcal{O}_{T_0^*{\mathbb{P}}^1}],\\ \operatorname{stab}^{+, {\nabla_-}}_{s_\alpha}={-}q^{-{1/2}}e^{-\alpha}[\mathcal{O}_{{\mathbb{P}}^1}]+ \big({-}q^{{1/2}}e^{{-}2\alpha}+(q^{-{1/2}}-q^{{1/2}})e^{-\alpha}\big)[\mathcal{O}_{T_0^*{\mathbb{P}}^1}]. \end{gather*}

So their localizations are given by

\begin{gather*} \operatorname{stab}^{+, {\nabla_-}}_{e}|_e=1-e^{\alpha}, \\ \operatorname{stab}^{+, {\nabla_-}}_{s_\alpha}|_e=q^{{1/2}}-q^{-{1/2}}, \quad \operatorname{stab}^{+, {\nabla_-}}_{s_\alpha}|_{s_\alpha}=q^{{1/2}}-q^{-{1/2}}e^{-\alpha}. \end{gather*}

By (15), for any $y,w\in W$,

\[ \operatorname{stab}^{+, {\nabla_+}}_{y}|_w = e^{-{1/2}y\alpha+{1/2}w\alpha}\operatorname{stab}^{+, {\nabla_-}}_{y}|_w. \]

Therefore,

\begin{gather*} \operatorname{stab}^{+, {\nabla_+}}_{e}|_e=1-e^{\alpha},\\ \operatorname{stab}^{+, {\nabla_+}}_{s_\alpha}|_e =(q^{{1/2}}-q^{-{1/2}})e^\alpha, \quad \operatorname{stab}^{+, {\nabla_+}}_{s_\alpha}|_{s_\alpha}=q^{{1/2}}-q^{-{1/2}}e^{-\alpha}. \end{gather*}

By Lemma 3.5,

\[ \operatorname{stab}^{+, {\nabla_-}}_{s_\alpha}=\operatorname{stab}^{+, {\nabla_+}}_{s_\alpha}+f_{s_\alpha}^{\nabla_-\leftarrow \nabla_+}\operatorname{stab}^{+, {\nabla_+}}_{e}. \]

Restricting both sides to the fixed point $e$, we get

\[ q^{{1/2}}-q^{-{1/2}}=(q^{{1/2}}-q^{-{1/2}})e^\alpha+f_{s_\alpha}^{\nabla_- \leftarrow \nabla_+}(1-e^{\alpha}). \]

Hence

\[ f_{s_\alpha}^{\nabla_-\leftarrow \nabla_+}=q^{{1/2}}-q^{-{1/2}}. \]

3.4 Translations

We consider the effect of translation by an integral weight $\mu \in \Lambda$ on a stable basis. By the uniqueness of stable basis, we have (see [Reference Aluffi, Mihalcea, Schürmann and SuAMSS19, Lemma 8.2.(c)])

(15)\begin{equation} \operatorname{stab}^{{\frak C},T^{1/2},\nabla+\mu}_y=e^{{-}y\mu}\mathcal{L}_\mu\otimes \operatorname{stab}^{\mathfrak{C},T^{1/2},\nabla}_y. \end{equation}

An immediate corollary of this fact is the following.

Lemma 3.7 Let $\nabla _1,\nabla _2$ are adjacent alcoves separated by $H_{\alpha ^\vee , n}$. For any integral weight $\mu \in \Lambda$, we let $\nabla +\mu$ be the alcove obtained by translating $\nabla$ by $\mu$. Then

\[ f_y^{\nabla_1+\mu\leftarrow \nabla_2+\mu}=e^{-(\mu,\alpha^\vee)y\alpha}f_y^{\nabla_1\leftarrow \nabla_2}. \]

Proof. By Lemma 3.5 and (15),

\[ f_y^{\nabla_1\leftarrow \nabla_2}=e^{y\mu-ys_\alpha\mu}f_y^{\nabla_1+\mu\leftarrow \nabla_2+\mu}. \]

Hence the conclusion follows from the identity $\mu -s_\alpha \mu =(\mu , \alpha ^\vee )\alpha$.

With this lemma, we can reduce the wall-crossing for $H_{\alpha ^\vee , n}$ to $H_{\alpha ^\vee , 0}$ as follows. Let $\nabla _1,\nabla _2$ be adjacent alcoves separated by $H_{\alpha ^\vee , n}$. If $\alpha$ is simple, then $\nabla _1-n\varpi _\alpha ,\nabla _2-n\varpi _\alpha$ are adjacent alcoves separated by $H_{\alpha ^\vee , 0}$, where $\varpi _\alpha$ is the corresponding fundamental weight. Otherwise, $\alpha =w\beta$ for some $w\in W$ and simple root $\beta$. Then we can shift the alcoves $\nabla _i$ by $-nw(\varpi _\beta )$ to get adjacent alcoves separated by $H_{\alpha ^\vee , 0}$.

Therefore, we only need to cross the walls on the root hyperplanes $H_{\alpha ^\vee , 0}$. These are divided into two cases depending on whether $\alpha$ is simple or not.

4.1 The formulas

The following is the main result of this section.

Theorem 4.1 Let $\alpha ^\vee$ be a simple coroot. Suppose $\nabla$ has a wall on $H_{\alpha ^\vee , 0}$, and $(\lambda , \alpha ^\vee )>0$, $\forall \, \lambda \in \nabla$. Then

(16)\begin{align} \operatorname{stab}^{+, {s_\alpha\nabla}}_{y}&= \left\{ \begin{array}{@{}ll} \operatorname{stab}^{+, {\nabla}}_{y}+(q^{{1/2}}-q^{-{1/2}})\cdot\operatorname{stab}^{+, {\nabla}}_{ys_{\alpha}} & \text{if } ys_{\alpha}< y,\\ \operatorname{stab}^{+, {\nabla}}_{y} & \text{if } ys_{\alpha}>y, \end{array}\right.\end{align}

(17)\begin{align} \operatorname{stab}^{-,{s_\alpha\nabla}}_{y}&= \left\{ \begin{array}{@{}ll} \operatorname{stab}^{-,{\nabla}}_{y}+(q^{{1/2}}-q^{-{1/2}})\cdot\operatorname{stab}^{-,{\nabla}}_{ys_{\alpha}} & \text{if } ys_{\alpha}> y,\\ \operatorname{stab}^{-,{\nabla}}_{y} & \text{if } ys_{\alpha}< y. \end{array}\right. \end{align}

Proof. We first show that (16) follows from (17) via duality (3). By Lemma 3.5, we can assume $ys_\alpha < y$ and it suffices to compute

\begin{align*} &\langle \operatorname{stab}^{+, {s_\alpha\nabla}}_{y}, \operatorname{stab}^{-,{-\nabla}}_{ys_{\alpha}}\rangle\\ &\quad = \langle \operatorname{stab}^{+, {s_\alpha\nabla}}_{y}, \operatorname{stab}^{-,{-s_\alpha\nabla}}_{ys_{\alpha}} +(q^{{1/2}}-q^{-{1/2}})\operatorname{stab}^{-,{-s_\alpha\nabla}}_{y}\rangle\\ &\quad = q^{{1/2}}-q^{-{1/2}}, \end{align*}

where the first equality follows from (17) with the alcove being $s_\alpha (-\nabla )$ satisfying the condition of theorem.

Now we prove (17). From Lemma 3.5 it suffices to assume that $ys_\alpha >y$. Applying the operator $D_\alpha$ from § 2.3 to the identity

\[ \operatorname{stab}^{-,{\nabla}}_{y}+f^{s_\alpha\nabla\leftarrow\nabla}_y\operatorname{stab}^{-,{\nabla}}_{ys_\alpha}=\operatorname{stab}^{-,{s_\alpha \nabla}}_{y}, \]

and use (18) and (19) below, we get

\begin{align*} &-\operatorname{stab}^{-,{\nabla}}_{y}-q^{1/2}\operatorname{stab}^{-,{\nabla}}_{ys_\alpha}+\sum_{w\not< ys_\alpha }a_w\operatorname{stab}^{-,{\nabla}}_{w}\\ &\qquad +f^{s_\alpha\nabla\leftarrow\nabla}_y\bigg({-}q\operatorname{stab}^{-,{\nabla}}_{ys_\alpha}-q^{1/2}\operatorname{stab}^{-,{\nabla}}_{y} +\sum_{w\not< ys_\alpha }a_w\operatorname{stab}^{-,{\nabla}}_{w}\bigg)\\ &\quad ={-}q\operatorname{stab}^{-,{s_\alpha\nabla}}_{y}-q^{1/2}\operatorname{stab}^{-,{s_\alpha\nabla}}_{ys_\alpha}+\sum_{w\not< ys_\alpha}a_w\operatorname{stab}^{-,{s_\alpha\nabla}}_{w}\\ &\quad \overset{\text{Lemma 3.5}}={-}q(\operatorname{stab}^{-,{\nabla}}_{y}+f^{s_\alpha\nabla \leftarrow\nabla}_y\operatorname{stab}^{-,{\nabla}}_{ys_\alpha})-q^{1/2}\operatorname{stab}^{-,{\nabla}}_{ys_\alpha}+\sum_{w \not< ys_\alpha}a_w\operatorname{stab}^{-,{s_\alpha\nabla}}_{w}, \end{align*}

where the $a_w$ are some coefficients. These coefficients are determined by (18) and (19) in Lemma 4.4 below. By comparing the coefficients of $\operatorname {stab}^{-,{\nabla }}_{y}$, we see that $f^{s_\alpha \nabla \leftarrow \nabla }_y=q^{1/2}-q^{-1/2}$. Note that the sums in the above identity will have no contribution to such coefficients. The first two are obvious since $ys_\alpha >y$. The reason for the last one is that $\operatorname {stab}^{-,{s_\alpha} \nabla }_w$ is a linear combination of $\operatorname {stab}^{-,\nabla }_{w}$ and $\operatorname {stab}^{-,{\nabla }}_{ws_\alpha }$, and both will not be equal to $\operatorname {stab}^{-,{\nabla }}_{y}$ when $w\not < ys_\alpha$.

4.2 Some technical lemmas

In this subsection we prove Lemma 4.4, which is used in the proof of Theorem 4.1. First we recall a well-known fact.

Lemma 4.2 [Reference DeodharDeo77, Theorem 1.1]

Let $w_1, w_2\in W$, and $\alpha$ be a simple root. Assume $w_1s_\alpha < w_1$ and $w_2s_\alpha < w_2$. Then

\[ w_1\leq w_2\Leftrightarrow w_1s_\alpha\leq w_2\Leftrightarrow w_1s_\alpha\leq w_2s_\alpha. \]

In preparation for the proof of Lemma 4.4, we have the following lemma.

Proof. (1a)–(1d) are obvious. For (1e), we have $ys_\alpha =w\ge z\ge y$. So $z=y$ or $z=ys_\alpha$.

(2a) $w< y$ and $w\neq ys_\alpha$ imply that $z\ge y>w\ge zs_\alpha$, so

\[ \ell(z)\ge \ell(y)>\ell(w)\ge \ell(zs_\alpha)\ge \ell(z)-1. \]

So $\ell (z)=\ell (y)$ and $\ell (w)=\ell (zs_\alpha )$. That is, $z=y$ and $w=zs_\alpha$. Hence, $w=ys_\alpha$, which is a contradiction. So no such $z$ exists.

(2b) If $z=y$, then $w\ge ys_\alpha$, which contradicts $ys_\alpha >w$. So $y< z$.

If $zs_\alpha >z$, then $\ell (y)+1=\ell (ys_\alpha )> \ell (w)\geq \ell (zs_\alpha )=\ell (z)+1$. So $\ell (y)>\ell (z)$. This contradicts the assumption that $y\le z$.

If $zs_\alpha < z$, then $\ell (y)+1=\ell (ys_\alpha )>\ell (w)\geq \ell (zs_\alpha )=\ell (z)-1$. So $\ell (y)+2>\ell (z)> \ell (y)$ which forces $\ell (z)=\ell (y)+1$, i.e. $z=ys_\beta$ for some positive root $\beta$ such that $y\beta >0$. We claim $\alpha =\beta$. If not, then $s_\alpha \beta >0$. But $\ell (ys_\alpha )=\ell (y)+1=\ell (z)>\ell (zs_\alpha )=\ell (ys_\beta s_\alpha )=\ell (ys_\alpha s_{s_\alpha \beta })$, which implies $ys_\alpha (s_\alpha (\beta ))=y\beta <0$. This contradicts the assumption that $y\beta >0$. Therefore $\alpha =\beta$, and $z=ys_\alpha$.

Then $ys_\alpha >w\geq zs_\alpha =y$. Since $w\neq y$, such $w$ does not exist.

(2c) We have $z\ge y\ge zs_\alpha$. So $z=y$ or $z=ys_\alpha$. Since $z\geq y$, $z$ must be equal to $y$.

(2d) The proof is similar as that of (2c).

(2e) Since $zs_\alpha \le ys_\alpha < y\le z$, $z=y$.

(2f) It is easy to see that $z=y$ and $z=ys_\alpha$ satisfy the conditions. Now assume that $ys_\alpha >y, z>y$ and $ys_\alpha >zs_\alpha$. Then $\ell (y)+1=\ell (ys_\alpha )>\ell (zs_\alpha )\ge \ell (z)-1$. Hence, $\ell (y)+2>\ell (z)>\ell (y)$, which implies that $\ell (z)=\ell (y)+1$. It follows similarly as the proof of (2b) that $z=ys_\alpha$.

(3) This follows from a case-by-case study, using Lemma 4.2.

Now we are ready to prove the following technical lemma, which has been used in the proof of Theorem 4.1.

Proof. The proof is similar to that for [Reference Su, Zhao and ZhongSZZ20, Proposition 4.3].

Step 1. By duality (3),

\[ D_\alpha(\operatorname{stab}^{-,{\nabla}}_{y}) =\sum_{w\in W}a_w\operatorname{stab}^{-,{\nabla}}_{w}, \text{with }a_w=\langle D_\alpha(\operatorname{stab}^{-,{\nabla}}_{y}), \operatorname{stab}^{+, {-\nabla}}_{w}\rangle. \]

By the support condition of the stable basis, $a_w=\langle D_\alpha (\operatorname {stab}^{-,{\nabla }}_{y}), \operatorname {stab}^{+, {-\nabla }}_{w}\rangle$ is a proper integral. Hence $a_w\in K_T(\operatorname {pt})$. By localization formula and the support condition (see the first several lines in the proof of [Reference Su, Zhao and ZhongSZZ20, Proposition 4.3]),

(20)\begin{align} a_w=\sum_{y\leq z\leq w}\frac{\operatorname{stab}^{+, {-\nabla}}_{w}|_z\operatorname{stab}^{-,{\nabla}}_{y}|_z}{\bigwedge^\bullet T_z(T^*\mathcal{B})}\frac{e^{z\alpha}-q}{1-e^{z\alpha}} +\sum_{w\geq zs_\alpha\text{ and } y\leq z}\frac{\operatorname{stab}^{+, {-\nabla}}_{w}|_{zs_\alpha}\operatorname{stab}^{-,{\nabla}}_{y}|_z}{\bigwedge^\bullet T_z(T^*\mathcal{B})}\frac{1-qe^{{-}z\alpha}}{1-e^{{-}z\alpha}}e^{z\alpha}. \end{align}

From Lemma 4.3 above, we know that if $ys_\alpha < y$, $w< y$ and $w\not \in \{y, ys_\alpha \}$, or if $ys_\alpha >y$, $w< ys_\alpha$ and $w\not \in \{y, ys_\alpha \}$, then the two sums are empty, so $a_w=0$. Therefore, we just need to consider $a_y$ and $a_{ys_\alpha }$.

In the following Steps 2–5, we assume $ys_\alpha < y$, and starting from Step 6, we assume $ys_\alpha >y$.

Step 2. We compute $a_{ys_\alpha }$, assuming $ys_\alpha < y$. By Lemma 4.3, there is only one term in the second sum with $z=y$, then

\begin{align*} a_{ys_\alpha}&=\frac{\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{ys_\alpha}\operatorname{stab}^{-,{\nabla}}_{y}|_y}{\bigwedge^\bullet T_y(T^*\mathcal{B})}\frac{1-qe^{{-}y\alpha}}{1-e^{{-}y\alpha}}e^{y\alpha}\\ &\overset{\text{Lemma 2.3}}=\frac{\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{ys_\alpha}}{\operatorname{stab}^{+, {-\nabla}}_{y}|_y} \frac{1-qe^{{-}y\alpha}}{1-e^{{-}y\alpha}}e^{y\alpha}\\ &\overset{(4)}=\frac{q^{-\ell(ys_\alpha)/2}\prod_{\beta>0, ys_\alpha\beta<0}(q-e^{ys_\alpha\beta})\prod_{\beta>0, ys_\alpha\beta>0}(1-e^{ys_\alpha \beta})}{q^{-\ell(y)/2}\prod_{\beta>0, y\beta<0}(q-e^{y\beta})\prod_{\beta>0, y\beta>0}(1-e^{y\beta})} \frac{1-qe^{{-}y\alpha}}{1-e^{{-}y\alpha}}e^{y\alpha}\\ &\overset{\sharp}=q^{1/2}\frac{1-e^{{-}y\alpha}}{q-e^{y\alpha}} \frac{1-qe^{{-}y\alpha}}{1-e^{{-}y\alpha}}e^{y\alpha}\\ &={-}q^{{1/2}}, \end{align*}

where $\sharp$ follows from the facts $y\alpha <0$ and $s_\alpha \{\beta >0|\beta \neq \alpha \}=\{\beta >0|\beta \neq \alpha \}$, since $\alpha$ is simple.

Step 3. We consider $a_y$, assuming $ys_\alpha < y$. By Lemma 4.3 above, there is only one term in each sum with both $z=y$, and we have

\begin{align*} a_y&=\frac{\operatorname{stab}^{+, {-\nabla}}_{y}|_y\operatorname{stab}^{-,{\nabla}}_{y}|_y}{\bigwedge^\bullet T_y(T^*\mathcal{B})}\frac{e^{y\alpha}-q}{1-e^{y\alpha}} +\frac{\operatorname{stab}^{+, {-\nabla}}_{y}|_{ys_\alpha}\operatorname{stab}^{-,{\nabla}}_{y}|_y}{\bigwedge^\bullet T_y(T^*\mathcal{B})}\frac{1-qe^{{-}y\alpha}}{1-e^{{-}y\alpha}}e^{y\alpha}\\ &\overset{\text{Lemma 2.3}}=\frac{e^{y\alpha}-q}{1-e^{y\alpha}}+ \frac{\operatorname{stab}^{+, {-\nabla}}_{y}|_{ys_\alpha}\operatorname{stab}^{-,{\nabla}}_{y}|_y}{\bigwedge^\bullet T_y(T^*\mathcal{B})} \frac{1-qe^{{-}y\alpha}}{1-e^{{-}y\alpha}}e^{y\alpha}\\ &=\frac{e^{y\alpha}-q}{1-e^{y\alpha}}\bigg(1-\frac{\operatorname{stab}^{+, {-\nabla}}_{y}|_{ys_\alpha}\operatorname{stab}^{-,{\nabla}}_{y}|_y\cdot e^{y\alpha}}{\bigwedge^\bullet T_y(T^*\mathcal{B})}\bigg). \end{align*}

Step 4. In this step we compute $a_y$, assuming $(\lambda , \alpha ^\vee )>0, \lambda \in \nabla$. For any Laurent polynomial $g=\sum _{\mu \in I}c_\mu e^\mu$ and $\xi \in {\frak h}$, write

\[ g(\xi)=\sum_{\mu\in I}c_\mu e^{(\mu, \xi)}, \quad \operatorname{Max}_\xi(g)=\max_{\mu\in I}\{(\mu, \xi)\}, \quad \operatorname{Min}_\xi(g)=\min_{\mu\in I}\{(\mu, \xi)\}. \]

For example, for any $y\in W$,

(21)\begin{equation} \operatorname{Max}_\xi\Big(\bigwedge\nolimits^\bullet T_y(T^*\mathcal{B})\Big)={-}\operatorname{Min}_\xi\Big(\bigwedge\nolimits^\bullet T_y(T^*\mathcal{B})\Big)=\left\{\begin{array}{@{}ll} (2\rho, \xi) & \text{if }\xi\in {\frak C}_+,\\ -(2\rho, \xi) & \text{if }\xi\in {\frak C}_-. \end{array}\right. \end{equation}

Write

\[ f=\operatorname{stab}^{+, {-\nabla}}_{y}|_{ys_\alpha} \operatorname{stab}^{-,{\nabla}}_{y}|_y \cdot e^{y\alpha},\quad \text{and}\quad K_T(pt)\ni a_y=\sum_{\mu\in I}c_\mu e^\mu, c_\mu\in {\mathbb{Z}}[q^{1/2}, q^{{-}1/2}]. \]

Here $I\subset X^*(T)$ is a finite subset. Following [Reference Su, Zhao and ZhongSZZ20, Lemma 3.1], let $\xi$ be in ${\frak C}_+$ (so that $(2\rho , \xi )>0$) such that

(22)\begin{equation} (\mu, \xi)\in {\mathbb{Z}}\backslash\{0\}, \quad (\mu, \xi)\neq (\mu', \xi), \quad \forall\,\mu\neq \mu'\in I. \end{equation}

Since $ys_\alpha < y$, $y\alpha <0$, so $(y\alpha , \xi )<0$. Since $(\lambda , \alpha ^\vee )>0$, we adjust $\lambda \in \nabla$ close to $H_{\alpha ^\vee , 0}$ so that

\[{-}1<(y\lambda-ys_\alpha\lambda, \xi)=(\lambda, \alpha^\vee)(y\alpha, \xi)<0. \]

The degree condition together with [Reference Su, Zhao and ZhongSZZ20, (9)] implies that

\begin{align*} \operatorname{Max}_\xi(f)&\le \operatorname{Max}_\xi(\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{ys_\alpha})+( \mathcal{L}_{-\lambda}|_{ys_\alpha}-\mathcal{L}_{-\lambda}|_y, \xi)+\operatorname{Max}_\xi(\operatorname{stab}^{-,{\nabla}}_{y}|_y)+(y\alpha, \xi)\\ &=(2\rho, \xi)+(\lambda, \alpha^\vee)(y\alpha, \xi). \end{align*}

Since $\operatorname {Max}_\xi (f)\in {\mathbb {Z}}$, $\operatorname {Max}_\xi (f)\le (2\rho , \xi )-1$. Therefore, by using (21), we have

\[ \underset{t\to \infty}\lim \frac{f(t\xi)}{\bigwedge^\bullet T_y(T^*\mathcal{B})(t\xi)}=0. \]

So

\[ \underset{t\to \infty}\lim a_y(t\xi)= \underset{t\to \infty}\lim \bigg(\frac{e^{(y\alpha, t\xi)}-q}{1-e^{(y\alpha, t\xi)}}\bigg(1-\frac{f(t\xi)}{\bigwedge^\bullet T_y(T^*\mathcal{B})(t\xi)}\bigg)\bigg)=\frac{0-q}{1-0}\cdot (1-0)={-}q. \]

Similarly, we have [Reference Su, Zhao and ZhongSZZ20, (10)]

\begin{align*} \operatorname{Min} _\xi(f)&\ge \operatorname{Min}_\xi(\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{ys_\alpha})+(\mathcal{L}_{-\lambda}|_{ys_\alpha} -\mathcal{L}_{-\lambda}|_{y},\xi)+\operatorname{Min}_\xi(\operatorname{stab}^{-,{\nabla}}_{y}|_y)+(y\alpha, \xi)\\ &={-}(2\rho, \xi)+(\lambda, \alpha^\vee)(y\alpha, \xi). \end{align*}

Since $\operatorname {Min}_\xi (f_1)\in {\mathbb {Z}}$, so $\operatorname {Min}_\xi (f_1)\ge -(2\rho , \xi )$. From (21), we get that

\[ \underset{t\to -\infty}\lim\frac{f(t\xi)}{\bigwedge^\bullet T_y(T^*\mathcal{B})(t\xi)} \text{ is bounded}. \]

Therefore,

\[ \underset{t\to -\infty}\lim a_y(t\xi)= \underset{t\to -\infty}\lim\biggl(\frac{e^{(y\alpha, t\xi)}-q}{1-e^{(y\alpha, t\xi)}}\bigg(1-\frac{f(t\xi)}{\bigwedge^\bullet T_y(T^*\mathcal{B})(t\xi)}\bigg)\biggr), \]

which is bounded. Since $a_y\in K_T(\operatorname {pt})$, $a_y(t\xi )$ is a Laurent polynomial in variable $e^t$. The above two limits show that $a_y=-q$ is a constant Laurent polynomial.

Step 5. In this step we compute $a_y$ assuming $(\lambda , \alpha ^\vee )<0, \lambda \in \nabla$. We can pick $\xi \in {\frak C}_-$ (so that $(2\rho , \xi )<0$) satisfying (22). In this case, $(y\alpha , \xi )>0$, and we adjust $\lambda \in \nabla$ close to $H_{\alpha ^\vee , 0}$ so that

\[{-}1<(y\lambda-ys_\alpha\lambda, \xi)=(\lambda, \alpha^\vee)(y\alpha, \xi)<0. \]

Then similarly as in Step 4, we can show that

\[ \underset{t\to \infty}\lim a_y(t\xi)=\underset{t\to \infty}\lim \biggl(\frac{e^{(y\alpha, t\xi)}-q}{1-e^{(y\alpha, t\xi)}}\bigg(1-\frac{f(t\xi)}{\bigwedge^\bullet T_y(T^*\mathcal{B})(t\xi)}\bigg)\biggr)={-}1, \]

and $\lim _{t\to -\infty } a_y(t\xi )$ is bounded. Hence, $a_y=-1$.

Step 6. We now consider the case $ys_\alpha >y$. For $a_y$ with $ys_\alpha >y$, then in (20), $z=y$ in the first sum and $z=ys_\alpha$ in the second, so

\begin{align*} a_y&=\frac{\operatorname{stab}^{+, {-\nabla}}_{y}|_y\operatorname{stab}^{-,{\nabla}}_{y}|_y}{\bigwedge^\bullet T_y(T^*\mathcal{B})}\frac{e^{y\alpha}-q}{1-e^{y\alpha}}+ \frac{\operatorname{stab}^{+, {-\nabla}}_{y}|_{y}\operatorname{stab}^{-,{\nabla}}_{y}|_{ys_\alpha}}{\bigwedge^\bullet T_{ys_\alpha}(T^*\mathcal{B})}\frac{1-qe^{y\alpha}}{1-e^{y\alpha}}e^{{-}y\alpha}\\ &=\frac{e^{y\alpha}-q}{1-e^{y\alpha}}+\frac{\operatorname{stab}^{+, {-\nabla}}_{y}|_{y} \operatorname{stab}^{-,{\nabla}}_{y}|_{ys_\alpha}}{\bigwedge^\bullet T_{ys_\alpha}(T^*\mathcal{B})}\frac{1-qe^{y\alpha}}{1-e^{y\alpha}}e^{{-}y\alpha}. \end{align*}

Now let $k=\operatorname {stab}^{+, {-\nabla }}_{y}|_{y}\operatorname {stab}^{-,{\nabla }}_{y}|_{ys_\alpha }e^{-y\alpha }$.

Step 7. Suppose $(\lambda , \alpha ^\vee )>0, \forall \, \lambda \in \nabla$. We choose $\xi \in {\frak C}_+$ (so that $(2\rho , \xi )>0$ and $(y\alpha , \xi )>0$) satisfying properties similar as (22). We pick $\lambda \in \nabla$, so that

\[{-}1<{-}(\lambda, \alpha^\vee)(y\alpha, \xi)<0. \]

Then

\begin{align*} \operatorname{Max}_\xi(k)&\le \operatorname{Max}_\xi(\operatorname{stab}^{+, {-\nabla}}_{y}|_y)+\operatorname{Max}_\xi(\operatorname{stab}^{-,{\nabla}}_{ys_\alpha}|_{ys_\alpha}) +(\mathcal{L}_\lambda|_{ys_\alpha}-\mathcal{L}_\lambda|_y, \xi)-(y\alpha, \xi)\\ &=(2\rho, \xi)-(\lambda,\alpha^\vee)(y\alpha, \xi).\end{align*}

So

\[ \underset{t\to \infty}\lim a_y(t\xi)={-}1. \]

On the other hand, we have

\begin{align*} \operatorname{Min}_\xi(k)&\ge \operatorname{Min}_\xi(\operatorname{stab}^{+, {-\nabla}}_{y}|_y)+\operatorname{Min}_\xi(\operatorname{stab}^{-,{\nabla}}_{ys_\alpha}|_{ys_\alpha}) +(\mathcal{L}_\lambda|_{ys_\alpha}-\mathcal{L}_\lambda|_y, \xi)-(y\alpha, \xi) \\ &\ge ({-}2\rho, \xi)-(\lambda, \alpha^\vee)(y\alpha, \xi). \end{align*}

Since $\operatorname {Min}_\xi (k)\in {\mathbb {Z}}$, so $\operatorname {Min}_\xi (k)\ge (-2\rho , \xi )$, therefore,

\[ \underset{t\to -\infty}\lim a_y(t\xi) \text{ is bounded}. \]

So $a_y=-1$.

Step 8. If $(\lambda , \alpha ^\vee )<0, \forall \, \lambda \in \nabla$, then we can mimic Step 7 by picking $\xi \in {\frak C}_-$ (so that $(2\rho , \xi )<0$ and $(y\alpha , \xi )<0$), and pick $\lambda \in \nabla$ so that $-1<-(\lambda , \alpha ^\vee )(y\alpha , \xi )<0$; then

\[ \underset{t\to \infty}\lim a_y(t\xi)={-}q, \quad \text{and}\quad \underset{t\to -\infty}\lim a_y(t\xi) \text{ is bounded.} \]

So $a_y=-q.$

Step 9. We then compute $a_{ys_\alpha }$ assuming $ys_\alpha >y$. In (20), $z=y$ and $z=ys_\alpha$ in both sums, so we have

\begin{align*} a_{ys_\alpha}&=\frac{\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_y\operatorname{stab}^{-,{\nabla}}_{y}|_y}{\bigwedge^\bullet T_y(T^*\mathcal{B})}\frac{e^{y\alpha}-q}{1-e^{y\alpha}}+\frac{\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{ys_\alpha} \operatorname{stab}^{-,{\nabla}}_{y}|_{ys_\alpha}}{\bigwedge^\bullet T_{ys_\alpha}(T^*\mathcal{B})} \frac{e^{{-}y\alpha}-q}{1-e^{{-}y\alpha}}\\ &\quad+\frac{\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{ys_\alpha}\operatorname{stab}^{-,{\nabla}}_{y}|_{y}}{\bigwedge^\bullet T_{y}(T^*\mathcal{B})}\frac{1-qe^{{-}y\alpha}}{1-e^{{-}y\alpha}}e^{y\alpha}+ \frac{\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{y}\operatorname{stab}^{-,{\nabla}}_{y}|_{ys_\alpha}}{\bigwedge^\bullet T_{ys_\alpha}T^*\mathcal{B}}\frac{1-qe^{y\alpha}}{1-e^{y\alpha}}e^{{-}y\alpha}. \end{align*}

Write

\begin{gather*} f_1 =\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_y\operatorname{stab}^{-,{\nabla}}_{y}|_y, \quad f_2=\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{ys_\alpha}\operatorname{stab}^{-,{\nabla}}_{y}|_{ys_\alpha}, \\ f_3 =\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{ys_\alpha}\operatorname{stab}^{-,{\nabla}}_{y}|_{y}e^{y\alpha}, \quad f_4=\operatorname{stab}^{+, {-\nabla}}_{ys_\alpha}|_{y}\operatorname{stab}^{-,{\nabla}}_{y}|_{ys_\alpha}e^{{-}y\alpha}. \end{gather*}

Step 10. If $(\lambda , \alpha ^\vee )>0, \forall \,\lambda \in \nabla$, then we can choose $\xi \in {\frak C}_+$ satisfying (22) and $(y\alpha , \xi )>0$, and pick $\lambda \in \nabla$ satisfying $-1<-2(\lambda , \alpha ^\vee )(y\alpha , \xi )<0$; then

\[ \underset{t\to \infty}\lim f_1(t\xi)=\underset{t\to \infty}\lim f_2(t\xi)=\underset{t\to \infty}\lim f_4(t\xi)=0, \]

and it is easy to calculate from (5) and (4) that

\[ \frac{f_3}{\bigwedge^\bullet T_y(T^*\mathcal{B})}=q^{{-}1/2}\frac{q-e^{{-}y\alpha}}{1-e^{y\alpha}}. \]

Therefore,

\[ \underset{t\to \infty}\lim a_{ys_\alpha}(t\xi)={-}q^{1/2}, \quad \underset{t\to -\infty}\lim a_{ys_\alpha}(t\xi) \text{ is bounded. } \]

Similarly, if $(\lambda , \alpha ^\vee )<0, \forall \, \lambda \in \nabla$, then we can choose $\xi \in {\frak C}_-$ (so that $(y\alpha , \xi )<0$), and $\lambda \in \nabla$ satisfying $-1<-2(\lambda , \alpha ^\vee )(y\alpha , \xi )<0$; then

\[ \underset{t\to \infty}\lim a_{ys_\alpha}(t\xi)={-}q^{1/2}, \quad \underset{t\to -\infty}\lim a_{ys_\alpha}(t\xi) \text{ is bounded.} \]

Therefore, $a_{ys_\alpha }=-q^{1/2}$.

4.3 Arbitrary chamber

We generalize Theorem 4.1 to the setting of an arbitrary chamber. The choice of a chamber ${\frak C}$ determines a set of simple roots. Replacing the positive chamber $+$ in Theorem 4.1 by the chamber ${\frak C}$, we get the following result.

Theorem 4.5 Let $\alpha ^\vee$ be a simple coroot determined by the chamber ${\frak C}$. Suppose $\nabla$ has a wall on $H_{\alpha ^\vee , 0}$, and $(\lambda , \alpha ^\vee )>0, \forall \, \lambda \in \nabla$. Then

\begin{align*} \operatorname{stab}^{{\frak C}, T\mathcal{B}, s_\alpha\nabla}_{y}&= \left\{ \begin{array}{@{}ll} \operatorname{stab}^{{\frak C}, T\mathcal{B}, \nabla}_{y}+(q^{{1/2}}-q^{-{1/2}})\cdot\operatorname{stab}^{{\frak C}, T\mathcal{B}, \nabla}_{ys_{\alpha}} & \text{if } ys_{\alpha}\prec_{\frak C} y,\\ \operatorname{stab}^{{\frak C}, T\mathcal{B},\nabla}_{y} & \text{if } ys_{\alpha}\succ_{\frak C} y. \end{array}\right. \\ \operatorname{stab}^{-{\frak C}, T^*\mathcal{B}, s_\alpha\nabla}_{y}&= \left\{ \begin{array}{@{}ll} \operatorname{stab}^{-{\frak C}, T^*\mathcal{B}, \nabla}_{y}+(q^{{1/2}}-q^{-{1/2}})\cdot\operatorname{stab}^{-{\frak C}, T^*\mathcal{B}, \nabla}_{ys_{\alpha}} & \text{if } ys_{\alpha}\succ_{\frak C} y,\\ \operatorname{stab}^{-{\frak C}, T^*\mathcal{B}, \nabla}_{y} & \text{if } ys_{\alpha}\prec_{\frak C} y. \end{array}\right. \end{align*}

Thanks to this result, in the rest of the paper we focus on stable bases associated to the positive/negative chamber.

5.1 Crossing the non-simple walls

Theorem 5.1 Suppose $\nabla$ has a wall on $H_{\alpha ^\vee , 0}$ (so that $\nabla$ and $s_\alpha \nabla$ share a wall on $H_{\alpha ^\vee , 0}$), and $(\lambda ,\alpha ^\vee )>0, \forall \, \lambda \in \nabla$. Then

\begin{align*} \operatorname{stab}^{{\frak C}, T\mathcal{B},s_\alpha\nabla}_{y}&= \left\{ \begin{array}{@{}ll} \operatorname{stab}^{{\frak C}, T\mathcal{B},\nabla}_{y}+(q^{{1/2}}-q^{-{1/2}})\cdot\operatorname{stab}^{{\frak C}, T\mathcal{B}, \nabla}_{ys_{\alpha}} & \text{if } ys_{\alpha}\prec_{\frak C} y,\\ \operatorname{stab}^{{\frak C}, T\mathcal{B}, \nabla}_{y} & \text{if } ys_{\alpha}\succ_{\frak C} y. \end{array}\right. \\ \operatorname{stab}^{-{\frak C}, T^*\mathcal{B}, s_\alpha\nabla}_{y}&= \left\{ \begin{array}{@{}ll} \operatorname{stab}^{-{\frak C}, T^*\mathcal{B}, \nabla}_{y}+(q^{{1/2}}-q^{-{1/2}})\cdot\operatorname{stab}^{-{\frak C}, T^*\mathcal{B}, \nabla}_{ys_{\alpha}} & \text{if } ys_{\alpha}\succ_{\frak C} y,\\ \operatorname{stab}^{-{\frak C}, T^*\mathcal{B}, \nabla}_{y} & \text{if } ys_{\alpha}\prec_{\frak C} y. \end{array}\right. \end{align*}

Proof. It suffices to prove the first one, since the other one will follow from it via duality as in Theorem 4.1. Assume first that ${\frak C}={\frak C}_+$, then we just need to consider the case when $ys_\alpha < y$ due to Lemma 3.5. We have

\[ \operatorname{stab}^{{\frak C}_+, T\mathcal{B}, s_\alpha\nabla}_{y}=\operatorname{stab}^{{\frak C}_+, T\mathcal{B}, \nabla}_{y}+f_y^{s_\alpha\nabla\leftarrow\nabla}\cdot\operatorname{stab}^{{\frak C}_+, T\mathcal{B}, \nabla}_{ys_{\alpha}}, \]

where $f^{s_\alpha \nabla \leftarrow \nabla }\in K_{\mathbb {C}^*}(pt)$ due to Lemma 3.5. Suppose $\alpha ^\vee =w\beta ^\vee$ for some simple coroot $\beta ^\vee$ and $w\in W$. Applying $w$ to the above identity and using Lemma 3.3, we get

\[ \operatorname{stab}^{w({\frak C}_+),T\mathcal{B}, s_\alpha\nabla}_{wy}=\operatorname{stab}^{w({\frak C}_+),T\mathcal{B}, \nabla}_{wy}+f_y^{s_\alpha\nabla\leftarrow\nabla}\cdot\operatorname{stab}^{w({\frak C}_+),T\mathcal{B}, \nabla}_{wys_{\alpha}}. \]

Here we have used the fact that $w(T\mathcal {B})=T\mathcal {B}$ and $w(f_y^{s_\alpha \nabla \leftarrow \nabla })=f_y^{s_\alpha \nabla \leftarrow \nabla }$. From definition, we see that $ys_\alpha < y$ implies that $wys_\alpha \prec _{w{\frak C}_+}wy$. Moreover, since $\beta ^\vee$ is a simple coroot with respect to ${\frak C}_+$, $\alpha ^\vee =w\beta ^\vee$ is a simple coroot with respect to $w{\frak C}_+$, and $s_\alpha \nabla , \nabla$ are separated by the simple wall $H_{\alpha ^\vee , 0}$. Therefore, Theorem 4.5 implies that $f_y^{s_\alpha \nabla \leftarrow \nabla }=q^{1/2}-q^{-1/2}$.

Now if ${\frak C}$ is general, we need to compute $f^{s_\alpha \nabla \leftarrow \nabla }_y\in K_{\mathbb {C}^*}(\operatorname {pt})$ in

\[ \operatorname{stab}^{{\frak C}, T\mathcal{B}, s_\alpha\nabla}_y=\operatorname{stab}^{{\frak C}, T\mathcal{B}, \nabla}_y+f^{s_\alpha\nabla\leftarrow \nabla}_y\operatorname{stab}^{{\frak C}, T\mathcal{B}, \nabla}_{ys_\alpha}. \]

Write ${\frak C}_+=v{\frak C}$; then $ys_\alpha \prec _{\frak C} y$ implies $vys_\alpha < vy$. Applying $v$ to the above identity, we get

\[ \operatorname{stab}^{{\frak C}_+, T\mathcal{B}, s_\alpha\nabla}_{vy}=\operatorname{stab}^{{\frak C}_+, T\mathcal{B}, \nabla}_{vy}+f^{s_\alpha\nabla\leftarrow\nabla}_y\operatorname{stab}^{{\frak C}_+, T\mathcal{B}, \nabla}_{vys_\alpha}. \]

By using the first part of the proof, we see that $f_{y}^{s_\alpha \nabla \leftarrow \nabla }=q^{1/2}-q^{-1/2}$.

By translation, we can compute the wall-crossing coefficients for any wall $H_{\alpha ^\vee , n}$.

Proof. There exists an integral weight $\varpi _\alpha \in \Lambda$ such that, $(\varpi _\alpha , \alpha ^\vee )=1$. Indeed, if $\alpha$ is simple, take $\varpi _\alpha$ to be the corresponding fundamental weight. If $\alpha$ is not simple, there exists $v\in W$ such that $v\alpha$ is simple. We can take $\varpi _\alpha =v^{-1}\varpi _{v\alpha }$, where $\varpi _{v\alpha }$ is the fundamental weight corresponding to the simple root $v\alpha$. Then $\nabla _1':=\nabla _1-n\varpi _\alpha$ and $\nabla _2'=\nabla _2-n\varpi _\alpha$ are adjacent and separated by $H_{\alpha ^\vee , 0}$. Lemma 3.7 and Theorem 5.1 imply that

\[ f_y^{\nabla_1\leftarrow \nabla_2}=e^{{-}ny\alpha}f_y^{\nabla_1'\leftarrow \nabla_2'}=e^{{-}ny\alpha}(q^{1/2}-q^{{-}1/2}). \]

5.2 The affine Hecke algebra action

We can further express the wall-crossing formulas in terms of the affine Hecke algebra action. Let $q_y=q^{\ell (y)}, \epsilon _y=(-1)^{\ell (y)}$.

Theorem 5.4 For any $x, y\in W$, we have

(23)\begin{align} \operatorname{stab}^{-,{x\nabla_+}}_{y}&= q_x^{{-}1/2}T_x(\operatorname{stab}^{-,{\nabla_+}}_{yx}), \end{align}

(24)\begin{align} \operatorname{stab}^{+, {x\nabla_-}}_{y}&= q_x^{1/2}(T'_{x^{{-}1}})^{{-}1}(\operatorname{stab}^{+, {\nabla_-}}_{yx}). \end{align}

Proof. We first show that (23) implies (24). By the duality of stable bases (3) and the adjointness between $T_\alpha$ and $T'_\alpha$ (8), we get

\begin{align*} q_x^{1/2}(T_{x^{{-}1}}')^{{-}1}(\operatorname{stab}^{+, {\nabla_-}}_{yx})&=\sum_{v\in W}\langle q_x^{1/2}(T_{x^{{-}1}}')^{{-}1}(\operatorname{stab}^{+, {\nabla_-}}_{yx}), \operatorname{stab}^{-,{x\nabla_+}}_{v}\rangle \operatorname{stab}^{+, {x\nabla_-}}_{v}\\ &=\sum_{v\in W} \langle \operatorname{stab}^{+, {\nabla_-}}_{yx}, q_x^{1/2}T_{x}^{{-}1}(\operatorname{stab}^{-,{x\nabla_+}}_{v})\rangle\operatorname{stab}^{+, {x\nabla_-}}_{v}\\ &\overset{(23)}=\sum_{v\in W}\langle \operatorname{stab}^{+, {\nabla_-}}_{yx}, \operatorname{stab}^{-,{\nabla_+}}_{vx}\rangle\operatorname{stab}^{+, {x\nabla_-}}_{v}\\ &=\operatorname{stab}^{+, {x\nabla_-}}_{y}. \end{align*}

Now we prove (23) by induction on $\ell (x)$. If $x$ is a simple reflection, it follows from (17) and (10). Now assume it holds for $x$. Let $\alpha$ be a simple root such that $xs_\alpha >x$. Then $\beta :=x\alpha >0$ and $s_\beta =x s_\alpha x^{-1}$. Notice that the two alcoves $x\nabla _+$ and $xs_\alpha \nabla _+=s_{\beta }x\nabla _+$ are adjacent alcoves separated by a wall on $H_{\beta ^\vee ,0}$, and for any $\lambda \in \nabla _+$,

\[ (xs_\alpha\lambda,\beta^\vee)=(\lambda,s_\alpha x^{{-}1}\beta^\vee)={-}(\lambda,\alpha^\vee)<0<(x\lambda,\beta^\vee). \]

So we can use Theorem 5.1 with respect to the wall-crossing $xs_\alpha \nabla _+\leftarrow x\nabla _+$, which corresponds to $s_\beta x\nabla _+\leftarrow x\nabla _+.$

If $y\beta <0$, then $ys_\beta < y$, and $ys_\beta \succ _{{\frak C}_-} y$. Since $yx \alpha =y\beta <0$, $yxs_\alpha \succ _{{\frak C}_-}yx$. Then

\begin{align*} \operatorname{stab}^{-,{xs_\alpha\nabla_+}}_{y} &\overset{\text{Theorem 5.1}}=\operatorname{stab}^{-,{x\nabla_+}}_{y}\\ &\overset{\text{induction}}=q_x^{{-}1/2}T_x(\operatorname{stab}^{-,{\nabla_+}}_{yx})\\ &\overset{\text{Theorem 2.5}}=q_x^{{-}1/2}T_x \big(q^{{-}1/2}T_\alpha(\operatorname{stab}^{-,{\nabla_+}}_{yxs_\alpha})\big)\\ &=q_{xs_\alpha}^{{-}1/2}T_{x s_\alpha}(\operatorname{stab}^{-,{\nabla_+}}_{yxs_\alpha}). \end{align*}

If $y\beta >0$, then $ys_\beta \prec _{{\frak C}_-}y$, and $yxs_\alpha \prec _{{\frak C}_-}yx$. We have

\begin{align*} q_{xs_\alpha}^{{-}1/2}T_{xs_\alpha}(\operatorname{stab}^{-,{\nabla_+}}_{yxs_\alpha}) &=q_{xs_\alpha}^{{-}1/2}T_xT_\alpha(\operatorname{stab}^{-,{\nabla_+}}_{yxs_\alpha})\\ &\overset{\text{Theorem 2.5}}=q_{xs_\alpha}^{{-}1/2} T_xT_\alpha[q^{{-}1/2}T_\alpha(\operatorname{stab}^{-,{\nabla_+}}_{yx})]\\ &=q_{xs_\alpha}^{{-}1/2}T_x\big(q^{1/2}+(q^{1/2}-q^{{-}1/2})T_\alpha \big)(\operatorname{stab}^{-,{\nabla_+}}_{yx})\\ &\overset{\text{Theorem 2.5}}=q_{xs_\alpha}^{{-}1/2}T_x \big(q^{1/2}\operatorname{stab}^{-,{\nabla_+}}_{yx}+(q^{1/2}-q^{{-}1/2})q^{1/2}\operatorname{stab}^{-,{\nabla_+}}_{yxs_\alpha}\big)\\ &=q_x^{{-}1/2}\big(T_x(\operatorname{stab}^{-,{\nabla_+}}_{yx})+(q^{1/2}-q^{{-}1/2})T_x(\operatorname{stab}^{-,{\nabla_+}}_{yxs_\alpha})\big) \\ &\overset{\text{induction}}=\operatorname{stab}^{-,{x\nabla_+}}_{y}+(q^{{1/2}}-q^{-{1/2}}) \operatorname{stab}^{-,{x\nabla_+}}_{yxs_\alpha x^{{-}1}}\\ &\overset{\text{Theorem 5.1}}=\operatorname{stab}^{-,{xs_\alpha\nabla_+}}_{y}. \end{align*}

Corollary 5.6 If $\alpha$ is a simple root and $s_\alpha x>x$, then we have

\[ q^{{-}1/2}T_\alpha(\operatorname{stab}^{-,{x\nabla_+}}_{y})=\operatorname{stab}^{-,{s_\alpha x\nabla_+}}_{ys_\alpha}, \quad q^{{-}1/2}T'_\alpha(\operatorname{stab}^{+, {s_\alpha x\nabla_-}}_{ys_\alpha})=\operatorname{stab}^{+, { x\nabla_-}}_{y}. \]

Proof. We have $(T_{s_\alpha x})^{-1}=(T_x)^{-1}T_\alpha ^{-1}$. By Theorem 5.4, we have

\begin{align*} q_x^{1/2}(T_x)^{{-}1}\operatorname{stab}^{-,{x\nabla_+}}_{yx^{{-}1}}&=\operatorname{stab}^{-,{\nabla_+}}_{y}=q_{s_\alpha x}^{1/2}(T_{s_\alpha x})^{{-}1}(\operatorname{stab}^{-,{s_\alpha x\nabla_+}}_{yx^{{-}1}s_\alpha})\\&=q_{ x}^{1/2}q^{1/2}(T_{ x})^{{-}1}T_\alpha^{{-}1}(\operatorname{stab}^{-,{s_\alpha x\nabla_+}}_{yx^{{-}1}s_\alpha}). \end{align*}

Cancelling $q_x^{1/2}(T_x)^{-1}$, we get the first identity. The second one can be proved similarly.

6.1 The affine braid group

We first recall the level-$p$ affine Weyl group action. In $\Lambda$, we consider the level-$p$ configuration of $\rho$-shifted affine hyperplanes. That is, for a coroot $\alpha ^\vee$ and $n\in \mathbb {Z}$, let the hyperplanes $H^p_{\alpha ^\vee ,n}$ be $\{\lambda \in \Lambda \mid {\langle \alpha ^\vee ,\lambda +\rho \rangle }= np\}$. The open facets are called $p$-alcoves, and the codimension one facets are called faces.

There is a special alcove, denoted by $A_0$, which is the alcove containing $(\epsilon -1)\rho$ for small $\epsilon >0$. It consists of those weights $\lambda$ such that $0< {\langle \lambda +\rho ,\alpha ^\vee \rangle }< p$ for all $\alpha \in \Phi ^+$. This alcove is referred as the fundamental alcove in [Reference Bezrukavnikov, Mirković and RumyninBMR06], which, to avoid confusion, will not be called the fundamental alcove in the present paper.

Let $Q$ be the root lattice. We consider the level-$p$ affine Weyl group. It is the usual affine Weyl group $W_{\operatorname {aff}}:= W\ltimes Q$ as an abstract group, with a different action on the lattice $\Lambda$, the level-$p$ dot-action, recalled as follows. Elements in $W$ acts via the usual dot-action $w:\lambda \mapsto w\bullet \lambda :=w(\lambda +\rho )-\rho$. Elements $\nu$ in the lattice act by $\lambda \mapsto \lambda +p\nu$. The group $W_{\operatorname {aff}}$ is generated by reflections in affine hyperplanes $H_{\alpha ^\vee ,n}^p$. The set of faces in the closure of $A_0$ will be denoted by $I_{\operatorname {aff}}$. The $(W_{\operatorname {aff}},\bullet )$-orbits in the set of faces are canonically identified with $I_{\operatorname {aff}}$. Via this identification, the (Coxeter) generators of the group $W_{\operatorname {aff}}$ can be chosen to be the reflections in the faces of the alcove $A_0$.

We will consider the extended affine Weyl group Footnote ⁴ $W_{\operatorname {aff}}':=W\ltimes \Lambda$. Let $\Omega$ denote the stabilizer subgroup of $W'_{\operatorname {aff}}$ that fixes $A_0$. The group $W_{\operatorname {aff}}'$ is the semi-direct product $\Omega \ltimes W_{\operatorname {aff}}$ [Reference CartierCar79, § 3.5(a)]. For example, for type $A_2$ root system (see Figure A.1 below) the subgroup $\Omega$ is a cyclic group of order 3, cyclically permuting the 3 walls of the fundamental alcove. A fundamental domain of $\Lambda$ can be chosen to be $(W,\bullet )$-orbit of $A_0$. This means, for any $\lambda \in \Lambda$, there exists $\mu \in \Lambda$ and a Weyl group element $w\in W,$ such that $\lambda -\mu \in w\bullet A_0$.

Note that in the definition above, as abstract groups, $W_{\operatorname {aff}}$ and $W'_{\operatorname {aff}}$ do not depend on the level $p$. When $p$ is equal to 1, this action is the $\rho$-shift of the action used in § 2, that is, $H^{p=1}_{\alpha ^\vee ,n}$ is the $\rho$-shift of $H_{\alpha ^\vee ,n}$. In §§ 6.2 and 6.3 below, we will focus on the $p=1$ case. The case when $p$ is a prime number will be relevant in § 7.

We refer the readers to [Reference LusztigLus80, § 1.1] for the local action of $W_{\operatorname {aff}}$, where for each $\alpha \in I_{\operatorname {aff}}$ and on each alcove $s_\alpha$ acts via reflection along the wall labelled by $\alpha$. We follow [Reference Bezrukavnikov, Mirković and RumyninBMR06, § 2.1.2] to define the local action of $W_{\operatorname {aff}}'$. This depends on the choice of a character $\lambda _0\in A_0$, so that $W_{\operatorname {aff}}' \bullet \lambda _0$ is a free $W_{\operatorname {aff}}' \bullet$-orbit. Given $v\in W'_{\operatorname {aff}}$ and $\lambda '=w\bullet \lambda _0\in W_{\operatorname {aff}}' \bullet \lambda _0$ with $w\in W'_{\operatorname {aff}}$, define $\lambda '\star v:=wv \bullet \lambda _0$. Note that $\bullet$ is a left action of $W_{\operatorname {aff}}'$ on $\Lambda$, and $\star$ is a right action of $W_{\operatorname {aff}}'$ only on the subset $W_{\operatorname {aff}}' \bullet \lambda _0$. For $\lambda \in W_{\operatorname {aff}}' \bullet \lambda _0$ and $w\in W'_{\operatorname {aff}}$, we say $w$ increases $\lambda$ if the following holds. Write $w=s_1\cdots s_n \omega$, with $s_i\in I_{\operatorname {aff}}$ and $\omega \in \Omega$. Then, $\lambda \star (s_1\cdots s_{i-1})<\lambda \star (s_1\cdots s_{i})$ in the standard order of $\Lambda$ for any $i=1,\ldots , n$ [Reference Bezrukavnikov, Mirković and RumyninBMR06, § 2.1.3].

To be consistent with [Reference Su, Zhao and ZhongSZZ20] and § 2, the roots of the Borel subalgebra ${\frak b}$ are positive roots $\Delta ^+$. Note that this is the opposite of the convention used in [Reference Bezrukavnikov, Mirković and RumyninBMR06, § 1.1.1], and hence results in an opposite lifting of the affine Weyl group into the affine braid group, as will be discussed in detail in Remark 6.3(a).

Now we recall the affine braid group. The group $B_{\operatorname {aff}}$ is generated by $\widetilde {w}$ for $w\in W_{\operatorname {aff}}$ subject to the relation $\widetilde {wu}=\widetilde {w}\widetilde {u}$ when $l(wu)=l(w)+l(u)$ [Reference MacDonaldMac96, § 3]. There is a set theoretical lifting $C :W_{\operatorname {aff}}\to B_{\operatorname {aff}}$, sending the simple reflection $s_\alpha$ for $\alpha \in I_{\operatorname {aff}}$ to $\tilde {s}_\alpha$, and any reduced decomposition $w=s_{\alpha _1}\cdots s_{\alpha _{k}}$ to $\widetilde {w}=\tilde {s}_{\alpha _1}\cdots \tilde {s}_{\alpha _{k}}$.

On the extended affine Weyl group $W_{\operatorname {aff}}'$, the length function is given by $l(w\omega )=l(w)$ for $\omega \in \Omega$. The extended affine Braid group $B_{\operatorname {aff}}'$ is generated by $\widetilde {w}$ for $w\in W_{\operatorname {aff}}'$, subject to the relation $\widetilde {wu}=\widetilde {w}\widetilde {u}$ when $l(wu)=l(w)+l(u)$. As $\Omega$ permutes $I_{\operatorname {aff}}$, we have $B_{\operatorname {aff}}'=B_{\operatorname {aff}}\rtimes \Omega$. A smaller set of generators of $B_{\operatorname {aff}}'$ can be chosen to be $\{\widetilde {s}_\alpha \mid \alpha \in I_{\operatorname {aff}}\}$ and $\Omega$.

The set of finite simple roots $\Sigma$ can be naturally identified with a subset of $I_{\operatorname {aff}}$ via their corresponding simple reflections. Alternatively, the group $B_{\operatorname {aff}}'$ is generated by simple finite reflections $\widetilde {s}_\alpha$ for $\alpha \in \Sigma$ and translations $\Lambda ^+\subset \Lambda$ consisting of $\lambda \in \Lambda$ with $\langle \alpha ^\vee ,\lambda \rangle >0$ for all positive coroots $\alpha ^\vee$.

For any $w\in W$, let $w=s_1\dots s_k$ be a reduced decomposition of $w$. Then, in $B_{\operatorname {aff}}'$ we have the element $\tilde {s}_1\dots \tilde {s}_k$ via $\Sigma \subseteq I_{\operatorname {aff}}$. This element is independent of the reduced word decomposition, hence is a well-defined lifting $\widetilde {w}\in B'_{\operatorname {aff}}$. Sending $w\in W$ to $\widetilde {w}$ defines a set-theoretical lifting $W\to B_{\operatorname {aff}}\subseteq B'_{\operatorname {aff}}$. Since $w^{-1}=s_k\dots s_1$, we have $\widetilde {w^{-1}}=\tilde {s}_k\dots \tilde {s}_1$, which is different from $(\widetilde {w})^{-1}=(\tilde {s}_k)^{-1}\dots (\tilde {s}_1)^{-1}$. On the other hand, the lifting of $\Lambda \subseteq W'_{\operatorname {aff}}$ is a subgroup of $B_{\operatorname {aff}}'$ [Reference MacDonaldMac96, § 3]. Therefore for any $\lambda \in A$, we will not distinguish between the group inverse taken in $\Lambda$ and the group inverse taken in $B_{\operatorname {aff}}'$. For this reason, we will denote the lifting of $\lambda$ still by $\lambda$. A similar property holds for the subgroup $\Omega \subseteq W'_{\operatorname {aff}}$.

6.2 A categorical affine braid group action

Let $G_{\mathbb {Z}}$ be a split ${\mathbb {Z}}$-form of the complex algebraic group $G$. Let $A_{\mathbb {Z}}\subseteq B_{\mathbb {Z}}\subseteq G_{\mathbb {Z}}$ be a maximal torus and a Borel subgroup, which we assume to have base changes $A\subset B\subset G$. Let ${\frak h}_{\mathbb {Z}}\subset {\frak b}_{\mathbb {Z}}\subset {\frak g}_{\mathbb {Z}}$ be the corresponding Lie algebras. Similar to the setup of § 2, we have the integral forms $\mathcal {B}_{\mathbb {Z}}$ and $T^*\mathcal {B}_{\mathbb {Z}}$, etc.

In [Reference Bezrukavnikov and RicheBR13], an action of the extended affine braid group on the category $D^{\operatorname {b}}_{T_{\mathbb {Z}}}(T^*\mathcal {B}_{\mathbb {Z}})$ is constructed, where elements of $B'_{\operatorname {aff}}$ act by Fourier–Mukai transforms. Moreover, for any algebraically closed field $k$, this affine braid group action can be base changed to $k$. More precisely, we have the following. The base change of the groups and varieties from ${\mathbb {Z}}$ to $k$ are denoted by $A_k\subseteq B_k\subseteq G_k$, $T^*\mathcal {B}_k$, etc. When $k=\mathbb {C}$, we will omit the subscripts. This is consistent with the notations introduced in § 2.

6.3 Integral form of stable bases

Now we define the objects ${\frak s}{\frak t}{\frak a}{\frak b}^{\mathbb {Z}}_{\lambda }(y)\in D^{\operatorname {b}}_{T_{\mathbb {Z}}}(T^*\mathcal {B}_{\mathbb {Z}}), y\in W, \lambda \in \Lambda _\mathbb {Q}$ in the interior of an alcove, inductively as follows:

(25)\begin{align} {\frak s}{\frak t}{\frak a}{\frak b}^{{\mathbb{Z}}}_{\lambda_0}(id)&=\mathcal{L}_{-\rho} \otimes\mathcal{O}_{T^*_{id}\mathcal{B}_{\mathbb{Z}}}, \quad \lambda_0\in \nabla_-, \end{align}

(26)\begin{align} {\frak s}{\frak t}{\frak a}{\frak b}^{{\mathbb{Z}}}_{\lambda_0}(w)&=J_{\widetilde{w}}^{\operatorname{R}}{\frak s}{\frak t}{\frak a}{\frak b}^{{\mathbb{Z}}}_{\lambda_0}(id), \quad \lambda_0\in \nabla_-, \end{align}

(27)\begin{align} {\frak s}{\frak t}{\frak a}{\frak b}^{\mathbb{Z}}_{\lambda}(y)&=(J_{\widetilde{x}}^{\operatorname{R}})^{{-}1}{\frak s}{\frak t}{\frak a}{\frak b}^{\mathbb{Z}}_{\lambda_0}(yx), \quad y,x\in W,\ x\lambda_0=\lambda, \ \lambda_0\in \nabla_-, \end{align}

(28)\begin{align} {\frak s}{\frak t}{\frak a}{\frak b}^{\mathbb{Z}}_{\lambda}(y)&=e^{{-}y\mu}J^{\operatorname{R}}_{\mu}{\frak s}{\frak t}{\frak a}{\frak b}^{\mathbb{Z}}_{\lambda_1}(y), \quad y\in W, \ \mu+\lambda_1=\lambda,\ \mu\in Q,\ \lambda_1\in W\nabla_-. \end{align}

According to 6.2, the functor $J^{\operatorname {R}}_{\mu }$ in (28) is twisting by $\mathcal {L}_\mu$. From the definitions, ${\frak s}{\frak t}{\frak a}{\frak b}^{\mathbb {Z}}_{\lambda }(y)$ only depends on the alcove $\nabla$ containing $\lambda$, so sometimes it is also denoted by ${\frak s}{\frak t}{\frak a}{\frak b}^{\mathbb {Z}}_{\nabla }(y)$.

Now we are ready to state the main result of this section.

7.1 Reminder on localization in prime characteristic

We start by a brief review of [Reference Bezrukavnikov, Mirković and RumyninBMR08]. Now we assume $k$ is an algebraically closed field of characteristic $p$ greater than the Coxeter number of $G$. For any $k$-variety $X$, its Frobenius twist is denoted by $X^{(1)}$. If $X$ is defined over ${\mathbb {F}}_p$, then $X^{(1)}\cong X$ as abstract $k$-varieties. However, the natural map $X\to X^{(1)}$ is not an isomorphism. Below we will take $X$ to be $\mathcal {B}_k$, $T^*\mathcal {B}_k$, etc. We will often omit the subscript $k$, when there is a Frobenius twist. For example, ${\mathcal {B}_k}^{(1)}$ will be simply denoted by $\mathcal {B}^{(1)}$.

Differentiating a character of $A_k$ defines a map $d:\Lambda \to {\frak h}_k^*$. We call a character $\lambda \in {\frak h}^*_k$ integral if it lies in the image of $d:\Lambda \to {\frak h}^*_k$. An integral character is said to be regular if it does not lie on any walls $H^p_{\alpha ^\vee ,n}$. For $\lambda \in \Lambda$, on the flag variety $\mathcal {B}_k$, consider the ring of $\mathcal {L}_\lambda$-twisted differential operators $\mathcal {D}^\lambda$, which is a sheaf of algebras. Its central subalgebra, i.e. the sheaf of functions on $T^*\mathcal {B}^{(1)}$, makes any coherent $\mathcal {D}^\lambda$-module naturally a coherent sheaf on $T^*\mathcal {B}^{(1)}$, and moreover, $\mathcal {D}^\lambda$ is an Azumaya algebra on $T^*\mathcal {B}^{(1)}_k$ [Reference Bezrukavnikov, Mirković and RumyninBMR08, § 2.3].

There exists a vector bundle $\mathcal {E}^s$ with the following properties.

1. (1) $\mathcal {E} nd_{T^*\mathcal {B}^{(1)\wedge }}(\mathcal {E}^s)\cong \mathcal {D}^{\lambda _0}|_{T^*\mathcal {B}^{(1)\wedge }}$, where $T^*\mathcal {B}^{(1)\wedge }$ is the formal neighbourhood of the zero-section $\mathcal {B}^{(1)}$. That is, the Azumaya algebra $\mathcal {D}^{\lambda _0}$ splits on this formal neighbourhood [Reference Bezrukavnikov, Mirković and RumyninBMR08, Theorem 5.1.1].

2. (2) $\mathcal {E}^s$ is $T_k$-equivariant [Reference Bezrukavnikov and MirkovićBM13, § 5.2.4], where $T_k=A_k\times k^*.$

3. (3) Define $\mathcal {A}^{\lambda _0}$ to be $End_{T^*\mathcal {B}^{(1)}}(\mathcal {E}^s)$, a $\mathcal {O}({\mathcal {N}^{(1)}})$-algebra endowed with a compatible $A_k\times (\mathbb {G}_m)_k$-action. Then there exists a localization functor $\gamma ^{\lambda _0}: D^{\operatorname {b}}\operatorname {Mod}^{\operatorname {gr}}(\mathcal {A}^{\lambda _0},A_k)\cong D^{\operatorname {b}}_{T_k}\operatorname {Coh}(T^*\mathcal {B}^{(1)}_k)$ [Reference Bezrukavnikov and MirkovićBM13, Theorem 5.1.1].Footnote ⁵

4. (4) The completion $(\mathcal {A}^{\lambda _0})^\wedge _0$ is related to the Lie algebra ${\frak g}_k$ (see (29) below).

The algebra $U({\frak g}_k)$ has two central subalgebras: the Frobenius center, which is naturally identified with $\mathcal {O}({\frak g}^{*(1)})$; and the Harish-Chandra center, which is identified with $\mathcal {O}({\frak h}^*_k/(W,\bullet ))$ under the choice of the Borel subgroup $B_k$ and the maximal torus $A_k$. A point $W\bullet d\lambda \in {\frak h}_k^*/(W,\bullet )$ defines a maximal ideal in $\mathcal {O}({\frak h}^*_k/(W,\bullet ))$, which is a central subalgebra of $U({\frak g}_k)$. The quotient of $U({\frak g}_k)$ by the central ideal is denote by $U({\frak g}_k)^\lambda$, that is $U({\frak g}_k)^\lambda :=U({\frak g}_k)\otimes _{\mathcal {O}({\frak h}^*_k/(W,\bullet ))}k_\lambda$. Similarly $\chi \in \mathcal {N}^{(1)}$ defines a central ideal, and the completion of $U({\frak g}_k)^{\lambda _0}$ at this ideal is denoted by $U({\frak g}_k)^{\lambda _0}_\chi$. Then

(29)\begin{equation} (\mathcal{A}^{\lambda_0})^\wedge_0\cong U({\frak g}_k)_0^{\lambda_0}. \end{equation}

The vector bundle $\mathcal {E}^s$ above is not unique. Different choices are related by $\mathcal {D}^\lambda$ [Reference Bezrukavnikov, Mirković and RumyninBMR08, Remark 5.2.2]. Nevertheless, as has been done in [Reference Bezrukavnikov and RicheBR13, Reference Bezrukavnikov and MirkovićBM13], the property (30) below about baby Verma modules fixes the choice [Reference Bezrukavnikov, Mirković and RumyninBMR06, Remark 1.3.5].

For the Borel subalgebra ${\frak b}_k$ and for any $\chi \in \mathcal {N}^{(1)}$ and $\lambda \in \Lambda$, consider the $U({\frak b}_k)$-character $k_\lambda$ obtained via the projection ${\frak b}\to {\frak h}$ composing with the ${\frak h}$-character $k_\lambda$. Recall that the Verma module of $U({\frak g}_k)$ associated to the Borel ${\frak b}$ and $\lambda \in \Lambda$ is $Z^{{\frak b}}(\lambda ):=U({\frak g}_k)\otimes _{U({\frak b}_k)}k_\lambda$, and the baby Verma module is $Z^{{\frak b}}_\chi (\lambda ):=U_\chi ({\frak g}_k)\otimes _{U({\frak b}_k)}k_{\lambda }$, where $U_\chi ({\frak g}_k)$ is the quotient of $U({\frak g}_k)$ by the central ideal $\chi \in \mathcal {N}^{(1)}\subseteq {\frak g}^{*(1)}$.

Recall that under the identification $\mathcal {B}_k\cong G_k/B_k$, the $T_k$-fixed points on $T^*\mathcal {B}^{(1)}$ are identified with the Weyl group elements, with $B_k$ corresponding to $id$, whose skyscraper sheaf is denoted by $k_{id}$. Then [Reference Bezrukavnikov, Mirković and RumyninBMR08, Example 5.3.3.(0)]

(30)\begin{equation} \gamma^{\lambda_0} Z_0^{{\frak b}}(\lambda_0+2\rho)\cong k_{id}. \end{equation}

The isomorphism (29) defines a $T_k$-action on $U({\frak g}_k)^{\lambda _0}_0$ making it an equivariant isomorphism. Taking completion of $Z^{{\frak b}}(\lambda _0+2\rho )$ with respect to the maximal ideal $W\bullet d\lambda \in {\frak h}_k^*/(W,\bullet )$ of the central subalgebra $\mathcal {O}({\frak h}_k^*/(W,\bullet ))$ defines a module over $U({\frak g}_k)^{\lambda _0}_0$. The $T_k$-action on $U({\frak g}_k)^{\lambda _0}_0$ provides $T_k$-actions on $Z_0^{{\frak b}}(\lambda _0+2\rho )$ and $Z^{{\frak b}}(\lambda _0+2\rho )$, making (30) equivariant. Taking $\mathbb {G}_m$-finite vectors in $Z^{{\frak b}}(\lambda _0+2\rho )$, we get an $A_k$-Koszul graded module of $\mathcal {A}^{\lambda _0}$, which, without causing confusions, will still be denoted by $Z^{{\frak b}}(\lambda _0+2\rho )$. Then, under the equivalence in property (3) of § 7.1 above, we have the isomorphism

(31)\begin{equation} \gamma^{\lambda_0} Z^{\frak b}(\lambda_0+2\rho)\cong \mathcal{O}_{T^*_{id}\mathcal{B}^{(1)}}. \end{equation}

7.2 Localization and the affine braid group action

We collect a few preliminary results, which are direct consequences of [Reference Bezrukavnikov, Mirković and RumyninBMR06] reviewed above. Let us consider the category $\operatorname {Mod}_0 U({\frak g}_k)^\lambda$ of finitely generated modules of $U({\frak g}_k)^\lambda$ on which the Frobenius center $\mathcal {O}({\frak g}^{*(1)})$ acts by the generalized character $0\in {\frak g}^{*(1)}$. For $\lambda$, $\mu \in \Lambda$, we define $T_\lambda ^\mu :\operatorname {Mod}_0 U({\frak g}_k)^\lambda \to \operatorname {Mod}_0 U({\frak g}_k)^\mu$ sending $M$ to $[V_{\mu -\lambda }\otimes M]_\mu$. Here $V_{\mu -\lambda }$ is a finite-dimensional representation with extremal weight $\mu -\lambda$, and $[-]_\mu$ means taking the component supported on the point $\mu$ in ${\frak h}^*//W$. Assume $\nu$ lies in a codimension-$1$ wall of the facet of the alcove containing $\mu$. Define

\[ R_{\mu|\nu}:=T^\mu_\nu T_\mu^\nu: \operatorname{Mod}_0 U({\frak g}_k)^\mu\to \operatorname{Mod}_0 U({\frak g}_k)^\mu. \]

Indeed, $R_{\mu |\nu }$ depends only on the wall, not the character $\nu$ itself. Taking mapping cone of the adjunction, we define

\[ \Theta_{\mu|\nu}:=cone(id\to R_{\mu|\nu}). \]

When $\mu$ is in the alcove $A_0$, then the wall on which $\nu$ lies is labelled by some $\alpha \in I_{\operatorname {aff}}$, therefore we will also denote $\Theta _{\mu |\nu }$ by $\Theta _\alpha$ in this case. For $\omega \in \Omega$, we write $T_{\lambda _0}^{\lambda _0\star \omega }:\operatorname {Mod}_0 U({\frak g}_k)^{\lambda _0}\to \operatorname {Mod}_0 U({\frak g}_k)^{\lambda _0}$ simply as $\Theta _{\omega }$. Note that this is possible due to the fact that $\lambda _0$ and $\lambda _0\star \omega$ have the same central character in ${\frak h}^*_k/(W,\bullet )$.

The following is essentially [Reference Bezrukavnikov, Mirković and RumyninBMR06, Theorem 2.1.4] and [Reference Bezrukavnikov and RicheBR13, Theorem 2.5.4], taking into account the correction in [Reference Bezrukavnikov and RicheBR13, Remark 2.5.2] and the translation between the left and right actions in Remark 6.3.

Note that under this identification of the two affine braid group actions, $\operatorname {Pic}(T^*\mathcal {B})\cong \Lambda$ acts on $\Lambda$ as in the level-$p$ affine Weyl group action $\bullet$.

The un-graded version of the following has been well known (see, e.g. [Reference JanztenJan00, Lemma 4.7]). Here we give a different proof, which also holds in the $A_k$-Koszul-bigraded setting.

Lemma 7.3 Assuming $\mu$ is regular. Let $\alpha \in I_{\operatorname {aff}}$ be such that $\dot {\mu }:=\mu \star s_\alpha >\mu$, then we have as objects in $D^{\operatorname {b}}\operatorname {Mod}^{\operatorname {gr}}(\mathcal {A}^{\lambda _0},A_k)$

\begin{align*} \Theta_{\widetilde{s}_\alpha}^{{-}1}(Z^{{\frak b}}_0(\mu+2\rho))&\cong Z^{{\frak b}}_0(\dot{\mu}+2\rho),\\ \Theta_{\widetilde{s}_\alpha}^{{-}1}(Z^{{\frak b}}(\mu+2\rho))&\cong Z^{{\frak b}}(\dot{\mu}+2\rho). \end{align*}

Proof. The usual equivariant D-module calculation as in [Reference Beilinson and BernsteinBB81] (see also [Reference Ben-Zvi and FrenkelBF04, § 11.2.11]) yields that the global section of the delta-D-module at $id\in \mathcal {B}_k$ gives the Verma module over $U({\frak g}_k)$. More precisely, for any $\lambda$ regular and integral, let $\delta _{\frak b}^\lambda$ be the module of $\mathcal {D}^\lambda$ given by the $\delta$-distributions at $id\in \mathcal {B}_k$. Under the equivalence $R\Gamma _{\mathcal {D}^\lambda ,0}:D^{\operatorname {b}}(\operatorname {Coh}_{0}\mathcal {D}^\lambda ) \overset {\cong }\longrightarrow D^{\operatorname {b}}(\operatorname {Mod}_{0}U({\frak g}_k)^\lambda )$ [Reference Bezrukavnikov, Mirković and RumyninBMR08, Theorem 3.2],

(32)\begin{equation} R\Gamma_{\mathcal{D}^\lambda,0}(\delta_{\frak b}^\lambda)\cong Z^{\frak b}(\lambda+2\rho). \end{equation}

Similarly, let $\delta ^\lambda _{{\frak b},0}$ be the central reduction of $\delta _{\frak b}^\lambda$ with respect to the sheaf of ideals corresponding to $\mathcal {B}^{(1)}\subseteq T^*\mathcal {B}^{(1)}$ in the center of $\mathcal {D}^\lambda$. Then we have [Reference Bezrukavnikov, Mirković and RumyninBMR08, § 3.1.4]

(33)\begin{equation} R\Gamma_{\mathcal{D}^\lambda,0}(\delta_{{\frak b},0}^\lambda)\cong Z^{\frak b}_0(\lambda+2\rho). \end{equation}

By [Reference Bezrukavnikov, Mirković and RumyninBMR06, Theorem 2.2.8] (taking into account the correction in [Reference Bezrukavnikov and RicheBR13] in Remark 2.5.2 and the paragraph afterwards), the following diagram commutes.

Here we have used the assumption that $\dot {\mu }=\mu \star s_\alpha >\mu$. By definition,

\[ \delta_{{\frak b}, 0}^{\dot{\mu}}\cong \delta_{{\frak b}, 0}^{\mu}\otimes\mathcal{L}_{\dot{\mu}-\mu}. \]

By (33), $R\Gamma _{\mathcal {D}^\mu ,0}(\delta _{{\frak b}, 0}^{{\mu }})=Z^{\frak b}_0(\mu +2\rho )$, and $R\Gamma _{\mathcal {D}^{\dot {\mu }},0}(\delta _{{\frak b}, 0}^{\dot {\mu }})=Z^{\frak b}_0(\dot {\mu }+2\rho )$. Therefore, we have $\Theta _{\alpha }^{-1}(Z^{{\frak b}}_0(\mu +2\rho ))\cong Z^{{\frak b}}_0(\dot {\mu }+2\rho )$. Note that this isomorphism is equivariant with respect to the $A_k$ and Koszul gradings.

About the assertion on Verma modules, we follow the same argument using (32), with the commutative square above replaced by their counterparts for the completed algebras. We get an isomorphism between the completions of $\Theta _{\alpha }^{-1}(Z^{{\frak b}}(\mu +2\rho ))$ and $Z^{{\frak b}}(\dot {\mu }+2\rho )$, which is equivariant with respect to the $A_k$ and Koszul gradings. Then, taking the subspaces of finite vectors with respect to the Koszul grading gives the required isomorphism.

The following is a direct consequence of the existence of the affine braid group action on the representation category.

Motivated by [Reference Bezrukavnikov and MirkovićBM13, p.870], for any $\lambda$ in the $W_{\operatorname {aff}}'$-orbit of $\lambda _0$, we define the localization functor

\[ \gamma^\lambda:D^{\operatorname{b}}\operatorname{Mod}_0U({\frak g}_k)^{\lambda_0}\to D^{\operatorname{b}}\operatorname{Coh}_{\mathcal{B}^{(1)}}T^*\mathcal{B}^{(1)}, \]

which is determined by the property that $\gamma ^{\lambda \star w}=\gamma ^{\lambda }\circ \Theta _{\widetilde {w^{^{-1}}}}^{\operatorname {R}}$ for $w\in W'_{\operatorname {aff}}$, as long as $w$ increases $\lambda$. Localization functor associated to singular $\lambda$ has also been studied in [Reference Bezrukavnikov, Mirković and RumyninBMR06], although we will not explicitly use this. In particular, if $\lambda _0\in A_0$, and $w\in W$ such that $\lambda =\lambda _0\star w$, then using the fact that $\lambda _0=\lambda _0\star w\star w^{-1}$ and that $w^{-1}$ increases $\lambda _0\star w$, we get

(34)\begin{equation} \gamma^{\lambda_0}= \gamma^{\lambda}\circ \Theta_{\tilde{w}}^{\operatorname{R}}=J_{\tilde{w}}^{\operatorname{R}}\circ\gamma^{\lambda}. \end{equation}

7.3 A categorification of the stable basis via the Verma modules

Now we are ready to state and prove the main theorem of this section.

Theorem 7.5 Let $k$ be an algebraically closed field of characteristic $p$, and $p$ is greater than the Coxeter number. Assume $\lambda$ is regular and integral, then in $D^{\operatorname {b}}_{T_k}(T^*\mathcal {B}_k^{(1)})$, we have isomorphisms

\[ e^{\rho}{\frak s}{\frak t}{\frak a}{\frak b}^k_{-({\lambda+\rho})/{p}}(y)\cong \gamma^{\lambda} Z^{{\frak b}}(y\bullet \lambda+2\rho). \]

The left-hand side is the base changed integral form of the stable basis defined in § 6.3, not to be confused with the complex one. The relation between these two is given by Theorem 6.4. Using the isomorphism of $k$-varieties $T^*\mathcal {B}_k\cong T^*\mathcal {B}^{(1)}$, we consider $T^*\mathcal {B}^{(1)}$ as the base-change of $T^*\mathcal {B}_{\mathbb {Z}}$ to the field $k$.

Proof. As the objects ${\frak s}{\frak t}{\frak a}{\frak b}^k_{-({\lambda +\rho })/{p}}(y)$ are defined inductively, we prove this theorem inductively.

First, we prove the case when $\lambda =\lambda _0\in A_0$. We have $\gamma ^{\lambda _0}Z^{\frak b}(\lambda _0+2\rho )=\mathcal {O}_{T^*_{id}\mathcal {B}}= \mathcal {L}_\rho \otimes {\frak s}{\frak t}{\frak a}{\frak b}^k_{-({\lambda _0+\rho })/{p}}(id)=e^\rho {\frak s}{\frak t}{\frak a}{\frak b}^k_{-({\lambda _0+\rho })/{p}}(id)$. Here $id\in \mathcal {B}_k$ corresponds to the Borel subalgebra ${\frak b}$ that has been used to define the Verma modules. For $w\in W$, Remark 6.1(c) yields that $Z^{\frak b}(w\bullet \lambda _0+2\rho )=Z^{\frak b}(\lambda _0\star w+2\rho )$, so

\begin{align*} \gamma^{\lambda_0}Z^{\frak b}(\lambda_0\star w+2\rho) &\overset{\text{Corollary 7.4}}= \gamma^{\lambda_0}\Theta^{\operatorname{R}}_{\widetilde w}Z^{\frak b}(\lambda_0+2\rho) \overset{\text{Theorem 7.2 (ii)}}=J^{\operatorname{R}}_{\widetilde w} \gamma^{\lambda_0}Z^{\frak b}(\lambda_0+2\rho)\\ & = J^{\operatorname{R}}_{\widetilde w}e^\rho{\frak s}{\frak t}{\frak a}{\frak b}^k_{-({\lambda_0+\rho})/{p}}(id) \overset{(26)}=e^{\rho}{\frak s}{\frak t}{\frak a}{\frak b}^k_{-({\lambda_0+\rho})/{p}}(w). \end{align*}

Here the third equality follows from the initial case above.

Second, we prove this for general $\lambda \in W\bullet A_0$, a fundamental domain of the action of the Picard group. We assume $\lambda =x\bullet \lambda _0$ with $\lambda _0\in A_0$ and $x\in W$. We have

\begin{align*} {\frak s}{\frak t}{\frak a}{\frak b}^k_{-({\lambda+\rho})/{p}}(y) &\overset{\text{Remark 6.1(b)}}= {\frak s}{\frak t}{\frak a}{\frak b}^k_{{-}x(({\lambda_0+\rho})/{p})}(y) \overset{(27)}=(J^{\operatorname{R}}_{\widetilde{x}})^{{-}1} {\frak s}{\frak t}{\frak a}{\frak b}_{-({\lambda_0+\rho})/{p}}(yx)\\ &= (J^{\operatorname{R}}_{\widetilde x})^{{-}1}e^{-\rho}\gamma^{\lambda_0} Z^{{\frak b}}((yx)\bullet\lambda_0+2\rho)\\ &\overset{(34)}=e^{-\rho}\gamma^{\lambda}Z^{{\frak b}}((yx) \bullet\lambda_0+2\rho) =e^{-\rho}\gamma^{\lambda}Z^{{\frak b}}(y\bullet \lambda+2\rho). \end{align*}

Here the third equality follows from the first step above; The last equality follows from $(yx)\bullet \lambda _0=y\bullet \lambda$.

Finally, we consider the case of general $\lambda$. There is an $\mu \in \Lambda$, and $\lambda _1\in W\bullet A_0$ so that $\lambda =\lambda _1\star (-\mu )$. If $\lambda _1=x\bullet \lambda _0$ for $x\in W$ and $\lambda _0\in A_0$, the above is equivalent to

\[ \lambda=\lambda_1\star (-\mu)=(x\bullet\lambda_0)\star (-\mu)=x\bullet((-\mu)\bullet \lambda_0)={-}px(\mu)+x\bullet \lambda_0. \]

Without loss of generality, we assume $-\mu$ increases $\lambda _1$. That is, $-x\mu \in \Lambda ^+$. We have

\begin{align*} {\frak s}{\frak t}{\frak a}{\frak b}^k_{-({\lambda+\rho})/{p}}(y)&= {\frak s}{\frak t}{\frak a}{\frak b}^k_{x(\mu)-({\lambda_1+\rho})/{p}}(y)\\ &\overset{(28)}= e^{{-}yx(\mu)}\mathcal{L}_{x(\mu)}\otimes{\frak s}{\frak t}{\frak a}{\frak b}^k_{-({\lambda_1+\rho})/{p}}(y)\\ &\overset{\text{Theorem 6.2}}= e^{{-}yx(\mu)}J_{x(\mu)}^{\operatorname{R}}e^{-\rho}\gamma^{\lambda_1}Z^{\frak b}(y\bullet\lambda_1 +2\rho)\\ &\overset{(34)}= e^{{-}yx(\mu)}e^{-\rho}J_{x(\mu)}^{\operatorname{R}} \gamma^{\lambda_0}(\Theta_{\widetilde x}^{\operatorname{R}})^{{-}1}Z^{\frak b}(y\bullet\lambda_1+2\rho)\\ &\overset{\text{Theorem 7.2(ii)}}= e^{{-}yx(\mu)}e^{-\rho}\gamma^{\lambda_0}\Theta^{\operatorname{R}}_{x(\mu)}(\Theta_{\widetilde x}^{\operatorname{R}})^{{-}1}Z^{\frak b}(y\bullet\lambda_1+2\rho)\\ &= e^{{-}yx(\mu)}e^{-\rho}\gamma^{\lambda_0-px(\mu)}(\Theta_{\widetilde x}^{\operatorname{R}})^{{-}1}Z^{\frak b}(y\bullet\lambda_1+2\rho)\\ &= e^{{-}yx(\mu)}e^{-\rho}\gamma^{\lambda}Z^{\frak b}(y\bullet\lambda_1+2\rho). \end{align*}

Here the third equality follows from Theorem 6.2 and the second step. The second to last equality follows from the fact that $-x\mu$ increases $\lambda _0$, since $x\mu \in \Lambda ^-$ and that the inverse of $x\mu$ is $-x\mu$. The last equality follows from the fact that $\lambda \star x^{-1}=\lambda _0-px(\mu )$, and that $x^{-1}$ increases $\lambda$, which in turn is equivalent to $x^{-1}$ increases $\lambda _0\star x$. Finally, note that the definition of the Verma modules only depends on an element in ${\frak h}^*_k$, which in particular is invariant under shifting by $p\lambda '$ for any $\lambda '\in \Lambda$. Hence , $Z^{\frak b}(y\bullet \lambda _1+2\rho )$ and $Z^{\frak b}(y\bullet \lambda +2\rho )$ only differ by a $A_k$-grading, namely, $e^{yx(\mu )}$. Therefore, the proof is finished.

7.4 Restriction formula and Lie algebra cohomology

In general, for any variety $X$, let $a\in X$ be a closed point with residue field $k_a$ and embedding $i:\{a\}\hookrightarrow X$, and $\mathcal {F}\in D^{\operatorname {b}}\operatorname {Coh}(X)$. Then $\mathcal {F}|_a^\vee \cong \operatorname {Hom}_{k_a}(i^*\mathcal {F},k_a)\cong \operatorname {Hom}_X(\mathcal {F},i_*k_a)=\operatorname {Hom}_X(\mathcal {F},\mathcal {O}_a)$.

We have $k_{x{\frak b}}={\frak L}^{\lambda } Z_0^{x{\frak b}}(\lambda +2\rho )$ for any $x\in W$. Recall that here ${\frak b}$ is labelled by $id\in W$. Putting these together with Theorems 6.4 and 7.5, we get