text:
\section{introduction}\label{1}

\subsection{Main results}\label{1a}
Let $G$ be a simple simply-connected complex algebraic group and $P$ a parabolic subgroup of $G$. We prove the following results expressing the equivariant quantum cohomology $QH_T^{\bl}(G/P)$ and its localization in terms of the Langlands dual group scheme $G^{\vee}$. Our coefficient ring is $\mathbb{Z}[\ell_G^{-1}]$ where $\ell_G$ is equal to 1 for type ADE, 2 for type BCF and 3 for type G.
\begin{customtheorem}{A}\label{1athm} There exists an isomorphism of $H_T^{\bl}(\pt)[q_1,\ldots,q_k]$-algebras
\[ \Phi:\mathcal{O}(\mathcal{Y}_P)\xrightarrow{\sim}QH_T^{\bl}(G/P)\]
which induces a monomorphism
\[ \Phi^*:\mathcal{O}(\mathcal{Y}_P^*)\rightarrow  QH_T^{\bl}(G/P)[\ecc].\]
The latter is also an isomorphism if we set all equivariant parameters to zero.
\end{customtheorem}

The schemes $\mathcal{Y}_P$ and $\mathcal{Y}_P^*$ are defined as follows. Define first $e^T=e+f\in \dd{\mathfrak{b}}\otimes H_T^{\bl}(\pt)$ where
\[ e:= \sum_{i=1}^r|\ad_i|^2 e_{\alpha^{\vee}_i}\quad\text{ and }\quad f:\alpha^{\vee}_i\mapsto -|\alpha^{\vee}_i|^2\alpha_i,\]
and
\[ \mathcal{Y}:=\left\{(gB_-^{\vee},h)\inG^{\vee}/B^{\vee}_-\times\specH_T^{\bl}(\pt)\left|~g^{-1}\cdot e^T(h)\in \dd{\mathfrak{b}}_-\oplus\bigoplus_{i=1}^r\dd{\mathfrak{g}}_{\alpha^{\vee}_i} \right.\right\}.\]
Here $e_{\alpha^{\vee}_i}$ is a generator of $\dd{\mathfrak{g}}_{\alpha^{\vee}_i}$ and $|\alpha^{\vee}_i|^2=1$ or $\ell_G$ depending on whether $\alpha^{\vee}_i$ is short or long. Then
\[ \mathcal{Y}_P:=\mathcal{Y}\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}\]
and
\[ \mathcal{Y}_P^*:=\mathcal{Y}_P\times_{G^{\vee}/B^{\vee}_-}\UdB_-^{\vee}/B_-^{\vee}.\]
The regular functions
\begin{equation}\label{1aeq}
\overline{q}_i: (u\wpdB_-^{\vee},h)\mapsto  |\ad_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i},(u\dot{w}_P\dot{w}_0)^{-1}\cdot e^T(h)\rangle
\end{equation}
make $\mathcal{Y}_P$ and $\mathcal{Y}_P^*$ schemes over $\specH_T^{\bl}(\pt)[q_1,\ldots,q_k]$. The isomorphism $\Phi$ stated in Theorem \ref{1athm} is required to send these regular functions to the quantum parameters $q_1,\ldots,q_k$.

In general, $\mathcal{O}(\mathcal{Y}_P^*)$ is not isomorphic to $QH_T^{\bl}(G/P)[\ecc]$. A counterexample arises already when $G=\mathbf{SL}_{2}$. See Section \ref{1ab}. To obtain a presentation of $QH_T^{\bl}(G/P)[\ecc]$, we consider another scheme. Define $B_{e^T}^{\vee}$ to be the centralizer group scheme of $e^T$ in $B^{\vee}\times\specH_T^{\bl}(\pt)$. This group scheme was introduced by Yun and Zhu \cite{YZ} who proved that $\mathcal{O}(B_{e^T}^{\vee})$ is isomorphic to the equivariant homology $H^T_{-\bl}(\ag)$ of the affine Grassmannian $\mathcal{G}r$ associated to $G$. The morphism $B^{\vee}\rightarrow G^{\vee}/B^{\vee}_-:b\mapsto bB_-^{\vee}$ makes $B_{e^T}^{\vee}$ a scheme over $G^{\vee}/B^{\vee}_-$.
\begin{customtheorem}{B}\label{1athmb} There exists an isomorphism of $H_T^{\bl}(\pt)[q_1,\ldots, q_k]$-algebras
\[ \Phi_{loc}:\mathcal{O}(B_{e^T}^{\vee}\times_{G^{\vee}/B^{\vee}_-} U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee})\xrightarrow{\sim}QH_T^{\bl}(G/P)[\ecc].\]
\end{customtheorem}

Now suppose the coefficient ring is a field of characteristic dividing $\ell_G$. As already remarked by Yun and Zhu \cite{YZ}, the scheme $B_{e^T}^{\vee}$ is too big to be isomorphic to $\spec H^T_{-\bl}(\ag)$. By the same reason, $\mathcal{Y}_B$ and $\mathcal{Y}^*_B$ are too big for Theorem \ref{1athm} to hold.

It is natural to ask what if all $|\alpha^{\vee}_i|^2$ are replaced by $1$, i.e. $e$ is replaced by a principal nilpotent. We prove that Theorem \ref{1athm} still does not hold at least for the classical types.
\begin{customtheorem}{C}\label{1athmc} Suppose $G$ is of type B or C and we define $e:=\sum_{i=1}^re_{\alpha^{\vee}_i}$ instead. Then there exists a parabolic subgroup $P$ of $G$ such that
\[ \mathcal{O}((\mathcal{Y}_P)_{(0,0)})\not\simeq H^{\bullet}(G/P;\overline{\mathbb{F}}_2) \]
where $(0,0)\in\specH_T^{\bl}(\pt)[q_1,\ldots,q_k]$ is the point at which both the equivariant and quantum parameters vanish.
\end{customtheorem}
\subsection{History}\label{1h}
In 1997, Dale Peterson gave a lecture course at MIT where he announced the following four results about the quantum cohomology of flag varieties:
\begin{enumerate}
\item \textbf{Quantum Chevalley formula}, an explicit formula expressing the quantum product of a Schubert class and a divisor class in terms of other Schubert classes.

\item \textbf{Comparison formula}, an explicit formula expressing the Schubert structure constants of $QH^{\bullet}(G/P)$ in terms of those of $QH^{\bullet}(G/B)$.

\item \textbf{``Quantum equals affine'' and its generalization to the parabolic case}, which states that the affine Schubert calculus determines the quantum Schubert calculus completely and vice versa for $P=B$, through an explicit ring homomorphism from the Pontryagin homology $H_{-\bullet}(\mathcal{G}r)$ of the affine Grassmannian $\mathcal{G}r$ to $QH^{\bullet}(G/P)[q^{-1}_1,\ldots,q^{-1}_k]$.

\item \textbf{Peterson variety presentation}, which is the non-equivariant version of our Theorem \ref{1athm}.
\end{enumerate}

However, he did not publish any of these results. Some years later, complete proofs of the first three results were given by other people: (1) was proved by Fulton and Woodward \cite{FW} and generalized later by Mihalcea \cite{Mihalcea} to the equivariant settings; (2) was proved by Woodward \cite{Woodward}; and (3) was proved by Lam and Shimozono \cite{LS}. The proof in \cite{LS} is combinatorial. Recently, Chow \cite{me} has given a new proof of (3) which is geometric. Both proofs work for the equivariant settings.

The present work provides the final piece of the puzzle to Peterson's theory: it is (4) which has not been solved completely but only partially. The case $P=B$ is complete. The quantum cohomology ring $QH^{\bullet}(G/B)$ has another presentation formulated in terms of the Toda lattice. It was established by Ciocan-Fontanine \cite{CF}, Givental and Kim \cite{GK} for type A, and Kim \cite{Kim} for the general type. Although it is different from Peterson's presentation, they are equivalent after a change of variables discovered by Kostant \cite{Kostant}. Beyond the Borel case, Rietsch verified (4) for Grassmannians \cite{Pa} and later all type A flag varieties \cite{Pab}; Pech, Rietsch and Williams for quadrics \cite{Pb}; Cheong for Lagrangian and orthogonal Grassmannians \cite{Pc}; and Lam and Templier \cite{LT} for all minuscule flag varieties.

\subsection{Applications to mirror symmetry}\label{1b} The coefficient ring is taken to be $\mathbb{C}$.

\subsubsection{Teleman's proposal for three-dimensional mirror symmetry}\label{1bb}
Denote by $\bfm(G^{\vee})$ the universal centralizer of $G^{\vee}$, i.e. $\bfm(G^{\vee}):=\mathcal{J}_{reg}//G^{\vee}$ where
\[\mathcal{J}_{reg}:=\{ (g,\xi)\in G^{\vee}\times\dd{\mathfrak{g}}|~g\cdot \xi=\xi\text{ and }\xi\text{ is regular}\} . \]
It is a smooth affine variety admitting a holomorphic symplectic structure, and is related to $B_{e^T}^{\vee}$ by
\[ \mathcal{O}(\bfm(G^{\vee}))\simeq \mathcal{O}(B_{e^T}^{\vee})^W.\]
The word ``$\bfm$'' stands for Bezrukavnikov, Finkelberg and Mirkovi\'c who constructed in \cite{BFM} an isomorphism of $H_G^{\bullet}(\pt)$-algebras
\begin{equation}\nonumber\label{1beq2}
\Phi_{BFM}:\mathcal{O}(\bfm(G^{\vee}))\xrightarrow{\sim} H_{-\bullet}^G(\mathcal{G}r).
\end{equation}

Fix a maximal compact subgroup $K$ of $G$. Teleman \cite{T1} proposed that the topological gauge theory in three dimensions for $K$ be equivalent to the Rozansky-Witten theory for $\bfm(G^{\vee})$. A possible approach would be to associate to every compact Hamiltonian $K$-manifold $(M,\omega)$ an object of the Rozansky-Witten 2-category of $\bfm(G^{\vee})$. While this 2-category is so far undefined, its objects should be holomorphic Lagrangian subvarieties of $\bfm(G^{\vee})$ equipped with ``coherent sheaves of linear categories'' over them. Forgetting the extra data, one could simply ask how to associate the underlying Lagrangian $\mathbb{L}_K^{\mathbf{q}_0}(M)$ where $\mathbf{q}_0$ denotes the quantum parameters.

A recipe for defining $\mathbb{L}_K^{\mathbf{q}_0}(M)$ is already known to Teleman himself. In \cite{T2}, he defined it to be the reduced closed subscheme associated to the pull-back, via $\Phi_{BFM}$, of the kernel of a conjectural homomorphism of $H_G^{\bullet}(\pt)$-algebras
\[ \Phi_M^{\mathbf{q}_0}: H_{-\bullet}^G(\mathcal{G}r)\rightarrow QH_K^{\bullet}(M)|_{\mathbf{q}=\mathbf{q}_0}\]
which is required to admit a non-commutative deformation. He showed that the Lagrangian property follows from a theorem of Gabber \cite{Gabber}. Recently, Gonz\'alez, Mak and Pomerleano \cite{Mak} have constructed $\Phi_M^{\mathbf{q}_0}$ and its deformation (and thus $\mathbb{L}_K^{\mathbf{q}_0}(M)$) by combining Savelyev's work \cite{Savelyev1, Savelyev2, Savelyev3} and the theory of shift operators \cite{Shift1, Shift2, Shift3, Shift4, Shift5}, both generalizing Seidel representations \cite{Seidel}. Around the same time, Chow \cite{me}, who was motivated by a different problem, has constructed the same map for $M=G/P$ (without however the deformation) and proved that it coincides with the Peterson-Lam-Shimozono homomorphism which will be discussed in Section \ref{3}.

Specializing to the case $M=G/B$, Gonz\'alez, Mak and Pomerleano conjectured that $\mathbb{L}_K^{\mathbf{q}_0}(G/B)$ is the cotangent fiber $T^*_{\mathbf{q}_0}\dd{T}$ of the cotangent bundle $T^*\dd{T}$ embedded in $\bfm(G^{\vee})$ as an open symplectic submanifold via the Toda isomorphism. They expected that the result in \cite{me} would be useful. Inspired by this, we are able to verify their conjecture and generalize it to the parabolic case. It turns out that in addition to the aforementioned result, we also need Theorem \ref{1athmb}.

Let us give the precise statement. Recall the Toda space $\toda(G^{\vee})$ (see e.g. \cite[Section 6.2]{T1}) which is a symplectic quotient of $T^*G^{\vee}$ by the natural $U^{\vee}_-\timesU^{\vee}_-$-action:
\[ \toda(G^{\vee}):= U^{\vee}_-\setminus \mu^{-1}(\chi\times\chi)/U^{\vee}_-\]
where
\[
\begin{array}{ccccc}
\mu &:& T^*G^{\vee}\simeqG^{\vee}\times(\dd{\mathfrak{g}})^*&\rightarrow&(\mathfrak{n}^{\vee}_-\times\mathfrak{n}^{\vee}_-)^*\\ [.5em]
& & (g,\zeta) &\mapsto & ((g\cdot\zeta)|_{\mathfrak{n}^{\vee}_-},\zeta|_{\mathfrak{n}^{\vee}_-})
\end{array}
\]
and $\chi:=\kil{e,-}|_{\mathfrak{n}^{\vee}_-}\in (\mathfrak{n}^{\vee}_-)^*$. Here $e:=\sum_{i=1}^r|\alpha^{\vee}_i|^2 e_{\alpha^{\vee}_i}$ and $\kil{-,-}$ is any non-zero multiple of the Killing form. The Toda isomorphism is an isomorphism of schemes
\begin{equation}\nonumber
\Toda:\bfm(G^{\vee})\xrightarrow{\sim}\toda(G^{\vee})
\end{equation}
which we will describe in Section \ref{6d}.

\begin{customtheorem}{D}\label{1bthmb} For any $\mathbf{q}_0\in\spec\mathbb{C}[q_1^{\pm 1},\ldots,q_k^{\pm 1}]\subseteq T^{\vee}$, the closed subscheme $\Toda\left(\mathbb{L}_K^{\mathbf{q}_0}(G/P)\right)$ is equal to the fiber over $\mathbf{q}_0$ of the morphism
\[ \toda(G^{\vee}) \supset U^{\vee}_-\setminus \left(\mu^{-1}(\chi\times\chi)\times_{G^{\vee}} U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}T^{\vee}U^{\vee}_-\right)/U^{\vee}_-\rightarrow T^{\vee}\]
induced by the canonical projection $U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}T^{\vee}U^{\vee}_-\rightarrow T^{\vee}$.
\end{customtheorem}
These fibers happen to be the Lagrangian leaves considered in \cite[Section 6.4]{T1}.

\subsubsection{The Rietsch mirror}\label{1ba}
Building on the work of Batyrev, Ciocan-Fontanine, Kim and van Straten \cite{BCFKS}, Givental \cite{Givental}, and Joe and Kim \cite{JK}, Rietsch \cite{Rietsch} constructed, for any $X:=G/P$, a pair $(\dd{X},W)$ consisting of an affine variety $\dd{X}$ over $Z(\dd{L}):=\spec\mathbb{C}[q_1^{\pm 1},\ldots, q_k^{\pm 1}]$ and a regular function $W\in\mathcal{O}(\dd{X})$. She proved that $(\dd{X},W)$ fulfils the following mirror symmetry prediction:
\begin{customtheorem}{E}\label{1bthm} There exists an isomorphism of rings
\[ QH^{\bullet}(G/P)[q^{-1}_1,\ldots,q^{-1}_k]\simeq \jac(\dd{X},W)\]
where $\jac(\dd{X},W)$ is the Jacobi ring of $W$ relative to $Z(\dd{L})$, i.e. the quotient of $\mathcal{O}(\dd{X})$ by the ideal generated by all partial derivatives $\partial W/\partial x_i$ in the fiber direction. There is also a $T$-equivariant analogue.
\end{customtheorem}

Her proof, however, assumes our Theorem \ref{1athmb}. For completeness, we give an alternative proof based on Teleman's interpretation of the Rietsch mirror \cite[Section 6.6]{T1}. We emphasize that our contribution to the proof is verifying Theorem \ref{1athmb} only. The other arguments are essentially due to Rietsch and Teleman.
\subsection{An example}\label{1ab}
As a warm up, let us prove by hand Theorem \ref{1athm} and Theorem \ref{1athmb} for the case $G=\mathbf{SL}_{2}$ and $P=B$. We have $G/P\simeq\mathbb{P}^1_{\mathbb{C}}$ and $H_T^{\bl}(\pt)\simeq \mathbb{Z}[h]$. The Langlands dual group scheme $G^{\vee}$ is isomorphic to $\mathbf{PGL}_{2}$. Take $T^{\vee}\subsetG^{\vee}$ to be the diagonal torus and $B^{\vee}$ (resp. $B_-^{\vee}$) to be the subgroups of upper (resp. lower) triangular matrices. Then $G^{\vee}/B^{\vee}_-\simeq\mathbb{P}^1_{\mathbb{Z}}$. Denote by $U^{\vee}$ (resp. $U^{\vee}_-$) the unipotent radical of $B^{\vee}$ (resp. $B_-^{\vee}$). Let $\alpha$ (resp. $\alpha^{\vee}$)  be the unique positive root (resp. coroot) determined by $B^{\vee}$. Define
\[ e_{\alpha^{\vee}}:=\left(\begin{matrix}
0&1\\0&0
\end{matrix}\right) \in \dd{\mathfrak{g}}_{\alpha^{\vee}}\quad \text{ and }\quad e_{-\alpha^{\vee}}:=\left(\begin{matrix}
0&0\\1&0
\end{matrix}\right) \in \dd{\mathfrak{g}}_{-\alpha^{\vee}}.\]
We have
\[ e^T(h)= -h\alpha +e_{\alpha^{\vee}}.\]

The equivariant quantum cohomology $QH_T^{\bullet}(\mathbb{P}^1_{\mathbb{C}})$ is generated by the equivariant parameter $h$, the quantum parameter $q$ and the unique Schubert class $\sigma$ which is not equal to 1, subject to
\[ \sigma^2=2h\sigma +q.\]
(The presence of the factor ``2'' in the above expression is due to the fact that we are using the maximal torus of $\mathbf{SL}_{2}$ instead of $\mathbf{PGL}_{2}$.) In other words,
\[ QH_T^{\bullet}(\mathbb{P}^1_{\mathbb{C}})\simeq \mathbb{Z}[h,q,\sigma]/\langle \sigma^2-2h\sigma -q\rangle \simeq \mathbb{Z}[h,\sigma]. \]

Observe that the condition for the definition of $\mathcal{Y}$ is always satisfied, and hence
\[ \mathcal{Y}=G^{\vee}/B^{\vee}_-\times\specH_T^{\bl}(\pt)\simeq \mathbb{P}^1_{\mathbb{Z}}\times \mathbb{A}^1_{\mathbb{Z}}.\]
The projective line $\mathbb{P}^1_{\mathbb{Z}}$ is covered by two open subschemes
\[ U^{\vee}(e):=\UdB_-^{\vee}/B_-^{\vee} \simeq U^{\vee}\quad\text{ and }\quad U^{\vee}(w_0):=U^{\vee}_-\dot{w}_0B_-^{\vee}/B_-^{\vee}\simeq U^{\vee}_-\]
where $w_0\in W\simeq S_2$ is the longest element. Then we have
\[ \mathcal{Y}_P\simeq U^{\vee}(w_0)\times\mathbb{A}^1_{\mathbb{Z}}\quad\text{ and }\quad\mathcal{Y}_P^*\simeq \left(U^{\vee}(w_0)\capU^{\vee}(e)\right)\times\mathbb{A}^1_{\mathbb{Z}},\]
and hence
\[  \mathcal{O}(\mathcal{Y}_P)\simeq\mathbb{Z}[h,y_-]\quad\text{ and }\quad\mathcal{O}(\mathcal{Y}_P^*)\simeq\mathcal{O}(\mathcal{Y}_P)[y_-^{-1}]\simeq\mathbb{Z}[h,y_-^{\pm 1}]\]
where $y_-:=\chi_-(u_-)$ is the composition of the $U^{\vee}(w_0)$-coordinate $u_-$ of $\mathcal{Y}_P$ and the canonical isomorphism $\chi_-:U^{\vee}(\omega_0)\xrightarrow{\sim}\mathbb{G}_a$. The homomorphism
\[
\begin{array}{ccccc}
\Phi& :&\mathcal{O}(\mathcal{Y}_P) &\rightarrow & QH_T^{\bullet}(\mathbb{P}^1_{\mathbb{C}})\\ [.5em]
& & y_- & \mapsto & \sigma
\end{array}\]
is obviously an isomorphism of $H_T^{\bl}(\pt)$-algebras.

We also need to show that $\Phi$ is $\mathbb{Z}[q]$-linear. Let $e_{\alpha^{\vee}}^*\in (\dd{\mathfrak{g}})^*$ be the unique element satisfying $\langle e_{\alpha^{\vee}}^*,e_{\alpha^{\vee}}\rangle = 1$ and $e_{\alpha^{\vee}}^*|_{\dd{\mathfrak{b}}_-}=0$. Let $\dot{w}_0$ be the $\mathbb{Z}$-point of $N_{G^{\vee}}(T^{\vee})$ represented by the matrix $e_{\alpha^{\vee}}-e_{-\alpha^{\vee}}$. It represents $w_0$ in the Weyl group. Let $\overline{q}\in\mathcal{O}(\mathcal{Y}_P)$ be the regular function in \eqref{1aeq}. We have
\begin{align*}
\overline{q} &= \langle e_{\alpha^{\vee}}^*, (u_-\dot{w}_0)^{-1}\cdot (-h\alpha+e_{\alpha^{\vee}})\rangle\\
& = \langle e_{\alpha^{\vee}}^*, \dot{w}_0^{-1}\cdot [(2hy_--y_-^2) e_{-\alpha^{\vee}} + (y_--h)\alpha+e_{\alpha^{\vee}}]\rangle\\
& = y_-^2-2hy_-.
\end{align*}
It follows that $\Phi(\overline{q}) = \sigma^2-2h\sigma =q$ as desired.

Next we look at $\Phi^*$. Define
\[ \Phi^*:=\Phi[y_-^{-1}]:\mathcal{O}(\mathcal{Y}_P^*)\xrightarrow{\sim} QH_T^{\bullet}(\mathbb{P}^1_{\mathbb{C}})[\sigma^{-1}].\]
Observe that $\sigma^{-1}=q^{-1}(\sigma-2h)$, and hence
\[ QH_T^{\bullet}(\mathbb{P}^1_{\mathbb{C}})[\sigma^{-1}]\subseteq QH_T^{\bullet}(\mathbb{P}^1_{\mathbb{C}})[q^{-1}].\]
But this inclusion is proper unless we set $h=0$. In fact, if $\Bbbk$ is an integral domain such that $2\ne 0$, then $\Bbbk[h,\sigma^{\pm1}]$ and $\Bbbk[h,\sigma,(\sigma^2-2h\sigma)^{-1}]$ are not isomorphic as $\Bbbk$-algebras. Indeed, after inverting 2, they are isomorphic to $\Bbbk_1[X,Y^{\pm 1}]$ and $\Bbbk_1[X^{\pm 1},Y^{\pm 1}]$ respectively, where $\Bbbk_1:=\Bbbk[2^{-1}]$. But
\[\Bbbk_1[X,Y^{\pm 1}]^{\times}/\Bbbk_1^{\times}\simeq\mathbb{Z}\quad\text{ while }\quad\Bbbk_1[X^{\pm 1},Y^{\pm 1}]^{\times}/\Bbbk_1^{\times}\simeq\mathbb{Z}\oplus\mathbb{Z}. \]

We now look at $\Phi_{loc}$. Let $u_+$ and $t$ be the $U^{\vee}$- and $T^{\vee}$-coordinates of $B^{\vee}\simeqU^{\vee}\timesT^{\vee}$ respectively. Then
\[ (u_+t)\cdot e^T(h) = -h\alpha +(\alpha^{\vee}(t)+2h\chi_+(u_+))e_{\alpha^{\vee}}\]
where $\chi_+:U^{\vee}\xrightarrow{\sim} \mathbb{G}_a$ is the canonical isomorphism. Put $y_+:=\chi_+(u_+)$ and $z:=\alpha^{\vee}(t)$. It follows that
\[ \mathcal{O}(B_{e^T}^{\vee})\simeq \mathbb{Z}[h,y_+,z^{\pm 1}]/\langle z+2hy_+-1\rangle\simeq \mathbb{Z}[h,y_+,(1-2hy_+)^{-1}]\]
and
\[ \mathcal{O}(B_{e^T}^{\vee}\tGBU^{\vee}(w_0))\simeq \mathbb{Z}[h,y_+^{\pm 1}, (1-2hy_+)^{-1}].\]
It is straightforward to show that the homomorphism
\[
\begin{array}{ccccc}
\Phi_{loc}& :&\mathcal{O}(B_{e^T}^{\vee}\tGBU^{\vee}(w_0)) &\rightarrow & QH_T^{\bullet}(\mathbb{P}^1_{\mathbb{C}})[q^{-1}]\\ [.5em]
& & y_+ & \mapsto & \sigma^{-1}=q^{-1}(\sigma-2h)
\end{array}\]
is an isomorphism of $H_T^{\bl}(\pt)$-algebras. Finally, observe that $y_-=y_+^{-1}$, and hence
\[ \Phi_{loc}(\overline{q}) = \Phi_{loc}(y_+^{-1}-2hy_+^{-1})=\sigma^2-2h\sigma=q.\]
Therefore, $\Phi_{loc}$ is $\mathbb{Z}[q]$-linear.
\section{Preliminaries}\label{2}
\subsection{Basic notations}\label{2a}
Let $G$ be a complex connected reductive group. Fix a maximal torus $T$ and a Borel subgroup $B$ containing $T$. Let $B_-$ be the unique Borel subgroup containing $T$ such that $B\cap B_-=T$. Denote by $R$ the set of roots and by $R^+$ the set of positive roots determined by $B$. The simple roots are denoted by $\alpha_1,\ldots,\alpha_r$. For any $\alpha\in R$, denote by $\alpha^{\vee}$ the corresponding coroot. Let $W$ be the Weyl group of $G$ and $w_0$ the longest element of $W$. Let $\mathbb{X}^{\bl}(T)$ (resp. $\mathbb{X}_{\bl}(T)$) be the lattice of the characters (resp. cocharacters) of $T$. Let $\mathbb{X}^+_{\bl}(T)$ be the subset of $\mathbb{X}_{\bl}(T)$ consisting of elements which are dominant. Let $Q^{\vee}$ be the coroot lattice. In general, $Q^{\vee}\subseteq\mathbb{X}_{\bl}(T)$, and the equality holds precisely when $G$ is simply-connected. Define
\[ 2\rho:=\sum_{\alpha\in R^+}\alpha\in\mathbb{X}^{\bl}(T)\quad\text{ and }\quad 2\rho^{\vee}:=\sum_{\alpha\in R^+}\alpha^{\vee}\in\mathbb{X}_{\bl}(T).\]

Fix a parabolic subgroup $P$ of $G$ containing $B$. Denote by $R_P\subseteq R$ (resp. $R^+_P\subseteq R^+$) the set of (resp. positive) roots of $P$. WLOG, we assume
\[R_P\cap\{\alpha_1,\ldots,\alpha_r\}=\{\alpha_{k+1},\ldots,\alpha_r\}\]
for some integer $0\leqslant k\leqslant r$. Let $W_P\subseteq W$ be the Weyl group of P and $w_P\in W_P$ the longest element of $W_P$. Define $Q^{\vee}_P$ to be the sublattice of $Q^{\vee}$ generated by $\alpha^{\vee}$ with $\alpha\in R_P$.
\subsection{Equivariant quantum cohomology of flag variety}\label{2bc}
Let $P$ be the standard parabolic subgroup of $G$ fixed in Section \ref{2a}. Denote by $W^P$ the set of minimal length representatives of $W/W_P$. For each $v\in W^P$, define
\begin{align*}
 \sigma_v &:=\pd\bk{\overline{B^-\dot{v}P/P}} \in H_T^{2\ell(v)}(G/P)\\
  \sigma^v & :=\pd\bk{\overline{B\dot{v}P/P}}\in H_T^{\dim_{\mathbb{R}}(G/P)-2\ell(v)}(G/P).
\end{align*}
The families $\{\sigma_v\}_{v\in W^P}$ and $\{\sigma^v\}_{v\in W^P}$ are dual bases of $H_T^{\bullet}(G/P)$ with respect to $\int_{G/P}-\cup -$.

Let $\ec$ be the semi-group of effective curve classes in $G/P$ and $\mathbb{Z}[\ec]$ the associated group ring. Since $\ec$ is naturally identified with the semi-subgroup of $Q^{\vee}/Q^{\vee}_P\simeq \bigoplus_{i=1}^k\mathbb{Z}\cdot\alpha^{\vee}_i$ generated by $\alpha^{\vee}_1,\ldots,\alpha^{\vee}_k$, the ring $\mathbb{Z}[\ec]$ is isomorphic to the polynomial ring $\mathbb{Z}[q_1,\ldots,q_k]$ where $q_i:=q^{[\alpha^{\vee}_i]}$. Define the equivariant quantum cohomology of $G/P$
\[ QH_T^{\bullet}(G/P):= H_T^{\bullet}(G/P)\otimes\mathbb{Z}[\ec].\]
We grade $QH_T^{\bullet}(G/P)$ by declaring each quantum parameter $q^{\beta}$ to have degree $2\langle c_1(G/P),\beta\rangle$. The equivariant quantum cup product $\star$ is a deformation of the equivariant cup product, defined by
\[\sigma_u\star \sigma_v := \sum_{w\in W^P}\sum_{\beta\in\ec} q^{\beta} \left(\int_{\overline{\mathcal{M}}_{0,3}(G/P,\beta)} \ev_1^*\sigma_u\cup \ev_2^*\sigma_v\cup  \ev_3^*\sigma^w \right) \sigma_w,\]
where
\begin{enumerate}[(i)]
\item $\overline{\mathcal{M}}_{0,3}(G/P,\beta)$ is the moduli of genus-zero stable maps to $G/P$ of degree $\beta$ which have three marked points in the domain and
\[\ev_1,\ev_2,\ev_3:\overline{\mathcal{M}}_{0,3}(G/P,\beta)\rightarrow G/P\]
are the evaluation maps at these marked points; and

\item the integral $\int_{\overline{\mathcal{M}}_{0,3}(G/P,\beta)}$ is the $T$-equivariant integral.
\end{enumerate}

\noindent It is well-known (see e.g. \cite{BM, FP}) that
\begin{enumerate}
\item $\overline{\mathcal{M}}_{0,3}(G/P,\beta)$ is a smooth projective Deligne-Mumford stack of expected dimension. In particular, the above integral is well-defined.

\item $(QH_T^{\bullet}(G/P),\star)$ is a graded commutative $H_T^{\bl}(\pt)$-algebra.
\end{enumerate}
\subsection{Affine Grassmannian}\label{2b}
Let $\mathcal{G}r:=\mathcal{G}r_G:=G(\mathcal{K})/G(\mathcal{O})$ be the affine Grassmannian, where $\mathcal{K}:=\mathbb{C}((z))$ and $\mathcal{O}:=\mathbb{C}[[z]]$. Every $\lambda\in\mathbb{X}_{\bl}(T)$ gives rise to a point $t^{\lambda}$ of $\mathcal{G}r$. The set $\{t^{\lambda}\}_{\lambda\in\mathbb{X}_{\bl}(T)}$ is the set of $T$-fixed points of $\mathcal{G}r$. Suppose $\lambda$ is dominant. Define $\agl:=G(\mathcal{O})\cdot t^{\lambda}$ which is an affine bundle over $G\cdot t^{\lambda}$. Its closure $\overline{\agl}$ is called a spherical affine Schubert variety. It is known that $\overline{\agl}=\bigcup_{\mathbb{X}^+_{\bl}(T)\ni\mu\leqslant\lambda}\mathcal{G}r^{\mu}$ where $\mu\leqslant\lambda$ if and only if $\lambda-\mu$ is equal to a $\mathbb{Z}_{\geqslant 0}$-linear combination of $\alpha^{\vee}_1,\ldots,\alpha^{\vee}_r$. Because of this equality, we will also denote the closure by $\agll$.

The Cartan decomposition tells us that $\mathcal{G}r=\bigcup_{\lambda\in\mathbb{X}^+_{\bl}(T)}\agl$. We can further decompose each $\agl$ into smaller strata. Define $\mathcal{B}:=\ev^{-1}_{z=0}(B^-)$ where $\ev_{z=0}:G(\mathcal{O})\rightarrow G$ is the evaluation map at $z=0$. For any $\mu\in\mathbb{X}_{\bl}(T)$ not necessarily dominant, the orbit $\mathcal{B}\cdot t^{\mu}$ is isomorphic to an affine space and called an affine Schubert cell. Its closure $\overline{\mathcal{B}\cdot t^{\mu}}$ is called an affine Schubert variety. We have $\agl=\bigcup_{\mu\in W\cdot \lambda}\mathcal{B}\cdot t^{\mu}$.

Let $K$ be a maximal compact subgroup of $G$ such that $T\cap K$ is a maximal torus of $K$. It is known that $\mathcal{G}r$ is homeomorphic, as $T\cap K$-spaces, to the polynomial based loop group $\Omega K$. The group structure on $\Omega K$ thus induces an $H_T^{\bl}(\pt)$-algebra structure on $H_{\bullet}^{T\cap K}(\Omega K)\simeq H^T_{\bl}(\ag)$. This product is called the Pontryagin product and denoted by $\bulletsmall$. It is very simple when expressed in terms of the point classes:
\begin{equation}\nonumber \label{2beq1}
[t^{\mu_1}]\bulletsmall [t^{\mu_2}] = [t^{\mu_1+\mu_2}]\quad \mu_1,\mu_2\in \mathbb{X}_{\bl}(T).
\end{equation}
\subsection{Geometric Satake equivalence}\label{2c}
We give a brief review on the geometric Satake equivalence proved by Mirkovi\'c and Vilonen \cite{MV}. See also the work \cite{BD, Ginzburg, Lu}. Let $\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$ be the category of $G(\mathcal{O})$-equivariant perverse sheaves on $\mathcal{G}r$ with finite dimensional support. The coefficient ring we consider here is $\mathbb{Z}$. It is an abelian category equipped with a tensor product structure given by the convolution product. The hypercohomology functor $H^{\bullet}(-)$ defines a fiber functor on $\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$. There exists a functorial isomorphism
\begin{equation}\label{4aweight}
H^{\bullet}(\mathcal{F})\simeq \bigoplus_{\mu\in\mathbb{X}_{\bl}(T)} H^{2\rho(\mu)}_{S_{\mu}^-}(\mathcal{F})
\end{equation}
for any object $\mathcal{F}$, where
\[  S^-_{\mu}:= \left\{ y\in\mathcal{G}r\left|~\lim_{s\to\infty}2\rho^{\vee}(s)\cdot y= t^{\mu} \right.\right\}.\]
Using the information about $\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$, Mirkovi\'c and Vilonen constructed a group scheme $\widetilde{G}$ over $\spec\mathbb{Z}$ which contains the dual split torus $T^{\vee} :=\spec\mathbb{Z}[\mathbb{X}_{\bl}(T)]$ as a maximal torus and satisfies
\begin{equation}\label{4agse}
\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)\simeq \rep(\widetilde{G})
\end{equation}
as tensor categories, where $\rep(\widetilde{G})$ is the category of representations of $\widetilde{G}$ which are finitely generated over $\mathbb{Z}$. Moreover, the fiber functor $H^{\bullet}(-)$ on $\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$ corresponds to the forgetful functor on $\rep(\widetilde{G})$. They showed that $\widetilde{G}$ is split reductive and the root datum associated to $(\widetilde{G},T^{\vee})$ is dual to that associated to $(G,T)$, and hence $\widetilde{G}\simeq G^{\vee}$, the Langlands dual group scheme of $G$.

We will consider a particular class of objects of $\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$ called costandard sheaves. Let $\lambda\in\mathbb{X}^+_{\bl}(T)$. Define
\[\mathcal{J}_{\lambda}:={}^p\mathcal{H}^0((j_{\lambda})_*\mathbb{Z}[2\rho(\lambda)])\]
where ${}^p\mathcal{H}^0$ is the zeroth perverse cohomology functor and $j_{\lambda}:\agl\hookrightarrow \agll$ is the inclusion map.
\begin{lemma}\label{2clemma2} \cite[Proposition 3.10]{MV}
For any $\mu\in\mathbb{X}_{\bl}(T)$, there exists an isomorphism between $H^{2\rho(\mu)}_{S^-_{\mu}}(\mathcal{J}_{\lambda})$ and the free abelian group generated by the irreducible components of $S^-_{\mu}\cap \agl$. \hfill$\square$
\end{lemma}
\begin{lemma}\label{2clemma1}
There exists a functorial isomorphism
\begin{equation}\label{2cuniversal}
\home_{\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)}(\mathcal{F},\mathcal{J}_{\lambda})\simeq \home_{\mathbb{Z}}\left(H^{2\rho(\lambda)}_{S_{\lambda}^-}(\mathcal{F}), \mathbb{Z}\right)
\end{equation}
for any $\mathcal{F}\in\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$ which has support in $\agll$.
\end{lemma}
\begin{proof}
By the definition of the perverse t-structure on $\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$, the complex $(j_{\lambda})_*\mathbb{Z}[2\rho(\lambda)]$ is concentrated in non-negative perverse degrees. Hence we have
\begin{equation}\label{2ceq}
\home_{\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)}(\mathcal{F},\mathcal{J}_{\lambda})\simeq \home_{D_{G(\mathcal{O})}(\mathcal{G}r)}(\mathcal{F},(j_{\lambda})_*\mathbb{Z}[2\rho(\lambda)]).
\end{equation}
By adjunction and Verdier duality, the RHS of \eqref{2ceq} is isomorphic to $\home_{\mathbb{Z}}\left( H_c^{2\rho(\lambda)}(\agl;\mathcal{F}),\mathbb{Z}\right)$. By \cite[Theorem 3.5]{MV} and an excision argument, we have
\[H_c^{2\rho(\lambda)}(\agl;\mathcal{F})\simeq H^{2\rho(\lambda)}_{S_{\lambda}^-}(\mathcal{F}).\] The result follows.
\end{proof}
\subsection{Scheme of Borel subgroups}\label{2d}
Let $(G^{\vee},T^{\vee})$ be as in Section \ref{2c}, i.e. $G^{\vee}$ is the Chevalley-Demazure group scheme defined over $\spec\mathbb{Z}$ and $T^{\vee}$ is a maximal torus of $G^{\vee}$ such that the root datum associated to $(G^{\vee},T^{\vee})$ is dual to that associated to $(G,T)$. Let $B^{\vee}$ be the Borel subgroup of $G^{\vee}$ associated to the set of positive coroots and $B_-^{\vee}$ the unique Borel subgroup containing $T^{\vee}$ such that $B^{\vee}\capB_-^{\vee}=T^{\vee}$. Let $U^{\vee}$ (resp. $U^{\vee}_-$) denote the unipotent radical of $B^{\vee}$ (resp. $B_-^{\vee}$). By \cite[Corollary 5.2.8]{BC}, the \'etale quotient sheaf $G^{\vee}/B^{\vee}_-$, i.e. the \'etale sheafification of the presheaf $R\mapsto G^{\vee}(R)/B_-^{\vee}(R)$, is represented by a smooth projective scheme.

Let $\lambda\in\mathbb{X}^+_{\bl}(T)$ and $\mathcal{J}_{\lambda}$ be the costandard sheaf defined in Section \ref{2c}. Define
\[S(\lambda):=H^{\bullet}(\mathcal{J}_{\lambda})\] which is a $G^{\vee}$-module whose $T^{\vee}$-weights are all $\leqslant\lambda$. For any $\mu\in\mathbb{X}_{\bl}(T)$, put $S(\lambda)_{\mu}:=H^{2\rho(\mu)}_{S^-_{\mu}}(\mathcal{J}_{\lambda})$. Let $v_{\lambda}^*\in S(\lambda)^*$ be the unique element vanishing on $S(\lambda)_{\mu} $ for any $\mu\ne\lambda$ and sending the generator of $S(\lambda)_{\lambda}\simeq \mathbb{Z}$ corresponding to $S^-_{\lambda}\cap\agl =\{t^{\lambda}\}$ to 1. See Lemma \ref{2clemma2}.

Define $M_{\lambda}$ to be the rank-one $B_-^{\vee}$-module satisfying
\[ \res^{B_-^{\vee}}_{U^{\vee}_-}(M_{\lambda})\text{ is trivial}\quad\text{and}\quad\res^{B_-^{\vee}}_{T^{\vee}}(M_{\lambda})\text{ has weight }\lambda. \]
Define
\[\mathcal{L}(\lambda):=G^{\vee}\times_{B_-^{\vee}}M_{\lambda}\]
to be the fppf descent, along the quotient morphism $G^{\vee}\rightarrowG^{\vee}/B^{\vee}_-$, of the $B_-^{\vee}$-equivariant line bundle $\mathcal{O}_{G^{\vee}}\otimes M_{\lambda}$. It is a line bundle over $G^{\vee}/B^{\vee}_-$. The element $v_{\lambda}^*$ defined above can be regarded as a morphism $S(\lambda)\rightarrow M_{\lambda}$ of $B_-^{\vee}$-modules. Hence it gives rise to a morphism
\begin{equation}\label{2deq1}
\mathcal{O}_{G^{\vee}/B^{\vee}_-}\otimes S(\lambda)\simeq G^{\vee}\times_{B_-^{\vee}} S(\lambda)\rightarrow \mathcal{L}(\lambda)
\end{equation}
of $\mathcal{O}_{G^{\vee}/B^{\vee}_-}$-modules, where the trivialization of $G^{\vee}\times_{B_-^{\vee}} S(\lambda)$ is induced by $(g,v)\mapsto g^{-1}\cdot v$.
\begin{lemma}\label{2dlemma1}
For any $\lambda\in\mathbb{X}^+_{\bl}(T)$, the linear map
\begin{equation}\label{2deq15}
 S(\lambda)\simeq H^0(G^{\vee}/B^{\vee}_-;G^{\vee}\times_{B_-^{\vee}} S(\lambda))\rightarrow H^0(G^{\vee}/B^{\vee}_-;\mathcal{L}(\lambda))
\end{equation}
induced by \eqref{2deq1} is bijective.
\end{lemma}
\begin{proof}
The following arguments can be found in \cite[Section 13]{MV}. Notice that
\[H^0(G^{\vee}/B^{\vee}_-;\mathcal{L}(\lambda))\simeq \ind_{B_-^{\vee}}^{G^{\vee}}(M_{\lambda})\]
as $G^{\vee}$-modules. It is well-known that the latter $G^{\vee}$-module is an object of $\rep^{\leqslant\lambda}(G^{\vee})$, the full subcategory of $\rep(G^{\vee})$ consisting of objects whose $T^{\vee}$-weights are all $\leqslant\lambda$, and that its $\lambda$-weight space is isomorphic to $\mathbb{Z}$. Moreover, for any $V\in\rep^{\leqslant\lambda}(G^{\vee})$, the Frobenius reciprocity gives
\begin{equation}\label{2deq2}
\home_{\rep(G^{\vee})}\left(V,\ind_{B_-^{\vee}}^{G^{\vee}}(M_{\lambda})\right)\simeq \home_{\rep(B_-^{\vee})}\left(\res_{B_-^{\vee}}^{G^{\vee}}(V),M_{\lambda}\right)\simeq \home_{\mathbb{Z}}(V_{\lambda},\mathbb{Z}).
\end{equation}
Take $V=S(\lambda)$. Since $S(\lambda)$ satisfies the same universal property, namely \eqref{2cuniversal}, we have
\[S(\lambda)\simeq \ind_{B_-^{\vee}}^{G^{\vee}}(M_{\lambda})\]
as $G^{\vee}$-modules. The proof is complete if we can show that the element of the RHS of \eqref{2deq2} corresponding to our map \eqref{2deq15} is an isomorphism. This follows from the assumption that $v_{\lambda}^*$ sends a generator of $S(\lambda)_{\lambda}$ to $1$.
\end{proof}

Recall $W$ is the Weyl group of $G$. It is also the Weyl group of $G^{\vee}$ considered as a constant group scheme over $\spec\mathbb{Z}$. For any $w\in W$, choose a representative $\dot{w}\in N_{G^{\vee}}(T^{\vee})(\spec\mathbb{Z})$ of $w$. (In later sections, we will consider specific choices for some $w$.) Define $U^{\vee}(w):=\dot{w}\UdB_-^{\vee}/B_-^{\vee}$. It is an open Bruhat cell of $G^{\vee}/B^{\vee}_-$ with respect to the Borel subgroup $B^{\vee}_w:=\dot{w}B^{\vee}\dot{w}^{-1}$. Define $v_w\in S(\lambda)_{w(\lambda)}\simeq \mathbb{Z}$ to be the generator corresponding to the unique irreducible component of $S^-_{w(\lambda)}\cap \agl$, via the isomorphism in Lemma \ref{2clemma2}.

For any $v\in S(\lambda)$, denote by $s_{\lambda,v}\in H^0(G^{\vee}/B^{\vee}_-;\mathcal{L}(\lambda))$ its image under \eqref{2deq15}. An important fact about $s_{\lambda,v}$ is that the pull-back of $s_{\lambda,v}$ via the quotient morphism $G^{\vee}\rightarrowG^{\vee}/B^{\vee}_-$ is equal to the regular function $\theta(\mathcal{J}_{\lambda},v,v_{\lambda}^*)\in \mathcal{O}(G^{\vee})\otimes M_{\lambda}\simeq\mathcal{O}(G^{\vee})$ defined by
\[\theta(\mathcal{J}_{\lambda},v,v_{\lambda}^*):g\mapsto \langle v_{\lambda}^*,g^{-1}\cdot v\rangle.\]

\begin{lemma}\label{2dlemma3}
For any $w\in W$ and $\lambda\in\mathbb{X}^+_{\bl}(T)$, the section $s_{\lambda,v_w}$ is non-vanishing on $U^{\vee}(w)$. If $\lambda$ is regular dominant, then $s_{\lambda,v_w}$ is non-vanishing precisely on $U^{\vee}(w)$.
\end{lemma}
\begin{proof}
It suffices to verify the assertions for the geometric points so we assume that the base scheme is $\spec \Bbbk$ for some algebraically closed field $\Bbbk$. In this case, we have the Bruhat decomposition for $G^{\vee}/B^{\vee}_-$, and since $ \{s_{\lambda,v_w}\ne 0\}$ is $B^{\vee}_w$-invariant, we only need to show $s_{\lambda,v_w}(\dot{w}B_-^{\vee})\ne 0$, and if $\lambda$ is regular dominant, then $s_{\lambda,v_w}(\dot{v}B_-^{\vee})=0$ for any $v\in W\setminus\{w\}$. This follows from the equality
\[s_{\lambda,v_w}(\dot{v}B_-^{\vee})=\langle v_{\lambda}^*,\dot{v}^{-1}\cdot v_w \rangle\]
and the fact that $\dot{v}^{-1}\cdot v_w $ is a generator of $S(\lambda)_{v^{-1}w(\lambda)}$.
\end{proof}

\begin{lemma}\label{2dlemma2}
There exists $\lambda_0\in\mathbb{Z}_{>0}\cdot 2\rho^{\vee}$ such that $\mathcal{L}(\lambda_0)$ is very ample.
\end{lemma}
\begin{proof}
Put $X:=G^{\vee}/B^{\vee}_-$ and let $f:G^{\vee}\rightarrow X$ be the quotient morphism. By \cite[Remark 2.3.7]{BC} and \cite[Theorem 5.2.11]{BC}, the determinant $\det(\mathcal{T}_X)$ of the tangent bundle $\mathcal{T}_X$ is ample. Hence it suffices to show $\det(\mathcal{T}_X)\simeq \mathcal{L}(2\rho^{\vee})$. By descent, it amounts to showing
\[ f^*\det(\mathcal{T}_X)\simeq \mathcal{O}_{G^{\vee}}\otimes M_{2\rho^{\vee}}\]
as $B_-^{\vee}$-equivariant sheaves.

Since $f$ is smooth, we have a short exact sequence
\[ 0\rightarrow \mathcal{T}_{G^{\vee}/X}\rightarrow \mathcal{T}_{G^{\vee}}\rightarrow f^*\mathcal{T}_X\rightarrow 0.\]
Let us determine the second arrow. Recall it is induced by the canonical morphism
\begin{equation}\label{2deeq1}
G^{\vee}\times_XG^{\vee}\rightarrow G^{\vee}\timesG^{\vee}.
\end{equation}
There are isomorphisms
\[ G^{\vee}\times_XG^{\vee}\simeq G^{\vee}\timesB_-^{\vee}\quad\text{ and }\quad G^{\vee}\timesG^{\vee}\simeq  G^{\vee}\timesG^{\vee} \]
under which \eqref{2deeq1} is identified with $\id_{G^{\vee}}\times\iota$ where $\iota:B_-^{\vee}\hookrightarrow G^{\vee}$ is the inclusion, and the diagonal morphisms become $g\mapsto (g,e)$. It follows that the second arrow is the monomorphism $\mathcal{O}_{G^{\vee}}\otimes\dd{\mathfrak{b}}_-\rightarrow \mathcal{O}_{G^{\vee}}\otimes\dd{\mathfrak{g}}$ induced by the inclusion $\dd{\mathfrak{b}}_-\hookrightarrow\dd{\mathfrak{g}}$, and hence
\[  f^*\det(\mathcal{T}_X)\simeq \det(f^*\mathcal{T}_X)\simeq \det(\mathcal{O}_{G^{\vee}}\otimes \dd{\mathfrak{g}}/\dd{\mathfrak{b}}_-)\simeq \mathcal{O}_{G^{\vee}}\otimes \det( \dd{\mathfrak{g}}/\dd{\mathfrak{b}}_-).\]

The result will follow if we can show that $\det( \dd{\mathfrak{g}}/\dd{\mathfrak{b}}_-)\simeq M_{2\rho^{\vee}}$ as $B_-^{\vee}$-modules. It is clear that they are isomorphic as $T^{\vee}$-modules. By definition, $M_{2\rho^{\vee}}$ is a trivial $U^{\vee}_-$-module. The same holds for $\det( \dd{\mathfrak{g}}/\dd{\mathfrak{b}}_-)$ as well because every graded piece associated to the $2\rho^{\vee}$-filtration on $\dd{\mathfrak{g}}/\dd{\mathfrak{b}}_-$ is a trivial $U^{\vee}_-$-module. The proof is complete.
\end{proof}

\begin{proposition} \label{2dcor}
Let $\lambda_0$ be the element in Lemma \ref{2dlemma2}. For any $w\in W$, the ring $\mathcal{O}(U^{\vee}(w))$ is generated by regular functions $s_{\lambda_0,v}/s_{\lambda_0,v_w}$ with $v\in S(\lambda_0)$.
\end{proposition}
\begin{proof}
This follows from Lemma \ref{2dlemma1}, Lemma \ref{2dlemma3} and Lemma \ref{2dlemma2}.
\end{proof}
\section{Peterson-Lam-Shimozono's theorem}\label{3}
\subsection{Statement}\label{3a}
Assume $G$ is simple and simply-connected. Denote by $W_{af}$ the affine Weyl group and by $W_{af}^-$ the set of minimal length representatives of the quotient $W_{af}/W$. Elements of $W_{af}$ are denoted by $wt_{\lambda}$ where $w\in W$ and $t_{\lambda}$ ($\lambda\inQ^{\vee}$) are translations $x\mapsto x+\lambda$. Notice that the map $wt_{\lambda}\mapsto w(\lambda)$ defines a bijection $W_{af}^-\simeq Q^{\vee}$. For any $wt_{\lambda}\in W_{af}^-$, define
\[ \xi_{wt_{\lambda}} := \bk{\overline{\mathcal{B}\cdot t^{w(\lambda)}}}\in H^T_{2\ell(wt_{\lambda})}(\mathcal{G}r). \]
It is known that $\{\xi_{wt_{\lambda}}\}_{wt_{\lambda}\in W_{af}^-}$ forms an $H_T^{\bl}(\pt)$-basis of $H^T_{\bl}(\ag)$. Define $(W^P)_{af}$ to be the set of $wt_{\lambda}\in W_{af}$ such that
\[\left\{
\begin{array}{rcl}
\alpha\in R_P^+\cap w^{-1}R^+&\Longrightarrow& \alpha(\lambda)=0\\
\alpha\in R_P^+\cap (-w^{-1}R^+) &\Longrightarrow& \alpha(\lambda)=-1
\end{array}
\right. .\]
For any $w\in W$, denote by $\widetilde{w}$ the unique element of $W^P$ satisfying $wW_P=\widetilde{w}W_P$.
\begin{theorem}\cite{me, LS}\label{3aPLS}
The $H_T^{\bl}(\pt)$-linear map
\[\Phi_{PLS}: H_{-\bullet}^T(\mathcal{G}r)\rightarrow QH^{\bullet}_T(G/P)[q^{-1}_1,\ldots,q^{-1}_k]\]
defined by
\[\Phi_{PLS}(\xi_{wt_{\lambda}}):=\left\{
\begin{array}{cc}
q^{[\lambda]}\sigma_{\widetilde{w}}& wt_{\lambda}\in (W^P)_{af}\\ [.5em]
0& \text{otherwise}
\end{array}
\right.
\]
is a graded homomorphism of $H_T^{\bl}(\pt)$-algebras. \hfill$\square$
\end{theorem}
\subsection{As Savelyev-Seidel homomorphism}\label{3b}
We briefly recall a result of Chow \cite{me} which interprets $\Phi_{PLS}$ as the Savelyev-Seidel homomorphism \cite{Savelyev1, Savelyev2, Savelyev3, Seidel}. This point of view will be crucial for the construction of the isomorphism $\Phi$ stated in Theorem \ref{1athm}.

By Beauville-Laszlo's theorem, the affine Grassmannian $\mathcal{G}r$ parametrizes isomorphism classes of principal $G$-bundles over $\mathbb{P}^1$ with a trivialization over $\mathbb{P}^1\setminus\{0\}$. For any morphism $f:\Gamma\rightarrow \mathcal{G}r$, denote by $\fib{f}$ the associated object which is a principal $G$-bundle over $\mathbb{P}^1\times \Gamma$ with a trivialization over $(\mathbb{P}^1\setminus\{0\})\times \Gamma$. For any $G$-variety $X$, define $\fib{f}(X):=\fib{f}\times_G X$. We will mainly consider the case $X=G/P$. Let $\eta\in (Q^{\vee}/Q^{\vee}_P)^*\simeq H^2(G/P)$. By the same construction, there exists a line bundle $\mathcal{L}_{\eta}$ over $\ffib{f}$ whose restriction to each fiber of the natural projection $\pi_f:\ffib{f}\rightarrow \mathbb{P}^1\times \Gamma$ is isomorphic to the line bundle $G\times_P\mathbb{C}_{-\eta}$ over $G/P$.

We consider the following two classes of the morphism $f$:
\begin{enumerate}
\item \textit{The inclusion map $\{t^{\mu}\}\hookrightarrow \mathcal{G}r$, $\mu\inQ^{\vee}$.} Denote by $\ffib{t^{\mu}}$ the corresponding fiber bundle. Then
\[ \ffib{t^{\mu}}\simeq (\mathbb{C}\times G/P)\cup (\mathbb{C}\times G/P)/_{(z,y)\sim (z^{-1}, t^{\mu}(z)\cdot y)}.\]

\item \textit{The inclusion map $\agll\hookrightarrow \mathcal{G}r$, $\lambda\inQ^{\vee}$ dominant.} Denote by $\ffib{\agll}$ the corresponding fiber bundle.

\item \textit{$G(\mathcal{O})$-equivariant resolutions $r_{\lambda}:\Gamma_{\lambda}\rightarrow \agll$, $\lambda\inQ^{\vee}$ dominant.} Denote by $\ffib{r_{\lambda}}$ the corresponding fiber bundle and by $\pi_{\lambda}:\ffib{r_{\lambda}}\rightarrow \mathbb{P}^1\times \Gamma_{\lambda}$ the natural projection. Define $D_{\lambda}:=\pi_{\lambda}^{-1}(\{\infty\}\times \Gamma_{\lambda})$ and denote by $\iota_{\lambda}:D_{\lambda}\hookrightarrow \ffib{r_{\lambda}}$ the inclusion map. The trivialization of $\ffib{r_{\lambda}}$ over $(\mathbb{P}^1\setminus\{0\})\times\Gamma_{\lambda}$ induces an isomorphism $D_{\lambda}\simeq \Gamma_{\lambda}\times G/P$.
\end{enumerate}

\begin{definition}\label{3bdef} Let $\lambda\inQ^{\vee}$ be dominant and $[\nu]\in Q^{\vee}/Q^{\vee}_P$.
\begin{enumerate}
\item Define
\[ \ol{\mathcal{M}}(\lambda,[\nu]):=\bigcup_{\beta}\overline{\mathcal{M}}_{0,1}(\ffib{r_{\lambda}},\beta)\times_{(\ev_1,\iota_{\lambda})}D_{\lambda}\]
where $\beta$ runs over all curve classes in $\ffib{r_{\lambda}}$ such that $(\pi_{\lambda})_*\beta=[\mathbb{P}^1\times\{\pt\}]$ and $\langle c_1(\mathcal{L}_{\eta}),\beta\rangle=\eta(\nu)$ for any $\eta\in (Q^{\vee}/Q^{\vee}_P)^*$.

\item Define
\[ \ev_{\lambda,[\nu]}: \ol{\mathcal{M}}(\lambda,[\nu])\rightarrow G/P\]
to be the composition of the morphisms
\[ \ol{\mathcal{M}}(\lambda,[\nu])\xrightarrow{\ev_1} D_{\lambda}\simeq \Gamma_{\lambda}\times G/P\rightarrow G/P\]
where the last arrow is the natural projection.

\item Define
\[ \Pi_{\lambda,[\nu]}:\ol{\mathcal{M}}(\lambda,[\nu])\rightarrow \Gamma_{\lambda}\]
to be the morphism sending every stable map $u$ to the point in $\Gamma_{\lambda}$ over which $u$ lies, i.e. $\im(\pi_{\lambda}\circ u)=\mathbb{P}^1\times\{\Pi_{\lambda,[\nu]}(u)\}$.
\end{enumerate}
\end{definition}

\begin{proposition}\label{3bDM} \cite[Proposition 4.5]{me} $\ol{\mathcal{M}}(\lambda,[\nu])$ is a smooth projective Deligne-Mumford stack of expected dimension such that
\[ \dim\ol{\mathcal{M}}(\lambda,[\nu]) = 2\rho(\lambda)+\dim G/P+\sum_{\alpha\in R^+\setminus R^+_P}\alpha(\nu).\]
Moreover, the $G$-action on $\Gamma_{\lambda}$ induces a $G$-action on $\ol{\mathcal{M}}(\lambda,[\nu])$ such that $\ev_{\lambda,[\nu]}$ is $G$-equivariant. In particular, $\ev_{\lambda,[\nu]}$ is smooth. \hfill$\square$
\end{proposition}

\begin{definition} Define the Savelyev-Seidel homomorphism
\[ \Phi_{SS}: H^T_{-\bl}(\ag)\rightarrow QH_T^{\bl}(G/P)[\ecc]\]
characterized by the equation
\begin{equation}\label{3beq1}
\Phi_{SS}((r_{\lambda})_*x)=\sum_{v\in W^P}\sum_{[\nu]\in Q^{\vee}/Q^{\vee}_P} q^{[\nu]}\left( x\bullet (\Pi_{\lambda,[\nu]})_*\ev_{\lambda,[\nu]}^*\sigma^v\right) \sigma_v
\end{equation}
for any dominant $\lambda\inQ^{\vee}$ and $x\in H^T_{\bl}(\Gamma_{\lambda})$, where $\bullet$ is the intersection product on $\Gamma_{\lambda}$. One can show, e.g. by the localization formula, that \eqref{3beq1} does yield a well-defined map.
\end{definition}

Theorem \ref{3aPLS} follows from \cite[Proposition 3.10]{me} and \cite[Theorem 4.7]{me}, summarized as follows.
\begin{theorem}\label{3bthm} $~$
\begin{enumerate}
\item $\Phi_{SS}$ is a homomorphism of $H_T^{\bl}(\pt)$-algebras.
\item $\Phi_{SS}=\Phi_{PLS}$.
\end{enumerate}
\hfill $\square$
\end{theorem}

Notice that both $\mathcal{G}r$ and $G/P$ are $G$-spaces. Hence there are natural $W$-actions on $H^T_{\bl}(\ag)$ and $QH_T^{\bl}(G/P)$. (In fact, the rings of the invariant elements are $H^G_{\bullet}(\mathcal{G}r)$ and $QH_G^{\bullet}(G/P)$ respectively.) Clearly, $\Phi_{SS}$ is $W$-equivariant. Therefore, we have
\begin{corollary}\label{3bequiv}
$\Phi_{PLS}$ is $W$-equivariant. \hfill$\square$
\end{corollary}

\begin{lemma}\label{3bnoempty}
The stack $\ol{\mathcal{M}}(\lambda,[\nu])$ is non-empty if and only if $[\nu]\in [w_0(\lambda)]+\ec$.
\end{lemma}
\begin{proof}
It suffices to look at the $T$-invariant sections of $\ffib{t^{\mu}}$ with $t^{\mu}\in \agll$. By \cite[Lemma 4.2]{me}, these sections are in bijective correspondence with the $T$-fixed points of $G/P$, and for each $T$-fixed point $\dot{v}P\in G/P$, the degree of the corresponding section is equal to $[v^{-1}(\mu)]$. The rest is clear.
\end{proof}

\begin{proposition}\label{3bzero}
For any dominant $\lambda\inQ^{\vee}$, $\Phi_{PLS}$ vanishes on $H^T_{>2\rho(\lambda-w_Pw_0(\lambda))}(\agll)$.
\end{proposition}
\begin{proof}
Let $x\in H^T_{>2\rho(\lambda-w_Pw_0(\lambda))}(\agll)$. We may assume $x$ is an affine Schubert class $
\bk{\overline{\mathcal{B}\cdot t^{\mu}}}$ for some $\mu\in W\cdot\lambda$. (If not, replace $\lambda$ by another $\lambda'$ with $\lambda'\leqslant\lambda$.) Then $x\in \im(r_{\lambda})_*$ so that we can apply the characterization \eqref{3beq1}. Observe that for any $v\in W^P$ and $[\nu]\in Q^{\vee}/Q^{\vee}_P$ for which $\ol{\mathcal{M}}(\lambda,[\nu])\ne\emptyset$, the complex degree of $(\Pi_{\lambda,[\nu]})_*\ev_{\lambda,[\nu]}^*\sigma^v$ is greater than or equal to $2\rho(\lambda)+\sum_{\alpha\in R^+\setminus R^+_P}\alpha(\nu)$ which is greater than or equal to $2\rho(\lambda)+\sum_{\alpha\in R^+\setminus R^+_P}\alpha(w_0(\lambda))$ by Lemma \ref{3bnoempty}. The last expression is equal to $2\rho(\lambda)-\rho(\lambda-w_Pw_0(\lambda))$. It follows that
\[ \deg\left((\Pi_{\lambda,[\nu]})_*\ev_{\lambda,[\nu]}^*\sigma^v\right) + \deg x> 2\rho(\lambda) = \dim \Gamma_{\lambda} ,\]
and hence the intersection product in \eqref{3beq1} is zero. The result now follows from Theorem \ref{3bthm}.
\end{proof}

\begin{lemma}\label{3blevi}
For any dominant $\lambda\inQ^{\vee}$, the restriction of $\Pi_{\lambda,[w_0(\lambda)]}$ to $\ev_{\lambda,[w_0(\lambda)]}^{-1}(eP)$ is an isomorphism onto $r_{\lambda}^{-1}(\agp)$ where
\[\agp:=\overline{L(\mathcal{O})\cdot t^{w_Pw_0(\lambda)}} \subseteq \agll\] and $L$ is the Levi subgroup of $P$.
\end{lemma}
\begin{proof}
Observe that the structure group of the principal $G$-bundle $\fib{\agp}$ is reduced to $L$. In fact, the associated principal $L$-bundle is nothing but $\fibl{\agp}$. Hence
\[ \left. \ffib{\agll}\right|_{\mathbb{P}^1\times \agp}\simeq \ffibl{\agp}.\]
On the other hand, the $L$-equivariant morphism $\{eP\}\hookrightarrow G/P$ induces a morphism
\begin{equation}\label{3beq2}
\mathbb{P}^1\times \agp\simeq \fibl{\agp}(\{eP\})\hookrightarrow \ffibl{\agp}.
\end{equation}
In other words, every point in $\agp$ gives rise to a section of $\ffib{\agll}$ lying over the same point. It is easy to see that the degrees of these sections are equal to $[w_0(\lambda)]$. Thus, \eqref{3beq2} defines a closed immersion
\begin{equation}\label{3beeq1}
r_{\lambda}^{-1}(\agp)\hookrightarrow \overline{\mathcal{M}}(\lambda,[w_0(\lambda)]).
\end{equation}
By a dimension argument and the smoothness of $\overline{\mathcal{M}}(\lambda,[w_0(\lambda)])$ (Proposition \ref{3bDM}), the image of \eqref{3beeq1} is a connected component of $\overline{\mathcal{M}}(\lambda,[w_0(\lambda)])$.

It remains to show that $\overline{\mathcal{M}}(\lambda,[w_0(\lambda)])$ has no other connected component. Every other component has a $T$-fixed point, and every $T$-fixed point is represented by a $T$-invariant section of $\ffib{t^{\mu}}$ for some $\mu\in \conv(W\cdot \lambda)$. Notice that it has no bubbles by Lemma \ref{3bnoempty}. Since the marked point on this section (over $\infty$) is required to hit $eP\in G/P$, its degree is equal to $[\mu]$. It follows that $[\mu]=[w_0(\lambda)]$, and hence $\mu\in\conv(W_P\cdot w_0(\lambda))$ or equivalently $t^{\mu}\in \agp$. Thus, this section, as a point of the component of $\overline{\mathcal{M}}(\lambda,[w_0(\lambda)])$ we are looking at, belongs to the component we have already determined, a contradiction.
\end{proof}

\begin{proposition}\label{3bone}
Let $\lambda\inQ^{\vee}$ be dominant. For any $x\in H^T_{2\rho(\lambda-w_Pw_0(\lambda))}(\Gamma_{\lambda})$, we have
\[ \Phi_{PLS}((r_{\lambda})_*x)=q^{[w_0(\lambda)]} x\bullet \left[ r_{\lambda}^{-1}\left(\agp\right)\right] .\]
The RHS depends only on the non-equivariant limit of $x$.
\end{proposition}
\begin{proof}
By the degree argument in the proof of Proposition \ref{3bzero}, the summation \eqref{3beq1} contains only one possibly non-zero summand, namely the term corresponding to $v=e$ and $[\nu]=[w_0(\lambda)]$. The first assertion follows from Theorem \ref{3bthm} and Lemma \ref{3blevi}, and the second from Proposition \ref{3bzero}.
\end{proof}
\section{Yun-Zhu's theorem}\label{4}
\subsection{Statement}\label{4a}
Recall the notations introduced in Section \ref{2c}. Assume $G$ is simple.

Consider the $T$-equivariant hypercohomology functor $H^{\bullet}_T(-)$ on $\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$ and the endomorphism $\sigma_{can}$ of the functor $H^{\bullet}_T(-)\otimes_{H_T^{\bl}(\pt)} H^T_{\bl}(\ag)$ defined by
\[ \sigma_{can}(v\otimes h):= \sum_i (h^i\cup v)\otimes (h_i\bulletsmall h)\]
for any $v\in H^{\bullet}_T(\mathcal{F})$ with $\mathcal{F}\in \mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$ and $h\in H^T_{\bl}(\ag)$, where $\{h_i\}$ and $\{h^i\}$ are dual bases of $H^T_{\bl}(\ag)$ and $H_T^{\bullet}(\mathcal{G}r)$ respectively. Yun and Zhu \cite{YZ} showed that it is a tensor automorphism, and hence it induces, via the Tannakian formalism, a homomorphism of $H_T^{\bl}(\pt)$-algebras
\[ \widetilde{\Phi}_{YZ}:\mathcal{O}(G^{\vee}\times \specH_T^{\bl}(\pt) )\rightarrow H^T_{-\bl}(\ag)\]
characterized by property that for any object $\mathcal{F}$ of $\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$, $v\in H^{\bullet}(\mathcal{F})$ and $v^*\in H^{\bullet}(\mathcal{F})^*$,
\begin{equation}\label{4aeq}
\widetilde{\Phi}_{YZ}(\theta(\mathcal{F},v,v^*)) = \sum_i\langle v^*, h^i\cup v\rangle h_i
\end{equation}
where $\theta(\mathcal{F},v,v^*)\in\mathcal{O}(G^{\vee})$ is defined by
\[ \theta(\mathcal{F},v,v^*)(g):=\langle v^*, g^{-1}\cdot v\rangle.\]

\begin{remark}\label{4armk1}
Recall $G^{\vee}$ is constructed from certain projective generators of $\mathcal{P}_{G(\mathcal{O})}(\mathcal{G}r)$. See \cite[Section 9-11]{MV}. One can write down $\widetilde{\Phi}_{YZ}$ explicitly in terms of $\sigma_{can}$ and these projective generators.
\end{remark}

\begin{remark}\label{4armk2}
Notice that our convention is different from that in \cite{YZ}, namely the regular function $\theta(\mathcal{F},v,v^*)$ in \textit{loc. cit.} is defined to be $g\mapsto \langle v^*, g\cdot v\rangle$. Our $\widetilde{\Phi}_{YZ}$ thus differs from theirs by the inverse morphism $\mathcal{O}(G^{\vee})\xrightarrow{\sim}\mathcal{O}(G^{\vee})$.
\end{remark}

$\widetilde{\Phi}_{YZ}$ is graded if we grade $\mathcal{O}(G^{\vee})$ using the conjugate action of $2\rho:\mathbb{G}_m\rightarrow T^{\vee}$. Moreover, by showing that $\sigma_{can}$ preserves the weight filtration induced by the $T$-equivariant analogue of the decomposition \eqref{4aweight}, $\widetilde{\Phi}_{YZ}$ factors through $\mathcal{O}(B^{\vee}\times\specH_T^{\bl}(\pt))$.

Yun and Zhu further considered an element $c\in H^2_T(\mathcal{G}r;\mathbb{Q})$ which is the unique primitive $T$-equivariant lift of the positive generator of $H^2(\mathcal{G}r;\mathbb{Z})$. By the facts that $c$ is primitive and $\sigma_{can}$ commutes with the cup product $c\cup -$, $\widetilde{\Phi}_{YZ}$ further factors through the coordinate ring $\mathcal{O}(B_{e^T}^{\vee})$ of the centralizer group scheme in $B^{\vee}\times\specH_T^{\bl}(\pt)$ of an element $e^T\in \dd{\mathfrak{g}}\otimesH_T^{\bl}(\pt)\otimes\mathbb{Q}$ which is induced by the endomorphism functor
\[c\cup -:H^{\bullet}_T(-)\rightarrow H^{\bullet+2}_T(-)\]
via the Tannakian formalism. They determined $e^T$ explicitly, namely
\[e^T=e+f\]
where
\[ e:= \sum_{i=1}^r|\ad_i|^2 e_{\alpha^{\vee}_i}\quad\text{ and }\quad f:=-2\frac{\langle-,-\rangle_{Kil}}{\langle\alpha^{\vee}_0,\alpha^{\vee}_0\rangle_{Kil}}.\]
Here,
\begin{enumerate}
\item $|\cdot|$ is a $W$-invariant norm on $\mathbb{X}_{\bl}(T)_\mathbb{R}$, normalized such that $|\ad_i|^2=1$ if $\ad_i$ is short;

\item $e_{\alpha^{\vee}_i}$ is a generator of $\dd{\mathfrak{g}}_{\alpha^{\vee}_i}\simeq\mathbb{Z}$; and

\item $\langle-,-\rangle_{Kil}$ is the Killing form as an element of $\mathbb{X}^{\bl}(T)\otimes \mathbb{X}^{\bl}(T)\subset\dd{\mathfrak{g}}\otimesH_T^{\bl}(\pt)$.
\end{enumerate}

Observe that $f$, regarded as an element of $\home_{\mathbb{Q}}(\mathbb{X}_{\bl}(T)_\mathbb{Q},\mathbb{X}^{\bl}(T)_\mathbb{Q})$, sends each $\alpha^{\vee}_i$ to $-|\alpha^{\vee}_i|^2 \alpha_i$. Suppose $G$ is simply-connected. Then $\mathbb{X}_{\bl}(T)=Q^{\vee}$. Since $Q^{\vee}$ is generated by the simple coroots over $\mathbb{Z}$, we conclude that $f$ is defined over $\mathbb{Z}$.

\begin{theorem}\cite[Theorem 6.1]{YZ}\label{4aYZ}
Assume $G$ is simple and simply-connected. The homomorphism
\[ \Phi_{YZ}:\mathcal{O}(B_{e^T}^{\vee})\rightarrow H^T_{-\bl}(\ag)\]
induced by $\widetilde{\Phi}_{YZ}$ is bijective after inverting $\ell_G\in\mathbb{Z}$, the square of the ratio of the lengths of the long and short roots of $G$. \hfill$\square$
\end{theorem}

\begin{remark}\label{4armk} We will need the surjectivity of $\Phi_{YZ}$ only. Yun-Zhu's proof of the surjectivity is to show first that the image of $\Phi_{YZ}$ contains the set $\mathcal{S}$ of equivariant homology classes from $G\cdot t^{\mu}$ for arbitrary $\mu\in\mathbb{X}_{\bl}(T)$, and then apply a theorem of Bott \cite{Bott}. Bott's theorem was originally stated for the non-equivariant homology of $\Omega K$ but can be generalized to the equivariant settings. The following version is sufficient for our need.
\begin{lemma}\label{4alemma}
Every subset of $H^T_{\bl}(\ag)$ preserved by all Demazure operators $D_i:=\alpha_i^{-1}(s_{\alpha_i}-\id)$ and containing $\{[t^{\mu}]\}_{\mu\in\mathbb{X}_{\bl}(T)}$ and $([t^{\alpha^{\vee}_0}]-1)/\alpha_0$ generates $H^T_{\bl}(\ag)$ multiplicatively.
\end{lemma}
\begin{proof}
Since our subset contains $\{[t^{\mu}]\}_{\mu\in\mathbb{X}_{\bl}(T)}$, we may assume $G$ is simply-connected. Additively, $H^T_{\bl}(\ag)$ is generated by the fundamental classes represented by cycles
\[
\begin{array}{ccccc}
\bs(j_1,\ldots,j_{\ell}) &:& K_{\alpha_{j_1}}\times_{T'}\cdots\times_{T'}K_{\alpha_{j_{\ell}}}/T' &\rightarrow &LK/K\simeq \Omega K\simeq \mathcal{G}r\\ [.5em]
& & [\varphi_1:\cdots:\varphi_{\ell}] &\mapsto & \varphi_1\cdots \varphi_{\ell} K
\end{array}
\]
where $T':=T\cap K$, $(j_1,\ldots,j_{\ell})$ is any sequence of integers from $\{0,\ldots,r\}$ and $K_{\alpha_0},\ldots, K_{\alpha_r}$ are some compact semi-simple-rank-one subgroups of the free loop group $LK$ associated to the affine simple roots. See e.g. \cite{Magyar} for the definition of these subgroups.

One can show that $\bs(j_1,\ldots,j_{\ell}) $ factors through
\begin{equation}\label{4aeq1}
 (K\times_{T'} K_{\alpha_0}/T')^N\rightarrow \Omega K^N\rightarrow \Omega K
\end{equation}
for some $N\in\mathbb{N}$ where the first arrow is the $N$-fold product of the map
\[
\begin{array}{ccccc}
\bs_0&:& K\times_{T'} K_{\alpha_0}/T' &\rightarrow &\Omega K \\ [.5em]
& & [k:\varphi] &\mapsto & k\varphi\vp(1)^{-1}k^{-1}
\end{array}
\]
and the second arrow is the group product. For example, if $(j_1,\ldots,j_{\ell})=(1,2,0,1,0)$, then $\bs(j_1,\ldots,j_{\ell})$ is the composition of the map
\[
\begin{array}{ccc}
K_{\alpha_1}\times_{T'} K_{\alpha_2}\times_{T'}\cdots \times_{T'}K_{\alpha_0}/T'&\rightarrow &\left(K\times_{T'} K_{\alpha_0}/T'\right)^2 \\ [.5em]
 [\varphi_1:\varphi_2:\varphi_3:\varphi_4:\varphi_5] &\mapsto & \left( [\varphi_1(1)\varphi_2(1):\varphi_3],[\varphi_1(1)\varphi_2(1)\varphi_3(1)\varphi_4(1):\varphi_5]\right)
\end{array}
\]
and \eqref{4aeq1}. (Notice that $\varphi_1$, $\varphi_2$ and $\varphi_4$ are constant loops.)

It follows that the fundamental class represented by $\bs(j_1,\ldots,j_{\ell}) $ is generated multiplicatively by the equivariant homology classes from the image of $(\bs_0)_*$. Observe that $\bs_0$ lands in $\mathcal{G}r^{\leqslant \alpha^{\vee}_0}$, and hence these homology classes are elements of $H^T_{\bl}( \mathcal{G}r^{\leqslant \alpha^{\vee}_0})$ which are additively generated by elements of the form $D_{i_1}\cdots D_{i_k}( ([t^{\alpha^{\vee}_0}]-1)/\alpha_0)$. The result follows.
\end{proof}

The surjectivity of $\Phi_{YZ}$ follows if we can show that the sub-$H_T^{\bl}(\pt)$-algebra of $H^T_{\bl}(\ag)$ generated by $\mathcal{S}$ satisfies the condition in Lemma \ref{4alemma}. This holds if $G$ is of adjoint type. The proof is left to the reader. Finally, the surjectivity for this case evidently implies the surjectivity for the general case.
\end{remark}
\subsection{Description in terms of Mirkovi\'c-Vilonen cycles}\label{4b}
The main goal of this subsection is to prove Proposition \ref{4bmv2} which describes the non-equivariant limit $\Phi^0_{YZ}$ of $\Phi_{YZ}$ in terms of Mirkovi\'c-Vilonen cycles (MV cycles) in $\mathcal{G}r$. When the coefficient ring is $\mathbb{C}$, this was discussed in \cite[Section 10.2]{Acta}. The integral case is similar. We reproduce the proof in \textit{loc. cit.} for the convenience of the reader.

Let $\mu\in \mathbb{X}_{\bl}(T)$. Recall
\[  S^-_{\mu}:= \left\{ y\in\mathcal{G}r\left|~\lim_{s\to\infty}2\rho^{\vee}(s)\cdot y= t^{\mu}\right. \right\}.\]
It is a subset of $\mathcal{G}r$ whose dimension and codimension are both infinite.
\begin{lemma}\cite[Proposition 3.1]{MV} \label{4bclosure}
$\overline{S^-_{\mu}}=\bigcup_{\nu\geqslant\mu}S_{\nu}^-$. \hfill $\square$
\end{lemma}

\begin{lemma}\cite[Theorem 3.2]{MV}\label{4bdim}
Let $\lambda\in\mathbb{X}^+_{\bl}(T)$ and $\mu\in\mathbb{X}_{\bl}(T)$. Every irreducible component of $S^-_{\mu}\cap \agl$ has dimension $\rho(\lambda-\mu)$.  \hfill $\square$
\end{lemma}

\begin{definition}\label{4bdef}
The closure in $\agll$ of any irreducible component of $S^-_{\mu}\cap \agl$ is called an MV cycle of type $\lambda$ and weight $\mu$.
\end{definition}

The following lemma is well-known.
\begin{lemma}\label{4bequal}
The following two sets are equal:
\begin{enumerate}
\item The set of MV cycles of type $\lambda$ and weight $\mu$.

\item The set of irreducible components of $\overline{S^-_{\mu}}\cap\agll$ which have dimension $\rho(\lambda-\mu)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Denote by $A$ and $B$ the sets in (1) and (2) respectively. By $\agll=\bigcup_{\mathbb{X}^+_{\bl}(T)\ni \nu \leqslant\lambda}\mathcal{G}r^{\nu}$, Lemma \ref{4bclosure} and Lemma \ref{4bdim}, the dimension of every irreducible component of $\overline{S^-_{\mu}}\cap\agll$ is less than or equal to $\rho(\lambda-\mu)$. This shows $A\subseteq B$. Conversely, let $Z\in B$. Put $Z':=Z\cap S^-_{\mu}\cap\agl$. By the same reasoning, $Z'$ is a non-empty open dense subset of $Z$. Thus, it is contained in an irreducible component $Z''$ of $S^-_{\mu}\cap\agl$, and hence $Z\subseteq\overline{Z''}$. By Lemma \ref{4bdim} again, we have $Z=\overline{Z''}$, and hence $Z\in A$.
\end{proof}

Let $\lambda\in\mathbb{X}^+_{\bl}(T)$. Consider the costandard sheaf $\mathcal{J}_{\lambda}$ defined in Section \ref{2c}. Put $S(\lambda):=H^{\bullet}(\mathcal{J}_{\lambda})$. Let $v_{\lambda}^*\in S(\lambda)^*$ be the element defined in Section \ref{2d}, i.e. the unique element which vanishes on $S(\lambda)_{\nu} $ for any $\nu\ne\lambda$ and sends the generator $v_e$ of $S(\lambda)_{\lambda}\simeq \mathbb{Z}$ corresponding to $S^-_{\lambda}\cap\agl =\{t^{\lambda}\}$ to 1. Recall given $v\in S(\lambda)$, $\theta(\mathcal{J}_{\lambda},v,v_{\lambda}^*)$ is the regular function $g\mapsto \langle v_{\lambda}^*,g^{-1}\cdot v\rangle$.
\begin{proposition}\label{4bmv2}
Let $\mu\in\mathbb{X}_{\bl}(T)$ and $v_Z\in S(\lambda)_{\mu}$ be the element corresponding to an MV cycle $Z$ via the isomorphism in Lemma \ref{2clemma2}. Then $\Phi_{YZ}(\theta(\mathcal{J}_{\lambda},v_Z,v_{\lambda}^*))\in H^T_{2\rho(\lambda-\mu)}(\agll;\mathbb{Z})$ and
\[ \Phi^0_{YZ}(\theta(\mathcal{J}_{\lambda},v_Z,v_{\lambda}^*))=[Z]\in H_{2\rho(\lambda-\mu)}(\agll;\mathbb{Z}). \]
\end{proposition}
\begin{proof}
The first assertion follows from the facts that $\Phi_{YZ}$ is graded and the support of $\mathcal{J}_{\lambda}$ is $\agll$.

To prove the second assertion, let us first recall how the isomorphism in Lemma \ref{2clemma2} is established. As we will see, it is in fact $v_{\lambda}^*$ which determines this isomorphism. For any space $X$, denote by $\mathbb{D}_X$ its dualizing complex. The element $v_{\lambda}^*\in S(\lambda)^*$ induces, by adjunction and the fact $H^{>2\rho(\lambda)}(\mathcal{J}_{\lambda})=0$, a morphism
\[ \mathcal{J}_{\lambda}\rightarrow \mathbb{D}_{\agll}[-2\rho(\lambda)],\]
giving rise to the following commutative diagram
\begin{equation}\nonumber
\begin{tikzpicture}
\tikzmath{\x1 = 4; \x2 = 1.5;}
\node (A) at (0,0) {$\iota_*\iota^!\mathcal{J}_{\lambda}$} ;
\node (B) at (\x1,0) {$\mathcal{J}_{\lambda}$} ;
\node (C) at (0,-\x2) {$\iota_*\iota^!\mathbb{D}_{\agll}[-2\rho(\lambda)]$} ;
\node (D) at (\x1,-\x2) {$ \mathbb{D}_{\agll}[-2\rho(\lambda)]$} ;

\path[->, font=\small] (A) edge node[above]{} (B);
\path[->, font=\small] (A) edge node[left]{} (C);
\path[->, font=\small] (B) edge node[right]{} (D);
\path[->]
(C) edge node[left]{} (D);
\end{tikzpicture}
\end{equation}
where $\iota:\overline{S^-_{\mu}}\cap\agll\hookrightarrow \agll$ is the inclusion and the horizontal arrows are the counit maps. Apply $H^{2\rho(\mu)}(-)$, and we get
\begin{equation}\label{4beqnew}
\begin{tikzpicture}
\tikzmath{\x1 = 6; \x2 = 2;}
\node (A) at (0,0) {$H^{2\rho(\mu)}_{\overline{S^-_{\mu}}}(\mathcal{J}_{\lambda})$} ;
\node (B) at (\x1,0) {$H^{2\rho(\mu)}(\mathcal{J}_{\lambda})$} ;
\node (C) at (0,-\x2) {$H^{2\rho(\mu-\lambda)}(\overline{S^-_{\mu}}\cap\agll;\mathbb{D}_{\overline{S^-_{\mu}}\cap\agll})$} ;
\node (D) at (\x1,-\x2) {$H^{2\rho(\mu-\lambda)}(\agll;\mathbb{D}_{\agll})$} ;

\path[->, font=\tiny] (A) edge node[above]{$\phi_1$} (B);
\path[->, font=\tiny] (A) edge node[left]{$\phi_2$} (C);
\path[->, font=\tiny] (B) edge node[right]{$\phi_3$} (D);
\path[->, font=\tiny] (C) edge node[above]{$\phi_4$} (D);
\end{tikzpicture}.
\end{equation}
One can show that $\phi_1$ identifies $H^{2\rho(\mu)}_{\overline{S^-_{\mu}}}(\mathcal{J}_{\lambda})$ with the direct summand $S(\lambda)_{\mu}= H^{2\rho(\mu)}_{S^-_{\mu}}(\mathcal{J}_{\lambda})$ of $S(\lambda)=H^{\bullet}(\mathcal{J}_{\lambda})$, and $\phi_2$ is an isomorphism. The isomorphism in Lemma \ref{2clemma2} is established by noticing that $H^{2\rho(\mu-\lambda)}(\overline{S^-_{\mu}}\cap\agll;\mathbb{D}_{\overline{S^-_{\mu}}\cap\agll})$ is freely generated by the fundamental classes of the irreducible components of $\overline{S^-_{\mu}}\cap\agll$ which have dimension $\rho(\lambda-\mu)$, i.e. the MV cycles of type $\lambda$ and weight $\mu$ by Lemma \ref{4bequal}.

Back to the proof of the second assertion, we have, by the discussion in the previous paragraph,
\[ v_Z=\phi_1\circ \phi_2^{-1}[Z].\]
By the characterization \eqref{4aeq}, we have
\begin{equation}\label{4beq2}
\Phi^0_{YZ}(\theta(\mathcal{J}_{\lambda},v_Z,v_{\lambda}^*)) = \sum_i\langle v_{\lambda}^*, h^i\cup v_Z\rangle h_i.
\end{equation}
Observe that the RHS of \eqref{4beq2} is equal to $\phi_3(v_Z)$, after we identify $H^{2\rho(\mu-\lambda)}(\agll;\mathbb{D}_{\agll})$ with $H_{2\rho(\lambda-\mu)}(\agll;\mathbb{Z})$. Therefore,
\[ \Phi^0_{YZ}(\theta(\mathcal{J}_{\lambda},v_Z,v_{\lambda}^*)) = \phi_3(v_Z) = \phi_3\circ \phi_1\circ\phi_2^{-1}[Z] = \phi_4[Z]\]
as desired. (We have put $\phi_4[Z]=[Z]$ by an abuse of notation.)
\end{proof}

\begin{corollary}\label{4bcor}
If $v$ corresponds to the unique MV cycle of type $\lambda$ and weight $w_0(\lambda)$, i.e. the spherical affine Schubert variety $\agll$, then
\[ \Phi_{YZ}(\theta(\mathcal{J}_{\lambda},v,v_{\lambda}^*))=\bk{\agll}.\]
\end{corollary}
\begin{proof}
This follows from Proposition \ref{4bmv2} and the fact that $H^T_{>4\rho(\lambda)}(\agll)=0$.
\end{proof}

Next we look at divisors on $\agll$. The following lemma will be useful later.
\begin{lemma}\label{4bdivisor}
Let $0\leqslant k\leqslant r$ be an integer. There exists dominant $\lambda\inQ^{\vee}$ satisfying
\begin{enumerate}
\item $\alpha_i(w_0(\lambda))<0$ if $1\leqslant i\leqslant k$ and $\alpha_i(w_0(\lambda))=0$ otherwise; and
\item the image of the $H_T^{\bl}(\pt)$-linear map
\begin{equation}\label{4bdivisoreq1}
\begin{array}{ccc}
S(\lambda)\otimesH_T^{\bl}(\pt) & \rightarrow & H^T_{\bl}(\agll)\\ [.5em]
v\otimes 1 &\mapsto & \Phi_{YZ}(\theta(\mathcal{J}_{\lambda},v,v_{\lambda}^*))
\end{array}
\end{equation}
contains the direct summand $H^T_{4\rho(\lambda)-2}(\agll)$.
\end{enumerate}
\end{lemma}
\begin{proof}
WLOG, we assume $G$ is of adjoint type. Let $\{\omega^{\vee}_1,\ldots,\omega^{\vee}_r\}$ be the dual basis of $\{\alpha_1,\ldots,\alpha_r\}$. Take $\lambda'\inQ^{\vee}$ lying in the $\mathbb{Z}_{>0}$-span of $-\omega^{\vee}_i,\ldots,-\omega^{\vee}_k$. Then $\lambda'+\omega^{\vee}_i$ is anti-dominant for any $1\leqslant i\leqslant k$. Define $\lambda:=w_0(\lambda')$. Clearly, $\lambda$ satisfies (1). It remains to verify (2). We have
\[ H^T_{4\rho(\lambda)-2}(\agll)=H^2_T(\pt)\cdot\bk{\agll}\oplus\bigoplus_{i=1}^k\mathbb{Z}\cdot\bk{\overline{\mathcal{B}\cdot t^{s_{\alpha_i}(\lambda')}}}.\]
Hence, by Corollary \ref{4bcor}, the result will follow if we can show that $\bk{\overline{\mathcal{B}\cdot t^{s_{\alpha_i}(\lambda')}}}$ belongs to the image of \eqref{4bdivisoreq1} for each $1\leqslant i\leqslant k$.

Observe that $\overline{\mathcal{B}\cdot t^{-s_{\alpha_i}(\omega^{\vee}_i)}}$ is the unique MV cycle of type $-w_0(\omega^{\vee}_i)$ and weight $-s_{\alpha_i}(\omega^{\vee}_i)=-\omega^{\vee}_i+\alpha^{\vee}_i$, and $\mathcal{G}r^{\leqslant w_0(\lambda'+\omega^{\vee}_i)}$ is the unique MV cycle of type $w_0(\lambda'+\omega^{\vee}_i)$ and weight $\lambda'+\omega^{\vee}_i$. It follows that, by Proposition \ref{4bmv2}, their fundamental classes lie in the images of \eqref{4bdivisoreq1} with $\lambda$ replaced by $-w_0(\omega^{\vee}_i)$ and $w_0(\lambda'+\omega^{\vee}_i)$ respectively. Strictly speaking, Proposition \ref{4bmv2} holds for the non-equivariant settings only. But we can use Corollary \ref{4bcor} and a simple degree argument to verify our claim. By Lemma \ref{4bnewlemma} below, we have
\begin{equation}\label{4bdivisoreq2}
\bk{\overline{\mathcal{B}\cdot t^{s_{\alpha_i}(\lambda')}}} =\bk{\overline{\mathcal{B}\cdot t^{-s_{\alpha_i}(\omega^{\vee}_i)}}}\bulletsmall\bk{\mathcal{G}r^{\leqslant w_0(\lambda'+\omega^{\vee}_i)}}.
\end{equation}

On the other hand, for any $\lambda_1,\lambda_2\in\mathbb{X}^+_{\bl}(T)$, the linear map
\begin{equation}\label{4bdivisoreq25}
v_{\lambda_1}^*\otimes v_{\lambda_2}^*: S(\lambda_1)_{\lambda_1}\otimes S(\lambda_2)_{\lambda_2}\rightarrow\mathbb{Z}
\end{equation}
induces, via \eqref{2cuniversal}, a morphism $\mathcal{J}_{\lambda_1}*\mathcal{J}_{\lambda_2}\rightarrow\mathcal{J}_{\lambda_1+\lambda_2}$ and hence a $G^{\vee}$-equivariant linear map $S(\lambda_1)\otimes S(\lambda_2)\rightarrow S(\lambda_1+\lambda_2)$ extending \eqref{4bdivisoreq25}. This implies that for any $v_i\in S(\lambda_i)$, there exists $v\in S(\lambda_1+\lambda_2)$ such that
\begin{equation}\label{4bdivisoreq3}
\theta(\mathcal{J}_{\lambda_1},v_1,v_{\lambda_1}^*)\theta(\mathcal{J}_{\lambda_2},v_2,v_{\lambda_2}^*) = \theta(\mathcal{J}_{\lambda_1+\lambda_2},v,v_{\lambda_1+\lambda_2}^*).
\end{equation}
The result now follows from Corollary \ref{4bcor}, \eqref{4bdivisoreq2} and \eqref{4bdivisoreq3}.
\end{proof}

\begin{lemma}\label{4bnewlemma} The equality \eqref{4bdivisoreq2} is true.
\end{lemma}
\begin{proof}
Observe that
\[ \lambda'\not\in\left\{ \mu_1+\mu_2\left| ~t^{\mu_1}\in\overline{\mathcal{B}\cdot t^{-s_{\alpha_i}(\omega^{\vee}_i)}} ,~t^{\mu_2}\in \mathcal{G}r^{\leqslant w_0(\lambda'+\omega^{\vee}_i)}\right.\right\}\subseteq \conv(W\cdot\lambda')\cap (\lambda'+Q^{\vee}).\]
Hence, there exist integers $m_1,\ldots,m_k$ such that
\[ \bk{\overline{ \mathcal{B}\cdot t^{-s_{\alpha_i}(\omega^{\vee}_i)}}}\bulletsmall \bk{\mathcal{G}r^{\leqslant w_0(\lambda'+\omega^{\vee}_i)} } =\sum_{j=1}^k m_j\bk{\overline{\mathcal{B}\cdot t^{s_{\alpha_j}(\lambda')}}}.\]
Let $D_j:=\alpha_j^{-1}(s_{\alpha_j}-\id)$ be the Demazure operators on $H^T_{\bl}(\ag)$. Then $D_{j'}\left(\bk{\overline{\mathcal{B}\cdot t^{s_{\alpha_j}(\lambda')}}}\right)$ is equal to $\bk{\mathcal{G}r^{\leqslant w_0(\lambda')}}$ if $j'=j$ and zero otherwise. If follows that $m_j=0$ if $j\ne i$ and
\begin{equation}\label{4beeq}
D_i\left( \bk{\overline{ \mathcal{B}\cdot t^{-s_{\alpha_i}(\omega^{\vee}_i)}}}\bulletsmall \bk{\mathcal{G}r^{\leqslant w_0(\lambda'+\omega^{\vee}_i)} }\right) = m_i \bk{\mathcal{G}r^{\leqslant w_0(\lambda')}}.
\end{equation}
The LHS of \eqref{4beeq} is equal to $\bk{\mathcal{G}r^{\leqslant -w_0(\omega^{\vee}_i)}}\bulletsmall\bk{\mathcal{G}r^{\leqslant w_0(\lambda'+\omega^{\vee}_i)}}$. One can deduce from Corollary \ref{4bcor} that the last expression is equal to $\bk{\mathcal{G}r^{\leqslant w_0(\lambda')}}$. The proof is complete.
\end{proof}
\section{The Peterson schemes}\label{5}
Assume $G$ is simple and simply-connected. We are going to define $\mathcal{Y}_P$ and $\mathcal{Y}_P^*$ in Section \ref{5a} and construct the homomorphisms $\Phi$, $\Phi^*$ and $\Phi_{loc}$ in Section \ref{5b}. Unless otherwise specified, the coefficient ring $\Bbbk$ is $\mathbb{Z}[\ell_G^{-1}]$ where $\ell_G$ is equal to 1 for type ADE, 2 for type BCF and 3 for type G. Put $S:=H_T^{\bl}(\pt)[q_1,\ldots,q_k]$. The spectrum of $S$ will serve as the base scheme of $\mathcal{Y}_P$ and $\mathcal{Y}_P^*$.

In this section and the next, we will complete the following commutative diagram
\begin{equation}\label{1adiag}
\begin{tikzpicture}
\tikzmath{\x1 = 2.5; \x2 = 4.3; \x3=4.3; \x4=2.5; \x5=6;}
\node (A) at (0,0) {$\mathcal{O}(\mathcal{Y}^*)$} ;
\node (B) at (0,-\x1) {$\mathcal{O}(B_{e^T}^{\vee})$} ;
\node (C) at (-\x2,0) {$\mathcal{O}(\mathcal{Y}^*\tGBU^{\vee}(w_Pw_0))$} ;
\node (D) at (-\x2,-\x1) {$\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]$} ;
\node (E) at (-\x2-\x3,0) {$\mathcal{O}(\mathcal{Y}_P^*)$};
\node (F) at (-\x2-\x3,-\x1) {$\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle$};
\node (G) at (0,-\x1-\x4) {$H^T_{-\bl}(\ag)$};
\node (H) at (-\x2-\x3,-\x1-\x4) {$QH_T^{\bl}(G/P)[\ecc]$};
\node (I) at (-\x2-\x3-\x5,-\x1-\x4) {$QH_T^{\bl}(G/P)$};
\node (J) at (-\x2-\x3-\x5,0) {$\mathcal{O}(\mathcal{Y}_P)$};
\node (K) at (-\x2-\x3-0.5*\x5,-0.5*\x1) {$\mathcal{O}(\mathcal{Y}_P)[q^{-1}_1,\ldots,q^{-1}_k]$};

\node (L) at (-\x2-\x3-0.83*\x5,-0.29*\x1) {{\tiny $\eta_2$}};

\node (M) at (-\x2-\x3-0.3*\x5,-0.18*\x1) {{\small  $\varphi_1$}};

\node (N) at (-\x2-\x3-0.3*\x5,-0.82*\x1) {{\small  $\varphi_4$}};

\path[->, font=\tiny]
(A) edge node[left]{$\eta_4$} (B);

\path[->, font=\tiny]
(A) edge node[above]{$\eta_5$} (C);

\path[->, font=\tiny]
(B) edge node[above]{$\eta_6$} (D);

\path[->]
(B) edge node[left]{$\Phi_{YZ}$} (G);

\path[->, font=\tiny]
(C) edge node[right]{$\eta_7$} (D);

\path[->, font=\tiny]
(C) edge node[above]{$\eta_8$} (E);

\path[->, font=\tiny]
(D) edge node[above]{$\eta_9$} (F);

\path[->, font=\small]
(E) edge node[right]{$\varphi_2$} (F);

\path[->, font=\tiny]
(E) edge node[left]{} (K);

\path[->, font=\small]
(F) edge node[right]{$\varphi_3$} (H);

\path[->]
(G) edge node[above]{$\Phi_{PLS}$} (H);

\path[->, font=\tiny]
(I) edge node[above]{$\eta_3$} (H);

\path[->, font=\tiny]
(J) edge node[above]{$\eta_1$} (E);

\path[->]
(J) edge node[left]{$\Phi$} (I);

\path[->, font=\tiny]
(J) edge node[below]{} (K);

\path[->, font=\tiny]
(K) edge node[below]{} (F);
\end{tikzpicture}
\end{equation}
\noindent The labellings of all ring homomorphisms below will be compatible with this diagram.
\subsection{Definitions and some properties}\label{5a}
Let $e^T$ be the element defined in Section \ref{4a}. Define
\[ \widetilde{\mathcal{Y}}:=\left\{(g,h)\inG^{\vee}\times\specH_T^{\bl}(\pt)\left|~g^{-1}\cdot e^T(h)\in \dd{\mathfrak{b}}_-\oplus\bigoplus_{i=1}^r\dd{\mathfrak{g}}_{\alpha^{\vee}_i} \right.\right\}.\]
One checks easily that $\widetilde{\mathcal{Y}}$ descends to a closed subscheme $\mathcal{Y}$ of $G^{\vee}/B^{\vee}_-\times\specH_T^{\bl}(\pt)$. Define
\[ \mathcal{Y}^*:=\mathcal{Y}\times_{G^{\vee}/B^{\vee}_-} \UdB_-^{\vee}/B_-^{\vee}.\]
Let $P$ be the parabolic subgroup of $G$ fixed in Section \ref{2a}. Denote by $U^{\vee}_{P^{\vee}_-}$ the product of the $\alpha^{\vee}$-root groups $U^{\vee}_{\alpha^{\vee}}\simeq\mathbb{G}_a$ where $\alpha$ runs over $-(R^+\setminus R^+_P)$. It is a unipotent subgroup of $G^{\vee}$ and satisfies
\[U^{\vee}_{P^{\vee}_-}\simeq U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}=U^{\vee}_-\wpdB_-^{\vee}/B_-^{\vee}.\]
 Define
\[ \mathcal{Y}_P:=\mathcal{Y}\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}\]
and
\[ \mathcal{Y}_P^*:=\mathcal{Y}_P\times_{G^{\vee}/B^{\vee}_-}\UdB_-^{\vee}/B_-^{\vee}\simeq\mathcal{Y}^*\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}.\]
Notice that $\mathcal{Y}^*$, $\mathcal{Y}_P$ and $\mathcal{Y}_P^*$ are affine schemes over $\specH_T^{\bl}(\pt)$.

Recall we have fixed generators $e_{\alpha^{\vee}_i}$ of $\dd{\mathfrak{g}}_{\alpha^{\vee}_i}$ when we introduced $e^T$ in Section \ref{4a}. For every other coroot $\alpha^{\vee}$, we pick a generator $e_{\alpha^{\vee}}$ of $\dd{\mathfrak{g}}_{\alpha^{\vee}}$. Let $e^*_{\alpha^{\vee}}\in (\dd{\mathfrak{g}})^*$ be the unique element satisfying
\[ e^*_{\alpha^{\vee}}|_{\lie(T^{\vee})}=0,\quad \langle e^*_{\alpha^{\vee}},e_{\alpha^{\vee}}\rangle=1\quad\text{ and }\quad \forall \overline{\alpha}\in R\setminus\{\alpha\},~\langle e^*_{\alpha^{\vee}},e_{\overline{\alpha}^{\vee}}\rangle=0. \]

For any $1\leqslant i\leqslant r$, define $\overline{q}_i\in\mathcal{O}(\mathcal{Y}_P)$ to be the pull-back, via the natural morphism
\[\mathcal{Y}_P\rightarrow U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}\times\specH_T^{\bl}(\pt),\]
of the regular function
\[ (u\wpdB_-^{\vee},h)\mapsto  |\ad_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i},(u\dot{w}_P\dot{w}_0)^{-1}\cdot e^T(h)\rangle.\]
In order for $\overline{q}_i$ to be well-defined, it is necessary to specify the representatives $\dot{w}_P$ and $\dot{w}_0$. We will do it in Appendix \ref{app}.

The map $q_i\mapsto \overline{q}_i$, $1\leqslant i\leqslant k$, makes $\mathcal{O}(\mathcal{Y}_P)$ an algebra over $S:=H_T^{\bl}(\pt)[q_1,\ldots,q_k]$, or equivalently, $\mathcal{Y}_P$ a scheme over $\spec S$. The other $\overline{q}_i$'s are not so useful because of the following lemma.
\begin{lemma}\label{5aq=1}
For any $k+1\leqslant i\leqslant r$, we have $\overline{q}_i=1$.
\end{lemma}
\begin{proof}
For such $i$, $-w_P\alpha^{\vee}_i$ is equal to a simple coroot $\alpha^{\vee}_j$ which has the same length as $\alpha^{\vee}_i$. Since $U^{\vee}_{P^{\vee}_-}\subseteqU^{\vee}_-$, we have $u^{-1}\cdot e^T(h)\in e+\dd{\mathfrak{b}}_-$, and hence
\[  \overline{q}_i(u\wpdB_-^{\vee},h)= \pm|\ad_i|^{-2}\langle e^*_{\alpha^{\vee}_j},u^{-1}\cdot e^T(h)\rangle = \pm|\ad_i|^{-2}|\ad_j|^{2}=\pm 1.\]
The sign is $+ 1$ by Proposition \ref{appprop1}.
\end{proof}

\begin{lemma}\label{5alocal}
The localization map
\[\eta_2:\mathcal{O}(\mathcal{Y}_P)\rightarrow \mathcal{O}(\mathcal{Y}_P)[q^{-1}_1,\ldots,q^{-1}_k]\]
factors through a map
\[ \varphi_1:\mathcal{O}(\mathcal{Y}_P^*)\rightarrow \mathcal{O}(\mathcal{Y}_P)[q^{-1}_1,\ldots,q^{-1}_k].\]
\end{lemma}
\begin{proof}
By Lemma \ref{2dlemma3}, $\mathcal{O}(\mathcal{Y}_P^*)$ is a localization of $\mathcal{O}(\mathcal{Y}_P)$. Hence it suffices to show that $\overline{q}_1\cdots\overline{q}_k$ vanishes on $\mathcal{Y}_P\setminus\mathcal{Y}_P^*$. We may assume $\Bbbk$ is an algebraically closed field (of characteristic not dividing $\ell_G$). Let $(y,h)$ be a closed point of $\mathcal{Y}_P\setminus\mathcal{Y}_P^*$. Write $y=u\wpdB_-^{\vee}$ with $u\inU^{\vee}_{P^{\vee}_-}$. By the Bruhat decomposition, we can write $u\dot{w}_P\dot{w}_0=b\dot{w}u_1$ for some $b\inB^{\vee}$, $w\in W$ and $u_1\inU^{\vee}_-$. Since $(y,h)\not\in\mathcal{Y}_P^*$, we have $w\ne e$, and hence there exists $1\leqslant i\leqslant r$ such that $-ww_0\alpha^{\vee}_i<0$. It follows that
\begin{align*}
\overline{q}_i(y,h)&=|\ad_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i},(u\dot{w}_P\dot{w}_0)^{-1}\cdot e^T(h)\rangle\\
&= |\ad_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i},(u\dot{w}_P\dot{w}_0 u_1^{-1} )^{-1}\cdot e^T(h)\rangle\quad \because (y,h)\in\mathcal{Y}\\
&= \pm |\ad_i|^{-2}\langle e^*_{-ww_0\alpha^{\vee}_i},b^{-1}\cdot e^T(h)\rangle\\
&=0.
\end{align*}
By Lemma \ref{5aq=1}, we must have $1\leqslant i\leqslant k$, and hence $\overline{q}_1\cdots\overline{q}_k(y,h)=0$ as desired.
\end{proof}

Introduce a $\mathbb{G}_m$-action on $G^{\vee}/B^{\vee}_-$ defined by restricting the natural $G^{\vee}$-action to the cocharacter $2\rho:\mathbb{G}_m\rightarrow T^{\vee}$. We also let $\mathbb{G}_m$ act on $\spec S$ by taking the usual grading on $H_T^{\bl}(\pt)$ and declaring each $q_i$ to have degree $2\sum_{\alpha\in R^+\setminus R^+_P}\alpha(\alpha^{\vee}_i)$.
\begin{lemma}\label{5aeq}
The $\mathbb{G}_m$-action on $G^{\vee}/B^{\vee}_-\times \specH_T^{\bl}(\pt)$ preserves and hence induces a $\mathbb{G}_m$-action on the schemes $\mathcal{Y}$, $\mathcal{Y}^*$, $\mathcal{Y}_P$ and $\mathcal{Y}_P^*$. The structure morphism $\mathcal{Y}_P\rightarrow\spec S$ is $\mathbb{G}_m$-equivariant.
\end{lemma}
\begin{proof}
This is straightforward and left to the reader.
\end{proof}

This $\mathbb{G}_m$-action is crucial for verifying the following properties of $\mathcal{Y}_P$.
\begin{lemma}\label{5astratum}
Suppose $\Bbbk$ is an algebraically closed field of characteristic not dividing $\ell_G$. Then
\[ |\mathcal{Y}|_{\Bbbk} =\bigcup_Q |\mathcal{Y}_Q|_{\Bbbk}\]
where $|\cdot|_{\Bbbk}$ denotes the set of $\Bbbk$-points and $Q$ runs over all standard parabolic subgroups of $G$.
\end{lemma}
\begin{proof}
By flowing every point of $\mathcal{Y}$ to a $\mathbb{G}_m$-fixed point and the Bruhat decomposition, it suffices to show
\[ (\dot{w}B_-^{\vee},0)\in \mathcal{Y}\implies w=w_Qw_0\text{ for some }Q.\]
By the definition of $\mathcal{Y}$, $(\dot{w}B_-^{\vee},0)\in \mathcal{Y}$ implies that for any $1\leqslant i\leqslant r$, $w^{-1}\alpha^{\vee}_i$ is either negative or equal to a simple coroot. Denote by $I$ the set of $1\leqslant i\leqslant r$ for which the second case holds, and by $Q$ the associated standard parabolic subgroup. WLOG, assume $I=\{\ell+1,\ldots,r\}$. Put $w':=w_0w^{-1}w_Q$. Let $1\leqslant i\leqslant \ell$. Write $w_Q\alpha^{\vee}_i=\sum_{j=1}^rc_j\alpha^{\vee}_j$. Then $c_j\geqslant 0$ for each $j$ and is non-zero for some $1\leqslant j_0\leqslant \ell$. Suppose $w'\alpha^{\vee}_i$ is negative. Then, by the assumption, $w_0w^{-1}\alpha^{\vee}_{j_0}$ lies in the span of $w_0w^{-1}\alpha^{\vee}_j$ with $\ell +1\leqslant j\leqslant r$, or equivalently, $\alpha^{\vee}_{j_0}$ lies in the span of $\alpha^{\vee}_j$ for these $j$, a contradiction. It follows that $w'\alpha^{\vee}_i>0$ for any $1\leqslant i\leqslant \ell$. On the other hand, it is easy to see that $w'\alpha^{\vee}_i>0$ for any $\ell+1\leqslant i\leqslant r$. Therefore, $w'=e$, and hence $w=w_Qw_0$.
\end{proof}

\begin{lemma}\label{5avf}
Suppose $\Bbbk$ is an algebraically closed field of characteristic not dividing $\ell_G$. The fiber $(\mathcal{Y}_P)_x$ of $\mathcal{Y}_P$ over $x:=(0,0)\in\spec S$ is isomorphic to the zero locus of a vector field on $G^{\vee}/\ol{P}^{\vee}_-$, where
$\ol{P}^{\vee}_-$ is the parabolic subgroup of $G^{\vee}$ containing $B_-^{\vee}$ such that
\[ \ol{\mathfrak{p}}^{\vee}_-:=\lie(\ol{P}^{\vee}_-)=\dd{\mathfrak{b}}_-\oplus\bigoplus_{\alpha\in -w_0R^+_P}\dd{\mathfrak{g}}_{\alpha^{\vee}}.\]
Set-theoretically, it is equal to $\{(\wpdB_-^{\vee},0)\}$.
\end{lemma}
\begin{proof}
We have $\mathcal{T}_{G^{\vee}/\ol{P}^{\vee}_-}\simeq G^{\vee}\times_{\ol{P}^{\vee}_-}(\dd{\mathfrak{g}}/\ol{\mathfrak{p}}^{\vee}_-)$ so the morphism $g\mapsto g^{-1}\cdot e^T(0)\bmod\ol{\mathfrak{p}}^{\vee}_-$ defines a section $s_{e^T(0)}\in H^0(G^{\vee}/\ol{P}^{\vee}_-;\mathcal{T}_{G^{\vee}/\ol{P}^{\vee}_-})$. Observe that $U^{\vee}_{P^{\vee}_-}\simeq U^{\vee}_{P^{\vee}_-}\dot{w}_P\dot{w}_0\ol{P}^{\vee}_-/\ol{P}^{\vee}_-$ is an open subscheme of $G^{\vee}/\ol{P}^{\vee}_-$. We first show that $(\mathcal{Y}_P)_x$ is isomorphic to the zero locus of the restriction of $s_{e^T(0)}$ to this subscheme. The latter scheme is by definition the closed subscheme of $U^{\vee}_{P^{\vee}_-}$ cut by the polynomials $\langle e^*_{\alpha^{\vee}}, (u\dot{w}_P\dot{w}_0)^{-1}\cdot e^T(0)\rangle$ for $\alpha\in -w_0(R^+\setminus R^+_P)$ which are part of the defining polynomials of $(\mathcal{Y}_P)_x$. The other polynomials are those corresponding to $\alpha\in -w_0R^+_P\setminus\{\alpha_1,\ldots,\alpha_r\}$. It is easy to see that they are all equal to zero. (We have to use the assumption $u\inU^{\vee}_{P^{\vee}_-}$.) This justifies our claim.

To complete the proof, it suffices to show that the set of points of the zero locus of $s_{e^T(0)}$ is equal to $\{\dot{w}_P\dot{w}_0\ol{P}^{\vee}_-\}$. This amounts to showing that $\dot{w}_P\dot{w}_0\ol{P}^{\vee}_-$ is the only $\mathbb{G}_m$-fixed point of $G^{\vee}/\ol{P}^{\vee}_-$ at which $s_{e^T(0)}$ vanishes. Let $w\in W$. Pick a point $b$ in the dominant chamber such that $\alpha_i(b)=0$ if $\alpha_i\in -w_0R^+_P$ and $\alpha_i(b)>0$ otherwise. If $w\not\in w_Pw_0 W_{\ol{P}^{\vee}_-}=w_0W_{\ol{P}^{\vee}_-}$, then $w(b)\ne w_0(b)$, and hence $w(b)$ is away from the anti-dominant chamber. It follows that there exists $1\leqslant i\leqslant r$ such that $\alpha_i(w(b))>0$. This implies $w^{-1}\alpha_i\in -w_0(R^+\setminus R^+_P)$, and hence $s_{e^T(0)}(\dot{w}\ol{P}^{\vee}_-)\ne 0$.
\end{proof}

\begin{lemma}\label{5aflat}
$\mathcal{Y}_P$ is flat over $\spec S$.
\end{lemma}
\begin{proof}
It suffices to verify the flatness at every closed point of $\mathcal{Y}_P$. Let $y\in \mathcal{Y}_P$ be a closed point lying over a point $x\in\spec S$. Then $x$ is a closed point. By \cite[Proposition 6.1.5]{EGA2}, $\mathcal{O}_{\mathcal{Y}_P,y}$ is flat over $\mathcal{O}_{\spec S,x}$ if the following conditions are satisfied:
\begin{enumerate}
\item $\mathcal{O}_{\spec S,x}$ is regular;

\item $\mathcal{O}_{\mathcal{Y}_P,y}$ is Cohen-Macaulay; and

\item $\dim\mathcal{O}_{\mathcal{Y}_P,y}=\dim_y(\mathcal{Y}_P)_x + \dim\mathcal{O}_{\spec S,x}$ (in general LHS $\leqslant$ RHS).
\end{enumerate}
(1) is obvious. Observe that $\mathcal{Y}_P$ is a closed subscheme of $U^{\vee}_{P^{\vee}_-}\times\spec S\simeq \mathbb{A}^{N}_S$ ($N:=\dimU^{\vee}_{P^{\vee}_-}$) defined by $N$ polynomials, namely $\langle e^*_{\alpha^{\vee}},(u\dot{w}_P\dot{w}_0)^{-1}\cdot e^T(h)\rangle$ for $\alpha\in -w_0(R^+\setminus R^+_P)\setminus\{\alpha_1,\ldots,\alpha_r\}$ and $|\alpha^{\vee}_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i},(u\dot{w}_P\dot{w}_0)^{-1}\cdot e^T(h)\rangle -q_i$ for $1\leqslant i\leqslant k$. It follows that
\[\dim\mathcal{O}_{\mathcal{Y}_P,y}\geqslant \dim S=\dim\mathcal{O}_{\spec S,x}.\] To complete the proof, it suffices to show that $\mathcal{Y}_P$ has zero-dimensional fibers, for we will have $\dim\mathcal{O}_{\mathcal{Y}_P,y}=\dim\mathcal{O}_{\spec S,x}$, yielding (2) and (3) simultaneously.

We may now assume $\Bbbk$ is an algebraically closed field of characteristic not dividing $\ell_G$. Consider $y_0:=(\dot{w}_P\dot{w}_0 B_-^{\vee},0)\in \mathcal{Y}_P$ which lies over $x_0:=(0,0)\in\spec S$. By Lemma \ref{5avf}, $(\mathcal{Y}_P)_{x_0}$ is set-theoretically equal to $\{y_0\}$, and hence, by \cite[Th\'eor\`eme 13.1.3]{EGA3}, the fiber dimension at every point near $y_0$ is equal to zero. The result then follows from Lemma \ref{5aeq} and the observation that the $\mathbb{G}_m$-action takes every closed point of $\mathcal{Y}_P$ to $y_0$.
\end{proof}

\begin{lemma}\label{5aproper}
Suppose $\Bbbk=\mathbb{C}$. $\mathcal{Y}_P$ is proper over $\spec S$.
\end{lemma}
\begin{proof}
Let $Z\subset U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}\times\spec S$ be the graph of the morphism
\begin{equation}\nonumber
\begin{array}{ccc}
U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}\times\specH_T^{\bl}(\pt) & \rightarrow & \spec\Bbbk[q_1,\ldots,q_k] \\
(u\wpdB_-^{\vee}, h) &\mapsto & \left(\overline{q}_1(u\wpdB_-^{\vee},h),\ldots,\overline{q}_k(u\wpdB_-^{\vee},h) \right)
\end{array}
\end{equation}
and $\overline{Z}$ the reduced closed subscheme associated to the closure of $Z$ in $G^{\vee}/B^{\vee}_-\times\spec S$. Then
\[\mathcal{Y}_P\simeq Z\times_{G^{\vee}/B^{\vee}_-\times\spec S}\left(\mathcal{Y}\times \spec\Bbbk[q_1,\ldots,q_k]\right).\]
Define
\[\mathcal{Y}_P':=\overline{Z}\times_{G^{\vee}/B^{\vee}_-\times\spec S}\left(\mathcal{Y}\times \spec\Bbbk[q_1,\ldots,q_k]\right).\]
To prove that $\mathcal{Y}_P$ is proper over $\spec S$, it suffices to show that the natural morphism $\mathcal{Y}_P\rightarrow\mathcal{Y}_P'$ is a closed immersion. Observe that $U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}$ is reduced and is a closed subscheme of $\dot{w}_P\dot{w}_0\UdB_-^{\vee}/B_-^{\vee}$ which is a divisor complement of $G^{\vee}/B^{\vee}_-$. It follows that $\mathcal{Y}_P$ is a divisor complement of $\mathcal{Y}_P'$. Hence it suffices to show that $\mathcal{Y}_P$ is closed in $\mathcal{Y}_P'$.

Let $\overline{\mathcal{Y}}_P$ be the closure of $\mathcal{Y}_P$ in $\mathcal{Y}_P'$. Suppose $\overline{\mathcal{Y}}_P\ne\mathcal{Y}_P$. Clearly, there is a natural $\mathbb{G}_m$-action on $\overline{\mathcal{Y}}_P$ extending $\mathcal{Y}_P$.  Consider a $\mathbb{G}_m$-fixed point of $\overline{\mathcal{Y}}_P\setminus\mathcal{Y}_P$ which, by Lemma \ref{5aeq} and Lemma \ref{5astratum}, must be of the form $(\dot{w}_Q\dot{w}_0B_-^{\vee},0,0)$ for some standard parabolic subgroup $Q$ of $G$. Since $\dot{w}_Q\dot{w}_0B_-^{\vee}\in \overline{U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}}\setminus\{\wpdB_-^{\vee}\}$, we have $P\subsetneq Q$. WLOG, assume that the simple roots of $Q$ are $\alpha^{\vee}_{\ell+1},\ldots,\alpha^{\vee}_r$. Notice that $\ell< k$.

Let $(u\wpdB_-^{\vee},h)\in \mathcal{Y}_P\times_{G^{\vee}/B^{\vee}_-}\dot{w}_Q\dot{w}_0\UdB_-^{\vee}/B_-^{\vee}$ with $u\inU^{\vee}_{P^{\vee}_-}$. There are unique $u_1\inU^{\vee}$, $u_2\inU^{\vee}_-$ and $t\in T^{\vee}$ such that
\begin{equation}\label{5apropereq1}
u\dot{w}_P\dot{w}_0=\dot{w}_Q\dot{w}_0u_1tu_2.
\end{equation}
For any $1\leqslant i\leqslant r$, we have
\begin{align}\label{5aeeqa}
\overline{q}_i(u\wpdB_-^{\vee},h)&=|\ad_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i},(u\dot{w}_P\dot{w}_0)^{-1}\cdot e^T(h)\rangle \nonumber\\
&=|\ad_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i},(u\dot{w}_P\dot{w}_0 u_2^{-1})^{-1}\cdot e^T(h)\rangle\quad\because (u\wpdB_-^{\vee},h)\in \mathcal{Y}_P \nonumber\\
&=|\ad_i|^{-2}\alpha^{\vee}_i(w_0(t))\langle e^*_{-w_0\alpha^{\vee}_i},(\dot{w}_Q\dot{w}_0u_1)^{-1}\cdot e^T(h)\rangle \nonumber\\
&=\alpha^{\vee}_i(w_0(t))f_i(u_1,h)
\end{align}
where
\[f_i:(u_1,h)\mapsto |\ad_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i},(\dot{w}_Q\dot{w}_0u_1)^{-1}\cdot e^T(h)\rangle\] is a regular function on $U^{\vee}\times\specH_T^{\bl}(\pt)$. Notice that $f_i(e,0)\ne 0$ if $\ell+1\leqslant i\leqslant r$.

Let $\lambda\inQ^{\vee}$ be dominant. Recall the elements $v_{w_Pw_0}\in S(\lambda):=H^{\bullet}(\mathcal{J}_{\lambda})$ and $v_{\lambda}^*\in S(\lambda)^*$ defined in Section \ref{2d}. Define $g_{\lambda}\in\mathcal{O}(U^{\vee}\times\specH_T^{\bl}(\pt))$ by
\[ g_{\lambda}(u_1,h):=\langle v_{\lambda}^*, (\dot{w}_Q\dot{w}_0u_1)^{-1}\cdot v_{w_Pw_0}\rangle.\]
Suppose \eqref{5apropereq1} holds. Put $u_3:=(\dot{w}_P\dot{w}_0)^{-1}u(\dot{w}_P\dot{w}_0)$. We have $u_3\inU^{\vee}$ and
\begin{equation}\label{5aeeqb}
g_{\lambda}(u_1,h)=\langle v_{\lambda}^*, tu_2u_3^{-1}(\dot{w}_P\dot{w}_0)^{-1}\cdot v_{w_Pw_0}\rangle=\lambda(t).
\end{equation}
(Strictly speaking, we need Proposition \ref{appprop2} to handle the sign issue. But the sign is not important here.) Let $\{\omega^{\vee}_1,\ldots,\omega^{\vee}_r\}$ be the dual basis of $\{\alpha_1,\ldots,\alpha_r\}$. For each $1\leqslant i\leqslant k$, pick $\lambda_i\inQ^{\vee}\cap\mathbb{R}_{>0}\cdot\omega^{\vee}_i$. Observe that
\begin{align*}
g_{-w_0(\lambda_i)}(e,0)\neq 0~  \Longleftrightarrow~&~ w_Qw_P(\lambda_i)=\lambda_i\\
\Longleftrightarrow~&~ s_{\alpha_i}\text{ does not appear in a reduced}\\
&~\text{word decomposition of }w_Qw_P
\\
\Longleftrightarrow~&~ 1\leqslant i\leqslant\ell.
\end{align*}

Now write
\begin{equation}\label{5aeeqc}
\lambda_{\ell+1}=\sum_{i=1}^{\ell}b_i\lambda_i+\sum_{j=\ell+1}^rc_j\alpha^{\vee}_j
\end{equation}
where $b_i$ and $c_j$ are rational numbers. By rescaling $\lambda_{\ell+1}$, we may assume these numbers are integers. Notice that they are necessarily non-negative. (We determine them by pairing \eqref{5aeeqc} with the simple roots. Our claim then follows from the facts that all non-diagonal entries of a Cartan matrix are $\leqslant 0$ and all entries of the inverse of a Cartan matrix are $\geqslant 0$.) By \eqref{5aeeqa}, \eqref{5aeeqb} and \eqref{5aeeqc}, along $\mathcal{Y}_P\times_{G^{\vee}/B^{\vee}_-}\dot{w}_Q\dot{w}_0\UdB_-^{\vee}/B_-^{\vee}$, we have
\[ g_{-w_0(\lambda_{\ell+1})} = \prod_{i=1}^{\ell} (-w_0(\lambda_i))(t)^{b_i}\times \prod_{j=\ell+1}^r\alpha^{\vee}_j(w_0(t^{-1}))^{c_j}= \prod_{i=1}^{\ell} g_{-w_0(\lambda_i)}^{b_i}\times \prod_{j=\ell+1}^r \frac{f_j^{c_j}}{\overline{q}_j^{c_j}} .\]
It follows that $g_{-w_0(\lambda_{\ell+1})}$ is either non-zero or undefined at the limit point $(\dot{w}_Q\dot{w}_0B_-^{\vee},0,0)$ (depending on whether $c_{\ell+1}=\cdots=c_k=0$ or not), in contradiction to the observation made at the end of the previous paragraph. The proof of Lemma \ref{5aproper} is complete.
\end{proof}

\begin{proposition}\label{5arank}
The dimension of $\mathcal{O}(\mathcal{Y}_P)\otimes_S\fof S$ is equal to $|W/W_P|$.
\end{proposition}
\begin{proof}
We may assume $\Bbbk=\mathbb{C}$. Since $\mathcal{Y}_P$ is quasi-finite (proof of Lemma \ref{5aflat}) and proper (Lemma \ref{5aproper}) over $\spec S$, it follows that, by \cite[Th\'eor\`eme 8.11.1]{EGA3}, $\mathcal{O}(\mathcal{Y}_P)$ is a finitely generated $S$-module and hence a projective $S$-module by the flatness (Lemma \ref{5aflat}). The dimension in question is thus equal to the dimension of the coordinate ring of the fiber of $\mathcal{Y}_P$ over $x:=(0,0)\in\spec S$. By Lemma \ref{5avf}, $(\mathcal{Y}_P)_x$ is isomorphic to the zero locus of a vector field on $G^{\vee}/\ol{P}^{\vee}_-$, and hence
\[ \dim_{\mathbb{C}}\mathcal{O}((\mathcal{Y}_P)_x)=\chi( G^{\vee}/\ol{P}^{\vee}_-) = |W/W_{\ol{P}^{\vee}_-}| = |W/W_P|. \]
\end{proof}
\subsection{Construction of homomorphisms}\label{5b}
The goal of this subsection is to define three homomorphisms of $S$-algebras
\[ \Phi:\mathcal{O}(\mathcal{Y}_P)\rightarrow QH_T^{\bl}(G/P),\]

\[ \Phi^*:\mathcal{O}(\mathcal{Y}_P^*)\rightarrow QH_T^{\bl}(G/P)[\ecc]\]
and
\[\Phi_{loc}:\mathcal{O}(B_{e^T}^{\vee}\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee})\rightarrow QH_T^{\bl}(G/P)[\ecc] .\]

Let us start with a morphism of schemes
\begin{equation}\label{5beq1}
B_{e^T}^{\vee}\rightarrow G^{\vee}/B^{\vee}_-\times\specH_T^{\bl}(\pt)
\end{equation}
defined by $(b,h)\mapsto (bB_-^{\vee},h)$. Clearly, it factors through the immersion $\mathcal{Y}^*\hookrightarrowG^{\vee}/B^{\vee}_-\times\specH_T^{\bl}(\pt)$. Let $\lambda_0\in\mathbb{Z}_{>0}\cdot 2\rho^{\vee}$ be the element in Proposition \ref{2dcor}. By Lemma \ref{2dlemma3}, the open Bruhat cell $U^{\vee}(w_Pw_0)=\dot{w}_P\dot{w}_0\UdB_-^{\vee}/B_-^{\vee}$ is the complement of the divisor $\{s_{\lambda_0,v_{w_Pw_0}}=0\}$. Since the pull-back of $s_{\lambda_0,v_{w_Pw_0}}$ via \eqref{5beq1} is equal to $g_P:=g_{P,\lambda_0}:=\theta(\mathcal{J}_{\lambda_0},v_{w_Pw_0},v_{\lambda_0}^*)$ (see Section \ref{2d}), we obtain a homomorphism of $H_T^{\bl}(\pt)$-algebras
\[ \eta_7: \mathcal{O}(\mathcal{Y}^*\tGBU^{\vee}(w_Pw_0))\rightarrow \mathcal{O}(B_{e^T}^{\vee}\tGBU^{\vee}(w_Pw_0))\simeq\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}].\]
Define $J_P$ to be the ideal of $\mathcal{O}(B_{e^T}^{\vee})$ generated by $\theta(\mathcal{J}_{\lambda_0},v,v_{\lambda_0}^*)$ with $v\in S(\lambda_0)_{\mu}$ where $\mu\in \conv(W_P\cdot w_0(\lambda_0))\setminus\{w_Pw_0(\lambda_0)\}$.

\begin{lemma}\label{5bexist1}
We have
\[\mathcal{O}(B_{e^T}^{\vee}\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}) \simeq \mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle,\]
and hence $\eta_7$ induces a homomorphism
\[ \varphi_2:\mathcal{O}(\mathcal{Y}_P^*)\rightarrow \mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle.\]
\end{lemma}
\begin{proof}
Notice that $B_{e^T}^{\vee}\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}$ is a closed subscheme of $B_{e^T}^{\vee}\tGBU^{\vee}(w_Pw_0)$ whose defining ideal is generated by the pull-back of the regular functions on $U^{\vee}(w_Pw_0)$ belonging to the defining ideal of $U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}$. There exists a $T^{\vee}$-equivariant isomorphism
\[U^{\vee}(w_Pw_0) \simeq\prod_{\alpha\inw_Pw_0 R^+}U^{\vee}_{\alpha^{\vee}}\]
under which $U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}$ is identified with $\prod_{\alpha\inw_Pw_0 R^+\cap (-R^+)}U^{\vee}_{\alpha^{\vee}}$. Observe that
\[ w_Pw_0 R^+\cap (-R^+) = -(R^+\setminus R^+_P)\quad\text{ and } \quadw_Pw_0 R^+\cap R^+= R^+_P.\]
It follows that the defining ideal of $U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}$ is generated by the homogeneous elements of $\mathcal{O}(U^{\vee}(w_Pw_0))$ whose $T^{\vee}$-weights belong to $\left(\sum_{\alpha\in R^+_P}\mathbb{Z}_{\leqslant 0}\cdot \alpha^{\vee}\right)\setminus\{0\}$.

Now, by Proposition \ref{2dcor}, the ring $\mathcal{O}(U^{\vee}(w_Pw_0))$ is generated by regular functions of the form $s_{\lambda_0,v}/s_{\lambda_0,v_{w_Pw_0}}$ with $v\in S(\lambda_0)$. If $v\in S(\lambda_0)_{\mu}$, then the $T^{\vee}$-weight of $s_{\lambda_0,v}/s_{\lambda_0,v_{w_Pw_0}}$ is equal to $\mu-w_Pw_0(\lambda_0)$. Notice that
\[\mu-w_Pw_0(\lambda_0)\in \left(\sum_{\alpha\in R^+_P}\mathbb{Z}_{\leqslant 0}\cdot\alpha^{\vee}\right)\setminus\{0\}\Longleftrightarrow \mu\in\conv(W_P\cdot w_0(\lambda_0))\setminus\{w_Pw_0(\lambda_0)\}. \]
Therefore, each $s_{\lambda_0,v}/s_{\lambda_0,v_{w_Pw_0}}$ with $v\in S(\lambda_0)_{\mu}$ where $\mu\in\conv(W_P\cdot w_0(\lambda_0))\setminus\{w_Pw_0(\lambda_0)\}$ belongs to the defining ideal of $U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}$. It is not hard to see that these elements in fact generate the ideal. The proof is complete by noticing that the pull-back of $s_{\lambda_0,v}/s_{\lambda_0,v_{w_Pw_0}}$ to $\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]$ is equal to $g_P^{-1}\cdot\theta(\mathcal{J}_{\lambda_0},v,v_{\lambda_0}^*)$.
\end{proof}

For any dominant $\lambda\inQ^{\vee}$, define $g_{P,\lambda}:=\theta(\mathcal{J}_{\lambda},v_{w_Pw_0},v_{\lambda}^*)\in \mathcal{O}(B_{e^T}^{\vee})$.
\begin{lemma}\label{5bgp}
We have $\Phi_{PLS}\circ\Phi_{YZ}(g_{P,\lambda})=q^{[w_0(\lambda)]}$.
\end{lemma}
\begin{proof}
Let $Z$ be the unique MV cycle of type $\lambda$ and weight $w_Pw_0(\lambda)$. By Proposition \ref{4bmv2},
\[ \Phi_{YZ}(g_{P,\lambda})\in H^T_{2\rho(\lambda-w_Pw_0(\lambda))}(\agll)\]
and
\[ \Phi_{YZ}^0(g_{P,\lambda})=[Z]\in H_{2\rho(\lambda-w_Pw_0(\lambda))}(\agll).\]
Observe that the subvarieties $Z$ and $\agp:=\overline{L(\mathcal{O})\cdot t^{w_Pw_0(\lambda)}}$ of $\agll$ are regular at $t^{w_Pw_0(\lambda)}$, and the intersection at this point is transverse. Let $t^{\mu}\in Z$. Then $\mu\geqslant w_Pw_0(\lambda)$ by Lemma \ref{4bclosure}, and hence $t^{\mu}\not\in \agp$ unless $\mu=w_Pw_0(\lambda)$. This gives $Z\cap \agp=\{t^{w_Pw_0(\lambda)}\}$. The result then follows from Proposition \ref{3bone}.
\end{proof}

\begin{lemma}\label{5bjp}
We have $\Phi_{PLS}\circ\Phi_{YZ}(J_P)=\{0\}.$
\end{lemma}
\begin{proof}
Recall the ideal $J_P$ is generated by $\theta(\mathcal{J}_{\lambda_0},v,v_{\lambda_0}^*)$ with $v\in S(\lambda_0)_{\mu}$ where $\mu\in\conv(W_P\cdot w_0(\lambda_0))\setminus\{w_Pw_0(\lambda_0)\}$. By Proposition \ref{4bmv2}, $\Phi_{YZ}(\theta(\mathcal{J}_{\lambda_0},v,v_{\lambda_0}^*))$ lies in $H^T_{2\rho(\lambda_0-\mu)}(\mathcal{G}r^{\leqslant\lambda_0})$. By the assumption on these $\mu$, we have
\[2\rho(\lambda_0-\mu)>2\rho(\lambda_0-w_Pw_0(\lambda_0)).\]
The result then follows from Proposition \ref{3bzero}.
\end{proof}

\begin{lemma}\label{5bexist2}
The composition $\Phi_{PLS}\circ\Phi_{YZ}$ induces a homomorphism of $H_T^{\bl}(\pt)$-algebras
\[ \varphi_3: \mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle \rightarrowQH_T^{\bl}(G/P)[\ecc].\]
\end{lemma}
\begin{proof}
This follows immediately from Lemma \ref{5bgp} and Lemma \ref{5bjp}.
\end{proof}

\begin{lemma}\label{5bland}
The composition $\varphi_3\circ\varphi_2\circ\eta_1$
\[  \mathcal{O}(\mathcal{Y}_P)\xrightarrow{\eta_1} \mathcal{O}(\mathcal{Y}_P^*) \xrightarrow{\varphi_2} \mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle \xrightarrow{\varphi_3} QH_T^{\bl}(G/P)[\ecc]\]
lands in $QH_T^{\bl}(G/P)$. (Recall $\varphi_2$ and $\varphi_3$ come from Lemma \ref{5bexist1} and Lemma \ref{5bexist2} respectively.)
\end{lemma}
\begin{proof}
By Proposition \ref{2dcor}, $\mathcal{O}(U^{\vee}(w_Pw_0))$ is generated by regular functions $s_{\lambda_0,v}/s_{\lambda_0,v_{w_Pw_0}}$ with $v\in S(\lambda_0)$. Hence it suffices to show that all these regular functions are mapped into $QH_T^{\bl}(G/P)$. Their images in $\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]$ are equal to $g_P^{-1}\cdot\theta(\mathcal{J}_{\lambda_0},v,v_{\lambda_0}^*)$. By Lemma \ref{5bgp}, $g_P^{-1}$ goes to $q^{-[w_0(\lambda_0)]}$. Since $\Phi_{YZ}(\theta(\mathcal{J}_{\lambda_0},v,v_{\lambda_0}^*))\in H^T_{\bl}(\mathcal{G}r^{\leqslant \lambda_0})$ by Proposition \ref{4bmv2}, we have, by the explicit description of $\Phi_{PLS}$,
\[  \varphi_3(g_P^{-1}\cdot\theta(\mathcal{J}_{\lambda_0},v,v_{\lambda_0}^*))\in\sum_{\mu} H_T^{\bullet}(G/P)\otimes q^{[\mu-w_0(\lambda_0)]} \]
where the sum is taken over all anti-dominant $\mu$ lying in $Q^{\vee}\cap \conv(W\cdot\lambda_0)$. Clearly, for each of these $\mu$, $\mu-w_0(\lambda_0)$ is a $\mathbb{Z}_{\geqslant 0}$-linear combination of the simple coroots. The proof is complete.
\end{proof}

\begin{definition}\label{5bphi} $~$
\begin{enumerate}
\item Define
\[ \Phi:\mathcal{O}(\mathcal{Y}_P)\rightarrow QH_T^{\bl}(G/P) \]
to be the unique map satisfying $\eta_3\circ\Phi =\varphi_3\circ\varphi_2\circ\eta_1$. See Lemma \ref{5bland} and diagram \eqref{1adiag}.
\item Define
\[ \Phi^*:\mathcal{O}(\mathcal{Y}_P^*)\rightarrow QH_T^{\bl}(G/P)[\ecc] \]
to be the composition $\varphi_3\circ\varphi_2$.
\item Define
\[\Phi_{loc}:\mathcal{O}(B_{e^T}^{\vee}\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee})\rightarrow QH_T^{\bl}(G/P)[\ecc]\]
to be $\varphi_3$ defined in Lemma \ref{5bexist2}. (We apply Lemma \ref{5bexist1} to identify their domains.)
\end{enumerate}
\end{definition}

\begin{lemma}\label{5bhomo}
$\Phi$, $\Phi^*$ and $\Phi_{loc}$ are homomorphisms of $S$-algebras.
\end{lemma}
\begin{proof}
It is clear that they are homomorphisms of $H_T^{\bl}(\pt)$-algebras. To complete the proof , it suffices to show that $\Phi(q_i\cdot 1)=q^{[\alpha^{\vee}_i]}$ for any $1\leqslant i\leqslant k$. Recall $q_i$ acts on $\mathcal{O}(\mathcal{Y}_P)$ by the regular function
\[ \overline{q}_i: (u\wpdB_-^{\vee},h)\mapsto |\ad_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i},(u\dot{w}_P\dot{w}_0)^{-1}\cdot e^T(h)\rangle.\]
Denote by $b$ and $u$ the composites
\[ \spec\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle \hookrightarrow B^{\vee}\times\specH_T^{\bl}(\pt) \rightarrow B^{\vee}\]
and
\[ \spec\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle\xrightarrow{\spec(\varphi_2\circ\eta_1)} \mathcal{Y}_P\rightarrow U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}\simeq U^{\vee}_{P^{\vee}_-} \]
respectively, where the unlabelled arrows are the obvious morphisms. There exist morphisms $u_1$ and $t$ from $\spec\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle$ to $U^{\vee}_-$ and $T^{\vee}$ respectively such that $u\dot{w}_P\dot{w}_0 = bu_1 t$. It is easy to see that
\[ \varphi_2\circ\eta_1(\overline{q}_i)= |\alpha^{\vee}_i|^{-2} \langle e^*_{-w_0\alpha^{\vee}_i}, t^{-1}u_1^{-1}b^{-1}\cdot e^T(h)\rangle =\alpha^{\vee}_i(w_0(t)).\]

On the other hand, for any dominant $\lambda\inQ^{\vee}$, we have
\[g_{P,\lambda}=\theta(\mathcal{J}_{\lambda},v_{w_Pw_0},v_{\lambda}^*)=\langle v_{\lambda}^*, u_1t(\dot{w}_P\dot{w}_0)^{-1}u^{-1}\cdot v_{w_Pw_0}\rangle=\langle v_{\lambda}^*, u_1t(\dot{w}_P\dot{w}_0)^{-1}u^{-1}(\dot{w}_P\dot{w}_0)\cdot v_{e}\rangle\]
by Proposition \ref{appprop2}. Since $(\dot{w}_P\dot{w}_0)^{-1}u^{-1}(\dot{w}_P\dot{w}_0)\inU^{\vee}$, it follows that
\[g_{P,\lambda}=\lambda(t). \]
Pick dominant $\lambda_1, \lambda_2\inQ^{\vee}$ such that $\lambda_1=\lambda_2+w_0\alpha^{\vee}_i$. It follows that
\[ \varphi_2\circ\eta_1(\overline{q}_i)=\alpha^{\vee}_i(w_0(t))=\lambda_1(t)/\lambda_2(t)=g_{P,\lambda_1}/g_{P,\lambda_2},\]
and hence, by Lemma \ref{5bgp},
\[ \Phi(q_i\cdot 1)=\Phi_{PLS}\circ\Phi_{YZ}(g_{P,\lambda_1}/g_{P,\lambda_2}) = q^{[w_0(\lambda_1-\lambda_2)]}=q^{[\alpha^{\vee}_i]}.\]
\end{proof}
\section{Proof of main results}\label{6}
\subsection{Proof of Theorem \ref{1athm}}\label{6a}
Recall we have defined $\Phi$ and $\Phi^*$ in Definition \ref{5bphi}. Theorem \ref{1athm} follows immediately from
\begin{theorem}\label{6athma}
$\Phi$ is bijective.
\end{theorem}

\begin{theorem}\label{6athmb}
$\Phi^*$ is injective and is bijective at the non-equivariant limit.
\end{theorem}

We prove some lemmas first.
\begin{lemma}\label{6asurj1}
The localization
\[\Phi[q^{-1}_1,\ldots,q^{-1}_k]:\mathcal{O}(\mathcal{Y}_P)[q^{-1}_1,\ldots,q^{-1}_k]\rightarrow QH_T^{\bl}(G/P)[\ecc]\]
is surjective.
\end{lemma}
\begin{proof}
Let $\lambda\inQ^{\vee}$ be dominant. By the proof of Lemma \ref{5bhomo}, the image of $q^{-[w_0(\lambda)]}\cdot 1\in\mathcal{O}(\mathcal{Y}_P)$ in $\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle$ (through $\varphi_2\circ\eta_1$) is equal to $g_{P,\lambda}^{-1}$ which is a unit. Hence there exists
\[ \varphi_4:\mathcal{O}(\mathcal{Y}_P)[q^{-1}_1,\ldots,q^{-1}_k]\rightarrow \mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle\]
such that $\varphi_4\circ\eta_2=\varphi_2\circ\eta_1$. It follows that $\Phi[q^{-1}_1,\ldots,q^{-1}_k]=\varphi_3\circ\varphi_4$. By the explicit description of $\Phi_{PLS}$, the image of $\Phi_{PLS}$ generates the $S[q^{-1}_1,\ldots,q^{-1}_k]$-module $QH_T^{\bl}(G/P)[\ecc]$. Since $\Phi_{YZ}$ is surjective (Theorem \ref{4aYZ}), $\Phi[q^{-1}_1,\ldots,q^{-1}_k]$ is surjective if we can show that $\varphi_4$ is.

By definition, $B_{e^T}^{\vee}$ is a closed subscheme of $B^{\vee}\times\specH_T^{\bl}(\pt)\simeq U^{\vee}\timesT^{\vee}\times\specH_T^{\bl}(\pt)$. Hence it suffices to show that, when placed in $\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle$, the regular functions on $U^{\vee}$ and $T^{\vee}$ belong to $\im\varphi_4$. Since the projection $B_{e^T}^{\vee}\rightarrow U^{\vee}\simeq \UdB_-^{\vee}/B_-^{\vee}$ factors through $\mathcal{Y}^*$, the image of the composite
\[ \mathcal{O}(U^{\vee}) \rightarrow \mathcal{O}(B_{e^T}^{\vee}) \rightarrow \mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle\]
is contained in $\im\varphi_2$ and hence in $\im\varphi_4$ by Lemma \ref{5alocal}.

It remains to show that $\im\varphi_4$ contains all regular functions coming from $T^{\vee}$. Since $\mathcal{O}(T^{\vee})$ is generated by the characters of $T^{\vee}$ and the character of weight equal to any dominant $\lambda\inQ^{\vee}$ corresponds to $\theta(\mathcal{J}_{\lambda},v_e,v_{\lambda}^*)$, it suffices to show that $\im\varphi_4$ contains these regular functions and their inverses. By Lemma \ref{2dlemma3}, both $s_{\lambda,v_e}/s_{\lambda,v_{w_Pw_0}}$ and $s_{\lambda,v_{w_Pw_0}}/s_{\lambda,v_e}$ are regular functions on $U^{\vee}(w_Pw_0)\cap\UdB_-^{\vee}/B_-^{\vee}$. They give rise to elements of $\mathcal{O}(\mathcal{Y}_P^*)$ which are sent by $\varphi_2$ to $g_{P,\lambda}^{-1}\cdot\theta(\mathcal{J}_{\lambda},v_e,v_{\lambda}^*)$ and $g_{P,\lambda}\cdot\theta(\mathcal{J}_{\lambda},v_e,v_{\lambda}^*)^{-1}$ respectively. It follows that, by Lemma \ref{5alocal}, we have $g_{P,\lambda}^{\pm 1}\cdot\theta(\mathcal{J}_{\lambda},v_e,v_{\lambda}^*)^{\mp 1}\in\im\varphi_4$. But we have seen at the beginning of the proof that $g_{P,\lambda}^{\pm 1}\in\im\varphi_4$. Hence $\theta(\mathcal{J}_{\lambda},v_e,v_{\lambda}^*)^{\pm 1}\in\im\varphi_4$ as desired.
\end{proof}

\begin{lemma}\label{6asurj2}
There exists $\eta\in S$ which is coprime to $q_1\cdots q_k$ such that $\Phi[\eta^{-1}]$ is surjective.
\end{lemma}
\begin{proof}
Let $\lambda\inQ^{\vee}$ be the element in Lemma \ref{4bdivisor}.
By Theorem \ref{3aPLS},
\[ q^{-[w_0(\lambda)]}\Phi_{PLS}\left(\bk{\overline{\mathcal{B}\cdot t^{s_{\alpha_i}(w_0(\lambda))}}}\right)=\sigma_{s_{\alpha_i}}\]
for any $1\leqslant i\leqslant k$. It follows that the image of the linear map
\[ \begin{array}{ccc}
S(\lambda)\otimesH_T^{\bl}(\pt) & \rightarrow & QH_T^{\bl}(G/P)[\ecc]\\ [.5em]
v\otimes 1 &\mapsto & q^{-[w_0(\lambda)]}\Phi_{PLS}\circ\Phi_{YZ}(\theta(\mathcal{J}_{\lambda},v,v_{\lambda}^*))
\end{array} \]
contains $H_T^2(G/P)$. For any $v\in S(\lambda)$, the regular function $s_{\lambda,v}/s_{\lambda,v_{w_Pw_0}}$ on $U^{\vee}(w_Pw_0)$ induces a regular function on $\mathcal{Y}_P$ whose pull-back to $\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]$ is equal to $g_{P,\lambda}^{-1}\cdot \theta(\mathcal{J}_{\lambda},v,v_{\lambda}^*)$. Hence, by Lemma \ref{5bgp}, the image of $\Phi$ contains $H^2_T(G/P)$.

By \cite[Lemma 4.1.3]{Invent}, the $\fofH_T^{\bl}(\pt)$-algebra $H_T^{\bullet}(G/P)\otimes_{H_T^{\bl}(\pt)}\fofH_T^{\bl}(\pt)$ is generated by $H_T^2(G/P)$. (We take a Pl\"ucker embedding of $G/P$ into $\mathbb{P}(V)$ for some representation $V$ of $G$. The condition for this lemma is satisfied because the $T$-fixed points of $G/P$ correspond to the extremal weights of $V$ which all have multiplicities one.) By linear algebra, there exists $\eta\in S$ which is coprime to $q_1\cdots q_k$ such that the $S[\eta^{-1}]$-algebra $QH_T^{\bl}(G/P)[\eta^{-1}]$ is generated by $H_T^2(G/P)$.
\end{proof}

\begin{lemma}\label{6ainj}
$\Phi$ is injective.
\end{lemma}
\begin{proof}
Since $\mathcal{O}(\mathcal{Y}_P)$ is a flat $S$-module by Lemma \ref{5aflat}, it suffices to show that $\Phi\otimes_S\fof S$ is injective. We have proved that it is surjective (Lemma \ref{6asurj1} or Lemma \ref{6asurj2}), and the dimension of its domain is equal to $|W/W_P|$ (Lemma \ref{5arank}) which is equal to the rank of the free $S$-module $QH_T^{\bl}(G/P)$. The result follows.
\end{proof}

\begin{myproof}{Theorem}{\ref{6athma}}
We have
\begin{enumerate}
\item $\mathcal{O}(\mathcal{Y}_P)$ is a flat $S$-module (Lemma \ref{5aflat});

\item $QH_T^{\bl}(G/P)$ is a flat $S$-module (obvious);

\item $\Phi[q^{-1}_1,\ldots,q^{-1}_k]$ is surjective (Lemma \ref{6asurj1});

\item $\Phi[\eta^{-1}]$ is surjective for some $\eta\in S$ which is coprime to $q_1\cdots q_k$ (Lemma \ref{6asurj2}); and

\item $\Phi$ is injective (Lemma \ref{6ainj}).
\end{enumerate}
It is easy to see that (1) to (5) imply the bijectivity of $\Phi$.
\end{myproof}

\vspace{.5cm}
\begin{myproof}{Theorem}{\ref{6athmb}}
Let $\lambda_0\in\mathbb{Z}_{>0}\cdot 2\rho^{\vee}$ be the element in Proposition \ref{2dcor}. Since $\mathcal{O}(\mathcal{Y}_P^*)$ is the localization of $\mathcal{O}(\mathcal{Y}_P)$ by $s_{\lambda_0,v_e}/s_{\lambda_0,v_{w_Pw_0}}$, the injectivity of $\Phi^*$ follows from the injectivity of $\Phi$ (Theorem \ref{6athma}), and we have
\[ \im\Phi^*=QH_T^{\bl}(G/P)[\Phi(s_{\lambda_0,v_e}/s_{\lambda_0,v_{w_Pw_0}})^{-1}] \]
by the surjectivity of $\Phi$ (Theorem \ref{6athma}). Notice that
\[ \varphi_2\circ\eta_1(s_{\lambda_0,v_e}/s_{\lambda_0,v_{w_Pw_0}})^{-1}=g_P\cdot \theta(\mathcal{J}_{\lambda_0},v_e,v_{\lambda_0}^*)^{-1}, \]
and $\theta(\mathcal{J}_{\lambda_0},v_e,v_{\lambda_0}^*)$ is the pull-back of a character of $T^{\vee}$ via the natural morphism
\[B_{e^T}^{\vee}\rightarrow B^{\vee}\simeqU^{\vee}\timesT^{\vee}\rightarrowT^{\vee}.\]
By \cite[Proposition 3.3]{YZ}, we have $B^{\vee}_{e^T(0)}\subsetU^{\vee}$, and hence $\theta(\mathcal{J}_{\lambda_0},v_e,v_{\lambda_0}^*)=1$ in the non-equivariant settings. On the other hand, we have $\varphi_3(g_P)=q^{[w_0(\lambda_0)]}$ by Lemma \ref{5bgp}. Therefore,
\[ \im\Phi^*|_{h=0} = QH^{\bullet}(G/P)[q^{[w_0(\lambda_0)]}]=QH^{\bullet}(G/P)[q^{-1}_1,\ldots,q^{-1}_k].\]
\end{myproof}
\subsection{Proof of Theorem \ref{1athmb}} \label{6b}
Recall $\Phi[q^{-1}_1,\ldots,q^{-1}_k]=\varphi_3\circ\varphi_4$ where $\varphi_4$ is defined in the proof of Lemma \ref{6asurj1}. We proved there that $\varphi_4$ is surjective. Since $\Phi[q^{-1}_1,\ldots,q^{-1}_k]$ is bijective by Theorem \ref{6athma}, it follows that $\varphi_3=\Phi_{loc}$ and $\varphi_4$ are bijective as well. Theorem \ref{1athmb} is proved.
\subsection{Proof of Theorem \ref{1athmc}} \label{6c} Put $\Bbbk:=\overline{\mathbb{F}}_2$.
\subsubsection{Type $B_{\ell}$}
We have $G=\mathbf{Spin}_{2\ell+1}$. Let $P$ be the standard parabolic subgroup of $G$ associated to the subset consisting of all but the first node. Then $G/P$ is isomorphic to the $(2\ell-1)$-dimensional quadric $Q^{2\ell-1}_{\mathbb{C}}$. The abelian group $H^{\bullet}(G/P;\mathbb{Z})$ is freely generated by the Schubert classes $x_i\in H^{2i}(G/P;\mathbb{Z})$, $i=0,\ldots,2\ell-1$. The Chevalley formula gives
\begin{equation}\label{6ceq1}
x_1\cup x_i=\left\{
\begin{array}{cl}
x_{i+1}& i\ne\ell-1,2\ell-1\\
2x_{\ell}& i=\ell -1\\
0&i=2\ell -1
\end{array}\right. .
\end{equation}
Since $\chari\Bbbk=2$, we have $x_1^{\ell}=0$, and hence
\[ H^{\bullet}(G/P;\Bbbk)\simeq \Bbbk[x_1,x_{\ell}]/\langle x_1^{\ell}, x_{\ell}^2\rangle.\]

We have $G^{\vee}_{\Bbbk}=\mathbf{PSp}_{2\ell}$. Its Lie algebra consists of $2\ell\times 2\ell$ matrices of the form
\[ \left(\begin{matrix}
A& B\\ C&A^T
\end{matrix}\right)
\]
where $A,B,C\in M_{\ell\times\ell}(\Bbbk)$ such that $B^T=B$ and $C^T=C$. We take the principal nilpotent $e$ to be the matrix $M_1$ for which $A=\sum_{i=1}^{\ell -1}E_{i,i+1}$, $B=E_{\ell,\ell}$ and $C=0$.

The flag variety $G^{\vee}_{\Bbbk}/\ol{P}^{\vee}_-$ defined in Lemma \ref{5avf} is isomorphic to $\mathbb{P}^{2\ell-1}_{\Bbbk}$. The section $s_{e^T(0)}\in H^0(G^{\vee}_{\Bbbk}/\ol{P}^{\vee}_-;\mathcal{T}_{G^{\vee}_{\Bbbk}/\ol{P}^{\vee}_-})$ introduced in the proof of the same lemma is induced by $M_1$. For convenience, let us permute the indices $1,\ldots,2\ell +1$ such that $M_1$ sends each $e_i$ to $e_{i-1}$ (where $e_0:=0$). Consider the standard open neighbourhood $U\simeq \mathbb{A}^{2\ell-1}_{\Bbbk}$ of the unique zero $p:=[1:0:\cdots :0]$ of $s_{e^T(0)}$. Then $s_{e^T(0)}|_U$ is given by the morphism
\[
\begin{array}{ccc}
\mathbb{A}^{2\ell-1}_{\Bbbk} & \rightarrow & \mathbb{A}^{2\ell-1}_{\Bbbk}\\
(t_2,\ldots,t_{2\ell}) &\mapsto & (t_3+t_2^2,t_4+t_2t_3,\ldots, t_{2\ell}+t_2t_{2\ell-1},t_2t_{2\ell})
\end{array}.
\]
It follows that, by Lemma \ref{5avf},
\[ \mathcal{O}((\mathcal{Y}_P)_{(0,0)})\simeq \Bbbk[t_2,\ldots,t_{2\ell}]/\langle t_3+t_2^2,\ldots,t_{2\ell}+t_2t_{2\ell-1},t_2t_{2\ell} \rangle.\]
The latter $\Bbbk$-algebra is isomorphic to $\Bbbk[t_2]/\langle t_2^{2\ell}\rangle$ which is not isomorphic to $H^{\bullet}(G/P;\Bbbk)$.

\subsubsection{Type $C_{\ell}$}
We have $G=\mathbf{Sp}_{2\ell}$. Let $P$ be the standard parabolic subgroup of $G$ associated to the subset consisting of all but the first node. Then $G/P\simeq \mathbb{P}_{\mathbb{C}}^{2\ell -1}$, and hence
\[ H^{\bullet}(G/P;\Bbbk)\simeq \Bbbk[x]/\langle x^{2\ell}\rangle. \]

We have $G^{\vee}_{\Bbbk}=\mathbf{SO}_{2\ell +1}$. Recall it is constructed as follows. Let $Q$ be the quadratic form on $\Bbbk^{2\ell+1}$ defined by
\[ Q(v_1,\ldots,v_{2\ell+1}):= v_{2\ell+1}^2+\sum_{i=1}^{\ell} v_iv_{i+\ell}.\]
Then
\[G^{\vee}_{\Bbbk} = \left\{g\in \mathbf{GL}_{2\ell+1}|~\forall \mathbf{v}\in\Bbbk^{2\ell+1},~Q(g\cdot\mathbf{v}) = Q(\mathbf{v})\right\}.\]
The Lie algebra of $G^{\vee}_{\Bbbk}$ is the space of $(2\ell+1)\times (2\ell+1)$ matrices of the form
\[ \left(\begin{matrix}
A& B&0\\ C&A^T&0\\ x&y&0
\end{matrix}\right)
\]
where $A,B,C\in M_{\ell\times\ell}(\Bbbk)$ and $x,y\in M_{1\times\ell}(\Bbbk)$ such that $B^T=B$, $C^T=C$ and the diagonal entries of $B$ and $C$ are all equal to zero. We take the principal nilpotent $e$ to be the matrix $M_2$ for which $A=\sum_{i=1}^{\ell -1}E_{i,i+1}$, $B=C=x=0$ and $y=(0,\ldots,0,1)$.

The flag variety $G^{\vee}_{\Bbbk}/\ol{P}^{\vee}_-$ is isomorphic to the quadric $Q_{\Bbbk}^{2\ell-1}$ defined over $\Bbbk$
\[  Q_{\Bbbk}^{2\ell-1}:=\{ [\mathbf{v}]\in \mathbb{P}_{\Bbbk}^{2\ell}|~Q(\mathbf{v})=0 \}.\]
The section $s_{e^T(0)}\in H^0(G^{\vee}_{\Bbbk}/\ol{P}^{\vee}_-;\mathcal{T}_{G^{\vee}_{\Bbbk}/\ol{P}^{\vee}_-})$ is the restriction of the vector field on $\mathbb{P}_{\Bbbk}^{2\ell}$ induced by $M_2$. As in the above case, consider the standard open neighbourhood $U$ of the unique zero $p=[1:0:\cdots:0]$ of $s_{e^T(0)}$. By a straightforward computation, we get
\[ \mathcal{O}((\mathcal{Y}_P)_{(0,0)})\simeq \Bbbk[t_2,\ldots,t_{2\ell+1}]/I \]
where $I$ is the ideal generated by
\[ t_2t_{\ell},~ Q(1,t_2,\ldots, t_{2\ell+1}),~ t_j+t_2t_{j-1}~ (3\leqslant j\leqslant \ell)~\text{ and }~t_k+t_2t_{k+1}~ (\ell+1\leqslant k\leqslant 2\ell). \]
By some algebra, we see that $I$ is also generated by
\[ t_2^{\ell}, ~t_{2\ell+1}^2,~ t_j+t_2^{j-1}~ (3\leqslant j\leqslant \ell)~\text{ and }~t_k+t_2^{2\ell+1 -k}t_{2\ell+1}~ (\ell+1\leqslant k\leqslant 2\ell),\]
and hence $\Bbbk[t_2,\ldots,t_{2\ell+1}]/I\simeq \Bbbk[t_2,t_{2\ell+1}]/\langle t_2^{\ell}, t_{2\ell +1}^2\rangle$ which is not isomorphic to $H^{\bullet}(G/P;\Bbbk)$.
\subsection{Proof of Theorem \ref{1bthmb}} \label{6d}
Let us first recall some basic facts about $\bfm(G^{\vee})$ and $\toda(G^{\vee})$. See \cite[Section 5 \& 6]{T1} for more details.

Let $\dd{\mathfrak{g}}_{reg}\subset\dd{\mathfrak{g}}$ be the set of regular elements of $\dd{\mathfrak{g}}$. We have $\bfm(G^{\vee}):=\mathcal{J}_{reg}//G^{\vee}$ where
\[\mathcal{J}_{reg}:=\{ (g,\xi)\in G^{\vee}\times\dd{\mathfrak{g}}_{reg}|~g\cdot \xi=\xi \} . \]
The Chevalley polynomials define an isomorphism of schemes
\[  \dd{\mathfrak{g}}_{reg}// G^{\vee}\simeq\spec\left(\sym^{\bullet}\mathfrak{t} \right)^W.\]
Identify $\mathfrak{t} $ with $\mathfrak{t}^{\vee}$ using the rescaled Killing form $f=-2\langle -,-\rangle_{Kil}/\langle \alpha^{\vee}_0,\alpha^{\vee}_0\rangle_{Kil}$ so that
\[  \left(\sym^{\bullet}\mathfrak{t} \right)^W\simeq \left(\sym^{\bullet}\mathfrak{t}^{\vee} \right)^W\simeq H_G^{\bullet}(\pt).\]
Therefore, $\bfm(G^{\vee})$ is a group scheme over $\dd{\mathfrak{g}}_{reg}// G^{\vee}\simeq \spec H_G^{\bullet}(\pt)$.
In \cite{BFM}, Bezrukavnikov, Finkelberg and Mirkovi\'c constructed an isomorphism of $H_G^{\bullet}(\pt)$-algebras
\[\Phi_{BFM}:\mathcal{O}(\bfm(G^{\vee}))\xrightarrow{\sim} H_{-\bullet}^G(\mathcal{G}r).\]

The following lemma is standard. See e.g. \cite[Section 6.5]{YZ}.
\begin{lemma}\label{6dlemma0}
The morphism
\[
\begin{array}{ccccc}
\Pi&:&B_{e^T}^{\vee}&\rightarrow &\bfm(G^{\vee})\\ [.5em]
&&(b,h)&\mapsto & [b^{-1},e^T(h)]
\end{array}
\]
is the base change of the quotient morphism $\specH_T^{\bl}(\pt)\rightarrow \spec H_G^{\bullet}(\pt)$ along the structure morphism $\bfm(G^{\vee})\rightarrow \spec H_G^{\bullet}(\pt)$. In particular, $B_{e^T}^{\vee}$ admits a $W$-action such that $\Pi$ induces an isomorphism
\[ \mathcal{O}(\Pi):\mathcal{O}(\bfm(G^{\vee}))\xrightarrow{\sim} \mathcal{O}(B_{e^T}^{\vee})^W.\]
Moreover, the Yun-Zhu isomorphism $\Phi_{YZ}$ considered in Section \ref{4} is $W$-equivariant and
\[ \Phi_{BFM}=\Phi_{YZ}^W\circ \mathcal{O}(\Pi).\]
\hfill$\square$
\end{lemma}
\begin{remark}\label{6drmk}
Recall our $\Phi_{YZ}$ differs from the one defined in \cite{YZ} by the inverse morphism of the group scheme $B_{e^T}^{\vee}$. See Remark \ref{4armk2}. That is why we take the inverse $b^{-1}$ in the definition of $\Pi$.
\end{remark}

Next, we consider the Toda space $\toda(G^{\vee})$. Fix a non-degenerate $G^{\vee}$-invariant bilinear form $\kil{-,-}$ on $\dd{\mathfrak{g}}$, i.e. a non-zero multiple of the Killing form. Put $\chi:=\kil{e,-}|_{\mathfrak{n}^{\vee}_-}\in (\mathfrak{n}^{\vee}_-)^*$. (Recall $e=\sum_{i=1}^r |\alpha^{\vee}_i|^2e_{\alpha^{\vee}_i}$. See Section \ref{4a}.) We have
\[ \toda(G^{\vee}):= U^{\vee}_-\setminus \mu^{-1}(\chi\times\chi)/U^{\vee}_-\]
where
\[
\begin{array}{ccccc}
\mu &:& T^*G^{\vee}\simeqG^{\vee}\times(\dd{\mathfrak{g}})^*&\rightarrow&(\mathfrak{n}^{\vee}_-\times\mathfrak{n}^{\vee}_-)^*\\ [.5em]
& & (g,\zeta) &\mapsto & ((g\cdot\zeta)|_{\mathfrak{n}^{\vee}_-},\zeta|_{\mathfrak{n}^{\vee}_-})
\end{array}
\]
is the moment map. The Toda isomorphism is an isomorphism of schemes
\begin{equation}\nonumber
\Toda:\bfm(G^{\vee})\xrightarrow{\sim}\toda(G^{\vee})
\end{equation}
defined as follows. By Kostant's slice theorem, there is an isomorphism
\[ \bfm(G^{\vee})\simeq (\mathcal{J}_{reg}\times_{\dd{\mathfrak{g}}} (e+\dd{\mathfrak{b}}_-))/U^{\vee}_- .\]
Let $(g,\xi)\in (\mathcal{J}_{reg}\times_{\dd{\mathfrak{g}}} (e+\dd{\mathfrak{b}}_-))/U^{\vee}_-$. We have $\kil{\xi, g^{-1}\cdot \eta} = \kil{\xi,\eta} =\kil{e,\eta}$ for any $\eta\in\mathfrak{n}^{\vee}_-$, and hence $(g,\kil{\xi,-})\in\mu^{-1}(\chi\times\chi)$. Define
\[ \Toda([g,\xi]) := [g, \kil{ \xi,-} ].\]

For simplicity, put
\begin{align*}
X:= &~B_{e^T}^{\vee}\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}\\
Y:=&~ U^{\vee}_-\setminus \left(\mu^{-1}(\chi\times\chi)\times_{G^{\vee}} U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}T^{\vee}U^{\vee}_-\right)/U^{\vee}_-.
\end{align*}
\begin{lemma}\label{6dlemma1}
The composition $\Toda\circ\Pi$ induces a morphism
\[ f_1: X \rightarrow Y \]
which induces an isomorphism
\[ \mathcal{O}(f_1):\mathcal{O}\left(Y\right)\xrightarrow{\sim} \mathcal{O}\left(X\right)^W.\]
\end{lemma}
\begin{proof}
We have
\[ \Toda\circ\Pi (b,h) = [b^{-1},\kil{e^T(h),-}]\]
for any $(b,h)\inB_{e^T}^{\vee}$. The first assertion follows from the isomorphism
\[ B_{e^T}^{\vee}\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee} \simeq B_{e^T}^{\vee}\times_{G^{\vee}}U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}\]
and the equality
\[\left(U^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}\right)^{-1}= U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}T^{\vee}U^{\vee}_-,\]
and the second follows from $\mathcal{O}(\bfm(G^{\vee}))\simeq\mathcal{O}(B_{e^T}^{\vee})^W$ and the standard averaging trick.
\end{proof}

\begin{lemma}\label{6dlemma2}
The diagram
\begin{center}
\begin{tikzpicture}
\tikzmath{\x1 = 7; \x2 = 2;}
\node (A) at (0,0) {$X$} ;
\node (B) at (\x1,0) {$Y$} ;
\node (C) at (0,-\x2) {$\spec\mathbb{C}[q_1^{\pm 1},\ldots,q_k^{\pm 1}]\simeq\spec\mathbb{C}[Q^{\vee}/Q^{\vee}_P]$} ;
\node (D) at (\x1,-\x2) {$ \spec\mathbb{C}[Q^{\vee}]\simeqT^{\vee}$} ;

\path[->, font=\small] (A) edge node[above]{$f_1$} (B);
\path[->, font=\small] (A) edge node[left]{$(\overline{q}_1,\ldots,\overline{q}_k)$} (C);
\path[->, font=\small] (B) edge node[right]{$f_2$} (D);
\path[right hook->]
(C) edge node[left]{} (D);
\end{tikzpicture}.
\end{center}
is commutative where $f_2$ is induced by the canonical projection $U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}T^{\vee}U^{\vee}_-\rightarrow T^{\vee}$.
\end{lemma}
\begin{proof}
Let $(b,h)\in X$. Write $b=u_1\dot{w}_P\dot{w}_0 tu_2$ with $u_1\inU^{\vee}_{P^{\vee}_-}$, $t\inT^{\vee}$ and $u_2\inU^{\vee}_-$. (Strictly speaking, we should regard $b$, $u_1$, $t$ and $u_2$ as morphisms from $X$ to the corresponding group schemes. See the proof of Lemma \ref{5bhomo}.) Then
\begin{equation}\label{6deq3}
 b^{-1}=u_2^{-1}t^{-1}(\dot{w}_P\dot{w}_0)^{-1}u_1^{-1}=u_2^{-1}(\dot{w}_P\dot{w}_0)^{-1} w_Pw_0(t^{-1})u_1^{-1}.
\end{equation}
For any $1\leqslant i\leqslant r$, we have
\begin{align}\label{6deq4}
\overline{q}_i(b,h) &=|\alpha^{\vee}_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i}, (u_1\dot{w}_P\dot{w}_0)^{-1}\cdot e^T(h) \rangle \nonumber \\
&=|\alpha^{\vee}_i|^{-2}\langle e^*_{-w_0\alpha^{\vee}_i}, tu_2b^{-1}\cdot e^T(h) \rangle \nonumber\\
&=\alpha^{\vee}_i(w_0(t^{-1})).
\end{align}
Since for any $k+1\leqslant i\leqslant r$, the LHS of \eqref{6deq4} is equal to 1 (Lemma \ref{5aq=1}), we have $\alpha^{\vee}_i(w_0(t^{-1}))=1$, and hence $ w_0(t^{-1})\in\spec\mathbb{C}[Q^{\vee}/Q^{\vee}_P]\subseteq T^{\vee}$. By definition and \eqref{6deq3}, we have $f_2\circ f_1(b,h) = w_Pw_0(t^{-1}) =w_0(t^{-1})$. Therefore, by \eqref{6deq4}, $ \alpha^{\vee}_i \circ f_2\circ f_1(b,h)=\overline{q}_i(b,h)$.
\end{proof}

\begin{lemma}\label{6dlemma3}
For any $\mathbf{q}_0\in\spec\mathbb{C}[q_1^{\pm 1},\ldots,q_k^{\pm 1}]$, the fiber product $\mu^{-1}(\chi\times\chi)\times_{G^{\vee}} Z_{\mathbf{q}_0}$, where $Z_{\mathbf{q}_0}:= U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}\mathbf{q}_0U^{\vee}_-$, is isomorphic to the pull-back of the graph of a 1-form on $Z_{\mathbf{q}_0}$ via the canonical projection
\[ T^*G^{\vee}\times_{G^{\vee}}Z_{\mathbf{q}_0}\rightarrow T^*Z_{\mathbf{q}_0}.\]
In particular, the fibers of $Y$ over $\spec\mathbb{C}[q_1^{\pm 1},\ldots,q_k^{\pm 1}]$ are regular.
\end{lemma}
\begin{proof}
By definition,
\begin{equation}\label{6deq6}
\mu^{-1}(\chi\times\chi)\times_{G^{\vee}} Z_{\mathbf{q}_0} \simeq \{ (g,\zeta)\in Z_{\mathbf{q}_0}\times (\dd{\mathfrak{g}})^*|~\forall \eta\in \mathfrak{n}^{\vee}_-,~\langle \zeta, g^{-1}\cdot\eta\rangle = \langle \zeta,\eta\rangle = \kil{e,\eta}\}.
\end{equation}
Since for any $g\in Z_{\mathbf{q}_0}$, the tangent space $T_gZ_{\mathbf{q}_0}$ is the sum of $g^{-1}\cdot\mathfrak{n}^{\vee}_-$ and $\mathfrak{n}^{\vee}_-$, the result will follow if we can show that the equations in \eqref{6deq6} are compatible over $g^{-1}\cdot\mathfrak{n}^{\vee}_-\cap\mathfrak{n}^{\vee}_-$, i.e. for any $\eta\in\mathfrak{n}^{\vee}_-$ such that $g^{-1}\cdot\eta\in\mathfrak{n}^{\vee}_-$, we have $\kil{e,\eta}=\kil{e,g^{-1}\cdot\eta}$. This is straightforward and left to the reader. Notice that it is crucial to assume $\mathbf{q}_0\in\spec\mathbb{C}[q_1^{\pm 1},\ldots,q_k^{\pm 1}]$, otherwise our assertion will be false and $\mu^{-1}(\chi\times\chi)\times_{G^{\vee}} Z_{\mathbf{q}_0}$ will be empty.
\end{proof}
\begin{myproof2}{Theorem \ref{1bthmb}}
We have the following commutative diagram
\begin{equation}\label{6ddiag1}
\begin{tikzpicture}
\tikzmath{\x1 = 5; \x2 = 5; \x3=2;}
\node (A) at (0,0) {$\mathcal{O}(X_{\mathbf{q}_0})$} ;
\node (B) at (\x1,0) {$\mathcal{O}(X)$} ;
\node (C) at (\x1+\x2,0) {$\mathcal{O}(B_{e^T}^{\vee})$} ;
\node (D) at (0,-\x3) {$ QH_T^{\bl}(G/P)|_{\mathbf{q}=\mathbf{q}_0}$} ;
\node (E) at (\x1,-\x3) {$ QH_T^{\bl}(G/P)[\ecc]$} ;
\node (F) at (\x1+\x2,-\x3) {$ H^T_{-\bl}(\ag) $} ;

\path[->, font=\small] (B) edge node[above]{$\eta'$} (A);
\path[->, font=\small] (C) edge node[above]{$\eta_9\circ\eta_6$} (B);
\path[->, font=\small] (E) edge node[above]{$\eta''$} (D);
\path[->, font=\small] (F) edge node[above]{$\Phi_{PLS}$} (E);
\path[->, font=\small] (A) edge node[left]{$\simeq$} (D);
\path[->, font=\small] (B) edge node[left]{$\simeq$} (E);
\path[->, font=\small] (C) edge node[left]{$\simeq$} (F);
\path[->, font=\small] (A) edge node[right]{} (D);
\path[->, font=\small] (B) edge node[right]{$\Phi_{loc}$} (E);
\path[->, font=\small] (C) edge node[right]{$\Phi_{YZ}$} (F);
\end{tikzpicture}
\end{equation}
where the right square comes from diagram \eqref{1adiag} and the left square is induced by the right square. (Recall $\mathcal{O}(X)\simeq\mathcal{O}(B_{e^T}^{\vee})[g_P^{-1}]/\langle J_P\rangle$ and $\Phi_{loc}=\varphi_3$. See Lemma \ref{5bexist1} and Definition \ref{5bphi}.) By Lemma \ref{6dlemma0} and Lemma \ref{6dlemma1}, there is a $W$-action on $\mathcal{O}(X)$ such that $\eta_9\circ\eta_6$ is $W$-equivariant and the following diagram is commutative
\begin{equation}\label{6ddiag2}
\begin{tikzpicture}
\tikzmath{\x1 = 4; \x2 = 4; \x3=2;}
\node (A) at (0,0) {$\mathcal{O}(Y)$} ;
\node (B) at (\x1,0) {$\mathcal{O}(\toda(G^{\vee}))$} ;
\node (C) at (\x1+\x2,0) {$\mathcal{O}(\bfm(G^{\vee}))$} ;
\node (D) at (0,-\x3) {$ \mathcal{O}(X)^W$} ;
\node (E) at (\x1,-\x3) {} ;
\node (F) at (\x1+\x2,-\x3) {$\mathcal{O}(B_{e^T}^{\vee})^W $} ;

\path[->, font=\small] (B) edge node[above]{$\mathcal{O}(\inc)$} (A);
\path[->, font=\small] (B) edge node[above]{$\mathcal{O}(\Toda)$} (C);
\path[->, font=\small] (B) edge node[below]{$\simeq$} (C);
\path[->, font=\small] (F) edge node[above]{$(\eta_9\circ\eta_6)^W$} (D);
\path[->, font=\small] (A) edge node[left]{$\mathcal{O}(f_1)$} (D);
\path[->, font=\small] (C) edge node[left]{$\simeq$} (F);
\path[->, font=\small] (A) edge node[right]{$\simeq$} (D);
\path[->, font=\small] (C) edge node[right]{$\mathcal{O}(\Pi)$} (F);
\end{tikzpicture}.
\end{equation}

We show that all arrows in diagram \eqref{6ddiag1} are $W$-equivariant. We have seen that this is true for $\eta_9\circ\eta_6$. For $\Phi_{YZ}$ and $\Phi_{PLS}$, this follows from Lemma \ref{6dlemma0} and Corollary \ref{3bequiv} respectively. It follows that the same is true for $\Phi_{loc}$ because $\Phi_{PLS}$ is surjective after localization (Theorem \ref{3aPLS}). As for the arrows in the left square, we actually have to show that there are $W$-actions on the rings in the leftmost column before we can talk about whether the arrows are $W$-equivariant. What we need to do is to show that the defining ideals of these rings are $W$-invariant. This is obvious for the bottom ring. The case for the other ring follows from the fact that $\Phi_{loc}$ is a $W$-equivariant isomorphism of $\mathbb{C}[q_1^{\pm 1},\ldots, q_k^{\pm 1}]$-algebras.

By chasing diagrams \eqref{6ddiag1} and \eqref{6ddiag2}, we have
\begin{equation}\label{6deq5}
\ker\left( \left(\eta''\circ\Phi_{PLS}\circ\Phi_{YZ}\right)^W\circ\mathcal{O}(\Pi)\circ\mathcal{O}(\Toda)\right) = \ker\left( (\eta')^W\circ\mathcal{O}(f_1)\circ\mathcal{O}(\inc)\right).
\end{equation}
Observe that $(\eta')^W\circ\mathcal{O}(f_1)\circ\mathcal{O}(\inc)$, or equivalently $(\eta''\circ\Phi_{PLS})^W$, is surjective, since
\[ \im(\Phi_{PLS})[q_1,\ldots,q_k]=QH_T^{\bl}(G/P)[\ecc]\]
by Theorem \ref{3aPLS} and $\eta''$ sends each quantum parameter $q_i$ to a non-zero complex number. Hence, the image of $(\eta')^W\circ\mathcal{O}(f_1)\circ\mathcal{O}(\inc)$ is equal to $\mathcal{O}(X_{\mathbf{q}_0})^W$ which is isomorphic to $\mathcal{O}(Y_{\mathbf{q}_0})$ by Lemma \ref{6dlemma1}, Lemma \ref{6dlemma2} and the standard averaging trick. Since the $W$-invariant subrings of the rings in the bottom row of diagram \eqref{6ddiag1} are their $G$-equivariant analogues, the defining ideal of the closed subscheme $\Toda\left(\mathbb{L}_K^{\mathbf{q}_0}(G/P)\right) $ of $\toda(G^{\vee})$ is equal to the radical of the LHS of \eqref{6deq5}. Therefore, we have
\[\Toda\left(\mathbb{L}_K^{\mathbf{q}_0}(G/P)\right)\simeq (Y_{\mathbf{q}_0})_{red}. \]
But we know $(Y_{\mathbf{q}_0})_{red}\simeq Y_{\mathbf{q}_0}$ by Lemma \ref{6dlemma3}. The proof of Theorem \ref{1bthmb} is complete.
\end{myproof2}
\subsection{Proof of Theorem \ref{1bthm}} \label{6e}
Let us first recall the definition of the Rietsch mirror. For more details, see \cite[Section 6]{LT}, \cite[Section 4]{Rietsch} or \cite[Section 6.6]{T1}.

Put $Z(\dd{L}):=\spec\mathbb{C}[q_1^{\pm 1},\ldots, q_k^{\pm 1}]\subseteq T^{\vee}$. Define $\psi:U^{\vee}_-\rightarrow\mathbb{C}$ to be the composite
\begin{equation}\nonumber\label{6eeq1}
U^{\vee}_-\rightarrow U^{\vee}_-/(U^{\vee}_-,U^{\vee}_-)\simeq\prod_{i=1}^rU^{\vee}_{-\alpha^{\vee}_i}\xrightarrow{\sum_{i=1}^r|\alpha^{\vee}_i|^2\chi_i}\mathbb{G}_a
\end{equation}
where each $\chi_i:U^{\vee}_{-\alpha^{\vee}_i}\xrightarrow{\sim}\mathbb{G}_a$ is the isomorphism such that $\lie(\chi_i)=\kil{e_{\alpha^{\vee}_i},-}|_{\dd{\mathfrak{g}}_{-\alpha^{\vee}_i}}\in (\dd{\mathfrak{g}}_{-\alpha^{\vee}_i})^*$. (Here $\kil{-,-}$ is the bilinear form on $\dd{\mathfrak{g}}$ fixed in Section \ref{6d}.)
\begin{definition}\label{6edef1}
Define
\begin{enumerate}
\item (mirror space) $~ X^{\vee}:=B^{\vee}\cap U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}Z(\dd{L})U^{\vee}_-$;

\item[]

\item (superpotential) $~W:X^{\vee}\rightarrow\mathbb{C}$ by
\[W(u_1(\dot{w}_P\dot{w}_0)^{-1}tu_2):=\psi(u_1)+\psi(u_2);\]

\item[]

\item (projection) $~\mathbf{q}:X^{\vee}\rightarrow Z(\dd{L})$ by
\[\mathbf{q}(u_1(\dot{w}_P\dot{w}_0)^{-1}tu_2):=t.\]
\end{enumerate}
\end{definition}

\begin{lemma}\label{6elemma1}$~$
\begin{enumerate}
\item Given $x=u_1(\dot{w}_P\dot{w}_0)^{-1}tu_2\in U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}Z(\dd{L})U^{\vee}_-$. The point $t\in Z(\dd{L})$ is unique, and although $u_1,u_2\inU^{\vee}_-$ are not unique, the expression $\psi(u_1)+\psi(u_2)$ depends only on $x$ but not $u_1,u_2$. In particular, $W$ and $\mathbf{q}$ are well-defined.

\item For any $\mathbf{q}_0\in Z(\dd{L})$, the intersection $X^{\vee}_{\mathbf{q}_0}:=B^{\vee}\cap U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}\mathbf{q}_0U^{\vee}_-$ is transverse and is mapped isomorphically onto the double Bruhat call
\[\mathcal{R}_{e,w_0w_P}:=\UdB_-^{\vee}/B_-^{\vee}\cap U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}B_-^{\vee}/B_-^{\vee}\]
via the morphism $B^{\vee}\rightarrow G^{\vee}/B^{\vee}_-:b\mapsto bB_-^{\vee}$. In particular, we have
\[ X^{\vee}\simeq Z(\dd{L})\times \mathcal{R}_{e,w_0w_P}\]
as $Z(\dd{L})$-schemes.
\end{enumerate}
\end{lemma}
\begin{proof}
Let us prove the assertion about $\psi(u_1)+\psi(u_2)$ only. The others are straightforward and left to the reader. Fix $\mathbf{q}_0\in Z(\dd{L})$. One can show that for $u_1$, $u_2$, $u_1'$, $u_2'\in U^{\vee}_-$,
\[ u_1(\dot{w}_P\dot{w}_0)^{-1}\mathbf{q}_0 u_2= u'_1(\dot{w}_P\dot{w}_0)^{-1}\mathbf{q}_0 u'_2\]
if and only if there exists $u\in \exp\left(\bigoplus_{\alpha\in w_0 R_P^+}\dd{\mathfrak{g}}_{\alpha^{\vee}}\right)$ such that
\[ u'_1 = u_1u\quad\text{ and }\quad u'_2=(\dot{w}_P\dot{w}_0)u^{-1}(\dot{w}_P\dot{w}_0)^{-1} u_2.\]
Since $\psi$ is a group homomorphism, it suffices to show
\[ \psi(u)+\psi((\dot{w}_P\dot{w}_0)u^{-1}(\dot{w}_P\dot{w}_0)^{-1} )=0\]
for any $ u\in \exp\left(\bigoplus_{\alpha\in w_0 R_P^+}\dd{\mathfrak{g}}_{\alpha^{\vee}}\right)$. There exists unique $\xi\in \bigoplus_{i=k+1}^r \dd{\mathfrak{g}}_{w_0\alpha^{\vee}_i}$ such that
\[ u\equiv\exp(\xi)\bmod(U^{\vee}_-,U^{\vee}_-).\]
Then
\[ (\dot{w}_P\dot{w}_0)u^{-1}(\dot{w}_P\dot{w}_0)^{-1}\equiv \exp(-\dot{w}_P\dot{w}_0\cdot\xi)\bmod(U^{\vee}_-,U^{\vee}_-). \]
It follows that
\begin{align*}
\psi(u) &= \sum_{i=k+1}^r|\alpha^{\vee}_i|^2 \kil{e_{-w_0\alpha^{\vee}_i},\xi} = \sum_{i=k+1}^r|\alpha^{\vee}_i|^2\kil{ (\dot{w}_P\dot{w}_0)^{-1}\cdot e_{-w_P\alpha^{\vee}_i},\xi}\\
&=  \sum_{i=k+1}^r|\alpha^{\vee}_i|^2 \kil{ e_{-w_P\alpha^{\vee}_i},\dot{w}_P\dot{w}_0\cdot \xi} = -\psi((\dot{w}_P\dot{w}_0)u^{-1}(\dot{w}_P\dot{w}_0)^{-1}).
\end{align*}
(The second equality follows from Proposition \ref{appprop1}.)
\end{proof}

The triple $(X^{\vee},W,\mathbf{q})$ is sufficient to state Theorem \ref{1bthm} for the non-equivariant version. In order to generalize to the equivariant settings, we also need a family of 1-forms on $X^{\vee}$ parametrized by $\specH_T^{\bl}(\pt)\simeq (\mathfrak{t}^{\vee})^*$. Consider the composite
\[ p:X^{\vee}\hookrightarrow B^{\vee}\simeq U^{\vee}\timesT^{\vee}\rightarrow T^{\vee}\]
where all arrows are the canonical morphisms. Let $h_0\in(\mathfrak{t}^{\vee})^*$. Define a 1-form $\Theta_{h_0}$ on $T^{\vee}$ by
\[ (\Theta_{h_0})_t(\xi) :=\kil{ f(h_0),\xi}\]
for any $t\in T^{\vee}$ and $\xi\in\dd{\mathfrak{t}}$. (Recall $f:\alpha^{\vee}_i\mapsto -|\alpha^{\vee}_i|^2\alpha_i$ for each $1\leqslant i\leqslant r$.) Define
\[ \theta_{h_0}:= p^*\Theta_{h_0}\in \Omega^1(X^{\vee}).\]

The following theorem is the precise formulation of Theorem \ref{1bthm}.
\begin{theorem}\label{6ethm}
For any $(h_0,\mathbf{q}_0)\in (\dd{\mathfrak{t}})^*\times Z(\dd{L})$, there is an isomorphism of $\mathbb{C}$-algebras
\[ \mathcal{O}(\{dW_{\mathbf{q}_0} = \iota_{\mathbf{q}_0}^*\theta_{h_0}\})\simeq QH_T^{\bl}(G/P)|_{(h_0,\mathbf{q}_0)}\]
where $W_{\mathbf{q}_0}:=W|_{X^{\vee}_{\mathbf{q}_0}}$, $\iota_{\mathbf{q}_0}:X^{\vee}_{\mathbf{q}_0}\hookrightarrow X^{\vee}$ is the inclusion and $\{dW_{\mathbf{q}_0} = \iota_{\mathbf{q}_0}^*\theta_{h_0}\}$ is the scheme-theoretic zero locus of the section $dW_{\mathbf{q}_0} - \iota_{\mathbf{q}_0}^*\theta_{h_0}$ of $\mathcal{T}_{X^{\vee}_{\mathbf{q}_0}}^*$.
\end{theorem}
\begin{proof}
Consider the closed immersion
\begin{equation}\label{6eeq2}
\begin{array}{ccc}
B_{e^T}^{\vee}&\hookrightarrow & T^*G^{\vee}\simeq G^{\vee}\times (\dd{\mathfrak{g}})^* \\ [.5em]
(b,h) &\mapsto & (b^{-1},\kil{e^T(h),-} )
\end{array}.
\end{equation}
Put $\mathcal{Z}:=B_{e^T}^{\vee}\tGBU^{\vee}_{P^{\vee}_-}\wpdB_-^{\vee}/B_-^{\vee}$. By Theorem \ref{1athmb}, we have
\[ \mathcal{O}(\mathcal{Z})\simeqQH_T^{\bl}(G/P)[\ecc]\]
as $H_T^{\bl}(\pt)[q_1^{\pm 1},\ldots, q_k^{\pm 1}]$-algebras. In particular, we have
\[ \mathcal{O}(\mathcal{Z}_{(h_0,\mathbf{q}_0)})\simeq QH_T^{\bl}(G/P)|_{(h_0,\mathbf{q}_0)}\]
as $\mathbb{C}$-algebras. Our strategy is to write down the defining equations of $\mathcal{Z}_{(h_0,\mathbf{q}_0)}$ as a subscheme of $T^*G^{\vee}$ and show that they are exactly those which define $\{dW_{\mathbf{q}_0} = \iota_{\mathbf{q}_0}^*\theta_{h_0}\}$.

The scheme $\mathcal{Z}_{(h_0,\mathbf{q}_0)}$ satisfies the centralizing condition $b\cdot e^T(h)=e^T(h)$ and the condition $(h,\mathbf{q})=(h_0,\mathbf{q}_0)$. By transferring these conditions through \eqref{6eeq2}, we see that $\mathcal{Z}_{(h_0,\mathbf{q}_0)}$ is isomorphic to a subscheme of the scheme $\mathcal{Z}'_{(h_0,\mathbf{q}_0)}\subset T^*G^{\vee}$ defined by
\begin{equation}\label{6eeq3}
\mathcal{Z}'_{(h_0,\mathbf{q}_0)}:=\left\{ (g,\zeta)\in T^*G^{\vee} \left|~\begin{array}{l}
(1) ~\langle\zeta,g^{-1}\cdot -  \rangle|_{\mathfrak{n}^{\vee}_-} =\langle\zeta,- \rangle|_{\mathfrak{n}^{\vee}_-}  = \kil{ e,- }|_{\mathfrak{n}^{\vee}_-} \\[.5em]
(2) ~\langle \zeta,-\rangle|_{\dd{\mathfrak{b}}}= \kil{ f(h_0),- }|_{\dd{\mathfrak{b}}} \\ [.5em]
(3) ~g\in X^{\vee}_{\mathbf{q}_0}= B^{\vee}\capU^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}\mathbf{q}_0U^{\vee}_-
\end{array} \right.\right\}.
\end{equation}
(We have used Lemma \ref{6dlemma2} to handle the condition $\mathbf{q}=\mathbf{q}_0$.) We show that $\mathcal{Z}_{(h_0,\mathbf{q}_0)}$ is in fact isomorphic to $\mathcal{Z}'_{(h_0,\mathbf{q}_0)}$, i.e. conditions (1) to (3) in \eqref{6eeq3} suffice to determine $\mathcal{Z}_{(h_0,\mathbf{q}_0)}$. The non-trivial step is to obtain the centralizing condition. Let $(g,\zeta)\in \mathcal{Z}'_{(h_0,\mathbf{q}_0)}$. Write $\zeta=\kil{ \xi,-}$ where $\xi\in \dd{\mathfrak{g}}$. Observe that (1) and (2) imply $\xi, g\cdot\xi\in e+\dd{\mathfrak{b}}_-$ and $\xi\in f(h_0)+\mathfrak{n}^{\vee}$ respectively. Thus, we have $\xi=e+f(h_0)=e^T(h_0)$. By Kostant's slice theorem, there exists $u\in U^{\vee}_-$ such that $u\cdot e^T(h_0)= g\cdot e^T(h_0)$. It follows that $u^{-1}g$ centralizes $e^T(h_0)$. But then $u^{-1}g\in B^{\vee}$, and hence $u=e$ because $g\inB^{\vee}$ by (3) and $B^{\vee}\capU^{\vee}_-=\{e\}$. Therefore, $g$ centralizes $\xi=e^T(h_0)$.

It remains to show that conditions (1) to (3) in \eqref{6eeq3} define $\{dW_{\mathbf{q}_0} = \iota_{\mathbf{q}_0}^*\theta_{h_0}\}$. Put $Z_{\mathbf{q}_0}:=U^{\vee}_-(\dot{w}_P\dot{w}_0)^{-1}\mathbf{q}_0U^{\vee}_-$. By Lemma \ref{6dlemma3}, condition (1) defines a closed subscheme of $T^*G^{\vee}\times_{G^{\vee}}Z_{\mathbf{q}_0}$ which is the pull-back of the graph of a 1-form $\theta_1\in\Omega^1(Z_{\mathbf{q}_0})$ via the canonical projection $T^*G^{\vee}\times_{G^{\vee}}Z_{\mathbf{q}_0}\rightarrow T^*Z_{\mathbf{q}_0}$. It is easy to see that condition (2) also defines a closed subscheme of $T^*G^{\vee}\times_{G^{\vee}}B^{\vee}$ corresponding to a 1-form $\theta_2\in\Omega^1(B^{\vee})$ in a similar manner. In fact, $\theta_2$ is the constant 1-form $\kil{ f(h_0),-}$, and hence its pull-back to $X^{\vee}_{\mathbf{q}_0}$ coincides with $\iota_{\mathbf{q}_0}^*\theta_{\mathbf{q}_0}$. Let us determine $\theta_1$. Denote by $\widetilde{\theta}_1\in\Omega^1(U^{\vee}_-\timesU^{\vee}_-)$ the pull-back of $\theta_1$ via the morphism
\[
\begin{array}{ccccc}
\phi &:& U^{\vee}_-\timesU^{\vee}_- &\rightarrow&Z_{\mathbf{q}_0}\\ [.5em]
 && (u_1,u_2)&\mapsto & u_1(\dot{w}_P\dot{w}_0)^{-1}\mathbf{q}_0 u_2
\end{array}.
\]
Then
\[ (\widetilde{\theta}_1)_{(u_1,u_2)}(\eta_1,\eta_2) = \kil{ e, \eta_1} + \kil{ e,\eta_2} \]
for any $u_1,u_2\inU^{\vee}_-$ and $\eta_1,\eta_2\in\mathfrak{n}^{\vee}_-$. It is easy to see that the regular function $(u_1,u_2)\mapsto \psi(u_1)+\psi(u_2)$ is a primitive of $\widetilde{\theta}_1$ (recall $\psi:U^{\vee}_-\rightarrow\mathbb{C}$ is defined at the beginning of this subsection), and that it descends to a regular function on $Z_{\mathbf{q}_0}$ whose restriction to $X^{\vee}_{\mathbf{q}_0}$ is equal to $W_{\mathbf{q}_0}$.

Finally, we apply Lemma \ref{6elemma2} below to complete the proof.
\end{proof}

\begin{lemma}\label{6elemma2}
Let $Z$ be a smooth affine variety over $\mathbb{C}$ and $Z_1$, $Z_2$ two smooth affine subvarieties of $Z$. Suppose the intersection $Z_1\cap Z_2$ is transverse. Let $\theta_1\in \Omega^1(Z_1)$ and $\theta_2\in\Omega^1(Z_2)$ be given, and let $\widetilde{\Gamma}_{\theta_i}\subset T^*Z\times_Z Z_i$ be the pull-back of the graph $\Gamma_{\theta_i}$ of $\theta_i$ via the canonical projection $T^*Z\times_Z Z_i\rightarrow T^*Z_i$. Then
\[ \widetilde{\Gamma}_{\theta_1}\times_{T^* Z}\widetilde{\Gamma}_{\theta_2}\simeq \{\theta_1|_{Z_1\cap Z_2} =\theta_2|_{Z_1\cap Z_2} \}. \]
\end{lemma}
\begin{proof}
This is straightforward and left to the reader.
\end{proof}