2.1 Quantum groups

Fix a finite index set $\mathrm {I}$ , and a Cartan matrix $A=(a_{ij})_{i,j\in \mathrm {I}}$ . Let $D=diag(\epsilon _i)_{i\in \mathrm {I}}$ be the diagonal matrix such that $DA$ is symmetric positive definite, with $\epsilon _i\in \mathbb {Z}_{>0}$ and $\{\epsilon _i\mid i\in \mathrm {I}\}$ relatively prime.

Let $\mathfrak {g}$ be the complex semi-simple Lie algebra associated with the Cartan matrix A, and $\mathfrak {h}$ be a Cartan subalgebra of $\mathfrak {g}$ . Take Borel subalgebras $\mathfrak {b}^+$ and $\mathfrak {b}^-$ of $\mathfrak {g}$ , such that $\mathfrak {b}^+\cap \mathfrak {b}^-=\mathfrak {h}$ . Denote by $\Phi \subset \mathfrak {h}^*$ the root system of $\mathfrak {g}$ relative to $\mathfrak {h}$ . Let $\mathcal {R}^+\subset \Phi $ be the set of positive roots associated with $\mathfrak {b}^+$ , and $\Delta =\{\alpha _i\mid i\in \mathrm {I}\}$ be the subset of simple roots. Set $\mathcal {R}^-=-\mathcal {R}^+$ . Let

$$ \begin{align*}\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_\alpha\end{align*} $$

be the root space decomposition. Set $\mathfrak {u}^+=\oplus _{\alpha \in \mathcal {R}^+}\mathfrak {g}_\alpha $ and $\mathfrak {u}^-=\oplus _{\alpha \in \mathcal {R}^-}\mathfrak {g}_{\alpha }$ . Let Q be the root lattice, and W be the Weyl group of $\mathfrak {g}$ with generators $\{s_i\mid i\in \mathrm {I}\}$ . The group W acts on Q in a natural way. For an element w in W, let $|w|$ be the length of w. Let $w_0$ be the longest element in W, and write $n=|w_0|$ .

Let q be an indeterminate, and $q^{1/2}\in \overline {\mathbb {Q}(q)}$ be a fixed square root of q. Let $q_i=q^{\epsilon _i}$ , for $i\in \mathrm {I}$ . Let $\mathrm {U}$ be the Drinfeld–Jimbo quantum group associated with $\mathfrak {g}$ over the base field $\mathbb {C}(q^{1/2})$ (cf. [Reference Jantzen18, 4.3]). Let $\varepsilon :\mathrm {U}\rightarrow \mathbb {C}(q^{1/2})$ be the counit. For $i\in \mathrm {I}$ , let $\mathbf {E}_i$ , $\mathbf {F}_i$ and $\mathbf {K}_i$ be the generators as in loc. cit. For our purpose, we consider the rescaled generators

$$ \begin{align*}E_i= q_i^{1/2}(q_i^{-1}-q_i)\mathbf{E}_i,\quad F_i= q_i^{-1/2}(q_i-q_i^{-1})\mathbf{F_i},\quad K_i= \mathbf{K}_i,\quad \text{for }i\in\mathrm{I}.\end{align*} $$

Let $\mathrm {U}^-$ (resp., $\mathrm {U}^+$ ) be the unital $\mathbb {C}(q^{1/2})$ -subalgebra of $\mathrm {U}$ generated by $F_i$ (resp., $E_i$ ), for various $i\in \mathrm {I}$ . Let $\mathrm {U}^0$ be the unital $\mathbb {C}(q^{1/2})$ -subalgebra of $\mathrm {U}$ generated by $K_i^{\pm 1}$ , for $i\in \mathrm {I}$ . For any $\mu =\sum _{i\in \mathrm {I}}a_i\alpha _i\in Q$ , we write $K_\mu =\prod _{i\in \mathrm {I}}K_i^{a_i}$ .

For $i\in \mathrm {I}$ , let $\mathrm {T}_i$ be the $\mathbb {C}(q^{1/2})$ -algebra automorphism of $\mathrm {U}$ , denoted by $T_{i,-1}''$ in [Reference Lusztig21, 37.1.3]. It is known that automorphisms $\{\mathrm {T}_i\mid i\in \mathrm {I}\}$ satisfy the braid group relations. For $w\in W$ , we write $\mathrm {T}_w=\mathrm {T}_{i_1}\mathrm {T}_{i_2}\cdots \mathrm {T}_{i_r}$ , where $w=s_{i_1}s_{i_2}\cdots s_{i_r}$ and $r=|w|$ .

Let $\mathbf {i}=(i_1,i_2,\cdots ,i_n)\in \mathrm {I}^n$ be a reduced expression of $w_0$ . For $\mathbf {a}=(a_1,\cdots ,a_n)\in \mathbb {N}^n$ , set

$$ \begin{align*} &F_{\mathbf{i}}(\mathbf{a})=E_{i_1}^{a_1}\mathrm{T}_{i_1}(E_{i_2}^{a_2})\cdots \mathrm{T}_{i_1}\cdots\mathrm{T}_{i_{n-1}}(E_{i_n}^{a_n}),\\ &E_{\mathbf{i}}(\mathbf{a})=F_{i_1}^{a_1}\mathrm{T}_{i_1}(F_{i_2}^{a_2})\cdots \mathrm{T}_{i_1}\cdots\mathrm{T}_{i_{n-1}}(F_{i_n}^{a_n}). \end{align*} $$

For $1\leq k\leq n$ , we write $F_{\mathbf {i},k}=F_{\mathbf {i}}(\mathbf {e}_k)$ and $E_{\mathbf {i},k}=E_{\mathbf {i}}(\mathbf {e}_k)$ , where $\mathbf {e}_k=(\delta _{kj})_{j=1}^n\in \mathbb {N}^n$ .

Set $\mathcal {A}=\mathbb {C}[q^{1/2},q^{-1/2}]$ to be the subring of $\mathbb {C}(q^{1/2})$ . Let $_{\mathcal {A}}\mathrm {U}^-$ (resp., $_{\mathcal {A}}\mathrm {U}^+$ ) be the $\mathcal {A}$ -submodule of $\mathrm {U}^-$ (resp., $\mathrm {U}^+$ ), spanned by $F_{\mathbf {i}}(\mathbf {a})$ (resp., $E_{\mathbf {i}}(\mathbf {a})$ ), for various $\mathbf {a}\in \mathbb {N}^n$ . It is known that $_{\mathcal {A}}\mathrm {U}^-$ and $_{\mathcal {A}}\mathrm {U}^+$ are independent of the choice of $\mathbf {i}$ , and are closed under multiplications. Let $_{\mathcal {A}}\mathrm {U}^0$ be the unital $\mathcal {A}$ -subalgebra of $\mathrm {U}^0$ , generated by $K_i^{\pm 1}$ , for $i\in \mathrm {I}$ . Let $_{\mathcal {A}}\mathrm {U}$ be the $\mathcal {A}$ -subalgebra of $\mathrm {U}$ , generated by $_{\mathcal {A}}\mathrm {U}^-$ , $_{\mathcal {A}}\mathrm {U}^0$ and $_{\mathcal {A}}\mathrm {U}^+$ . It is different from the integral form defined by Lusztig [Reference Lusztig21, 3.1.13], and is the same as the one studied by Berenstein–Greenstein [Reference Berenstein and Greenstein4]. By [Reference Berenstein and Greenstein4, Corollary 3.13 & Theorem 3.11], the multiplication gives an isomorphism as $\mathcal {A}$ -modules

(2.1) $$ \begin{align} {}_{\mathcal{A}}\mathrm{U}\cong {}_{\mathcal{A}}\mathrm{U}^+\otimes{}{_{\mathcal{A}}\mathrm{U}^0}\otimes{} {_{\mathcal{A}}\mathrm{U}^-}. \end{align} $$

Here the tensor products are over $\mathcal {A}$ . It is easy to see that the $\mathcal {A}$ -subalgebra $_{\mathcal {A}}\mathrm {U}$ is invariant under the braid group action $\mathrm {T}_i$ , for $i\in \mathrm {I}$ .

2.2 Quantum symmetric pairs

Let $(\mathrm {I}=\mathrm {I}_{\circ }\sqcup \mathrm {I}_{\bullet },\tau )$ be a Satake diagram (cf. [Reference Bao and Wang8]). Recall that $\tau $ is a graph involution on $\mathrm {I}$ , which leaves $\mathrm {I}_\circ $ and $\mathrm {I}_{\bullet }$ invariant. We fix once for all a subset $\mathrm {I}^{\prime }_\circ \subset \mathrm {I}_\circ $ , consisting of exactly one element in each $\tau $ -orbits of $\mathrm {I}_\circ $ . Let $\Phi _{\bullet }\subset \Phi $ be the sub-root system with simple roots $\{\alpha _i\mid i\in \mathrm {I}_{\bullet }\}$ , and $W_{\bullet }=\langle s_i\mid i\in \mathrm {I}_{\bullet }\rangle $ be the parabolic subgroup. Let $w_{\bullet }$ be the longest element in $W_{\bullet }$ . Let $\mathcal {R}_{\bullet }^+=\mathcal {R}^+\cap \Phi _{\bullet }$ and $\mathcal {R}_{\bullet }^-=\mathcal {R}^-\cap \Phi _{\bullet }$ . Let $\theta =-w_{\bullet }\tau :Q\rightarrow Q$ be the group involution on Q, and $Q^\theta =\{\mu \in Q\mid \theta (\mu )=\mu \}$ .

We fix parameters $\varsigma _i\in \pm q^{\mathbb {Z}}$ , for $i\in \mathrm {I}_\circ $ , which satisfy the conditions in [Reference Bao and Wang8, Definition 3.5]. The associated $\imath $ quantum group $\mathrm {U}^\imath $ is the unital $\mathbb {C}(q^{1/2})$ -subalgebra of $\mathrm {U}$ , generated by the following elements,

$$ \begin{align*} &B_{i}=F_{i}-\varsigma_{i}\mathrm{T}_{w_{\bullet}}(E_{\tau i})K_{i}^{-1},\; k_{i}=K_{i}K_{\tau i}^{-1},\qquad\text{for } i\in \mathrm{I}_\circ;\\ & E_{i},\;F_{i},\;K_{i}^{\pm1}\qquad\text{ for } i\in \mathrm{I}_{\bullet}. \end{align*} $$

We write $B_i=F_i$ , for $i\in \mathrm {I}_{\bullet }$ . The pair $(\mathrm {U},\mathrm {U}^\imath )$ is called a quantum symmetric pair.

2.3 Poisson structures

For any complex affine algebraic variety V, we write $\mathcal {O}(V)$ to denote the $\mathbb {C}$ -algebra of regular functions on V.

Let G be the complex semi-simple adjoint group with the Lie algebra $\mathfrak {g}$ . Let $B^+$ , $B^-$ and H be the connected closed subgroups of G, whose Lie algebras are $\mathfrak {b}^+$ , $\mathfrak {b}^-$ and $\mathfrak {h}$ , respectively. Let $U^+$ (resp., $U^-$ ) be the unipotent radical of $B^+$ (resp., $B^-$ ). For $\alpha \in \Phi $ , let $U_\alpha $ be the root subgroup of G associated with $\alpha $ .

Let us briefly recall the standard Poisson structure associated with G, and the dual Poisson group $G^*$ . We refer to [Reference Chari and Pressley10, §1] for a detailed exposition. Let $\langle \,,\,\rangle $ be the Killing form of $\mathfrak {g}$ , and $\mathfrak {g}^*$ be the space of $\mathbb {C}$ -linear forms on $\mathfrak {g}$ . The Drinfeld double $\mathfrak {d}=\mathfrak {g}\oplus \mathfrak {g}$ admits a non-degenerate invariant bilinear form $\langle \langle \,,\,\rangle \rangle $ , given by

$$ \begin{align*} \langle\langle(X_1,Y_1),(X_2,Y_2)\rangle\rangle=\langle X_1,X_2\rangle-\langle Y_1,Y_2\rangle,\quad \forall\, X_1,X_2,Y_1,Y_2\in\mathfrak{g}. \end{align*} $$

Let us identify $\mathfrak {g}$ with the diagonal Lie subalgebra of $\mathfrak {d}$ , and identify $\mathfrak {g}^*$ with another Lie subalgebra,

(2.2) $$ \begin{align} \mathfrak{g}^*\cong\{(X,Y)\in\mathfrak{b}^+\times\mathfrak{b}^-\mid \text{the }\mathfrak{h}\text{-component of } X+Y\ \text{is}\ 0\}, \end{align} $$

via the bilinear form $\langle \langle \,,\,\rangle \rangle $ . Under these identifications the triple $(\mathfrak {d},\mathfrak {g},\mathfrak {g}^*)$ forms a Manin triple, which determines a Lie bialgebra structure on $\mathfrak {g}$ , and determines the standard Poisson structure on G.

By a dual construction, one gets a Lie bialgebra structure on $\mathfrak {g}^*$ , which determines a Poisson structure on the dual Poisson group $G^*$ , where

$$ \begin{align*}G^*=\{(u_+t,t^{-1}u_-)\mid u_\pm\in U^\pm,t\in H\}\end{align*} $$

is the connected closed subgroup of $B^+\times B^-$ with the Lie algebra $\mathfrak {g}^*$ .

We have an isomorphism as varieties

(2.3) $$ \begin{align} U^+\times H\times U^-\overset{\sim}{\longrightarrow} G^*,\qquad (u_+,t,u_-)\mapsto (u_+t,t^{-1}u_-), \end{align} $$

which determines an isomorphism as $\mathbb {C}$ -algebras

(2.4) $$ \begin{align} \mathcal{O}(G^*)\cong\mathcal{O}(U^+)\otimes\mathcal{O}(H)\otimes\mathcal{O}(U^-). \end{align} $$

2.4 Specialization of quantum algebras

Let $R_q$ be a non-commutative unital $\mathcal {A}$ -algebra, which satisfies the condition

(2.5) $$ \begin{align} [R_q,R_q]\subseteq (q^{1/2}-1)R_q. \end{align} $$

Let $R=\mathbb {C}\otimes _{\mathcal {A}}R_q$ where the tensor product is given by $q^{1/2}\mapsto 1$ . The condition (2.5) is equivalent to saying that the $\mathbb {C}$ -algebra R is commutative. Denote by $f\mapsto \overline {f}$ the canonical map $R_q\rightarrow R$ . Then R carries a Poisson bracket $\{\,,\,\}:R\times R\rightarrow R$ , defined by

(2.6) $$ \begin{align} \{\overline{f},\overline{g}\}=\overline{\frac{1}{2(q^{1/2}-1)}(fg-gf)},\quad \forall f,g\in R_q. \end{align} $$

2.5 Specialization of $_{\mathcal {A}}\mathrm {U}$

Let us fix a pinning $\{x_i,y_i\mid i\in \mathrm {I}\}$ of the group G, where $x_i:\mathbb {C}\rightarrow U_{\alpha _i}$ and $y_i:\mathbb {C}\rightarrow U_{-\alpha _i}$ are one-parameter subgroups.

For $i\in \mathrm {I}$ , set

$$ \begin{align*} \dot{s_i}=x_i(1)y_i(-1)x_i(1)\in G. \end{align*} $$

It is well-known that elements $\{\dot {s}_i\mid i\in \mathrm {I}\}$ satisfy the braid group relations. For $w\in W$ , denote by $\dot {w}=\dot {s}_{i_1}\cdot \dot {s}_{i_2}\cdots \dot {s}_{i_k}$ , for one (and hence for all) reduced expression $w=s_{i_1}s_{i_2}\cdots s_{i_k}$ .

Take a reduced expression $\mathbf {i}=(i_1,\cdots , i_n)$ of the longest $w_0$ of W. For $1\leq k\leq n$ , set ${\beta _{\mathbf {i},k}=s_{i_1}\cdots s_{i_{k-1}}(\alpha _{i_k})\in \mathcal {R}^+}$ . Set $x_{\mathbf {i},k}=\text {Ad}_{\dot {s}_{i_1}\cdots \dot {s}_{i_{k-1}}}\circ x_{i_k}:\mathbb {C}\rightarrow U_{\beta _{\mathbf {i},k}}$ , and $y_{\mathbf {i},k}=\text {Ad}_{\dot {s}_{i_1}\cdots \dot {s}_{i_{k-1}}}\circ y_{i_k}:\mathbb {C}\rightarrow U_{-\beta _{\mathbf {i},k}}$ . We have $x_{\mathbf {i},k}=x_j$ and $y_{\mathbf {i},k}=y_j$ , if $\beta _{\mathbf {i},k}=\alpha _j$ , for some $j\in \mathrm {I}$ .

By [Reference Springer25, Lemma 8.3.5], elements $a\in U^+$ , $b\in U^-$ can be factored as

(2.7) $$ \begin{align} a=x_{\mathbf{i},n}(a_n)x_{\mathbf{i},n-1}(a_{n-1})\cdots x_{\mathbf{i},1}(a_1)\quad \text{and}\quad b=y_{\mathbf{i},1}(b_1)y_{\mathbf{i},{2}}(b_{2})\cdots y_{\mathbf{i},n}(b_n), \end{align} $$

for unique tuples $(a_1,\cdots ,a_n)\in \mathbb {C}^n$ and $(b_1,\cdots ,b_n)\in \mathbb {C}^n$ . We define $\chi _{\mathbf {i},k}^+\in \mathcal {O}(U^+)$ by $\chi _{\mathbf {i},k}^+(a)=a_k$ , and define $\chi _{\mathbf {i},k}^-\in \mathcal {O}(U^-)$ by $\chi _{\mathbf {i},k}(b)=b_k$ . If $\beta _{\mathbf {i},k}=\alpha _j$ is a simple root, elements $\chi ^+_{\mathrm {i},k}$ and $\chi _{\mathrm {i},k}^-$ are independent of the reduced expression, in which case we denote by $\chi ^+_j=\chi ^+_{\mathrm {i},k}$ and $\chi _j^-=\chi _{\mathrm {i},k}^-$ .

Recall the $\mathcal {A}$ -algebras $_{\mathcal {A}}\mathrm {U}$ , $_{\mathcal {A}}\mathrm {U}^+$ , $_{\mathcal {A}}\mathrm {U}^-$ and $_{\mathcal {A}}\mathrm {U}^0$ in §2.1. Let $_{\mathbb {C}}\mathrm {U}$ , $_{\mathbb {C}}\mathrm {U}^+$ , $_{\mathbb {C}}\mathrm {U}^-$ and $_{\mathbb {C}}\mathrm {U}^0$ be the $\mathbb {C}$ -algebras obtained by the base change $q^{1/2}\mapsto 1$ of the corresponding $\mathcal {A}$ -algebras, which are known to be commutative [Reference Berenstein and Greenstein4, Corollary 3.15]. Moreover the isomorphism (2.1) induces the isomorphism $_{\mathbb {C}}\mathrm {U}\cong {}_{\mathbb {C}}\mathrm {U}^+\otimes {}_{\mathbb {C}}\mathrm {U}^0\otimes {}_{\mathbb {C}}\mathrm {U}^-$ as $\mathbb {C}$ -algebras. Here tensor products are over $\mathbb {C}$ .

Take a reduced expression $\mathbf {i}$ of $w_0$ . One has $\mathbb {C}$ -algebra isomorphisms

(2.8) $$ \begin{align} \varphi^+:{}_{\mathbb{C}}\mathrm{U}^+\overset{\sim}{\longrightarrow}\mathcal{O}(U^+)\qquad\text{and}\qquad\varphi^-:{}_{\mathbb{C}}\mathrm{U}^-\overset{\sim}{\longrightarrow}\mathcal{O}(U^-), \end{align} $$

given by $\varphi ^+(\overline {E_{\mathbf {i},k}})=\chi _{\mathbf {i},k}^+$ and $\varphi ^-(\overline {F_{\mathbf {i},k}})=\chi _{\mathbf {i},k}^-$ , for $1\leq k\leq n$ . In particular $\varphi ^+(\overline {E_i})=\chi _i^+$ and $\varphi ^-(\overline {F_i})=\chi _i^-$ , for $i\in \mathrm {I}$ . The maps $\varphi ^\pm $ are independent of the choice of $\mathbf {i}$ .

Since G has trivial center, one has the canonical $\mathbb {C}$ -algebra isomorphism $\mathcal {O}(H)\cong \mathbb {C}[Q]$ , which induces the $\mathbb {C}$ -algebra isomorphism $\varphi ^0:{}_{\mathbb {C}}\mathrm {U}^0\overset {\sim }{\longrightarrow }\mathcal {O}(H)$ , given by $\varphi ^0(\overline {K_\mu })=\mu $ , for $\mu \in Q$ .

The following theorem asserts that $_{\mathcal {A}}\mathrm {U}$ specializes to the coordinate algebra $\mathcal {O}(G^*)$ . The proof essentially follows from the work of De Concini–Kac–Procesi.

Theorem 2.1 [Reference De Concini, Kac and Procesi15, Theorem 7.6].

The map

(2.9) $$ \begin{align} \varphi=\varphi^+\otimes\varphi^0\otimes\varphi^-:{_{\mathbb{C}}\mathrm{U}}\overset{\sim}{\longrightarrow}\mathcal{O}(G^*) \end{align} $$

is an isomorphism as Poisson algebras.

Since $_{\mathcal {A}}\mathrm {U}$ is invariant under the action $\mathrm {T}_i$ , for $i\in \mathrm {I}$ , each $\mathrm {T}_i$ descends to a Poisson algebra automorphism on $_{\mathbb {C}}\mathrm {U}\cong \mathcal {O}(G^*)$ . We use the same notation $\mathrm {T}_i$ to denote the induced action.

3.1 Geometric preparations

For $i\in \mathrm {I}_\circ $ , recall the parameter $\varsigma _i$ in §2.2. Set $c_i=\overline {\varsigma _i}\in \{\pm 1\}$ . By [Reference Springer24], there is a unique complex group involution $\theta :G\rightarrow G$ , such that,

$$ \begin{align*} \theta(x_i(\xi))=x_i(\xi),\quad \theta(y_i(\xi))=y_i(\xi),\qquad\text{for } i\in\mathrm{I}_{\bullet}, \text{ and } \xi\in\mathbb{C};\\ \theta(x_i(\xi))=\text{Ad}_{\dot{w}_{\bullet}}(y_{\tau i}(c_i\xi)),\quad \theta(y_i(\xi))=\text{Ad}_{\dot{w}_{\bullet}}(x_{\tau i}(c_i\xi)),\qquad \text{for } i\in\mathrm{I}_\circ, \text{ and } \xi\in\mathbb{C}. \end{align*} $$

Let $K=G^\theta $ be the fixed-point subgroup, and $\mathfrak {k}$ be the Lie algebra of K. We use the same notation $\theta $ to denote the induced Lie algebra involution on $\mathfrak {g}$ .

Proposition 3.1. The closed subgroup K is coisotropic, that is, the complementary dual

$$\begin{align*}\mathfrak{k}^\perp=\{f\in\mathfrak{g}^*\mid f(X)=0,\;\forall X\in\mathfrak{k}\} \end{align*}$$

is a Lie subalgebra of $\mathfrak {g}^*$ .

Proof. We identify $\mathfrak {g}^*$ with the subalgebra of the double $\mathfrak {d}$ as in (2.2). We claim that under the identification one has

(3.1) $$ \begin{align} \mathfrak{k}^\perp=\{(X_1,X_2)\in\mathfrak{g}^*\mid X_1=\theta(X_2)\}. \end{align} $$

Let $\mathfrak {p}=\{X\in \mathfrak {g}\mid \theta (X)=-X\}$ . Then $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ as vector spaces. Since the involution $\theta :\mathfrak {g}\rightarrow \mathfrak {g}$ preserves the Killing form, we have $\mathfrak {p}=\{X\in \mathfrak {g}\mid \langle X,Y\rangle =0,\;\forall \;Y\in \mathfrak {k}\}$ .

Take any $x=(X_1,X_2)\in \mathfrak {g}^*$ . We have

(3.2) $$ \begin{align} x\in\mathfrak{k} &\iff 0=\langle\langle x,y\rangle\rangle=\langle X_1-X_2,Y\rangle,\quad \text{for any } y=(Y,Y) \text{ with } Y\in\mathfrak{k},\notag\\ & \iff X_1-X_2\in\mathfrak{g}_1,\notag\\ & \iff \theta(X_1)-X_2=\theta(X_2)-X_1. \end{align} $$

Let us write $X_1=X_1'+h$ and $X_2=X_2'-h$ , where $X_1'\in \mathfrak {u}^+$ , $X_2'\in \mathfrak {u}^-$ and $h\in \mathfrak {h}$ . Then the condition (3.2) holds if and only if $\theta (X_1')-X_2'=\theta (X_2')-X_1'$ and $\theta (h)=-h$ . Therefore $\theta (X_1')-X_2'\in \mathfrak {p}$ . On the other hand, by the construction of $\theta $ , we have

$$ \begin{align*}\theta(X_1')-X_2'\in\bigoplus_{\alpha\in\mathcal{R}^--\Phi_{\bullet}}\mathfrak{g}_\alpha.\end{align*} $$

Note that $\mathfrak {g}_\alpha \subset \mathfrak {k}$ , for $\alpha \in \Phi _{\bullet }$ , and $\theta (\mathcal {R}^--\Phi _{\bullet })\cap \mathcal {R}^-=\emptyset $ . We conclude that $\theta (X_1')-X_2'=0$ . Therefore we have $\theta (X_1)=X_2$ . This proves the claim.

The proposition follows immediately.

Let $K^\perp $ be the connected closed subgroup of $G^*$ with the Lie algebra $\mathfrak {k}^\perp $ . It is also a coisotropic subgroup. Hence the quotient $K^\perp \backslash G^*$ is automatically a Poisson homogeneous space of $G^*$ [Reference Drinfeld17].

In order to study the geometry of $K^\perp \backslash G^*$ , let us introduce another closed subgroup

$$\begin{align*}P=\{(u_+t,t^{-1}u_-)\mid u_+\in U^+_{w_{\bullet}},u_-\in U^-,t\in H'\}\subset G^*, \end{align*}$$

where $U_{w_{\bullet }}^+$ is the closed subgroup of $U^+$ with the Lie algebra $\oplus _{\alpha \in \mathcal {R}_{\bullet }^+}\mathfrak {g}_\alpha $ , and

$$ \begin{align*}H'=\{t\in H\mid \alpha_i(t)=1,\,\forall i\in \mathrm{I}_\circ'\}.\end{align*} $$

It follows immediately from the definition that the map

(3.3) $$ \begin{align} U_{w_{\bullet}}^+\times H'\times U^-\overset{\sim}{\longrightarrow} P,\qquad (u_+,t,u_-)\mapsto (u_+t,t^{-1}u_-), \end{align} $$

defines an isomorphism as varieties.

Proposition 3.2. The map

(3.4) $$ \begin{align} K^\perp\times P\overset{\sim}{\longrightarrow} G^*,\qquad (g_1,g_2)\mapsto g_1g_2, \end{align} $$

is an isomorphism as varieties. Therefore we have $K^\perp \backslash G^*\cong P$ as varieties.

Proof. Let $U_{w_0w_{\bullet }}^+$ be the closed subgroups of $U^+$ , with the Lie algebra $\oplus _{\alpha \in \mathcal {R}^+-\mathcal {R}_{\bullet }^+}\mathfrak {g}_\alpha $ . Let $H_1$ be the closed connected subgroup of H, with the Lie algebra $\{X\in \mathfrak {h}\mid \theta (X)=-X\}$ . By the proof of Proposition 3.1, the subgroup $K^\perp $ is the identity component of the subgroup

$$ \begin{align*}\widetilde{K}=\{(b_+,b_-)\in G^*\mid b_+=\theta(b_-)\}.\end{align*} $$

Also note that $U^+_{w_\circ w_{\bullet }}=U^+\cap \theta (U^-).$ Therefore the map

(3.5) $$ \begin{align} U^+_{w_0w_{\bullet}}\times H_1 \overset{\sim}{\longrightarrow} K^\perp,\qquad (u_+,t)\mapsto (u_+t,\theta(u_+t)), \end{align} $$

defines an isomorphism as varieties.

By [Reference Springer25, Lemma 8.3.5] the group multiplication $U^+_{w_0w_{\bullet }}\times U^+_{w_{\bullet }}\rightarrow U^+$ is an isomorphism as varieties. Since G is of adjoint type, the multiplication $H_{1}\times H'\rightarrow H$ also defines an isomorphism as varieties.

Since the adjoint action of H preserves the subgroups $U^+_{w_0w_{\bullet }}$ and $U^+_{w_{\bullet }}$ , the proposition then follows from the isomorphism (2.3).

By Proposition 3.2, the quotient variety $K^\perp \backslash G^*$ is actually affine. Therefore, by [Reference Springer25, Exercise 5.5.9 (8)], the coordinate ring $\mathcal {O}(K^\perp \backslash G^*)$ is isomorphic to the subalgebra

$$\begin{align*}\{f\in\mathcal{O}(G^*)\mid f(kg)=f(g),\;\forall\,k\in K^\perp,g\in G^*\}\end{align*}$$

of $K^\perp $ -invariant functions in $\mathcal {O}(G^*)$ . We shall make this identification throughout the paper.

Let $\iota ^*:\mathcal {O}(G^*)\rightarrow \mathcal {O}(P)$ be the comorphism of the embedding $\iota :P\hookrightarrow G^*$ . Thanks to Proposition 3.2, we conclude that the restriction map

(3.6) $$ \begin{align} \iota^*\mid_{\mathcal{O}(K^\perp\backslash G^*)}:\mathcal{O}(K^\perp\backslash G^*)\overset{\sim}{\longrightarrow}\mathcal{O}(P) \end{align} $$

defines an isomorphism as $\mathbb {C}$ -algebras.

In light of the isomorphism (3.6), the map $\iota ^*$ provides a tool to control the size of the algebra $\mathcal {O}(K^\perp \backslash G^*)$ . We also need a similar map on the quantum level.

3.2 Quantization of the group P

We firstly describe a quantization of the algebra $\mathcal {O}(P)$ . Following [Reference Kolb and Yakimov19, 2.3], define the partial parabolic subalgebra $\mathrm {U}_P$ to be the unital $\mathbb {C}(q^{1/2})$ -subalgebra of $\mathrm {U}$ , generated by $F_i$ ( $i\in \mathrm {I}$ ), $E_i$ ( $i\in \mathrm {I}_{\bullet }$ ) and $K_\mu $ ( $\mu \in Q^\theta $ ). Let $\mathrm {U}^+_{w_{\bullet }}$ be the unital subalgebra of $\mathrm {U}^+$ generated by $E_i$ , for various $i\in \mathrm {I}_{\bullet }$ . Let $\mathrm {U}^{0\theta }$ be the unital subalgebra of $\mathrm {U}^0$ generated by $K_\mu $ , for $\mu \in Q^\theta $ . Then the multiplication gives an isomorphism as $\mathbb {C}(q^{1/2})$ -vector spaces

(3.7) $$ \begin{align} \mathrm{U}_P\cong \mathrm{U}_{w_{\bullet}}^+\otimes \mathrm{U}^{0\theta}\otimes \mathrm{U}^-. \end{align} $$

Recall the $\mathcal {A}$ -subalgebra $_{\mathcal {A}}\mathrm {U}$ in §2.1. Set $_{\mathcal {A}}\mathrm {U}_P=\mathrm {U}_P\cap {}_{\mathcal {A}}\mathrm {U}$ . Similarly define $_{\mathcal {A}}\mathrm {U}^{0\theta }$ and $_{\mathcal {A}}\mathrm {U}_{w_{\bullet }}^+$ . By the PBW-bases, the multiplication gives an isomorphism as $\mathcal {A}$ -modules

(3.8) $$ \begin{align} _{\mathcal{A}}\mathrm{U}_P\cong {}_{\mathcal{A}}\mathrm{U}_{w_{\bullet}}^+\otimes {}_{\mathcal{A}}\mathrm{U}^{0\theta}\otimes{}_{\mathcal{A}}\mathrm{U}^-, \end{align} $$

where the tensor products are over $\mathcal {A}$ .

Let $_{\mathbb {C}}\mathrm {U}_P=\mathbb {C}\otimes {}_{\mathcal {A}}\mathrm {U}_P$ be the commutative $\mathbb {C}$ -algebra obtained by the base change $q^{1/2}\mapsto 1$ . Similarly define the $\mathbb {C}$ -algebras $_{\mathbb {C}}\mathrm {U}^+_{w_{\bullet }}$ , $_{\mathbb {C}}\mathrm {U}^{0\theta }$ and $_{\mathbb {C}}\mathrm {U}^-$ . Thanks to (3.8), we have the isomorphism as $\mathbb {C}$ -algebras

(3.9) $$ \begin{align} _{\mathbb{C}}\mathrm{U}_P\cong {}_{\mathbb{C}}{\mathrm{U}}^+_{w_{\bullet}}\otimes {}_{\mathbb{C}}\mathrm{U}^{0\theta}\otimes {}_{\mathbb{C}}\mathrm{U}^-. \end{align} $$

Recall the $\mathbb {C}$ -algebra isomorphisms $\varphi ^\pm :{}_{\mathbb {C}}\mathrm {U}^\pm \overset {\sim }{\rightarrow }\mathcal {O}(U^\pm )$ in (2.8). The map $\varphi ^+$ induces an isomorphism $ \varphi ^+_{w_{\bullet }}:{}_{\mathbb {C}}\mathrm {U}_{w_{\bullet }}^+\overset {\sim }{\rightarrow }\mathcal {O}(U_{w_{\bullet }}^+). $ Let $ \varphi ^0: {}_{\mathbb {C}}\mathrm {U}^{0\theta }\overset {\sim }{\rightarrow }\mathcal {O}(H') $ be the $\mathbb {C}$ -algebra isomorphism given by $\overline {k_i}\mapsto \alpha _i\mid _{H'}$ , for $i\in \mathrm {I}_\circ -\mathrm {I}_\circ '$ , and $\overline {K_i}\mapsto \alpha _i\mid _{H'}$ , for $i\in \mathrm {I}_{\bullet }$ .

By (3.3), we have the natural isomorphism $\mathcal {O}(P)\cong \mathcal {O}(U^+_{w_{\bullet }})\otimes \mathcal {O}(H')\otimes \mathcal {O}(U^-)$ as $\mathbb {C}$ -algebras. Under this isomorphism, one gets the $\mathbb {C}$ -algebra isomorphism

$$\begin{align*}\varphi_P=\varphi_{w_{\bullet}}^+\otimes\varphi^0\otimes\varphi^-:{}_{\mathbb{C}}\mathrm{U}_P\overset{\sim}{\longrightarrow} \mathcal{O}(P). \end{align*}$$

Therefore the algebra $\mathrm {U}_P$ provides a quantization of $\mathcal {O}(P)$ .

3.3 Quantization of the map $\iota ^*$

We next describe a quantization of the map $\iota ^*:\mathcal {O}(G^*)\rightarrow \mathcal {O}(P)$ . Note that $\mathrm {U}_{w_{\bullet }}^+=\mathrm {U}^+\cap \mathrm {T}_{w_{\bullet }}(\mathrm {U}^-)$ . Set $\mathrm {U}^{+\prime }_{w_{\bullet }}=\mathrm {U}^+\cap \mathrm {T}_{w_{\bullet }}(\mathrm {U}^+)$ to be another subalgebra of $\mathrm {U}^+$ . By the PBW-bases we have the tensor product decomposition $\mathrm {U}^+\cong \mathrm {U}_{w_{\bullet }}^+\otimes {\mathrm {U}_{w_{\bullet }} ^{+\prime }}$ . Under this isomorphism, set

$$ \begin{align*}\pi_{w_{\bullet}}^+=id\mid_{\mathrm{U}^+_{w_{\bullet}}}\otimes\varepsilon:\mathrm{U}^+\longrightarrow\mathrm{U}^+_{w_{\bullet}}.\end{align*} $$

Set $Q'=\mathbb {Z}[\alpha _i\mid i\in \mathrm {I}_\circ ']$ to be the sublattice of Q. Let $\mathrm {U}^{0}{'}$ be the unital subalgebra of $\mathrm {U}^0$ , generated by $K_i^{\pm 1}$ , for $i\in \mathrm {I}_\circ '$ . Then we have the natural isomorphism $\mathrm {U}^0\cong \mathrm {U}^{0\theta }\otimes \mathrm {U}^{0}{}'$ . Under this isomorphism, set

$$ \begin{align*}\pi^0=id\mid_{\mathrm{U}^{0\theta}}\otimes \varepsilon:\mathrm{U}^0\longrightarrow\mathrm{U}^{0\theta}.\end{align*} $$

Finally, under the isomorphisms (3.7), set

(3.10) $$ \begin{align} \pi=\pi_{w_{\bullet}}^+\otimes\pi^0\otimes id\mid_{\mathrm{U}^-}:\mathrm{U}\cong\mathrm{U}^+\otimes\mathrm{U}^0\times\mathrm{U}^-\longrightarrow \mathrm{U}_P. \end{align} $$

The map $\pi $ restricts to integral forms $_{\mathcal {A}}\pi :{}_{\mathcal {A}}\mathrm {U}\rightarrow {}_{\mathcal {A}}\mathrm {U}_P$ .

The following proposition asserts that the map $_{\mathcal {A}}\pi $ quantize the map $\iota ^*$ .

Proposition 3.3. Let $_{\mathbb {C}}\pi :{}_{\mathbb {C}}\mathrm {U}\rightarrow {}_{\mathbb {C}}\mathrm {U}_P$ be the base change of the map $_{\mathcal {A}}\pi $ . Then the diagram

(3.11)

commutes.

3.4 Poisson generators

The last ingredient that is needed for the proof is a set of Poisson generators for the algebra $\mathcal {O}(K^\perp \backslash G^*)$ .

Let us consider the following elements in $\mathcal {O}(G^*)$ :

1. (i) $\chi _i^+$ , $\chi _i^-$ , for $i\in \mathrm {I}_{\bullet }$ ;

2. (ii) $\chi _i^--c_i\mathrm {T}_{w_{\bullet }}(\chi _{\tau i}^+)\alpha _i^{-1}$ , for $i\in \mathrm {I}_\circ $ ;

3. (iii) $\alpha _i^{\pm 1}$ , for $i\in \mathrm {I}_{\bullet }$ ;

4. (iv) $(\alpha _i\alpha _{\tau i}^{-1})^{\pm 1}$ , for $i\in \mathrm {I}_\circ -\mathrm {I}_\circ '$ .

Proof. Let $\pi _2:G^*\rightarrow P$ be the projection map onto P under the isomorphism (3.4). By Proposition 3.2, the image of the comorphism $\pi _2^*:\mathcal {O}(P)\rightarrow \mathcal {O}(G^*)$ is exactly $\mathcal {O}(K^\perp \backslash G^*)$ .

In order to compute images of various functions under $\pi _2^*$ , we give a precise description on the map $\pi _2$ . Take any element $(u_+t,t^{-1}u_-)$ in $G^*$ . Decompose $u_+=u_+'u_+''$ , with $u_+'\in U^+_{w_0w_{\bullet }}$ and $u_+''\in U^+_{w_{\bullet }}$ . Decompose $t=t't''$ , with $t'\in H_0$ and $t''\in H'$ . It is direct to verify that

(3.12) $$ \begin{align} \pi_2\big((u_+t,t^{-1}u_-)\big)=\big(\text{Ad}_{t^{\prime-1}}(u_+")t",t^{\prime\prime-1}\text{Ad}_t(\theta(u_+'))^{-1}u_-\big). \end{align} $$

Let us identify regular functions on $U^+$ , $U_{w_{\bullet }}^-$ and $H'$ as regular functions on P, via the isomorphism $P\cong U^+_{w_{\bullet }}\times H'\times U^-$ (3.3).

1. (a) For $i\in \mathrm {I}_{\bullet }$ , we have $\pi _2^*(\chi _{i}^+\mid _{U_{w_{\bullet }}^+})=\chi _i^{+}$ and $\pi _2^*(\chi _{i}^-)=\chi _i^{-}$ .

   Take $i\in \mathrm {I}_{\bullet }$ , and $(u_+t,t^{-1}u_-)\in G^*$ . By (3.12), we have

   $$\begin{align*}\pi_2^*(\chi_{i}^+\mid_{U_{w_{\bullet}}^+})\big((u_+t,t^{-1}u_-)\big)=\chi_{i}^+\big(\text{Ad}_{t^{\prime-1}}(u_+")\big)=\alpha_i(t^{\prime-1})\chi_i^+(u_+). \end{align*}$$

   Note that $\alpha _i(t')=1$ , because $\theta (\alpha _i)=\alpha _i$ and $\theta (t')=t^{\prime -1}$ . Therefore we have $\pi _2^*(\chi _{P,i}^+\mid _{U_{w_{\bullet }}^+})=\chi _i^+$ .

   Since $\theta (U_{w_0w_{\bullet }}^+)\cap U^-_{w_{\bullet }}=\{e\}$ , we deduce that

   $$\begin{align*}\pi_2^*(\chi_{i}^-)\big((u_+t,t^{-1}u_-)\big)=\chi_{i}^-\big(\text{Ad}_t(\theta(u_+'))^{-1}u_-\big)=\chi_i^{-}(u_-). \end{align*}$$

   This implies $\pi _2^*(\chi _{i}^-)=\chi _i^-$ .

2. (b) For $i\in \mathrm {I}_\circ $ , we have $\pi _2^*(\chi _{P,i}^-)=\chi _i^--c_i\mathrm {T}_{w_{\bullet }}(\chi _{\tau i}^+)\alpha _i^{-1}$ .

   Take $i\in \mathrm {I}_\circ $ , and $(u_+t,t^{-1}u_-)\in G^*$ . Then by (3.12) one has

   $$\begin{align*}\pi_2^*(\chi_{i}^-)\big((u_+t,t^{-1}u_-)\big)=\chi_{i}^-\big(\text{Ad}_t(\theta(u_+'))^{-1}u_-\big)=\chi_i^-(u_-)+\chi_i^-(\text{Ad}_t(\theta(u_+^{\prime-1}))). \end{align*}$$

   Take a reduced expression $\mathbf {i}=(i_1,i_2,\cdots , i_n)$ of $w_0$ , such that $(i_1,i_2,\cdots , i_{n'})$ is a reduced expression of $w_{\bullet }$ , and $i_{n'+1}=\tau i$ . Write $u_+'=x_{\mathbf {i},n}(a_n)x_{\mathbf {i},n-1}(a_{n-1})\cdots x_{\mathbf {i},1}(a_1)$ , for $a_k\in \mathbb {C}$ . Then $u_+'\in U_{w_0w_{\bullet }}^+$ . Hence we have $a_k=0$ for $1\leq k\leq n'$ . Therefore

   $$ \begin{align*} \theta(u_+^{\prime-1})&=\theta(x_{\mathbf{i},n'+1}(a_{n'+1}))\theta(x_{\mathbf{i},n'+2}(a_{n'+2}))\cdots \theta(x_{\mathbf{i},n}(a_n))\\ &=y_i(c_ia_{n'+1})\theta(x_{\mathbf{i},n'+2}(a_{n'+2}))\cdots \theta(x_{\mathbf{i},n}(a_n)) \end{align*} $$

   is a factorisation as in (2.7). Hence

   $$\begin{align*}\chi_i^-(\text{Ad}_t(\theta(u_+^{\prime-1})))=\chi_i^-(y_i(\alpha_i^{-1}(t)c_ia_{n'+1}))=c_i\alpha_i^{-1}(t)a_{n'+1}=c_i\alpha_i^{-1}(t)\mathrm{T}_{w_{\bullet}}(\chi_{\tau i}^+)(u_+). \end{align*}$$

   This implies that $\pi _2^*(\chi _{P,i}^-)=\chi _i^--c_i\mathrm {T}_{w_{\bullet }}(\chi ^+_{\tau i})\alpha _i^{-1}$ .

3. (c) For $i\in \mathrm {I}_{\bullet }$ , we have $\pi _2^*(\alpha _{i}\mid _{H'})=\alpha _i$ . For $i\in \mathrm {I}_\circ -\mathrm {I}_\circ '$ , we have $\pi _2^*(\alpha _{i}\mid _{H'})=\alpha _i\alpha _{\tau i}^{-1}A_i$ , where $A_i$ is a monomial of the form $\prod _{i\in \mathrm {I}_{\bullet }}\alpha _i^{n_i}$ with $n_i\in \mathbb {Z}$ .

It follows from the similar argument. Statement (c) is actually easier to prove since we only need to consider elements in the torus. We skip the details.

Thanks to the statements (a)–(c), we conclude that elements in (i)–(iv) belong to the image of $\pi _2^*$ and hence belong to $\mathcal {O}(K^\perp \backslash G^*)$ .

Next we show that these elements form a set of Poisson generators.

Let $\mathcal {O}'$ be the Poisson subalgebra of $\mathcal {O}(G^*)$ generated by elements in (i)–(iv). By the previous argument we have $\mathcal {O}'\subset \mathcal {O}(K^\perp \backslash G^*)$ . Recall that the restriction of the map $\iota ^*:\mathcal {O}(G^*)\rightarrow \mathcal {O}(P)$ to $\mathcal {O}(K^\perp \backslash G^*)$ is an isomorphism. In order to show $\mathcal {O}'=\mathcal {O}(K^\perp \backslash G^*)$ , it suffices to show that $\iota ^*(\mathcal {O}')=\mathcal {O}(P)$ . Still let us identify $\mathcal {O}(U_{w_{\bullet }}^+)$ , $\mathcal {O}(H')$ and $\mathcal {O}(U^-)$ as the subalgebras of $\mathcal {O}(P)$ . It suffices to show that $\iota ^*(\mathcal {O}')$ contains these subalgebras.

Firstly, by the proof of [Reference De Concini, Kac and Procesi15, Theorem 7.6], the algebra $\mathcal {O}(U_{w_{\bullet }}^+)$ is generated by functions $\chi _{i}^+\mid _{U_{w_{\bullet }}^+}$ , for $i\in \mathrm {I}_{\bullet }$ , as a Poisson algebra. Therefore as subalgebras of $\mathcal {O}(G^*)$ , we have $\mathcal {O}'\supset \mathcal {O}(U_{w_{\bullet }}^+)$ , thanks to (a). This implies $\iota ^*(\mathcal {O}')\supset \mathcal {O}(U_{w_{\bullet }})$ , where $\mathcal {O}(U_{w_{\bullet }})$ on the right hand side is viewed as a subalgebra of $\mathcal {O}(P)$ .

Next, the algebra $\mathcal {O}(H')$ is generated by $(\alpha _{i}\mid _{H'})^{\pm 1}$ , for $i\in \mathrm {I}-\mathrm {I}_\circ '$ . Thanks to (c) we conclude that $\iota ^*(\mathcal {O}')$ contains $\mathcal {O}(H')$ .

Lastly, still by the proof of [Reference De Concini, Kac and Procesi15, Theorem 7.6], as a Poisson algebra $\mathcal {O}(U^-)$ is generated by $\chi _i^-$ , for $i\in \mathrm {I}$ . Write $b_i=\chi _i^-$ , for $i\in \mathrm {I}_{\bullet }$ , and $b_i=\chi _i^--c_i\mathrm {T}_{w_{\bullet }}(\chi _{\tau i}^+)\alpha _i^{-1}$ , for $i\in \mathrm {I}_\circ $ , to be elements in $\mathcal {O}(G^*)$ . Let $f(x_1,x_2,\cdots ,x_k)$ be a Poisson polynomial, that is, a polynomial possibly with Poisson brackets. Note that there is a $\mathbb {Z}[\mathrm {I}]$ -grading on the algebra $\mathcal {O}(G^*)$ respecting the Poisson brackets, where $\text {deg}(\chi _i^-)=i$ , $\text {deg}(\chi _i^+)=-i$ and $\text {deg}(\mu )=0$ , for $i\in \mathrm {I}$ and $\mu \in Q$ . We define a partial order on $\mathbb {Z}[\mathrm {I}]$ by setting $\mu \leq \mu '$ if $\mu '-\mu $ is a non-negative combination of various $i\in \mathrm {I}$ . Then for any $i_1,i_2,\cdots ,i_k$ in $ \mathrm {I}$ , we have

$$ \begin{align*}f(b_{i_1},b_{i_2},\cdots,b_{i_k})=f(\chi_{i_1}^-,\chi_{i_2}^-,\cdots,\chi_{i_k}^-)+\text{lower terms}\end{align*} $$

in $\mathcal {O}(G^*).$ Therefore we deduce that $\iota ^*(\widetilde {\mathcal {O}})$ contains $\mathcal {O}(U^-)\subset \mathcal {O}(P)$ by induction on the degree.

We complete the proof of the lemma.

3.5 The final proof

We are now ready to prove the main theorem of the paper.

Proof of Theorem 1.

The uniqueness of $\varphi ^\imath $ is clear. We show the existence. Consider the following diagram

(3.13)

Let $\widetilde {\mathcal {O}}\subset \mathcal {O}(G^*)$ be the image of the map $\varphi \circ {}_{\mathbb {C}}\iota :{}_{\mathbb {C}}\mathrm {U}^\imath \rightarrow \mathcal {O}(G^*)$ .

1. (a) We have $\widetilde {\mathcal {O}}\supset {}\mathcal {O}(K^\perp \backslash G^*)$ .

   Recall the generators $B_i$ ( $i\in \mathrm {I}$ ), $E_i$ ( $i\in \mathrm {I}_{\bullet }$ ), $K_i^{\pm 1}$ ( $i\in \mathrm {I}_{\bullet }$ ) and $k_i$ ( $i\in \mathrm {I}_{\bullet }$ ) of the subalgebra $\mathrm {U}^\imath $ in §2.2. It is clear that these elements belong to the integral form $_{\mathcal {A}}\mathrm {U}^\imath $ . Moreover these elements specialize exactly to the elements (i)–(iv) in §3.4. Since the maps $_{\mathbb {C}}\iota $ and $\varphi $ are Poisson, $\widetilde {\mathcal {O}}$ is a Poisson subalgebra of $\mathcal {O}(G^*)$ . Then (a) follows from Lemma 3.1.

2. (b) The map $_{\mathbb {C}}\pi \circ {}_{\mathbb {C}}\iota :{}_{\mathbb {C}}\mathrm {U}\rightarrow {}_{\mathbb {C}}\mathrm {U}_P$ is surjective.

   Recall that $\varphi _P$ is an isomorphism. Therefore it suffices to show that $\varphi _P\circ {}_{\mathbb {C}}\pi \circ {}_{\mathbb {C}}\iota $ is surjective. By Proposition 3.3, we have $\varphi _P\circ {}_{\mathbb {C}}\pi \circ {}_{\mathbb {C}}\iota =\iota ^*\circ \varphi \circ {}_{\mathbb {C}}\iota $ . Recall that $\iota ^*({}\mathcal {O}(K^\perp \backslash G^*))=\mathcal {O}(P)$ . Then (b) follows from (a).

3. (c) The map $_{\mathbb {C}}\pi \circ {}_{\mathbb {C}}\iota :{}_{\mathbb {C}}\mathrm {U}\rightarrow {}_{\mathbb {C}}\mathrm {U}_P$ is an isomorphism.

   Since $_{\mathcal {A}}\mathrm {U}_P$ is a free $\mathcal {A}$ -module, we take an $\mathcal {A}$ -basis $\mathrm {B_P}$ of $_{\mathcal {A}}\mathrm {U}_P$ . Then $\mathrm {B}_P$ is a $\mathbb {C}(q^{1/2})$ -basis of $\mathrm {U}_P$ . Thanks to [Reference Kolb and Yakimov19, Lemma 2.10], the Letzter’s map $\pi ^\imath =\pi \mid _{\mathrm {U}^\imath }:\mathrm {U}^\imath \rightarrow \mathrm {U}_P$ is an isomorphism as vector spaces. Therefore, the set $(\pi ^\imath )^{-1}(\mathrm {B}_P)\subset \mathrm {U}^\imath $ gives a $\mathbb {C}(q^{1/2})$ -basis of $\mathrm {U}^\imath $ . For $b\in \mathrm {B}_P$ , let $d_b\in \mathbb {C}[q^{1/2}]$ be the monic polynomial with the lowest degree, such that $d_b(\pi ^\imath )^{-1}(b)\in {}_{\mathcal {A}}\mathrm {U}^\imath $ . Then $\{d_b(\pi ^\imath )^{-1}(b)\mid b\in \mathrm {B}\}$ is an $\mathcal {A}$ -basis of $_{\mathcal {A}}\mathrm {U}^\imath $ . Hence the map $_{\mathbb {C}}\pi ^\imath ={}_{\mathbb {C}}\pi \circ {}_{\mathbb {C}}\iota :{}_{\mathbb {C}}\mathrm {U}^\imath \rightarrow {}_{\mathbb {C}}\mathrm {U}_P$ , obtained by the base change of the map $\pi ^\imath \mid _{{}_{\mathcal {A}}\mathrm {U}^\imath }:{}_{\mathcal {A}}\mathrm {U}^\imath \rightarrow {}_{\mathcal {A}}\mathrm {U}_P$ , is an isomorphism if $d_b(1)\neq 0$ , for any $b\in \mathrm {B}$ . Suppose otherwise $d_b(1)=0$ for some $b\in \mathrm {B}$ . Then $\overline {b}\in {}_{\mathbb {C}}\mathrm {U}_P$ does not belong to the image of $_{\mathbb {C}}\pi ^\imath $ , which contradicts with the surjectivity established in (b). This completes the proof of (c).

4. (d) We have $\widetilde {\mathcal {O}}={}\mathcal {O}(K^\perp \backslash G^*)$ . Hence there exists $\varphi ^\imath $ which makes the diagram (3.13) commute.

   By (c) and the Proposition 3.3, we deduce that $\iota ^*\circ \varphi \circ {}_{\mathbb {C}}\iota =\varphi _P\circ {}_{\mathbb {C}}\pi \circ {}_{\mathbb {C}}\iota $ is an isomorphism. In particular $\iota ^*\mid _{\widetilde {\mathcal {O}}}$ is injective. Since $\iota ^*\mid _{{}^K\mathcal {O}(G^*)}$ is an isomorphism, (d) follows from (a).

5. (e) The map $\varphi ^\imath $ is an isomorphism as Poisson algebras.

By the commuting diagram (3.13), we have $\varphi ^\imath =\pi _2^*\circ \varphi _P\circ {}_{\mathbb {C}}\pi \circ {}_{\mathbb {C}}\iota $ . Since the maps $\pi _2^*$ , $\varphi _P$ and $_{\mathbb {C}}\pi \circ {}_{\mathbb {C}}\iota $ are isomorphisms, we deduce $\varphi ^\imath $ is an isomorphism. It is moreover Poisson since $_{\mathbb {C}}\iota $ and $\varphi $ are Poisson.

We complete the proof of the theorem.