$(u,(X_{\gamma_{1}}^{+})^{(m_{1})}\ldots(X_{\gamma_{n}}^{+})^{(m_{n})}v)=(u,\sum_{q_{1},\ldots,q_{p}}K_{q_{1},\ldots,q_{p}}T_{\gamma_{1}}\ldots
T_{\gamma_{n}}(T_{i_{1}}\ldots
T_{i_{k_{n}}})((Y_{\phi_{k_{n}+1}})^{(q_{k_{n}+1})}(Y_{\phi_{k_{n}+2}})^{(q_{k_{n}+2})}\ldots$
$\ldots(Y_{\phi_{p}})^{(q_{p})}(Y_{\phi_{1}})^{(q_{1})}\ldots(Y_{\phi_{k_{n}}})^{(q_{k_{n}})})v),$
(4.1.31)

where $K_{q_{1},\ldots,q_{p}}\in\mathbb{C}[q,q^{-1}]$.

Now for $r=k_{n}+1,\ldots p$

$(T_{i_{1}}\ldots T_{i_{k_{n}}})(Y_{\phi_{r}})=T_{i_{p}}^{-1}\ldots
T_{i_{r+1}}^{-1}X_{i_{r}}^{+},$ (4.1.32)

and for $r=1,\ldots k_{n}$

$(T_{i_{1}}\ldots T_{i_{k_{n}}})(Y_{\phi_{r}})=T_{i_{1}}\ldots
T_{i_{r}}X_{i_{r}}^{+}=-T_{i_{1}}\ldots
T_{i_{r-1}}X_{i_{r}}^{-}K_{i_{r}}=-X_{\beta_{r}}^{-}K_{\beta_{r}},$ (4.1.33)

where $K_{\beta_{r}}=T_{i_{1}}\ldots T_{i_{r-1}}K_{i_{r}}$.

Since $u$ is a highest weight vector, by Lemma 2.7.4 we have
$T_{\gamma_{1}}\ldots T_{\gamma_{1}}u=\omega(c)T_{i_{p}}\ldots T_{i_{1}}u$,
where $c\in\mathbb{C}[q,q^{-1}]^{*}$. Therefore for any element $w\in V$,
$x\in U_{h}(\mathfrak{g})$ we can write

$(u,T_{\gamma_{1}}\ldots T_{\gamma_{n}}xw)=c(u,T_{i_{1}}\ldots T_{i_{p}}xw).$

Combining this with (4.1.32) yields for $r=k_{n}+1,\ldots p$

$\displaystyle(u,T_{\gamma_{1}}\ldots T_{\gamma_{n}}(T_{i_{1}}\ldots
T_{i_{k_{n}}})(Y_{\phi_{r}})xw)=c(u,T_{i_{1}}\ldots
T_{i_{p}}(T_{i_{p}}^{-1}\ldots T_{i_{r+1}}^{-1})(X_{i_{r}}^{+})xw)=$
$\displaystyle=c(u,(T_{i_{1}}\ldots T_{i_{r}})(X_{i_{r}}^{+})T_{i_{1}}\ldots
T_{i_{p}}xw)=-c(u,(T_{i_{1}}\ldots
T_{i_{r-1}})(X_{i_{r}}^{-}K_{i_{r}})T_{i_{1}}\ldots T_{i_{p}}xw)=$
$\displaystyle=-c(u,X_{\beta_{r}}^{-}K_{\beta_{r}}T_{i_{1}}\ldots
T_{i_{p}}xw)=0.$

From this formula, (4.1.31), (4.1.33) and the commutation relations between
the elements $K_{\beta_{r}}$ and the quantum root vectors we deduce

$(u,(X_{\gamma_{1}}^{+})^{(m_{1})}\ldots(X_{\gamma_{n}}^{+})^{(m_{n})}v)=$
$=(u,\sum_{q_{1},\ldots,q_{k_{n}}}C_{q_{1},\ldots,q_{k_{n}}}T_{\gamma_{1}}\ldots
T_{\gamma_{n}}(X_{\beta_{1}}^{-})^{(q_{1})}\ldots(X_{\beta_{k_{n}}}^{-})^{(q_{k_{n}})}v),$

where $C_{q_{1},\ldots,q_{k_{n}}}\in\mathbb{C}[q,q^{-1}]$.

∎

Recall now that according to the results obtained in the proof of Proposition
5.1 in [104] $s^{2}$ is the longest element in the Weyl group $W_{2}$ of the
semisimple part $\mathfrak{m}^{2}$ of the Levi subalgebra the positive roots
of which in $\Delta_{+}$ form the set

$\displaystyle\Delta_{+}^{2}=\\{\gamma_{n+1},\beta_{q+2}^{2},\ldots,\beta_{q+m_{1}}^{2},\gamma_{n+2},\beta_{q+m_{1}+2}^{2},\ldots,\beta_{q+m_{2}}^{2},\gamma_{n+3},\ldots,$
(4.1.34)
$\displaystyle\gamma_{l^{\prime}},\beta_{q+m_{l(s^{2})}+1}^{2},\ldots,\beta_{2q+2m_{l(s^{2})}-(l^{\prime}-n)}^{2}\\},$

and $s^{2}$ acts on them by multiplication by $-1$. The roots in (4.1.34) are
ordered as in the normal ordering of $\Delta_{+}$ associated to $s$.

As above we denote $\gamma_{m}=s_{i_{1}}\ldots s_{i_{k_{n+1}}}\ldots
s_{i_{k_{m}-1}}\alpha_{i_{k_{m}}}$, $m=n+1,\ldots,l^{\prime}$.

Let $w=s_{i_{1}}\ldots s_{i_{k_{n+1}-1}}$, $T_{w}=T_{i_{1}}\ldots
T_{i_{k_{n+1}-1}}$. Then $\widetilde{\Delta}_{+}^{2}=w^{-1}(\Delta_{+}^{2})$
is the set of positive roots of the semisimple part
$\widetilde{\mathfrak{m}}^{2}$ of a standard Levi subalgebra of
$\mathfrak{g}$, $\widetilde{s}^{2}=w^{-1}s^{2}w$ is the longest element in the
Weyl group $\widetilde{W}_{2}$ of $\widetilde{\mathfrak{m}}^{2}$.

For any root $\beta\in\Delta_{+}$ denote

$\widetilde{X}_{\beta}^{\pm}=T_{w}^{-1}(X_{\beta}^{\pm}),~{}~{}\widetilde{\overline{X}}_{\beta}^{\pm}=\overline{T}_{w}^{-1}(\overline{X}_{\beta}^{\pm}),~{}~{}\widetilde{T}_{\beta}=T_{w}^{-1}T_{\beta}T_{w}.$

Similarly to Lemma 4.1.4 we have

######  Lemma 4.1.6.

Let $V$ be a finite rank representation of $U_{h}(\mathfrak{g})$, $u,v\in V$.
Then for any $m_{n+1},\ldots,m_{l^{\prime}}\in\mathbb{N}$

$(u,(\widetilde{X}_{\gamma_{n+1}}^{+})^{(m_{n+1})}\ldots(\widetilde{X}_{\gamma_{l^{\prime}}}^{+})^{(m_{l^{\prime}})}v)=(u,\widetilde{K}\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}v),$
(4.1.35)

where $\widetilde{K}$ is of the form

$\widetilde{K}=\sum_{\tiny\begin{array}[]{l}p_{1},\ldots,p_{k}\\\
p_{1}^{\prime},\ldots,p_{m}^{\prime}\end{array}}M^{p_{1},\ldots,p_{k}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}(\widetilde{{X}}_{\delta_{1}}^{-})^{(p_{1})}\ldots(\widetilde{{X}}_{\delta_{k}}^{-})^{(p_{k})}(\widetilde{\overline{X}}_{\beta_{1}^{\prime}}^{+})^{(p_{1}^{\prime})}\ldots(\widetilde{\overline{X}}_{\beta_{m}^{\prime}}^{+})^{(p_{m}^{\prime})},$
(4.1.36)

the finite sum in the right hand side is over all
$\delta_{i},\beta_{j}\in\Delta_{+}^{2}$ such that
$\delta_{1}<\delta_{2}<\ldots<\delta_{k}$,
$\gamma_{n+1}\leq\beta_{1}^{\prime}<\beta_{2}^{\prime}<\ldots<\beta_{m}^{\prime}\leq\gamma_{l^{\prime}}$,
$M^{p_{1},\ldots,p_{n}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}\in
U_{q}^{res}(H)$.

###### Proof.

Similarly to Lemma 4.1.4 we deduce that formula (4.1.35) holds with

$\widetilde{K}=\sum_{\tiny\begin{array}[]{l}p_{1},\ldots,p_{k}\\\
p_{1}^{\prime},\ldots,p_{m}^{\prime}\end{array}}L^{p_{1},\ldots,p_{k}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}(\widetilde{\overline{X}}_{\delta_{1}}^{-})^{(p_{1})}\ldots(\widetilde{\overline{X}}_{\delta_{k}}^{-})^{(p_{k})}(\widetilde{\overline{X}}_{\beta_{1}^{\prime}}^{+})^{(p_{1}^{\prime})}\ldots(\widetilde{\overline{X}}_{\beta_{m}^{\prime}}^{+})^{(p_{m}^{\prime})},$
(4.1.37)

where $L^{p_{1},\ldots,p_{n}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}\in
U_{q}^{res}(H)$.

Let $U_{q}^{res}(\widetilde{\mathfrak{m}}^{2})$ be the restricted
specialization of $U_{h}(\widetilde{\mathfrak{m}}^{2})$,
$U_{q}^{res}(\widetilde{\mathfrak{m}}^{2}_{+})$ the subalgebra of
$U_{q}^{res}(\widetilde{\mathfrak{m}}^{2})$ generated by
$({X}_{\beta}^{+})^{(k)}$ for simple roots
$\beta\in\widetilde{\Delta}_{+}^{2}$ and $k\geq 0$. Then the monomials
$(\widetilde{\overline{X}}_{\delta_{1}}^{-})^{(p_{1})}\ldots(\widetilde{\overline{X}}_{\delta_{k}}^{-})^{(p_{k})}$,
$\delta_{i},\in\Delta_{+}^{2}$, $\delta_{1}<\delta_{2}<\ldots<\delta_{k}$ form
a linear basis of $U_{q}^{res}(\widetilde{\mathfrak{m}}^{2}_{+})$ and the
monomials
$(\widetilde{{X}}_{\delta_{1}}^{-})^{(p_{1})}\ldots(\widetilde{{X}}_{\delta_{k}}^{-})^{(p_{k})}$
$\delta_{i},\in\Delta_{+}^{2}$, $\delta_{1}<\delta_{2}<\ldots<\delta_{k}$ form
a linear basis of $U_{q}^{res}(\widetilde{\mathfrak{m}}^{2}_{+})$ as well.
Thus $\widetilde{K}$ can be rewritten in form (4.1.36) using the latter basis.

∎

######  Lemma 4.1.7.

Let $V$ be a finite rank representation of $U_{h}(\mathfrak{g})$, $u,v\in V$.
Suppose that $u$ is a highest weight vector. Then for any
$m_{1},\ldots,m_{l^{\prime}}\in\mathbb{N}$

$(u,(X_{\gamma_{1}}^{+})^{(m_{1})}\ldots(X_{\gamma_{n}}^{+})^{(m_{n})}({X}_{\gamma_{n+1}}^{+})^{(m_{n+1})}\ldots({X}_{\gamma_{l^{\prime}}}^{+})^{(m_{l^{\prime}})}v)=$
$=\sum_{n_{1},\ldots,n_{D}}F_{n_{1},\ldots,n_{D}}(u,T_{\gamma_{1}}\ldots
T_{\gamma_{n}}{T}_{\gamma_{n+1}}\ldots{T}_{\gamma_{l^{\prime}}}({X}^{-}_{\beta_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots({X}^{-}_{\beta_{1}})^{(n_{1})}(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta_{D}})^{(n_{D})}v)=$
$=c\sum_{n_{1},\ldots,n_{D}}F_{n_{1},\ldots,n_{D}}(u,T_{s}({X}^{-}_{\beta_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots({X}^{-}_{\beta_{1}})^{(n_{1})}(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta_{D}})^{(n_{D})}v)=$
$=c^{\prime}\sum_{n_{1},\ldots,n_{D}}F_{n_{1},\ldots,n_{D}}(u,\overline{T}_{s}({X}^{-}_{\beta_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots({X}^{-}_{\beta_{1}})^{(n_{1})}(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta_{D}})^{(n_{D})}v),$

where $F_{n_{1},\ldots,n_{D}}\in\mathbb{C}[q,q^{-1}]$,
$c,c^{\prime}\in\mathbb{C}[q,q^{-1}]^{*}$ are integer powers of $q$ up to
constant factors.

###### Proof.

By Lemmas 4.1.5 and 4.1.6 we have

$(u,(X_{\gamma_{1}}^{+})^{(m_{1})}\ldots(X_{\gamma_{n}}^{+})^{(m_{n})}({X}_{\gamma_{n+1}}^{+})^{(m_{n+1})}\ldots({X}_{\gamma_{l^{\prime}}}^{+})^{(m_{l^{\prime}})}v)=$
$=(u,\sum_{q_{1},\ldots,q_{k_{n}}}C_{q_{1},\ldots,q_{k_{n}}}T_{\gamma_{1}}\ldots
T_{\gamma_{n}}(X_{\beta_{1}}^{-})^{(q_{1})}\ldots(X_{\beta_{k_{n}}}^{-})^{(q_{k_{n}})}\times$
$\times T_{w}\sum_{\tiny\begin{array}[]{l}p_{1},\ldots,p_{k}\\\
p_{1}^{\prime},\ldots,p_{m}^{\prime}\end{array}}M^{p_{1},\ldots,p_{k}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}(\widetilde{{X}}_{\delta_{1}}^{-})^{(p_{1})}\ldots(\widetilde{{X}}_{\delta_{k}}^{-})^{(p_{k})}(\widetilde{\overline{X}}_{\beta_{1}^{\prime}}^{+})^{(p_{1}^{\prime})}\ldots(\widetilde{\overline{X}}_{\beta_{m}^{\prime}}^{+})^{(p_{m}^{\prime})}\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}T_{w}^{-1}v)=$
$=(u,\sum_{q_{1},\ldots,q_{k_{n}}}C_{q_{1},\ldots,q_{k_{n}}}T_{\gamma_{1}}\ldots
T_{\gamma_{n}}(X_{\beta_{1}}^{-})^{(q_{1})}\ldots(X_{\beta_{k_{n}}}^{-})^{(q_{k_{n}})}\times$
$\times\sum_{\tiny\begin{array}[]{l}p_{1},\ldots,p_{k}\\\
p_{1}^{\prime},\ldots,p_{m}^{\prime}\end{array}}M^{p_{1},\ldots,p_{k}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}({{X}}_{\delta_{1}}^{-})^{(p_{1})}\ldots({{X}}_{\delta_{k}}^{-})^{(p_{k})}T_{w}(\widetilde{\overline{X}}_{\beta_{1}^{\prime}}^{+})^{(p_{1}^{\prime})}\ldots(\widetilde{\overline{X}}_{\beta_{m}^{\prime}}^{+})^{(p_{m}^{\prime})}\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}T_{w}^{-1}v).$

Observe that $(X_{\beta}^{-})^{(k)}$,
$\beta_{1}\leq\beta\leq\gamma_{l^{\prime}}$ generate a
$\mathbb{C}[q,q^{-1}]$–subalgebra for which the monomials

$(X_{\beta_{k_{l^{\prime}}}}^{-})^{(q_{k_{l^{\prime}}})}\ldots(X_{\beta_{1}}^{-})^{(q_{1})}$

form a linear basis. Rewriting the monomials

$(X_{\beta_{1}}^{-})^{(q_{1})}\ldots(X_{\beta_{k_{n}}}^{-})^{(q_{k_{n}})}({{X}}_{\delta_{1}}^{-})^{(p_{1})}\ldots({{X}}_{\delta_{k}}^{-})^{(p_{k})}$

in terms of this basis we arrive at

$(u,(X_{\gamma_{1}}^{+})^{(k_{1})}\ldots(X_{\gamma_{n}}^{+})^{(k_{n})}({X}_{\gamma_{n+1}}^{+})^{(k_{n+1})}\ldots({X}_{\gamma_{l^{\prime}}}^{+})^{(k_{l^{\prime}})}v)=$
(4.1.38) $=(u,T_{\gamma_{1}}\ldots
T_{\gamma_{n}}\sum_{\tiny\begin{array}[]{l}q_{1},\ldots,q_{k_{l^{\prime}}}\\\
p_{1}^{\prime},\ldots,p_{m}^{\prime}\end{array}}N^{q_{1},\ldots,q_{k_{l^{\prime}}}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}(X_{\beta_{k_{l^{\prime}}}}^{-})^{(q_{k_{l^{\prime}}})}\ldots(X_{\beta_{1}}^{-})^{(q_{1})}T_{w}(\widetilde{\overline{X}}_{\beta_{1}^{\prime}}^{+})^{(p_{1}^{\prime})}\ldots$
$\ldots(\widetilde{\overline{X}}_{\beta_{m}^{\prime}}^{+})^{(p_{m}^{\prime})}\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}T_{w}^{-1}v),$

where
$N^{q_{1},\ldots,q_{k_{l^{\prime}}}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}\in\mathbb{C}[q,q^{-1}]$.

Now observe that if $\gamma_{n}<\beta\leq\gamma_{l^{\prime}}$ then
$\beta\not\in\Delta_{+}^{1}$. The root $\beta$ can be written in the form
$\beta=n\alpha+c$, where $\alpha\not\in\Delta_{+}^{1}$ is a simple root, $n>0$
and $c$ is a linear combination of other simple roots. Since $W_{1}$ is
parabolic in $W$, $s^{1}\in W_{1}$ and $\mathfrak{m}_{1}\subset\mathfrak{g}$
is a standard Levi subalgebra we deduce $s^{1}\beta=n\alpha+c^{\prime}$, where
$c^{\prime}$ is a linear combination of other simple roots. This implies
$s^{1}\beta\in\Delta_{+}$. Thus $\omega(T_{\gamma_{1}}\ldots
T_{\gamma_{n}}(X_{\beta}^{-}))u=0$ for otherwise the vector
$\omega(T_{\gamma_{1}}\ldots T_{\gamma_{n}}(X_{\beta}^{-}))u$ has weight
greater than that of $u$ which is impossible as $u$ is a highest weight
vector. Applying this observation in (4.1.38) we deduce

$(u,(X_{\gamma_{1}}^{+})^{(m_{1})}\ldots(X_{\gamma_{n}}^{+})^{(m_{n})}({X}_{\gamma_{n+1}}^{+})^{(m_{n+1})}\ldots({X}_{\gamma_{l^{\prime}}}^{+})^{(m_{l^{\prime}})}v)=$
(4.1.39) $=(u,T_{\gamma_{1}}\ldots
T_{\gamma_{n}}\sum_{\tiny\begin{array}[]{l}q_{1},\ldots,q_{k_{l^{\prime}}}\\\
p_{1}^{\prime},\ldots,p_{m}^{\prime}\end{array}}K^{q_{1},\ldots,q_{k_{n}}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}(X_{\beta_{k_{n}}}^{-})^{(q_{k_{n}})}\ldots(X_{\beta_{1}}^{-})^{(q_{1})}T_{w}(\widetilde{\overline{X}}_{\beta_{1}^{\prime}}^{+})^{(p_{1}^{\prime})}\ldots(\widetilde{\overline{X}}_{\beta_{m}^{\prime}}^{+})^{(p_{m}^{\prime})}\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}T_{w}^{-1}v),$

where
$K^{q_{1},\ldots,q_{k_{n}}}_{p_{1}^{\prime},\ldots,p_{m}^{\prime}}\in\mathbb{C}[q,q^{-1}]$.

Denote
$\widetilde{T}^{2}=\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}$.
For $\gamma_{n+1}\leq\beta_{q}\leq\gamma_{l^{\prime}}$ we have
$\beta_{q}=ws_{k_{n+1}}\ldots s_{i_{q-1}}\alpha_{i_{q}}$,
$q=k_{n+1},\ldots,k_{l^{\prime}}$. Similarly to (4.1.29) we infer

$(\widetilde{T}^{2})^{-1}\widetilde{\overline{X}}_{\beta_{q}}^{+}=T_{i_{k_{n+1}}}\ldots
T_{i_{k_{l^{\prime}}}}T_{i_{k_{l^{\prime}}}}\ldots T_{i_{q+1}}X_{i_{q}}^{+}.$
(4.1.40)

Denote $\widetilde{X}_{\delta_{q}}^{+}=T_{i_{k_{l^{\prime}}}}\ldots
T_{i_{q+1}}X_{i_{q}}^{+}$, $q=k_{n+1},\ldots k_{l^{\prime}}$. Then from
(4.1.39) and (4.1.40) we obtain

$(u,(X_{\gamma_{1}}^{+})^{(m_{1})}\ldots(X_{\gamma_{n}}^{+})^{(m_{n})}({X}_{\gamma_{n+1}}^{+})^{(m_{n+1})}\ldots({X}_{\gamma_{l^{\prime}}}^{+})^{(m_{l^{\prime}})}v)=$
(4.1.41) $=(u,T_{\gamma_{1}}\ldots
T_{\gamma_{n}}\sum_{\tiny\begin{array}[]{l}q_{1},\ldots,q_{k_{l^{\prime}}}\\\
p_{k_{n+1}}^{\prime},\ldots,p_{k_{l^{\prime}}}^{\prime}\end{array}}K^{q_{1},\ldots,q_{k_{n}}}_{p_{k_{n+1}}^{\prime},\ldots,p_{k_{l^{\prime}}}^{\prime}}(X_{\beta_{k_{n}}}^{-})^{(q_{k_{n}})}\ldots(X_{\beta_{1}}^{-})^{(q_{1})}T_{w}\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}(T_{i_{k_{n+1}}}\ldots
T_{i_{k_{l^{\prime}}}})\times$
$\times((\widetilde{X}_{\delta_{k_{n+1}}}^{+})^{(p_{k_{n+1}}^{\prime})}\ldots(\widetilde{{X}}_{\delta_{k_{l^{\prime}}}}^{+})^{(p_{k_{l^{\prime}}}^{\prime})})T_{w}^{-1}v)=$
$=(u,T_{\gamma_{1}}\ldots
T_{\gamma_{n}}\sum_{\tiny\begin{array}[]{l}q_{1},\ldots,q_{k_{l^{\prime}}}\\\
p_{k_{n+1}}^{\prime},\ldots,p_{k_{l^{\prime}}}^{\prime}\end{array}}K^{q_{1},\ldots,q_{k_{n}}}_{p_{k_{n+1}}^{\prime},\ldots,p_{k_{l^{\prime}}}^{\prime}}(X_{\beta_{k_{n}}}^{-})^{(q_{k_{n}})}\ldots(X_{\beta_{1}}^{-})^{(q_{1})}T_{w}\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}T_{i_{k_{n+1}}}\ldots
T_{i_{k_{l^{\prime}}}}\times$
$\times(\widetilde{X}_{\delta_{k_{n+1}}}^{+})^{(p_{k_{n+1}}^{\prime})}\ldots(\widetilde{{X}}_{\delta_{k_{l^{\prime}}}}^{+})^{(p_{k_{l^{\prime}}}^{\prime})}T_{w^{\prime}}^{-1}v),$

where
$K^{q_{1},\ldots,q_{k_{n}}}_{p_{k_{n+1}^{\prime}},\ldots,p_{k_{l^{\prime}}}^{\prime}}\in\mathbb{C}[q,q^{-1}]$
and

$T_{w^{\prime}}=T_{w}T_{i_{k_{n+1}}}\ldots
T_{i_{k_{l^{\prime}}}}=T_{i_{1}}\ldots T_{i_{k_{l^{\prime}}}}.$

Now if

$\overline{w}=s_{i_{1}}\ldots s_{i_{k_{n+1}}}\ldots
s_{i_{k_{l^{\prime}}}}s_{i_{k_{l^{\prime}}+1}}\ldots s_{i_{p^{\prime}}}\ldots
s_{i_{D}}$

is the reduced decomposition of the longest element of the Weyl group
corresponding to the normal ordering in $\Delta_{+}$ associated to $s$, so
that $\widetilde{s}^{2}=s_{i_{k_{n+1}}}\ldots
s_{i_{k_{l^{\prime}}}}s_{i_{k_{l^{\prime}}+1}}\ldots s_{i_{p^{\prime}}}$ is
the corresponding reduced decomposition of $\widetilde{s}^{2}$, then,
similarly to (4.1.27), we have

$\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}=T_{i_{p^{\prime}}}\ldots
T_{i_{k_{l^{\prime}}+1}}T_{i_{k_{l^{\prime}}}}^{-1}T_{i_{k_{l^{\prime}}-1}}^{-1}\ldots
T_{i_{k_{n+1}}}^{-1},$

and hence

$\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}T_{i_{k_{n+1}}}\ldots
T_{i_{k_{l^{\prime}}}}=T_{i_{p^{\prime}}}\ldots T_{i_{k_{l^{\prime}}+1}}.$
(4.1.42)

Observe that

$T=T_{w}\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}T_{i_{k_{n+1}}}\ldots
T_{i_{k_{l^{\prime}}}}=T_{i_{1}}\ldots
T_{i_{k_{n+1}-1}}T_{i_{p^{\prime}}}\ldots T_{i_{k_{l^{\prime}}+1}}$

is the braid group element corresponding to the reduced decomposition

$s_{i_{1}}\ldots s_{i_{k_{n+1}-1}}s_{i_{p^{\prime}}}\ldots
s_{i_{k_{l^{\prime}}+1}}$ (4.1.43)

which is a part of the reduced decomposition obtained from the reduced
decomposition

$s_{i_{1}}\ldots s_{i_{k_{n+1}-1}}s_{i_{k_{n+1}}}\ldots
s_{i_{k_{l^{\prime}}}}s_{i_{k_{l^{\prime}}+1}}\ldots s_{i_{p^{\prime}}}$

by inverting the part $\widetilde{s}^{2}=s_{i_{k_{n+1}}}\ldots
s_{i_{k_{l^{\prime}}}}s_{i_{k_{l^{\prime}}+1}}\ldots s_{i_{p^{\prime}}}$. This
inversion gives a reduced decomposition again because $\widetilde{s}^{2}=-1$
is the longest element in $\widetilde{W}_{2}$.

Now for $\beta_{1}\leq\beta_{q}\leq\beta_{k_{n}}$ we have

$T^{-1}(X_{\beta_{q}}^{-})=T_{i_{k_{l^{\prime}}+1}}^{-1}\ldots
T_{i_{p^{\prime}}}^{-1}T_{i_{k_{n+1}-1}}^{-1}\ldots
T_{i_{q+1}}^{-1}T_{i_{q}}^{-1}\ldots T_{i_{1}}^{-1}T_{i_{1}}\ldots
T_{i_{q-1}}X_{i_{q}}^{-}=$ (4.1.44) $=T_{i_{k_{l^{\prime}}+1}}^{-1}\ldots
T_{i_{p^{\prime}}}^{-1}T_{i_{k_{n+1}-1}}^{-1}\ldots
T_{i_{q+1}}^{-1}T_{i_{q}}^{-1}X_{i_{q}}^{-}=$
$=-T_{i_{k_{l^{\prime}}+1}}^{-1}\ldots
T_{i_{p^{\prime}}}^{-1}T_{i_{k_{n+1}-1}}^{-1}\ldots
T_{i_{q+1}}^{-1}(X_{i_{q}}^{+}K_{i_{q}})=T_{i_{k_{l^{\prime}}+1}}^{-1}\ldots
T_{i_{p^{\prime}}}^{-1}T_{i_{k_{n+1}-1}}^{-1}\ldots
T_{i_{q+1}}^{-1}(X_{i_{q}}^{+})R_{q},$

where

$R_{q}=T_{i_{k_{l^{\prime}}+1}}^{-1}\ldots
T_{i_{p^{\prime}}}^{-1}T_{i_{k_{n+1}-1}}^{-1}\ldots
T_{i_{q+1}}^{-1}(K_{i_{q}}).$

Since $s_{i_{1}}\ldots s_{i_{k_{n+1}-1}}s_{i_{p^{\prime}}}\ldots
s_{i_{k_{l^{\prime}}+1}}$ is a reduced decomposition

$T_{i_{k_{l^{\prime}}+1}}^{-1}\ldots
T_{i_{p^{\prime}}}^{-1}T_{i_{k_{n+1}-1}}^{-1}\ldots
T_{i_{q+1}}^{-1}(X_{i_{q}}^{+})^{(k)}\in U_{q}^{res}(\mathfrak{n}_{+}),$

where $U_{q}^{res}(\mathfrak{n}_{+})$ is the subalgebra of
$U_{q}^{res}(\mathfrak{g})$ generated by $(X_{i}^{+})^{(k)}$ $i=1,\ldots,l$,
$k=0,1,\ldots$. Applying this observation, (4.1.44), commutation relations
between elements $R_{q}$ and the quantum root vectors, and the fact that $u$
is a weight vector we obtain from (4.1.41)

$\displaystyle(u,(X_{\gamma_{1}}^{+})^{(m_{1})}\ldots(X_{\gamma_{n}}^{+})^{(m_{n})}({X}_{\gamma_{n+1}}^{+})^{(m_{n+1})}\ldots({X}_{\gamma_{l^{\prime}}}^{+})^{(m_{l^{\prime}})}v)=$
$\displaystyle=(u,T_{\gamma_{1}}\ldots
T_{\gamma_{n}}T_{w}\widetilde{T}_{\gamma_{n+1}}\ldots\widetilde{T}_{\gamma_{l^{\prime}}}T_{i_{k_{n+1}}}\ldots
T_{i_{k_{l^{\prime}}}}XT_{w^{\prime}}^{-1}v)=$ (4.1.45)
$\displaystyle=(u,T_{\gamma_{1}}\ldots
T_{\gamma_{n}}{T}_{\gamma_{n+1}}\ldots{T}_{\gamma_{l^{\prime}}}T_{w}T_{i_{k_{n+1}}}\ldots
T_{i_{k_{l^{\prime}}}}XT_{w^{\prime}}^{-1}v)=$
$\displaystyle=(u,T_{\gamma_{1}}\ldots
T_{\gamma_{n}}{T}_{\gamma_{n+1}}\ldots{T}_{\gamma_{l^{\prime}}}T_{w^{\prime}}XT_{w^{\prime}}^{-1}v),X\in
U_{q}^{res}(\mathfrak{n}_{+}),$

where we also used the fact that all monomials
$(\widetilde{X}_{\delta_{k_{n+1}}}^{+})^{(p_{k_{n+1}}^{\prime})}\ldots(\widetilde{{X}}_{\delta_{k_{l^{\prime}}}}^{+})^{(p_{k_{l^{\prime}}}^{\prime})}$
in (4.1.39) belong to $U_{q}^{res}(\mathfrak{n}_{+})$ since
$s_{i_{k_{n+1}}}\ldots s_{i_{k_{l^{\prime}}}}$ is a reduced decomposition.

Now consider the reduced decomposition
$\overline{w}=s_{i_{k_{l^{\prime}}}}\ldots
s_{i_{1}}s_{p_{k_{l^{\prime}}+1}}\ldots s_{p_{D}}$, the corresponding normal
ordering
$\beta_{k_{l^{\prime}}}^{\prime},\ldots,\beta_{1}^{\prime}\beta_{p_{k_{l^{\prime}}+1}}^{\prime},\ldots,\beta_{p_{D}}^{\prime}$
of $\Delta_{+}$, the quantum root vectors

$\overline{X}_{\beta_{q}^{\prime}}^{+}=\left\\{\begin{array}[]{ll}T^{-1}_{i_{k_{l^{\prime}}}}\ldots
T^{-1}_{i_{q-1}}X_{i_{q}}^{+}&1\leq q\leq k_{l^{\prime}}\\\
T_{w^{\prime}}^{-1}T_{p_{k_{l^{\prime}}+1}}^{-1}\ldots
T_{p_{q-1}}^{-1}X_{p_{q}}^{+}&k_{l^{\prime}}+1\leq q\leq D\end{array}\right.,$

and the linear basis of $U_{q}^{res}(\mathfrak{n}_{+})$,

$(\overline{X}^{+}_{\beta^{\prime}_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots(\overline{X}^{+}_{\beta^{\prime}_{1}})^{(n_{1})}(\overline{X}^{+}_{\beta^{\prime}_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(\overline{X}^{+}_{\beta^{\prime}_{D}})^{(n_{D})}.$
(4.1.46)

Consider also another reduced decomposition
$\overline{w}=s_{i_{k_{l^{\prime}}+1}}\ldots s_{i_{D}}s_{p_{1}}\ldots
s_{p_{k_{l^{\prime}}}}$, the corresponding normal ordering
$\beta_{k_{l^{\prime}}}^{\prime},\ldots,\beta_{1}^{\prime}\beta_{p_{k_{l^{\prime}}+1}}^{\prime},\ldots,\beta_{p_{D}}^{\prime}$
of $\Delta_{+}$, the quantum root vectors

$X_{\beta_{q}^{\prime\prime}}^{+}=\left\\{\begin{array}[]{ll}T_{w^{\prime\prime}}T_{i_{p_{1}}}\ldots
T_{p_{q-1}}X_{p_{q}}^{+}&1\leq q\leq k_{l^{\prime}}\\\
T_{i_{k_{l^{\prime}}+1}}\ldots T_{i_{q-1}}X_{i_{q}}^{+}&k_{l^{\prime}}+1\leq
q\leq D\end{array}\right.,$

where $T_{w^{\prime\prime}}=T_{i_{k_{l^{\prime}}+1}}\ldots T_{i_{D}}$, and the
linear basis of $U_{q}^{res}(\mathfrak{n}_{+})$,

$(X^{+}_{\beta^{\prime\prime}_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta^{\prime\prime}_{D}})^{(n_{D})}(X^{+}_{\beta^{\prime\prime}_{1}})^{(n_{1})}\ldots(X^{+}_{\beta^{\prime\prime}_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}.$
(4.1.47)

We can represent the monomials
$(\overline{X}^{+}_{\beta^{\prime}_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(\overline{X}^{+}_{\beta^{\prime}_{D}})^{(n_{D})}$
using basis (4.1.47),

$(\overline{X}^{+}_{\beta^{\prime}_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(\overline{X}^{+}_{\beta^{\prime}_{D}})^{(n_{D})}=$
(4.1.48) $=\sum_{q_{1},\ldots q_{D}}C_{q_{1},\ldots
q_{D}}(X^{+}_{\beta^{\prime\prime}_{k_{l^{\prime}}+1}})^{(q_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta^{\prime\prime}_{D}})^{(q_{D})}(X^{+}_{\beta^{\prime\prime}_{1}})^{(q_{1})}\ldots(X^{+}_{\beta^{\prime\prime}_{k_{l^{\prime}}}})^{(q_{k_{l^{\prime}}})},$

where $C_{q_{1},\ldots q_{D}}\in\mathbb{C}[q,q^{-1}]$. Applying
$T_{w^{\prime}}$ to this identity we get

$(\widetilde{\overline{X}}^{+}_{\beta^{\prime}_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(\widetilde{\overline{X}}^{+}_{\beta^{\prime}_{D}})^{(n_{D})}=$
(4.1.49) $=\sum_{q_{1},\ldots q_{D}}C_{q_{1},\ldots
q_{D}}(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(q_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta_{D}})^{(q_{D})}(\widetilde{X}^{+}_{\beta^{\prime\prime}_{1}})^{(q_{1})}\ldots(\widetilde{X}^{+}_{\beta^{\prime\prime}_{k_{l^{\prime}}}})^{(q_{k_{l^{\prime}}})},$

where for $k_{l^{\prime}}+1\leq q\leq D$

$\widetilde{\overline{X}}_{\beta_{q}^{\prime}}^{+}=T_{w^{\prime}}\overline{X}_{\beta_{q}^{\prime}}^{+}=T_{p_{k_{l^{\prime}}+1}}^{-1}\ldots
T_{p_{q-1}}^{-1}X_{p_{q}}^{+}\in U_{q}^{res}(\mathfrak{n}_{+}),$ (4.1.50)

and for $1\leq q\leq k_{l^{\prime}}$

$\widetilde{X}_{\beta_{q}^{\prime\prime}}^{+}=T_{w^{\prime}}X_{\beta_{q}^{\prime\prime}}^{+}=T_{w^{\prime}}T_{w^{\prime\prime}}T_{i_{p_{1}}}\ldots
T_{p_{q-1}}X_{p_{q}}^{+}=T_{\overline{w}}T_{i_{p_{1}}}\ldots
T_{p_{q-1}}X_{p_{q}}^{+}\in U_{q}^{res}(\mathfrak{n}_{-})U_{q}(H),$ (4.1.51)

$U_{q}^{res}(\mathfrak{n}_{-})U_{q}(H)$ is the algebra generated by
$U_{q}^{res}(\mathfrak{n}_{-})$ and by $U_{q}(H)$, and we used the fact that
$T_{i_{p_{1}}}\ldots T_{p_{q-1}}X_{p_{q}}^{+}\in
U_{q}^{res}(\mathfrak{n}_{+})$ and
$T_{\overline{w}}U_{q}^{res}(\mathfrak{n}_{+})\subset
U_{q}^{res}(\mathfrak{n}_{-})U_{q}(H)$. Observing that the elements
$\widetilde{X}_{\beta_{q}^{\prime\prime}}^{+}\in
U_{q}^{res}(\mathfrak{n}_{-})U_{q}(H)$, $1\leq q\leq k_{l^{\prime}}$ have
strictly negative weights we deduce from (4.1.49), (4.1.50), (4.1.51) and
(2.4.1) in Lemma 2.4.2 that (4.1.48) takes the form

$(\overline{X}^{+}_{\beta^{\prime}_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(\overline{X}^{+}_{\beta^{\prime}_{D}})^{(n_{D})}=\sum_{q_{k_{l^{\prime}}+1},\ldots
q_{D}}C_{q_{k_{l^{\prime}}+1},\ldots
q_{D}}(X^{+}_{\beta^{\prime\prime}_{k_{l^{\prime}}+1}})^{(q_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta^{\prime\prime}_{D}})^{(q_{D})},$
(4.1.52)

where $C_{q_{k_{l^{\prime}}+1},\ldots q_{D}}\in\mathbb{C}[q,q^{-1}]$.
Recalling basis (4.1.46) we infer that every element of
$U_{q}^{res}(\mathfrak{n}_{+})$ is a $\mathbb{C}[q,q^{-1}]$–linear combination
of monomials of the form

$(\overline{X}^{+}_{\beta^{\prime}_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots(\overline{X}^{+}_{\beta^{\prime}_{1}})^{(n_{1})}(X^{+}_{\beta^{\prime\prime}_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta^{\prime\prime}_{D}})^{(n_{D})}.$
(4.1.53)

Kostant’s formula shows that they form a linear basis of
$U_{q}^{res}(\mathfrak{n}_{+})$.

Now for $1\leq q\leq k_{l^{\prime}}$ we have

$T_{w^{\prime}}\overline{X}_{\beta_{q}^{\prime}}^{+}=T_{w^{\prime}}T^{-1}_{i_{k_{l^{\prime}}}}\ldots
T^{-1}_{i_{q-1}}X_{i_{q}}^{+}=T_{i_{1}}\ldots
T_{i_{q}}X_{i_{q}}^{+}=-T_{i_{1}}\ldots
T_{i_{q-1}}X_{i_{q}}^{-}K_{i_{q}}=-X_{\beta_{q}}^{-}K_{\beta_{q}},$

where

$K_{\beta_{q}}=T_{i_{1}}\ldots T_{i_{q-1}}K_{i_{q}},$

and for $k_{l^{\prime}}+1\leq q\leq D$

$T_{w^{\prime}}X_{\beta_{q}^{\prime\prime}}^{+}=T_{w^{\prime}}T_{i_{k_{l^{\prime}}+1}}\ldots
T_{i_{q-1}}X_{i_{q}}^{+}=X_{\beta_{q}}^{+}.$

The last two identities and commutation relations between elements
$K_{\beta_{q}}$ and quantum root vectors imply

$T_{w^{\prime}}\left((\overline{X}^{+}_{\beta^{\prime}_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots(\overline{X}^{+}_{\beta^{\prime}_{1}})^{(n_{1})}(X^{+}_{\beta^{\prime\prime}_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta^{\prime\prime}_{D}})^{(n_{D})}\right)=$
$=Q_{n_{1},\ldots,n_{k_{l^{\prime}}}}({X}^{-}_{\beta_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots({X}^{-}_{\beta_{1}})^{(n_{1})}(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta_{D}})^{(n_{D})},$

where $Q_{n_{1},\ldots,n_{k_{l^{\prime}}}}$ is a monomial in $K_{1}^{\pm
1},\ldots K_{l}^{\pm 1}$.

Recalling basis (4.1.53), and using the previous formula we deduce that in
(4.1.45)

$T_{w^{\prime}}XT_{w^{\prime}}^{-1}=\sum_{n_{1},\ldots,n_{D}}F^{\prime}_{n_{1},\ldots,n_{D}}({X}^{-}_{\beta_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots({X}^{-}_{\beta_{1}})^{(n_{1})}(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta_{D}})^{(n_{D})},$

where $F^{\prime}_{n_{1},\ldots,n_{D}}\in U_{q}(H)$, and (4.1.45) takes the
form

$(u,(X_{\gamma_{1}}^{+})^{(m_{1})}\ldots(X_{\gamma_{n}}^{+})^{(m_{n})}({X}_{\gamma_{n+1}}^{+})^{(m_{n+1})}\ldots({X}_{\gamma_{l^{\prime}}}^{+})^{(m_{l^{\prime}})}v)=$
(4.1.54)
$=\sum_{n_{1},\ldots,n_{D}}F_{n_{1},\ldots,n_{D}}(u,T_{\gamma_{1}}\ldots
T_{\gamma_{n}}{T}_{\gamma_{n+1}}\ldots{T}_{\gamma_{l^{\prime}}}({X}^{-}_{\beta_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots({X}^{-}_{\beta_{1}})^{(n_{1})}(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta_{D}})^{(n_{D})}v),$

where $F_{n_{1},\ldots,n_{D}}\in\mathbb{C}[q,q^{-1}]$. This proves the first
formula in the statement of this lemma.

To justify the last two formulas in the statement of the lemma we observe that
${T}_{\gamma_{l^{\prime}}}\ldots{T}_{\gamma_{n+1}}T_{\gamma_{n}}\ldots
T_{\gamma_{1}}$, $T_{s}^{-1}$ and $\overline{T}_{s}^{-1}$ act as the same
transformations of $\mathfrak{h}\subset U_{h}(\mathfrak{h})$ and apply Lemma
2.7.4. This completes the proof.

∎

In the course of the proof of the previous lemma we obtained the following
result.

######  Corollary 4.1.8.

The products

$({X}^{-}_{\beta_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots({X}^{-}_{\beta_{1}})^{(n_{1})}(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta_{D}})^{(n_{D})}$

or

$(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})}\ldots(X^{+}_{\beta_{D}})^{(n_{D})}({X}^{-}_{\beta_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})}\ldots({X}^{-}_{\beta_{1}})^{(n_{1})}$

form a $U_{q}^{res}(H)$–basis in the subalgebra
$U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$ in $U_{q}^{res}(\mathfrak{g})$
generated over $U_{q}^{res}(H)$ by the elements

$({X}^{-}_{\beta_{k_{l^{\prime}}}})^{(n_{k_{l^{\prime}}})},\ldots,({X}^{-}_{\beta_{1}})^{(n_{1})},(X^{+}_{\beta_{k_{l^{\prime}}+1}})^{(n_{k_{l^{\prime}}+1})},\ldots,(X^{+}_{\beta_{D}})^{(n_{D})},n_{i}\in\mathbb{N},i=1,\ldots,D.$

###  4.2 Some auxiliary results on the quantized algebra of regular functions
on an algebraic Poisson–Lie group

In this short section we give several formulas related to the adjoint action
and commutation relations in the algebra $\mathbb{C}_{\mathcal{B}}[G]$.

Firstly, following [12], Theorem I.8.16 we recall the commutation relations in
the algebra $\mathbb{C}_{\mathcal{B}}[G]$ which follow from the fact that
$U_{h}^{s}(\mathfrak{g})$ is quasitriangular. Namely, if $V$, $V^{\prime}$ are
finite rank representations of $U_{h}(\mathfrak{g})$,
$(V)_{\eta},(V^{\prime})_{\rho}$, $(V)_{\beta},(V^{\prime})_{\gamma}$ their
weight subspaces of weights $\eta,\rho,\beta$ and $\gamma$, respectively, and
$v\in(V)_{\eta},v_{1}\in(V^{\prime})_{\rho}$,
$u\in(V)_{\beta},u_{1}\in(V^{\prime})_{\gamma}$ then evaluating the identity
$\Delta_{s}^{opp}(x)\mathcal{R}^{s}=\mathcal{R}^{s}\Delta_{s}(x)$ on the
matrix element $(u,\cdot v)\otimes(u_{1},\cdot v_{1})$ and recalling formula
(2.6.11) we obtain

$q^{((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)\eta^{\vee},\rho^{\vee})}\left((u_{1},\cdot
v_{1})\otimes(u,\cdot v)+\sum_{\nu,i}(u_{1},\cdot
u_{\nu,i}v_{1})\otimes(u,\cdot u_{-\nu,i}v)\right)=$ (4.2.1)
$=q^{((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)\beta^{\vee},\gamma^{\vee})}(u,\cdot
v)\otimes(u_{1},\cdot v_{1})+\sum_{\nu,i}q^{((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)(\beta^{\vee}+\nu^{\vee}),\gamma^{\vee}-\nu^{\vee})}(\omega(u_{-\nu,i})u,\cdot
v))\otimes(\omega(u_{\nu,i})u_{1},\cdot v_{1}),$

where

$u_{-\nu,i}=c_{-\nu,i}f_{\beta_{1}}^{(n_{1})}\ldots
f_{\beta_{D}}^{(n_{D})},n_{1}\beta_{1}+\ldots+n_{D}\beta_{D}=\nu,$
$u_{\nu,i}=c_{\nu,i}e_{\beta_{1}}^{n_{1}}\ldots
e_{\beta_{D}}^{n_{D}},n_{1}\beta_{1}+\ldots+n_{D}\beta_{D}=\nu,$

$c_{\pm\nu,i}\in\mathcal{B}$, and similarly evaluating the identity
$\Delta_{s}^{opp}(x){\mathcal{R}^{s}_{21}}^{-1}={\mathcal{R}^{s}_{21}}^{-1}\Delta_{s}(x)$
on the matrix element $(u,\cdot v)\otimes(u_{1},\cdot v_{1})$ we get

$q^{((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-id)\eta^{\vee},\rho^{\vee})}\left((u_{1},\cdot
v_{1})\otimes(u,\cdot v)+\sum_{\nu,i}(u_{1},\cdot
u_{-\nu,i}^{\prime}v_{1})\otimes(u,\cdot u_{\nu,i}^{\prime}v)\right)=$
$=q^{((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-id)\beta^{\vee},\gamma^{\vee})}(u,\cdot
v)\otimes(u_{1},\cdot v_{1})+\sum_{\nu,i}q^{((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-id)(\beta^{\vee}+\nu^{\vee}),\gamma^{\vee}-\nu^{\vee})}(\omega(u_{\nu,i}^{\prime})u,\cdot
v))\otimes(\omega(u_{-\nu,i}^{\prime})u_{1},\cdot v_{1}),$

where

$u_{-\nu,i}^{\prime}=c_{-\nu,i}^{\prime}f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{1}}^{(n_{1})},n_{1}\beta_{1}+\ldots+n_{D}\beta_{D}=\nu,$
$u_{\nu,i}^{\prime}=c_{\nu,i}^{\prime}e_{\beta_{D}}^{n_{D}}\ldots
e_{\beta_{1}}^{n_{1}},n_{1}\beta_{1}+\ldots+n_{D}\beta_{D}=\nu,$

$c_{\pm\nu,i}^{\prime}\in\mathcal{B}$.

If
$v=T_{s^{-1}}v_{\lambda}\in(V)_{s^{-1}\lambda},v_{1}=T_{s^{-1}}v_{\mu}\in(V^{\prime})_{s^{-1}\mu}$,
$u\in(V)_{\beta},u_{1}=v_{\mu}\in(V^{\prime})_{\mu}$, where $v_{\lambda}\in V$
and $v_{\mu}\in V^{\prime}$ are highest weight vectors, then the previous
identity yields

$q^{((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-id)\lambda^{\vee},\mu^{\vee})}(v_{\mu},\cdot
T_{s^{-1}}v_{\mu})\otimes(u,\cdot T_{s^{-1}}v_{\lambda})=$ (4.2.2)
$=q^{((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-id)\beta^{\vee},\mu^{\vee})}(u,\cdot
T_{s^{-1}}v_{\lambda})\otimes(v_{\mu},\cdot T_{s^{-1}}v_{\mu}).$

The next lemma shows how the adjoint action behaves with respect to the
multiplication in $\mathbb{C}_{\mathcal{B}}[G]$.

######  Lemma 4.2.1.

For any $f,g\in\mathbb{C}_{\mathcal{B}}[G],x\in
U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ we have

${\rm Ad}_{0}x(f\otimes g)={\rm Ad}_{0}x^{2}f\otimes
g((\omega_{0}S_{s}^{-1})(x^{1})\cdot\omega_{0}x^{3}),$

where $\Delta_{s}^{2}x=(\Delta_{s}\otimes id)\Delta_{s}x=x^{1}\otimes
x^{2}\otimes x^{3}$ in the Sweedler notation.

In particular,

${\rm Ad}_{0}f_{\beta}(f\otimes g)=f\otimes
g((\omega_{0}S_{s}^{-1})(f_{\beta})\cdot)+{\rm Ad}_{0}f_{\beta}f\otimes
g(G_{\beta}\cdot)+\sum_{i}{\rm Ad}_{0}x_{i}f\otimes
g((\omega_{0}S_{s}^{-1})(y_{i})\cdot)+$ (4.2.3) $+f(G_{\beta}\cdot
G_{\beta}^{-1})\otimes g(G_{\beta}\cdot\omega_{0}(f_{\beta}))+\sum_{i}{\rm
Ad}_{0}y_{i}^{2}f\otimes
g((\omega_{0}S_{s}^{-1})(y_{i}^{1})\cdot\omega_{0}x_{i})=$ $=f\otimes
g((\omega_{0}S_{s}^{-1})(f_{\beta})\cdot)+{\rm Ad}_{0}f_{\beta}f\otimes
g(G_{\beta}\cdot)+f(G_{\beta}\cdot G_{\beta}^{-1})\otimes
g(G_{\beta}\cdot\omega_{0}(f_{\beta}))+$ $+\sum_{i}{\rm Ad}_{0}y_{i}f\otimes
g(G_{\beta}\cdot\omega_{0}x_{i})+\sum_{i}{\rm Ad}_{0}x_{i}^{1}f\otimes
g((\omega_{0}S_{s}^{-1})(y_{i})\cdot\omega_{0}x_{i}^{2})$

and

${\rm Ad}_{0}f_{\beta}^{(n)}(f\otimes
g)=\sum_{k=0}^{n}\sum_{p=0}^{n-k}q_{\beta}^{-k(n-k)-p(n-k-p)}{\rm
Ad}_{0}(G_{\beta}^{-k}f_{\beta}^{(p)})(f)\otimes
g(\omega_{0}S_{s}^{-1}(G_{\beta}^{-k-p}f_{\beta}^{(n-k-p)})\cdot\omega_{0}(f_{\beta}^{(k)}))+$
$+\sum_{k=0}^{n-1}\sum_{i}q_{\beta}^{-k(n-k)}{\rm
Ad}_{0}(G_{\beta}^{-k}x_{i}^{(n-k)})(f)\otimes
g((\omega_{0}S_{s}^{-1})(G_{\beta}^{-k}y_{i}^{(n-k)})\cdot\omega_{0}(f_{\beta}^{(k)}))+$
(4.2.4) $+\sum_{i}{\rm Ad}_{0}({y_{i}^{(n)}}^{2})(f)\otimes
g((\omega_{0}S_{s}^{-1})({y_{i}^{(n)}}^{1})\cdot\omega_{0}(x_{i}^{(n)})),$

where $G_{\beta}$, $x_{i}$, $y_{i}$, $x_{i}^{(p)}$, $y_{i}^{(p)}$ are defined
in (2.7.12) and (2.7.13), and $\Delta_{s}(x_{i})=x_{i}^{1}\otimes x_{i}^{2}$,
$\Delta_{s}(y_{i})=y_{i}^{1}\otimes y_{i}^{2}$,
$\Delta_{s}(y_{i}^{(p)})={y_{i}^{(p)}}^{1}\otimes{y_{i}^{(p)}}^{2}$ in the
Sweedler notation.

###### Proof.

Denote using the Sweedler notation

$\Delta_{s}^{3}x=(\Delta_{s}\otimes id\otimes id)(\Delta_{s}\otimes
id)\Delta_{s}x=x^{1}\otimes x^{2}\otimes x^{3}\otimes x^{4}$

and observe that the definition of ${\rm Ad}^{\prime}$ implies that for any
$x,z\in U_{\mathcal{B}}^{s,res}(\mathfrak{g})$

$\Delta_{s}^{opp}({\rm Ad}^{\prime}xz)=(x^{2}\otimes x^{1})(z^{2}\otimes
z^{1})(S_{s}x^{3}\otimes S_{s}x^{4})={\rm Ad}^{\prime}x^{2}z^{2}\otimes
x^{1}z^{1}S_{s}x^{3}.$

Let $z=S_{s}\omega_{0}y$, $y\in U_{\mathcal{B}}^{s,res}(\mathfrak{g})$. Then,
since $\omega_{0}S_{s}^{-1}$ is an algebra homomorphism and a coalgebra anti-
homomorphism, we deduce

$\Delta_{s}\omega_{0}S_{s}^{-1}({\rm
Ad}^{\prime}xS_{s}\omega_{0}y)=(\omega_{0}S_{s}^{-1}\otimes\omega_{0}S_{s}^{-1})\Delta_{s}^{opp}({\rm
Ad}^{\prime}xz)=(\omega_{0}S_{s}^{-1}\otimes\omega_{0}S_{s}^{-1}){\rm
Ad}^{\prime}x^{2}z^{2}\otimes x^{1}z^{1}S_{s}x^{3}=$
$=(\omega_{0}S_{s}^{-1}\otimes\omega_{0}S_{s}^{-1}){\rm
Ad}^{\prime}x^{2}(S_{s}\omega_{0})(y^{1})\otimes
x^{1}(S_{s}\omega_{0})(y^{2})S_{s}x^{3}=\omega_{0}S_{s}^{-1}{\rm
Ad}^{\prime}x^{2}((S_{s}\omega_{0})(y^{1}))\otimes(\omega_{0}S_{s}^{-1})(x^{1})y^{2}\omega_{0}x^{3}.$

Evaluating the last identity on $f\otimes g$ we get the first formula in the
statement of the lemma. (4.2.3) and (4.2.4) are obtained from it using
(2.7.12) and (2.7.13).

∎

###  4.3 Properties of the quantized vanishing ideal of the level surface of
the moment map for q-W–algebras

In this section we study the properties of the quantum counterpart
${\overline{J}^{1}_{\mathcal{B}}}\subset\mathbb{C}_{\mathcal{B}}[G]$ of the
vanishing ideal of the variety $\overline{N_{-}ZHsM_{-}^{s}}$, and the
properties of a $\mathcal{B}$–submodule
$J^{1}_{\mathcal{B}}\subset{\overline{J}^{1}_{\mathcal{B}}}$ which is a
quantum counterpart of the vanishing ideal of
$\overline{N_{-}sZHN_{-}^{\prime}}\supset\overline{N_{-}ZHsM_{-}^{s}}$.

We start with a technical lemma which will allow us to use properties of the
vanishing ideals of the varieties $\overline{N_{-}ZHsM_{-}^{s}}$ and
$\overline{N_{-}sZHN_{-}^{\prime}}$ to prove some properties of their quantum
counterparts ${\overline{J}^{1}_{\mathcal{B}}}$ and $J^{1}_{\mathcal{B}}$.

######  Lemma 4.3.1.

Let $X$ be a free $\mathcal{B}$–module, $V\subset V^{\prime}\subset X$ two its
submodules such that $V=V^{\prime}$ mod $(q^{\frac{1}{dr^{2}}}-1)V^{\prime}$.
Let $V_{q}=V\otimes_{\mathcal{B}}{\mathbb{C}}(q^{\frac{1}{dr^{2}}})$,
$V_{q}^{\prime}=V^{\prime}\otimes_{\mathcal{B}}{\mathbb{C}}(q^{\frac{1}{dr^{2}}})$.
Then $V_{q}=V^{\prime}_{q}$.

###### Proof.

First observe that by Theorem 6.5 in [92] $V^{\prime}\subset X$ is
$\mathcal{B}$–free, and $V\subset V^{\prime}$ is a free submodule of
$V^{\prime}$. Let $e_{a}$, $a\in A$ be a basis of $V^{\prime}$. Then as shown
in the proof of Theorem 6.5 in [92] $V$ has a basis elements of which has the
form $f_{a}=\sum_{b\leq a}c_{a}^{b}e_{b}$, $a\in B\subset A$, where
$c_{a}^{a}\neq 0$, $c_{a}^{b}\in\mathcal{B}$. Since $V=V^{\prime}$ mod
$(q^{\frac{1}{dr^{2}}}-1)V^{\prime}$ we must have $B=A$, and $c_{a}^{a}\neq 0$
mod $(q^{\frac{1}{dr^{2}}}-1)$, i.e.
$c_{a}^{a}=1+(q^{\frac{1}{dr^{2}}}-1)h_{a}$, $h_{a}\in\mathcal{B}$.

Now using a simple transfinite induction we can express the elements of the
basis $e_{a}$ regarded as elements of $V^{\prime}_{q}$ in terms of $f_{a}$.
Indeed, if $a_{0}\in A$ is the minimal element then
$e_{a_{0}}={c_{a_{0}}^{a_{0}}}^{-1}f_{a_{0}}$, and assuming that
$e_{d}=\sum_{b\leq d}g_{d}^{b}f_{b}$,
$g_{d}^{b}\in{\mathbb{C}}(q^{\frac{1}{dr^{2}}})$, $g_{d}^{d}\neq 0$ holds for
all $d<a$ we get
$e_{a}={c_{a}^{a}}^{-1}(f_{a}-\sum_{b<a}c_{a}^{b}e_{b})={c_{a}^{a}}^{-1}(f_{a}-\sum_{b<a}c_{a}^{b}\sum_{h\leq
b}g_{b}^{h}f_{h})$, where all sums are finite. This establishes the base of
the induction and completes the proof.

∎

The following lemma describes ${\overline{I}^{1}_{\mathcal{B}}}$ as a
vanishing ideal, in complete analogy with the classical situation.

######  Lemma 4.3.2.

Let ${\overline{J}^{1}_{\mathcal{B}}}$ be the left ideal in
$\mathbb{C}_{\mathcal{B}}[G]$ generated by the elements $(u,\cdot
v)\in\mathbb{C}_{\mathcal{B}}[G]$, where $u$ is a highest weight vector in a
finite rank representation $V$ of $U_{h}(\mathfrak{g})$, and $v\in V$ is such
that $(u,T_{s}xv)=0$ for any $x\in U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$.

Let $U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$ be the subalgebra in
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ generated by the elements
$f_{\beta}^{(n)}$, $\beta\in\Delta_{\mathfrak{m}_{+}}$, $n\in\mathbb{N}$. Then
${\overline{J}^{1}_{\mathcal{B}}}$ and
${\overline{I}^{1}_{\mathcal{B}}}\supset{\overline{J}^{1}_{\mathcal{B}}}$ are
stable under the ${\rm Ad}_{0}$-action of
$U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$. Thus
$\mathbb{C}_{\mathcal{B}}[G]/{\overline{J}^{1}_{\mathcal{B}}}$ and
$\mathbb{C}_{\mathcal{B}}[G]/{\overline{I}^{1}_{\mathcal{B}}}$ are naturally
equipped with the $U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$–action induced
by the ${\rm Ad}_{0}$-action of $U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$ on
$\mathbb{C}_{\mathcal{B}}[G]$.

Let
$\Delta_{\mathfrak{m}_{+}}^{s}=\Delta_{\mathfrak{m}_{+}}\cap\Delta_{s}^{s}$
and $\Delta_{0}^{+}=\Delta_{0}\cap\Delta_{+}$. Both
$\Delta_{\mathfrak{m}_{+}}^{s},\Delta_{0}^{+}\subset\Delta_{+}$ are minimal
segments. Denote by $U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s})$ and
$U_{q}^{res}(\Delta_{0}^{+})$ the subalgebras of $U_{q}^{res}(\mathfrak{g})$
corresponding to $-\Delta_{\mathfrak{m}_{+}}^{s}$ and $\Delta_{0}^{+}$,
respectively. Let ${\overline{J}^{1}_{\mathcal{B}}}^{\prime}$ be the left
ideal in $\mathbb{C}_{\mathcal{B}}[G]$ generated by the elements vanishing on
$xT_{s}hz_{+}x^{\prime}$ with arbitrary $x\in
U_{q}^{res}(\mathfrak{n}_{-}),z_{+}\in
U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U_{q}^{res}(H)$. Denote
${\overline{J}^{1}_{q}}={\overline{J}^{1}_{\mathcal{B}}}\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$,
${\overline{J}^{1}_{q}}^{\prime}={\overline{J}^{1}_{\mathcal{B}}}^{\prime}\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$.
Then
${\overline{J}^{1}_{\mathcal{B}}}\subset{\overline{J}^{1}_{\mathcal{B}}}^{\prime}$
and ${\overline{J}^{1}_{q}}={\overline{J}^{1}_{q}}^{\prime}$. Thus
${\overline{I}^{1}_{\mathcal{B}}}={\overline{J}^{1}_{q}}\cap\mathbb{C}_{\mathcal{B}}[G]={\overline{J}^{1}_{q}}^{\prime}\cap\mathbb{C}_{\mathcal{B}}[G]$.

Moreover,
${\overline{J}^{1}_{\mathcal{B}}}={\overline{J}^{1}_{\mathcal{B}}}^{\prime}=\overline{J}$
mod $(q^{\frac{1}{dr^{2}}}-1){\overline{J}^{1}_{\mathcal{B}}}$, where
$\overline{J}$ is the vanishing ideal of $\overline{N_{-}sZHM_{-}^{s}}$.

###### Proof.

From formula (4.2.4) with $g(\cdot)=(u,\cdot
v)\in\mathbb{C}_{\mathcal{B}}[G]$, where $u$ is a highest weight vector in a
finite rank representation $V$ of $U_{h}(\mathfrak{g})$, and $v\in V$ is such
that $(u,T_{s}xv)=0$ for any $x\in U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$,
and arbitrary $f$ we have, since $u$ is a highest weight vector,

${\rm Ad}_{0}f_{\beta}^{(n)}(f\otimes
g)=\sum_{k=0}^{n}\sum_{p=0}^{n-k}q_{\beta}^{-k(n-k)-p(n-k-p)}{\rm
Ad}_{0}(G_{\beta}^{-k}f_{\beta}^{(p)})(f)\otimes
g(\omega_{0}S_{s}^{-1}(G_{\beta}^{-k-p}f_{\beta}^{(n-k-p)})\cdot\omega_{0}(f_{\beta}^{(k)}))+$
$+\sum_{k=0}^{n-1}\sum_{i}q_{\beta}^{-k(n-k)}{\rm
Ad}_{0}(G_{\beta}^{-k}x_{i}^{(n-k)})(f)\otimes
g((\omega_{0}S_{s}^{-1})(G_{\beta}^{-k}y_{i}^{(n-k)})\cdot\omega_{0}(f_{\beta}^{(k)}))+$
(4.3.1) $+\sum_{i}{\rm Ad}_{0}({y_{i}^{(n)}}^{2})(f)\otimes
g((\omega_{0}S_{s}^{-1})({y_{i}^{(n)}}^{1})\cdot\omega_{0}(x_{i}^{(n)}))=$
$=\sum_{k=0}^{n}q_{\beta}^{-k(n-k)}{\rm
Ad}_{0}(G_{\beta}^{-k}f_{\beta}^{(n-k)})(f)\otimes
g(G_{\beta}^{n}\cdot\omega_{0}(f_{\beta}^{(k)})).$

Since $u$ is a highest weight vector and
$U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$ is invariant under multiplication
by $\omega_{0}((X^{-}_{\beta})^{(k)})$ from the right,
$f_{\beta}=q^{-K\beta^{\vee}}X^{-}_{\beta}$, and $q^{-K\beta^{\vee}}$
normalizes $w^{\prime}(\mathfrak{b}_{+})$, elements
$g(G_{\beta}^{n}\cdot\omega_{0}(f_{\beta}^{(k)}))$ have the same properties as
$g$. Now from the definition of ${\overline{J}^{1}_{\mathcal{B}}}$ it follows
that ${\rm Ad}_{0}f_{\beta}^{(n)}(f\otimes
g)\in{\overline{J}^{1}_{\mathcal{B}}}$. Since elements $f\otimes g$, with
$f,g\in\mathbb{C}_{\mathcal{B}}[G]$ as above, span
${\overline{J}^{1}_{\mathcal{B}}}$ we deduce ${\rm
Ad}_{0}f_{\beta}^{(n)}({\overline{J}^{1}_{\mathcal{B}}})\subset{\overline{J}^{1}_{\mathcal{B}}}$.

If we naturally extend the ${\rm Ad}_{0}$-action of
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ to $\mathbb{C}_{q}[G]$ the last
inclusion implies ${\rm
Ad}_{0}f_{\beta}^{(n)}({\overline{I}^{1}_{q}})\subset{\overline{I}^{1}_{q}}$,
and hence ${\rm
Ad}_{0}f_{\beta}^{(n)}({\overline{I}^{1}_{\mathcal{B}}})\subset{\overline{I}^{1}_{\mathcal{B}}}$
as clearly ${\rm
Ad}_{0}f_{\beta}^{(n)}(\mathbb{C}_{\mathcal{B}}[G])\subset\mathbb{C}_{\mathcal{B}}[G]$
and ${\rm Ad}_{0}f_{\beta}^{(n)}({\overline{I}^{1}_{\mathcal{B}}})={\rm
Ad}_{0}f_{\beta}^{(n)}({\overline{I}^{1}_{q}}\cap\mathbb{C}_{\mathcal{B}}[G])\subset{\overline{I}^{1}_{q}}\cap\mathbb{C}_{\mathcal{B}}[G]={\overline{I}^{1}_{\mathcal{B}}}$.

Now we justify the second claim of the lemma. Let $(u,\cdot
v)\in\mathbb{C}_{\mathcal{B}}[G]$ be such that $u$ is a highest weight vector
in a finite rank representation $V$ of $U_{h}(\mathfrak{g})$, and $v\in V$.
First note that if
$\alpha\in\Delta_{\mathfrak{m}_{+}}\setminus\Delta_{\mathfrak{m}_{+}}^{s}$
then $s\alpha\in\Delta_{+}$ and if $\alpha\in\Delta_{0}^{+}$ then
$s\alpha=\alpha$. Therefore applying the Poincaré–Birkhoff–Witt theorem for
$U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$ as stated in Corollary 4.1.8 and
Lemma 2.4.2 we obtain that for $x\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$

$(u,T_{s}xv)=(u,T_{s}\sum_{i}z^{+}_{i}x^{\prime}_{i}v),z_{i}^{+}\in
U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}_{i}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}).$

Since for all $i$ $z^{+}_{i}x^{\prime}_{i}\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$ we deduce that $(u,T_{s}xv)=0$ for
any $x\in U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$ if and only if
$(u,T_{s}z_{+}x^{\prime}v)=0$ for any $z_{+}\in U_{q}^{res}(\Delta_{0}^{+})$,
$x^{\prime}\in U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s})$, and hence
${\overline{J}^{1}_{\mathcal{B}}}$ coincides with the left ideal generated by
the elements $(u,\cdot v)\in\mathbb{C}_{\mathcal{B}}[G]$ such that $u$ is a
highest weight vector in a finite rank representation $V$ of
$U_{h}(\mathfrak{g})$, and $v\in V$ satisfies $(u,T_{s}z_{+}x^{\prime}v)=0$
for any $z_{+}\in U_{q}^{res}(\Delta_{0}^{+})$, $x^{\prime}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s})$.

Obviously
${\overline{J}^{1}_{\mathcal{B}}}\subset{\overline{J}^{1}_{\mathcal{B}}}^{\prime}$.

On the other hand ${\overline{J}^{1}_{\mathcal{B}}}=\overline{J}$ mod
$(q^{\frac{1}{dr^{2}}}-1){\overline{J}^{1}_{\mathcal{B}}}$, where
$\overline{J}$ is the ideal in $\mathbb{C}[G]$ generated by matrix elements of
the form $(u,\cdot v)\in\mathbb{C}[G]$, where $u$ is a highest weight vector
in a finite-dimensional representation $V$ of $U(\mathfrak{g})$, and $v\in V$
is such that $(u,sz_{+}x^{\prime}v)=0$ for any $z_{+}\in
U(\Delta_{0}^{+})=U_{q}^{res}(\Delta_{0}^{+})/(q^{\frac{1}{dr^{2}}}-1)U_{q}^{res}(\Delta_{0}^{+})$,
$x^{\prime}\in
U(-\Delta_{\mathfrak{m}_{+}}^{s})=U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s})/(q^{\frac{1}{dr^{2}}}-1)U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s})$.
We claim that $\overline{J}$ coincides with the ideal of regular functions
vanishing on the closure of $N_{-}sZHM_{-}^{s}$.

The proof of this claim is based on the description of closed subvarieties in
$B_{-}\setminus G$ in terms of the so–called generalized Plücker coordinates.

Recall that matrix elements of the form $(u,\cdot v)$, where $u$ is a highest
weight vector in a finite-dimensional representation $V$ of $U(\mathfrak{g})$,
and $v\in V$, can be regarded as sections of line bundles on $B_{-}\setminus
G$ (see [35]). They are also called generalized Plücker coordinates on
$B_{-}\setminus G$.

######  Lemma 4.3.3.

Any closed subvariety in $B_{-}\setminus G$ is the zero locus of a finite set
of generalized Plücker coordinates.

###### Proof.

Indeed, the flag variety $B_{-}\setminus G$ can be realized as the $G$–orbit
$\mathcal{O}$ of the line $[u]$ defined by a non–zero lowest weight vector $u$
in the projectivisation $P(V^{*})$ of a finite-dimensional irreducible
representation $V^{*}$ of $G$ dual to a highest weight irreducible
representation $V$ with a regular dominant highest weight (see e.g. [9],
Section 2 or [43], §4). If we identify $V$ with $V^{*}$ using the
contravariant bilinear form then a highest weight vector $u$ of $V$ becomes a
lowest weight vector in $V^{*}$ and the class $[g]\in B_{-}\setminus G$ of an
element $g\in G$ corresponds to
$\omega(g)[u]=[\omega(g)u]\in\mathcal{O}\subset P(V^{*})$.

Now let $v\in V$. Then

$(u,gv)=(\omega(g)u,v)=(x,v),x=\omega(g)u\in V^{*},$ (4.3.2)

so that $[x]=[\omega(g)u]\in\mathcal{O}\subset P(V^{*})$.

Any $y\in V^{*}$ can be written in the form $y=\sum_{n=1}^{S}c_{n}e_{n}\in
V^{*}$, where $e_{n}$, $n=1,\ldots,S$ is a weight basis of $V^{*}$. The
functions $\phi_{n}(y)=c_{n}$ are linear coordinates on $V^{*}$. Moreover,
since $V\simeq V^{*}$ one can find elements $v_{n}\in V$ such that
$\phi_{n}(y)=(y,v_{n})$. In particular, $\phi_{n}$ generate the algebra of
polynomial functions on $V^{*}$, and any closed subvariety in $P(V^{*})$, and
hence in $\mathcal{O}$, is the zero locus of a finite collection of some
polynomials homogeneous in $\phi_{n}$ (see [47], Ch. 1, §2).

If $f(\phi_{1},\ldots\phi_{S})$ is such a polynomial of degree $d$ then the
equation $f(\phi_{1}(y),\ldots\phi_{S}(y))=0$ is well–defined in $P(V^{*})$,
i.e. if $y\in V^{*}$ is its solution then any element of the line $[y]\in
P(V^{*})$ is also its solution. Now using (4.3.2), the definition of
$\phi_{n}$ and the homogeneity of $f$ we deduce that for $[x]\in\mathcal{O}$
$f(\phi_{1}(x),\ldots\phi_{S}(x))=0$ if and only if
$f((u,gv_{1}),\ldots,(u,gv_{S}))=0$ for any $g\in G$, where $[g]\in
B_{-}\setminus G$ corresponds to $[x]$ under the isomorphism
$\mathcal{O}\simeq B_{-}\setminus G$. Note also that using algebraic rules for
matrix elements we immediately obtain that $f((u,\cdot v_{1}),\ldots,(u,\cdot
v_{S}))$ is a generalized Plücker coordinate defined using the representation
$V^{\otimes d}$ and the highest weight vector

$u^{\otimes d}=\underbrace{u\otimes\ldots\otimes u}_{d~{}{\rm times}}\in
V^{\otimes d},$

i.e. $f((u,\cdot v_{1}),\ldots,(u,\cdot v_{S}))=(u^{\otimes d},\cdot v)$ for
some $v\in V^{\otimes d}$. The last two facts imply that any closed subvariety
in $B_{-}\setminus G$ is the zero locus of a finite set of generalized Plücker
coordinates.

∎

We proceed with the proof of Lemma 4.3.2. Let $Z_{\pm}=Z\cap N_{\pm}$. Then
$Z_{-}HZ_{+}$ is dense in $ZH$, and hence the closure of $N_{-}sZHM_{-}^{s}$
coincides with the closure of $N_{-}sZ_{-}HZ_{+}M_{-}^{s}$. Since $s$ fixes
$\Delta_{0}$ we can also write
$N_{-}sZ_{-}HZ_{+}M_{-}^{s}=B_{-}sZ_{+}M_{-}^{s}$ and

$\overline{N_{-}sZHM_{-}^{s}}=\overline{B_{-}sZ_{+}M_{-}^{s}}.$ (4.3.3)

Since $B_{-}sZ_{+}M_{-}^{s}$ is the preimage of $B_{-}\setminus
B_{-}sZ_{+}M_{-}^{s}$ with respect to the natural projection $G\rightarrow
B_{-}\setminus G$, $(u,gv)=0$ for any $g\in B_{-}sZ_{+}M_{-}^{s}$ if and only
if $(u,\cdot v)$ regarded as a Plücker coordinate vanishes on $B_{-}\setminus
B_{-}sZ_{+}M_{-}^{s}$. By Lemma 4.3.3 the closure of $B_{-}\setminus
B_{-}sZ_{+}M_{-}^{s}$ is the zero locus of a finite set of generalized Plücker
coordinates vanishing on $B_{-}\setminus B_{-}sZ_{+}M_{-}^{s}$ which is dense
in its closure. Hence the vanishing ideal of $\overline{B_{-}sZ_{+}M_{-}^{s}}$
is generated by the matrix elements $(u,\cdot v)$ which correspond to these
Plücker coordinates.

If $(u,\cdot v)$ is any generalized Plücker coordinate vanishing on
$B_{-}\setminus B_{-}sZ_{+}M_{-}^{s}$ all its derivatives in the directions
tangent to $B_{-}\setminus B_{-}sZ_{+}M_{-}^{s}$ must vanish as well.
Therefore $(u,sz_{+}x^{\prime}v)=0$ for any $x^{\prime}\in
U(-\Delta_{\mathfrak{m}_{+}}^{s})$ and $z_{+}\in U(\Delta_{0}^{+})$.

Conversely, any generalized Plücker coordinate of the form $(u,\cdot
v)\in\mathbb{C}[G]$, where $u$ is a highest weight vector in a finite-
dimensional representation $V$ of $U(\mathfrak{g})$ and $v\in V$, obeying the
last property gives rise to a generalized Plücker coordinate vanishing on
$B_{-}\setminus B_{-}sZ_{+}M_{-}^{s}$ and hence on the closure of
$B_{-}\setminus B_{-}sZ_{+}M_{-}^{s}$.

Thus the closure of $B_{-}\setminus B_{-}sZ_{+}M_{-}^{s}$ is the zero locus of
the set of the generalized Plücker coordinates of the above described type,
and by the results proved above the corresponding matrix elements also
generate the vanishing ideal of $\overline{B_{-}sZ_{+}M_{-}^{s}}$. By (4.3.3)
we deduce that this ideal also coincides with the vanishing ideal of
$\overline{N_{-}sZHM_{-}^{s}}$. Hence the vanishing ideal of
$\overline{N_{-}sZHM_{-}^{s}}$ is generated by functions of the form $(u,\cdot
v)\in\mathbb{C}[G]$, where where $u$ is a highest weight vector in a finite-
dimensional representation $V$ of $U(\mathfrak{g})$, and $v\in V$ satisfies
$(u,sz_{+}x^{\prime}v)=0$ for any $x^{\prime}\in
U(-\Delta_{\mathfrak{m}_{+}}^{s})$ and $z_{+}\in U(\Delta_{0}^{+})$, i.e.
$\overline{J}$ coincides with the ideal of regular functions vanishing on the
closure of $N_{-}sZHM_{-}^{s}$.

All derivatives of elements of $\overline{J}$ in the directions tangent to
$N_{-}sZHM_{-}^{s}$ must vanish. Therefore every element of $\overline{J}$
must vanish on elements $xshz_{+}x^{\prime}$ for any $x\in
U(\mathfrak{n}_{-}),z_{+}\in U(\Delta_{0}^{+}),x^{\prime}\in
U(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U(\mathfrak{h})$.

Conversely, any element of $\mathbb{C}[G]$ vanishing on on elements
$xshz_{+}x^{\prime}$ for any $x\in U(\mathfrak{n}_{-}),z_{+}\in
U(\Delta_{0}^{+}),x^{\prime}\in U(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in
U(\mathfrak{h})$ must also vanish on the closure of $N_{-}sZHM_{-}^{s}$, and
hence from the definition of $\overline{J}$ we deduce that $\overline{J}$
coincides with the ideal generated by the elements vanishing on
$xshz_{+}x^{\prime}$ for arbitrary $x\in U(\mathfrak{n}_{-}),z_{+}\in
U(\Delta_{0}^{+}),x^{\prime}\in U(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in
U(\mathfrak{h})$. Therefore
${\overline{J}^{1}_{\mathcal{B}}}^{\prime}={\overline{J}^{1}_{\mathcal{B}}}=\overline{J}$
mod $(q^{\frac{1}{dr^{2}}}-1){\overline{J}^{1}_{\mathcal{B}}}^{\prime}$.

Note that
${\overline{J}^{1}_{\mathcal{B}}}\subset{\overline{J}^{1}_{\mathcal{B}}}^{\prime}$
are submodules of the $\mathcal{B}$–module $\mathbb{C}_{\mathcal{B}}[G]$ which
is free by Proposition 3.2.6. Therefore by Lemma 4.3.1
${\overline{J}^{1}_{q}}={\overline{J}^{1}_{q}}^{\prime}$. This completes the
proof.

∎

The following lemma will be needed later to cancel some terms in formula
(4.2.4) for the adjoint action.

######  Lemma 4.3.4.

For any $u$ from the augmentation ideal of
$U_{\mathcal{B}}^{s,res}(\mathfrak{n}_{-})$, $u\not\in
U_{\mathcal{B}}^{s,res}(\Delta_{0}^{-})$ we have $(v_{\mu},\cdot
uT_{s^{-1}}v_{\mu})\in{\overline{J}^{1}_{\mathcal{B}}}^{\prime}$.

###### Proof.

It suffices to verify the statement for $u=f_{\beta_{1}}^{(n_{1})}\ldots
f_{\beta_{D}}^{(n_{D})}\neq 1$, $u\not\in
U_{\mathcal{B}}^{s,res}(\Delta_{0}^{-})$. For any $x\in
U_{q}^{res}(\mathfrak{n}_{-}),z_{+}\in
U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U_{q}^{res}(H)$ we have

$(v_{\mu},xT_{s}hz_{+}x^{\prime}uT_{s^{-1}}v_{\mu})=0$

if $x$ is from the augmentation ideal of $U_{q}^{res}(\mathfrak{n}_{-})$. Thus
without loss of generality we can assume that $x=1$. In this case

$(v_{\mu},xT_{s}hz_{+}x^{\prime}uT_{s^{-1}}v_{\mu})=(v_{\mu},T_{s}hz_{+}x^{\prime}uT_{s^{-1}}v_{\mu})=k(v_{\mu},T_{s}z_{+}x^{\prime}uT_{s^{-1}}v_{\mu})=$
$=k(v_{\mu},T_{s}z_{+}x^{\prime}f_{\beta_{1}}^{(n_{1})}\ldots
f_{\beta_{m}}^{(n_{m})}T_{s^{-1}}v_{\mu}),k\in\mathcal{B}.$ (4.3.4)

Let $\\{\beta_{1},\ldots,\beta_{m}\\}=\Delta_{+}\cap\Delta_{+}^{s}$. Since
$s(\beta_{p})\in\Delta_{-}$ for $p>m$, the right hand side of (4.3.4) vanishes
if there exists $p>m$ such that $n_{p}\neq 0$.

If $n_{p}=0$ for all $p>m$ then, assuming without loss of generality that
$z_{+}$ and $x^{\prime}$ are monomials in quantum root vectors, we infer that
the weight of the element $z_{+}x^{\prime}f_{\beta_{1}}^{(n_{1})}\ldots
f_{\beta_{m}}^{(n_{m})}$ is not zero as
$(\overline{h}_{0},wt(z_{+}x^{\prime}f_{\beta_{1}}^{(n_{1})}\ldots
f_{\beta_{m}}^{(n_{m})}))<0$. Hence the right hand side of (4.3.4) vanishes in
this case as well.

Thus $(v_{\mu},T_{s}hz_{+}x^{\prime}uT_{s^{-1}}v_{\mu})=0$ in all cases and by
the definition of ${\overline{J}^{1}_{\mathcal{B}}}^{\prime}$ given in Lemma
4.3.2 $(v_{\mu},\cdot
uT_{s^{-1}}v_{\mu})\in{\overline{J}^{1}_{\mathcal{B}}}^{\prime}$.

∎

Next, as it was promised in the beginning of this section, we define a
$\mathcal{B}$–submodule
${J^{1}_{\mathcal{B}}}\subset{\overline{J}^{1}_{\mathcal{B}}}$ which is a
quantum counterpart of the vanishing ideal of
$\overline{N_{-}sZHN_{-}^{\prime}}\supset\overline{N_{-}ZHsM_{-}^{s}}$. The
next three lemmas are similar to Lemmas 4.3.2 and 4.3.4 for
${\overline{J}^{1}_{\mathcal{B}}}$.

######  Lemma 4.3.5.

Let ${J^{1}_{\mathcal{B}}}$ be the left ideal in $\mathbb{C}_{\mathcal{B}}[G]$
generated by the elements $(u,\cdot v)\in\mathbb{C}_{\mathcal{B}}[G]$, where
$u$ is a highest weight vector in a finite rank representation $V$ of
$U_{h}(\mathfrak{g})$, and $v\in V$ is such that $(u,T_{s}xv)=0$ for any $x\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))U_{q}^{res}(\mathfrak{n}_{-})$.
Denote
$J^{1}_{q}={J^{1}_{\mathcal{B}}}\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$
and ${I^{1}_{\mathcal{B}}}=J^{1}_{q}\cap\mathbb{C}_{\mathcal{B}}[G]$. Then
${J^{1}_{\mathcal{B}}}\subset{\overline{J}^{1}_{\mathcal{B}}}$ and
${I^{1}_{\mathcal{B}}}\subset{\overline{I}^{1}_{\mathcal{B}}}$ are two sided
ideals in $\mathbb{C}_{\mathcal{B}}[G]$ stable under the ${\rm Ad}_{0}$-action
of $U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$. Thus
$\mathbb{C}_{\mathcal{B}}[G]/{J^{1}_{\mathcal{B}}}$ and
$\mathbb{C}_{\mathcal{B}}[G]/{I^{1}_{\mathcal{B}}}$ are naturally equipped
with the $U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$–action induced by the
${\rm Ad}_{0}$-action of $U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$ on
$\mathbb{C}_{\mathcal{B}}[G]$.

###### Proof.

From commutation relations (4.2.1) it follows that ${J^{1}_{\mathcal{B}}}$ is
in fact a two sided ideal. First without loss of generality we can assume that
$v$ belongs to a weight subspace. Indeed, for any given $V$ and $u\in V$ as in
the statement of this lemma $U_{q}(H)$ naturally acts on the subspace of $V$
spanned by elements $v\in V$ such that $(u,T_{s}xv)=0$ for any $x\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))U_{q}^{res}(\mathfrak{n}_{-})$.
Therefore this subspace is $P$–graded.

From (4.2.1) with $u,v$ as in the statement of the lemma and arbitrary
$(u_{1},\cdot v_{1})\in\mathbb{C}_{\mathcal{B}}[G]$ such that
$v\in(V)_{\eta},v_{1}\in(V^{\prime})_{\rho}$,
$u\in(V)_{\beta},u_{1}\in(V^{\prime})_{\gamma}$ we have

$q^{((\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}+id)\eta^{\vee},\rho^{\vee})}\left((u_{1},\cdot
v_{1})\otimes(u,\cdot v)+\sum_{\nu,i}(u_{1},\cdot
u_{\nu,i}v_{1})\otimes(u,\cdot u_{-\nu,i}v)\right)=$ (4.3.5)
$=q^{((\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}+id)\beta^{\vee},\gamma^{\vee})}(u,\cdot
v)\otimes(u_{1},\cdot v_{1})$

since $u$ is a highest weight vector. Observe that
$U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))U_{q}^{res}(\mathfrak{n}_{-})$ is
invariant with respect to the right multiplication by the elements
$u_{-\nu,i}$, and hence $(u,\cdot u_{-\nu,i}v)$ vanishes on elements of the
form $T_{s}x$ for any $x\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))U_{q}^{res}(\mathfrak{n}_{-})$. We
deduce that $(u_{1},\cdot u_{\nu,i}v_{1})\otimes(u,\cdot
u_{-\nu,i}v)\in{J^{1}_{\mathcal{B}}}$, and hence (4.3.5) implies $(u,\cdot
v)\otimes(u_{1},\cdot v_{1})\in{J^{1}_{\mathcal{B}}}$, i.e.
${J^{1}_{\mathcal{B}}}$ is a right ideal as well.

This immediately implies that $J^{1}_{q}$ is a two sided ideal in
$\mathbb{C}_{q}[G]$, and hence
${I^{1}_{\mathcal{B}}}=J^{1}_{q}\cap\mathbb{C}_{\mathcal{B}}[G]$ is a two
sided ideal in $\mathbb{C}_{\mathcal{B}}[G]$.

Similarly to the proof of an analogous property for
${\overline{J}^{1}_{\mathcal{B}}}$ in Lemma 4.3.2 (see formula (4.3.1)) one
can deduce that ${J^{1}_{\mathcal{B}}}$ and ${I^{1}_{\mathcal{B}}}$ are stable
with respect to the ${\rm Ad}_{0}$-action of
$U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$.

∎

Similarly to Lemma 4.3.2 one can describe ${J^{1}_{\mathcal{B}}}$ as a
vanishing ideal.

######  Lemma 4.3.6.

Let
$\Delta_{s}^{\prime}=(\Delta_{\mathfrak{m}_{+}}\cap\Delta_{s}^{s})\cup\\{\alpha\in\Delta_{+}:\alpha>\gamma_{l^{\prime}}\\}$
and $\Delta_{0}^{+}=\Delta_{0}\cap\Delta_{+}$. Both
$\Delta_{s}^{\prime},\Delta_{0}^{+}\subset\Delta_{+}$ are minimal segments.
Denote by $U_{q}^{res}(-\Delta_{s}^{\prime})$ and
$U_{q}^{res}(\Delta_{0}^{+})$ the subalgebras of $U_{q}^{res}(\mathfrak{g})$
corresponding to $-\Delta_{s}^{\prime}$ and $\Delta_{0}^{+}$, respectively.
Let ${J^{1}_{\mathcal{B}}}^{\prime}$ be the left ideal in
$\mathbb{C}_{\mathcal{B}}[G]$ generated by the elements vanishing on
$xT_{s}hz_{+}x^{\prime}$ with arbitrary $x\in
U_{q}^{res}(\mathfrak{n}_{-}),z_{+}\in
U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{s}^{\prime}),h\in U_{q}^{res}(H)$. Denote
${J^{1}_{q}}={J^{1}_{\mathcal{B}}}\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$,
${J^{1}_{q}}^{\prime}={J^{1}_{\mathcal{B}}}^{\prime}\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$.
Then ${J^{1}_{\mathcal{B}}}\subset{J^{1}_{\mathcal{B}}}^{\prime}$ and
${J^{1}_{q}}={J^{1}_{q}}^{\prime}$. In particular,
${I^{1}_{\mathcal{B}}}={J^{1}_{q}}\cap\mathbb{C}_{\mathcal{B}}[G]={J^{1}_{q}}^{\prime}\cap\mathbb{C}_{\mathcal{B}}[G]$.

Moreover, ${J^{1}_{\mathcal{B}}}={J^{1}_{\mathcal{B}}}^{\prime}=J$ mod
$(q^{\frac{1}{dr^{2}}}-1){J^{1}_{\mathcal{B}}}$, where $J$ is the vanishing
ideal of the variety
$\overline{N_{-}sZHN_{-}}=\overline{N_{-}sZHN_{-}^{\prime}}$.

###### Proof.

Let $u$ be a highest weight vector in a finite rank representation $V$ of
$U_{h}(\mathfrak{g})$, $v\in V$.

First note that if
$\alpha\in\Delta_{\mathfrak{m}_{+}}\setminus\Delta_{s}^{\prime}$ then
$s\alpha\in\Delta_{+}$ and if $\alpha\in\Delta_{0}^{+}$ then $s\alpha=\alpha$.
Therefore applying the Poincaré–Birkhoff–Witt theorem for
$U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$ as stated in Corollary 4.1.8 and
Lemma 2.4.2 we obtain that for any $x\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))U_{q}^{res}(\mathfrak{n}_{-})$

$(u,T_{s}xv)=(u,T_{s}\sum_{i}z_{i}^{+}x^{\prime}v),z_{i}^{+}\in
U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}_{i}\in
U_{q}^{res}(-\Delta_{s}^{\prime}).$

Since for all $i$ $z^{+}_{i}x^{\prime}_{i}\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))U_{q}^{res}(\mathfrak{n}_{-})$ we
deduce that $(u,T_{s}xv)=0$ for any $x\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))U_{q}^{res}(\mathfrak{n}_{-})$ if and
only if $(u,T_{s}z_{+}x^{\prime}v)=0$ for any $x^{\prime}\in
U_{q}^{res}(-\Delta_{s}^{\prime})$, $z_{+}\in U_{q}^{res}(\Delta_{0}^{+})$,
and hence ${J^{1}_{\mathcal{B}}}$ coincides with the left ideal generated by
the elements $(u,\cdot v)$, where $u$ is a highest weight vector in a finite
rank representation $V$ of $U_{h}(\mathfrak{g})$, $v\in V$ is such that
$(u,T_{s}z_{+}x^{\prime}v)=0$ for any $z_{+}\in U_{q}^{res}(\Delta_{0}^{+})$,
$x^{\prime}\in U_{q}^{res}(-\Delta_{s}^{\prime})$.

Obviously ${J^{1}_{\mathcal{B}}}\subset{J^{1}_{\mathcal{B}}}^{\prime}$.

On the other hand ${J^{1}_{\mathcal{B}}}=J$ mod
$(q^{\frac{1}{dr^{2}}}-1){J^{1}_{\mathcal{B}}}$, where $J$ is the ideal in
$\mathbb{C}[G]$ generated by matrix elements of the form $(u,\cdot
v)\in\mathbb{C}[G]$, where $u$ is a highest weight vector in a finite-
dimensional representation $V$ of $U(\mathfrak{g})$, and $v\in V$ is such that
$(u,sz_{+}x^{\prime}v)=0$ for any $z_{+}\in
U(\Delta_{0}^{+})=U_{q}^{res}(\Delta_{0}^{+})/(q^{\frac{1}{dr^{2}}}-1)U_{q}^{res}(\Delta_{0}^{+})$,
$x^{\prime}\in
U(-\Delta_{s}^{\prime})=U_{q}^{res}(-\Delta_{s}^{\prime})/(q^{\frac{1}{dr^{2}}}-1)U_{q}^{res}(-\Delta_{s}^{\prime})$.
Similarly to the proof of the second claim of Lemma 4.3.2 one can show that
$J$ coincides with the ideal of regular functions vanishing on
$\overline{N_{-}sZHN_{-}}=\overline{N_{-}sZHN_{-}^{\prime}}=\overline{B_{-}sZ_{+}N_{-}^{\prime}}$.

All derivatives of elements of ${J}$ in the directions tangent to
$N_{-}sZHN_{-}^{\prime}$ must vanish. Therefore every element of $J$ must
vanish on elements $xshz_{+}x^{\prime}$ with arbitrary $x\in
U(\mathfrak{n}_{-}),z_{+}\in U(\Delta_{0}^{+}),x^{\prime}\in
U(-\Delta_{s}^{\prime}),h\in U(\mathfrak{h})$. Conversely, if
$(u^{\prime},\cdot v^{\prime})$ is a regular function such that
$(u^{\prime},xshz_{+}x^{\prime}v^{\prime})=0$ for any $x\in
U(\mathfrak{n}_{-})$, $h\in U(\mathfrak{h})$, $x^{\prime}\in
U(-\Delta_{s}^{\prime})$ and $z_{-}\in U(-\Delta_{0}^{+})$ then
$(u^{\prime},\cdot v^{\prime})$ vanishes on $N_{-}sZHN_{-}^{\prime}$, and
hence on its closure. Thus $J$ coincides with the ideal of regular functions
vanishing on elements $xshz_{+}x^{\prime}$ with arbitrary $x\in
U(\mathfrak{n}_{-}),z_{+}\in U(\Delta_{0}^{+}),x^{\prime}\in
U(-\Delta_{s}^{\prime}),h\in U(\mathfrak{h})$. Therefore
${J^{1}_{\mathcal{B}}}^{\prime}=J$ mod
$(q^{\frac{1}{dr^{2}}}-1){J^{1}_{\mathcal{B}}}^{\prime}$.

Recall that ${J^{1}_{\mathcal{B}}}=J$ mod
$(q^{\frac{1}{dr^{2}}}-1){J^{1}_{\mathcal{B}}}$ and
${J^{1}_{\mathcal{B}}}\subset{J^{1}_{\mathcal{B}}}^{\prime}$. Note also that
${J^{1}_{\mathcal{B}}}$ and ${J^{1}_{\mathcal{B}}}$ are submodules of the
$\mathcal{B}$–module $\mathbb{C}_{\mathcal{B}}[G]$ which is free by
Proposition 3.2.6. Therefore by Lemma 4.3.1
${J^{1}_{q}}={J^{1}_{q}}^{\prime}$. This completes the proof.

∎

The following lemma will be needed later to cancel some terms in formula
(4.2.4) for the adjoint action.

######  Lemma 4.3.7.

For any $\beta\in\Delta_{\mathfrak{m}_{+}}\setminus\Delta_{s^{-1}}^{s}$ and
$n>0$ we have $(v_{\mu},\cdot
f_{\beta}^{(n)}T_{s^{-1}}v_{\mu})\in{J^{1}_{\mathcal{B}}}^{\prime}$.

###### Proof.

For any $x\in U_{q}^{res}(\mathfrak{n}_{-}),z_{+}\in
U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{s}^{\prime}),h\in U_{q}^{res}(H)$ we have

$(v_{\mu},xT_{s}hz_{+}x^{\prime}f_{\beta}^{(n)}T_{s^{-1}}v_{\mu})=0$

if $x$ is from the augmentation ideal of $U_{q}^{res}(\mathfrak{n}_{-})$. Thus
without loss of generality we can assume that $x=1$. We can also assume
without loss of generality that $z_{+}$ and $x^{\prime}$ are monomials in
quantum root vectors. In the proof of Lemma 4.4.3 we shall consider a more
general situation and show that in this case the weight of the element
$T_{s}z_{+}x^{\prime}f_{\beta}^{(n)}T_{s^{-1}}$ is not zero (see the arguments
after formula (4.4.7)). Thus

$(v_{\mu},xT_{s}hz_{+}x^{\prime}f_{\beta}^{(n)}T_{s^{-1}}v_{\mu})=(v_{\mu},T_{s}hz_{+}x^{\prime}f_{\beta}^{(n)}T_{s^{-1}}v_{\mu})=k(v_{\mu},T_{s}z_{+}x^{\prime}f_{\beta}^{(n)}T_{s^{-1}}v_{\mu})=0,$

where $k\in\mathcal{B}$, and hence by Lemma 4.3.6 $(v_{\mu},\cdot
f_{\beta}^{(n)}T_{s^{-1}}v_{\mu})\in{J^{1}_{\mathcal{B}}}^{\prime}$.

∎

###  4.4 Higher quantized vanishing ideals

The next step in our approach is, as it was mentioned in the introduction to
this chapter, the definition and the study of quantum analogues
${\overline{J}^{p}_{\mathcal{B}}}$ and ${J^{p}_{\mathcal{B}}}$ of the
vanishing ideals of the varieties

$\overline{N_{[-\beta_{p},-\beta_{D}]}sZHM_{-}^{s}}\subset\overline{N_{-}sZHM_{-}^{s}},\overline{N_{[-\beta_{p},-\beta_{D}]}sZHN_{-}^{\prime}}\subset\overline{N_{-}sZHN_{-}^{\prime}}.$

Recall that according to Corollary 3.5.5 the vanishing ideals of these
varieties are defined in terms of functions $\phi_{k}$ introduced in Corollary
3.5.4. Therefore firstly we have to define quantum analogues $A_{p}$ of these
functions. This is done with the help of quantum counterparts of matrix
elements (3.5.4). More precisely, by Proposition 8.3 in [45] there exist
integral dominant weights $\mu_{p}$, $p=1,\ldots,D$, and elements $v_{p}\in
V_{\mu_{p}}$ such that $(v_{p},\cdot
v_{\mu_{p}})\in\mathbb{C}_{\mathcal{B}}[G]$, and

$(v_{p},\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{1}}^{(n_{1})})v_{\mu_{p}})=\left\\{\begin{array}[]{ll}1&{\rm
if}~{}f_{\beta_{D}}^{(n_{D})}\ldots f_{\beta_{1}}^{(n_{1})}=f_{\beta_{p}}\\\
0&{\rm otherwise}\end{array}\right.,$ (4.4.1)

where the highest weight vectors $v_{\mu_{p}}\in V_{\mu_{p}}$ are normalized
by the condition $(v_{\mu_{p}},v_{\mu_{p}})=1$.

This definition implies
$\omega(\omega_{0}S_{s}^{-1}(f_{\beta_{p}}))v_{p}=v_{\mu_{p}}$, i.e. $v_{p}$
has weight $\mu_{p}-\beta_{p}$, $(v_{\mu_{p}},\cdot
v_{\mu_{p}})\in\mathbb{C}_{\mathcal{B}}[G]$, and

$(v_{p},\omega_{0}S_{s}^{-1}(f_{\beta_{p}})\cdot
T_{s^{-1}}v_{\mu_{p}})=(v_{\mu_{p}},\cdot T_{s^{-1}}v_{\mu_{p}}).$ (4.4.2)

Let

$A_{p}=(v_{p},\cdot T_{s^{-1}}v_{\mu_{p}}),A_{p}^{0}=(v_{\mu_{p}},\cdot
T_{s^{-1}}v_{\mu_{p}}).$ (4.4.3)

From (4.2.2) we deduce

$(v_{\mu_{p}},\cdot T_{s^{-1}}v_{\mu_{p}})\otimes(v_{p},\cdot
T_{s^{-1}}v_{\mu_{p}})=q^{((-\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)\beta_{p}^{\vee},\mu_{p}^{\vee})}(v_{p},\cdot
T_{s^{-1}}v_{\mu_{p}})\otimes(v_{\mu_{p}},\cdot T_{s^{-1}}v_{\mu_{p}}),$

i.e.

$A_{p}^{0}\otimes A_{p}=q^{((-\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)\beta_{p}^{\vee},\mu_{p}^{\vee})}A_{p}\otimes
A_{p}^{0}.$ (4.4.4)

Now we can define ${\overline{I}^{p}_{\mathcal{B}}}$ and characterize it as a
vanishing ideal.

######  Lemma 4.4.1.

For $p=1,\ldots,c+1$, let ${\overline{J}^{p}_{\mathcal{B}}}$ be the left ideal
in $\mathbb{C}_{\mathcal{B}}[G]$ generated by
${\overline{J}^{1}_{\mathcal{B}}}$ and by $A_{1},\ldots,A_{p-1}$. Let
${\overline{J}^{p}_{\mathcal{B}}}^{\prime}$ be the left ideal in
$\mathbb{C}_{\mathcal{B}}[G]$ generated by the elements vanishing on
$xT_{s}hz_{+}x^{\prime}$ with arbitrary
$x=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}),z_{+}\in U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U_{q}^{res}(H)$,
$n_{i}\in\mathbb{N}$. Let ${\overline{J}^{p}_{\mathcal{B}}}^{\prime\prime}$ be
the left ideal in $\mathbb{C}_{\mathcal{B}}[G]$ generated by the elements
vanishing on $x\overline{T}_{s}hz_{+}x^{\prime}$ with arbitrary
$x=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}),z_{+}\in U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U_{q}^{res}(H)$,
$n_{i}\in\mathbb{N}$.

Denote
${\overline{J}^{p}_{q}}={\overline{J}^{p}_{\mathcal{B}}}\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$,
${\overline{J}^{p}_{q}}^{\prime}={\overline{J}^{p}_{\mathcal{B}}}\otimes_{\mathcal{B}}^{\prime}\mathbb{C}(q^{\frac{1}{dr^{2}}})$,
${\overline{J}^{p}_{q}}^{\prime\prime}={\overline{J}^{p}_{\mathcal{B}}}^{\prime\prime}\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$,
${\overline{I}^{p}_{\mathcal{B}}}={\overline{J}^{p}_{q}}\cap\mathbb{C}_{\mathcal{B}}[G]$.
Then ${\overline{I}^{p}_{\mathcal{B}}}$ is a left ideal in
$\mathbb{C}_{\mathcal{B}}[G]$,
${\overline{J}^{p}_{\mathcal{B}}}\subset{\overline{J}^{p}_{\mathcal{B}}}^{\prime}$,
${\overline{J}^{p}_{\mathcal{B}}}\subset{\overline{J}^{p}_{\mathcal{B}}}^{\prime\prime}$,
${\overline{J}^{p}_{q}}={\overline{J}^{p}_{q}}^{\prime}={\overline{J}^{p}_{q}}^{\prime\prime}$,
and hence
${\overline{I}^{p}_{\mathcal{B}}}={\overline{J}^{p}_{q}}^{\prime}\cap\mathbb{C}_{\mathcal{B}}[G]={\overline{J}^{p}_{q}}^{\prime\prime}\cap\mathbb{C}_{\mathcal{B}}[G]$.

Moreover,
${\overline{J}^{p}_{\mathcal{B}}}={\overline{J}^{p}_{\mathcal{B}}}^{\prime}=\overline{J}^{p}_{1}$
mod $(q^{\frac{1}{dr^{2}}}-1){\overline{J}^{p}_{\mathcal{B}}}^{\prime}$,
${\overline{J}^{p}_{\mathcal{B}}}={\overline{J}^{p}_{\mathcal{B}}}^{\prime\prime}=\overline{J}^{p}_{1}$
mod $(q^{\frac{1}{dr^{2}}}-1){\overline{J}^{p}_{\mathcal{B}}}^{\prime\prime}$,
where $\overline{J}^{p}_{1}$ is the vanishing ideal of
$\overline{N_{[-\beta_{p},-\beta_{D}]}sZHM_{-}^{s}}$.

###### Proof.

Firstly we claim that
${\overline{J}^{p}_{\mathcal{B}}}\subset{\overline{J}^{p}_{\mathcal{B}}}^{\prime}$.
Indeed, any element from ${\overline{J}^{1}_{\mathcal{B}}}$ belongs to
${\overline{J}^{p}_{\mathcal{B}}}^{\prime}\supset{\overline{J}^{1}_{\mathcal{B}}}^{\prime}$
as by Lemma 4.3.2
${\overline{J}^{1}_{\mathcal{B}}}\subset{\overline{J}^{1}_{\mathcal{B}}}^{\prime}$.

Next, for $q<p$ and $x=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}),z_{+}\in U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U_{q}^{res}(H)$,
$n_{i}\in\mathbb{N}$ we have, since $v_{\mu_{q}}$ is a highest weight vector,

$A_{q}(xT_{s}hz_{+}x^{\prime})=(v_{q},xT_{s}hz_{+}x^{\prime}T_{s^{-1}}v_{\mu_{q}})=(v_{q},x\omega_{0}S_{s}^{-1}(y)v_{\mu_{q}})$
(4.4.5)

for some $y\in U_{\mathcal{B}}^{s,res}(\mathfrak{n}_{-})$. We claim that that
the right hand side of (4.4.5) vanishes.

Indeed, if $x\neq 1$ then
$x\omega_{0}S_{s}^{-1}(y)=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}y)$ and $f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}y$ belongs to the right ideal in
$U_{\mathcal{B}}^{s,res}(\mathfrak{n}_{-})$ generated by
$f_{\beta_{u}}^{(n_{u})}$, $u\geq p>q$ and $n_{u}>0$. So by (4.4.1) and by
Lemma 2.7.2 we have

$A_{q}(xT_{s}hz_{+}x^{\prime})=(v_{q},\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}y)v_{\mu_{q}})=0$

in this case.

If $x=1$ we can assume without loss of generality that $z_{+}$ and
$x^{\prime}$ are monomials in quantum root vectors. From (4.4.1) it follows
that $A_{q}(xT_{s}hz_{+}x^{\prime})\neq 0$ only if
$A_{q}(xT_{s}hz_{+}x^{\prime})=(v_{q},T_{s}hz_{+}x^{\prime}T_{s^{-1}}v_{\mu_{q}})=(v_{q},c\omega_{0}S_{s}^{-1}(f_{\beta_{q}})v_{\mu_{q}})$
for some $c\in\mathcal{B}$, $c\neq 0$.

Now observe that from the definitions of $\Delta_{0}$ and
$-\Delta_{\mathfrak{m}_{+}}^{s}$ it follows that the weight of the element
$T_{s}hz_{+}x^{\prime}T_{s^{-1}}$ is a linear combination of simple roots from
$\Delta_{+}^{s}$ with non-negative integer coefficients, and this collection
of weights does not contain $-\beta_{q}\in\Delta_{-}^{s}$, the weight of
$\omega_{0}S_{s}^{-1}(f_{\beta_{q}})$. Therefore
$A_{q}(xT_{s}hz_{+}x^{\prime})=0$ in all cases.

We deduce that the elements generating ${\overline{J}^{p}_{\mathcal{B}}}$ obey
the defining property of ${\overline{J}^{p}_{\mathcal{B}}}^{\prime}$, and
hence
${\overline{J}^{p}_{\mathcal{B}}}\subset{\overline{J}^{p}_{\mathcal{B}}}^{\prime}$.

Observe that by Lemma 4.3.2 ${\overline{J}^{1}_{\mathcal{B}}}=\overline{J}$
mod $(q^{\frac{1}{dr^{2}}}-1){\overline{J}^{1}_{\mathcal{B}}}$, where
$\overline{J}$ is the vanishing ideal of $\overline{N_{-}sZHM_{-}^{s}}$, and
by the definition $A_{n}=\phi_{n}$ mod
$(q^{\frac{1}{dr^{2}}}-1)\mathbb{C}_{\mathcal{B}}[G]$ for any $n$. Hence,
since ${\overline{J}^{p}_{\mathcal{B}}}$ is generated by
${\overline{J}^{1}_{\mathcal{B}}}$ and by $A_{1},\ldots,A_{p-1}$, Corollary
3.5.5 implies that ${\overline{J}^{p}_{\mathcal{B}}}=\overline{J}^{p}_{1}$ mod
$(q^{\frac{1}{dr^{2}}}-1){\overline{J}^{p}_{\mathcal{B}}}$, where
$\overline{J}^{p}_{1}$ is the vanishing ideal of
$\overline{N_{[-\beta_{p},-\beta_{D}]}sZHM_{-}^{s}}$,

All derivatives of elements of $\overline{J}^{p}_{1}$ in the directions
tangent to $N_{[-\beta_{p},-\beta_{D}]}sZHM_{-}^{s}$ must vanish. Therefore
every element of $\overline{J}^{p}_{1}$ must vanish on elements
$xshz_{+}x^{\prime}$ with arbitrary $x=X_{-\beta_{D}}^{(n_{D})}\ldots
X_{-\beta_{p}}^{(n_{p})}$, $z_{+}\in U(\Delta_{0}^{+}),x^{\prime}\in
U(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U(\mathfrak{h})$.

Conversely, any element of $\mathbb{C}[G]$ vanishing on $xshz_{+}x^{\prime}$
with arbitrary $x=X_{-\beta_{D}}^{(n_{D})}\ldots
X_{-\beta_{p}}^{(n_{p})},z_{+}\in U(\Delta_{0}^{+}),x^{\prime}\in
U(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U(\mathfrak{h})$ must vanish on
$N_{[-\beta_{p},-\beta_{D}]}sZHM_{-}^{s}$, and hence on
$\overline{N_{[-\beta_{p},-\beta_{D}]}sZHM_{-}^{s}}$. Thus
$\overline{J}^{p}_{1}$ coincides with the ideal in $\mathbb{C}[G]$ generated
by the regular functions vanishing on $xshz_{+}x^{\prime}$ with arbitrary
$x=X_{-\beta_{D}}^{(n_{D})}\ldots X_{-\beta_{p}}^{(n_{p})},z_{+}\in
U(\Delta_{0}^{+}),x^{\prime}\in U(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in
U(\mathfrak{h})$. Therefore
${\overline{J}^{p}_{\mathcal{B}}}^{\prime}={\overline{J}^{p}_{\mathcal{B}}}=\overline{J}^{p}_{1}$
mod $(q^{\frac{1}{dr^{2}}}-1){\overline{J}^{p}_{\mathcal{B}}}^{\prime}$.
Recalling that
${\overline{J}^{p}_{\mathcal{B}}}\subset{\overline{J}^{p}_{\mathcal{B}}}^{\prime}$
we deduce with the help of Lemma 4.3.1 that
${\overline{J}^{p}_{q}}={\overline{J}^{p}_{q}}^{\prime}$.

The case of the left ideal generated by the elements of
$\mathbb{C}_{\mathcal{B}}[G]$ vanishing on elements
$x\overline{T}_{s}hz_{+}x^{\prime}$ with arbitrary
$x=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}),z_{+}\in U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U_{q}^{res}(H)$,
$n_{i}\in\mathbb{N}$ is treated in a similar way since by Lemma 2.7.4 for a
highest weight vector $v_{\mu}\in V_{\mu}$ there is a non–zero multiple $k$ of
a power of $q$ such that $T_{s^{-1}}v_{\mu}=k\overline{T}_{s^{-1}}v_{\mu}$.

Finally observe that ${\overline{I}^{p}_{\mathcal{B}}}$ is a left ideal in
$\mathbb{C}_{\mathcal{B}}[G]$ as clearly $\overline{J}^{p}_{q}$ is a left
ideal in $\mathbb{C}_{q}[G]$ and
${\overline{I}^{p}_{\mathcal{B}}}={\overline{J}^{p}_{\mathcal{B}}}\cap\mathbb{C}_{\mathcal{B}}[G]$.
This completes the proof.

∎

Similarly one can define quantum analogues ${J^{p}_{\mathcal{B}}}$ of the
vanishing ideals of the varieties
$\overline{N_{[-\beta_{p},-\beta_{D}]}sZHN_{-}^{\prime}}$.

######  Lemma 4.4.2.

For $p=1,\ldots,c+1$, let ${J^{p}_{\mathcal{B}}}$ be the left ideal in
$\mathbb{C}_{\mathcal{B}}[G]$ generated by ${J^{1}_{\mathcal{B}}}$ and by
$A_{1},\ldots,A_{p-1}$. Let ${J^{p}_{\mathcal{B}}}^{\prime}$ be the left ideal
in $\mathbb{C}_{\mathcal{B}}[G]$ generated by the elements vanishing on
$xT_{s}hz_{+}x^{\prime}$ with arbitrary
$x=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}),z_{+}\in U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{s}^{\prime}),h\in U_{q}^{res}(H)$, $n_{i}\in\mathbb{N}$.
Denote
${J^{p}_{q}}={J^{p}_{\mathcal{B}}}\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$,
${J^{p}_{q}}^{\prime}={J^{p}_{\mathcal{B}}}^{\prime}\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$
and let ${I^{p}_{\mathcal{B}}}={J^{p}_{q}}\cap\mathbb{C}_{\mathcal{B}}[G]$.
Then ${I^{p}_{\mathcal{B}}}$ is a left ideal in $\mathbb{C}_{\mathcal{B}}[G]$,
${J^{p}_{q}}={J^{p}_{q}}^{\prime}$, and hence
${I^{p}_{\mathcal{B}}}={J^{p}_{q}}^{\prime}\cap\mathbb{C}_{\mathcal{B}}[G]$.

Moreover, ${J^{p}_{\mathcal{B}}}={J^{p}_{\mathcal{B}}}^{\prime}=J^{p}_{1}$ mod
$(q^{\frac{1}{dr^{2}}}-1){J^{p}_{\mathcal{B}}}^{\prime}$, where $J^{p}_{1}$ is
the vanishing ideal of
$\overline{N_{[-\beta_{p},-\beta_{D}]}sZHN_{-}^{\prime}}$.

###### Proof.

Firstly as in the proof of Lemma 4.4.1 we deduce that
${J^{p}_{\mathcal{B}}}\subset{J^{p}_{\mathcal{B}}}^{\prime}$.

Similarly to the proof of Lemma 4.4.1 we also obtain, using the second
assertion in Corollary 3.5.5, that
${J^{p}_{\mathcal{B}}}={J^{p}_{\mathcal{B}}}^{\prime}=J^{p}_{1}$ mod
$(q^{\frac{1}{dr^{2}}}-1){J^{p}_{\mathcal{B}}}^{\prime}$, where $J^{p}_{1}$ is
the vanishing ideal of
$\overline{N_{[-\beta_{p},-\beta_{D}]}sZHN_{-}^{\prime}}$, and hence by Lemma
4.3.1 ${J^{p}_{q}}={J^{p}_{q}}^{\prime}$.

Finally using the same arguments as in the proof of Lemma 4.4.1 we deduce that
${I^{p}_{\mathcal{B}}}$ is a left ideal in $\mathbb{C}_{\mathcal{B}}[G]$. ∎

We conclude this section with a technical lemma which will be used to cancel
some terms in formula (4.2.4) for the adjoint action.

######  Lemma 4.4.3.

For any $n>0$, $A_{p}(\cdot
f_{\beta_{q}}^{(n)})\in{J^{p}_{\mathcal{B}}}^{\prime}$ for $q\geq p$,
$\beta_{p}\not\in\Delta_{s^{1}}^{s}$, $q\in\\{1,\ldots,c\\}$ and $A_{p}(\cdot
f_{\beta_{q}}^{(n)})\in{\overline{J}^{p}_{\mathcal{B}}}^{\prime}$ for $q\geq
p$, $q\in\\{1,\ldots,c\\}$.

###### Proof.

We prove the first statement. The second one can be justified in a similar way
with the help of Lemma 4.4.1.

For $q\geq p$, $\beta_{p}\not\in\Delta_{s^{1}}^{s}$ and any
$x=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}),z_{+}\in U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{s}^{\prime}),h\in U_{q}^{res}(H)$ we have, since
$v_{\mu_{p}}$ is a highest weight vector,

$(v_{p},xT_{s}hz_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}})=(v_{p},x\omega_{0}S_{s}^{-1}(y)v_{\mu_{p}})$

for some $y\in U_{\mathcal{B}}^{s,res}(\mathfrak{n}_{-})$. We claim that

$(v_{p},xT_{s}hz_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}})=0$

if $x\neq 1,\omega_{0}S_{s}^{-1}(f_{\beta_{p}})$.

Indeed, let $x\neq 1,\omega_{0}S_{s}^{-1}(f_{\beta_{p}})$, so
$x\omega_{0}S_{s}^{-1}(y)=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}y)$. Using commutation relations (2.7.11) the element
$f_{\beta_{D}}^{(n_{D})}\ldots f_{\beta_{p}}^{(n_{p})}y$ can be uniquely
represented as a $\mathbb{C}(q^{\frac{1}{dr^{2}}})$–linear combination of
monomials $f_{\beta_{D}}^{(n_{D})}\ldots f_{\beta_{1}}^{(n_{1})}$ with
$n_{p}>1$ or at least one $n_{t}>0$ for $t>p$. This presentation is unique by
the Poincaré–Birkhoff–Witt decomposition for $U_{q}^{s}({\mathfrak{n}}_{-})$,
and by the uniqueness of the Poincaré–Birkhoff–Witt decomposition for
$U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{-})$ the coefficients in this
decomposition belong to $\mathcal{B}$ (see Lemma 2.7.2).

So by (4.4.1) we have

$A_{q}(xT_{s}hz_{+}x^{\prime})=(v_{q},\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}y)v_{\mu_{q}})=0$

in this case.

If $x=1$ then

$(v_{p},xT_{s}hz_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}})=(v_{p},T_{s}hz_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}})=c(v_{p},T_{s}z_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}}),c\in\mathcal{B},$
(4.4.6)

and if $x=S_{s}^{-1}(f_{\beta_{p}})$ then

$(v_{p},xT_{s}hz_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}})=(v_{\mu_{p}},T_{s}hz_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}})=c^{\prime}(v_{\mu_{p}},T_{s}z_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}}),c^{\prime}\in\mathcal{B}.$
(4.4.7)

We claim that in both cases the right hand side in the formulas above
vanishes. This is obviously the case if $s\beta_{q}\in\Delta_{-}$. Suppose
that $\beta_{p}\in\Delta_{i_{j}}$ and $s\beta_{q}\in\Delta_{+}$ . Without loss
of generality we can assume that $z_{+}$ and $x^{\prime}$ are monomials in
quantum root vectors corresponding to roots from $\Delta_{0}^{+}$ and
$-\Delta_{s}^{\prime}$, respectively, so
$z_{+}x^{\prime}=({X}^{+}_{\delta_{a}})^{(r_{a})}\ldots({X}^{+}_{\delta_{b}})^{(r_{b})}({X}^{-}_{\epsilon_{g}})^{(t_{g})}\ldots({X}^{-}_{\epsilon_{h}})^{(t_{h})}$,
$\delta_{a},\ldots,\delta_{b}\in\Delta_{0}^{+}\subset\Delta_{0}$,
$\epsilon_{g},\ldots,\epsilon_{h}\in\Delta_{s}^{\prime}$. If
$(v_{p},T_{s}z_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}})$ is not
zero then the weight of the element
$T_{s}z_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}$ must be equal to
$-\beta_{p}$ as the weight of $v_{p}$ is $\mu_{p}-\beta_{p}$ and if
$(v_{\mu_{p}},T_{s}z_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}v_{\mu_{p}})$
is not zero then the weight of the element
$T_{s}z_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}$ must be equal to zero.

Let $k$ be minimal possible such that
$\beta_{q},\epsilon_{g},\ldots\epsilon_{h}\in\overline{\Delta}_{i_{k}}$.
Denote by
$\lambda=r_{a}\delta_{a}+\ldots+r_{b}\delta_{b}-t_{g}\epsilon_{g}-\ldots-
t_{h}\epsilon_{h}-n\beta_{q}$ the weight of
$z_{+}x^{\prime}f_{\beta_{q}}^{(n)}$. Note that $i_{k}>0$ as
$q\in\\{1,\ldots,c\\}$ and hence $\beta_{q}\not\in\Delta_{0}$.

The corresponding plane $\mathfrak{h}_{i_{k}}$ is shown at Figure 6 where we
use the same notation as at Figure 2, Section 1.2. We denote by
$\Delta_{s^{-1}}^{s}$ the open sector which is transformed into the lower half
plane under the action of $s^{-1}$, by $s\Delta_{s^{-1}}^{s}$ its image under
the action of $s$, and by $-\Delta_{s^{-1}}^{s}$, $-s\Delta_{s^{-1}}^{s}$
their images under the multiplication by $-1$. We also denote by
$\Delta_{i_{k}}^{1}$ the sector which appeared at Figure 2 under the same
name, by $s\Delta_{i_{k}}^{1}$ its image under the action of $s$, and by
$-s\Delta_{i_{k}}^{1}$ its image under the multiplication by $-1$. The
orthogonal projections of the roots from the set $\Delta_{0}$ onto
$\mathfrak{h}_{i_{k}}$ are zeroes and by the definition of the set
$\Delta_{s}^{\prime}$ and by the properties of normal ordering (1.6.3) of the
root system $\Delta_{+}^{s}$ in Proposition 1.6.1 the orthogonal projections
of the roots $-s\epsilon_{g},\ldots,-s\epsilon_{h}\in s(-\Delta_{s}^{\prime})$
onto $\mathfrak{h}_{i_{k}}$ belong to the sectors labeled
$\Delta_{s^{-1}}^{s}$ and $s\Delta_{i_{k}}^{1}$ at Figure 6.

$\textstyle{h_{i_{k}}}$$\textstyle{s\Delta_{s^{-1}}^{s}}$$\textstyle{s\Delta_{i_{k}}^{1}}$$\textstyle{-s\Delta_{i_{k}}^{1}}$$\textstyle{\Delta_{s^{-1}}^{s}}$$\textstyle{-s\Delta_{s^{-1}}^{s}}$$\textstyle{-\Delta_{i_{k}}^{1}}$$\textstyle{-\Delta_{s^{-1}}^{s}}$$\textstyle{-s\beta_{q}}$$\textstyle{-\beta_{p}~{}{\text{if}}~{}k=j}$$\textstyle{s\lambda}$

Fig. 6

Since $\beta_{q}\geq\beta_{p}\not\in\Delta_{s^{1}}^{s}$ and
$\Delta_{s^{1}}^{s}$ is an initial segment in $\Delta_{+}^{s}$ we also have
$\beta_{q}\not\in\Delta_{s^{1}}^{s}$ and also by the assumption
$s\beta_{q}\in\Delta_{+}$. Therefore the orthogonal projection of
$-s\beta_{q}$ onto $\mathfrak{h}_{i_{k}}$ belongs to the lower half plane or
possibly to the sector labeled $\Delta_{s^{-1}}^{s}$ at Figure 6 and does not
belong to the sectors labeled $-\Delta_{s^{-1}}$ and $-s\Delta_{i_{k}}^{1}$ at
Figure 6 This projection is labeled $-s\beta_{q}$ at Figure 6.

If the orthogonal projection of $-s\beta_{q}$ onto $\mathfrak{h}_{i_{k}}$
belongs to the sector labeled $\Delta_{s^{-1}}^{s}$ at Figure 6 then the
orthogonal projection of $s\lambda$ onto $\mathfrak{h}_{i_{k}}$ belongs to the
interior of the closure of the union of the sectors $\Delta_{s^{-1}}^{s}$ and
$s\Delta_{i_{k}}^{1}$, and the intersection of this union with the lower half
plane is at most $-\Delta_{i_{k}}^{1}$. Remark that in this case the
orthogonal projection of $s\lambda$ onto $\mathfrak{h}_{i_{k}}$ is not zero,
and hence $\lambda\neq 0$.

If $k=j$ note that the orthogonal projection of $-\beta_{p}$ onto
$\mathfrak{h}_{i_{j}}$ belongs to the open lower half plane and does not
belong to $-\Delta_{i_{k}}^{1}$. Thus in this case $s\lambda\neq-\beta_{p}$.

If the orthogonal projection of $-s\beta_{q}$ onto $\mathfrak{h}_{i_{k}}$
belongs to the lower half plane and does not belong to the sectors labeled
$-\Delta_{s^{-1}}^{s}$ and $-s\Delta_{i_{k}}^{1}$ at Figure 6 then the
orthogonal projection of $s\lambda$ onto $\mathfrak{h}_{i_{j}}$ belongs to the
closed left half plane bounded by the dashed line at Figure 6. This projection
labeled $s\lambda$ is shown at Figure 6 as well. Observe that in this case the
orthogonal projection of $s\lambda$ onto $\mathfrak{h}_{i_{k}}$ is not zero,
and hence $\lambda\neq 0$.

If $k=j$, recall that $\beta_{p}\leq\beta_{q}$ where
$\beta_{p},\beta_{q}\in{\Delta}_{i_{j}}$, and hence by the last property of
the normal ordering in $\Delta_{+}^{s}$ in Proposition 1.6.1 the orthogonal
projection of $-\beta_{p}$ onto $\mathfrak{h}_{i_{j}}$ labeled by the same
letter at Figure 6 belongs to the other open half plane bounded by the dashed
line at Figure 6. Thus again $s\lambda\neq-\beta_{p}$.

In all cases above, if $k>j$ then the orthogonal projection of $s\lambda$ onto
$\mathfrak{h}_{i_{k}}$ is not zero while the orthogonal projection of
$-\beta_{p}$ onto $\mathfrak{h}_{i_{k}}$ is zero, and hence
$s\lambda\neq-\beta_{p}$. If $k<j$ then the orthogonal projection of
$s\lambda$ onto $\mathfrak{h}_{i_{j}}$ is zero while the orthogonal projection
of $-\beta_{p}$ onto $\mathfrak{h}_{i_{j}}$ is zero, and hence again
$s\lambda\neq-\beta_{p}$.

We deduce that in all cases the weight of the element
$T_{s}z_{+}x^{\prime}f_{\beta_{q}}^{(n)}T_{s^{-1}}$ is not equal to
$-\beta_{p}$ or to zero. Therefore the right hand side in formulas (4.4.6),
(4.4.7) vanishes, and by the definition of ${J^{p}_{\mathcal{B}}}^{\prime}$ in
Lemma 4.4.2 $A_{p}(\cdot
f_{\beta_{q}}^{(n)})\in{J^{p}_{\mathcal{B}}}^{\prime}$ for $q\geq p$,
$\beta_{p}\not\in\Delta_{s^{1}}^{s}$, $q\in\\{1,\ldots,c\\}$.

∎

###  4.5 Localizations

In order co define quantum counterparts of formulas (3.5.10) we have to
introduce a certain localization $\mathbb{C}_{\mathcal{B}}^{loc}[G]$ of
$\mathbb{C}_{\mathcal{B}}[G]$. However, $\mathbb{C}_{\mathcal{B}}^{loc}[G]$ is
not a localization in the sense of localization for non–commutative algebras,
and it is rather impossible to define a proper localization of the algebra
$\mathbb{C}_{\mathcal{B}}[G]$ which contains the quantum counterparts of the
denominators in formula (3.5.10) given by (4.4.3). But in fact we shall only
need right denominators and $\mathbb{C}_{\mathcal{B}}^{loc}[G]$ will be
defined as a “right” localization of $\mathbb{C}_{\mathcal{B}}[G]$. The exact
meaning of this term will be explained below. Along with
$\mathbb{C}_{\mathcal{B}}^{loc}[G]$ we shall also define “right” localizations
${\overline{I}^{p}_{\mathcal{B}}}^{loc}$ and ${I^{p}_{\mathcal{B}}}^{loc}$ of
$\overline{I}^{p}_{\mathcal{B}}$ and $I^{p}_{\mathcal{B}}$ and study their
properties.

We start by introducing a subalgebra
$\mathbb{C}_{\mathcal{B}}[G]_{0}\subset\mathbb{C}_{\mathcal{B}}[G]$ a proper
localization of which contains all the required denominators.

######  Lemma 4.5.1.

The set of elements $(u,\cdot T_{s^{-1}}v)\in\mathbb{C}_{\mathcal{B}}[G]$,
where $v$ is a highest weight vector in a finite rank representation $V$ of
$U_{h}(\mathfrak{g})$ and $u\in V$ form a subalgebra
$\mathbb{C}_{\mathcal{B}}[G]_{0}$ in $\mathbb{C}_{\mathcal{B}}[G]$.

###### Proof.

It suffices to show that the product of two elements from
$\mathbb{C}_{\mathcal{B}}[G]_{0}$ belongs to
$\mathbb{C}_{\mathcal{B}}[G]_{0}$. Let $s^{-1}=s_{i_{1}}\ldots s_{i_{k}}$ be a
reduced decomposition of $s^{-1}$. Then by (2.6.12)

$\displaystyle\Delta_{s}(T_{s^{-1}})=T_{s^{-1}}\otimes
T_{s^{-1}}q^{\sum_{i=1}^{l}(-T_{s}Y_{i}\otimes T_{s}KH_{i}+Y_{i}\otimes
KH_{i})}\prod_{p=1}^{k}\overline{\theta}_{\beta_{p}^{\prime}}^{s},$ (4.5.1)

where for $p=1,\ldots,k$

$\beta_{p}^{\prime}=s_{i_{k}}\ldots
s_{i_{p+1}}\alpha_{i_{p}},{\overline{X}_{\beta_{p}^{\prime}}^{\pm}}=T_{i_{k}}^{-1}\ldots
T_{i_{p+1}}^{-1}X_{i_{p}}^{\pm},$
$\overline{e}_{\beta_{p}^{\prime}}=\psi_{\\{n_{ij}\\}}^{-1}(\overline{X}_{\beta_{p}^{\prime}}^{+}e^{hK{\beta_{p}^{\prime}}^{\vee}}),\overline{f}_{\beta_{p}^{\prime}}=\psi_{\\{n_{ij}\\}}^{-1}(e^{-hK{\beta_{p}^{\prime}}^{\vee}}\overline{X}_{\beta_{p}^{\prime}}^{-}),~{}K_{\beta_{p}^{\prime}}=T_{i_{k}}^{-1}\ldots
T_{i_{p+1}}^{-1}K_{i_{p}},$
$\overline{\theta}_{\beta_{p}^{\prime}}^{s}=exp_{q_{\beta_{p}^{\prime}}}[(1-q_{\beta_{p}^{\prime}}^{-2})K_{\beta_{p}^{\prime}}^{-1}e^{h\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}{\beta_{p}^{\prime}}^{\vee}}{\overline{f}_{\beta_{p}^{\prime}}}\otimes{\overline{e}_{\beta_{p}^{\prime}}}K_{\beta_{p}^{\prime}}].$

Since $Kh=\frac{\kappa}{2}{1+s\over 1-s}P_{{\mathfrak{h}^{\prime}}}h$ for any
$h\in\mathfrak{h}$, we have $Ks=sK$, and hence

$q^{\sum_{i=1}^{l}(-T_{s}Y_{i}\otimes T_{s}KH_{i}+Y_{i}\otimes
KH_{i})}=q^{\sum_{i=1}^{l}(-T_{s}Y_{i}\otimes KT_{s}H_{i}+Y_{i}\otimes
KH_{i})}=q^{\sum_{i=1}^{l}(-Y_{i}\otimes KH_{i}+Y_{i}\otimes KH_{i})}=1$
(4.5.2)

where we also used the identity $\sum_{i=1}^{l}T_{s}Y_{i}\otimes
KT_{s}H_{i}=\sum_{i=1}^{l}Y_{i}\otimes KH_{i}$ which holds since $T_{s}Y_{i}$,
$T_{s}H_{i}$, $i=1,\ldots,l$ is a pair of dual bases in $\mathfrak{h}$.

Now (4.5.1) takes the form

$\Delta_{s}(T_{s^{-1}})=T_{s^{-1}}\otimes
T_{s^{-1}}\prod_{p=1}^{k}\overline{\theta}_{\beta_{p}^{\prime}}^{s},$

and for two highest weight vectors $v\in V,v^{\prime}\in V^{\prime}$ we have

$\Delta_{s}(T_{s^{-1}})v\otimes v^{\prime}=T_{s^{-1}}v\otimes
T_{s^{-1}}v^{\prime}.$ (4.5.3)

Therefore for any $u\in V,u^{\prime}\in V^{\prime}$ we have by (4.5.3)

$(u,\cdot T_{s^{-1}}v)\otimes(u^{\prime},\cdot T_{s^{-1}}v^{\prime})=(u\otimes
u^{\prime},\cdot T_{s^{-1}}v\otimes T_{s^{-1}}v^{\prime})=(u\otimes
u^{\prime},\cdot T_{s^{-1}}(v\otimes v^{\prime})).$ (4.5.4)

Since $v\otimes v^{\prime}$ is a highest weight vector in $V\otimes
V^{\prime}$, the last identity implies $(u,\cdot
T_{s^{-1}}v)\otimes(u^{\prime},\cdot
T_{s^{-1}}v^{\prime})\in\mathbb{C}_{\mathcal{B}}[G]_{0}$. This completes the
proof.

∎

From formulas (4.2.2) and (4.5.4) it follows that the set
$S=\\{\Delta_{\mu}=(v_{\mu},\cdot T_{s^{-1}}v_{\mu})\\}|\mu\in P_{+}\\}$,
where $v_{\mu}\in V_{\mu}$ are highest weight vectors normalized by
$(v_{\mu},v_{\mu})=1$, is a multiplicative set of normal elements in
$\mathbb{C}_{\mathcal{B}}[G]_{0}$. Let $\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$
be the localization of $\mathbb{C}_{\mathcal{B}}[G]_{0}$ by $S$.

We shall need more information on the structure of subalgebras $N_{k}$ in
$\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$ generated by the elements

$B_{p}={A_{p}^{0}}^{-1}\otimes A_{p}\in\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0},$
(4.5.5)

$p=1,\ldots,k$. The elements $B_{p}$ are quantum analogues of functions
(3.5.10). The following Lemma is similar to Proposition 8.3 in [45].

######  Lemma 4.5.2.

Let $N_{k}$ be the subalgebra in $\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$
generated by the elements $B_{p}={A_{p}^{0}}^{-1}\otimes
A_{p}\in\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$, $p=1,\ldots,k$, and
$U_{\mathcal{B}}^{k}$ the subalgebra in ${\mathbb{C}}_{\mathcal{B}}[G_{*}]$
generated by the elements
$e_{p}=(q_{\beta_{p}}^{-1}-q_{\beta_{p}})q^{\beta_{p}^{\vee}}e_{\beta_{p}}$,
$p=1,\ldots,k$. Then the map

$B_{p}\mapsto e_{p}$

gives rise to an algebra isomorphism $\vartheta:N_{k}\rightarrow
U_{\mathcal{B}}^{k}$.

In particular, the elements $B_{p}$ satisfy the following commutation
relations

$B_{p}B_{r}-q^{(\beta_{p},\beta_{r})+(\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime*}}\beta_{p},\beta_{r})}B_{r}B_{p}=\sum_{k_{p+1},\ldots,k_{r-1}}C(k_{p+1},\ldots,k_{r-1})B_{p+1}^{k_{p+1}}\ldots
B_{r-1}^{k_{r-1}},~{}p<r,$ (4.5.6)

where $C(k_{p+1},\ldots,k_{r-1})\in\mathcal{B}$.

###### Proof.

First observe that the map

$\psi:\mathbb{C}_{\mathcal{B}}[G]\rightarrow{\mathbb{C}}_{\mathcal{B}}[B_{-}],(u,\cdot
v)\mapsto(u\otimes id,(\omega_{0}\otimes id)(\mathcal{R})v\otimes id)$

is an algebra homomorphism. Indeed, since $\omega_{0}$ is a coautomorphism

$(\Delta_{s}\otimes id)((\omega_{0}\otimes
id)(\mathcal{R}^{s}))=(\omega_{0}\otimes\omega_{0}\otimes
id)(\Delta_{s}\otimes id)\mathcal{R}^{s}=(\omega_{0}\otimes\omega_{0}\otimes
id)(\mathcal{R}_{13}^{s}\mathcal{R}_{23}^{s})=(\omega_{0}\otimes
id)(\mathcal{R}_{13}^{s})(\omega_{0}\otimes id)(\mathcal{R}_{23}^{s}),$

and hence

$\psi((u,\cdot v)\otimes(u^{\prime},\cdot v^{\prime}))=(u\otimes
id,(\omega_{0}\otimes id)(\mathcal{R}_{13}^{s})v\otimes id)(u^{\prime}\otimes
id,(\omega_{0}\otimes id)(\mathcal{R}_{23}^{s})v^{\prime}\otimes
id)=\psi((u,\cdot v))\psi((u^{\prime},\cdot v^{\prime})),$

i.e. $\psi$ is an algebra homomorphism.

Recall that by Lemma 4.5.1 elements of the form $(u,\cdot
T_{s^{-1}}v)\in\mathbb{C}_{\mathcal{B}}[G]$, where $v$ is a highest weight
vector in a finite rank representation $V$ of $U_{h}(\mathfrak{g})$ and $u\in
V$, form a subalgebra $\mathbb{C}_{\mathcal{B}}[G]_{0}$ in
$\mathbb{C}_{\mathcal{B}}[G]$.

From (4.5.3) we also deduce that the map

$\mathbb{C}_{\mathcal{B}}[G]_{0}\rightarrow\mathbb{C}_{\mathcal{B}}[G],(u,\cdot
T_{s^{-1}}v)\mapsto(u,\cdot v)$

is an algebra homomorphism. Composing this map with $\psi$ we obtain another
algebra homomorphism $\psi^{0}$.

Next using (4.1.5) and the definition of $A_{p}$ we obtain

$\psi^{0}(A_{p})=(v_{p}\otimes id,(\omega_{0}S_{s}^{-1}\otimes
id)(\mathcal{R})v_{\mu_{p}}\otimes id)=(v_{p}\otimes id,(\omega_{0}\otimes
id)({{\mathcal{R}}^{s}}^{-1})v_{\mu_{p}}\otimes id)=$
$=(q_{\beta_{p}}^{-1}-q_{\beta_{p}})q^{(\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)\mu_{p}^{\vee}}q^{\beta_{p}^{\vee}}e_{\beta_{p}}.$

Similarly

$\psi^{0}(A_{p}^{0})=q^{(\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)\mu_{p}^{\vee}}.$

From the last two formulas we deduce that $\psi^{0}$ gives rise to an algebra
homomorphism $\vartheta:N_{k}\rightarrow U_{\mathcal{B}}^{k}$ such that

$\vartheta(B_{p})=\vartheta({A_{p}^{0}}^{-1}\otimes
A_{p}))=(q_{\beta_{p}}^{-1}-q_{\beta_{p}})q^{\beta_{p}^{\vee}}e_{\beta_{p}}=e_{p}.$

This homomorphism is surjective by construction. $\vartheta$ is also injective
as $\psi^{0}$ is injective (see Proposition 8.3 in [45] and [118], Theorem 2.6
for the proof).

Commutation relations (4.5.6) follow from (3.2.10) by applying $\omega$ and by
multiplying by $q^{\beta_{p}^{\vee}}q^{\beta_{r}^{\vee}}$.

∎

Denote
$\mathbb{C}_{\mathcal{B}}^{loc}[G]=\mathbb{C}_{\mathcal{B}}[G]\otimes_{\mathbb{C}_{\mathcal{B}}[G]_{0}}\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$.
$\mathbb{C}_{\mathcal{B}}^{loc}[G]$ is naturally a left
$\mathbb{C}_{\mathcal{B}}[G]$–module and a right
$\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$–module. We denote by $\otimes$ both
the left $\mathbb{C}_{\mathcal{B}}[G]$–action and the right
$\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$–action on
$\mathbb{C}_{\mathcal{B}}^{loc}[G]$ and call these action multiplications.

Let
$S^{*}=\\{\Delta_{\lambda}\otimes\Delta_{\mu}^{-1}\in\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}|\lambda,\mu\in
P_{+}\\}$,
$S^{-1}=\\{\Delta_{\mu}^{-1}\in\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}|\mu\in
P_{+}\\}$, ${\overline{I}^{p}_{\mathcal{B}}}^{loc}$ and
${I^{p}_{\mathcal{B}}}^{loc}$ the images of
${\overline{I}^{p}_{\mathcal{B}}}\otimes
S^{*}\subset\mathbb{C}_{\mathcal{B}}[G]\otimes\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$
and of ${I^{p}_{\mathcal{B}}}\otimes
S^{*}\subset\mathbb{C}_{\mathcal{B}}[G]\otimes\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$,
respectively, in $\mathbb{C}_{\mathcal{B}}^{loc}[G]$ under the projection
$\mathbb{C}_{\mathcal{B}}[G]\otimes\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}\rightarrow\mathbb{C}_{\mathcal{B}}[G]\otimes_{\mathbb{C}_{\mathcal{B}}[G]_{0}}\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}=\mathbb{C}_{\mathcal{B}}^{loc}[G]$.

For $p=1,\ldots,c+1$ let

$\mathbb{C}_{p}^{loc}[G]=\left\\{\begin{array}[]{l}\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{p}_{\mathcal{B}}}^{loc}~{}~{}{\rm
if}~{}~{}\beta_{p}\not\in\Delta_{s^{1}}^{s}\\\
\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}^{p}_{\mathcal{B}}}^{loc}~{}~{}{\rm
if}~{}~{}\beta_{p}\in\Delta_{s^{1}}^{s}\end{array}\right..$

$\mathbb{C}_{p}^{loc}[G]$ is naturally a left
$\mathbb{C}_{\mathcal{B}}[G]$–module.

Note that
${\overline{I}^{p}_{\mathcal{B}}}^{loc}\subset{\overline{I}^{p+1}_{\mathcal{B}}}^{loc}$,
${I^{p}_{\mathcal{B}}}^{loc}\subset{I^{p+1}_{\mathcal{B}}}^{loc}$ for
$p=1,\ldots,c$, and
${I^{p}_{\mathcal{B}}}^{loc}\subset{\overline{I}^{p}_{\mathcal{B}}}^{loc}$ for
$p=1,\ldots,c+1$.

Here and everywhere below for the technical reasons which already became
manifest in Lemma 4.3.7 we have to distinguish two cases in our consideration,
$\beta_{p}\not\in\Delta_{s^{1}}^{s}$ and $\beta_{p}\in\Delta_{s^{-1}}^{s}$.
Conceptually this complication is related to the non–commutativity of the
multiplication in $\mathbb{C}_{\mathcal{B}}[G]$.

By the definition ${\overline{I}^{p}_{\mathcal{B}}}^{loc}$ and
${I^{p}_{\mathcal{B}}}^{loc}$ are left
$\mathbb{C}_{\mathcal{B}}[G]$–submodules in
$\mathbb{C}_{\mathcal{B}}^{loc}[G]$. The following lemma shows that
${\overline{I}^{p}_{\mathcal{B}}}$, ${I^{p}_{\mathcal{B}}}$,
${\overline{I}^{p}_{\mathcal{B}}}^{loc}$ and ${I^{p}_{\mathcal{B}}}^{loc}$
have some nice properties with respect to multiplication by the elements
$A_{r},\Delta_{\mu},B_{r}\in\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$ from the
right as well.

######  Lemma 4.5.3.

For any $p\in\\{1,\ldots,c+1\\}$ ${I^{p}_{\mathcal{B}}}^{loc}$ is invariant
with respect to multiplication from the right by $A_{p}$ and $B_{p}$, and
${I^{p}_{\mathcal{B}}}^{loc}A_{r}={I^{p}_{\mathcal{B}}}^{loc}B_{r}\subset{I^{r}_{\mathcal{B}}}^{loc}$
for any $r>p$, $r\in\\{1,\ldots,c+1\\}$.

For any $p\in\\{1,\ldots,c+1\\}$ and any $\mu\in P_{+}$
${I^{p}_{\mathcal{B}}}$ is invariant with respect to multiplication from the
right by $\Delta_{\mu}$.

For any $p\in\\{1,\ldots,c+1\\}$ such that $\beta_{p}\in\Delta_{s^{-1}}^{s}$
${\overline{I}^{p}_{\mathcal{B}}}^{loc}$ is invariant with respect to
multiplication from the right by $A_{p}$ and $B_{p}$, and for
$\beta_{r}\in\Delta_{s^{-1}}^{s}$ and any $r>p$, $r\in\\{1,\ldots,c+1\\}$ we
have
${\overline{I}^{p}_{\mathcal{B}}}^{loc}A_{r}={\overline{I}^{p}_{\mathcal{B}}}^{loc}B_{r}\subset{\overline{I}^{r}_{\mathcal{B}}}^{loc}$.

For any $p\in\\{1,\ldots,c+1\\}$ such that $\beta_{p}\in\Delta_{s^{-1}}^{s}$
and any $\mu\in P_{+}$ ${\overline{I}^{p}_{\mathcal{B}}}$ is invariant with
respect to multiplication from the right by $\Delta_{\mu}$.

Thus multiplication from the right induces a natural action of $A_{p}$ and
$B_{p}$ on $\mathbb{C}_{p}^{loc}[G]$ and for $r\geq p$,
$r\in\\{1,\ldots,c+1\\}$ right multiplication by $A_{r}$ or $B_{r}$ gives rise
to well–defined homomorphisms of left $\mathbb{C}_{\mathcal{B}}[G]$–modules
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{p}_{\mathcal{B}}}^{loc}\rightarrow\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{r}_{\mathcal{B}}}^{loc}$,
and, if in addition $\beta_{r}\in\Delta_{s^{-1}}^{s}$, right multiplication by
$A_{r}$ or $B_{r}$ gives rise to well–defined homomorphisms of left
$\mathbb{C}_{\mathcal{B}}[G]$–modules
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}^{p}_{\mathcal{B}}}^{loc}\rightarrow\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}^{r}_{\mathcal{B}}}^{loc}$.

###### Proof.

The proof of the first claim follows from the definitions of
${I^{p}_{\mathcal{B}}}$ and ${I^{p}_{\mathcal{B}}}^{loc}$, commutation
relations (4.2.2), (4.5.6) and from the fact that ${I^{1}_{\mathcal{B}}}$ is a
two–sided ideal in $\mathbb{C}_{\mathcal{B}}[G]$. For the same reason
${I^{p}_{\mathcal{B}}}$ is invariant with respect to the multiplication from
the right by $\Delta_{\mu}$ for any $\mu\in P_{+}$.

From commutation relations (4.2.2), (4.5.6) it also follows that to establish
the second claim it suffices to show that for $r\leq p$ with
$\beta_{p}\in\Delta_{s^{-1}}^{s}$ we have
${\overline{J}^{1}_{q}}A_{r}\subset\overline{J}^{p}_{q}$, and to establish the
last claim it suffices to verify that
${\overline{J}^{1}_{q}}\Delta_{\mu}\subset{\overline{J}^{1}_{q}}$ for any
$\mu\in P_{+}$.

Indeed, recall that ${\overline{J}^{1}_{q}}$ is the left ideal in
$\mathbb{C}_{q}[G]$ generated by the elements $(u,\cdot
v)\in\mathbb{C}_{\mathcal{B}}[G]$, where $u$ is a highest weight vector in a
finite rank representation $V$ of $U_{h}(\mathfrak{g})$, and $v\in V$ is such
that $(u,\overline{T}_{s}xv)=0$ for any $x\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$.

From formula (2.6.13) with $k$ equal to the length of $s$ we have

$\Delta_{s}(\overline{T}_{s})=\prod^{k}_{p=1}{\overline{\theta}_{\beta_{p}}^{s}}^{\prime}q^{\sum_{i=1}^{l}(-Y_{i}\otimes
KH_{i}+T_{s}Y_{i}\otimes
T_{s}KH_{i})}\overline{T}_{s}\otimes\overline{T}_{s}=\prod^{k}_{p=1}{\overline{\theta}_{\beta_{p}}^{s}}^{\prime}\overline{T}_{s}\otimes\overline{T}_{s}$
(4.5.7)

where we used (4.5.2).

Let $y=x\overline{T}_{s}hzx^{\prime}$,
$x=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})})$, $z\in U_{q}^{res}(\Delta_{0}^{+})$, $x^{\prime}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U_{q}^{res}(H)$. Since $u$ is
a highest weight vector we obtain from (4.5.7), the definition of
${\overline{\theta}_{\beta_{p}}^{s}}^{\prime}$ after (2.6.13), (2.7.12) and
the fact that $\omega_{0}S_{s}^{-1}$ is an anticoautomorphism that

$((u,\cdot v)\otimes
A_{r})(y)=(u,\overline{T}_{s}h_{1}z_{1}x^{\prime}_{1}v)(v_{r},x\overline{T}_{s}h_{2}z_{2}x^{\prime}_{2}T_{s}^{-1}v_{\mu_{r}})=$
$=t(u,\overline{T}_{s}h_{1}z_{1}x^{\prime}_{1}v)(v_{r},x\overline{T}_{s}h_{2}z_{2}x^{\prime}_{2}\overline{T}_{s}^{-1}v_{\mu_{r}}),$

where $t$ is a non–zero multiple of a power of $q$ defined in Lemma 2.7.4, and
we use the Sweedler notation for the coproducts, so for any $a\in
U_{q}^{res}(\mathfrak{g})\subset U_{\mathcal{B}}^{res}(\mathfrak{g})\simeq
U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ $\Delta_{s}(a)=a_{1}\otimes a_{2}$.

As in the proof of Lemma 4.4.1 (see the discussion after formula (4.4.5)) we
also deduce that
$(v_{r},x\overline{T}_{s}h_{2}z_{2}x^{\prime}_{2}\overline{T}_{s}^{-1}v_{\mu_{r}})=0$
if $x\neq 1,\omega_{0}S_{s}^{-1}(f_{\beta_{r}})$, i.e $((u,\cdot v)\otimes
A_{r})(y)=0$ if $x\neq 1,\omega_{0}S_{s}^{-1}(f_{\beta_{r}})$.

Let $x=1$. Recall that $e_{\beta}=X_{\beta}^{+}q^{K\beta^{\vee}}$ and
$f_{\beta}=q^{-K\beta^{\vee}}X_{\beta}^{-}$, and hence

$\Delta_{s}(X_{\beta}^{+})=\Delta_{s}(e_{\beta})q^{-K\beta^{\vee}}\otimes
q^{-K\beta^{\vee}},\Delta_{s}(X_{\beta}^{-})=q^{K\beta^{\vee}}\otimes
q^{K\beta^{\vee}}\Delta_{s}(f_{\beta}).$ (4.5.8)

Combining these expressions with formulas (2.7.12) and (2.7.14) we deduce that

$h_{2}z_{2}x^{\prime}_{2}\in
U_{\mathcal{B}}^{s,res}(\mathfrak{h})U_{q}^{res}([\Delta_{0}^{+},-\gamma_{l^{\prime}}]),$

where $[\Delta_{0}^{+},-\gamma_{l^{\prime}}]$ is the minimal segment in
$\Delta$ containing $\Delta_{0}^{+}$ as its initial segment and
$-\gamma_{l^{\prime}}$ is its last element. But
$\beta_{r}\in\Delta_{s^{-1}}^{s}$, so $s^{-1}(-\beta_{r})\in\Delta_{s}^{s}$.
Note also that $[\Delta_{0}^{+},-\gamma_{l^{\prime}}]$ is a subset of a system
of positive roots $\Delta_{0}^{+}\cup(\Delta_{-}^{s}\setminus\Delta_{0}^{-})$
and
$\Delta_{s}^{s}\cap(\Delta_{0}^{+}\cup(\Delta_{-}^{s}\setminus\Delta_{0}^{-}))=\emptyset$.
Therefore the element
$\overline{T}_{s}h_{2}z_{2}x_{2}^{\prime}\overline{T}_{s^{-1}}$ has no
components of weight $-\beta_{r}$ in its weight decomposition. Since the
weight of $v_{r}$ is $\mu_{r}-\beta_{r}$ we deduce that

$((u,\cdot v)\otimes
A_{r})(y)=(u,\overline{T}_{s}h_{1}z_{1}x^{\prime}_{1}v)(v_{r},\overline{T}_{s}h_{2}z_{2}x^{\prime}_{2}\overline{T}_{s}^{-1}v_{\mu_{r}})=0$

in this case.

If $x=\omega_{0}S_{s}^{-1}(f_{\beta_{r}})$ we have by the definition of
$v_{r}$

$((u,\cdot v)\otimes
A_{r})(y)=(u,\overline{T}_{s}h_{1}z_{1}x^{\prime}_{1}v)(v_{\mu_{r}},\overline{T}_{s}h_{2}z_{2}x_{2}^{\prime}\overline{T}_{s}^{-1}v_{\mu_{r}}),$
(4.5.9)

and hence only zero weight components of the element
$h_{2}z_{2}x_{2}^{\prime}$ will contribute to the right hand side of the
formula above. Recall that $h_{2}z_{2}x^{\prime}_{2}\in
U_{\mathcal{B}}^{s,res}(\mathfrak{h})U_{q}^{res}([\Delta_{0}^{+},-\gamma_{l^{\prime}}])$
and as we observed above $[\Delta_{0}^{+},-\gamma_{l^{\prime}}]$ is a subset
of the system of positive roots
$\Delta_{0}^{+}\cup(\Delta_{-}^{s}\setminus\Delta_{0}^{-})$. Therefore only
zero degree monomials in quantum root vectors in the decomposition of
$h_{2}z_{2}x_{2}^{\prime}$ will contribute to the right hand side of (4.5.9).
If we assume, without loss of generality, that $z$ and $x^{\prime}$ are
monomials in quantum root vectors then from (4.5.8), (2.7.12) and (2.7.14) one
can see that the above mentioned contribution amounts to

$((u,\cdot v)\otimes
A_{r})(y)=(u,\overline{T}_{s}h_{1}z_{1}x^{\prime}_{1}v)(v_{\mu_{r}},\overline{T}_{s}h_{2}z_{2}x_{2}^{\prime}\overline{T}_{s}^{-1}v_{\mu_{r}})=c^{\prime}(u,\overline{T}_{s}zx^{\prime}v),c^{\prime}\in\mathcal{B}.$

Finally observe that $zx^{\prime}\in
U_{q}^{res}(w^{\prime}(\mathfrak{b}_{+}))$, and hence
$(u,\overline{T}_{s}zx^{\prime}v)=t^{\prime}(u,{T}_{s}zx^{\prime}v)=0$ by the
choice of $u$ and $v$. Here $t^{\prime}$ is a non–zero multiple of a power of
$q$ given by Lemma 2.7.4.

So in all cases $((u,\cdot v)\otimes A_{r})(y)=0$, and hence
${\overline{J}^{1}_{q}}A_{r}\subset\overline{J}^{p}_{q}$ by Lemma 4.4.1.

Similar arguments show that for any $\mu\in P_{+}$ and
$y=x\overline{T}_{s}hzx^{\prime}$ with any $x\in
U_{q}^{res}(\mathfrak{n}_{-})$, $z\in U_{q}^{res}(\Delta_{0}^{+})$,
$x^{\prime}\in U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in
U_{q}^{res}(H)$ one has

$((u,\cdot v)\otimes\Delta_{\mu})(y)=0,$

and hence ${\overline{J}^{1}_{q}}$ is invariant with respect to multiplication
from the right by $\Delta_{\mu}$ for any $\mu\in P_{+}$ by Lemma 4.3.2.

∎

In the next two lemmas we study the properties of
${\overline{I}^{p}_{\mathcal{B}}}$, ${I^{p}_{\mathcal{B}}}$, and
$\mathbb{C}_{p}^{loc}[G]$ with respect to the adjoint action. These properties
will be needed to study properties of quantum analogues of the operators
$\Pi_{p}$ defined in (3.5.11).

######  Lemma 4.5.4.

For any $p=1,\ldots,c$, $n\in\mathbb{N}$ and $f\in\mathbb{C}_{\mathcal{B}}[G]$
we have

${\rm Ad}_{0}f_{\beta_{p}}^{(n)}(f\otimes\Delta_{\mu})=q^{n((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-id)\beta_{p}^{\vee},\mu^{\vee})}{\rm
Ad}_{0}f_{\beta_{p}}^{(n)}(f)\otimes\Delta_{\mu}$ (4.5.10)

in $\mathbb{C}_{\mathcal{B}}[G]/{\overline{I}^{1}_{\mathcal{B}}}$. Thus the
adjoint action of $U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$ on
$\mathbb{C}_{\mathcal{B}}[G]/{\overline{I}^{1}_{\mathcal{B}}}$ defined in
Lemma 4.3.2 induces an action on $\mathbb{C}_{1}^{loc}[G]$ satisfying

${\rm
Ad}_{0}f_{\beta_{p}}^{(n)}(f\otimes\Delta_{\mu}^{-1})=q^{-n((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-id)\beta_{p}^{\vee},\mu^{\vee})}{\rm
Ad}_{0}f_{\beta_{p}}^{(n)}(f)\otimes\Delta_{\mu}^{-1},f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}^{1}_{\mathcal{B}}}^{loc}=\mathbb{C}_{1}^{loc}[G].$
(4.5.11)

This action is locally finite.

For any $p=1,\ldots,c$ with $\beta_{p}\not\in\Delta_{s^{1}}^{s}$,
$n\in\mathbb{N}$ and $f\in\mathbb{C}_{\mathcal{B}}[G]$ we have

${\rm Ad}_{0}f_{\beta_{p}}^{(n)}(f\otimes\Delta_{\mu})=q^{n((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-id)\beta_{p}^{\vee},\mu^{\vee})}{\rm
Ad}_{0}f_{\beta_{p}}^{(n)}(f)\otimes\Delta_{\mu}$ (4.5.12)

in $\mathbb{C}_{\mathcal{B}}[G]/{I^{1}_{\mathcal{B}}}$. Thus the adjoint
action of $f_{\beta_{p}}^{(n)}$ on
$\mathbb{C}_{\mathcal{B}}[G]/{I^{1}_{\mathcal{B}}}$ defined in Lemma 4.3.5
induces a locally finite action on
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{1}_{\mathcal{B}}}^{loc}$ satisfying

${\rm
Ad}_{0}f_{\beta_{p}}^{(n)}(f\otimes\Delta_{\mu}^{-1})=q^{-n((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-id)\beta_{p}^{\vee},\mu^{\vee})}{\rm
Ad}_{0}f_{\beta_{p}}^{(n)}(f)\otimes\Delta_{\mu}^{-1},f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{1}_{\mathcal{B}}}^{loc}.$
(4.5.13)

###### Proof.

The proof of the first part follows from formula (4.2.4) applied to
$g=\Delta_{\mu}$. All terms in the r.h.s. of (4.2.4), except for the ones with
$p=n-k$ in the first sum, vanish since $v_{\mu}$ is a highest weight vector.
Lemma 4.3.4 implies vanishing all the remaining terms in the first sum in the
r.h.s. of (4.2.4) modulo ${\overline{I}^{1}_{\mathcal{B}}}$, except for $k=0$.
The only non-vanishing term in the first sum in the r.h.s. of (4.2.4)
corresponds to $p=n$, $k=0$. It gives the r.h.s. of (4.5.10).

Local finiteness of the action of $U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$
on $\mathbb{C}_{1}^{loc}[G]$ follows from the local finiteness of the action
of $U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$ on
$\mathbb{C}_{\mathcal{B}}[G]/{\overline{I}^{1}_{\mathcal{B}}}$ and from
formula (4.5.11).

The second part is proved similarly using Lemma 4.3.7.

∎

######  Lemma 4.5.5.

For any $r\in\\{1,\ldots,c\\}$, $p\leq r$ and $n\in\mathbb{N}$ the left ideal
${\overline{I}^{p}_{\mathcal{B}}}\subset\mathbb{C}_{\mathcal{B}}[G]$ is ${\rm
Ad}_{0}f_{\beta_{r}}^{(n)}$–stable, and, if in addition
$\beta_{r}\not\in\Delta_{s^{1}}^{s}$, then ${I^{p}_{\mathcal{B}}}$ is ${\rm
Ad}_{0}f_{\beta_{r}}^{(n)}$–stable.

For any $r\in\\{1,\ldots,c\\}$, $p\leq r$ and $n\in\mathbb{N}$
${\overline{I}^{p}_{\mathcal{B}}}^{loc}/{\overline{I}^{1}_{\mathcal{B}}}^{loc}\subset\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}^{1}_{\mathcal{B}}}^{loc}\simeq\mathbb{C}_{1}^{loc}[G]$
is ${\rm Ad}_{0}f_{\beta_{r}}^{(n)}$–stable, and, if in addition
$\beta_{r}\not\in\Delta_{s^{1}}^{s}$, then
${I^{p}_{\mathcal{B}}}^{loc}/{I^{1}_{\mathcal{B}}}^{loc}\subset\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{1}_{\mathcal{B}}}^{loc}$
is ${\rm Ad}_{0}f_{\beta_{r}}^{(n)}$–stable.

Thus for any $r\in\\{1,\ldots,c\\}$, $p\leq r$ and $n\in\mathbb{N}$
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}^{p}_{\mathcal{B}}}^{loc}$ is
naturally equipped with an ${\rm Ad}_{0}f_{\beta_{r}}^{(n)}$–action, and, if
in addition $\beta_{r}\not\in\Delta_{s^{1}}^{s}$
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{p}_{\mathcal{B}}}^{loc}$ is naturally
equipped with an ${\rm Ad}_{0}f_{\beta_{r}}^{(n)}$–action.

In particular, $\mathbb{C}_{p}^{loc}[G]$ is naturally equipped with an ${\rm
Ad}_{0}f_{\beta_{r}}^{(n)}$–action.

###### Proof.

By (4.5.11) and (4.5.13) to establish the first two claims it suffices to show
that ${\overline{J}^{p}_{q}}$ is ${\rm Ad}_{0}f_{\beta_{r}}^{(n)}$–stable for
any $r\in\\{1,\ldots,c\\}$, $p\leq r$ and $n\in\mathbb{N}$ and ${J^{p}_{q}}$
is ${\rm Ad}_{0}f_{\beta_{r}}^{(n)}$–stable for any $r\in\\{1,\ldots,c\\}$
with $\beta_{r}\not\in\Delta_{s^{1}}^{s}$, $p\leq r$ and $n\in\mathbb{N}$.

We show that ${\overline{J}^{p}_{q}}$ is ${\rm
Ad}_{0}f_{\beta_{r}}^{(n)}$–stable for any $r\in\\{1,\ldots,c\\}$, $p\leq r$
and $n\in\mathbb{N}$. Since ${\overline{J}^{p}_{q}}$ is generated by
${\overline{J}^{1}_{q}}$ and by $A_{1},\ldots,A_{p-1}$ and by Lemma 4.3.2
${\overline{J}^{1}_{q}}$ is ${\rm Ad}_{0}f_{\beta_{r}}^{(n)}$–stable it
suffices to prove that ${\rm Ad}_{0}f_{\beta_{r}}^{(n)}(f\otimes
A_{t})\in{\overline{J}^{p}_{q}}$, for any $f\in\mathbb{C}_{\mathcal{B}}[G]$,
$t=1,\ldots,p-1$. By (4.2.3) we have

${\rm Ad}_{0}f_{\beta_{r}}^{(n)}(f\otimes
A_{t})=\sum_{k=0}^{n}\sum_{p=0}^{n-k}q_{{\beta_{r}}}^{-k(n-k)-p(n-k-p)}{\rm
Ad}_{0}(G_{\beta_{r}}^{-k}f_{\beta_{r}}^{(p)})(f)\otimes
A_{t}(\omega_{0}S_{s}^{-1}(G_{\beta_{r}}^{-k-p}f_{\beta_{r}}^{(n-k-p)})\cdot\omega_{0}(f_{\beta_{r}}^{(k)}))+$
$+\sum_{k=0}^{n-1}\sum_{i}q_{\beta_{r}}^{-k(n-k)}{\rm
Ad}_{0}(G_{\beta_{r}}^{-k}x_{i}^{(n-k)})(f)\otimes
A_{t}((\omega_{0}S_{s}^{-1})(G_{\beta_{r}}^{-k}y_{i}^{(n-k)})\cdot\omega_{0}(f_{\beta_{r}}^{(k)}))+$
(4.5.14) $+\sum_{i}{\rm Ad}_{0}({y_{i}^{(n)}}^{2})(f)\otimes
A_{t}((\omega_{0}S_{s}^{-1})({y_{i}^{(n)}}^{1})\cdot\omega_{0}(x_{i}^{(n)}))=$
$=\sum_{k=0}^{n}q_{{\beta_{r}}}^{-k(n-k)}q^{n(\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}\beta_{r}-\beta_{r},\mu_{t}-\beta_{t})}{\rm
Ad}_{0}(G_{\beta_{r}}^{-k}f_{\beta_{r}}^{(n-k)})(f)\otimes
A_{t}(\cdot\omega_{0}(f_{\beta_{r}}^{(k)}))+$
$+\sum_{k=0}^{n}\sum_{p=0}^{n-k-1}q_{{\beta_{r}}}^{-k(n-k)-p(n-k-p)}{\rm
Ad}_{0}(G_{\beta_{r}}^{-k}f_{\beta_{r}}^{(p)})(f)\otimes
A_{t}(\omega_{0}S_{s}^{-1}(G_{\beta_{r}}^{-k-p}f_{\beta_{r}}^{(n-k-p)})\cdot\omega_{0}(f_{\beta_{r}}^{(k)}))+$
$+\sum_{k=0}^{n-1}\sum_{i}q_{\beta_{r}}^{-k(n-k)}{\rm
Ad}_{0}(G_{\beta_{r}}^{-k}x_{i}^{(n-k)})(f)\otimes
A_{t}((\omega_{0}S_{s}^{-1})(G_{\beta_{r}}^{-k}y_{i}^{(n-k)})\cdot\omega_{0}(f_{\beta_{r}}^{(k)}))+$
$+\sum_{i}{\rm Ad}_{0}({y_{i}^{(n)}}^{2})(f)\otimes
A_{t}((\omega_{0}S_{s}^{-1})({y_{i}^{(n)}}^{1})\cdot\omega_{0}(x_{i}^{(n)})).$

We claim that all terms in the right hand side of the formula above belong to
${\overline{J}^{p}_{q}}$. Indeed, by Lemma 4.4.3

$q_{{\beta_{r}}}^{-k(n-k)}q^{n(\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}\beta_{r}-\beta_{r},\mu_{t}-\beta_{t})}{\rm
Ad}_{0}(G_{\beta_{r}}^{-k}f_{\beta_{r}}^{(n-k)})(f)\otimes
A_{t}(\cdot\omega_{0}(f_{\beta_{r}}^{(k)}))\in\overline{J}^{t}_{q}\subset\overline{J}^{p}_{q}$

for $k>0$. If $k=0$

$q^{n(\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}\beta_{r}-\beta_{r},\mu_{t}-\beta_{t})}{\rm
Ad}_{0}(f_{\beta_{r}}^{(n)})(f)\otimes
A_{t}\in\overline{J}^{t+1}_{q}\subset\overline{J}^{p}_{q}.$

To establish the same claim for all the other terms in the right hand side of
(4.5.14) we shall apply Lemma 4.4.1. The arguments are similar for all these
terms. Consider, for instance, the terms in the last sum. For
$xT_{s}hz_{+}x^{\prime}$ with arbitrary
$x=\omega_{0}S_{s}^{-1}(f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}),z_{+}\in U_{q}^{res}(\Delta_{0}^{+}),x^{\prime}\in
U_{q}^{res}(-\Delta_{\mathfrak{m}_{+}}^{s}),h\in U_{q}^{res}(H)$,
$n_{i}\in\mathbb{N}_{0}$ we have

$A_{t}((\omega_{0}S_{s}^{-1})({y_{i}^{(n)}}^{1})xT_{s}hz_{+}x^{\prime}\omega_{0}x_{i}^{(n)})=(v_{t},(\omega_{0}S_{s}^{-1})({y_{i}^{(n)}}^{1})xT_{s}hz_{+}x^{\prime}\omega_{0}x_{i}^{(n)}T_{s^{-1}}v_{\mu_{t}})=$
$=c(v_{t},(\omega_{0}S_{s}^{-1})({\overline{y}_{i}^{(n)}}^{1}f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})})T_{s}hz_{+}x^{\prime}\omega_{0}x_{i}^{(n)}T_{s^{-1}}v_{\mu_{t}})=$
$=c(v_{t},(\omega_{0}S_{s}^{-1})({\overline{y}_{i}^{(n)}}^{1}f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})})T_{s}hz_{+}x^{\prime}\omega_{0}x_{i}^{(n)}T_{s^{-1}}v_{\mu_{t}})=(v_{t},(\omega_{0}S_{s}^{-1})({\overline{y}_{i}^{(n)}}^{1}f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}y)v_{\mu_{t}}),$

where $c\in\mathcal{B}$, $y\in U_{\mathcal{B}}^{s,res}(\mathfrak{n}_{-})$ and
for the last implication we used the fact that $v_{t}$ is a weight vector and
$v_{\mu_{t}}$ is a highest weight vector. Since
${\overline{y}_{i}^{(n)}}^{1}\in I_{r}^{>}$, $r\geq p$ and the weight of
${\overline{y}_{i}^{(n)}}^{1}$ is not zero, Lemma 2.7.2 applied to
${\overline{y}_{i}^{(n)}}^{1}f_{\beta_{D}}^{(n_{D})}\ldots
f_{\beta_{p}}^{(n_{p})}y$ implies that this element belongs to the right ideal
in $U_{\mathcal{B}}^{s,res}(\mathfrak{n}_{-})$ generated by the elements
$f_{\beta_{D}}^{(n_{D})}\ldots f_{\beta_{r+1}}^{(n_{r+1})}$, where at least
one $n_{i}>0$. This ideal does not contain elements proportional to
$f_{\beta_{t}}$ as $1\leq t<p\leq r$. Therefore by the definition of $A_{t}$
(see (4.4.1))
$A_{t}((\omega_{0}S_{s}^{-1})({y_{i}^{(n)}}^{1})xT_{s}hz_{+}x^{\prime}\omega_{0}x_{i}^{(n)})=0$
and hence by Lemma 4.4.1 $f\otimes
A_{t}((\omega_{0}S_{s}^{-1})({y_{i}^{(n)}}^{1})\cdot\omega_{0}x_{i}^{(n)})\in{\overline{J}^{p}_{q}}$.
Note that this statement holds for $t=p$ as well. This proves that
${\overline{J}^{p}_{q}}$ is ${\rm Ad}_{0}f_{\beta_{r}}$–stable for $r\geq p$.

To show that ${J^{p}_{q}}$ is ${\rm Ad}_{0}f_{\beta_{r}}$–stable for any
$r\in\\{1,\ldots,c\\}$ with $\beta_{r}\not\in\Delta_{s^{1}}^{s}$, $p\leq r$
and $n\in\mathbb{N}$ one has to apply similar arguments in conjunction with
Lemmas 4.3.5, 4.4.2 and 4.4.3.

∎

###  4.6 Zhelobenko type operators for q-W–algebras

This section is central in this part and in the whole book. We are going to
introduce and study some quantum analogues of the operators $\Pi_{p}$ and
$\Pi$ defined in (3.5.11) and (3.5.12). It turns out that the analogues
$P_{p}$ of $\Pi_{p}$ can be obtained by proper extrapolation of the expansion
of the conjugation operator in (3.5.11) in terms of the adjoint action
operator and by replacing the coefficients $G_{p}$ with their quantum
counterparts $B_{p}$ introduced in (4.5.5). However, the proof of Proposition
4.6.1 which asserts that the image of their composition $P$ consists of
invariant elements with respect to the adjoint action of
$U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$ is rather complicated. It entirely
relies on the properties of the subspaces
${\overline{I}^{p}_{\mathcal{B}}}^{loc},{I^{p}_{\mathcal{B}}}^{loc}\subset\mathbb{C}_{\mathcal{B}}^{loc}[G]$,
of the quotients $\mathbb{C}_{p}^{loc}[G]$ and of the adjoint action, which
were obtained in the previous sections of this chapter.

The point is that in the quantum case we do not have in our disposal the
isomorphism (3.4.29) which plays a crucial role in the proof of a similar
property for the operator $\Pi$ given by (3.5.11), (3.5.12) as one can see
from the proofs of Propositions 3.5.1 and 3.5.2.

We start with the definition of quantum analogues $P_{p}$ of the operators
$\Pi_{p}$. For technical reasons we shall also need more general operators
$P_{p}^{k}$. More precisely, by Lemmas 4.5.3 and 4.5.5 for
$\beta_{p}\not\in\Delta_{s^{1}}^{s}$,
$f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{p}}^{loc}$,
$p=1,\ldots,c$, $n,k=0,1,\ldots$, $n\geq k$, we have ${\rm
Ad}_{0}f_{\beta_{p}}^{(n-k)}(f)\otimes
B_{p}^{n}\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{p}}^{loc}$.
Note that by Lemma 4.5.4 for each
$f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{p}}^{loc}$ ${\rm
Ad}_{0}f_{\beta_{p}}^{(n-k)}(f)=0$ for $n>N(f)$, where $N(f)\in\mathbb{N}$
depends on $f$. Therefore we can define define an element
$P_{p}^{k}(f)\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{p}}^{loc}$
by

$P_{p}^{k}(f)=\sum_{n=k}^{\infty}(-1)^{n}q_{\beta_{p}}^{\frac{(n-1)(n-2k)}{2}}{\rm
Ad}_{0}f_{\beta_{p}}^{(n-k)}(f)\otimes B_{p}^{n}.$ (4.6.1)

Similarly, for $\beta_{p}\in\Delta_{s^{-1}}^{s}$,
$f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{p}}^{loc}$,
$p=1,\ldots,c$, $n,k=0,1,\ldots$, $n\geq k$, we have ${\rm
Ad}_{0}f_{\beta_{p}}^{(n-k)}(f)\otimes
B_{p}^{n}\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{p}}^{loc}$.
Note that by Lemma 4.5.4 for each
$f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{p}}^{loc}$
${\rm Ad}_{0}f_{\beta_{p}}^{(n-k)}(f)=0$ for $n>N(f)$, where
$N(f)\in\mathbb{N}$ depends on $f$. Therefore we can define define an element
$P_{p}^{k}(f)\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{p}}^{loc}$
by formula (4.6.1).

Let $\beta_{e}$ be the first root of $\Delta_{s^{-1}}^{s}$ in normal ordering
(1.6.3) which does not belong to $\Delta_{s^{1}}^{s}$. Then, using the
imbedding
${I_{\mathcal{B}}^{e}}^{loc}\subset{\overline{I}_{\mathcal{B}}^{e}}^{loc}$ we
can define the canonical projection
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{e}}^{loc}\rightarrow\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{e}}^{loc}$,
and hence in this case for
$f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{e}}^{loc}=\mathbb{C}_{e}^{loc}[G]$
projecting
$P_{e}^{k}(f)\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{e}}^{loc}$
defined in the former situation above onto
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{e}}^{loc}$ we
can introduce an operator

$P_{e}^{k}:\mathbb{C}_{e}^{loc}[G]\rightarrow\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{e}}^{loc}.$

From the discussion above it follows that for all other indexes
$p=1,\ldots,c$, $p\neq e$ formula (4.6.1) can be used to define operators

$P_{p}^{k}:\mathbb{C}_{p}^{loc}[G]\rightarrow\mathbb{C}_{p}^{loc}[G].$

Denote $P_{p}^{0}=P_{p}$.

The following proposition summarizes the main properties of the operators
$P_{p}$.

######  Proposition 4.6.1.

For any $p=1,\ldots,c$ the composition $P_{\leq p}=P_{1}\ldots
P_{p}:\mathbb{C}_{p}^{loc}[G]\rightarrow\mathbb{C}_{1}^{loc}[G]$ is
well–defined. In fact for $p\neq e-1$ $P_{\leq p}$ is well–defined as an
operator with the domain $\mathbb{C}_{p+1}^{loc}[G]$ and $P_{\leq e-1}$ is
well–defined as an operator with the domain
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{e}}^{loc}$.

For $q\leq p$, $n>0$ and for any $f\in\mathbb{C}_{p}^{loc}[G]$ we have

${\rm Ad}_{0}f_{\beta_{q}}^{(n)}(P_{\leq p}(f))=0.$ (4.6.2)

Moreover,

${\rm Ad}_{0}f_{\beta_{p+1}}^{(n)}(P_{\leq p}(f))=P_{\leq p}({\rm
Ad}_{0}f_{\beta_{p+1}}^{(n)}f).$ (4.6.3)

For $\beta_{p}\in\Delta_{s^{1}}^{s}$ and any $f\in\mathbb{C}_{1}^{loc}[G]$
such that ${\rm Ad}_{0}x(f)=\varepsilon_{s}(x)f~{}~{}\forall x\in
U_{\mathcal{B}}^{s,res}([-\beta_{1},-\beta_{p}])$ we have in
$\mathbb{C}_{1}^{loc}[G]$

$P_{\leq p}(\bar{f})=f,$ (4.6.4)

where $\bar{f}$ is the class of $f$ in $\mathbb{C}_{p}^{loc}[G]$.

For $\beta_{p}\not\in\Delta_{s^{1}}^{s}$ and any
$f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{1}_{\mathcal{B}}}^{loc}$ such that
${\rm Ad}_{0}x(f)=\varepsilon_{s}(x)f~{}~{}\forall x\in
U_{\mathcal{B}}^{s,res}([-\beta_{1},-\beta_{p}])$ we have

$P_{\leq
p}(\bar{f})=f~{}(\text{mod}~{}{\overline{I}^{1}_{\mathcal{B}}}^{loc}),$
(4.6.5)

where $\bar{f}$ is the class of $f$ in $\mathbb{C}_{p}^{loc}[G]$.

In particular, if $P=P_{\leq c}$ then for any
$f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{1}_{\mathcal{B}}}^{loc}$ such that
${\rm Ad}_{0}x(f)=\varepsilon_{s}(x)f~{}~{}\forall x\in
U_{\mathcal{B}}^{s,res}(\mathfrak{m}_{-})$ we have

$P(\bar{f})=f~{}(\text{mod}~{}{\overline{I}^{1}_{\mathcal{B}}}^{loc}),$
(4.6.6)

where $\bar{f}$ is the class of $f$ in $\mathbb{C}_{c}^{loc}[G]$.

The proof of Proposition 4.6.1 is quite long. It will be split into several
lemmas. The strategy of the proof can be explained by looking at a similar
situation in the classical case.

Namely, from the definition of the operators $\Pi_{p}$ in (3.5.11),
Proposition 3.5.2 and Corollary 3.5.4 it follows that the kernel of the
operator
$\Pi_{1}\ldots\Pi_{p}:\mathbb{C}[N_{-}sZM_{-}^{s}]\rightarrow\mathbb{C}[N_{-}sZM_{-}^{s}]$
coincides with the vanishing ideal of
${N_{[-\beta_{p+1},-\beta_{D}]}sZM_{-}^{s}}\subset{N_{-}sZM_{-}^{s}}$.
Similarly, in the quantum case we shall show that $P_{\leq p}$ is well–defined
as an operator with the domain of definition $\mathbb{C}_{p+1}^{loc}[G]$, i.e.
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}^{p+1}_{\mathcal{B}}}^{loc}$
or $\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I^{p+1}_{\mathcal{B}}}^{loc}$ depending
on $p$.

Denote by $L_{p}$ the the derivative of the action induced by the conjugation
action by the elements of the one–parameter subgroup $\exp(-tX_{-\beta_{p}})$
at $t=0$. From Propositions 3.5.1, 3.5.2 and Corollary 3.5.4 one can
immediately deduce that
$L_{p}\circ\Pi_{p}:\mathbb{C}[N_{-}sZM_{-}^{s}]\rightarrow\mathbb{C}[N_{-}sZM_{-}^{s}]$
is the zero operator modulo the vanishing ideal of
${N_{[-\beta_{p},-\beta_{D}]}sZM_{-}^{s}}\subset{N_{-}sZM_{-}^{s}}$. In the
quantum case we shall show that for $n>0$ ${\rm
Ad}_{0}f_{\beta_{p}}^{(n)}P_{p}$ is the zero operator modulo
${\overline{I}^{p}_{\mathcal{B}}}^{loc}$ or ${I^{p}_{\mathcal{B}}}^{loc}$
depending on $p$. This is done using an explicit calculation and observing
that ${\rm Ad}_{0}f_{\beta_{p}}^{(n)}$ has some properties analogues to
properties of derivatives of order $n$.

Finally to show that for $q\leq p$ and $n>0$ ${\rm
Ad}_{0}f_{\beta_{q}}^{(n)}(P_{\leq p}(f))=0$ modulo
${\overline{I}^{1}_{\mathcal{B}}}^{loc}$ we shall use the above mentioned
property of the images of operators $P_{p}$ and commutation relations between
$P_{p}$ and ${\rm Ad}_{0}f_{\beta_{q}}^{(n)}$ for $q>p$ which, in particular,
lead to commutation relations (4.6.3).

Firstly, we are going to show by an explicit calculation that for $n>0$ ${\rm
Ad}_{0}f_{\beta_{p}}^{(n)}P_{p}$ is the zero operator modulo
${\overline{I}^{p}_{\mathcal{B}}}^{loc}$ (or ${I^{p}_{\mathcal{B}}}^{loc}$).
Since the right hand side of formula (4.6.1) contains products of elements
from $\mathbb{C}_{q}^{loc}[G]$ and of $B_{p}^{n}$, we have to study the
adjoint action on such products.

######  Lemma 4.6.2.

Let $B_{p}={A_{p}^{0}}^{-1}\otimes
A_{p}\in\mathbb{C}_{\mathcal{B}}^{loc}[G]_{0}$,
$f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]$, $p=1,\ldots,c$. Let either
$\beta_{p}\in\Delta_{s^{-1}}^{s}$ or $\beta_{p}\not\in\Delta_{s^{1}}^{s}$.
Then the identity

${\rm Ad}_{0}f_{\beta_{p}}^{(n)}(\bar{f}\otimes B_{p}^{k})=\sum_{r=0}^{{\rm
min}(n,k)}q_{\beta_{p}}^{(2n-r)k+\frac{r(r-1)}{2}-rn}\left[\begin{array}[]{c}k\\\
r\end{array}\right]_{q_{\beta_{p}}}{\rm
Ad}_{0}f_{\beta_{p}}^{(n-r)}(\bar{f})\otimes B_{p}^{k-r}$ (4.6.7)

holds in
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{p}}^{loc}$ in
the former case or in
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{p}}^{loc}$ in the latter
case, where $\bar{f}$ is the class of $f$ in
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{p}}^{loc}$ or
in $\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{p}}^{loc}$,
respectively.

For any $f\in\mathbb{C}_{\mathcal{B}}^{loc}[G]$, $p,q=1,\ldots,c$, $q>p$ let
either $\beta_{p}\in\Delta_{s^{-1}}^{s}$ or
$\beta_{q}\not\in\Delta_{s^{1}}^{s}$. For any two roots
$\alpha,\beta\in\Delta$ we denote

$c_{\alpha\beta}=q^{((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)\alpha,\beta)}.$

Then the identity

${\rm Ad}_{0}f_{\beta_{q}}^{(m)}(\bar{f}\otimes
B_{p}^{n})=c_{\beta_{p}\beta_{q}}^{mn}{\rm
Ad}_{0}f_{\beta_{q}}^{(m)}(\bar{f})\otimes B_{p}^{n}$ (4.6.8)

holds in
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{p}}^{loc}$ in
the former case or in
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{p}}^{loc}$ in the latter
case, where $\bar{f}$ is the class of $f$ in
$\mathbb{C}_{\mathcal{B}}^{loc}[G]/{\overline{I}_{\mathcal{B}}^{p}}^{loc}$ or
in $\mathbb{C}_{\mathcal{B}}^{loc}[G]/{I_{\mathcal{B}}^{p}}^{loc}$,
respectively.

###### Proof.

We prove (4.6.7) by induction over $k$. Firstly by (4.2.4), similarly to
(4.5.14), we have for any $f\in\mathbb{C}_{\mathcal{B}}[G]$

${\rm Ad}_{0}f_{\beta_{p}}^{(n)}(f\otimes A_{p})=q^{n(\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}\beta_{p}-\beta_{p},\mu_{p}-\beta_{p})}{\rm
Ad}_{0}(f_{\beta_{p}}^{(n)})(f)\otimes A_{p}+$ $+q^{(n-1)(\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}\beta_{p}-\beta_{p},\mu_{p}-\beta_{p})}q_{{\beta_{p}}}^{-(n-1)}{\rm
Ad}_{0}(f_{\beta_{p}}^{(n-1)})(f)\otimes
A_{p}(\omega_{0}S_{s}^{-1}(f_{\beta_{p}})\cdot)+$
$+\sum_{k=1}^{n}q_{{\beta_{p}}}^{-k(n-k)}q^{n(\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}\beta_{p}-\beta_{p},\mu_{p}-\beta_{p})}{\rm
Ad}_{0}(G_{\beta_{p}}^{-k}f_{\beta_{p}}^{(n-k)})(f)\otimes
A_{p}(\cdot\omega_{0}(f_{\beta_{p}}^{(k)}))+$
$+\sum_{k=1}^{n}q_{{\beta_{p}}}^{-k(n-k)-(n-k-1)}{\rm
Ad}_{0}(G_{\beta_{p}}^{-k}f_{\beta_{p}}^{(n-k-1)})(f)\otimes
A_{p}(\omega_{0}S_{s}^{-1}(G_{\beta_{p}}^{-n+1}f_{\beta_{p}})\cdot\omega_{0}(f_{\beta_{p}}^{(k)}))+$
(4.6.9)
$+\sum_{k=0}^{n}\sum_{p=0}^{n-k-2}q_{{\beta_{p}}}^{-k(n-k)-p(n-k-p)}{\rm
Ad}_{0}(G_{\beta_{p}}^{-k}f_{\beta_{p}}^{(p)})(f)\otimes
A_{p}(\omega_{0}S_{s}^{-1}(G_{\beta_{p}}^{-k-p}f_{\beta_{p}}^{(n-k-p)})\cdot\omega_{0}(f_{\beta_{p}}^{(k)}))+$
$+\sum_{k=0}^{n-1}\sum_{i}q_{\beta_{p}}^{-k(n-k)}{\rm
Ad}_{0}(G_{\beta_{p}}^{-k}x_{i}^{(n-k)})(f)\otimes